LMFDB - Embedded newform 722.6.a.p.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,6,Mod(1,722)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("722.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 722 = 2 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	722.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$115.797117905$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 1707 x^{10} - 2102 x^{9} + 996375 x^{8} + 3286746 x^{7} - 234371509 x^{6} - 1273502622 x^{5} + \cdots + 125341931625 $ x^12 - 1707x^10 - 2102x^9 + 996375x^8 + 3286746x^7 - 234371509x^6 - 1273502622x^5 + 17215098171x^4 + 126150154718x^3 + 110842127625x^2 - 356868836100x + 125341931625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 19^{3} $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-4.63163$ of defining polynomial
Character	$\chi$	$=$	722.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +4.97893 q^{3} +16.0000 q^{4} +0.812786 q^{5} +19.9157 q^{6} +105.563 q^{7} +64.0000 q^{8} -218.210 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} +4.97893 q^{3} +16.0000 q^{4} +0.812786 q^{5} +19.9157 q^{6} +105.563 q^{7} +64.0000 q^{8} -218.210 q^{9} +3.25115 q^{10} -290.679 q^{11} +79.6628 q^{12} -328.687 q^{13} +422.253 q^{14} +4.04681 q^{15} +256.000 q^{16} +589.620 q^{17} -872.841 q^{18} +13.0046 q^{20} +525.591 q^{21} -1162.71 q^{22} +1925.74 q^{23} +318.651 q^{24} -3124.34 q^{25} -1314.75 q^{26} -2296.33 q^{27} +1689.01 q^{28} -271.975 q^{29} +16.1872 q^{30} +648.509 q^{31} +1024.00 q^{32} -1447.27 q^{33} +2358.48 q^{34} +85.8003 q^{35} -3491.36 q^{36} -5823.98 q^{37} -1636.51 q^{39} +52.0183 q^{40} -4060.95 q^{41} +2102.37 q^{42} -5472.63 q^{43} -4650.86 q^{44} -177.358 q^{45} +7702.96 q^{46} -20375.1 q^{47} +1274.61 q^{48} -5663.42 q^{49} -12497.4 q^{50} +2935.68 q^{51} -5258.99 q^{52} +3684.03 q^{53} -9185.33 q^{54} -236.260 q^{55} +6756.04 q^{56} -1087.90 q^{58} -22923.1 q^{59} +64.7489 q^{60} -34271.6 q^{61} +2594.03 q^{62} -23035.0 q^{63} +4096.00 q^{64} -267.152 q^{65} -5789.07 q^{66} -20642.7 q^{67} +9433.93 q^{68} +9588.12 q^{69} +343.201 q^{70} -22120.9 q^{71} -13965.5 q^{72} +80283.9 q^{73} -23295.9 q^{74} -15555.9 q^{75} -30685.0 q^{77} -6546.03 q^{78} +6756.41 q^{79} +208.073 q^{80} +41591.8 q^{81} -16243.8 q^{82} +60676.0 q^{83} +8409.46 q^{84} +479.236 q^{85} -21890.5 q^{86} -1354.14 q^{87} -18603.4 q^{88} +42096.0 q^{89} -709.433 q^{90} -34697.2 q^{91} +30811.9 q^{92} +3228.88 q^{93} -81500.2 q^{94} +5098.42 q^{96} -34067.5 q^{97} -22653.7 q^{98} +63429.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 48 q^{2} + 192 q^{4} - 42 q^{5} - 438 q^{7} + 768 q^{8} + 546 q^{9}+O(q^{10}) $ 12 * q + 48 * q^2 + 192 * q^4 - 42 * q^5 - 438 * q^7 + 768 * q^8 + 546 * q^9	\( 12 q + 48 q^{2} + 192 q^{4} - 42 q^{5} - 438 q^{7} + 768 q^{8} + 546 q^{9} - 168 q^{10} - 600 q^{11} + 444 q^{13} - 1752 q^{14} + 3072 q^{16} - 2220 q^{17} + 2184 q^{18} - 672 q^{20} - 11046 q^{21} - 2400 q^{22} - 7464 q^{23} + 4146 q^{25} + 1776 q^{26} - 5130 q^{27} - 7008 q^{28} - 6606 q^{29} + 294 q^{31} + 12288 q^{32} - 14886 q^{33} - 8880 q^{34} - 2064 q^{35} + 8736 q^{36} - 14304 q^{37} - 19434 q^{39} - 2688 q^{40} - 19512 q^{41} - 44184 q^{42} - 8634 q^{43} - 9600 q^{44} + 34344 q^{45} - 29856 q^{46} - 17964 q^{47} - 1422 q^{49} + 16584 q^{50} + 27456 q^{51} + 7104 q^{52} - 44112 q^{53} - 20520 q^{54} - 121596 q^{55} - 28032 q^{56} - 26424 q^{58} - 34002 q^{59} - 27582 q^{61} + 1176 q^{62} - 204222 q^{63} + 49152 q^{64} - 202674 q^{65} - 59544 q^{66} - 33702 q^{67} - 35520 q^{68} - 19938 q^{69} - 8256 q^{70} - 2904 q^{71} + 34944 q^{72} - 336072 q^{73} - 57216 q^{74} - 87480 q^{75} - 57528 q^{77} - 77736 q^{78} - 40098 q^{79} - 10752 q^{80} + 97476 q^{81} - 78048 q^{82} + 148440 q^{83} - 176736 q^{84} - 252912 q^{85} - 34536 q^{86} - 182874 q^{87} - 38400 q^{88} - 123774 q^{89} + 137376 q^{90} - 178866 q^{91} - 119424 q^{92} - 118584 q^{93} - 71856 q^{94} - 317346 q^{97} - 5688 q^{98} - 409860 q^{99}+O(q^{100}) \) 12 * q + 48 * q^2 + 192 * q^4 - 42 * q^5 - 438 * q^7 + 768 * q^8 + 546 * q^9 - 168 * q^10 - 600 * q^11 + 444 * q^13 - 1752 * q^14 + 3072 * q^16 - 2220 * q^17 + 2184 * q^18 - 672 * q^20 - 11046 * q^21 - 2400 * q^22 - 7464 * q^23 + 4146 * q^25 + 1776 * q^26 - 5130 * q^27 - 7008 * q^28 - 6606 * q^29 + 294 * q^31 + 12288 * q^32 - 14886 * q^33 - 8880 * q^34 - 2064 * q^35 + 8736 * q^36 - 14304 * q^37 - 19434 * q^39 - 2688 * q^40 - 19512 * q^41 - 44184 * q^42 - 8634 * q^43 - 9600 * q^44 + 34344 * q^45 - 29856 * q^46 - 17964 * q^47 - 1422 * q^49 + 16584 * q^50 + 27456 * q^51 + 7104 * q^52 - 44112 * q^53 - 20520 * q^54 - 121596 * q^55 - 28032 * q^56 - 26424 * q^58 - 34002 * q^59 - 27582 * q^61 + 1176 * q^62 - 204222 * q^63 + 49152 * q^64 - 202674 * q^65 - 59544 * q^66 - 33702 * q^67 - 35520 * q^68 - 19938 * q^69 - 8256 * q^70 - 2904 * q^71 + 34944 * q^72 - 336072 * q^73 - 57216 * q^74 - 87480 * q^75 - 57528 * q^77 - 77736 * q^78 - 40098 * q^79 - 10752 * q^80 + 97476 * q^81 - 78048 * q^82 + 148440 * q^83 - 176736 * q^84 - 252912 * q^85 - 34536 * q^86 - 182874 * q^87 - 38400 * q^88 - 123774 * q^89 + 137376 * q^90 - 178866 * q^91 - 119424 * q^92 - 118584 * q^93 - 71856 * q^94 - 317346 * q^97 - 5688 * q^98 - 409860 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	4.97893	0.319398	0.159699	−	0.987166i	$-0.448948\pi$
$3$	4.97893	0.319398	0.159699	+	0.987166i	$0.448948\pi$
$4$	16.0000	0.500000
$5$	0.812786	0.0145396	0.00726978	−	0.999974i	$-0.497686\pi$
$5$	0.812786	0.0145396	0.00726978	+	0.999974i	$0.497686\pi$
$6$	19.9157	0.225849
$7$	105.563	0.814268	0.407134	−	0.913368i	$-0.366528\pi$
$7$	105.563	0.814268	0.407134	+	0.913368i	$0.366528\pi$
$8$	64.0000	0.353553
$9$	−218.210	−0.897985
$10$	3.25115	0.0102810
$11$	−290.679	−0.724322	−0.362161	−	0.932116i	$-0.617961\pi$
$11$	−290.679	−0.724322	−0.362161	+	0.932116i	$0.617961\pi$
$12$	79.6628	0.159699
$13$	−328.687	−0.539416	−0.269708	−	0.962942i	$-0.586927\pi$
$13$	−328.687	−0.539416	−0.269708	+	0.962942i	$0.586927\pi$
$14$	422.253	0.575774
$15$	4.04681	0.00464391
$16$	256.000	0.250000
$17$	589.620	0.494823	0.247412	−	0.968910i	$-0.420420\pi$
$17$	589.620	0.494823	0.247412	+	0.968910i	$0.420420\pi$
$18$	−872.841	−0.634971
$19$	0	0
$19$	0	0
$20$	13.0046	0.00726978
$21$	525.591	0.260076
$22$	−1162.71	−0.512173
$23$	1925.74	0.759064	0.379532	−	0.925179i	$-0.376085\pi$
