LMFDB - Embedded newform 2175.2.c.n.349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(349,2175)] 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("2175.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-10,0,6,0,0,-8,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1267360000.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 37x^{4} + 44x^{2} + 16 $ x^8 + 11x^6 + 37x^4 + 44*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.75660i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.n.349.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.75660i q^{2} +1.00000i q^{3} -5.59883 q^{4} +2.75660 q^{6} +0.393832i q^{7} +9.92054i q^{8} -1.00000 q^{9} -0.393832 q^{11} -5.59883i q^{12} +2.56511i q^{13} +1.08564 q^{14} +16.1493 q^{16} +2.07830i q^{17} +2.75660i q^{18} +0.958939 q^{19} -0.393832 q^{21} +1.08564i q^{22} -6.15661i q^{23} -9.92054 q^{24} +7.07097 q^{26} -1.00000i q^{27} -2.20500i q^{28} +1.00000 q^{29} -10.1566 q^{31} -24.6760i q^{32} -0.393832i q^{33} +5.72905 q^{34} +5.59883 q^{36} -7.34192i q^{37} -2.64341i q^{38} -2.56511 q^{39} -1.65745 q^{41} +1.08564i q^{42} -10.3279i q^{43} +2.20500 q^{44} -16.9713 q^{46} -11.5915i q^{47} +16.1493i q^{48} +6.84490 q^{49} -2.07830 q^{51} -14.3616i q^{52} +12.3279i q^{53} -2.75660 q^{54} -3.90703 q^{56} +0.958939i q^{57} -2.75660i q^{58} +9.54022 q^{59} -6.25340 q^{61} +27.9977i q^{62} -0.393832i q^{63} -35.7232 q^{64} -1.08564 q^{66} -7.42023i q^{67} -11.6361i q^{68} +6.15661 q^{69} +5.98533 q^{71} -9.92054i q^{72} -3.34192i q^{73} -20.2387 q^{74} -5.36894 q^{76} -0.155104i q^{77} +7.07097i q^{78} +2.06745 q^{79} +1.00000 q^{81} +4.56892i q^{82} -6.41000i q^{83} +2.20500 q^{84} -28.4698 q^{86} +1.00000i q^{87} -3.90703i q^{88} -15.8302 q^{89} -1.01022 q^{91} +34.4698i q^{92} -10.1566i q^{93} -31.9531 q^{94} +24.6760 q^{96} -18.4575i q^{97} -18.8686i q^{98} +0.393832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{11} + 6 q^{14} + 22 q^{16} + 4 q^{19} - 4 q^{21} - 24 q^{24} - 14 q^{26} + 8 q^{29} - 8 q^{31} + 2 q^{34} + 10 q^{36} - 16 q^{39} - 24 q^{41} - 18 q^{44} - 16 q^{46}+ \cdots + 4 q^{99}+O(q^{100}) $ 8 * q - 10 * q^4 + 6 * q^6 - 8 * q^9 - 4 * q^11 + 6 * q^14 + 22 * q^16 + 4 * q^19 - 4 * q^21 - 24 * q^24 - 14 * q^26 + 8 * q^29 - 8 * q^31 + 2 * q^34 + 10 * q^36 - 16 * q^39 - 24 * q^41 - 18 * q^44 - 16 * q^46 - 12 * q^49 + 20 * q^51 - 6 * q^54 - 4 * q^59 - 52 * q^61 - 68 * q^64 - 6 * q^66 - 24 * q^69 - 20 * q^71 - 96 * q^74 + 32 * q^76 - 44 * q^79 + 8 * q^81 - 18 * q^84 - 8 * q^86 + 8 * q^89 - 16 * q^91 - 78 * q^94 + 34 * q^96 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.75660i	− 1.94921i	−0.223932	−	0.974605i	$-0.571890\pi$
$2$	− 2.75660i	− 1.94921i	0.223932	−	0.974605i	$-0.428110\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−5.59883	−2.79942
$5$	0	0
$5$	0	0
$6$	2.75660	1.12538
$7$	0.393832i	0.148855i	0.997226	+	0.0744273i	$0.0237129\pi$
$7$	0.393832i	0.148855i	−0.997226	+	0.0744273i	$0.976287\pi$
$8$	9.92054i	3.50744i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.393832	−0.118745	−0.0593725	−	0.998236i	$-0.518910\pi$
$11$	−0.393832	−0.118745	−0.0593725	+	0.998236i	$0.518910\pi$
$12$	− 5.59883i	− 1.61624i
$13$	2.56511i	0.711433i	0.934594	+	0.355716i	$0.115763\pi$
$13$	2.56511i	0.711433i	−0.934594	+	0.355716i	$0.884237\pi$
$14$	1.08564	0.290149
$15$	0	0
$16$	16.1493	4.03732
$17$	2.07830i	0.504063i	0.967719	+	0.252031i	$0.0810986\pi$
$17$	2.07830i	0.504063i	−0.967719	+	0.252031i	$0.918901\pi$
$18$	2.75660i	0.649736i
$19$	0.958939	0.219996	0.109998	−	0.993932i	$-0.464916\pi$
$19$	0.958939	0.219996	0.109998	+	0.993932i	$0.464916\pi$
$20$	0	0
$21$	−0.393832	−0.0859413
$22$	1.08564i	0.231459i
$23$	− 6.15661i	− 1.28374i	−0.766813	−	0.641871i	$-0.778159\pi$
$23$	− 6.15661i	− 1.28374i	0.766813	−	0.641871i	$-0.221841\pi$
$24$	−9.92054	−2.02502
$25$	0	0
$26$	7.07097	1.38673
$27$	− 1.00000i	− 0.192450i
$28$	− 2.20500i	− 0.416706i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−10.1566	−1.82418	−0.912090	−	0.409989i	$-0.865532\pi$
$31$	−10.1566	−1.82418	−0.912090	+	0.409989i	$0.865532\pi$
$32$	− 24.6760i	− 4.36214i
$33$	− 0.393832i	− 0.0685574i
$34$	5.72905	0.982524
$35$	0	0
$36$	5.59883	0.933139
$37$	− 7.34192i	− 1.20700i	−0.797361	−	0.603502i	$-0.793771\pi$
$37$	− 7.34192i	− 1.20700i	0.797361	−	0.603502i	$-0.206229\pi$
$38$	− 2.64341i	− 0.428818i
$39$	−2.56511	−0.410746
$40$	0	0
$41$	−1.65745	−0.258850	−0.129425	−	0.991589i	$-0.541313\pi$
$41$	−1.65745	−0.258850	−0.129425	+	0.991589i	$0.541313\pi$
$42$	1.08564i	0.167517i
$43$	− 10.3279i	− 1.57499i	−0.616323	−	0.787494i	$-0.711378\pi$
$43$	− 10.3279i	− 1.57499i	0.616323	−	0.787494i	$-0.288622\pi$
$44$	2.20500	0.332417
$45$	0	0
$46$	−16.9713	−2.50228
$47$	− 11.5915i	− 1.69079i	−0.534138	−	0.845397i	$-0.679364\pi$
$47$	− 11.5915i	− 1.69079i	0.534138	−	0.845397i	$-0.320636\pi$
$48$	16.1493i	2.33095i
$49$	6.84490	0.977842
$50$	0	0
$51$	−2.07830	−0.291021
$52$	− 14.3616i	− 1.99160i
$53$	12.3279i	1.69336i	0.532099	+	0.846682i	$0.321404\pi$
$53$	12.3279i	1.69336i	−0.532099	+	0.846682i	$0.678596\pi$
$54$	−2.75660	−0.375126
$55$	0	0
$56$	−3.90703	−0.522099
$57$	0.958939i	0.127015i
$58$	− 2.75660i	− 0.361959i
$59$	9.54022	1.24203	0.621015	−	0.783798i	$-0.286720\pi$
$59$	9.54022	1.24203	0.621015	+	0.783798i	$0.286720\pi$
$60$	0	0
$61$	−6.25340	−0.800665	−0.400333	−	0.916370i	$-0.631105\pi$
$61$	−6.25340	−0.800665	−0.400333	+	0.916370i	$0.631105\pi$
$62$	27.9977i	3.55571i
$63$	− 0.393832i	− 0.0496182i
