LMFDB - Embedded newform 1445.2.d.i.866.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1445,2,Mod(866,1445)] 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("1445.866"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1445.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,-6,0,42,0,0,0,-24,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$11.5383830921$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			866.5
Character	$\chi$	$=$	1445.866
Dual form			1445.2.d.i.866.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.27524 q^{2} -1.70271i q^{3} +3.17670 q^{4} +1.00000i q^{5} +3.87408i q^{6} +2.70724i q^{7} -2.67727 q^{8} +0.100765 q^{9} -2.27524i q^{10} -2.17632i q^{11} -5.40901i q^{12} +5.37970 q^{13} -6.15962i q^{14} +1.70271 q^{15} -0.261974 q^{16} -0.229265 q^{18} -8.53588 q^{19} +3.17670i q^{20} +4.60966 q^{21} +4.95164i q^{22} -5.32570i q^{23} +4.55863i q^{24} -1.00000 q^{25} -12.2401 q^{26} -5.27972i q^{27} +8.60011i q^{28} -1.31247i q^{29} -3.87408 q^{30} -0.757667i q^{31} +5.95060 q^{32} -3.70565 q^{33} -2.70724 q^{35} +0.320102 q^{36} +8.20380i q^{37} +19.4212 q^{38} -9.16010i q^{39} -2.67727i q^{40} +0.0130696i q^{41} -10.4881 q^{42} +0.330596 q^{43} -6.91352i q^{44} +0.100765i q^{45} +12.1172i q^{46} +11.3395 q^{47} +0.446067i q^{48} -0.329174 q^{49} +2.27524 q^{50} +17.0897 q^{52} +8.04151 q^{53} +12.0126i q^{54} +2.17632 q^{55} -7.24803i q^{56} +14.5342i q^{57} +2.98619i q^{58} +13.1546 q^{59} +5.40901 q^{60} +7.20639i q^{61} +1.72387i q^{62} +0.272797i q^{63} -13.0151 q^{64} +5.37970i q^{65} +8.43123 q^{66} +2.67143 q^{67} -9.06814 q^{69} +6.15962 q^{70} +1.87106i q^{71} -0.269777 q^{72} -11.9780i q^{73} -18.6656i q^{74} +1.70271i q^{75} -27.1160 q^{76} +5.89183 q^{77} +20.8414i q^{78} -17.0477i q^{79} -0.261974i q^{80} -8.68755 q^{81} -0.0297365i q^{82} +3.35585 q^{83} +14.6435 q^{84} -0.752183 q^{86} -2.23477 q^{87} +5.82660i q^{88} -10.8909 q^{89} -0.229265i q^{90} +14.5642i q^{91} -16.9182i q^{92} -1.29009 q^{93} -25.8001 q^{94} -8.53588i q^{95} -10.1322i q^{96} +10.9119i q^{97} +0.748948 q^{98} -0.219298i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{2} + 42 q^{4} - 24 q^{8} - 42 q^{9} + 18 q^{13} - 6 q^{15} + 78 q^{16} - 18 q^{18} - 54 q^{19} + 12 q^{21} - 24 q^{25} - 12 q^{26} - 18 q^{30} - 24 q^{32} + 12 q^{35} - 96 q^{36} - 6 q^{38}+ \cdots - 84 q^{98}+O(q^{100}) $ 24 * q - 6 * q^2 + 42 * q^4 - 24 * q^8 - 42 * q^9 + 18 * q^13 - 6 * q^15 + 78 * q^16 - 18 * q^18 - 54 * q^19 + 12 * q^21 - 24 * q^25 - 12 * q^26 - 18 * q^30 - 24 * q^32 + 12 * q^35 - 96 * q^36 - 6 * q^38 - 60 * q^42 - 90 * q^43 + 6 * q^47 - 24 * q^49 + 6 * q^50 + 102 * q^52 + 24 * q^53 - 12 * q^55 - 72 * q^59 - 12 * q^60 + 84 * q^64 + 78 * q^66 + 156 * q^67 - 132 * q^69 + 36 * q^70 - 24 * q^72 - 54 * q^76 + 78 * q^77 + 24 * q^81 - 54 * q^83 + 54 * q^84 + 36 * q^86 - 48 * q^87 + 12 * q^89 + 24 * q^93 - 42 * q^94 - 84 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.27524	−1.60884	−0.804418	−	0.594064i	$-0.797522\pi$
$2$	−2.27524	−1.60884	−0.804418	+	0.594064i	$0.797522\pi$
$3$	− 1.70271i	− 0.983062i	−0.870860	−	0.491531i	$-0.836437\pi$
$3$	− 1.70271i	− 0.983062i	0.870860	−	0.491531i	$-0.163563\pi$
$4$	3.17670	1.58835
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	3.87408i	1.58159i
$7$	2.70724i	1.02324i	0.859211	+	0.511621i	$0.170955\pi$
$7$	2.70724i	1.02324i	−0.859211	+	0.511621i	$0.829045\pi$
$8$	−2.67727	−0.946559
$9$	0.100765	0.0335885
$10$	− 2.27524i	− 0.719493i
$11$	− 2.17632i	− 0.656185i	−0.944646	−	0.328093i	$-0.893594\pi$
$11$	− 2.17632i	− 0.656185i	0.944646	−	0.328093i	$-0.106406\pi$
$12$	− 5.40901i	− 1.56145i
$13$	5.37970	1.49206	0.746031	−	0.665912i	$-0.231957\pi$
$13$	5.37970	1.49206	0.746031	+	0.665912i	$0.231957\pi$
$14$	− 6.15962i	− 1.64623i
$15$	1.70271	0.439639
$16$	−0.261974	−0.0654935
$17$	0	0
$17$	0	0
$18$	−0.229265	−0.0540384
$19$	−8.53588	−1.95827	−0.979133	−	0.203220i	$-0.934859\pi$
$19$	−8.53588	−1.95827	−0.979133	+	0.203220i	$0.934859\pi$
$20$	3.17670i	0.710332i
$21$	4.60966	1.00591
$22$	4.95164i	1.05569i
$23$	− 5.32570i	− 1.11049i	−0.831688	−	0.555243i	$-0.812625\pi$
$23$	− 5.32570i	− 1.11049i	0.831688	−	0.555243i	$-0.187375\pi$
$24$	4.55863i	0.930526i
$25$	−1.00000	−0.200000
$26$	−12.2401	−2.40048
$27$	− 5.27972i	− 1.01608i
$28$	8.60011i	1.62527i
$29$	− 1.31247i	− 0.243720i	−0.992547	−	0.121860i	$-0.961114\pi$
$29$	− 1.31247i	− 0.243720i	0.992547	−	0.121860i	$-0.0388860\pi$
$30$	−3.87408	−0.707306
$31$	− 0.757667i	− 0.136081i	−0.997683	−	0.0680405i	$-0.978325\pi$
$31$	− 0.757667i	− 0.136081i	0.997683	−	0.0680405i	$-0.0216747\pi$
$32$	5.95060	1.05193
$33$	−3.70565	−0.645071
$34$	0	0
$35$	−2.70724	−0.457608
$36$	0.320102	0.0533503
$37$	8.20380i	1.34870i	0.738414	+	0.674348i	$0.235575\pi$
$37$	8.20380i	1.34870i	−0.738414	+	0.674348i	$0.764425\pi$
$38$	19.4212	3.15053
$39$	− 9.16010i	− 1.46679i
$40$	− 2.67727i	− 0.423314i
$41$	0.0130696i	0.00204113i	0.999999	+	0.00102057i	$0.000324857\pi$
$41$	0.0130696i	0.00204113i	−0.999999	+	0.00102057i	$0.999675\pi$
$42$	−10.4881	−1.61834
$43$	0.330596	0.0504154	0.0252077	−	0.999682i	$-0.491975\pi$
$43$	0.330596	0.0504154	0.0252077	+	0.999682i	$0.491975\pi$
$44$	− 6.91352i	− 1.04225i
$45$	0.100765i	0.0150212i
$46$	12.1172i	1.78659i
$47$	11.3395	1.65404	0.827019	−	0.562175i	$-0.190035\pi$
$47$	11.3395	1.65404	0.827019	+	0.562175i	$0.190035\pi$
$48$	0.446067i	0.0643842i
$49$	−0.329174	−0.0470248
$50$	2.27524	0.321767
$51$	0	0
$52$	17.0897	2.36992
$53$	8.04151	1.10459	0.552293	−	0.833650i	$-0.313753\pi$
$53$	8.04151	1.10459	0.552293	+	0.833650i	$0.313753\pi$
$54$	12.0126i	1.63471i
$55$	2.17632	0.293455
$56$	− 7.24803i	− 0.968559i
$57$	14.5342i	1.92510i
$58$	2.98619i	0.392106i
$59$	13.1546	1.71259	0.856293	−	0.516490i	$-0.172762\pi$
$59$	13.1546	1.71259	0.856293	+	0.516490i	$0.172762\pi$
$60$	5.40901	0.698300
$61$	7.20639i	0.922684i	0.887222	+	0.461342i	$0.152632\pi$
