LMFDB - Embedded newform 338.4.a.n.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.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:	$19.9426455819$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6681389953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951 $ x^6 - x^5 - 107x^4 + 85x^3 + 3703x^2 - 1659x - 41951
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$5.84973$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +10.0086 q^{3} +4.00000 q^{4} +13.8136 q^{5} -20.0171 q^{6} +0.262285 q^{7} -8.00000 q^{8} +73.1713 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +10.0086 q^{3} +4.00000 q^{4} +13.8136 q^{5} -20.0171 q^{6} +0.262285 q^{7} -8.00000 q^{8} +73.1713 q^{9} -27.6271 q^{10} -40.0159 q^{11} +40.0343 q^{12} -0.524571 q^{14} +138.254 q^{15} +16.0000 q^{16} +79.7361 q^{17} -146.343 q^{18} +22.5864 q^{19} +55.2543 q^{20} +2.62510 q^{21} +80.0318 q^{22} -65.5291 q^{23} -80.0685 q^{24} +65.8147 q^{25} +462.109 q^{27} +1.04914 q^{28} +40.1368 q^{29} -276.508 q^{30} +113.748 q^{31} -32.0000 q^{32} -400.502 q^{33} -159.472 q^{34} +3.62310 q^{35} +292.685 q^{36} -121.080 q^{37} -45.1727 q^{38} -110.509 q^{40} -396.184 q^{41} -5.25020 q^{42} -275.900 q^{43} -160.064 q^{44} +1010.76 q^{45} +131.058 q^{46} +440.483 q^{47} +160.137 q^{48} -342.931 q^{49} -131.629 q^{50} +798.043 q^{51} -615.108 q^{53} -924.217 q^{54} -552.762 q^{55} -2.09828 q^{56} +226.057 q^{57} -80.2736 q^{58} +230.908 q^{59} +553.016 q^{60} -109.524 q^{61} -227.495 q^{62} +19.1918 q^{63} +64.0000 q^{64} +801.003 q^{66} +221.132 q^{67} +318.944 q^{68} -655.852 q^{69} -7.24620 q^{70} +322.203 q^{71} -585.371 q^{72} +323.664 q^{73} +242.160 q^{74} +658.711 q^{75} +90.3455 q^{76} -10.4956 q^{77} +743.263 q^{79} +221.017 q^{80} +2649.42 q^{81} +792.369 q^{82} -539.296 q^{83} +10.5004 q^{84} +1101.44 q^{85} +551.800 q^{86} +401.712 q^{87} +320.127 q^{88} +1264.14 q^{89} -2021.51 q^{90} -262.116 q^{92} +1138.45 q^{93} -880.967 q^{94} +311.998 q^{95} -320.274 q^{96} -326.080 q^{97} +685.862 q^{98} -2928.02 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) $ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * 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
$3$	10.0086	1.92615	0.963074	−	0.269235i	$-0.0867710\pi$
$3$	10.0086	1.92615	0.963074	+	0.269235i	$0.0867710\pi$
$4$	4.00000	0.500000
$5$	13.8136	1.23552	0.617762	−	0.786365i	$-0.288040\pi$
$5$	13.8136	1.23552	0.617762	+	0.786365i	$0.288040\pi$
$6$	−20.0171	−1.36199
$7$	0.262285	0.0141621	0.00708104	−	0.999975i	$-0.497746\pi$
$7$	0.262285	0.0141621	0.00708104	+	0.999975i	$0.497746\pi$
$8$	−8.00000	−0.353553
$9$	73.1713	2.71005
$10$	−27.6271	−0.873647
$11$	−40.0159	−1.09684	−0.548420	−	0.836203i	$-0.684771\pi$
$11$	−40.0159	−1.09684	−0.548420	+	0.836203i	$0.684771\pi$
$12$	40.0343	0.963074
$13$	0	0
$13$	0	0
$14$	−0.524571	−0.0100141
$15$	138.254	2.37980
$16$	16.0000	0.250000
$17$	79.7361	1.13758	0.568789	−	0.822483i	$-0.307412\pi$
$17$	79.7361	1.13758	0.568789	+	0.822483i	$0.307412\pi$
$18$	−146.343	−1.91629
$19$	22.5864	0.272719	0.136360	−	0.990659i	$-0.456460\pi$
$19$	22.5864	0.272719	0.136360	+	0.990659i	$0.456460\pi$
$20$	55.2543	0.617762
$21$	2.62510	0.0272783
$22$	80.0318	0.775583
$23$	−65.5291	−0.594077	−0.297038	−	0.954866i	$-0.595999\pi$
$23$	−65.5291	−0.594077	−0.297038	+	0.954866i	$0.595999\pi$
$24$	−80.0685	−0.680996
$25$	65.8147	0.526518
$26$	0	0
$27$	462.109	3.29381
$28$	1.04914	0.00708104
$29$	40.1368	0.257007	0.128504	−	0.991709i	$-0.458983\pi$
$29$	40.1368	0.257007	0.128504	+	0.991709i	$0.458983\pi$
$30$	−276.508	−1.68277
$31$	113.748	0.659022	0.329511	−	0.944152i	$-0.393116\pi$
$31$	113.748	0.659022	0.329511	+	0.944152i	$0.393116\pi$
$32$	−32.0000	−0.176777
$33$	−400.502	−2.11268
$34$	−159.472	−0.804389
$35$	3.62310	0.0174976
$36$	292.685	1.35502
$37$	−121.080	−0.537984	−0.268992	−	0.963142i	$-0.586690\pi$
$37$	−121.080	−0.537984	−0.268992	+	0.963142i	$0.586690\pi$
$38$	−45.1727	−0.192842
$39$	0	0
$40$	−110.509	−0.436823
$41$	−396.184	−1.50911	−0.754556	−	0.656236i	$-0.772148\pi$
$41$	−396.184	−1.50911	−0.754556	+	0.656236i	$0.772148\pi$
$42$	−5.25020	−0.0192887
$43$	−275.900	−0.978473	−0.489237	−	0.872151i	$-0.662725\pi$
$43$	−275.900	−0.978473	−0.489237	+	0.872151i	$0.662725\pi$
$44$	−160.064	−0.548420
$45$	1010.76	3.34833
$46$	131.058	0.420076
$47$	440.483	1.36704	0.683522	−	0.729930i	$-0.260447\pi$
$47$	440.483	1.36704	0.683522	+	0.729930i	$0.260447\pi$
$48$	160.137	0.481537
$49$	−342.931	−0.999799
$50$	−131.629	−0.372304
$51$	798.043	2.19115
$52$	0	0
$53$	−615.108	−1.59418	−0.797091	−	0.603860i	$-0.793629\pi$
$53$	−615.108	−1.59418	−0.797091	+	0.603860i	$0.793629\pi$
$54$	−924.217	−2.32907
$55$	−552.762	−1.35517
$56$	−2.09828	−0.00500705
$57$	226.057	0.525298
$58$	−80.2736	−0.181732
$59$	230.908	0.509520	0.254760	−	0.967004i	$-0.418004\pi$
$59$	230.908	0.509520	0.254760	+	0.967004i	$0.418004\pi$
$60$	553.016	1.18990
$61$	−109.524	−0.229886	−0.114943	−	0.993372i	$-0.536669\pi$
$61$	−109.524	−0.229886	−0.114943	+	0.993372i	$0.536669\pi$
$62$	−227.495	−0.465999
$63$	19.1918	0.0383799
$64$	64.0000	0.125000
$65$	0	0
$66$	801.003	1.49389
$67$	221.132	0.403218	0.201609	−	0.979466i	$-0.435383\pi$
$67$	221.132	0.403218	0.201609	+	0.979466i	$0.435383\pi$
$68$	318.944	0.568789
$69$	−655.852	−1.14428
$70$	−7.24620	−0.0123727
$71$	322.203	0.538571	0.269285	−	0.963060i	$-0.413213\pi$
$71$	322.203	0.538571	0.269285	+	0.963060i	$0.413213\pi$
