LMFDB - Embedded newform 575.4.b.f.24.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [575,4,Mod(24,575)] 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("575.24"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,18] 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:	no
Analytic conductor:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{109})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 55x^{2} + 729 $ x^4 + 55*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.1
Root			$4.72015i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.4.b.f.24.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.00000i q^{2} -3.72015i q^{3} -1.00000 q^{4} -11.1605 q^{6} -25.6008i q^{7} -21.0000i q^{8} +13.1605 q^{9} +12.6008 q^{11} +3.72015i q^{12} -8.16046i q^{13} -76.8023 q^{14} -71.0000 q^{16} -76.0411i q^{17} -39.4814i q^{18} +103.362 q^{19} -95.2388 q^{21} -37.8023i q^{22} +23.0000i q^{23} -78.1232 q^{24} -24.4814 q^{26} -149.403i q^{27} +25.6008i q^{28} +267.202 q^{29} -63.7574 q^{31} +45.0000i q^{32} -46.8768i q^{33} -228.123 q^{34} -13.1605 q^{36} +112.164i q^{37} -310.086i q^{38} -30.3582 q^{39} -239.078 q^{41} +285.716i q^{42} -282.239i q^{43} -12.6008 q^{44} +69.0000 q^{46} +577.291i q^{47} +264.131i q^{48} -312.399 q^{49} -282.884 q^{51} +8.16046i q^{52} +2.31326i q^{53} -448.209 q^{54} -537.616 q^{56} -384.522i q^{57} -801.605i q^{58} -272.888 q^{59} +294.049 q^{61} +191.272i q^{62} -336.918i q^{63} -433.000 q^{64} -140.630 q^{66} +426.732i q^{67} +76.0411i q^{68} +85.5635 q^{69} -1020.85 q^{71} -276.370i q^{72} +286.650i q^{73} +336.493 q^{74} -103.362 q^{76} -322.589i q^{77} +91.0745i q^{78} +551.866 q^{79} -200.470 q^{81} +717.235i q^{82} -21.7021i q^{83} +95.2388 q^{84} -846.716 q^{86} -994.031i q^{87} -264.616i q^{88} +1049.63 q^{89} -208.914 q^{91} -23.0000i q^{92} +237.187i q^{93} +1731.87 q^{94} +167.407 q^{96} +1729.19i q^{97} +937.198i q^{98} +165.832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 18 q^{6} - 10 q^{9} - 54 q^{11} + 6 q^{14} - 284 q^{16} + 142 q^{19} - 548 q^{21} + 126 q^{24} + 90 q^{26} + 860 q^{29} - 610 q^{31} - 474 q^{34} + 10 q^{36} - 372 q^{39} - 1186 q^{41} + 54 q^{44}+ \cdots + 1770 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 18 * q^6 - 10 * q^9 - 54 * q^11 + 6 * q^14 - 284 * q^16 + 142 * q^19 - 548 * q^21 + 126 * q^24 + 90 * q^26 + 860 * q^29 - 610 * q^31 - 474 * q^34 + 10 * q^36 - 372 * q^39 - 1186 * q^41 + 54 * q^44 + 276 * q^46 - 1354 * q^49 - 526 * q^51 - 540 * q^54 + 42 * q^56 + 36 * q^59 - 14 * q^61 - 1732 * q^64 - 1878 * q^66 - 138 * q^69 - 2058 * q^71 - 408 * q^74 - 142 * q^76 - 1384 * q^79 - 2180 * q^81 + 548 * q^84 - 3888 * q^86 + 440 * q^89 - 1650 * q^91 + 2292 * q^94 - 270 * q^96 + 1770 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 3.00000i	− 1.06066i	−0.847791	−	0.530330i	$-0.822068\pi$
$2$	− 3.00000i	− 1.06066i	0.847791	−	0.530330i	$-0.177932\pi$
$3$	− 3.72015i	− 0.715944i	−0.933732	−	0.357972i	$-0.883468\pi$
$3$	− 3.72015i	− 0.715944i	0.933732	−	0.357972i	$-0.116532\pi$
$4$	−1.00000	−0.125000
$5$	0	0
$5$	0	0
$6$	−11.1605	−0.759373
$7$	− 25.6008i	− 1.38231i	−0.722706	−	0.691156i	$-0.757102\pi$
$7$	− 25.6008i	− 1.38231i	0.722706	−	0.691156i	$-0.242898\pi$
$8$	− 21.0000i	− 0.928078i
$9$	13.1605	0.487424
$10$	0	0
$11$	12.6008	0.345389	0.172694	−	0.984975i	$-0.444753\pi$
$11$	12.6008	0.345389	0.172694	+	0.984975i	$0.444753\pi$
$12$	3.72015i	0.0894930i
$13$	− 8.16046i	− 0.174100i	−0.996204	−	0.0870502i	$-0.972256\pi$
$13$	− 8.16046i	− 0.174100i	0.996204	−	0.0870502i	$-0.0277440\pi$
$14$	−76.8023	−1.46616
$15$	0	0
$16$	−71.0000	−1.10938
$17$	− 76.0411i	− 1.08486i	−0.840100	−	0.542431i	$-0.817504\pi$
$17$	− 76.0411i	− 1.08486i	0.840100	−	0.542431i	$-0.182496\pi$
$18$	− 39.4814i	− 0.516992i
$19$	103.362	1.24805	0.624023	−	0.781406i	$-0.285497\pi$
$19$	103.362	1.24805	0.624023	+	0.781406i	$0.285497\pi$
$20$	0	0
$21$	−95.2388	−0.989657
$22$	− 37.8023i	− 0.366340i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	−78.1232	−0.664451
$25$	0	0
$26$	−24.4814	−0.184661
$27$	− 149.403i	− 1.06491i
$28$	25.6008i	0.172789i
$29$	267.202	1.71097	0.855484	−	0.517829i	$-0.173260\pi$
$29$	267.202	1.71097	0.855484	+	0.517829i	$0.173260\pi$
$30$	0	0
$31$	−63.7574	−0.369392	−0.184696	−	0.982796i	$-0.559130\pi$
$31$	−63.7574	−0.369392	−0.184696	+	0.982796i	$0.559130\pi$
$32$	45.0000i	0.248592i
$33$	− 46.8768i	− 0.247279i
$34$	−228.123	−1.15067
$35$	0	0
$36$	−13.1605	−0.0609281
$37$	112.164i	0.498370i	0.968456	+	0.249185i	$0.0801627\pi$
$37$	112.164i	0.498370i	−0.968456	+	0.249185i	$0.919837\pi$
$38$	− 310.086i	− 1.32375i
$39$	−30.3582	−0.124646
$40$	0	0
$41$	−239.078	−0.910677	−0.455339	−	0.890318i	$-0.650482\pi$
$41$	−239.078	−0.910677	−0.455339	+	0.890318i	$0.650482\pi$
$42$	285.716i	1.04969i
$43$	− 282.239i	− 1.00095i	−0.865750	−	0.500477i	$-0.833158\pi$
$43$	− 282.239i	− 1.00095i	0.865750	−	0.500477i	$-0.166842\pi$
$44$	−12.6008	−0.0431736
$45$	0	0
$46$	69.0000	0.221163
$47$	577.291i	1.79163i	0.444427	+	0.895815i	$0.353407\pi$
$47$	577.291i	1.79163i	−0.444427	+	0.895815i	$0.646593\pi$
$48$	264.131i	0.794250i
$49$	−312.399	−0.910785
$50$	0	0
$51$	−282.884	−0.776701
$52$	8.16046i	0.0217625i
$53$	2.31326i	0.00599529i	0.999996	+	0.00299764i	$0.000954181\pi$
$53$	2.31326i	0.00599529i	−0.999996	+	0.00299764i	$0.999046\pi$
$54$	−448.209	−1.12951
$55$	0	0
$56$	−537.616	−1.28289
$57$	− 384.522i	− 0.893531i
$58$	− 801.605i	− 1.81476i
$59$	−272.888	−0.602153	−0.301077	−	0.953600i	$-0.597346\pi$
