LMFDB - Embedded newform 231.4.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [231,4,Mod(1,231)] 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("231.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 231 = 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	231.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-1,-15,21,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.6294412113$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162 $ x^5 - x^4 - 30x^3 + 11x^2 + 185*x + 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.28053$ of defining polynomial
Character	$\chi$	$=$	231.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.28053 q^{2} -3.00000 q^{3} -6.36023 q^{4} +16.8824 q^{5} -3.84160 q^{6} +7.00000 q^{7} -18.3888 q^{8} +9.00000 q^{9} +21.6185 q^{10} +11.0000 q^{11} +19.0807 q^{12} -68.0397 q^{13} +8.96374 q^{14} -50.6472 q^{15} +27.3344 q^{16} +119.045 q^{17} +11.5248 q^{18} +16.8404 q^{19} -107.376 q^{20} -21.0000 q^{21} +14.0859 q^{22} +199.722 q^{23} +55.1663 q^{24} +160.016 q^{25} -87.1272 q^{26} -27.0000 q^{27} -44.5216 q^{28} +181.053 q^{29} -64.8555 q^{30} -31.5474 q^{31} +182.113 q^{32} -33.0000 q^{33} +152.441 q^{34} +118.177 q^{35} -57.2421 q^{36} +75.9047 q^{37} +21.5647 q^{38} +204.119 q^{39} -310.447 q^{40} +408.485 q^{41} -26.8912 q^{42} -97.8817 q^{43} -69.9626 q^{44} +151.942 q^{45} +255.750 q^{46} +41.8437 q^{47} -82.0033 q^{48} +49.0000 q^{49} +204.906 q^{50} -357.136 q^{51} +432.748 q^{52} -563.375 q^{53} -34.5744 q^{54} +185.707 q^{55} -128.721 q^{56} -50.5212 q^{57} +231.845 q^{58} -224.425 q^{59} +322.128 q^{60} -622.237 q^{61} -40.3975 q^{62} +63.0000 q^{63} +14.5263 q^{64} -1148.67 q^{65} -42.2576 q^{66} -280.572 q^{67} -757.155 q^{68} -599.165 q^{69} +151.329 q^{70} -807.229 q^{71} -165.499 q^{72} +1038.49 q^{73} +97.1985 q^{74} -480.047 q^{75} -107.109 q^{76} +77.0000 q^{77} +261.382 q^{78} +710.954 q^{79} +461.471 q^{80} +81.0000 q^{81} +523.079 q^{82} -191.854 q^{83} +133.565 q^{84} +2009.77 q^{85} -125.341 q^{86} -543.159 q^{87} -202.276 q^{88} +1562.10 q^{89} +194.566 q^{90} -476.278 q^{91} -1270.28 q^{92} +94.6421 q^{93} +53.5822 q^{94} +284.307 q^{95} -546.338 q^{96} +816.513 q^{97} +62.7462 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - 15 q^{3} + 21 q^{4} + 21 q^{5} + 3 q^{6} + 35 q^{7} - 42 q^{8} + 45 q^{9} - 23 q^{10} + 55 q^{11} - 63 q^{12} + 101 q^{13} - 7 q^{14} - 63 q^{15} - 7 q^{16} - 20 q^{17} - 9 q^{18} + 237 q^{19}+ \cdots + 495 q^{99}+O(q^{100}) $ 5 * q - q^2 - 15 * q^3 + 21 * q^4 + 21 * q^5 + 3 * q^6 + 35 * q^7 - 42 * q^8 + 45 * q^9 - 23 * q^10 + 55 * q^11 - 63 * q^12 + 101 * q^13 - 7 * q^14 - 63 * q^15 - 7 * q^16 - 20 * q^17 - 9 * q^18 + 237 * q^19 + 85 * q^20 - 105 * q^21 - 11 * q^22 - 80 * q^23 + 126 * q^24 + 486 * q^25 + 165 * q^26 - 135 * q^27 + 147 * q^28 - 11 * q^29 + 69 * q^30 + 316 * q^31 + 453 * q^32 - 165 * q^33 + 936 * q^34 + 147 * q^35 + 189 * q^36 + 319 * q^37 + 89 * q^38 - 303 * q^39 + 624 * q^40 + 1190 * q^41 + 21 * q^42 + 88 * q^43 + 231 * q^44 + 189 * q^45 + 1000 * q^46 + 377 * q^47 + 21 * q^48 + 245 * q^49 - 644 * q^50 + 60 * q^51 + 1001 * q^52 - 992 * q^53 + 27 * q^54 + 231 * q^55 - 294 * q^56 - 711 * q^57 + 721 * q^58 + 71 * q^59 - 255 * q^60 - 574 * q^61 + 272 * q^62 + 315 * q^63 - 1380 * q^64 + 589 * q^65 + 33 * q^66 - 527 * q^67 - 2974 * q^68 + 240 * q^69 - 161 * q^70 - 1156 * q^71 - 378 * q^72 + 1061 * q^73 - 1609 * q^74 - 1458 * q^75 + 2399 * q^76 + 385 * q^77 - 495 * q^78 + 588 * q^79 - 1643 * q^80 + 405 * q^81 - 2602 * q^82 - 212 * q^83 - 441 * q^84 + 1918 * q^85 - 4760 * q^86 + 33 * q^87 - 462 * q^88 + 1030 * q^89 - 207 * q^90 + 707 * q^91 - 1174 * q^92 - 948 * q^93 - 1799 * q^94 - 3593 * q^95 - 1359 * q^96 + 2488 * q^97 - 49 * q^98 + 495 * 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 $

$2$	1.28053	0.452737	0.226369	−	0.974042i	$-0.427315\pi$
$2$	1.28053	0.452737	0.226369	+	0.974042i	$0.427315\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−6.36023	−0.795029
$5$	16.8824	1.51001	0.755004	−	0.655720i	$-0.227635\pi$
$5$	16.8824	1.51001	0.755004	+	0.655720i	$0.227635\pi$
$6$	−3.84160	−0.261388
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−18.3888	−0.812676
$9$	9.00000	0.333333
$10$	21.6185	0.683637
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	19.0807	0.459010
$13$	−68.0397	−1.45160	−0.725801	−	0.687905i	$-0.758531\pi$
$13$	−68.0397	−1.45160	−0.725801	+	0.687905i	$0.758531\pi$
$14$	8.96374	0.171119
$15$	−50.6472	−0.871804
$16$	27.3344	0.427100
$17$	119.045	1.69839	0.849197	−	0.528076i	$-0.177087\pi$
$17$	119.045	1.69839	0.849197	+	0.528076i	$0.177087\pi$
$18$	11.5248	0.150912
$19$	16.8404	0.203340	0.101670	−	0.994818i	$-0.467581\pi$
$19$	16.8404	0.203340	0.101670	+	0.994818i	$0.467581\pi$
$20$	−107.376	−1.20050
$21$	−21.0000	−0.218218
$22$	14.0859	0.136505
$23$	199.722	1.81065	0.905323	−	0.424724i	$-0.139629\pi$
$23$	199.722	1.81065	0.905323	+	0.424724i	$0.139629\pi$
$24$	55.1663	0.469199
$25$	160.016	1.28013
$26$	−87.1272	−0.657194
$27$	−27.0000	−0.192450
$28$	−44.5216	−0.300493
$29$	181.053	1.15933	0.579667	−	0.814853i	$-0.303183\pi$
$29$	181.053	1.15933	0.579667	+	0.814853i	$0.303183\pi$
$30$	−64.8555	−0.394698
$31$	−31.5474	−0.182776	−0.0913882	−	0.995815i	$-0.529130\pi$
$31$	−31.5474	−0.182776	−0.0913882	+	0.995815i	$0.529130\pi$
$32$	182.113	1.00604
$33$	−33.0000	−0.174078
$34$	152.441	0.768926
$35$	118.177	0.570730
$36$	−57.2421	−0.265010
$37$	75.9047	0.337261	0.168630	−	0.985679i	$-0.446066\pi$
$37$	75.9047	0.337261	0.168630	+	0.985679i	$0.446066\pi$
$38$	21.5647	0.0920594
$39$	204.119	0.838083
$40$	−310.447	−1.22715
$41$	408.485	1.55597	0.777983	−	0.628285i	$-0.216243\pi$
$41$	408.485	1.55597	0.777983	+	0.628285i	$0.216243\pi$
$42$	−26.8912	−0.0987953
$43$	−97.8817	−0.347135	−0.173568	−	0.984822i	$-0.555530\pi$
$43$	−97.8817	−0.347135	−0.173568	+	0.984822i	$0.555530\pi$
$44$	−69.9626	−0.239710
