LMFDB - Embedded newform 3174.2.a.z.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3174,2,Mod(1,3174)] 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("3174.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3174, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3174 = 2 \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3174.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-5,5,5,11,-5,-1,-5,5,-11,11,5,10,1,11,5,11,-5,-1,11,-1,-11, 0,-5,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.3445176016$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.284630$ of defining polynomial
Character	$\chi$	$=$	3174.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.95185 q^{5} -1.00000 q^{6} -0.458044 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.95185 q^{10} +1.25954 q^{11} +1.00000 q^{12} +5.60149 q^{13} +0.458044 q^{14} +1.95185 q^{15} +1.00000 q^{16} +6.54436 q^{17} -1.00000 q^{18} +2.11970 q^{19} +1.95185 q^{20} -0.458044 q^{21} -1.25954 q^{22} -1.00000 q^{24} -1.19028 q^{25} -5.60149 q^{26} +1.00000 q^{27} -0.458044 q^{28} +5.06465 q^{29} -1.95185 q^{30} -0.109003 q^{31} -1.00000 q^{32} +1.25954 q^{33} -6.54436 q^{34} -0.894034 q^{35} +1.00000 q^{36} -0.926128 q^{37} -2.11970 q^{38} +5.60149 q^{39} -1.95185 q^{40} -9.83955 q^{41} +0.458044 q^{42} +6.77428 q^{43} +1.25954 q^{44} +1.95185 q^{45} -2.50740 q^{47} +1.00000 q^{48} -6.79020 q^{49} +1.19028 q^{50} +6.54436 q^{51} +5.60149 q^{52} -2.64612 q^{53} -1.00000 q^{54} +2.45843 q^{55} +0.458044 q^{56} +2.11970 q^{57} -5.06465 q^{58} -4.27686 q^{59} +1.95185 q^{60} -8.58595 q^{61} +0.109003 q^{62} -0.458044 q^{63} +1.00000 q^{64} +10.9333 q^{65} -1.25954 q^{66} -15.0332 q^{67} +6.54436 q^{68} +0.894034 q^{70} -0.303444 q^{71} -1.00000 q^{72} +3.34620 q^{73} +0.926128 q^{74} -1.19028 q^{75} +2.11970 q^{76} -0.576924 q^{77} -5.60149 q^{78} -8.04741 q^{79} +1.95185 q^{80} +1.00000 q^{81} +9.83955 q^{82} +10.4920 q^{83} -0.458044 q^{84} +12.7736 q^{85} -6.77428 q^{86} +5.06465 q^{87} -1.25954 q^{88} +5.95710 q^{89} -1.95185 q^{90} -2.56573 q^{91} -0.109003 q^{93} +2.50740 q^{94} +4.13734 q^{95} -1.00000 q^{96} +18.2589 q^{97} +6.79020 q^{98} +1.25954 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 11 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9} - 11 q^{10} + 11 q^{11} + 5 q^{12} + 10 q^{13} + q^{14} + 11 q^{15} + 5 q^{16} + 11 q^{17} - 5 q^{18} - q^{19} + 11 q^{20}+ \cdots + 11 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 11 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9 - 11 * q^10 + 11 * q^11 + 5 * q^12 + 10 * q^13 + q^14 + 11 * q^15 + 5 * q^16 + 11 * q^17 - 5 * q^18 - q^19 + 11 * q^20 - q^21 - 11 * q^22 - 5 * q^24 + 30 * q^25 - 10 * q^26 + 5 * q^27 - q^28 + 3 * q^29 - 11 * q^30 + 2 * q^31 - 5 * q^32 + 11 * q^33 - 11 * q^34 - 11 * q^35 + 5 * q^36 + 16 * q^37 + q^38 + 10 * q^39 - 11 * q^40 - 8 * q^41 + q^42 - 12 * q^43 + 11 * q^44 + 11 * q^45 + 5 * q^48 - 4 * q^49 - 30 * q^50 + 11 * q^51 + 10 * q^52 + 27 * q^53 - 5 * q^54 + q^56 - q^57 - 3 * q^58 - 6 * q^59 + 11 * q^60 + 5 * q^61 - 2 * q^62 - q^63 + 5 * q^64 + 22 * q^65 - 11 * q^66 - 7 * q^67 + 11 * q^68 + 11 * q^70 - 11 * q^71 - 5 * q^72 - 12 * q^73 - 16 * q^74 + 30 * q^75 - q^76 + 11 * q^77 - 10 * q^78 + 16 * q^79 + 11 * q^80 + 5 * q^81 + 8 * q^82 + 22 * q^83 - q^84 - 11 * q^85 + 12 * q^86 + 3 * q^87 - 11 * q^88 + 38 * q^89 - 11 * q^90 + 9 * q^91 + 2 * q^93 + 11 * q^95 - 5 * q^96 + 5 * q^97 + 4 * q^98 + 11 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.95185	0.872894	0.436447	−	0.899730i	$-0.356237\pi$
$5$	1.95185	0.872894	0.436447	+	0.899730i	$0.356237\pi$
$6$	−1.00000	−0.408248
$7$	−0.458044	−0.173125	−0.0865623	−	0.996246i	$-0.527588\pi$
$7$	−0.458044	−0.173125	−0.0865623	+	0.996246i	$0.527588\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.95185	−0.617229
$11$	1.25954	0.379765	0.189882	−	0.981807i	$-0.439189\pi$
$11$	1.25954	0.379765	0.189882	+	0.981807i	$0.439189\pi$
$12$	1.00000	0.288675
$13$	5.60149	1.55357	0.776787	−	0.629763i	$-0.216848\pi$
$13$	5.60149	1.55357	0.776787	+	0.629763i	$0.216848\pi$
$14$	0.458044	0.122418
$15$	1.95185	0.503965
$16$	1.00000	0.250000
$17$	6.54436	1.58724	0.793621	−	0.608413i	$-0.208194\pi$
$17$	6.54436	1.58724	0.793621	+	0.608413i	$0.208194\pi$
$18$	−1.00000	−0.235702
$19$	2.11970	0.486294	0.243147	−	0.969990i	$-0.421820\pi$
$19$	2.11970	0.486294	0.243147	+	0.969990i	$0.421820\pi$
$20$	1.95185	0.436447
$21$	−0.458044	−0.0999535
$22$	−1.25954	−0.268534
$23$	0	0
$23$	0	0
$24$	−1.00000	−0.204124
$25$	−1.19028	−0.238057
$26$	−5.60149	−1.09854
$27$	1.00000	0.192450
$28$	−0.458044	−0.0865623
$29$	5.06465	0.940481	0.470241	−	0.882538i	$-0.344167\pi$
$29$	5.06465	0.940481	0.470241	+	0.882538i	$0.344167\pi$
$30$	−1.95185	−0.356357
$31$	−0.109003	−0.0195775	−0.00978875	−	0.999952i	$-0.503116\pi$
$31$	−0.109003	−0.0195775	−0.00978875	+	0.999952i	$0.503116\pi$
$32$	−1.00000	−0.176777
$33$	1.25954	0.219257
$34$	−6.54436	−1.12235
$35$	−0.894034	−0.151119
$36$	1.00000	0.166667
$37$	−0.926128	−0.152254	−0.0761272	−	0.997098i	$-0.524256\pi$
$37$	−0.926128	−0.152254	−0.0761272	+	0.997098i	$0.524256\pi$
$38$	−2.11970	−0.343861
$39$	5.60149	0.896957
$40$	−1.95185	−0.308614
$41$	−9.83955	−1.53668	−0.768340	−	0.640042i	$-0.778917\pi$
$41$	−9.83955	−1.53668	−0.768340	+	0.640042i	$0.778917\pi$
$42$	0.458044	0.0706778
$43$	6.77428	1.03307	0.516534	−	0.856267i	$-0.327222\pi$
$43$	6.77428	1.03307	0.516534	+	0.856267i	$0.327222\pi$
$44$	1.25954	0.189882
$45$	1.95185	0.290965
$46$	0	0
$47$	−2.50740	−0.365742	−0.182871	−	0.983137i	$-0.558539\pi$
$47$	−2.50740	−0.365742	−0.182871	+	0.983137i	$0.558539\pi$
$48$	1.00000	0.144338
$49$	−6.79020	−0.970028
$50$	1.19028	0.168332
$51$	6.54436	0.916394
$52$	5.60149	0.776787
$53$	−2.64612	−0.363472	−0.181736	−	0.983347i	$-0.558172\pi$
$53$	−2.64612	−0.363472	−0.181736	+	0.983347i	$0.558172\pi$
