LMFDB - Embedded newform 2175.2.c.m.349.6

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 = [6,0,0,-10,0,-2,0,0,-6,0,18] 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:	$6$
Coefficient field:	6.0.14077504.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 11x^{4} + 33x^{2} + 16 $ x^6 + 11x^4 + 33x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.6
Root			$2.39138i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.m.349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.39138i q^{2} +1.00000i q^{3} -3.71871 q^{4} -2.39138 q^{6} -1.32733i q^{7} -4.11009i q^{8} -1.00000 q^{9} +3.00000 q^{11} -3.71871i q^{12} -6.11009i q^{13} +3.17415 q^{14} +2.39138 q^{16} +0.672673i q^{17} -2.39138i q^{18} +5.43742 q^{19} +1.32733 q^{21} +7.17415i q^{22} +7.89286i q^{23} +4.11009 q^{24} +14.6116 q^{26} -1.00000i q^{27} +4.93594i q^{28} +1.00000 q^{29} -2.50147i q^{32} +3.00000i q^{33} -1.60862 q^{34} +3.71871 q^{36} -3.89286i q^{37} +13.0029i q^{38} +6.11009 q^{39} +4.32733 q^{41} +3.17415i q^{42} +8.45544i q^{43} -11.1561 q^{44} -18.8748 q^{46} +4.76475i q^{47} +2.39138i q^{48} +5.23820 q^{49} -0.672673 q^{51} +22.7217i q^{52} +13.3303i q^{53} +2.39138 q^{54} -5.45544 q^{56} +5.43742i q^{57} +2.39138i q^{58} -3.21724 q^{59} +7.43742 q^{61} +1.32733i q^{63} +10.7647 q^{64} -7.17415 q^{66} -1.10714i q^{67} -2.50147i q^{68} -7.89286 q^{69} +12.3483 q^{71} +4.11009i q^{72} -9.20217i q^{73} +9.30931 q^{74} -20.2202 q^{76} -3.98198i q^{77} +14.6116i q^{78} -11.6576 q^{79} +1.00000 q^{81} +10.3483i q^{82} +0.889908i q^{83} -4.93594 q^{84} -20.2202 q^{86} +1.00000i q^{87} -12.3303i q^{88} +2.33028 q^{89} -8.11009 q^{91} -29.3512i q^{92} -11.3943 q^{94} +2.50147 q^{96} -12.6756i q^{97} +12.5265i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 18 q^{11} - 18 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{21} + 26 q^{26} + 6 q^{29} - 22 q^{34} + 10 q^{36} + 12 q^{39} + 26 q^{41} - 30 q^{44} - 64 q^{46} - 18 q^{49} - 4 q^{51}+ \cdots - 18 q^{99}+O(q^{100}) $ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 18 * q^11 - 18 * q^14 + 2 * q^16 + 8 * q^19 + 8 * q^21 + 26 * q^26 + 6 * q^29 - 22 * q^34 + 10 * q^36 + 12 * q^39 + 26 * q^41 - 30 * q^44 - 64 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 44 * q^59 + 20 * q^61 + 40 * q^64 - 6 * q^66 + 2 * q^69 + 56 * q^74 - 72 * q^76 + 4 * q^79 + 6 * q^81 - 42 * q^84 - 72 * q^86 - 60 * q^89 - 24 * q^91 + 18 * q^94 - 22 * q^96 - 18 * 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 $

$2$	2.39138i	1.69096i	0.534005	+	0.845481i	$0.320686\pi$
$2$	2.39138i	1.69096i	−0.534005	+	0.845481i	$0.679314\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−3.71871	−1.85935
$5$	0	0
$5$	0	0
$6$	−2.39138	−0.976278
$7$	− 1.32733i	− 0.501683i	−0.968028	−	0.250841i	$-0.919293\pi$
$7$	− 1.32733i	− 0.501683i	0.968028	−	0.250841i	$-0.0807072\pi$
$8$	− 4.11009i	− 1.45314i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	− 3.71871i	− 1.07350i
$13$	− 6.11009i	− 1.69463i	−0.531087	−	0.847317i	$-0.678216\pi$
$13$	− 6.11009i	− 1.69463i	0.531087	−	0.847317i	$-0.321784\pi$
$14$	3.17415	0.848327
$15$	0	0
$16$	2.39138	0.597846
$17$	0.672673i	0.163147i	0.996667	+	0.0815735i	$0.0259945\pi$
$17$	0.672673i	0.163147i	−0.996667	+	0.0815735i	$0.974005\pi$
$18$	− 2.39138i	− 0.563654i
$19$	5.43742	1.24743	0.623715	−	0.781652i	$-0.285623\pi$
$19$	5.43742	1.24743	0.623715	+	0.781652i	$0.285623\pi$
$20$	0	0
$21$	1.32733	0.289647
$22$	7.17415i	1.52953i
$23$	7.89286i	1.64577i	0.568205	+	0.822887i	$0.307638\pi$
$23$	7.89286i	1.64577i	−0.568205	+	0.822887i	$0.692362\pi$
$24$	4.11009	0.838969
$25$	0	0
$26$	14.6116	2.86556
$27$	− 1.00000i	− 0.192450i
$28$	4.93594i	0.932806i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 2.50147i	− 0.442202i
$33$	3.00000i	0.522233i
$34$	−1.60862	−0.275876
$35$	0	0
$36$	3.71871	0.619785
$37$	− 3.89286i	− 0.639982i	−0.947421	−	0.319991i	$-0.896320\pi$
$37$	− 3.89286i	− 0.639982i	0.947421	−	0.319991i	$-0.103680\pi$
$38$	13.0029i	2.10936i
$39$	6.11009	0.978398
$40$	0	0
$41$	4.32733	0.675815	0.337907	−	0.941179i	$-0.390281\pi$
$41$	4.32733	0.675815	0.337907	+	0.941179i	$0.390281\pi$
$42$	3.17415i	0.489782i
$43$	8.45544i	1.28944i	0.764418	+	0.644721i	$0.223026\pi$
$43$	8.45544i	1.28944i	−0.764418	+	0.644721i	$0.776974\pi$
$44$	−11.1561	−1.68185
$45$	0	0
$46$	−18.8748	−2.78294
$47$	4.76475i	0.695010i	0.937678	+	0.347505i	$0.112971\pi$
$47$	4.76475i	0.695010i	−0.937678	+	0.347505i	$0.887029\pi$
$48$	2.39138i	0.345166i
$49$	5.23820	0.748315
$50$	0	0
$51$	−0.672673	−0.0941930
$52$	22.7217i	3.15093i
$53$	13.3303i	1.83105i	0.402256	+	0.915527i	$0.368226\pi$
$53$	13.3303i	1.83105i	−0.402256	+	0.915527i	$0.631774\pi$
$54$	2.39138	0.325426
$55$	0	0
$56$	−5.45544	−0.729013
$57$	5.43742i	0.720204i
$58$	2.39138i	0.314004i
$59$	−3.21724	−0.418848	−0.209424	−	0.977825i	$-0.567159\pi$
$59$	−3.21724	−0.418848	−0.209424	+	0.977825i	$0.567159\pi$
$60$	0	0
$61$	7.43742	0.952264	0.476132	−	0.879374i	$-0.342038\pi$
$61$	7.43742	0.952264	0.476132	+	0.879374i	$0.342038\pi$
$62$	0	0
$63$	1.32733i	0.167228i
$64$	10.7647	1.34559
$65$	0	0
$66$	−7.17415	−0.883076
$67$	− 1.10714i	− 0.135259i	−0.997711	−	0.0676295i	$-0.978456\pi$
$67$	− 1.10714i	− 0.135259i	0.997711	−	0.0676295i	$-0.0215436\pi$
$68$	− 2.50147i	− 0.303348i
$69$	−7.89286	−0.950188
$70$	0	0
$71$	12.3483	1.46547	0.732736	−	0.680513i	$-0.238243\pi$
