LMFDB - Embedded newform 423.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [423,4,Mod(1,423)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(423, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("423.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 423 = 3^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	423.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$24.9578079324$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 47)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.68740$ of defining polynomial
Character	$\chi$	$=$	423.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.90952 q^{2} +16.1033 q^{4} -9.19383 q^{5} -30.3255 q^{7} +39.7835 q^{8} +O(q^{10})$	\(q+4.90952 q^{2} +16.1033 q^{4} -9.19383 q^{5} -30.3255 q^{7} +39.7835 q^{8} -45.1373 q^{10} -22.4298 q^{11} -62.0257 q^{13} -148.883 q^{14} +66.4910 q^{16} +72.1639 q^{17} -25.0550 q^{19} -148.051 q^{20} -110.119 q^{22} -103.176 q^{23} -40.4735 q^{25} -304.516 q^{26} -488.341 q^{28} +234.381 q^{29} +198.714 q^{31} +8.17058 q^{32} +354.290 q^{34} +278.807 q^{35} -203.083 q^{37} -123.008 q^{38} -365.763 q^{40} -210.889 q^{41} +111.430 q^{43} -361.194 q^{44} -506.542 q^{46} -47.0000 q^{47} +576.634 q^{49} -198.705 q^{50} -998.822 q^{52} -499.576 q^{53} +206.215 q^{55} -1206.45 q^{56} +1150.70 q^{58} +562.752 q^{59} +548.091 q^{61} +975.591 q^{62} -491.814 q^{64} +570.254 q^{65} -760.831 q^{67} +1162.08 q^{68} +1368.81 q^{70} +668.059 q^{71} -1145.92 q^{73} -997.040 q^{74} -403.469 q^{76} +680.193 q^{77} +975.010 q^{79} -611.307 q^{80} -1035.37 q^{82} -698.827 q^{83} -663.463 q^{85} +547.067 q^{86} -892.335 q^{88} -451.477 q^{89} +1880.96 q^{91} -1661.47 q^{92} -230.747 q^{94} +230.351 q^{95} -390.906 q^{97} +2830.99 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 45 q^{7} + 39 q^{8}+O(q^{10}) $ 3 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 45 * q^7 + 39 * q^8	$ 3 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 45 q^{7} + 39 q^{8} - 40 q^{10} - 2 q^{11} - 80 q^{13} - 162 q^{14} + 89 q^{16} + 39 q^{17} - 24 q^{19} - 232 q^{20} - 14 q^{22} - 120 q^{23} - 171 q^{25} - 316 q^{26} - 408 q^{28} + 184 q^{29} - 4 q^{31} + 7 q^{32} + 218 q^{34} + 156 q^{35} - 589 q^{37} - 42 q^{38} - 432 q^{40} + 92 q^{41} - 250 q^{43} - 466 q^{44} - 816 q^{46} - 141 q^{47} + 30 q^{49} - 137 q^{50} - 900 q^{52} - 459 q^{53} + 448 q^{55} - 1032 q^{56} + 684 q^{58} - 579 q^{59} + 267 q^{61} + 244 q^{62} - 87 q^{64} + 424 q^{65} - 540 q^{67} + 1334 q^{68} + 1236 q^{70} - 749 q^{71} - 1924 q^{73} - 950 q^{74} - 402 q^{76} + 288 q^{77} + 805 q^{79} - 448 q^{80} - 938 q^{82} - 712 q^{83} - 1038 q^{85} + 1294 q^{86} - 2190 q^{88} - 835 q^{89} + 2040 q^{91} - 1596 q^{92} - 235 q^{94} + 312 q^{95} - 2243 q^{97} + 2989 q^{98}+O(q^{100}) $ 3 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 45 * q^7 + 39 * q^8 - 40 * q^10 - 2 * q^11 - 80 * q^13 - 162 * q^14 + 89 * q^16 + 39 * q^17 - 24 * q^19 - 232 * q^20 - 14 * q^22 - 120 * q^23 - 171 * q^25 - 316 * q^26 - 408 * q^28 + 184 * q^29 - 4 * q^31 + 7 * q^32 + 218 * q^34 + 156 * q^35 - 589 * q^37 - 42 * q^38 - 432 * q^40 + 92 * q^41 - 250 * q^43 - 466 * q^44 - 816 * q^46 - 141 * q^47 + 30 * q^49 - 137 * q^50 - 900 * q^52 - 459 * q^53 + 448 * q^55 - 1032 * q^56 + 684 * q^58 - 579 * q^59 + 267 * q^61 + 244 * q^62 - 87 * q^64 + 424 * q^65 - 540 * q^67 + 1334 * q^68 + 1236 * q^70 - 749 * q^71 - 1924 * q^73 - 950 * q^74 - 402 * q^76 + 288 * q^77 + 805 * q^79 - 448 * q^80 - 938 * q^82 - 712 * q^83 - 1038 * q^85 + 1294 * q^86 - 2190 * q^88 - 835 * q^89 + 2040 * q^91 - 1596 * q^92 - 235 * q^94 + 312 * q^95 - 2243 * q^97 + 2989 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	4.90952	1.73578	0.867888	−	0.496760i	$-0.165477\pi$
$2$	4.90952	1.73578	0.867888	+	0.496760i	$0.165477\pi$
$3$	0	0
$3$	0	0
$4$	16.1033	2.01292
$5$	−9.19383	−0.822321	−0.411161	−	0.911563i	$-0.634876\pi$
$5$	−9.19383	−0.822321	−0.411161	+	0.911563i	$0.634876\pi$
$6$	0	0
$7$	−30.3255	−1.63742	−0.818711	−	0.574206i	$-0.805311\pi$
$7$	−30.3255	−1.63742	−0.818711	+	0.574206i	$0.805311\pi$
$8$	39.7835	1.75820
$9$	0	0
$10$	−45.1373	−1.42737
$11$	−22.4298	−0.614803	−0.307401	−	0.951580i	$-0.599459\pi$
$11$	−22.4298	−0.614803	−0.307401	+	0.951580i	$0.599459\pi$
$12$	0	0
$13$	−62.0257	−1.32330	−0.661648	−	0.749815i	$-0.730143\pi$
$13$	−62.0257	−1.32330	−0.661648	+	0.749815i	$0.730143\pi$
$14$	−148.883	−2.84220
$15$	0	0
$16$	66.4910	1.03892
$17$	72.1639	1.02955	0.514774	−	0.857326i	$-0.327876\pi$
$17$	72.1639	1.02955	0.514774	+	0.857326i	$0.327876\pi$
$18$	0	0
$19$	−25.0550	−0.302526	−0.151263	−	0.988494i	$-0.548334\pi$
$19$	−25.0550	−0.302526	−0.151263	+	0.988494i	$0.548334\pi$
$20$	−148.051	−1.65527
$21$	0	0
$22$	−110.119	−1.06716
$23$	−103.176	−0.935373	−0.467687	−	0.883894i	$-0.654912\pi$
$23$	−103.176	−0.935373	−0.467687	+	0.883894i	$0.654912\pi$
$24$	0	0
$25$	−40.4735	−0.323788
$26$	−304.516	−2.29694
$27$	0	0
$28$	−488.341	−3.29600
$29$	234.381	1.50081	0.750405	−	0.660978i	$-0.229858\pi$
$29$	234.381	1.50081	0.750405	+	0.660978i	$0.229858\pi$
$30$	0	0
$31$	198.714	1.15129	0.575647	−	0.817698i	$-0.304750\pi$
$31$	198.714	1.15129	0.575647	+	0.817698i	$0.304750\pi$
$32$	8.17058	0.0451365
$33$	0	0
$34$	354.290	1.78707
$35$	278.807	1.34649
$36$	0	0
$37$	−203.083	−0.902343	−0.451171	−	0.892437i	$-0.648994\pi$