$54$	−1.00000	−0.136083
$55$	2.45843	0.331494
$56$	0.458044	0.0612088
$57$	2.11970	0.280762
$58$	−5.06465	−0.665021
$59$	−4.27686	−0.556800	−0.278400	−	0.960465i	$-0.589804\pi$
$59$	−4.27686	−0.556800	−0.278400	+	0.960465i	$0.589804\pi$
$60$	1.95185	0.251983
$61$	−8.58595	−1.09932	−0.549659	−	0.835389i	$-0.685242\pi$
$61$	−8.58595	−1.09932	−0.549659	+	0.835389i	$0.685242\pi$
$62$	0.109003	0.0138434
$63$	−0.458044	−0.0577082
$64$	1.00000	0.125000
$65$	10.9333	1.35611
$66$	−1.25954	−0.155038
$67$	−15.0332	−1.83660	−0.918298	−	0.395890i	$-0.870436\pi$
$67$	−15.0332	−1.83660	−0.918298	+	0.395890i	$0.870436\pi$
$68$	6.54436	0.793621
$69$	0	0
$70$	0.894034	0.106857
$71$	−0.303444	−0.0360122	−0.0180061	−	0.999838i	$-0.505732\pi$
$71$	−0.303444	−0.0360122	−0.0180061	+	0.999838i	$0.505732\pi$
$72$	−1.00000	−0.117851
$73$	3.34620	0.391643	0.195822	−	0.980640i	$-0.437263\pi$
$73$	3.34620	0.391643	0.195822	+	0.980640i	$0.437263\pi$
$74$	0.926128	0.107660
$75$	−1.19028	−0.137442
$76$	2.11970	0.243147
$77$	−0.576924	−0.0657466
$78$	−5.60149	−0.634244
$79$	−8.04741	−0.905405	−0.452702	−	0.891662i	$-0.649540\pi$
$79$	−8.04741	−0.905405	−0.452702	+	0.891662i	$0.649540\pi$
$80$	1.95185	0.218223
$81$	1.00000	0.111111
$82$	9.83955	1.08660
$83$	10.4920	1.15165	0.575825	−	0.817573i	$-0.304681\pi$
$83$	10.4920	1.15165	0.575825	+	0.817573i	$0.304681\pi$
$84$	−0.458044	−0.0499767
$85$	12.7736	1.38549
$86$	−6.77428	−0.730489
$87$	5.06465	0.542987
$88$	−1.25954	−0.134267
$89$	5.95710	0.631451	0.315726	−	0.948851i	$-0.397752\pi$
$89$	5.95710	0.631451	0.315726	+	0.948851i	$0.397752\pi$
$90$	−1.95185	−0.205743
$91$	−2.56573	−0.268962
$92$	0	0
$93$	−0.109003	−0.0113031
$94$	2.50740	0.258619
$95$	4.13734	0.424482
$96$	−1.00000	−0.102062
$97$	18.2589	1.85391	0.926953	−	0.375177i	$-0.122418\pi$
$97$	18.2589	1.85391	0.926953	+	0.375177i	$0.122418\pi$
$98$	6.79020	0.685913
$99$	1.25954	0.126588
$100$	−1.19028	−0.119028
$101$	10.6597	1.06068	0.530341	−	0.847785i	$-0.322064\pi$
$101$	10.6597	1.06068	0.530341	+	0.847785i	$0.322064\pi$
$102$	−6.54436	−0.647988
$103$	7.07465	0.697086	0.348543	−	0.937293i	$-0.386677\pi$
$103$	7.07465	0.697086	0.348543	+	0.937293i	$0.386677\pi$
$104$	−5.60149	−0.549272
$105$	−0.894034	−0.0872488
$106$	2.64612	0.257014
$107$	−2.76376	−0.267182	−0.133591	−	0.991037i	$-0.542651\pi$
$107$	−2.76376	−0.267182	−0.133591	+	0.991037i	$0.542651\pi$
$108$	1.00000	0.0962250
$109$	16.4960	1.58003	0.790013	−	0.613090i	$-0.210074\pi$
$109$	16.4960	1.58003	0.790013	+	0.613090i	$0.210074\pi$
$110$	−2.45843	−0.234402
$111$	−0.926128	−0.0879042
$112$	−0.458044	−0.0432811
$113$	15.3359	1.44268	0.721342	−	0.692579i	$-0.243526\pi$
$113$	15.3359	1.44268	0.721342	+	0.692579i	$0.243526\pi$
$114$	−2.11970	−0.198529
$115$	0	0
$116$	5.06465	0.470241
$117$	5.60149	0.517858
$118$	4.27686	0.393717
$119$	−2.99761	−0.274790
$120$	−1.95185	−0.178179
$121$	−9.41356	−0.855779
$122$	8.58595	0.777335
$123$	−9.83955	−0.887203
$124$	−0.109003	−0.00978875
$125$	−12.0825	−1.08069
$126$	0.458044	0.0408058
$127$	−21.6633	−1.92231	−0.961155	−	0.276010i	$-0.910988\pi$
$127$	−21.6633	−1.92231	−0.961155	+	0.276010i	$0.910988\pi$
$128$	−1.00000	−0.0883883
$129$	6.77428	0.596442
$130$	−10.9333	−0.958911
$131$	17.1468	1.49812	0.749062	−	0.662500i	$-0.230505\pi$
$131$	17.1468	1.49812	0.749062	+	0.662500i	$0.230505\pi$
$132$	1.25954	0.109629
$133$	−0.970919	−0.0841893
$134$	15.0332	1.29867
$135$	1.95185	0.167988
$136$	−6.54436	−0.561174
$137$	−3.10334	−0.265136	−0.132568	−	0.991174i	$-0.542322\pi$
$137$	−3.10334	−0.265136	−0.132568	+	0.991174i	$0.542322\pi$
$138$	0	0
$139$	−22.4485	−1.90406	−0.952028	−	0.306011i	$-0.901006\pi$
$139$	−22.4485	−1.90406	−0.952028	+	0.306011i	$0.901006\pi$
$140$	−0.894034	−0.0755596
$141$	−2.50740	−0.211161
$142$	0.303444	0.0254644
$143$	7.05529	0.589993
$144$	1.00000	0.0833333
$145$	9.88543	0.820940
$146$	−3.34620	−0.276933
$147$	−6.79020	−0.560046
$148$	−0.926128	−0.0761272
$149$	15.7100	1.28701	0.643507	−	0.765440i	$-0.277479\pi$
$149$	15.7100	1.28701	0.643507	+	0.765440i	$0.277479\pi$
$150$	1.19028	0.0971863
$151$	−5.57516	−0.453700	−0.226850	−	0.973930i	$-0.572843\pi$
$151$	−5.57516	−0.453700	−0.226850	+	0.973930i	$0.572843\pi$
$152$	−2.11970	−0.171931
$153$	6.54436	0.529080
$154$	0.576924	0.0464899
$155$	−0.212757	−0.0170891
$156$	5.60149	0.448478
$157$	−8.16701	−0.651799	−0.325899	−	0.945404i	$-0.605667\pi$
$157$	−8.16701	−0.651799	−0.325899	+	0.945404i	$0.605667\pi$
$158$	8.04741	0.640218
$159$	−2.64612	−0.209851
$160$	−1.95185	−0.154307
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−17.0615	−1.33636	−0.668181	−	0.743999i	$-0.732927\pi$
$163$	−17.0615	−1.33636	−0.668181	+	0.743999i	$0.732927\pi$
$164$	−9.83955	−0.768340
$165$	2.45843	0.191388
$166$	−10.4920	−0.814340
$167$	18.0693	1.39825	0.699123	−	0.715001i	$-0.253574\pi$
$167$	18.0693	1.39825	0.699123	+	0.715001i	$0.253574\pi$
$168$	0.458044	0.0353389
$169$	18.3767	1.41359
$170$	−12.7736	−0.979691
$171$	2.11970	0.162098
$172$	6.77428	0.516534
$173$	−15.9612	−1.21350	−0.606752	−	0.794891i	$-0.707528\pi$
$173$	−15.9612	−1.21350	−0.606752	+	0.794891i	$0.707528\pi$
$174$	−5.06465	−0.383950
$175$	0.545203	0.0412135
$176$	1.25954	0.0949412
$177$	−4.27686	−0.321468
$178$	−5.95710	−0.446503
$179$	−15.6916	−1.17284	−0.586421	−	0.810006i	$-0.699464\pi$
$179$	−15.6916	−1.17284	−0.586421	+	0.810006i	$0.699464\pi$
$180$	1.95185	0.145482
$181$	−1.81930	−0.135228	−0.0676139	−	0.997712i	$-0.521539\pi$
$181$	−1.81930	−0.135228	−0.0676139	+	0.997712i	$0.521539\pi$
$182$	2.56573	0.190185
$183$	−8.58595	−0.634692
$184$	0	0
$185$	−1.80766	−0.132902
$186$	0.109003	0.00799248
$187$	8.24287	0.602779
$188$	−2.50740	−0.182871
$189$	−0.458044	−0.0333178
$190$	−4.13734	−0.300154
$191$	4.24849	0.307410	0.153705	−	0.988117i	$-0.450880\pi$
