LMFDB - Embedded newform 2450.4.a.cx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 92x^{4} + 26x^{3} + 2116x^{2} + 80x - 7800 $ x^6 - x^5 - 92x^4 + 26x^3 + 2116x^2 + 80x - 7800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.18765$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -9.18765 q^{3} +4.00000 q^{4} -18.3753 q^{6} +8.00000 q^{8} +57.4130 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -9.18765 q^{3} +4.00000 q^{4} -18.3753 q^{6} +8.00000 q^{8} +57.4130 q^{9} +35.4764 q^{11} -36.7506 q^{12} +45.5531 q^{13} +16.0000 q^{16} -93.7447 q^{17} +114.826 q^{18} -44.8775 q^{19} +70.9527 q^{22} -122.473 q^{23} -73.5012 q^{24} +91.1063 q^{26} -279.424 q^{27} -17.8390 q^{29} -27.1587 q^{31} +32.0000 q^{32} -325.945 q^{33} -187.489 q^{34} +229.652 q^{36} -78.0182 q^{37} -89.7550 q^{38} -418.526 q^{39} +21.0352 q^{41} +467.300 q^{43} +141.905 q^{44} -244.946 q^{46} -578.861 q^{47} -147.002 q^{48} +861.294 q^{51} +182.213 q^{52} -161.262 q^{53} -558.848 q^{54} +412.319 q^{57} -35.6781 q^{58} +559.436 q^{59} +108.776 q^{61} -54.3174 q^{62} +64.0000 q^{64} -651.889 q^{66} +407.643 q^{67} -374.979 q^{68} +1125.24 q^{69} +1159.37 q^{71} +459.304 q^{72} -256.677 q^{73} -156.036 q^{74} -179.510 q^{76} -837.053 q^{78} -853.583 q^{79} +1017.10 q^{81} +42.0703 q^{82} +828.152 q^{83} +934.599 q^{86} +163.899 q^{87} +283.811 q^{88} -164.983 q^{89} -489.892 q^{92} +249.525 q^{93} -1157.72 q^{94} -294.005 q^{96} -38.6575 q^{97} +2036.80 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} - 7 q^{3} + 24 q^{4} - 14 q^{6} + 48 q^{8} + 31 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 - 7 * q^3 + 24 * q^4 - 14 * q^6 + 48 * q^8 + 31 * q^9	$ 6 q + 12 q^{2} - 7 q^{3} + 24 q^{4} - 14 q^{6} + 48 q^{8} + 31 q^{9} + 31 q^{11} - 28 q^{12} - 59 q^{13} + 96 q^{16} - 68 q^{17} + 62 q^{18} - 93 q^{19} + 62 q^{22} - 94 q^{23} - 56 q^{24} - 118 q^{26} - 385 q^{27} + 169 q^{29} - 326 q^{31} + 192 q^{32} - 400 q^{33} - 136 q^{34} + 124 q^{36} - 253 q^{37} - 186 q^{38} - 434 q^{39} - 198 q^{41} - 99 q^{43} + 124 q^{44} - 188 q^{46} - 901 q^{47} - 112 q^{48} + 724 q^{51} - 236 q^{52} + 233 q^{53} - 770 q^{54} - 518 q^{57} + 338 q^{58} - 668 q^{59} - 157 q^{61} - 652 q^{62} + 384 q^{64} - 800 q^{66} + 1193 q^{67} - 272 q^{68} + 45 q^{69} + 1108 q^{71} + 248 q^{72} - 1458 q^{73} - 506 q^{74} - 372 q^{76} - 868 q^{78} - 886 q^{79} - 614 q^{81} - 396 q^{82} - 1799 q^{83} - 198 q^{86} - 789 q^{87} + 248 q^{88} - 3047 q^{89} - 376 q^{92} + 1902 q^{93} - 1802 q^{94} - 224 q^{96} - 3404 q^{97} + 4273 q^{99}+O(q^{100}) $ 6 * q + 12 * q^2 - 7 * q^3 + 24 * q^4 - 14 * q^6 + 48 * q^8 + 31 * q^9 + 31 * q^11 - 28 * q^12 - 59 * q^13 + 96 * q^16 - 68 * q^17 + 62 * q^18 - 93 * q^19 + 62 * q^22 - 94 * q^23 - 56 * q^24 - 118 * q^26 - 385 * q^27 + 169 * q^29 - 326 * q^31 + 192 * q^32 - 400 * q^33 - 136 * q^34 + 124 * q^36 - 253 * q^37 - 186 * q^38 - 434 * q^39 - 198 * q^41 - 99 * q^43 + 124 * q^44 - 188 * q^46 - 901 * q^47 - 112 * q^48 + 724 * q^51 - 236 * q^52 + 233 * q^53 - 770 * q^54 - 518 * q^57 + 338 * q^58 - 668 * q^59 - 157 * q^61 - 652 * q^62 + 384 * q^64 - 800 * q^66 + 1193 * q^67 - 272 * q^68 + 45 * q^69 + 1108 * q^71 + 248 * q^72 - 1458 * q^73 - 506 * q^74 - 372 * q^76 - 868 * q^78 - 886 * q^79 - 614 * q^81 - 396 * q^82 - 1799 * q^83 - 198 * q^86 - 789 * q^87 + 248 * q^88 - 3047 * q^89 - 376 * q^92 + 1902 * q^93 - 1802 * q^94 - 224 * q^96 - 3404 * q^97 + 4273 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−9.18765	−1.76816	−0.884082	−	0.467331i	$-0.845216\pi$
$3$	−9.18765	−1.76816	−0.884082	+	0.467331i	$0.845216\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−18.3753	−1.25028
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	57.4130	2.12641
$10$	0	0
$11$	35.4764	0.972411	0.486206	−	0.873844i	$-0.338381\pi$
$11$	35.4764	0.972411	0.486206	+	0.873844i	$0.338381\pi$
$12$	−36.7506	−0.884082
$13$	45.5531	0.971859	0.485930	−	0.873998i	$-0.338481\pi$
$13$	45.5531	0.971859	0.485930	+	0.873998i	$0.338481\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−93.7447	−1.33744	−0.668718	−	0.743516i	$-0.733157\pi$
$17$	−93.7447	−1.33744	−0.668718	+	0.743516i	$0.733157\pi$
$18$	114.826	1.50360
$19$	−44.8775	−0.541874	−0.270937	−	0.962597i	$-0.587334\pi$
$19$	−44.8775	−0.541874	−0.270937	+	0.962597i	$0.587334\pi$
$20$	0	0
$21$	0	0
$22$	70.9527	0.687599
$23$	−122.473	−1.11032	−0.555161	−	0.831743i	$-0.687343\pi$
$23$	−122.473	−1.11032	−0.555161	+	0.831743i	$0.687343\pi$
$24$	−73.5012	−0.625141
$25$	0	0
$26$	91.1063	0.687208
$27$	−279.424	−1.99167
$28$	0	0
$29$	−17.8390	−0.114228	−0.0571142	−	0.998368i	$-0.518190\pi$
$29$	−17.8390	−0.114228	−0.0571142	+	0.998368i	$0.518190\pi$
$30$	0	0
$31$	−27.1587	−0.157350	−0.0786749	−	0.996900i	$-0.525069\pi$
$31$	−27.1587	−0.157350	−0.0786749	+	0.996900i	$0.525069\pi$
$32$	32.0000	0.176777
$33$	−325.945	−1.71938
$34$	−187.489	−0.945711
$35$	0	0
$36$	229.652	1.06320
$37$	−78.0182	−0.346652	−0.173326	−	0.984865i	$-0.555451\pi$
$37$	−78.0182	−0.346652	−0.173326	+	0.984865i	$0.555451\pi$
$38$	−89.7550	−0.383163
$39$	−418.526	−1.71841
$40$	0	0
$41$	21.0352	0.0801254	0.0400627	−	0.999197i	$-0.487244\pi$
$41$	21.0352	0.0801254	0.0400627	+	0.999197i	$0.487244\pi$
$42$	0	0
$43$	467.300	1.65727	0.828634	−	0.559791i	$-0.189118\pi$
$43$	467.300	1.65727	0.828634	+	0.559791i	$0.189118\pi$
$44$	141.905	0.486206
$45$	0	0
$46$	−244.946	−0.785116
$47$	−578.861	−1.79650	−0.898250	−	0.439484i	$-0.855161\pi$
