LMFDB - Embedded newform 1125.2.b.f.874.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1125,2,Mod(874,1125)] 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("1125.874"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1125 = 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1125.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,0,0,0,0,0,0,12] 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:	$8.98317022739$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 125)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			874.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1125.874
Dual form			1125.2.b.f.874.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803i q^{2} -0.618034 q^{4} +3.00000i q^{7} -2.23607i q^{8} +3.00000 q^{11} -4.85410i q^{13} +4.85410 q^{14} -4.85410 q^{16} +4.23607i q^{17} +3.61803 q^{19} -4.85410i q^{22} +1.23607i q^{23} -7.85410 q^{26} -1.85410i q^{28} +6.70820 q^{29} +5.09017 q^{31} +3.38197i q^{32} +6.85410 q^{34} -3.70820i q^{37} -5.85410i q^{38} +3.00000 q^{41} -9.00000i q^{43} -1.85410 q^{44} +2.00000 q^{46} -8.32624i q^{47} -2.00000 q^{49} +3.00000i q^{52} -4.61803i q^{53} +6.70820 q^{56} -10.8541i q^{58} +4.14590 q^{59} -6.09017 q^{61} -8.23607i q^{62} -4.23607 q^{64} +13.8541i q^{67} -2.61803i q^{68} +3.00000 q^{71} +1.85410i q^{73} -6.00000 q^{74} -2.23607 q^{76} +9.00000i q^{77} -0.527864 q^{79} -4.85410i q^{82} -0.472136i q^{83} -14.5623 q^{86} -6.70820i q^{88} -13.4164 q^{89} +14.5623 q^{91} -0.763932i q^{92} -13.4721 q^{94} -7.85410i q^{97} +3.23607i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 12 q^{11} + 6 q^{14} - 6 q^{16} + 10 q^{19} - 18 q^{26} - 2 q^{31} + 14 q^{34} + 12 q^{41} + 6 q^{44} + 8 q^{46} - 8 q^{49} + 30 q^{59} - 2 q^{61} - 8 q^{64} + 12 q^{71} - 24 q^{74} - 20 q^{79}+ \cdots - 36 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 + 12 * q^11 + 6 * q^14 - 6 * q^16 + 10 * q^19 - 18 * q^26 - 2 * q^31 + 14 * q^34 + 12 * q^41 + 6 * q^44 + 8 * q^46 - 8 * q^49 + 30 * q^59 - 2 * q^61 - 8 * q^64 + 12 * q^71 - 24 * q^74 - 20 * q^79 - 18 * q^86 + 18 * q^91 - 36 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1125\mathbb{Z}\right)^\times$.

$n$	$127$	$1001$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.61803i	− 1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$	− 1.61803i	− 1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$	0	0
$3$	0	0
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 2.23607i	− 0.790569i
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	− 4.85410i	− 1.34629i	−0.739512	−	0.673143i	$-0.764944\pi$
$13$	− 4.85410i	− 1.34629i	0.739512	−	0.673143i	$-0.235056\pi$
$14$	4.85410	1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	4.23607i	1.02740i	0.857971	+	0.513699i	$0.171725\pi$
$17$	4.23607i	1.02740i	−0.857971	+	0.513699i	$0.828275\pi$
$18$	0	0
$19$	3.61803	0.830034	0.415017	−	0.909814i	$-0.363776\pi$
$19$	3.61803	0.830034	0.415017	+	0.909814i	$0.363776\pi$
$20$	0	0
$21$	0	0
$22$	− 4.85410i	− 1.03490i
$23$	1.23607i	0.257738i	0.991662	+	0.128869i	$0.0411347\pi$
$23$	1.23607i	0.257738i	−0.991662	+	0.128869i	$0.958865\pi$
$24$	0	0
$25$	0	0
$26$	−7.85410	−1.54032
$27$	0	0
$28$	− 1.85410i	− 0.350392i
$29$	6.70820	1.24568	0.622841	−	0.782348i	$-0.285978\pi$
$29$	6.70820	1.24568	0.622841	+	0.782348i	$0.285978\pi$
$30$	0	0
$31$	5.09017	0.914222	0.457111	−	0.889410i	$-0.348884\pi$
$31$	5.09017	0.914222	0.457111	+	0.889410i	$0.348884\pi$
$32$	3.38197i	0.597853i
$33$	0	0
$34$	6.85410	1.17547
$35$	0	0
$36$	0	0
$37$	− 3.70820i	− 0.609625i	−0.952412	−	0.304812i	$-0.901406\pi$
$37$	− 3.70820i	− 0.609625i	0.952412	−	0.304812i	$-0.0985938\pi$
$38$	− 5.85410i	− 0.949661i
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	−1.85410	−0.279516
$45$	0	0
