LMFDB - Embedded newform 415.4.a.c.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [415,4,Mod(1,415)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("415.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 415 = 5 \cdot 83 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	415.a (trivial)

Newform invariants

comment:select newform

sage:traces = [21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.4857926524$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	415.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.70189 q^{2} -4.13473 q^{3} -0.699769 q^{4} -5.00000 q^{5} +11.1716 q^{6} +4.29122 q^{7} +23.5059 q^{8} -9.90398 q^{9} +13.5095 q^{10} -34.9071 q^{11} +2.89336 q^{12} +7.72746 q^{13} -11.5944 q^{14} +20.6737 q^{15} -57.9122 q^{16} +41.4349 q^{17} +26.7595 q^{18} +98.8477 q^{19} +3.49885 q^{20} -17.7431 q^{21} +94.3153 q^{22} +35.8174 q^{23} -97.1904 q^{24} +25.0000 q^{25} -20.8788 q^{26} +152.588 q^{27} -3.00287 q^{28} +101.422 q^{29} -55.8581 q^{30} +195.195 q^{31} -31.5743 q^{32} +144.332 q^{33} -111.953 q^{34} -21.4561 q^{35} +6.93050 q^{36} -70.6428 q^{37} -267.076 q^{38} -31.9510 q^{39} -117.529 q^{40} +268.214 q^{41} +47.9399 q^{42} -277.083 q^{43} +24.4269 q^{44} +49.5199 q^{45} -96.7749 q^{46} -543.139 q^{47} +239.451 q^{48} -324.585 q^{49} -67.5473 q^{50} -171.322 q^{51} -5.40744 q^{52} -369.045 q^{53} -412.277 q^{54} +174.536 q^{55} +100.869 q^{56} -408.709 q^{57} -274.032 q^{58} -410.547 q^{59} -14.4668 q^{60} +751.190 q^{61} -527.397 q^{62} -42.5002 q^{63} +548.608 q^{64} -38.6373 q^{65} -389.969 q^{66} +78.1829 q^{67} -28.9949 q^{68} -148.096 q^{69} +57.9721 q^{70} +208.576 q^{71} -232.801 q^{72} -105.398 q^{73} +190.869 q^{74} -103.368 q^{75} -69.1706 q^{76} -149.794 q^{77} +86.3282 q^{78} -968.969 q^{79} +289.561 q^{80} -363.504 q^{81} -724.685 q^{82} +83.0000 q^{83} +12.4161 q^{84} -207.175 q^{85} +748.649 q^{86} -419.354 q^{87} -820.521 q^{88} +222.017 q^{89} -133.797 q^{90} +33.1603 q^{91} -25.0639 q^{92} -807.080 q^{93} +1467.50 q^{94} -494.238 q^{95} +130.551 q^{96} -541.705 q^{97} +876.995 q^{98} +345.719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 5 q^{2} - 12 q^{3} + 87 q^{4} - 105 q^{5} - 7 q^{6} - 11 q^{7} - 84 q^{8} + 153 q^{9} + 25 q^{10} - 30 q^{11} - 244 q^{12} - 89 q^{13} - 191 q^{14} + 60 q^{15} + 583 q^{16} - 357 q^{17} - 281 q^{18}+ \cdots - 5369 q^{99}+O(q^{100}) $ 21 * q - 5 * q^2 - 12 * q^3 + 87 * q^4 - 105 * q^5 - 7 * q^6 - 11 * q^7 - 84 * q^8 + 153 * q^9 + 25 * q^10 - 30 * q^11 - 244 * q^12 - 89 * q^13 - 191 * q^14 + 60 * q^15 + 583 * q^16 - 357 * q^17 - 281 * q^18 - 175 * q^19 - 435 * q^20 - 41 * q^21 - 122 * q^22 - 554 * q^23 - 136 * q^24 + 525 * q^25 - 933 * q^26 - 555 * q^27 - 773 * q^28 - 96 * q^29 + 35 * q^30 + 99 * q^31 - 1307 * q^32 - 1097 * q^33 + 393 * q^34 + 55 * q^35 - 746 * q^36 - 756 * q^37 - 1192 * q^38 + 40 * q^39 + 420 * q^40 - 1436 * q^41 - 1112 * q^42 - 1003 * q^43 - 706 * q^44 - 765 * q^45 - 775 * q^46 - 858 * q^47 - 2837 * q^48 + 1384 * q^49 - 125 * q^50 - 279 * q^51 - 1092 * q^52 - 1625 * q^53 - 269 * q^54 + 150 * q^55 - 3806 * q^56 - 1740 * q^57 - 1087 * q^58 - 1224 * q^59 + 1220 * q^60 - 794 * q^61 - 2979 * q^62 - 1236 * q^63 + 2570 * q^64 + 445 * q^65 - 1094 * q^66 - 571 * q^67 - 4064 * q^68 + 774 * q^69 + 955 * q^70 - 1078 * q^71 - 2610 * q^72 - 1986 * q^73 - 1872 * q^74 - 300 * q^75 - 2854 * q^76 - 2519 * q^77 - 2449 * q^78 - 432 * q^79 - 2915 * q^80 - 219 * q^81 - 2143 * q^82 + 1743 * q^83 - 377 * q^84 + 1785 * q^85 + 1556 * q^86 - 5681 * q^87 - 3158 * q^88 - 6252 * q^89 + 1405 * q^90 + 684 * q^91 - 7623 * q^92 - 3314 * q^93 - 3877 * q^94 + 875 * q^95 - 954 * q^96 - 4880 * q^97 - 5072 * q^98 - 5369 * q^99

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 $

$2$	−2.70189	−0.955264	−0.477632	−	0.878560i	$-0.658505\pi$
$2$	−2.70189	−0.955264	−0.477632	+	0.878560i	$0.658505\pi$
$3$	−4.13473	−0.795730	−0.397865	−	0.917444i	$-0.630249\pi$
$3$	−4.13473	−0.795730	−0.397865	+	0.917444i	$0.630249\pi$
$4$	−0.699769	−0.0874712
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	11.1716	0.760132
$7$	4.29122	0.231704	0.115852	−	0.993266i	$-0.463040\pi$
$7$	4.29122	0.231704	0.115852	+	0.993266i	$0.463040\pi$
$8$	23.5059	1.03882
$9$	−9.90398	−0.366814
$10$	13.5095	0.427207
$11$	−34.9071	−0.956808	−0.478404	−	0.878140i	$-0.658785\pi$
$11$	−34.9071	−0.956808	−0.478404	+	0.878140i	$0.658785\pi$
$12$	2.89336	0.0696034
$13$	7.72746	0.164862	0.0824312	−	0.996597i	$-0.473732\pi$
$13$	7.72746	0.164862	0.0824312	+	0.996597i	$0.473732\pi$
$14$	−11.5944	−0.221339
$15$	20.6737	0.355861
$16$	−57.9122	−0.904878
$17$	41.4349	0.591144	0.295572	−	0.955320i	$-0.404490\pi$
$17$	41.4349	0.591144	0.295572	+	0.955320i	$0.404490\pi$
$18$	26.7595	0.350404
$19$	98.8477	1.19354	0.596769	−	0.802413i	$-0.296451\pi$
$19$	98.8477	1.19354	0.596769	+	0.802413i	$0.296451\pi$
$20$	3.49885	0.0391183
$21$	−17.7431	−0.184374
$22$	94.3153	0.914004
$23$	35.8174	0.324715	0.162358	−	0.986732i	$-0.448090\pi$
$23$	35.8174	0.324715	0.162358	+	0.986732i	$0.448090\pi$
$24$	−97.1904	−0.826622
$25$	25.0000	0.200000
$26$	−20.8788	−0.157487
$27$	152.588	1.08761
$28$	−3.00287	−0.0202674
$29$	101.422	0.649436	0.324718	−	0.945811i	$-0.394731\pi$
$29$	101.422	0.649436	0.324718	+	0.945811i	$0.394731\pi$
$30$	−55.8581	−0.339941
$31$	195.195	1.13091	0.565453	−	0.824780i	$-0.308701\pi$
$31$	195.195	1.13091	0.565453	+	0.824780i	$0.308701\pi$
$32$	−31.5743	−0.174425
$33$	144.332	0.761361
$34$	−111.953	−0.564699
$35$	−21.4561	−0.103621
$36$	6.93050	0.0320856
$37$	−70.6428	−0.313881	−0.156941	−	0.987608i	$-0.550163\pi$
$37$	−70.6428	−0.313881	−0.156941	+	0.987608i	$0.550163\pi$
$38$	−267.076	−1.14014
$39$	−31.9510	−0.131186
$40$	−117.529	−0.464575
$41$	268.214	1.02166	0.510829	−	0.859682i	$-0.329339\pi$
$41$	268.214	1.02166	0.510829	+	0.859682i	$0.329339\pi$
$42$	47.9399	0.176126
$43$	−277.083	−0.982669	−0.491335	−	0.870971i	$-0.663491\pi$