$47$	−578.861	−1.79650	−0.898250	+	0.439484i	$0.855161\pi$
$48$	−147.002	−0.442041
$49$	0	0
$50$	0	0
$51$	861.294	2.36481
$52$	182.213	0.485930
$53$	−161.262	−0.417944	−0.208972	−	0.977922i	$-0.567012\pi$
$53$	−161.262	−0.417944	−0.208972	+	0.977922i	$0.567012\pi$
$54$	−558.848	−1.40833
$55$	0	0
$56$	0	0
$57$	412.319	0.958123
$58$	−35.6781	−0.0807717
$59$	559.436	1.23445	0.617223	−	0.786788i	$-0.288258\pi$
$59$	559.436	1.23445	0.617223	+	0.786788i	$0.288258\pi$
$60$	0	0
$61$	108.776	0.228318	0.114159	−	0.993463i	$-0.463583\pi$
$61$	108.776	0.228318	0.114159	+	0.993463i	$0.463583\pi$
$62$	−54.3174	−0.111263
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−651.889	−1.21579
$67$	407.643	0.743307	0.371653	−	0.928372i	$-0.378791\pi$
$67$	407.643	0.743307	0.371653	+	0.928372i	$0.378791\pi$
$68$	−374.979	−0.668718
$69$	1125.24	1.96323
$70$	0	0
$71$	1159.37	1.93791	0.968954	−	0.247243i	$-0.0795245\pi$
$71$	1159.37	1.93791	0.968954	+	0.247243i	$0.0795245\pi$
$72$	459.304	0.751798
$73$	−256.677	−0.411531	−0.205766	−	0.978601i	$-0.565968\pi$
$73$	−256.677	−0.411531	−0.205766	+	0.978601i	$0.565968\pi$
$74$	−156.036	−0.245120
$75$	0	0
$76$	−179.510	−0.270937
$77$	0	0
$78$	−837.053	−1.21510
$79$	−853.583	−1.21564	−0.607820	−	0.794075i	$-0.707956\pi$
$79$	−853.583	−1.21564	−0.607820	+	0.794075i	$0.707956\pi$
$80$	0	0
$81$	1017.10	1.39520
$82$	42.0703	0.0566572
$83$	828.152	1.09520	0.547599	−	0.836741i	$-0.315542\pi$
$83$	828.152	1.09520	0.547599	+	0.836741i	$0.315542\pi$
$84$	0	0
$85$	0	0
$86$	934.599	1.17187
$87$	163.899	0.201975
$88$	283.811	0.343799
$89$	−164.983	−0.196497	−0.0982483	−	0.995162i	$-0.531324\pi$
$89$	−164.983	−0.196497	−0.0982483	+	0.995162i	$0.531324\pi$
$90$	0	0
$91$	0	0
$92$	−489.892	−0.555161
$93$	249.525	0.278220
$94$	−1157.72	−1.27032
$95$	0	0
$96$	−294.005	−0.312570
$97$	−38.6575	−0.0404647	−0.0202324	−	0.999795i	$-0.506441\pi$
$97$	−38.6575	−0.0404647	−0.0202324	+	0.999795i	$0.506441\pi$
$98$	0	0
$99$	2036.80	2.06774
$100$	0	0
$101$	−766.628	−0.755270	−0.377635	−	0.925954i	$-0.623263\pi$
$101$	−766.628	−0.755270	−0.377635	+	0.925954i	$0.623263\pi$
$102$	1722.59	1.67217
$103$	−529.223	−0.506271	−0.253136	−	0.967431i	$-0.581462\pi$
$103$	−529.223	−0.506271	−0.253136	+	0.967431i	$0.581462\pi$
$104$	364.425	0.343604
$105$	0	0
$106$	−322.524	−0.295531
$107$	713.613	0.644743	0.322372	−	0.946613i	$-0.395520\pi$
$107$	713.613	0.644743	0.322372	+	0.946613i	$0.395520\pi$
$108$	−1117.70	−0.995837
$109$	−318.229	−0.279640	−0.139820	−	0.990177i	$-0.544652\pi$
$109$	−318.229	−0.279640	−0.139820	+	0.990177i	$0.544652\pi$
$110$	0	0
$111$	716.804	0.612937
$112$	0	0
$113$	−506.300	−0.421493	−0.210747	−	0.977541i	$-0.567589\pi$
$113$	−506.300	−0.421493	−0.210747	+	0.977541i	$0.567589\pi$
$114$	824.638	0.677495
$115$	0	0
$116$	−71.3561	−0.0571142
$117$	2615.34	2.06657
$118$	1118.87	0.872885
$119$	0	0
$120$	0	0
$121$	−72.4280	−0.0544163
$122$	217.552	0.161445
$123$	−193.264	−0.141675
$124$	−108.635	−0.0786749
$125$	0	0
$126$	0	0
$127$	−2225.10	−1.55469	−0.777345	−	0.629074i	$-0.783434\pi$
$127$	−2225.10	−1.55469	−0.777345	+	0.629074i	$0.783434\pi$
$128$	128.000	0.0883883
$129$	−4293.39	−2.93032
$130$	0	0
$131$	1809.22	1.20666	0.603330	−	0.797492i	$-0.293840\pi$
$131$	1809.22	1.20666	0.603330	+	0.797492i	$0.293840\pi$
$132$	−1303.78	−0.859692
$133$	0	0
$134$	815.287	0.525597
$135$	0	0
$136$	−749.958	−0.472855
$137$	−2323.86	−1.44920	−0.724600	−	0.689170i	$-0.757975\pi$
$137$	−2323.86	−1.44920	−0.724600	+	0.689170i	$0.757975\pi$
$138$	2250.48	1.38821
$139$	−1939.15	−1.18328	−0.591642	−	0.806200i	$-0.701520\pi$
$139$	−1939.15	−1.18328	−0.591642	+	0.806200i	$0.701520\pi$
$140$	0	0
$141$	5318.37	3.17651
$142$	2318.73	1.37031
$143$	1616.06	0.945047
$144$	918.608	0.531602
$145$	0	0
$146$	−513.354	−0.290997
$147$	0	0
$148$	−312.073	−0.173326
$149$	−360.298	−0.198099	−0.0990497	−	0.995082i	$-0.531580\pi$
$149$	−360.298	−0.198099	−0.0990497	+	0.995082i	$0.531580\pi$
$150$	0	0
$151$	353.760	0.190653	0.0953263	−	0.995446i	$-0.469611\pi$
$151$	353.760	0.190653	0.0953263	+	0.995446i	$0.469611\pi$
$152$	−359.020	−0.191581
$153$	−5382.16	−2.84394
$154$	0	0
$155$	0	0
$156$	−1674.11	−0.859204
$157$	−1406.80	−0.715127	−0.357563	−	0.933889i	$-0.616392\pi$
$157$	−1406.80	−0.715127	−0.357563	+	0.933889i	$0.616392\pi$
$158$	−1707.17	−0.859587
$159$	1481.62	0.738994
$160$	0	0
$161$	0	0
$162$	2034.20	0.986555
$163$	−269.618	−0.129559	−0.0647794	−	0.997900i	$-0.520634\pi$
$163$	−269.618	−0.129559	−0.0647794	+	0.997900i	$0.520634\pi$
$164$	84.1406	0.0400627
$165$	0	0
$166$	1656.30	0.774422
$167$	−248.313	−0.115060	−0.0575300	−	0.998344i	$-0.518322\pi$
$167$	−248.313	−0.115060	−0.0575300	+	0.998344i	$0.518322\pi$
$168$	0	0
$169$	−121.912	−0.0554900
$170$	0	0
$171$	−2576.55	−1.15224
$172$	1869.20	0.828634
$173$	−113.927	−0.0500679	−0.0250339	−	0.999687i	$-0.507969\pi$
$173$	−113.927	−0.0500679	−0.0250339	+	0.999687i	$0.507969\pi$
$174$	327.798	0.142818
$175$	0	0
$176$	567.622	0.243103
$177$	−5139.90	−2.18270
$178$	−329.967	−0.138944
$179$	1564.68	0.653349	0.326674	−	0.945137i	$-0.394072\pi$
$179$	1564.68	0.653349	0.326674	+	0.945137i	$0.394072\pi$
$180$	0	0
$181$	4811.12	1.97573	0.987867	−	0.155300i	$-0.0496343\pi$
$181$	4811.12	1.97573	0.987867	+	0.155300i	$0.0496343\pi$
$182$	0	0
$183$	−999.398	−0.403703
$184$	−979.784	−0.392558
$185$	0	0
$186$	499.049	0.196732
$187$	−3325.72	−1.30054
$188$	−2315.44	−0.898250