$37$	−203.083	−0.902343	−0.451171	+	0.892437i	$0.648994\pi$
$38$	−123.008	−0.525118
$39$	0	0
$40$	−365.763	−1.44580
$41$	−210.889	−0.803303	−0.401651	−	0.915793i	$-0.631564\pi$
$41$	−210.889	−0.803303	−0.401651	+	0.915793i	$0.631564\pi$
$42$	0	0
$43$	111.430	0.395184	0.197592	−	0.980284i	$-0.436688\pi$
$43$	111.430	0.395184	0.197592	+	0.980284i	$0.436688\pi$
$44$	−361.194	−1.23755
$45$	0	0
$46$	−506.542	−1.62360
$47$	−47.0000	−0.145865
$47$	−47.0000	−0.145865
$48$	0	0
$49$	576.634	1.68115
$50$	−198.705	−0.562023
$51$	0	0
$52$	−998.822	−2.66369
$53$	−499.576	−1.29476	−0.647378	−	0.762169i	$-0.724134\pi$
$53$	−499.576	−1.29476	−0.647378	+	0.762169i	$0.724134\pi$
$54$	0	0
$55$	206.215	0.505565
$56$	−1206.45	−2.87891
$57$	0	0
$58$	1150.70	2.60507
$59$	562.752	1.24176	0.620882	−	0.783904i	$-0.286775\pi$
$59$	562.752	1.24176	0.620882	+	0.783904i	$0.286775\pi$
$60$	0	0
$61$	548.091	1.15042	0.575212	−	0.818004i	$-0.304919\pi$
$61$	548.091	1.15042	0.575212	+	0.818004i	$0.304919\pi$
$62$	975.591	1.99839
$63$	0	0
$64$	−491.814	−0.960575
$65$	570.254	1.08817
$66$	0	0
$67$	−760.831	−1.38732	−0.693659	−	0.720304i	$-0.744002\pi$
$67$	−760.831	−1.38732	−0.693659	+	0.720304i	$0.744002\pi$
$68$	1162.08	2.07240
$69$	0	0
$70$	1368.81	2.33720
$71$	668.059	1.11668	0.558338	−	0.829613i	$-0.311439\pi$
$71$	668.059	1.11668	0.558338	+	0.829613i	$0.311439\pi$
$72$	0	0
$73$	−1145.92	−1.83726	−0.918629	−	0.395121i	$-0.870703\pi$
$73$	−1145.92	−1.83726	−0.918629	+	0.395121i	$0.870703\pi$
$74$	−997.040	−1.56626
$75$	0	0
$76$	−403.469	−0.608961
$77$	680.193	1.00669
$78$	0	0
$79$	975.010	1.38857	0.694286	−	0.719699i	$-0.255720\pi$
$79$	975.010	1.38857	0.694286	+	0.719699i	$0.255720\pi$
$80$	−611.307	−0.854328
$81$	0	0
$82$	−1035.37	−1.39435
$83$	−698.827	−0.924171	−0.462086	−	0.886835i	$-0.652899\pi$
$83$	−698.827	−0.924171	−0.462086	+	0.886835i	$0.652899\pi$
$84$	0	0
$85$	−663.463	−0.846620
$86$	547.067	0.685950
$87$	0	0
$88$	−892.335	−1.08095
$89$	−451.477	−0.537713	−0.268857	−	0.963180i	$-0.586646\pi$
$89$	−451.477	−0.537713	−0.268857	+	0.963180i	$0.586646\pi$
$90$	0	0
$91$	1880.96	2.16679
$92$	−1661.47	−1.88283
$93$	0	0
$94$	−230.747	−0.253189
$95$	230.351	0.248774
$96$	0	0
$97$	−390.906	−0.409180	−0.204590	−	0.978848i	$-0.565586\pi$
$97$	−390.906	−0.409180	−0.204590	+	0.978848i	$0.565586\pi$
$98$	2830.99	2.91810
$99$	0	0
$100$	−651.758	−0.651758
$101$	1112.66	1.09618	0.548089	−	0.836420i	$-0.315356\pi$
$101$	1112.66	1.09618	0.548089	+	0.836420i	$0.315356\pi$
$102$	0	0
$103$	−882.574	−0.844298	−0.422149	−	0.906527i	$-0.638724\pi$
$103$	−882.574	−0.844298	−0.422149	+	0.906527i	$0.638724\pi$
$104$	−2467.60	−2.32662
$105$	0	0
$106$	−2452.68	−2.24741
$107$	−840.742	−0.759604	−0.379802	−	0.925068i	$-0.624008\pi$
$107$	−840.742	−0.759604	−0.379802	+	0.925068i	$0.624008\pi$
$108$	0	0
$109$	−1321.06	−1.16086	−0.580432	−	0.814309i	$-0.697116\pi$
$109$	−1321.06	−1.16086	−0.580432	+	0.814309i	$0.697116\pi$
$110$	1012.42	0.877548
$111$	0	0
$112$	−2016.37	−1.70115
$113$	−701.293	−0.583824	−0.291912	−	0.956445i	$-0.594291\pi$
$113$	−701.293	−0.583824	−0.291912	+	0.956445i	$0.594291\pi$
$114$	0	0
$115$	948.579	0.769177
$116$	3774.32	3.02101
$117$	0	0
$118$	2762.84	2.15542
$119$	−2188.40	−1.68580
$120$	0	0
$121$	−827.906	−0.622018
$122$	2690.86	1.99688
$123$	0	0
$124$	3199.96	2.31746
$125$	1521.34	1.08858
$126$	0	0
$127$	1142.65	0.798374	0.399187	−	0.916869i	$-0.369292\pi$
$127$	1142.65	0.798374	0.399187	+	0.916869i	$0.369292\pi$
$128$	−2479.94	−1.71248
$129$	0	0
$130$	2799.67	1.88883
$131$	−104.451	−0.0696639	−0.0348319	−	0.999393i	$-0.511090\pi$
$131$	−104.451	−0.0696639	−0.0348319	+	0.999393i	$0.511090\pi$
$132$	0	0
$133$	759.803	0.495363
$134$	−3735.31	−2.40807
$135$	0	0
$136$	2870.93	1.81015
$137$	543.914	0.339195	0.169598	−	0.985513i	$-0.445753\pi$
$137$	543.914	0.339195	0.169598	+	0.985513i	$0.445753\pi$
$138$	0	0
$139$	−41.8374	−0.0255295	−0.0127647	−	0.999919i	$-0.504063\pi$
$139$	−41.8374	−0.0255295	−0.0127647	+	0.999919i	$0.504063\pi$
$140$	4489.73	2.71037
$141$	0	0
$142$	3279.85	1.93830
$143$	1391.22	0.813565
$144$	0	0
$145$	−2154.86	−1.23415
$146$	−5625.91	−3.18907
$147$	0	0
$148$	−3270.32	−1.81634
$149$	−790.139	−0.434435	−0.217217	−	0.976123i	$-0.569698\pi$
$149$	−790.139	−0.434435	−0.217217	+	0.976123i	$0.569698\pi$
$150$	0	0
$151$	−2073.79	−1.11763	−0.558816	−	0.829292i	$-0.688744\pi$
$151$	−2073.79	−1.11763	−0.558816	+	0.829292i	$0.688744\pi$
$152$	−996.774	−0.531902
$153$	0	0
$154$	3339.42	1.74739
$155$	−1826.95	−0.946734
$156$	0	0
$157$	2501.12	1.27141	0.635704	−	0.771933i	$-0.280710\pi$
$157$	2501.12	1.27141	0.635704	+	0.771933i	$0.280710\pi$
$158$	4786.83	2.41025
$159$	0	0
$160$	−75.1189	−0.0371167
$161$	3128.85	1.53160
$162$	0	0
$163$	−97.0569	−0.0466386	−0.0233193	−	0.999728i	$-0.507423\pi$
$163$	−97.0569	−0.0466386	−0.0233193	+	0.999728i	$0.507423\pi$
$164$	−3396.03	−1.61698
$165$	0	0
$166$	−3430.90	−1.60415
$167$	1826.71	0.846437	0.423218	−	0.906028i	$-0.360900\pi$
$167$	1826.71	0.846437	0.423218	+	0.906028i	$0.360900\pi$
$168$	0	0
$169$	1650.19	0.751111
$170$	−3257.28	−1.46954
$171$	0	0
$172$	1794.39	0.795473
$173$	1843.63	0.810223	0.405111	−	0.914267i	$-0.367233\pi$
$173$	1843.63	0.810223	0.405111	+	0.914267i	$0.367233\pi$
$174$	0	0
$175$	1227.38	0.530177
$176$	−1491.38	−0.638732