$23$	1925.74	0.759064	0.379532	+	0.925179i	$0.376085\pi$
$24$	318.651	0.112924
$25$	−3124.34	−0.999789
$26$	−1314.75	−0.381425
$27$	−2296.33	−0.606213
$28$	1689.01	0.407134
$29$	−271.975	−0.0600529	−0.0300264	−	0.999549i	$-0.509559\pi$
$29$	−271.975	−0.0600529	−0.0300264	+	0.999549i	$0.509559\pi$
$30$	16.1872	0.00328374
$31$	648.509	0.121202	0.0606012	−	0.998162i	$-0.480698\pi$
$31$	648.509	0.121202	0.0606012	+	0.998162i	$0.480698\pi$
$32$	1024.00	0.176777
$33$	−1447.27	−0.231347
$34$	2358.48	0.349893
$35$	85.8003	0.0118391
$36$	−3491.36	−0.448992
$37$	−5823.98	−0.699384	−0.349692	−	0.936865i	$-0.613714\pi$
$37$	−5823.98	−0.699384	−0.349692	+	0.936865i	$0.613714\pi$
$38$	0	0
$39$	−1636.51	−0.172289
$40$	52.0183	0.00514051
$41$	−4060.95	−0.377284	−0.188642	−	0.982046i	$-0.560409\pi$
$41$	−4060.95	−0.377284	−0.188642	+	0.982046i	$0.560409\pi$
$42$	2102.37	0.183901
$43$	−5472.63	−0.451362	−0.225681	−	0.974201i	$-0.572461\pi$
$43$	−5472.63	−0.451362	−0.225681	+	0.974201i	$0.572461\pi$
$44$	−4650.86	−0.362161
$45$	−177.358	−0.0130563
$46$	7702.96	0.536739
$47$	−20375.1	−1.34541	−0.672704	−	0.739911i	$-0.734867\pi$
$47$	−20375.1	−1.34541	−0.672704	+	0.739911i	$0.734867\pi$
$48$	1274.61	0.0798496
$49$	−5663.42	−0.336968
$50$	−12497.4	−0.706957
$51$	2935.68	0.158046
$52$	−5258.99	−0.269708
$53$	3684.03	0.180149	0.0900747	−	0.995935i	$-0.471289\pi$
$53$	3684.03	0.180149	0.0900747	+	0.995935i	$0.471289\pi$
$54$	−9185.33	−0.428657
$55$	−236.260	−0.0105313
$56$	6756.04	0.287887
$57$	0	0
$58$	−1087.90	−0.0424638
$59$	−22923.1	−0.857320	−0.428660	−	0.903466i	$-0.641014\pi$
$59$	−22923.1	−0.857320	−0.428660	+	0.903466i	$0.641014\pi$
$60$	64.7489	0.00232196
$61$	−34271.6	−1.17926	−0.589630	−	0.807673i	$-0.700726\pi$
$61$	−34271.6	−1.17926	−0.589630	+	0.807673i	$0.700726\pi$
$62$	2594.03	0.0857031
$63$	−23035.0	−0.731200
$64$	4096.00	0.125000
$65$	−267.152	−0.00784287
$66$	−5789.07	−0.163587
$67$	−20642.7	−0.561798	−0.280899	−	0.959737i	$-0.590633\pi$
$67$	−20642.7	−0.561798	−0.280899	+	0.959737i	$0.590633\pi$
$68$	9433.93	0.247412
$69$	9588.12	0.242444
$70$	343.201	0.00837151
$71$	−22120.9	−0.520784	−0.260392	−	0.965503i	$-0.583852\pi$
$71$	−22120.9	−0.520784	−0.260392	+	0.965503i	$0.583852\pi$
$72$	−13965.5	−0.317486
$73$	80283.9	1.76328	0.881640	−	0.471923i	$-0.156440\pi$
$73$	80283.9	1.76328	0.881640	+	0.471923i	$0.156440\pi$
$74$	−23295.9	−0.494539
$75$	−15555.9	−0.319331
$76$	0	0
$77$	−30685.0	−0.589792
$78$	−6546.03	−0.121826
$79$	6756.41	0.121800	0.0609001	−	0.998144i	$-0.480603\pi$
$79$	6756.41	0.121800	0.0609001	+	0.998144i	$0.480603\pi$
$80$	208.073	0.00363489
$81$	41591.8	0.704361
$82$	−16243.8	−0.266780
$83$	60676.0	0.966767	0.483384	−	0.875409i	$-0.339408\pi$
$83$	60676.0	0.966767	0.483384	+	0.875409i	$0.339408\pi$
$84$	8409.46	0.130038
$85$	479.236	0.00719452
$86$	−21890.5	−0.319161
$87$	−1354.14	−0.0191808
$88$	−18603.4	−0.256086
$89$	42096.0	0.563334	0.281667	−	0.959512i	$-0.409113\pi$
$89$	42096.0	0.563334	0.281667	+	0.959512i	$0.409113\pi$
$90$	−709.433	−0.00923220
$91$	−34697.2	−0.439229
$92$	30811.9	0.379532
$93$	3228.88	0.0387119
$94$	−81500.2	−0.951348
$95$	0	0
$96$	5098.42	0.0564622
$97$	−34067.5	−0.367630	−0.183815	−	0.982961i	$-0.558845\pi$
$97$	−34067.5	−0.367630	−0.183815	+	0.982961i	$0.558845\pi$
$98$	−22653.7	−0.238272
$99$	63429.1	0.650430
$100$	−49989.4	−0.499894
$101$	−98550.3	−0.961290	−0.480645	−	0.876915i	$-0.659597\pi$
$101$	−98550.3	−0.961290	−0.480645	+	0.876915i	$0.659597\pi$
$102$	11742.7	0.111755
$103$	126768.	1.17738	0.588689	−	0.808360i	$-0.299644\pi$
$103$	126768.	1.17738	0.588689	+	0.808360i	$0.299644\pi$
$104$	−21035.9	−0.190712
$105$	427.194	0.00378139
$106$	14736.1	0.127385
$107$	−202209.	−1.70742	−0.853712	−	0.520746i	$-0.825654\pi$
$107$	−202209.	−1.70742	−0.853712	+	0.520746i	$0.825654\pi$
$108$	−36741.3	−0.303107
$109$	147664.	1.19045	0.595223	−	0.803560i	$-0.297064\pi$
$109$	147664.	1.19045	0.595223	+	0.803560i	$0.297064\pi$
$110$	−945.039	−0.00744677
$111$	−28997.2	−0.223382
$112$	27024.2	0.203567
$113$	−234697.	−1.72907	−0.864533	−	0.502577i	$-0.832385\pi$
$113$	−234697.	−1.72907	−0.864533	+	0.502577i	$0.832385\pi$
$114$	0	0
$115$	1565.22	0.0110365
$116$	−4351.60	−0.0300264
$117$	71722.8	0.484387
$118$	−91692.3	−0.606217
$119$	62242.2	0.402919
$120$	258.996	0.00164187
$121$	−76556.9	−0.475358
$122$	−137086.	−0.833863
$123$	−20219.2	−0.120504
$124$	10376.1	0.0606012
$125$	−5079.38	−0.0290761
$126$	−92139.9	−0.517036
$127$	118131.	0.649910	0.324955	−	0.945729i	$-0.394651\pi$
$127$	118131.	0.649910	0.324955	+	0.945729i	$0.394651\pi$
$128$	16384.0	0.0883883
$129$	−27247.8	−0.144164
$130$	−1068.61	−0.00554575
$131$	−209351.	−1.06585	−0.532926	−	0.846162i	$-0.678908\pi$
$131$	−209351.	−1.06585	−0.532926	+	0.846162i	$0.678908\pi$
$132$	−23156.3	−0.115674
$133$	0	0
$134$	−82570.9	−0.397251
$135$	−1866.43	−0.00881408
$136$	37735.7	0.174947
$137$	−111543.	−0.507741	−0.253871	−	0.967238i	$-0.581704\pi$
$137$	−111543.	−0.507741	−0.253871	+	0.967238i	$0.581704\pi$
$138$	38352.5	0.171434
$139$	−237298.	−1.04173	−0.520867	−	0.853638i	$-0.674391\pi$
$139$	−237298.	−1.04173	−0.520867	+	0.853638i	$0.674391\pi$
$140$	1372.80	0.00591955
$141$	−101446.	−0.429721
$142$	−88483.8	−0.368250
$143$	95542.2	0.390711
$144$	−55861.8	−0.224496
$145$	−221.057	−0.000873142 0
$146$	321135.	1.24683
$147$	−28197.8	−0.107627
$148$	−93183.7	−0.349692
$149$	15690.9	0.0579006	0.0289503	−	0.999581i	$-0.490784\pi$
$149$	15690.9	0.0579006	0.0289503	+	0.999581i	$0.490784\pi$
$150$	−62223.4	−0.225801
$151$	−507697.	−1.81202	−0.906008	−	0.423260i	$-0.860886\pi$
$151$	−507697.	−1.81202	−0.906008	+	0.423260i	$0.860886\pi$
$152$	0	0
$153$	−128661.	−0.444344
$154$	−122740.	−0.417046
$155$	527.099	0.00176223
$156$	−26184.1	−0.0861443
$157$	−46077.9	−0.149191	−0.0745956	−	0.997214i	$-0.523767\pi$
$157$	−46077.9	−0.149191	−0.0745956	+	0.997214i	$0.523767\pi$
$158$	27025.6	0.0861258
$159$	18342.5	0.0575394
$160$	832.293	0.00257026
$161$	203287.	0.618081
$162$	166367.	0.498059
$163$	−289027.	−0.852057	−0.426029	−	0.904710i	$-0.640088\pi$
$163$	−289027.	−0.852057	−0.426029	+	0.904710i	$0.640088\pi$
$164$	−64975.2	−0.188642
$165$	−1176.32	−0.00336369
$166$	242704.	0.683608
$167$	38518.9	0.106877	0.0534383	−	0.998571i	$-0.482982\pi$
$167$	38518.9	0.106877	0.0534383	+	0.998571i	$0.482982\pi$