$189$	0	0
$190$	0	0
$191$	2465.11	0.933871	0.466935	−	0.884291i	$-0.345358\pi$
$191$	2465.11	0.933871	0.466935	+	0.884291i	$0.345358\pi$
$192$	−588.010	−0.221021
$193$	−2353.62	−0.877811	−0.438905	−	0.898533i	$-0.644634\pi$
$193$	−2353.62	−0.877811	−0.438905	+	0.898533i	$0.644634\pi$
$194$	−77.3151	−0.0286129
$195$	0	0
$196$	0	0
$197$	−1552.27	−0.561393	−0.280697	−	0.959797i	$-0.590565\pi$
$197$	−1552.27	−0.561393	−0.280697	+	0.959797i	$0.590565\pi$
$198$	4073.61	1.46211
$199$	760.404	0.270872	0.135436	−	0.990786i	$-0.456756\pi$
$199$	760.404	0.270872	0.135436	+	0.990786i	$0.456756\pi$
$200$	0	0
$201$	−3745.29	−1.31429
$202$	−1533.26	−0.534057
$203$	0	0
$204$	3445.18	1.18240
$205$	0	0
$206$	−1058.45	−0.357988
$207$	−7031.54	−2.36099
$208$	728.850	0.242965
$209$	−1592.09	−0.526924
$210$	0	0
$211$	−2942.38	−0.960008	−0.480004	−	0.877266i	$-0.659365\pi$
$211$	−2942.38	−0.960008	−0.480004	+	0.877266i	$0.659365\pi$
$212$	−645.048	−0.208972
$213$	−10651.9	−3.42654
$214$	1427.23	0.455902
$215$	0	0
$216$	−2235.39	−0.704163
$217$	0	0
$218$	−636.458	−0.197736
$219$	2358.26	0.727655
$220$	0	0
$221$	−4270.37	−1.29980
$222$	1433.61	0.433412
$223$	1219.27	0.366136	0.183068	−	0.983100i	$-0.441397\pi$
$223$	1219.27	0.366136	0.183068	+	0.983100i	$0.441397\pi$
$224$	0	0
$225$	0	0
$226$	−1012.60	−0.298041
$227$	−4573.67	−1.33729	−0.668646	−	0.743581i	$-0.733126\pi$
$227$	−4573.67	−1.33729	−0.668646	+	0.743581i	$0.733126\pi$
$228$	1649.28	0.479061
$229$	−3084.27	−0.890020	−0.445010	−	0.895526i	$-0.646800\pi$
$229$	−3084.27	−0.890020	−0.445010	+	0.895526i	$0.646800\pi$
$230$	0	0
$231$	0	0
$232$	−142.712	−0.0403858
$233$	5932.87	1.66813	0.834067	−	0.551663i	$-0.186006\pi$
$233$	5932.87	1.66813	0.834067	+	0.551663i	$0.186006\pi$
$234$	5230.68	1.46128
$235$	0	0
$236$	2237.74	0.617223
$237$	7842.42	2.14945
$238$	0	0
$239$	−807.121	−0.218445	−0.109222	−	0.994017i	$-0.534836\pi$
$239$	−807.121	−0.218445	−0.109222	+	0.994017i	$0.534836\pi$
$240$	0	0
$241$	−4261.56	−1.13905	−0.569526	−	0.821974i	$-0.692873\pi$
$241$	−4261.56	−1.13905	−0.569526	+	0.821974i	$0.692873\pi$
$242$	−144.856	−0.0384781
$243$	−1800.32	−0.475270
$244$	435.105	0.114159
$245$	0	0
$246$	−386.528	−0.100179
$247$	−2044.31	−0.526625
$248$	−217.270	−0.0556316
$249$	−7608.77	−1.93649
$250$	0	0
$251$	−2962.48	−0.744980	−0.372490	−	0.928036i	$-0.621496\pi$
$251$	−2962.48	−0.744980	−0.372490	+	0.928036i	$0.621496\pi$
$252$	0	0
$253$	−4344.90	−1.07969
$254$	−4450.20	−1.09933
$255$	0	0
$256$	256.000	0.0625000
$257$	−6170.94	−1.49779	−0.748896	−	0.662687i	$-0.769416\pi$
$257$	−6170.94	−1.49779	−0.748896	+	0.662687i	$0.769416\pi$
$258$	−8586.78	−2.07205
$259$	0	0
$260$	0	0
$261$	−1024.19	−0.242896
$262$	3618.44	0.853237
$263$	129.912	0.0304589	0.0152294	−	0.999884i	$-0.495152\pi$
$263$	129.912	0.0304589	0.0152294	+	0.999884i	$0.495152\pi$
$264$	−2607.56	−0.607894
$265$	0	0
$266$	0	0
$267$	1515.81	0.347438
$268$	1630.57	0.371653
$269$	−2473.67	−0.560678	−0.280339	−	0.959901i	$-0.590447\pi$
$269$	−2473.67	−0.560678	−0.280339	+	0.959901i	$0.590447\pi$
$270$	0	0
$271$	−2878.92	−0.645321	−0.322660	−	0.946515i	$-0.604577\pi$
$271$	−2878.92	−0.645321	−0.322660	+	0.946515i	$0.604577\pi$
$272$	−1499.92	−0.334359
$273$	0	0
$274$	−4647.71	−1.02474
$275$	0	0
$276$	4500.96	0.981615
$277$	544.478	0.118103	0.0590514	−	0.998255i	$-0.481192\pi$
$277$	544.478	0.118103	0.0590514	+	0.998255i	$0.481192\pi$
$278$	−3878.30	−0.836709
$279$	−1559.26	−0.334590
$280$	0	0
$281$	−97.2982	−0.0206560	−0.0103280	−	0.999947i	$-0.503288\pi$
$281$	−97.2982	−0.0206560	−0.0103280	+	0.999947i	$0.503288\pi$
$282$	10636.7	2.24613
$283$	−5087.62	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$283$	−5087.62	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$284$	4637.46	0.968954
$285$	0	0
$286$	3232.12	0.668249
$287$	0	0
$288$	1837.22	0.375899
$289$	3875.07	0.788738
$290$	0	0
$291$	355.172	0.0715483
$292$	−1026.71	−0.205766
$293$	4413.85	0.880068	0.440034	−	0.897981i	$-0.354966\pi$
$293$	4413.85	0.880068	0.440034	+	0.897981i	$0.354966\pi$
$294$	0	0
$295$	0	0
$296$	−624.145	−0.122560
$297$	−9912.95	−1.93673
$298$	−720.597	−0.140077
$299$	−5579.03	−1.07908
$300$	0	0
$301$	0	0
$302$	707.519	0.134812
$303$	7043.51	1.33544
$304$	−718.040	−0.135469
$305$	0	0
$306$	−10764.3	−2.01097
$307$	−1193.21	−0.221824	−0.110912	−	0.993830i	$-0.535377\pi$
$307$	−1193.21	−0.221824	−0.110912	+	0.993830i	$0.535377\pi$
$308$	0	0
$309$	4862.32	0.895171
$310$	0	0
$311$	9729.55	1.77399	0.886997	−	0.461775i	$-0.152787\pi$
$311$	9729.55	1.77399	0.886997	+	0.461775i	$0.152787\pi$
$312$	−3348.21	−0.607549
$313$	2794.49	0.504646	0.252323	−	0.967643i	$-0.418805\pi$
$313$	2794.49	0.504646	0.252323	+	0.967643i	$0.418805\pi$
$314$	−2813.60	−0.505671
$315$	0	0
$316$	−3414.33	−0.607820
$317$	−6231.90	−1.10416	−0.552079	−	0.833792i	$-0.686165\pi$
$317$	−6231.90	−1.10416	−0.552079	+	0.833792i	$0.686165\pi$
$318$	2963.24	0.522548
$319$	−632.864	−0.111077
$320$	0	0
$321$	−6556.43	−1.14001
$322$	0	0
$323$	4207.03	0.724722
$324$	4068.40	0.697600
$325$	0	0
$326$	−539.235	−0.0916119
$327$	2923.78	0.494450
$328$	168.281	0.0283286
$329$	0	0
$330$	0	0
$331$	−10892.5	−1.80878	−0.904391	−	0.426704i	$-0.859675\pi$
$331$	−10892.5	−1.80878	−0.904391	+	0.426704i	$0.859675\pi$
$332$	3312.61	0.547599
$333$	−4479.26	−0.737122
$334$	−496.625	−0.0813597
$335$	0	0
$336$	0	0
$337$	−413.950	−0.0669119	−0.0334560	−	0.999440i	$-0.510651\pi$