$177$	0	0
$178$	−2216.53	−0.933350
$179$	370.515	0.154713	0.0773565	−	0.997003i	$-0.475352\pi$
$179$	370.515	0.154713	0.0773565	+	0.997003i	$0.475352\pi$
$180$	0	0
$181$	−866.586	−0.355872	−0.177936	−	0.984042i	$-0.556942\pi$
$181$	−866.586	−0.355872	−0.177936	+	0.984042i	$0.556942\pi$
$182$	9234.60	3.76107
$183$	0	0
$184$	−4104.69	−1.64457
$185$	1867.11	0.742016
$186$	0	0
$187$	−1618.62	−0.632969
$188$	−756.857	−0.293614
$189$	0	0
$190$	1130.91	0.431816
$191$	3667.49	1.38937	0.694687	−	0.719312i	$-0.255543\pi$
$191$	3667.49	1.38937	0.694687	+	0.719312i	$0.255543\pi$
$192$	0	0
$193$	−4676.91	−1.74431	−0.872154	−	0.489231i	$-0.837277\pi$
$193$	−4676.91	−1.74431	−0.872154	+	0.489231i	$0.837277\pi$
$194$	−1919.16	−0.710245
$195$	0	0
$196$	9285.73	3.38401
$197$	−3700.99	−1.33850	−0.669251	−	0.743037i	$-0.733385\pi$
$197$	−3700.99	−1.33850	−0.669251	+	0.743037i	$0.733385\pi$
$198$	0	0
$199$	−1361.63	−0.485042	−0.242521	−	0.970146i	$-0.577974\pi$
$199$	−1361.63	−0.485042	−0.242521	+	0.970146i	$0.577974\pi$
$200$	−1610.18	−0.569283
$201$	0	0
$202$	5462.63	1.90272
$203$	−7107.72	−2.45746
$204$	0	0
$205$	1938.88	0.660573
$206$	−4333.01	−1.46551
$207$	0	0
$208$	−4124.15	−1.37480
$209$	561.977	0.185994
$210$	0	0
$211$	−2873.21	−0.937442	−0.468721	−	0.883346i	$-0.655285\pi$
$211$	−2873.21	−0.937442	−0.468721	+	0.883346i	$0.655285\pi$
$212$	−8044.85	−2.60624
$213$	0	0
$214$	−4127.64	−1.31850
$215$	−1024.47	−0.324968
$216$	0	0
$217$	−6026.10	−1.88515
$218$	−6485.74	−2.01500
$219$	0	0
$220$	3320.76	1.01766
$221$	−4476.02	−1.36240
$222$	0	0
$223$	3356.14	1.00782	0.503909	−	0.863757i	$-0.331895\pi$
$223$	3356.14	1.00782	0.503909	+	0.863757i	$0.331895\pi$
$224$	−247.777	−0.0739074
$225$	0	0
$226$	−3443.01	−1.01339
$227$	3991.01	1.16693	0.583464	−	0.812139i	$-0.301697\pi$
$227$	3991.01	1.16693	0.583464	+	0.812139i	$0.301697\pi$
$228$	0	0
$229$	−968.028	−0.279341	−0.139670	−	0.990198i	$-0.544604\pi$
$229$	−968.028	−0.279341	−0.139670	+	0.990198i	$0.544604\pi$
$230$	4657.06	1.33512
$231$	0	0
$232$	9324.51	2.63872
$233$	1227.70	0.345189	0.172595	−	0.984993i	$-0.444785\pi$
$233$	1227.70	0.345189	0.172595	+	0.984993i	$0.444785\pi$
$234$	0	0
$235$	432.110	0.119948
$236$	9062.19	2.49957
$237$	0	0
$238$	−10744.0	−2.92618
$239$	−647.534	−0.175253	−0.0876265	−	0.996153i	$-0.527928\pi$
$239$	−647.534	−0.175253	−0.0876265	+	0.996153i	$0.527928\pi$
$240$	0	0
$241$	−4174.42	−1.11576	−0.557880	−	0.829922i	$-0.688385\pi$
$241$	−4174.42	−1.11576	−0.557880	+	0.829922i	$0.688385\pi$
$242$	−4064.62	−1.07968
$243$	0	0
$244$	8826.10	2.31571
$245$	−5301.47	−1.38244
$246$	0	0
$247$	1554.05	0.400332
$248$	7905.55	2.02421
$249$	0	0
$250$	7469.02	1.88953
$251$	1368.39	0.344112	0.172056	−	0.985087i	$-0.444959\pi$
$251$	1368.39	0.344112	0.172056	+	0.985087i	$0.444959\pi$
$252$	0	0
$253$	2314.20	0.575070
$254$	5609.84	1.38580
$255$	0	0
$256$	−8240.77	−2.01191
$257$	−1709.94	−0.415031	−0.207515	−	0.978232i	$-0.566538\pi$
$257$	−1709.94	−0.415031	−0.207515	+	0.978232i	$0.566538\pi$
$258$	0	0
$259$	6158.59	1.47752
$260$	9183.00	2.19041
$261$	0	0
$262$	−512.806	−0.120921
$263$	−5368.95	−1.25880	−0.629399	−	0.777082i	$-0.716699\pi$
$263$	−5368.95	−1.25880	−0.629399	+	0.777082i	$0.716699\pi$
$264$	0	0
$265$	4593.02	1.06471
$266$	3730.27	0.859840
$267$	0	0
$268$	−12251.9	−2.79256
$269$	4213.70	0.955070	0.477535	−	0.878613i	$-0.341530\pi$
$269$	4213.70	0.955070	0.477535	+	0.878613i	$0.341530\pi$
$270$	0	0
$271$	4769.83	1.06918	0.534588	−	0.845113i	$-0.320467\pi$
$271$	4769.83	1.06918	0.534588	+	0.845113i	$0.320467\pi$
$272$	4798.25	1.06962
$273$	0	0
$274$	2670.36	0.588767
$275$	907.810	0.199066
$276$	0	0
$277$	5651.38	1.22584	0.612922	−	0.790144i	$-0.289994\pi$
$277$	5651.38	1.22584	0.612922	+	0.790144i	$0.289994\pi$
$278$	−205.401	−0.0443135
$279$	0	0
$280$	11091.9	2.36739
$281$	−3264.83	−0.693109	−0.346554	−	0.938030i	$-0.612648\pi$
$281$	−3264.83	−0.693109	−0.346554	+	0.938030i	$0.612648\pi$
$282$	0	0
$283$	2019.38	0.424169	0.212084	−	0.977251i	$-0.431975\pi$
$283$	2019.38	0.424169	0.212084	+	0.977251i	$0.431975\pi$
$284$	10758.0	2.24778
$285$	0	0
$286$	6830.23	1.41217
$287$	6395.32	1.31534
$288$	0	0
$289$	294.634	0.0599703
$290$	−10579.3	−2.14221
$291$	0	0
$292$	−18453.2	−3.69825
$293$	−3432.28	−0.684354	−0.342177	−	0.939636i	$-0.611164\pi$
$293$	−3432.28	−0.684354	−0.342177	+	0.939636i	$0.611164\pi$
$294$	0	0
$295$	−5173.85	−1.02113
$296$	−8079.36	−1.58650
$297$	0	0
$298$	−3879.20	−0.754081
$299$	6399.54	1.23778
$300$	0	0
$301$	−3379.16	−0.647082
$302$	−10181.3	−1.93996
$303$	0	0
$304$	−1665.93	−0.314301
$305$	−5039.06	−0.946019
$306$	0	0
$307$	−6327.92	−1.17640	−0.588198	−	0.808717i	$-0.700162\pi$
$307$	−6327.92	−1.17640	−0.588198	+	0.808717i	$0.700162\pi$
$308$	10953.4	2.02639
$309$	0	0
$310$	−8969.42	−1.64332
$311$	−9118.69	−1.66262	−0.831308	−	0.555812i	$-0.812408\pi$
$311$	−9118.69	−1.66262	−0.831308	+	0.555812i	$0.812408\pi$
$312$	0	0
$313$	4611.70	0.832808	0.416404	−	0.909180i	$-0.363290\pi$
$313$	4611.70	0.832808	0.416404	+	0.909180i	$0.363290\pi$
$314$	12279.3	2.20688
$315$	0	0
$316$	15700.9	2.79508
$317$	4243.78	0.751907	0.375954	−	0.926638i	$-0.377315\pi$
$317$	4243.78	0.751907	0.375954	+	0.926638i	$0.377315\pi$
$318$	0	0
$319$	−5257.12	−0.922702
$320$	4521.66	0.789901
$321$	0	0
$322$	15361.1	2.65851
