LMFDB - Embedded newform 4805.2.a.bb.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4805,2,Mod(1,4805)] 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("4805.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4805 = 5 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4805.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,0,0,32,-24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.3681181712$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4805.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.43563 q^{2} +1.06814 q^{3} +0.0610318 q^{4} -1.00000 q^{5} +1.53345 q^{6} -3.78195 q^{7} -2.78364 q^{8} -1.85907 q^{9} -1.43563 q^{10} +0.820218 q^{11} +0.0651906 q^{12} +5.36168 q^{13} -5.42947 q^{14} -1.06814 q^{15} -4.11834 q^{16} +2.50630 q^{17} -2.66894 q^{18} -1.94218 q^{19} -0.0610318 q^{20} -4.03965 q^{21} +1.17753 q^{22} +2.37775 q^{23} -2.97332 q^{24} +1.00000 q^{25} +7.69739 q^{26} -5.19018 q^{27} -0.230819 q^{28} -2.61532 q^{29} -1.53345 q^{30} -0.345129 q^{32} +0.876109 q^{33} +3.59811 q^{34} +3.78195 q^{35} -0.113463 q^{36} +9.89738 q^{37} -2.78825 q^{38} +5.72703 q^{39} +2.78364 q^{40} -5.50338 q^{41} -5.79944 q^{42} +6.41683 q^{43} +0.0500594 q^{44} +1.85907 q^{45} +3.41357 q^{46} -9.22762 q^{47} -4.39897 q^{48} +7.30311 q^{49} +1.43563 q^{50} +2.67708 q^{51} +0.327233 q^{52} +13.4992 q^{53} -7.45117 q^{54} -0.820218 q^{55} +10.5276 q^{56} -2.07452 q^{57} -3.75463 q^{58} +13.1993 q^{59} -0.0651906 q^{60} -8.13940 q^{61} +7.03092 q^{63} +7.74120 q^{64} -5.36168 q^{65} +1.25777 q^{66} +6.13051 q^{67} +0.152964 q^{68} +2.53978 q^{69} +5.42947 q^{70} -8.59800 q^{71} +5.17499 q^{72} -0.169452 q^{73} +14.2090 q^{74} +1.06814 q^{75} -0.118535 q^{76} -3.10202 q^{77} +8.22190 q^{78} +9.07408 q^{79} +4.11834 q^{80} +0.0333819 q^{81} -7.90082 q^{82} +9.90962 q^{83} -0.246547 q^{84} -2.50630 q^{85} +9.21219 q^{86} -2.79353 q^{87} -2.28319 q^{88} -6.37980 q^{89} +2.66894 q^{90} -20.2776 q^{91} +0.145119 q^{92} -13.2474 q^{94} +1.94218 q^{95} -0.368646 q^{96} +13.9270 q^{97} +10.4846 q^{98} -1.52485 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} - 24 q^{5} + 8 q^{7} + 40 q^{9} + 8 q^{14} + 88 q^{16} - 64 q^{18} + 40 q^{19} - 32 q^{20} + 24 q^{25} + 72 q^{28} + 56 q^{33} - 8 q^{35} + 88 q^{36} - 72 q^{38} + 64 q^{39} - 56 q^{41}+ \cdots - 8 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 - 24 * q^5 + 8 * q^7 + 40 * q^9 + 8 * q^14 + 88 * q^16 - 64 * q^18 + 40 * q^19 - 32 * q^20 + 24 * q^25 + 72 * q^28 + 56 * q^33 - 8 * q^35 + 88 * q^36 - 72 * q^38 + 64 * q^39 - 56 * q^41 - 40 * q^45 - 32 * q^47 + 56 * q^49 + 88 * q^51 + 104 * q^56 + 8 * q^59 + 56 * q^63 + 160 * q^64 - 8 * q^66 + 96 * q^67 + 8 * q^69 - 8 * q^70 - 64 * q^71 - 120 * q^72 + 128 * q^76 - 128 * q^78 - 88 * q^80 + 72 * q^81 + 32 * q^82 - 32 * q^87 + 64 * q^90 + 56 * q^94 - 40 * q^95 + 48 * q^97 - 8 * q^98

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 $

$2$	1.43563	1.01514	0.507572	−	0.861610i	$-0.330543\pi$
$2$	1.43563	1.01514	0.507572	+	0.861610i	$0.330543\pi$
$3$	1.06814	0.616692	0.308346	−	0.951274i	$-0.400225\pi$
$3$	1.06814	0.616692	0.308346	+	0.951274i	$0.400225\pi$
$4$	0.0610318	0.0305159
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.53345	0.626030
$7$	−3.78195	−1.42944	−0.714721	−	0.699410i	$-0.753446\pi$
$7$	−3.78195	−1.42944	−0.714721	+	0.699410i	$0.753446\pi$
$8$	−2.78364	−0.984165
$9$	−1.85907	−0.619692
$10$	−1.43563	−0.453986
$11$	0.820218	0.247305	0.123653	−	0.992326i	$-0.460539\pi$
$11$	0.820218	0.247305	0.123653	+	0.992326i	$0.460539\pi$
$12$	0.0651906	0.0188189
$13$	5.36168	1.48706	0.743531	−	0.668701i	$-0.233149\pi$
$13$	5.36168	1.48706	0.743531	+	0.668701i	$0.233149\pi$
$14$	−5.42947	−1.45109
$15$	−1.06814	−0.275793
$16$	−4.11834	−1.02958
$17$	2.50630	0.607866	0.303933	−	0.952693i	$-0.401700\pi$
$17$	2.50630	0.607866	0.303933	+	0.952693i	$0.401700\pi$
$18$	−2.66894	−0.629076
$19$	−1.94218	−0.445567	−0.222784	−	0.974868i	$-0.571514\pi$
$19$	−1.94218	−0.445567	−0.222784	+	0.974868i	$0.571514\pi$
$20$	−0.0610318	−0.0136471
$21$	−4.03965	−0.881524
$22$	1.17753	0.251050
$23$	2.37775	0.495796	0.247898	−	0.968786i	$-0.420260\pi$
$23$	2.37775	0.495796	0.247898	+	0.968786i	$0.420260\pi$
$24$	−2.97332	−0.606926
$25$	1.00000	0.200000
$26$	7.69739	1.50958
$27$	−5.19018	−0.998850
$28$	−0.230819	−0.0436207
$29$	−2.61532	−0.485652	−0.242826	−	0.970070i	$-0.578074\pi$
$29$	−2.61532	−0.485652	−0.242826	+	0.970070i	$0.578074\pi$
$30$	−1.53345	−0.279969
$31$	0	0
$31$	0	0
$32$	−0.345129	−0.0610107
$33$	0.876109	0.152511
$34$	3.59811	0.617071
$35$	3.78195	0.639265
$36$	−0.113463	−0.0189105
$37$	9.89738	1.62712	0.813559	−	0.581482i	$-0.197527\pi$
$37$	9.89738	1.62712	0.813559	+	0.581482i	$0.197527\pi$
$38$	−2.78825	−0.452315
$39$	5.72703	0.917059
$40$	2.78364	0.440132
$41$	−5.50338	−0.859484	−0.429742	−	0.902952i	$-0.641396\pi$
$41$	−5.50338	−0.859484	−0.429742	+	0.902952i	$0.641396\pi$
$42$	−5.79944	−0.894873
$43$	6.41683	0.978557	0.489279	−	0.872128i	$-0.337260\pi$
$43$	6.41683	0.978557	0.489279	+	0.872128i	$0.337260\pi$
$44$	0.0500594	0.00754674
$45$	1.85907	0.277134
$46$	3.41357	0.503304
$47$	−9.22762	−1.34599	−0.672994	−	0.739648i	$-0.734992\pi$
$47$	−9.22762	−1.34599	−0.672994	+	0.739648i	$0.734992\pi$
$48$	−4.39897	−0.634936
$49$	7.30311	1.04330
$50$	1.43563	0.203029
$51$	2.67708	0.374866
$52$	0.327233	0.0453791
$53$	13.4992	1.85425	0.927126	−	0.374749i	$-0.122271\pi$
$53$	13.4992	1.85425	0.927126	+	0.374749i	$0.122271\pi$
$54$	−7.45117	−1.01398
$55$	−0.820218	−0.110598
$56$	10.5276	1.40681
$57$	−2.07452	−0.274778
$58$	−3.75463	−0.493007
$59$	13.1993	1.71840	0.859198	−	0.511643i	$-0.170963\pi$
$59$	13.1993	1.71840	0.859198	+	0.511643i	$0.170963\pi$