$45$	151.942	0.503336
$46$	255.750	0.819747
$47$	41.8437	0.129862	0.0649311	−	0.997890i	$-0.479317\pi$
$47$	41.8437	0.129862	0.0649311	+	0.997890i	$0.479317\pi$
$48$	−82.0033	−0.246586
$49$	49.0000	0.142857
$50$	204.906	0.579561
$51$	−357.136	−0.980568
$52$	432.748	1.15407
$53$	−563.375	−1.46010	−0.730052	−	0.683392i	$-0.760504\pi$
$53$	−563.375	−1.46010	−0.730052	+	0.683392i	$0.760504\pi$
$54$	−34.5744	−0.0871293
$55$	185.707	0.455285
$56$	−128.721	−0.307163
$57$	−50.5212	−0.117398
$58$	231.845	0.524874
$59$	−224.425	−0.495213	−0.247607	−	0.968861i	$-0.579644\pi$
$59$	−224.425	−0.495213	−0.247607	+	0.968861i	$0.579644\pi$
$60$	322.128	0.693109
$61$	−622.237	−1.30605	−0.653027	−	0.757335i	$-0.726501\pi$
$61$	−622.237	−1.30605	−0.653027	+	0.757335i	$0.726501\pi$
$62$	−40.3975	−0.0827497
$63$	63.0000	0.125988
$64$	14.5263	0.0283716
$65$	−1148.67	−2.19193
$66$	−42.2576	−0.0788114
$67$	−280.572	−0.511603	−0.255801	−	0.966729i	$-0.582339\pi$
$67$	−280.572	−0.511603	−0.255801	+	0.966729i	$0.582339\pi$
$68$	−757.155	−1.35027
$69$	−599.165	−1.04538
$70$	151.329	0.258390
$71$	−807.229	−1.34930	−0.674651	−	0.738137i	$-0.735706\pi$
$71$	−807.229	−1.34930	−0.674651	+	0.738137i	$0.735706\pi$
$72$	−165.499	−0.270892
$73$	1038.49	1.66501	0.832505	−	0.554017i	$-0.186906\pi$
$73$	1038.49	1.66501	0.832505	+	0.554017i	$0.186906\pi$
$74$	97.1985	0.152691
$75$	−480.047	−0.739081
$76$	−107.109	−0.161661
$77$	77.0000	0.113961
$78$	261.382	0.379431
$79$	710.954	1.01251	0.506257	−	0.862383i	$-0.331029\pi$
$79$	710.954	1.01251	0.506257	+	0.862383i	$0.331029\pi$
$80$	461.471	0.644925
$81$	81.0000	0.111111
$82$	523.079	0.704444
$83$	−191.854	−0.253719	−0.126860	−	0.991921i	$-0.540490\pi$
$83$	−191.854	−0.253719	−0.126860	+	0.991921i	$0.540490\pi$
$84$	133.565	0.173490
$85$	2009.77	2.56459
$86$	−125.341	−0.157161
$87$	−543.159	−0.669342
$88$	−202.276	−0.245031
$89$	1562.10	1.86047	0.930236	−	0.366962i	$-0.119602\pi$
$89$	1562.10	1.86047	0.930236	+	0.366962i	$0.119602\pi$
$90$	194.566	0.227879
$91$	−476.278	−0.548654
$92$	−1270.28	−1.43952
$93$	94.6421	0.105526
$94$	53.5822	0.0587935
$95$	284.307	0.307045
$96$	−546.338	−0.580838
$97$	816.513	0.854684	0.427342	−	0.904090i	$-0.359450\pi$
$97$	816.513	0.854684	0.427342	+	0.904090i	$0.359450\pi$
$98$	62.7462	0.0646767
$99$	99.0000	0.100504
$100$	−1017.74	−1.01774
$101$	−990.633	−0.975957	−0.487979	−	0.872856i	$-0.662266\pi$
$101$	−990.633	−0.975957	−0.487979	+	0.872856i	$0.662266\pi$
$102$	−457.324	−0.443940
$103$	−1947.55	−1.86308	−0.931542	−	0.363634i	$-0.881536\pi$
$103$	−1947.55	−1.86308	−0.931542	+	0.363634i	$0.881536\pi$
$104$	1251.17	1.17968
$105$	−354.531	−0.329511
$106$	−721.421	−0.661043
$107$	−479.820	−0.433513	−0.216757	−	0.976226i	$-0.569548\pi$
$107$	−479.820	−0.433513	−0.216757	+	0.976226i	$0.569548\pi$
$108$	171.726	0.153003
$109$	−474.762	−0.417192	−0.208596	−	0.978002i	$-0.566889\pi$
$109$	−474.762	−0.417192	−0.208596	+	0.978002i	$0.566889\pi$
$110$	237.803	0.206124
$111$	−227.714	−0.194718
$112$	191.341	0.161429
$113$	486.828	0.405283	0.202641	−	0.979253i	$-0.435047\pi$
$113$	486.828	0.405283	0.202641	+	0.979253i	$0.435047\pi$
$114$	−64.6942	−0.0531505
$115$	3371.78	2.73409
$116$	−1151.54	−0.921704
$117$	−612.357	−0.483867
$118$	−287.383	−0.224201
$119$	833.316	0.641933
$120$	931.340	0.708494
$121$	121.000	0.0909091
$122$	−796.795	−0.591299
$123$	−1225.45	−0.898338
$124$	200.648	0.145313
$125$	591.150	0.422993
$126$	80.6736	0.0570395
$127$	−1060.45	−0.740943	−0.370471	−	0.928844i	$-0.620804\pi$
$127$	−1060.45	−0.740943	−0.370471	+	0.928844i	$0.620804\pi$
$128$	−1438.30	−0.993196
$129$	293.645	0.200419
$130$	−1470.92	−0.992369
$131$	1988.88	1.32648	0.663242	−	0.748405i	$-0.269180\pi$
$131$	1988.88	1.32648	0.663242	+	0.748405i	$0.269180\pi$
$132$	209.888	0.138397
$133$	117.883	0.0768552
$134$	−359.283	−0.231622
$135$	−455.825	−0.290601
$136$	−2189.09	−1.38024
$137$	−1270.95	−0.792590	−0.396295	−	0.918123i	$-0.629704\pi$
$137$	−1270.95	−0.792590	−0.396295	+	0.918123i	$0.629704\pi$
$138$	−767.251	−0.473281
$139$	−1086.86	−0.663210	−0.331605	−	0.943418i	$-0.607590\pi$
$139$	−1086.86	−0.663210	−0.331605	+	0.943418i	$0.607590\pi$
$140$	−751.632	−0.453747
$141$	−125.531	−0.0749760
$142$	−1033.68	−0.610879
$143$	−748.437	−0.437674
$144$	246.010	0.142367
$145$	3056.61	1.75060
$146$	1329.82	0.753812
$147$	−147.000	−0.0824786
$148$	−482.772	−0.268132
$149$	−811.871	−0.446383	−0.223192	−	0.974775i	$-0.571648\pi$
$149$	−811.871	−0.446383	−0.223192	+	0.974775i	$0.571648\pi$
$150$	−614.717	−0.334609
$151$	−719.203	−0.387602	−0.193801	−	0.981041i	$-0.562082\pi$
$151$	−719.203	−0.387602	−0.193801	+	0.981041i	$0.562082\pi$
$152$	−309.674	−0.165249
$153$	1071.41	0.566131
$154$	98.6011	0.0515942
$155$	−532.595	−0.275994
$156$	−1298.25	−0.666300
$157$	3411.37	1.73412	0.867061	−	0.498201i	$-0.166006\pi$
$157$	3411.37	1.73412	0.867061	+	0.498201i	$0.166006\pi$
$158$	910.401	0.458402
$159$	1690.12	0.842991
$160$	3074.50	1.51913
$161$	1398.05	0.684360
$162$	103.723	0.0503041
$163$	−2691.36	−1.29327	−0.646637	−	0.762798i	$-0.723825\pi$
$163$	−2691.36	−1.29327	−0.646637	+	0.762798i	$0.723825\pi$
$164$	−2598.06	−1.23704
$165$	−557.120	−0.262859
$166$	−245.675	−0.114868
$167$	−2449.16	−1.13486	−0.567429	−	0.823422i	$-0.692062\pi$
$167$	−2449.16	−1.13486	−0.567429	+	0.823422i	$0.692062\pi$
$168$	386.164	0.177341
$169$	2432.40	1.10715
$170$	2573.58	1.16108
$171$	151.564	0.0677799
$172$	622.550	0.275983
$173$	914.239	0.401782	0.200891	−	0.979614i	$-0.435616\pi$
$173$	914.239	0.401782	0.200891	+	0.979614i	$0.435616\pi$
$174$	−695.534	−0.303036
$175$	1120.11	0.483842
$176$	300.679	0.128776
$177$	673.274	0.285912
$178$	2000.32	0.842305
$179$	−3651.00	−1.52452	−0.762258	−	0.647273i	$-0.775909\pi$
$179$	−3651.00	−1.52452	−0.762258	+	0.647273i	$0.775909\pi$
$180$	−966.384	−0.400167
$181$	−193.523	−0.0794722	−0.0397361	−	0.999210i	$-0.512652\pi$