$323$	−1808.06	−0.311466
$324$	0	0
$325$	2510.40	0.428467
$326$	−476.502	−0.0809541
$327$	0	0
$328$	−8389.92	−1.41237
$329$	1425.30	0.238842
$330$	0	0
$331$	−3803.59	−0.631613	−0.315807	−	0.948824i	$-0.602275\pi$
$331$	−3803.59	−0.631613	−0.315807	+	0.948824i	$0.602275\pi$
$332$	−11253.5	−1.86028
$333$	0	0
$334$	8968.25	1.46922
$335$	6994.95	1.14082
$336$	0	0
$337$	−6419.12	−1.03760	−0.518801	−	0.854895i	$-0.673621\pi$
$337$	−6419.12	−1.03760	−0.518801	+	0.854895i	$0.673621\pi$
$338$	8101.64	1.30376
$339$	0	0
$340$	−10684.0	−1.70418
$341$	−4457.11	−0.707819
$342$	0	0
$343$	−7085.05	−1.11533
$344$	4433.07	0.694812
$345$	0	0
$346$	9051.33	1.40637
$347$	3495.57	0.540783	0.270392	−	0.962750i	$-0.412847\pi$
$347$	3495.57	0.540783	0.270392	+	0.962750i	$0.412847\pi$
$348$	0	0
$349$	−435.838	−0.0668477	−0.0334239	−	0.999441i	$-0.510641\pi$
$349$	−435.838	−0.0668477	−0.0334239	+	0.999441i	$0.510641\pi$
$350$	6025.83	0.920268
$351$	0	0
$352$	−183.264	−0.0277500
$353$	−1941.73	−0.292771	−0.146385	−	0.989228i	$-0.546764\pi$
$353$	−1941.73	−0.292771	−0.146385	+	0.989228i	$0.546764\pi$
$354$	0	0
$355$	−6142.03	−0.918267
$356$	−7270.29	−1.08237
$357$	0	0
$358$	1819.05	0.268547
$359$	−7188.88	−1.05687	−0.528433	−	0.848975i	$-0.677220\pi$
$359$	−7188.88	−1.05687	−0.528433	+	0.848975i	$0.677220\pi$
$360$	0	0
$361$	−6231.25	−0.908478
$362$	−4254.52	−0.617714
$363$	0	0
$364$	30289.7	4.36158
$365$	10535.4	1.51082
$366$	0	0
$367$	1674.78	0.238209	0.119105	−	0.992882i	$-0.461998\pi$
$367$	1674.78	0.238209	0.119105	+	0.992882i	$0.461998\pi$
$368$	−6860.25	−0.971780
$369$	0	0
$370$	9166.62	1.28797
$371$	15149.9	2.12006
$372$	0	0
$373$	6028.31	0.836820	0.418410	−	0.908258i	$-0.362588\pi$
$373$	6028.31	0.836820	0.418410	+	0.908258i	$0.362588\pi$
$374$	−7946.64	−1.09869
$375$	0	0
$376$	−1869.83	−0.256460
$377$	−14537.7	−1.98602
$378$	0	0
$379$	−2065.63	−0.279959	−0.139979	−	0.990154i	$-0.544704\pi$
$379$	−2065.63	−0.279959	−0.139979	+	0.990154i	$0.544704\pi$
$380$	3709.42	0.500762
$381$	0	0
$382$	18005.6	2.41164
$383$	−7634.28	−1.01852	−0.509260	−	0.860613i	$-0.670081\pi$
$383$	−7634.28	−1.01852	−0.509260	+	0.860613i	$0.670081\pi$
$384$	0	0
$385$	−6253.58	−0.827823
$386$	−22961.4	−3.02773
$387$	0	0
$388$	−6294.89	−0.823646
$389$	2814.13	0.366792	0.183396	−	0.983039i	$-0.441291\pi$
$389$	2814.13	0.366792	0.183396	+	0.983039i	$0.441291\pi$
$390$	0	0
$391$	−7445.55	−0.963012
$392$	22940.5	2.95579
$393$	0	0
$394$	−18170.1	−2.32334
$395$	−8964.08	−1.14185
$396$	0	0
$397$	8990.52	1.13658	0.568288	−	0.822829i	$-0.307606\pi$
$397$	8990.52	1.13658	0.568288	+	0.822829i	$0.307606\pi$
$398$	−6684.94	−0.841924
$399$	0	0
$400$	−2691.12	−0.336390
$401$	4829.93	0.601484	0.300742	−	0.953706i	$-0.402766\pi$
$401$	4829.93	0.601484	0.300742	+	0.953706i	$0.402766\pi$
$402$	0	0
$403$	−12325.4	−1.52350
$404$	17917.6	2.20652
$405$	0	0
$406$	−34895.5	−4.26560
$407$	4555.11	0.554763
$408$	0	0
$409$	−6551.78	−0.792089	−0.396045	−	0.918231i	$-0.629617\pi$
$409$	−6551.78	−0.792089	−0.396045	+	0.918231i	$0.629617\pi$
$410$	9518.97	1.14661
$411$	0	0
$412$	−14212.4	−1.69950
$413$	−17065.7	−2.03329
$414$	0	0
$415$	6424.90	0.759966
$416$	−506.786	−0.0597289
$417$	0	0
$418$	2759.03	0.322844
$419$	−10660.5	−1.24296	−0.621481	−	0.783429i	$-0.713469\pi$
$419$	−10660.5	−1.24296	−0.621481	+	0.783429i	$0.713469\pi$
$420$	0	0
$421$	−8654.47	−1.00188	−0.500942	−	0.865481i	$-0.667013\pi$
$421$	−8654.47	−1.00188	−0.500942	+	0.865481i	$0.667013\pi$
$422$	−14106.1	−1.62719
$423$	0	0
$424$	−19874.9	−2.27644
$425$	−2920.73	−0.333355
$426$	0	0
$427$	−16621.1	−1.88373
$428$	−13538.8	−1.52902
$429$	0	0
$430$	−5029.64	−0.564072
$431$	−4990.22	−0.557703	−0.278852	−	0.960334i	$-0.589954\pi$
$431$	−4990.22	−0.557703	−0.278852	+	0.960334i	$0.589954\pi$
$432$	0	0
$433$	−7921.49	−0.879174	−0.439587	−	0.898200i	$-0.644875\pi$
$433$	−7921.49	−0.879174	−0.439587	+	0.898200i	$0.644875\pi$
$434$	−29585.2	−3.27220
$435$	0	0
$436$	−21273.4	−2.33673
$437$	2585.06	0.282975
$438$	0	0
$439$	11034.1	1.19961	0.599807	−	0.800145i	$-0.295244\pi$
$439$	11034.1	1.19961	0.599807	+	0.800145i	$0.295244\pi$
$440$	8203.97	0.888884
$441$	0	0
$442$	−21975.1	−2.36482
$443$	−9160.66	−0.982474	−0.491237	−	0.871026i	$-0.663455\pi$
$443$	−9160.66	−0.982474	−0.491237	+	0.871026i	$0.663455\pi$
$444$	0	0
$445$	4150.80	0.442173
$446$	16477.0	1.74935
$447$	0	0
$448$	14914.5	1.57287
$449$	10071.1	1.05854	0.529271	−	0.848453i	$-0.322465\pi$
$449$	10071.1	1.05854	0.529271	+	0.848453i	$0.322465\pi$
$450$	0	0
$451$	4730.20	0.493872
$452$	−11293.2	−1.17519
$453$	0	0
$454$	19593.9	2.02553
$455$	−17293.2	−1.78180
$456$	0	0
$457$	−2988.54	−0.305904	−0.152952	−	0.988234i	$-0.548878\pi$
$457$	−2988.54	−0.305904	−0.152952	+	0.988234i	$0.548878\pi$
$458$	−4752.55	−0.484873
$459$	0	0
$460$	15275.3	1.54829
$461$	−956.638	−0.0966487	−0.0483244	−	0.998832i	$-0.515388\pi$
$461$	−956.638	−0.0966487	−0.0483244	+	0.998832i	$0.515388\pi$
$462$	0	0
$463$	−4352.75	−0.436910	−0.218455	−	0.975847i	$-0.570102\pi$
$463$	−4352.75	−0.436910	−0.218455	+	0.975847i	$0.570102\pi$
$464$	15584.2	1.55923
$465$	0	0
$466$	6027.40	0.599171
$467$	−8131.35	−0.805725	−0.402863	−	0.915260i	$-0.631985\pi$
$467$	−8131.35	−0.805725	−0.402863	+	0.915260i	$0.631985\pi$
$468$	0	0
$469$	23072.5	2.27162
$470$	2121.45	0.208203
$471$	0	0