$60$	−0.0651906	−0.00841607
$61$	−8.13940	−1.04214	−0.521072	−	0.853513i	$-0.674468\pi$
$61$	−8.13940	−1.04214	−0.521072	+	0.853513i	$0.674468\pi$
$62$	0	0
$63$	7.03092	0.885813
$64$	7.74120	0.967650
$65$	−5.36168	−0.665035
$66$	1.25777	0.154820
$67$	6.13051	0.748961	0.374480	−	0.927235i	$-0.377821\pi$
$67$	6.13051	0.748961	0.374480	+	0.927235i	$0.377821\pi$
$68$	0.152964	0.0185496
$69$	2.53978	0.305753
$70$	5.42947	0.648946
$71$	−8.59800	−1.02039	−0.510197	−	0.860057i	$-0.670428\pi$
$71$	−8.59800	−1.02039	−0.510197	+	0.860057i	$0.670428\pi$
$72$	5.17499	0.609879
$73$	−0.169452	−0.0198328	−0.00991642	−	0.999951i	$-0.503157\pi$
$73$	−0.169452	−0.0198328	−0.00991642	+	0.999951i	$0.503157\pi$
$74$	14.2090	1.65176
$75$	1.06814	0.123338
$76$	−0.118535	−0.0135969
$77$	−3.10202	−0.353508
$78$	8.22190	0.930946
$79$	9.07408	1.02091	0.510457	−	0.859903i	$-0.329476\pi$
$79$	9.07408	1.02091	0.510457	+	0.859903i	$0.329476\pi$
$80$	4.11834	0.460444
$81$	0.0333819	0.00370910
$82$	−7.90082	−0.872500
$83$	9.90962	1.08772	0.543861	−	0.839175i	$-0.316962\pi$
$83$	9.90962	1.08772	0.543861	+	0.839175i	$0.316962\pi$
$84$	−0.246547	−0.0269005
$85$	−2.50630	−0.271846
$86$	9.21219	0.993376
$87$	−2.79353	−0.299498
$88$	−2.28319	−0.243389
$89$	−6.37980	−0.676258	−0.338129	−	0.941100i	$-0.609794\pi$
$89$	−6.37980	−0.676258	−0.338129	+	0.941100i	$0.609794\pi$
$90$	2.66894	0.281331
$91$	−20.2776	−2.12567
$92$	0.145119	0.0151297
$93$	0	0
$94$	−13.2474	−1.36637
$95$	1.94218	0.199264
$96$	−0.368646	−0.0376248
$97$	13.9270	1.41407	0.707035	−	0.707179i	$-0.250032\pi$
$97$	13.9270	1.41407	0.707035	+	0.707179i	$0.250032\pi$
$98$	10.4846	1.05910
$99$	−1.52485	−0.153253
$100$	0.0610318	0.00610318
$101$	2.41226	0.240029	0.120014	−	0.992772i	$-0.461706\pi$
$101$	2.41226	0.240029	0.120014	+	0.992772i	$0.461706\pi$
$102$	3.84329	0.380542
$103$	8.60350	0.847728	0.423864	−	0.905726i	$-0.360673\pi$
$103$	8.60350	0.847728	0.423864	+	0.905726i	$0.360673\pi$
$104$	−14.9250	−1.46352
$105$	4.03965	0.394230
$106$	19.3798	1.88233
$107$	8.78208	0.848995	0.424498	−	0.905429i	$-0.360451\pi$
$107$	8.78208	0.848995	0.424498	+	0.905429i	$0.360451\pi$
$108$	−0.316766	−0.0304808
$109$	4.01557	0.384622	0.192311	−	0.981334i	$-0.438402\pi$
$109$	4.01557	0.384622	0.192311	+	0.981334i	$0.438402\pi$
$110$	−1.17753	−0.112273
$111$	10.5718	1.00343
$112$	15.5753	1.47173
$113$	17.6814	1.66333	0.831665	−	0.555278i	$-0.187388\pi$
$113$	17.6814	1.66333	0.831665	+	0.555278i	$0.187388\pi$
$114$	−2.97825	−0.278939
$115$	−2.37775	−0.221727
$116$	−0.159618	−0.0148201
$117$	−9.96776	−0.921520
$118$	18.9492	1.74442
$119$	−9.47867	−0.868908
$120$	2.97332	0.271426
$121$	−10.3272	−0.938840
$122$	−11.6852	−1.05793
$123$	−5.87839	−0.530037
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	10.0938	0.899227
$127$	18.4032	1.63302	0.816509	−	0.577333i	$-0.195907\pi$
$127$	18.4032	1.63302	0.816509	+	0.577333i	$0.195907\pi$
$128$	11.8038	1.04331
$129$	6.85408	0.603468
$130$	−7.69739	−0.675105
$131$	−19.8446	−1.73383	−0.866914	−	0.498458i	$-0.833900\pi$
$131$	−19.8446	−1.73383	−0.866914	+	0.498458i	$0.833900\pi$
$132$	0.0534705	0.00465401
$133$	7.34523	0.636912
$134$	8.80114	0.760302
$135$	5.19018	0.446699
$136$	−6.97662	−0.598240
$137$	−12.0925	−1.03313	−0.516567	−	0.856247i	$-0.672790\pi$
$137$	−12.0925	−1.03313	−0.516567	+	0.856247i	$0.672790\pi$
$138$	3.64618	0.310383
$139$	8.38192	0.710945	0.355473	−	0.934687i	$-0.384320\pi$
$139$	8.38192	0.710945	0.355473	+	0.934687i	$0.384320\pi$
$140$	0.230819	0.0195078
$141$	−9.85641	−0.830059
$142$	−12.3435	−1.03585
$143$	4.39775	0.367758
$144$	7.65630	0.638025
$145$	2.61532	0.217190
$146$	−0.243270	−0.0201332
$147$	7.80076	0.643395
$148$	0.604055	0.0496530
$149$	−1.11588	−0.0914168	−0.0457084	−	0.998955i	$-0.514555\pi$
$149$	−1.11588	−0.0914168	−0.0457084	+	0.998955i	$0.514555\pi$
$150$	1.53345	0.125206
$151$	13.1547	1.07051	0.535257	−	0.844689i	$-0.320215\pi$
$151$	13.1547	1.07051	0.535257	+	0.844689i	$0.320215\pi$
$152$	5.40634	0.438512
$153$	−4.65939	−0.376689
$154$	−4.45335	−0.358861
$155$	0	0
$156$	0.349531	0.0279849
$157$	−24.3348	−1.94213	−0.971065	−	0.238817i	$-0.923241\pi$
$157$	−24.3348	−1.94213	−0.971065	+	0.238817i	$0.923241\pi$
$158$	13.0270	1.03637
$159$	14.4190	1.14350
$160$	0.345129	0.0272848
$161$	−8.99253	−0.708711
$162$	0.0479240	0.00376527
$163$	−10.8405	−0.849091	−0.424545	−	0.905407i	$-0.639566\pi$
$163$	−10.8405	−0.849091	−0.424545	+	0.905407i	$0.639566\pi$
$164$	−0.335882	−0.0262280
$165$	−0.876109	−0.0682050
$166$	14.2265	1.10419
$167$	16.6980	1.29213	0.646067	−	0.763281i	$-0.276413\pi$
$167$	16.6980	1.29213	0.646067	+	0.763281i	$0.276413\pi$
$168$	11.2449	0.867566
$169$	15.7476	1.21136
$170$	−3.59811	−0.275963
$171$	3.61066	0.276114
$172$	0.391631	0.0298616
$173$	−8.92663	−0.678679	−0.339340	−	0.940664i	$-0.610204\pi$
$173$	−8.92663	−0.678679	−0.339340	+	0.940664i	$0.610204\pi$
$174$	−4.01047	−0.304033
$175$	−3.78195	−0.285888
$176$	−3.37794	−0.254622
$177$	14.0987	1.05972
$178$	−9.15903	−0.686498
$179$	−0.108542	−0.00811285	−0.00405642	−	0.999992i	$-0.501291\pi$
$179$	−0.108542	−0.00811285	−0.00405642	+	0.999992i	$0.501291\pi$
$180$	0.113463	0.00845701
$181$	15.5247	1.15394	0.576970	−	0.816766i	$-0.304235\pi$
$181$	15.5247	1.15394	0.576970	+	0.816766i	$0.304235\pi$
$182$	−29.1111	−2.15786
$183$	−8.69403	−0.642681
$184$	−6.61881	−0.487945
$185$	−9.89738	−0.727670
$186$	0	0
$187$	2.05571	0.150328
$188$	−0.563179	−0.0410740
$189$	19.6290	1.42780
$190$	2.78825	0.202281
$191$	7.38103	0.534073	0.267036	−	0.963686i	$-0.413956\pi$
$191$	7.38103	0.534073	0.267036	+	0.963686i	$0.413956\pi$
$192$	8.26869	0.596742
$193$	−20.4545	−1.47234	−0.736172	−	0.676795i	$-0.763369\pi$