$61$	7.20639i	0.922684i	−0.887222	+	0.461342i	$0.847368\pi$
$62$	1.72387i	0.218932i
$63$	0.272797i	0.0343692i
$64$	−13.0151	−1.62688
$65$	5.37970i	0.667270i
$66$	8.43123	1.03781
$67$	2.67143	0.326367	0.163184	−	0.986596i	$-0.447824\pi$
$67$	2.67143	0.326367	0.163184	+	0.986596i	$0.447824\pi$
$68$	0	0
$69$	−9.06814	−1.09168
$70$	6.15962	0.736216
$71$	1.87106i	0.222054i	0.993817	+	0.111027i	$0.0354140\pi$
$71$	1.87106i	0.222054i	−0.993817	+	0.111027i	$0.964586\pi$
$72$	−0.269777	−0.0317935
$73$	− 11.9780i	− 1.40192i	−0.713199	−	0.700962i	$-0.752754\pi$
$73$	− 11.9780i	− 1.40192i	0.713199	−	0.700962i	$-0.247246\pi$
$74$	− 18.6656i	− 2.16983i
$75$	1.70271i	0.196612i
$76$	−27.1160	−3.11041
$77$	5.89183	0.671437
$78$	20.8414i	2.35982i
$79$	− 17.0477i	− 1.91802i	−0.283375	−	0.959009i	$-0.591454\pi$
$79$	− 17.0477i	− 1.91802i	0.283375	−	0.959009i	$-0.408546\pi$
$80$	− 0.261974i	− 0.0292896i
$81$	−8.68755	−0.965283
$82$	− 0.0297365i	− 0.00328385i
$83$	3.35585	0.368352	0.184176	−	0.982893i	$-0.441038\pi$
$83$	3.35585	0.368352	0.184176	+	0.982893i	$0.441038\pi$
$84$	14.6435	1.59774
$85$	0	0
$86$	−0.752183	−0.0811100
$87$	−2.23477	−0.239592
$88$	5.82660i	0.621118i
$89$	−10.8909	−1.15444	−0.577218	−	0.816590i	$-0.695862\pi$
$89$	−10.8909	−1.15444	−0.577218	+	0.816590i	$0.695862\pi$
$90$	− 0.229265i	− 0.0241667i
$91$	14.5642i	1.52674i
$92$	− 16.9182i	− 1.76384i
$93$	−1.29009	−0.133776
$94$	−25.8001	−2.66107
$95$	− 8.53588i	− 0.875763i
$96$	− 10.1322i	− 1.03411i
$97$	10.9119i	1.10793i	0.832540	+	0.553965i	$0.186886\pi$
$97$	10.9119i	1.10793i	−0.832540	+	0.553965i	$0.813114\pi$
$98$	0.748948	0.0756552
$99$	− 0.219298i	− 0.0220403i
$100$	−3.17670	−0.317670
$101$	−3.95599	−0.393636	−0.196818	−	0.980440i	$-0.563061\pi$
$101$	−3.95599	−0.393636	−0.196818	+	0.980440i	$0.563061\pi$
$102$	0	0
$103$	7.16779	0.706263	0.353132	−	0.935574i	$-0.385117\pi$
$103$	7.16779	0.706263	0.353132	+	0.935574i	$0.385117\pi$
$104$	−14.4029	−1.41232
$105$	4.60966i	0.449857i
$106$	−18.2963	−1.77710
$107$	3.70041i	0.357732i	0.983873	+	0.178866i	$0.0572429\pi$
$107$	3.70041i	0.357732i	−0.983873	+	0.178866i	$0.942757\pi$
$108$	− 16.7721i	− 1.61389i
$109$	− 2.73869i	− 0.262319i	−0.991361	−	0.131160i	$-0.958130\pi$
$109$	− 2.73869i	− 0.262319i	0.991361	−	0.131160i	$-0.0418701\pi$
$110$	−4.95164	−0.472121
$111$	13.9687	1.32585
$112$	− 0.709227i	− 0.0670157i
$113$	− 1.51487i	− 0.142507i	−0.997458	−	0.0712536i	$-0.977300\pi$
$113$	− 1.51487i	− 0.142507i	0.997458	−	0.0712536i	$-0.0227000\pi$
$114$	− 33.0687i	− 3.09716i
$115$	5.32570	0.496624
$116$	− 4.16934i	− 0.387114i
$117$	0.542088	0.0501161
$118$	−29.9299	−2.75527
$119$	0	0
$120$	−4.55863	−0.416144
$121$	6.26363	0.569421
$122$	− 16.3963i	− 1.48445i
$123$	0.0222538	0.00200656
$124$	− 2.40688i	− 0.216144i
$125$	− 1.00000i	− 0.0894427i
$126$	− 0.620677i	− 0.0552943i
$127$	−4.15113	−0.368353	−0.184176	−	0.982893i	$-0.558962\pi$
$127$	−4.15113	−0.368353	−0.184176	+	0.982893i	$0.558962\pi$
$128$	17.7112	1.56546
$129$	− 0.562910i	− 0.0495614i
$130$	− 12.2401i	− 1.07353i
$131$	− 12.3436i	− 1.07847i	−0.842156	−	0.539234i	$-0.818714\pi$
$131$	− 12.3436i	− 1.07847i	0.842156	−	0.539234i	$-0.181286\pi$
$132$	−11.7717	−1.02460
$133$	− 23.1087i	− 2.00378i
$134$	−6.07814	−0.525071
$135$	5.27972	0.454406
$136$	0	0
$137$	13.1700	1.12519	0.562594	−	0.826733i	$-0.309803\pi$
$137$	13.1700	1.12519	0.562594	+	0.826733i	$0.309803\pi$
$138$	20.6322	1.75633
$139$	0.706325i	0.0599097i	0.999551	+	0.0299548i	$0.00953635\pi$
$139$	0.706325i	0.0599097i	−0.999551	+	0.0299548i	$0.990464\pi$
$140$	−8.60011	−0.726842
$141$	− 19.3079i	− 1.62602i
$142$	− 4.25710i	− 0.357248i
$143$	− 11.7080i	− 0.979069i
$144$	−0.0263979	−0.00219983
$145$	1.31247	0.108995
$146$	27.2529i	2.25546i
$147$	0.560489i	0.0462283i
$148$	26.0610i	2.14220i
$149$	7.41327	0.607319	0.303659	−	0.952781i	$-0.401792\pi$
$149$	7.41327	0.607319	0.303659	+	0.952781i	$0.401792\pi$
$150$	− 3.87408i	− 0.316317i
$151$	2.19280	0.178448	0.0892238	−	0.996012i	$-0.471561\pi$
$151$	2.19280	0.178448	0.0892238	+	0.996012i	$0.471561\pi$
$152$	22.8529	1.85361
$153$	0	0
$154$	−13.4053	−1.08023
$155$	0.757667	0.0608573
$156$	− 29.0989i	− 2.32978i
$157$	5.33731	0.425963	0.212982	−	0.977056i	$-0.431683\pi$
$157$	5.33731	0.425963	0.212982	+	0.977056i	$0.431683\pi$
$158$	38.7876i	3.08578i
$159$	− 13.6924i	− 1.08588i
$160$	5.95060i	0.470436i
$161$	14.4180	1.13630
$162$	19.7662	1.55298
$163$	7.78483i	0.609755i	0.952392	+	0.304878i	$0.0986155\pi$
$163$	7.78483i	0.609755i	−0.952392	+	0.304878i	$0.901384\pi$
$164$	0.0415183i	0.00324204i
$165$	− 3.70565i	− 0.288485i
$166$	−7.63534	−0.592617
$167$	− 4.42165i	− 0.342157i	−0.985257	−	0.171079i	$-0.945275\pi$
$167$	− 4.42165i	− 0.342157i	0.985257	−	0.171079i	$-0.0547252\pi$
$168$	−12.3413	−0.952154
$169$	15.9412	1.22625
$170$	0	0
$171$	−0.860123	−0.0657752
$172$	1.05020	0.0800773
$173$	− 14.6188i	− 1.11145i	−0.831367	−	0.555724i	$-0.812441\pi$
$173$	− 14.6188i	− 1.11145i	0.831367	−	0.555724i	$-0.187559\pi$
$174$	5.08463	0.385465
$175$	− 2.70724i	− 0.204648i
$176$	0.570139i	0.0429759i
$177$	− 22.3986i	− 1.68358i
$178$	24.7794	1.85730
$179$	5.42912	0.405791	0.202896	−	0.979200i	$-0.434965\pi$
$179$	5.42912	0.405791	0.202896	+	0.979200i	$0.434965\pi$
$180$	0.320102i	0.0238590i
$181$	18.3483i	1.36382i	0.731437	+	0.681909i	$0.238850\pi$
$181$	18.3483i	1.36382i	−0.731437	+	0.681909i	$0.761150\pi$
$182$	− 33.1369i	− 2.45627i
$183$	12.2704	0.907056
$184$	14.2583i	1.05114i
$185$	−8.20380	−0.603155
$186$	2.93526	0.215224
$187$	0	0
$188$	36.0222	2.62719
$189$	14.2935	1.03970
$190$	19.4212i	1.40896i
$191$	4.91654	0.355748	0.177874	−	0.984053i	$-0.443078\pi$
$191$	4.91654	0.355748	0.177874	+	0.984053i	$0.443078\pi$
$192$	22.1609i	1.59933i
$193$	17.5487i	1.26318i	0.775302	+	0.631591i	$0.217598\pi$
$193$	17.5487i	1.26318i	−0.775302	+	0.631591i	$0.782402\pi$
$194$	− 24.8270i	− 1.78248i