$59$	−272.888	−0.602153	−0.301077	+	0.953600i	$0.597346\pi$
$60$	0	0
$61$	294.049	0.617198	0.308599	−	0.951192i	$-0.400140\pi$
$61$	294.049	0.617198	0.308599	+	0.951192i	$0.400140\pi$
$62$	191.272i	0.391800i
$63$	− 336.918i	− 0.673772i
$64$	−433.000	−0.845703
$65$	0	0
$66$	−140.630	−0.262279
$67$	426.732i	0.778113i	0.921214	+	0.389056i	$0.127199\pi$
$67$	426.732i	0.778113i	−0.921214	+	0.389056i	$0.872801\pi$
$68$	76.0411i	0.135608i
$69$	85.5635	0.149285
$70$	0	0
$71$	−1020.85	−1.70638	−0.853191	−	0.521598i	$-0.825336\pi$
$71$	−1020.85	−1.70638	−0.853191	+	0.521598i	$0.825336\pi$
$72$	− 276.370i	− 0.452368i
$73$	286.650i	0.459586i	0.973240	+	0.229793i	$0.0738049\pi$
$73$	286.650i	0.459586i	−0.973240	+	0.229793i	$0.926195\pi$
$74$	336.493	0.528601
$75$	0	0
$76$	−103.362	−0.156006
$77$	− 322.589i	− 0.477435i
$78$	91.0745i	0.132207i
$79$	551.866	0.785947	0.392974	−	0.919550i	$-0.371446\pi$
$79$	551.866	0.785947	0.392974	+	0.919550i	$0.371446\pi$
$80$	0	0
$81$	−200.470	−0.274993
$82$	717.235i	0.965919i
$83$	− 21.7021i	− 0.0287001i	−0.999897	−	0.0143501i	$-0.995432\pi$
$83$	− 21.7021i	− 0.0287001i	0.999897	−	0.0143501i	$-0.00456793\pi$
$84$	95.2388	0.123707
$85$	0	0
$86$	−846.716	−1.06167
$87$	− 994.031i	− 1.22496i
$88$	− 264.616i	− 0.320547i
$89$	1049.63	1.25012	0.625058	−	0.780578i	$-0.285075\pi$
$89$	1049.63	1.25012	0.625058	+	0.780578i	$0.285075\pi$
$90$	0	0
$91$	−208.914	−0.240661
$92$	− 23.0000i	− 0.0260643i
$93$	237.187i	0.264464i
$94$	1731.87	1.90031
$95$	0	0
$96$	167.407	0.177978
$97$	1729.19i	1.81003i	0.425381	+	0.905014i	$0.360140\pi$
$97$	1729.19i	1.81003i	−0.425381	+	0.905014i	$0.639860\pi$
$98$	937.198i	0.966033i
$99$	165.832	0.168351
$100$	0	0
$101$	855.874	0.843195	0.421597	−	0.906783i	$-0.361470\pi$
$101$	855.874	0.843195	0.421597	+	0.906783i	$0.361470\pi$
$102$	848.653i	0.823816i
$103$	632.534i	0.605101i	0.953133	+	0.302551i	$0.0978381\pi$
$103$	632.534i	0.605101i	−0.953133	+	0.302551i	$0.902162\pi$
$104$	−171.370	−0.161579
$105$	0	0
$106$	6.93977	0.00635896
$107$	1309.62i	1.18323i	0.806220	+	0.591616i	$0.201510\pi$
$107$	1309.62i	1.18323i	−0.806220	+	0.591616i	$0.798490\pi$
$108$	149.403i	0.133114i
$109$	−1726.21	−1.51689	−0.758443	−	0.651739i	$-0.774040\pi$
$109$	−1726.21	−1.51689	−0.758443	+	0.651739i	$0.774040\pi$
$110$	0	0
$111$	417.268	0.356805
$112$	1817.65i	1.53350i
$113$	1492.24i	1.24228i	0.783698	+	0.621142i	$0.213331\pi$
$113$	1492.24i	1.24228i	−0.783698	+	0.621142i	$0.786669\pi$
$114$	−1153.57	−0.947732
$115$	0	0
$116$	−267.202	−0.213871
$117$	− 107.395i	− 0.0848608i
$118$	818.665i	0.638680i
$119$	−1946.71	−1.49962
$120$	0	0
$121$	−1172.22	−0.880707
$122$	− 882.146i	− 0.654637i
$123$	889.408i	0.651994i
$124$	63.7574	0.0461741
$125$	0	0
$126$	−1010.75	−0.714644
$127$	− 1368.29i	− 0.956034i	−0.878351	−	0.478017i	$-0.841356\pi$
$127$	− 1368.29i	− 0.956034i	0.878351	−	0.478017i	$-0.158644\pi$
$128$	1659.00i	1.14560i
$129$	−1049.97	−0.716627
$130$	0	0
$131$	1984.24	1.32339	0.661694	−	0.749774i	$-0.269838\pi$
$131$	1984.24	1.32339	0.661694	+	0.749774i	$0.269838\pi$
$132$	46.8768i	0.0309098i
$133$	− 2646.15i	− 1.72519i
$134$	1280.19	0.825313
$135$	0	0
$136$	−1596.86	−1.00684
$137$	273.331i	0.170455i	0.996362	+	0.0852273i	$0.0271616\pi$
$137$	273.331i	0.170455i	−0.996362	+	0.0852273i	$0.972838\pi$
$138$	− 256.691i	− 0.158340i
$139$	557.889	0.340428	0.170214	−	0.985407i	$-0.445554\pi$
$139$	557.889	0.340428	0.170214	+	0.985407i	$0.445554\pi$
$140$	0	0
$141$	2147.61	1.28271
$142$	3062.56i	1.80989i
$143$	− 102.828i	− 0.0601323i
$144$	−934.393	−0.540736
$145$	0	0
$146$	859.949	0.487465
$147$	1162.17i	0.652071i
$148$	− 112.164i	− 0.0622963i
$149$	1354.16	0.744545	0.372272	−	0.928124i	$-0.378579\pi$
$149$	1354.16	0.744545	0.372272	+	0.928124i	$0.378579\pi$
$150$	0	0
$151$	162.652	0.0876586	0.0438293	−	0.999039i	$-0.486044\pi$
$151$	162.652	0.0876586	0.0438293	+	0.999039i	$0.486044\pi$
$152$	− 2170.60i	− 1.15828i
$153$	− 1000.74i	− 0.528789i
$154$	−967.768	−0.506396
$155$	0	0
$156$	30.3582	0.0155808
$157$	− 2364.83i	− 1.20213i	−0.799201	−	0.601063i	$-0.794744\pi$
$157$	− 2364.83i	− 1.20213i	0.799201	−	0.601063i	$-0.205256\pi$
$158$	− 1655.60i	− 0.833623i
$159$	8.60567	0.00429229
$160$	0	0
$161$	588.818	0.288232
$162$	601.410i	0.291674i
$163$	− 471.162i	− 0.226406i	−0.993572	−	0.113203i	$-0.963889\pi$
$163$	− 471.162i	− 0.226406i	0.993572	−	0.113203i	$-0.0361111\pi$
$164$	239.078	0.113835
$165$	0	0
$166$	−65.1062	−0.0304411
$167$	− 3735.73i	− 1.73102i	−0.500895	−	0.865508i	$-0.666996\pi$
$167$	− 3735.73i	− 1.73102i	0.500895	−	0.865508i	$-0.333004\pi$
$168$	2000.01i	0.918479i
$169$	2130.41	0.969689
$170$	0	0
$171$	1360.29	0.608328
$172$	282.239i	0.125119i
$173$	2709.33i	1.19067i	0.803477	+	0.595336i	$0.202981\pi$
$173$	2709.33i	1.19067i	−0.803477	+	0.595336i	$0.797019\pi$
$174$	−2982.09	−1.29926
$175$	0	0
$176$	−894.654	−0.383165
$177$	1015.19i	0.431108i
$178$	− 3148.88i	− 1.32595i
$179$	−3300.43	−1.37813	−0.689066	−	0.724699i	$-0.741979\pi$
$179$	−3300.43	−1.37813	−0.689066	+	0.724699i	$0.741979\pi$
$180$	0	0
$181$	−583.868	−0.239771	−0.119886	−	0.992788i	$-0.538253\pi$
$181$	−583.868	−0.239771	−0.119886	+	0.992788i	$0.538253\pi$
$182$	626.742i	0.255259i
$183$	− 1093.91i	− 0.441879i
$184$	483.000	0.193518
$185$	0	0
$186$	711.562	0.280507
$187$	− 958.176i	− 0.374699i
$188$	− 577.291i	− 0.223954i
$189$	−3824.83	−1.47204
$190$	0	0
$191$	−3058.98	−1.15885	−0.579424	−	0.815026i	$-0.696722\pi$
$191$	−3058.98	−1.15885	−0.579424	+	0.815026i	$0.696722\pi$
$192$	1610.83i	0.605476i
$193$	1336.93i	0.498625i	0.968423	+	0.249313i	$0.0802046\pi$