$64$	−35.7232	−4.46540
$65$	0	0
$66$	−1.08564	−0.133633
$67$	− 7.42023i	− 0.906525i	−0.891377	−	0.453262i	$-0.850260\pi$
$67$	− 7.42023i	− 0.906525i	0.891377	−	0.453262i	$-0.149740\pi$
$68$	− 11.6361i	− 1.41108i
$69$	6.15661	0.741168
$70$	0	0
$71$	5.98533	0.710328	0.355164	−	0.934804i	$-0.384425\pi$
$71$	5.98533	0.710328	0.355164	+	0.934804i	$0.384425\pi$
$72$	− 9.92054i	− 1.16915i
$73$	− 3.34192i	− 0.391142i	−0.980690	−	0.195571i	$-0.937344\pi$
$73$	− 3.34192i	− 0.391142i	0.980690	−	0.195571i	$-0.0626561\pi$
$74$	−20.2387	−2.35270
$75$	0	0
$76$	−5.36894	−0.615860
$77$	− 0.155104i	− 0.0176757i
$78$	7.07097i	0.800630i
$79$	2.06745	0.232607	0.116303	−	0.993214i	$-0.462896\pi$
$79$	2.06745	0.232607	0.116303	+	0.993214i	$0.462896\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.56892i	0.504553i
$83$	− 6.41000i	− 0.703589i	−0.936077	−	0.351795i	$-0.885572\pi$
$83$	− 6.41000i	− 0.703589i	0.936077	−	0.351795i	$-0.114428\pi$
$84$	2.20500	0.240585
$85$	0	0
$86$	−28.4698	−3.06998
$87$	1.00000i	0.107211i
$88$	− 3.90703i	− 0.416491i
$89$	−15.8302	−1.67800	−0.839000	−	0.544131i	$-0.816860\pi$
$89$	−15.8302	−1.67800	−0.839000	+	0.544131i	$0.816860\pi$
$90$	0	0
$91$	−1.01022	−0.105900
$92$	34.4698i	3.59373i
$93$	− 10.1566i	− 1.05319i
$94$	−31.9531	−3.29571
$95$	0	0
$96$	24.6760	2.51848
$97$	− 18.4575i	− 1.87407i	−0.349233	−	0.937036i	$-0.613558\pi$
$97$	− 18.4575i	− 1.87407i	0.349233	−	0.937036i	$-0.386442\pi$
$98$	− 18.8686i	− 1.90602i
$99$	0.393832	0.0395816
$100$	0	0
$101$	−12.8038	−1.27403	−0.637015	−	0.770852i	$-0.719831\pi$
$101$	−12.8038	−1.27403	−0.637015	+	0.770852i	$0.719831\pi$
$102$	5.72905i	0.567260i
$103$	− 4.86979i	− 0.479834i	−0.970793	−	0.239917i	$-0.922880\pi$
$103$	− 4.86979i	− 0.479834i	0.970793	−	0.239917i	$-0.0771203\pi$
$104$	−25.4472	−2.49531
$105$	0	0
$106$	33.9830	3.30072
$107$	− 2.34255i	− 0.226463i	−0.993569	−	0.113231i	$-0.963880\pi$
$107$	− 2.34255i	− 0.226463i	0.993569	−	0.113231i	$-0.0361201\pi$
$108$	5.59883i	0.538748i
$109$	−8.55044	−0.818984	−0.409492	−	0.912314i	$-0.634294\pi$
$109$	−8.55044	−0.818984	−0.409492	+	0.912314i	$0.634294\pi$
$110$	0	0
$111$	7.34192	0.696864
$112$	6.36011i	0.600973i
$113$	11.2085i	1.05441i	0.849739	+	0.527204i	$0.176760\pi$
$113$	11.2085i	1.05441i	−0.849739	+	0.527204i	$0.823240\pi$
$114$	2.64341	0.247578
$115$	0	0
$116$	−5.59883	−0.519839
$117$	− 2.56511i	− 0.237144i
$118$	− 26.2985i	− 2.42098i
$119$	−0.818503	−0.0750321
$120$	0	0
$121$	−10.8449	−0.985900
$122$	17.2381i	1.56066i
$123$	− 1.65745i	− 0.149447i
$124$	56.8652	5.10664
$125$	0	0
$126$	−1.08564	−0.0967163
$127$	20.1566i	1.78861i	0.447458	+	0.894305i	$0.352329\pi$
$127$	20.1566i	1.78861i	−0.447458	+	0.894305i	$0.647671\pi$
$128$	49.1226i	4.34187i
$129$	10.3279	0.909319
$130$	0	0
$131$	−5.50235	−0.480742	−0.240371	−	0.970681i	$-0.577269\pi$
$131$	−5.50235	−0.480742	−0.240371	+	0.970681i	$0.577269\pi$
$132$	2.20500i	0.191921i
$133$	0.377661i	0.0327474i
$134$	−20.4546	−1.76701
$135$	0	0
$136$	−20.6179	−1.76797
$137$	7.49853i	0.640643i	0.947309	+	0.320321i	$0.103791\pi$
$137$	7.49853i	0.640643i	−0.947309	+	0.320321i	$0.896209\pi$
$138$	− 16.9713i	− 1.44469i
$139$	9.35277	0.793292	0.396646	−	0.917972i	$-0.370174\pi$
$139$	9.35277	0.793292	0.396646	+	0.917972i	$0.370174\pi$
$140$	0	0
$141$	11.5915	0.976180
$142$	− 16.4992i	− 1.38458i
$143$	− 1.01022i	− 0.0844790i
$144$	−16.1493	−1.34577
$145$	0	0
$146$	−9.21234	−0.762418
$147$	6.84490i	0.564558i
$148$	41.1062i	3.37891i
$149$	11.5402	0.945411	0.472706	−	0.881220i	$-0.343277\pi$
$149$	11.5402	0.945411	0.472706	+	0.881220i	$0.343277\pi$
$150$	0	0
$151$	6.08212	0.494956	0.247478	−	0.968894i	$-0.420398\pi$
$151$	6.08212	0.494956	0.247478	+	0.968894i	$0.420398\pi$
$152$	9.51320i	0.771622i
$153$	− 2.07830i	− 0.168021i
$154$	−0.427559	−0.0344537
$155$	0	0
$156$	14.3616	1.14985
$157$	− 11.5953i	− 0.925407i	−0.886513	−	0.462704i	$-0.846879\pi$
$157$	− 11.5953i	− 0.925407i	0.886513	−	0.462704i	$-0.153121\pi$
$158$	− 5.69914i	− 0.453399i
$159$	−12.3279	−0.977665
$160$	0	0
$161$	2.42467	0.191091
$162$	− 2.75660i	− 0.216579i
$163$	0.855118i	0.0669780i	0.999439	+	0.0334890i	$0.0106619\pi$
$163$	0.855118i	0.0669780i	−0.999439	+	0.0334890i	$0.989338\pi$
$164$	9.27979	0.724630
$165$	0	0
$166$	−17.6698	−1.37144
$167$	7.73194i	0.598315i	0.954204	+	0.299158i	$0.0967056\pi$
$167$	7.73194i	0.598315i	−0.954204	+	0.299158i	$0.903294\pi$
$168$	− 3.90703i	− 0.301434i
$169$	6.42023	0.493863
$170$	0	0
$171$	−0.958939	−0.0733319
$172$	57.8241i	4.40905i
$173$	− 8.07449i	− 0.613892i	−0.951727	−	0.306946i	$-0.900693\pi$
$173$	− 8.07449i	− 0.613892i	0.951727	−	0.306946i	$-0.0993071\pi$
$174$	2.75660	0.208977
$175$	0	0
$176$	−6.36011	−0.479411
$177$	9.54022i	0.717087i
$178$	43.6376i	3.27077i
$179$	−12.4100	−0.927567	−0.463784	−	0.885949i	$-0.653508\pi$
$179$	−12.4100	−0.927567	−0.463784	+	0.885949i	$0.653508\pi$
$180$	0	0
$181$	−2.49765	−0.185649	−0.0928246	−	0.995682i	$-0.529590\pi$
$181$	−2.49765	−0.185649	−0.0928246	+	0.995682i	$0.529590\pi$
$182$	2.78478i	0.206421i
$183$	− 6.25340i	− 0.462264i
$184$	61.0769	4.50265
$185$	0	0
$186$	−27.9977	−2.05289
$187$	− 0.818503i	− 0.0598549i
$188$	64.8989i	4.73324i
$189$	0.393832	0.0286471
$190$	0	0
$191$	−6.32085	−0.457361	−0.228680	−	0.973502i	$-0.573441\pi$
$191$	−6.32085	−0.457361	−0.228680	+	0.973502i	$0.573441\pi$
$192$	− 35.7232i	− 2.57810i
$193$	− 25.2921i	− 1.82057i	−0.413984	−	0.910284i	$-0.635863\pi$
$193$	− 25.2921i	− 1.82057i	0.413984	−	0.910284i	$-0.364137\pi$