$71$	12.3483	1.46547	0.732736	+	0.680513i	$0.238243\pi$
$72$	4.11009i	0.484379i
$73$	− 9.20217i	− 1.07703i	−0.842615	−	0.538516i	$-0.818985\pi$
$73$	− 9.20217i	− 1.07703i	0.842615	−	0.538516i	$-0.181015\pi$
$74$	9.30931	1.08219
$75$	0	0
$76$	−20.2202	−2.31941
$77$	− 3.98198i	− 0.453789i
$78$	14.6116i	1.65443i
$79$	−11.6576	−1.31158	−0.655791	−	0.754942i	$-0.727665\pi$
$79$	−11.6576	−1.31158	−0.655791	+	0.754942i	$0.727665\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.3483i	1.14278i
$83$	0.889908i	0.0976801i	0.998807	+	0.0488400i	$0.0155525\pi$
$83$	0.889908i	0.0976801i	−0.998807	+	0.0488400i	$0.984448\pi$
$84$	−4.93594	−0.538556
$85$	0	0
$86$	−20.2202	−2.18040
$87$	1.00000i	0.107211i
$88$	− 12.3303i	− 1.31441i
$89$	2.33028	0.247009	0.123504	−	0.992344i	$-0.460587\pi$
$89$	2.33028	0.247009	0.123504	+	0.992344i	$0.460587\pi$
$90$	0	0
$91$	−8.11009	−0.850169
$92$	− 29.3512i	− 3.06008i
$93$	0	0
$94$	−11.3943	−1.17524
$95$	0	0
$96$	2.50147	0.255306
$97$	− 12.6756i	− 1.28701i	−0.765440	−	0.643507i	$-0.777479\pi$
$97$	− 12.6756i	− 1.28701i	0.765440	−	0.643507i	$-0.222521\pi$
$98$	12.5265i	1.26537i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	3.56258	0.354490	0.177245	−	0.984167i	$-0.443281\pi$
$101$	3.56258	0.354490	0.177245	+	0.984167i	$0.443281\pi$
$102$	− 1.60862i	− 0.159277i
$103$	12.9109i	1.27215i	0.771629	+	0.636073i	$0.219442\pi$
$103$	12.9109i	1.27215i	−0.771629	+	0.636073i	$0.780558\pi$
$104$	−25.1130	−2.46254
$105$	0	0
$106$	−31.8778	−3.09624
$107$	4.22018i	0.407981i	0.978973	+	0.203990i	$0.0653911\pi$
$107$	4.22018i	0.407981i	−0.978973	+	0.203990i	$0.934609\pi$
$108$	3.71871i	0.357833i
$109$	11.8748	1.13740	0.568702	−	0.822544i	$-0.307446\pi$
$109$	11.8748	1.13740	0.568702	+	0.822544i	$0.307446\pi$
$110$	0	0
$111$	3.89286	0.369494
$112$	− 3.17415i	− 0.299929i
$113$	− 14.2382i	− 1.33942i	−0.742624	−	0.669709i	$-0.766419\pi$
$113$	− 14.2382i	− 1.33942i	0.742624	−	0.669709i	$-0.233581\pi$
$114$	−13.0029	−1.21784
$115$	0	0
$116$	−3.71871	−0.345274
$117$	6.11009i	0.564878i
$118$	− 7.69364i	− 0.708257i
$119$	0.892857	0.0818481
$120$	0	0
$121$	−2.00000	−0.181818
$122$	17.7857i	1.61024i
$123$	4.32733i	0.390182i
$124$	0	0
$125$	0	0
$126$	−3.17415	−0.282776
$127$	− 7.20217i	− 0.639089i	−0.947571	−	0.319544i	$-0.896470\pi$
$127$	− 7.20217i	− 0.639089i	0.947571	−	0.319544i	$-0.103530\pi$
$128$	20.7397i	1.83315i
$129$	−8.45544	−0.744460
$130$	0	0
$131$	14.2382	1.24400	0.621999	−	0.783018i	$-0.286321\pi$
$131$	14.2382	1.24400	0.621999	+	0.783018i	$0.286321\pi$
$132$	− 11.1561i	− 0.971016i
$133$	− 7.21724i	− 0.625814i
$134$	2.64760	0.228718
$135$	0	0
$136$	2.76475	0.237075
$137$	15.0950i	1.28965i	0.764328	+	0.644827i	$0.223071\pi$
$137$	15.0950i	1.28965i	−0.764328	+	0.644827i	$0.776929\pi$
$138$	− 18.8748i	− 1.60673i
$139$	6.43742	0.546015	0.273007	−	0.962012i	$-0.411982\pi$
$139$	6.43742	0.546015	0.273007	+	0.962012i	$0.411982\pi$
$140$	0	0
$141$	−4.76475	−0.401264
$142$	29.5295i	2.47806i
$143$	− 18.3303i	− 1.53285i
$144$	−2.39138	−0.199282
$145$	0	0
$146$	22.0059	1.82122
$147$	5.23820i	0.432040i
$148$	14.4764i	1.18995i
$149$	−1.21724	−0.0997198	−0.0498599	−	0.998756i	$-0.515877\pi$
$149$	−1.21724	−0.0997198	−0.0498599	+	0.998756i	$0.515877\pi$
$150$	0	0
$151$	−2.32733	−0.189395	−0.0946976	−	0.995506i	$-0.530188\pi$
$151$	−2.32733	−0.189395	−0.0946976	+	0.995506i	$0.530188\pi$
$152$	− 22.3483i	− 1.81269i
$153$	− 0.672673i	− 0.0543824i
$154$	9.52244	0.767340
$155$	0	0
$156$	−22.7217	−1.81919
$157$	9.43742i	0.753188i	0.926378	+	0.376594i	$0.122905\pi$
$157$	9.43742i	0.753188i	−0.926378	+	0.376594i	$0.877095\pi$
$158$	− 27.8778i	− 2.21784i
$159$	−13.3303	−1.05716
$160$	0	0
$161$	10.4764	0.825656
$162$	2.39138i	0.187885i
$163$	21.3303i	1.67072i	0.549706	+	0.835358i	$0.314740\pi$
$163$	21.3303i	1.67072i	−0.549706	+	0.835358i	$0.685260\pi$
$164$	−16.0921	−1.25658
$165$	0	0
$166$	−2.12811	−0.165173
$167$	5.30931i	0.410847i	0.978673	+	0.205423i	$0.0658571\pi$
$167$	5.30931i	0.410847i	−0.978673	+	0.205423i	$0.934143\pi$
$168$	− 5.45544i	− 0.420896i
$169$	−24.3332	−1.87179
$170$	0	0
$171$	−5.43742	−0.415810
$172$	− 31.4433i	− 2.39753i
$173$	− 22.7677i	− 1.73100i	−0.500913	−	0.865498i	$-0.667002\pi$
$173$	− 22.7677i	− 1.73100i	0.500913	−	0.865498i	$-0.332998\pi$
$174$	−2.39138	−0.181290
$175$	0	0
$176$	7.17415	0.540772
$177$	− 3.21724i	− 0.241822i
$178$	5.57258i	0.417683i
$179$	−17.6576	−1.31979	−0.659896	−	0.751357i	$-0.729400\pi$
$179$	−17.6576	−1.31979	−0.659896	+	0.751357i	$0.729400\pi$
$180$	0	0
$181$	10.5655	0.785330	0.392665	−	0.919682i	$-0.371553\pi$
$181$	10.5655	0.785330	0.392665	+	0.919682i	$0.371553\pi$
$182$	− 19.3943i	− 1.43760i
$183$	7.43742i	0.549790i
$184$	32.4404	2.39154
$185$	0	0
$186$	0	0
$187$	2.01802i	0.147572i
$188$	− 17.7187i	− 1.29227i
$189$	−1.32733	−0.0965489
$190$	0	0
$191$	−3.23820	−0.234308	−0.117154	−	0.993114i	$-0.537377\pi$
$191$	−3.23820	−0.234308	−0.117154	+	0.993114i	$0.537377\pi$
$192$	10.7647i	0.776879i
$193$	3.30931i	0.238209i	0.992882	+	0.119105i	$0.0380024\pi$
$193$	3.30931i	0.238209i	−0.992882	+	0.119105i	$0.961998\pi$
$194$	30.3123	2.17629
$195$	0	0
$196$	−19.4794	−1.39138
$197$	− 1.89286i	− 0.134860i	−0.997724	−	0.0674302i	$-0.978520\pi$
$197$	− 1.89286i	− 0.134860i	0.997724	−	0.0674302i	$-0.0214800\pi$
$198$	− 7.17415i	− 0.509844i
$199$	5.56258	0.394321	0.197160	−	0.980371i	$-0.436828\pi$