$195$	9.16010	0.655968
$196$	−1.04569	−0.0746919
$197$	− 21.7620i	− 1.55048i	−0.631668	−	0.775239i	$-0.717629\pi$
$197$	− 21.7620i	− 1.55048i	0.631668	−	0.775239i	$-0.282371\pi$
$198$	0.498955i	0.0354592i
$199$	0.241280i	0.0171039i	0.999963	+	0.00855196i	$0.00272221\pi$
$199$	0.241280i	0.0171039i	−0.999963	+	0.00855196i	$0.997278\pi$
$200$	2.67727	0.189312
$201$	− 4.54869i	− 0.320840i
$202$	9.00082	0.633295
$203$	3.55319	0.249385
$204$	0	0
$205$	−0.0130696	−0.000912823 0
$206$	−16.3084	−1.13626
$207$	− 0.536647i	− 0.0372995i
$208$	−1.40934	−0.0977203
$209$	18.5768i	1.28499i
$210$	− 10.4881i	− 0.723746i
$211$	− 9.41485i	− 0.648145i	−0.946032	−	0.324072i	$-0.894948\pi$
$211$	− 9.41485i	− 0.648145i	0.946032	−	0.324072i	$-0.105052\pi$
$212$	25.5455	1.75447
$213$	3.18588	0.218293
$214$	− 8.41931i	− 0.575532i
$215$	0.330596i	0.0225464i
$216$	14.1352i	0.961781i
$217$	2.05119	0.139244
$218$	6.23118i	0.422029i
$219$	−20.3952	−1.37818
$220$	6.91352	0.466109
$221$	0	0
$222$	−31.7821	−2.13308
$223$	14.8338	0.993345	0.496672	−	0.867938i	$-0.334555\pi$
$223$	14.8338	0.993345	0.496672	+	0.867938i	$0.334555\pi$
$224$	16.1097i	1.07638i
$225$	−0.100765	−0.00671770
$226$	3.44669i	0.229271i
$227$	− 9.83158i	− 0.652545i	−0.945276	−	0.326272i	$-0.894207\pi$
$227$	− 9.83158i	− 0.652545i	0.945276	−	0.326272i	$-0.105793\pi$
$228$	46.1707i	3.05773i
$229$	12.4315	0.821500	0.410750	−	0.911748i	$-0.365267\pi$
$229$	12.4315	0.821500	0.410750	+	0.911748i	$0.365267\pi$
$230$	−12.1172	−0.798986
$231$	− 10.0321i	− 0.660064i
$232$	3.51385i	0.230696i
$233$	12.5812i	0.824224i	0.911133	+	0.412112i	$0.135209\pi$
$233$	12.5812i	0.824224i	−0.911133	+	0.412112i	$0.864791\pi$
$234$	−1.23338	−0.0806285
$235$	11.3395i	0.739708i
$236$	41.7883	2.72019
$237$	−29.0274	−1.88553
$238$	0	0
$239$	−15.7966	−1.02179	−0.510897	−	0.859642i	$-0.670687\pi$
$239$	−15.7966	−1.02179	−0.510897	+	0.859642i	$0.670687\pi$
$240$	−0.446067	−0.0287935
$241$	− 8.45284i	− 0.544496i	−0.962227	−	0.272248i	$-0.912233\pi$
$241$	− 8.45284i	− 0.544496i	0.962227	−	0.272248i	$-0.0877670\pi$
$242$	−14.2512	−0.916104
$243$	− 1.04674i	− 0.0671482i
$244$	22.8926i	1.46555i
$245$	− 0.329174i	− 0.0210301i
$246$	−0.0506328	−0.00322823
$247$	−45.9205	−2.92185
$248$	2.02848i	0.128809i
$249$	− 5.71404i	− 0.362113i
$250$	2.27524i	0.143899i
$251$	14.7900	0.933540	0.466770	−	0.884379i	$-0.345418\pi$
$251$	14.7900	0.933540	0.466770	+	0.884379i	$0.345418\pi$
$252$	0.866594i	0.0545903i
$253$	−11.5904	−0.728684
$254$	9.44480	0.592619
$255$	0	0
$256$	−14.2669	−0.891684
$257$	−13.8681	−0.865070	−0.432535	−	0.901617i	$-0.642381\pi$
$257$	−13.8681	−0.865070	−0.432535	+	0.901617i	$0.642381\pi$
$258$	1.28075i	0.0797362i
$259$	−22.2097	−1.38004
$260$	17.0897i	1.05986i
$261$	− 0.132252i	− 0.00818620i
$262$	28.0847i	1.73508i
$263$	−0.382638	−0.0235945	−0.0117972	−	0.999930i	$-0.503755\pi$
$263$	−0.382638	−0.0235945	−0.0117972	+	0.999930i	$0.503755\pi$
$264$	9.92104	0.610598
$265$	8.04151i	0.493986i
$266$	52.5778i	3.22375i
$267$	18.5441i	1.13488i
$268$	8.48634	0.518386
$269$	− 21.8946i	− 1.33494i	−0.744638	−	0.667468i	$-0.767378\pi$
$269$	− 21.8946i	− 1.33494i	0.744638	−	0.667468i	$-0.232622\pi$
$270$	−12.0126	−0.731064
$271$	16.9450	1.02933	0.514667	−	0.857390i	$-0.327916\pi$
$271$	16.9450	1.02933	0.514667	+	0.857390i	$0.327916\pi$
$272$	0	0
$273$	24.7986	1.50088
$274$	−29.9648	−1.81024
$275$	2.17632i	0.131237i
$276$	−28.8068	−1.73396
$277$	− 24.4581i	− 1.46954i	−0.678315	−	0.734771i	$-0.737290\pi$
$277$	− 24.4581i	− 1.46954i	0.678315	−	0.734771i	$-0.262710\pi$
$278$	− 1.60706i	− 0.0963848i
$279$	− 0.0763467i	− 0.00457076i
$280$	7.24803	0.433153
$281$	13.2014	0.787529	0.393765	−	0.919211i	$-0.371173\pi$
$281$	13.2014	0.787529	0.393765	+	0.919211i	$0.371173\pi$
$282$	43.9301i	2.61600i
$283$	− 13.7732i	− 0.818730i	−0.912371	−	0.409365i	$-0.865750\pi$
$283$	− 13.7732i	− 0.818730i	0.912371	−	0.409365i	$-0.134250\pi$
$284$	5.94379i	0.352699i
$285$	−14.5342	−0.860930
$286$	26.6384i	1.57516i
$287$	−0.0353827	−0.00208857
$288$	0.599615	0.0353326
$289$	0	0
$290$	−2.98619	−0.175355
$291$	18.5798	1.08916
$292$	− 38.0507i	− 2.22675i
$293$	21.1370	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$293$	21.1370	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$294$	− 1.27524i	− 0.0743737i
$295$	13.1546i	0.765892i
$296$	− 21.9638i	− 1.27662i
$297$	−11.4904	−0.666738
$298$	−16.8669	−0.977076
$299$	− 28.6507i	− 1.65691i
$300$	5.40901i	0.312289i
$301$	0.895003i	0.0515871i
$302$	−4.98914	−0.287093
$303$	6.73592i	0.386969i
$304$	2.23618	0.128254
$305$	−7.20639	−0.412637
$306$	0	0
$307$	−23.2068	−1.32448	−0.662242	−	0.749290i	$-0.730395\pi$
$307$	−23.2068	−1.32448	−0.662242	+	0.749290i	$0.730395\pi$
$308$	18.7166	1.06648
$309$	− 12.2047i	− 0.694301i
$310$	−1.72387	−0.0979094
$311$	− 19.0929i	− 1.08266i	−0.840810	−	0.541330i	$-0.817921\pi$
$311$	− 19.0929i	− 1.08266i	0.840810	−	0.541330i	$-0.182079\pi$
$312$	24.5241i	1.38840i
$313$	31.9061i	1.80344i	0.432320	+	0.901720i	$0.357695\pi$
$313$	31.9061i	1.80344i	−0.432320	+	0.901720i	$0.642305\pi$
$314$	−12.1436	−0.685305
$315$	−0.272797	−0.0153704
$316$	− 54.1555i	− 3.04649i
$317$	7.35179i	0.412917i	0.978455	+	0.206459i	$0.0661939\pi$
$317$	7.35179i	0.412917i	−0.978455	+	0.206459i	$0.933806\pi$
$318$	31.1534i	1.74700i
$319$	−2.85637	−0.159926
$320$	− 13.0151i	− 0.727564i
$321$	6.30074	0.351673
$322$	−32.8043	−1.82811
$323$	0	0
$324$	−27.5977	−1.53321
$325$	−5.37970	−0.298412
$326$	− 17.7123i	− 0.980995i
$327$	−4.66321	−0.257876
$328$	− 0.0349910i	− 0.00193205i
$329$	30.6988i	1.69248i
$330$	8.43123i	0.464124i
$331$	−16.3546	−0.898930	−0.449465	−	0.893298i	$-0.648385\pi$
$331$	−16.3546	−0.898930	−0.449465	+	0.893298i	$0.648385\pi$
$332$	10.6605	0.585072
$333$	0.826660i	0.0453007i
$334$	10.0603i	0.550475i
$335$	2.67143i	0.145956i
$336$	−1.20761	−0.0658806
$337$	2.73232i	0.148839i	0.997227	+	0.0744196i	$0.0237104\pi$