$46$	2.00000	0.294884
$47$	− 8.32624i	− 1.21451i	−0.794508	−	0.607253i	$-0.792271\pi$
$47$	− 8.32624i	− 1.21451i	0.794508	−	0.607253i	$-0.207729\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	3.00000i	0.416025i
$53$	− 4.61803i	− 0.634336i	−0.948369	−	0.317168i	$-0.897268\pi$
$53$	− 4.61803i	− 0.634336i	0.948369	−	0.317168i	$-0.102732\pi$
$54$	0	0
$55$	0	0
$56$	6.70820	0.896421
$57$	0	0
$58$	− 10.8541i	− 1.42521i
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	0	0
$61$	−6.09017	−0.779766	−0.389883	−	0.920864i	$-0.627485\pi$
$61$	−6.09017	−0.779766	−0.389883	+	0.920864i	$0.627485\pi$
$62$	− 8.23607i	− 1.04598i
$63$	0	0
$64$	−4.23607	−0.529508
$65$	0	0
$66$	0	0
$67$	13.8541i	1.69255i	0.532748	+	0.846274i	$0.321159\pi$
$67$	13.8541i	1.69255i	−0.532748	+	0.846274i	$0.678841\pi$
$68$	− 2.61803i	− 0.317483i
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	1.85410i	0.217006i	0.994096	+	0.108503i	$0.0346057\pi$
$73$	1.85410i	0.217006i	−0.994096	+	0.108503i	$0.965394\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−2.23607	−0.256495
$77$	9.00000i	1.02565i
$78$	0	0
$79$	−0.527864	−0.0593893	−0.0296947	−	0.999559i	$-0.509453\pi$
$79$	−0.527864	−0.0593893	−0.0296947	+	0.999559i	$0.509453\pi$
$80$	0	0
$81$	0	0
$82$	− 4.85410i	− 0.536046i
$83$	− 0.472136i	− 0.0518237i	−0.999664	−	0.0259118i	$-0.991751\pi$
$83$	− 0.472136i	− 0.0518237i	0.999664	−	0.0259118i	$-0.00824891\pi$
$84$	0	0
$85$	0	0
$86$	−14.5623	−1.57029
$87$	0	0
$88$	− 6.70820i	− 0.715097i
$89$	−13.4164	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$90$	0	0
$91$	14.5623	1.52654
$92$	− 0.763932i	− 0.0796454i
$93$	0	0
$94$	−13.4721	−1.38954
$95$	0	0
$96$	0	0
$97$	− 7.85410i	− 0.797463i	−0.917068	−	0.398732i	$-0.869451\pi$
$97$	− 7.85410i	− 0.797463i	0.917068	−	0.398732i	$-0.130549\pi$
$98$	3.23607i	0.326892i
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	12.7082i	1.25218i	0.779752	+	0.626088i	$0.215345\pi$
$103$	12.7082i	1.25218i	−0.779752	+	0.626088i	$0.784655\pi$
$104$	−10.8541	−1.06433
$105$	0	0
$106$	−7.47214	−0.725758
$107$	0.0901699i	0.00871706i	0.999991	+	0.00435853i	$0.00138737\pi$
$107$	0.0901699i	0.00871706i	−0.999991	+	0.00435853i	$0.998613\pi$
$108$	0	0
$109$	−5.32624	−0.510161	−0.255081	−	0.966920i	$-0.582102\pi$
$109$	−5.32624	−0.510161	−0.255081	+	0.966920i	$0.582102\pi$
$110$	0	0
$111$	0	0
$112$	− 14.5623i	− 1.37601i
$113$	− 2.05573i	− 0.193387i	−0.995314	−	0.0966933i	$-0.969173\pi$
$113$	− 2.05573i	− 0.193387i	0.995314	−	0.0966933i	$-0.0308266\pi$
$114$	0	0
$115$	0	0
$116$	−4.14590	−0.384937
$117$	0	0
$118$	− 6.70820i	− 0.617540i
$119$	−12.7082	−1.16496
$120$	0	0
$121$	−2.00000	−0.181818
$122$	9.85410i	0.892148i
$123$	0	0
$124$	−3.14590	−0.282510
$125$	0	0
$126$	0	0
$127$	9.70820i	0.861464i	0.902480	+	0.430732i	$0.141745\pi$
$127$	9.70820i	0.861464i	−0.902480	+	0.430732i	$0.858255\pi$
$128$	13.6180i	1.20368i
$129$	0	0
$130$	0	0
$131$	12.2705	1.07208	0.536040	−	0.844193i	$-0.319920\pi$
$131$	12.2705	1.07208	0.536040	+	0.844193i	$0.319920\pi$
$132$	0	0
$133$	10.8541i	0.941170i
$134$	22.4164	1.93648
$135$	0	0
$136$	9.47214	0.812229
$137$	5.61803i	0.479981i	0.970775	+	0.239991i	$0.0771443\pi$
$137$	5.61803i	0.479981i	−0.970775	+	0.239991i	$0.922856\pi$
$138$	0	0
$139$	12.2361	1.03785	0.518925	−	0.854820i	$-0.326332\pi$
$139$	12.2361	1.03785	0.518925	+	0.854820i	$0.326332\pi$
$140$	0	0
$141$	0	0
$142$	− 4.85410i	− 0.407347i
$143$	− 14.5623i	− 1.21776i
$144$	0	0
$145$	0	0
$146$	3.00000	0.248282
$147$	0	0
$148$	2.29180i	0.188384i
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−21.0902	−1.71629	−0.858147	−	0.513404i	$-0.828384\pi$
$151$	−21.0902	−1.71629	−0.858147	+	0.513404i	$0.828384\pi$
$152$	− 8.09017i	− 0.656199i
$153$	0	0
$154$	14.5623	1.17346
$155$	0	0
$156$	0	0