$193$	−20.4545	−1.47234	−0.736172	+	0.676795i	$0.763369\pi$
$194$	19.9940	1.43548
$195$	−5.72703	−0.410121
$196$	0.445722	0.0318373
$197$	10.7832	0.768271	0.384135	−	0.923277i	$-0.374500\pi$
$197$	10.7832	0.768271	0.384135	+	0.923277i	$0.374500\pi$
$198$	−2.18911	−0.155574
$199$	11.1747	0.792157	0.396078	−	0.918217i	$-0.370371\pi$
$199$	11.1747	0.792157	0.396078	+	0.918217i	$0.370371\pi$
$200$	−2.78364	−0.196833
$201$	6.54825	0.461878
$202$	3.46311	0.243664
$203$	9.89099	0.694211
$204$	0.163387	0.0114394
$205$	5.50338	0.384373
$206$	12.3514	0.860566
$207$	−4.42042	−0.307240
$208$	−22.0812	−1.53106
$209$	−1.59301	−0.110191
$210$	5.79944	0.400200
$211$	13.2764	0.913986	0.456993	−	0.889470i	$-0.348926\pi$
$211$	13.2764	0.913986	0.456993	+	0.889470i	$0.348926\pi$
$212$	0.823879	0.0565842
$213$	−9.18388	−0.629269
$214$	12.6078	0.861852
$215$	−6.41683	−0.437624
$216$	14.4476	0.983034
$217$	0	0
$218$	5.76487	0.390446
$219$	−0.180998	−0.0122307
$220$	−0.0500594	−0.00337501
$221$	13.4380	0.903935
$222$	15.1772	1.01863
$223$	0.335583	0.0224723	0.0112362	−	0.999937i	$-0.496423\pi$
$223$	0.335583	0.0224723	0.0112362	+	0.999937i	$0.496423\pi$
$224$	1.30526	0.0872113
$225$	−1.85907	−0.123938
$226$	25.3840	1.68852
$227$	9.97977	0.662381	0.331190	−	0.943564i	$-0.392550\pi$
$227$	9.97977	0.662381	0.331190	+	0.943564i	$0.392550\pi$
$228$	−0.126612	−0.00838509
$229$	−11.0510	−0.730269	−0.365135	−	0.930955i	$-0.618977\pi$
$229$	−11.0510	−0.730269	−0.365135	+	0.930955i	$0.618977\pi$
$230$	−3.41357	−0.225084
$231$	−3.31340	−0.218005
$232$	7.28010	0.477962
$233$	−5.55089	−0.363651	−0.181825	−	0.983331i	$-0.558201\pi$
$233$	−5.55089	−0.363651	−0.181825	+	0.983331i	$0.558201\pi$
$234$	−14.3100	−0.935475
$235$	9.22762	0.601944
$236$	0.805575	0.0524385
$237$	9.69240	0.629589
$238$	−13.6079	−0.882067
$239$	−14.1087	−0.912617	−0.456309	−	0.889822i	$-0.650829\pi$
$239$	−14.1087	−0.912617	−0.456309	+	0.889822i	$0.650829\pi$
$240$	4.39897	0.283952
$241$	2.28476	0.147174	0.0735872	−	0.997289i	$-0.476555\pi$
$241$	2.28476	0.147174	0.0735872	+	0.997289i	$0.476555\pi$
$242$	−14.8261	−0.953057
$243$	15.6062	1.00114
$244$	−0.496763	−0.0318020
$245$	−7.30311	−0.466579
$246$	−8.43919	−0.538063
$247$	−10.4134	−0.662586
$248$	0	0
$249$	10.5849	0.670789
$250$	−1.43563	−0.0907972
$251$	−12.5526	−0.792310	−0.396155	−	0.918184i	$-0.629656\pi$
$251$	−12.5526	−0.792310	−0.396155	+	0.918184i	$0.629656\pi$
$252$	0.429110	0.0270314
$253$	1.95028	0.122613
$254$	26.4201	1.65775
$255$	−2.67708	−0.167645
$256$	1.46341	0.0914634
$257$	−17.2138	−1.07377	−0.536884	−	0.843656i	$-0.680399\pi$
$257$	−17.2138	−1.07377	−0.536884	+	0.843656i	$0.680399\pi$
$258$	9.83992	0.612606
$259$	−37.4313	−2.32587
$260$	−0.327233	−0.0202941
$261$	4.86207	0.300955
$262$	−28.4894	−1.76008
$263$	12.6266	0.778589	0.389294	−	0.921113i	$-0.372719\pi$
$263$	12.6266	0.778589	0.389294	+	0.921113i	$0.372719\pi$
$264$	−2.43877	−0.150096
$265$	−13.4992	−0.829247
$266$	10.5450	0.646557
$267$	−6.81453	−0.417042
$268$	0.374156	0.0228552
$269$	15.3716	0.937225	0.468612	−	0.883404i	$-0.344754\pi$
$269$	15.3716	0.937225	0.468612	+	0.883404i	$0.344754\pi$
$270$	7.45117	0.453464
$271$	−12.7303	−0.773309	−0.386654	−	0.922225i	$-0.626369\pi$
$271$	−12.7303	−0.773309	−0.386654	+	0.922225i	$0.626369\pi$
$272$	−10.3218	−0.625849
$273$	−21.6593	−1.31088
$274$	−17.3604	−1.04878
$275$	0.820218	0.0494610
$276$	0.155007	0.00933034
$277$	−10.8740	−0.653358	−0.326679	−	0.945135i	$-0.605930\pi$
$277$	−10.8740	−0.653358	−0.326679	+	0.945135i	$0.605930\pi$
$278$	12.0333	0.721711
$279$	0	0
$280$	−10.5276	−0.629143
$281$	17.7639	1.05971	0.529853	−	0.848089i	$-0.322247\pi$
$281$	17.7639	1.05971	0.529853	+	0.848089i	$0.322247\pi$
$282$	−14.1501	−0.842629
$283$	−14.8889	−0.885052	−0.442526	−	0.896756i	$-0.645918\pi$
$283$	−14.8889	−0.885052	−0.442526	+	0.896756i	$0.645918\pi$
$284$	−0.524752	−0.0311383
$285$	2.07452	0.122884
$286$	6.31354	0.373327
$287$	20.8135	1.22858
$288$	0.641620	0.0378078
$289$	−10.7185	−0.630499
$290$	3.75463	0.220479
$291$	14.8760	0.872045
$292$	−0.0103420	−0.000605217 0
$293$	−8.63771	−0.504621	−0.252310	−	0.967646i	$-0.581190\pi$
$293$	−8.63771	−0.504621	−0.252310	+	0.967646i	$0.581190\pi$
$294$	11.1990	0.653139
$295$	−13.1993	−0.768490
$296$	−27.5507	−1.60135
$297$	−4.25708	−0.247021
$298$	−1.60200	−0.0928012
$299$	12.7487	0.737279
$300$	0.0651906	0.00376378
$301$	−24.2681	−1.39879
$302$	18.8853	1.08673
$303$	2.57663	0.148024
$304$	7.99856	0.458749
$305$	8.13940	0.466061
$306$	−6.68916	−0.382394
$307$	28.9790	1.65392	0.826961	−	0.562260i	$-0.190068\pi$
$307$	28.9790	1.65392	0.826961	+	0.562260i	$0.190068\pi$
$308$	−0.189322	−0.0107876
$309$	9.18976	0.522787
$310$	0	0
$311$	−22.3890	−1.26956	−0.634781	−	0.772692i	$-0.718910\pi$
$311$	−22.3890	−1.26956	−0.634781	+	0.772692i	$0.718910\pi$
$312$	−15.9420	−0.902538
$313$	17.7118	1.00113	0.500565	−	0.865699i	$-0.333126\pi$
$313$	17.7118	1.00113	0.500565	+	0.865699i	$0.333126\pi$
$314$	−34.9358	−1.97154
$315$	−7.03092	−0.396147
$316$	0.553808	0.0311541
$317$	−3.63158	−0.203970	−0.101985	−	0.994786i	$-0.532519\pi$
$317$	−3.63158	−0.203970	−0.101985	+	0.994786i	$0.532519\pi$
$318$	20.7004	1.16082
$319$	−2.14513	−0.120104
$320$	−7.74120	−0.432746
$321$	9.38050	0.523568
$322$	−12.9099	−0.719443
$323$	−4.86768	−0.270845
$324$	0.00203736	0.000113187 0
$325$	5.36168	0.297413
$326$	−15.5629	−0.861949
$327$	4.28920	0.237193
$328$	15.3194	0.845875
$329$	34.8984	1.92401
$330$	−1.25777	−0.0692378
$331$	25.1631	1.38309	0.691543	−	0.722335i	$-0.256931\pi$
$331$	25.1631	1.38309	0.691543	+	0.722335i	$0.256931\pi$
$332$	0.604803	0.0331929
$333$	−18.4000	−1.00831
$334$	23.9722	1.31170
$335$	−6.13051	−0.334945