$72$	−585.371	−0.958147
$73$	323.664	0.518932	0.259466	−	0.965752i	$-0.416453\pi$
$73$	323.664	0.518932	0.259466	+	0.965752i	$0.416453\pi$
$74$	242.160	0.380412
$75$	658.711	1.01415
$76$	90.3455	0.136360
$77$	−10.4956	−0.0155335
$78$	0	0
$79$	743.263	1.05853	0.529263	−	0.848458i	$-0.322468\pi$
$79$	743.263	1.05853	0.529263	+	0.848458i	$0.322468\pi$
$80$	221.017	0.308881
$81$	2649.42	3.63432
$82$	792.369	1.06710
$83$	−539.296	−0.713198	−0.356599	−	0.934258i	$-0.616064\pi$
$83$	−539.296	−0.713198	−0.356599	+	0.934258i	$0.616064\pi$
$84$	10.5004	0.0136391
$85$	1101.44	1.40550
$86$	551.800	0.691885
$87$	401.712	0.495035
$88$	320.127	0.387792
$89$	1264.14	1.50560	0.752801	−	0.658248i	$-0.228702\pi$
$89$	1264.14	1.50560	0.752801	+	0.658248i	$0.228702\pi$
$90$	−2021.51	−2.36763
$91$	0	0
$92$	−262.116	−0.297038
$93$	1138.45	1.26937
$94$	−880.967	−0.966646
$95$	311.998	0.336951
$96$	−320.274	−0.340498
$97$	−326.080	−0.341324	−0.170662	−	0.985330i	$-0.554591\pi$
$97$	−326.080	−0.341324	−0.170662	+	0.985330i	$0.554591\pi$
$98$	685.862	0.706965
$99$	−2928.02	−2.97249
$100$	263.259	0.263259
$101$	−1435.16	−1.41390	−0.706948	−	0.707266i	$-0.749928\pi$
$101$	−1435.16	−1.41390	−0.706948	+	0.707266i	$0.749928\pi$
$102$	−1596.09	−1.54937
$103$	1853.77	1.77337	0.886684	−	0.462375i	$-0.153003\pi$
$103$	1853.77	1.77337	0.886684	+	0.462375i	$0.153003\pi$
$104$	0	0
$105$	36.2620	0.0337029
$106$	1230.22	1.12726
$107$	−153.925	−0.139070	−0.0695352	−	0.997580i	$-0.522152\pi$
$107$	−153.925	−0.139070	−0.0695352	+	0.997580i	$0.522152\pi$
$108$	1848.43	1.64690
$109$	−1788.06	−1.57124	−0.785621	−	0.618708i	$-0.787657\pi$
$109$	−1788.06	−1.57124	−0.785621	+	0.618708i	$0.787657\pi$
$110$	1105.52	0.958251
$111$	−1211.83	−1.03624
$112$	4.19657	0.00354052
$113$	−929.781	−0.774039	−0.387020	−	0.922071i	$-0.626495\pi$
$113$	−929.781	−0.774039	−0.387020	+	0.922071i	$0.626495\pi$
$114$	−452.114	−0.371442
$115$	−905.191	−0.733995
$116$	160.547	0.128504
$117$	0	0
$118$	−461.816	−0.360285
$119$	20.9136	0.0161105
$120$	−1106.03	−0.841387
$121$	270.272	0.203059
$122$	219.047	0.162554
$123$	−3965.24	−2.90677
$124$	454.991	0.329511
$125$	−817.560	−0.584998
$126$	−38.3835	−0.0271387
$127$	−785.484	−0.548822	−0.274411	−	0.961612i	$-0.588483\pi$
$127$	−785.484	−0.548822	−0.274411	+	0.961612i	$0.588483\pi$
$128$	−128.000	−0.0883883
$129$	−2761.36	−1.88469
$130$	0	0
$131$	−2851.17	−1.90159	−0.950793	−	0.309826i	$-0.899729\pi$
$131$	−2851.17	−1.90159	−0.950793	+	0.309826i	$0.899729\pi$
$132$	−1602.01	−1.05634
$133$	5.92408	0.00386228
$134$	−442.264	−0.285118
$135$	6383.37	4.06958
$136$	−637.888	−0.402195
$137$	402.079	0.250744	0.125372	−	0.992110i	$-0.459988\pi$
$137$	402.079	0.250744	0.125372	+	0.992110i	$0.459988\pi$
$138$	1311.70	0.809128
$139$	860.223	0.524915	0.262457	−	0.964944i	$-0.415467\pi$
$139$	860.223	0.524915	0.262457	+	0.964944i	$0.415467\pi$
$140$	14.4924	0.00874879
$141$	4408.60	2.63313
$142$	−644.407	−0.380827
$143$	0	0
$144$	1170.74	0.677512
$145$	554.433	0.317539
$146$	−647.328	−0.366940
$147$	−3432.25	−1.92576
$148$	−484.319	−0.268992
$149$	1430.01	0.786249	0.393125	−	0.919485i	$-0.371394\pi$
$149$	1430.01	0.786249	0.393125	+	0.919485i	$0.371394\pi$
$150$	−1317.42	−0.717114
$151$	941.478	0.507393	0.253697	−	0.967284i	$-0.418354\pi$
$151$	941.478	0.507393	0.253697	+	0.967284i	$0.418354\pi$
$152$	−180.691	−0.0964209
$153$	5834.39	3.08289
$154$	20.9912	0.0109839
$155$	1571.26	0.814237
$156$	0	0
$157$	−182.395	−0.0927179	−0.0463590	−	0.998925i	$-0.514762\pi$
$157$	−182.395	−0.0927179	−0.0463590	+	0.998925i	$0.514762\pi$
$158$	−1486.53	−0.748492
$159$	−6156.35	−3.07063
$160$	−442.034	−0.218412
$161$	−17.1873	−0.00841336
$162$	−5298.83	−2.56985
$163$	−887.221	−0.426335	−0.213167	−	0.977016i	$-0.568378\pi$
$163$	−887.221	−0.426335	−0.213167	+	0.977016i	$0.568378\pi$
$164$	−1584.74	−0.754556
$165$	−5532.36	−2.61026
$166$	1078.59	0.504307
$167$	−2898.67	−1.34315	−0.671574	−	0.740938i	$-0.734381\pi$
$167$	−2898.67	−1.34315	−0.671574	+	0.740938i	$0.734381\pi$
$168$	−21.0008	−0.00964433
$169$	0	0
$170$	−2202.88	−0.993842
$171$	1652.67	0.739083
$172$	−1103.60	−0.489237
$173$	2756.98	1.21162	0.605808	−	0.795611i	$-0.292850\pi$
$173$	2756.98	1.21162	0.605808	+	0.795611i	$0.292850\pi$
$174$	−803.423	−0.350042
$175$	17.2622	0.00745659
$176$	−640.254	−0.274210
$177$	2311.06	0.981411
$178$	−2528.28	−1.06462
$179$	−2106.68	−0.879668	−0.439834	−	0.898079i	$-0.644963\pi$
$179$	−2106.68	−0.879668	−0.439834	+	0.898079i	$0.644963\pi$
$180$	4043.03	1.67416
$181$	778.123	0.319544	0.159772	−	0.987154i	$-0.448924\pi$
$181$	778.123	0.319544	0.159772	+	0.987154i	$0.448924\pi$
$182$	0	0
$183$	−1096.17	−0.442795
$184$	524.233	0.210038
$185$	−1672.54	−0.664691
$186$	−2276.90	−0.897583
$187$	−3190.71	−1.24774
$188$	1761.93	0.683522
$189$	121.204	0.0466472
$190$	−623.997	−0.238260
$191$	−1791.06	−0.678517	−0.339259	−	0.940693i	$-0.610176\pi$
$191$	−1791.06	−0.678517	−0.339259	+	0.940693i	$0.610176\pi$
$192$	640.548	0.240769
$193$	−1876.99	−0.700043	−0.350022	−	0.936742i	$-0.613826\pi$
$193$	−1876.99	−0.700043	−0.350022	+	0.936742i	$0.613826\pi$
$194$	652.161	0.241353
$195$	0	0
$196$	−1371.72	−0.499900
$197$	−1584.04	−0.572885	−0.286443	−	0.958097i	$-0.592473\pi$
$197$	−1584.04	−0.572885	−0.286443	+	0.958097i	$0.592473\pi$
$198$	5856.03	2.10187
$199$	4968.18	1.76977	0.884887	−	0.465807i	$-0.154236\pi$
$199$	4968.18	1.76977	0.884887	+	0.465807i	$0.154236\pi$
$200$	−526.518	−0.186152