$337$	2.73232i	0.148839i	−0.997227	+	0.0744196i	$0.976290\pi$
$338$	−36.2700	−1.97283
$339$	−2.57939	−0.140093
$340$	0	0
$341$	−1.64893	−0.0892944
$342$	1.95698	0.105821
$343$	18.0596i	0.975125i
$344$	−0.885095	−0.0477211
$345$	− 9.06814i	− 0.488212i
$346$	33.2613i	1.78814i
$347$	18.9609i	1.01788i	0.860803	+	0.508938i	$0.169962\pi$
$347$	18.9609i	1.01788i	−0.860803	+	0.508938i	$0.830038\pi$
$348$	−7.09919	−0.380557
$349$	32.3382	1.73102	0.865511	−	0.500890i	$-0.166994\pi$
$349$	32.3382	1.73102	0.865511	+	0.500890i	$0.166994\pi$
$350$	6.15962i	0.329246i
$351$	− 28.4033i	− 1.51606i
$352$	− 12.9504i	− 0.690259i
$353$	−13.0916	−0.696796	−0.348398	−	0.937347i	$-0.613274\pi$
$353$	−13.0916	−0.696796	−0.348398	+	0.937347i	$0.613274\pi$
$354$	50.9620i	2.70860i
$355$	−1.87106	−0.0993054
$356$	−34.5972	−1.83365
$357$	0	0
$358$	−12.3525	−0.652851
$359$	−8.64264	−0.456141	−0.228071	−	0.973645i	$-0.573242\pi$
$359$	−8.64264	−0.456141	−0.228071	+	0.973645i	$0.573242\pi$
$360$	− 0.269777i	− 0.0142185i
$361$	53.8613	2.83481
$362$	− 41.7467i	− 2.19416i
$363$	− 10.6652i	− 0.559776i
$364$	46.2660i	2.42500i
$365$	11.9780	0.626960
$366$	−27.9181	−1.45930
$367$	6.61035i	0.345057i	0.985005	+	0.172529i	$0.0551937\pi$
$367$	6.61035i	0.345057i	−0.985005	+	0.172529i	$0.944806\pi$
$368$	1.39519i	0.0727295i
$369$	0.00131697i	0 6.85586e-5i
$370$	18.6656	0.970377
$371$	21.7703i	1.13026i
$372$	−4.09823	−0.212483
$373$	−7.14349	−0.369876	−0.184938	−	0.982750i	$-0.559208\pi$
$373$	−7.14349	−0.369876	−0.184938	+	0.982750i	$0.559208\pi$
$374$	0	0
$375$	−1.70271	−0.0879278
$376$	−30.3589	−1.56564
$377$	− 7.06073i	− 0.363646i
$378$	−32.5211	−1.67270
$379$	7.19225i	0.369441i	0.982791	+	0.184720i	$0.0591380\pi$
$379$	7.19225i	0.369441i	−0.982791	+	0.184720i	$0.940862\pi$
$380$	− 27.1160i	− 1.39102i
$381$	7.06818i	0.362114i
$382$	−11.1863	−0.572340
$383$	12.2903	0.628006	0.314003	−	0.949422i	$-0.398330\pi$
$383$	12.2903	0.628006	0.314003	+	0.949422i	$0.398330\pi$
$384$	− 30.1570i	− 1.53895i
$385$	5.89183i	0.300276i
$386$	− 39.9274i	− 2.03225i
$387$	0.0333126	0.00169338
$388$	34.6637i	1.75978i
$389$	−10.6413	−0.539536	−0.269768	−	0.962925i	$-0.586947\pi$
$389$	−10.6413	−0.539536	−0.269768	+	0.962925i	$0.586947\pi$
$390$	−20.8414	−1.05534
$391$	0	0
$392$	0.881288	0.0445117
$393$	−21.0177	−1.06020
$394$	49.5137i	2.49446i
$395$	17.0477	0.857764
$396$	− 0.696644i	− 0.0350077i
$397$	− 24.0554i	− 1.20731i	−0.797247	−	0.603654i	$-0.793711\pi$
$397$	− 24.0554i	− 1.20731i	0.797247	−	0.603654i	$-0.206289\pi$
$398$	− 0.548970i	− 0.0275174i
$399$	−39.3476	−1.96984
$400$	0.261974	0.0130987
$401$	22.9193i	1.14453i	0.820067	+	0.572267i	$0.193936\pi$
$401$	22.9193i	1.14453i	−0.820067	+	0.572267i	$0.806064\pi$
$402$	10.3493i	0.516178i
$403$	− 4.07603i	− 0.203041i
$404$	−12.5670	−0.625232
$405$	− 8.68755i	− 0.431688i
$406$	−8.08435	−0.401220
$407$	17.8541	0.884995
$408$	0	0
$409$	−7.50913	−0.371303	−0.185651	−	0.982616i	$-0.559439\pi$
$409$	−7.50913	−0.371303	−0.185651	+	0.982616i	$0.559439\pi$
$410$	0.0297365	0.00146858
$411$	− 22.4247i	− 1.10613i
$412$	22.7699	1.12179
$413$	35.6128i	1.75239i
$414$	1.22100i	0.0600088i
$415$	3.35585i	0.164732i
$416$	32.0124	1.56954
$417$	1.20267	0.0588949
$418$	− 42.2667i	− 2.06733i
$419$	13.3993i	0.654598i	0.944921	+	0.327299i	$0.106138\pi$
$419$	13.3993i	0.654598i	−0.944921	+	0.327299i	$0.893862\pi$
$420$	14.6435i	0.714531i
$421$	−20.5078	−0.999491	−0.499745	−	0.866172i	$-0.666573\pi$
$421$	−20.5078	−0.999491	−0.499745	+	0.866172i	$0.666573\pi$
$422$	21.4210i	1.04276i
$423$	1.14263	0.0555566
$424$	−21.5293	−1.04556
$425$	0	0
$426$	−7.24862	−0.351197
$427$	−19.5095	−0.944130
$428$	11.7551i	0.568204i
$429$	−19.9353	−0.962486
$430$	− 0.752183i	− 0.0362735i
$431$	8.98646i	0.432863i	0.976298	+	0.216431i	$0.0694417\pi$
$431$	8.98646i	0.432863i	−0.976298	+	0.216431i	$0.930558\pi$
$432$	1.38315i	0.0665467i
$433$	−26.3738	−1.26744	−0.633721	−	0.773562i	$-0.718473\pi$
$433$	−26.3738	−1.26744	−0.633721	+	0.773562i	$0.718473\pi$
$434$	−4.66694	−0.224020
$435$	− 2.23477i	− 0.107149i
$436$	− 8.70001i	− 0.416655i
$437$	45.4596i	2.17463i
$438$	46.4039	2.21726
$439$	29.3558i	1.40108i	0.713615	+	0.700538i	$0.247057\pi$
$439$	29.3558i	1.40108i	−0.713615	+	0.700538i	$0.752943\pi$
$440$	−5.82660	−0.277772
$441$	−0.0331693	−0.00157949
$442$	0	0
$443$	−23.4483	−1.11406	−0.557030	−	0.830492i	$-0.688059\pi$
$443$	−23.4483	−1.11406	−0.557030	+	0.830492i	$0.688059\pi$
$444$	44.3744	2.10592
$445$	− 10.8909i	− 0.516279i
$446$	−33.7504	−1.59813
$447$	− 12.6227i	− 0.597032i
$448$	− 35.2350i	− 1.66470i
$449$	− 11.8222i	− 0.557926i	−0.960302	−	0.278963i	$-0.910009\pi$
$449$	− 11.8222i	− 0.557926i	0.960302	−	0.278963i	$-0.0899907\pi$
$450$	0.229265	0.0108077
$451$	0.0284437	0.00133936
$452$	− 4.81230i	− 0.226351i
$453$	− 3.73371i	− 0.175425i
$454$	22.3692i	1.04984i
$455$	−14.5642	−0.682779
$456$	− 38.9119i	− 1.82222i
$457$	3.31478	0.155059	0.0775293	−	0.996990i	$-0.475297\pi$
$457$	3.31478	0.155059	0.0775293	+	0.996990i	$0.475297\pi$
$458$	−28.2847	−1.32166
$459$	0	0
$460$	16.9182	0.788813
$461$	−36.2559	−1.68860	−0.844302	−	0.535868i	$-0.819984\pi$
$461$	−36.2559	−1.68860	−0.844302	+	0.535868i	$0.819984\pi$
$462$	22.8254i	1.06193i
$463$	10.9581	0.509268	0.254634	−	0.967038i	$-0.418045\pi$
$463$	10.9581	0.509268	0.254634	+	0.967038i	$0.418045\pi$
$464$	0.343834i	0.0159621i
$465$	− 1.29009i	− 0.0598265i
$466$	− 28.6253i	− 1.32604i
$467$	34.3919	1.59147	0.795734	−	0.605647i	$-0.207086\pi$
$467$	34.3919	1.59147	0.795734	+	0.605647i	$0.207086\pi$
$468$	1.72205	0.0796019
$469$	7.23222i	0.333953i
$470$	− 25.8001i	− 1.19007i
$471$	− 9.08791i	− 0.418749i
$472$	−35.2185	−1.62106
$473$	− 0.719482i	− 0.0330818i
$474$	66.0442	3.03351
$475$	8.53588	0.391653
$476$	0	0
$477$	0.810307	0.0371014
$478$	35.9409	1.64390
$479$	− 27.5154i	− 1.25721i	−0.777724	−	0.628606i	$-0.783626\pi$