$193$	1336.93i	0.498625i	−0.968423	+	0.249313i	$0.919795\pi$
$194$	5187.57	1.91983
$195$	0	0
$196$	312.399	0.113848
$197$	− 2449.21i	− 0.885783i	−0.896575	−	0.442892i	$-0.853953\pi$
$197$	− 2449.21i	− 0.885783i	0.896575	−	0.442892i	$-0.146047\pi$
$198$	− 497.496i	− 0.178563i
$199$	−1138.66	−0.405614	−0.202807	−	0.979219i	$-0.565006\pi$
$199$	−1138.66	−0.405614	−0.202807	+	0.979219i	$0.565006\pi$
$200$	0	0
$201$	1587.51	0.557085
$202$	− 2567.62i	− 0.894343i
$203$	− 6840.56i	− 2.36509i
$204$	282.884	0.0970876
$205$	0	0
$206$	1897.60	0.641807
$207$	302.691i	0.101635i
$208$	579.393i	0.193143i
$209$	1302.44	0.431061
$210$	0	0
$211$	5596.81	1.82607	0.913034	−	0.407884i	$-0.133733\pi$
$211$	5596.81	1.82607	0.913034	+	0.407884i	$0.133733\pi$
$212$	− 2.31326i	0 0.000749411i
$213$	3797.74i	1.22167i
$214$	3928.86	1.25501
$215$	0	0
$216$	−3137.46	−0.988321
$217$	1632.24i	0.510615i
$218$	5178.62i	1.60890i
$219$	1066.38	0.329038
$220$	0	0
$221$	−620.530	−0.188875
$222$	− 1251.81i	− 0.378449i
$223$	− 5580.60i	− 1.67581i	−0.545820	−	0.837903i	$-0.683782\pi$
$223$	− 5580.60i	− 1.67581i	0.545820	−	0.837903i	$-0.316218\pi$
$224$	1152.03	0.343632
$225$	0	0
$226$	4476.72	1.31764
$227$	− 566.323i	− 0.165587i	−0.996567	−	0.0827934i	$-0.973616\pi$
$227$	− 566.323i	− 0.165587i	0.996567	−	0.0827934i	$-0.0263841\pi$
$228$	384.522i	0.111691i
$229$	693.702	0.200180	0.100090	−	0.994978i	$-0.468087\pi$
$229$	693.702	0.200180	0.100090	+	0.994978i	$0.468087\pi$
$230$	0	0
$231$	−1200.08	−0.341816
$232$	− 5611.23i	− 1.58791i
$233$	2208.68i	0.621011i	0.950572	+	0.310506i	$0.100498\pi$
$233$	2208.68i	0.621011i	−0.950572	+	0.310506i	$0.899502\pi$
$234$	−322.186	−0.0900084
$235$	0	0
$236$	272.888	0.0752691
$237$	− 2053.03i	− 0.562694i
$238$	5840.13i	1.59059i
$239$	3876.79	1.04924	0.524621	−	0.851336i	$-0.324207\pi$
$239$	3876.79	1.04924	0.524621	+	0.851336i	$0.324207\pi$
$240$	0	0
$241$	−2024.57	−0.541136	−0.270568	−	0.962701i	$-0.587211\pi$
$241$	−2024.57	−0.541136	−0.270568	+	0.962701i	$0.587211\pi$
$242$	3516.66i	0.934131i
$243$	− 3288.10i	− 0.868033i
$244$	−294.049	−0.0771498
$245$	0	0
$246$	2668.22	0.691544
$247$	− 843.481i	− 0.217285i
$248$	1338.91i	0.342825i
$249$	−80.7350	−0.0205477
$250$	0	0
$251$	−733.444	−0.184441	−0.0922203	−	0.995739i	$-0.529396\pi$
$251$	−733.444	−0.184441	−0.0922203	+	0.995739i	$0.529396\pi$
$252$	336.918i	0.0842215i
$253$	289.818i	0.0720185i
$254$	−4104.88	−1.01403
$255$	0	0
$256$	1513.00	0.369385
$257$	2367.83i	0.574713i	0.957824	+	0.287356i	$0.0927764\pi$
$257$	2367.83i	0.574713i	−0.957824	+	0.287356i	$0.907224\pi$
$258$	3149.91i	0.760097i
$259$	2871.49	0.688903
$260$	0	0
$261$	3516.50	0.833968
$262$	− 5952.72i	− 1.40366i
$263$	− 6632.76i	− 1.55511i	−0.628816	−	0.777554i	$-0.716460\pi$
$263$	− 6632.76i	− 1.55511i	0.628816	−	0.777554i	$-0.283540\pi$
$264$	−984.412	−0.229494
$265$	0	0
$266$	−7938.44	−1.82984
$267$	− 3904.78i	− 0.895013i
$268$	− 426.732i	− 0.0972641i
$269$	6092.97	1.38102	0.690511	−	0.723322i	$-0.257386\pi$
$269$	6092.97	1.38102	0.690511	+	0.723322i	$0.257386\pi$
$270$	0	0
$271$	−1981.30	−0.444115	−0.222058	−	0.975034i	$-0.571277\pi$
$271$	−1981.30	−0.444115	−0.222058	+	0.975034i	$0.571277\pi$
$272$	5398.92i	1.20352i
$273$	777.192i	0.172300i
$274$	819.994	0.180794
$275$	0	0
$276$	−85.5635	−0.0186606
$277$	− 4097.42i	− 0.888773i	−0.895835	−	0.444386i	$-0.853422\pi$
$277$	− 4097.42i	− 0.888773i	0.895835	−	0.444386i	$-0.146578\pi$
$278$	− 1673.67i	− 0.361079i
$279$	−839.077	−0.180051
$280$	0	0
$281$	−3866.40	−0.820819	−0.410409	−	0.911901i	$-0.634614\pi$
$281$	−3866.40	−0.820819	−0.410409	+	0.911901i	$0.634614\pi$
$282$	− 6442.84i	− 1.36052i
$283$	− 3982.00i	− 0.836415i	−0.908351	−	0.418208i	$-0.862658\pi$
$283$	− 3982.00i	− 0.836415i	0.908351	−	0.418208i	$-0.137342\pi$
$284$	1020.85	0.213298
$285$	0	0
$286$	−308.484	−0.0637799
$287$	6120.59i	1.25884i
$288$	592.221i	0.121170i
$289$	−869.245	−0.176927
$290$	0	0
$291$	6432.86	1.29588
$292$	− 286.650i	− 0.0574483i
$293$	7491.70i	1.49375i	0.664962	+	0.746877i	$0.268448\pi$
$293$	7491.70i	1.49375i	−0.664962	+	0.746877i	$0.731552\pi$
$294$	3486.52	0.691626
$295$	0	0
$296$	2355.45	0.462526
$297$	− 1882.59i	− 0.367809i
$298$	− 4062.48i	− 0.789709i
$299$	187.691	0.0363024
$300$	0	0
$301$	−7225.53	−1.38363
$302$	− 487.957i	− 0.0929760i
$303$	− 3183.98i	− 0.603680i
$304$	−7338.70	−1.38455
$305$	0	0
$306$	−3002.21	−0.560865
$307$	− 5813.01i	− 1.08067i	−0.841450	−	0.540335i	$-0.818297\pi$
$307$	− 5813.01i	− 1.08067i	0.841450	−	0.540335i	$-0.181703\pi$
$308$	322.589i	0.0596793i
$309$	2353.12	0.433218
$310$	0	0
$311$	4769.43	0.869613	0.434807	−	0.900524i	$-0.356817\pi$
$311$	4769.43	0.869613	0.434807	+	0.900524i	$0.356817\pi$
$312$	637.521i	0.115681i
$313$	− 7053.43i	− 1.27375i	−0.770967	−	0.636875i	$-0.780227\pi$
$313$	− 7053.43i	− 1.27375i	0.770967	−	0.636875i	$-0.219773\pi$
$314$	−7094.49	−1.27505
$315$	0	0
$316$	−551.866	−0.0982434
$317$	8845.36i	1.56721i	0.621261	+	0.783604i	$0.286621\pi$
$317$	8845.36i	1.56721i	−0.621261	+	0.783604i	$0.713379\pi$
$318$	− 25.8170i	− 0.00455266i
$319$	3366.94	0.590949
$320$	0	0
$321$	4871.99	0.847127
$322$	− 1766.45i	− 0.305716i
$323$	− 7859.76i	− 1.35396i
$324$	200.470	0.0343741
$325$	0	0
$326$	−1413.48	−0.240140
$327$	6421.75i	1.08601i
$328$	5020.64i	0.845179i
$329$	14779.1	2.47659
$330$	0	0
$331$	4264.07	0.708079	0.354040	−	0.935230i	$-0.384808\pi$
$331$	4264.07	0.708079	0.354040	+	0.935230i	$0.384808\pi$
$332$	21.7021i	0.00358752i
$333$	1476.13i	0.242918i
$334$	−11207.2	−1.83602