$201$	2213.21	0.776657
$202$	2870.31	0.999775
$203$	10.5273	0.00363976
$204$	3192.17	1.09557
$205$	−5472.72	−1.86454
$206$	−3707.53	−1.25396
$207$	−4794.85	−1.60998
$208$	0	0
$209$	−903.814	−0.299130
$210$	−72.5240	−0.0238316
$211$	3662.67	1.19502	0.597509	−	0.801862i	$-0.296157\pi$
$211$	3662.67	1.19502	0.597509	+	0.801862i	$0.296157\pi$
$212$	−2460.43	−0.797091
$213$	3224.79	1.03737
$214$	307.851	0.0983376
$215$	−3811.16	−1.20893
$216$	−3696.87	−1.16454
$217$	29.8344	0.00933312
$218$	3576.13	1.11104
$219$	3239.41	0.999540
$220$	−2211.05	−0.677586
$221$	0	0
$222$	2423.67	0.732730
$223$	4495.24	1.34988	0.674940	−	0.737872i	$-0.264169\pi$
$223$	4495.24	1.34988	0.674940	+	0.737872i	$0.264169\pi$
$224$	−8.39313	−0.00250353
$225$	4815.75	1.42689
$226$	1859.56	0.547328
$227$	−3601.61	−1.05307	−0.526536	−	0.850153i	$-0.676509\pi$
$227$	−3601.61	−1.05307	−0.526536	+	0.850153i	$0.676509\pi$
$228$	904.229	0.262649
$229$	−5162.11	−1.48962	−0.744808	−	0.667279i	$-0.767459\pi$
$229$	−5162.11	−1.48962	−0.744808	+	0.667279i	$0.767459\pi$
$230$	1810.38	0.519013
$231$	−105.046	−0.0299199
$232$	−321.094	−0.0908659
$233$	−2022.92	−0.568781	−0.284390	−	0.958709i	$-0.591791\pi$
$233$	−2022.92	−0.568781	−0.284390	+	0.958709i	$0.591791\pi$
$234$	0	0
$235$	6084.65	1.68902
$236$	923.633	0.254760
$237$	7438.99	2.03888
$238$	−41.8272	−0.0113918
$239$	−1423.12	−0.385163	−0.192581	−	0.981281i	$-0.561686\pi$
$239$	−1423.12	−0.385163	−0.192581	+	0.981281i	$0.561686\pi$
$240$	2212.06	0.594950
$241$	3554.34	0.950021	0.475011	−	0.879980i	$-0.342444\pi$
$241$	3554.34	0.950021	0.475011	+	0.879980i	$0.342444\pi$
$242$	−540.544	−0.143585
$243$	14039.9	3.70643
$244$	−438.094	−0.114943
$245$	−4737.10	−1.23528
$246$	7930.47	2.05540
$247$	0	0
$248$	−909.981	−0.232999
$249$	−5397.58	−1.37373
$250$	1635.12	0.413656
$251$	−1152.40	−0.289797	−0.144899	−	0.989447i	$-0.546286\pi$
$251$	−1152.40	−0.289797	−0.144899	+	0.989447i	$0.546286\pi$
$252$	76.7671	0.0191900
$253$	2622.21	0.651607
$254$	1570.97	0.388076
$255$	11023.8	2.70721
$256$	256.000	0.0625000
$257$	−1849.01	−0.448787	−0.224393	−	0.974499i	$-0.572040\pi$
$257$	−1849.01	−0.448787	−0.224393	+	0.974499i	$0.572040\pi$
$258$	5522.73	1.33267
$259$	−31.7575	−0.00761897
$260$	0	0
$261$	2936.86	0.696503
$262$	5702.34	1.34462
$263$	−3350.88	−0.785643	−0.392822	−	0.919615i	$-0.628501\pi$
$263$	−3350.88	−0.785643	−0.392822	+	0.919615i	$0.628501\pi$
$264$	3204.01	0.746945
$265$	−8496.84	−1.96965
$266$	−11.8482	−0.00273104
$267$	12652.2	2.90001
$268$	884.529	0.201609
$269$	−834.127	−0.189062	−0.0945309	−	0.995522i	$-0.530135\pi$
$269$	−834.127	−0.189062	−0.0945309	+	0.995522i	$0.530135\pi$
$270$	−12766.7	−2.87763
$271$	8061.06	1.80692	0.903459	−	0.428675i	$-0.141019\pi$
$271$	8061.06	1.80692	0.903459	+	0.428675i	$0.141019\pi$
$272$	1275.78	0.284395
$273$	0	0
$274$	−804.157	−0.177303
$275$	−2633.64	−0.577506
$276$	−2623.41	−0.572140
$277$	−4709.54	−1.02155	−0.510774	−	0.859715i	$-0.670641\pi$
$277$	−4709.54	−1.02155	−0.510774	+	0.859715i	$0.670641\pi$
$278$	−1720.45	−0.371171
$279$	8323.07	1.78598
$280$	−28.9848	−0.00618633
$281$	−2110.40	−0.448028	−0.224014	−	0.974586i	$-0.571916\pi$
$281$	−2110.40	−0.448028	−0.224014	+	0.974586i	$0.571916\pi$
$282$	−8817.21	−1.86190
$283$	1093.59	0.229708	0.114854	−	0.993382i	$-0.463360\pi$
$283$	1093.59	0.229708	0.114854	+	0.993382i	$0.463360\pi$
$284$	1288.81	0.269285
$285$	3122.66	0.649018
$286$	0	0
$287$	−103.913	−0.0213722
$288$	−2341.48	−0.479074
$289$	1444.84	0.294085
$290$	−1108.87	−0.224534
$291$	−3263.60	−0.657441
$292$	1294.66	0.259466
$293$	−9605.18	−1.91516	−0.957578	−	0.288174i	$-0.906952\pi$
$293$	−9605.18	−1.91516	−0.957578	+	0.288174i	$0.906952\pi$
$294$	6864.50	1.36172
$295$	3189.67	0.629524
$296$	968.638	0.190206
$297$	−18491.7	−3.61278
$298$	−2860.02	−0.555962
$299$	0	0
$300$	2634.84	0.507076
$301$	−72.3646	−0.0138572
$302$	−1882.96	−0.358781
$303$	−14363.9	−2.72337
$304$	361.382	0.0681799
$305$	−1512.91	−0.284030
$306$	−11668.8	−2.17993
$307$	4387.82	0.815721	0.407860	−	0.913044i	$-0.366275\pi$
$307$	4387.82	0.815721	0.407860	+	0.913044i	$0.366275\pi$
$308$	−41.9823	−0.00776677
$309$	18553.5	3.41577
$310$	−3142.52	−0.575752
$311$	2875.57	0.524304	0.262152	−	0.965027i	$-0.415568\pi$
$311$	2875.57	0.524304	0.262152	+	0.965027i	$0.415568\pi$
$312$	0	0
$313$	5715.61	1.03216	0.516078	−	0.856541i	$-0.327391\pi$
$313$	5715.61	1.03216	0.516078	+	0.856541i	$0.327391\pi$
$314$	364.790	0.0655615
$315$	265.107	0.0474193
$316$	2973.05	0.529263
$317$	7381.87	1.30791	0.653954	−	0.756534i	$-0.273109\pi$
$317$	7381.87	1.30791	0.653954	+	0.756534i	$0.273109\pi$
$318$	12312.7	2.17126
$319$	−1606.11	−0.281896
$320$	884.069	0.154440
$321$	−1540.57	−0.267870
$322$	34.3747	0.00594915
$323$	1800.95	0.310240
$324$	10597.7	1.81716
$325$	0	0
$326$	1774.44	0.301464
$327$	−17895.9	−3.02645
$328$	3169.47	0.533552
$329$	115.532	0.0193602
$330$	11064.7	1.84573
$331$	−6054.59	−1.00541	−0.502704	−	0.864458i	$-0.667662\pi$
$331$	−6054.59	−1.00541	−0.502704	+	0.864458i	$0.667662\pi$
$332$	−2157.18	−0.356599
$333$	−8859.57	−1.45796
$334$	5797.34	0.949749
$335$	3054.62	0.498185
$336$	42.0016	0.00681957
$337$	−3686.95	−0.595967	−0.297984	−	0.954571i	$-0.596314\pi$
$337$	−3686.95	−0.595967	−0.297984	+	0.954571i	$0.596314\pi$
$338$	0	0
$339$	−9305.77	−1.49091
$340$	4405.76	0.702752
$341$	−4551.71	−0.722842
$342$	−3305.35	−0.522611
$343$	−179.910	−0.0283213
$344$	2207.20	0.345943
$345$	−9059.66	−1.41378