$191$	4.24849	0.307410	0.153705	+	0.988117i	$0.450880\pi$
$192$	1.00000	0.0721688
$193$	13.1121	0.943828	0.471914	−	0.881645i	$-0.343563\pi$
$193$	13.1121	0.943828	0.471914	+	0.881645i	$0.343563\pi$
$194$	−18.2589	−1.31091
$195$	10.9333	0.782948
$196$	−6.79020	−0.485014
$197$	−9.92381	−0.707042	−0.353521	−	0.935427i	$-0.615016\pi$
$197$	−9.92381	−0.707042	−0.353521	+	0.935427i	$0.615016\pi$
$198$	−1.25954	−0.0895115
$199$	6.30555	0.446989	0.223494	−	0.974705i	$-0.428254\pi$
$199$	6.30555	0.446989	0.223494	+	0.974705i	$0.428254\pi$
$200$	1.19028	0.0841658
$201$	−15.0332	−1.06036
$202$	−10.6597	−0.750015
$203$	−2.31983	−0.162820
$204$	6.54436	0.458197
$205$	−19.2053	−1.34136
$206$	−7.07465	−0.492914
$207$	0	0
$208$	5.60149	0.388394
$209$	2.66985	0.184677
$210$	0.894034	0.0616942
$211$	25.0546	1.72483	0.862415	−	0.506202i	$-0.168951\pi$
$211$	25.0546	1.72483	0.862415	+	0.506202i	$0.168951\pi$
$212$	−2.64612	−0.181736
$213$	−0.303444	−0.0207916
$214$	2.76376	0.188926
$215$	13.2224	0.901758
$216$	−1.00000	−0.0680414
$217$	0.0499282	0.00338935
$218$	−16.4960	−1.11725
$219$	3.34620	0.226115
$220$	2.45843	0.165747
$221$	36.6582	2.46590
$222$	0.926128	0.0621576
$223$	1.78108	0.119270	0.0596348	−	0.998220i	$-0.481006\pi$
$223$	1.78108	0.119270	0.0596348	+	0.998220i	$0.481006\pi$
$224$	0.458044	0.0306044
$225$	−1.19028	−0.0793523
$226$	−15.3359	−1.02013
$227$	−20.7093	−1.37453	−0.687264	−	0.726408i	$-0.741188\pi$
$227$	−20.7093	−1.37453	−0.687264	+	0.726408i	$0.741188\pi$
$228$	2.11970	0.140381
$229$	8.62288	0.569816	0.284908	−	0.958555i	$-0.408037\pi$
$229$	8.62288	0.569816	0.284908	+	0.958555i	$0.408037\pi$
$230$	0	0
$231$	−0.576924	−0.0379588
$232$	−5.06465	−0.332510
$233$	4.77637	0.312910	0.156455	−	0.987685i	$-0.449993\pi$
$233$	4.77637	0.312910	0.156455	+	0.987685i	$0.449993\pi$
$234$	−5.60149	−0.366181
$235$	−4.89407	−0.319254
$236$	−4.27686	−0.278400
$237$	−8.04741	−0.522736
$238$	2.99761	0.194306
$239$	−23.9090	−1.54655	−0.773273	−	0.634073i	$-0.781382\pi$
$239$	−23.9090	−1.54655	−0.773273	+	0.634073i	$0.781382\pi$
$240$	1.95185	0.125991
$241$	−23.0500	−1.48478	−0.742389	−	0.669969i	$-0.766307\pi$
$241$	−23.0500	−1.48478	−0.742389	+	0.669969i	$0.766307\pi$
$242$	9.41356	0.605127
$243$	1.00000	0.0641500
$244$	−8.58595	−0.549659
$245$	−13.2534	−0.846731
$246$	9.83955	0.627347
$247$	11.8735	0.755493
$248$	0.109003	0.00692169
$249$	10.4920	0.664906
$250$	12.0825	0.764165
$251$	−0.221255	−0.0139655	−0.00698273	−	0.999976i	$-0.502223\pi$
$251$	−0.221255	−0.0139655	−0.00698273	+	0.999976i	$0.502223\pi$
$252$	−0.458044	−0.0288541
$253$	0	0
$254$	21.6633	1.35928
$255$	12.7736	0.799914
$256$	1.00000	0.0625000
$257$	−0.544992	−0.0339957	−0.0169978	−	0.999856i	$-0.505411\pi$
$257$	−0.544992	−0.0339957	−0.0169978	+	0.999856i	$0.505411\pi$
$258$	−6.77428	−0.421748
$259$	0.424208	0.0263590
$260$	10.9333	0.678053
$261$	5.06465	0.313494
$262$	−17.1468	−1.05933
$263$	14.7270	0.908104	0.454052	−	0.890975i	$-0.349978\pi$
$263$	14.7270	0.908104	0.454052	+	0.890975i	$0.349978\pi$
$264$	−1.25954	−0.0775192
$265$	−5.16482	−0.317272
$266$	0.970919	0.0595309
$267$	5.95710	0.364568
$268$	−15.0332	−0.918298
$269$	1.00716	0.0614078	0.0307039	−	0.999529i	$-0.490225\pi$
$269$	1.00716	0.0614078	0.0307039	+	0.999529i	$0.490225\pi$
$270$	−1.95185	−0.118786
$271$	−19.9145	−1.20972	−0.604860	−	0.796332i	$-0.706771\pi$
$271$	−19.9145	−1.20972	−0.604860	+	0.796332i	$0.706771\pi$
$272$	6.54436	0.396810
$273$	−2.56573	−0.155285
$274$	3.10334	0.187479
$275$	−1.49921	−0.0904057
$276$	0	0
$277$	−24.4934	−1.47167	−0.735834	−	0.677162i	$-0.763209\pi$
$277$	−24.4934	−1.47167	−0.735834	+	0.677162i	$0.763209\pi$
$278$	22.4485	1.34637
$279$	−0.109003	−0.00652584
$280$	0.894034	0.0534287
$281$	22.2104	1.32496	0.662480	−	0.749080i	$-0.269504\pi$
$281$	22.2104	1.32496	0.662480	+	0.749080i	$0.269504\pi$
$282$	2.50740	0.149314
$283$	16.4220	0.976185	0.488093	−	0.872792i	$-0.337693\pi$
$283$	16.4220	0.976185	0.488093	+	0.872792i	$0.337693\pi$
$284$	−0.303444	−0.0180061
$285$	4.13734	0.245075
$286$	−7.05529	−0.417188
$287$	4.50695	0.266037
$288$	−1.00000	−0.0589256
$289$	25.8287	1.51933
$290$	−9.88543	−0.580492
$291$	18.2589	1.07035
$292$	3.34620	0.195822
$293$	24.4969	1.43112	0.715561	−	0.698550i	$-0.246171\pi$
$293$	24.4969	1.43112	0.715561	+	0.698550i	$0.246171\pi$
$294$	6.79020	0.396012
$295$	−8.34778	−0.486027
$296$	0.926128	0.0538301
$297$	1.25954	0.0730858
$298$	−15.7100	−0.910057
$299$	0	0
$300$	−1.19028	−0.0687211
$301$	−3.10292	−0.178849
$302$	5.57516	0.320815
$303$	10.6597	0.612385
$304$	2.11970	0.121573
$305$	−16.7585	−0.959588
$306$	−6.54436	−0.374116
$307$	14.6749	0.837539	0.418769	−	0.908093i	$-0.362462\pi$
$307$	14.6749	0.837539	0.418769	+	0.908093i	$0.362462\pi$
$308$	−0.576924	−0.0328733
$309$	7.07465	0.402463
$310$	0.212757	0.0120838
$311$	33.2393	1.88483	0.942414	−	0.334449i	$-0.108550\pi$
$311$	33.2393	1.88483	0.942414	+	0.334449i	$0.108550\pi$
$312$	−5.60149	−0.317122
$313$	−20.4935	−1.15836	−0.579181	−	0.815199i	$-0.696627\pi$
$313$	−20.4935	−1.15836	−0.579181	+	0.815199i	$0.696627\pi$
$314$	8.16701	0.460891
$315$	−0.894034	−0.0503731
$316$	−8.04741	−0.452702
$317$	22.9004	1.28622	0.643108	−	0.765775i	$-0.277645\pi$
$317$	22.9004	1.28622	0.643108	+	0.765775i	$0.277645\pi$
$318$	2.64612	0.148387
$319$	6.37911	0.357162
$320$	1.95185	0.109112
$321$	−2.76376	−0.154258
$322$	0	0
$323$	13.8721	0.771865
$324$	1.00000	0.0555556
$325$	−6.66737	−0.369839
$326$	17.0615	0.944950
$327$	16.4960	0.912228
$328$	9.83955	0.543298
$329$	1.14850	0.0633190
$330$	−2.45843	−0.135332
$331$	−10.2762	−0.564831	−0.282416	−	0.959292i	$-0.591136\pi$
$331$	−10.2762	−0.564831	−0.282416	+	0.959292i	$0.591136\pi$
$332$	10.4920	0.575825
$333$	−0.926128	−0.0507515
$334$	−18.0693	−0.988709
$335$	−29.3425	−1.60315
$336$	−0.458044	−0.0249884
$337$	11.6162	0.632773	0.316386	−	0.948630i	$-0.397530\pi$