$43$	−277.083	−0.982669	−0.491335	+	0.870971i	$0.663491\pi$
$44$	24.4269	0.0836931
$45$	49.5199	0.164044
$46$	−96.7749	−0.310189
$47$	−543.139	−1.68564	−0.842819	−	0.538197i	$-0.819106\pi$
$47$	−543.139	−1.68564	−0.842819	+	0.538197i	$0.819106\pi$
$48$	239.451	0.720038
$49$	−324.585	−0.946313
$50$	−67.5473	−0.191053
$51$	−171.322	−0.470391
$52$	−5.40744	−0.0144207
$53$	−369.045	−0.956457	−0.478229	−	0.878235i	$-0.658721\pi$
$53$	−369.045	−0.956457	−0.478229	+	0.878235i	$0.658721\pi$
$54$	−412.277	−1.03896
$55$	174.536	0.427898
$56$	100.869	0.240699
$57$	−408.709	−0.949733
$58$	−274.032	−0.620383
$59$	−410.547	−0.905909	−0.452955	−	0.891534i	$-0.649630\pi$
$59$	−410.547	−0.905909	−0.452955	+	0.891534i	$0.649630\pi$
$60$	−14.4668	−0.0311276
$61$	751.190	1.57672	0.788361	−	0.615213i	$-0.210930\pi$
$61$	751.190	1.57672	0.788361	+	0.615213i	$0.210930\pi$
$62$	−527.397	−1.08031
$63$	−42.5002	−0.0849924
$64$	548.608	1.07150
$65$	−38.6373	−0.0737287
$66$	−389.969	−0.727300
$67$	78.1829	0.142561	0.0712803	−	0.997456i	$-0.477292\pi$
$67$	78.1829	0.142561	0.0712803	+	0.997456i	$0.477292\pi$
$68$	−28.9949	−0.0517081
$69$	−148.096	−0.258386
$70$	57.9721	0.0989857
$71$	208.576	0.348640	0.174320	−	0.984689i	$-0.444227\pi$
$71$	208.576	0.348640	0.174320	+	0.984689i	$0.444227\pi$
$72$	−232.801	−0.381054
$73$	−105.398	−0.168984	−0.0844921	−	0.996424i	$-0.526927\pi$
$73$	−105.398	−0.168984	−0.0844921	+	0.996424i	$0.526927\pi$
$74$	190.869	0.299839
$75$	−103.368	−0.159146
$76$	−69.1706	−0.104400
$77$	−149.794	−0.221697
$78$	86.3282	0.125317
$79$	−968.969	−1.37997	−0.689985	−	0.723824i	$-0.742383\pi$
$79$	−968.969	−1.37997	−0.689985	+	0.723824i	$0.742383\pi$
$80$	289.561	0.404674
$81$	−363.504	−0.498634
$82$	−724.685	−0.975952
$83$	83.0000	0.109764
$83$	83.0000	0.109764
$84$	12.4161	0.0161274
$85$	−207.175	−0.264368
$86$	748.649	0.938708
$87$	−419.354	−0.516776
$88$	−820.521	−0.993953
$89$	222.017	0.264425	0.132212	−	0.991221i	$-0.457792\pi$
$89$	222.017	0.264425	0.132212	+	0.991221i	$0.457792\pi$
$90$	−133.797	−0.156705
$91$	33.1603	0.0381993
$92$	−25.0639	−0.0284032
$93$	−807.080	−0.899896
$94$	1467.50	1.61023
$95$	−494.238	−0.533766
$96$	130.551	0.138795
$97$	−541.705	−0.567029	−0.283515	−	0.958968i	$-0.591500\pi$
$97$	−541.705	−0.567029	−0.283515	+	0.958968i	$0.591500\pi$
$98$	876.995	0.903979
$99$	345.719	0.350971
$100$	−17.4942	−0.0174942
$101$	−613.770	−0.604677	−0.302339	−	0.953201i	$-0.597767\pi$
$101$	−613.770	−0.604677	−0.302339	+	0.953201i	$0.597767\pi$
$102$	462.895	0.449347
$103$	−116.700	−0.111638	−0.0558192	−	0.998441i	$-0.517777\pi$
$103$	−116.700	−0.111638	−0.0558192	+	0.998441i	$0.517777\pi$
$104$	181.641	0.171263
$105$	88.7153	0.0824546
$106$	997.121	0.913669
$107$	−412.532	−0.372720	−0.186360	−	0.982482i	$-0.559669\pi$
$107$	−412.532	−0.372720	−0.186360	+	0.982482i	$0.559669\pi$
$108$	−106.776	−0.0951349
$109$	−824.001	−0.724083	−0.362041	−	0.932162i	$-0.617920\pi$
$109$	−824.001	−0.724083	−0.362041	+	0.932162i	$0.617920\pi$
$110$	−471.576	−0.408755
$111$	292.089	0.249765
$112$	−248.514	−0.209664
$113$	−1702.27	−1.41714	−0.708568	−	0.705642i	$-0.750659\pi$
$113$	−1702.27	−1.41714	−0.708568	+	0.705642i	$0.750659\pi$
$114$	1104.29	0.907246
$115$	−179.087	−0.145217
$116$	−70.9722	−0.0568070
$117$	−76.5326	−0.0604739
$118$	1109.25	0.865382
$119$	177.807	0.136971
$120$	485.952	0.369676
$121$	−112.494	−0.0845184
$122$	−2029.64	−1.50619
$123$	−1108.99	−0.812963
$124$	−136.592	−0.0989217
$125$	−125.000	−0.0894427
$126$	114.831	0.0811901
$127$	699.691	0.488878	0.244439	−	0.969665i	$-0.421396\pi$
$127$	699.691	0.488878	0.244439	+	0.969665i	$0.421396\pi$
$128$	−1229.69	−0.849140
$129$	1145.66	0.781939
$130$	104.394	0.0704304
$131$	646.088	0.430908	0.215454	−	0.976514i	$-0.430877\pi$
$131$	646.088	0.430908	0.215454	+	0.976514i	$0.430877\pi$
$132$	−100.999	−0.0665971
$133$	424.177	0.276548
$134$	−211.242	−0.136183
$135$	−762.941	−0.486396
$136$	973.964	0.614093
$137$	−337.668	−0.210576	−0.105288	−	0.994442i	$-0.533576\pi$
$137$	−337.668	−0.210576	−0.105288	+	0.994442i	$0.533576\pi$
$138$	400.139	0.246826
$139$	1893.93	1.15569	0.577845	−	0.816146i	$-0.303894\pi$
$139$	1893.93	1.15569	0.577845	+	0.816146i	$0.303894\pi$
$140$	15.0143	0.00906388
$141$	2245.74	1.34131
$142$	−563.551	−0.333043
$143$	−269.743	−0.157742
$144$	573.561	0.331922
$145$	−507.112	−0.290437
$146$	284.773	0.161425
$147$	1342.07	0.753010
$148$	49.4337	0.0274555
$149$	−2773.00	−1.52465	−0.762324	−	0.647195i	$-0.775942\pi$
$149$	−2773.00	−1.52465	−0.762324	+	0.647195i	$0.775942\pi$
$150$	279.290	0.152026
$151$	2072.88	1.11714	0.558571	−	0.829457i	$-0.311350\pi$
$151$	2072.88	1.11714	0.558571	+	0.829457i	$0.311350\pi$
$152$	2323.50	1.23987
$153$	−410.371	−0.216840
$154$	404.728	0.211779
$155$	−975.976	−0.505757
$156$	22.3583	0.0114750
$157$	201.042	0.102197	0.0510984	−	0.998694i	$-0.483728\pi$
$157$	201.042	0.102197	0.0510984	+	0.998694i	$0.483728\pi$
$158$	2618.05	1.31823
$159$	1525.90	0.761082
$160$	157.871	0.0780052
$161$	153.701	0.0752379
$162$	982.149	0.476327
$163$	1061.03	0.509852	0.254926	−	0.966961i	$-0.417949\pi$
$163$	1061.03	0.509852	0.254926	+	0.966961i	$0.417949\pi$
$164$	−187.688	−0.0893656
$165$	−721.658	−0.340491
$166$	−224.257	−0.104854
$167$	780.416	0.361619	0.180810	−	0.983518i	$-0.442128\pi$
$167$	780.416	0.361619	0.180810	+	0.983518i	$0.442128\pi$
$168$	−417.066	−0.191532
$169$	−2137.29	−0.972820
$170$	559.764	0.252541
$171$	−978.985	−0.437806
$172$	193.894	0.0859552
$173$	2190.71	0.962755	0.481377	−	0.876513i	$-0.340137\pi$
$173$	2190.71	0.962755	0.481377	+	0.876513i	$0.340137\pi$
$174$	1133.05	0.493657
$175$	107.281	0.0463409
$176$	2021.55	0.865794
$177$	1697.50	0.720859
$178$	−599.867	−0.252595
$179$	699.942	0.292269	0.146135	−	0.989265i	$-0.453317\pi$
$179$	699.942	0.292269	0.146135	+	0.989265i	$0.453317\pi$
$180$	−34.6525	−0.0143491
$181$	2777.32	1.14053	0.570267	−	0.821459i	$-0.306840\pi$
$181$	2777.32	1.14053	0.570267	+	0.821459i	$0.306840\pi$