$194$	−50.8798	−3.65296
$195$	0	0
$196$	−38.3234	−2.73739
$197$	− 21.2047i	− 1.51077i	−0.655280	−	0.755386i	$-0.727449\pi$
$197$	− 21.2047i	− 1.51077i	0.655280	−	0.755386i	$-0.272551\pi$
$198$	− 1.08564i	− 0.0771529i
$199$	2.64723	0.187657	0.0938285	−	0.995588i	$-0.470089\pi$
$199$	2.64723	0.187657	0.0938285	+	0.995588i	$0.470089\pi$
$200$	0	0
$201$	7.42023	0.523382
$202$	35.2950i	2.48335i
$203$	0.393832i	0.0276416i
$204$	11.6361	0.814688
$205$	0	0
$206$	−13.4240	−0.935297
$207$	6.15661i	0.427914i
$208$	41.4246i	2.87228i
$209$	−0.377661	−0.0261234
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	− 69.0218i	− 4.74043i
$213$	5.98533i	0.410108i
$214$	−6.45747	−0.441423
$215$	0	0
$216$	9.92054	0.675007
$217$	− 4.00000i	− 0.271538i
$218$	23.5701i	1.59637i
$219$	3.34192	0.225826
$220$	0	0
$221$	−5.33107	−0.358607
$222$	− 20.2387i	− 1.35833i
$223$	− 12.5504i	− 0.840440i	−0.907422	−	0.420220i	$-0.861953\pi$
$223$	− 12.5504i	− 0.840440i	0.907422	−	0.420220i	$-0.138047\pi$
$224$	9.71820	0.649324
$225$	0	0
$226$	30.8974	2.05526
$227$	1.30149i	0.0863829i	0.999067	+	0.0431914i	$0.0137525\pi$
$227$	1.30149i	0.0863829i	−0.999067	+	0.0431914i	$0.986247\pi$
$228$	− 5.36894i	− 0.355567i
$229$	−3.28682	−0.217199	−0.108600	−	0.994086i	$-0.534637\pi$
$229$	−3.28682	−0.217199	−0.108600	+	0.994086i	$0.534637\pi$
$230$	0	0
$231$	0.155104	0.0102051
$232$	9.92054i	0.651315i
$233$	− 17.7115i	− 1.16032i	−0.814503	−	0.580159i	$-0.802990\pi$
$233$	− 17.7115i	− 1.16032i	0.814503	−	0.580159i	$-0.197010\pi$
$234$	−7.07097	−0.462244
$235$	0	0
$236$	−53.4141	−3.47696
$237$	2.06745i	0.134296i
$238$	2.25628i	0.146253i
$239$	−21.2651	−1.37553	−0.687763	−	0.725935i	$-0.741407\pi$
$239$	−21.2651	−1.37553	−0.687763	+	0.725935i	$0.741407\pi$
$240$	0	0
$241$	15.3177	0.986697	0.493349	−	0.869832i	$-0.335773\pi$
$241$	15.3177	0.986697	0.493349	+	0.869832i	$0.335773\pi$
$242$	29.8950i	1.92172i
$243$	1.00000i	0.0641500i
$244$	35.0117	2.24140
$245$	0	0
$246$	−4.56892	−0.291304
$247$	2.45978i	0.156512i
$248$	− 100.759i	− 6.39820i
$249$	6.41000	0.406217
$250$	0	0
$251$	26.8917	1.69739	0.848696	−	0.528882i	$-0.177388\pi$
$251$	26.8917	1.69739	0.848696	+	0.528882i	$0.177388\pi$
$252$	2.20500i	0.138902i
$253$	2.42467i	0.152438i
$254$	55.5637	3.48637
$255$	0	0
$256$	63.9648	3.99780
$257$	− 6.85512i	− 0.427611i	−0.976876	−	0.213805i	$-0.931414\pi$
$257$	− 6.85512i	− 0.427611i	0.976876	−	0.213805i	$-0.0685858\pi$
$258$	− 28.4698i	− 1.77245i
$259$	2.89149	0.179668
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	15.1678i	0.937067i
$263$	13.6294i	0.840423i	0.907426	+	0.420212i	$0.138044\pi$
$263$	13.6294i	0.840423i	−0.907426	+	0.420212i	$0.861956\pi$
$264$	3.90703	0.240461
$265$	0	0
$266$	1.04106	0.0638315
$267$	− 15.8302i	− 0.968794i
$268$	41.5446i	2.53774i
$269$	8.46129	0.515894	0.257947	−	0.966159i	$-0.416954\pi$
$269$	8.46129	0.515894	0.257947	+	0.966159i	$0.416954\pi$
$270$	0	0
$271$	6.34255	0.385282	0.192641	−	0.981269i	$-0.438295\pi$
$271$	6.34255	0.385282	0.192641	+	0.981269i	$0.438295\pi$
$272$	33.5631i	2.03506i
$273$	− 1.01022i	− 0.0611414i
$274$	20.6704	1.24875
$275$	0	0
$276$	−34.4698	−2.07484
$277$	− 17.1464i	− 1.03023i	−0.857122	−	0.515113i	$-0.827750\pi$
$277$	− 17.1464i	− 1.03023i	0.857122	−	0.515113i	$-0.172250\pi$
$278$	− 25.7818i	− 1.54629i
$279$	10.1566	0.608060
$280$	0	0
$281$	−12.2985	−0.733670	−0.366835	−	0.930286i	$-0.619559\pi$
$281$	−12.2985	−0.733670	−0.366835	+	0.930286i	$0.619559\pi$
$282$	− 31.9531i	− 1.90278i
$283$	− 25.1830i	− 1.49697i	−0.663149	−	0.748487i	$-0.730781\pi$
$283$	− 25.1830i	− 1.49697i	0.663149	−	0.748487i	$-0.269219\pi$
$284$	−33.5109	−1.98851
$285$	0	0
$286$	−2.78478	−0.164667
$287$	− 0.652757i	− 0.0385311i
$288$	24.6760i	1.45405i
$289$	12.6807	0.745921
$290$	0	0
$291$	18.4575	1.08200
$292$	18.7109i	1.09497i
$293$	− 12.3170i	− 0.719569i	−0.933035	−	0.359784i	$-0.882850\pi$
$293$	− 12.3170i	− 0.719569i	0.933035	−	0.359784i	$-0.117150\pi$
$294$	18.8686	1.10044
$295$	0	0
$296$	72.8358	4.23350
$297$	0.393832i	0.0228525i
$298$	− 31.8117i	− 1.84280i
$299$	15.7924	0.913296
$300$	0	0
$301$	4.06745	0.234444
$302$	− 16.7660i	− 0.964773i
$303$	− 12.8038i	− 0.735561i
$304$	15.4862	0.888193
$305$	0	0
$306$	−5.72905	−0.327508
$307$	− 22.7379i	− 1.29772i	−0.760908	−	0.648860i	$-0.775246\pi$
$307$	− 22.7379i	− 1.29772i	0.760908	−	0.648860i	$-0.224754\pi$
$308$	0.868401i	0.0494817i
$309$	4.86979	0.277032
$310$	0	0
$311$	−5.33810	−0.302696	−0.151348	−	0.988481i	$-0.548361\pi$
$311$	−5.33810	−0.302696	−0.151348	+	0.988481i	$0.548361\pi$
$312$	− 25.4472i	− 1.44067i
$313$	1.96338i	0.110977i	0.998459	+	0.0554885i	$0.0176716\pi$
$313$	1.96338i	0.110977i	−0.998459	+	0.0554885i	$0.982328\pi$
$314$	−31.9636	−1.80381
$315$	0	0
$316$	−11.5753	−0.651163
$317$	1.29064i	0.0724895i	0.999343	+	0.0362448i	$0.0115396\pi$
$317$	1.29064i	0.0724895i	−0.999343	+	0.0362448i	$0.988460\pi$
$318$	33.9830i	1.90567i
$319$	−0.393832	−0.0220504
$320$	0	0
$321$	2.34255	0.130748
$322$	− 6.68384i	− 0.372476i
$323$	1.99297i	0.110892i
$324$	−5.59883	−0.311046
$325$	0	0
$326$	2.35722	0.130554
$327$	− 8.55044i	− 0.472840i
$328$	− 16.4428i	− 0.907902i
$329$	4.56511	0.251683
$330$	0	0
$331$	28.9971	1.59382	0.796911	−	0.604096i	$-0.206466\pi$
$331$	28.9971	1.59382	0.796911	+	0.604096i	$0.206466\pi$
$332$	35.8885i	1.96964i
$333$	7.34192i	0.402335i
$334$	21.3138	1.16624
$335$	0	0
$336$	−6.36011	−0.346972
$337$	− 16.7854i	− 0.914356i	−0.889375	−	0.457178i	$-0.848860\pi$