$479$	− 27.5154i	− 1.25721i	0.777724	−	0.628606i	$-0.216374\pi$
$480$	10.1322	0.462468
$481$	44.1340i	2.01234i
$482$	19.2322i	0.876004i
$483$	− 24.5497i	− 1.11705i
$484$	19.8977	0.904440
$485$	−10.9119	−0.495482
$486$	2.38158i	0.108030i
$487$	10.6229i	0.481368i	0.970604	+	0.240684i	$0.0773718\pi$
$487$	10.6229i	0.481368i	−0.970604	+	0.240684i	$0.922628\pi$
$488$	− 19.2935i	− 0.873375i
$489$	13.2553	0.599427
$490$	0.748948i	0.0338340i
$491$	−36.2103	−1.63415	−0.817074	−	0.576533i	$-0.804405\pi$
$491$	−36.2103	−1.63415	−0.817074	+	0.576533i	$0.804405\pi$
$492$	0.0706938	0.00318712
$493$	0	0
$494$	104.480	4.70078
$495$	0.219298	0.00985671
$496$	0.198489i	0.00891242i
$497$	−5.06541	−0.227215
$498$	13.0008i	0.582580i
$499$	− 20.8740i	− 0.934447i	−0.884139	−	0.467224i	$-0.845254\pi$
$499$	− 20.8740i	− 0.934447i	0.884139	−	0.467224i	$-0.154746\pi$
$500$	− 3.17670i	− 0.142066i
$501$	−7.52880	−0.336362
$502$	−33.6509	−1.50191
$503$	7.63988i	0.340645i	0.985388	+	0.170323i	$0.0544810\pi$
$503$	7.63988i	0.340645i	−0.985388	+	0.170323i	$0.945519\pi$
$504$	− 0.730351i	− 0.0325324i
$505$	− 3.95599i	− 0.176039i
$506$	26.3710	1.17233
$507$	− 27.1433i	− 1.20548i
$508$	−13.1869	−0.585074
$509$	−20.9805	−0.929945	−0.464972	−	0.885325i	$-0.653936\pi$
$509$	−20.9805	−0.929945	−0.464972	+	0.885325i	$0.653936\pi$
$510$	0	0
$511$	32.4275	1.43451
$512$	−2.96165	−0.130888
$513$	45.0671i	1.98976i
$514$	31.5533	1.39175
$515$	7.16779i	0.315851i
$516$	− 1.78820i	− 0.0787209i
$517$	− 24.6784i	− 1.08536i
$518$	50.5323	2.22026
$519$	−24.8917	−1.09262
$520$	− 14.4029i	− 0.631610i
$521$	− 6.11774i	− 0.268023i	−0.990980	−	0.134012i	$-0.957214\pi$
$521$	− 6.11774i	− 0.268023i	0.990980	−	0.134012i	$-0.0427859\pi$
$522$	0.300905i	0.0131703i
$523$	−5.40916	−0.236526	−0.118263	−	0.992982i	$-0.537733\pi$
$523$	−5.40916	−0.236526	−0.118263	+	0.992982i	$0.537733\pi$
$524$	− 39.2120i	− 1.71298i
$525$	−4.60966	−0.201182
$526$	0.870591	0.0379596
$527$	0	0
$528$	0.970784	0.0422480
$529$	−5.36307	−0.233177
$530$	− 18.2963i	− 0.794742i
$531$	1.32553	0.0575232
$532$	− 73.4095i	− 3.18271i
$533$	0.0703107i	0.00304550i
$534$	− 42.1923i	− 1.82584i
$535$	−3.70041	−0.159983
$536$	−7.15215	−0.308926
$537$	− 9.24423i	− 0.398918i
$538$	49.8154i	2.14769i
$539$	0.716388i	0.0308570i
$540$	16.7721	0.721755
$541$	20.4944i	0.881124i	0.897722	+	0.440562i	$0.145221\pi$
$541$	20.4944i	0.881124i	−0.897722	+	0.440562i	$0.854779\pi$
$542$	−38.5538	−1.65603
$543$	31.2419	1.34072
$544$	0	0
$545$	2.73869	0.117313
$546$	−56.4227	−2.41467
$547$	− 10.1545i	− 0.434175i	−0.976152	−	0.217088i	$-0.930344\pi$
$547$	− 10.1545i	− 0.434175i	0.976152	−	0.217088i	$-0.0696557\pi$
$548$	41.8371	1.78719
$549$	0.726156i	0.0309916i
$550$	− 4.95164i	− 0.211139i
$551$	11.2031i	0.477270i
$552$	24.2779	1.03334
$553$	46.1523	1.96260
$554$	55.6479i	2.36425i
$555$	13.9687i	0.592939i
$556$	2.24378i	0.0951575i
$557$	30.1117	1.27587	0.637937	−	0.770088i	$-0.279788\pi$
$557$	30.1117	1.27587	0.637937	+	0.770088i	$0.279788\pi$
$558$	0.173707i	0.00735360i
$559$	1.77851	0.0752228
$560$	0.709227	0.0299703
$561$	0	0
$562$	−30.0363	−1.26700
$563$	−12.8482	−0.541486	−0.270743	−	0.962652i	$-0.587269\pi$
$563$	−12.8482	−0.541486	−0.270743	+	0.962652i	$0.587269\pi$
$564$	− 61.3355i	− 2.58269i
$565$	1.51487	0.0637312
$566$	31.3372i	1.31720i
$567$	− 23.5193i	− 0.987719i
$568$	− 5.00933i	− 0.210187i
$569$	−19.6602	−0.824196	−0.412098	−	0.911139i	$-0.635204\pi$
$569$	−19.6602	−0.824196	−0.412098	+	0.911139i	$0.635204\pi$
$570$	33.0687	1.38509
$571$	− 20.2253i	− 0.846403i	−0.906036	−	0.423202i	$-0.860906\pi$
$571$	− 20.2253i	− 0.846403i	0.906036	−	0.423202i	$-0.139094\pi$
$572$	− 37.1927i	− 1.55510i
$573$	− 8.37146i	− 0.349723i
$574$	0.0805040	0.00336017
$575$	5.32570i	0.222097i
$576$	−1.31147	−0.0546446
$577$	34.5597	1.43874	0.719369	−	0.694628i	$-0.244431\pi$
$577$	34.5597	1.43874	0.719369	+	0.694628i	$0.244431\pi$
$578$	0	0
$579$	29.8804	1.24179
$580$	4.16934	0.173122
$581$	9.08509i	0.376913i
$582$	−42.2734	−1.75229
$583$	− 17.5009i	− 0.724814i
$584$	32.0685i	1.32700i
$585$	0.542088i	0.0224126i
$586$	−48.0916	−1.98665
$587$	−23.7081	−0.978540	−0.489270	−	0.872132i	$-0.662737\pi$
$587$	−23.7081	−0.978540	−0.489270	+	0.872132i	$0.662737\pi$
$588$	1.78050i	0.0734268i
$589$	6.46736i	0.266483i
$590$	− 29.9299i	− 1.23219i
$591$	−37.0545	−1.52422
$592$	− 2.14918i	− 0.0883308i
$593$	11.5371	0.473770	0.236885	−	0.971538i	$-0.423873\pi$
$593$	11.5371	0.473770	0.236885	+	0.971538i	$0.423873\pi$
$594$	26.1433	1.07267
$595$	0	0
$596$	23.5497	0.964635
$597$	0.410831	0.0168142
$598$	65.1871i	2.66570i
$599$	−9.83844	−0.401988	−0.200994	−	0.979593i	$-0.564417\pi$
$599$	−9.83844	−0.401988	−0.200994	+	0.979593i	$0.564417\pi$
$600$	− 4.55863i	− 0.186105i
$601$	10.1246i	0.412991i	0.978448	+	0.206496i	$0.0662059\pi$
$601$	10.1246i	0.412991i	−0.978448	+	0.206496i	$0.933794\pi$
$602$	− 2.03634i	− 0.0829952i
$603$	0.269188	0.0109622
$604$	6.96587	0.283437
$605$	6.26363i	0.254653i
$606$	− 15.3258i	− 0.622569i
$607$	0.707519i	0.0287173i	0.999897	+	0.0143587i	$0.00457066\pi$
$607$	0.707519i	0.0287173i	−0.999897	+	0.0143587i	$0.995429\pi$
$608$	−50.7936	−2.05995
$609$	− 6.05007i	− 0.245161i
$610$	16.3963	0.663865
$611$	61.0032	2.46792
$612$	0	0
$613$	−28.1775	−1.13808	−0.569039	−	0.822310i	$-0.692685\pi$
$613$	−28.1775	−1.13808	−0.569039	+	0.822310i	$0.692685\pi$
$614$	52.8010	2.13088
$615$	0.0222538i	0 0.000897362i
$616$	−15.7740	−0.635554
$617$	39.8676i	1.60501i	0.596645	+	0.802505i	$0.296500\pi$
$617$	39.8676i	1.60501i	−0.596645	+	0.802505i	$0.703500\pi$
$618$	27.7686i	1.11702i
$619$	37.4591i	1.50561i	0.658244	+	0.752804i	$0.271299\pi$
$619$	37.4591i	1.50561i	−0.658244	+	0.752804i	$0.728701\pi$
$620$	2.40688	0.0966627
$621$	−28.1182	−1.12834
$622$	43.4409i	1.74182i
$623$	− 29.4844i	− 1.18127i
$624$	2.39971i	0.0960651i
$625$	1.00000	0.0400000
$626$	− 72.5940i	− 2.90144i
$627$	31.6310	1.26322
$628$	16.9550	0.676579