$157$	12.2705i	0.979293i	0.871921	+	0.489647i	$0.162874\pi$
$157$	12.2705i	0.979293i	−0.871921	+	0.489647i	$0.837126\pi$
$158$	0.854102i	0.0679487i
$159$	0	0
$160$	0	0
$161$	−3.70820	−0.292247
$162$	0	0
$163$	− 19.8541i	− 1.55509i	−0.628825	−	0.777547i	$-0.716464\pi$
$163$	− 19.8541i	− 1.55509i	0.628825	−	0.777547i	$-0.283536\pi$
$164$	−1.85410	−0.144781
$165$	0	0
$166$	−0.763932	−0.0592926
$167$	9.23607i	0.714708i	0.933969	+	0.357354i	$0.116321\pi$
$167$	9.23607i	0.714708i	−0.933969	+	0.357354i	$0.883679\pi$
$168$	0	0
$169$	−10.5623	−0.812485
$170$	0	0
$171$	0	0
$172$	5.56231i	0.424122i
$173$	0.0557281i	0.00423693i	0.999998	+	0.00211846i	$0.000674329\pi$
$173$	0.0557281i	0.00423693i	−0.999998	+	0.00211846i	$0.999326\pi$
$174$	0	0
$175$	0	0
$176$	−14.5623	−1.09768
$177$	0	0
$178$	21.7082i	1.62710i
$179$	6.70820	0.501395	0.250697	−	0.968066i	$-0.419340\pi$
$179$	6.70820	0.501395	0.250697	+	0.968066i	$0.419340\pi$
$180$	0	0
$181$	18.1803	1.35133	0.675667	−	0.737207i	$-0.263856\pi$
$181$	18.1803	1.35133	0.675667	+	0.737207i	$0.263856\pi$
$182$	− 23.5623i	− 1.74655i
$183$	0	0
$184$	2.76393	0.203760
$185$	0	0
$186$	0	0
$187$	12.7082i	0.929316i
$188$	5.14590i	0.375303i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	15.2705i	1.09920i	0.835429	+	0.549598i	$0.185219\pi$
$193$	15.2705i	1.09920i	−0.835429	+	0.549598i	$0.814781\pi$
$194$	−12.7082	−0.912396
$195$	0	0
$196$	1.23607	0.0882906
$197$	− 4.05573i	− 0.288959i	−0.989508	−	0.144479i	$-0.953849\pi$
$197$	− 4.05573i	− 0.288959i	0.989508	−	0.144479i	$-0.0461507\pi$
$198$	0	0
$199$	−16.1803	−1.14699	−0.573497	−	0.819208i	$-0.694414\pi$
$199$	−16.1803	−1.14699	−0.573497	+	0.819208i	$0.694414\pi$
$200$	0	0
$201$	0	0
$202$	− 4.85410i	− 0.341533i
$203$	20.1246i	1.41247i
$204$	0	0
$205$	0	0
$206$	20.5623	1.43264
$207$	0	0
$208$	23.5623i	1.63375i
$209$	10.8541	0.750794
$210$	0	0
$211$	−4.18034	−0.287786	−0.143893	−	0.989593i	$-0.545962\pi$
$211$	−4.18034	−0.287786	−0.143893	+	0.989593i	$0.545962\pi$
$212$	2.85410i	0.196021i
$213$	0	0
$214$	0.145898	0.00997338
$215$	0	0
$216$	0	0
$217$	15.2705i	1.03663i
$218$	8.61803i	0.583687i
$219$	0	0
$220$	0	0
$221$	20.5623	1.38317
$222$	0	0
$223$	1.85410i	0.124160i	0.998071	+	0.0620799i	$0.0197734\pi$
$223$	1.85410i	0.124160i	−0.998071	+	0.0620799i	$0.980227\pi$
$224$	−10.1459	−0.677901
$225$	0	0
$226$	−3.32624	−0.221258
$227$	− 6.61803i	− 0.439254i	−0.975584	−	0.219627i	$-0.929516\pi$
$227$	− 6.61803i	− 0.439254i	0.975584	−	0.219627i	$-0.0704841\pi$
$228$	0	0
$229$	1.38197	0.0913229	0.0456614	−	0.998957i	$-0.485460\pi$
$229$	1.38197	0.0913229	0.0456614	+	0.998957i	$0.485460\pi$
$230$	0	0
$231$	0	0
$232$	− 15.0000i	− 0.984798i
$233$	26.8885i	1.76153i	0.473556	+	0.880764i	$0.342970\pi$
$233$	26.8885i	1.76153i	−0.473556	+	0.880764i	$0.657030\pi$
$234$	0	0
$235$	0	0
$236$	−2.56231	−0.166792
$237$	0	0
$238$	20.5623i	1.33286i
$239$	25.8541	1.67236	0.836181	−	0.548453i	$-0.184783\pi$
$239$	25.8541	1.67236	0.836181	+	0.548453i	$0.184783\pi$
$240$	0	0
$241$	−19.1803	−1.23551	−0.617757	−	0.786369i	$-0.711959\pi$
$241$	−19.1803	−1.23551	−0.617757	+	0.786369i	$0.711959\pi$
$242$	3.23607i	0.208022i
$243$	0	0
$244$	3.76393	0.240961
$245$	0	0
$246$	0	0
$247$	− 17.5623i	− 1.11746i
$248$	− 11.3820i	− 0.722756i
$249$	0	0
$250$	0	0
$251$	−6.27051	−0.395791	−0.197896	−	0.980223i	$-0.563411\pi$
$251$	−6.27051	−0.395791	−0.197896	+	0.980223i	$0.563411\pi$
$252$	0	0
$253$	3.70820i	0.233133i
$254$	15.7082	0.985620
$255$	0	0
$256$	13.5623	0.847644
$257$	29.3607i	1.83147i	0.401784	+	0.915734i	$0.368390\pi$
$257$	29.3607i	1.83147i	−0.401784	+	0.915734i	$0.631610\pi$
$258$	0	0
$259$	11.1246	0.691250
$260$	0	0
$261$	0	0
$262$	− 19.8541i	− 1.22659i
$263$	− 16.3262i	− 1.00672i	−0.864077	−	0.503359i	$-0.832097\pi$