$337$	− 16.7854i	− 0.914356i	0.889375	−	0.457178i	$-0.151140\pi$
$338$	− 17.6980i	− 0.962643i
$339$	−11.2085	−0.608763
$340$	0	0
$341$	4.00000	0.216612
$342$	2.64341i	0.142939i
$343$	5.45257i	0.294411i
$344$	102.458	5.52417
$345$	0	0
$346$	−22.2581	−1.19660
$347$	− 17.8681i	− 0.959210i	−0.877485	−	0.479605i	$-0.840780\pi$
$347$	− 17.8681i	− 0.959210i	0.877485	−	0.479605i	$-0.159220\pi$
$348$	− 5.59883i	− 0.300129i
$349$	−8.73789	−0.467728	−0.233864	−	0.972269i	$-0.575137\pi$
$349$	−8.73789	−0.467728	−0.233864	+	0.972269i	$0.575137\pi$
$350$	0	0
$351$	2.56511	0.136915
$352$	9.71820i	0.517982i
$353$	22.6017i	1.20297i	0.798885	+	0.601484i	$0.205424\pi$
$353$	22.6017i	1.20297i	−0.798885	+	0.601484i	$0.794576\pi$
$354$	26.2985	1.39775
$355$	0	0
$356$	88.6308	4.69742
$357$	− 0.818503i	− 0.0433198i
$358$	34.2094i	1.80802i
$359$	13.1830	0.695772	0.347886	−	0.937537i	$-0.386900\pi$
$359$	13.1830	0.695772	0.347886	+	0.937537i	$0.386900\pi$
$360$	0	0
$361$	−18.0804	−0.951602
$362$	6.88503i	0.361869i
$363$	− 10.8449i	− 0.569209i
$364$	5.65607	0.296458
$365$	0	0
$366$	−17.2381	−0.901050
$367$	17.4511i	0.910938i	0.890252	+	0.455469i	$0.150528\pi$
$367$	17.4511i	0.910938i	−0.890252	+	0.455469i	$0.849472\pi$
$368$	− 99.4247i	− 5.18287i
$369$	1.65745	0.0862834
$370$	0	0
$371$	−4.85512	−0.252065
$372$	56.8652i	2.94832i
$373$	32.2018i	1.66734i	0.552260	+	0.833672i	$0.313766\pi$
$373$	32.2018i	1.66734i	−0.552260	+	0.833672i	$0.686234\pi$
$374$	−2.25628	−0.116670
$375$	0	0
$376$	114.994	5.93036
$377$	2.56511i	0.132110i
$378$	− 1.08564i	− 0.0558392i
$379$	−32.3660	−1.66253	−0.831265	−	0.555876i	$-0.812383\pi$
$379$	−32.3660	−1.66253	−0.831265	+	0.555876i	$0.812383\pi$
$380$	0	0
$381$	−20.1566	−1.03265
$382$	17.4240i	0.891492i
$383$	25.8739i	1.32209i	0.750345	+	0.661047i	$0.229887\pi$
$383$	25.8739i	1.32209i	−0.750345	+	0.661047i	$0.770113\pi$
$384$	−49.1226	−2.50678
$385$	0	0
$386$	−69.7203	−3.54867
$387$	10.3279i	0.524996i
$388$	103.340i	5.24631i
$389$	−32.6103	−1.65341	−0.826703	−	0.562639i	$-0.809786\pi$
$389$	−32.6103	−1.65341	−0.826703	+	0.562639i	$0.809786\pi$
$390$	0	0
$391$	12.7953	0.647086
$392$	67.9051i	3.42972i
$393$	− 5.50235i	− 0.277557i
$394$	−58.4528	−2.94481
$395$	0	0
$396$	−2.20500	−0.110806
$397$	4.39534i	0.220596i	0.993899	+	0.110298i	$0.0351805\pi$
$397$	4.39534i	0.220596i	−0.993899	+	0.110298i	$0.964820\pi$
$398$	− 7.29734i	− 0.365783i
$399$	−0.377661	−0.0189067
$400$	0	0
$401$	−19.0658	−0.952099	−0.476049	−	0.879418i	$-0.657932\pi$
$401$	−19.0658	−0.952099	−0.476049	+	0.879418i	$0.657932\pi$
$402$	− 20.4546i	− 1.02018i
$403$	− 26.0528i	− 1.29778i
$404$	71.6865	3.56654
$405$	0	0
$406$	1.08564	0.0538793
$407$	2.89149i	0.143326i
$408$	− 20.6179i	− 1.02074i
$409$	1.38235	0.0683530	0.0341765	−	0.999416i	$-0.489119\pi$
$409$	1.38235	0.0683530	0.0341765	+	0.999416i	$0.489119\pi$
$410$	0	0
$411$	−7.49853	−0.369875
$412$	27.2651i	1.34326i
$413$	3.75725i	0.184882i
$414$	16.9713	0.834094
$415$	0	0
$416$	63.2965	3.10337
$417$	9.35277i	0.458007i
$418$	1.04106i	0.0509199i
$419$	−2.86979	−0.140198	−0.0700991	−	0.997540i	$-0.522332\pi$
$419$	−2.86979	−0.140198	−0.0700991	+	0.997540i	$0.522332\pi$
$420$	0	0
$421$	13.6440	0.664970	0.332485	−	0.943109i	$-0.392113\pi$
$421$	13.6440	0.664970	0.332485	+	0.943109i	$0.392113\pi$
$422$	− 5.51320i	− 0.268378i
$423$	11.5915i	0.563598i
$424$	−122.299	−5.93938
$425$	0	0
$426$	16.4992	0.799387
$427$	− 2.46279i	− 0.119183i
$428$	13.1155i	0.633964i
$429$	1.01022	0.0487740
$430$	0	0
$431$	−11.9707	−0.576607	−0.288303	−	0.957539i	$-0.593091\pi$
$431$	−11.9707	−0.576607	−0.288303	+	0.957539i	$0.593091\pi$
$432$	− 16.1493i	− 0.776982i
$433$	− 17.9907i	− 0.864576i	−0.901736	−	0.432288i	$-0.857706\pi$
$433$	− 17.9907i	− 0.864576i	0.901736	−	0.432288i	$-0.142294\pi$
$434$	−11.0264	−0.529284
$435$	0	0
$436$	47.8725	2.29268
$437$	− 5.90381i	− 0.282418i
$438$	− 9.21234i	− 0.440182i
$439$	16.2226	0.774260	0.387130	−	0.922025i	$-0.373466\pi$
$439$	16.2226	0.774260	0.387130	+	0.922025i	$0.373466\pi$
$440$	0	0
$441$	−6.84490	−0.325947
$442$	14.6956i	0.698999i
$443$	35.3762i	1.68078i	0.541986	+	0.840388i	$0.317673\pi$
$443$	35.3762i	1.68078i	−0.541986	+	0.840388i	$0.682327\pi$
$444$	−41.1062	−1.95081
$445$	0	0
$446$	−34.5965	−1.63819
$447$	11.5402i	0.545834i
$448$	− 14.0690i	− 0.664696i
$449$	−41.6971	−1.96781	−0.983903	−	0.178702i	$-0.942810\pi$
$449$	−41.6971	−1.96781	−0.983903	+	0.178702i	$0.942810\pi$
$450$	0	0
$451$	0.652757	0.0307371
$452$	− 62.7546i	− 2.95173i
$453$	6.08212i	0.285763i
$454$	3.58768	0.168378
$455$	0	0
$456$	−9.51320	−0.445496
$457$	− 16.1111i	− 0.753646i	−0.926285	−	0.376823i	$-0.877017\pi$
$457$	− 16.1111i	− 0.753646i	0.926285	−	0.376823i	$-0.122983\pi$
$458$	9.06045i	0.423367i
$459$	2.07830	0.0970069
$460$	0	0
$461$	17.3396	0.807586	0.403793	−	0.914850i	$-0.367692\pi$
$461$	17.3396	0.807586	0.403793	+	0.914850i	$0.367692\pi$
$462$	− 0.427559i	− 0.0198918i
$463$	− 19.3177i	− 0.897768i	−0.893590	−	0.448884i	$-0.851822\pi$
$463$	− 19.3177i	− 0.897768i	0.893590	−	0.448884i	$-0.148178\pi$
$464$	16.1493	0.749711
$465$	0	0
$466$	−48.8235	−2.26170
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	14.3616i	0.663866i
$469$	2.92232	0.134940
$470$	0	0
$471$	11.5953	0.534284
$472$	94.6441i	4.35635i
$473$	4.06745i	0.187022i
$474$	5.69914	0.261770
$475$	0	0
$476$	4.58266	0.210046
$477$	− 12.3279i	− 0.564455i
$478$	58.6194i	2.68119i
$479$	40.3877	1.84536	0.922681	−	0.385565i	$-0.125994\pi$