$335$	0	0
$336$	6761.95	1.09790
$337$	− 1290.58i	− 0.208613i	−0.994545	−	0.104306i	$-0.966738\pi$
$337$	− 1290.58i	− 0.208613i	0.994545	−	0.104306i	$-0.0332622\pi$
$338$	− 6391.22i	− 1.02851i
$339$	5551.36	0.889405
$340$	0	0
$341$	−803.392	−0.127584
$342$	− 4080.87i	− 0.645229i
$343$	− 783.403i	− 0.123323i
$344$	−5927.01	−0.928963
$345$	0	0
$346$	8127.98	1.26290
$347$	− 1805.09i	− 0.279257i	−0.990204	−	0.139629i	$-0.955409\pi$
$347$	− 1805.09i	− 0.279257i	0.990204	−	0.139629i	$-0.0445909\pi$
$348$	994.031i	0.153120i
$349$	3586.81	0.550136	0.275068	−	0.961425i	$-0.411300\pi$
$349$	3586.81	0.550136	0.275068	+	0.961425i	$0.411300\pi$
$350$	0	0
$351$	−1219.20	−0.185402
$352$	567.034i	0.0858609i
$353$	2292.90i	0.345719i	0.984947	+	0.172859i	$0.0553006\pi$
$353$	2292.90i	0.345719i	−0.984947	+	0.172859i	$0.944699\pi$
$354$	3045.56	0.457259
$355$	0	0
$356$	−1049.63	−0.156264
$357$	7242.06i	1.07364i
$358$	9901.28i	1.46173i
$359$	3077.18	0.452388	0.226194	−	0.974082i	$-0.427372\pi$
$359$	3077.18	0.452388	0.226194	+	0.974082i	$0.427372\pi$
$360$	0	0
$361$	3824.70	0.557618
$362$	1751.60i	0.254316i
$363$	4360.84i	0.630537i
$364$	208.914	0.0300826
$365$	0	0
$366$	−3281.72	−0.468684
$367$	− 1378.56i	− 0.196076i	−0.995183	−	0.0980382i	$-0.968743\pi$
$367$	− 1378.56i	− 0.196076i	0.995183	−	0.0980382i	$-0.0312567\pi$
$368$	− 1633.00i	− 0.231321i
$369$	−3146.38	−0.443886
$370$	0	0
$371$	59.2211	0.00828736
$372$	− 237.187i	− 0.0330580i
$373$	− 11182.3i	− 1.55227i	−0.630567	−	0.776135i	$-0.717178\pi$
$373$	− 11182.3i	− 1.55227i	0.630567	−	0.776135i	$-0.282822\pi$
$374$	−2874.53	−0.397429
$375$	0	0
$376$	12123.1	1.66277
$377$	− 2180.49i	− 0.297880i
$378$	11474.5i	1.56133i
$379$	−12021.9	−1.62935	−0.814677	−	0.579915i	$-0.803086\pi$
$379$	−12021.9	−1.62935	−0.814677	+	0.579915i	$0.803086\pi$
$380$	0	0
$381$	−5090.26	−0.684467
$382$	9176.93i	1.22914i
$383$	13274.6i	1.77102i	0.464617	+	0.885512i	$0.346192\pi$
$383$	13274.6i	1.77102i	−0.464617	+	0.885512i	$0.653808\pi$
$384$	6171.73	0.820182
$385$	0	0
$386$	4010.80	0.528872
$387$	− 3714.39i	− 0.487889i
$388$	− 1729.19i	− 0.226254i
$389$	−11016.4	−1.43587	−0.717934	−	0.696112i	$-0.754912\pi$
$389$	−11016.4	−1.43587	−0.717934	+	0.696112i	$0.754912\pi$
$390$	0	0
$391$	1748.94	0.226210
$392$	6560.38i	0.845279i
$393$	− 7381.67i	− 0.947471i
$394$	−7347.64	−0.939515
$395$	0	0
$396$	−165.832	−0.0210439
$397$	− 1893.18i	− 0.239335i	−0.992814	−	0.119668i	$-0.961817\pi$
$397$	− 1893.18i	− 0.239335i	0.992814	−	0.119668i	$-0.0381829\pi$
$398$	3415.97i	0.430219i
$399$	−9844.07	−1.23514
$400$	0	0
$401$	11281.8	1.40495	0.702475	−	0.711709i	$-0.252078\pi$
$401$	11281.8	1.40495	0.702475	+	0.711709i	$0.252078\pi$
$402$	− 4762.52i	− 0.590878i
$403$	520.290i	0.0643113i
$404$	−855.874	−0.105399
$405$	0	0
$406$	−20521.7	−2.50856
$407$	1413.36i	0.172131i
$408$	5940.57i	0.720839i
$409$	14677.6	1.77447	0.887236	−	0.461316i	$-0.152623\pi$
$409$	14677.6	1.77447	0.887236	+	0.461316i	$0.152623\pi$
$410$	0	0
$411$	1016.83	0.122036
$412$	− 632.534i	− 0.0756376i
$413$	6986.15i	0.832363i
$414$	908.072	0.107800
$415$	0	0
$416$	367.221	0.0432800
$417$	− 2075.43i	− 0.243728i
$418$	− 3907.32i	− 0.457209i
$419$	−2196.50	−0.256100	−0.128050	−	0.991768i	$-0.540872\pi$
$419$	−2196.50	−0.256100	−0.128050	+	0.991768i	$0.540872\pi$
$420$	0	0
$421$	−6571.28	−0.760723	−0.380362	−	0.924838i	$-0.624200\pi$
$421$	−6571.28	−0.760723	−0.380362	+	0.924838i	$0.624200\pi$
$422$	− 16790.4i	− 1.93684i
$423$	7597.42i	0.873284i
$424$	48.5784	0.00556409
$425$	0	0
$426$	11393.2	1.29578
$427$	− 7527.87i	− 0.853160i
$428$	− 1309.62i	− 0.147904i
$429$	−382.536	−0.0430513
$430$	0	0
$431$	−583.607	−0.0652235	−0.0326118	−	0.999468i	$-0.510382\pi$
$431$	−583.607	−0.0652235	−0.0326118	+	0.999468i	$0.510382\pi$
$432$	10607.6i	1.18139i
$433$	2677.60i	0.297176i	0.988899	+	0.148588i	$0.0474728\pi$
$433$	2677.60i	0.297176i	−0.988899	+	0.148588i	$0.952527\pi$
$434$	4896.71	0.541589
$435$	0	0
$436$	1726.21	0.189611
$437$	2377.33i	0.260236i
$438$	− 3199.14i	− 0.348997i
$439$	5509.18	0.598949	0.299475	−	0.954104i	$-0.403189\pi$
$439$	5509.18	0.598949	0.299475	+	0.954104i	$0.403189\pi$
$440$	0	0
$441$	−4111.32	−0.443939
$442$	1861.59i	0.200332i
$443$	13054.6i	1.40009i	0.714097	+	0.700047i	$0.246838\pi$
$443$	13054.6i	1.40009i	−0.714097	+	0.700047i	$0.753162\pi$
$444$	−417.268	−0.0446006
$445$	0	0
$446$	−16741.8	−1.77746
$447$	− 5037.68i	− 0.533052i
$448$	11085.1i	1.16903i
$449$	11819.1	1.24227	0.621135	−	0.783704i	$-0.286672\pi$
$449$	11819.1	1.24227	0.621135	+	0.783704i	$0.286672\pi$
$450$	0	0
$451$	−3012.57	−0.314537
$452$	− 1492.24i	− 0.155285i
$453$	− 605.091i	− 0.0627587i
$454$	−1698.97	−0.175631
$455$	0	0
$456$	−8074.97	−0.829266
$457$	− 4144.87i	− 0.424264i	−0.977241	−	0.212132i	$-0.931959\pi$
$457$	− 4144.87i	− 0.424264i	0.977241	−	0.212132i	$-0.0680407\pi$
$458$	− 2081.11i	− 0.212323i
$459$	−11360.8	−1.15528
$460$	0	0
$461$	−6061.38	−0.612379	−0.306189	−	0.951971i	$-0.599054\pi$
$461$	−6061.38	−0.612379	−0.306189	+	0.951971i	$0.599054\pi$
$462$	3600.24i	0.362551i
$463$	5729.13i	0.575066i	0.957771	+	0.287533i	$0.0928350\pi$
$463$	5729.13i	0.575066i	−0.957771	+	0.287533i	$0.907165\pi$
$464$	−18971.3	−1.89811
$465$	0	0
$466$	6626.05	0.658682
$467$	8958.05i	0.887643i	0.896115	+	0.443821i	$0.146378\pi$
$467$	8958.05i	0.887643i	−0.896115	+	0.443821i	$0.853622\pi$
$468$	107.395i	0.0106076i
$469$	10924.7	1.07559
$470$	0	0
$471$	−8797.53	−0.860655
$472$	5730.65i	0.558845i
$473$	− 3556.42i	− 0.345718i
$474$	−6159.08	−0.596827
$475$	0	0