$337$	11.6162	0.632773	0.316386	+	0.948630i	$0.397530\pi$
$338$	−18.3767	−0.999562
$339$	15.3359	0.832934
$340$	12.7736	0.692746
$341$	−0.137293	−0.00743485
$342$	−2.11970	−0.114620
$343$	6.31652	0.341060
$344$	−6.77428	−0.365245
$345$	0	0
$346$	15.9612	0.858077
$347$	−12.8984	−0.692424	−0.346212	−	0.938156i	$-0.612532\pi$
$347$	−12.8984	−0.692424	−0.346212	+	0.938156i	$0.612532\pi$
$348$	5.06465	0.271494
$349$	15.8619	0.849070	0.424535	−	0.905412i	$-0.360438\pi$
$349$	15.8619	0.849070	0.424535	+	0.905412i	$0.360438\pi$
$350$	−0.545203	−0.0291423
$351$	5.60149	0.298986
$352$	−1.25954	−0.0671336
$353$	−1.79503	−0.0955397	−0.0477698	−	0.998858i	$-0.515211\pi$
$353$	−1.79503	−0.0955397	−0.0477698	+	0.998858i	$0.515211\pi$
$354$	4.27686	0.227312
$355$	−0.592277	−0.0314348
$356$	5.95710	0.315726
$357$	−2.99761	−0.158650
$358$	15.6916	0.829325
$359$	19.0479	1.00531	0.502655	−	0.864487i	$-0.332357\pi$
$359$	19.0479	1.00531	0.502655	+	0.864487i	$0.332357\pi$
$360$	−1.95185	−0.102871
$361$	−14.5069	−0.763519
$362$	1.81930	0.0956205
$363$	−9.41356	−0.494084
$364$	−2.56573	−0.134481
$365$	6.53128	0.341863
$366$	8.58595	0.448795
$367$	−3.88864	−0.202985	−0.101493	−	0.994836i	$-0.532362\pi$
$367$	−3.88864	−0.202985	−0.101493	+	0.994836i	$0.532362\pi$
$368$	0	0
$369$	−9.83955	−0.512227
$370$	1.80766	0.0939759
$371$	1.21204	0.0629259
$372$	−0.109003	−0.00565154
$373$	−31.2555	−1.61835	−0.809175	−	0.587567i	$-0.800086\pi$
$373$	−31.2555	−1.61835	−0.809175	+	0.587567i	$0.800086\pi$
$374$	−8.24287	−0.426229
$375$	−12.0825	−0.623938
$376$	2.50740	0.129309
$377$	28.3696	1.46111
$378$	0.458044	0.0235593
$379$	−7.32134	−0.376072	−0.188036	−	0.982162i	$-0.560212\pi$
$379$	−7.32134	−0.376072	−0.188036	+	0.982162i	$0.560212\pi$
$380$	4.13734	0.212241
$381$	−21.6633	−1.10985
$382$	−4.24849	−0.217371
$383$	19.2426	0.983253	0.491627	−	0.870806i	$-0.336402\pi$
$383$	19.2426	0.983253	0.491627	+	0.870806i	$0.336402\pi$
$384$	−1.00000	−0.0510310
$385$	−1.12607	−0.0573898
$386$	−13.1121	−0.667387
$387$	6.77428	0.344356
$388$	18.2589	0.926953
$389$	8.25813	0.418704	0.209352	−	0.977840i	$-0.432865\pi$
$389$	8.25813	0.418704	0.209352	+	0.977840i	$0.432865\pi$
$390$	−10.9333	−0.553628
$391$	0	0
$392$	6.79020	0.342957
$393$	17.1468	0.864942
$394$	9.92381	0.499954
$395$	−15.7073	−0.790322
$396$	1.25954	0.0632942
$397$	−11.3695	−0.570620	−0.285310	−	0.958435i	$-0.592096\pi$
$397$	−11.3695	−0.570620	−0.285310	+	0.958435i	$0.592096\pi$
$398$	−6.30555	−0.316069
$399$	−0.970919	−0.0486067
$400$	−1.19028	−0.0595142
$401$	15.2135	0.759728	0.379864	−	0.925042i	$-0.375971\pi$
$401$	15.2135	0.759728	0.379864	+	0.925042i	$0.375971\pi$
$402$	15.0332	0.749787
$403$	−0.610579	−0.0304151
$404$	10.6597	0.530341
$405$	1.95185	0.0969882
$406$	2.31983	0.115131
$407$	−1.16649	−0.0578209
$408$	−6.54436	−0.323994
$409$	1.53365	0.0758340	0.0379170	−	0.999281i	$-0.487928\pi$
$409$	1.53365	0.0758340	0.0379170	+	0.999281i	$0.487928\pi$
$410$	19.2053	0.948483
$411$	−3.10334	−0.153076
$412$	7.07465	0.348543
$413$	1.95899	0.0963956
$414$	0	0
$415$	20.4789	1.00527
$416$	−5.60149	−0.274636
$417$	−22.4485	−1.09931
$418$	−2.66985	−0.130587
$419$	4.63094	0.226236	0.113118	−	0.993582i	$-0.463916\pi$
$419$	4.63094	0.226236	0.113118	+	0.993582i	$0.463916\pi$
$420$	−0.894034	−0.0436244
$421$	−2.59683	−0.126562	−0.0632809	−	0.997996i	$-0.520156\pi$
$421$	−2.59683	−0.126562	−0.0632809	+	0.997996i	$0.520156\pi$
$422$	−25.0546	−1.21964
$423$	−2.50740	−0.121914
$424$	2.64612	0.128507
$425$	−7.78965	−0.377854
$426$	0.303444	0.0147019
$427$	3.93275	0.190319
$428$	−2.76376	−0.133591
$429$	7.05529	0.340633
$430$	−13.2224	−0.637639
$431$	−9.94166	−0.478873	−0.239436	−	0.970912i	$-0.576963\pi$
$431$	−9.94166	−0.478873	−0.239436	+	0.970912i	$0.576963\pi$
$432$	1.00000	0.0481125
$433$	−21.5133	−1.03386	−0.516931	−	0.856027i	$-0.672926\pi$
$433$	−21.5133	−1.03386	−0.516931	+	0.856027i	$0.672926\pi$
$434$	−0.0499282	−0.00239663
$435$	9.88543	0.473970
$436$	16.4960	0.790013
$437$	0	0
$438$	−3.34620	−0.159888
$439$	−25.3531	−1.21004	−0.605019	−	0.796211i	$-0.706834\pi$
$439$	−25.3531	−1.21004	−0.605019	+	0.796211i	$0.706834\pi$
$440$	−2.45843	−0.117201
$441$	−6.79020	−0.323343
$442$	−36.6582	−1.74365
$443$	−6.75984	−0.321170	−0.160585	−	0.987022i	$-0.551338\pi$
$443$	−6.75984	−0.321170	−0.160585	+	0.987022i	$0.551338\pi$
$444$	−0.926128	−0.0439521
$445$	11.6274	0.551190
$446$	−1.78108	−0.0843364
$447$	15.7100	0.743058
$448$	−0.458044	−0.0216406
$449$	7.80490	0.368336	0.184168	−	0.982895i	$-0.441041\pi$
$449$	7.80490	0.368336	0.184168	+	0.982895i	$0.441041\pi$
$450$	1.19028	0.0561106
$451$	−12.3933	−0.583577
$452$	15.3359	0.721342
$453$	−5.57516	−0.261944
$454$	20.7093	0.971938
$455$	−5.00792	−0.234775
$456$	−2.11970	−0.0992643
$457$	9.29867	0.434973	0.217487	−	0.976063i	$-0.430214\pi$
$457$	9.29867	0.434973	0.217487	+	0.976063i	$0.430214\pi$
$458$	−8.62288	−0.402921
$459$	6.54436	0.305465
$460$	0	0
$461$	−17.2422	−0.803049	−0.401525	−	0.915848i	$-0.631520\pi$
$461$	−17.2422	−0.803049	−0.401525	+	0.915848i	$0.631520\pi$
$462$	0.576924	0.0268409
$463$	−9.03520	−0.419901	−0.209951	−	0.977712i	$-0.567330\pi$
$463$	−9.03520	−0.419901	−0.209951	+	0.977712i	$0.567330\pi$
$464$	5.06465	0.235120
$465$	−0.212757	−0.00986638
$466$	−4.77637	−0.221261
$467$	−10.5457	−0.487996	−0.243998	−	0.969776i	$-0.578459\pi$
$467$	−10.5457	−0.487996	−0.243998	+	0.969776i	$0.578459\pi$
$468$	5.60149	0.258929
$469$	6.88587	0.317960
$470$	4.89407	0.225747
$471$	−8.16701	−0.376316
$472$	4.27686	0.196858
$473$	8.53246	0.392323
$474$	8.04741	0.369630
$475$	−2.52305	−0.115766
$476$	−2.99761	−0.137395
$477$	−2.64612	−0.121157
$478$	23.9090	1.09357
$479$	−13.1682	−0.601670	−0.300835	−	0.953676i	$-0.597265\pi$
$479$	−13.1682	−0.601670	−0.300835	+	0.953676i	$0.597265\pi$
$480$	−1.95185	−0.0890893