$181$	−193.523	−0.0794722	−0.0397361	+	0.999210i	$0.512652\pi$
$182$	−609.890	−0.248396
$183$	1866.71	0.754050
$184$	−3672.64	−1.47147
$185$	1281.45	0.509267
$186$	121.192	0.0477756
$187$	1309.50	0.512085
$188$	−266.135	−0.103244
$189$	−189.000	−0.0727393
$190$	364.064	0.139011
$191$	3300.22	1.25024	0.625119	−	0.780529i	$-0.285050\pi$
$191$	3300.22	1.25024	0.625119	+	0.780529i	$0.285050\pi$
$192$	−43.5788	−0.0163804
$193$	2659.32	0.991824	0.495912	−	0.868373i	$-0.334834\pi$
$193$	2659.32	0.991824	0.495912	+	0.868373i	$0.334834\pi$
$194$	1045.57	0.386947
$195$	3446.02	1.26551
$196$	−311.651	−0.113576
$197$	1318.61	0.476890	0.238445	−	0.971156i	$-0.423362\pi$
$197$	1318.61	0.476890	0.238445	+	0.971156i	$0.423362\pi$
$198$	126.773	0.0455018
$199$	−1213.60	−0.432309	−0.216155	−	0.976359i	$-0.569352\pi$
$199$	−1213.60	−0.432309	−0.216155	+	0.976359i	$0.569352\pi$
$200$	−2942.49	−1.04033
$201$	841.717	0.295374
$202$	−1268.54	−0.441852
$203$	1267.37	0.438187
$204$	2271.46	0.779580
$205$	6896.21	2.34952
$206$	−2493.90	−0.843487
$207$	1797.50	0.603549
$208$	−1859.83	−0.619979
$209$	185.244	0.0613092
$210$	−453.988	−0.149182
$211$	−999.365	−0.326062	−0.163031	−	0.986621i	$-0.552127\pi$
$211$	−999.365	−0.326062	−0.163031	+	0.986621i	$0.552127\pi$
$212$	3583.20	1.16083
$213$	2421.69	0.779020
$214$	−614.425	−0.196268
$215$	−1652.48	−0.524177
$216$	496.497	0.156400
$217$	−220.831	−0.0690830
$218$	−607.949	−0.188878
$219$	−3115.46	−0.961294
$220$	−1181.14	−0.361965
$221$	−8099.80	−2.46539
$222$	−291.596	−0.0881559
$223$	594.639	0.178565	0.0892825	−	0.996006i	$-0.471543\pi$
$223$	594.639	0.178565	0.0892825	+	0.996006i	$0.471543\pi$
$224$	1274.79	0.380248
$225$	1440.14	0.426709
$226$	623.400	0.183487
$227$	3854.66	1.12706	0.563531	−	0.826095i	$-0.309443\pi$
$227$	3854.66	1.12706	0.563531	+	0.826095i	$0.309443\pi$
$228$	321.327	0.0933350
$229$	−1878.04	−0.541942	−0.270971	−	0.962588i	$-0.587345\pi$
$229$	−1878.04	−0.541942	−0.270971	+	0.962588i	$0.587345\pi$
$230$	4317.68	1.23782
$231$	−231.000	−0.0657952
$232$	−3329.34	−0.942164
$233$	−2960.57	−0.832418	−0.416209	−	0.909269i	$-0.636641\pi$
$233$	−2960.57	−0.832418	−0.416209	+	0.909269i	$0.636641\pi$
$234$	−784.145	−0.219065
$235$	706.422	0.196093
$236$	1427.39	0.393709
$237$	−2132.86	−0.584575
$238$	1067.09	0.290627
$239$	−1857.27	−0.502664	−0.251332	−	0.967901i	$-0.580869\pi$
$239$	−1857.27	−0.502664	−0.251332	+	0.967901i	$0.580869\pi$
$240$	−1384.41	−0.372348
$241$	3597.60	0.961583	0.480792	−	0.876835i	$-0.340349\pi$
$241$	3597.60	0.961583	0.480792	+	0.876835i	$0.340349\pi$
$242$	154.945	0.0411579
$243$	−243.000	−0.0641500
$244$	3957.57	1.03835
$245$	827.238	0.215716
$246$	−1569.24	−0.406711
$247$	−1145.82	−0.295168
$248$	580.117	0.148538
$249$	575.562	0.146485
$250$	756.988	0.191504
$251$	910.320	0.228920	0.114460	−	0.993428i	$-0.463486\pi$
$251$	910.320	0.228920	0.114460	+	0.993428i	$0.463486\pi$
$252$	−400.695	−0.100164
$253$	2196.94	0.545930
$254$	−1357.94	−0.335452
$255$	−6029.31	−1.48067
$256$	−1958.00	−0.478028
$257$	−669.516	−0.162503	−0.0812515	−	0.996694i	$-0.525892\pi$
$257$	−669.516	−0.162503	−0.0812515	+	0.996694i	$0.525892\pi$
$258$	376.022	0.0907369
$259$	531.333	0.127473
$260$	7305.84	1.74265
$261$	1629.48	0.386445
$262$	2546.83	0.600549
$263$	5046.95	1.18330	0.591651	−	0.806195i	$-0.298477\pi$
$263$	5046.95	1.18330	0.591651	+	0.806195i	$0.298477\pi$
$264$	606.829	0.141469
$265$	−9511.13	−2.20477
$266$	150.953	0.0347952
$267$	−4686.29	−1.07414
$268$	1784.51	0.406739
$269$	7226.13	1.63786	0.818931	−	0.573892i	$-0.194567\pi$
$269$	7226.13	1.63786	0.818931	+	0.573892i	$0.194567\pi$
$270$	−583.699	−0.131566
$271$	−4741.05	−1.06272	−0.531362	−	0.847145i	$-0.678320\pi$
$271$	−4741.05	−1.06272	−0.531362	+	0.847145i	$0.678320\pi$
$272$	3254.03	0.725384
$273$	1428.83	0.316765
$274$	−1627.50	−0.358835
$275$	1760.17	0.385972
$276$	3810.83	0.831105
$277$	−7997.89	−1.73482	−0.867412	−	0.497590i	$-0.834218\pi$
$277$	−7997.89	−1.73482	−0.867412	+	0.497590i	$0.834218\pi$
$278$	−1391.76	−0.300260
$279$	−283.926	−0.0609255
$280$	−2173.13	−0.463818
$281$	−304.490	−0.0646418	−0.0323209	−	0.999478i	$-0.510290\pi$
$281$	−304.490	−0.0646418	−0.0323209	+	0.999478i	$0.510290\pi$
$282$	−160.747	−0.0339444
$283$	6994.49	1.46919	0.734593	−	0.678508i	$-0.237373\pi$
$283$	6994.49	1.46919	0.734593	+	0.678508i	$0.237373\pi$
$284$	5134.16	1.07273
$285$	−852.920	−0.177272
$286$	−958.399	−0.198151
$287$	2859.39	0.588100
$288$	1639.02	0.335347
$289$	9258.75	1.88454
$290$	3914.09	0.792564
$291$	−2449.54	−0.493452
$292$	−6605.02	−1.32373
$293$	3090.16	0.616141	0.308070	−	0.951364i	$-0.400317\pi$
$293$	3090.16	0.616141	0.308070	+	0.951364i	$0.400317\pi$
$294$	−188.239	−0.0373411
$295$	−3788.83	−0.747776
$296$	−1395.79	−0.274084
$297$	−297.000	−0.0580259
$298$	−1039.63	−0.202094
$299$	−13589.0	−2.62834
$300$	3053.21	0.587591
$301$	−685.172	−0.131205
$302$	−920.964	−0.175482
$303$	2971.90	0.563469
$304$	460.323	0.0868465
$305$	−10504.9	−1.97215
$306$	1371.97	0.256309
$307$	2816.61	0.523624	0.261812	−	0.965119i	$-0.415680\pi$
$307$	2816.61	0.523624	0.261812	+	0.965119i	$0.415680\pi$
$308$	−489.738	−0.0906020
$309$	5842.64	1.07565
$310$	−682.006	−0.124953
$311$	−1962.61	−0.357844	−0.178922	−	0.983863i	$-0.557261\pi$
$311$	−1962.61	−0.357844	−0.178922	+	0.983863i	$0.557261\pi$
$312$	−3753.50	−0.681090
$313$	−9270.94	−1.67420	−0.837100	−	0.547050i	$-0.815751\pi$
$313$	−9270.94	−1.67420	−0.837100	+	0.547050i	$0.815751\pi$
$314$	4368.38	0.785102
$315$	1063.59	0.190243
$316$	−4521.83	−0.804978
$317$	−2929.73	−0.519084	−0.259542	−	0.965732i	$-0.583572\pi$
$317$	−2929.73	−0.519084	−0.259542	+	0.965732i	$0.583572\pi$
$318$	2164.26	0.381654
$319$	1991.58	0.349552
$320$	245.238	0.0428414
$321$	1439.46	0.250289
$322$	1790.25	0.309835
$323$	2004.77	0.345351
$324$	−515.179	−0.0883366
$325$	−10887.4	−1.85823
$326$	−3446.38	−0.585513
$327$	1424.29	0.240866
$328$	−7511.53	−1.26450