$182$	−89.5955	−0.0364904
$183$	−3105.97	−1.25465
$184$	841.920	0.337321
$185$	353.214	0.140372
$186$	2180.65	0.859638
$187$	−1446.37	−0.565611
$188$	380.072	0.147445
$189$	654.790	0.252005
$190$	1335.38	0.509887
$191$	−3071.79	−1.16370	−0.581851	−	0.813296i	$-0.697671\pi$
$191$	−3071.79	−1.16370	−0.581851	+	0.813296i	$0.697671\pi$
$192$	−2268.35	−0.852624
$193$	−698.123	−0.260373	−0.130187	−	0.991490i	$-0.541558\pi$
$193$	−698.123	−0.260373	−0.130187	+	0.991490i	$0.541558\pi$
$194$	1463.63	0.541663
$195$	159.755	0.0586682
$196$	227.135	0.0827751
$197$	180.409	0.0652469	0.0326234	−	0.999468i	$-0.489614\pi$
$197$	180.409	0.0652469	0.0326234	+	0.999468i	$0.489614\pi$
$198$	−934.097	−0.335269
$199$	65.4047	0.0232986	0.0116493	−	0.999932i	$-0.496292\pi$
$199$	65.4047	0.0232986	0.0116493	+	0.999932i	$0.496292\pi$
$200$	587.646	0.207764
$201$	−323.266	−0.113440
$202$	1658.34	0.577626
$203$	435.226	0.150477
$204$	119.886	0.0411457
$205$	−1341.07	−0.456899
$206$	315.310	0.106644
$207$	−354.735	−0.119110
$208$	−447.514	−0.149180
$209$	−3450.49	−1.14199
$210$	−239.699	−0.0787659
$211$	4409.01	1.43852	0.719262	−	0.694739i	$-0.244480\pi$
$211$	4409.01	1.43852	0.719262	+	0.694739i	$0.244480\pi$
$212$	258.246	0.0836625
$213$	−862.407	−0.277423
$214$	1114.62	0.356045
$215$	1385.42	0.439463
$216$	3586.71	1.12984
$217$	837.626	0.262036
$218$	2226.36	0.691690
$219$	435.791	0.134466
$220$	−122.135	−0.0374287
$221$	320.187	0.0974575
$222$	−789.194	−0.238591
$223$	931.924	0.279849	0.139924	−	0.990162i	$-0.455314\pi$
$223$	931.924	0.279849	0.139924	+	0.990162i	$0.455314\pi$
$224$	−135.492	−0.0404150
$225$	−247.599	−0.0733628
$226$	4599.36	1.35374
$227$	−5299.76	−1.54959	−0.774796	−	0.632211i	$-0.782147\pi$
$227$	−5299.76	−1.54959	−0.774796	+	0.632211i	$0.782147\pi$
$228$	286.002	0.0830743
$229$	−1638.18	−0.472724	−0.236362	−	0.971665i	$-0.575955\pi$
$229$	−1638.18	−0.472724	−0.236362	+	0.971665i	$0.575955\pi$
$230$	483.875	0.138721
$231$	619.359	0.176411
$232$	2384.02	0.674649
$233$	−4858.00	−1.36592	−0.682958	−	0.730458i	$-0.739307\pi$
$233$	−4858.00	−1.36592	−0.682958	+	0.730458i	$0.739307\pi$
$234$	206.783	0.0577685
$235$	2715.70	0.753840
$236$	287.288	0.0792409
$237$	4006.43	1.09808
$238$	−480.414	−0.130843
$239$	−7178.62	−1.94287	−0.971435	−	0.237305i	$-0.923736\pi$
$239$	−7178.62	−1.94287	−0.971435	+	0.237305i	$0.923736\pi$
$240$	−1197.26	−0.322011
$241$	386.962	0.103429	0.0517145	−	0.998662i	$-0.483531\pi$
$241$	386.962	0.103429	0.0517145	+	0.998662i	$0.483531\pi$
$242$	303.947	0.0807373
$243$	−2616.89	−0.690837
$244$	−525.660	−0.137918
$245$	1622.93	0.423204
$246$	2996.38	0.776594
$247$	763.841	0.196770
$248$	4588.23	1.17481
$249$	−343.183	−0.0873427
$250$	337.737	0.0854414
$251$	−6661.72	−1.67524	−0.837618	−	0.546257i	$-0.816052\pi$
$251$	−6661.72	−1.67524	−0.837618	+	0.546257i	$0.816052\pi$
$252$	29.7403	0.00743438
$253$	−1250.28	−0.310690
$254$	−1890.49	−0.467008
$255$	856.612	0.210365
$256$	−1066.38	−0.260347
$257$	5071.91	1.23104	0.615519	−	0.788122i	$-0.288946\pi$
$257$	5071.91	1.23104	0.615519	+	0.788122i	$0.288946\pi$
$258$	−3095.46	−0.746958
$259$	−303.144	−0.0727276
$260$	27.0372	0.00644914
$261$	−1004.48	−0.238222
$262$	−1745.66	−0.411631
$263$	−4324.87	−1.01400	−0.507001	−	0.861945i	$-0.669246\pi$
$263$	−4324.87	−1.01400	−0.507001	+	0.861945i	$0.669246\pi$
$264$	3392.64	0.790918
$265$	1845.23	0.427741
$266$	−1146.08	−0.264176
$267$	−917.982	−0.210411
$268$	−54.7100	−0.0124699
$269$	461.214	0.104538	0.0522689	−	0.998633i	$-0.483355\pi$
$269$	461.214	0.104538	0.0522689	+	0.998633i	$0.483355\pi$
$270$	2061.38	0.464637
$271$	6164.37	1.38177	0.690884	−	0.722966i	$-0.257222\pi$
$271$	6164.37	1.38177	0.690884	+	0.722966i	$0.257222\pi$
$272$	−2399.59	−0.534913
$273$	−137.109	−0.0303964
$274$	912.344	0.201156
$275$	−872.678	−0.191362
$276$	103.633	0.0226013
$277$	−4943.06	−1.07220	−0.536100	−	0.844154i	$-0.680103\pi$
$277$	−4943.06	−1.07220	−0.536100	+	0.844154i	$0.680103\pi$
$278$	−5117.20	−1.10399
$279$	−1933.21	−0.414832
$280$	−504.344	−0.107644
$281$	4380.40	0.929939	0.464970	−	0.885327i	$-0.346065\pi$
$281$	4380.40	0.929939	0.464970	+	0.885327i	$0.346065\pi$
$282$	−6067.74	−1.28131
$283$	−4396.33	−0.923444	−0.461722	−	0.887025i	$-0.652768\pi$
$283$	−4396.33	−0.923444	−0.461722	+	0.887025i	$0.652768\pi$
$284$	−145.955	−0.0304960
$285$	2043.54	0.424734
$286$	728.818	0.150685
$287$	1150.97	0.236722
$288$	312.711	0.0639815
$289$	−3196.15	−0.650549
$290$	1370.16	0.277444
$291$	2239.81	0.451202
$292$	73.7540	0.0147813
$293$	826.604	0.164815	0.0824074	−	0.996599i	$-0.473739\pi$
$293$	826.604	0.164815	0.0824074	+	0.996599i	$0.473739\pi$
$294$	−3626.14	−0.719323
$295$	2052.73	0.405135
$296$	−1660.52	−0.326067
$297$	−5326.41	−1.04064
$298$	7492.34	1.45644
$299$	276.778	0.0535334
$300$	72.3340	0.0139207
$301$	−1189.03	−0.227689
$302$	−5600.69	−1.06716
$303$	2537.78	0.481160
$304$	−5724.48	−1.08001
$305$	−3755.95	−0.705132
$306$	1108.78	0.207139
$307$	5818.77	1.08174	0.540871	−	0.841106i	$-0.318095\pi$
$307$	5818.77	1.08174	0.540871	+	0.841106i	$0.318095\pi$
$308$	104.821	0.0193921
$309$	482.522	0.0888340
$310$	2636.98	0.483131
$311$	6201.04	1.13064	0.565319	−	0.824872i	$-0.308753\pi$
$311$	6201.04	1.13064	0.565319	+	0.824872i	$0.308753\pi$
$312$	−751.035	−0.136279
$313$	−1646.68	−0.297368	−0.148684	−	0.988885i	$-0.547504\pi$
$313$	−1646.68	−0.297368	−0.148684	+	0.988885i	$0.547504\pi$
$314$	−543.195	−0.0976250
$315$	212.501	0.0380097
$316$	678.055	0.120708
$317$	−10701.5	−1.89608	−0.948042	−	0.318146i	$-0.896940\pi$
$317$	−10701.5	−1.89608	−0.948042	+	0.318146i	$0.896940\pi$
$318$	−4122.83	−0.727034
$319$	−3540.36	−0.621386
$320$	−2743.04	−0.479189
$321$	1705.71	0.296584
$322$	−415.283	−0.0718721
$323$	4095.75	0.705552
$324$	254.369	0.0436161
$325$	193.187	0.0329725
$326$	−2866.78	−0.487043
$327$	3407.03	0.576174
$328$	6304.59	1.06132
$329$	−2330.73	−0.390569
$330$	1949.84	0.325259
$331$	5391.95	0.895374	0.447687	−	0.894190i	$-0.352248\pi$