$481$	−5.18770	−0.236539
$482$	23.0500	1.04990
$483$	0	0
$484$	−9.41356	−0.427889
$485$	35.6385	1.61826
$486$	−1.00000	−0.0453609
$487$	−3.03122	−0.137358	−0.0686789	−	0.997639i	$-0.521878\pi$
$487$	−3.03122	−0.137358	−0.0686789	+	0.997639i	$0.521878\pi$
$488$	8.58595	0.388668
$489$	−17.0615	−0.771549
$490$	13.2534	0.598729
$491$	6.30567	0.284571	0.142285	−	0.989826i	$-0.454555\pi$
$491$	6.30567	0.284571	0.142285	+	0.989826i	$0.454555\pi$
$492$	−9.83955	−0.443601
$493$	33.1449	1.49277
$494$	−11.8735	−0.534214
$495$	2.45843	0.110498
$496$	−0.109003	−0.00489438
$497$	0.138991	0.00623459
$498$	−10.4920	−0.470160
$499$	−6.78493	−0.303735	−0.151868	−	0.988401i	$-0.548529\pi$
$499$	−6.78493	−0.303735	−0.151868	+	0.988401i	$0.548529\pi$
$500$	−12.0825	−0.540346
$501$	18.0693	0.807278
$502$	0.221255	0.00987508
$503$	−35.9779	−1.60418	−0.802088	−	0.597206i	$-0.796277\pi$
$503$	−35.9779	−1.60418	−0.802088	+	0.597206i	$0.796277\pi$
$504$	0.458044	0.0204029
$505$	20.8062	0.925862
$506$	0	0
$507$	18.3767	0.816139
$508$	−21.6633	−0.961155
$509$	12.0660	0.534817	0.267409	−	0.963583i	$-0.413833\pi$
$509$	12.0660	0.534817	0.267409	+	0.963583i	$0.413833\pi$
$510$	−12.7736	−0.565625
$511$	−1.53271	−0.0678030
$512$	−1.00000	−0.0441942
$513$	2.11970	0.0935872
$514$	0.544992	0.0240386
$515$	13.8086	0.608481
$516$	6.77428	0.298221
$517$	−3.15817	−0.138896
$518$	−0.424208	−0.0186386
$519$	−15.9612	−0.700617
$520$	−10.9333	−0.479456
$521$	−0.654256	−0.0286635	−0.0143317	−	0.999897i	$-0.504562\pi$
$521$	−0.654256	−0.0286635	−0.0143317	+	0.999897i	$0.504562\pi$
$522$	−5.06465	−0.221674
$523$	−37.4241	−1.63644	−0.818220	−	0.574905i	$-0.805039\pi$
$523$	−37.4241	−1.63644	−0.818220	+	0.574905i	$0.805039\pi$
$524$	17.1468	0.749062
$525$	0.545203	0.0237946
$526$	−14.7270	−0.642127
$527$	−0.713355	−0.0310742
$528$	1.25954	0.0548144
$529$	0	0
$530$	5.16482	0.224346
$531$	−4.27686	−0.185600
$532$	−0.970919	−0.0420947
$533$	−55.1162	−2.38735
$534$	−5.95710	−0.257789
$535$	−5.39443	−0.233222
$536$	15.0332	0.649335
$537$	−15.6916	−0.677141
$538$	−1.00716	−0.0434219
$539$	−8.55251	−0.368383
$540$	1.95185	0.0839942
$541$	26.4762	1.13830	0.569151	−	0.822233i	$-0.307272\pi$
$541$	26.4762	1.13830	0.569151	+	0.822233i	$0.307272\pi$
$542$	19.9145	0.855401
$543$	−1.81930	−0.0780738
$544$	−6.54436	−0.280587
$545$	32.1976	1.37919
$546$	2.56573	0.109803
$547$	−6.74099	−0.288224	−0.144112	−	0.989561i	$-0.546033\pi$
$547$	−6.74099	−0.288224	−0.144112	+	0.989561i	$0.546033\pi$
$548$	−3.10334	−0.132568
$549$	−8.58595	−0.366439
$550$	1.49921	0.0639265
$551$	10.7356	0.457350
$552$	0	0
$553$	3.68607	0.156748
$554$	24.4934	1.04063
$555$	−1.80766	−0.0767310
$556$	−22.4485	−0.952028
$557$	−18.8827	−0.800087	−0.400043	−	0.916496i	$-0.631005\pi$
$557$	−18.8827	−0.800087	−0.400043	+	0.916496i	$0.631005\pi$
$558$	0.109003	0.00461446
$559$	37.9461	1.60495
$560$	−0.894034	−0.0377798
$561$	8.24287	0.348014
$562$	−22.2104	−0.936888
$563$	−3.81476	−0.160773	−0.0803864	−	0.996764i	$-0.525615\pi$
$563$	−3.81476	−0.160773	−0.0803864	+	0.996764i	$0.525615\pi$
$564$	−2.50740	−0.105581
$565$	29.9335	1.25931
$566$	−16.4220	−0.690267
$567$	−0.458044	−0.0192361
$568$	0.303444	0.0127322
$569$	−18.6062	−0.780014	−0.390007	−	0.920812i	$-0.627527\pi$
$569$	−18.6062	−0.780014	−0.390007	+	0.920812i	$0.627527\pi$
$570$	−4.13734	−0.173294
$571$	−42.9221	−1.79623	−0.898117	−	0.439757i	$-0.855065\pi$
$571$	−42.9221	−1.79623	−0.898117	+	0.439757i	$0.855065\pi$
$572$	7.05529	0.294997
$573$	4.24849	0.177483
$574$	−4.50695	−0.188117
$575$	0	0
$576$	1.00000	0.0416667
$577$	−11.4723	−0.477599	−0.238799	−	0.971069i	$-0.576754\pi$
$577$	−11.4723	−0.477599	−0.238799	+	0.971069i	$0.576754\pi$
$578$	−25.8287	−1.07433
$579$	13.1121	0.544919
$580$	9.88543	0.410470
$581$	−4.80582	−0.199379
$582$	−18.2589	−0.756854
$583$	−3.33288	−0.138034
$584$	−3.34620	−0.138467
$585$	10.9333	0.452035
$586$	−24.4969	−1.01196
$587$	11.0101	0.454436	0.227218	−	0.973844i	$-0.427037\pi$
$587$	11.0101	0.454436	0.227218	+	0.973844i	$0.427037\pi$
$588$	−6.79020	−0.280023
$589$	−0.231054	−0.00952041
$590$	8.34778	0.343673
$591$	−9.92381	−0.408211
$592$	−0.926128	−0.0380636
$593$	17.4334	0.715904	0.357952	−	0.933740i	$-0.383475\pi$
$593$	17.4334	0.715904	0.357952	+	0.933740i	$0.383475\pi$
$594$	−1.25954	−0.0516795
$595$	−5.85088	−0.239863
$596$	15.7100	0.643507
$597$	6.30555	0.258069
$598$	0	0
$599$	−11.8571	−0.484470	−0.242235	−	0.970218i	$-0.577880\pi$
$599$	−11.8571	−0.484470	−0.242235	+	0.970218i	$0.577880\pi$
$600$	1.19028	0.0485932
$601$	5.00627	0.204210	0.102105	−	0.994774i	$-0.467442\pi$
$601$	5.00627	0.204210	0.102105	+	0.994774i	$0.467442\pi$
$602$	3.10292	0.126466
$603$	−15.0332	−0.612199
$604$	−5.57516	−0.226850
$605$	−18.3739	−0.747004
$606$	−10.6597	−0.433021
$607$	31.9481	1.29673	0.648367	−	0.761328i	$-0.275452\pi$
$607$	31.9481	1.29673	0.648367	+	0.761328i	$0.275452\pi$
$608$	−2.11970	−0.0859654
$609$	−2.31983	−0.0940044
$610$	16.7585	0.678531
$611$	−14.0452	−0.568208
$612$	6.54436	0.264540
$613$	−30.3307	−1.22504	−0.612522	−	0.790454i	$-0.709845\pi$
$613$	−30.3307	−1.22504	−0.612522	+	0.790454i	$0.709845\pi$
$614$	−14.6749	−0.592229
$615$	−19.2053	−0.774433
$616$	0.576924	0.0232449
$617$	10.9864	0.442294	0.221147	−	0.975241i	$-0.429020\pi$
$617$	10.9864	0.442294	0.221147	+	0.975241i	$0.429020\pi$
$618$	−7.07465	−0.284584
$619$	−11.7786	−0.473422	−0.236711	−	0.971580i	$-0.576070\pi$
$619$	−11.7786	−0.473422	−0.236711	+	0.971580i	$0.576070\pi$
$620$	−0.212757	−0.00854454
$621$	0	0
$622$	−33.2393	−1.33277
$623$	−2.72862	−0.109320
$624$	5.60149	0.224239
$625$	−17.6318	−0.705272
$626$	20.4935	0.819085
$627$	2.66985	0.106623
$628$	−8.16701	−0.325899
$629$	−6.06092	−0.241665
$630$	0.894034	0.0356192
$631$	−11.6774	−0.464870	−0.232435	−	0.972612i	$-0.574669\pi$
$631$	−11.6774	−0.464870	−0.232435	+	0.972612i	$0.574669\pi$