$329$	292.906	0.0490833
$330$	−713.410	−0.119006
$331$	−6024.72	−1.00045	−0.500224	−	0.865896i	$-0.666749\pi$
$331$	−6024.72	−1.00045	−0.500224	+	0.865896i	$0.666749\pi$
$332$	1220.24	0.201714
$333$	683.142	0.112420
$334$	−3136.23	−0.513793
$335$	−4736.74	−0.772524
$336$	−574.023	−0.0932009
$337$	1851.77	0.299324	0.149662	−	0.988737i	$-0.452181\pi$
$337$	1851.77	0.299324	0.149662	+	0.988737i	$0.452181\pi$
$338$	3114.77	0.501247
$339$	−1460.48	−0.233990
$340$	−12782.6	−2.03892
$341$	−347.021	−0.0551092
$342$	194.082	0.0306865
$343$	343.000	0.0539949
$344$	1799.92	0.282109
$345$	−10115.4	−1.57853
$346$	1170.71	0.181902
$347$	3931.79	0.608269	0.304134	−	0.952629i	$-0.401633\pi$
$347$	3931.79	0.608269	0.304134	+	0.952629i	$0.401633\pi$
$348$	3454.62	0.532146
$349$	−746.311	−0.114467	−0.0572337	−	0.998361i	$-0.518228\pi$
$349$	−746.311	−0.114467	−0.0572337	+	0.998361i	$0.518228\pi$
$350$	1434.34	0.219053
$351$	1837.07	0.279361
$352$	2003.24	0.303333
$353$	−8623.77	−1.30027	−0.650137	−	0.759817i	$-0.725289\pi$
$353$	−8623.77	−1.30027	−0.650137	+	0.759817i	$0.725289\pi$
$354$	862.150	0.129443
$355$	−13628.0	−2.03746
$356$	−9935.30	−1.47913
$357$	−2499.95	−0.370620
$358$	−4675.23	−0.690205
$359$	4195.46	0.616791	0.308395	−	0.951258i	$-0.400208\pi$
$359$	4195.46	0.616791	0.308395	+	0.951258i	$0.400208\pi$
$360$	−2794.02	−0.409049
$361$	−6575.40	−0.958653
$362$	−247.813	−0.0359800
$363$	−363.000	−0.0524864
$364$	3029.24	0.436196
$365$	17532.2	2.51418
$366$	2390.39	0.341386
$367$	−11009.4	−1.56590	−0.782948	−	0.622087i	$-0.786285\pi$
$367$	−11009.4	−1.56590	−0.782948	+	0.622087i	$0.786285\pi$
$368$	5459.28	0.773328
$369$	3676.36	0.518655
$370$	1640.95	0.230564
$371$	−3943.62	−0.551867
$372$	−601.945	−0.0838963
$373$	−647.517	−0.0898852	−0.0449426	−	0.998990i	$-0.514310\pi$
$373$	−647.517	−0.0898852	−0.0449426	+	0.998990i	$0.514310\pi$
$374$	1676.86	0.231840
$375$	−1773.45	−0.244215
$376$	−769.453	−0.105536
$377$	−12318.8	−1.68289
$378$	−242.021	−0.0329318
$379$	10934.0	1.48191	0.740955	−	0.671555i	$-0.234373\pi$
$379$	10934.0	1.48191	0.740955	+	0.671555i	$0.234373\pi$
$380$	−1808.26	−0.244109
$381$	3181.35	0.427783
$382$	4226.04	0.566029
$383$	−662.006	−0.0883210	−0.0441605	−	0.999024i	$-0.514061\pi$
$383$	−662.006	−0.0883210	−0.0441605	+	0.999024i	$0.514061\pi$
$384$	4314.90	0.573422
$385$	1299.95	0.172081
$386$	3405.35	0.449036
$387$	−880.935	−0.115712
$388$	−5193.21	−0.679499
$389$	46.1336	0.00601302	0.00300651	−	0.999995i	$-0.499043\pi$
$389$	46.1336	0.00601302	0.00300651	+	0.999995i	$0.499043\pi$
$390$	4412.75	0.572944
$391$	23775.9	3.07519
$392$	−901.050	−0.116097
$393$	−5966.64	−0.765846
$394$	1688.53	0.215906
$395$	12002.6	1.52890
$396$	−629.663	−0.0799034
$397$	11516.2	1.45587	0.727936	−	0.685645i	$-0.240480\pi$
$397$	11516.2	1.45587	0.727936	+	0.685645i	$0.240480\pi$
$398$	−1554.05	−0.195723
$399$	−353.649	−0.0443724
$400$	4373.94	0.546742
$401$	−9201.87	−1.14593	−0.572967	−	0.819578i	$-0.694208\pi$
$401$	−9201.87	−1.14593	−0.572967	+	0.819578i	$0.694208\pi$
$402$	1077.85	0.133727
$403$	2146.47	0.265319
$404$	6300.66	0.775914
$405$	1367.48	0.167779
$406$	1622.91	0.198384
$407$	834.952	0.101688
$408$	6567.28	0.796884
$409$	5529.75	0.668529	0.334265	−	0.942479i	$-0.391512\pi$
$409$	5529.75	0.668529	0.334265	+	0.942479i	$0.391512\pi$
$410$	8830.83	1.06372
$411$	3812.86	0.457602
$412$	12386.9	1.48121
$413$	−1570.97	−0.187173
$414$	2301.75	0.273249
$415$	−3238.96	−0.383118
$416$	−12390.9	−1.46037
$417$	3260.58	0.382905
$418$	237.212	0.0277570
$419$	−13372.9	−1.55921	−0.779604	−	0.626273i	$-0.784580\pi$
$419$	−13372.9	−1.55921	−0.779604	+	0.626273i	$0.784580\pi$
$420$	2254.90	0.261971
$421$	−6100.40	−0.706213	−0.353106	−	0.935583i	$-0.614875\pi$
$421$	−6100.40	−0.706213	−0.353106	+	0.935583i	$0.614875\pi$
$422$	−1279.72	−0.147620
$423$	376.593	0.0432874
$424$	10359.8	1.18659
$425$	19049.1	2.17416
$426$	3101.05	0.352691
$427$	−4355.66	−0.493642
$428$	3051.76	0.344656
$429$	2245.31	0.252691
$430$	−2116.05	−0.237314
$431$	−4465.34	−0.499043	−0.249522	−	0.968369i	$-0.580273\pi$
$431$	−4465.34	−0.499043	−0.249522	+	0.968369i	$0.580273\pi$
$432$	−738.029	−0.0821955
$433$	−9331.67	−1.03568	−0.517842	−	0.855476i	$-0.673265\pi$
$433$	−9331.67	−1.03568	−0.517842	+	0.855476i	$0.673265\pi$
$434$	−282.782	−0.0312765
$435$	−9169.83	−1.01071
$436$	3019.60	0.331680
$437$	3363.40	0.368176
$438$	−3989.45	−0.435214
$439$	2725.68	0.296332	0.148166	−	0.988963i	$-0.452663\pi$
$439$	2725.68	0.296332	0.148166	+	0.988963i	$0.452663\pi$
$440$	−3414.91	−0.369999
$441$	441.000	0.0476190
$442$	−10372.1	−1.11617
$443$	−14731.7	−1.57996	−0.789980	−	0.613132i	$-0.789909\pi$
$443$	−14731.7	−1.57996	−0.789980	+	0.613132i	$0.789909\pi$
$444$	1448.31	0.154806
$445$	26372.0	2.80933
$446$	761.456	0.0808430
$447$	2435.61	0.257719
$448$	101.684	0.0107235
$449$	−1591.42	−0.167269	−0.0836345	−	0.996497i	$-0.526653\pi$
$449$	−1591.42	−0.167269	−0.0836345	+	0.996497i	$0.526653\pi$
$450$	1844.15	0.193187
$451$	4493.33	0.469141
$452$	−3096.34	−0.322212
$453$	2157.61	0.223782
$454$	4936.03	0.510263
$455$	−8040.72	−0.828472
$456$	929.023	0.0954068
$457$	−9627.76	−0.985487	−0.492744	−	0.870175i	$-0.664006\pi$
$457$	−9627.76	−0.985487	−0.492744	+	0.870175i	$0.664006\pi$
$458$	−2404.90	−0.245357
$459$	−3214.22	−0.326856
$460$	−21445.3	−2.17368
$461$	15234.5	1.53913	0.769567	−	0.638566i	$-0.220472\pi$
$461$	15234.5	1.53913	0.769567	+	0.638566i	$0.220472\pi$
$462$	−295.803	−0.0297879
$463$	5534.70	0.555549	0.277775	−	0.960646i	$-0.410403\pi$
$463$	5534.70	0.555549	0.277775	+	0.960646i	$0.410403\pi$
$464$	4948.98	0.495152
$465$	1597.79	0.159345
$466$	−3791.11	−0.376866
$467$	−2265.02	−0.224438	−0.112219	−	0.993683i	$-0.535796\pi$
$467$	−2265.02	−0.224438	−0.112219	+	0.993683i	$0.535796\pi$
$468$	3894.74	0.384689
$469$	−1964.01	−0.193368
$470$	904.597	0.0887786
$471$	−10234.1	−1.00120
$472$	4126.89	0.402448
$473$	−1076.70	−0.104665