$168$	33637.8	0.0919507
$169$	−263258.	−0.709031
$170$	1916.94	0.00508729
$171$	0	0
$172$	−87562.0	−0.225681
$173$	−433744.	−1.10184	−0.550920	−	0.834558i	$-0.685723\pi$
$173$	−433744.	−1.10184	−0.550920	+	0.834558i	$0.685723\pi$
$174$	−5416.57	−0.0135629
$175$	−329815.	−0.814096
$176$	−74413.7	−0.181080
$177$	−114132.	−0.273826
$178$	168384.	0.398337
$179$	613877.	1.43202	0.716010	−	0.698090i	$-0.245966\pi$
$179$	613877.	1.43202	0.716010	+	0.698090i	$0.245966\pi$
$180$	−2837.73	−0.00652815
$181$	445528.	1.01083	0.505415	−	0.862876i	$-0.331339\pi$
$181$	445528.	1.01083	0.505415	+	0.862876i	$0.331339\pi$
$182$	−138789.	−0.310582
$183$	−170636.	−0.376654
$184$	123247.	0.268370
$185$	−4733.65	−0.0101687
$186$	12915.5	0.0273734
$187$	−171390.	−0.358411
$188$	−326001.	−0.672704
$189$	−242408.	−0.493620
$190$	0	0
$191$	−328924.	−0.652397	−0.326199	−	0.945301i	$-0.605768\pi$
$191$	−328924.	−0.652397	−0.326199	+	0.945301i	$0.605768\pi$
$192$	20393.7	0.0399248
$193$	598118.	1.15583	0.577914	−	0.816098i	$-0.303867\pi$
$193$	598118.	1.15583	0.577914	+	0.816098i	$0.303867\pi$
$194$	−136270.	−0.259954
$195$	−1330.13	−0.00250500
$196$	−90614.7	−0.168484
$197$	941928.	1.72923	0.864614	−	0.502436i	$-0.167563\pi$
$197$	941928.	1.72923	0.864614	+	0.502436i	$0.167563\pi$
$198$	253716.	0.459923
$199$	456247.	0.816709	0.408354	−	0.912823i	$-0.366103\pi$
$199$	456247.	0.816709	0.408354	+	0.912823i	$0.366103\pi$
$200$	−199958.	−0.353479
$201$	−102779.	−0.179437
$202$	−394201.	−0.679735
$203$	−28710.5	−0.0488991
$204$	46970.8	0.0790229
$205$	−3300.69	−0.00548554
$206$	507071.	0.832532
$207$	−420216.	−0.681628
$208$	−84143.8	−0.134854
$209$	0	0
$210$	1708.77	0.00267385
$211$	1.21414e6	1.87743	0.938713	−	0.344701i	$-0.112020\pi$
$211$	1.21414e6	1.87743	0.938713	+	0.344701i	$0.112020\pi$
$212$	58944.4	0.0900747
$213$	−110139.	−0.166338
$214$	−808836.	−1.20733
$215$	−4448.08	−0.00656260
$216$	−146965.	−0.214329
$217$	68458.6	0.0986913
$218$	590658.	0.841773
$219$	399728.	0.563188
$220$	−3780.16	−0.00526566
$221$	−193800.	−0.266916
$222$	−115989.	−0.157955
$223$	−789931.	−1.06372	−0.531860	−	0.846832i	$-0.678507\pi$
$223$	−789931.	−1.06372	−0.531860	+	0.846832i	$0.678507\pi$
$224$	108097.	0.143944
$225$	681763.	0.897795
$226$	−938788.	−1.22263
$227$	−1.08757e6	−1.40085	−0.700426	−	0.713725i	$-0.747007\pi$
$227$	−1.08757e6	−1.40085	−0.700426	+	0.713725i	$0.747007\pi$
$228$	0	0
$229$	−1.57844e6	−1.98902	−0.994508	−	0.104662i	$-0.966624\pi$
$229$	−1.57844e6	−1.98902	−0.994508	+	0.104662i	$0.966624\pi$
$230$	6260.86	0.00780395
$231$	−152778.	−0.188379
$232$	−17406.4	−0.0212319
$233$	−456945.	−0.551409	−0.275705	−	0.961242i	$-0.588911\pi$
$233$	−456945.	−0.551409	−0.275705	+	0.961242i	$0.588911\pi$
$234$	286891.	0.342513
$235$	−16560.6	−0.0195617
$236$	−366769.	−0.428660
$237$	33639.7	0.0389028
$238$	248969.	0.284907
$239$	−862241.	−0.976414	−0.488207	−	0.872728i	$-0.662349\pi$
$239$	−862241.	−0.976414	−0.488207	+	0.872728i	$0.662349\pi$
$240$	1035.98	0.00116098
$241$	796433.	0.883297	0.441648	−	0.897188i	$-0.354394\pi$
$241$	796433.	0.883297	0.441648	+	0.897188i	$0.354394\pi$
$242$	−306228.	−0.336129
$243$	765092.	0.831185
$244$	−548345.	−0.589630
$245$	−4603.15	−0.00489937
$246$	−80876.7	−0.0852091
$247$	0	0
$248$	41504.6	0.0428515
$249$	302102.	0.308784
$250$	−20317.5	−0.0205599
$251$	−1.01685e6	−1.01876	−0.509379	−	0.860542i	$-0.670125\pi$
$251$	−1.01685e6	−1.01876	−0.509379	+	0.860542i	$0.670125\pi$
$252$	−368559.	−0.365600
$253$	−559772.	−0.549806
$254$	472523.	0.459556
$255$	2386.08	0.00229792
$256$	65536.0	0.0625000
$257$	1.45872e6	1.37765	0.688827	−	0.724926i	$-0.258126\pi$
$257$	1.45872e6	1.37765	0.688827	+	0.724926i	$0.258126\pi$
$258$	−108991.	−0.101939
$259$	−614798.	−0.569486
$260$	−4274.43	−0.00392144
$261$	59347.7	0.0539265
$262$	−837404.	−0.753671
$263$	168445.	0.150165	0.0750826	−	0.997177i	$-0.476078\pi$
$263$	168445.	0.150165	0.0750826	+	0.997177i	$0.476078\pi$
$264$	−92625.2	−0.0817936
$265$	2994.33	0.00261929
$266$	0	0
$267$	209593.	0.179928
$268$	−330283.	−0.280899
$269$	1.68333e6	1.41837	0.709185	−	0.705023i	$-0.249063\pi$
$269$	1.68333e6	1.41837	0.709185	+	0.705023i	$0.249063\pi$
$270$	−7465.71	−0.00623249
$271$	−517950.	−0.428415	−0.214208	−	0.976788i	$-0.568717\pi$
$271$	−517950.	−0.428415	−0.214208	+	0.976788i	$0.568717\pi$
$272$	150943.	0.123706
$273$	−172755.	−0.140289
$274$	−446174.	−0.359027
$275$	908179.	0.724169
$276$	153410.	0.121222
$277$	−2.38324e6	−1.86624	−0.933122	−	0.359560i	$-0.882927\pi$
$277$	−2.38324e6	−1.86624	−0.933122	+	0.359560i	$0.882927\pi$
$278$	−949191.	−0.736617
$279$	−141511.	−0.108838
$280$	5491.22	0.00418575
$281$	2.56528e6	1.93807	0.969034	−	0.246929i	$-0.0794215\pi$
$281$	2.56528e6	1.93807	0.969034	+	0.246929i	$0.0794215\pi$
$282$	−405784.	−0.303859
$283$	−335501.	−0.249016	−0.124508	−	0.992219i	$-0.539735\pi$
$283$	−335501.	−0.249016	−0.124508	+	0.992219i	$0.539735\pi$
$284$	−353935.	−0.260392
$285$	0	0
$286$	382169.	0.276274
$287$	−428687.	−0.307210
$288$	−223447.	−0.158743
$289$	−1.07220e6	−0.755150
$290$	−884.230	−0.000617405 0
$291$	−169620.	−0.117421
$292$	1.28454e6	0.881640
$293$	450390.	0.306493	0.153246	−	0.988188i	$-0.451027\pi$
$293$	450390.	0.306493	0.153246	+	0.988188i	$0.451027\pi$
$294$	−112791.	−0.0761038
$295$	−18631.6	−0.0124651
$296$	−372735.	−0.247270
$297$	667495.	0.439093
$298$	62763.7	0.0409419
$299$	−632965.	−0.409451
$300$	−248894.	−0.159665
$301$	−577708.	−0.367529
$302$	−2.03079e6	−1.28129
$303$	−490675.	−0.307035
$304$	0	0
$305$	−27855.5	−0.0171459
$306$	−514645.	−0.314199
$307$	1.92768e6	1.16732	0.583658	−	0.812000i	$-0.301621\pi$
$307$	1.92768e6	1.16732	0.583658	+	0.812000i	$0.301621\pi$
$308$	−490959.	−0.294896
$309$	631168.	0.376053
$310$	2108.40	0.00124609
$311$	−1.70392e6	−0.998958	−0.499479	−	0.866326i	$-0.666475\pi$
$311$	−1.70392e6	−0.998958	−0.499479	+	0.866326i	$0.666475\pi$
$312$	−104736.	−0.0609132
$313$	1.61683e6	0.932831	0.466416	−	0.884566i	$-0.345545\pi$
$313$	1.61683e6	0.932831	0.466416	+	0.884566i	$0.345545\pi$
$314$	−184312.	−0.105494
$315$	−18722.5	−0.0106313
$316$	108103.	0.0609001
$317$	−578761.	−0.323483	−0.161741	−	0.986833i	$-0.551711\pi$
$317$	−578761.	−0.323483	−0.161741	+	0.986833i	$0.551711\pi$
$318$	73370.0	0.0406865
$319$	79057.3	0.0434976
$320$	3329.17	0.00181745
$321$	−1.00678e6	−0.545348
$322$	813149.	0.437049
$323$	0	0