$632$	8.04741	0.320109
$633$	25.0546	0.995831
$634$	−22.9004	−0.909492
$635$	−42.2836	−1.67797
$636$	−2.64612	−0.104925
$637$	−38.0352	−1.50701
$638$	−6.37911	−0.252552
$639$	−0.303444	−0.0120041
$640$	−1.95185	−0.0771536
$641$	−4.70626	−0.185886	−0.0929431	−	0.995671i	$-0.529627\pi$
$641$	−4.70626	−0.185886	−0.0929431	+	0.995671i	$0.529627\pi$
$642$	2.76376	0.109077
$643$	38.9574	1.53633	0.768165	−	0.640252i	$-0.221170\pi$
$643$	38.9574	1.53633	0.768165	+	0.640252i	$0.221170\pi$
$644$	0	0
$645$	13.2224	0.520630
$646$	−13.8721	−0.545791
$647$	42.8026	1.68274	0.841371	−	0.540457i	$-0.181749\pi$
$647$	42.8026	1.68274	0.841371	+	0.540457i	$0.181749\pi$
$648$	−1.00000	−0.0392837
$649$	−5.38686	−0.211453
$650$	6.66737	0.261516
$651$	0.0499282	0.00195684
$652$	−17.0615	−0.668181
$653$	−21.2091	−0.829976	−0.414988	−	0.909827i	$-0.636214\pi$
$653$	−21.2091	−0.829976	−0.414988	+	0.909827i	$0.636214\pi$
$654$	−16.4960	−0.645043
$655$	33.4680	1.30770
$656$	−9.83955	−0.384170
$657$	3.34620	0.130548
$658$	−1.14850	−0.0447733
$659$	−10.2226	−0.398215	−0.199107	−	0.979978i	$-0.563804\pi$
$659$	−10.2226	−0.398215	−0.199107	+	0.979978i	$0.563804\pi$
$660$	2.45843	0.0956942
$661$	39.9796	1.55503	0.777513	−	0.628867i	$-0.216481\pi$
$661$	39.9796	1.55503	0.777513	+	0.628867i	$0.216481\pi$
$662$	10.2762	0.399396
$663$	36.6582	1.42369
$664$	−10.4920	−0.407170
$665$	−1.89509	−0.0734883
$666$	0.926128	0.0358867
$667$	0	0
$668$	18.0693	0.699123
$669$	1.78108	0.0688604
$670$	29.3425	1.13360
$671$	−10.8143	−0.417483
$672$	0.458044	0.0176694
$673$	−6.90220	−0.266060	−0.133030	−	0.991112i	$-0.542471\pi$
$673$	−6.90220	−0.266060	−0.133030	+	0.991112i	$0.542471\pi$
$674$	−11.6162	−0.447438
$675$	−1.19028	−0.0458141
$676$	18.3767	0.706797
$677$	17.8035	0.684245	0.342122	−	0.939655i	$-0.388854\pi$
$677$	17.8035	0.684245	0.342122	+	0.939655i	$0.388854\pi$
$678$	−15.3359	−0.588973
$679$	−8.36337	−0.320957
$680$	−12.7736	−0.489846
$681$	−20.7093	−0.793584
$682$	0.137293	0.00525723
$683$	−51.9690	−1.98854	−0.994268	−	0.106913i	$-0.965903\pi$
$683$	−51.9690	−1.98854	−0.994268	+	0.106913i	$0.965903\pi$
$684$	2.11970	0.0810489
$685$	−6.05725	−0.231435
$686$	−6.31652	−0.241166
$687$	8.62288	0.328983
$688$	6.77428	0.258267
$689$	−14.8222	−0.564681
$690$	0	0
$691$	−26.8796	−1.02255	−0.511274	−	0.859418i	$-0.670826\pi$
$691$	−26.8796	−1.02255	−0.511274	+	0.859418i	$0.670826\pi$
$692$	−15.9612	−0.606752
$693$	−0.576924	−0.0219155
$694$	12.8984	0.489618
$695$	−43.8161	−1.66204
$696$	−5.06465	−0.191975
$697$	−64.3936	−2.43908
$698$	−15.8619	−0.600383
$699$	4.77637	0.180659
$700$	0.545203	0.0206067
$701$	5.09731	0.192523	0.0962614	−	0.995356i	$-0.469312\pi$
$701$	5.09731	0.192523	0.0962614	+	0.995356i	$0.469312\pi$
$702$	−5.60149	−0.211415
$703$	−1.96312	−0.0740404
$704$	1.25954	0.0474706
$705$	−4.89407	−0.184321
$706$	1.79503	0.0675567
$707$	−4.88262	−0.183630
$708$	−4.27686	−0.160734
$709$	0.828261	0.0311060	0.0155530	−	0.999879i	$-0.495049\pi$
$709$	0.828261	0.0311060	0.0155530	+	0.999879i	$0.495049\pi$
$710$	0.592277	0.0222277
$711$	−8.04741	−0.301802
$712$	−5.95710	−0.223252
$713$	0	0
$714$	2.99761	0.112183
$715$	13.7709	0.515001
$716$	−15.6916	−0.586421
$717$	−23.9090	−0.892899
$718$	−19.0479	−0.710862
$719$	4.75003	0.177146	0.0885731	−	0.996070i	$-0.471769\pi$
$719$	4.75003	0.177146	0.0885731	+	0.996070i	$0.471769\pi$
$720$	1.95185	0.0727411
$721$	−3.24050	−0.120683
$722$	14.5069	0.539889
$723$	−23.0500	−0.857237
$724$	−1.81930	−0.0676139
$725$	−6.02837	−0.223888
$726$	9.41356	0.349370
$727$	−31.3593	−1.16305	−0.581526	−	0.813528i	$-0.697544\pi$
$727$	−31.3593	−1.16305	−0.581526	+	0.813528i	$0.697544\pi$
$728$	2.56573	0.0950924
$729$	1.00000	0.0370370
$730$	−6.53128	−0.241733
$731$	44.3333	1.63973
$732$	−8.58595	−0.317346
$733$	11.3462	0.419081	0.209541	−	0.977800i	$-0.432803\pi$
$733$	11.3462	0.419081	0.209541	+	0.977800i	$0.432803\pi$
$734$	3.88864	0.143532
$735$	−13.2534	−0.488860
$736$	0	0
$737$	−18.9349	−0.697475
$738$	9.83955	0.362199
$739$	27.4253	1.00885	0.504427	−	0.863454i	$-0.331704\pi$
$739$	27.4253	1.00885	0.504427	+	0.863454i	$0.331704\pi$
$740$	−1.80766	−0.0664510
$741$	11.8735	0.436184
$742$	−1.21204	−0.0444954
$743$	−52.7833	−1.93643	−0.968216	−	0.250117i	$-0.919531\pi$
$743$	−52.7833	−1.93643	−0.968216	+	0.250117i	$0.919531\pi$
$744$	0.109003	0.00399624
$745$	30.6636	1.12343
$746$	31.2555	1.14435
$747$	10.4920	0.383884
$748$	8.24287	0.301389
$749$	1.26592	0.0462558
$750$	12.0825	0.441191
$751$	17.4346	0.636197	0.318098	−	0.948058i	$-0.396956\pi$
$751$	17.4346	0.636197	0.318098	+	0.948058i	$0.396956\pi$
$752$	−2.50740	−0.0914356
$753$	−0.221255	−0.00806297
$754$	−28.3696	−1.03316
$755$	−10.8819	−0.396032
$756$	−0.458044	−0.0166589
$757$	18.1347	0.659118	0.329559	−	0.944135i	$-0.393100\pi$
$757$	18.1347	0.659118	0.329559	+	0.944135i	$0.393100\pi$
$758$	7.32134	0.265923
$759$	0	0
$760$	−4.13734	−0.150077
$761$	4.37269	0.158510	0.0792550	−	0.996854i	$-0.474746\pi$
$761$	4.37269	0.158510	0.0792550	+	0.996854i	$0.474746\pi$
$762$	21.6633	0.784780
$763$	−7.55588	−0.273541
$764$	4.24849	0.153705
$765$	12.7736	0.461831
$766$	−19.2426	−0.695265
$767$	−23.9568	−0.865030
$768$	1.00000	0.0360844
$769$	27.7635	1.00118	0.500588	−	0.865686i	$-0.333117\pi$
$769$	27.7635	1.00118	0.500588	+	0.865686i	$0.333117\pi$
$770$	1.12607	0.0405807
$771$	−0.544992	−0.0196274
$772$	13.1121	0.471914
$773$	0.160762	0.00578221	0.00289111	−	0.999996i	$-0.499080\pi$
$773$	0.160762	0.00578221	0.00289111	+	0.999996i	$0.499080\pi$
$774$	−6.77428	−0.243496
$775$	0.129745	0.00466056
$776$	−18.2589	−0.655455
$777$	0.424208	0.0152184
$778$	−8.25813	−0.296068
$779$	−20.8569	−0.747278
$780$	10.9333	0.391474
$781$	−0.382199	−0.0136762
$782$	0	0
$783$	5.06465	0.180996
$784$	−6.79020	−0.242507
$785$	−15.9408	−0.568951
$786$	−17.1468	−0.611606