$629$	0	0
$630$	0.620677	0.0247284
$631$	10.5005	0.418020	0.209010	−	0.977913i	$-0.432976\pi$
$631$	10.5005	0.418020	0.209010	+	0.977913i	$0.432976\pi$
$632$	45.6414i	1.81552i
$633$	−16.0308	−0.637167
$634$	− 16.7271i	− 0.664316i
$635$	− 4.15113i	− 0.164732i
$636$	− 43.4966i	− 1.72475i
$637$	−1.77086	−0.0701639
$638$	6.49891	0.257294
$639$	0.188538i	0.00745845i
$640$	17.7112i	0.700095i
$641$	− 44.9746i	− 1.77639i	−0.459466	−	0.888195i	$-0.651959\pi$
$641$	− 44.9746i	− 1.77639i	0.459466	−	0.888195i	$-0.348041\pi$
$642$	−14.3357	−0.565784
$643$	− 3.17835i	− 0.125342i	−0.998034	−	0.0626710i	$-0.980038\pi$
$643$	− 3.17835i	− 0.125342i	0.998034	−	0.0626710i	$-0.0199619\pi$
$644$	45.8016	1.80484
$645$	0.562910	0.0221645
$646$	0	0
$647$	3.28494	0.129144	0.0645721	−	0.997913i	$-0.479432\pi$
$647$	3.28494	0.129144	0.0645721	+	0.997913i	$0.479432\pi$
$648$	23.2589	0.913697
$649$	− 28.6287i	− 1.12377i
$650$	12.2401	0.480096
$651$	− 3.49259i	− 0.136885i
$652$	24.7301i	0.968505i
$653$	− 33.9120i	− 1.32708i	−0.748140	−	0.663540i	$-0.769053\pi$
$653$	− 33.9120i	− 1.32708i	0.748140	−	0.663540i	$-0.230947\pi$
$654$	10.6099	0.414880
$655$	12.3436	0.482305
$656$	− 0.00342390i	0 0.000133681i
$657$	− 1.20697i	− 0.0470885i
$658$	− 69.8471i	− 2.72292i
$659$	27.5756	1.07419	0.537097	−	0.843521i	$-0.319521\pi$
$659$	27.5756	1.07419	0.537097	+	0.843521i	$0.319521\pi$
$660$	− 11.7717i	− 0.458215i
$661$	19.4561	0.756756	0.378378	−	0.925651i	$-0.376482\pi$
$661$	19.4561	0.756756	0.378378	+	0.925651i	$0.376482\pi$
$662$	37.2106	1.44623
$663$	0	0
$664$	−8.98451	−0.348667
$665$	23.1087	0.896118
$666$	− 1.88085i	− 0.0728813i
$667$	−6.98985	−0.270648
$668$	− 14.0463i	− 0.543466i
$669$	− 25.2577i	− 0.976520i
$670$	− 6.07814i	− 0.234819i
$671$	15.6834	0.605452
$672$	27.4302	1.05814
$673$	− 2.96580i	− 0.114323i	−0.998365	−	0.0571616i	$-0.981795\pi$
$673$	− 2.96580i	− 0.114323i	0.998365	−	0.0571616i	$-0.0182050\pi$
$674$	− 6.21668i	− 0.239458i
$675$	5.27972i	0.203216i
$676$	50.6404	1.94771
$677$	− 4.50634i	− 0.173193i	−0.996243	−	0.0865964i	$-0.972401\pi$
$677$	− 4.50634i	− 0.173193i	0.996243	−	0.0865964i	$-0.0275991\pi$
$678$	5.86873	0.225387
$679$	−29.5411	−1.13368
$680$	0	0
$681$	−16.7404	−0.641492
$682$	3.75170	0.143660
$683$	− 48.5495i	− 1.85769i	−0.370464	−	0.928847i	$-0.620801\pi$
$683$	− 48.5495i	− 1.85769i	0.370464	−	0.928847i	$-0.379199\pi$
$684$	−2.73235	−0.104474
$685$	13.1700i	0.503199i
$686$	− 41.0898i	− 1.56881i
$687$	− 21.1674i	− 0.807585i
$688$	−0.0866074	−0.00330188
$689$	43.2609	1.64811
$690$	20.6322i	0.785453i
$691$	− 1.63196i	− 0.0620827i	−0.999518	−	0.0310413i	$-0.990118\pi$
$691$	− 1.63196i	− 0.0620827i	0.999518	−	0.0310413i	$-0.00988235\pi$
$692$	− 46.4396i	− 1.76537i
$693$	0.593693	0.0225525
$694$	− 43.1406i	− 1.63759i
$695$	−0.706325	−0.0267924
$696$	5.98309	0.226788
$697$	0	0
$698$	−73.5769	−2.78493
$699$	21.4222	0.810264
$700$	− 8.60011i	− 0.325053i
$701$	−40.4449	−1.52758	−0.763792	−	0.645463i	$-0.776664\pi$
$701$	−40.4449	−1.52758	−0.763792	+	0.645463i	$0.776664\pi$
$702$	64.6242i	2.43908i
$703$	− 70.0267i	− 2.64111i
$704$	28.3250i	1.06754i
$705$	19.3079	0.727179
$706$	29.7865	1.12103
$707$	− 10.7098i	− 0.402785i
$708$	− 71.1535i	− 2.67411i
$709$	47.3049i	1.77657i	0.459291	+	0.888286i	$0.348103\pi$
$709$	47.3049i	1.77657i	−0.459291	+	0.888286i	$0.651897\pi$
$710$	4.25710	0.159766
$711$	− 1.71782i	− 0.0644234i
$712$	29.1580	1.09274
$713$	−4.03511	−0.151116
$714$	0	0
$715$	11.7080	0.437853
$716$	17.2467	0.644539
$717$	26.8970i	1.00449i
$718$	19.6640	0.733856
$719$	− 8.17117i	− 0.304733i	−0.988324	−	0.152367i	$-0.951311\pi$
$719$	− 8.17117i	− 0.304733i	0.988324	−	0.152367i	$-0.0486894\pi$
$720$	− 0.0263979i	0 0.000983793i
$721$	19.4050i	0.722679i
$722$	−122.547	−4.56074
$723$	−14.3928	−0.535273
$724$	58.2871i	2.16622i
$725$	1.31247i	0.0487441i
$726$	24.2658i	0.900587i
$727$	−18.8237	−0.698133	−0.349067	−	0.937098i	$-0.613501\pi$
$727$	−18.8237	−0.698133	−0.349067	+	0.937098i	$0.613501\pi$
$728$	− 38.9923i	− 1.44515i
$729$	−27.8449	−1.03129
$730$	−27.2529	−1.00867
$731$	0	0
$732$	38.9795	1.44072
$733$	−25.9444	−0.958278	−0.479139	−	0.877739i	$-0.659051\pi$
$733$	−25.9444	−0.958278	−0.479139	+	0.877739i	$0.659051\pi$
$734$	− 15.0401i	− 0.555140i
$735$	−0.560489	−0.0206739
$736$	− 31.6911i	− 1.16815i
$737$	− 5.81389i	− 0.214158i
$738$	− 0.00299641i	0 0.000110299i
$739$	−18.9248	−0.696162	−0.348081	−	0.937465i	$-0.613167\pi$
$739$	−18.9248	−0.696162	−0.348081	+	0.937465i	$0.613167\pi$
$740$	−26.0610	−0.958022
$741$	78.1895i	2.87236i
$742$	− 49.5327i	− 1.81840i
$743$	0.0747338i	0.00274172i	0.999999	+	0.00137086i	$0.000436358\pi$
$743$	0.0747338i	0.00274172i	−0.999999	+	0.00137086i	$0.999564\pi$
$744$	3.45392	0.126627
$745$	7.41327i	0.271601i
$746$	16.2531	0.595069
$747$	0.338153	0.0123724
$748$	0	0
$749$	−10.0179	−0.366047
$750$	3.87408	0.141461
$751$	31.3032i	1.14227i	0.820856	+	0.571135i	$0.193497\pi$
$751$	31.3032i	1.14227i	−0.820856	+	0.571135i	$0.806503\pi$
$752$	−2.97066	−0.108329
$753$	− 25.1832i	− 0.917728i
$754$	16.0648i	0.585046i
$755$	2.19280i	0.0798042i
$756$	45.4061	1.65140
$757$	10.9290	0.397223	0.198612	−	0.980078i	$-0.436357\pi$
$757$	10.9290	0.397223	0.198612	+	0.980078i	$0.436357\pi$
$758$	− 16.3641i	− 0.594369i
$759$	19.7352i	0.716342i
$760$	22.8529i	0.828961i
$761$	−34.7125	−1.25833	−0.629164	−	0.777272i	$-0.716603\pi$
$761$	−34.7125	−1.25833	−0.629164	+	0.777272i	$0.716603\pi$
$762$	− 16.0818i	− 0.582582i
$763$	7.41431	0.268416
$764$	15.6184	0.565053
$765$	0	0
$766$	−27.9634	−1.01036
$767$	70.7680	2.55528
$768$	24.2925i	0.876581i
$769$	28.5727	1.03036	0.515178	−	0.857083i	$-0.327726\pi$
$769$	28.5727	1.03036	0.515178	+	0.857083i	$0.327726\pi$
$770$	− 13.4053i	− 0.483094i
$771$	23.6135i	0.850418i
$772$	55.7469i	2.00638i
$773$	−31.3773	−1.12856	−0.564281	−	0.825583i	$-0.690847\pi$
$773$	−31.3773	−1.12856	−0.564281	+	0.825583i	$0.690847\pi$