$199$	5.56258	0.394321	0.197160	+	0.980371i	$0.436828\pi$
$200$	0	0
$201$	1.10714	0.0780919
$202$	8.51949i	0.599429i
$203$	− 1.32733i	− 0.0931601i
$204$	2.50147	0.175138
$205$	0	0
$206$	−30.8748	−2.15115
$207$	− 7.89286i	− 0.548591i
$208$	− 14.6116i	− 1.01313i
$209$	16.3123	1.12834
$210$	0	0
$211$	15.7497	1.08425	0.542126	−	0.840297i	$-0.317619\pi$
$211$	15.7497	1.08425	0.542126	+	0.840297i	$0.317619\pi$
$212$	− 49.5714i	− 3.40458i
$213$	12.3483i	0.846091i
$214$	−10.0921	−0.689880
$215$	0	0
$216$	−4.11009	−0.279656
$217$	0	0
$218$	28.3973i	1.92331i
$219$	9.20217	0.621825
$220$	0	0
$221$	4.11009	0.276475
$222$	9.30931i	0.624800i
$223$	2.01802i	0.135136i	0.997715	+	0.0675682i	$0.0215240\pi$
$223$	2.01802i	0.135136i	−0.997715	+	0.0675682i	$0.978476\pi$
$224$	−3.32028	−0.221845
$225$	0	0
$226$	34.0490	2.26490
$227$	− 20.6756i	− 1.37229i	−0.727465	−	0.686145i	$-0.759302\pi$
$227$	− 20.6756i	− 1.37229i	0.727465	−	0.686145i	$-0.240698\pi$
$228$	− 20.2202i	− 1.33911i
$229$	−14.2562	−0.942078	−0.471039	−	0.882113i	$-0.656121\pi$
$229$	−14.2562	−0.942078	−0.471039	+	0.882113i	$0.656121\pi$
$230$	0	0
$231$	3.98198	0.261995
$232$	− 4.11009i	− 0.269841i
$233$	− 20.7677i	− 1.36054i	−0.732963	−	0.680268i	$-0.761863\pi$
$233$	− 20.7677i	− 1.36054i	0.732963	−	0.680268i	$-0.238137\pi$
$234$	−14.6116	−0.955188
$235$	0	0
$236$	11.9640	0.778788
$237$	− 11.6576i	− 0.757243i
$238$	2.13516i	0.138402i
$239$	−3.30931	−0.214061	−0.107031	−	0.994256i	$-0.534134\pi$
$239$	−3.30931	−0.214061	−0.107031	+	0.994256i	$0.534134\pi$
$240$	0	0
$241$	−8.78571	−0.565938	−0.282969	−	0.959129i	$-0.591319\pi$
$241$	−8.78571	−0.565938	−0.282969	+	0.959129i	$0.591319\pi$
$242$	− 4.78276i	− 0.307448i
$243$	1.00000i	0.0641500i
$244$	−27.6576	−1.77060
$245$	0	0
$246$	−10.3483	−0.659783
$247$	− 33.2231i	− 2.11394i
$248$	0	0
$249$	−0.889908	−0.0563956
$250$	0	0
$251$	23.8037	1.50248	0.751239	−	0.660030i	$-0.229457\pi$
$251$	23.8037	1.50248	0.751239	+	0.660030i	$0.229457\pi$
$252$	− 4.93594i	− 0.310935i
$253$	23.6786i	1.48866i
$254$	17.2231	1.08068
$255$	0	0
$256$	−28.0670	−1.75419
$257$	− 27.3303i	− 1.70482i	−0.522877	−	0.852408i	$-0.675141\pi$
$257$	− 27.3303i	− 1.70482i	0.522877	−	0.852408i	$-0.324859\pi$
$258$	− 20.2202i	− 1.25885i
$259$	−5.16710	−0.321068
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	34.0490i	2.10355i
$263$	21.7497i	1.34114i	0.741845	+	0.670571i	$0.233951\pi$
$263$	21.7497i	1.34114i	−0.741845	+	0.670571i	$0.766049\pi$
$264$	12.3303	0.758876
$265$	0	0
$266$	17.2592	1.05823
$267$	2.33028i	0.142611i
$268$	4.11714i	0.251495i
$269$	−10.3303	−0.629848	−0.314924	−	0.949117i	$-0.601979\pi$
$269$	−10.3303	−0.629848	−0.314924	+	0.949117i	$0.601979\pi$
$270$	0	0
$271$	−6.25622	−0.380038	−0.190019	−	0.981780i	$-0.560855\pi$
$271$	−6.25622	−0.380038	−0.190019	+	0.981780i	$0.560855\pi$
$272$	1.60862i	0.0975368i
$273$	− 8.11009i	− 0.490845i
$274$	−36.0980	−2.18076
$275$	0	0
$276$	29.3512	1.76674
$277$	31.6815i	1.90356i	0.306784	+	0.951779i	$0.400747\pi$
$277$	31.6815i	1.90356i	−0.306784	+	0.951779i	$0.599253\pi$
$278$	15.3943i	0.923291i
$279$	0	0
$280$	0	0
$281$	27.9138	1.66520	0.832600	−	0.553875i	$-0.186852\pi$
$281$	27.9138	1.66520	0.832600	+	0.553875i	$0.186852\pi$
$282$	− 11.3943i	− 0.678523i
$283$	− 23.5295i	− 1.39868i	−0.714788	−	0.699342i	$-0.753477\pi$
$283$	− 23.5295i	− 1.39868i	0.714788	−	0.699342i	$-0.246523\pi$
$284$	−45.9197	−2.72483
$285$	0	0
$286$	43.8347	2.59200
$287$	− 5.74378i	− 0.339045i
$288$	2.50147i	0.147401i
$289$	16.5475	0.973383
$290$	0	0
$291$	12.6756	0.743058
$292$	34.2202i	2.00258i
$293$	6.23820i	0.364440i	0.983258	+	0.182220i	$0.0583283\pi$
$293$	6.23820i	0.364440i	−0.983258	+	0.182220i	$0.941672\pi$
$294$	−12.5265	−0.730563
$295$	0	0
$296$	−16.0000	−0.929981
$297$	− 3.00000i	− 0.174078i
$298$	− 2.91087i	− 0.168622i
$299$	48.2261	2.78899
$300$	0	0
$301$	11.2231	0.646891
$302$	− 5.56553i	− 0.320260i
$303$	3.56258i	0.204665i
$304$	13.0029	0.745770
$305$	0	0
$306$	1.60862	0.0919585
$307$	3.45249i	0.197044i	0.995135	+	0.0985220i	$0.0314115\pi$
$307$	3.45249i	0.197044i	−0.995135	+	0.0985220i	$0.968589\pi$
$308$	14.8078i	0.843755i
$309$	−12.9109	−0.734474
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	− 25.1130i	− 1.42175i
$313$	32.9908i	1.86475i	0.361490	+	0.932376i	$0.382268\pi$
$313$	32.9908i	1.86475i	−0.361490	+	0.932376i	$0.617732\pi$
$314$	−22.5685	−1.27361
$315$	0	0
$316$	43.3512	2.43870
$317$	4.63664i	0.260419i	0.991486	+	0.130210i	$0.0415651\pi$
$317$	4.63664i	0.260419i	−0.991486	+	0.130210i	$0.958435\pi$
$318$	− 31.8778i	− 1.78762i
$319$	3.00000	0.167968
$320$	0	0
$321$	−4.22018	−0.235548
$322$	25.0531i	1.39615i
$323$	3.65760i	0.203515i
$324$	−3.71871	−0.206595
$325$	0	0
$326$	−51.0088	−2.82512
$327$	11.8748i	0.656680i
$328$	− 17.7857i	− 0.982052i
$329$	6.32438	0.348674
$330$	0	0
$331$	−22.2261	−1.22166	−0.610828	−	0.791763i	$-0.709163\pi$
$331$	−22.2261	−1.22166	−0.610828	+	0.791763i	$0.709163\pi$
$332$	− 3.30931i	− 0.181622i
$333$	3.89286i	0.213327i
$334$	−12.6966	−0.694726
$335$	0	0
$336$	3.17415	0.173164
$337$	− 27.5655i	− 1.50159i	−0.660535	−	0.750795i	$-0.729671\pi$
$337$	− 27.5655i	− 1.50159i	0.660535	−	0.750795i	$-0.270329\pi$
$338$	− 58.1900i	− 3.16512i
$339$	14.2382	0.773313
$340$	0	0
$341$	0	0
$342$	− 13.0029i	− 0.703119i
$343$	− 16.2441i	− 0.877099i
$344$	34.7526	1.87374
$345$	0	0
$346$	54.4463	2.92705
$347$	− 22.3273i	− 1.19859i	−0.800527	−	0.599297i	$-0.795447\pi$