$331$	5391.95	0.895374	0.447687	+	0.894190i	$0.352248\pi$
$332$	−58.0809	−0.00960121
$333$	699.644	0.115136
$334$	−2108.60	−0.345442
$335$	−390.915	−0.0637551
$336$	1027.54	0.166836
$337$	−945.945	−0.152905	−0.0764524	−	0.997073i	$-0.524359\pi$
$337$	−945.945	−0.152905	−0.0764524	+	0.997073i	$0.524359\pi$
$338$	5774.72	0.929300
$339$	7038.45	1.12766
$340$	144.975	0.0231246
$341$	−6813.70	−1.08206
$342$	2645.11	0.418220
$343$	−2864.76	−0.450969
$344$	−6513.07	−1.02082
$345$	740.478	0.115554
$346$	−5919.07	−0.919685
$347$	−7733.16	−1.19636	−0.598182	−	0.801361i	$-0.704110\pi$
$347$	−7733.16	−1.19636	−0.598182	+	0.801361i	$0.704110\pi$
$348$	293.451	0.0452030
$349$	1012.14	0.155240	0.0776198	−	0.996983i	$-0.475268\pi$
$349$	1012.14	0.155240	0.0776198	+	0.996983i	$0.475268\pi$
$350$	−289.861	−0.0442677
$351$	1179.12	0.179307
$352$	1102.17	0.166891
$353$	−1253.87	−0.189056	−0.0945278	−	0.995522i	$-0.530134\pi$
$353$	−1253.87	−0.189056	−0.0945278	+	0.995522i	$0.530134\pi$
$354$	−4586.47	−0.688610
$355$	−1042.88	−0.155917
$356$	−155.361	−0.0231295
$357$	−735.183	−0.108992
$358$	−1891.17	−0.279194
$359$	−8185.81	−1.20343	−0.601714	−	0.798712i	$-0.705515\pi$
$359$	−8185.81	−1.20343	−0.601714	+	0.798712i	$0.705515\pi$
$360$	1164.01	0.170413
$361$	2911.86	0.424531
$362$	−7504.03	−1.08951
$363$	465.132	0.0672538
$364$	−23.2045	−0.00334134
$365$	526.988	0.0755721
$366$	8392.01	1.19852
$367$	−4895.39	−0.696287	−0.348143	−	0.937441i	$-0.613188\pi$
$367$	−4895.39	−0.696287	−0.348143	+	0.937441i	$0.613188\pi$
$368$	−2074.27	−0.293828
$369$	−2656.38	−0.374758
$370$	−954.346	−0.134092
$371$	−1583.65	−0.221615
$372$	564.770	0.0787150
$373$	−2564.38	−0.355975	−0.177987	−	0.984033i	$-0.556959\pi$
$373$	−2564.38	−0.355975	−0.177987	+	0.984033i	$0.556959\pi$
$374$	3907.95	0.540308
$375$	516.842	0.0711722
$376$	−12766.9	−1.75108
$377$	783.737	0.107068
$378$	−1769.17	−0.240731
$379$	5914.78	0.801641	0.400820	−	0.916157i	$-0.368725\pi$
$379$	5914.78	0.801641	0.400820	+	0.916157i	$0.368725\pi$
$380$	345.853	0.0466891
$381$	−2893.04	−0.389015
$382$	8299.65	1.11164
$383$	−9394.94	−1.25342	−0.626709	−	0.779254i	$-0.715598\pi$
$383$	−9394.94	−1.25342	−0.626709	+	0.779254i	$0.715598\pi$
$384$	5084.42	0.675686
$385$	748.971	0.0991457
$386$	1886.25	0.248725
$387$	2744.22	0.360457
$388$	379.069	0.0495987
$389$	−13960.2	−1.81957	−0.909784	−	0.415082i	$-0.863753\pi$
$389$	−13960.2	−1.81957	−0.909784	+	0.415082i	$0.863753\pi$
$390$	−431.641	−0.0560436
$391$	1484.09	0.191954
$392$	−7629.66	−0.983051
$393$	−2671.40	−0.342886
$394$	−487.447	−0.0623280
$395$	4844.85	0.617141
$396$	−241.924	−0.0306998
$397$	1146.97	0.144999	0.0724997	−	0.997368i	$-0.476902\pi$
$397$	1146.97	0.144999	0.0724997	+	0.997368i	$0.476902\pi$
$398$	−176.717	−0.0222563
$399$	−1753.86	−0.220057
$400$	−1447.80	−0.180976
$401$	9147.37	1.13915	0.569573	−	0.821940i	$-0.307108\pi$
$401$	9147.37	1.13915	0.569573	+	0.821940i	$0.307108\pi$
$402$	873.429	0.108365
$403$	1508.36	0.186444
$404$	429.497	0.0528918
$405$	1817.52	0.222996
$406$	−1175.93	−0.143745
$407$	2465.93	0.300324
$408$	−4027.08	−0.488652
$409$	1583.57	0.191449	0.0957245	−	0.995408i	$-0.469483\pi$
$409$	1583.57	0.191449	0.0957245	+	0.995408i	$0.469483\pi$
$410$	3623.43	0.436459
$411$	1396.17	0.167562
$412$	81.6629	0.00976515
$413$	−1761.75	−0.209903
$414$	958.457	0.113782
$415$	−415.000	−0.0490881
$416$	−243.989	−0.0287561
$417$	−7830.90	−0.919618
$418$	9322.85	1.09090
$419$	10673.4	1.24447	0.622233	−	0.782832i	$-0.286225\pi$
$419$	10673.4	1.24447	0.622233	+	0.782832i	$0.286225\pi$
$420$	−62.0803	−0.00721240
$421$	1218.36	0.141043	0.0705214	−	0.997510i	$-0.477534\pi$
$421$	1218.36	0.141043	0.0705214	+	0.997510i	$0.477534\pi$
$422$	−11912.7	−1.37417
$423$	5379.24	0.618316
$424$	−8674.72	−0.993589
$425$	1035.87	0.118229
$426$	2330.13	0.265012
$427$	3223.53	0.365333
$428$	288.677	0.0326022
$429$	1115.32	0.125520
$430$	−3743.25	−0.419803
$431$	−12037.4	−1.34529	−0.672646	−	0.739964i	$-0.734842\pi$
$431$	−12037.4	−1.34529	−0.672646	+	0.739964i	$0.734842\pi$
$432$	−8836.71	−0.984158
$433$	−12055.7	−1.33801	−0.669006	−	0.743257i	$-0.733280\pi$
$433$	−12055.7	−1.33801	−0.669006	+	0.743257i	$0.733280\pi$
$434$	−2263.18	−0.250313
$435$	2096.77	0.231109
$436$	576.611	0.0633363
$437$	3540.47	0.387560
$438$	−1177.46	−0.128450
$439$	10406.7	1.13140	0.565698	−	0.824613i	$-0.308607\pi$
$439$	10406.7	1.13140	0.565698	+	0.824613i	$0.308607\pi$
$440$	4102.61	0.444509
$441$	3214.69	0.347121
$442$	−865.111	−0.0930976
$443$	14079.4	1.51000	0.755002	−	0.655722i	$-0.227636\pi$
$443$	14079.4	1.51000	0.755002	+	0.655722i	$0.227636\pi$
$444$	−204.395	−0.0218472
$445$	−1110.09	−0.118254
$446$	−2517.96	−0.267329
$447$	11465.6	1.21321
$448$	2354.20	0.248271
$449$	3961.90	0.416423	0.208211	−	0.978084i	$-0.433236\pi$
$449$	3961.90	0.416423	0.208211	+	0.978084i	$0.433236\pi$
$450$	668.987	0.0700808
$451$	−9362.57	−0.977530
$452$	1191.20	0.123959
$453$	−8570.80	−0.888943
$454$	14319.4	1.48027
$455$	−165.801	−0.0170833
$456$	−9607.05	−0.986604
$457$	−19434.5	−1.98929	−0.994646	−	0.103344i	$-0.967046\pi$
$457$	−19434.5	−1.98929	−0.994646	+	0.103344i	$0.967046\pi$
$458$	4426.18	0.451576
$459$	6322.48	0.642937
$460$	125.320	0.0127023
$461$	−5885.64	−0.594624	−0.297312	−	0.954780i	$-0.596090\pi$
$461$	−5885.64	−0.594624	−0.297312	+	0.954780i	$0.596090\pi$
$462$	−1673.44	−0.168519
$463$	−654.532	−0.0656991	−0.0328495	−	0.999460i	$-0.510458\pi$
$463$	−654.532	−0.0656991	−0.0328495	+	0.999460i	$0.510458\pi$
$464$	−5873.59	−0.587660
$465$	4035.40	0.402446
$466$	13125.8	1.30481
$467$	−4551.81	−0.451034	−0.225517	−	0.974239i	$-0.572407\pi$
$467$	−4551.81	−0.451034	−0.225517	+	0.974239i	$0.572407\pi$
$468$	53.5552	0.00528972
$469$	335.500	0.0330319
$470$	−7337.52	−0.720116
$471$	−831.256	−0.0813211
$472$	−9650.25	−0.941078
$473$	9672.17	0.940226
$474$	−10824.9	−1.04896
$475$	2471.19	0.238707
$476$	−124.424	−0.0119810
$477$	3655.01	0.350842
$478$	19395.9	1.85595