$476$	1946.71	0.187452
$477$	30.4435i	0.00292225i
$478$	− 11630.4i	− 1.11289i
$479$	−6356.60	−0.606347	−0.303174	−	0.952935i	$-0.598046\pi$
$479$	−6356.60	−0.606347	−0.303174	+	0.952935i	$0.598046\pi$
$480$	0	0
$481$	915.312	0.0867664
$482$	6073.70i	0.573961i
$483$	− 2190.49i	− 0.206358i
$484$	1172.22	0.110088
$485$	0	0
$486$	−9864.31	−0.920688
$487$	13991.2i	1.30185i	0.759141	+	0.650926i	$0.225619\pi$
$487$	13991.2i	1.30185i	−0.759141	+	0.650926i	$0.774381\pi$
$488$	− 6175.02i	− 0.572808i
$489$	−1752.79	−0.162094
$490$	0	0
$491$	−5301.96	−0.487320	−0.243660	−	0.969861i	$-0.578348\pi$
$491$	−5301.96	−0.487320	−0.243660	+	0.969861i	$0.578348\pi$
$492$	− 889.408i	− 0.0814992i
$493$	− 20318.3i	− 1.85617i
$494$	−2530.44	−0.230466
$495$	0	0
$496$	4526.78	0.409795
$497$	26134.7i	2.35875i
$498$	242.205i	0.0217941i
$499$	8013.45	0.718900	0.359450	−	0.933164i	$-0.382964\pi$
$499$	8013.45	0.718900	0.359450	+	0.933164i	$0.382964\pi$
$500$	0	0
$501$	−13897.5	−1.23931
$502$	2200.33i	0.195629i
$503$	11881.9i	1.05325i	0.850097	+	0.526627i	$0.176544\pi$
$503$	11881.9i	1.05325i	−0.850097	+	0.526627i	$0.823456\pi$
$504$	−7075.27	−0.625313
$505$	0	0
$506$	869.453	0.0763871
$507$	− 7925.44i	− 0.694243i
$508$	1368.29i	0.119504i
$509$	12113.8	1.05488	0.527441	−	0.849592i	$-0.323152\pi$
$509$	12113.8	1.05488	0.527441	+	0.849592i	$0.323152\pi$
$510$	0	0
$511$	7338.45	0.635291
$512$	8733.00i	0.753804i
$513$	− 15442.6i	− 1.32906i
$514$	7103.49	0.609575
$515$	0	0
$516$	1049.97	0.0895783
$517$	7274.31i	0.618808i
$518$	− 8614.48i	− 0.730692i
$519$	10079.1	0.852454
$520$	0	0
$521$	−15327.6	−1.28890	−0.644449	−	0.764647i	$-0.722913\pi$
$521$	−15327.6	−1.28890	−0.644449	+	0.764647i	$0.722913\pi$
$522$	− 10549.5i	− 0.884556i
$523$	− 14545.8i	− 1.21614i	−0.793883	−	0.608070i	$-0.791944\pi$
$523$	− 14545.8i	− 1.21614i	0.793883	−	0.608070i	$-0.208056\pi$
$524$	−1984.24	−0.165423
$525$	0	0
$526$	−19898.3	−1.64944
$527$	4848.18i	0.400740i
$528$	3328.25i	0.274325i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−3591.34	−0.293504
$532$	2646.15i	0.215648i
$533$	1950.99i	0.158549i
$534$	−11714.3	−0.949304
$535$	0	0
$536$	8961.36	0.722149
$537$	12278.1i	0.986665i
$538$	− 18278.9i	− 1.46479i
$539$	−3936.47	−0.314575
$540$	0	0
$541$	−5468.93	−0.434617	−0.217308	−	0.976103i	$-0.569728\pi$
$541$	−5468.93	−0.434617	−0.217308	+	0.976103i	$0.569728\pi$
$542$	5943.89i	0.471055i
$543$	2172.08i	0.171663i
$544$	3421.85	0.269688
$545$	0	0
$546$	2331.58	0.182751
$547$	− 19566.3i	− 1.52943i	−0.644371	−	0.764713i	$-0.722881\pi$
$547$	− 19566.3i	− 1.52943i	0.644371	−	0.764713i	$-0.277119\pi$
$548$	− 273.331i	− 0.0213068i
$549$	3869.82	0.300837
$550$	0	0
$551$	27618.5	2.13537
$552$	− 1796.83i	− 0.138548i
$553$	− 14128.2i	− 1.08642i
$554$	−12292.3	−0.942686
$555$	0	0
$556$	−557.889	−0.0425536
$557$	− 9803.11i	− 0.745729i	−0.927886	−	0.372865i	$-0.878376\pi$
$557$	− 9803.11i	− 0.745729i	0.927886	−	0.372865i	$-0.121624\pi$
$558$	2517.23i	0.190973i
$559$	−2303.20	−0.174266
$560$	0	0
$561$	−3564.56	−0.268264
$562$	11599.2i	0.870610i
$563$	4909.73i	0.367532i	0.982970	+	0.183766i	$0.0588288\pi$
$563$	4909.73i	0.367532i	−0.982970	+	0.183766i	$0.941171\pi$
$564$	−2147.61	−0.160338
$565$	0	0
$566$	−11946.0	−0.887153
$567$	5132.18i	0.380126i
$568$	21438.0i	1.58366i
$569$	−11498.2	−0.847155	−0.423578	−	0.905860i	$-0.639226\pi$
$569$	−11498.2	−0.847155	−0.423578	+	0.905860i	$0.639226\pi$
$570$	0	0
$571$	14766.6	1.08225	0.541124	−	0.840943i	$-0.317999\pi$
$571$	14766.6	1.08225	0.541124	+	0.840943i	$0.317999\pi$
$572$	102.828i	0.00751653i
$573$	11379.9i	0.829670i
$574$	18361.8	1.33520
$575$	0	0
$576$	−5698.48	−0.412216
$577$	13364.0i	0.964212i	0.876113	+	0.482106i	$0.160128\pi$
$577$	13364.0i	0.964212i	−0.876113	+	0.482106i	$0.839872\pi$
$578$	2607.73i	0.187660i
$579$	4973.60	0.356988
$580$	0	0
$581$	−555.590	−0.0396725
$582$	− 19298.6i	− 1.37449i
$583$	29.1488i	0.00207070i
$584$	6019.64	0.426532
$585$	0	0
$586$	22475.1	1.58437
$587$	12524.7i	0.880664i	0.897835	+	0.440332i	$0.145139\pi$
$587$	12524.7i	0.880664i	−0.897835	+	0.440332i	$0.854861\pi$
$588$	− 1162.17i	− 0.0815089i
$589$	−6590.09	−0.461019
$590$	0	0
$591$	−9111.45	−0.634171
$592$	− 7963.66i	− 0.552879i
$593$	17938.6i	1.24224i	0.783714	+	0.621121i	$0.213323\pi$
$593$	17938.6i	1.24224i	−0.783714	+	0.621121i	$0.786677\pi$
$594$	−5647.78	−0.390120
$595$	0	0
$596$	−1354.16	−0.0930681
$597$	4235.98i	0.290397i
$598$	− 563.072i	− 0.0385045i
$599$	26735.8	1.82370	0.911848	−	0.410528i	$-0.134656\pi$
$599$	26735.8	1.82370	0.911848	+	0.410528i	$0.134656\pi$
$600$	0	0
$601$	−21043.6	−1.42826	−0.714131	−	0.700012i	$-0.753178\pi$
$601$	−21043.6	−1.42826	−0.714131	+	0.700012i	$0.753178\pi$
$602$	21676.6i	1.46756i
$603$	5615.98i	0.379271i
$604$	−162.652	−0.0109573
$605$	0	0
$606$	−9551.95	−0.640299
$607$	5098.83i	0.340947i	0.985362	+	0.170474i	$0.0545298\pi$
$607$	5098.83i	0.340947i	−0.985362	+	0.170474i	$0.945470\pi$
$608$	4651.29i	0.310254i
$609$	−25447.9	−1.69327
$610$	0	0
$611$	4710.96	0.311923
$612$	1000.74i	0.0660986i
$613$	12784.8i	0.842369i	0.906975	+	0.421185i	$0.138386\pi$
$613$	12784.8i	0.842369i	−0.906975	+	0.421185i	$0.861614\pi$
$614$	−17439.0	−1.14622
$615$	0	0
$616$	−6774.37	−0.443096
$617$	− 8940.73i	− 0.583372i	−0.956514	−	0.291686i	$-0.905784\pi$
$617$	− 8940.73i	− 0.583372i	0.956514	−	0.291686i	$-0.0942162\pi$
$618$	− 7059.37i	− 0.459498i
$619$	−1333.26	−0.0865725	−0.0432863	−	0.999063i	$-0.513783\pi$
$619$	−1333.26	−0.0865725	−0.0432863	+	0.999063i	$0.513783\pi$
$620$	0	0
$621$	3436.27	0.222050