$336$	16.6367	0.907604
$337$	−33.2513	−1.81131	−0.905657	−	0.424012i	$-0.860622\pi$
$337$	−33.2513	−1.81131	−0.905657	+	0.424012i	$0.860622\pi$
$338$	22.6078	1.22970
$339$	18.8863	1.02576
$340$	−0.152964	−0.00829563
$341$	0	0
$342$	5.18357	0.280295
$343$	−1.14636	−0.0618975
$344$	−17.8621	−0.963062
$345$	−2.53978	−0.136737
$346$	−12.8153	−0.688957
$347$	−23.6403	−1.26908	−0.634540	−	0.772890i	$-0.718810\pi$
$347$	−23.6403	−1.26908	−0.634540	+	0.772890i	$0.718810\pi$
$348$	−0.170494	−0.00913945
$349$	−36.4213	−1.94959	−0.974793	−	0.223111i	$-0.928379\pi$
$349$	−36.4213	−1.94959	−0.974793	+	0.223111i	$0.928379\pi$
$350$	−5.42947	−0.290218
$351$	−27.8281	−1.48535
$352$	−0.283081	−0.0150883
$353$	−6.59235	−0.350875	−0.175438	−	0.984491i	$-0.556134\pi$
$353$	−6.59235	−0.350875	−0.175438	+	0.984491i	$0.556134\pi$
$354$	20.2405	1.07577
$355$	8.59800	0.456334
$356$	−0.389371	−0.0206366
$357$	−10.1246	−0.535848
$358$	−0.155827	−0.00823570
$359$	13.6504	0.720439	0.360220	−	0.932868i	$-0.382702\pi$
$359$	13.6504	0.720439	0.360220	+	0.932868i	$0.382702\pi$
$360$	−5.17499	−0.272746
$361$	−15.2279	−0.801470
$362$	22.2877	1.17141
$363$	−11.0310	−0.578975
$364$	−1.23758	−0.0648667
$365$	0.169452	0.00886951
$366$	−12.4814	−0.652414
$367$	−23.0600	−1.20372	−0.601862	−	0.798600i	$-0.705574\pi$
$367$	−23.0600	−1.20372	−0.601862	+	0.798600i	$0.705574\pi$
$368$	−9.79239	−0.510464
$369$	10.2312	0.532615
$370$	−14.2090	−0.738689
$371$	−51.0531	−2.65054
$372$	0	0
$373$	−3.83488	−0.198562	−0.0992812	−	0.995059i	$-0.531654\pi$
$373$	−3.83488	−0.198562	−0.0992812	+	0.995059i	$0.531654\pi$
$374$	2.95124	0.152605
$375$	−1.06814	−0.0551586
$376$	25.6864	1.32467
$377$	−14.0225	−0.722195
$378$	28.1799	1.44942
$379$	23.7430	1.21960	0.609798	−	0.792557i	$-0.291251\pi$
$379$	23.7430	1.21960	0.609798	+	0.792557i	$0.291251\pi$
$380$	0.118535	0.00608072
$381$	19.6572	1.00707
$382$	10.5964	0.542160
$383$	−15.2682	−0.780170	−0.390085	−	0.920779i	$-0.627554\pi$
$383$	−15.2682	−0.780170	−0.390085	+	0.920779i	$0.627554\pi$
$384$	12.6081	0.643403
$385$	3.10202	0.158094
$386$	−29.3650	−1.49464
$387$	−11.9294	−0.606404
$388$	0.849989	0.0431516
$389$	−34.0822	−1.72804	−0.864018	−	0.503460i	$-0.832060\pi$
$389$	−34.0822	−1.72804	−0.864018	+	0.503460i	$0.832060\pi$
$390$	−8.22190	−0.416332
$391$	5.95935	0.301377
$392$	−20.3292	−1.02678
$393$	−21.1968	−1.06924
$394$	15.4807	0.779905
$395$	−9.07408	−0.456566
$396$	−0.0930642	−0.00467665
$397$	−19.2729	−0.967280	−0.483640	−	0.875267i	$-0.660686\pi$
$397$	−19.2729	−0.967280	−0.483640	+	0.875267i	$0.660686\pi$
$398$	16.0428	0.804153
$399$	7.84574	0.392778
$400$	−4.11834	−0.205917
$401$	30.3512	1.51566	0.757832	−	0.652450i	$-0.226259\pi$
$401$	30.3512	1.51566	0.757832	+	0.652450i	$0.226259\pi$
$402$	9.40085	0.468872
$403$	0	0
$404$	0.147225	0.00732470
$405$	−0.0333819	−0.00165876
$406$	14.1998	0.704724
$407$	8.11801	0.402395
$408$	−7.45202	−0.368930
$409$	−3.66946	−0.181443	−0.0907215	−	0.995876i	$-0.528917\pi$
$409$	−3.66946	−0.181443	−0.0907215	+	0.995876i	$0.528917\pi$
$410$	7.90082	0.390194
$411$	−12.9165	−0.637125
$412$	0.525088	0.0258692
$413$	−49.9189	−2.45635
$414$	−6.34608	−0.311893
$415$	−9.90962	−0.486444
$416$	−1.85047	−0.0907268
$417$	8.95307	0.438434
$418$	−2.28698	−0.111860
$419$	15.7437	0.769130	0.384565	−	0.923098i	$-0.374351\pi$
$419$	15.7437	0.769130	0.384565	+	0.923098i	$0.374351\pi$
$420$	0.246547	0.0120303
$421$	−36.7926	−1.79316	−0.896581	−	0.442879i	$-0.853957\pi$
$421$	−36.7926	−1.79316	−0.896581	+	0.442879i	$0.853957\pi$
$422$	19.0600	0.927827
$423$	17.1548	0.834097
$424$	−37.5768	−1.82489
$425$	2.50630	0.121573
$426$	−13.1846	−0.638798
$427$	30.7828	1.48968
$428$	0.535986	0.0259079
$429$	4.69742	0.226793
$430$	−9.21219	−0.444251
$431$	−5.38386	−0.259331	−0.129666	−	0.991558i	$-0.541390\pi$
$431$	−5.38386	−0.259331	−0.129666	+	0.991558i	$0.541390\pi$
$432$	21.3749	1.02840
$433$	11.4151	0.548574	0.274287	−	0.961648i	$-0.411558\pi$
$433$	11.4151	0.548574	0.274287	+	0.961648i	$0.411558\pi$
$434$	0	0
$435$	2.79353	0.133939
$436$	0.245078	0.0117371
$437$	−4.61803	−0.220910
$438$	−0.259847	−0.0124160
$439$	−2.45916	−0.117369	−0.0586847	−	0.998277i	$-0.518691\pi$
$439$	−2.45916	−0.117369	−0.0586847	+	0.998277i	$0.518691\pi$
$440$	2.28319	0.108847
$441$	−13.5770	−0.646525
$442$	19.2919	0.917623
$443$	−0.738016	−0.0350642	−0.0175321	−	0.999846i	$-0.505581\pi$
$443$	−0.738016	−0.0350642	−0.0175321	+	0.999846i	$0.505581\pi$
$444$	0.645216	0.0306206
$445$	6.37980	0.302432
$446$	0.481773	0.0228126
$447$	−1.19192	−0.0563760
$448$	−29.2768	−1.38320
$449$	16.1798	0.763570	0.381785	−	0.924251i	$-0.375309\pi$
$449$	16.1798	0.763570	0.381785	+	0.924251i	$0.375309\pi$
$450$	−2.66894	−0.125815
$451$	−4.51398	−0.212555
$452$	1.07913	0.0507580
$453$	14.0511	0.660177
$454$	14.3273	0.672411
$455$	20.2776	0.950628
$456$	5.77473	0.270426
$457$	10.9858	0.513896	0.256948	−	0.966425i	$-0.417283\pi$
$457$	10.9858	0.513896	0.256948	+	0.966425i	$0.417283\pi$
$458$	−15.8651	−0.741328
$459$	−13.0081	−0.607167
$460$	−0.145119	−0.00676619
$461$	31.5081	1.46748	0.733739	−	0.679431i	$-0.237773\pi$
$461$	31.5081	1.46748	0.733739	+	0.679431i	$0.237773\pi$
$462$	−4.75681	−0.221307
$463$	13.5310	0.628840	0.314420	−	0.949284i	$-0.398190\pi$
$463$	13.5310	0.628840	0.314420	+	0.949284i	$0.398190\pi$
$464$	10.7708	0.500020
$465$	0	0
$466$	−7.96902	−0.369158
$467$	−28.5922	−1.32309	−0.661544	−	0.749907i	$-0.730098\pi$
$467$	−28.5922	−1.32309	−0.661544	+	0.749907i	$0.730098\pi$
$468$	−0.608351	−0.0281210
$469$	−23.1852	−1.07060
$470$	13.2474	0.611059
$471$	−25.9930	−1.19769
$472$	−36.7420	−1.69119
$473$	5.26320	0.242002
$474$	13.9147	0.639123
$475$	−1.94218	−0.0891134