$337$	−413.950	−0.0669119	−0.0334560	+	0.999440i	$0.510651\pi$
$338$	−243.823	−0.0392374
$339$	4651.71	0.745270
$340$	0	0
$341$	−963.492	−0.153009
$342$	−5153.10	−0.814760
$343$	0	0
$344$	3738.40	0.585933
$345$	0	0
$346$	−227.855	−0.0354033
$347$	−1709.53	−0.264474	−0.132237	−	0.991218i	$-0.542216\pi$
$347$	−1709.53	−0.264474	−0.132237	+	0.991218i	$0.542216\pi$
$348$	655.595	0.100987
$349$	6269.06	0.961534	0.480767	−	0.876848i	$-0.340358\pi$
$349$	6269.06	0.961534	0.480767	+	0.876848i	$0.340358\pi$
$350$	0	0
$351$	−12728.6	−1.93563
$352$	1135.24	0.171900
$353$	6723.74	1.01379	0.506896	−	0.862007i	$-0.330793\pi$
$353$	6723.74	1.01379	0.506896	+	0.862007i	$0.330793\pi$
$354$	−10279.8	−1.54340
$355$	0	0
$356$	−659.933	−0.0982483
$357$	0	0
$358$	3129.35	0.461988
$359$	1762.95	0.259178	0.129589	−	0.991568i	$-0.458634\pi$
$359$	1762.95	0.259178	0.129589	+	0.991568i	$0.458634\pi$
$360$	0	0
$361$	−4845.01	−0.706373
$362$	9622.25	1.39706
$363$	665.444	0.0962169
$364$	0	0
$365$	0	0
$366$	−1998.80	−0.285461
$367$	6177.31	0.878618	0.439309	−	0.898336i	$-0.355223\pi$
$367$	6177.31	0.878618	0.439309	+	0.898336i	$0.355223\pi$
$368$	−1959.57	−0.277580
$369$	1207.69	0.170379
$370$	0	0
$371$	0	0
$372$	998.099	0.139110
$373$	−8193.45	−1.13738	−0.568688	−	0.822554i	$-0.692549\pi$
$373$	−8193.45	−1.13738	−0.568688	+	0.822554i	$0.692549\pi$
$374$	−6651.44	−0.919620
$375$	0	0
$376$	−4630.89	−0.635159
$377$	−812.624	−0.111014
$378$	0	0
$379$	−10828.4	−1.46760	−0.733799	−	0.679367i	$-0.762255\pi$
$379$	−10828.4	−1.46760	−0.733799	+	0.679367i	$0.762255\pi$
$380$	0	0
$381$	20443.4	2.74895
$382$	4930.23	0.660346
$383$	−7453.03	−0.994339	−0.497170	−	0.867653i	$-0.665627\pi$
$383$	−7453.03	−0.994339	−0.497170	+	0.867653i	$0.665627\pi$
$384$	−1176.02	−0.156285
$385$	0	0
$386$	−4707.25	−0.620706
$387$	26829.1	3.52403
$388$	−154.630	−0.0202324
$389$	−4059.01	−0.529049	−0.264524	−	0.964379i	$-0.585215\pi$
$389$	−4059.01	−0.529049	−0.264524	+	0.964379i	$0.585215\pi$
$390$	0	0
$391$	11481.2	1.48498
$392$	0	0
$393$	−16622.5	−2.13357
$394$	−3104.54	−0.396965
$395$	0	0
$396$	8147.21	1.03387
$397$	10247.9	1.29554	0.647769	−	0.761837i	$-0.275702\pi$
$397$	10247.9	1.29554	0.647769	+	0.761837i	$0.275702\pi$
$398$	1520.81	0.191536
$399$	0	0
$400$	0	0
$401$	2745.99	0.341965	0.170983	−	0.985274i	$-0.445306\pi$
$401$	2745.99	0.341965	0.170983	+	0.985274i	$0.445306\pi$
$402$	−7490.57	−0.929343
$403$	−1237.16	−0.152922
$404$	−3066.51	−0.377635
$405$	0	0
$406$	0	0
$407$	−2767.80	−0.337088
$408$	6890.35	0.836086
$409$	10406.5	1.25812	0.629059	−	0.777357i	$-0.283440\pi$
$409$	10406.5	1.25812	0.629059	+	0.777357i	$0.283440\pi$
$410$	0	0
$411$	21350.8	2.56242
$412$	−2116.89	−0.253136
$413$	0	0
$414$	−14063.1	−1.66948
$415$	0	0
$416$	1457.70	0.171802
$417$	17816.2	2.09224
$418$	−3184.18	−0.372592
$419$	3901.99	0.454951	0.227476	−	0.973784i	$-0.426953\pi$
$419$	3901.99	0.454951	0.227476	+	0.973784i	$0.426953\pi$
$420$	0	0
$421$	−7780.63	−0.900724	−0.450362	−	0.892846i	$-0.648705\pi$
$421$	−7780.63	−0.900724	−0.450362	+	0.892846i	$0.648705\pi$
$422$	−5884.76	−0.678828
$423$	−33234.1	−3.82009
$424$	−1290.10	−0.147766
$425$	0	0
$426$	−21303.7	−2.42293
$427$	0	0
$428$	2854.45	0.322372
$429$	−14847.8	−1.67100
$430$	0	0
$431$	−13353.0	−1.49232	−0.746161	−	0.665766i	$-0.768105\pi$
$431$	−13353.0	−1.49232	−0.746161	+	0.665766i	$0.768105\pi$
$432$	−4470.78	−0.497918
$433$	−12859.5	−1.42723	−0.713613	−	0.700540i	$-0.752942\pi$
$433$	−12859.5	−1.42723	−0.713613	+	0.700540i	$0.752942\pi$
$434$	0	0
$435$	0	0
$436$	−1272.92	−0.139820
$437$	5496.28	0.601654
$438$	4716.52	0.514530
$439$	12233.5	1.33001	0.665006	−	0.746838i	$-0.268429\pi$
$439$	12233.5	1.33001	0.665006	+	0.746838i	$0.268429\pi$
$440$	0	0
$441$	0	0
$442$	−8540.73	−0.919098
$443$	8620.21	0.924511	0.462255	−	0.886747i	$-0.347040\pi$
$443$	8620.21	0.924511	0.462255	+	0.886747i	$0.347040\pi$
$444$	2867.22	0.306469
$445$	0	0
$446$	2438.54	0.258897
$447$	3310.30	0.350272
$448$	0	0
$449$	53.3141	0.00560367	0.00280183	−	0.999996i	$-0.499108\pi$
$449$	53.3141	0.00560367	0.00280183	+	0.999996i	$0.499108\pi$
$450$	0	0
$451$	746.251	0.0779148
$452$	−2025.20	−0.210747
$453$	−3250.22	−0.337105
$454$	−9147.35	−0.945608
$455$	0	0
$456$	3298.55	0.338747
$457$	−6421.65	−0.657313	−0.328656	−	0.944450i	$-0.606596\pi$
$457$	−6421.65	−0.657313	−0.328656	+	0.944450i	$0.606596\pi$
$458$	−6168.55	−0.629339
$459$	26194.5	2.66374
$460$	0	0
$461$	−13775.0	−1.39168	−0.695840	−	0.718197i	$-0.744968\pi$
$461$	−13775.0	−1.39168	−0.695840	+	0.718197i	$0.744968\pi$
$462$	0	0
$463$	−6286.78	−0.631040	−0.315520	−	0.948919i	$-0.602179\pi$
$463$	−6286.78	−0.631040	−0.315520	+	0.948919i	$0.602179\pi$
$464$	−285.424	−0.0285571
$465$	0	0
$466$	11865.7	1.17955
$467$	−6945.27	−0.688198	−0.344099	−	0.938933i	$-0.611816\pi$
$467$	−6945.27	−0.688198	−0.344099	+	0.938933i	$0.611816\pi$
$468$	10461.4	1.03328
$469$	0	0
$470$	0	0
$471$	12925.2	1.26446
$472$	4475.48	0.436443
$473$	16578.1	1.61155
$474$	15684.8	1.51989
$475$	0	0
$476$	0	0
$477$	−9258.54	−0.888720
$478$	−1614.24	−0.154464
$479$	7228.76	0.689542	0.344771	−	0.938687i	$-0.387957\pi$
$479$	7228.76	0.689542	0.344771	+	0.938687i	$0.387957\pi$
$480$	0	0
$481$	−3553.97	−0.336897
$482$	−8523.12	−0.805431
$483$	0	0
$484$	−289.712	−0.0272081
$485$	0	0
$486$	−3600.64	−0.336067
$487$	1423.58	0.132461	0.0662306	−	0.997804i	$-0.478903\pi$
$487$	1423.58	0.132461	0.0662306	+	0.997804i	$0.478903\pi$