$479$	−2908.50	−0.277438	−0.138719	−	0.990332i	$-0.544298\pi$
$479$	−2908.50	−0.277438	−0.138719	+	0.990332i	$0.544298\pi$
$480$	−652.757	−0.0620711
$481$	−545.889	−0.0517472
$482$	−1045.53	−0.0988019
$483$	−635.511	−0.0598691
$484$	78.7198	0.00739292
$485$	2708.53	0.253583
$486$	7070.55	0.659932
$487$	6664.89	0.620154	0.310077	−	0.950711i	$-0.399645\pi$
$487$	6664.89	0.620154	0.310077	+	0.950711i	$0.399645\pi$
$488$	17657.4	1.63793
$489$	−4387.06	−0.405705
$490$	−4384.98	−0.404272
$491$	−8035.09	−0.738531	−0.369265	−	0.929324i	$-0.620391\pi$
$491$	−8035.09	−0.738531	−0.369265	+	0.929324i	$0.620391\pi$
$492$	776.039	0.0711109
$493$	4202.43	0.383910
$494$	−2063.82	−0.187967
$495$	−1728.60	−0.156959
$496$	−11304.2	−1.02333
$497$	895.047	0.0807814
$498$	927.244	0.0834353
$499$	17863.7	1.60258	0.801292	−	0.598274i	$-0.204147\pi$
$499$	17863.7	1.60258	0.801292	+	0.598274i	$0.204147\pi$
$500$	87.4712	0.00782366
$501$	−3226.81	−0.287751
$502$	17999.3	1.60029
$503$	−21.6941	−0.00192305	−0.000961524	−	1.00000i	$-0.500306\pi$
$503$	−21.6941	−0.00192305	−0.000961524		1.00000i	$0.500306\pi$
$504$	−999.003	−0.0882919
$505$	3068.85	0.270420
$506$	3378.13	0.296791
$507$	8837.11	0.774102
$508$	−489.622	−0.0427627
$509$	2233.11	0.194462	0.0972309	−	0.995262i	$-0.469001\pi$
$509$	2233.11	0.194462	0.0972309	+	0.995262i	$0.469001\pi$
$510$	−2314.48	−0.200954
$511$	−452.285	−0.0391544
$512$	12718.7	1.09784
$513$	15083.0	1.29811
$514$	−13703.8	−1.17597
$515$	583.498	0.0499262
$516$	−801.701	−0.0683971
$517$	18959.4	1.61283
$518$	819.063	0.0694740
$519$	−9058.00	−0.766093
$520$	−908.203	−0.0765910
$521$	17882.6	1.50375	0.751874	−	0.659307i	$-0.229150\pi$
$521$	17882.6	1.50375	0.751874	+	0.659307i	$0.229150\pi$
$522$	2714.01	0.227565
$523$	−1377.75	−0.115191	−0.0575953	−	0.998340i	$-0.518343\pi$
$523$	−1377.75	−0.115191	−0.0575953	+	0.998340i	$0.518343\pi$
$524$	−452.113	−0.0376920
$525$	−443.577	−0.0368748
$526$	11685.3	0.968640
$527$	8087.90	0.668529
$528$	−8358.56	−0.688938
$529$	−10884.1	−0.894560
$530$	−4985.60	−0.408605
$531$	4066.05	0.332300
$532$	−296.826	−0.0241899
$533$	2072.61	0.168433
$534$	2480.29	0.200998
$535$	2062.66	0.166685
$536$	1837.76	0.148095
$537$	−2894.08	−0.232567
$538$	−1246.15	−0.0998612
$539$	11330.3	0.905440
$540$	533.882	0.0425456
$541$	−20403.2	−1.62145	−0.810723	−	0.585430i	$-0.800926\pi$
$541$	−20403.2	−1.62145	−0.810723	+	0.585430i	$0.800926\pi$
$542$	−16655.5	−1.31995
$543$	−11483.5	−0.907558
$544$	−1308.28	−0.103110
$545$	4120.01	0.323820
$546$	370.454	0.0290365
$547$	20662.2	1.61509	0.807543	−	0.589808i	$-0.200797\pi$
$547$	20662.2	1.61509	0.807543	+	0.589808i	$0.200797\pi$
$548$	236.290	0.0184194
$549$	−7439.77	−0.578364
$550$	2357.88	0.182801
$551$	10025.4	0.775126
$552$	−3481.11	−0.268417
$553$	−4158.06	−0.319745
$554$	13355.6	1.02423
$555$	−1460.45	−0.111698
$556$	−1325.31	−0.101090
$557$	−17223.8	−1.31023	−0.655115	−	0.755530i	$-0.727380\pi$
$557$	−17223.8	−1.31023	−0.655115	+	0.755530i	$0.727380\pi$
$558$	5223.33	0.396274
$559$	−2141.15	−0.162005
$560$	1242.57	0.0937646
$561$	5980.37	0.450074
$562$	−11835.4	−0.888337
$563$	12743.4	0.953947	0.476974	−	0.878918i	$-0.341734\pi$
$563$	12743.4	0.953947	0.476974	+	0.878918i	$0.341734\pi$
$564$	−1571.50	−0.117326
$565$	8511.37	0.633763
$566$	11878.4	0.882133
$567$	−1559.88	−0.115536
$568$	4902.76	0.362175
$569$	−26985.6	−1.98822	−0.994108	−	0.108391i	$-0.965430\pi$
$569$	−26985.6	−1.98822	−0.994108	+	0.108391i	$0.965430\pi$
$570$	−5521.44	−0.405733
$571$	11153.0	0.817403	0.408701	−	0.912668i	$-0.365982\pi$
$571$	11153.0	0.817403	0.408701	+	0.912668i	$0.365982\pi$
$572$	188.758	0.0137979
$573$	12701.0	0.925992
$574$	−3109.79	−0.226132
$575$	895.436	0.0649431
$576$	−5433.40	−0.393041
$577$	−12032.8	−0.868164	−0.434082	−	0.900873i	$-0.642927\pi$
$577$	−12032.8	−0.868164	−0.434082	+	0.900873i	$0.642927\pi$
$578$	8635.65	0.621446
$579$	2886.55	0.207187
$580$	354.861	0.0254048
$581$	356.172	0.0254328
$582$	−6051.72	−0.431017
$583$	12882.3	0.915146
$584$	−2477.46	−0.175545
$585$	382.663	0.0270447
$586$	−2233.40	−0.157442
$587$	−12326.6	−0.866737	−0.433369	−	0.901217i	$-0.642675\pi$
$587$	−12326.6	−0.866737	−0.433369	+	0.901217i	$0.642675\pi$
$588$	−939.142	−0.0658666
$589$	19294.6	1.34978
$590$	−5546.27	−0.387011
$591$	−745.945	−0.0519189
$592$	4091.08	0.284024
$593$	−20938.7	−1.45000	−0.724998	−	0.688751i	$-0.758159\pi$
$593$	−20938.7	−1.45000	−0.724998	+	0.688751i	$0.758159\pi$
$594$	14391.4	0.994084
$595$	−889.033	−0.0612551
$596$	1940.46	0.133363
$597$	−270.431	−0.0185394
$598$	−747.825	−0.0511385
$599$	14441.0	0.985045	0.492523	−	0.870300i	$-0.336075\pi$
$599$	14441.0	0.985045	0.492523	+	0.870300i	$0.336075\pi$
$600$	−2429.76	−0.165324
$601$	−3102.85	−0.210596	−0.105298	−	0.994441i	$-0.533580\pi$
$601$	−3102.85	−0.210596	−0.105298	+	0.994441i	$0.533580\pi$
$602$	3212.62	0.217503
$603$	−774.322	−0.0522932
$604$	−1450.54	−0.0977177
$605$	562.470	0.0377978
$606$	−6856.80	−0.459634
$607$	4582.40	0.306415	0.153208	−	0.988194i	$-0.451040\pi$
$607$	4582.40	0.306415	0.153208	+	0.988194i	$0.451040\pi$
$608$	−3121.05	−0.208183
$609$	−1799.54	−0.119739
$610$	10148.2	0.673587
$611$	−4197.09	−0.277898
$612$	287.165	0.0189672
$613$	8219.91	0.541597	0.270799	−	0.962636i	$-0.412712\pi$
$613$	8219.91	0.541597	0.270799	+	0.962636i	$0.412712\pi$
$614$	−15721.7	−1.03335
$615$	5544.96	0.363568
$616$	−3521.04	−0.230303
$617$	6422.32	0.419049	0.209524	−	0.977803i	$-0.432808\pi$
$617$	6422.32	0.419049	0.209524	+	0.977803i	$0.432808\pi$
$618$	−1303.72	−0.0848599
$619$	−189.749	−0.0123209	−0.00616046	−	0.999981i	$-0.501961\pi$
$619$	−189.749	−0.0123209	−0.00616046	+	0.999981i	$0.501961\pi$
$620$	682.958	0.0442391
$621$	5465.32	0.353165
$622$	−16754.5	−1.08006
$623$	952.726	0.0612683
$624$	1850.35	0.118707
$625$	625.000	0.0400000
$626$	4449.16	0.284064
$627$	14266.8	0.908712
$628$	−140.683	−0.00893928
$629$	−2927.08	−0.185549
$630$	−574.155	−0.0363093