$476$	−0.578501	−0.0265155
$477$	−25.0959	−1.14906
$478$	−20.2549	−0.926437
$479$	12.6025	0.575823	0.287911	−	0.957657i	$-0.407039\pi$
$479$	12.6025	0.575823	0.287911	+	0.957657i	$0.407039\pi$
$480$	0.368646	0.0168263
$481$	53.0666	2.41963
$482$	3.28007	0.149403
$483$	−9.60529	−0.437056
$484$	−0.630291	−0.0286496
$485$	−13.9270	−0.632391
$486$	22.4047	1.01630
$487$	27.2541	1.23500	0.617501	−	0.786570i	$-0.288145\pi$
$487$	27.2541	1.23500	0.617501	+	0.786570i	$0.288145\pi$
$488$	22.6572	1.02564
$489$	−11.5791	−0.523627
$490$	−10.4846	−0.473644
$491$	−3.99429	−0.180260	−0.0901298	−	0.995930i	$-0.528728\pi$
$491$	−3.99429	−0.180260	−0.0901298	+	0.995930i	$0.528728\pi$
$492$	−0.358769	−0.0161746
$493$	−6.55476	−0.295211
$494$	−14.9497	−0.672620
$495$	1.52485	0.0685368
$496$	0	0
$497$	32.5172	1.45859
$498$	15.1960	0.680947
$499$	16.0297	0.717588	0.358794	−	0.933417i	$-0.383188\pi$
$499$	16.0297	0.717588	0.358794	+	0.933417i	$0.383188\pi$
$500$	−0.0610318	−0.00272943
$501$	17.8359	0.796848
$502$	−18.0208	−0.804309
$503$	4.16642	0.185771	0.0928857	−	0.995677i	$-0.470391\pi$
$503$	4.16642	0.185771	0.0928857	+	0.995677i	$0.470391\pi$
$504$	−19.5715	−0.871786
$505$	−2.41226	−0.107344
$506$	2.79987	0.124470
$507$	16.8207	0.747033
$508$	1.12318	0.0498330
$509$	10.0885	0.447164	0.223582	−	0.974685i	$-0.428225\pi$
$509$	10.0885	0.447164	0.223582	+	0.974685i	$0.428225\pi$
$510$	−3.84329	−0.170184
$511$	0.640858	0.0283499
$512$	−21.5066	−0.950466
$513$	10.0803	0.445055
$514$	−24.7127	−1.09003
$515$	−8.60350	−0.379116
$516$	0.418317	0.0184154
$517$	−7.56867	−0.332870
$518$	−53.7375	−2.36109
$519$	−9.53491	−0.418536
$520$	14.9250	0.654504
$521$	−13.4020	−0.587150	−0.293575	−	0.955936i	$-0.594845\pi$
$521$	−13.4020	−0.587150	−0.293575	+	0.955936i	$0.594845\pi$
$522$	6.98013	0.305512
$523$	19.3755	0.847230	0.423615	−	0.905842i	$-0.360761\pi$
$523$	19.3755	0.847230	0.423615	+	0.905842i	$0.360761\pi$
$524$	−1.21115	−0.0529094
$525$	−4.03965	−0.176305
$526$	18.1271	0.790379
$527$	0	0
$528$	−3.60811	−0.157023
$529$	−17.3463	−0.754187
$530$	−19.3798	−0.841804
$531$	−24.5384	−1.06488
$532$	0.448293	0.0194360
$533$	−29.5074	−1.27811
$534$	−9.78314	−0.423358
$535$	−8.78208	−0.379682
$536$	−17.0651	−0.737101
$537$	−0.115939	−0.00500312
$538$	22.0680	0.951418
$539$	5.99015	0.258014
$540$	0.316766	0.0136314
$541$	25.3043	1.08792	0.543959	−	0.839112i	$-0.316925\pi$
$541$	25.3043	1.08792	0.543959	+	0.839112i	$0.316925\pi$
$542$	−18.2760	−0.785019
$543$	16.5825	0.711625
$544$	−0.864995	−0.0370864
$545$	−4.01557	−0.172008
$546$	−31.0948	−1.33073
$547$	39.8870	1.70545	0.852723	−	0.522364i	$-0.174950\pi$
$547$	39.8870	1.70545	0.852723	+	0.522364i	$0.174950\pi$
$548$	−0.738029	−0.0315270
$549$	15.1318	0.645808
$550$	1.17753	0.0502100
$551$	5.07942	0.216391
$552$	−7.06982	−0.300911
$553$	−34.3177	−1.45934
$554$	−15.6111	−0.663252
$555$	−10.5718	−0.448748
$556$	0.511564	0.0216952
$557$	18.1799	0.770307	0.385153	−	0.922853i	$-0.374149\pi$
$557$	18.1799	0.770307	0.385153	+	0.922853i	$0.374149\pi$
$558$	0	0
$559$	34.4050	1.45518
$560$	−15.5753	−0.658178
$561$	2.19579	0.0927062
$562$	25.5024	1.07575
$563$	10.4562	0.440674	0.220337	−	0.975424i	$-0.429284\pi$
$563$	10.4562	0.440674	0.220337	+	0.975424i	$0.429284\pi$
$564$	−0.601555	−0.0253300
$565$	−17.6814	−0.743864
$566$	−21.3749	−0.898454
$567$	−0.126249	−0.00530194
$568$	23.9337	1.00424
$569$	1.46280	0.0613236	0.0306618	−	0.999530i	$-0.490239\pi$
$569$	1.46280	0.0613236	0.0306618	+	0.999530i	$0.490239\pi$
$570$	2.97825	0.124745
$571$	−17.9873	−0.752747	−0.376374	−	0.926468i	$-0.622829\pi$
$571$	−17.9873	−0.752747	−0.376374	+	0.926468i	$0.622829\pi$
$572$	0.268403	0.0112225
$573$	7.88399	0.329358
$574$	29.8805	1.24719
$575$	2.37775	0.0991591
$576$	−14.3915	−0.599645
$577$	43.9743	1.83067	0.915336	−	0.402690i	$-0.131925\pi$
$577$	43.9743	1.83067	0.915336	+	0.402690i	$0.131925\pi$
$578$	−15.3878	−0.640047
$579$	−21.8482	−0.907982
$580$	0.159618	0.00662776
$581$	−37.4777	−1.55484
$582$	21.3564	0.885250
$583$	11.0723	0.458566
$584$	0.471693	0.0195188
$585$	9.96776	0.412116
$586$	−12.4006	−0.512262
$587$	35.1511	1.45084	0.725420	−	0.688307i	$-0.241646\pi$
$587$	35.1511	1.45084	0.725420	+	0.688307i	$0.241646\pi$
$588$	0.476095	0.0196338
$589$	0	0
$590$	−18.9492	−0.780128
$591$	11.5180	0.473786
$592$	−40.7608	−1.67526
$593$	−8.33526	−0.342288	−0.171144	−	0.985246i	$-0.554746\pi$
$593$	−8.33526	−0.342288	−0.171144	+	0.985246i	$0.554746\pi$
$594$	−6.11159	−0.250761
$595$	9.47867	0.388588
$596$	−0.0681045	−0.00278967
$597$	11.9362	0.488516
$598$	18.3025	0.748444
$599$	16.7036	0.682490	0.341245	−	0.939974i	$-0.389151\pi$
$599$	16.7036	0.682490	0.341245	+	0.939974i	$0.389151\pi$
$600$	−2.97332	−0.121385
$601$	27.4085	1.11801	0.559007	−	0.829163i	$-0.311182\pi$
$601$	27.4085	1.11801	0.559007	+	0.829163i	$0.311182\pi$
$602$	−34.8400	−1.41997
$603$	−11.3971	−0.464125
$604$	0.802856	0.0326677
$605$	10.3272	0.419862
$606$	3.69909	0.150265
$607$	12.4835	0.506691	0.253346	−	0.967376i	$-0.418469\pi$
$607$	12.4835	0.506691	0.253346	+	0.967376i	$0.418469\pi$
$608$	0.670303	0.0271844
$609$	10.5650	0.428114
$610$	11.6852	0.473119
$611$	−49.4756	−2.00157
$612$	−0.284371	−0.0114950
$613$	18.2344	0.736480	0.368240	−	0.929731i	$-0.379960\pi$
$613$	18.2344	0.736480	0.368240	+	0.929731i	$0.379960\pi$
$614$	41.6032	1.67897
$615$	5.87839	0.237040
$616$	8.63491	0.347910
$617$	7.49052	0.301557	0.150779	−	0.988568i	$-0.451822\pi$
$617$	7.49052	0.301557	0.150779	+	0.988568i	$0.451822\pi$
$618$	13.1931	0.530704
$619$	27.4044	1.10148	0.550739	−	0.834678i	$-0.314346\pi$
$619$	27.4044	1.10148	0.550739	+	0.834678i	$0.314346\pi$
$620$	0	0
$621$	−12.3410	−0.495226
$622$	−32.1423	−1.28879