$324$	665469.	0.352181
$325$	1.02693e6	0.539302
$326$	−1.15611e6	−0.602495
$327$	735211.	0.380227
$328$	−259901.	−0.133390
$329$	−2.15086e6	−1.09552
$330$	−4705.28	−0.00237849
$331$	311892.	0.156471	0.0782355	−	0.996935i	$-0.475071\pi$
$331$	311892.	0.156471	0.0782355	+	0.996935i	$0.475071\pi$
$332$	970816.	0.483384
$333$	1.27085e6	0.628036
$334$	154076.	0.0755731
$335$	−16778.1	−0.00816829
$336$	134551.	0.0650190
$337$	−19953.9	−0.00957092	−0.00478546	−	0.999989i	$-0.501523\pi$
$337$	−19953.9	−0.00957092	−0.00478546	+	0.999989i	$0.501523\pi$
$338$	−1.05303e6	−0.501360
$339$	−1.16854e6	−0.552261
$340$	7667.77	0.00359726
$341$	−188508.	−0.0877896
$342$	0	0
$343$	−2.37205e6	−1.08865
$344$	−350248.	−0.159580
$345$	7793.10	0.00352503
$346$	−1.73498e6	−0.779118
$347$	235610.	0.105044	0.0525219	−	0.998620i	$-0.483274\pi$
$347$	235610.	0.105044	0.0525219	+	0.998620i	$0.483274\pi$
$348$	−21666.3	−0.00959039
$349$	518804.	0.228003	0.114001	−	0.993481i	$-0.463633\pi$
$349$	518804.	0.228003	0.114001	+	0.993481i	$0.463633\pi$
$350$	−1.31926e6	−0.575653
$351$	754774.	0.327001
$352$	−297655.	−0.128043
$353$	2.71982e6	1.16172	0.580862	−	0.814002i	$-0.302715\pi$
$353$	2.71982e6	1.16172	0.580862	+	0.814002i	$0.302715\pi$
$354$	−456529.	−0.193625
$355$	−17979.6	−0.00757197
$356$	673536.	0.281667
$357$	309899.	0.128692
$358$	2.45551e6	1.01259
$359$	3.72097e6	1.52377	0.761886	−	0.647711i	$-0.224274\pi$
$359$	3.72097e6	1.52377	0.761886	+	0.647711i	$0.224274\pi$
$360$	−11350.9	−0.00461610
$361$	0	0
$362$	1.78211e6	0.714765
$363$	−381171.	−0.151829
$364$	−555155.	−0.219614
$365$	65253.6	0.0256373
$366$	−682543.	−0.266334
$367$	1.27638e6	0.494669	0.247335	−	0.968930i	$-0.420445\pi$
$367$	1.27638e6	0.494669	0.247335	+	0.968930i	$0.420445\pi$
$368$	492990.	0.189766
$369$	886141.	0.338795
$370$	−18934.6	−0.00719039
$371$	388897.	0.146690
$372$	51662.0	0.0193559
$373$	−2.06567e6	−0.768757	−0.384378	−	0.923176i	$-0.625584\pi$
$373$	−2.06567e6	−0.768757	−0.384378	+	0.923176i	$0.625584\pi$
$374$	−685560.	−0.253435
$375$	−25289.9	−0.00928685
$376$	−1.30400e6	−0.475674
$377$	89394.5	0.0323935
$378$	−969632.	−0.349042
$379$	−775646.	−0.277374	−0.138687	−	0.990336i	$-0.544288\pi$
$379$	−775646.	−0.277374	−0.138687	+	0.990336i	$0.544288\pi$
$380$	0	0
$381$	588164.	0.207580
$382$	−1.31570e6	−0.461314
$383$	2.65307e6	0.924170	0.462085	−	0.886836i	$-0.347102\pi$
$383$	2.65307e6	0.924170	0.462085	+	0.886836i	$0.347102\pi$
$384$	81574.8	0.0282311
$385$	−24940.3	−0.00857532
$386$	2.39247e6	0.817294
$387$	1.19418e6	0.405316
$388$	−545081.	−0.183815
$389$	−1.87108e6	−0.626928	−0.313464	−	0.949600i	$-0.601489\pi$
$389$	−1.87108e6	−0.626928	−0.313464	+	0.949600i	$0.601489\pi$
$390$	−5320.52	−0.00177130
$391$	1.13546e6	0.375603
$392$	−362459.	−0.119136
$393$	−1.04234e6	−0.340431
$394$	3.76771e6	1.22275
$395$	5491.52	0.00177092
$396$	1.01487e6	0.325215
$397$	5.29294e6	1.68547	0.842735	−	0.538329i	$-0.180944\pi$
$397$	5.29294e6	1.68547	0.842735	+	0.538329i	$0.180944\pi$
$398$	1.82499e6	0.577500
$399$	0	0
$400$	−799831.	−0.249947
$401$	3.09763e6	0.961985	0.480993	−	0.876725i	$-0.340276\pi$
$401$	3.09763e6	0.961985	0.480993	+	0.876725i	$0.340276\pi$
$402$	−411114.	−0.126881
$403$	−213156.	−0.0653785
$404$	−1.57680e6	−0.480645
$405$	33805.3	0.0102411
$406$	−114842.	−0.0345769
$407$	1.69291e6	0.506579
$408$	187883.	0.0558776
$409$	−3.46566e6	−1.02442	−0.512210	−	0.858860i	$-0.671173\pi$
$409$	−3.46566e6	−1.02442	−0.512210	+	0.858860i	$0.671173\pi$
$410$	−13202.7	−0.00387886
$411$	−555367.	−0.162172
$412$	2.02828e6	0.588689
$413$	−2.41983e6	−0.698088
$414$	−1.68087e6	−0.481984
$415$	49316.7	0.0140564
$416$	−336575.	−0.0953561
$417$	−1.18149e6	−0.332728
$418$	0	0
$419$	−1.19240e6	−0.331808	−0.165904	−	0.986142i	$-0.553054\pi$
$419$	−1.19240e6	−0.331808	−0.165904	+	0.986142i	$0.553054\pi$
$420$	6835.10	0.00189069
$421$	−4.80219e6	−1.32049	−0.660244	−	0.751051i	$-0.729547\pi$
$421$	−4.80219e6	−1.32049	−0.660244	+	0.751051i	$0.729547\pi$
$422$	4.85656e6	1.32754
$423$	4.44605e6	1.20816
$424$	235778.	0.0636924
$425$	−1.84217e6	−0.494719
$426$	−440554.	−0.117618
$427$	−3.61782e6	−0.960234
$428$	−3.23534e6	−0.853712
$429$	475698.	0.124792
$430$	−17792.3	−0.00464046
$431$	278669.	0.0722597	0.0361299	−	0.999347i	$-0.488497\pi$
$431$	278669.	0.0722597	0.0361299	+	0.999347i	$0.488497\pi$
$432$	−587861.	−0.151553
$433$	2.31602e6	0.593640	0.296820	−	0.954934i	$-0.404074\pi$
$433$	2.31602e6	0.593640	0.296820	+	0.954934i	$0.404074\pi$
$434$	273834.	0.0697853
$435$	−1100.63	−0.000278880 0
$436$	2.36263e6	0.595223
$437$	0	0
$438$	1.59891e6	0.398234
$439$	990192.	0.245221	0.122611	−	0.992455i	$-0.460873\pi$
$439$	990192.	0.245221	0.122611	+	0.992455i	$0.460873\pi$
$440$	−15120.6	−0.00372338
$441$	1.23582e6	0.302592
$442$	−775202.	−0.188738
$443$	−1.75365e6	−0.424555	−0.212277	−	0.977209i	$-0.568088\pi$
$443$	−1.75365e6	−0.424555	−0.212277	+	0.977209i	$0.568088\pi$
$444$	−463955.	−0.111691
$445$	34215.0	0.00819063
$446$	−3.15973e6	−0.752163
$447$	78124.0	0.0184933
$448$	432387.	0.101783
$449$	7.07068e6	1.65518	0.827591	−	0.561332i	$-0.189711\pi$
$449$	7.07068e6	1.65518	0.827591	+	0.561332i	$0.189711\pi$
$450$	2.72705e6	0.634837
$451$	1.18043e6	0.273275
$452$	−3.75515e6	−0.864533
$453$	−2.52779e6	−0.578755
$454$	−4.35028e6	−0.990552
$455$	−28201.4	−0.00638620
$456$	0	0
$457$	2.87879e6	0.644791	0.322396	−	0.946605i	$-0.395512\pi$
$457$	2.87879e6	0.644791	0.322396	+	0.946605i	$0.395512\pi$
$458$	−6.31374e6	−1.40645
$459$	−1.35396e6	−0.299968
$460$	25043.5	0.00551823
$461$	4.56572e6	1.00059	0.500296	−	0.865854i	$-0.333224\pi$
$461$	4.56572e6	1.00059	0.500296	+	0.865854i	$0.333224\pi$
$462$	−611113.	−0.133204
$463$	−7.35800e6	−1.59517	−0.797586	−	0.603206i	$-0.793890\pi$
$463$	−7.35800e6	−1.59517	−0.797586	+	0.603206i	$0.793890\pi$
$464$	−69625.5	−0.0150132
$465$	2624.39	0.000562854 0
$466$	−1.82778e6	−0.389905
$467$	1.60478e6	0.340506	0.170253	−	0.985400i	$-0.445542\pi$
$467$	1.60478e6	0.340506	0.170253	+	0.985400i	$0.445542\pi$
$468$	1.14756e6	0.242194
$469$	−2.17911e6	−0.457454
$470$	−66242.3	−0.0138322
$471$	−229418.	−0.0476515
$472$	−1.46708e6	−0.303108
$473$	1.59078e6	0.326931
$474$	134559.	0.0275084
$475$	0	0
$476$	995875.	0.201459
$477$	−803892.	−0.161771
$478$	−3.44896e6	−0.690429
$479$	3.93159e6	0.782943	0.391471	−	0.920190i	$-0.371966\pi$