$622$	− 14308.3i	− 0.922364i
$623$	− 26871.3i	− 1.72805i
$624$	2155.43	0.138279
$625$	0	0
$626$	−21160.3	−1.35102
$627$	− 4845.28i	− 0.308615i
$628$	2364.83i	0.150266i
$629$	8529.09	0.540663
$630$	0	0
$631$	−16667.9	−1.05157	−0.525784	−	0.850618i	$-0.676228\pi$
$631$	−16667.9	−1.05157	−0.525784	+	0.850618i	$0.676228\pi$
$632$	− 11589.2i	− 0.729420i
$633$	− 20821.0i	− 1.30736i
$634$	26536.1	1.66228
$635$	0	0
$636$	−8.60567	−0.000536536 0
$637$	2549.32i	0.158568i
$638$	− 10100.8i	− 0.626796i
$639$	−13434.9	−0.831733
$640$	0	0
$641$	27431.9	1.69032	0.845158	−	0.534516i	$-0.179506\pi$
$641$	27431.9	1.69032	0.845158	+	0.534516i	$0.179506\pi$
$642$	− 14616.0i	− 0.898514i
$643$	− 9618.00i	− 0.589886i	−0.955515	−	0.294943i	$-0.904699\pi$
$643$	− 9618.00i	− 0.589886i	0.955515	−	0.294943i	$-0.0953007\pi$
$644$	−588.818	−0.0360290
$645$	0	0
$646$	−23579.3	−1.43609
$647$	− 12548.6i	− 0.762496i	−0.924473	−	0.381248i	$-0.875494\pi$
$647$	− 12548.6i	− 0.762496i	0.924473	−	0.381248i	$-0.124506\pi$
$648$	4209.87i	0.255215i
$649$	−3438.60	−0.207977
$650$	0	0
$651$	6072.18	0.365572
$652$	471.162i	0.0283008i
$653$	8829.79i	0.529152i	0.964365	+	0.264576i	$0.0852320\pi$
$653$	8829.79i	0.529152i	−0.964365	+	0.264576i	$0.914768\pi$
$654$	19265.3	1.15188
$655$	0	0
$656$	16974.6	1.01028
$657$	3772.44i	0.224014i
$658$	− 44337.3i	− 2.62682i
$659$	−5935.56	−0.350860	−0.175430	−	0.984492i	$-0.556132\pi$
$659$	−5935.56	−0.350860	−0.175430	+	0.984492i	$0.556132\pi$
$660$	0	0
$661$	−1182.03	−0.0695549	−0.0347775	−	0.999395i	$-0.511072\pi$
$661$	−1182.03	−0.0695549	−0.0347775	+	0.999395i	$0.511072\pi$
$662$	− 12792.2i	− 0.751032i
$663$	2308.47i	0.135224i
$664$	−455.743	−0.0266360
$665$	0	0
$666$	4428.40	0.257653
$667$	6145.64i	0.356762i
$668$	3735.73i	0.216377i
$669$	−20760.7	−1.19978
$670$	0	0
$671$	3705.24	0.213173
$672$	− 4285.74i	− 0.246021i
$673$	6053.25i	0.346710i	0.984859	+	0.173355i	$0.0554608\pi$
$673$	6053.25i	0.346710i	−0.984859	+	0.173355i	$0.944539\pi$
$674$	−3871.74	−0.221267
$675$	0	0
$676$	−2130.41	−0.121211
$677$	16694.2i	0.947728i	0.880598	+	0.473864i	$0.157141\pi$
$677$	16694.2i	0.947728i	−0.880598	+	0.473864i	$0.842859\pi$
$678$	− 16654.1i	− 0.943357i
$679$	44268.6	2.50202
$680$	0	0
$681$	−2106.81	−0.118551
$682$	2410.18i	0.135323i
$683$	4150.03i	0.232499i	0.993220	+	0.116249i	$0.0370871\pi$
$683$	4150.03i	0.232499i	−0.993220	+	0.116249i	$0.962913\pi$
$684$	−1360.29	−0.0760410
$685$	0	0
$686$	−2350.21	−0.130804
$687$	− 2580.68i	− 0.143317i
$688$	20039.0i	1.11043i
$689$	18.8772	0.00104378
$690$	0	0
$691$	33382.4	1.83781	0.918904	−	0.394481i	$-0.129076\pi$
$691$	33382.4	1.83781	0.918904	+	0.394481i	$0.129076\pi$
$692$	− 2709.33i	− 0.148834i
$693$	− 4245.42i	− 0.232713i
$694$	−5415.27	−0.296197
$695$	0	0
$696$	−20874.6	−1.13686
$697$	18179.8i	0.987960i
$698$	− 10760.4i	− 0.583508i
$699$	8216.64	0.444609
$700$	0	0
$701$	−29528.7	−1.59099	−0.795494	−	0.605961i	$-0.792789\pi$
$701$	−29528.7	−1.59099	−0.795494	+	0.605961i	$0.792789\pi$
$702$	3657.59i	0.196648i
$703$	11593.5i	0.621989i
$704$	−5456.13	−0.292096
$705$	0	0
$706$	6878.69	0.366690
$707$	− 21911.0i	− 1.16556i
$708$	− 1015.19i	− 0.0538885i
$709$	−27016.3	−1.43106	−0.715529	−	0.698583i	$-0.753814\pi$
$709$	−27016.3	−1.43106	−0.715529	+	0.698583i	$0.753814\pi$
$710$	0	0
$711$	7262.82	0.383090
$712$	− 22042.2i	− 1.16020i
$713$	− 1466.42i	− 0.0770237i
$714$	21726.2	1.13877
$715$	0	0
$716$	3300.43	0.172266
$717$	− 14422.3i	− 0.751198i
$718$	− 9231.54i	− 0.479830i
$719$	−24752.9	−1.28390	−0.641952	−	0.766745i	$-0.721875\pi$
$719$	−24752.9	−1.28390	−0.641952	+	0.766745i	$0.721875\pi$
$720$	0	0
$721$	16193.4	0.836438
$722$	− 11474.1i	− 0.591443i
$723$	7531.69i	0.387423i
$724$	583.868	0.0299714
$725$	0	0
$726$	13082.5	0.668785
$727$	− 17574.5i	− 0.896565i	−0.893892	−	0.448283i	$-0.852036\pi$
$727$	− 17574.5i	− 0.896565i	0.893892	−	0.448283i	$-0.147964\pi$
$728$	4387.19i	0.223352i
$729$	−17644.9	−0.896456
$730$	0	0
$731$	−21461.7	−1.08590
$732$	1093.91i	0.0552349i
$733$	− 27739.6i	− 1.39780i	−0.715221	−	0.698898i	$-0.753674\pi$
$733$	− 27739.6i	− 1.39780i	0.715221	−	0.698898i	$-0.246326\pi$
$734$	−4135.67	−0.207970
$735$	0	0
$736$	−1035.00	−0.0518351
$737$	5377.15i	0.268751i
$738$	9439.14i	0.470812i
$739$	3140.78	0.156340	0.0781701	−	0.996940i	$-0.475092\pi$
$739$	3140.78	0.156340	0.0781701	+	0.996940i	$0.475092\pi$
$740$	0	0
$741$	−3137.88	−0.155564
$742$	− 177.663i	− 0.00879007i
$743$	35881.5i	1.77169i	0.463984	+	0.885844i	$0.346420\pi$
$743$	35881.5i	1.77169i	−0.463984	+	0.885844i	$0.653580\pi$
$744$	4980.93	0.245443
$745$	0	0
$746$	−33546.9	−1.64643
$747$	− 285.609i	− 0.0139891i
$748$	958.176i	0.0468374i
$749$	33527.3	1.63559
$750$	0	0
$751$	−7009.03	−0.340563	−0.170282	−	0.985395i	$-0.554468\pi$
$751$	−7009.03	−0.340563	−0.170282	+	0.985395i	$0.554468\pi$
$752$	− 40987.7i	− 1.98759i
$753$	2728.52i	0.132049i
$754$	−6541.46	−0.315950
$755$	0	0
$756$	3824.83	0.184005
$757$	29998.3i	1.44030i	0.693818	+	0.720150i	$0.255927\pi$
$757$	29998.3i	1.44030i	−0.693818	+	0.720150i	$0.744073\pi$
$758$	36065.8i	1.72819i
$759$	1078.17	0.0515612
$760$	0	0
$761$	5283.49	0.251677	0.125839	−	0.992051i	$-0.459838\pi$
$761$	5283.49	0.251677	0.125839	+	0.992051i	$0.459838\pi$
$762$	15270.8i	0.725987i
$763$	44192.2i	2.09681i
$764$	3058.98	0.144856
$765$	0	0
$766$	39823.9	1.87845
$767$	2226.89i	0.104835i
$768$	− 5628.59i	− 0.264459i
$769$	15216.2	0.713538	0.356769	−	0.934193i	$-0.383878\pi$
$769$	15216.2	0.713538	0.356769	+	0.934193i	$0.383878\pi$
$770$	0	0
$771$	8808.69	0.411462