$488$	870.210	0.0807224
$489$	2477.15	0.229081
$490$	0	0
$491$	−7574.62	−0.696207	−0.348104	−	0.937456i	$-0.613174\pi$
$491$	−7574.62	−0.696207	−0.348104	+	0.937456i	$0.613174\pi$
$492$	−773.055	−0.0708374
$493$	1672.31	0.152773
$494$	−4088.62	−0.372380
$495$	0	0
$496$	−434.539	−0.0393375
$497$	0	0
$498$	−15217.5	−1.36931
$499$	−13841.1	−1.24171	−0.620854	−	0.783926i	$-0.713214\pi$
$499$	−13841.1	−1.24171	−0.620854	+	0.783926i	$0.713214\pi$
$500$	0	0
$501$	2281.41	0.203445
$502$	−5924.96	−0.526780
$503$	−10917.4	−0.967763	−0.483882	−	0.875134i	$-0.660773\pi$
$503$	−10917.4	−0.967763	−0.483882	+	0.875134i	$0.660773\pi$
$504$	0	0
$505$	0	0
$506$	−8689.79	−0.763455
$507$	1120.08	0.0981155
$508$	−8900.40	−0.777345
$509$	−8831.06	−0.769018	−0.384509	−	0.923121i	$-0.625629\pi$
$509$	−8831.06	−0.769018	−0.384509	+	0.923121i	$0.625629\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	12539.9	1.07924
$514$	−12341.9	−1.05910
$515$	0	0
$516$	−17173.6	−1.46516
$517$	−20535.9	−1.74694
$518$	0	0
$519$	1046.73	0.0885283
$520$	0	0
$521$	−20972.0	−1.76353	−0.881767	−	0.471684i	$-0.843646\pi$
$521$	−20972.0	−1.76353	−0.881767	+	0.471684i	$0.843646\pi$
$522$	−2048.38	−0.171754
$523$	536.004	0.0448142	0.0224071	−	0.999749i	$-0.492867\pi$
$523$	536.004	0.0448142	0.0224071	+	0.999749i	$0.492867\pi$
$524$	7236.89	0.603330
$525$	0	0
$526$	259.823	0.0215377
$527$	2545.98	0.210446
$528$	−5215.11	−0.429846
$529$	2832.64	0.232813
$530$	0	0
$531$	32118.9	2.62493
$532$	0	0
$533$	958.218	0.0778706
$534$	3031.62	0.245676
$535$	0	0
$536$	3261.15	0.262799
$537$	−14375.7	−1.15523
$538$	−4947.34	−0.396459
$539$	0	0
$540$	0	0
$541$	8203.34	0.651921	0.325960	−	0.945383i	$-0.394312\pi$
$541$	8203.34	0.651921	0.325960	+	0.945383i	$0.394312\pi$
$542$	−5757.84	−0.456311
$543$	−44202.9	−3.49342
$544$	−2999.83	−0.236428
$545$	0	0
$546$	0	0
$547$	23851.1	1.86435	0.932173	−	0.362013i	$-0.117910\pi$
$547$	23851.1	1.86435	0.932173	+	0.362013i	$0.117910\pi$
$548$	−9295.42	−0.724600
$549$	6245.17	0.485496
$550$	0	0
$551$	800.571	0.0618974
$552$	9001.92	0.694107
$553$	0	0
$554$	1088.96	0.0835114
$555$	0	0
$556$	−7756.60	−0.591642
$557$	−25438.3	−1.93511	−0.967555	−	0.252660i	$-0.918695\pi$
$557$	−25438.3	−1.93511	−0.967555	+	0.252660i	$0.918695\pi$
$558$	−3118.52	−0.236591
$559$	21287.0	1.61063
$560$	0	0
$561$	30555.6	2.29957
$562$	−194.596	−0.0146060
$563$	−7219.84	−0.540462	−0.270231	−	0.962796i	$-0.587100\pi$
$563$	−7219.84	−0.540462	−0.270231	+	0.962796i	$0.587100\pi$
$564$	21273.5	1.58825
$565$	0	0
$566$	−10175.2	−0.755649
$567$	0	0
$568$	9274.93	0.685154
$569$	−6801.74	−0.501131	−0.250566	−	0.968100i	$-0.580617\pi$
$569$	−6801.74	−0.501131	−0.250566	+	0.968100i	$0.580617\pi$
$570$	0	0
$571$	17178.3	1.25900	0.629502	−	0.776999i	$-0.283259\pi$
$571$	17178.3	1.25900	0.629502	+	0.776999i	$0.283259\pi$
$572$	6464.24	0.472523
$573$	−22648.6	−1.65124
$574$	0	0
$575$	0	0
$576$	3674.43	0.265801
$577$	−17496.5	−1.26237	−0.631187	−	0.775631i	$-0.717432\pi$
$577$	−17496.5	−1.26237	−0.631187	+	0.775631i	$0.717432\pi$
$578$	7750.14	0.557722
$579$	21624.3	1.55211
$580$	0	0
$581$	0	0
$582$	710.344	0.0505923
$583$	−5720.99	−0.406414
$584$	−2053.42	−0.145498
$585$	0	0
$586$	8827.70	0.622302
$587$	−9824.21	−0.690781	−0.345390	−	0.938459i	$-0.612254\pi$
$587$	−9824.21	−0.690781	−0.345390	+	0.938459i	$0.612254\pi$
$588$	0	0
$589$	1218.81	0.0852638
$590$	0	0
$591$	14261.7	0.992636
$592$	−1248.29	−0.0866629
$593$	3309.32	0.229170	0.114585	−	0.993413i	$-0.463446\pi$
$593$	3309.32	0.229170	0.114585	+	0.993413i	$0.463446\pi$
$594$	−19825.9	−1.36947
$595$	0	0
$596$	−1441.19	−0.0990497
$597$	−6986.33	−0.478947
$598$	−11158.1	−0.763022
$599$	−10718.0	−0.731096	−0.365548	−	0.930792i	$-0.619118\pi$
$599$	−10718.0	−0.731096	−0.365548	+	0.930792i	$0.619118\pi$
$600$	0	0
$601$	8655.81	0.587484	0.293742	−	0.955885i	$-0.405099\pi$
$601$	8655.81	0.587484	0.293742	+	0.955885i	$0.405099\pi$
$602$	0	0
$603$	23404.0	1.58057
$604$	1415.04	0.0953263
$605$	0	0
$606$	14087.0	0.944300
$607$	−16498.6	−1.10322	−0.551611	−	0.834101i	$-0.685987\pi$
$607$	−16498.6	−1.10322	−0.551611	+	0.834101i	$0.685987\pi$
$608$	−1436.08	−0.0957907
$609$	0	0
$610$	0	0
$611$	−26368.9	−1.74595
$612$	−21528.7	−1.42197
$613$	−17687.2	−1.16538	−0.582692	−	0.812693i	$-0.698001\pi$
$613$	−17687.2	−1.16538	−0.582692	+	0.812693i	$0.698001\pi$
$614$	−2386.42	−0.156853
$615$	0	0
$616$	0	0
$617$	−19665.9	−1.28318	−0.641589	−	0.767049i	$-0.721724\pi$
$617$	−19665.9	−1.28318	−0.641589	+	0.767049i	$0.721724\pi$
$618$	9724.64	0.632981
$619$	−19963.6	−1.29629	−0.648147	−	0.761516i	$-0.724456\pi$
$619$	−19963.6	−1.29629	−0.648147	+	0.761516i	$0.724456\pi$
$620$	0	0
$621$	34221.9	2.21140
$622$	19459.1	1.25440
$623$	0	0
$624$	−6696.42	−0.429602
$625$	0	0
$626$	5588.99	0.356839
$627$	14627.6	0.931689
$628$	−5627.20	−0.357563
$629$	7313.79	0.463625
$630$	0	0
$631$	16085.0	1.01479	0.507395	−	0.861713i	$-0.330608\pi$
$631$	16085.0	1.01479	0.507395	+	0.861713i	$0.330608\pi$
$632$	−6828.66	−0.429794
$633$	27033.5	1.69745
$634$	−12463.8	−0.780758
$635$	0	0
$636$	5926.48	0.369497
$637$	0	0
$638$	−1265.73	−0.0785433
$639$	66562.7	4.12078
$640$	0	0
$641$	−23172.5	−1.42786	−0.713931	−	0.700216i	$-0.753087\pi$
$641$	−23172.5	−1.42786	−0.713931	+	0.700216i	$0.753087\pi$
$642$	−13112.9	−0.806111
$643$	12677.6	0.777535	0.388768	−	0.921336i	$-0.372901\pi$
$643$	12677.6	0.777535	0.388768	+	0.921336i	$0.372901\pi$
$644$	0	0
$645$	0	0