$479$	3.93159e6	0.782943	0.391471	+	0.920190i	$0.371966\pi$
$480$	4143.93	0.000820936 0
$481$	1.91427e6	0.377259
$482$	3.18573e6	0.624585
$483$	1.01215e6	0.197414
$484$	−1.22491e6	−0.237679
$485$	−27689.6	−0.00534519
$486$	3.06037e6	0.587737
$487$	7.36984e6	1.40811	0.704053	−	0.710147i	$-0.251372\pi$
$487$	7.36984e6	1.40811	0.704053	+	0.710147i	$0.251372\pi$
$488$	−2.19338e6	−0.416931
$489$	−1.43904e6	−0.272146
$490$	−18412.6	−0.00346438
$491$	326410.	0.0611026	0.0305513	−	0.999533i	$-0.490274\pi$
$491$	326410.	0.0611026	0.0305513	+	0.999533i	$0.490274\pi$
$492$	−323507.	−0.0602519
$493$	−160362.	−0.0297156
$494$	0	0
$495$	51554.3	0.00945697
$496$	166018.	0.0303006
$497$	−2.33516e6	−0.424058
$498$	1.20841e6	0.218343
$499$	5.67715e6	1.02065	0.510327	−	0.859980i	$-0.329524\pi$
$499$	5.67715e6	1.02065	0.510327	+	0.859980i	$0.329524\pi$
$500$	−81270.1	−0.0145380
$501$	191783.	0.0341362
$502$	−4.06738e6	−0.720370
$503$	−9.40425e6	−1.65731	−0.828656	−	0.559758i	$-0.810894\pi$
$503$	−9.40425e6	−1.65731	−0.828656	+	0.559758i	$0.810894\pi$
$504$	−1.47424e6	−0.258518
$505$	−80100.4	−0.0139767
$506$	−2.23909e6	−0.388772
$507$	−1.31074e6	−0.226463
$508$	1.89009e6	0.324955
$509$	−4.22529e6	−0.722873	−0.361436	−	0.932397i	$-0.617714\pi$
$509$	−4.22529e6	−0.722873	−0.361436	+	0.932397i	$0.617714\pi$
$510$	9544.32	0.00162487
$511$	8.47502e6	1.43578
$512$	262144.	0.0441942
$513$	0	0
$514$	5.83489e6	0.974148
$515$	103035.	0.0171186
$516$	−435965.	−0.0720821
$517$	5.92259e6	0.974509
$518$	−2.45919e6	−0.402687
$519$	−2.15958e6	−0.351926
$520$	−17097.7	−0.00277287
$521$	110002.	0.0177544	0.00887721	−	0.999961i	$-0.497174\pi$
$521$	110002.	0.0177544	0.00887721	+	0.999961i	$0.497174\pi$
$522$	237391.	0.0381318
$523$	37312.4	0.00596485	0.00298242	−	0.999996i	$-0.499051\pi$
$523$	37312.4	0.00596485	0.00298242	+	0.999996i	$0.499051\pi$
$524$	−3.34962e6	−0.532926
$525$	−1.64213e6	−0.260021
$526$	673781.	0.106183
$527$	382374.	0.0599738
$528$	−370501.	−0.0578368
$529$	−2.72787e6	−0.423822
$530$	11977.3	0.00185212
$531$	5.00205e6	0.769860
$532$	0	0
$533$	1.33478e6	0.203513
$534$	838371.	0.127228
$535$	−164353.	−0.0248252
$536$	−1.32113e6	−0.198625
$537$	3.05645e6	0.457385
$538$	6.73333e6	1.00294
$539$	1.64624e6	0.244073
$540$	−29862.9	−0.00440704
$541$	−9.75474e6	−1.43292	−0.716461	−	0.697627i	$-0.754239\pi$
$541$	−9.75474e6	−1.43292	−0.716461	+	0.697627i	$0.754239\pi$
$542$	−2.07180e6	−0.302935
$543$	2.21825e6	0.322858
$544$	603771.	0.0874733
$545$	120020.	0.0173086
$546$	−691019.	−0.0991993
$547$	8.06321e6	1.15223	0.576116	−	0.817368i	$-0.304568\pi$
$547$	8.06321e6	1.15223	0.576116	+	0.817368i	$0.304568\pi$
$548$	−1.78469e6	−0.253871
$549$	7.47841e6	1.05896
$550$	3.63272e6	0.512064
$551$	0	0
$552$	613640.	0.0857168
$553$	713228.	0.0991780
$554$	−9.53296e6	−1.31963
$555$	−23568.5	−0.00324788
$556$	−3.79677e6	−0.520867
$557$	−2.35589e6	−0.321748	−0.160874	−	0.986975i	$-0.551431\pi$
$557$	−2.35589e6	−0.321748	−0.160874	+	0.986975i	$0.551431\pi$
$558$	−566045.	−0.0769601
$559$	1.79878e6	0.243472
$560$	21964.9	0.00295978
$561$	−853339.	−0.114476
$562$	1.02611e7	1.37042
$563$	1.41310e7	1.87890	0.939448	−	0.342692i	$-0.111339\pi$
$563$	1.41310e7	1.87890	0.939448	+	0.342692i	$0.111339\pi$
$564$	−1.62314e6	−0.214861
$565$	−190759.	−0.0251399
$566$	−1.34200e6	−0.176081
$567$	4.39056e6	0.573539
$568$	−1.41574e6	−0.184125
$569$	6.61092e6	0.856015	0.428007	−	0.903775i	$-0.359216\pi$
$569$	6.61092e6	0.856015	0.428007	+	0.903775i	$0.359216\pi$
$570$	0	0
$571$	−6.81165e6	−0.874303	−0.437151	−	0.899388i	$-0.644013\pi$
$571$	−6.81165e6	−0.874303	−0.437151	+	0.899388i	$0.644013\pi$
$572$	1.52868e6	0.195355
$573$	−1.63769e6	−0.208375
$574$	−1.71475e6	−0.217230
$575$	−6.01667e6	−0.758903
$576$	−893789.	−0.112248
$577$	−8.42895e6	−1.05398	−0.526992	−	0.849870i	$-0.676680\pi$
$577$	−8.42895e6	−1.05398	−0.526992	+	0.849870i	$0.676680\pi$
$578$	−4.28882e6	−0.533972
$579$	2.97798e6	0.369170
$580$	−3536.92	−0.000436571 0
$581$	6.40515e6	0.787207
$582$	−678479.	−0.0830289
$583$	−1.07087e6	−0.130486
$584$	5.13817e6	0.623413
$585$	58295.3	0.00704278
$586$	1.80156e6	0.216723
$587$	7.01281e6	0.840034	0.420017	−	0.907516i	$-0.362024\pi$
$587$	7.01281e6	0.840034	0.420017	+	0.907516i	$0.362024\pi$
$588$	−451164.	−0.0538135
$589$	0	0
$590$	−74526.2	−0.00881413
$591$	4.68979e6	0.552313
$592$	−1.49094e6	−0.174846
$593$	1.27944e7	1.49411	0.747057	−	0.664760i	$-0.231466\pi$
$593$	1.27944e7	1.49411	0.747057	+	0.664760i	$0.231466\pi$
$594$	2.66998e6	0.310486
$595$	50589.6	0.00585826
$596$	251055.	0.0289503
$597$	2.27162e6	0.260855
$598$	−2.53186e6	−0.289526
$599$	−1.56884e7	−1.78653	−0.893267	−	0.449526i	$-0.851593\pi$
$599$	−1.56884e7	−1.78653	−0.893267	+	0.449526i	$0.851593\pi$
$600$	−995575.	−0.112901
$601$	−8.53094e6	−0.963409	−0.481705	−	0.876334i	$-0.659982\pi$
$601$	−8.53094e6	−0.963409	−0.481705	+	0.876334i	$0.659982\pi$
$602$	−2.31083e6	−0.259882
$603$	4.50445e6	0.504486
$604$	−8.12315e6	−0.906008
$605$	−62224.4	−0.00691150
$606$	−1.96270e6	−0.217106
$607$	1.00742e7	1.10979	0.554893	−	0.831921i	$-0.312759\pi$
$607$	1.00742e7	1.10979	0.554893	+	0.831921i	$0.312759\pi$
$608$	0	0
$609$	−142948.	−0.0156183
$610$	−111422.	−0.0121240
$611$	6.69701e6	0.725735
$612$	−2.05858e6	−0.222172
$613$	7.04420e6	0.757148	0.378574	−	0.925571i	$-0.376415\pi$
$613$	7.04420e6	0.757148	0.378574	+	0.925571i	$0.376415\pi$
$614$	7.71070e6	0.825416
$615$	−16433.9	−0.00175207
$616$	−1.96384e6	−0.208523
$617$	1.74561e7	1.84601	0.923004	−	0.384790i	$-0.125726\pi$
$617$	1.74561e7	1.84601	0.923004	+	0.384790i	$0.125726\pi$
$618$	2.52467e6	0.265909
$619$	1.08464e7	1.13779	0.568893	−	0.822411i	$-0.307372\pi$
$619$	1.08464e7	1.13779	0.568893	+	0.822411i	$0.307372\pi$
$620$	8433.58	0.000881116 0
$621$	−4.42214e6	−0.460154
$622$	−6.81566e6	−0.706370
$623$	4.44378e6	0.458704
$624$	−418946.	−0.0430721
$625$	9.75943e6	0.999366
$626$	6.46731e6	0.659611
$627$	0	0
$628$	−737246.	−0.0745956
$629$	−3.43394e6	−0.346072
$630$	−74890.0	−0.00751749
$631$	−3.84348e6	−0.384283	−0.192141	−	0.981367i	$-0.561543\pi$
$631$	−3.84348e6	−0.384283	−0.192141	+	0.981367i	$0.561543\pi$
$632$	432410.	0.0430629
$633$	6.04512e6	0.599646
$634$	−2.31504e6	−0.228737
$635$	96015.0	0.00944942
$636$	293480.	0.0287697
$637$	1.86149e6	0.181766
$638$	316229.	0.0307574
$639$	4.82702e6	0.467656
$640$	13316.7	0.00128513