$772$	− 1336.93i	− 0.0623281i
$773$	− 15651.3i	− 0.728250i	−0.931350	−	0.364125i	$-0.881368\pi$
$773$	− 15651.3i	− 0.728250i	0.931350	−	0.364125i	$-0.118632\pi$
$774$	−11143.2	−0.517485
$775$	0	0
$776$	36313.0	1.67985
$777$	− 10682.4i	− 0.493216i
$778$	33049.1i	1.52297i
$779$	−24711.6	−1.13657
$780$	0	0
$781$	−12863.6	−0.589365
$782$	− 5246.83i	− 0.239931i
$783$	− 39920.7i	− 1.82203i
$784$	22180.3	1.01040
$785$	0	0
$786$	−22145.0	−1.00494
$787$	9401.09i	0.425810i	0.977073	+	0.212905i	$0.0682925\pi$
$787$	9401.09i	0.425810i	−0.977073	+	0.212905i	$0.931707\pi$
$788$	2449.21i	0.110723i
$789$	−24674.9	−1.11337
$790$	0	0
$791$	38202.5	1.71722
$792$	− 3482.47i	− 0.156243i
$793$	− 2399.57i	− 0.107454i
$794$	−5679.55	−0.253854
$795$	0	0
$796$	1138.66	0.0507018
$797$	37388.5i	1.66169i	0.556504	+	0.830845i	$0.312143\pi$
$797$	37388.5i	1.66169i	−0.556504	+	0.830845i	$0.687857\pi$
$798$	29532.2i	1.31006i
$799$	43897.9	1.94367
$800$	0	0
$801$	13813.6	0.609337
$802$	− 33845.3i	− 1.49017i
$803$	3612.00i	0.158736i
$804$	−1587.51	−0.0696356
$805$	0	0
$806$	1560.87	0.0682125
$807$	− 22666.8i	− 0.988734i
$808$	− 17973.4i	− 0.782550i
$809$	−24390.6	−1.05999	−0.529993	−	0.848002i	$-0.677805\pi$
$809$	−24390.6	−1.05999	−0.529993	+	0.848002i	$0.677805\pi$
$810$	0	0
$811$	23750.8	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$811$	23750.8	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$812$	6840.56i	0.295636i
$813$	7370.73i	0.317962i
$814$	4240.07	0.182573
$815$	0	0
$816$	20084.8	0.861653
$817$	− 29172.8i	− 1.24924i
$818$	− 44032.7i	− 1.88211i
$819$	−2749.40	−0.117304
$820$	0	0
$821$	3041.59	0.129296	0.0646481	−	0.997908i	$-0.479408\pi$
$821$	3041.59	0.129296	0.0646481	+	0.997908i	$0.479408\pi$
$822$	− 3050.50i	− 0.129439i
$823$	− 28411.6i	− 1.20336i	−0.798737	−	0.601681i	$-0.794498\pi$
$823$	− 28411.6i	− 1.20336i	0.798737	−	0.601681i	$-0.205502\pi$
$824$	13283.2	0.561581
$825$	0	0
$826$	20958.4	0.882854
$827$	− 24351.5i	− 1.02392i	−0.859008	−	0.511962i	$-0.828919\pi$
$827$	− 24351.5i	− 1.02392i	0.859008	−	0.511962i	$-0.171081\pi$
$828$	− 302.691i	− 0.0127044i
$829$	47240.8	1.97918	0.989590	−	0.143914i	$-0.0459688\pi$
$829$	47240.8	1.97918	0.989590	+	0.143914i	$0.0459688\pi$
$830$	0	0
$831$	−15243.0	−0.636311
$832$	3533.48i	0.147237i
$833$	23755.2i	0.988077i
$834$	−6226.30	−0.258512
$835$	0	0
$836$	−1302.44	−0.0538826
$837$	9525.55i	0.393371i
$838$	6589.49i	0.271635i
$839$	−2646.33	−0.108893	−0.0544466	−	0.998517i	$-0.517339\pi$
$839$	−2646.33	−0.108893	−0.0544466	+	0.998517i	$0.517339\pi$
$840$	0	0
$841$	47007.7	1.92741
$842$	19713.8i	0.806869i
$843$	14383.6i	0.587660i
$844$	−5596.81	−0.228258
$845$	0	0
$846$	22792.3	0.926258
$847$	30009.7i	1.21741i
$848$	− 164.241i	− 0.00665102i
$849$	−14813.7	−0.598827
$850$	0	0
$851$	−2579.78	−0.103917
$852$	− 3797.74i	− 0.152709i
$853$	40826.4i	1.63877i	0.573244	+	0.819385i	$0.305685\pi$
$853$	40826.4i	1.63877i	−0.573244	+	0.819385i	$0.694315\pi$
$854$	−22583.6	−0.904913
$855$	0	0
$856$	27502.0	1.09813
$857$	1516.61i	0.0604510i	0.999543	+	0.0302255i	$0.00962254\pi$
$857$	1516.61i	0.0604510i	−0.999543	+	0.0302255i	$0.990377\pi$
$858$	1147.61i	0.0456628i
$859$	−6514.74	−0.258766	−0.129383	−	0.991595i	$-0.541300\pi$
$859$	−6514.74	−0.258766	−0.129383	+	0.991595i	$0.541300\pi$
$860$	0	0
$861$	22769.5	0.901258
$862$	1750.82i	0.0691800i
$863$	7120.18i	0.280850i	0.990091	+	0.140425i	$0.0448469\pi$
$863$	7120.18i	0.280850i	−0.990091	+	0.140425i	$0.955153\pi$
$864$	6723.14	0.264729
$865$	0	0
$866$	8032.79	0.315202
$867$	3233.72i	0.126670i
$868$	− 1632.24i	− 0.0638269i
$869$	6953.94	0.271457
$870$	0	0
$871$	3482.33	0.135470
$872$	36250.3i	1.40779i
$873$	22757.0i	0.882252i
$874$	7131.98	0.276021
$875$	0	0
$876$	−1066.38	−0.0411297
$877$	− 34013.4i	− 1.30964i	−0.755787	−	0.654818i	$-0.772745\pi$
$877$	− 34013.4i	− 1.30964i	0.755787	−	0.654818i	$-0.227255\pi$
$878$	− 16527.5i	− 0.635282i
$879$	27870.3	1.06944
$880$	0	0
$881$	6704.11	0.256376	0.128188	−	0.991750i	$-0.459084\pi$
$881$	6704.11	0.256376	0.128188	+	0.991750i	$0.459084\pi$
$882$	12334.0i	0.470868i
$883$	− 47808.5i	− 1.82207i	−0.412333	−	0.911033i	$-0.635286\pi$
$883$	− 47808.5i	− 1.82207i	0.412333	−	0.911033i	$-0.364714\pi$
$884$	620.530	0.0236094
$885$	0	0
$886$	39163.7	1.48502
$887$	− 38955.7i	− 1.47464i	−0.675544	−	0.737320i	$-0.736091\pi$
$887$	− 38955.7i	− 1.47464i	0.675544	−	0.737320i	$-0.263909\pi$
$888$	− 8762.64i	− 0.331143i
$889$	−35029.3	−1.32154
$890$	0	0
$891$	−2526.07	−0.0949794
$892$	5580.60i	0.209476i
$893$	59670.0i	2.23604i
$894$	−15113.1	−0.565387
$895$	0	0
$896$	42471.7	1.58357
$897$	− 698.238i	− 0.0259905i
$898$	− 35457.4i	− 1.31763i
$899$	−17036.1	−0.632019
$900$	0	0
$901$	175.903	0.00650407
$902$	9037.71i	0.333617i
$903$	26880.1i	0.990601i
$904$	31337.0	1.15294
$905$	0	0
$906$	−1815.27	−0.0665656
$907$	7725.66i	0.282830i	0.989950	+	0.141415i	$0.0451651\pi$
$907$	7725.66i	0.282830i	−0.989950	+	0.141415i	$0.954835\pi$
$908$	566.323i	0.0206983i
$909$	11263.7	0.410994
$910$	0	0
$911$	−13150.7	−0.478269	−0.239135	−	0.970986i	$-0.576864\pi$
$911$	−13150.7	−0.478269	−0.239135	+	0.970986i	$0.576864\pi$
$912$	27301.1i	0.991260i
$913$	− 273.463i	− 0.00991270i
$914$	−12434.6	−0.450000
$915$	0	0
$916$	−693.702	−0.0250224
$917$	− 50798.0i	− 1.82933i
$918$	34082.3i	1.22536i
$919$	−33083.7	−1.18752	−0.593759	−	0.804643i	$-0.702357\pi$
$919$	−33083.7	−1.18752	−0.593759	+	0.804643i	$0.702357\pi$
$920$	0	0
$921$	−21625.3	−0.773699
$922$	18184.1i	0.649526i
$923$	8330.65i	0.297082i
$924$	1200.08	0.0427270