$623$	24.1281	0.966671
$624$	−23.5859	−0.944190
$625$	1.00000	0.0400000
$626$	25.4276	1.01629
$627$	−1.70156	−0.0679539
$628$	−1.48520	−0.0592659
$629$	24.8058	0.989070
$630$	−10.0938	−0.402146
$631$	11.9115	0.474188	0.237094	−	0.971487i	$-0.423805\pi$
$631$	11.9115	0.474188	0.237094	+	0.971487i	$0.423805\pi$
$632$	−25.2590	−1.00475
$633$	14.1811	0.563648
$634$	−5.21361	−0.207059
$635$	−18.4032	−0.730308
$636$	0.880019	0.0348950
$637$	39.1570	1.55146
$638$	−3.07961	−0.121923
$639$	15.9843	0.632330
$640$	−11.8038	−0.466584
$641$	28.4409	1.12335	0.561674	−	0.827359i	$-0.310157\pi$
$641$	28.4409	1.12335	0.561674	+	0.827359i	$0.310157\pi$
$642$	13.4669	0.531497
$643$	14.7022	0.579800	0.289900	−	0.957057i	$-0.406378\pi$
$643$	14.7022	0.579800	0.289900	+	0.957057i	$0.406378\pi$
$644$	−0.548831	−0.0216270
$645$	−6.85408	−0.269879
$646$	−6.98819	−0.274947
$647$	17.5279	0.689095	0.344547	−	0.938769i	$-0.388032\pi$
$647$	17.5279	0.689095	0.344547	+	0.938769i	$0.388032\pi$
$648$	−0.0929232	−0.00365037
$649$	10.8263	0.424968
$650$	7.69739	0.301916
$651$	0	0
$652$	−0.661614	−0.0259108
$653$	29.0373	1.13632	0.568159	−	0.822919i	$-0.307656\pi$
$653$	29.0373	1.13632	0.568159	+	0.822919i	$0.307656\pi$
$654$	6.15769	0.240785
$655$	19.8446	0.775391
$656$	22.6648	0.884912
$657$	0.315024	0.0122902
$658$	50.1011	1.95315
$659$	34.6269	1.34887	0.674436	−	0.738334i	$-0.264387\pi$
$659$	34.6269	1.34887	0.674436	+	0.738334i	$0.264387\pi$
$660$	−0.0534705	−0.00208134
$661$	−10.5264	−0.409431	−0.204715	−	0.978822i	$-0.565627\pi$
$661$	−10.5264	−0.409431	−0.204715	+	0.978822i	$0.565627\pi$
$662$	36.1248	1.40403
$663$	14.3536	0.557449
$664$	−27.5848	−1.07050
$665$	−7.34523	−0.284836
$666$	−26.4155	−1.02358
$667$	−6.21858	−0.240784
$668$	1.01911	0.0394307
$669$	0.358450	0.0138585
$670$	−8.80114	−0.340018
$671$	−6.67609	−0.257727
$672$	1.39420	0.0537825
$673$	32.4815	1.25207	0.626034	−	0.779795i	$-0.284677\pi$
$673$	32.4815	1.25207	0.626034	+	0.779795i	$0.284677\pi$
$674$	−47.7365	−1.83874
$675$	−5.19018	−0.199770
$676$	0.961107	0.0369656
$677$	−27.0788	−1.04072	−0.520362	−	0.853946i	$-0.674203\pi$
$677$	−27.0788	−1.04072	−0.520362	+	0.853946i	$0.674203\pi$
$678$	27.1137	1.04129
$679$	−52.6710	−2.02133
$680$	6.97662	0.267541
$681$	10.6598	0.408485
$682$	0	0
$683$	−6.58687	−0.252040	−0.126020	−	0.992028i	$-0.540220\pi$
$683$	−6.58687	−0.252040	−0.126020	+	0.992028i	$0.540220\pi$
$684$	0.220365	0.00842588
$685$	12.0925	0.462031
$686$	−1.64574	−0.0628348
$687$	−11.8040	−0.450351
$688$	−26.4267	−1.00751
$689$	72.3782	2.75739
$690$	−3.64618	−0.138808
$691$	−28.0920	−1.06867	−0.534335	−	0.845273i	$-0.679438\pi$
$691$	−28.0920	−1.06867	−0.534335	+	0.845273i	$0.679438\pi$
$692$	−0.544809	−0.0207105
$693$	5.76689	0.219066
$694$	−33.9388	−1.28830
$695$	−8.38192	−0.317944
$696$	7.77618	0.294755
$697$	−13.7931	−0.522451
$698$	−52.2875	−1.97911
$699$	−5.92913	−0.224260
$700$	−0.230819	−0.00872414
$701$	−17.2242	−0.650548	−0.325274	−	0.945620i	$-0.605457\pi$
$701$	−17.2242	−0.650548	−0.325274	+	0.945620i	$0.605457\pi$
$702$	−39.9508	−1.50785
$703$	−19.2225	−0.724991
$704$	6.34947	0.239305
$705$	9.85641	0.371214
$706$	−9.46417	−0.356189
$707$	−9.12304	−0.343107
$708$	0.860468	0.0323384
$709$	41.9460	1.57532	0.787658	−	0.616113i	$-0.211293\pi$
$709$	41.9460	1.57532	0.787658	+	0.616113i	$0.211293\pi$
$710$	12.3435	0.463245
$711$	−16.8694	−0.632651
$712$	17.7591	0.665549
$713$	0	0
$714$	−14.5351	−0.543963
$715$	−4.39775	−0.164466
$716$	−0.00662455	−0.000247571 0
$717$	−15.0701	−0.562803
$718$	19.5969	0.731349
$719$	−5.66219	−0.211164	−0.105582	−	0.994411i	$-0.533671\pi$
$719$	−5.66219	−0.211164	−0.105582	+	0.994411i	$0.533671\pi$
$720$	−7.65630	−0.285333
$721$	−32.5380	−1.21178
$722$	−21.8617	−0.813607
$723$	2.44045	0.0907612
$724$	0.947499	0.0352135
$725$	−2.61532	−0.0971304
$726$	−15.8364	−0.587742
$727$	5.21279	0.193332	0.0966659	−	0.995317i	$-0.469182\pi$
$727$	5.21279	0.193332	0.0966659	+	0.995317i	$0.469182\pi$
$728$	56.4455	2.09201
$729$	16.5695	0.613684
$730$	0.243270	0.00900383
$731$	16.0825	0.594831
$732$	−0.530613	−0.0196120
$733$	19.1226	0.706308	0.353154	−	0.935565i	$-0.385109\pi$
$733$	19.1226	0.706308	0.353154	+	0.935565i	$0.385109\pi$
$734$	−33.1056	−1.22195
$735$	−7.80076	−0.287735
$736$	−0.820631	−0.0302489
$737$	5.02835	0.185222
$738$	14.6882	0.540681
$739$	−41.7015	−1.53401	−0.767007	−	0.641639i	$-0.778255\pi$
$739$	−41.7015	−1.53401	−0.767007	+	0.641639i	$0.778255\pi$
$740$	−0.604055	−0.0222055
$741$	−11.1229	−0.408611
$742$	−73.2933	−2.69068
$743$	−31.6205	−1.16004	−0.580022	−	0.814601i	$-0.696956\pi$
$743$	−31.6205	−1.16004	−0.580022	+	0.814601i	$0.696956\pi$
$744$	0	0
$745$	1.11588	0.0408829
$746$	−5.50546	−0.201569
$747$	−18.4227	−0.674052
$748$	0.125464	0.00458741
$749$	−33.2133	−1.21359
$750$	−1.53345	−0.0559939
$751$	33.6927	1.22946	0.614732	−	0.788736i	$-0.289264\pi$
$751$	33.6927	1.22946	0.614732	+	0.788736i	$0.289264\pi$
$752$	38.0025	1.38581
$753$	−13.4079	−0.488611
$754$	−20.1311	−0.733132
$755$	−13.1547	−0.478748
$756$	1.19799	0.0435706
$757$	−19.3895	−0.704723	−0.352362	−	0.935864i	$-0.614621\pi$
$757$	−19.3895	−0.704723	−0.352362	+	0.935864i	$0.614621\pi$
$758$	34.0861	1.23806
$759$	2.08317	0.0756143
$760$	−5.40634	−0.196108
$761$	6.06617	0.219899	0.109949	−	0.993937i	$-0.464931\pi$
$761$	6.06617	0.219899	0.109949	+	0.993937i	$0.464931\pi$
$762$	28.2204	1.02232
$763$	−15.1867	−0.549794
$764$	0.450478	0.0162977
$765$	4.65939	0.168461
$766$	−21.9195	−0.791984
$767$	70.7702	2.55536
$768$	1.56313	0.0564047
$769$	−5.93757	−0.214114	−0.107057	−	0.994253i	$-0.534143\pi$
$769$	−5.93757	−0.214114	−0.107057	+	0.994253i	$0.534143\pi$
$770$	4.45335	0.160488
$771$	−18.3868	−0.662184