$774$	−0.0757941	−0.00272436
$775$	0.757667i	0.0272162i
$776$	− 29.2140i	− 1.04872i
$777$	37.8167i	1.35667i
$778$	24.2115	0.868025
$779$	− 0.111561i	− 0.00399708i
$780$	29.0989	1.04191
$781$	4.07202	0.145708
$782$	0	0
$783$	−6.92950	−0.247640
$784$	0.0862349	0.00307982
$785$	5.33731i	0.190497i
$786$	47.8201	1.70569
$787$	19.5079i	0.695380i	0.937610	+	0.347690i	$0.113034\pi$
$787$	19.5079i	0.695380i	−0.937610	+	0.347690i	$0.886966\pi$
$788$	− 69.1314i	− 2.46270i
$789$	0.651523i	0.0231948i
$790$	−38.7876	−1.38000
$791$	4.10113	0.145819
$792$	0.587121i	0.0208624i
$793$	38.7683i	1.37670i
$794$	54.7318i	1.94236i
$795$	13.6924	0.485619
$796$	0.766476i	0.0271670i
$797$	16.1734	0.572892	0.286446	−	0.958096i	$-0.407526\pi$
$797$	16.1734	0.572892	0.286446	+	0.958096i	$0.407526\pi$
$798$	89.5250	3.16915
$799$	0	0
$800$	−5.95060	−0.210385
$801$	−1.09743	−0.0387757
$802$	− 52.1468i	− 1.84137i
$803$	−26.0681	−0.919922
$804$	− 14.4498i	− 0.509606i
$805$	14.4180i	0.508167i
$806$	9.27392i	0.326660i
$807$	−37.2802	−1.31233
$808$	10.5913	0.372600
$809$	37.5852i	1.32143i	0.750639	+	0.660713i	$0.229746\pi$
$809$	37.5852i	1.32143i	−0.750639	+	0.660713i	$0.770254\pi$
$810$	19.7662i	0.694515i
$811$	− 34.8967i	− 1.22539i	−0.790320	−	0.612695i	$-0.790085\pi$
$811$	− 34.8967i	− 1.22539i	0.790320	−	0.612695i	$-0.209915\pi$
$812$	11.2874	0.396111
$813$	− 28.8524i	− 1.01190i
$814$	−40.6223	−1.42381
$815$	−7.78483	−0.272691
$816$	0	0
$817$	−2.82193	−0.0987267
$818$	17.0850	0.597365
$819$	1.46757i	0.0512809i
$820$	−0.0415183	−0.00144988
$821$	− 30.8990i	− 1.07838i	−0.842184	−	0.539191i	$-0.818730\pi$
$821$	− 30.8990i	− 1.07838i	0.842184	−	0.539191i	$-0.181270\pi$
$822$	51.0215i	1.77958i
$823$	42.4684i	1.48035i	0.672412	+	0.740177i	$0.265258\pi$
$823$	42.4684i	1.48035i	−0.672412	+	0.740177i	$0.734742\pi$
$824$	−19.1901	−0.668520
$825$	3.70565	0.129014
$826$	− 81.0275i	− 2.81931i
$827$	20.4596i	0.711449i	0.934591	+	0.355724i	$0.115766\pi$
$827$	20.4596i	0.711449i	−0.934591	+	0.355724i	$0.884234\pi$
$828$	− 1.70477i	− 0.0592447i
$829$	−38.1368	−1.32455	−0.662273	−	0.749263i	$-0.730408\pi$
$829$	−38.1368	−1.32455	−0.662273	+	0.749263i	$0.730408\pi$
$830$	− 7.63534i	− 0.265027i
$831$	−41.6451	−1.44465
$832$	−70.0172	−2.42741
$833$	0	0
$834$	−2.73636	−0.0947522
$835$	4.42165	0.153017
$836$	59.0130i	2.04101i
$837$	−4.00027	−0.138270
$838$	− 30.4866i	− 1.05314i
$839$	37.7486i	1.30323i	0.758552	+	0.651613i	$0.225907\pi$
$839$	37.7486i	1.30323i	−0.758552	+	0.651613i	$0.774093\pi$
$840$	− 12.3413i	− 0.425816i
$841$	27.2774	0.940600
$842$	46.6602	1.60802
$843$	− 22.4782i	− 0.774190i
$844$	− 29.9082i	− 1.02948i
$845$	15.9412i	0.548394i
$846$	−2.59976	−0.0893814
$847$	16.9572i	0.582655i
$848$	−2.10667	−0.0723432
$849$	−23.4518	−0.804863
$850$	0	0
$851$	43.6910	1.49771
$852$	10.1206	0.346725
$853$	19.5859i	0.670608i	0.942110	+	0.335304i	$0.108839\pi$
$853$	19.5859i	0.670608i	−0.942110	+	0.335304i	$0.891161\pi$
$854$	44.3887	1.51895
$855$	− 0.860123i	− 0.0294156i
$856$	− 9.90701i	− 0.338615i
$857$	− 14.2754i	− 0.487639i	−0.969821	−	0.243820i	$-0.921599\pi$
$857$	− 14.2754i	− 0.487639i	0.969821	−	0.243820i	$-0.0784005\pi$
$858$	45.3575	1.54848
$859$	−50.2428	−1.71426	−0.857130	−	0.515099i	$-0.827755\pi$
$859$	−50.2428	−1.71426	−0.857130	+	0.515099i	$0.827755\pi$
$860$	1.05020i	0.0358116i
$861$	0.0602466i	0.00205320i
$862$	− 20.4463i	− 0.696405i
$863$	13.1930	0.449095	0.224548	−	0.974463i	$-0.427910\pi$
$863$	13.1930	0.449095	0.224548	+	0.974463i	$0.427910\pi$
$864$	− 31.4175i	− 1.06884i
$865$	14.6188	0.497055
$866$	60.0065	2.03910
$867$	0	0
$868$	6.51602	0.221168
$869$	−37.1013	−1.25858
$870$	5.08463i	0.172385i
$871$	14.3715	0.486960
$872$	7.33223i	0.248301i
$873$	1.09954i	0.0372137i
$874$	− 103.431i	− 3.49861i
$875$	2.70724	0.0915216
$876$	−64.7894	−2.18903
$877$	6.22918i	0.210345i	0.994454	+	0.105172i	$0.0335394\pi$
$877$	6.22918i	0.210345i	−0.994454	+	0.105172i	$0.966461\pi$
$878$	− 66.7914i	− 2.25410i
$879$	− 35.9902i	− 1.21392i
$880$	−0.570139	−0.0192194
$881$	30.3408i	1.02221i	0.859519	+	0.511104i	$0.170763\pi$
$881$	30.3408i	1.02221i	−0.859519	+	0.511104i	$0.829237\pi$
$882$	0.0754681	0.00254114
$883$	27.0762	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$883$	27.0762	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$884$	0	0
$885$	22.3986	0.752920
$886$	53.3504	1.79234
$887$	28.1506i	0.945205i	0.881276	+	0.472603i	$0.156685\pi$
$887$	28.1506i	0.945205i	−0.881276	+	0.472603i	$0.843315\pi$
$888$	−37.3981	−1.25500
$889$	− 11.2381i	− 0.376914i
$890$	24.7794i	0.830608i
$891$	18.9069i	0.633405i
$892$	47.1225	1.57778
$893$	−96.7927	−3.23905
$894$	28.7196i	0.960526i
$895$	5.42912i	0.181475i
$896$	47.9484i	1.60185i
$897$	−48.7839	−1.62885
$898$	26.8984i	0.897611i
$899$	−0.994419	−0.0331657
$900$	−0.320102	−0.0106701
$901$	0	0
$902$	−0.0647162	−0.00215481
$903$	1.52393	0.0507134
$904$	4.05573i	0.134891i
$905$	−18.3483	−0.609918
$906$	8.49507i	0.282230i
$907$	− 37.7225i	− 1.25256i	−0.779600	−	0.626278i	$-0.784577\pi$
$907$	− 37.7225i	− 1.25256i	0.779600	−	0.626278i	$-0.215423\pi$
$908$	− 31.2320i	− 1.03647i
$909$	−0.398628	−0.0132216
$910$	33.1369	1.09848
$911$	17.9517i	0.594765i	0.954758	+	0.297383i	$0.0961137\pi$
$911$	17.9517i	0.594765i	−0.954758	+	0.297383i	$0.903886\pi$
$912$	− 3.80757i	− 0.126081i
$913$	− 7.30340i	− 0.241707i
$914$	−7.54190	−0.249464
$915$	12.2704i	0.405648i
$916$	39.4913	1.30483
$917$	33.4172	1.10353
$918$	0	0
$919$	−25.0436	−0.826112	−0.413056	−	0.910706i	$-0.635539\pi$
$919$	−25.0436	−0.826112	−0.413056	+	0.910706i	$0.635539\pi$
$920$	−14.2583	−0.470084
$921$	39.5146i	1.30205i
$922$	82.4906	2.71668
$923$	10.0657i	0.331318i
$924$	− 31.8690i	− 1.04841i
$925$	− 8.20380i	− 0.269739i
$926$	−24.9324	−0.819328
$927$	0.722266	0.0237223
$928$	− 7.81001i	− 0.256376i
$929$	21.1573i	0.694148i	0.937838	+	0.347074i	$0.112825\pi$