$479$	40.3877	1.84536	0.922681	+	0.385565i	$0.125994\pi$
$480$	0	0
$481$	18.8328	0.858702
$482$	− 42.2246i	− 1.92328i
$483$	2.42467i	0.110326i
$484$	60.7188	2.75994
$485$	0	0
$486$	2.75660	0.125042
$487$	2.84171i	0.128770i	0.997925	+	0.0643850i	$0.0205086\pi$
$487$	2.84171i	0.128770i	−0.997925	+	0.0643850i	$0.979491\pi$
$488$	− 62.0371i	− 2.80829i
$489$	−0.855118	−0.0386698
$490$	0	0
$491$	0.157863	0.00712428	0.00356214	−	0.999994i	$-0.498866\pi$
$491$	0.157863	0.00712428	0.00356214	+	0.999994i	$0.498866\pi$
$492$	9.27979i	0.418365i
$493$	2.07830i	0.0936021i
$494$	6.78063	0.305075
$495$	0	0
$496$	−164.022	−7.36480
$497$	2.35722i	0.105736i
$498$	− 17.6698i	− 0.791803i
$499$	−2.64723	−0.118506	−0.0592531	−	0.998243i	$-0.518872\pi$
$499$	−2.64723	−0.118506	−0.0592531	+	0.998243i	$0.518872\pi$
$500$	0	0
$501$	−7.73194	−0.345437
$502$	− 74.1297i	− 3.30857i
$503$	− 13.9545i	− 0.622200i	−0.950377	−	0.311100i	$-0.899303\pi$
$503$	− 13.9545i	− 0.622200i	0.950377	−	0.311100i	$-0.100697\pi$
$504$	3.90703	0.174033
$505$	0	0
$506$	6.68384	0.297133
$507$	6.42023i	0.285132i
$508$	− 112.853i	− 5.00706i
$509$	16.4921	0.731001	0.365500	−	0.930811i	$-0.380898\pi$
$509$	16.4921	0.731001	0.365500	+	0.930811i	$0.380898\pi$
$510$	0	0
$511$	1.31616	0.0582233
$512$	− 78.0802i	− 3.45069i
$513$	− 0.958939i	− 0.0423382i
$514$	−18.8968	−0.833502
$515$	0	0
$516$	−57.8241	−2.54556
$517$	4.56511i	0.200773i
$518$	− 7.97066i	− 0.350211i
$519$	8.07449	0.354431
$520$	0	0
$521$	−12.2253	−0.535601	−0.267800	−	0.963474i	$-0.586297\pi$
$521$	−12.2253	−0.535601	−0.267800	+	0.963474i	$0.586297\pi$
$522$	2.75660i	0.120653i
$523$	− 1.66659i	− 0.0728748i	−0.999336	−	0.0364374i	$-0.988399\pi$
$523$	− 1.66659i	− 0.0728748i	0.999336	−	0.0364374i	$-0.0116010\pi$
$524$	30.8067	1.34580
$525$	0	0
$526$	37.5707	1.63816
$527$	− 21.1085i	− 0.919501i
$528$	− 6.36011i	− 0.276788i
$529$	−14.9038	−0.647992
$530$	0	0
$531$	−9.54022	−0.414010
$532$	− 2.11446i	− 0.0916736i
$533$	− 4.25154i	− 0.184155i
$534$	−43.6376	−1.88838
$535$	0	0
$536$	73.6126	3.17958
$537$	− 12.4100i	− 0.535531i
$538$	− 23.3244i	− 1.00558i
$539$	−2.69574	−0.116114
$540$	0	0
$541$	−34.6558	−1.48997	−0.744984	−	0.667083i	$-0.767543\pi$
$541$	−34.6558	−1.48997	−0.744984	+	0.667083i	$0.767543\pi$
$542$	− 17.4839i	− 0.750996i
$543$	− 2.49765i	− 0.107185i
$544$	51.2842	2.19879
$545$	0	0
$546$	−2.78478	−0.119177
$547$	38.2530i	1.63558i	0.575515	+	0.817791i	$0.304801\pi$
$547$	38.2530i	1.63558i	−0.575515	+	0.817791i	$0.695199\pi$
$548$	− 41.9830i	− 1.79343i
$549$	6.25340	0.266888
$550$	0	0
$551$	0.958939	0.0408522
$552$	61.0769i	2.59960i
$553$	0.814230i	0.0346246i
$554$	−47.2657	−2.00813
$555$	0	0
$556$	−52.3646	−2.22075
$557$	− 28.7760i	− 1.21928i	−0.792679	−	0.609639i	$-0.791314\pi$
$557$	− 28.7760i	− 1.21928i	0.792679	−	0.609639i	$-0.208686\pi$
$558$	− 27.9977i	− 1.18524i
$559$	26.4921	1.12050
$560$	0	0
$561$	0.818503	0.0345572
$562$	33.9022i	1.43008i
$563$	32.9809i	1.38998i	0.719020	+	0.694989i	$0.244591\pi$
$563$	32.9809i	1.38998i	−0.719020	+	0.694989i	$0.755409\pi$
$564$	−64.8989	−2.73274
$565$	0	0
$566$	−69.4194	−2.91792
$567$	0.393832i	0.0165394i
$568$	59.3777i	2.49143i
$569$	−16.8038	−0.704453	−0.352227	−	0.935915i	$-0.614575\pi$
$569$	−16.8038	−0.704453	−0.352227	+	0.935915i	$0.614575\pi$
$570$	0	0
$571$	−6.21703	−0.260175	−0.130087	−	0.991503i	$-0.541526\pi$
$571$	−6.21703	−0.260175	−0.130087	+	0.991503i	$0.541526\pi$
$572$	5.65607i	0.236492i
$573$	− 6.32085i	− 0.264057i
$574$	−1.79939	−0.0751051
$575$	0	0
$576$	35.7232	1.48847
$577$	11.9977i	0.499470i	0.968314	+	0.249735i	$0.0803435\pi$
$577$	11.9977i	0.499470i	−0.968314	+	0.249735i	$0.919656\pi$
$578$	− 34.9555i	− 1.45396i
$579$	25.2921	1.05111
$580$	0	0
$581$	2.52447	0.104733
$582$	− 50.8798i	− 2.10904i
$583$	− 4.85512i	− 0.201078i
$584$	33.1537	1.37191
$585$	0	0
$586$	−33.9531	−1.40259
$587$	− 11.2194i	− 0.463073i	−0.972826	−	0.231536i	$-0.925625\pi$
$587$	− 11.2194i	− 0.463073i	0.972826	−	0.231536i	$-0.0743753\pi$
$588$	− 38.3234i	− 1.58043i
$589$	−9.73957	−0.401312
$590$	0	0
$591$	21.2047	0.872245
$592$	− 118.567i	− 4.87306i
$593$	7.69682i	0.316071i	0.987433	+	0.158035i	$0.0505160\pi$
$593$	7.69682i	0.316071i	−0.987433	+	0.158035i	$0.949484\pi$
$594$	1.08564	0.0445442
$595$	0	0
$596$	−64.6118	−2.64660
$597$	2.64723i	0.108344i
$598$	− 43.5332i	− 1.78020i
$599$	−20.0543	−0.819396	−0.409698	−	0.912221i	$-0.634366\pi$
$599$	−20.0543	−0.819396	−0.409698	+	0.912221i	$0.634366\pi$
$600$	0	0
$601$	3.10851	0.126799	0.0633995	−	0.997988i	$-0.479806\pi$
$601$	3.10851	0.126799	0.0633995	+	0.997988i	$0.479806\pi$
$602$	− 11.2123i	− 0.456981i
$603$	7.42023i	0.302175i
$604$	−34.0528	−1.38559
$605$	0	0
$606$	−35.2950	−1.43376
$607$	37.0441i	1.50357i	0.659407	+	0.751786i	$0.270807\pi$
$607$	37.0441i	1.50357i	−0.659407	+	0.751786i	$0.729193\pi$
$608$	− 23.6628i	− 0.959652i
$609$	−0.393832	−0.0159589
$610$	0	0
$611$	29.7334	1.20289
$612$	11.6361i	0.470361i
$613$	13.3839i	0.540569i	0.962781	+	0.270284i	$0.0871177\pi$
$613$	13.3839i	0.540569i	−0.962781	+	0.270284i	$0.912882\pi$
$614$	−62.6792	−2.52953
$615$	0	0
$616$	1.53871	0.0619966
$617$	− 12.6364i	− 0.508721i	−0.967109	−	0.254361i	$-0.918135\pi$
$617$	− 12.6364i	− 0.508721i	0.967109	−	0.254361i	$-0.0818650\pi$
$618$	− 13.4240i	− 0.539994i
$619$	41.2447	1.65776	0.828882	−	0.559424i	$-0.188978\pi$
$619$	41.2447	1.65776	0.828882	+	0.559424i	$0.188978\pi$
$620$	0	0
$621$	−6.15661	−0.247056
$622$	14.7150i	0.590018i
$623$	− 6.23446i	− 0.249778i