$631$	−804.709	−0.0507685	−0.0253843	−	0.999678i	$-0.508081\pi$
$631$	−804.709	−0.0507685	−0.0253843	+	0.999678i	$0.508081\pi$
$632$	−22776.4	−1.43354
$633$	−18230.1	−1.14468
$634$	28914.4	1.81126
$635$	−3498.45	−0.218633
$636$	−1067.78	−0.0665727
$637$	−2508.22	−0.156012
$638$	9565.68	0.593587
$639$	−2065.73	−0.127886
$640$	6148.43	0.379747
$641$	−29738.1	−1.83243	−0.916214	−	0.400690i	$-0.868771\pi$
$641$	−29738.1	−1.83243	−0.916214	+	0.400690i	$0.868771\pi$
$642$	−4608.65	−0.283316
$643$	−24675.8	−1.51340	−0.756701	−	0.653762i	$-0.773190\pi$
$643$	−24675.8	−1.51340	−0.756701	+	0.653762i	$0.773190\pi$
$644$	−107.555	−0.00658115
$645$	−5728.32	−0.349694
$646$	−11066.3	−0.673989
$647$	−5733.00	−0.348358	−0.174179	−	0.984714i	$-0.555727\pi$
$647$	−5733.00	−0.348358	−0.174179	+	0.984714i	$0.555727\pi$
$648$	−8544.47	−0.517991
$649$	14331.0	0.866781
$650$	−521.970	−0.0314974
$651$	−3463.36	−0.208510
$652$	−742.473	−0.0445974
$653$	1179.74	0.0706993	0.0353496	−	0.999375i	$-0.488746\pi$
$653$	1179.74	0.0706993	0.0353496	+	0.999375i	$0.488746\pi$
$654$	−9205.42	−0.550398
$655$	−3230.44	−0.192708
$656$	−15532.8	−0.924475
$657$	1043.86	0.0619858
$658$	6297.39	0.373097
$659$	14538.5	0.859391	0.429696	−	0.902974i	$-0.358621\pi$
$659$	14538.5	0.859391	0.429696	+	0.902974i	$0.358621\pi$
$660$	504.994	0.0297831
$661$	5617.64	0.330561	0.165281	−	0.986247i	$-0.447147\pi$
$661$	5617.64	0.330561	0.165281	+	0.986247i	$0.447147\pi$
$662$	−14568.5	−0.855318
$663$	−1323.89	−0.0775498
$664$	1950.99	0.114026
$665$	−2120.89	−0.123676
$666$	−1890.37	−0.109985
$667$	3632.69	0.210882
$668$	−546.112	−0.0316313
$669$	−3853.26	−0.222684
$670$	1056.21	0.0609029
$671$	−26221.9	−1.50862
$672$	560.225	0.0321594
$673$	−14475.7	−0.829118	−0.414559	−	0.910022i	$-0.636064\pi$
$673$	−14475.7	−0.829118	−0.414559	+	0.910022i	$0.636064\pi$
$674$	2555.84	0.146064
$675$	3814.70	0.217523
$676$	1495.61	0.0850937
$677$	−17886.9	−1.01544	−0.507718	−	0.861523i	$-0.669511\pi$
$677$	−17886.9	−1.01544	−0.507718	+	0.861523i	$0.669511\pi$
$678$	−19017.1	−1.07721
$679$	−2324.58	−0.131383
$680$	−4869.82	−0.274631
$681$	21913.1	1.23306
$682$	18409.9	1.03365
$683$	18163.2	1.01756	0.508782	−	0.860896i	$-0.330096\pi$
$683$	18163.2	1.01756	0.508782	+	0.860896i	$0.330096\pi$
$684$	685.064	0.0382954
$685$	1688.34	0.0941726
$686$	7740.27	0.430794
$687$	6773.43	0.376161
$688$	16046.5	0.889195
$689$	−2851.78	−0.157684
$690$	−2000.69	−0.110384
$691$	15759.1	0.867591	0.433795	−	0.901011i	$-0.357174\pi$
$691$	15759.1	0.867591	0.433795	+	0.901011i	$0.357174\pi$
$692$	−1532.99	−0.0842133
$693$	1483.56	0.0813214
$694$	20894.2	1.14284
$695$	−9469.65	−0.516841
$696$	−9857.28	−0.536838
$697$	11113.4	0.603947
$698$	−2734.70	−0.148295
$699$	20086.5	1.08690
$700$	−75.0717	−0.00405349
$701$	13059.9	0.703658	0.351829	−	0.936064i	$-0.385560\pi$
$701$	13059.9	0.703658	0.351829	+	0.936064i	$0.385560\pi$
$702$	−3185.85	−0.171285
$703$	−6982.87	−0.374629
$704$	−19150.3	−1.02522
$705$	−11228.7	−0.599853
$706$	3387.82	0.180598
$707$	−2633.82	−0.140106
$708$	−1187.86	−0.0630544
$709$	−4348.41	−0.230336	−0.115168	−	0.993346i	$-0.536741\pi$
$709$	−4348.41	−0.230336	−0.115168	+	0.993346i	$0.536741\pi$
$710$	2817.75	0.148941
$711$	9596.65	0.506192
$712$	5218.71	0.274690
$713$	6991.39	0.367223
$714$	1986.39	0.104116
$715$	1348.72	0.0705443
$716$	−489.798	−0.0255651
$717$	29681.7	1.54600
$718$	22117.2	1.14959
$719$	25468.2	1.32101	0.660504	−	0.750822i	$-0.270343\pi$
$719$	25468.2	1.32101	0.660504	+	0.750822i	$0.270343\pi$
$720$	−2867.80	−0.148440
$721$	−500.784	−0.0258671
$722$	−7867.53	−0.405539
$723$	−1599.98	−0.0823015
$724$	−1943.49	−0.0997639
$725$	2535.56	0.129887
$726$	−1256.74	−0.0642451
$727$	3070.74	0.156654	0.0783270	−	0.996928i	$-0.475042\pi$
$727$	3070.74	0.156654	0.0783270	+	0.996928i	$0.475042\pi$
$728$	779.460	0.0396823
$729$	20634.7	1.04835
$730$	−1423.87	−0.0721913
$731$	−11480.9	−0.580899
$732$	2173.46	0.109745
$733$	−23632.2	−1.19083	−0.595413	−	0.803420i	$-0.703011\pi$
$733$	−23632.2	−1.19083	−0.595413	+	0.803420i	$0.703011\pi$
$734$	13226.8	0.665137
$735$	−6710.37	−0.336756
$736$	−1130.91	−0.0566385
$737$	−2729.14	−0.136403
$738$	7177.27	0.357993
$739$	−14914.1	−0.742388	−0.371194	−	0.928555i	$-0.621051\pi$
$739$	−14914.1	−0.742388	−0.371194	+	0.928555i	$0.621051\pi$
$740$	−247.168	−0.0122785
$741$	−3158.28	−0.156575
$742$	4278.87	0.211701
$743$	−1679.54	−0.0829292	−0.0414646	−	0.999140i	$-0.513202\pi$
$743$	−1679.54	−0.0829292	−0.0414646	+	0.999140i	$0.513202\pi$
$744$	−18971.1	−0.934832
$745$	13865.0	0.681844
$746$	6928.68	0.340050
$747$	−822.030	−0.0402631
$748$	1012.13	0.0494747
$749$	−1770.27	−0.0863607
$750$	−1396.45	−0.0679883
$751$	3454.58	0.167856	0.0839278	−	0.996472i	$-0.473253\pi$
$751$	3454.58	0.167856	0.0839278	+	0.996472i	$0.473253\pi$
$752$	31454.4	1.52530
$753$	27544.4	1.33303
$754$	−2117.57	−0.102278
$755$	−10364.4	−0.499601
$756$	−458.202	−0.0220432
$757$	26211.2	1.25847	0.629236	−	0.777214i	$-0.283368\pi$
$757$	26211.2	1.25847	0.629236	+	0.777214i	$0.283368\pi$
$758$	−15981.1	−0.765778
$759$	5169.59	0.247226
$760$	−11617.5	−0.554488
$761$	−19099.6	−0.909802	−0.454901	−	0.890542i	$-0.650325\pi$
$761$	−19099.6	−0.909802	−0.454901	+	0.890542i	$0.650325\pi$
$762$	7816.67	0.371612
$763$	−3535.97	−0.167773
$764$	2149.55	0.101790
$765$	2051.85	0.0969738
$766$	25384.1	1.19734
$767$	−3172.48	−0.149350
$768$	4409.21	0.207166
$769$	−23151.4	−1.08565	−0.542823	−	0.839847i	$-0.682645\pi$
$769$	−23151.4	−1.08565	−0.542823	+	0.839847i	$0.682645\pi$
$770$	−2023.64	−0.0947103
$771$	−20971.0	−0.979574
$772$	488.525	0.0227751
$773$	−1481.99	−0.0689564	−0.0344782	−	0.999405i	$-0.510977\pi$
$773$	−1481.99	−0.0689564	−0.0344782	+	0.999405i	$0.510977\pi$
$774$	−7414.60	−0.344331
$775$	4879.88	0.226181
$776$	−12733.2	−0.589043
$777$	1253.42	0.0578715
$778$	37719.1	1.73817
$779$	26512.3	1.21939
$780$	−111.792	−0.00513177
$781$	−7280.79	−0.333582
$782$	−4009.86	−0.183366
$783$	15475.8	0.706337