$474$	−2731.20	−0.264659
$475$	2694.73	0.260300
$476$	−5300.08	−0.510355
$477$	−5070.37	−0.486701
$478$	−2378.29	−0.227575
$479$	−2150.36	−0.205120	−0.102560	−	0.994727i	$-0.532703\pi$
$479$	−2150.36	−0.205120	−0.102560	+	0.994727i	$0.532703\pi$
$480$	−9223.51	−0.877070
$481$	−5164.53	−0.489569
$482$	4606.84	0.435344
$483$	−4194.16	−0.395115
$484$	−769.588	−0.0722754
$485$	13784.7	1.29058
$486$	−311.170	−0.0290431
$487$	−19739.0	−1.83667	−0.918336	−	0.395801i	$-0.870467\pi$
$487$	−19739.0	−1.83667	−0.918336	+	0.395801i	$0.870467\pi$
$488$	11442.2	1.06140
$489$	8074.08	0.746672
$490$	1059.31	0.0976624
$491$	18494.8	1.69992	0.849958	−	0.526850i	$-0.176627\pi$
$491$	18494.8	1.69992	0.849958	+	0.526850i	$0.176627\pi$
$492$	7794.18	0.714204
$493$	21553.5	1.96901
$494$	−1467.26	−0.133634
$495$	1671.36	0.151762
$496$	−862.329	−0.0780639
$497$	−5650.60	−0.509988
$498$	737.026	0.0663191
$499$	1228.36	0.110199	0.0550993	−	0.998481i	$-0.482452\pi$
$499$	1228.36	0.110199	0.0550993	+	0.998481i	$0.482452\pi$
$500$	−3759.85	−0.336291
$501$	7347.47	0.655211
$502$	1165.70	0.103641
$503$	17887.6	1.58562	0.792812	−	0.609466i	$-0.208616\pi$
$503$	17887.6	1.58562	0.792812	+	0.609466i	$0.208616\pi$
$504$	−1158.49	−0.102388
$505$	−16724.3	−1.47370
$506$	2813.26	0.247163
$507$	−7297.21	−0.639212
$508$	6744.71	0.589071
$509$	6755.97	0.588317	0.294158	−	0.955757i	$-0.404961\pi$
$509$	6755.97	0.588317	0.294158	+	0.955757i	$0.404961\pi$
$510$	−7720.73	−0.670353
$511$	7269.41	0.629315
$512$	8999.12	0.776775
$513$	−454.691	−0.0391327
$514$	−857.339	−0.0735712
$515$	−32879.3	−2.81327
$516$	−1867.65	−0.159339
$517$	460.280	0.0391549
$518$	680.390	0.0577116
$519$	−2742.72	−0.231969
$520$	21122.7	1.78133
$521$	10300.4	0.866158	0.433079	−	0.901356i	$-0.357427\pi$
$521$	10300.4	0.866158	0.433079	+	0.901356i	$0.357427\pi$
$522$	2086.60	0.174958
$523$	−1205.75	−0.100811	−0.0504053	−	0.998729i	$-0.516051\pi$
$523$	−1205.75	−0.100811	−0.0504053	+	0.998729i	$0.516051\pi$
$524$	−12649.7	−1.05459
$525$	−3360.33	−0.279346
$526$	6462.79	0.535724
$527$	−3755.56	−0.310426
$528$	−902.036	−0.0743486
$529$	27721.8	2.27844
$530$	−12179.3	−0.998181
$531$	−2019.82	−0.165071
$532$	−749.762	−0.0611021
$533$	−27793.2	−2.25864
$534$	−6000.96	−0.486305
$535$	−8100.51	−0.654609
$536$	5159.38	0.415767
$537$	10953.0	0.880180
$538$	9253.31	0.741521
$539$	539.000	0.0430730
$540$	2899.15	0.231036
$541$	3970.16	0.315509	0.157755	−	0.987478i	$-0.449575\pi$
$541$	3970.16	0.315509	0.157755	+	0.987478i	$0.449575\pi$
$542$	−6071.08	−0.481135
$543$	580.570	0.0458833
$544$	21679.6	1.70865
$545$	−8015.13	−0.629964
$546$	1829.67	0.143411
$547$	−24218.7	−1.89308	−0.946541	−	0.322584i	$-0.895449\pi$
$547$	−24218.7	−1.89308	−0.946541	+	0.322584i	$0.895449\pi$
$548$	8083.56	0.630132
$549$	−5600.13	−0.435351
$550$	2253.96	0.174744
$551$	3049.01	0.235739
$552$	11017.9	0.849553
$553$	4976.68	0.382694
$554$	−10241.6	−0.785420
$555$	−3844.36	−0.294025
$556$	6912.68	0.527271
$557$	−15753.5	−1.19838	−0.599190	−	0.800607i	$-0.704511\pi$
$557$	−15753.5	−1.19838	−0.599190	+	0.800607i	$0.704511\pi$
$558$	−363.577	−0.0275832
$559$	6659.84	0.503902
$560$	3230.30	0.243759
$561$	−3928.49	−0.295652
$562$	−389.910	−0.0292658
$563$	−7797.63	−0.583714	−0.291857	−	0.956462i	$-0.594273\pi$
$563$	−7797.63	−0.583714	−0.291857	+	0.956462i	$0.594273\pi$
$564$	798.406	0.0596081
$565$	8218.83	0.611980
$566$	8956.69	0.665155
$567$	567.000	0.0419961
$568$	14843.9	1.09655
$569$	5074.55	0.373877	0.186939	−	0.982372i	$-0.440143\pi$
$569$	5074.55	0.373877	0.186939	+	0.982372i	$0.440143\pi$
$570$	−1092.19	−0.0802578
$571$	19247.4	1.41065	0.705323	−	0.708886i	$-0.250802\pi$
$571$	19247.4	1.41065	0.705323	+	0.708886i	$0.250802\pi$
$572$	4760.23	0.347964
$573$	−9900.66	−0.721826
$574$	3661.55	0.266255
$575$	31958.6	2.31786
$576$	130.736	0.00945720
$577$	13981.0	1.00873	0.504364	−	0.863491i	$-0.331727\pi$
$577$	13981.0	1.00873	0.504364	+	0.863491i	$0.331727\pi$
$578$	11856.1	0.853202
$579$	−7977.96	−0.572630
$580$	−19440.8	−1.39178
$581$	−1342.98	−0.0958969
$582$	−3136.72	−0.223404
$583$	−6197.12	−0.440238
$584$	−19096.5	−1.35311
$585$	−10338.1	−0.730644
$586$	3957.06	0.278950
$587$	−864.831	−0.0608099	−0.0304049	−	0.999538i	$-0.509680\pi$
$587$	−864.831	−0.0608099	−0.0304049	+	0.999538i	$0.509680\pi$
$588$	934.954	0.0655729
$589$	−531.270	−0.0371657
$590$	−4851.72	−0.338546
$591$	−3955.84	−0.275333
$592$	2074.81	0.144044
$593$	3157.59	0.218662	0.109331	−	0.994005i	$-0.465129\pi$
$593$	3157.59	0.218662	0.109331	+	0.994005i	$0.465129\pi$
$594$	−380.319	−0.0262705
$595$	14068.4	0.969324
$596$	5163.69	0.354888
$597$	3640.79	0.249594
$598$	−17401.2	−1.18995
$599$	11270.2	0.768758	0.384379	−	0.923175i	$-0.374416\pi$
$599$	11270.2	0.768758	0.384379	+	0.923175i	$0.374416\pi$
$600$	8827.48	0.600634
$601$	−22222.4	−1.50827	−0.754135	−	0.656719i	$-0.771944\pi$
$601$	−22222.4	−1.50827	−0.754135	+	0.656719i	$0.771944\pi$
$602$	−877.386	−0.0594013
$603$	−2525.15	−0.170534
$604$	4574.30	0.308155
$605$	2042.77	0.137274
$606$	3805.62	0.255103
$607$	1829.17	0.122312	0.0611562	−	0.998128i	$-0.480521\pi$
$607$	1829.17	0.122312	0.0611562	+	0.998128i	$0.480521\pi$
$608$	3066.85	0.204568
$609$	−3802.11	−0.252987
$610$	−13451.8	−0.892866
$611$	−2847.03	−0.188508
$612$	−6814.39	−0.450091
$613$	11172.5	0.736136	0.368068	−	0.929799i	$-0.380019\pi$
$613$	11172.5	0.736136	0.368068	+	0.929799i	$0.380019\pi$
$614$	3606.77	0.237064
$615$	−20688.6	−1.35650
$616$	−1415.94	−0.0926131
$617$	−14278.2	−0.931634	−0.465817	−	0.884881i	$-0.654240\pi$
$617$	−14278.2	−0.931634	−0.465817	+	0.884881i	$0.654240\pi$
$618$	7481.70	0.486988
$619$	−12918.1	−0.838809	−0.419405	−	0.907799i	$-0.637761\pi$
$619$	−12918.1	−0.838809	−0.419405	+	0.907799i	$0.637761\pi$
$620$	3387.43	0.219423
$621$	−5392.49	−0.348459
$622$	−2513.19	−0.162009
$623$	10934.7	0.703192
$624$	5579.48	0.357945
$625$	−10021.9	−0.641404