$347$	− 22.3273i	− 1.19859i	0.800527	−	0.599297i	$-0.204553\pi$
$348$	− 3.71871i	− 0.199344i
$349$	20.5115	1.09795	0.548977	−	0.835837i	$-0.315017\pi$
$349$	20.5115	1.09795	0.548977	+	0.835837i	$0.315017\pi$
$350$	0	0
$351$	−6.11009	−0.326133
$352$	− 7.50442i	− 0.399987i
$353$	28.8748i	1.53685i	0.639938	+	0.768426i	$0.278960\pi$
$353$	28.8748i	1.53685i	−0.639938	+	0.768426i	$0.721040\pi$
$354$	7.69364	0.408912
$355$	0	0
$356$	−8.66562	−0.459277
$357$	0.892857i	0.0472550i
$358$	− 42.2261i	− 2.23172i
$359$	−2.76770	−0.146073	−0.0730367	−	0.997329i	$-0.523269\pi$
$359$	−2.76770	−0.146073	−0.0730367	+	0.997329i	$0.523269\pi$
$360$	0	0
$361$	10.5655	0.556081
$362$	25.2662i	1.32796i
$363$	− 2.00000i	− 0.104973i
$364$	30.1591	1.58077
$365$	0	0
$366$	−17.7857	−0.929674
$367$	25.4584i	1.32892i	0.747325	+	0.664458i	$0.231338\pi$
$367$	25.4584i	1.32892i	−0.747325	+	0.664458i	$0.768662\pi$
$368$	18.8748i	0.983919i
$369$	−4.32733	−0.225272
$370$	0	0
$371$	17.6936	0.918608
$372$	0	0
$373$	4.03604i	0.208978i	0.994526	+	0.104489i	$0.0333207\pi$
$373$	4.03604i	0.208978i	−0.994526	+	0.104489i	$0.966679\pi$
$374$	−4.82585	−0.249539
$375$	0	0
$376$	19.5835	1.00994
$377$	− 6.11009i	− 0.314686i
$378$	− 3.17415i	− 0.163261i
$379$	−20.4043	−1.04810	−0.524050	−	0.851687i	$-0.675580\pi$
$379$	−20.4043	−1.04810	−0.524050	+	0.851687i	$0.675580\pi$
$380$	0	0
$381$	7.20217	0.368978
$382$	− 7.74378i	− 0.396206i
$383$	5.80078i	0.296406i	0.988957	+	0.148203i	$0.0473489\pi$
$383$	5.80078i	0.296406i	−0.988957	+	0.148203i	$0.952651\pi$
$384$	−20.7397	−1.05837
$385$	0	0
$386$	−7.91382	−0.402803
$387$	− 8.45544i	− 0.429814i
$388$	47.1370i	2.39302i
$389$	32.2592	1.63560	0.817802	−	0.575499i	$-0.195192\pi$
$389$	32.2592	1.63560	0.817802	+	0.575499i	$0.195192\pi$
$390$	0	0
$391$	−5.30931	−0.268503
$392$	− 21.5295i	− 1.08740i
$393$	14.2382i	0.718222i
$394$	4.52654	0.228044
$395$	0	0
$396$	11.1561	0.560617
$397$	22.1841i	1.11339i	0.830717	+	0.556695i	$0.187931\pi$
$397$	22.1841i	1.11339i	−0.830717	+	0.556695i	$0.812069\pi$
$398$	13.3023i	0.666782i
$399$	7.21724	0.361314
$400$	0	0
$401$	−38.6045	−1.92782	−0.963909	−	0.266233i	$-0.914221\pi$
$401$	−38.6045	−1.92782	−0.963909	+	0.266233i	$0.914221\pi$
$402$	2.64760i	0.132050i
$403$	0	0
$404$	−13.2482	−0.659123
$405$	0	0
$406$	3.17415	0.157530
$407$	− 11.6786i	− 0.578885i
$408$	2.76475i	0.136875i
$409$	−28.0980	−1.38936	−0.694678	−	0.719321i	$-0.744453\pi$
$409$	−28.0980	−1.38936	−0.694678	+	0.719321i	$0.744453\pi$
$410$	0	0
$411$	−15.0950	−0.744583
$412$	− 48.0118i	− 2.36537i
$413$	4.27032i	0.210129i
$414$	18.8748	0.927648
$415$	0	0
$416$	−15.2842	−0.749371
$417$	6.43742i	0.315242i
$418$	39.0088i	1.90799i
$419$	−3.56553	−0.174188	−0.0870938	−	0.996200i	$-0.527758\pi$
$419$	−3.56553	−0.174188	−0.0870938	+	0.996200i	$0.527758\pi$
$420$	0	0
$421$	−17.6216	−0.858823	−0.429411	−	0.903109i	$-0.641279\pi$
$421$	−17.6216	−0.858823	−0.429411	+	0.903109i	$0.641279\pi$
$422$	37.6635i	1.83343i
$423$	− 4.76475i	− 0.231670i
$424$	54.7887	2.66077
$425$	0	0
$426$	−29.5295	−1.43071
$427$	− 9.87189i	− 0.477734i
$428$	− 15.6936i	− 0.758581i
$429$	18.3303	0.884994
$430$	0	0
$431$	−33.7916	−1.62768	−0.813842	−	0.581086i	$-0.802628\pi$
$431$	−33.7916	−1.62768	−0.813842	+	0.581086i	$0.802628\pi$
$432$	− 2.39138i	− 0.115055i
$433$	− 23.3663i	− 1.12291i	−0.827506	−	0.561457i	$-0.810241\pi$
$433$	− 23.3663i	− 1.12291i	0.827506	−	0.561457i	$-0.189759\pi$
$434$	0	0
$435$	0	0
$436$	−44.1591	−2.11484
$437$	42.9168i	2.05299i
$438$	22.0059i	1.05148i
$439$	16.1461	0.770613	0.385306	−	0.922789i	$-0.374096\pi$
$439$	16.1461	0.770613	0.385306	+	0.922789i	$0.374096\pi$
$440$	0	0
$441$	−5.23820	−0.249438
$442$	9.82880i	0.467508i
$443$	− 9.38337i	− 0.445817i	−0.974839	−	0.222909i	$-0.928445\pi$
$443$	− 9.38337i	− 0.445817i	0.974839	−	0.222909i	$-0.0715551\pi$
$444$	−14.4764	−0.687020
$445$	0	0
$446$	−4.82585	−0.228511
$447$	− 1.21724i	− 0.0575733i
$448$	− 14.2883i	− 0.675061i
$449$	−17.5685	−0.829108	−0.414554	−	0.910025i	$-0.636062\pi$
$449$	−17.5685	−0.829108	−0.414554	+	0.910025i	$0.636062\pi$
$450$	0	0
$451$	12.9820	0.611298
$452$	52.9477i	2.49045i
$453$	− 2.32733i	− 0.109347i
$454$	49.4433	2.32049
$455$	0	0
$456$	22.3483	1.04655
$457$	− 17.4194i	− 0.814845i	−0.913240	−	0.407423i	$-0.866428\pi$
$457$	− 17.4194i	− 0.814845i	0.913240	−	0.407423i	$-0.133572\pi$
$458$	− 34.0921i	− 1.59302i
$459$	0.672673	0.0313977
$460$	0	0
$461$	−0.113041	−0.00526484	−0.00263242	−	0.999997i	$-0.500838\pi$
$461$	−0.113041	−0.00526484	−0.00263242	+	0.999997i	$0.500838\pi$
$462$	9.52244i	0.443024i
$463$	26.8568i	1.24814i	0.781368	+	0.624071i	$0.214522\pi$
$463$	26.8568i	1.24814i	−0.781368	+	0.624071i	$0.785478\pi$
$464$	2.39138	0.111017
$465$	0	0
$466$	49.6635	2.30062
$467$	0.726727i	0.0336289i	0.999859	+	0.0168145i	$0.00535246\pi$
$467$	0.726727i	0.0336289i	−0.999859	+	0.0168145i	$0.994648\pi$
$468$	− 22.7217i	− 1.05031i
$469$	−1.46954	−0.0678571
$470$	0	0
$471$	−9.43742	−0.434853
$472$	13.2231i	0.608644i
$473$	25.3663i	1.16634i
$474$	27.8778	1.28047
$475$	0	0
$476$	−3.32028	−0.152185
$477$	− 13.3303i	− 0.610351i
$478$	− 7.91382i	− 0.361970i
$479$	9.12516	0.416939	0.208470	−	0.978029i	$-0.433152\pi$
$479$	9.12516	0.416939	0.208470	+	0.978029i	$0.433152\pi$
$480$	0	0
$481$	−23.7857	−1.08454
$482$	− 21.0100i	− 0.956979i
$483$	10.4764i	0.476693i
$484$	7.43742	0.338065