$263$	− 16.3262i	− 1.00672i	0.864077	−	0.503359i	$-0.167903\pi$
$264$	0	0
$265$	0	0
$266$	17.5623	1.07681
$267$	0	0
$268$	− 8.56231i	− 0.523026i
$269$	2.56231	0.156227	0.0781133	−	0.996944i	$-0.475110\pi$
$269$	2.56231	0.156227	0.0781133	+	0.996944i	$0.475110\pi$
$270$	0	0
$271$	20.0902	1.22039	0.610195	−	0.792251i	$-0.291091\pi$
$271$	20.0902	1.22039	0.610195	+	0.792251i	$0.291091\pi$
$272$	− 20.5623i	− 1.24677i
$273$	0	0
$274$	9.09017	0.549157
$275$	0	0
$276$	0	0
$277$	− 5.29180i	− 0.317953i	−0.987282	−	0.158977i	$-0.949181\pi$
$277$	− 5.29180i	− 0.317953i	0.987282	−	0.158977i	$-0.0508194\pi$
$278$	− 19.7984i	− 1.18743i
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	− 15.7082i	− 0.933756i	−0.884322	−	0.466878i	$-0.845379\pi$
$283$	− 15.7082i	− 0.933756i	0.884322	−	0.466878i	$-0.154621\pi$
$284$	−1.85410	−0.110021
$285$	0	0
$286$	−23.5623	−1.39327
$287$	9.00000i	0.531253i
$288$	0	0
$289$	−0.944272	−0.0555454
$290$	0	0
$291$	0	0
$292$	− 1.14590i	− 0.0670586i
$293$	21.3607i	1.24790i	0.781463	+	0.623952i	$0.214474\pi$
$293$	21.3607i	1.24790i	−0.781463	+	0.623952i	$0.785526\pi$
$294$	0	0
$295$	0	0
$296$	−8.29180	−0.481951
$297$	0	0
$298$	24.2705i	1.40595i
$299$	6.00000	0.346989
$300$	0	0
$301$	27.0000	1.55625
$302$	34.1246i	1.96365i
$303$	0	0
$304$	−17.5623	−1.00727
$305$	0	0
$306$	0	0
$307$	− 6.27051i	− 0.357877i	−0.983860	−	0.178938i	$-0.942734\pi$
$307$	− 6.27051i	− 0.357877i	0.983860	−	0.178938i	$-0.0572663\pi$
$308$	− 5.56231i	− 0.316942i
$309$	0	0
$310$	0	0
$311$	−21.2705	−1.20614	−0.603070	−	0.797688i	$-0.706056\pi$
$311$	−21.2705	−1.20614	−0.603070	+	0.797688i	$0.706056\pi$
$312$	0	0
$313$	19.4164i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$313$	19.4164i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$314$	19.8541	1.12043
$315$	0	0
$316$	0.326238	0.0183523
$317$	− 26.9443i	− 1.51334i	−0.653796	−	0.756671i	$-0.726825\pi$
$317$	− 26.9443i	− 1.51334i	0.653796	−	0.756671i	$-0.273175\pi$
$318$	0	0
$319$	20.1246	1.12676
$320$	0	0
$321$	0	0
$322$	6.00000i	0.334367i
$323$	15.3262i	0.852775i
$324$	0	0
$325$	0	0
$326$	−32.1246	−1.77922
$327$	0	0
$328$	− 6.70820i	− 0.370399i
$329$	24.9787	1.37712
$330$	0	0
$331$	−21.0902	−1.15922	−0.579610	−	0.814894i	$-0.696795\pi$
$331$	−21.0902	−1.15922	−0.579610	+	0.814894i	$0.696795\pi$
$332$	0.291796i	0.0160144i
$333$	0	0
$334$	14.9443	0.817714
$335$	0	0
$336$	0	0
$337$	29.8328i	1.62510i	0.582894	+	0.812549i	$0.301920\pi$
$337$	29.8328i	1.62510i	−0.582894	+	0.812549i	$0.698080\pi$
$338$	17.0902i	0.929583i
$339$	0	0
$340$	0	0
$341$	15.2705	0.826944
$342$	0	0
$343$	15.0000i	0.809924i
$344$	−20.1246	−1.08505
$345$	0	0
$346$	0.0901699	0.00484757
$347$	15.9443i	0.855933i	0.903795	+	0.427967i	$0.140770\pi$
$347$	15.9443i	0.855933i	−0.903795	+	0.427967i	$0.859230\pi$
$348$	0	0
$349$	17.3607	0.929296	0.464648	−	0.885496i	$-0.346181\pi$
$349$	17.3607	0.929296	0.464648	+	0.885496i	$0.346181\pi$
$350$	0	0
$351$	0	0
$352$	10.1459i	0.540778i
$353$	− 3.88854i	− 0.206966i	−0.994631	−	0.103483i	$-0.967001\pi$
$353$	− 3.88854i	− 0.206966i	0.994631	−	0.103483i	$-0.0329988\pi$
$354$	0	0
$355$	0	0
$356$	8.29180	0.439464
$357$	0	0
$358$	− 10.8541i	− 0.573657i
$359$	−10.8541	−0.572858	−0.286429	−	0.958102i	$-0.592468\pi$
$359$	−10.8541	−0.572858	−0.286429	+	0.958102i	$0.592468\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	− 29.4164i	− 1.54609i
$363$	0	0
$364$	−9.00000	−0.471728
$365$	0	0
$366$	0	0
$367$	− 1.14590i	− 0.0598154i	−0.999553	−	0.0299077i	$-0.990479\pi$
$367$	− 1.14590i	− 0.0598154i	0.999553	−	0.0299077i	$-0.00952133\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	0	0
$371$	13.8541	0.719269
$372$	0	0
$373$	1.85410i	0.0960018i	0.998847	+	0.0480009i	$0.0152850\pi$
$373$	1.85410i	0.0960018i	−0.998847	+	0.0480009i	$0.984715\pi$
$374$	20.5623	1.06325