$646$	8414.06	0.512456
$647$	3398.92	0.206530	0.103265	−	0.994654i	$-0.467071\pi$
$647$	3398.92	0.206530	0.103265	+	0.994654i	$0.467071\pi$
$648$	8136.80	0.493278
$649$	19846.7	1.20039
$650$	0	0
$651$	0	0
$652$	−1078.47	−0.0647794
$653$	−13753.6	−0.824224	−0.412112	−	0.911133i	$-0.635209\pi$
$653$	−13753.6	−0.824224	−0.412112	+	0.911133i	$0.635209\pi$
$654$	5847.56	0.349629
$655$	0	0
$656$	336.563	0.0200313
$657$	−14736.6	−0.875083
$658$	0	0
$659$	3871.62	0.228857	0.114429	−	0.993431i	$-0.463496\pi$
$659$	3871.62	0.228857	0.114429	+	0.993431i	$0.463496\pi$
$660$	0	0
$661$	−8392.02	−0.493815	−0.246908	−	0.969039i	$-0.579414\pi$
$661$	−8392.02	−0.493815	−0.246908	+	0.969039i	$0.579414\pi$
$662$	−21785.0	−1.27900
$663$	39234.6	2.29826
$664$	6625.21	0.387211
$665$	0	0
$666$	−8958.51	−0.521224
$667$	2184.80	0.126830
$668$	−993.251	−0.0575300
$669$	−11202.2	−0.647388
$670$	0	0
$671$	3858.98	0.222019
$672$	0	0
$673$	−18527.8	−1.06121	−0.530604	−	0.847620i	$-0.678035\pi$
$673$	−18527.8	−1.06121	−0.530604	+	0.847620i	$0.678035\pi$
$674$	−827.901	−0.0473139
$675$	0	0
$676$	−487.646	−0.0277450
$677$	19787.2	1.12332	0.561659	−	0.827369i	$-0.310163\pi$
$677$	19787.2	1.12332	0.561659	+	0.827369i	$0.310163\pi$
$678$	9303.43	0.526985
$679$	0	0
$680$	0	0
$681$	42021.3	2.36455
$682$	−1926.98	−0.108194
$683$	27984.2	1.56777	0.783883	−	0.620908i	$-0.213236\pi$
$683$	27984.2	1.56777	0.783883	+	0.620908i	$0.213236\pi$
$684$	−10306.2	−0.576122
$685$	0	0
$686$	0	0
$687$	28337.2	1.57370
$688$	7476.80	0.414317
$689$	−7345.99	−0.406183
$690$	0	0
$691$	−10860.2	−0.597891	−0.298946	−	0.954270i	$-0.596635\pi$
$691$	−10860.2	−0.597891	−0.298946	+	0.954270i	$0.596635\pi$
$692$	−455.710	−0.0250339
$693$	0	0
$694$	−3419.07	−0.187012
$695$	0	0
$696$	1311.19	0.0714088
$697$	−1971.93	−0.107163
$698$	12538.1	0.679907
$699$	−54509.2	−2.94954
$700$	0	0
$701$	−5808.02	−0.312933	−0.156466	−	0.987683i	$-0.550010\pi$
$701$	−5808.02	−0.312933	−0.156466	+	0.987683i	$0.550010\pi$
$702$	−25457.3	−1.36869
$703$	3501.26	0.187842
$704$	2270.49	0.121551
$705$	0	0
$706$	13447.5	0.716859
$707$	0	0
$708$	−20559.6	−1.09135
$709$	7207.83	0.381800	0.190900	−	0.981610i	$-0.438859\pi$
$709$	7207.83	0.381800	0.190900	+	0.981610i	$0.438859\pi$
$710$	0	0
$711$	−49006.7	−2.58494
$712$	−1319.87	−0.0694720
$713$	3326.21	0.174709
$714$	0	0
$715$	0	0
$716$	6258.71	0.326674
$717$	7415.55	0.386247
$718$	3525.90	0.183266
$719$	−4956.92	−0.257110	−0.128555	−	0.991702i	$-0.541034\pi$
$719$	−4956.92	−0.257110	−0.128555	+	0.991702i	$0.541034\pi$
$720$	0	0
$721$	0	0
$722$	−9690.02	−0.499481
$723$	39153.8	2.01403
$724$	19244.5	0.987867
$725$	0	0
$726$	1330.89	0.0680356
$727$	3302.14	0.168459	0.0842295	−	0.996446i	$-0.473157\pi$
$727$	3302.14	0.168459	0.0842295	+	0.996446i	$0.473157\pi$
$728$	0	0
$729$	−10921.0	−0.554844
$730$	0	0
$731$	−43806.9	−2.21649
$732$	−3997.59	−0.201852
$733$	−28053.6	−1.41362	−0.706809	−	0.707405i	$-0.749866\pi$
$733$	−28053.6	−1.41362	−0.706809	+	0.707405i	$0.749866\pi$
$734$	12354.6	0.621277
$735$	0	0
$736$	−3919.14	−0.196279
$737$	14461.7	0.722800
$738$	2415.38	0.120476
$739$	−20132.1	−1.00213	−0.501064	−	0.865410i	$-0.667058\pi$
$739$	−20132.1	−1.00213	−0.501064	+	0.865410i	$0.667058\pi$
$740$	0	0
$741$	18782.4	0.931160
$742$	0	0
$743$	4320.73	0.213341	0.106670	−	0.994294i	$-0.465981\pi$
$743$	4320.73	0.213341	0.106670	+	0.994294i	$0.465981\pi$
$744$	1996.20	0.0983658
$745$	0	0
$746$	−16386.9	−0.804246
$747$	47546.7	2.32884
$748$	−13302.9	−0.650269
$749$	0	0
$750$	0	0
$751$	−5222.83	−0.253773	−0.126887	−	0.991917i	$-0.540498\pi$
$751$	−5222.83	−0.253773	−0.126887	+	0.991917i	$0.540498\pi$
$752$	−9261.77	−0.449125
$753$	27218.2	1.31725
$754$	−1625.25	−0.0784987
$755$	0	0
$756$	0	0
$757$	−16291.5	−0.782198	−0.391099	−	0.920349i	$-0.627905\pi$
$757$	−16291.5	−0.782198	−0.391099	+	0.920349i	$0.627905\pi$
$758$	−21656.9	−1.03775
$759$	39919.4	1.90907
$760$	0	0
$761$	727.281	0.0346438	0.0173219	−	0.999850i	$-0.494486\pi$
$761$	727.281	0.0346438	0.0173219	+	0.999850i	$0.494486\pi$
$762$	40886.9	1.94380
$763$	0	0
$764$	9860.45	0.466935
$765$	0	0
$766$	−14906.1	−0.703104
$767$	25484.0	1.19971
$768$	−2352.04	−0.110510
$769$	31107.7	1.45874	0.729371	−	0.684118i	$-0.239813\pi$
$769$	31107.7	1.45874	0.729371	+	0.684118i	$0.239813\pi$
$770$	0	0
$771$	56696.5	2.64834
$772$	−9414.49	−0.438905
$773$	30354.3	1.41238	0.706189	−	0.708023i	$-0.250413\pi$
$773$	30354.3	1.41238	0.706189	+	0.708023i	$0.250413\pi$
$774$	53658.1	2.49186
$775$	0	0
$776$	−309.260	−0.0143064
$777$	0	0
$778$	−8118.02	−0.374094
$779$	−944.006	−0.0434179
$780$	0	0
$781$	41130.1	1.88444
$782$	22962.4	1.05004
$783$	4984.65	0.227506
$784$	0	0
$785$	0	0
$786$	−33245.0	−1.50866
$787$	30312.8	1.37298	0.686491	−	0.727139i	$-0.259150\pi$
$787$	30312.8	1.37298	0.686491	+	0.727139i	$0.259150\pi$
$788$	−6209.07	−0.280697
$789$	−1193.58	−0.0538563
$790$	0	0
$791$	0	0
$792$	16294.4	0.731057
$793$	4955.10	0.221892
$794$	20495.9	0.916084
$795$	0	0
$796$	3041.62	0.135436
$797$	34499.2	1.53328	0.766639	−	0.642078i	$-0.221927\pi$
$797$	34499.2	1.53328	0.766639	+	0.642078i	$0.221927\pi$
$798$	0	0
$799$	54265.1	2.40271
$800$	0	0
$801$	−9472.18	−0.417832
$802$	5491.98	0.241806
$803$	−9105.97	−0.400178
$804$	−14981.1	−0.657144
$805$	0	0
$806$	−2474.33	−0.108132
$807$	22727.2	0.991371
$808$	−6133.02	−0.267028
$809$	22849.8	0.993025	0.496513	−	0.868029i	$-0.334614\pi$
$809$	22849.8	0.993025	0.496513	+	0.868029i	$0.334614\pi$