$784$	18797.4	0.856298
$785$	−1005.21	−0.0457038
$786$	7217.84	0.327547
$787$	−34437.7	−1.55981	−0.779906	−	0.625897i	$-0.784733\pi$
$787$	−34437.7	−1.55981	−0.779906	+	0.625897i	$0.784733\pi$
$788$	−126.245	−0.00570722
$789$	17882.2	0.806872
$790$	−13090.3	−0.589532
$791$	−7304.84	−0.328357
$792$	8126.42	0.364596
$793$	5804.79	0.259942
$794$	−3098.99	−0.138513
$795$	−7629.52	−0.340366
$796$	−45.7682	−0.00203795
$797$	8944.29	0.397519	0.198760	−	0.980048i	$-0.436309\pi$
$797$	8944.29	0.397519	0.198760	+	0.980048i	$0.436309\pi$
$798$	4738.74	0.210213
$799$	−22504.9	−0.996455
$800$	−789.357	−0.0348850
$801$	−2198.85	−0.0969946
$802$	−24715.2	−1.08819
$803$	3679.12	0.161686
$804$	226.211	0.00992271
$805$	−768.503	−0.0336474
$806$	−4075.44	−0.178103
$807$	−1907.00	−0.0831839
$808$	−14427.2	−0.628152
$809$	8554.01	0.371747	0.185873	−	0.982574i	$-0.440489\pi$
$809$	8554.01	0.371747	0.185873	+	0.982574i	$0.440489\pi$
$810$	−4910.74	−0.213020
$811$	10051.6	0.435215	0.217608	−	0.976036i	$-0.430175\pi$
$811$	10051.6	0.435215	0.217608	+	0.976036i	$0.430175\pi$
$812$	−304.558	−0.0131624
$813$	−25488.0	−1.09951
$814$	−6662.69	−0.286889
$815$	−5305.13	−0.228013
$816$	9921.65	0.425646
$817$	−27389.0	−1.17285
$818$	−4278.65	−0.182884
$819$	−328.418	−0.0140121
$820$	938.439	0.0399655
$821$	−10118.1	−0.430113	−0.215057	−	0.976602i	$-0.568994\pi$
$821$	−10118.1	−0.430113	−0.215057	+	0.976602i	$0.568994\pi$
$822$	−3772.30	−0.160066
$823$	−38789.2	−1.64290	−0.821449	−	0.570282i	$-0.806834\pi$
$823$	−38789.2	−1.64290	−0.821449	+	0.570282i	$0.806834\pi$
$824$	−2743.13	−0.115972
$825$	3608.29	0.152272
$826$	4760.06	0.200513
$827$	4171.29	0.175393	0.0876965	−	0.996147i	$-0.472049\pi$
$827$	4171.29	0.175393	0.0876965	+	0.996147i	$0.472049\pi$
$828$	248.233	0.0104187
$829$	−38792.4	−1.62523	−0.812615	−	0.582801i	$-0.801957\pi$
$829$	−38792.4	−1.62523	−0.812615	+	0.582801i	$0.801957\pi$
$830$	1121.29	0.0468921
$831$	20438.2	0.853182
$832$	4239.35	0.176650
$833$	−13449.2	−0.559407
$834$	21158.2	0.878478
$835$	−3902.08	−0.161721
$836$	2414.54	0.0998909
$837$	29784.5	1.22999
$838$	−28838.5	−1.18879
$839$	−30724.6	−1.26428	−0.632140	−	0.774854i	$-0.717823\pi$
$839$	−30724.6	−1.26428	−0.632140	+	0.774854i	$0.717823\pi$
$840$	2085.33	0.0856556
$841$	−14102.5	−0.578232
$842$	−3291.87	−0.134733
$843$	−18111.8	−0.739981
$844$	−3085.29	−0.125829
$845$	10686.4	0.435058
$846$	−14534.1	−0.590654
$847$	−482.737	−0.0195833
$848$	21372.2	0.865477
$849$	18177.7	0.734812
$850$	−2798.82	−0.112940
$851$	−2530.24	−0.101922
$852$	603.486	0.0242665
$853$	−13541.7	−0.543564	−0.271782	−	0.962359i	$-0.587613\pi$
$853$	−13541.7	−0.543564	−0.271782	+	0.962359i	$0.587613\pi$
$854$	−8709.62	−0.348990
$855$	4894.92	0.195793
$856$	−9696.92	−0.387189
$857$	12515.7	0.498867	0.249433	−	0.968392i	$-0.419756\pi$
$857$	12515.7	0.498867	0.249433	+	0.968392i	$0.419756\pi$
$858$	−3013.47	−0.119905
$859$	8297.54	0.329579	0.164789	−	0.986329i	$-0.447306\pi$
$859$	8297.54	0.329579	0.164789	+	0.986329i	$0.447306\pi$
$860$	−969.471	−0.0384403
$861$	−4758.93	−0.188367
$862$	32523.8	1.28511
$863$	4006.21	0.158022	0.0790110	−	0.996874i	$-0.474824\pi$
$863$	4006.21	0.158022	0.0790110	+	0.996874i	$0.474824\pi$
$864$	−4817.86	−0.189707
$865$	−10953.6	−0.430557
$866$	32573.2	1.27815
$867$	13215.2	0.517661
$868$	−586.145	−0.0229206
$869$	33823.9	1.32037
$870$	−5665.25	−0.220770
$871$	604.156	0.0235029
$872$	−19368.9	−0.752193
$873$	5365.04	0.207994
$874$	−9565.97	−0.370222
$875$	−536.403	−0.0207243
$876$	−304.953	−0.0117619
$877$	11533.6	0.444085	0.222042	−	0.975037i	$-0.428728\pi$
$877$	11533.6	0.444085	0.222042	+	0.975037i	$0.428728\pi$
$878$	−28117.7	−1.08078
$879$	−3417.79	−0.131148
$880$	−10107.7	−0.387195
$881$	13409.5	0.512803	0.256401	−	0.966570i	$-0.417463\pi$
$881$	13409.5	0.512803	0.256401	+	0.966570i	$0.417463\pi$
$882$	−8685.74	−0.331592
$883$	42702.9	1.62748	0.813740	−	0.581229i	$-0.197428\pi$
$883$	42702.9	1.62748	0.813740	+	0.581229i	$0.197428\pi$
$884$	−224.057	−0.00852472
$885$	−8487.51	−0.322378
$886$	−38041.0	−1.44245
$887$	179.541	0.00679640	0.00339820	−	0.999994i	$-0.498918\pi$
$887$	179.541	0.00679640	0.00339820	+	0.999994i	$0.498918\pi$
$888$	6865.80	0.259461
$889$	3002.53	0.113275
$890$	2999.34	0.112964
$891$	12688.9	0.477097
$892$	−652.132	−0.0244787
$893$	−53688.0	−2.01187
$894$	−30978.8	−1.15893
$895$	−3499.71	−0.130707
$896$	−5276.85	−0.196749
$897$	−1144.40	−0.0425981
$898$	−10704.6	−0.397793
$899$	19797.2	0.734452
$900$	173.263	0.00641713
$901$	−15291.4	−0.565404
$902$	25296.7	0.933799
$903$	4916.30	0.181179
$904$	−40013.4	−1.47215
$905$	−13886.6	−0.510063
$906$	23157.4	0.849175
$907$	19252.7	0.704825	0.352413	−	0.935845i	$-0.385361\pi$
$907$	19252.7	0.704825	0.352413	+	0.935845i	$0.385361\pi$
$908$	3708.61	0.135545
$909$	6078.76	0.221804
$910$	447.978	0.0163190
$911$	−9052.76	−0.329233	−0.164617	−	0.986358i	$-0.552639\pi$
$911$	−9052.76	−0.329233	−0.164617	+	0.986358i	$0.552639\pi$
$912$	23669.2	0.859392
$913$	−2897.29	−0.105023
$914$	52509.9	1.90030
$915$	15529.9	0.561094
$916$	1146.35	0.0413497
$917$	2772.51	0.0998432
$918$	−17082.7	−0.614174
$919$	−32293.2	−1.15915	−0.579573	−	0.814921i	$-0.696780\pi$
$919$	−32293.2	−1.15915	−0.579573	+	0.814921i	$0.696780\pi$
$920$	−4209.60	−0.150855
$921$	−24059.0	−0.860774
$922$	15902.4	0.568022
$923$	1611.76	0.0574777
$924$	−433.408	−0.0154308
$925$	−1766.07	−0.0627762
$926$	1768.48	0.0627600
$927$	1155.79	0.0409505
$928$	−3202.34	−0.113278
$929$	17991.9	0.635409	0.317704	−	0.948190i	$-0.397088\pi$
$929$	17991.9	0.635409	0.317704	+	0.948190i	$0.397088\pi$
$930$	−10903.2	−0.384442
$931$	−32084.5	−1.12946
$932$	3399.48	0.119478
$933$	−25639.6	−0.899683
$934$	12298.5	0.430856
$935$	7231.87	0.252949
$936$	−1798.96	−0.0628216
$937$	20624.3	0.719068	0.359534	−	0.933132i	$-0.382936\pi$
$937$	20624.3	0.719068	0.359534	+	0.933132i	$0.382936\pi$
$938$	−906.486	−0.0315542
$939$	6808.60	0.236624
$940$	−1900.36	−0.0659393