$375$	0	0
$376$	−18.6180	−0.960152
$377$	− 32.5623i	− 1.67704i
$378$	0	0
$379$	7.56231	0.388450	0.194225	−	0.980957i	$-0.437781\pi$
$379$	7.56231	0.388450	0.194225	+	0.980957i	$0.437781\pi$
$380$	0	0
$381$	0	0
$382$	19.4164i	0.993430i
$383$	− 35.4721i	− 1.81254i	−0.422698	−	0.906271i	$-0.638917\pi$
$383$	− 35.4721i	− 1.81254i	0.422698	−	0.906271i	$-0.361083\pi$
$384$	0	0
$385$	0	0
$386$	24.7082	1.25761
$387$	0	0
$388$	4.85410i	0.246430i
$389$	1.58359	0.0802913	0.0401457	−	0.999194i	$-0.487218\pi$
$389$	1.58359	0.0802913	0.0401457	+	0.999194i	$0.487218\pi$
$390$	0	0
$391$	−5.23607	−0.264799
$392$	4.47214i	0.225877i
$393$	0	0
$394$	−6.56231	−0.330604
$395$	0	0
$396$	0	0
$397$	16.4164i	0.823916i	0.911203	+	0.411958i	$0.135155\pi$
$397$	16.4164i	0.823916i	−0.911203	+	0.411958i	$0.864845\pi$
$398$	26.1803i	1.31230i
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	− 24.7082i	− 1.23080i
$404$	−1.85410	−0.0922450
$405$	0	0
$406$	32.5623	1.61604
$407$	− 11.1246i	− 0.551427i
$408$	0	0
$409$	22.7639	1.12560	0.562802	−	0.826592i	$-0.309723\pi$
$409$	22.7639	1.12560	0.562802	+	0.826592i	$0.309723\pi$
$410$	0	0
$411$	0	0
$412$	− 7.85410i	− 0.386944i
$413$	12.4377i	0.612019i
$414$	0	0
$415$	0	0
$416$	16.4164	0.804881
$417$	0	0
$418$	− 17.5623i	− 0.859000i
$419$	−6.70820	−0.327717	−0.163859	−	0.986484i	$-0.552394\pi$
$419$	−6.70820	−0.327717	−0.163859	+	0.986484i	$0.552394\pi$
$420$	0	0
$421$	8.90983	0.434239	0.217119	−	0.976145i	$-0.430334\pi$
$421$	8.90983	0.434239	0.217119	+	0.976145i	$0.430334\pi$
$422$	6.76393i	0.329263i
$423$	0	0
$424$	−10.3262	−0.501486
$425$	0	0
$426$	0	0
$427$	− 18.2705i	− 0.884172i
$428$	− 0.0557281i	− 0.00269372i
$429$	0	0
$430$	0	0
$431$	12.2705	0.591050	0.295525	−	0.955335i	$-0.404505\pi$
$431$	12.2705	0.591050	0.295525	+	0.955335i	$0.404505\pi$
$432$	0	0
$433$	14.2918i	0.686820i	0.939186	+	0.343410i	$0.111582\pi$
$433$	14.2918i	0.686820i	−0.939186	+	0.343410i	$0.888418\pi$
$434$	24.7082	1.18603
$435$	0	0
$436$	3.29180	0.157648
$437$	4.47214i	0.213931i
$438$	0	0
$439$	−30.1246	−1.43777	−0.718885	−	0.695129i	$-0.755347\pi$
$439$	−30.1246	−1.43777	−0.718885	+	0.695129i	$0.755347\pi$
$440$	0	0
$441$	0	0
$442$	− 33.2705i	− 1.58252i
$443$	− 7.18034i	− 0.341148i	−0.985345	−	0.170574i	$-0.945438\pi$
$443$	− 7.18034i	− 0.341148i	0.985345	−	0.170574i	$-0.0545622\pi$
$444$	0	0
$445$	0	0
$446$	3.00000	0.142054
$447$	0	0
$448$	− 12.7082i	− 0.600406i
$449$	−39.2705	−1.85329	−0.926645	−	0.375938i	$-0.877321\pi$
$449$	−39.2705	−1.85329	−0.926645	+	0.375938i	$0.877321\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	1.27051i	0.0597598i
$453$	0	0
$454$	−10.7082	−0.502561
$455$	0	0
$456$	0	0
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	− 2.23607i	− 0.104485i
$459$	0	0
$460$	0	0
$461$	−36.2705	−1.68929	−0.844643	−	0.535330i	$-0.820187\pi$
$461$	−36.2705	−1.68929	−0.844643	+	0.535330i	$0.820187\pi$
$462$	0	0
$463$	− 14.1246i	− 0.656426i	−0.944604	−	0.328213i	$-0.893554\pi$
$463$	− 14.1246i	− 0.656426i	0.944604	−	0.328213i	$-0.106446\pi$
$464$	−32.5623	−1.51167
$465$	0	0
$466$	43.5066	2.01540
$467$	39.2361i	1.81563i	0.419372	+	0.907814i	$0.362250\pi$
$467$	39.2361i	1.81563i	−0.419372	+	0.907814i	$0.637750\pi$
$468$	0	0
$469$	−41.5623	−1.91917
$470$	0	0
$471$	0	0
$472$	− 9.27051i	− 0.426710i
$473$	− 27.0000i	− 1.24146i
$474$	0	0
$475$	0	0
$476$	7.85410	0.359992
$477$	0	0
$478$	− 41.8328i	− 1.91339i
$479$	−32.5623	−1.48781	−0.743905	−	0.668286i	$-0.767028\pi$
$479$	−32.5623	−1.48781	−0.743905	+	0.668286i	$0.767028\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	31.0344i	1.41358i
$483$	0	0
$484$	1.23607	0.0561849
$485$	0	0
$486$	0	0
$487$	− 3.70820i	− 0.168035i	−0.996464	−	0.0840174i	$-0.973225\pi$
$487$	− 3.70820i	− 0.168035i	0.996464	−	0.0840174i	$-0.0267751\pi$