$810$	0	0
$811$	−5593.76	−0.242199	−0.121100	−	0.992640i	$-0.538642\pi$
$811$	−5593.76	−0.242199	−0.121100	+	0.992640i	$0.538642\pi$
$812$	0	0
$813$	26450.5	1.14103
$814$	−5535.60	−0.238357
$815$	0	0
$816$	13780.7	0.591202
$817$	−20971.2	−0.898031
$818$	20813.1	0.889624
$819$	0	0
$820$	0	0
$821$	34682.4	1.47433	0.737164	−	0.675714i	$-0.236165\pi$
$821$	34682.4	1.47433	0.737164	+	0.675714i	$0.236165\pi$
$822$	42701.6	1.81191
$823$	22182.3	0.939522	0.469761	−	0.882794i	$-0.344340\pi$
$823$	22182.3	0.939522	0.469761	+	0.882794i	$0.344340\pi$
$824$	−4233.79	−0.178994
$825$	0	0
$826$	0	0
$827$	−42247.1	−1.77639	−0.888197	−	0.459463i	$-0.848042\pi$
$827$	−42247.1	−1.77639	−0.888197	+	0.459463i	$0.848042\pi$
$828$	−28126.2	−1.18050
$829$	−44662.3	−1.87115	−0.935577	−	0.353122i	$-0.885120\pi$
$829$	−44662.3	−1.87115	−0.935577	+	0.353122i	$0.885120\pi$
$830$	0	0
$831$	−5002.47	−0.208825
$832$	2915.40	0.121482
$833$	0	0
$834$	35632.5	1.47944
$835$	0	0
$836$	−6368.36	−0.263462
$837$	7588.79	0.313390
$838$	7803.97	0.321699
$839$	31559.5	1.29863	0.649317	−	0.760518i	$-0.275055\pi$
$839$	31559.5	1.29863	0.649317	+	0.760518i	$0.275055\pi$
$840$	0	0
$841$	−24070.8	−0.986952
$842$	−15561.3	−0.636908
$843$	893.942	0.0365231
$844$	−11769.5	−0.480004
$845$	0	0
$846$	−66468.3	−2.70121
$847$	0	0
$848$	−2580.19	−0.104486
$849$	46743.3	1.88955
$850$	0	0
$851$	9555.12	0.384895
$852$	−42607.4	−1.71327
$853$	−39859.6	−1.59996	−0.799980	−	0.600027i	$-0.795156\pi$
$853$	−39859.6	−1.59996	−0.799980	+	0.600027i	$0.795156\pi$
$854$	0	0
$855$	0	0
$856$	5708.90	0.227951
$857$	22061.1	0.879339	0.439669	−	0.898160i	$-0.355096\pi$
$857$	22061.1	0.879339	0.439669	+	0.898160i	$0.355096\pi$
$858$	−29695.6	−1.18157
$859$	−17384.6	−0.690518	−0.345259	−	0.938508i	$-0.612209\pi$
$859$	−17384.6	−0.690518	−0.345259	+	0.938508i	$0.612209\pi$
$860$	0	0
$861$	0	0
$862$	−26706.0	−1.05523
$863$	38430.3	1.51586	0.757928	−	0.652339i	$-0.226212\pi$
$863$	38430.3	1.51586	0.757928	+	0.652339i	$0.226212\pi$
$864$	−8941.57	−0.352081
$865$	0	0
$866$	−25719.0	−1.00920
$867$	−35602.8	−1.39462
$868$	0	0
$869$	−30282.0	−1.18210
$870$	0	0
$871$	18569.4	0.722389
$872$	−2545.83	−0.0988678
$873$	−2219.44	−0.0860445
$874$	10992.6	0.425434
$875$	0	0
$876$	9433.04	0.363828
$877$	4326.65	0.166591	0.0832956	−	0.996525i	$-0.473455\pi$
$877$	4326.65	0.166591	0.0832956	+	0.996525i	$0.473455\pi$
$878$	24467.1	0.940460
$879$	−40552.9	−1.55610
$880$	0	0
$881$	28322.7	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$881$	28322.7	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$882$	0	0
$883$	47353.8	1.80474	0.902369	−	0.430964i	$-0.141827\pi$
$883$	47353.8	1.80474	0.902369	+	0.430964i	$0.141827\pi$
$884$	−17081.5	−0.649900
$885$	0	0
$886$	17240.4	0.653728
$887$	38608.5	1.46150	0.730748	−	0.682648i	$-0.239172\pi$
$887$	38608.5	1.46150	0.730748	+	0.682648i	$0.239172\pi$
$888$	5734.43	0.216706
$889$	0	0
$890$	0	0
$891$	36083.0	1.35671
$892$	4877.07	0.183068
$893$	25977.8	0.973477
$894$	6620.60	0.247680
$895$	0	0
$896$	0	0
$897$	51258.2	1.90798
$898$	106.628	0.00396239
$899$	484.485	0.0179738
$900$	0	0
$901$	15117.5	0.558974
$902$	1492.50	0.0550941
$903$	0	0
$904$	−4050.40	−0.149020
$905$	0	0
$906$	−6500.44	−0.238369
$907$	5590.57	0.204666	0.102333	−	0.994750i	$-0.467369\pi$
$907$	5590.57	0.204666	0.102333	+	0.994750i	$0.467369\pi$
$908$	−18294.7	−0.668646
$909$	−44014.4	−1.60601
$910$	0	0
$911$	27415.0	0.997035	0.498517	−	0.866880i	$-0.333878\pi$
$911$	27415.0	0.997035	0.498517	+	0.866880i	$0.333878\pi$
$912$	6597.10	0.239531
$913$	29379.8	1.06498
$914$	−12843.3	−0.464790
$915$	0	0
$916$	−12337.1	−0.445010
$917$	0	0
$918$	52389.0	1.88355
$919$	−42634.8	−1.53035	−0.765176	−	0.643821i	$-0.777348\pi$
$919$	−42634.8	−1.53035	−0.765176	+	0.643821i	$0.777348\pi$
$920$	0	0
$921$	10962.8	0.392222
$922$	−27549.9	−0.984066
$923$	52812.8	1.88337
$924$	0	0
$925$	0	0
$926$	−12573.6	−0.446212
$927$	−30384.3	−1.07654
$928$	−570.849	−0.0201929
$929$	16027.2	0.566021	0.283011	−	0.959117i	$-0.408667\pi$
$929$	16027.2	0.566021	0.283011	+	0.959117i	$0.408667\pi$
$930$	0	0
$931$	0	0
$932$	23731.5	0.834067
$933$	−89391.8	−3.13671
$934$	−13890.5	−0.486630
$935$	0	0
$936$	20922.7	0.730642
$937$	20639.6	0.719601	0.359800	−	0.933029i	$-0.382845\pi$
$937$	20639.6	0.719601	0.359800	+	0.933029i	$0.382845\pi$
$938$	0	0
$939$	−25674.8	−0.892297
$940$	0	0
$941$	32957.5	1.14175	0.570873	−	0.821038i	$-0.306605\pi$
$941$	32957.5	1.14175	0.570873	+	0.821038i	$0.306605\pi$
$942$	25850.4	0.894109
$943$	−2576.24	−0.0889649
$944$	8950.97	0.308611
$945$	0	0
$946$	33156.2	1.13954
$947$	29530.6	1.01332	0.506660	−	0.862146i	$-0.330880\pi$
$947$	29530.6	1.01332	0.506660	+	0.862146i	$0.330880\pi$
$948$	31369.7	1.07473
$949$	−11692.4	−0.399950
$950$	0	0
$951$	57256.5	1.95233
$952$	0	0
$953$	−41951.6	−1.42596	−0.712982	−	0.701182i	$-0.752656\pi$
$953$	−41951.6	−1.42596	−0.712982	+	0.701182i	$0.752656\pi$
$954$	−18517.1	−0.628420
$955$	0	0
$956$	−3228.49	−0.109222
$957$	5814.53	0.196402
$958$	14457.5	0.487580
$959$	0	0
$960$	0	0
$961$	−29053.4	−0.975241
$962$	−7107.95	−0.238222
$963$	40970.6	1.37099
$964$	−17046.2	−0.569526
$965$	0	0
$966$	0	0
$967$	24197.6	0.804699	0.402349	−	0.915486i	$-0.368194\pi$
$967$	24197.6	0.804699	0.402349	+	0.915486i	$0.368194\pi$
$968$	−579.424	−0.0192391
$969$	−38652.7	−1.28143
$970$	0	0
$971$	9056.44	0.299315	0.149657	−	0.988738i	$-0.452183\pi$