$787$	37.6412	1.34176	0.670882	−	0.741564i	$-0.265916\pi$
$787$	37.6412	1.34176	0.670882	+	0.741564i	$0.265916\pi$
$788$	−9.92381	−0.353521
$789$	14.7270	0.524294
$790$	15.7073	0.558842
$791$	−7.02454	−0.249764
$792$	−1.25954	−0.0447557
$793$	−48.0941	−1.70787
$794$	11.3695	0.403489
$795$	−5.16482	−0.183177
$796$	6.30555	0.223494
$797$	−25.8591	−0.915976	−0.457988	−	0.888958i	$-0.651430\pi$
$797$	−25.8591	−0.915976	−0.457988	+	0.888958i	$0.651430\pi$
$798$	0.970919	0.0343702
$799$	−16.4094	−0.580521
$800$	1.19028	0.0420829
$801$	5.95710	0.210484
$802$	−15.2135	−0.537209
$803$	4.21467	0.148732
$804$	−15.0332	−0.530180
$805$	0	0
$806$	0.610579	0.0215067
$807$	1.00716	0.0354538
$808$	−10.6597	−0.375007
$809$	−11.4089	−0.401115	−0.200558	−	0.979682i	$-0.564275\pi$
$809$	−11.4089	−0.401115	−0.200558	+	0.979682i	$0.564275\pi$
$810$	−1.95185	−0.0685810
$811$	33.0446	1.16035	0.580176	−	0.814491i	$-0.302984\pi$
$811$	33.0446	1.16035	0.580176	+	0.814491i	$0.302984\pi$
$812$	−2.31983	−0.0814102
$813$	−19.9145	−0.698432
$814$	1.16649	0.0408856
$815$	−33.3015	−1.16650
$816$	6.54436	0.229099
$817$	14.3595	0.502374
$818$	−1.53365	−0.0536227
$819$	−2.56573	−0.0896540
$820$	−19.2053	−0.670679
$821$	−5.57488	−0.194565	−0.0972824	−	0.995257i	$-0.531015\pi$
$821$	−5.57488	−0.194565	−0.0972824	+	0.995257i	$0.531015\pi$
$822$	3.10334	0.108241
$823$	33.5946	1.17104	0.585518	−	0.810660i	$-0.300891\pi$
$823$	33.5946	1.17104	0.585518	+	0.810660i	$0.300891\pi$
$824$	−7.07465	−0.246457
$825$	−1.49921	−0.0521957
$826$	−1.95899	−0.0681620
$827$	−50.3483	−1.75078	−0.875391	−	0.483416i	$-0.839396\pi$
$827$	−50.3483	−1.75078	−0.875391	+	0.483416i	$0.839396\pi$
$828$	0	0
$829$	43.2992	1.50384	0.751922	−	0.659252i	$-0.229127\pi$
$829$	43.2992	1.50384	0.751922	+	0.659252i	$0.229127\pi$
$830$	−20.4789	−0.710832
$831$	−24.4934	−0.849667
$832$	5.60149	0.194197
$833$	−44.4375	−1.53967
$834$	22.4485	0.777328
$835$	35.2686	1.22052
$836$	2.66985	0.0923386
$837$	−0.109003	−0.00376769
$838$	−4.63094	−0.159973
$839$	47.6218	1.64409	0.822044	−	0.569425i	$-0.192834\pi$
$839$	47.6218	1.64409	0.822044	+	0.569425i	$0.192834\pi$
$840$	0.894034	0.0308471
$841$	−3.34936	−0.115495
$842$	2.59683	0.0894927
$843$	22.2104	0.764966
$844$	25.0546	0.862415
$845$	35.8686	1.23392
$846$	2.50740	0.0862063
$847$	4.31183	0.148156
$848$	−2.64612	−0.0908680
$849$	16.4220	0.563601
$850$	7.78965	0.267183
$851$	0	0
$852$	−0.303444	−0.0103958
$853$	−48.8622	−1.67301	−0.836504	−	0.547960i	$-0.815404\pi$
$853$	−48.8622	−1.67301	−0.836504	+	0.547960i	$0.815404\pi$
$854$	−3.93275	−0.134576
$855$	4.13734	0.141494
$856$	2.76376	0.0944632
$857$	1.52668	0.0521505	0.0260752	−	0.999660i	$-0.491699\pi$
$857$	1.52668	0.0521505	0.0260752	+	0.999660i	$0.491699\pi$
$858$	−7.05529	−0.240864
$859$	−1.19558	−0.0407928	−0.0203964	−	0.999792i	$-0.506493\pi$
$859$	−1.19558	−0.0407928	−0.0203964	+	0.999792i	$0.506493\pi$
$860$	13.2224	0.450879
$861$	4.50695	0.153597
$862$	9.94166	0.338614
$863$	49.0154	1.66851	0.834253	−	0.551383i	$-0.185900\pi$
$863$	49.0154	1.66851	0.834253	+	0.551383i	$0.185900\pi$
$864$	−1.00000	−0.0340207
$865$	−31.1538	−1.05926
$866$	21.5133	0.731051
$867$	25.8287	0.877188
$868$	0.0499282	0.00169467
$869$	−10.1360	−0.343841
$870$	−9.88543	−0.335147
$871$	−84.2083	−2.85329
$872$	−16.4960	−0.558624
$873$	18.2589	0.617969
$874$	0	0
$875$	5.53432	0.187094
$876$	3.34620	0.113058
$877$	1.08702	0.0367061	0.0183531	−	0.999832i	$-0.494158\pi$
$877$	1.08702	0.0367061	0.0183531	+	0.999832i	$0.494158\pi$
$878$	25.3531	0.855626
$879$	24.4969	0.826259
$880$	2.45843	0.0828736
$881$	−44.7765	−1.50856	−0.754279	−	0.656555i	$-0.772013\pi$
$881$	−44.7765	−1.50856	−0.754279	+	0.656555i	$0.772013\pi$
$882$	6.79020	0.228638
$883$	−48.4265	−1.62968	−0.814841	−	0.579685i	$-0.803176\pi$
$883$	−48.4265	−1.62968	−0.814841	+	0.579685i	$0.803176\pi$
$884$	36.6582	1.23295
$885$	−8.34778	−0.280608
$886$	6.75984	0.227101
$887$	−15.0113	−0.504029	−0.252015	−	0.967723i	$-0.581093\pi$
$887$	−15.0113	−0.504029	−0.252015	+	0.967723i	$0.581093\pi$
$888$	0.926128	0.0310788
$889$	9.92277	0.332799
$890$	−11.6274	−0.389750
$891$	1.25954	0.0421961
$892$	1.78108	0.0596348
$893$	−5.31495	−0.177858
$894$	−15.7100	−0.525422
$895$	−30.6276	−1.02377
$896$	0.458044	0.0153022
$897$	0	0
$898$	−7.80490	−0.260453
$899$	−0.552061	−0.0184123
$900$	−1.19028	−0.0396762
$901$	−17.3171	−0.576918
$902$	12.3933	0.412651
$903$	−3.10292	−0.103259
$904$	−15.3359	−0.510066
$905$	−3.55101	−0.118039
$906$	5.57516	0.185222
$907$	−23.6578	−0.785545	−0.392773	−	0.919636i	$-0.628484\pi$
$907$	−23.6578	−0.785545	−0.392773	+	0.919636i	$0.628484\pi$
$908$	−20.7093	−0.687264
$909$	10.6597	0.353560
$910$	5.00792	0.166011
$911$	−29.2129	−0.967866	−0.483933	−	0.875105i	$-0.660792\pi$
$911$	−29.2129	−0.967866	−0.483933	+	0.875105i	$0.660792\pi$
$912$	2.11970	0.0701904
$913$	13.2151	0.437357
$914$	−9.29867	−0.307573
$915$	−16.7585	−0.554018
$916$	8.62288	0.284908
$917$	−7.85400	−0.259362
$918$	−6.54436	−0.215996
$919$	−12.1150	−0.399637	−0.199818	−	0.979833i	$-0.564035\pi$
$919$	−12.1150	−0.399637	−0.199818	+	0.979833i	$0.564035\pi$
$920$	0	0
$921$	14.6749	0.483553
$922$	17.2422	0.567842
$923$	−1.69974	−0.0559476
$924$	−0.576924	−0.0189794
$925$	1.10236	0.0362452
$926$	9.03520	0.296915
$927$	7.07465	0.232362
$928$	−5.06465	−0.166255
$929$	−10.1569	−0.333236	−0.166618	−	0.986022i	$-0.553285\pi$
$929$	−10.1569	−0.333236	−0.166618	+	0.986022i	$0.553285\pi$
$930$	0.212757	0.00697659
$931$	−14.3932	−0.471718
$932$	4.77637	0.156455
$933$	33.2393	1.08821
$934$	10.5457	0.345065
$935$	16.0888	0.526161
$936$	−5.60149	−0.183091
$937$	−21.9518	−0.717133	−0.358567	−	0.933504i	$-0.616734\pi$
$937$	−21.9518	−0.717133	−0.358567	+	0.933504i	$0.616734\pi$
$938$	−6.88587	−0.224832
$939$	−20.4935	−0.668780
$940$	−4.89407	−0.159627
$941$	−37.9048	−1.23566	−0.617831	−	0.786311i	$-0.711988\pi$