$485$	0	0
$486$	−2.39138	−0.108475
$487$	35.5714i	1.61190i	0.591987	+	0.805948i	$0.298344\pi$
$487$	35.5714i	1.61190i	−0.591987	+	0.805948i	$0.701656\pi$
$488$	− 30.5685i	− 1.38377i
$489$	−21.3303	−0.964588
$490$	0	0
$491$	−4.44037	−0.200391	−0.100196	−	0.994968i	$-0.531947\pi$
$491$	−4.44037	−0.200391	−0.100196	+	0.994968i	$0.531947\pi$
$492$	− 16.0921i	− 0.725487i
$493$	0.672673i	0.0302957i
$494$	79.4492	3.57459
$495$	0	0
$496$	0	0
$497$	− 16.3902i	− 0.735202i
$498$	− 2.12811i	− 0.0953629i
$499$	40.3663	1.80704	0.903522	−	0.428541i	$-0.140972\pi$
$499$	40.3663	1.80704	0.903522	+	0.428541i	$0.140972\pi$
$500$	0	0
$501$	−5.30931	−0.237202
$502$	56.9238i	2.54063i
$503$	− 23.7117i	− 1.05725i	−0.848855	−	0.528625i	$-0.822708\pi$
$503$	− 23.7117i	− 1.05725i	0.848855	−	0.528625i	$-0.177292\pi$
$504$	5.45544	0.243004
$505$	0	0
$506$	−56.6245	−2.51727
$507$	− 24.3332i	− 1.08068i
$508$	26.7828i	1.18829i
$509$	0.568478	0.0251974	0.0125987	−	0.999921i	$-0.495990\pi$
$509$	0.568478	0.0251974	0.0125987	+	0.999921i	$0.495990\pi$
$510$	0	0
$511$	−12.2143	−0.540328
$512$	− 25.6396i	− 1.13312i
$513$	− 5.43742i	− 0.240068i
$514$	65.3571	2.88278
$515$	0	0
$516$	31.4433	1.38421
$517$	14.2942i	0.628660i
$518$	− 12.3565i	− 0.542913i
$519$	22.7677	0.999391
$520$	0	0
$521$	21.6576	0.948837	0.474418	−	0.880299i	$-0.342658\pi$
$521$	21.6576	0.948837	0.474418	+	0.880299i	$0.342658\pi$
$522$	− 2.39138i	− 0.104668i
$523$	− 14.0180i	− 0.612965i	−0.951876	−	0.306483i	$-0.900848\pi$
$523$	− 14.0180i	− 0.612965i	0.951876	−	0.306483i	$-0.0991521\pi$
$524$	−52.9477	−2.31303
$525$	0	0
$526$	−52.0118	−2.26782
$527$	0	0
$528$	7.17415i	0.312215i
$529$	−39.2972	−1.70857
$530$	0	0
$531$	3.21724	0.139616
$532$	26.8388i	1.16361i
$533$	− 26.4404i	− 1.14526i
$534$	−5.57258	−0.241149
$535$	0	0
$536$	−4.55046	−0.196550
$537$	− 17.6576i	− 0.761982i
$538$	− 24.7036i	− 1.06505i
$539$	15.7146	0.676876
$540$	0	0
$541$	20.8748	0.897479	0.448740	−	0.893663i	$-0.351873\pi$
$541$	20.8748	0.897479	0.448740	+	0.893663i	$0.351873\pi$
$542$	− 14.9610i	− 0.642631i
$543$	10.5655i	0.453410i
$544$	1.68267	0.0721440
$545$	0	0
$546$	19.3943	0.830001
$547$	4.45249i	0.190375i	0.995459	+	0.0951873i	$0.0303450\pi$
$547$	4.45249i	0.190375i	−0.995459	+	0.0951873i	$0.969655\pi$
$548$	− 56.1340i	− 2.39793i
$549$	−7.43742	−0.317421
$550$	0	0
$551$	5.43742	0.231642
$552$	32.4404i	1.38075i
$553$	15.4735i	0.657998i
$554$	−75.7626	−3.21885
$555$	0	0
$556$	−23.9389	−1.01524
$557$	34.4253i	1.45865i	0.684169	+	0.729323i	$0.260165\pi$
$557$	34.4253i	1.45865i	−0.684169	+	0.729323i	$0.739835\pi$
$558$	0	0
$559$	51.6635	2.18513
$560$	0	0
$561$	−2.01802	−0.0852008
$562$	66.7526i	2.81579i
$563$	− 19.6756i	− 0.829229i	−0.909997	−	0.414614i	$-0.863917\pi$
$563$	− 19.6756i	− 0.829229i	0.909997	−	0.414614i	$-0.136083\pi$
$564$	17.7187	0.746092
$565$	0	0
$566$	56.2680	2.36512
$567$	− 1.32733i	− 0.0557425i
$568$	− 50.7526i	− 2.12953i
$569$	−26.2942	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$569$	−26.2942	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$570$	0	0
$571$	37.0600	1.55091	0.775455	−	0.631402i	$-0.217520\pi$
$571$	37.0600	1.55091	0.775455	+	0.631402i	$0.217520\pi$
$572$	68.1650i	2.85012i
$573$	− 3.23820i	− 0.135278i
$574$	13.7356	0.573312
$575$	0	0
$576$	−10.7647	−0.448531
$577$	− 14.2202i	− 0.591994i	−0.955189	−	0.295997i	$-0.904348\pi$
$577$	− 14.2202i	− 0.591994i	0.955189	−	0.295997i	$-0.0956518\pi$
$578$	39.5714i	1.64595i
$579$	−3.30931	−0.137530
$580$	0	0
$581$	1.18120	0.0490044
$582$	30.3123i	1.25648i
$583$	39.9908i	1.65625i
$584$	−37.8217	−1.56508
$585$	0	0
$586$	−14.9179	−0.616254
$587$	32.0980i	1.32483i	0.749139	+	0.662413i	$0.230467\pi$
$587$	32.0980i	1.32483i	−0.749139	+	0.662413i	$0.769533\pi$
$588$	− 19.4794i	− 0.803315i
$589$	0	0
$590$	0	0
$591$	1.89286	0.0778617
$592$	− 9.30931i	− 0.382610i
$593$	27.1871i	1.11644i	0.829693	+	0.558220i	$0.188516\pi$
$593$	27.1871i	1.11644i	−0.829693	+	0.558220i	$0.811484\pi$
$594$	7.17415	0.294359
$595$	0	0
$596$	4.52654	0.185414
$597$	5.56258i	0.227661i
$598$	115.327i	4.71607i
$599$	−31.8037	−1.29947	−0.649733	−	0.760163i	$-0.725119\pi$
$599$	−31.8037	−1.29947	−0.649733	+	0.760163i	$0.725119\pi$
$600$	0	0
$601$	−13.3453	−0.544368	−0.272184	−	0.962245i	$-0.587746\pi$
$601$	−13.3453	−0.544368	−0.272184	+	0.962245i	$0.587746\pi$
$602$	26.8388i	1.09387i
$603$	1.10714i	0.0450864i
$604$	8.65465	0.352153
$605$	0	0
$606$	−8.51949	−0.346081
$607$	− 37.6635i	− 1.52871i	−0.644793	−	0.764357i	$-0.723056\pi$
$607$	− 37.6635i	− 1.52871i	0.644793	−	0.764357i	$-0.276944\pi$
$608$	− 13.6016i	− 0.551616i
$609$	1.32733	0.0537860
$610$	0	0
$611$	29.1130	1.17779
$612$	2.50147i	0.101116i
$613$	− 38.3663i	− 1.54960i	−0.632206	−	0.774800i	$-0.717850\pi$
$613$	− 38.3663i	− 1.54960i	0.632206	−	0.774800i	$-0.282150\pi$
$614$	−8.25622	−0.333194
$615$	0	0
$616$	−16.3663	−0.659418
$617$	− 34.6245i	− 1.39393i	−0.717105	−	0.696965i	$-0.754533\pi$
$617$	− 34.6245i	− 1.39393i	0.717105	−	0.696965i	$-0.245467\pi$
$618$	− 30.8748i	− 1.24197i
$619$	−0.00589781	−0.000237053 0	−0.000118526	−	1.00000i	$-0.500038\pi$
$619$	−0.00589781	−0.000237053 0	−0.000118526		1.00000i	$0.500038\pi$
$620$	0	0
$621$	7.89286	0.316729
$622$	26.3052i	1.05474i
$623$	− 3.09304i	− 0.123920i
$624$	14.6116	0.584931
$625$	0	0
$626$	−78.8937	−3.15323
$627$	16.3123i	0.651449i
$628$	− 35.0950i	− 1.40044i