$971$	9056.44	0.299315	0.149657	+	0.988738i	$0.452183\pi$
$972$	−7201.28	−0.237635
$973$	0	0
$974$	2847.16	0.0936642
$975$	0	0
$976$	1740.42	0.0570794
$977$	−22533.8	−0.737891	−0.368946	−	0.929451i	$-0.620281\pi$
$977$	−22533.8	−0.737891	−0.368946	+	0.929451i	$0.620281\pi$
$978$	4954.31	0.161985
$979$	−5853.01	−0.191075
$980$	0	0
$981$	−18270.5	−0.594629
$982$	−15149.2	−0.492293
$983$	−26648.2	−0.864645	−0.432322	−	0.901719i	$-0.642306\pi$
$983$	−26648.2	−0.864645	−0.432322	+	0.901719i	$0.642306\pi$
$984$	−1546.11	−0.0500896
$985$	0	0
$986$	3344.63	0.108027
$987$	0	0
$988$	−8177.25	−0.263313
$989$	−57231.6	−1.84010
$990$	0	0
$991$	−35170.4	−1.12737	−0.563686	−	0.825989i	$-0.690617\pi$
$991$	−35170.4	−1.12737	−0.563686	+	0.825989i	$0.690617\pi$
$992$	−869.078	−0.0278158
$993$	100077.	3.19823
$994$	0	0
$995$	0	0
$996$	−30435.1	−0.968245
$997$	3808.67	0.120985	0.0604923	−	0.998169i	$-0.480733\pi$
$997$	3808.67	0.120985	0.0604923	+	0.998169i	$0.480733\pi$
$998$	−27682.2	−0.878020
$999$	21800.2	0.690417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cx.1.1		6
5.2	odd	4		490.4.c.e.99.12		12
5.3	odd	4		490.4.c.e.99.1		12
5.4	even	2		2450.4.a.cw.1.6		6
7.3	odd	6		350.4.e.n.51.1		12
7.5	odd	6		350.4.e.n.151.1		12
7.6	odd	2		2450.4.a.cy.1.6		6
35.3	even	12		70.4.i.a.9.12	yes	24
35.12	even	12		70.4.i.a.39.12	yes	24
35.13	even	4		490.4.c.f.99.6		12
35.17	even	12		70.4.i.a.9.1	✓	24
35.19	odd	6		350.4.e.o.151.6		12
35.24	odd	6		350.4.e.o.51.6		12
35.27	even	4		490.4.c.f.99.7		12
35.33	even	12		70.4.i.a.39.1	yes	24
35.34	odd	2		2450.4.a.cv.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.i.a.9.1	✓	24	35.17	even	12
70.4.i.a.9.12	yes	24	35.3	even	12
70.4.i.a.39.1	yes	24	35.33	even	12
70.4.i.a.39.12	yes	24	35.12	even	12
350.4.e.n.51.1		12	7.3	odd	6
350.4.e.n.151.1		12	7.5	odd	6
350.4.e.o.51.6		12	35.24	odd	6
350.4.e.o.151.6		12	35.19	odd	6
490.4.c.e.99.1		12	5.3	odd	4
490.4.c.e.99.12		12	5.2	odd	4
490.4.c.f.99.6		12	35.13	even	4
490.4.c.f.99.7		12	35.27	even	4
2450.4.a.cv.1.1		6	35.34	odd	2
2450.4.a.cw.1.6		6	5.4	even	2
2450.4.a.cx.1.1		6	1.1	even	1	trivial
2450.4.a.cy.1.6		6	7.6	odd	2

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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -9.18765 q^{3} +4.00000 q^{4} -18.3753 q^{6} +8.00000 q^{8} +57.4130 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -9.18765 q^{3} +4.00000 q^{4} -18.3753 q^{6} +8.00000 q^{8} +57.4130 q^{9} +35.4764 q^{11} -36.7506 q^{12} +45.5531 q^{13} +16.0000 q^{16} -93.7447 q^{17} +114.826 q^{18} -44.8775 q^{19} +70.9527 q^{22} -122.473 q^{23} -73.5012 q^{24} +91.1063 q^{26} -279.424 q^{27} -17.8390 q^{29} -27.1587 q^{31} +32.0000 q^{32} -325.945 q^{33} -187.489 q^{34} +229.652 q^{36} -78.0182 q^{37} -89.7550 q^{38} -418.526 q^{39} +21.0352 q^{41} +467.300 q^{43} +141.905 q^{44} -244.946 q^{46} -578.861 q^{47} -147.002 q^{48} +861.294 q^{51} +182.213 q^{52} -161.262 q^{53} -558.848 q^{54} +412.319 q^{57} -35.6781 q^{58} +559.436 q^{59} +108.776 q^{61} -54.3174 q^{62} +64.0000 q^{64} -651.889 q^{66} +407.643 q^{67} -374.979 q^{68} +1125.24 q^{69} +1159.37 q^{71} +459.304 q^{72} -256.677 q^{73} -156.036 q^{74} -179.510 q^{76} -837.053 q^{78} -853.583 q^{79} +1017.10 q^{81} +42.0703 q^{82} +828.152 q^{83} +934.599 q^{86} +163.899 q^{87} +283.811 q^{88} -164.983 q^{89} -489.892 q^{92} +249.525 q^{93} -1157.72 q^{94} -294.005 q^{96} -38.6575 q^{97} +2036.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} - 7 q^{3} + 24 q^{4} - 14 q^{6} + 48 q^{8} + 31 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 - 7 * q^3 + 24 * q^4 - 14 * q^6 + 48 * q^8 + 31 * q^9	\( 6 q + 12 q^{2} - 7 q^{3} + 24 q^{4} - 14 q^{6} + 48 q^{8} + 31 q^{9} + 31 q^{11} - 28 q^{12} - 59 q^{13} + 96 q^{16} - 68 q^{17} + 62 q^{18} - 93 q^{19} + 62 q^{22} - 94 q^{23} - 56 q^{24} - 118 q^{26} - 385 q^{27} + 169 q^{29} - 326 q^{31} + 192 q^{32} - 400 q^{33} - 136 q^{34} + 124 q^{36} - 253 q^{37} - 186 q^{38} - 434 q^{39} - 198 q^{41} - 99 q^{43} + 124 q^{44} - 188 q^{46} - 901 q^{47} - 112 q^{48} + 724 q^{51} - 236 q^{52} + 233 q^{53} - 770 q^{54} - 518 q^{57} + 338 q^{58} - 668 q^{59} - 157 q^{61} - 652 q^{62} + 384 q^{64} - 800 q^{66} + 1193 q^{67} - 272 q^{68} + 45 q^{69} + 1108 q^{71} + 248 q^{72} - 1458 q^{73} - 506 q^{74} - 372 q^{76} - 868 q^{78} - 886 q^{79} - 614 q^{81} - 396 q^{82} - 1799 q^{83} - 198 q^{86} - 789 q^{87} + 248 q^{88} - 3047 q^{89} - 376 q^{92} + 1902 q^{93} - 1802 q^{94} - 224 q^{96} - 3404 q^{97} + 4273 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 7 * q^3 + 24 * q^4 - 14 * q^6 + 48 * q^8 + 31 * q^9 + 31 * q^11 - 28 * q^12 - 59 * q^13 + 96 * q^16 - 68 * q^17 + 62 * q^18 - 93 * q^19 + 62 * q^22 - 94 * q^23 - 56 * q^24 - 118 * q^26 - 385 * q^27 + 169 * q^29 - 326 * q^31 + 192 * q^32 - 400 * q^33 - 136 * q^34 + 124 * q^36 - 253 * q^37 - 186 * q^38 - 434 * q^39 - 198 * q^41 - 99 * q^43 + 124 * q^44 - 188 * q^46 - 901 * q^47 - 112 * q^48 + 724 * q^51 - 236 * q^52 + 233 * q^53 - 770 * q^54 - 518 * q^57 + 338 * q^58 - 668 * q^59 - 157 * q^61 - 652 * q^62 + 384 * q^64 - 800 * q^66 + 1193 * q^67 - 272 * q^68 + 45 * q^69 + 1108 * q^71 + 248 * q^72 - 1458 * q^73 - 506 * q^74 - 372 * q^76 - 868 * q^78 - 886 * q^79 - 614 * q^81 - 396 * q^82 - 1799 * q^83 - 198 * q^86 - 789 * q^87 + 248 * q^88 - 3047 * q^89 - 376 * q^92 + 1902 * q^93 - 1802 * q^94 - 224 * q^96 - 3404 * q^97 + 4273 * q^99

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