$346$	−5513.97	−0.856742
$347$	−6535.10	−1.01102	−0.505508	−	0.862822i	$-0.668695\pi$
$347$	−6535.10	−1.01102	−0.505508	+	0.862822i	$0.668695\pi$
$348$	1606.85	0.247517
$349$	2973.40	0.456053	0.228027	−	0.973655i	$-0.426773\pi$
$349$	2973.40	0.456053	0.228027	+	0.973655i	$0.426773\pi$
$350$	−34.5245	−0.00527260
$351$	0	0
$352$	1280.51	0.193896
$353$	−11092.2	−1.67246	−0.836232	−	0.548375i	$-0.815246\pi$
$353$	−11092.2	−1.67246	−0.836232	+	0.548375i	$0.815246\pi$
$354$	−4622.12	−0.693963
$355$	4450.78	0.665417
$356$	5056.56	0.752801
$357$	209.315	0.0310312
$358$	4213.36	0.622019
$359$	5263.11	0.773751	0.386875	−	0.922132i	$-0.373554\pi$
$359$	5263.11	0.773751	0.386875	+	0.922132i	$0.373554\pi$
$360$	−8086.06	−1.18381
$361$	−6348.86	−0.925624
$362$	−1556.25	−0.225951
$363$	2705.03	0.391122
$364$	0	0
$365$	4470.96	0.641152
$366$	2192.35	0.313103
$367$	8669.12	1.23304	0.616518	−	0.787341i	$-0.288543\pi$
$367$	8669.12	1.23304	0.616518	+	0.787341i	$0.288543\pi$
$368$	−1048.47	−0.148519
$369$	−28989.3	−4.08977
$370$	3345.09	0.470008
$371$	−161.334	−0.0225769
$372$	4553.80	0.634687
$373$	7859.73	1.09105	0.545524	−	0.838095i	$-0.316330\pi$
$373$	7859.73	1.09105	0.545524	+	0.838095i	$0.316330\pi$
$374$	6381.42	0.882287
$375$	−8182.60	−1.12679
$376$	−3523.87	−0.483323
$377$	0	0
$378$	−242.409	−0.0329846
$379$	3080.51	0.417508	0.208754	−	0.977968i	$-0.433059\pi$
$379$	3080.51	0.417508	0.208754	+	0.977968i	$0.433059\pi$
$380$	1247.99	0.168476
$381$	−7861.56	−1.05711
$382$	3582.13	0.479784
$383$	1431.12	0.190932	0.0954661	−	0.995433i	$-0.469566\pi$
$383$	1431.12	0.190932	0.0954661	+	0.995433i	$0.469566\pi$
$384$	−1281.10	−0.170249
$385$	−144.982	−0.0191921
$386$	3753.97	0.495005
$387$	−20188.0	−2.65171
$388$	−1304.32	−0.170662
$389$	11639.0	1.51701	0.758507	−	0.651665i	$-0.225929\pi$
$389$	11639.0	1.51701	0.758507	+	0.651665i	$0.225929\pi$
$390$	0	0
$391$	−5225.03	−0.675809
$392$	2743.45	0.353482
$393$	−28536.1	−3.66274
$394$	3168.09	0.405091
$395$	10267.1	1.30783
$396$	−11712.1	−1.48625
$397$	−1187.22	−0.150088	−0.0750438	−	0.997180i	$-0.523910\pi$
$397$	−1187.22	−0.150088	−0.0750438	+	0.997180i	$0.523910\pi$
$398$	−9936.36	−1.25142
$399$	59.2915	0.00743932
$400$	1053.04	0.131629
$401$	3348.27	0.416969	0.208485	−	0.978026i	$-0.433147\pi$
$401$	3348.27	0.416969	0.208485	+	0.978026i	$0.433147\pi$
$402$	−4426.43	−0.549180
$403$	0	0
$404$	−5740.63	−0.706948
$405$	36597.9	4.49028
$406$	−21.0546	−0.00257370
$407$	4845.12	0.590082
$408$	−6384.35	−0.774687
$409$	−16139.6	−1.95123	−0.975615	−	0.219491i	$-0.929560\pi$
$409$	−16139.6	−1.95123	−0.975615	+	0.219491i	$0.929560\pi$
$410$	10945.4	1.31843
$411$	4024.23	0.482970
$412$	7415.06	0.886684
$413$	60.5638	0.00721586
$414$	9589.70	1.13843
$415$	−7449.60	−0.881173
$416$	0	0
$417$	8609.60	1.01106
$418$	1807.63	0.211517
$419$	6346.58	0.739978	0.369989	−	0.929036i	$-0.379361\pi$
$419$	6346.58	0.739978	0.369989	+	0.929036i	$0.379361\pi$
$420$	145.048	0.0168515
$421$	5135.18	0.594473	0.297236	−	0.954804i	$-0.403935\pi$
$421$	5135.18	0.594473	0.297236	+	0.954804i	$0.403935\pi$
$422$	−7325.35	−0.845005
$423$	32230.7	3.70476
$424$	4920.87	0.563628
$425$	5247.81	0.598955
$426$	−6449.59	−0.733529
$427$	−28.7264	−0.00325567
$428$	−615.701	−0.0695352
$429$	0	0
$430$	7622.33	0.854840
$431$	−7044.54	−0.787293	−0.393647	−	0.919262i	$-0.628787\pi$
$431$	−7044.54	−0.787293	−0.393647	+	0.919262i	$0.628787\pi$
$432$	7393.74	0.823452
$433$	10173.8	1.12915	0.564576	−	0.825381i	$-0.309040\pi$
$433$	10173.8	1.12915	0.564576	+	0.825381i	$0.309040\pi$
$434$	−59.6687	−0.00659951
$435$	5549.07	0.611627
$436$	−7152.26	−0.785621
$437$	−1480.06	−0.162016
$438$	−6478.83	−0.706781
$439$	6385.59	0.694231	0.347116	−	0.937822i	$-0.387161\pi$
$439$	6385.59	0.694231	0.347116	+	0.937822i	$0.387161\pi$
$440$	4422.10	0.479126
$441$	−25092.7	−2.70951
$442$	0	0
$443$	2362.81	0.253410	0.126705	−	0.991940i	$-0.459560\pi$
$443$	2362.81	0.253410	0.126705	+	0.991940i	$0.459560\pi$
$444$	−4847.34	−0.518118
$445$	17462.3	1.86021
$446$	−8990.47	−0.954510
$447$	14312.4	1.51443
$448$	16.7863	0.00177026
$449$	−8148.20	−0.856431	−0.428216	−	0.903677i	$-0.640858\pi$
$449$	−8148.20	−0.856431	−0.428216	+	0.903677i	$0.640858\pi$
$450$	−9631.50	−1.00896
$451$	15853.7	1.65526
$452$	−3719.12	−0.387020
$453$	9422.84	0.977315
$454$	7203.22	0.744634
$455$	0	0
$456$	−1808.46	−0.185721
$457$	14034.4	1.43655	0.718273	−	0.695762i	$-0.244933\pi$
$457$	14034.4	1.43655	0.718273	+	0.695762i	$0.244933\pi$
$458$	10324.2	1.05332
$459$	36846.7	3.74697
$460$	−3620.76	−0.366998
$461$	7691.70	0.777089	0.388545	−	0.921430i	$-0.372978\pi$
$461$	7691.70	0.777089	0.388545	+	0.921430i	$0.372978\pi$
$462$	210.091	0.0211566
$463$	−15555.9	−1.56144	−0.780719	−	0.624883i	$-0.785147\pi$
$463$	−15555.9	−1.56144	−0.780719	+	0.624883i	$0.785147\pi$
$464$	642.189	0.0642519
$465$	15726.1	1.56834
$466$	4045.84	0.402189
$467$	−11941.8	−1.18330	−0.591648	−	0.806197i	$-0.701522\pi$
$467$	−11941.8	−1.18330	−0.591648	+	0.806197i	$0.701522\pi$
$468$	0	0
$469$	57.9997	0.00571040
$470$	−12169.3	−1.19431
$471$	−1825.51	−0.178589
$472$	−1847.27	−0.180143
$473$	11040.4	1.07323
$474$	−14878.0	−1.44171
$475$	1486.52	0.143592
$476$	83.6544	0.00805524
$477$	−45008.3	−4.32031
$478$	2846.24	0.272351
$479$	14856.4	1.41714	0.708568	−	0.705642i	$-0.249342\pi$
$479$	14856.4	1.41714	0.708568	+	0.705642i	$0.249342\pi$
$480$	−4424.13	−0.420693
$481$	0	0
$482$	−7108.68	−0.671767
$483$	−172.020	−0.0162054