$472$	22388.2	2.18327
$473$	−2499.35	−0.242960
$474$	0	0
$475$	1014.06	0.0979544
$476$	−35240.6	−3.39339
$477$	0	0
$478$	−3179.08	−0.304200
$479$	12912.4	1.23170	0.615849	−	0.787864i	$-0.288813\pi$
$479$	12912.4	1.23170	0.615849	+	0.787864i	$0.288813\pi$
$480$	0	0
$481$	12596.4	1.19407
$482$	−20494.4	−1.93671
$483$	0	0
$484$	−13332.1	−1.25207
$485$	3593.92	0.336477
$486$	0	0
$487$	−11010.0	−1.02446	−0.512228	−	0.858850i	$-0.671180\pi$
$487$	−11010.0	−1.02446	−0.512228	+	0.858850i	$0.671180\pi$
$488$	21805.0	2.02268
$489$	0	0
$490$	−26027.7	−2.39961
$491$	14636.7	1.34531	0.672655	−	0.739956i	$-0.265154\pi$
$491$	14636.7	1.34531	0.672655	+	0.739956i	$0.265154\pi$
$492$	0	0
$493$	16913.9	1.54516
$494$	7629.64	0.694887
$495$	0	0
$496$	13212.7	1.19611
$497$	−20259.2	−1.82847
$498$	0	0
$499$	−7549.78	−0.677304	−0.338652	−	0.940912i	$-0.609971\pi$
$499$	−7549.78	−0.677304	−0.338652	+	0.940912i	$0.609971\pi$
$500$	24498.6	2.19122
$501$	0	0
$502$	6718.14	0.597301
$503$	16150.9	1.43167	0.715837	−	0.698267i	$-0.246045\pi$
$503$	16150.9	1.43167	0.715837	+	0.698267i	$0.246045\pi$
$504$	0	0
$505$	−10229.6	−0.901410
$506$	11361.6	0.998193
$507$	0	0
$508$	18400.4	1.60706
$509$	−19183.7	−1.67053	−0.835266	−	0.549846i	$-0.814686\pi$
$509$	−19183.7	−1.67053	−0.835266	+	0.549846i	$0.814686\pi$
$510$	0	0
$511$	34750.6	3.00836
$512$	−20618.7	−1.77974
$513$	0	0
$514$	−8394.96	−0.720400
$515$	8114.24	0.694284
$516$	0	0
$517$	1054.20	0.0896782
$518$	30235.7	2.56464
$519$	0	0
$520$	22686.7	1.91323
$521$	−6867.69	−0.577503	−0.288751	−	0.957404i	$-0.593240\pi$
$521$	−6867.69	−0.577503	−0.288751	+	0.957404i	$0.593240\pi$
$522$	0	0
$523$	11205.2	0.936841	0.468420	−	0.883506i	$-0.344823\pi$
$523$	11205.2	0.936841	0.468420	+	0.883506i	$0.344823\pi$
$524$	−1682.02	−0.140228
$525$	0	0
$526$	−26359.0	−2.18499
$527$	14340.0	1.18531
$528$	0	0
$529$	−1521.81	−0.125076
$530$	22549.5	1.84809
$531$	0	0
$532$	12235.4	0.997126
$533$	13080.6	1.06301
$534$	0	0
$535$	7729.64	0.624639
$536$	−30268.5	−2.43918
$537$	0	0
$538$	20687.2	1.65779
$539$	−12933.8	−1.03357
$540$	0	0
$541$	14471.4	1.15004	0.575021	−	0.818139i	$-0.304994\pi$
$541$	14471.4	1.15004	0.575021	+	0.818139i	$0.304994\pi$
$542$	23417.6	1.85585
$543$	0	0
$544$	589.621	0.0464702
$545$	12145.6	0.954603
$546$	0	0
$547$	−13482.7	−1.05389	−0.526946	−	0.849899i	$-0.676663\pi$
$547$	−13482.7	−1.05389	−0.526946	+	0.849899i	$0.676663\pi$
$548$	8758.84	0.682772
$549$	0	0
$550$	4456.91	0.345533
$551$	−5872.41	−0.454035
$552$	0	0
$553$	−29567.6	−2.27368
$554$	27745.6	2.12779
$555$	0	0
$556$	−673.722	−0.0513888
$557$	−10194.0	−0.775466	−0.387733	−	0.921772i	$-0.626742\pi$
$557$	−10194.0	−0.775466	−0.387733	+	0.921772i	$0.626742\pi$
$558$	0	0
$559$	−6911.52	−0.522945
$560$	18538.2	1.39889
$561$	0	0
$562$	−16028.7	−1.20308
$563$	−3725.21	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$563$	−3725.21	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$564$	0	0
$565$	6447.57	0.480091
$566$	9914.18	0.736262
$567$	0	0
$568$	26577.7	1.96334
$569$	6274.57	0.462291	0.231145	−	0.972919i	$-0.425753\pi$
$569$	6274.57	0.462291	0.231145	+	0.972919i	$0.425753\pi$
$570$	0	0
$571$	20390.6	1.49443	0.747214	−	0.664584i	$-0.231391\pi$
$571$	20390.6	1.49443	0.747214	+	0.664584i	$0.231391\pi$
$572$	22403.3	1.63764
$573$	0	0
$574$	31397.9	2.28314
$575$	4175.87	0.302862
$576$	0	0
$577$	675.385	0.0487290	0.0243645	−	0.999703i	$-0.492244\pi$
$577$	675.385	0.0487290	0.0243645	+	0.999703i	$0.492244\pi$
$578$	1446.51	0.104095
$579$	0	0
$580$	−34700.5	−2.48424
$581$	21192.3	1.51326
$582$	0	0
$583$	11205.4	0.796019
$584$	−45588.7	−3.23027
$585$	0	0
$586$	−16850.8	−1.18788
$587$	5949.63	0.418343	0.209172	−	0.977879i	$-0.432923\pi$
$587$	5949.63	0.418343	0.209172	+	0.977879i	$0.432923\pi$
$588$	0	0
$589$	−4978.78	−0.348297
$590$	−25401.1	−1.77245
$591$	0	0
$592$	−13503.2	−0.937464
$593$	22071.9	1.52847	0.764235	−	0.644937i	$-0.223117\pi$
$593$	22071.9	1.52847	0.764235	+	0.644937i	$0.223117\pi$
$594$	0	0
$595$	20119.8	1.38627
$596$	−12723.9	−0.874481
$597$	0	0
$598$	31418.6	2.14850
$599$	9547.79	0.651272	0.325636	−	0.945495i	$-0.394422\pi$
$599$	9547.79	0.651272	0.325636	+	0.945495i	$0.394422\pi$
$600$	0	0
$601$	−12303.7	−0.835076	−0.417538	−	0.908660i	$-0.637107\pi$
$601$	−12303.7	−0.835076	−0.417538	+	0.908660i	$0.637107\pi$
$602$	−16590.1	−1.12319
$603$	0	0
$604$	−33394.9	−2.24970
$605$	7611.63	0.511498
$606$	0	0
$607$	18252.2	1.22049	0.610243	−	0.792214i	$-0.291072\pi$
$607$	18252.2	1.22049	0.610243	+	0.792214i	$0.291072\pi$
$608$	−204.714	−0.0136550
$609$	0	0
$610$	−24739.3	−1.64208
$611$	2915.21	0.193022
$612$	0	0
$613$	−13990.5	−0.921809	−0.460905	−	0.887450i	$-0.652475\pi$
$613$	−13990.5	−0.921809	−0.460905	+	0.887450i	$0.652475\pi$
$614$	−31067.0	−2.04196
$615$	0	0
$616$	27060.5	1.76996
$617$	5747.16	0.374995	0.187498	−	0.982265i	$-0.439962\pi$
$617$	5747.16	0.374995	0.187498	+	0.982265i	$0.439962\pi$
$618$	0	0
$619$	−3913.70	−0.254127	−0.127064	−	0.991895i	$-0.540555\pi$
$619$	−3913.70	−0.254127	−0.127064	+	0.991895i	$0.540555\pi$
$620$	−29419.9	−1.90570
$621$	0	0
$622$	−44768.4	−2.88593
$623$	13691.3	0.880463
$624$	0	0
$625$	−8927.71	−0.571374
$626$	22641.2	1.44557
$627$	0	0
$628$	40276.4	2.55924
$629$	−14655.3	−0.929006
$630$	0	0
$631$	26311.5	1.65998	0.829988	−	0.557781i	$-0.188347\pi$