$941$	7261.75	0.251569	0.125784	−	0.992058i	$-0.459855\pi$
$941$	7261.75	0.251569	0.125784	+	0.992058i	$0.459855\pi$
$942$	2245.97	0.0776831
$943$	9606.73	0.331748
$944$	23775.7	0.819737
$945$	−3273.95	−0.112700
$946$	−26133.2	−0.898163
$947$	1275.28	0.0437603	0.0218802	−	0.999761i	$-0.493035\pi$
$947$	1275.28	0.0437603	0.0218802	+	0.999761i	$0.493035\pi$
$948$	−2803.58	−0.0960506
$949$	−814.456	−0.0278592
$950$	−6676.90	−0.228029
$951$	44248.0	1.50877
$952$	4179.50	0.142288
$953$	−25371.9	−0.862410	−0.431205	−	0.902254i	$-0.641911\pi$
$953$	−25371.9	−0.862410	−0.431205	+	0.902254i	$0.641911\pi$
$954$	−9875.46	−0.335147
$955$	15359.0	0.520423
$956$	5023.38	0.169945
$957$	14638.4	0.494455
$958$	7858.45	0.265026
$959$	−1449.01	−0.0487914
$960$	11341.7	0.381305
$961$	8310.18	0.278949
$962$	1474.94	0.0494322
$963$	4085.71	0.136719
$964$	−270.784	−0.00904705
$965$	3490.62	0.116442
$966$	1717.08	0.0571907
$967$	−1269.50	−0.0422174	−0.0211087	−	0.999777i	$-0.506720\pi$
$967$	−1269.50	−0.0422174	−0.0211087	+	0.999777i	$0.506720\pi$
$968$	−2644.27	−0.0877995
$969$	−16934.8	−0.561429
$970$	−7318.15	−0.242239
$971$	−34501.7	−1.14028	−0.570139	−	0.821548i	$-0.693111\pi$
$971$	−34501.7	−1.14028	−0.570139	+	0.821548i	$0.693111\pi$
$972$	1831.22	0.0604283
$973$	8127.28	0.267779
$974$	−18007.8	−0.592411
$975$	−798.775	−0.0262372
$976$	−43503.1	−1.42674
$977$	28452.3	0.931699	0.465849	−	0.884864i	$-0.345749\pi$
$977$	28452.3	0.931699	0.465849	+	0.884864i	$0.345749\pi$
$978$	11853.4	0.387555
$979$	−7749.98	−0.253004
$980$	−1135.67	−0.0370182
$981$	8160.89	0.265604
$982$	21710.0	0.705492
$983$	26457.9	0.858470	0.429235	−	0.903193i	$-0.358783\pi$
$983$	26457.9	0.858470	0.429235	+	0.903193i	$0.358783\pi$
$984$	−26067.8	−0.844524
$985$	−902.047	−0.0291793
$986$	−11354.5	−0.366736
$987$	9636.95	0.310788
$988$	−534.513	−0.0172117
$989$	−9924.41	−0.319088
$990$	4670.48	0.149937
$991$	−8296.46	−0.265939	−0.132970	−	0.991120i	$-0.542451\pi$
$991$	−8296.46	−0.265939	−0.132970	+	0.991120i	$0.542451\pi$
$992$	−6163.15	−0.197258
$993$	−22294.3	−0.712475
$994$	−2418.32	−0.0771675
$995$	−327.023	−0.0104194
$996$	240.149	0.00763997
$997$	−20090.9	−0.638201	−0.319101	−	0.947721i	$-0.603381\pi$
$997$	−20090.9	−0.638201	−0.319101	+	0.947721i	$0.603381\pi$
$998$	−48265.8	−1.53089
$999$	−10779.2	−0.341382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	415.4.a.c.1.7	✓	21
5.4	even	2		2075.4.a.g.1.15		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.4.a.c.1.7	✓	21	1.1	even	1	trivial
2075.4.a.g.1.15		21	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 415 = 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	415.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70189 q^{2} -4.13473 q^{3} -0.699769 q^{4} -5.00000 q^{5} +11.1716 q^{6} +4.29122 q^{7} +23.5059 q^{8} -9.90398 q^{9} +13.5095 q^{10} -34.9071 q^{11} +2.89336 q^{12} +7.72746 q^{13} -11.5944 q^{14} +20.6737 q^{15} -57.9122 q^{16} +41.4349 q^{17} +26.7595 q^{18} +98.8477 q^{19} +3.49885 q^{20} -17.7431 q^{21} +94.3153 q^{22} +35.8174 q^{23} -97.1904 q^{24} +25.0000 q^{25} -20.8788 q^{26} +152.588 q^{27} -3.00287 q^{28} +101.422 q^{29} -55.8581 q^{30} +195.195 q^{31} -31.5743 q^{32} +144.332 q^{33} -111.953 q^{34} -21.4561 q^{35} +6.93050 q^{36} -70.6428 q^{37} -267.076 q^{38} -31.9510 q^{39} -117.529 q^{40} +268.214 q^{41} +47.9399 q^{42} -277.083 q^{43} +24.4269 q^{44} +49.5199 q^{45} -96.7749 q^{46} -543.139 q^{47} +239.451 q^{48} -324.585 q^{49} -67.5473 q^{50} -171.322 q^{51} -5.40744 q^{52} -369.045 q^{53} -412.277 q^{54} +174.536 q^{55} +100.869 q^{56} -408.709 q^{57} -274.032 q^{58} -410.547 q^{59} -14.4668 q^{60} +751.190 q^{61} -527.397 q^{62} -42.5002 q^{63} +548.608 q^{64} -38.6373 q^{65} -389.969 q^{66} +78.1829 q^{67} -28.9949 q^{68} -148.096 q^{69} +57.9721 q^{70} +208.576 q^{71} -232.801 q^{72} -105.398 q^{73} +190.869 q^{74} -103.368 q^{75} -69.1706 q^{76} -149.794 q^{77} +86.3282 q^{78} -968.969 q^{79} +289.561 q^{80} -363.504 q^{81} -724.685 q^{82} +83.0000 q^{83} +12.4161 q^{84} -207.175 q^{85} +748.649 q^{86} -419.354 q^{87} -820.521 q^{88} +222.017 q^{89} -133.797 q^{90} +33.1603 q^{91} -25.0639 q^{92} -807.080 q^{93} +1467.50 q^{94} -494.238 q^{95} +130.551 q^{96} -541.705 q^{97} +876.995 q^{98} +345.719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 5 q^{2} - 12 q^{3} + 87 q^{4} - 105 q^{5} - 7 q^{6} - 11 q^{7} - 84 q^{8} + 153 q^{9} + 25 q^{10} - 30 q^{11} - 244 q^{12} - 89 q^{13} - 191 q^{14} + 60 q^{15} + 583 q^{16} - 357 q^{17} - 281 q^{18}+ \cdots - 5369 q^{99}+O(q^{100}) \) 21 * q - 5 * q^2 - 12 * q^3 + 87 * q^4 - 105 * q^5 - 7 * q^6 - 11 * q^7 - 84 * q^8 + 153 * q^9 + 25 * q^10 - 30 * q^11 - 244 * q^12 - 89 * q^13 - 191 * q^14 + 60 * q^15 + 583 * q^16 - 357 * q^17 - 281 * q^18 - 175 * q^19 - 435 * q^20 - 41 * q^21 - 122 * q^22 - 554 * q^23 - 136 * q^24 + 525 * q^25 - 933 * q^26 - 555 * q^27 - 773 * q^28 - 96 * q^29 + 35 * q^30 + 99 * q^31 - 1307 * q^32 - 1097 * q^33 + 393 * q^34 + 55 * q^35 - 746 * q^36 - 756 * q^37 - 1192 * q^38 + 40 * q^39 + 420 * q^40 - 1436 * q^41 - 1112 * q^42 - 1003 * q^43 - 706 * q^44 - 765 * q^45 - 775 * q^46 - 858 * q^47 - 2837 * q^48 + 1384 * q^49 - 125 * q^50 - 279 * q^51 - 1092 * q^52 - 1625 * q^53 - 269 * q^54 + 150 * q^55 - 3806 * q^56 - 1740 * q^57 - 1087 * q^58 - 1224 * q^59 + 1220 * q^60 - 794 * q^61 - 2979 * q^62 - 1236 * q^63 + 2570 * q^64 + 445 * q^65 - 1094 * q^66 - 571 * q^67 - 4064 * q^68 + 774 * q^69 + 955 * q^70 - 1078 * q^71 - 2610 * q^72 - 1986 * q^73 - 1872 * q^74 - 300 * q^75 - 2854 * q^76 - 2519 * q^77 - 2449 * q^78 - 432 * q^79 - 2915 * q^80 - 219 * q^81 - 2143 * q^82 + 1743 * q^83 - 377 * q^84 + 1785 * q^85 + 1556 * q^86 - 5681 * q^87 - 3158 * q^88 - 6252 * q^89 + 1405 * q^90 + 684 * q^91 - 7623 * q^92 - 3314 * q^93 - 3877 * q^94 + 875 * q^95 - 954 * q^96 - 4880 * q^97 - 5072 * q^98 - 5369 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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