$484$	1081.09	0.101530
$485$	−4504.34	−0.421714
$486$	−28079.9	−2.62084
$487$	−14063.6	−1.30859	−0.654295	−	0.756239i	$-0.727035\pi$
$487$	−14063.6	−1.30859	−0.654295	+	0.756239i	$0.727035\pi$
$488$	876.188	0.0812770
$489$	−8879.81	−0.821184
$490$	9474.21	0.873472
$491$	−8218.07	−0.755349	−0.377675	−	0.925938i	$-0.623276\pi$
$491$	−8218.07	−0.755349	−0.377675	+	0.925938i	$0.623276\pi$
$492$	−15860.9	−1.45339
$493$	3200.35	0.292366
$494$	0	0
$495$	−40446.4	−3.67258
$496$	1819.96	0.164755
$497$	84.5093	0.00762728
$498$	10795.2	0.971370
$499$	6081.91	0.545619	0.272809	−	0.962068i	$-0.412047\pi$
$499$	6081.91	0.545619	0.272809	+	0.962068i	$0.412047\pi$
$500$	−3270.24	−0.292499
$501$	−29011.5	−2.58710
$502$	2304.81	0.204917
$503$	−3226.70	−0.286027	−0.143013	−	0.989721i	$-0.545679\pi$
$503$	−3226.70	−0.286027	−0.143013	+	0.989721i	$0.545679\pi$
$504$	−153.534	−0.0135694
$505$	−19824.6	−1.74690
$506$	−5244.41	−0.460756
$507$	0	0
$508$	−3141.93	−0.274411
$509$	−629.071	−0.0547801	−0.0273901	−	0.999625i	$-0.508720\pi$
$509$	−629.071	−0.0547801	−0.0273901	+	0.999625i	$0.508720\pi$
$510$	−22047.7	−1.91429
$511$	84.8924	0.00734915
$512$	−512.000	−0.0441942
$513$	10437.4	0.898286
$514$	3698.02	0.317340
$515$	25607.1	2.19104
$516$	−11045.5	−0.942343
$517$	−17626.3	−1.49943
$518$	63.5149	0.00538742
$519$	27593.4	2.33375
$520$	0	0
$521$	−785.154	−0.0660235	−0.0330117	−	0.999455i	$-0.510510\pi$
$521$	−785.154	−0.0660235	−0.0330117	+	0.999455i	$0.510510\pi$
$522$	−5873.73	−0.492502
$523$	−990.379	−0.0828035	−0.0414018	−	0.999143i	$-0.513182\pi$
$523$	−990.379	−0.0828035	−0.0414018	+	0.999143i	$0.513182\pi$
$524$	−11404.7	−0.950793
$525$	172.770	0.0143625
$526$	6701.76	0.555534
$527$	9069.79	0.749689
$528$	−6408.03	−0.528170
$529$	−7872.94	−0.647073
$530$	16993.7	1.39275
$531$	16895.9	1.38082
$532$	23.6963	0.00193114
$533$	0	0
$534$	−25304.5	−2.05062
$535$	−2126.26	−0.171825
$536$	−1769.06	−0.142559
$537$	−21084.8	−1.69437
$538$	1668.25	0.133687
$539$	13722.7	1.09662
$540$	25533.5	2.03479
$541$	−2509.00	−0.199390	−0.0996952	−	0.995018i	$-0.531787\pi$
$541$	−2509.00	−0.199390	−0.0996952	+	0.995018i	$0.531787\pi$
$542$	−16122.1	−1.27768
$543$	7787.89	0.615489
$544$	−2551.55	−0.201097
$545$	−24699.5	−1.94131
$546$	0	0
$547$	24461.5	1.91206	0.956032	−	0.293264i	$-0.0947415\pi$
$547$	24461.5	1.91206	0.956032	+	0.293264i	$0.0947415\pi$
$548$	1608.31	0.125372
$549$	−8013.98	−0.623003
$550$	5267.27	0.408359
$551$	906.545	0.0700909
$552$	5246.82	0.404564
$553$	194.947	0.0149909
$554$	9419.08	0.722344
$555$	−16739.8	−1.28029
$556$	3440.89	0.262457
$557$	12228.8	0.930250	0.465125	−	0.885245i	$-0.346009\pi$
$557$	12228.8	0.930250	0.465125	+	0.885245i	$0.346009\pi$
$558$	−16646.1	−1.26288
$559$	0	0
$560$	57.9696	0.00437440
$561$	−31934.4	−2.40334
$562$	4220.80	0.316804
$563$	−1932.20	−0.144641	−0.0723203	−	0.997381i	$-0.523040\pi$
$563$	−1932.20	−0.144641	−0.0723203	+	0.997381i	$0.523040\pi$
$564$	17634.4	1.31657
$565$	−12843.6	−0.956344
$566$	−2187.18	−0.162428
$567$	694.904	0.0514695
$568$	−2577.63	−0.190413
$569$	19743.1	1.45461	0.727307	−	0.686313i	$-0.240772\pi$
$569$	19743.1	1.45461	0.727307	+	0.686313i	$0.240772\pi$
$570$	−6245.31	−0.458925
$571$	1327.51	0.0972933	0.0486466	−	0.998816i	$-0.484509\pi$
$571$	1327.51	0.0972933	0.0486466	+	0.998816i	$0.484509\pi$
$572$	0	0
$573$	−17926.0	−1.30692
$574$	207.827	0.0151124
$575$	−4312.78	−0.312792
$576$	4682.97	0.338756
$577$	15077.5	1.08784	0.543921	−	0.839137i	$-0.316939\pi$
$577$	15077.5	1.08784	0.543921	+	0.839137i	$0.316939\pi$
$578$	−2889.68	−0.207949
$579$	−18785.9	−1.34839
$580$	2217.73	0.158769
$581$	−141.449	−0.0101004
$582$	6527.19	0.464881
$583$	24614.1	1.74856
$584$	−2589.31	−0.183470
$585$	0	0
$586$	19210.4	1.35422
$587$	7156.56	0.503207	0.251604	−	0.967830i	$-0.419042\pi$
$587$	7156.56	0.503207	0.251604	+	0.967830i	$0.419042\pi$
$588$	−13729.0	−0.962881
$589$	2569.15	0.179728
$590$	−6379.33	−0.445141
$591$	−15854.0	−1.10346
$592$	−1937.28	−0.134496
$593$	14729.2	1.01999	0.509997	−	0.860176i	$-0.329646\pi$
$593$	14729.2	1.01999	0.509997	+	0.860176i	$0.329646\pi$
$594$	36983.4	2.55462
$595$	288.892	0.0199049
$596$	5720.05	0.393125
$597$	49724.3	3.40885
$598$	0	0
$599$	28634.9	1.95324	0.976621	−	0.214970i	$-0.0689653\pi$
$599$	28634.9	1.95324	0.976621	+	0.214970i	$0.0689653\pi$
$600$	−5269.69	−0.358557
$601$	5515.59	0.374352	0.187176	−	0.982326i	$-0.440066\pi$
$601$	5515.59	0.374352	0.187176	+	0.982326i	$0.440066\pi$
$602$	144.729	0.00979854
$603$	16180.5	1.09274
$604$	3765.91	0.253697
$605$	3733.42	0.250885
$606$	28727.7	1.92572
$607$	12898.1	0.862470	0.431235	−	0.902240i	$-0.358078\pi$
$607$	12898.1	0.862470	0.431235	+	0.902240i	$0.358078\pi$
$608$	−722.764	−0.0482104
$609$	105.363	0.00701072
$610$	3025.82	0.200839
$611$	0	0
$612$	23337.6	1.54145
$613$	16080.3	1.05951	0.529754	−	0.848151i	$-0.322284\pi$
$613$	16080.3	1.05951	0.529754	+	0.848151i	$0.322284\pi$
$614$	−8775.64	−0.576802
$615$	−54774.1	−3.59139
$616$	83.9647	0.00549194
$617$	−923.247	−0.0602407	−0.0301203	−	0.999546i	$-0.509589\pi$
$617$	−923.247	−0.0602407	−0.0301203	+	0.999546i	$0.509589\pi$
$618$	−37107.1	−2.41532
$619$	−25650.3	−1.66555	−0.832773	−	0.553615i	$-0.813248\pi$
$619$	−25650.3	−1.66555	−0.832773	+	0.553615i	$0.813248\pi$
$620$	6285.04	0.407118
$621$	−30281.6	−1.95678
$622$	−5751.14	−0.370739
$623$	331.566	0.0213225
$624$	0	0
$625$	−19520.3	−1.24930
$626$	−11431.2	−0.729845
$627$	−9045.88	−0.576168