$631$	26311.5	1.65998	0.829988	+	0.557781i	$0.188347\pi$
$632$	38789.3	2.44139
$633$	0	0
$634$	20834.9	1.30514
$635$	−10505.3	−0.656520
$636$	0	0
$637$	−35766.1	−2.22466
$638$	−25809.9	−1.60160
$639$	0	0
$640$	22800.1	1.40821
$641$	−15538.7	−0.957476	−0.478738	−	0.877958i	$-0.658906\pi$
$641$	−15538.7	−0.957476	−0.478738	+	0.877958i	$0.658906\pi$
$642$	0	0
$643$	6125.10	0.375661	0.187831	−	0.982201i	$-0.439854\pi$
$643$	6125.10	0.375661	0.187831	+	0.982201i	$0.439854\pi$
$644$	50384.9	3.08299
$645$	0	0
$646$	−8876.72	−0.540635
$647$	−4635.86	−0.281692	−0.140846	−	0.990032i	$-0.544982\pi$
$647$	−4635.86	−0.281692	−0.140846	+	0.990032i	$0.544982\pi$
$648$	0	0
$649$	−12622.4	−0.763440
$650$	12324.8	0.743723
$651$	0	0
$652$	−1562.94	−0.0938796
$653$	11926.6	0.714740	0.357370	−	0.933963i	$-0.383674\pi$
$653$	11926.6	0.714740	0.357370	+	0.933963i	$0.383674\pi$
$654$	0	0
$655$	960.309	0.0572861
$656$	−14022.3	−0.834569
$657$	0	0
$658$	6997.52	0.414577
$659$	−881.384	−0.0520999	−0.0260500	−	0.999661i	$-0.508293\pi$
$659$	−881.384	−0.0520999	−0.0260500	+	0.999661i	$0.508293\pi$
$660$	0	0
$661$	−19161.6	−1.12753	−0.563766	−	0.825935i	$-0.690648\pi$
$661$	−19161.6	−1.12753	−0.563766	+	0.825935i	$0.690648\pi$
$662$	−18673.8	−1.09634
$663$	0	0
$664$	−27801.8	−1.62488
$665$	−6985.50	−0.407348
$666$	0	0
$667$	−24182.4	−1.40382
$668$	29416.1	1.70381
$669$	0	0
$670$	34341.8	1.98021
$671$	−12293.6	−0.707284
$672$	0	0
$673$	20156.0	1.15447	0.577235	−	0.816578i	$-0.304132\pi$
$673$	20156.0	1.15447	0.577235	+	0.816578i	$0.304132\pi$
$674$	−31514.8	−1.80104
$675$	0	0
$676$	26573.6	1.51192
$677$	−5567.34	−0.316056	−0.158028	−	0.987435i	$-0.550514\pi$
$677$	−5567.34	−0.316056	−0.158028	+	0.987435i	$0.550514\pi$
$678$	0	0
$679$	11854.4	0.670000
$680$	−26394.9	−1.48853
$681$	0	0
$682$	−21882.3	−1.22861
$683$	27835.8	1.55946	0.779728	−	0.626118i	$-0.215357\pi$
$683$	27835.8	1.55946	0.779728	+	0.626118i	$0.215357\pi$
$684$	0	0
$685$	−5000.66	−0.278927
$686$	−34784.2	−1.93596
$687$	0	0
$688$	7409.09	0.410565
$689$	30986.6	1.71334
$690$	0	0
$691$	17125.7	0.942824	0.471412	−	0.881913i	$-0.343745\pi$
$691$	17125.7	0.942824	0.471412	+	0.881913i	$0.343745\pi$
$692$	29688.6	1.63091
$693$	0	0
$694$	17161.5	0.938679
$695$	384.646	0.0209934
$696$	0	0
$697$	−15218.6	−0.827039
$698$	−2139.75	−0.116033
$699$	0	0
$700$	19764.9	1.06720
$701$	−5107.63	−0.275196	−0.137598	−	0.990488i	$-0.543938\pi$
$701$	−5107.63	−0.275196	−0.137598	+	0.990488i	$0.543938\pi$
$702$	0	0
$703$	5088.24	0.272983
$704$	11031.3	0.590564
$705$	0	0
$706$	−9532.97	−0.508184
$707$	−33742.0	−1.79490
$708$	0	0
$709$	11258.0	0.596335	0.298167	−	0.954514i	$-0.403625\pi$
$709$	11258.0	0.596335	0.298167	+	0.954514i	$0.403625\pi$
$710$	−30154.4	−1.59391
$711$	0	0
$712$	−17961.3	−0.945407
$713$	−20502.5	−1.07689
$714$	0	0
$715$	−12790.7	−0.669012
$716$	5966.54	0.311425
$717$	0	0
$718$	−35293.9	−1.83448
$719$	13735.6	0.712448	0.356224	−	0.934401i	$-0.384064\pi$
$719$	13735.6	0.712448	0.356224	+	0.934401i	$0.384064\pi$
$720$	0	0
$721$	26764.5	1.38247
$722$	−30592.4	−1.57691
$723$	0	0
$724$	−13954.9	−0.716341
$725$	−9486.22	−0.485944
$726$	0	0
$727$	14515.4	0.740502	0.370251	−	0.928932i	$-0.379272\pi$
$727$	14515.4	0.740502	0.370251	+	0.928932i	$0.379272\pi$
$728$	74831.1	3.80965
$729$	0	0
$730$	51723.7	2.62244
$731$	8041.22	0.406861
$732$	0	0
$733$	14218.5	0.716471	0.358236	−	0.933631i	$-0.383378\pi$
$733$	14218.5	0.716471	0.358236	+	0.933631i	$0.383378\pi$
$734$	8222.36	0.413478
$735$	0	0
$736$	−843.004	−0.0422195
$737$	17065.3	0.852926
$738$	0	0
$739$	8651.36	0.430643	0.215322	−	0.976543i	$-0.430920\pi$
$739$	8651.36	0.430643	0.215322	+	0.976543i	$0.430920\pi$
$740$	30066.8	1.49362
$741$	0	0
$742$	74378.6	3.67995
$743$	37263.7	1.83993	0.919967	−	0.391996i	$-0.128215\pi$
$743$	37263.7	1.83993	0.919967	+	0.391996i	$0.128215\pi$
$744$	0	0
$745$	7264.41	0.357245
$746$	29596.1	1.45253
$747$	0	0
$748$	−26065.2	−1.27412
$749$	25495.9	1.24379
$750$	0	0
$751$	−33980.7	−1.65110	−0.825548	−	0.564331i	$-0.809134\pi$
$751$	−33980.7	−1.65110	−0.825548	+	0.564331i	$0.809134\pi$
$752$	−3125.08	−0.151542
$753$	0	0
$754$	−71372.9	−3.44728
$755$	19066.0	0.919052
$756$	0	0
$757$	−33247.9	−1.59632	−0.798161	−	0.602445i	$-0.794193\pi$
$757$	−33247.9	−1.59632	−0.798161	+	0.602445i	$0.794193\pi$
$758$	−10141.2	−0.485945
$759$	0	0
$760$	9164.18	0.437394
$761$	13023.6	0.620376	0.310188	−	0.950675i	$-0.399608\pi$
$761$	13023.6	0.620376	0.310188	+	0.950675i	$0.399608\pi$
$762$	0	0
$763$	40061.6	1.90082
$764$	59058.9	2.79670
$765$	0	0
$766$	−37480.6	−1.76792
$767$	−34905.1	−1.64322
$768$	0	0
$769$	26526.2	1.24390	0.621949	−	0.783057i	$-0.286341\pi$
$769$	26526.2	1.24390	0.621949	+	0.783057i	$0.286341\pi$
$770$	−30702.0	−1.43692
$771$	0	0
$772$	−75313.9	−3.51115
$773$	−1143.17	−0.0531914	−0.0265957	−	0.999646i	$-0.508467\pi$
$773$	−1143.17	−0.0531914	−0.0265957	+	0.999646i	$0.508467\pi$
$774$	0	0
$775$	−8042.65	−0.372775
$776$	−15551.6	−0.719420
$777$	0	0
$778$	13816.0	0.636668
$779$	5283.83	0.243020
$780$	0	0
$781$	−14984.4	−0.686536
$782$	−36554.1	−1.67157
$783$	0	0
$784$	38341.0	1.74658
$785$	−22994.9	−1.04551
$786$	0	0
$787$	41354.8	1.87311	0.936556	−	0.350519i	$-0.113995\pi$
$787$	41354.8	1.87311	0.936556	+	0.350519i	$0.113995\pi$
$788$	−59598.4	−2.69429
$789$	0	0
$790$	−44009.3	−1.98200