$629$	2.61862	0.104411
$630$	0	0
$631$	42.3001	1.68394	0.841971	−	0.539523i	$-0.181395\pi$
$631$	42.3001	1.68394	0.841971	+	0.539523i	$0.181395\pi$
$632$	47.9138i	1.90591i
$633$	15.7497i	0.625993i
$634$	−11.0880	−0.440360
$635$	0	0
$636$	49.5714	1.96563
$637$	− 32.0059i	− 1.26812i
$638$	7.17415i	0.284027i
$639$	−12.3483	−0.488491
$640$	0	0
$641$	14.7297	0.581787	0.290894	−	0.956755i	$-0.406047\pi$
$641$	14.7297	0.581787	0.290894	+	0.956755i	$0.406047\pi$
$642$	− 10.0921i	− 0.398302i
$643$	− 37.9820i	− 1.49786i	−0.662647	−	0.748932i	$-0.730567\pi$
$643$	− 37.9820i	− 1.49786i	0.662647	−	0.748932i	$-0.269433\pi$
$644$	−38.9587	−1.53519
$645$	0	0
$646$	−8.74673	−0.344135
$647$	31.5144i	1.23896i	0.785012	+	0.619480i	$0.212656\pi$
$647$	31.5144i	1.23896i	−0.785012	+	0.619480i	$0.787344\pi$
$648$	− 4.11009i	− 0.161460i
$649$	−9.65171	−0.378863
$650$	0	0
$651$	0	0
$652$	− 79.3211i	− 3.10645i
$653$	38.8929i	1.52200i	0.648755	+	0.760998i	$0.275290\pi$
$653$	38.8929i	1.52200i	−0.648755	+	0.760998i	$0.724710\pi$
$654$	−28.3973	−1.11042
$655$	0	0
$656$	10.3483	0.404033
$657$	9.20217i	0.359011i
$658$	15.1240i	0.589595i
$659$	32.2792	1.25742	0.628709	−	0.777641i	$-0.283584\pi$
$659$	32.2792	1.25742	0.628709	+	0.777641i	$0.283584\pi$
$660$	0	0
$661$	−29.8447	−1.16082	−0.580412	−	0.814323i	$-0.697109\pi$
$661$	−29.8447	−1.16082	−0.580412	+	0.814323i	$0.697109\pi$
$662$	− 53.1511i	− 2.06577i
$663$	4.11009i	0.159623i
$664$	3.65760	0.141943
$665$	0	0
$666$	−9.30931	−0.360728
$667$	7.89286i	0.305613i
$668$	− 19.7438i	− 0.763910i
$669$	−2.01802	−0.0780211
$670$	0	0
$671$	22.3123	0.861355
$672$	− 3.32028i	− 0.128082i
$673$	− 10.5505i	− 0.406690i	−0.979107	−	0.203345i	$-0.934819\pi$
$673$	− 10.5505i	− 0.406690i	0.979107	−	0.203345i	$-0.0651814\pi$
$674$	65.9197	2.53913
$675$	0	0
$676$	90.4882	3.48032
$677$	40.0600i	1.53963i	0.638268	+	0.769815i	$0.279651\pi$
$677$	40.0600i	1.53963i	−0.638268	+	0.769815i	$0.720349\pi$
$678$	34.0490i	1.30764i
$679$	−16.8247	−0.645673
$680$	0	0
$681$	20.6756	0.792292
$682$	0	0
$683$	− 24.2913i	− 0.929480i	−0.885447	−	0.464740i	$-0.846148\pi$
$683$	− 24.2913i	− 0.929480i	0.885447	−	0.464740i	$-0.153852\pi$
$684$	20.2202	0.773138
$685$	0	0
$686$	38.8459	1.48314
$687$	− 14.2562i	− 0.543909i
$688$	20.2202i	0.770887i
$689$	81.4492	3.10297
$690$	0	0
$691$	−43.9017	−1.67010	−0.835050	−	0.550174i	$-0.814561\pi$
$691$	−43.9017	−1.67010	−0.835050	+	0.550174i	$0.814561\pi$
$692$	84.6665i	3.21854i
$693$	3.98198i	0.151263i
$694$	53.3932	2.02678
$695$	0	0
$696$	4.11009	0.155793
$697$	2.91087i	0.110257i
$698$	49.0508i	1.85660i
$699$	20.7677	0.785506
$700$	0	0
$701$	−39.9197	−1.50775	−0.753874	−	0.657020i	$-0.771817\pi$
$701$	−39.9197	−1.50775	−0.753874	+	0.657020i	$0.771817\pi$
$702$	− 14.6116i	− 0.551478i
$703$	− 21.1671i	− 0.798332i
$704$	32.2942	1.21713
$705$	0	0
$706$	−69.0508	−2.59876
$707$	− 4.72871i	− 0.177841i
$708$	11.9640i	0.449633i
$709$	−11.4885	−0.431461	−0.215730	−	0.976453i	$-0.569213\pi$
$709$	−11.4885	−0.431461	−0.215730	+	0.976453i	$0.569213\pi$
$710$	0	0
$711$	11.6576	0.437194
$712$	− 9.57765i	− 0.358938i
$713$	0	0
$714$	−2.13516	−0.0799064
$715$	0	0
$716$	65.6635	2.45396
$717$	− 3.30931i	− 0.123588i
$718$	− 6.61862i	− 0.247005i
$719$	1.00295	0.0374037	0.0187018	−	0.999825i	$-0.494047\pi$
$719$	1.00295	0.0374037	0.0187018	+	0.999825i	$0.494047\pi$
$720$	0	0
$721$	17.1370	0.638214
$722$	25.2662i	0.940311i
$723$	− 8.78571i	− 0.326744i
$724$	−39.2901	−1.46021
$725$	0	0
$726$	4.78276	0.177505
$727$	− 20.4404i	− 0.758091i	−0.925378	−	0.379046i	$-0.876252\pi$
$727$	− 20.4404i	− 0.758091i	0.925378	−	0.379046i	$-0.123748\pi$
$728$	33.3332i	1.23541i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−5.68774	−0.210369
$732$	− 27.6576i	− 1.02225i
$733$	− 30.9669i	− 1.14379i	−0.820327	−	0.571895i	$-0.806209\pi$
$733$	− 30.9669i	− 1.14379i	0.820327	−	0.571895i	$-0.193791\pi$
$734$	−60.8807	−2.24715
$735$	0	0
$736$	19.7438	0.727765
$737$	− 3.32143i	− 0.122346i
$738$	− 10.3483i	− 0.380926i
$739$	−3.62157	−0.133222	−0.0666108	−	0.997779i	$-0.521219\pi$
$739$	−3.62157	−0.133222	−0.0666108	+	0.997779i	$0.521219\pi$
$740$	0	0
$741$	33.2231	1.22048
$742$	42.3123i	1.55333i
$743$	25.1691i	0.923364i	0.887046	+	0.461682i	$0.152754\pi$
$743$	25.1691i	0.923364i	−0.887046	+	0.461682i	$0.847246\pi$
$744$	0	0
$745$	0	0
$746$	−9.65171	−0.353374
$747$	− 0.889908i	− 0.0325600i
$748$	− 7.50442i	− 0.274389i
$749$	5.60157	0.204677
$750$	0	0
$751$	−9.63760	−0.351681	−0.175841	−	0.984419i	$-0.556264\pi$
$751$	−9.63760	−0.351681	−0.175841	+	0.984419i	$0.556264\pi$
$752$	11.3943i	0.415509i
$753$	23.8037i	0.867456i
$754$	14.6116	0.532122
$755$	0	0
$756$	4.93594	0.179519
$757$	− 17.4282i	− 0.633440i	−0.948519	−	0.316720i	$-0.897418\pi$
$757$	− 17.4282i	− 0.633440i	0.948519	−	0.316720i	$-0.102582\pi$
$758$	− 48.7946i	− 1.77230i
$759$	−23.6786	−0.859478
$760$	0	0
$761$	2.27622	0.0825130	0.0412565	−	0.999149i	$-0.486864\pi$
$761$	2.27622	0.0825130	0.0412565	+	0.999149i	$0.486864\pi$
$762$	17.2231i	0.623928i
$763$	− 15.7618i	− 0.570615i
$764$	12.0419	0.435662
$765$	0	0
$766$	−13.8719	−0.501212
$767$	19.6576i	0.709795i
$768$	− 28.0670i	− 1.01278i
$769$	−29.4433	−1.06175	−0.530877	−	0.847449i	$-0.678137\pi$
$769$	−29.4433	−1.06175	−0.530877	+	0.847449i	$0.678137\pi$
$770$	0	0
$771$	27.3303	0.984276
$772$	− 12.3064i	− 0.442916i
$773$	− 14.2562i	− 0.512761i	−0.966576	−	0.256380i	$-0.917470\pi$