$941$	−37.9048	−1.23566	−0.617831	+	0.786311i	$0.711988\pi$
$942$	8.16701	0.266096
$943$	0	0
$944$	−4.27686	−0.139200
$945$	−0.894034	−0.0290829
$946$	−8.53246	−0.277414
$947$	34.5347	1.12223	0.561114	−	0.827739i	$-0.310373\pi$
$947$	34.5347	1.12223	0.561114	+	0.827739i	$0.310373\pi$
$948$	−8.04741	−0.261368
$949$	18.7437	0.608447
$950$	2.52305	0.0818586
$951$	22.9004	0.742597
$952$	2.99761	0.0971531
$953$	−46.1294	−1.49428	−0.747139	−	0.664668i	$-0.768573\pi$
$953$	−46.1294	−1.49428	−0.747139	+	0.664668i	$0.768573\pi$
$954$	2.64612	0.0856712
$955$	8.29240	0.268336
$956$	−23.9090	−0.773273
$957$	6.37911	0.206207
$958$	13.1682	0.425445
$959$	1.42147	0.0459015
$960$	1.95185	0.0629957
$961$	−30.9881	−0.999617
$962$	5.18770	0.167258
$963$	−2.76376	−0.0890608
$964$	−23.0500	−0.742389
$965$	25.5928	0.823861
$966$	0	0
$967$	32.2668	1.03763	0.518815	−	0.854886i	$-0.326373\pi$
$967$	32.2668	1.03763	0.518815	+	0.854886i	$0.326373\pi$
$968$	9.41356	0.302563
$969$	13.8721	0.445637
$970$	−35.6385	−1.14428
$971$	−30.5548	−0.980550	−0.490275	−	0.871568i	$-0.663104\pi$
$971$	−30.5548	−0.980550	−0.490275	+	0.871568i	$0.663104\pi$
$972$	1.00000	0.0320750
$973$	10.2824	0.329639
$974$	3.03122	0.0971266
$975$	−6.66737	−0.213527
$976$	−8.58595	−0.274830
$977$	−24.3056	−0.777604	−0.388802	−	0.921321i	$-0.627111\pi$
$977$	−24.3056	−0.777604	−0.388802	+	0.921321i	$0.627111\pi$
$978$	17.0615	0.545567
$979$	7.50319	0.239803
$980$	−13.2534	−0.423366
$981$	16.4960	0.526675
$982$	−6.30567	−0.201222
$983$	−58.2648	−1.85836	−0.929180	−	0.369628i	$-0.879485\pi$
$983$	−58.2648	−1.85836	−0.929180	+	0.369628i	$0.879485\pi$
$984$	9.83955	0.313674
$985$	−19.3698	−0.617172
$986$	−33.1449	−1.05555
$987$	1.14850	0.0365572
$988$	11.8735	0.377747
$989$	0	0
$990$	−2.45843	−0.0781340
$991$	35.2587	1.12003	0.560014	−	0.828483i	$-0.310796\pi$
$991$	35.2587	1.12003	0.560014	+	0.828483i	$0.310796\pi$
$992$	0.109003	0.00346085
$993$	−10.2762	−0.326106
$994$	−0.138991	−0.00440852
$995$	12.3075	0.390173
$996$	10.4920	0.332453
$997$	−44.2899	−1.40268	−0.701338	−	0.712829i	$-0.747413\pi$
$997$	−44.2899	−1.40268	−0.701338	+	0.712829i	$0.747413\pi$
$998$	6.78493	0.214773
$999$	−0.926128	−0.0293014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3174.2.a.z.1.2		5
3.2	odd	2		9522.2.a.bv.1.4		5
23.4	even	11		138.2.e.c.85.1	yes	10
23.6	even	11		138.2.e.c.13.1	✓	10
23.22	odd	2		3174.2.a.y.1.4		5
69.29	odd	22		414.2.i.b.289.1		10
69.50	odd	22		414.2.i.b.361.1		10
69.68	even	2		9522.2.a.ca.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.c.13.1	✓	10	23.6	even	11
138.2.e.c.85.1	yes	10	23.4	even	11
414.2.i.b.289.1		10	69.29	odd	22
414.2.i.b.361.1		10	69.50	odd	22
3174.2.a.y.1.4		5	23.22	odd	2
3174.2.a.z.1.2		5	1.1	even	1	trivial
9522.2.a.bv.1.4		5	3.2	odd	2
9522.2.a.ca.1.2		5	69.68	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:	\( N \)	\(=\)	\( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3174.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.95185 q^{5} -1.00000 q^{6} -0.458044 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.95185 q^{10} +1.25954 q^{11} +1.00000 q^{12} +5.60149 q^{13} +0.458044 q^{14} +1.95185 q^{15} +1.00000 q^{16} +6.54436 q^{17} -1.00000 q^{18} +2.11970 q^{19} +1.95185 q^{20} -0.458044 q^{21} -1.25954 q^{22} -1.00000 q^{24} -1.19028 q^{25} -5.60149 q^{26} +1.00000 q^{27} -0.458044 q^{28} +5.06465 q^{29} -1.95185 q^{30} -0.109003 q^{31} -1.00000 q^{32} +1.25954 q^{33} -6.54436 q^{34} -0.894034 q^{35} +1.00000 q^{36} -0.926128 q^{37} -2.11970 q^{38} +5.60149 q^{39} -1.95185 q^{40} -9.83955 q^{41} +0.458044 q^{42} +6.77428 q^{43} +1.25954 q^{44} +1.95185 q^{45} -2.50740 q^{47} +1.00000 q^{48} -6.79020 q^{49} +1.19028 q^{50} +6.54436 q^{51} +5.60149 q^{52} -2.64612 q^{53} -1.00000 q^{54} +2.45843 q^{55} +0.458044 q^{56} +2.11970 q^{57} -5.06465 q^{58} -4.27686 q^{59} +1.95185 q^{60} -8.58595 q^{61} +0.109003 q^{62} -0.458044 q^{63} +1.00000 q^{64} +10.9333 q^{65} -1.25954 q^{66} -15.0332 q^{67} +6.54436 q^{68} +0.894034 q^{70} -0.303444 q^{71} -1.00000 q^{72} +3.34620 q^{73} +0.926128 q^{74} -1.19028 q^{75} +2.11970 q^{76} -0.576924 q^{77} -5.60149 q^{78} -8.04741 q^{79} +1.95185 q^{80} +1.00000 q^{81} +9.83955 q^{82} +10.4920 q^{83} -0.458044 q^{84} +12.7736 q^{85} -6.77428 q^{86} +5.06465 q^{87} -1.25954 q^{88} +5.95710 q^{89} -1.95185 q^{90} -2.56573 q^{91} -0.109003 q^{93} +2.50740 q^{94} +4.13734 q^{95} -1.00000 q^{96} +18.2589 q^{97} +6.79020 q^{98} +1.25954 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 11 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9} - 11 q^{10} + 11 q^{11} + 5 q^{12} + 10 q^{13} + q^{14} + 11 q^{15} + 5 q^{16} + 11 q^{17} - 5 q^{18} - q^{19} + 11 q^{20}+ \cdots + 11 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 11 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9 - 11 * q^10 + 11 * q^11 + 5 * q^12 + 10 * q^13 + q^14 + 11 * q^15 + 5 * q^16 + 11 * q^17 - 5 * q^18 - q^19 + 11 * q^20 - q^21 - 11 * q^22 - 5 * q^24 + 30 * q^25 - 10 * q^26 + 5 * q^27 - q^28 + 3 * q^29 - 11 * q^30 + 2 * q^31 - 5 * q^32 + 11 * q^33 - 11 * q^34 - 11 * q^35 + 5 * q^36 + 16 * q^37 + q^38 + 10 * q^39 - 11 * q^40 - 8 * q^41 + q^42 - 12 * q^43 + 11 * q^44 + 11 * q^45 + 5 * q^48 - 4 * q^49 - 30 * q^50 + 11 * q^51 + 10 * q^52 + 27 * q^53 - 5 * q^54 + q^56 - q^57 - 3 * q^58 - 6 * q^59 + 11 * q^60 + 5 * q^61 - 2 * q^62 - q^63 + 5 * q^64 + 22 * q^65 - 11 * q^66 - 7 * q^67 + 11 * q^68 + 11 * q^70 - 11 * q^71 - 5 * q^72 - 12 * q^73 - 16 * q^74 + 30 * q^75 - q^76 + 11 * q^77 - 10 * q^78 + 16 * q^79 + 11 * q^80 + 5 * q^81 + 8 * q^82 + 22 * q^83 - q^84 - 11 * q^85 + 12 * q^86 + 3 * q^87 - 11 * q^88 + 38 * q^89 - 11 * q^90 + 9 * q^91 + 2 * q^93 + 11 * q^95 - 5 * q^96 + 5 * q^97 + 4 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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