$626$	−11871.8	−0.757973
$627$	−555.733	−0.0353969
$628$	−21697.1	−1.37868
$629$	9036.09	0.572802
$630$	1361.97	0.0861302
$631$	−17365.6	−1.09558	−0.547792	−	0.836614i	$-0.684532\pi$
$631$	−17365.6	−1.09558	−0.547792	+	0.836614i	$0.684532\pi$
$632$	−13073.6	−0.822846
$633$	2998.09	0.188252
$634$	−3751.61	−0.235009
$635$	−17902.9	−1.11883
$636$	−10749.6	−0.670203
$637$	−3333.95	−0.207372
$638$	2550.29	0.158255
$639$	−7265.06	−0.449767
$640$	−24282.0	−1.49973
$641$	−3234.39	−0.199299	−0.0996494	−	0.995023i	$-0.531772\pi$
$641$	−3234.39	−0.199299	−0.0996494	+	0.995023i	$0.531772\pi$
$642$	1843.28	0.113315
$643$	−4372.53	−0.268174	−0.134087	−	0.990970i	$-0.542810\pi$
$643$	−4372.53	−0.268174	−0.134087	+	0.990970i	$0.542810\pi$
$644$	−8891.94	−0.544086
$645$	4957.43	0.302634
$646$	2567.18	0.156353
$647$	9149.07	0.555930	0.277965	−	0.960591i	$-0.410340\pi$
$647$	9149.07	0.555930	0.277965	+	0.960591i	$0.410340\pi$
$648$	−1489.49	−0.0902974
$649$	−2468.67	−0.149312
$650$	−13941.7	−0.841291
$651$	662.494	0.0398851
$652$	17117.7	1.02819
$653$	−33267.8	−1.99367	−0.996837	−	0.0794734i	$-0.974676\pi$
$653$	−33267.8	−1.99367	−0.996837	+	0.0794734i	$0.974676\pi$
$654$	1823.85	0.109049
$655$	33577.1	2.00300
$656$	11165.7	0.664554
$657$	9346.38	0.555003
$658$	375.076	0.0222218
$659$	−15125.2	−0.894073	−0.447037	−	0.894516i	$-0.647521\pi$
$659$	−15125.2	−0.894073	−0.447037	+	0.894516i	$0.647521\pi$
$660$	3543.41	0.208980
$661$	12789.3	0.752568	0.376284	−	0.926504i	$-0.377202\pi$
$661$	12789.3	0.752568	0.376284	+	0.926504i	$0.377202\pi$
$662$	−7714.86	−0.452940
$663$	24299.4	1.42339
$664$	3527.96	0.206192
$665$	1990.15	0.116052
$666$	874.787	0.0508969
$667$	36160.2	2.09914
$668$	15577.2	0.902246
$669$	−1783.92	−0.103095
$670$	−6065.55	−0.349750
$671$	−6844.60	−0.393790
$672$	−3824.37	−0.219536
$673$	−18730.3	−1.07281	−0.536404	−	0.843962i	$-0.680217\pi$
$673$	−18730.3	−1.07281	−0.536404	+	0.843962i	$0.680217\pi$
$674$	2371.25	0.135515
$675$	−4320.42	−0.246360
$676$	−15470.6	−0.880214
$677$	−6160.79	−0.349747	−0.174873	−	0.984591i	$-0.555952\pi$
$677$	−6160.79	−0.349747	−0.174873	+	0.984591i	$0.555952\pi$
$678$	−1870.20	−0.105936
$679$	5715.59	0.323040
$680$	−36957.2	−2.08418
$681$	−11564.0	−0.650709
$682$	−444.372	−0.0249500
$683$	3050.21	0.170883	0.0854415	−	0.996343i	$-0.472770\pi$
$683$	3050.21	0.170883	0.0854415	+	0.996343i	$0.472770\pi$
$684$	−963.980	−0.0538870
$685$	−21456.7	−1.19682
$686$	439.223	0.0244455
$687$	5634.13	0.312890
$688$	−2675.54	−0.148262
$689$	38331.9	2.11949
$690$	−12953.1	−0.714658
$691$	−12418.3	−0.683668	−0.341834	−	0.939760i	$-0.611048\pi$
$691$	−12418.3	−0.683668	−0.341834	+	0.939760i	$0.611048\pi$
$692$	−5814.77	−0.319428
$693$	693.000	0.0379869
$694$	5034.79	0.275386
$695$	−18348.8	−1.00145
$696$	9988.02	0.543958
$697$	48628.1	2.64264
$698$	−955.676	−0.0518236
$699$	8881.71	0.480597
$700$	−7124.16	−0.384669
$701$	−15271.1	−0.822796	−0.411398	−	0.911456i	$-0.634959\pi$
$701$	−15271.1	−0.822796	−0.411398	+	0.911456i	$0.634959\pi$
$702$	2352.43	0.126477
$703$	1278.27	0.0685785
$704$	159.789	0.00855436
$705$	−2119.27	−0.113214
$706$	−11043.0	−0.588682
$707$	−6934.43	−0.368877
$708$	−4282.18	−0.227308
$709$	24174.5	1.28052	0.640262	−	0.768157i	$-0.278826\pi$
$709$	24174.5	1.28052	0.640262	+	0.768157i	$0.278826\pi$
$710$	−17451.1	−0.922433
$711$	6398.59	0.337504
$712$	−28725.0	−1.51196
$713$	−6300.69	−0.330944
$714$	−3201.27	−0.167793
$715$	−12635.4	−0.660892
$716$	23221.2	1.21203
$717$	5571.80	0.290213
$718$	5372.43	0.279244
$719$	21724.7	1.12683	0.563416	−	0.826173i	$-0.309487\pi$
$719$	21724.7	1.12683	0.563416	+	0.826173i	$0.309487\pi$
$720$	4153.24	0.214975
$721$	−13632.8	−0.704179
$722$	−8420.02	−0.434018
$723$	−10792.8	−0.555170
$724$	1230.85	0.0631827
$725$	28971.3	1.48409
$726$	−464.834	−0.0237625
$727$	−32751.2	−1.67081	−0.835403	−	0.549638i	$-0.814766\pi$
$727$	−32751.2	−1.67081	−0.835403	+	0.549638i	$0.814766\pi$
$728$	8758.16	0.445878
$729$	729.000	0.0370370
$730$	22450.5	1.13826
$731$	−11652.3	−0.589572
$732$	−11872.7	−0.599492
$733$	−5646.65	−0.284535	−0.142267	−	0.989828i	$-0.545439\pi$
$733$	−5646.65	−0.284535	−0.142267	+	0.989828i	$0.545439\pi$
$734$	−14097.9	−0.708940
$735$	−2481.71	−0.124543
$736$	36371.9	1.82158
$737$	−3086.30	−0.154254
$738$	4707.71	0.234815
$739$	34189.8	1.70188	0.850942	−	0.525260i	$-0.176032\pi$
$739$	34189.8	1.70188	0.850942	+	0.525260i	$0.176032\pi$
$740$	−8150.35	−0.404882
$741$	3437.45	0.170415
$742$	−5049.95	−0.249851
$743$	−15998.8	−0.789960	−0.394980	−	0.918690i	$-0.629249\pi$
$743$	−15998.8	−0.789960	−0.394980	+	0.918690i	$0.629249\pi$
$744$	−1740.35	−0.0857585
$745$	−13706.3	−0.674042
$746$	−829.168	−0.0406944
$747$	−1726.68	−0.0845731
$748$	−8328.70	−0.407122
$749$	−3358.74	−0.163853
$750$	−2270.96	−0.110565
$751$	−18097.7	−0.879354	−0.439677	−	0.898156i	$-0.644907\pi$
$751$	−18097.7	−0.879354	−0.439677	+	0.898156i	$0.644907\pi$
$752$	1143.77	0.0554642
$753$	−2730.96	−0.132167
$754$	−15774.6	−0.761908
$755$	−12141.9	−0.585282
$756$	1202.08	0.0578299
$757$	8597.91	0.412809	0.206404	−	0.978467i	$-0.433824\pi$
$757$	8597.91	0.412809	0.206404	+	0.978467i	$0.433824\pi$
$758$	14001.4	0.670915
$759$	−6590.82	−0.315193
$760$	−5228.05	−0.249528
$761$	31479.9	1.49953	0.749767	−	0.661702i	$-0.230165\pi$
$761$	31479.9	1.49953	0.749767	+	0.661702i	$0.230165\pi$
$762$	4073.83	0.193673
$763$	−3323.33	−0.157684
$764$	−20990.2	−0.993976
$765$	18087.9	0.854863
$766$	−847.721	−0.0399862
$767$	15269.8	0.718853
$768$	5874.01	0.275990
$769$	31854.0	1.49374	0.746868	−	0.664972i	$-0.231557\pi$
$769$	31854.0	1.49374	0.746868	+	0.664972i	$0.231557\pi$
$770$	1664.62	0.0779077
$771$	2008.55	0.0938212
$772$	−16913.9	−0.788529
$773$	−27.0929	−0.00126062	−0.000630312	−	1.00000i	$-0.500201\pi$
$773$	−27.0929	−0.00126062	−0.000630312		1.00000i	$0.500201\pi$
$774$	−1128.07	−0.0523870
$775$	−5048.07	−0.233977
$776$	−15014.7	−0.694582