$929$	21.1573i	0.694148i	−0.937838	+	0.347074i	$0.887175\pi$
$930$	2.93526i	0.0962510i
$931$	2.80979	0.0920871
$932$	39.9668i	1.30916i
$933$	−32.5098	−1.06432
$934$	−78.2497	−2.56041
$935$	0	0
$936$	−1.45132	−0.0474378
$937$	6.56252	0.214388	0.107194	−	0.994238i	$-0.465813\pi$
$937$	6.56252	0.214388	0.107194	+	0.994238i	$0.465813\pi$
$938$	− 16.4550i	− 0.537275i
$939$	54.3270	1.77289
$940$	36.0222i	1.17492i
$941$	18.5109i	0.603439i	0.953397	+	0.301719i	$0.0975605\pi$
$941$	18.5109i	0.603439i	−0.953397	+	0.301719i	$0.902439\pi$
$942$	20.6771i	0.673698i
$943$	0.0696049	0.00226665
$944$	−3.44617	−0.112163
$945$	14.2935i	0.464967i
$946$	1.63699i	0.0532232i
$947$	− 30.7922i	− 1.00061i	−0.865848	−	0.500307i	$-0.833221\pi$
$947$	− 30.7922i	− 1.00061i	0.865848	−	0.500307i	$-0.166779\pi$
$948$	−92.2113	−2.99488
$949$	− 64.4383i	− 2.09176i
$950$	−19.4212	−0.630106
$951$	12.5180	0.405924
$952$	0	0
$953$	−12.7261	−0.412240	−0.206120	−	0.978527i	$-0.566084\pi$
$953$	−12.7261	−0.412240	−0.206120	+	0.978527i	$0.566084\pi$
$954$	−1.84364	−0.0596900
$955$	4.91654i	0.159095i
$956$	−50.1809	−1.62297
$957$	4.86358i	0.157217i
$958$	62.6041i	2.02265i
$959$	35.6544i	1.15134i
$960$	−22.1609	−0.715241
$961$	30.4259	0.981482
$962$	− 100.415i	− 3.23752i
$963$	0.372874i	0.0120157i
$964$	− 26.8522i	− 0.864850i
$965$	−17.5487	−0.564912
$966$	55.8563i	1.79715i
$967$	−6.17919	−0.198709	−0.0993547	−	0.995052i	$-0.531678\pi$
$967$	−6.17919	−0.198709	−0.0993547	+	0.995052i	$0.531678\pi$
$968$	−16.7694	−0.538990
$969$	0	0
$970$	24.8270	0.797148
$971$	−8.79368	−0.282202	−0.141101	−	0.989995i	$-0.545064\pi$
$971$	−8.79368	−0.282202	−0.141101	+	0.989995i	$0.545064\pi$
$972$	− 3.32517i	− 0.106655i
$973$	−1.91219	−0.0613021
$974$	− 24.1695i	− 0.774442i
$975$	9.16010i	0.293358i
$976$	− 1.88789i	− 0.0604298i
$977$	14.5316	0.464906	0.232453	−	0.972608i	$-0.425325\pi$
$977$	14.5316	0.464906	0.232453	+	0.972608i	$0.425325\pi$
$978$	−30.1590	−0.964380
$979$	23.7021i	0.757524i
$980$	− 1.04569i	− 0.0334032i
$981$	− 0.275966i	− 0.00881091i
$982$	82.3870	2.62907
$983$	22.7184i	0.724603i	0.932061	+	0.362302i	$0.118009\pi$
$983$	22.7184i	0.724603i	−0.932061	+	0.362302i	$0.881991\pi$
$984$	−0.0595796	−0.00189933
$985$	21.7620	0.693395
$986$	0	0
$987$	52.2713	1.66381
$988$	−145.876	−4.64093
$989$	− 1.76065i	− 0.0559855i
$990$	−0.498955	−0.0158578
$991$	3.88859i	0.123525i	0.998091	+	0.0617626i	$0.0196722\pi$
$991$	3.88859i	0.123525i	−0.998091	+	0.0617626i	$0.980328\pi$
$992$	− 4.50857i	− 0.143147i
$993$	27.8472i	0.883705i
$994$	11.5250	0.365551
$995$	−0.241280	−0.00764910
$996$	− 18.1518i	− 0.575162i
$997$	31.1914i	0.987842i	0.869507	+	0.493921i	$0.164437\pi$
$997$	31.1914i	0.987842i	−0.869507	+	0.493921i	$0.835563\pi$
$998$	47.4932i	1.50337i
$999$	43.3137	1.37039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.d.i.866.5		24
17.4	even	4		1445.2.a.s.1.10	yes	12
17.13	even	4		1445.2.a.r.1.10	✓	12
17.16	even	2	inner	1445.2.d.i.866.6		24
85.4	even	4		7225.2.a.bn.1.3		12
85.64	even	4		7225.2.a.bo.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1445.2.a.r.1.10	✓	12	17.13	even	4
1445.2.a.s.1.10	yes	12	17.4	even	4
1445.2.d.i.866.5		24	1.1	even	1	trivial
1445.2.d.i.866.6		24	17.16	even	2	inner
7225.2.a.bn.1.3		12	85.4	even	4
7225.2.a.bo.1.3		12	85.64	even	4

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:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1445.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.27524 q^{2} -1.70271i q^{3} +3.17670 q^{4} +1.00000i q^{5} +3.87408i q^{6} +2.70724i q^{7} -2.67727 q^{8} +0.100765 q^{9} -2.27524i q^{10} -2.17632i q^{11} -5.40901i q^{12} +5.37970 q^{13} -6.15962i q^{14} +1.70271 q^{15} -0.261974 q^{16} -0.229265 q^{18} -8.53588 q^{19} +3.17670i q^{20} +4.60966 q^{21} +4.95164i q^{22} -5.32570i q^{23} +4.55863i q^{24} -1.00000 q^{25} -12.2401 q^{26} -5.27972i q^{27} +8.60011i q^{28} -1.31247i q^{29} -3.87408 q^{30} -0.757667i q^{31} +5.95060 q^{32} -3.70565 q^{33} -2.70724 q^{35} +0.320102 q^{36} +8.20380i q^{37} +19.4212 q^{38} -9.16010i q^{39} -2.67727i q^{40} +0.0130696i q^{41} -10.4881 q^{42} +0.330596 q^{43} -6.91352i q^{44} +0.100765i q^{45} +12.1172i q^{46} +11.3395 q^{47} +0.446067i q^{48} -0.329174 q^{49} +2.27524 q^{50} +17.0897 q^{52} +8.04151 q^{53} +12.0126i q^{54} +2.17632 q^{55} -7.24803i q^{56} +14.5342i q^{57} +2.98619i q^{58} +13.1546 q^{59} +5.40901 q^{60} +7.20639i q^{61} +1.72387i q^{62} +0.272797i q^{63} -13.0151 q^{64} +5.37970i q^{65} +8.43123 q^{66} +2.67143 q^{67} -9.06814 q^{69} +6.15962 q^{70} +1.87106i q^{71} -0.269777 q^{72} -11.9780i q^{73} -18.6656i q^{74} +1.70271i q^{75} -27.1160 q^{76} +5.89183 q^{77} +20.8414i q^{78} -17.0477i q^{79} -0.261974i q^{80} -8.68755 q^{81} -0.0297365i q^{82} +3.35585 q^{83} +14.6435 q^{84} -0.752183 q^{86} -2.23477 q^{87} +5.82660i q^{88} -10.8909 q^{89} -0.229265i q^{90} +14.5642i q^{91} -16.9182i q^{92} -1.29009 q^{93} -25.8001 q^{94} -8.53588i q^{95} -10.1322i q^{96} +10.9119i q^{97} +0.748948 q^{98} -0.219298i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} + 42 q^{4} - 24 q^{8} - 42 q^{9} + 18 q^{13} - 6 q^{15} + 78 q^{16} - 18 q^{18} - 54 q^{19} + 12 q^{21} - 24 q^{25} - 12 q^{26} - 18 q^{30} - 24 q^{32} + 12 q^{35} - 96 q^{36} - 6 q^{38}+ \cdots - 84 q^{98}+O(q^{100}) \) 24 * q - 6 * q^2 + 42 * q^4 - 24 * q^8 - 42 * q^9 + 18 * q^13 - 6 * q^15 + 78 * q^16 - 18 * q^18 - 54 * q^19 + 12 * q^21 - 24 * q^25 - 12 * q^26 - 18 * q^30 - 24 * q^32 + 12 * q^35 - 96 * q^36 - 6 * q^38 - 60 * q^42 - 90 * q^43 + 6 * q^47 - 24 * q^49 + 6 * q^50 + 102 * q^52 + 24 * q^53 - 12 * q^55 - 72 * q^59 - 12 * q^60 + 84 * q^64 + 78 * q^66 + 156 * q^67 - 132 * q^69 + 36 * q^70 - 24 * q^72 - 54 * q^76 + 78 * q^77 + 24 * q^81 - 54 * q^83 + 54 * q^84 + 36 * q^86 - 48 * q^87 + 12 * q^89 + 24 * q^93 - 42 * q^94 - 84 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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