$772$	−1.24837	−0.0449299
$773$	47.3416	1.70276	0.851379	−	0.524551i	$-0.175767\pi$
$773$	47.3416	1.70276	0.851379	+	0.524551i	$0.175767\pi$
$774$	−17.1261	−0.615586
$775$	0	0
$776$	−38.7677	−1.39168
$777$	−39.9820	−1.43434
$778$	−48.9294	−1.75421
$779$	10.6886	0.382958
$780$	−0.349531	−0.0125152
$781$	−7.05223	−0.252349
$782$	8.55542	0.305941
$783$	13.5740	0.485094
$784$	−30.0767	−1.07417
$785$	24.3348	0.868547
$786$	−30.4307	−1.08543
$787$	23.8753	0.851063	0.425531	−	0.904944i	$-0.360087\pi$
$787$	23.8753	0.851063	0.425531	+	0.904944i	$0.360087\pi$
$788$	0.658118	0.0234445
$789$	13.4870	0.480149
$790$	−13.0270	−0.463480
$791$	−66.8702	−2.37763
$792$	4.24462	0.150826
$793$	−43.6409	−1.54973
$794$	−27.6688	−0.981928
$795$	−14.4190	−0.511390
$796$	0.682015	0.0241734
$797$	−5.53857	−0.196186	−0.0980931	−	0.995177i	$-0.531274\pi$
$797$	−5.53857	−0.196186	−0.0980931	+	0.995177i	$0.531274\pi$
$798$	11.2636	0.398726
$799$	−23.1272	−0.818180
$800$	−0.345129	−0.0122021
$801$	11.8605	0.419071
$802$	43.5730	1.53862
$803$	−0.138987	−0.00490476
$804$	0.399652	0.0140946
$805$	8.99253	0.316945
$806$	0	0
$807$	16.4191	0.577979
$808$	−6.71486	−0.236228
$809$	18.1427	0.637863	0.318932	−	0.947778i	$-0.396676\pi$
$809$	18.1427	0.637863	0.318932	+	0.947778i	$0.396676\pi$
$810$	−0.0479240	−0.00168388
$811$	34.4710	1.21044	0.605220	−	0.796059i	$-0.293085\pi$
$811$	34.4710	1.21044	0.605220	+	0.796059i	$0.293085\pi$
$812$	0.603665	0.0211845
$813$	−13.5977	−0.476893
$814$	11.6545	0.408488
$815$	10.8405	0.379725
$816$	−11.0251	−0.385956
$817$	−12.4626	−0.436013
$818$	−5.26798	−0.184191
$819$	37.6975	1.31726
$820$	0.335882	0.0117295
$821$	−20.2220	−0.705754	−0.352877	−	0.935670i	$-0.614797\pi$
$821$	−20.2220	−0.705754	−0.352877	+	0.935670i	$0.614797\pi$
$822$	−18.5433	−0.646773
$823$	−23.3044	−0.812341	−0.406171	−	0.913797i	$-0.633136\pi$
$823$	−23.3044	−0.812341	−0.406171	+	0.913797i	$0.633136\pi$
$824$	−23.9491	−0.834305
$825$	0.876109	0.0305022
$826$	−71.6650	−2.49354
$827$	29.3940	1.02213	0.511065	−	0.859542i	$-0.329251\pi$
$827$	29.3940	1.02213	0.511065	+	0.859542i	$0.329251\pi$
$828$	−0.269786	−0.00937572
$829$	−31.5550	−1.09595	−0.547974	−	0.836495i	$-0.684601\pi$
$829$	−31.5550	−1.09595	−0.547974	+	0.836495i	$0.684601\pi$
$830$	−14.2265	−0.493811
$831$	−11.6150	−0.402920
$832$	41.5058	1.43896
$833$	18.3038	0.634188
$834$	12.8533	0.445073
$835$	−16.6980	−0.577860
$836$	−0.0972245	−0.00336258
$837$	0	0
$838$	22.6021	0.780777
$839$	49.6508	1.71414	0.857068	−	0.515203i	$-0.172283\pi$
$839$	49.6508	1.71414	0.857068	+	0.515203i	$0.172283\pi$
$840$	−11.2449	−0.387987
$841$	−22.1601	−0.764142
$842$	−52.8206	−1.82032
$843$	18.9744	0.653512
$844$	0.810284	0.0278911
$845$	−15.7476	−0.541735
$846$	24.6280	0.846728
$847$	39.0571	1.34202
$848$	−55.5941	−1.90911
$849$	−15.9034	−0.545804
$850$	3.59811	0.123414
$851$	23.5335	0.806719
$852$	−0.560509	−0.0192027
$853$	48.9477	1.67594	0.837969	−	0.545718i	$-0.183743\pi$
$853$	48.9477	1.67594	0.837969	+	0.545718i	$0.183743\pi$
$854$	44.1927	1.51224
$855$	−3.61066	−0.123482
$856$	−24.4461	−0.835552
$857$	−17.8176	−0.608637	−0.304319	−	0.952570i	$-0.598429\pi$
$857$	−17.8176	−0.608637	−0.304319	+	0.952570i	$0.598429\pi$
$858$	6.74375	0.230228
$859$	−25.5785	−0.872727	−0.436363	−	0.899771i	$-0.643734\pi$
$859$	−25.5785	−0.872727	−0.436363	+	0.899771i	$0.643734\pi$
$860$	−0.391631	−0.0133545
$861$	22.2318	0.757656
$862$	−7.72923	−0.263259
$863$	−38.4877	−1.31014	−0.655069	−	0.755569i	$-0.727360\pi$
$863$	−38.4877	−1.31014	−0.655069	+	0.755569i	$0.727360\pi$
$864$	1.79128	0.0609406
$865$	8.92663	0.303515
$866$	16.3878	0.556882
$867$	−11.4489	−0.388823
$868$	0	0
$869$	7.44272	0.252477
$870$	4.01047	0.135968
$871$	32.8698	1.11375
$872$	−11.1779	−0.378531
$873$	−25.8913	−0.876287
$874$	−6.62978	−0.224256
$875$	3.78195	0.127853
$876$	−0.0110467	−0.000373232 0
$877$	−55.6185	−1.87810	−0.939052	−	0.343775i	$-0.888294\pi$
$877$	−55.6185	−1.87810	−0.939052	+	0.343775i	$0.888294\pi$
$878$	−3.53045	−0.119147
$879$	−9.22630	−0.311195
$880$	3.37794	0.113870
$881$	4.62449	0.155803	0.0779016	−	0.996961i	$-0.475178\pi$
$881$	4.62449	0.155803	0.0779016	+	0.996961i	$0.475178\pi$
$882$	−19.4916	−0.656316
$883$	−30.9588	−1.04185	−0.520923	−	0.853603i	$-0.674412\pi$
$883$	−30.9588	−1.04185	−0.520923	+	0.853603i	$0.674412\pi$
$884$	0.820143	0.0275844
$885$	−14.0987	−0.473922
$886$	−1.05952	−0.0355952
$887$	−19.6048	−0.658265	−0.329132	−	0.944284i	$-0.606756\pi$
$887$	−19.6048	−0.658265	−0.329132	+	0.944284i	$0.606756\pi$
$888$	−29.4281	−0.987541
$889$	−69.5998	−2.33430
$890$	9.15903	0.307011
$891$	0.0273804	0.000917279 0
$892$	0.0204813	0.000685763 0
$893$	17.9217	0.599728
$894$	−1.71116	−0.0572297
$895$	0.108542	0.00362818
$896$	−44.6412	−1.49136
$897$	13.6175	0.454674
$898$	23.2281	0.775133
$899$	0	0
$900$	−0.113463	−0.00378209
$901$	33.8329	1.12714
$902$	−6.48040	−0.215774
$903$	−25.9217	−0.862622
$904$	−49.2187	−1.63699
$905$	−15.5247	−0.516057
$906$	20.1721	0.670174
$907$	−7.63790	−0.253612	−0.126806	−	0.991928i	$-0.540473\pi$
$907$	−7.63790	−0.253612	−0.126806	+	0.991928i	$0.540473\pi$
$908$	0.609084	0.0202132
$909$	−4.48457	−0.148744
$910$	29.1111	0.965024
$911$	13.1589	0.435975	0.217987	−	0.975952i	$-0.430051\pi$
$911$	13.1589	0.435975	0.217987	+	0.975952i	$0.430051\pi$
$912$	8.54360	0.282907
$913$	8.12805	0.268999
$914$	15.7716	0.521678
$915$	8.69403	0.287416
$916$	−0.674462	−0.0222848
$917$	75.0511	2.47840
$918$	−18.6748	−0.616361
$919$	17.6712	0.582919	0.291459	−	0.956583i	$-0.405859\pi$
$919$	17.6712	0.582919	0.291459	+	0.956583i	$0.405859\pi$
$920$	6.61881	0.218216
$921$	30.9537	1.01996
$922$	45.2340	1.48970
$923$	−46.0997	−1.51739
$924$	−0.202223	−0.00665264