$777$	−1594.00	−0.0735964
$778$	59.0756	0.00272232
$779$	6879.05	0.316390
$780$	−21917.5	−1.00612
$781$	−8879.52	−0.406830
$782$	30445.9	1.39225
$783$	−4888.43	−0.223114
$784$	1339.39	0.0610143
$785$	57592.2	2.61854
$786$	−7640.49	−0.346727
$787$	−6357.30	−0.287946	−0.143973	−	0.989582i	$-0.545988\pi$
$787$	−6357.30	−0.287946	−0.143973	+	0.989582i	$0.545988\pi$
$788$	−8386.69	−0.379142
$789$	−15140.8	−0.683179
$790$	15369.8	0.692192
$791$	3407.80	0.153182
$792$	−1820.49	−0.0816770
$793$	42336.8	1.89587
$794$	14746.9	0.659127
$795$	28533.4	1.27292
$796$	7718.76	0.343699
$797$	−9946.90	−0.442079	−0.221040	−	0.975265i	$-0.570945\pi$
$797$	−9946.90	−0.442079	−0.221040	+	0.975265i	$0.570945\pi$
$798$	−452.859	−0.0200890
$799$	4981.28	0.220557
$800$	29140.9	1.28786
$801$	14058.9	0.620157
$802$	−11783.3	−0.518807
$803$	11423.4	0.502019
$804$	−5353.52	−0.234831
$805$	23602.5	1.03339
$806$	2748.63	0.120120
$807$	−21678.4	−0.945620
$808$	18216.5	0.793137
$809$	21882.2	0.950972	0.475486	−	0.879723i	$-0.342272\pi$
$809$	21882.2	0.950972	0.475486	+	0.879723i	$0.342272\pi$
$810$	1751.10	0.0759597
$811$	−5407.69	−0.234143	−0.117071	−	0.993124i	$-0.537351\pi$
$811$	−5407.69	−0.234143	−0.117071	+	0.993124i	$0.537351\pi$
$812$	−8060.77	−0.348372
$813$	14223.2	0.613564
$814$	1069.18	0.0460379
$815$	−45436.7	−1.95285
$816$	−9762.09	−0.418801
$817$	−1648.37	−0.0705864
$818$	7081.03	0.302668
$819$	−4286.50	−0.182885
$820$	−43861.5	−1.86794
$821$	5294.69	0.225074	0.112537	−	0.993648i	$-0.464102\pi$
$821$	5294.69	0.225074	0.112537	+	0.993648i	$0.464102\pi$
$822$	4882.50	0.207173
$823$	7539.03	0.319312	0.159656	−	0.987173i	$-0.448961\pi$
$823$	7539.03	0.319312	0.159656	+	0.987173i	$0.448961\pi$
$824$	35813.0	1.51408
$825$	−5280.52	−0.222841
$826$	−2011.68	−0.0847402
$827$	−18827.2	−0.791640	−0.395820	−	0.918328i	$-0.629540\pi$
$827$	−18827.2	−0.791640	−0.395820	+	0.918328i	$0.629540\pi$
$828$	−11432.5	−0.479839
$829$	−5480.45	−0.229607	−0.114803	−	0.993388i	$-0.536624\pi$
$829$	−5480.45	−0.229607	−0.114803	+	0.993388i	$0.536624\pi$
$830$	−4147.59	−0.173452
$831$	23993.7	1.00160
$832$	−988.363	−0.0411843
$833$	5833.21	0.242628
$834$	4175.28	0.173355
$835$	−41347.7	−1.71365
$836$	−1178.20	−0.0487426
$837$	851.778	0.0351754
$838$	−17124.4	−0.705911
$839$	−32885.7	−1.35321	−0.676603	−	0.736348i	$-0.736549\pi$
$839$	−32885.7	−1.35321	−0.676603	+	0.736348i	$0.736549\pi$
$840$	6519.38	0.267786
$841$	8391.18	0.344056
$842$	−7811.78	−0.319729
$843$	913.470	0.0373210
$844$	6356.19	0.259229
$845$	41064.8	1.67180
$846$	482.240	0.0195978
$847$	847.000	0.0343604
$848$	−15399.5	−0.623611
$849$	−20983.5	−0.848235
$850$	24393.0	0.984322
$851$	15159.8	0.610660
$852$	−15402.5	−0.619344
$853$	−24852.9	−0.997593	−0.498797	−	0.866719i	$-0.666225\pi$
$853$	−24852.9	−0.997593	−0.498797	+	0.866719i	$0.666225\pi$
$854$	−5577.57	−0.223490
$855$	2558.76	0.102348
$856$	8823.29	0.352306
$857$	−26129.7	−1.04151	−0.520755	−	0.853706i	$-0.674349\pi$
$857$	−26129.7	−1.04151	−0.520755	+	0.853706i	$0.674349\pi$
$858$	2875.20	0.114403
$859$	3354.66	0.133248	0.0666238	−	0.997778i	$-0.478777\pi$
$859$	3354.66	0.133248	0.0666238	+	0.997778i	$0.478777\pi$
$860$	10510.1	0.416736
$861$	−8578.18	−0.339540
$862$	−5718.02	−0.225936
$863$	−37414.5	−1.47579	−0.737893	−	0.674918i	$-0.764179\pi$
$863$	−37414.5	−1.47579	−0.737893	+	0.674918i	$0.764179\pi$
$864$	−4917.05	−0.193613
$865$	15434.6	0.606694
$866$	−11949.5	−0.468893
$867$	−27776.3	−1.08804
$868$	1404.54	0.0549230
$869$	7820.49	0.305284
$870$	−11742.3	−0.457587
$871$	19090.1	0.742643
$872$	8730.29	0.339042
$873$	7348.62	0.284895
$874$	4306.94	0.166687
$875$	4138.05	0.159876
$876$	19815.1	0.764257
$877$	5518.99	0.212501	0.106250	−	0.994339i	$-0.466115\pi$
$877$	5518.99	0.212501	0.106250	+	0.994339i	$0.466115\pi$
$878$	3490.33	0.134160
$879$	−9270.49	−0.355729
$880$	5076.18	0.194452
$881$	35273.5	1.34892	0.674458	−	0.738313i	$-0.264377\pi$
$881$	35273.5	1.34892	0.674458	+	0.738313i	$0.264377\pi$
$882$	564.716	0.0215589
$883$	34154.4	1.30168	0.650842	−	0.759213i	$-0.274416\pi$
$883$	34154.4	1.30168	0.650842	+	0.759213i	$0.274416\pi$
$884$	51516.6	1.96006
$885$	11366.5	0.431729
$886$	−18864.4	−0.715307
$887$	−31925.4	−1.20851	−0.604257	−	0.796790i	$-0.706530\pi$
$887$	−31925.4	−1.20851	−0.604257	+	0.796790i	$0.706530\pi$
$888$	4187.38	0.158242
$889$	−7423.15	−0.280050
$890$	33770.2	1.27189
$891$	891.000	0.0335013
$892$	−3782.04	−0.141964
$893$	704.664	0.0264061
$894$	3118.89	0.116679
$895$	−61637.7	−2.30203
$896$	−10068.1	−0.375393
$897$	40767.0	1.51747
$898$	−2037.87	−0.0757288
$899$	−5711.74	−0.211899
$900$	−9159.64	−0.339246
$901$	−67067.1	−2.47983
$902$	5753.87	0.212398
$903$	2055.51	0.0757511
$904$	−8952.17	−0.329364
$905$	−3267.14	−0.120004
$906$	2762.89	0.101314
$907$	21299.4	0.779751	0.389875	−	0.920868i	$-0.372518\pi$
$907$	21299.4	0.779751	0.389875	+	0.920868i	$0.372518\pi$
$908$	−24516.5	−0.896046
$909$	−8915.70	−0.325319
$910$	−10296.4	−0.375080
$911$	25231.7	0.917631	0.458816	−	0.888531i	$-0.348274\pi$
$911$	25231.7	0.917631	0.458816	+	0.888531i	$0.348274\pi$
$912$	−1380.97	−0.0501408
$913$	−2110.39	−0.0764992
$914$	−12328.7	−0.446167
$915$	31514.6	1.13862
$916$	11944.8	0.430859
$917$	13922.2	0.501364
$918$	−4115.92	−0.147980
$919$	17335.9	0.622263	0.311131	−	0.950367i	$-0.399292\pi$
$919$	17335.9	0.622263	0.311131	+	0.950367i	$0.399292\pi$
$920$	−62003.0	−2.22193
$921$	−8449.84	−0.302314
$922$	19508.3	0.696824
$923$	54923.6	1.95865
$924$	1469.21	0.0523091
$925$	12145.9	0.431736
$926$	7087.37	0.251518
$927$	−17527.9	−0.621028
$928$	32972.1	1.16634
$929$	22981.2	0.811614	0.405807	−	0.913959i	$-0.366991\pi$
$929$	22981.2	0.811614	0.405807	+	0.913959i	$0.366991\pi$
$930$	2046.02	0.0721415
$931$	825.180	0.0290485
$932$	18829.9	0.661796
$933$	5887.83	0.206601
$934$	−2900.43	−0.101611
$935$	22107.5	0.773253
$936$	11260.5	0.393227
$937$	−15884.3	−0.553809	−0.276904	−	0.960897i	$-0.589309\pi$