$488$	13.6180i	0.616459i
$489$	0	0
$490$	0	0
$491$	−36.2705	−1.63687	−0.818433	−	0.574603i	$-0.805157\pi$
$491$	−36.2705	−1.63687	−0.818433	+	0.574603i	$0.805157\pi$
$492$	0	0
$493$	28.4164i	1.27981i
$494$	−28.4164	−1.27851
$495$	0	0
$496$	−24.7082	−1.10943
$497$	9.00000i	0.403705i
$498$	0	0
$499$	−42.3607	−1.89632	−0.948162	−	0.317787i	$-0.897060\pi$
$499$	−42.3607	−1.89632	−0.948162	+	0.317787i	$0.897060\pi$
$500$	0	0
$501$	0	0
$502$	10.1459i	0.452834i
$503$	− 25.7984i	− 1.15029i	−0.818051	−	0.575146i	$-0.804945\pi$
$503$	− 25.7984i	− 1.15029i	0.818051	−	0.575146i	$-0.195055\pi$
$504$	0	0
$505$	0	0
$506$	6.00000	0.266733
$507$	0	0
$508$	− 6.00000i	− 0.266207i
$509$	13.4164	0.594672	0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164	0.594672	0.297336	+	0.954773i	$0.403902\pi$
$510$	0	0
$511$	−5.56231	−0.246062
$512$	5.29180i	0.233867i
$513$	0	0
$514$	47.5066	2.09543
$515$	0	0
$516$	0	0
$517$	− 24.9787i	− 1.09856i
$518$	− 18.0000i	− 0.790875i
$519$	0	0
$520$	0	0
$521$	−6.27051	−0.274716	−0.137358	−	0.990521i	$-0.543861\pi$
$521$	−6.27051	−0.274716	−0.137358	+	0.990521i	$0.543861\pi$
$522$	0	0
$523$	− 37.4164i	− 1.63611i	−0.575143	−	0.818053i	$-0.695054\pi$
$523$	− 37.4164i	− 1.63611i	0.575143	−	0.818053i	$-0.304946\pi$
$524$	−7.58359	−0.331291
$525$	0	0
$526$	−26.4164	−1.15181
$527$	21.5623i	0.939269i
$528$	0	0
$529$	21.4721	0.933571
$530$	0	0
$531$	0	0
$532$	− 6.70820i	− 0.290838i
$533$	− 14.5623i	− 0.630763i
$534$	0	0
$535$	0	0
$536$	30.9787	1.33808
$537$	0	0
$538$	− 4.14590i	− 0.178742i
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−34.1803	−1.46953	−0.734764	−	0.678323i	$-0.762707\pi$
$541$	−34.1803	−1.46953	−0.734764	+	0.678323i	$0.762707\pi$
$542$	− 32.5066i	− 1.39628i
$543$	0	0
$544$	−14.3262	−0.614232
$545$	0	0
$546$	0	0
$547$	22.1459i	0.946890i	0.880823	+	0.473445i	$0.156990\pi$
$547$	22.1459i	0.946890i	−0.880823	+	0.473445i	$0.843010\pi$
$548$	− 3.47214i	− 0.148322i
$549$	0	0
$550$	0	0
$551$	24.2705	1.03396
$552$	0	0
$553$	− 1.58359i	− 0.0673412i
$554$	−8.56231	−0.363778
$555$	0	0
$556$	−7.56231	−0.320713
$557$	− 40.3607i	− 1.71014i	−0.518515	−	0.855068i	$-0.673515\pi$
$557$	− 40.3607i	− 1.71014i	0.518515	−	0.855068i	$-0.326485\pi$
$558$	0	0
$559$	−43.6869	−1.84776
$560$	0	0
$561$	0	0
$562$	19.4164i	0.819032i
$563$	− 7.05573i	− 0.297363i	−0.988885	−	0.148682i	$-0.952497\pi$
$563$	− 7.05573i	− 0.297363i	0.988885	−	0.148682i	$-0.0475030\pi$
$564$	0	0
$565$	0	0
$566$	−25.4164	−1.06833
$567$	0	0
$568$	− 6.70820i	− 0.281470i
$569$	26.8328	1.12489	0.562445	−	0.826835i	$-0.309861\pi$
$569$	26.8328	1.12489	0.562445	+	0.826835i	$0.309861\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	9.00000i	0.376309i
$573$	0	0
$574$	14.5623	0.607819
$575$	0	0
$576$	0	0
$577$	− 29.5623i	− 1.23069i	−0.788256	−	0.615347i	$-0.789016\pi$
$577$	− 29.5623i	− 1.23069i	0.788256	−	0.615347i	$-0.210984\pi$
$578$	1.52786i	0.0635508i
$579$	0	0
$580$	0	0
$581$	1.41641	0.0587625
$582$	0	0
$583$	− 13.8541i	− 0.573778i
$584$	4.14590	0.171558
$585$	0	0
$586$	34.5623	1.42776
$587$	− 2.47214i	− 0.102036i	−0.998698	−	0.0510180i	$-0.983753\pi$
$587$	− 2.47214i	− 0.102036i	0.998698	−	0.0510180i	$-0.0162466\pi$
$588$	0	0
$589$	18.4164	0.758835
$590$	0	0
$591$	0	0
$592$	18.0000i	0.739795i
$593$	41.3607i	1.69848i	0.528007	+	0.849240i	$0.322939\pi$
$593$	41.3607i	1.69848i	−0.528007	+	0.849240i	$0.677061\pi$
$594$	0	0
$595$	0	0
$596$	9.27051	0.379735
$597$	0	0
$598$	− 9.70820i	− 0.396998i
$599$	−24.2705	−0.991666	−0.495833	−	0.868418i	$-0.665137\pi$
$599$	−24.2705	−0.991666	−0.495833	+	0.868418i	$0.665137\pi$
$600$	0	0
$601$	−0.639320	−0.0260784	−0.0130392	−	0.999915i	$-0.504151\pi$
$601$	−0.639320	−0.0260784	−0.0130392	+	0.999915i	$0.504151\pi$
$602$	− 43.6869i	− 1.78055i
$603$	0	0