$791$	21267.0	0.955965
$792$	0	0
$793$	−33995.8	−1.52235
$794$	44139.1	1.97284
$795$	0	0
$796$	−21926.8	−0.976349
$797$	−8795.77	−0.390918	−0.195459	−	0.980712i	$-0.562620\pi$
$797$	−8795.77	−0.390918	−0.195459	+	0.980712i	$0.562620\pi$
$798$	0	0
$799$	−3391.71	−0.150175
$800$	−330.692	−0.0146146
$801$	0	0
$802$	23712.6	1.04404
$803$	25702.7	1.12955
$804$	0	0
$805$	−28766.1	−1.25947
$806$	−60511.7	−2.64446
$807$	0	0
$808$	44265.6	1.92730
$809$	−2152.09	−0.0935272	−0.0467636	−	0.998906i	$-0.514891\pi$
$809$	−2152.09	−0.0935272	−0.0467636	+	0.998906i	$0.514891\pi$
$810$	0	0
$811$	24307.1	1.05245	0.526225	−	0.850346i	$-0.323607\pi$
$811$	24307.1	1.05245	0.526225	+	0.850346i	$0.323607\pi$
$812$	−114458.	−4.94666
$813$	0	0
$814$	22363.4	0.962944
$815$	892.325	0.0383519
$816$	0	0
$817$	−2791.87	−0.119554
$818$	−32166.0	−1.37489
$819$	0	0
$820$	31222.5	1.32968
$821$	−45118.9	−1.91798	−0.958988	−	0.283446i	$-0.908522\pi$
$821$	−45118.9	−1.91798	−0.958988	+	0.283446i	$0.908522\pi$
$822$	0	0
$823$	38608.9	1.63526	0.817631	−	0.575742i	$-0.195287\pi$
$823$	38608.9	1.63526	0.817631	+	0.575742i	$0.195287\pi$
$824$	−35111.9	−1.48444
$825$	0	0
$826$	−83784.4	−3.52934
$827$	17908.8	0.753023	0.376512	−	0.926412i	$-0.377124\pi$
$827$	17908.8	0.753023	0.376512	+	0.926412i	$0.377124\pi$
$828$	0	0
$829$	42616.2	1.78543	0.892716	−	0.450620i	$-0.148797\pi$
$829$	42616.2	1.78543	0.892716	+	0.450620i	$0.148797\pi$
$830$	31543.1	1.31913
$831$	0	0
$832$	30505.2	1.27112
$833$	41612.2	1.73082
$834$	0	0
$835$	−16794.4	−0.696043
$836$	9049.71	0.374391
$837$	0	0
$838$	−52338.1	−2.15750
$839$	−9077.30	−0.373520	−0.186760	−	0.982406i	$-0.559799\pi$
$839$	−9077.30	−0.373520	−0.186760	+	0.982406i	$0.559799\pi$
$840$	0	0
$841$	30545.6	1.25243
$842$	−42489.2	−1.73905
$843$	0	0
$844$	−46268.4	−1.88699
$845$	−15171.6	−0.617654
$846$	0	0
$847$	25106.6	1.01851
$848$	−33217.3	−1.34515
$849$	0	0
$850$	−14339.3	−0.578630
$851$	20953.2	0.844027
$852$	0	0
$853$	−18315.0	−0.735163	−0.367581	−	0.929991i	$-0.619814\pi$
$853$	−18315.0	−0.735163	−0.367581	+	0.929991i	$0.619814\pi$
$854$	−81601.7	−3.26973
$855$	0	0
$856$	−33447.7	−1.33554
$857$	13738.7	0.547615	0.273807	−	0.961785i	$-0.411717\pi$
$857$	13738.7	0.547615	0.273807	+	0.961785i	$0.411717\pi$
$858$	0	0
$859$	9424.20	0.374330	0.187165	−	0.982328i	$-0.440070\pi$
$859$	9424.20	0.374330	0.187165	+	0.982328i	$0.440070\pi$
$860$	−16497.4	−0.654134
$861$	0	0
$862$	−24499.5	−0.968048
$863$	40557.8	1.59977	0.799887	−	0.600151i	$-0.204893\pi$
$863$	40557.8	1.59977	0.799887	+	0.600151i	$0.204893\pi$
$864$	0	0
$865$	−16950.0	−0.666263
$866$	−38890.7	−1.52605
$867$	0	0
$868$	−97040.4	−3.79466
$869$	−21869.2	−0.853698
$870$	0	0
$871$	47191.1	1.83583
$872$	−52556.2	−2.04103
$873$	0	0
$874$	12691.4	0.491182
$875$	−46135.2	−1.78246
$876$	0	0
$877$	−12966.5	−0.499258	−0.249629	−	0.968342i	$-0.580309\pi$
$877$	−12966.5	−0.499258	−0.249629	+	0.968342i	$0.580309\pi$
$878$	54172.2	2.08226
$879$	0	0
$880$	13711.5	0.525243
$881$	15640.0	0.598098	0.299049	−	0.954238i	$-0.403331\pi$
$881$	15640.0	0.598098	0.299049	+	0.954238i	$0.403331\pi$
$882$	0	0
$883$	−10326.2	−0.393548	−0.196774	−	0.980449i	$-0.563047\pi$
$883$	−10326.2	−0.393548	−0.196774	+	0.980449i	$0.563047\pi$
$884$	−72078.9	−2.74239
$885$	0	0
$886$	−44974.4	−1.70535
$887$	2173.72	0.0822846	0.0411423	−	0.999153i	$-0.486900\pi$
$887$	2173.72	0.0822846	0.0411423	+	0.999153i	$0.486900\pi$
$888$	0	0
$889$	−34651.3	−1.30727
$890$	20378.4	0.767513
$891$	0	0
$892$	54045.0	2.02866
$893$	1177.58	0.0441280
$894$	0	0
$895$	−3406.46	−0.127224
$896$	75205.2	2.80405
$897$	0	0
$898$	49444.3	1.83739
$899$	46574.9	1.72787
$900$	0	0
$901$	−36051.4	−1.33301
$902$	23223.0	0.857252
$903$	0	0
$904$	−27899.9	−1.02648
$905$	7967.24	0.292641
$906$	0	0
$907$	29981.6	1.09760	0.548800	−	0.835954i	$-0.315085\pi$
$907$	29981.6	1.09760	0.548800	+	0.835954i	$0.315085\pi$
$908$	64268.7	2.34893
$909$	0	0
$910$	−84901.3	−3.09280
$911$	725.040	0.0263684	0.0131842	−	0.999913i	$-0.495803\pi$
$911$	725.040	0.0263684	0.0131842	+	0.999913i	$0.495803\pi$
$912$	0	0
$913$	15674.5	0.568183
$914$	−14672.3	−0.530980
$915$	0	0
$916$	−15588.5	−0.562291
$917$	3167.54	0.114069
$918$	0	0
$919$	−1750.94	−0.0628488	−0.0314244	−	0.999506i	$-0.510004\pi$
$919$	−1750.94	−0.0628488	−0.0314244	+	0.999506i	$0.510004\pi$
$920$	37737.8	1.35237
$921$	0	0
$922$	−4696.63	−0.167761
$923$	−41436.9	−1.47769
$924$	0	0
$925$	8219.48	0.292168
$926$	−21369.9	−0.758378
$927$	0	0
$928$	1915.03	0.0677413
$929$	−5500.62	−0.194262	−0.0971311	−	0.995272i	$-0.530967\pi$
$929$	−5500.62	−0.194262	−0.0971311	+	0.995272i	$0.530967\pi$
$930$	0	0
$931$	−14447.5	−0.508592
$932$	19770.0	0.694838
$933$	0	0
$934$	−39921.0	−1.39856
$935$	14881.3	0.520504
$936$	0	0
$937$	−55288.1	−1.92762	−0.963811	−	0.266586i	$-0.914104\pi$
$937$	−55288.1	−1.92762	−0.963811	+	0.266586i	$0.914104\pi$
$938$	113275.	3.94303
$939$	0	0
$940$	6958.42	0.241445
$941$	−47729.6	−1.65350	−0.826748	−	0.562573i	$-0.809812\pi$
$941$	−47729.6	−1.65350	−0.826748	+	0.562573i	$0.809812\pi$
$942$	0	0
$943$	21758.6	0.751388
$944$	37417.9	1.29010
$945$	0	0
$946$	−12270.6	−0.421724
$947$	27191.6	0.933061	0.466530	−	0.884505i	$-0.345504\pi$
$947$	27191.6	0.933061	0.466530	+	0.884505i	$0.345504\pi$
$948$	0	0
$949$	71076.5	2.43124
$950$	4978.55	0.170027