$624$	−41.4246	−1.65831
$625$	0	0
$626$	5.41226	0.216318
$627$	− 0.377661i	− 0.0150823i
$628$	64.9203i	2.59060i
$629$	15.2587	0.608406
$630$	0	0
$631$	29.8009	1.18635	0.593177	−	0.805072i	$-0.297873\pi$
$631$	29.8009	1.18635	0.593177	+	0.805072i	$0.297873\pi$
$632$	20.5103i	0.815854i
$633$	2.00000i	0.0794929i
$634$	3.55777	0.141297
$635$	0	0
$636$	69.0218	2.73689
$637$	17.5579i	0.695669i
$638$	1.08564i	0.0429808i
$639$	−5.98533	−0.236776
$640$	0	0
$641$	17.7774	0.702167	0.351083	−	0.936344i	$-0.385813\pi$
$641$	17.7774	0.702167	0.351083	+	0.936344i	$0.385813\pi$
$642$	− 6.45747i	− 0.254856i
$643$	44.5534i	1.75702i	0.477727	+	0.878508i	$0.341461\pi$
$643$	44.5534i	1.75702i	−0.477727	+	0.878508i	$0.658539\pi$
$644$	−13.5753	−0.534943
$645$	0	0
$646$	5.49381	0.216151
$647$	42.9339i	1.68790i	0.536418	+	0.843952i	$0.319777\pi$
$647$	42.9339i	1.68790i	−0.536418	+	0.843952i	$0.680223\pi$
$648$	9.92054i	0.389716i
$649$	−3.75725	−0.147485
$650$	0	0
$651$	4.00000	0.156772
$652$	− 4.78766i	− 0.187499i
$653$	19.8426i	0.776500i	0.921554	+	0.388250i	$0.126920\pi$
$653$	19.8426i	0.776500i	−0.921554	+	0.388250i	$0.873080\pi$
$654$	−23.5701	−0.921665
$655$	0	0
$656$	−26.7666	−1.04506
$657$	3.34192i	0.130381i
$658$	− 12.5842i	− 0.490582i
$659$	−23.0483	−0.897836	−0.448918	−	0.893573i	$-0.648190\pi$
$659$	−23.0483	−0.897836	−0.448918	+	0.893573i	$0.648190\pi$
$660$	0	0
$661$	−36.2079	−1.40832	−0.704162	−	0.710040i	$-0.748677\pi$
$661$	−36.2079	−1.40832	−0.704162	+	0.710040i	$0.748677\pi$
$662$	− 79.9332i	− 3.10669i
$663$	− 5.33107i	− 0.207042i
$664$	63.5907	2.46780
$665$	0	0
$666$	20.2387	0.784235
$667$	− 6.15661i	− 0.238385i
$668$	− 43.2898i	− 1.67493i
$669$	12.5504	0.485228
$670$	0	0
$671$	2.46279	0.0950749
$672$	9.71820i	0.374888i
$673$	− 22.9000i	− 0.882731i	−0.897327	−	0.441365i	$-0.854494\pi$
$673$	− 22.9000i	− 0.882731i	0.897327	−	0.441365i	$-0.145506\pi$
$674$	−46.2705	−1.78227
$675$	0	0
$676$	−35.9458	−1.38253
$677$	36.0068i	1.38385i	0.721967	+	0.691927i	$0.243238\pi$
$677$	36.0068i	1.38385i	−0.721967	+	0.691927i	$0.756762\pi$
$678$	30.8974i	1.18661i
$679$	7.26915	0.278964
$680$	0	0
$681$	−1.30149	−0.0498732
$682$	− 11.0264i	− 0.422222i
$683$	− 9.12896i	− 0.349310i	−0.984630	−	0.174655i	$-0.944119\pi$
$683$	− 9.12896i	− 0.349310i	0.984630	−	0.174655i	$-0.0558810\pi$
$684$	5.36894	0.205287
$685$	0	0
$686$	15.0305	0.573869
$687$	− 3.28682i	− 0.125400i
$688$	− 166.788i	− 6.35873i
$689$	−31.6223	−1.20472
$690$	0	0
$691$	5.85956	0.222908	0.111454	−	0.993770i	$-0.464449\pi$
$691$	5.85956	0.222908	0.111454	+	0.993770i	$0.464449\pi$
$692$	45.2077i	1.71854i
$693$	0.155104i	0.00589191i
$694$	−49.2552	−1.86970
$695$	0	0
$696$	−9.92054	−0.376037
$697$	− 3.44469i	− 0.130477i
$698$	24.0868i	0.911700i
$699$	17.7115	0.669910
$700$	0	0
$701$	−7.56955	−0.285898	−0.142949	−	0.989730i	$-0.545658\pi$
$701$	−7.56955	−0.285898	−0.142949	+	0.989730i	$0.545658\pi$
$702$	− 7.07097i	− 0.266877i
$703$	− 7.04046i	− 0.265536i
$704$	14.0690	0.530244
$705$	0	0
$706$	62.3039	2.34484
$707$	− 5.04256i	− 0.189645i
$708$	− 53.4141i	− 2.00742i
$709$	−14.5209	−0.545342	−0.272671	−	0.962107i	$-0.587907\pi$
$709$	−14.5209	−0.545342	−0.272671	+	0.962107i	$0.587907\pi$
$710$	0	0
$711$	−2.06745	−0.0775356
$712$	− 157.044i	− 5.88549i
$713$	62.5302i	2.34178i
$714$	−2.25628	−0.0844393
$715$	0	0
$716$	69.4815	2.59665
$717$	− 21.2651i	− 0.794161i
$718$	− 36.3402i	− 1.35621i
$719$	−12.2164	−0.455596	−0.227798	−	0.973708i	$-0.573153\pi$
$719$	−12.2164	−0.455596	−0.227798	+	0.973708i	$0.573153\pi$
$720$	0	0
$721$	1.91788	0.0714255
$722$	49.8405i	1.85487i
$723$	15.3177i	0.569670i
$724$	13.9839	0.519709
$725$	0	0
$726$	−29.8950	−1.10951
$727$	− 40.8856i	− 1.51636i	−0.652044	−	0.758182i	$-0.726088\pi$
$727$	− 40.8856i	− 1.51636i	0.652044	−	0.758182i	$-0.273912\pi$
$728$	− 10.0219i	− 0.371438i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	21.4645	0.793892
$732$	35.0117i	1.29407i
$733$	− 25.4915i	− 0.941550i	−0.882253	−	0.470775i	$-0.843974\pi$
$733$	− 25.4915i	− 0.941550i	0.882253	−	0.470775i	$-0.156026\pi$
$734$	48.1056	1.77561
$735$	0	0
$736$	−151.920	−5.59986
$737$	2.92232i	0.107645i
$738$	− 4.56892i	− 0.168184i
$739$	−9.56830	−0.351975	−0.175988	−	0.984392i	$-0.556312\pi$
$739$	−9.56830	−0.351975	−0.175988	+	0.984392i	$0.556312\pi$
$740$	0	0
$741$	−2.45978	−0.0903624
$742$	13.3836i	0.491328i
$743$	18.0455i	0.662025i	0.943626	+	0.331013i	$0.107390\pi$
$743$	18.0455i	0.662025i	−0.943626	+	0.331013i	$0.892610\pi$
$744$	100.759	3.69400
$745$	0	0
$746$	88.7673	3.25000
$747$	6.41000i	0.234530i
$748$	4.58266i	0.167559i
$749$	0.922572	0.0337100
$750$	0	0
$751$	11.9255	0.435168	0.217584	−	0.976042i	$-0.430182\pi$
$751$	11.9255	0.435168	0.217584	+	0.976042i	$0.430182\pi$
$752$	− 187.194i	− 6.82627i
$753$	26.8917i	0.979989i
$754$	7.07097	0.257510
$755$	0	0
$756$	−2.20500	−0.0801951
$757$	5.86152i	0.213041i	0.994311	+	0.106520i	$0.0339709\pi$
$757$	5.86152i	0.213041i	−0.994311	+	0.106520i	$0.966029\pi$
$758$	89.2201i	3.24062i
$759$	−2.42467	−0.0880100
$760$	0	0
$761$	49.8739	1.80793	0.903963	−	0.427610i	$-0.140644\pi$
$761$	49.8739	1.80793	0.903963	+	0.427610i	$0.140644\pi$
$762$	55.5637i	2.01286i
$763$	− 3.36744i	− 0.121909i
$764$	35.3894	1.28034
$765$	0	0
$766$	71.3239	2.57704
$767$	24.4717i	0.883621i
$768$	63.9648i	2.30813i
$769$	−29.5121	−1.06423	−0.532117	−	0.846671i	$-0.678604\pi$
$769$	−29.5121	−1.06423	−0.532117	+	0.846671i	$0.678604\pi$
$770$	0	0
$771$	6.85512	0.246881
$772$	141.607i	5.09653i