$604$	13.0344	0.530364
$605$	0	0
$606$	0	0
$607$	− 30.5410i	− 1.23962i	−0.784751	−	0.619811i	$-0.787209\pi$
$607$	− 30.5410i	− 1.23962i	0.784751	−	0.619811i	$-0.212791\pi$
$608$	12.2361i	0.496238i
$609$	0	0
$610$	0	0
$611$	−40.4164	−1.63507
$612$	0	0
$613$	4.41641i	0.178377i	0.996015	+	0.0891885i	$0.0284274\pi$
$613$	4.41641i	0.178377i	−0.996015	+	0.0891885i	$0.971573\pi$
$614$	−10.1459	−0.409455
$615$	0	0
$616$	20.1246	0.810844
$617$	− 25.7639i	− 1.03722i	−0.855012	−	0.518608i	$-0.826450\pi$
$617$	− 25.7639i	− 1.03722i	0.855012	−	0.518608i	$-0.173550\pi$
$618$	0	0
$619$	20.5279	0.825085	0.412542	−	0.910938i	$-0.364641\pi$
$619$	20.5279	0.825085	0.412542	+	0.910938i	$0.364641\pi$
$620$	0	0
$621$	0	0
$622$	34.4164i	1.37997i
$623$	− 40.2492i	− 1.61255i
$624$	0	0
$625$	0	0
$626$	31.4164	1.25565
$627$	0	0
$628$	− 7.58359i	− 0.302618i
$629$	15.7082	0.626327
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	1.18034i	0.0469514i
$633$	0	0
$634$	−43.5967	−1.73145
$635$	0	0
$636$	0	0
$637$	9.70820i	0.384653i
$638$	− 32.5623i	− 1.28915i
$639$	0	0
$640$	0	0
$641$	8.72949	0.344794	0.172397	−	0.985028i	$-0.444849\pi$
$641$	8.72949	0.344794	0.172397	+	0.985028i	$0.444849\pi$
$642$	0	0
$643$	− 18.2705i	− 0.720519i	−0.932852	−	0.360259i	$-0.882688\pi$
$643$	− 18.2705i	− 0.720519i	0.932852	−	0.360259i	$-0.117312\pi$
$644$	2.29180	0.0903094
$645$	0	0
$646$	24.7984	0.975679
$647$	− 20.2361i	− 0.795562i	−0.917480	−	0.397781i	$-0.869780\pi$
$647$	− 20.2361i	− 0.795562i	0.917480	−	0.397781i	$-0.130220\pi$
$648$	0	0
$649$	12.4377	0.488222
$650$	0	0
$651$	0	0
$652$	12.2705i	0.480550i
$653$	4.65248i	0.182065i	0.995848	+	0.0910327i	$0.0290168\pi$
$653$	4.65248i	0.182065i	−0.995848	+	0.0910327i	$0.970983\pi$
$654$	0	0
$655$	0	0
$656$	−14.5623	−0.568563
$657$	0	0
$658$	− 40.4164i	− 1.57560i
$659$	−25.8541	−1.00713	−0.503566	−	0.863957i	$-0.667979\pi$
$659$	−25.8541	−1.00713	−0.503566	+	0.863957i	$0.667979\pi$
$660$	0	0
$661$	14.3607	0.558566	0.279283	−	0.960209i	$-0.409903\pi$
$661$	14.3607	0.558566	0.279283	+	0.960209i	$0.409903\pi$
$662$	34.1246i	1.32629i
$663$	0	0
$664$	−1.05573	−0.0409702
$665$	0	0
$666$	0	0
$667$	8.29180i	0.321060i
$668$	− 5.70820i	− 0.220857i
$669$	0	0
$670$	0	0
$671$	−18.2705	−0.705325
$672$	0	0
$673$	− 31.6869i	− 1.22144i	−0.791846	−	0.610720i	$-0.790880\pi$
$673$	− 31.6869i	− 1.22144i	0.791846	−	0.610720i	$-0.209120\pi$
$674$	48.2705	1.85931
$675$	0	0
$676$	6.52786	0.251072
$677$	4.11146i	0.158016i	0.996874	+	0.0790080i	$0.0251753\pi$
$677$	4.11146i	0.158016i	−0.996874	+	0.0790080i	$0.974825\pi$
$678$	0	0
$679$	23.5623	0.904238
$680$	0	0
$681$	0	0
$682$	− 24.7082i	− 0.946126i
$683$	43.0689i	1.64799i	0.566601	+	0.823993i	$0.308258\pi$
$683$	43.0689i	1.64799i	−0.566601	+	0.823993i	$0.691742\pi$
$684$	0	0
$685$	0	0
$686$	24.2705	0.926652
$687$	0	0
$688$	43.6869i	1.66555i
$689$	−22.4164	−0.853997
$690$	0	0
$691$	25.8197	0.982226	0.491113	−	0.871096i	$-0.336590\pi$
$691$	25.8197	0.982226	0.491113	+	0.871096i	$0.336590\pi$
$692$	− 0.0344419i	− 0.00130928i
$693$	0	0
$694$	25.7984	0.979293
$695$	0	0
$696$	0	0
$697$	12.7082i	0.481358i
$698$	− 28.0902i	− 1.06323i
$699$	0	0
$700$	0	0
$701$	−2.72949	−0.103091	−0.0515457	−	0.998671i	$-0.516415\pi$
$701$	−2.72949	−0.103091	−0.0515457	+	0.998671i	$0.516415\pi$
$702$	0	0
$703$	− 13.4164i	− 0.506009i
$704$	−12.7082	−0.478958
$705$	0	0
$706$	−6.29180	−0.236795
$707$	9.00000i	0.338480i
$708$	0	0
$709$	17.0344	0.639742	0.319871	−	0.947461i	$-0.396360\pi$
$709$	17.0344	0.639742	0.319871	+	0.947461i	$0.396360\pi$
$710$	0	0
$711$	0	0
$712$	30.0000i	1.12430i
$713$	6.29180i	0.235630i
$714$	0	0
$715$	0	0
$716$	−4.14590	−0.154939
$717$	0	0
$718$	17.5623i	0.655419i
$719$	47.5623	1.77377	0.886887	−	0.461986i	$-0.152863\pi$
$719$	47.5623	1.77377	0.886887	+	0.461986i	$0.152863\pi$