$628$	−729.581	−0.0463590
$629$	−9654.42	−0.611999
$630$	−530.214	−0.0335305
$631$	21747.9	1.37206	0.686029	−	0.727574i	$-0.259352\pi$
$631$	21747.9	1.37206	0.686029	+	0.727574i	$0.259352\pi$
$632$	−5946.10	−0.374246
$633$	36658.1	2.30178
$634$	−14763.7	−0.924831
$635$	−10850.3	−0.678082
$636$	−24625.4	−1.53532
$637$	0	0
$638$	3212.22	0.199331
$639$	23576.1	1.45955
$640$	−1768.14	−0.109206
$641$	22227.7	1.36964	0.684822	−	0.728711i	$-0.259880\pi$
$641$	22227.7	1.36964	0.684822	+	0.728711i	$0.259880\pi$
$642$	3081.14	0.189413
$643$	−2848.45	−0.174700	−0.0873498	−	0.996178i	$-0.527840\pi$
$643$	−2848.45	−0.174700	−0.0873498	+	0.996178i	$0.527840\pi$
$644$	−68.7493	−0.00420668
$645$	−38144.3	−2.32857
$646$	−3601.90	−0.219373
$647$	4148.86	0.252100	0.126050	−	0.992024i	$-0.459770\pi$
$647$	4148.86	0.252100	0.126050	+	0.992024i	$0.459770\pi$
$648$	−21195.3	−1.28493
$649$	−9240.00	−0.558862
$650$	0	0
$651$	298.599	0.0179770
$652$	−3548.88	−0.213167
$653$	12417.8	0.744173	0.372087	−	0.928198i	$-0.378642\pi$
$653$	12417.8	0.744173	0.372087	+	0.928198i	$0.378642\pi$
$654$	35791.9	2.14002
$655$	−39384.8	−2.34945
$656$	−6338.95	−0.377278
$657$	23682.9	1.40633
$658$	−231.065	−0.0136897
$659$	−15902.9	−0.940043	−0.470022	−	0.882655i	$-0.655754\pi$
$659$	−15902.9	−0.940043	−0.470022	+	0.882655i	$0.655754\pi$
$660$	−22129.4	−1.30513
$661$	−26324.6	−1.54903	−0.774514	−	0.632556i	$-0.782006\pi$
$661$	−26324.6	−1.54903	−0.774514	+	0.632556i	$0.782006\pi$
$662$	12109.2	0.710931
$663$	0	0
$664$	4314.37	0.252154
$665$	81.8327	0.00477193
$666$	17719.1	1.03093
$667$	−2630.13	−0.152682
$668$	−11594.7	−0.671574
$669$	44990.9	2.60007
$670$	−6109.25	−0.352270
$671$	4382.68	0.252148
$672$	−84.0032	−0.00482216
$673$	11586.6	0.663643	0.331821	−	0.943342i	$-0.392337\pi$
$673$	11586.6	0.663643	0.331821	+	0.943342i	$0.392337\pi$
$674$	7373.90	0.421413
$675$	30413.6	1.73425
$676$	0	0
$677$	−7512.76	−0.426498	−0.213249	−	0.976998i	$-0.568405\pi$
$677$	−7512.76	−0.426498	−0.213249	+	0.976998i	$0.568405\pi$
$678$	18611.5	1.05424
$679$	−85.5262	−0.00483386
$680$	−8811.52	−0.496921
$681$	−36047.0	−2.02837
$682$	9103.43	0.511126
$683$	−4354.81	−0.243971	−0.121986	−	0.992532i	$-0.538926\pi$
$683$	−4354.81	−0.243971	−0.121986	+	0.992532i	$0.538926\pi$
$684$	6610.70	0.369542
$685$	5554.14	0.309800
$686$	359.820	0.0200262
$687$	−51665.3	−2.86922
$688$	−4414.40	−0.244618
$689$	0	0
$690$	18119.3	0.999697
$691$	20534.9	1.13051	0.565255	−	0.824916i	$-0.308778\pi$
$691$	20534.9	1.13051	0.565255	+	0.824916i	$0.308778\pi$
$692$	11027.9	0.605808
$693$	−767.976	−0.0420967
$694$	13070.2	0.714896
$695$	11882.8	0.648544
$696$	−3213.69	−0.175021
$697$	−31590.2	−1.71673
$698$	−5946.81	−0.322478
$699$	−20246.5	−1.09556
$700$	69.0490	0.00372829
$701$	19666.5	1.05962	0.529810	−	0.848117i	$-0.322263\pi$
$701$	19666.5	1.05962	0.529810	+	0.848117i	$0.322263\pi$
$702$	0	0
$703$	−2734.75	−0.146719
$704$	−2561.02	−0.137105
$705$	60898.6	3.25329
$706$	22184.5	1.18261
$707$	−376.421	−0.0200237
$708$	9244.23	0.490706
$709$	7433.68	0.393763	0.196881	−	0.980427i	$-0.436919\pi$
$709$	7433.68	0.393763	0.196881	+	0.980427i	$0.436919\pi$
$710$	−8901.56	−0.470521
$711$	54385.5	2.86866
$712$	−10113.1	−0.532311
$713$	−7453.78	−0.391510
$714$	−418.630	−0.0219424
$715$	0	0
$716$	−8426.72	−0.439834
$717$	−14243.4	−0.741880
$718$	−10526.2	−0.547124
$719$	−24845.3	−1.28870	−0.644348	−	0.764733i	$-0.722871\pi$
$719$	−24845.3	−1.28870	−0.644348	+	0.764733i	$0.722871\pi$
$720$	16172.1	0.837082
$721$	486.216	0.0251146
$722$	12697.7	0.654515
$723$	35573.8	1.82988
$724$	3112.49	0.159772
$725$	2641.59	0.135319
$726$	−5410.07	−0.276565
$727$	8215.91	0.419135	0.209568	−	0.977794i	$-0.432794\pi$
$727$	8215.91	0.419135	0.209568	+	0.977794i	$0.432794\pi$
$728$	0	0
$729$	68985.2	3.50481
$730$	−8941.91	−0.453363
$731$	−21999.2	−1.11309
$732$	−4384.69	−0.221397
$733$	5235.91	0.263838	0.131919	−	0.991261i	$-0.457886\pi$
$733$	5235.91	0.263838	0.131919	+	0.991261i	$0.457886\pi$
$734$	−17338.2	−0.871888
$735$	−47411.6	−2.37932
$736$	2096.93	0.105019
$737$	−8848.80	−0.442266
$738$	57978.7	2.89190
$739$	11585.1	0.576678	0.288339	−	0.957528i	$-0.406897\pi$
$739$	11585.1	0.576678	0.288339	+	0.957528i	$0.406897\pi$
$740$	−6690.18	−0.332346
$741$	0	0
$742$	322.668	0.0159643
$743$	27000.0	1.33316	0.666578	−	0.745435i	$-0.267758\pi$
$743$	27000.0	1.33316	0.666578	+	0.745435i	$0.267758\pi$
$744$	−9107.60	−0.448792
$745$	19753.6	0.971429
$746$	−15719.5	−0.771488
$747$	−39461.0	−1.93280
$748$	−12762.8	−0.623871
$749$	−40.3724	−0.00196953
$750$	16365.2	0.796763
$751$	7972.51	0.387378	0.193689	−	0.981063i	$-0.437955\pi$
$751$	7972.51	0.387378	0.193689	+	0.981063i	$0.437955\pi$
$752$	7047.73	0.341761
$753$	−11533.9	−0.558192
$754$	0	0
$755$	13005.2	0.626896
$756$	484.817	0.0233236
$757$	20621.0	0.990068	0.495034	−	0.868873i	$-0.335156\pi$
$757$	20621.0	0.990068	0.495034	+	0.868873i	$0.335156\pi$
$758$	−6161.03	−0.295222
$759$	26244.5	1.25509
$760$	−2495.99	−0.119130
$761$	−25100.5	−1.19565	−0.597826	−	0.801626i	$-0.703969\pi$
$761$	−25100.5	−1.19565	−0.597826	+	0.801626i	$0.703969\pi$
$762$	15723.1	0.747492
$763$	−468.983	−0.0222521
$764$	−7164.25	−0.339259
$765$	80593.8	3.80899
$766$	−2862.25	−0.135009
$767$	0	0
$768$	2562.19	0.120384
$769$	−34363.4	−1.61141	−0.805705	−	0.592317i	$-0.798213\pi$
$769$	−34363.4	−1.61141	−0.805705	+	0.592317i	$0.798213\pi$
$770$	289.963	0.0135708
$771$	−18505.9	−0.864430
$772$	−7507.94	−0.350022