$641$	−7.80667e6	−0.750448	−0.375224	−	0.926934i	$-0.622434\pi$
$641$	−7.80667e6	−0.750448	−0.375224	+	0.926934i	$0.622434\pi$
$642$	−4.02714e6	−0.385619
$643$	−847815.	−0.0808675	−0.0404337	−	0.999182i	$-0.512874\pi$
$643$	−847815.	−0.0808675	−0.0404337	+	0.999182i	$0.512874\pi$
$644$	3.25260e6	0.309041
$645$	−22146.7	−0.00209609
$646$	0	0
$647$	2.07939e7	1.95288	0.976440	−	0.215791i	$-0.0692329\pi$
$647$	2.07939e7	1.95288	0.976440	+	0.215791i	$0.0692329\pi$
$648$	2.66188e6	0.249029
$649$	6.66325e6	0.620975
$650$	4.10771e6	0.381344
$651$	340850.	0.0315218
$652$	−4.62442e6	−0.426029
$653$	−1.46615e6	−0.134554	−0.0672769	−	0.997734i	$-0.521431\pi$
$653$	−1.46615e6	−0.134554	−0.0672769	+	0.997734i	$0.521431\pi$
$654$	2.94084e6	0.268861
$655$	−170158.	−0.0154970
$656$	−1.03960e6	−0.0943209
$657$	−1.75188e7	−1.58340
$658$	−8.60342e6	−0.774652
$659$	−8.43819e6	−0.756895	−0.378448	−	0.925623i	$-0.623542\pi$
$659$	−8.43819e6	−0.756895	−0.378448	+	0.925623i	$0.623542\pi$
$660$	−18821.1	−0.00168184
$661$	1.61700e6	0.143948	0.0719741	−	0.997407i	$-0.477070\pi$
$661$	1.61700e6	0.143948	0.0719741	+	0.997407i	$0.477070\pi$
$662$	1.24757e6	0.110642
$663$	−964918.	−0.0852524
$664$	3.88327e6	0.341804
$665$	0	0
$666$	5.08341e6	0.444089
$667$	−523753.	−0.0455839
$668$	616302.	0.0534383
$669$	−3.93301e6	−0.339750
$670$	−67112.5	−0.00577586
$671$	9.96202e6	0.854164
$672$	538206.	0.0459753
$673$	2.33424e6	0.198659	0.0993295	−	0.995055i	$-0.468330\pi$
$673$	2.33424e6	0.198659	0.0993295	+	0.995055i	$0.468330\pi$
$674$	−79815.7	−0.00676766
$675$	7.17452e6	0.606085
$676$	−4.21213e6	−0.354515
$677$	1.24604e7	1.04486	0.522432	−	0.852681i	$-0.325025\pi$
$677$	1.24604e7	1.04486	0.522432	+	0.852681i	$0.325025\pi$
$678$	−4.67416e6	−0.390507
$679$	−3.59628e6	−0.299350
$680$	30671.1	0.00254365
$681$	−5.41493e6	−0.447430
$682$	−754030.	−0.0620766
$683$	−4.76570e6	−0.390909	−0.195454	−	0.980713i	$-0.562618\pi$
$683$	−4.76570e6	−0.390909	−0.195454	+	0.980713i	$0.562618\pi$
$684$	0	0
$685$	−90661.0	−0.00738234
$686$	−9.48819e6	−0.769792
$687$	−7.85892e6	−0.635288
$688$	−1.40099e6	−0.112840
$689$	−1.21089e6	−0.0971755
$690$	31172.4	0.00249257
$691$	2.14576e6	0.170957	0.0854783	−	0.996340i	$-0.472758\pi$
$691$	2.14576e6	0.170957	0.0854783	+	0.996340i	$0.472758\pi$
$692$	−6.93991e6	−0.550920
$693$	6.69577e6	0.529624
$694$	942441.	0.0742772
$695$	−192873.	−0.0151464
$696$	−86665.1	−0.00678143
$697$	−2.39442e6	−0.186689
$698$	2.07522e6	0.161222
$699$	−2.27510e6	−0.176119
$700$	−5.27704e6	−0.407048
$701$	7.29852e6	0.560970	0.280485	−	0.959858i	$-0.409505\pi$
$701$	7.29852e6	0.560970	0.280485	+	0.959858i	$0.409505\pi$
$702$	3.01910e6	0.231225
$703$	0	0
$704$	−1.19062e6	−0.0905402
$705$	−82453.9	−0.00624796
$706$	1.08793e7	0.821463
$707$	−1.04033e7	−0.782748
$708$	−1.82612e6	−0.136913
$709$	2.91021e6	0.217424	0.108712	−	0.994073i	$-0.465327\pi$
$709$	2.91021e6	0.217424	0.108712	+	0.994073i	$0.465327\pi$
$710$	−71918.4	−0.00535419
$711$	−1.47432e6	−0.109375
$712$	2.69414e6	0.199168
$713$	1.24886e6	0.0920004
$714$	1.23960e6	0.0909987
$715$	77655.4	0.00568076
$716$	9.82204e6	0.716010
$717$	−4.29303e6	−0.311865
$718$	1.48839e7	1.07747
$719$	−2.09806e7	−1.51355	−0.756774	−	0.653677i	$-0.773226\pi$
$719$	−2.09806e7	−1.51355	−0.756774	+	0.653677i	$0.773226\pi$
$720$	−45403.7	−0.00326408
$721$	1.33820e7	0.958701
$722$	0	0
$723$	3.96538e6	0.282124
$724$	7.12845e6	0.505415
$725$	849741.	0.0600402
$726$	−1.52469e6	−0.107359
$727$	1.12006e7	0.785971	0.392986	−	0.919545i	$-0.371442\pi$
$727$	1.12006e7	0.785971	0.392986	+	0.919545i	$0.371442\pi$
$728$	−2.22062e6	−0.155291
$729$	−6.29748e6	−0.438882
$730$	261015.	0.0181283
$731$	−3.22677e6	−0.223344
$732$	−2.73017e6	−0.188327
$733$	−2.64626e6	−0.181917	−0.0909583	−	0.995855i	$-0.528993\pi$
$733$	−2.64626e6	−0.181917	−0.0909583	+	0.995855i	$0.528993\pi$
$734$	5.10552e6	0.349784
$735$	−22918.8	−0.00156485
$736$	1.97196e6	0.134185
$737$	6.00040e6	0.406922
$738$	3.54457e6	0.239564
$739$	−2.36780e7	−1.59490	−0.797451	−	0.603384i	$-0.793819\pi$
$739$	−2.36780e7	−1.59490	−0.797451	+	0.603384i	$0.793819\pi$
$740$	−75738.5	−0.00508437
$741$	0	0
$742$	1.55559e6	0.103725
$743$	9.25957e6	0.615345	0.307672	−	0.951492i	$-0.400450\pi$
$743$	9.25957e6	0.615345	0.307672	+	0.951492i	$0.400450\pi$
$744$	206648.	0.0136867
$745$	12753.4	0.000841849 0
$746$	−8.26268e6	−0.543593
$747$	−1.32401e7	−0.868142
$748$	−2.74224e6	−0.179206
$749$	−2.13458e7	−1.39030
$750$	−101159.	−0.00656679
$751$	1.14455e7	0.740515	0.370258	−	0.928929i	$-0.379269\pi$
$751$	1.14455e7	0.740515	0.370258	+	0.928929i	$0.379269\pi$
$752$	−5.21601e6	−0.336352
$753$	−5.06280e6	−0.325389
$754$	357578.	0.0229056
$755$	−412649.	−0.0263459
$756$	−3.87853e6	−0.246810
$757$	1.84236e7	1.16851	0.584257	−	0.811568i	$-0.301386\pi$
$757$	1.84236e7	1.16851	0.584257	+	0.811568i	$0.301386\pi$
$758$	−3.10259e6	−0.196133
$759$	−2.78706e6	−0.175607
$760$	0	0
$761$	−1.15998e7	−0.726085	−0.363043	−	0.931773i	$-0.618262\pi$
$761$	−1.15998e7	−0.726085	−0.363043	+	0.931773i	$0.618262\pi$
$762$	2.35266e6	0.146781
$763$	1.55879e7	0.969342
$764$	−5.26278e6	−0.326199
$765$	−104574.	−0.00646057
$766$	1.06123e7	0.653487
$767$	7.53451e6	0.462452
$768$	326299.	0.0199624
$769$	1.43657e6	0.0876015	0.0438007	−	0.999040i	$-0.486053\pi$
$769$	1.43657e6	0.0876015	0.0438007	+	0.999040i	$0.486053\pi$
$770$	−99761.3	−0.00606366
$771$	7.26288e6	0.440020
$772$	9.56988e6	0.577914
$773$	4.86891e6	0.293078	0.146539	−	0.989205i	$-0.453187\pi$
$773$	4.86891e6	0.293078	0.146539	+	0.989205i	$0.453187\pi$
$774$	4.77673e6	0.286602
$775$	−2.02616e6	−0.121177
$776$	−2.18032e6	−0.129977
$777$	−3.06103e6	−0.181893
$778$	−7.48431e6	−0.443305
$779$	0	0
$780$	−21282.1	−0.00125250
$781$	6.43009e6	0.377215
$782$	4.54183e6	0.265591
$783$	624545.	0.0364048
$784$	−1.44984e6	−0.0842420
$785$	−37451.5	−0.00216918
$786$	−4.16937e6	−0.240721
$787$	−2.95034e6	−0.169799	−0.0848996	−	0.996390i	$-0.527057\pi$
$787$	−2.95034e6	−0.169799	−0.0848996	+	0.996390i	$0.527057\pi$
$788$	1.50709e7	0.864614
$789$	838676.	0.0479625
$790$	21966.1	0.00125223
$791$	−2.47754e7	−1.40792
$792$	4.05946e6	0.229962
$793$	1.12646e7	0.636112
$794$	2.11718e7	1.19181
$795$	14908.5	0.000836598 0
$796$	7.29995e6	0.408354
$797$	−2.02684e7	−1.13025	−0.565123	−	0.825007i	$-0.691171\pi$
$797$	−2.02684e7	−1.13025	−0.565123	+	0.825007i	$0.691171\pi$
$798$	0	0
$799$	−1.20136e7	−0.665740