$925$	9.89738	0.325424
$926$	19.4255	0.638362
$927$	−15.9946	−0.525330
$928$	0.902622	0.0296300
$929$	27.6157	0.906041	0.453021	−	0.891500i	$-0.350346\pi$
$929$	27.6157	0.906041	0.453021	+	0.891500i	$0.350346\pi$
$930$	0	0
$931$	−14.1840	−0.464861
$932$	−0.338781	−0.0110971
$933$	−23.9146	−0.782929
$934$	−41.0477	−1.34312
$935$	−2.05571	−0.0672289
$936$	27.7467	0.906928
$937$	−25.8143	−0.843315	−0.421657	−	0.906755i	$-0.638552\pi$
$937$	−25.8143	−0.843315	−0.421657	+	0.906755i	$0.638552\pi$
$938$	−33.2854	−1.08681
$939$	18.9187	0.617388
$940$	0.563179	0.0183689
$941$	−3.18030	−0.103675	−0.0518375	−	0.998656i	$-0.516508\pi$
$941$	−3.18030	−0.103675	−0.0518375	+	0.998656i	$0.516508\pi$
$942$	−37.3163	−1.21583
$943$	−13.0857	−0.426129
$944$	−54.3590	−1.76924
$945$	−19.6290	−0.638530
$946$	7.55600	0.245667
$947$	37.3585	1.21399	0.606993	−	0.794707i	$-0.292375\pi$
$947$	37.3585	1.21399	0.606993	+	0.794707i	$0.292375\pi$
$948$	0.591545	0.0192125
$949$	−0.908547	−0.0294927
$950$	−2.78825	−0.0904629
$951$	−3.87904	−0.125787
$952$	26.3852	0.855150
$953$	−24.2246	−0.784712	−0.392356	−	0.919814i	$-0.628340\pi$
$953$	−24.2246	−0.784712	−0.392356	+	0.919814i	$0.628340\pi$
$954$	−36.0285	−1.16647
$955$	−7.38103	−0.238845
$956$	−0.861081	−0.0278494
$957$	−2.29130	−0.0740673
$958$	18.0925	0.584543
$959$	45.7332	1.47680
$960$	−8.26869	−0.266871
$961$	0	0
$962$	76.1840	2.45627
$963$	−16.3265	−0.526115
$964$	0.139443	0.00449116
$965$	20.4545	0.658452
$966$	−13.7896	−0.443674
$967$	−5.50389	−0.176993	−0.0884965	−	0.996076i	$-0.528206\pi$
$967$	−5.50389	−0.176993	−0.0884965	+	0.996076i	$0.528206\pi$
$968$	28.7473	0.923974
$969$	−5.19937	−0.167028
$970$	−19.9940	−0.641967
$971$	−24.0575	−0.772043	−0.386022	−	0.922490i	$-0.626151\pi$
$971$	−24.0575	−0.772043	−0.386022	+	0.922490i	$0.626151\pi$
$972$	0.952475	0.0305506
$973$	−31.7000	−1.01625
$974$	39.1268	1.25370
$975$	5.72703	0.183412
$976$	33.5208	1.07298
$977$	1.26599	0.0405026	0.0202513	−	0.999795i	$-0.493553\pi$
$977$	1.26599	0.0405026	0.0202513	+	0.999795i	$0.493553\pi$
$978$	−16.6234	−0.531556
$979$	−5.23283	−0.167242
$980$	−0.445722	−0.0142381
$981$	−7.46524	−0.238347
$982$	−5.73431	−0.182989
$983$	60.1842	1.91958	0.959789	−	0.280722i	$-0.0905738\pi$
$983$	60.1842	1.91958	0.959789	+	0.280722i	$0.0905738\pi$
$984$	16.3633	0.521644
$985$	−10.7832	−0.343581
$986$	−9.41020	−0.299682
$987$	37.2764	1.18652
$988$	−0.635547	−0.0202194
$989$	15.2576	0.485164
$990$	2.18911	0.0695746
$991$	−10.8943	−0.346070	−0.173035	−	0.984916i	$-0.555357\pi$
$991$	−10.8943	−0.346070	−0.173035	+	0.984916i	$0.555357\pi$
$992$	0	0
$993$	26.8777	0.852938
$994$	46.6826	1.48068
$995$	−11.1747	−0.354263
$996$	0.646014	0.0204698
$997$	43.8749	1.38953	0.694765	−	0.719236i	$-0.255508\pi$
$997$	43.8749	1.38953	0.694765	+	0.719236i	$0.255508\pi$
$998$	23.0127	0.728455
$999$	−51.3691	−1.62525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4805.2.a.bb.1.18	yes	24
31.30	odd	2	inner	4805.2.a.bb.1.17	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4805.2.a.bb.1.17	✓	24	31.30	odd	2	inner
4805.2.a.bb.1.18	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4805 = 5 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4805.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43563 q^{2} +1.06814 q^{3} +0.0610318 q^{4} -1.00000 q^{5} +1.53345 q^{6} -3.78195 q^{7} -2.78364 q^{8} -1.85907 q^{9} -1.43563 q^{10} +0.820218 q^{11} +0.0651906 q^{12} +5.36168 q^{13} -5.42947 q^{14} -1.06814 q^{15} -4.11834 q^{16} +2.50630 q^{17} -2.66894 q^{18} -1.94218 q^{19} -0.0610318 q^{20} -4.03965 q^{21} +1.17753 q^{22} +2.37775 q^{23} -2.97332 q^{24} +1.00000 q^{25} +7.69739 q^{26} -5.19018 q^{27} -0.230819 q^{28} -2.61532 q^{29} -1.53345 q^{30} -0.345129 q^{32} +0.876109 q^{33} +3.59811 q^{34} +3.78195 q^{35} -0.113463 q^{36} +9.89738 q^{37} -2.78825 q^{38} +5.72703 q^{39} +2.78364 q^{40} -5.50338 q^{41} -5.79944 q^{42} +6.41683 q^{43} +0.0500594 q^{44} +1.85907 q^{45} +3.41357 q^{46} -9.22762 q^{47} -4.39897 q^{48} +7.30311 q^{49} +1.43563 q^{50} +2.67708 q^{51} +0.327233 q^{52} +13.4992 q^{53} -7.45117 q^{54} -0.820218 q^{55} +10.5276 q^{56} -2.07452 q^{57} -3.75463 q^{58} +13.1993 q^{59} -0.0651906 q^{60} -8.13940 q^{61} +7.03092 q^{63} +7.74120 q^{64} -5.36168 q^{65} +1.25777 q^{66} +6.13051 q^{67} +0.152964 q^{68} +2.53978 q^{69} +5.42947 q^{70} -8.59800 q^{71} +5.17499 q^{72} -0.169452 q^{73} +14.2090 q^{74} +1.06814 q^{75} -0.118535 q^{76} -3.10202 q^{77} +8.22190 q^{78} +9.07408 q^{79} +4.11834 q^{80} +0.0333819 q^{81} -7.90082 q^{82} +9.90962 q^{83} -0.246547 q^{84} -2.50630 q^{85} +9.21219 q^{86} -2.79353 q^{87} -2.28319 q^{88} -6.37980 q^{89} +2.66894 q^{90} -20.2776 q^{91} +0.145119 q^{92} -13.2474 q^{94} +1.94218 q^{95} -0.368646 q^{96} +13.9270 q^{97} +10.4846 q^{98} -1.52485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} - 24 q^{5} + 8 q^{7} + 40 q^{9} + 8 q^{14} + 88 q^{16} - 64 q^{18} + 40 q^{19} - 32 q^{20} + 24 q^{25} + 72 q^{28} + 56 q^{33} - 8 q^{35} + 88 q^{36} - 72 q^{38} + 64 q^{39} - 56 q^{41}+ \cdots - 8 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 - 24 * q^5 + 8 * q^7 + 40 * q^9 + 8 * q^14 + 88 * q^16 - 64 * q^18 + 40 * q^19 - 32 * q^20 + 24 * q^25 + 72 * q^28 + 56 * q^33 - 8 * q^35 + 88 * q^36 - 72 * q^38 + 64 * q^39 - 56 * q^41 - 40 * q^45 - 32 * q^47 + 56 * q^49 + 88 * q^51 + 104 * q^56 + 8 * q^59 + 56 * q^63 + 160 * q^64 - 8 * q^66 + 96 * q^67 + 8 * q^69 - 8 * q^70 - 64 * q^71 - 120 * q^72 + 128 * q^76 - 128 * q^78 - 88 * q^80 + 72 * q^81 + 32 * q^82 - 32 * q^87 + 64 * q^90 + 56 * q^94 - 40 * q^95 + 48 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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