$773$	27266.7	1.26871	0.634357	−	0.773040i	$-0.281265\pi$
$773$	27266.7	1.26871	0.634357	+	0.773040i	$0.281265\pi$
$774$	40375.9	1.87504
$775$	7486.27	0.346987
$776$	2608.64	0.120676
$777$	−317.847	−0.0146753
$778$	−23277.9	−1.07269
$779$	−8948.37	−0.411564
$780$	0	0
$781$	−12893.3	−0.590726
$782$	10450.1	0.477869
$783$	18547.6	0.846534
$784$	−5486.90	−0.249950
$785$	−2519.53	−0.114555
$786$	57072.2	2.58995
$787$	−14620.7	−0.662225	−0.331112	−	0.943591i	$-0.607424\pi$
$787$	−14620.7	−0.662225	−0.331112	+	0.943591i	$0.607424\pi$
$788$	−6336.17	−0.286443
$789$	−33537.5	−1.51327
$790$	−20534.2	−0.924779
$791$	−243.868	−0.0109620
$792$	23424.1	1.05093
$793$	0	0
$794$	2374.44	0.106128
$795$	−85041.2	−3.79384
$796$	19872.7	0.884887
$797$	8273.19	0.367693	0.183846	−	0.982955i	$-0.441145\pi$
$797$	8273.19	0.367693	0.183846	+	0.982955i	$0.441145\pi$
$798$	−118.583	−0.00526039
$799$	35122.4	1.55512
$800$	−2106.07	−0.0930761
$801$	92498.9	4.08026
$802$	−6696.55	−0.294842
$803$	−12951.7	−0.569185
$804$	8852.86	0.388329
$805$	−237.418	−0.0103949
$806$	0	0
$807$	−8348.41	−0.364161
$808$	11481.3	0.499888
$809$	−45346.3	−1.97069	−0.985346	−	0.170567i	$-0.945440\pi$
$809$	−45346.3	−1.97069	−0.985346	+	0.170567i	$0.945440\pi$
$810$	−73195.8	−3.17511
$811$	−20946.4	−0.906940	−0.453470	−	0.891272i	$-0.649814\pi$
$811$	−20946.4	−0.906940	−0.453470	+	0.891272i	$0.649814\pi$
$812$	42.1092	0.00181988
$813$	80679.7	3.48039
$814$	−9690.23	−0.417251
$815$	−12255.7	−0.526746
$816$	12768.7	0.547786
$817$	−6231.58	−0.266849
$818$	32279.2	1.37973
$819$	0	0
$820$	−21890.9	−0.932272
$821$	21533.3	0.915366	0.457683	−	0.889115i	$-0.348679\pi$
$821$	21533.3	0.915366	0.457683	+	0.889115i	$0.348679\pi$
$822$	−8048.46	−0.341511
$823$	21993.9	0.931542	0.465771	−	0.884905i	$-0.345777\pi$
$823$	21993.9	0.931542	0.465771	+	0.884905i	$0.345777\pi$
$824$	−14830.1	−0.626980
$825$	−26358.9	−1.11236
$826$	−121.128	−0.00510239
$827$	−7954.97	−0.334488	−0.167244	−	0.985916i	$-0.553487\pi$
$827$	−7954.97	−0.334488	−0.167244	+	0.985916i	$0.553487\pi$
$828$	−19179.4	−0.804988
$829$	4576.45	0.191733	0.0958665	−	0.995394i	$-0.469438\pi$
$829$	4576.45	0.191733	0.0958665	+	0.995394i	$0.469438\pi$
$830$	14899.2	0.623083
$831$	−47135.7	−1.96765
$832$	0	0
$833$	−27344.0	−1.13735
$834$	−17219.2	−0.714930
$835$	−40041.0	−1.65949
$836$	−3615.26	−0.149565
$837$	52563.8	2.17069
$838$	−12693.2	−0.523244
$839$	−8134.37	−0.334719	−0.167360	−	0.985896i	$-0.553524\pi$
$839$	−8134.37	−0.334719	−0.167360	+	0.985896i	$0.553524\pi$
$840$	−290.096	−0.0119158
$841$	−22778.0	−0.933947
$842$	−10270.4	−0.420356
$843$	−21122.1	−0.862969
$844$	14650.7	0.597509
$845$	0	0
$846$	−64461.5	−2.61966
$847$	70.8884	0.00287574
$848$	−9841.73	−0.398545
$849$	10945.3	0.442451
$850$	−10495.6	−0.423525
$851$	7934.25	0.319604
$852$	12899.2	0.518684
$853$	44562.9	1.78875	0.894375	−	0.447317i	$-0.147620\pi$
$853$	44562.9	1.78875	0.894375	+	0.447317i	$0.147620\pi$
$854$	57.4528	0.00230210
$855$	22829.3	0.913154
$856$	1231.40	0.0491688
$857$	18917.8	0.754048	0.377024	−	0.926204i	$-0.376947\pi$
$857$	18917.8	0.754048	0.377024	+	0.926204i	$0.376947\pi$
$858$	0	0
$859$	−27153.9	−1.07855	−0.539277	−	0.842128i	$-0.681303\pi$
$859$	−27153.9	−1.07855	−0.539277	+	0.842128i	$0.681303\pi$
$860$	−15244.7	−0.604463
$861$	−1040.02	−0.0411660
$862$	14089.1	0.556700
$863$	33687.7	1.32879	0.664393	−	0.747384i	$-0.268690\pi$
$863$	33687.7	1.32879	0.664393	+	0.747384i	$0.268690\pi$
$864$	−14787.5	−0.582269
$865$	38083.8	1.49698
$866$	−20347.6	−0.798431
$867$	14460.8	0.566451
$868$	119.337	0.00466656
$869$	−29742.3	−1.16104
$870$	−11098.1	−0.432486
$871$	0	0
$872$	14304.5	0.555518
$873$	−23859.7	−0.925006
$874$	2960.13	0.114563
$875$	−214.434	−0.00828479
$876$	12957.7	0.499770
$877$	42475.9	1.63547	0.817737	−	0.575592i	$-0.195228\pi$
$877$	42475.9	1.63547	0.817737	+	0.575592i	$0.195228\pi$
$878$	−12771.2	−0.490895
$879$	−96134.1	−3.68888
$880$	−8844.20	−0.338793
$881$	−19806.6	−0.757438	−0.378719	−	0.925512i	$-0.623635\pi$
$881$	−19806.6	−0.757438	−0.378719	+	0.925512i	$0.623635\pi$
$882$	50185.5	1.91591
$883$	29417.9	1.12117	0.560584	−	0.828098i	$-0.310577\pi$
$883$	29417.9	1.12117	0.560584	+	0.828098i	$0.310577\pi$
$884$	0	0
$885$	31924.0	1.21256
$886$	−4725.62	−0.179188
$887$	−32286.2	−1.22217	−0.611084	−	0.791565i	$-0.709266\pi$
$887$	−32286.2	−1.22217	−0.611084	+	0.791565i	$0.709266\pi$
$888$	9694.68	0.366365
$889$	−206.021	−0.00777246
$890$	−34924.6	−1.31537
$891$	−106019.	−3.98627
$892$	17980.9	0.674940
$893$	9948.92	0.372820
$894$	−28624.7	−1.07087
$895$	−29100.8	−1.08685
$896$	−33.5725	−0.00125176
$897$	0	0
$898$	16296.4	0.605588
$899$	4565.47	0.169374
$900$	19263.0	0.713445
$901$	−49046.3	−1.81351
$902$	−31707.3	−1.17044
$903$	−724.265	−0.0266911
$904$	7438.25	0.273664
$905$	10748.7	0.394804
$906$	−18845.7	−0.691066
$907$	25746.5	0.942556	0.471278	−	0.881985i	$-0.343793\pi$
$907$	25746.5	0.942556	0.471278	+	0.881985i	$0.343793\pi$
$908$	−14406.4	−0.526536
$909$	−105012.	−3.83173
$910$	0	0
$911$	7802.68	0.283770	0.141885	−	0.989883i	$-0.454684\pi$
$911$	7802.68	0.283770	0.141885	+	0.989883i	$0.454684\pi$
$912$	3616.91	0.131325
$913$	21580.4	0.782264
$914$	−28068.8	−1.01579
$915$	−15142.1	−0.547083
$916$	−20648.4	−0.744808
$917$	−747.820	−0.0269304
$918$	−73693.4	−2.64951
$919$	−412.898	−0.0148207	−0.00741037	−	0.999973i	$-0.502359\pi$
$919$	−412.898	−0.0148207	−0.00741037	+	0.999973i	$0.502359\pi$
$920$	7241.53	0.259507