$800$	−3.19932e6	−0.176739
$801$	−9.18578e6	−0.505865
$802$	1.23905e7	0.680226
$803$	−2.33368e7	−1.27718
$804$	−1.64446e6	−0.0897186
$805$	165229.	0.00898663
$806$	−852624.	−0.0462296
$807$	8.38119e6	0.453025
$808$	−6.30722e6	−0.339867
$809$	−1.58392e7	−0.850865	−0.425433	−	0.904990i	$-0.639878\pi$
$809$	−1.58392e7	−0.850865	−0.425433	+	0.904990i	$0.639878\pi$
$810$	135221.	0.00724156
$811$	−9.54227e6	−0.509448	−0.254724	−	0.967014i	$-0.581985\pi$
$811$	−9.54227e6	−0.509448	−0.254724	+	0.967014i	$0.581985\pi$
$812$	−459368.	−0.0244496
$813$	−2.57884e6	−0.136835
$814$	6.77163e6	0.358205
$815$	−234917.	−0.0123885
$816$	751534.	0.0395114
$817$	0	0
$818$	−1.38626e7	−0.724374
$819$	7.57129e6	0.394421
$820$	−52811.0	−0.00274277
$821$	−431536.	−0.0223439	−0.0111720	−	0.999938i	$-0.503556\pi$
$821$	−431536.	−0.0223439	−0.0111720	+	0.999938i	$0.503556\pi$
$822$	−2.22147e6	−0.114673
$823$	1.60512e6	0.0826053	0.0413026	−	0.999147i	$-0.486849\pi$
$823$	1.60512e6	0.0826053	0.0413026	+	0.999147i	$0.486849\pi$
$824$	8.11314e6	0.416266
$825$	4.52176e6	0.231298
$826$	−9.67933e6	−0.493623
$827$	2.41526e7	1.22801	0.614004	−	0.789303i	$-0.289558\pi$
$827$	2.41526e7	1.22801	0.614004	+	0.789303i	$0.289558\pi$
$828$	−6.72346e6	−0.340814
$829$	−4.33004e6	−0.218829	−0.109415	−	0.993996i	$-0.534898\pi$
$829$	−4.33004e6	−0.218829	−0.109415	+	0.993996i	$0.534898\pi$
$830$	197267.	0.00993936
$831$	−1.18660e7	−0.596075
$832$	−1.34630e6	−0.0674270
$833$	−3.33927e6	−0.166740
$834$	−4.72596e6	−0.235274
$835$	31307.6	0.00155394
$836$	0	0
$837$	−1.48919e6	−0.0734745
$838$	−4.76960e6	−0.234623
$839$	9.66118e6	0.473833	0.236917	−	0.971530i	$-0.423863\pi$
$839$	9.66118e6	0.473833	0.236917	+	0.971530i	$0.423863\pi$
$840$	27340.4	0.00133692
$841$	−2.04372e7	−0.996394
$842$	−1.92088e7	−0.933726
$843$	1.27723e7	0.619015
$844$	1.94262e7	0.938713
$845$	−213973.	−0.0103090
$846$	1.77842e7	0.854296
$847$	−8.08159e6	−0.387069
$848$	943110.	0.0450374
$849$	−1.67044e6	−0.0795354
$850$	−7.36870e6	−0.349819
$851$	−1.12155e7	−0.530877
$852$	−1.76222e6	−0.0831688
$853$	−3.44413e7	−1.62072	−0.810359	−	0.585934i	$-0.800728\pi$
$853$	−3.44413e7	−1.62072	−0.810359	+	0.585934i	$0.800728\pi$
$854$	−1.44713e7	−0.678988
$855$	0	0
$856$	−1.29414e7	−0.603665
$857$	−1.34753e7	−0.626737	−0.313368	−	0.949632i	$-0.601457\pi$
$857$	−1.34753e7	−0.626737	−0.313368	+	0.949632i	$0.601457\pi$
$858$	1.90279e6	0.0882415
$859$	−1.70415e7	−0.787996	−0.393998	−	0.919111i	$-0.628908\pi$
$859$	−1.70415e7	−0.787996	−0.393998	+	0.919111i	$0.628908\pi$
$860$	−71169.2	−0.00328130
$861$	−2.13440e6	−0.0981224
$862$	1.11468e6	0.0510953
$863$	3.30421e7	1.51022	0.755110	−	0.655598i	$-0.227583\pi$
$863$	3.30421e7	1.51022	0.755110	+	0.655598i	$0.227583\pi$
$864$	−2.35144e6	−0.107164
$865$	−352541.	−0.0160203
$866$	9.26409e6	0.419767
$867$	−5.33843e6	−0.241194
$868$	1.09534e6	0.0493456
$869$	−1.96394e6	−0.0882226
$870$	−4402.52	−0.000197198 0
$871$	6.78499e6	0.303042
$872$	9.45053e6	0.420886
$873$	7.43389e6	0.330126
$874$	0	0
$875$	−536195.	−0.0236757
$876$	6.39564e6	0.281594
$877$	−3.40857e7	−1.49649	−0.748243	−	0.663425i	$-0.769102\pi$
$877$	−3.40857e7	−1.49649	−0.748243	+	0.663425i	$0.769102\pi$
$878$	3.96077e6	0.173398
$879$	2.24246e6	0.0978933
$880$	−60482.5	−0.00263283
$881$	3.86174e7	1.67627	0.838134	−	0.545465i	$-0.183647\pi$
$881$	3.86174e7	1.67627	0.838134	+	0.545465i	$0.183647\pi$
$882$	4.94327e6	0.213965
$883$	2.26575e6	0.0977935	0.0488967	−	0.998804i	$-0.484429\pi$
$883$	2.26575e6	0.0977935	0.0488967	+	0.998804i	$0.484429\pi$
$884$	−3.10081e6	−0.133458
$885$	−92765.2	−0.00398132
$886$	−7.01460e6	−0.300205
$887$	3.00801e7	1.28372	0.641860	−	0.766822i	$-0.278163\pi$
$887$	3.00801e7	1.28372	0.641860	+	0.766822i	$0.278163\pi$
$888$	−1.85582e6	−0.0789775
$889$	1.24703e7	0.529201
$890$	136860.	0.00579165
$891$	−1.20899e7	−0.510184
$892$	−1.26389e7	−0.531860
$893$	0	0
$894$	312496.	0.0130768
$895$	498951.	0.0208210
$896$	1.72955e6	0.0719718
$897$	−3.15149e6	−0.130778
$898$	2.82827e7	1.17039
$899$	−176378.	−0.00727855
$900$	1.09082e7	0.448897
$901$	2.17218e6	0.0891422
$902$	4.72173e6	0.193234
$903$	−2.87637e6	−0.117388
$904$	−1.50206e7	−0.611317
$905$	362119.	0.0146970
$906$	−1.01111e7	−0.409242
$907$	2.09629e7	0.846124	0.423062	−	0.906101i	$-0.360955\pi$
$907$	2.09629e7	0.846124	0.423062	+	0.906101i	$0.360955\pi$
$908$	−1.74011e7	−0.700426
$909$	2.15047e7	0.863224
$910$	−112806.	−0.00451572
$911$	−4.01654e7	−1.60345	−0.801726	−	0.597692i	$-0.796085\pi$
$911$	−4.01654e7	−1.60345	−0.801726	+	0.597692i	$0.796085\pi$
$912$	0	0
$913$	−1.76372e7	−0.700250
$914$	1.15151e7	0.455936
$915$	−138690.	−0.00547638
$916$	−2.52550e7	−0.994508
$917$	−2.20998e7	−0.867889
$918$	−5.41586e6	−0.212110
$919$	−1.92818e7	−0.753110	−0.376555	−	0.926394i	$-0.622891\pi$
$919$	−1.92818e7	−0.753110	−0.376555	+	0.926394i	$0.622891\pi$
$920$	100174.	0.00390198
$921$	9.59776e6	0.372839
$922$	1.82629e7	0.707526
$923$	7.27086e6	0.280919
$924$	−2.44445e6	−0.0941893
$925$	1.81961e7	0.699236
$926$	−2.94320e7	−1.12796
$927$	−2.76620e7	−1.05727
$928$	−278502.	−0.0106159
$929$	−2.49485e7	−0.948428	−0.474214	−	0.880410i	$-0.657268\pi$
$929$	−2.49485e7	−0.948428	−0.474214	+	0.880410i	$0.657268\pi$
$930$	10497.6	0.000397998 0
$931$	0	0
$932$	−7.31112e6	−0.275705
$933$	−8.48368e6	−0.319066
$934$	6.41914e6	0.240774
$935$	−139304.	−0.00521115
$936$	4.59026e6	0.171257
$937$	1.44200e6	0.0536556	0.0268278	−	0.999640i	$-0.491459\pi$
$937$	1.44200e6	0.0536556	0.0268278	+	0.999640i	$0.491459\pi$
$938$	−8.71644e6	−0.323469
$939$	8.05007e6	0.297945
$940$	−264969.	−0.00978083
$941$	−3.52604e7	−1.29811	−0.649057	−	0.760740i	$-0.724836\pi$
$941$	−3.52604e7	−1.29811	−0.649057	+	0.760740i	$0.724836\pi$
$942$	−917674.	−0.0336947
$943$	−7.82034e6	−0.286382
$944$	−5.86831e6	−0.214330
$945$	−197026.	−0.00717702
$946$	6.36310e6	0.231175
$947$	1.24037e7	0.449443	0.224722	−	0.974423i	$-0.427853\pi$
$947$	1.24037e7	0.449443	0.224722	+	0.974423i	$0.427853\pi$
$948$	538235.	0.0194514
$949$	−2.63882e7	−0.951141
$950$	0	0
$951$	−2.88161e6	−0.103320
$952$	3.98350e6	0.142453
$953$	2.06870e7	0.737847	0.368923	−	0.929460i	$-0.379726\pi$
$953$	2.06870e7	0.737847	0.368923	+	0.929460i	$0.379726\pi$
$954$	−3.21557e6	−0.114390
$955$	−267345.	−0.00948557
$956$	−1.37959e7	−0.488207
$957$	393620.	0.0138931
$958$	1.57264e7	0.553624
$959$	−1.17749e7	−0.413437