$773$	− 14.2562i	− 0.512761i	0.966576	−	0.256380i	$-0.0825299\pi$
$774$	20.2202	0.726800
$775$	0	0
$776$	−52.0980	−1.87021
$777$	− 5.16710i	− 0.185369i
$778$	77.1440i	2.76575i
$779$	23.5295	0.843032
$780$	0	0
$781$	37.0449	1.32557
$782$	− 12.6966i	− 0.454029i
$783$	− 1.00000i	− 0.0357371i
$784$	12.5265	0.447377
$785$	0	0
$786$	−34.0490	−1.21449
$787$	− 27.1311i	− 0.967118i	−0.875312	−	0.483559i	$-0.839344\pi$
$787$	− 27.1311i	− 0.967118i	0.875312	−	0.483559i	$-0.160656\pi$
$788$	7.03899i	0.250753i
$789$	−21.7497	−0.774309
$790$	0	0
$791$	−18.8988	−0.671962
$792$	12.3303i	0.438137i
$793$	− 45.4433i	− 1.61374i
$794$	−53.0508	−1.88270
$795$	0	0
$796$	−20.6856	−0.733182
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	17.2592i	0.610968i
$799$	−3.20511	−0.113389
$800$	0	0
$801$	−2.33028	−0.0823363
$802$	− 92.3182i	− 3.25987i
$803$	− 27.6065i	− 0.974212i
$804$	−4.11714	−0.145200
$805$	0	0
$806$	0	0
$807$	− 10.3303i	− 0.363643i
$808$	− 14.6425i	− 0.515123i
$809$	13.5324	0.475775	0.237888	−	0.971293i	$-0.423545\pi$
$809$	13.5324	0.475775	0.237888	+	0.971293i	$0.423545\pi$
$810$	0	0
$811$	−40.4492	−1.42036	−0.710182	−	0.704018i	$-0.751387\pi$
$811$	−40.4492	−1.42036	−0.710182	+	0.704018i	$0.751387\pi$
$812$	4.93594i	0.173218i
$813$	− 6.25622i	− 0.219415i
$814$	27.9279	0.978873
$815$	0	0
$816$	−1.60862	−0.0563129
$817$	45.9758i	1.60849i
$818$	− 67.1930i	− 2.34935i
$819$	8.11009	0.283390
$820$	0	0
$821$	−21.6016	−0.753900	−0.376950	−	0.926234i	$-0.623027\pi$
$821$	−21.6016	−0.753900	−0.376950	+	0.926234i	$0.623027\pi$
$822$	− 36.0980i	− 1.25906i
$823$	− 25.6995i	− 0.895830i	−0.894076	−	0.447915i	$-0.852167\pi$
$823$	− 25.6995i	− 0.895830i	0.894076	−	0.447915i	$-0.147833\pi$
$824$	53.0649	1.84860
$825$	0	0
$826$	−10.2120	−0.355320
$827$	− 19.6016i	− 0.681613i	−0.940133	−	0.340807i	$-0.889300\pi$
$827$	− 19.6016i	− 0.681613i	0.940133	−	0.340807i	$-0.110700\pi$
$828$	29.3512i	1.02003i
$829$	4.22608	0.146778	0.0733889	−	0.997303i	$-0.476619\pi$
$829$	4.22608	0.146778	0.0733889	+	0.997303i	$0.476619\pi$
$830$	0	0
$831$	−31.6815	−1.09902
$832$	− 65.7736i	− 2.28029i
$833$	3.52360i	0.122085i
$834$	−15.3943	−0.533062
$835$	0	0
$836$	−60.6606	−2.09799
$837$	0	0
$838$	− 8.52654i	− 0.294545i
$839$	2.45839	0.0848729	0.0424365	−	0.999099i	$-0.486488\pi$
$839$	2.45839	0.0848729	0.0424365	+	0.999099i	$0.486488\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 42.1399i	− 1.45224i
$843$	27.9138i	0.961403i
$844$	−58.5685	−2.01601
$845$	0	0
$846$	11.3943	0.391745
$847$	2.65465i	0.0912150i
$848$	31.8778i	1.09469i
$849$	23.5295	0.807530
$850$	0	0
$851$	30.7258	1.05327
$852$	− 45.9197i	− 1.57318i
$853$	− 39.8427i	− 1.36419i	−0.731264	−	0.682094i	$-0.761069\pi$
$853$	− 39.8427i	− 1.36419i	0.731264	−	0.682094i	$-0.238931\pi$
$854$	23.6075	0.807831
$855$	0	0
$856$	17.3453	0.592852
$857$	− 16.9518i	− 0.579064i	−0.957168	−	0.289532i	$-0.906500\pi$
$857$	− 16.9518i	− 0.579064i	0.957168	−	0.289532i	$-0.0934997\pi$
$858$	43.8347i	1.49649i
$859$	−38.3902	−1.30986	−0.654929	−	0.755691i	$-0.727301\pi$
$859$	−38.3902	−1.30986	−0.654929	+	0.755691i	$0.727301\pi$
$860$	0	0
$861$	5.74378	0.195747
$862$	− 80.8087i	− 2.75235i
$863$	− 49.6995i	− 1.69179i	−0.533348	−	0.845896i	$-0.679066\pi$
$863$	− 49.6995i	− 1.69179i	0.533348	−	0.845896i	$-0.320934\pi$
$864$	−2.50147	−0.0851019
$865$	0	0
$866$	55.8778	1.89880
$867$	16.5475i	0.561983i
$868$	0	0
$869$	−34.9728	−1.18637
$870$	0	0
$871$	−6.76475	−0.229215
$872$	− 48.8067i	− 1.65280i
$873$	12.6756i	0.429005i
$874$	−102.630	−3.47153
$875$	0	0
$876$	−34.2202	−1.15619
$877$	− 40.9168i	− 1.38166i	−0.723017	−	0.690831i	$-0.757245\pi$
$877$	− 40.9168i	− 1.38166i	0.723017	−	0.690831i	$-0.242755\pi$
$878$	38.6116i	1.30308i
$879$	−6.23820	−0.210409
$880$	0	0
$881$	−26.9079	−0.906551	−0.453276	−	0.891370i	$-0.649745\pi$
$881$	−26.9079	−0.906551	−0.453276	+	0.891370i	$0.649745\pi$
$882$	− 12.5265i	− 0.421791i
$883$	6.69069i	0.225160i	0.993643	+	0.112580i	$0.0359114\pi$
$883$	6.69069i	0.225160i	−0.993643	+	0.112580i	$0.964089\pi$
$884$	−15.2842	−0.514065
$885$	0	0
$886$	22.4392	0.753860
$887$	− 46.0859i	− 1.54741i	−0.633545	−	0.773706i	$-0.718401\pi$
$887$	− 46.0859i	− 1.54741i	0.633545	−	0.773706i	$-0.281599\pi$
$888$	− 16.0000i	− 0.536925i
$889$	−9.55963	−0.320620
$890$	0	0
$891$	3.00000	0.100504
$892$	− 7.50442i	− 0.251267i
$893$	25.9079i	0.866976i
$894$	2.91087	0.0973542
$895$	0	0
$896$	27.5283	0.919657
$897$	48.2261i	1.61022i
$898$	− 42.0129i	− 1.40199i
$899$	0	0
$900$	0	0
$901$	−8.96691	−0.298731
$902$	31.0449i	1.03368i
$903$	11.2231i	0.373482i
$904$	−58.5203	−1.94636
$905$	0	0
$906$	5.56553	0.184902
$907$	− 18.7117i	− 0.621310i	−0.950523	−	0.310655i	$-0.899452\pi$
$907$	− 18.7117i	− 0.621310i	0.950523	−	0.310655i	$-0.100548\pi$
$908$	76.8866i	2.55157i
$909$	−3.56258	−0.118163
$910$	0	0
$911$	25.2923	0.837970	0.418985	−	0.907993i	$-0.362386\pi$
$911$	25.2923	0.837970	0.418985	+	0.907993i	$0.362386\pi$
$912$	13.0029i	0.430571i
$913$	2.66972i	0.0883550i
$914$	41.6564	1.37787
$915$	0	0
$916$	53.0147	1.75166
$917$	− 18.8988i	− 0.624092i
$918$	1.60862i	0.0530923i
$919$	45.6396	1.50551	0.752756	−	0.658300i	$-0.228724\pi$
$919$	45.6396	1.50551	0.752756	+	0.658300i	$0.228724\pi$
$920$	0	0
$921$	−3.45249	−0.113763
$922$	− 0.270324i	− 0.00890265i
$923$	− 75.4492i	− 2.48344i
$924$	−14.8078	−0.487142
$925$	0	0
$926$	−64.2249	−2.11056