$921$	43915.8	1.57120
$922$	−15383.4	−0.549485
$923$	0	0
$924$	−420.183	−0.0149600
$925$	−7968.83	−0.283258
$926$	31111.9	1.10410
$927$	135643.	4.80592
$928$	−1284.38	−0.0454329
$929$	−5257.86	−0.185689	−0.0928443	−	0.995681i	$-0.529596\pi$
$929$	−5257.86	−0.185689	−0.0928443	+	0.995681i	$0.529596\pi$
$930$	−31452.1	−1.10898
$931$	−7745.57	−0.272665
$932$	−8091.68	−0.284390
$933$	28780.3	1.00989
$934$	23883.5	0.836716
$935$	−44075.1	−1.54161
$936$	0	0
$937$	−21763.7	−0.758793	−0.379396	−	0.925234i	$-0.623868\pi$
$937$	−21763.7	−0.758793	−0.379396	+	0.925234i	$0.623868\pi$
$938$	−115.999	−0.00403786
$939$	57205.0	1.98809
$940$	24338.6	0.844508
$941$	4630.07	0.160400	0.0801998	−	0.996779i	$-0.474444\pi$
$941$	4630.07	0.160400	0.0801998	+	0.996779i	$0.474444\pi$
$942$	3651.03	0.126281
$943$	25961.6	0.896528
$944$	3694.53	0.127380
$945$	1674.27	0.0576337
$946$	−22080.8	−0.758888
$947$	7631.66	0.261875	0.130937	−	0.991391i	$-0.458201\pi$
$947$	7631.66	0.261875	0.130937	+	0.991391i	$0.458201\pi$
$948$	29756.0	1.01944
$949$	0	0
$950$	−2973.03	−0.101535
$951$	73881.9	2.51923
$952$	−167.309	−0.00569592
$953$	26455.3	0.899234	0.449617	−	0.893221i	$-0.351560\pi$
$953$	26455.3	0.899234	0.449617	+	0.893221i	$0.351560\pi$
$954$	90016.6	3.05492
$955$	−24741.0	−0.838324
$956$	−5692.47	−0.192581
$957$	−16074.9	−0.542974
$958$	−29712.9	−1.00207
$959$	105.459	0.00355105
$960$	8848.26	0.297475
$961$	−16852.5	−0.565690
$962$	0	0
$963$	−11262.9	−0.376887
$964$	14217.4	0.475011
$965$	−25927.9	−0.864920
$966$	344.041	0.0114589
$967$	−18687.5	−0.621457	−0.310728	−	0.950499i	$-0.600573\pi$
$967$	−18687.5	−0.621457	−0.310728	+	0.950499i	$0.600573\pi$
$968$	−2162.18	−0.0717923
$969$	18024.9	0.597568
$970$	9008.67	0.298197
$971$	−6281.87	−0.207616	−0.103808	−	0.994597i	$-0.533103\pi$
$971$	−6281.87	−0.207616	−0.103808	+	0.994597i	$0.533103\pi$
$972$	56159.7	1.85321
$973$	225.624	0.00743389
$974$	28127.2	0.925313
$975$	0	0
$976$	−1752.38	−0.0574715
$977$	42266.0	1.38404	0.692021	−	0.721878i	$-0.256721\pi$
$977$	42266.0	1.38404	0.692021	+	0.721878i	$0.256721\pi$
$978$	17759.6	0.580665
$979$	−50585.7	−1.65141
$980$	−18948.4	−0.617638
$981$	−130835.	−4.25815
$982$	16436.1	0.534112
$983$	41111.7	1.33394	0.666968	−	0.745086i	$-0.267592\pi$
$983$	41111.7	1.33394	0.666968	+	0.745086i	$0.267592\pi$
$984$	31721.9	1.02770
$985$	−21881.3	−0.707813
$986$	−6400.70	−0.206734
$987$	1156.31	0.0372906
$988$	0	0
$989$	18079.5	0.581288
$990$	80892.7	2.59691
$991$	−40047.0	−1.28369	−0.641845	−	0.766835i	$-0.721831\pi$
$991$	−40047.0	−1.28369	−0.641845	+	0.766835i	$0.721831\pi$
$992$	−3639.92	−0.116500
$993$	−60597.7	−1.93657
$994$	−169.019	−0.00539330
$995$	68628.3	2.18660
$996$	−21590.3	−0.686863
$997$	23968.8	0.761383	0.380692	−	0.924702i	$-0.375686\pi$
$997$	23968.8	0.761383	0.380692	+	0.924702i	$0.375686\pi$
$998$	−12163.8	−0.385811
$999$	−55952.0	−1.77202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.n.1.6	✓	6
13.2	odd	12		338.4.e.i.147.1		24
13.3	even	3		338.4.c.p.191.1		12
13.4	even	6		338.4.c.o.315.1		12
13.5	odd	4		338.4.b.h.337.12		12
13.6	odd	12		338.4.e.i.23.10		24
13.7	odd	12		338.4.e.i.23.1		24
13.8	odd	4		338.4.b.h.337.6		12
13.9	even	3		338.4.c.p.315.1		12
13.10	even	6		338.4.c.o.191.1		12
13.11	odd	12		338.4.e.i.147.10		24
13.12	even	2		338.4.a.o.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.n.1.6	✓	6	1.1	even	1	trivial
338.4.a.o.1.6	yes	6	13.12	even	2
338.4.b.h.337.6		12	13.8	odd	4
338.4.b.h.337.12		12	13.5	odd	4
338.4.c.o.191.1		12	13.10	even	6
338.4.c.o.315.1		12	13.4	even	6
338.4.c.p.191.1		12	13.3	even	3
338.4.c.p.315.1		12	13.9	even	3
338.4.e.i.23.1		24	13.7	odd	12
338.4.e.i.23.10		24	13.6	odd	12
338.4.e.i.147.1		24	13.2	odd	12
338.4.e.i.147.10		24	13.11	odd	12

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +10.0086 q^{3} +4.00000 q^{4} +13.8136 q^{5} -20.0171 q^{6} +0.262285 q^{7} -8.00000 q^{8} +73.1713 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +10.0086 q^{3} +4.00000 q^{4} +13.8136 q^{5} -20.0171 q^{6} +0.262285 q^{7} -8.00000 q^{8} +73.1713 q^{9} -27.6271 q^{10} -40.0159 q^{11} +40.0343 q^{12} -0.524571 q^{14} +138.254 q^{15} +16.0000 q^{16} +79.7361 q^{17} -146.343 q^{18} +22.5864 q^{19} +55.2543 q^{20} +2.62510 q^{21} +80.0318 q^{22} -65.5291 q^{23} -80.0685 q^{24} +65.8147 q^{25} +462.109 q^{27} +1.04914 q^{28} +40.1368 q^{29} -276.508 q^{30} +113.748 q^{31} -32.0000 q^{32} -400.502 q^{33} -159.472 q^{34} +3.62310 q^{35} +292.685 q^{36} -121.080 q^{37} -45.1727 q^{38} -110.509 q^{40} -396.184 q^{41} -5.25020 q^{42} -275.900 q^{43} -160.064 q^{44} +1010.76 q^{45} +131.058 q^{46} +440.483 q^{47} +160.137 q^{48} -342.931 q^{49} -131.629 q^{50} +798.043 q^{51} -615.108 q^{53} -924.217 q^{54} -552.762 q^{55} -2.09828 q^{56} +226.057 q^{57} -80.2736 q^{58} +230.908 q^{59} +553.016 q^{60} -109.524 q^{61} -227.495 q^{62} +19.1918 q^{63} +64.0000 q^{64} +801.003 q^{66} +221.132 q^{67} +318.944 q^{68} -655.852 q^{69} -7.24620 q^{70} +322.203 q^{71} -585.371 q^{72} +323.664 q^{73} +242.160 q^{74} +658.711 q^{75} +90.3455 q^{76} -10.4956 q^{77} +743.263 q^{79} +221.017 q^{80} +2649.42 q^{81} +792.369 q^{82} -539.296 q^{83} +10.5004 q^{84} +1101.44 q^{85} +551.800 q^{86} +401.712 q^{87} +320.127 q^{88} +1264.14 q^{89} -2021.51 q^{90} -262.116 q^{92} +1138.45 q^{93} -880.967 q^{94} +311.998 q^{95} -320.274 q^{96} -326.080 q^{97} +685.862 q^{98} -2928.02 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more