$960$	16575.7	0.000580489 0
$961$	−2.82086e7	−0.985310
$962$	7.65706e6	0.266762
$963$	4.41241e7	1.53324
$964$	1.27429e7	0.441648
$965$	486142.	0.0168052
$966$	4.04861e6	0.139593
$967$	−3.48568e7	−1.19873	−0.599365	−	0.800476i	$-0.704580\pi$
$967$	−3.48568e7	−1.19873	−0.599365	+	0.800476i	$0.704580\pi$
$968$	−4.89964e6	−0.168064
$969$	0	0
$970$	−110759.	−0.00377962
$971$	−4.44520e7	−1.51301	−0.756507	−	0.653985i	$-0.773096\pi$
$971$	−4.44520e7	−1.51301	−0.756507	+	0.653985i	$0.773096\pi$
$972$	1.22415e7	0.415593
$973$	−2.50499e7	−0.848250
$974$	2.94794e7	0.995682
$975$	5.11300e6	0.172252
$976$	−8.77353e6	−0.294815
$977$	−3.81786e7	−1.27963	−0.639813	−	0.768531i	$-0.720988\pi$
$977$	−3.81786e7	−1.27963	−0.639813	+	0.768531i	$0.720988\pi$
$978$	−5.75617e6	−0.192436
$979$	−1.22364e7	−0.408035
$980$	−73650.4	−0.00244968
$981$	−3.22219e7	−1.06900
$982$	1.30564e6	0.0432061
$983$	−1.80150e7	−0.594636	−0.297318	−	0.954778i	$-0.596092\pi$
$983$	−1.80150e7	−0.594636	−0.297318	+	0.954778i	$0.596092\pi$
$984$	−1.29403e6	−0.0426045
$985$	765587.	0.0251422
$986$	−641448.	−0.0210121
$987$	−1.07090e7	−0.349908
$988$	0	0
$989$	−1.05389e7	−0.342612
$990$	206217.	0.00668709
$991$	2.09823e7	0.678685	0.339343	−	0.940663i	$-0.389795\pi$
$991$	2.09823e7	0.678685	0.339343	+	0.940663i	$0.389795\pi$
$992$	664073.	0.0214258
$993$	1.55289e6	0.0499766
$994$	−9.34062e6	−0.299854
$995$	370831.	0.0118746
$996$	4.83362e6	0.154392
$997$	−2.41182e7	−0.768434	−0.384217	−	0.923243i	$-0.625529\pi$
$997$	−2.41182e7	−0.768434	−0.384217	+	0.923243i	$0.625529\pi$
$998$	2.27086e7	0.721712
$999$	1.33738e7	0.423976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.6.a.p.1.8		12
19.3	odd	18		38.6.e.a.9.2	✓	24
19.13	odd	18		38.6.e.a.17.2	yes	24
19.18	odd	2		722.6.a.o.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.a.9.2	✓	24	19.3	odd	18
38.6.e.a.17.2	yes	24	19.13	odd	18
722.6.a.o.1.5		12	19.18	odd	2
722.6.a.p.1.8		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +4.97893 q^{3} +16.0000 q^{4} +0.812786 q^{5} +19.9157 q^{6} +105.563 q^{7} +64.0000 q^{8} -218.210 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} +4.97893 q^{3} +16.0000 q^{4} +0.812786 q^{5} +19.9157 q^{6} +105.563 q^{7} +64.0000 q^{8} -218.210 q^{9} +3.25115 q^{10} -290.679 q^{11} +79.6628 q^{12} -328.687 q^{13} +422.253 q^{14} +4.04681 q^{15} +256.000 q^{16} +589.620 q^{17} -872.841 q^{18} +13.0046 q^{20} +525.591 q^{21} -1162.71 q^{22} +1925.74 q^{23} +318.651 q^{24} -3124.34 q^{25} -1314.75 q^{26} -2296.33 q^{27} +1689.01 q^{28} -271.975 q^{29} +16.1872 q^{30} +648.509 q^{31} +1024.00 q^{32} -1447.27 q^{33} +2358.48 q^{34} +85.8003 q^{35} -3491.36 q^{36} -5823.98 q^{37} -1636.51 q^{39} +52.0183 q^{40} -4060.95 q^{41} +2102.37 q^{42} -5472.63 q^{43} -4650.86 q^{44} -177.358 q^{45} +7702.96 q^{46} -20375.1 q^{47} +1274.61 q^{48} -5663.42 q^{49} -12497.4 q^{50} +2935.68 q^{51} -5258.99 q^{52} +3684.03 q^{53} -9185.33 q^{54} -236.260 q^{55} +6756.04 q^{56} -1087.90 q^{58} -22923.1 q^{59} +64.7489 q^{60} -34271.6 q^{61} +2594.03 q^{62} -23035.0 q^{63} +4096.00 q^{64} -267.152 q^{65} -5789.07 q^{66} -20642.7 q^{67} +9433.93 q^{68} +9588.12 q^{69} +343.201 q^{70} -22120.9 q^{71} -13965.5 q^{72} +80283.9 q^{73} -23295.9 q^{74} -15555.9 q^{75} -30685.0 q^{77} -6546.03 q^{78} +6756.41 q^{79} +208.073 q^{80} +41591.8 q^{81} -16243.8 q^{82} +60676.0 q^{83} +8409.46 q^{84} +479.236 q^{85} -21890.5 q^{86} -1354.14 q^{87} -18603.4 q^{88} +42096.0 q^{89} -709.433 q^{90} -34697.2 q^{91} +30811.9 q^{92} +3228.88 q^{93} -81500.2 q^{94} +5098.42 q^{96} -34067.5 q^{97} -22653.7 q^{98} +63429.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 48 q^{2} + 192 q^{4} - 42 q^{5} - 438 q^{7} + 768 q^{8} + 546 q^{9}+O(q^{10}) \) 12 * q + 48 * q^2 + 192 * q^4 - 42 * q^5 - 438 * q^7 + 768 * q^8 + 546 * q^9	\( 12 q + 48 q^{2} + 192 q^{4} - 42 q^{5} - 438 q^{7} + 768 q^{8} + 546 q^{9} - 168 q^{10} - 600 q^{11} + 444 q^{13} - 1752 q^{14} + 3072 q^{16} - 2220 q^{17} + 2184 q^{18} - 672 q^{20} - 11046 q^{21} - 2400 q^{22} - 7464 q^{23} + 4146 q^{25} + 1776 q^{26} - 5130 q^{27} - 7008 q^{28} - 6606 q^{29} + 294 q^{31} + 12288 q^{32} - 14886 q^{33} - 8880 q^{34} - 2064 q^{35} + 8736 q^{36} - 14304 q^{37} - 19434 q^{39} - 2688 q^{40} - 19512 q^{41} - 44184 q^{42} - 8634 q^{43} - 9600 q^{44} + 34344 q^{45} - 29856 q^{46} - 17964 q^{47} - 1422 q^{49} + 16584 q^{50} + 27456 q^{51} + 7104 q^{52} - 44112 q^{53} - 20520 q^{54} - 121596 q^{55} - 28032 q^{56} - 26424 q^{58} - 34002 q^{59} - 27582 q^{61} + 1176 q^{62} - 204222 q^{63} + 49152 q^{64} - 202674 q^{65} - 59544 q^{66} - 33702 q^{67} - 35520 q^{68} - 19938 q^{69} - 8256 q^{70} - 2904 q^{71} + 34944 q^{72} - 336072 q^{73} - 57216 q^{74} - 87480 q^{75} - 57528 q^{77} - 77736 q^{78} - 40098 q^{79} - 10752 q^{80} + 97476 q^{81} - 78048 q^{82} + 148440 q^{83} - 176736 q^{84} - 252912 q^{85} - 34536 q^{86} - 182874 q^{87} - 38400 q^{88} - 123774 q^{89} + 137376 q^{90} - 178866 q^{91} - 119424 q^{92} - 118584 q^{93} - 71856 q^{94} - 317346 q^{97} - 5688 q^{98} - 409860 q^{99}+O(q^{100}) \) 12 * q + 48 * q^2 + 192 * q^4 - 42 * q^5 - 438 * q^7 + 768 * q^8 + 546 * q^9 - 168 * q^10 - 600 * q^11 + 444 * q^13 - 1752 * q^14 + 3072 * q^16 - 2220 * q^17 + 2184 * q^18 - 672 * q^20 - 11046 * q^21 - 2400 * q^22 - 7464 * q^23 + 4146 * q^25 + 1776 * q^26 - 5130 * q^27 - 7008 * q^28 - 6606 * q^29 + 294 * q^31 + 12288 * q^32 - 14886 * q^33 - 8880 * q^34 - 2064 * q^35 + 8736 * q^36 - 14304 * q^37 - 19434 * q^39 - 2688 * q^40 - 19512 * q^41 - 44184 * q^42 - 8634 * q^43 - 9600 * q^44 + 34344 * q^45 - 29856 * q^46 - 17964 * q^47 - 1422 * q^49 + 16584 * q^50 + 27456 * q^51 + 7104 * q^52 - 44112 * q^53 - 20520 * q^54 - 121596 * q^55 - 28032 * q^56 - 26424 * q^58 - 34002 * q^59 - 27582 * q^61 + 1176 * q^62 - 204222 * q^63 + 49152 * q^64 - 202674 * q^65 - 59544 * q^66 - 33702 * q^67 - 35520 * q^68 - 19938 * q^69 - 8256 * q^70 - 2904 * q^71 + 34944 * q^72 - 336072 * q^73 - 57216 * q^74 - 87480 * q^75 - 57528 * q^77 - 77736 * q^78 - 40098 * q^79 - 10752 * q^80 + 97476 * q^81 - 78048 * q^82 + 148440 * q^83 - 176736 * q^84 - 252912 * q^85 - 34536 * q^86 - 182874 * q^87 - 38400 * q^88 - 123774 * q^89 + 137376 * q^90 - 178866 * q^91 - 119424 * q^92 - 118584 * q^93 - 71856 * q^94 - 317346 * q^97 - 5688 * q^98 - 409860 * q^99

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