$927$	− 12.9109i	− 0.424049i
$928$	− 2.50147i	− 0.0821149i
$929$	21.2733	0.697953	0.348977	−	0.937131i	$-0.386529\pi$
$929$	21.2733	0.697953	0.348977	+	0.937131i	$0.386529\pi$
$930$	0	0
$931$	28.4823	0.933470
$932$	77.2290i	2.52972i
$933$	11.0000i	0.360124i
$934$	−1.73788	−0.0568652
$935$	0	0
$936$	25.1130	0.820845
$937$	− 46.9489i	− 1.53375i	−0.641794	−	0.766877i	$-0.721810\pi$
$937$	− 46.9489i	− 1.53375i	0.641794	−	0.766877i	$-0.278190\pi$
$938$	− 3.51424i	− 0.114744i
$939$	−32.9908	−1.07662
$940$	0	0
$941$	−18.5986	−0.606298	−0.303149	−	0.952943i	$-0.598038\pi$
$941$	−18.5986	−0.606298	−0.303149	+	0.952943i	$0.598038\pi$
$942$	− 22.5685i	− 0.735321i
$943$	34.1550i	1.11224i
$944$	−7.69364	−0.250407
$945$	0	0
$946$	−60.6606	−1.97224
$947$	40.0800i	1.30242i	0.758896	+	0.651212i	$0.225739\pi$
$947$	40.0800i	1.30242i	−0.758896	+	0.651212i	$0.774261\pi$
$948$	43.3512i	1.40798i
$949$	−56.2261	−1.82518
$950$	0	0
$951$	−4.63664	−0.150353
$952$	− 3.66972i	− 0.118936i
$953$	5.03309i	0.163038i	0.996672	+	0.0815188i	$0.0259771\pi$
$953$	5.03309i	0.163038i	−0.996672	+	0.0815188i	$0.974023\pi$
$954$	31.8778	1.03208
$955$	0	0
$956$	12.3064	0.398016
$957$	3.00000i	0.0969762i
$958$	21.8217i	0.705029i
$959$	20.0360	0.646997
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 56.8807i	− 1.83391i
$963$	− 4.22018i	− 0.135994i
$964$	32.6715	1.05228
$965$	0	0
$966$	−25.0531	−0.806070
$967$	1.72871i	0.0555916i	0.999614	+	0.0277958i	$0.00884882\pi$
$967$	1.72871i	0.0555916i	−0.999614	+	0.0277958i	$0.991151\pi$
$968$	8.22018i	0.264207i
$969$	−3.65760	−0.117499
$970$	0	0
$971$	−16.1071	−0.516903	−0.258451	−	0.966024i	$-0.583212\pi$
$971$	−16.1071	−0.516903	−0.258451	+	0.966024i	$0.583212\pi$
$972$	− 3.71871i	− 0.119278i
$973$	− 8.54456i	− 0.273926i
$974$	−85.0649	−2.72565
$975$	0	0
$976$	17.7857	0.569307
$977$	− 13.2323i	− 0.423339i	−0.977341	−	0.211669i	$-0.932110\pi$
$977$	− 13.2323i	− 0.423339i	0.977341	−	0.211669i	$-0.0678900\pi$
$978$	− 51.0088i	− 1.63108i
$979$	6.99083	0.223428
$980$	0	0
$981$	−11.8748	−0.379134
$982$	− 10.6186i	− 0.338854i
$983$	− 3.08913i	− 0.0985278i	−0.998786	−	0.0492639i	$-0.984312\pi$
$983$	− 3.08913i	− 0.0985278i	0.998786	−	0.0492639i	$-0.0156875\pi$
$984$	17.7857	0.566988
$985$	0	0
$986$	−1.60862	−0.0512288
$987$	6.32438i	0.201307i
$988$	123.547i	3.93056i
$989$	−66.7376	−2.12213
$990$	0	0
$991$	30.9138	0.982010	0.491005	−	0.871157i	$-0.336630\pi$
$991$	30.9138	0.982010	0.491005	+	0.871157i	$0.336630\pi$
$992$	0	0
$993$	− 22.2261i	− 0.705323i
$994$	39.1953	1.24320
$995$	0	0
$996$	3.30931	0.104859
$997$	− 25.3863i	− 0.803993i	−0.915641	−	0.401996i	$-0.868316\pi$
$997$	− 25.3863i	− 0.803993i	0.915641	−	0.401996i	$-0.131684\pi$
$998$	96.5313i	3.05564i
$999$	−3.89286	−0.123165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.m.349.6		6
5.2	odd	4		2175.2.a.u.1.1		3
5.3	odd	4		435.2.a.i.1.3	✓	3
5.4	even	2	inner	2175.2.c.m.349.1		6
15.2	even	4		6525.2.a.bf.1.3		3
15.8	even	4		1305.2.a.q.1.1		3
20.3	even	4		6960.2.a.cl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.i.1.3	✓	3	5.3	odd	4
1305.2.a.q.1.1		3	15.8	even	4
2175.2.a.u.1.1		3	5.2	odd	4
2175.2.c.m.349.1		6	5.4	even	2	inner
2175.2.c.m.349.6		6	1.1	even	1	trivial
6525.2.a.bf.1.3		3	15.2	even	4
6960.2.a.cl.1.2		3	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)

\(f(q)\)	\(=\)	\(q+2.39138i q^{2} +1.00000i q^{3} -3.71871 q^{4} -2.39138 q^{6} -1.32733i q^{7} -4.11009i q^{8} -1.00000 q^{9} +3.00000 q^{11} -3.71871i q^{12} -6.11009i q^{13} +3.17415 q^{14} +2.39138 q^{16} +0.672673i q^{17} -2.39138i q^{18} +5.43742 q^{19} +1.32733 q^{21} +7.17415i q^{22} +7.89286i q^{23} +4.11009 q^{24} +14.6116 q^{26} -1.00000i q^{27} +4.93594i q^{28} +1.00000 q^{29} -2.50147i q^{32} +3.00000i q^{33} -1.60862 q^{34} +3.71871 q^{36} -3.89286i q^{37} +13.0029i q^{38} +6.11009 q^{39} +4.32733 q^{41} +3.17415i q^{42} +8.45544i q^{43} -11.1561 q^{44} -18.8748 q^{46} +4.76475i q^{47} +2.39138i q^{48} +5.23820 q^{49} -0.672673 q^{51} +22.7217i q^{52} +13.3303i q^{53} +2.39138 q^{54} -5.45544 q^{56} +5.43742i q^{57} +2.39138i q^{58} -3.21724 q^{59} +7.43742 q^{61} +1.32733i q^{63} +10.7647 q^{64} -7.17415 q^{66} -1.10714i q^{67} -2.50147i q^{68} -7.89286 q^{69} +12.3483 q^{71} +4.11009i q^{72} -9.20217i q^{73} +9.30931 q^{74} -20.2202 q^{76} -3.98198i q^{77} +14.6116i q^{78} -11.6576 q^{79} +1.00000 q^{81} +10.3483i q^{82} +0.889908i q^{83} -4.93594 q^{84} -20.2202 q^{86} +1.00000i q^{87} -12.3303i q^{88} +2.33028 q^{89} -8.11009 q^{91} -29.3512i q^{92} -11.3943 q^{94} +2.50147 q^{96} -12.6756i q^{97} +12.5265i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 18 q^{11} - 18 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{21} + 26 q^{26} + 6 q^{29} - 22 q^{34} + 10 q^{36} + 12 q^{39} + 26 q^{41} - 30 q^{44} - 64 q^{46} - 18 q^{49} - 4 q^{51}+ \cdots - 18 q^{99}+O(q^{100}) \) 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 18 * q^11 - 18 * q^14 + 2 * q^16 + 8 * q^19 + 8 * q^21 + 26 * q^26 + 6 * q^29 - 22 * q^34 + 10 * q^36 + 12 * q^39 + 26 * q^41 - 30 * q^44 - 64 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 44 * q^59 + 20 * q^61 + 40 * q^64 - 6 * q^66 + 2 * q^69 + 56 * q^74 - 72 * q^76 + 4 * q^79 + 6 * q^81 - 42 * q^84 - 72 * q^86 - 60 * q^89 - 24 * q^91 + 18 * q^94 - 22 * q^96 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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