$925$	0	0
$926$	17187.4	0.609949
$927$	8324.44i	0.294941i
$928$	12024.1i	0.425333i
$929$	1310.12	0.0462686	0.0231343	−	0.999732i	$-0.492635\pi$
$929$	1310.12	0.0462686	0.0231343	+	0.999732i	$0.492635\pi$
$930$	0	0
$931$	−32290.2	−1.13670
$932$	− 2208.68i	− 0.0776264i
$933$	− 17743.0i	− 0.622594i
$934$	26874.1	0.941487
$935$	0	0
$936$	−2255.30	−0.0787574
$937$	40623.1i	1.41633i	0.706049	+	0.708163i	$0.250476\pi$
$937$	40623.1i	1.41633i	−0.706049	+	0.708163i	$0.749524\pi$
$938$	− 32774.0i	− 1.14084i
$939$	−26239.8	−0.911933
$940$	0	0
$941$	−10434.0	−0.361466	−0.180733	−	0.983532i	$-0.557847\pi$
$941$	−10434.0	−0.361466	−0.180733	+	0.983532i	$0.557847\pi$
$942$	26392.6i	0.912863i
$943$	− 5498.80i	− 0.189889i
$944$	19375.1	0.668013
$945$	0	0
$946$	−10669.3	−0.366689
$947$	− 51333.4i	− 1.76147i	−0.473612	−	0.880734i	$-0.657050\pi$
$947$	− 51333.4i	− 1.76147i	0.473612	−	0.880734i	$-0.342950\pi$
$948$	2053.03i	0.0703367i
$949$	2339.19	0.0800141
$950$	0	0
$951$	32906.1	1.12203
$952$	40880.9i	1.39176i
$953$	− 40263.6i	− 1.36859i	−0.729206	−	0.684294i	$-0.760110\pi$
$953$	− 40263.6i	− 1.36859i	0.729206	−	0.684294i	$-0.239890\pi$
$954$	91.3306	0.00309951
$955$	0	0
$956$	−3876.79	−0.131155
$957$	− 12525.5i	− 0.423086i
$958$	19069.8i	0.643128i
$959$	6997.49	0.235621
$960$	0	0
$961$	−25726.0	−0.863549
$962$	− 2745.94i	− 0.0920297i
$963$	17235.2i	0.576736i
$964$	2024.57	0.0676420
$965$	0	0
$966$	−6571.48	−0.218876
$967$	− 8353.12i	− 0.277785i	−0.990307	−	0.138893i	$-0.955646\pi$
$967$	− 8353.12i	− 0.277785i	0.990307	−	0.138893i	$-0.0443543\pi$
$968$	24616.6i	0.817364i
$969$	−29239.5	−0.969358
$970$	0	0
$971$	−28020.1	−0.926063	−0.463032	−	0.886342i	$-0.653238\pi$
$971$	−28020.1	−0.926063	−0.463032	+	0.886342i	$0.653238\pi$
$972$	3288.10i	0.108504i
$973$	− 14282.4i	− 0.470578i
$974$	41973.6	1.38082
$975$	0	0
$976$	−20877.5	−0.684704
$977$	− 50115.3i	− 1.64107i	−0.571593	−	0.820537i	$-0.693674\pi$
$977$	− 50115.3i	− 1.64107i	0.571593	−	0.820537i	$-0.306326\pi$
$978$	5258.38i	0.171927i
$979$	13226.1	0.431776
$980$	0	0
$981$	−22717.7	−0.739367
$982$	15905.9i	0.516881i
$983$	38113.0i	1.23664i	0.785926	+	0.618320i	$0.212186\pi$
$983$	38113.0i	1.23664i	−0.785926	+	0.618320i	$0.787814\pi$
$984$	18677.6	0.605101
$985$	0	0
$986$	−60954.9	−1.96876
$987$	− 54980.5i	− 1.77310i
$988$	843.481i	0.0271606i
$989$	6491.49	0.208713
$990$	0	0
$991$	−5245.02	−0.168127	−0.0840634	−	0.996460i	$-0.526790\pi$
$991$	−5245.02	−0.168127	−0.0840634	+	0.996460i	$0.526790\pi$
$992$	− 2869.08i	− 0.0918281i
$993$	− 15863.0i	− 0.506945i
$994$	78404.0	2.50183
$995$	0	0
$996$	80.7350	0.00256846
$997$	21896.1i	0.695544i	0.937579	+	0.347772i	$0.113062\pi$
$997$	21896.1i	0.695544i	−0.937579	+	0.347772i	$0.886938\pi$
$998$	− 24040.3i	− 0.762509i
$999$	16757.7	0.530721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.b.f.24.1		4
5.2	odd	4		575.4.a.h.1.1		2
5.3	odd	4		115.4.a.c.1.2	✓	2
5.4	even	2	inner	575.4.b.f.24.4		4
15.8	even	4		1035.4.a.g.1.1		2
20.3	even	4		1840.4.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.c.1.2	✓	2	5.3	odd	4
575.4.a.h.1.1		2	5.2	odd	4
575.4.b.f.24.1		4	1.1	even	1	trivial
575.4.b.f.24.4		4	5.4	even	2	inner
1035.4.a.g.1.1		2	15.8	even	4
1840.4.a.h.1.1		2	20.3	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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{2} -3.72015i q^{3} -1.00000 q^{4} -11.1605 q^{6} -25.6008i q^{7} -21.0000i q^{8} +13.1605 q^{9} +12.6008 q^{11} +3.72015i q^{12} -8.16046i q^{13} -76.8023 q^{14} -71.0000 q^{16} -76.0411i q^{17} -39.4814i q^{18} +103.362 q^{19} -95.2388 q^{21} -37.8023i q^{22} +23.0000i q^{23} -78.1232 q^{24} -24.4814 q^{26} -149.403i q^{27} +25.6008i q^{28} +267.202 q^{29} -63.7574 q^{31} +45.0000i q^{32} -46.8768i q^{33} -228.123 q^{34} -13.1605 q^{36} +112.164i q^{37} -310.086i q^{38} -30.3582 q^{39} -239.078 q^{41} +285.716i q^{42} -282.239i q^{43} -12.6008 q^{44} +69.0000 q^{46} +577.291i q^{47} +264.131i q^{48} -312.399 q^{49} -282.884 q^{51} +8.16046i q^{52} +2.31326i q^{53} -448.209 q^{54} -537.616 q^{56} -384.522i q^{57} -801.605i q^{58} -272.888 q^{59} +294.049 q^{61} +191.272i q^{62} -336.918i q^{63} -433.000 q^{64} -140.630 q^{66} +426.732i q^{67} +76.0411i q^{68} +85.5635 q^{69} -1020.85 q^{71} -276.370i q^{72} +286.650i q^{73} +336.493 q^{74} -103.362 q^{76} -322.589i q^{77} +91.0745i q^{78} +551.866 q^{79} -200.470 q^{81} +717.235i q^{82} -21.7021i q^{83} +95.2388 q^{84} -846.716 q^{86} -994.031i q^{87} -264.616i q^{88} +1049.63 q^{89} -208.914 q^{91} -23.0000i q^{92} +237.187i q^{93} +1731.87 q^{94} +167.407 q^{96} +1729.19i q^{97} +937.198i q^{98} +165.832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 18 q^{6} - 10 q^{9} - 54 q^{11} + 6 q^{14} - 284 q^{16} + 142 q^{19} - 548 q^{21} + 126 q^{24} + 90 q^{26} + 860 q^{29} - 610 q^{31} - 474 q^{34} + 10 q^{36} - 372 q^{39} - 1186 q^{41} + 54 q^{44}+ \cdots + 1770 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 18 * q^6 - 10 * q^9 - 54 * q^11 + 6 * q^14 - 284 * q^16 + 142 * q^19 - 548 * q^21 + 126 * q^24 + 90 * q^26 + 860 * q^29 - 610 * q^31 - 474 * q^34 + 10 * q^36 - 372 * q^39 - 1186 * q^41 + 54 * q^44 + 276 * q^46 - 1354 * q^49 - 526 * q^51 - 540 * q^54 + 42 * q^56 + 36 * q^59 - 14 * q^61 - 1732 * q^64 - 1878 * q^66 - 138 * q^69 - 2058 * q^71 - 408 * q^74 - 142 * q^76 - 1384 * q^79 - 2180 * q^81 + 548 * q^84 - 3888 * q^86 + 440 * q^89 - 1650 * q^91 + 2292 * q^94 - 270 * q^96 + 1770 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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