$773$	2.41935i	0.0870180i	0.999053	+	0.0435090i	$0.0138537\pi$
$773$	2.41935i	0.0870180i	−0.999053	+	0.0435090i	$0.986146\pi$
$774$	28.4698	1.02333
$775$	0	0
$776$	183.108	6.57320
$777$	2.89149i	0.103731i
$778$	89.8934i	3.22283i
$779$	−1.58939	−0.0569460
$780$	0	0
$781$	−2.35722	−0.0843479
$782$	− 35.2715i	− 1.26131i
$783$	− 1.00000i	− 0.0357371i
$784$	110.540	3.94786
$785$	0	0
$786$	−15.1678	−0.541016
$787$	− 51.1549i	− 1.82348i	−0.410772	−	0.911738i	$-0.634741\pi$
$787$	− 51.1549i	− 1.82348i	0.410772	−	0.911738i	$-0.365259\pi$
$788$	118.722i	4.22928i
$789$	−13.6294	−0.485218
$790$	0	0
$791$	−4.41428	−0.156954
$792$	3.90703i	0.138830i
$793$	− 16.0406i	− 0.569619i
$794$	12.1162	0.429987
$795$	0	0
$796$	−14.8214	−0.525330
$797$	28.8662i	1.02249i	0.859434	+	0.511247i	$0.170816\pi$
$797$	28.8662i	1.02249i	−0.859434	+	0.511247i	$0.829184\pi$
$798$	1.04106i	0.0368531i
$799$	24.0907	0.852266
$800$	0	0
$801$	15.8302	0.559334
$802$	52.5567i	1.85584i
$803$	1.31616i	0.0464462i
$804$	−41.5446	−1.46517
$805$	0	0
$806$	−71.8171	−2.52965
$807$	8.46129i	0.297851i
$808$	− 127.021i	− 4.46858i
$809$	−19.0426	−0.669501	−0.334750	−	0.942307i	$-0.608652\pi$
$809$	−19.0426	−0.669501	−0.334750	+	0.942307i	$0.608652\pi$
$810$	0	0
$811$	−20.8783	−0.733137	−0.366569	−	0.930391i	$-0.619467\pi$
$811$	−20.8783	−0.733137	−0.366569	+	0.930391i	$0.619467\pi$
$812$	− 2.20500i	− 0.0773804i
$813$	6.34255i	0.222443i
$814$	7.97066	0.279372
$815$	0	0
$816$	−33.5631	−1.17494
$817$	− 9.90381i	− 0.346491i
$818$	− 3.81060i	− 0.133234i
$819$	1.01022	0.0353000
$820$	0	0
$821$	3.60466	0.125804	0.0629018	−	0.998020i	$-0.479965\pi$
$821$	3.60466	0.125804	0.0629018	+	0.998020i	$0.479965\pi$
$822$	20.6704i	0.720964i
$823$	− 27.2288i	− 0.949135i	−0.880219	−	0.474567i	$-0.842605\pi$
$823$	− 27.2288i	− 0.949135i	0.880219	−	0.474567i	$-0.157395\pi$
$824$	48.3109	1.68299
$825$	0	0
$826$	10.3572	0.360374
$827$	47.3936i	1.64804i	0.566562	+	0.824019i	$0.308273\pi$
$827$	47.3936i	1.64804i	−0.566562	+	0.824019i	$0.691727\pi$
$828$	− 34.4698i	− 1.19791i
$829$	41.8388	1.45312	0.726560	−	0.687103i	$-0.241118\pi$
$829$	41.8388	1.45312	0.726560	+	0.687103i	$0.241118\pi$
$830$	0	0
$831$	17.1464	0.594802
$832$	− 91.6339i	− 3.17683i
$833$	14.2258i	0.492894i
$834$	25.7818	0.892752
$835$	0	0
$836$	2.11446	0.0731302
$837$	10.1566i	0.351064i
$838$	7.91085i	0.273276i
$839$	31.6385	1.09228	0.546141	−	0.837693i	$-0.316096\pi$
$839$	31.6385	1.09228	0.546141	+	0.837693i	$0.316096\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 37.6111i	− 1.29617i
$843$	− 12.2985i	− 0.423584i
$844$	−11.1977	−0.385440
$845$	0	0
$846$	31.9531	1.09857
$847$	− 4.27107i	− 0.146756i
$848$	199.086i	6.83665i
$849$	25.1830	0.864278
$850$	0	0
$851$	−45.2013	−1.54948
$852$	− 33.5109i	− 1.14806i
$853$	− 44.2000i	− 1.51338i	−0.653773	−	0.756690i	$-0.726815\pi$
$853$	− 44.2000i	− 1.51338i	0.653773	−	0.756690i	$-0.273185\pi$
$854$	−6.78892	−0.232312
$855$	0	0
$856$	23.2394	0.794305
$857$	14.4775i	0.494541i	0.968947	+	0.247270i	$0.0795335\pi$
$857$	14.4775i	0.494541i	−0.968947	+	0.247270i	$0.920466\pi$
$858$	− 2.78478i	− 0.0950707i
$859$	−22.5883	−0.770703	−0.385352	−	0.922770i	$-0.625920\pi$
$859$	−22.5883	−0.770703	−0.385352	+	0.922770i	$0.625920\pi$
$860$	0	0
$861$	0.652757	0.0222459
$862$	32.9983i	1.12393i
$863$	22.9900i	0.782590i	0.920265	+	0.391295i	$0.127973\pi$
$863$	22.9900i	0.782590i	−0.920265	+	0.391295i	$0.872027\pi$
$864$	−24.6760	−0.839494
$865$	0	0
$866$	−49.5930	−1.68524
$867$	12.6807i	0.430658i
$868$	22.3953i	0.760147i
$869$	−0.814230	−0.0276209
$870$	0	0
$871$	19.0337	0.644931
$872$	− 84.8250i	− 2.87254i
$873$	18.4575i	0.624691i
$874$	−16.2744	−0.550491
$875$	0	0
$876$	−18.7109	−0.632182
$877$	13.5741i	0.458364i	0.973384	+	0.229182i	$0.0736051\pi$
$877$	13.5741i	0.458364i	−0.973384	+	0.229182i	$0.926395\pi$
$878$	− 44.7191i	− 1.50920i
$879$	12.3170	0.415443
$880$	0	0
$881$	9.91235	0.333956	0.166978	−	0.985961i	$-0.446599\pi$
$881$	9.91235	0.333956	0.166978	+	0.985961i	$0.446599\pi$
$882$	18.8686i	0.635340i
$883$	− 39.3192i	− 1.32320i	−0.749859	−	0.661598i	$-0.769879\pi$
$883$	− 39.3192i	− 1.32320i	0.749859	−	0.661598i	$-0.230121\pi$
$884$	29.8478	1.00389
$885$	0	0
$886$	97.5180	3.27618
$887$	− 59.2520i	− 1.98949i	−0.102403	−	0.994743i	$-0.532653\pi$
$887$	− 59.2520i	− 1.98949i	0.102403	−	0.994743i	$-0.467347\pi$
$888$	72.8358i	2.44421i
$889$	−7.93832	−0.266243
$890$	0	0
$891$	−0.393832	−0.0131939
$892$	70.2678i	2.35274i
$893$	− 11.1155i	− 0.371968i
$894$	31.8117	1.06394
$895$	0	0
$896$	−19.3461	−0.646307
$897$	15.7924i	0.527291i
$898$	114.942i	3.83567i
$899$	−10.1566	−0.338742
$900$	0	0
$901$	−25.6211	−0.853562
$902$	− 1.79939i	− 0.0599131i
$903$	4.06745i	0.135356i
$904$	−111.195	−3.69828
$905$	0	0
$906$	16.7660	0.557012
$907$	− 5.91210i	− 0.196308i	−0.995171	−	0.0981541i	$-0.968706\pi$
$907$	− 5.91210i	− 0.196308i	0.995171	−	0.0981541i	$-0.0312938\pi$
$908$	− 7.28682i	− 0.241822i
$909$	12.8038	0.424676
$910$	0	0
$911$	−12.4185	−0.411445	−0.205722	−	0.978610i	$-0.565954\pi$
$911$	−12.4185	−0.411445	−0.205722	+	0.978610i	$0.565954\pi$
$912$	15.4862i	0.512799i
$913$	2.52447i	0.0835476i
$914$	−44.4118	−1.46901
$915$	0	0
$916$	18.4024	0.608031
$917$	− 2.16700i	− 0.0715607i
$918$	− 5.72905i	− 0.189087i
$919$	−16.4332	−0.542081	−0.271041	−	0.962568i	$-0.587368\pi$
$919$	−16.4332	−0.542081	−0.271041	+	0.962568i	$0.587368\pi$
$920$	0	0
$921$	22.7379	0.749239
$922$	− 47.7983i	− 1.57415i
$923$	15.3530i	0.505351i