$937$	−15884.3	−0.553809	−0.276904	+	0.960897i	$0.589309\pi$
$938$	−2514.98	−0.0875447
$939$	27812.8	0.966600
$940$	−4493.01	−0.155900
$941$	22366.2	0.774832	0.387416	−	0.921905i	$-0.373368\pi$
$941$	22366.2	0.774832	0.387416	+	0.921905i	$0.373368\pi$
$942$	−13105.1	−0.453279
$943$	81583.3	2.81730
$944$	−6134.51	−0.211506
$945$	−3190.78	−0.109837
$946$	−1378.75	−0.0473858
$947$	−26760.3	−0.918261	−0.459130	−	0.888369i	$-0.651839\pi$
$947$	−26760.3	−0.918261	−0.459130	+	0.888369i	$0.651839\pi$
$948$	13565.5	0.464754
$949$	−70658.4	−2.41693
$950$	3450.69	0.117848
$951$	8789.18	0.299694
$952$	−15323.7	−0.521683
$953$	−33171.7	−1.12753	−0.563766	−	0.825935i	$-0.690648\pi$
$953$	−33171.7	−1.12753	−0.563766	+	0.825935i	$0.690648\pi$
$954$	−6492.79	−0.220348
$955$	55715.7	1.88787
$956$	11812.7	0.399632
$957$	−5974.75	−0.201814
$958$	−2753.60	−0.0928652
$959$	−8896.67	−0.299571
$960$	−735.715	−0.0247345
$961$	−28795.8	−0.966593
$962$	−6613.36	−0.221646
$963$	−4318.38	−0.144504
$964$	−22881.6	−0.764487
$965$	44895.7	1.49766
$966$	−5370.76	−0.178883
$967$	7857.98	0.261319	0.130660	−	0.991427i	$-0.458291\pi$
$967$	7857.98	0.261319	0.130660	+	0.991427i	$0.458291\pi$
$968$	−2225.04	−0.0738797
$969$	−6014.31	−0.199388
$970$	17651.8	0.584294
$971$	−434.865	−0.0143723	−0.00718614	−	0.999974i	$-0.502287\pi$
$971$	−434.865	−0.0143723	−0.00718614	+	0.999974i	$0.502287\pi$
$972$	1545.54	0.0510011
$973$	−7608.02	−0.250670
$974$	−25276.5	−0.831530
$975$	32662.3	1.07285
$976$	−17008.5	−0.557816
$977$	19558.5	0.640463	0.320232	−	0.947339i	$-0.396239\pi$
$977$	19558.5	0.640463	0.320232	+	0.947339i	$0.396239\pi$
$978$	10339.1	0.338046
$979$	17183.1	0.560953
$980$	−5261.43	−0.171500
$981$	−4272.86	−0.139064
$982$	23683.2	0.769615
$983$	−59419.8	−1.92797	−0.963987	−	0.265950i	$-0.914315\pi$
$983$	−59419.8	−1.92797	−0.963987	+	0.265950i	$0.914315\pi$
$984$	22534.6	0.730058
$985$	22261.4	0.720108
$986$	27600.0	0.891442
$987$	−878.717	−0.0283383
$988$	7287.66	0.234667
$989$	−19549.1	−0.628539
$990$	2140.23	0.0687081
$991$	−25376.7	−0.813439	−0.406719	−	0.913553i	$-0.633327\pi$
$991$	−25376.7	−0.813439	−0.406719	+	0.913553i	$0.633327\pi$
$992$	−5745.18	−0.183881
$993$	18074.2	0.577609
$994$	−7235.79	−0.230891
$995$	−20488.4	−0.652791
$996$	−3660.71	−0.116460
$997$	−38621.4	−1.22683	−0.613416	−	0.789760i	$-0.710205\pi$
$997$	−38621.4	−1.22683	−0.613416	+	0.789760i	$0.710205\pi$
$998$	1572.96	0.0498910
$999$	−2049.43	−0.0649059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.4.a.k.1.3	✓	5
3.2	odd	2		693.4.a.p.1.3		5
7.6	odd	2		1617.4.a.n.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.3	✓	5	1.1	even	1	trivial
693.4.a.p.1.3		5	3.2	odd	2
1617.4.a.n.1.3		5	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28053 q^{2} -3.00000 q^{3} -6.36023 q^{4} +16.8824 q^{5} -3.84160 q^{6} +7.00000 q^{7} -18.3888 q^{8} +9.00000 q^{9} +21.6185 q^{10} +11.0000 q^{11} +19.0807 q^{12} -68.0397 q^{13} +8.96374 q^{14} -50.6472 q^{15} +27.3344 q^{16} +119.045 q^{17} +11.5248 q^{18} +16.8404 q^{19} -107.376 q^{20} -21.0000 q^{21} +14.0859 q^{22} +199.722 q^{23} +55.1663 q^{24} +160.016 q^{25} -87.1272 q^{26} -27.0000 q^{27} -44.5216 q^{28} +181.053 q^{29} -64.8555 q^{30} -31.5474 q^{31} +182.113 q^{32} -33.0000 q^{33} +152.441 q^{34} +118.177 q^{35} -57.2421 q^{36} +75.9047 q^{37} +21.5647 q^{38} +204.119 q^{39} -310.447 q^{40} +408.485 q^{41} -26.8912 q^{42} -97.8817 q^{43} -69.9626 q^{44} +151.942 q^{45} +255.750 q^{46} +41.8437 q^{47} -82.0033 q^{48} +49.0000 q^{49} +204.906 q^{50} -357.136 q^{51} +432.748 q^{52} -563.375 q^{53} -34.5744 q^{54} +185.707 q^{55} -128.721 q^{56} -50.5212 q^{57} +231.845 q^{58} -224.425 q^{59} +322.128 q^{60} -622.237 q^{61} -40.3975 q^{62} +63.0000 q^{63} +14.5263 q^{64} -1148.67 q^{65} -42.2576 q^{66} -280.572 q^{67} -757.155 q^{68} -599.165 q^{69} +151.329 q^{70} -807.229 q^{71} -165.499 q^{72} +1038.49 q^{73} +97.1985 q^{74} -480.047 q^{75} -107.109 q^{76} +77.0000 q^{77} +261.382 q^{78} +710.954 q^{79} +461.471 q^{80} +81.0000 q^{81} +523.079 q^{82} -191.854 q^{83} +133.565 q^{84} +2009.77 q^{85} -125.341 q^{86} -543.159 q^{87} -202.276 q^{88} +1562.10 q^{89} +194.566 q^{90} -476.278 q^{91} -1270.28 q^{92} +94.6421 q^{93} +53.5822 q^{94} +284.307 q^{95} -546.338 q^{96} +816.513 q^{97} +62.7462 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 15 q^{3} + 21 q^{4} + 21 q^{5} + 3 q^{6} + 35 q^{7} - 42 q^{8} + 45 q^{9} - 23 q^{10} + 55 q^{11} - 63 q^{12} + 101 q^{13} - 7 q^{14} - 63 q^{15} - 7 q^{16} - 20 q^{17} - 9 q^{18} + 237 q^{19}+ \cdots + 495 q^{99}+O(q^{100}) \) 5 * q - q^2 - 15 * q^3 + 21 * q^4 + 21 * q^5 + 3 * q^6 + 35 * q^7 - 42 * q^8 + 45 * q^9 - 23 * q^10 + 55 * q^11 - 63 * q^12 + 101 * q^13 - 7 * q^14 - 63 * q^15 - 7 * q^16 - 20 * q^17 - 9 * q^18 + 237 * q^19 + 85 * q^20 - 105 * q^21 - 11 * q^22 - 80 * q^23 + 126 * q^24 + 486 * q^25 + 165 * q^26 - 135 * q^27 + 147 * q^28 - 11 * q^29 + 69 * q^30 + 316 * q^31 + 453 * q^32 - 165 * q^33 + 936 * q^34 + 147 * q^35 + 189 * q^36 + 319 * q^37 + 89 * q^38 - 303 * q^39 + 624 * q^40 + 1190 * q^41 + 21 * q^42 + 88 * q^43 + 231 * q^44 + 189 * q^45 + 1000 * q^46 + 377 * q^47 + 21 * q^48 + 245 * q^49 - 644 * q^50 + 60 * q^51 + 1001 * q^52 - 992 * q^53 + 27 * q^54 + 231 * q^55 - 294 * q^56 - 711 * q^57 + 721 * q^58 + 71 * q^59 - 255 * q^60 - 574 * q^61 + 272 * q^62 + 315 * q^63 - 1380 * q^64 + 589 * q^65 + 33 * q^66 - 527 * q^67 - 2974 * q^68 + 240 * q^69 - 161 * q^70 - 1156 * q^71 - 378 * q^72 + 1061 * q^73 - 1609 * q^74 - 1458 * q^75 + 2399 * q^76 + 385 * q^77 - 495 * q^78 + 588 * q^79 - 1643 * q^80 + 405 * q^81 - 2602 * q^82 - 212 * q^83 - 441 * q^84 + 1918 * q^85 - 4760 * q^86 + 33 * q^87 - 462 * q^88 + 1030 * q^89 - 207 * q^90 + 707 * q^91 - 1174 * q^92 - 948 * q^93 - 1799 * q^94 - 3593 * q^95 - 1359 * q^96 + 2488 * q^97 - 49 * q^98 + 495 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more