$951$	0	0
$952$	−87062.4	−2.96398
$953$	−2941.45	−0.0999822	−0.0499911	−	0.998750i	$-0.515919\pi$
$953$	−2941.45	−0.0999822	−0.0499911	+	0.998750i	$0.515919\pi$
$954$	0	0
$955$	−33718.3	−1.14251
$956$	−10427.5	−0.352770
$957$	0	0
$958$	63393.8	2.13795
$959$	−16494.5	−0.555405
$960$	0	0
$961$	9696.35	0.325479
$962$	61842.2	2.07263
$963$	0	0
$964$	−67222.1	−2.24593
$965$	42998.7	1.43438
$966$	0	0
$967$	45733.8	1.52089	0.760445	−	0.649402i	$-0.224981\pi$
$967$	45733.8	1.52089	0.760445	+	0.649402i	$0.224981\pi$
$968$	−32937.0	−1.09363
$969$	0	0
$970$	17644.4	0.584050
$971$	−57658.9	−1.90562	−0.952812	−	0.303561i	$-0.901824\pi$
$971$	−57658.9	−1.90562	−0.952812	+	0.303561i	$0.901824\pi$
$972$	0	0
$973$	1268.74	0.0418025
$974$	−54053.7	−1.77823
$975$	0	0
$976$	36443.1	1.19520
$977$	−46156.4	−1.51144	−0.755718	−	0.654897i	$-0.772712\pi$
$977$	−46156.4	−1.51144	−0.755718	+	0.654897i	$0.772712\pi$
$978$	0	0
$979$	10126.5	0.330587
$980$	−85371.5	−2.78275
$981$	0	0
$982$	71859.3	2.33516
$983$	−10931.6	−0.354694	−0.177347	−	0.984148i	$-0.556752\pi$
$983$	−10931.6	−0.354694	−0.177347	+	0.984148i	$0.556752\pi$
$984$	0	0
$985$	34026.3	1.10068
$986$	83038.9	2.68205
$987$	0	0
$988$	25025.4	0.805835
$989$	−11496.8	−0.369644
$990$	0	0
$991$	−15279.8	−0.489788	−0.244894	−	0.969550i	$-0.578753\pi$
$991$	−15279.8	−0.489788	−0.244894	+	0.969550i	$0.578753\pi$
$992$	1623.61	0.0519654
$993$	0	0
$994$	−99462.9	−3.17382
$995$	12518.6	0.398860
$996$	0	0
$997$	−7648.81	−0.242969	−0.121485	−	0.992593i	$-0.538766\pi$
$997$	−7648.81	−0.242969	−0.121485	+	0.992593i	$0.538766\pi$
$998$	−37065.8	−1.17565
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	423.4.a.b.1.3		3
3.2	odd	2		47.4.a.a.1.1	✓	3
12.11	even	2		752.4.a.c.1.2		3
15.14	odd	2		1175.4.a.a.1.3		3
21.20	even	2		2303.4.a.a.1.1		3
141.140	even	2		2209.4.a.a.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
47.4.a.a.1.1	✓	3	3.2	odd	2
423.4.a.b.1.3		3	1.1	even	1	trivial
752.4.a.c.1.2		3	12.11	even	2
1175.4.a.a.1.3		3	15.14	odd	2
2209.4.a.a.1.1		3	141.140	even	2
2303.4.a.a.1.1		3	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 423 = 3^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	423.a (trivial)

\(f(q)\)	\(=\)	\(q+4.90952 q^{2} +16.1033 q^{4} -9.19383 q^{5} -30.3255 q^{7} +39.7835 q^{8} +O(q^{10})\)	\(q+4.90952 q^{2} +16.1033 q^{4} -9.19383 q^{5} -30.3255 q^{7} +39.7835 q^{8} -45.1373 q^{10} -22.4298 q^{11} -62.0257 q^{13} -148.883 q^{14} +66.4910 q^{16} +72.1639 q^{17} -25.0550 q^{19} -148.051 q^{20} -110.119 q^{22} -103.176 q^{23} -40.4735 q^{25} -304.516 q^{26} -488.341 q^{28} +234.381 q^{29} +198.714 q^{31} +8.17058 q^{32} +354.290 q^{34} +278.807 q^{35} -203.083 q^{37} -123.008 q^{38} -365.763 q^{40} -210.889 q^{41} +111.430 q^{43} -361.194 q^{44} -506.542 q^{46} -47.0000 q^{47} +576.634 q^{49} -198.705 q^{50} -998.822 q^{52} -499.576 q^{53} +206.215 q^{55} -1206.45 q^{56} +1150.70 q^{58} +562.752 q^{59} +548.091 q^{61} +975.591 q^{62} -491.814 q^{64} +570.254 q^{65} -760.831 q^{67} +1162.08 q^{68} +1368.81 q^{70} +668.059 q^{71} -1145.92 q^{73} -997.040 q^{74} -403.469 q^{76} +680.193 q^{77} +975.010 q^{79} -611.307 q^{80} -1035.37 q^{82} -698.827 q^{83} -663.463 q^{85} +547.067 q^{86} -892.335 q^{88} -451.477 q^{89} +1880.96 q^{91} -1661.47 q^{92} -230.747 q^{94} +230.351 q^{95} -390.906 q^{97} +2830.99 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 45 q^{7} + 39 q^{8}+O(q^{10}) \) 3 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 45 * q^7 + 39 * q^8	\( 3 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 45 q^{7} + 39 q^{8} - 40 q^{10} - 2 q^{11} - 80 q^{13} - 162 q^{14} + 89 q^{16} + 39 q^{17} - 24 q^{19} - 232 q^{20} - 14 q^{22} - 120 q^{23} - 171 q^{25} - 316 q^{26} - 408 q^{28} + 184 q^{29} - 4 q^{31} + 7 q^{32} + 218 q^{34} + 156 q^{35} - 589 q^{37} - 42 q^{38} - 432 q^{40} + 92 q^{41} - 250 q^{43} - 466 q^{44} - 816 q^{46} - 141 q^{47} + 30 q^{49} - 137 q^{50} - 900 q^{52} - 459 q^{53} + 448 q^{55} - 1032 q^{56} + 684 q^{58} - 579 q^{59} + 267 q^{61} + 244 q^{62} - 87 q^{64} + 424 q^{65} - 540 q^{67} + 1334 q^{68} + 1236 q^{70} - 749 q^{71} - 1924 q^{73} - 950 q^{74} - 402 q^{76} + 288 q^{77} + 805 q^{79} - 448 q^{80} - 938 q^{82} - 712 q^{83} - 1038 q^{85} + 1294 q^{86} - 2190 q^{88} - 835 q^{89} + 2040 q^{91} - 1596 q^{92} - 235 q^{94} + 312 q^{95} - 2243 q^{97} + 2989 q^{98}+O(q^{100}) \) 3 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 45 * q^7 + 39 * q^8 - 40 * q^10 - 2 * q^11 - 80 * q^13 - 162 * q^14 + 89 * q^16 + 39 * q^17 - 24 * q^19 - 232 * q^20 - 14 * q^22 - 120 * q^23 - 171 * q^25 - 316 * q^26 - 408 * q^28 + 184 * q^29 - 4 * q^31 + 7 * q^32 + 218 * q^34 + 156 * q^35 - 589 * q^37 - 42 * q^38 - 432 * q^40 + 92 * q^41 - 250 * q^43 - 466 * q^44 - 816 * q^46 - 141 * q^47 + 30 * q^49 - 137 * q^50 - 900 * q^52 - 459 * q^53 + 448 * q^55 - 1032 * q^56 + 684 * q^58 - 579 * q^59 + 267 * q^61 + 244 * q^62 - 87 * q^64 + 424 * q^65 - 540 * q^67 + 1334 * q^68 + 1236 * q^70 - 749 * q^71 - 1924 * q^73 - 950 * q^74 - 402 * q^76 + 288 * q^77 + 805 * q^79 - 448 * q^80 - 938 * q^82 - 712 * q^83 - 1038 * q^85 + 1294 * q^86 - 2190 * q^88 - 835 * q^89 + 2040 * q^91 - 1596 * q^92 - 235 * q^94 + 312 * q^95 - 2243 * q^97 + 2989 * q^98

Introduction

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