$924$	−0.868401	−0.0285683
$925$	0	0
$926$	−53.2510	−1.74994
$927$	4.86979i	0.159945i
$928$	− 24.6760i	− 0.810029i
$929$	−13.8358	−0.453936	−0.226968	−	0.973902i	$-0.572881\pi$
$929$	−13.8358	−0.453936	−0.226968	+	0.973902i	$0.572881\pi$
$930$	0	0
$931$	6.56384	0.215121
$932$	99.1637i	3.24822i
$933$	− 5.33810i	− 0.174762i
$934$	−22.0528	−0.721589
$935$	0	0
$936$	25.4472	0.831769
$937$	20.7528i	0.677964i	0.940793	+	0.338982i	$0.110083\pi$
$937$	20.7528i	0.677964i	−0.940793	+	0.338982i	$0.889917\pi$
$938$	− 8.05567i	− 0.263027i
$939$	−1.96338	−0.0640726
$940$	0	0
$941$	40.1666	1.30940	0.654698	−	0.755891i	$-0.272796\pi$
$941$	40.1666	1.30940	0.654698	+	0.755891i	$0.272796\pi$
$942$	− 31.9636i	− 1.04143i
$943$	10.2043i	0.332297i
$944$	154.068	5.01447
$945$	0	0
$946$	11.2123	0.364544
$947$	− 17.4144i	− 0.565894i	−0.959136	−	0.282947i	$-0.908688\pi$
$947$	− 17.4144i	− 0.565894i	0.959136	−	0.282947i	$-0.0913120\pi$
$948$	− 11.5753i	− 0.375949i
$949$	8.57239	0.278271
$950$	0	0
$951$	−1.29064	−0.0418518
$952$	− 8.11999i	− 0.263170i
$953$	− 13.5121i	− 0.437701i	−0.975758	−	0.218851i	$-0.929769\pi$
$953$	− 13.5121i	− 0.437701i	0.975758	−	0.218851i	$-0.0702307\pi$
$954$	−33.9830	−1.10024
$955$	0	0
$956$	119.060	3.85067
$957$	− 0.393832i	− 0.0127308i
$958$	− 111.333i	− 3.59700i
$959$	−2.95316	−0.0953626
$960$	0	0
$961$	72.1567	2.32763
$962$	− 51.9145i	− 1.67379i
$963$	2.34255i	0.0754876i
$964$	−85.7610	−2.76218
$965$	0	0
$966$	6.68384	0.215049
$967$	− 24.0598i	− 0.773712i	−0.922140	−	0.386856i	$-0.873561\pi$
$967$	− 24.0598i	− 0.773712i	0.922140	−	0.386856i	$-0.126439\pi$
$968$	− 107.587i	− 3.45798i
$969$	−1.99297	−0.0640233
$970$	0	0
$971$	34.7877	1.11639	0.558195	−	0.829710i	$-0.311494\pi$
$971$	34.7877	1.11639	0.558195	+	0.829710i	$0.311494\pi$
$972$	− 5.59883i	− 0.179583i
$973$	3.68342i	0.118085i
$974$	7.83344	0.251000
$975$	0	0
$976$	−100.988	−3.23254
$977$	35.7566i	1.14396i	0.820269	+	0.571978i	$0.193824\pi$
$977$	35.7566i	1.14396i	−0.820269	+	0.571978i	$0.806176\pi$
$978$	2.35722i	0.0753755i
$979$	6.23446	0.199254
$980$	0	0
$981$	8.55044	0.272995
$982$	− 0.435166i	− 0.0138867i
$983$	7.19064i	0.229346i	0.993403	+	0.114673i	$0.0365820\pi$
$983$	7.19064i	0.229346i	−0.993403	+	0.114673i	$0.963418\pi$
$984$	16.4428	0.524177
$985$	0	0
$986$	5.72905	0.182450
$987$	4.56511i	0.145309i
$988$	− 13.7719i	− 0.438143i
$989$	−63.5847	−2.02188
$990$	0	0
$991$	−19.9123	−0.632537	−0.316268	−	0.948670i	$-0.602430\pi$
$991$	−19.9123	−0.632537	−0.316268	+	0.948670i	$0.602430\pi$
$992$	250.624i	7.95733i
$993$	28.9971i	0.920194i
$994$	6.49790	0.206101
$995$	0	0
$996$	−35.8885	−1.13717
$997$	− 1.57302i	− 0.0498179i	−0.999690	−	0.0249089i	$-0.992070\pi$
$997$	− 1.57302i	− 0.0498179i	0.999690	−	0.0249089i	$-0.00792958\pi$
$998$	7.29734i	0.230993i
$999$	−7.34192	−0.232288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.n.349.1		8
5.2	odd	4		2175.2.a.v.1.4		4
5.3	odd	4		435.2.a.j.1.1	✓	4
5.4	even	2	inner	2175.2.c.n.349.8		8
15.2	even	4		6525.2.a.bi.1.1		4
15.8	even	4		1305.2.a.r.1.4		4
20.3	even	4		6960.2.a.co.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.j.1.1	✓	4	5.3	odd	4
1305.2.a.r.1.4		4	15.8	even	4
2175.2.a.v.1.4		4	5.2	odd	4
2175.2.c.n.349.1		8	1.1	even	1	trivial
2175.2.c.n.349.8		8	5.4	even	2	inner
6525.2.a.bi.1.1		4	15.2	even	4
6960.2.a.co.1.2		4	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.75660i q^{2} +1.00000i q^{3} -5.59883 q^{4} +2.75660 q^{6} +0.393832i q^{7} +9.92054i q^{8} -1.00000 q^{9} -0.393832 q^{11} -5.59883i q^{12} +2.56511i q^{13} +1.08564 q^{14} +16.1493 q^{16} +2.07830i q^{17} +2.75660i q^{18} +0.958939 q^{19} -0.393832 q^{21} +1.08564i q^{22} -6.15661i q^{23} -9.92054 q^{24} +7.07097 q^{26} -1.00000i q^{27} -2.20500i q^{28} +1.00000 q^{29} -10.1566 q^{31} -24.6760i q^{32} -0.393832i q^{33} +5.72905 q^{34} +5.59883 q^{36} -7.34192i q^{37} -2.64341i q^{38} -2.56511 q^{39} -1.65745 q^{41} +1.08564i q^{42} -10.3279i q^{43} +2.20500 q^{44} -16.9713 q^{46} -11.5915i q^{47} +16.1493i q^{48} +6.84490 q^{49} -2.07830 q^{51} -14.3616i q^{52} +12.3279i q^{53} -2.75660 q^{54} -3.90703 q^{56} +0.958939i q^{57} -2.75660i q^{58} +9.54022 q^{59} -6.25340 q^{61} +27.9977i q^{62} -0.393832i q^{63} -35.7232 q^{64} -1.08564 q^{66} -7.42023i q^{67} -11.6361i q^{68} +6.15661 q^{69} +5.98533 q^{71} -9.92054i q^{72} -3.34192i q^{73} -20.2387 q^{74} -5.36894 q^{76} -0.155104i q^{77} +7.07097i q^{78} +2.06745 q^{79} +1.00000 q^{81} +4.56892i q^{82} -6.41000i q^{83} +2.20500 q^{84} -28.4698 q^{86} +1.00000i q^{87} -3.90703i q^{88} -15.8302 q^{89} -1.01022 q^{91} +34.4698i q^{92} -10.1566i q^{93} -31.9531 q^{94} +24.6760 q^{96} -18.4575i q^{97} -18.8686i q^{98} +0.393832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{11} + 6 q^{14} + 22 q^{16} + 4 q^{19} - 4 q^{21} - 24 q^{24} - 14 q^{26} + 8 q^{29} - 8 q^{31} + 2 q^{34} + 10 q^{36} - 16 q^{39} - 24 q^{41} - 18 q^{44} - 16 q^{46}+ \cdots + 4 q^{99}+O(q^{100}) \) 8 * q - 10 * q^4 + 6 * q^6 - 8 * q^9 - 4 * q^11 + 6 * q^14 + 22 * q^16 + 4 * q^19 - 4 * q^21 - 24 * q^24 - 14 * q^26 + 8 * q^29 - 8 * q^31 + 2 * q^34 + 10 * q^36 - 16 * q^39 - 24 * q^41 - 18 * q^44 - 16 * q^46 - 12 * q^49 + 20 * q^51 - 6 * q^54 - 4 * q^59 - 52 * q^61 - 68 * q^64 - 6 * q^66 - 24 * q^69 - 20 * q^71 - 96 * q^74 + 32 * q^76 - 44 * q^79 + 8 * q^81 - 18 * q^84 - 8 * q^86 + 8 * q^89 - 16 * q^91 - 78 * q^94 + 34 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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