$720$	0	0
$721$	−38.1246	−1.41983
$722$	9.56231i	0.355872i
$723$	0	0
$724$	−11.2361	−0.417585
$725$	0	0
$726$	0	0
$727$	− 38.8328i	− 1.44023i	−0.693855	−	0.720115i	$-0.744089\pi$
$727$	− 38.8328i	− 1.44023i	0.693855	−	0.720115i	$-0.255911\pi$
$728$	− 32.5623i	− 1.20684i
$729$	0	0
$730$	0	0
$731$	38.1246	1.41009
$732$	0	0
$733$	23.5623i	0.870294i	0.900360	+	0.435147i	$0.143304\pi$
$733$	23.5623i	0.870294i	−0.900360	+	0.435147i	$0.856696\pi$
$734$	−1.85410	−0.0684362
$735$	0	0
$736$	−4.18034	−0.154089
$737$	41.5623i	1.53097i
$738$	0	0
$739$	−17.7639	−0.653457	−0.326728	−	0.945118i	$-0.605946\pi$
$739$	−17.7639	−0.653457	−0.326728	+	0.945118i	$0.605946\pi$
$740$	0	0
$741$	0	0
$742$	− 22.4164i	− 0.822932i
$743$	25.9098i	0.950539i	0.879840	+	0.475270i	$0.157650\pi$
$743$	25.9098i	0.950539i	−0.879840	+	0.475270i	$0.842350\pi$
$744$	0	0
$745$	0	0
$746$	3.00000	0.109838
$747$	0	0
$748$	− 7.85410i	− 0.287174i
$749$	−0.270510	−0.00988421
$750$	0	0
$751$	−15.3607	−0.560519	−0.280260	−	0.959924i	$-0.590421\pi$
$751$	−15.3607	−0.560519	−0.280260	+	0.959924i	$0.590421\pi$
$752$	40.4164i	1.47383i
$753$	0	0
$754$	−52.6869	−1.91874
$755$	0	0
$756$	0	0
$757$	− 27.0000i	− 0.981332i	−0.871348	−	0.490666i	$-0.836754\pi$
$757$	− 27.0000i	− 0.981332i	0.871348	−	0.490666i	$-0.163246\pi$
$758$	− 12.2361i	− 0.444434i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 15.9787i	− 0.578468i
$764$	7.41641	0.268316
$765$	0	0
$766$	−57.3951	−2.07377
$767$	− 20.1246i	− 0.726658i
$768$	0	0
$769$	12.8885	0.464773	0.232386	−	0.972624i	$-0.425347\pi$
$769$	12.8885	0.464773	0.232386	+	0.972624i	$0.425347\pi$
$770$	0	0
$771$	0	0
$772$	− 9.43769i	− 0.339670i
$773$	21.2361i	0.763808i	0.924202	+	0.381904i	$0.124732\pi$
$773$	21.2361i	0.763808i	−0.924202	+	0.381904i	$0.875268\pi$
$774$	0	0
$775$	0	0
$776$	−17.5623	−0.630450
$777$	0	0
$778$	− 2.56231i	− 0.0918631i
$779$	10.8541	0.388889
$780$	0	0
$781$	9.00000	0.322045
$782$	8.47214i	0.302963i
$783$	0	0
$784$	9.70820	0.346722
$785$	0	0
$786$	0	0
$787$	11.2918i	0.402509i	0.979539	+	0.201255i	$0.0645018\pi$
$787$	11.2918i	0.402509i	−0.979539	+	0.201255i	$0.935498\pi$
$788$	2.50658i	0.0892931i
$789$	0	0
$790$	0	0
$791$	6.16718	0.219280
$792$	0	0
$793$	29.5623i	1.04979i
$794$	26.5623	0.942661
$795$	0	0
$796$	10.0000	0.354441
$797$	35.2148i	1.24737i	0.781675	+	0.623686i	$0.214366\pi$
$797$	35.2148i	1.24737i	−0.781675	+	0.623686i	$0.785634\pi$
$798$	0	0
$799$	35.2705	1.24778
$800$	0	0
$801$	0	0
$802$	19.4164i	0.685617i
$803$	5.56231i	0.196290i
$804$	0	0
$805$	0	0
$806$	−39.9787	−1.40819
$807$	0	0
$808$	− 6.70820i	− 0.235994i
$809$	−5.12461	−0.180172	−0.0900859	−	0.995934i	$-0.528714\pi$
$809$	−5.12461	−0.180172	−0.0900859	+	0.995934i	$0.528714\pi$
$810$	0	0
$811$	−11.8197	−0.415044	−0.207522	−	0.978230i	$-0.566540\pi$
$811$	−11.8197	−0.415044	−0.207522	+	0.978230i	$0.566540\pi$
$812$	− 12.4377i	− 0.436477i
$813$	0	0
$814$	−18.0000	−0.630900
$815$	0	0
$816$	0	0
$817$	− 32.5623i	− 1.13921i
$818$	− 36.8328i	− 1.28783i
$819$	0	0
$820$	0	0
$821$	−21.2705	−0.742346	−0.371173	−	0.928564i	$-0.621044\pi$
$821$	−21.2705	−0.742346	−0.371173	+	0.928564i	$0.621044\pi$
$822$	0	0
$823$	− 22.4164i	− 0.781387i	−0.920521	−	0.390693i	$-0.872235\pi$
$823$	− 22.4164i	− 0.781387i	0.920521	−	0.390693i	$-0.127765\pi$
$824$	28.4164	0.989932
$825$	0	0
$826$	20.1246	0.700225
$827$	− 31.6180i	− 1.09947i	−0.835340	−	0.549733i	$-0.814729\pi$
$827$	− 31.6180i	− 1.09947i	0.835340	−	0.549733i	$-0.185271\pi$
$828$	0	0
$829$	−6.25735	−0.217327	−0.108663	−	0.994079i	$-0.534657\pi$
$829$	−6.25735	−0.217327	−0.108663	+	0.994079i	$0.534657\pi$
$830$	0	0
$831$	0	0
$832$	20.5623i	0.712870i
$833$	− 8.47214i	− 0.293542i
$834$	0	0
$835$	0	0
$836$	−6.70820	−0.232008
$837$	0	0
$838$	10.8541i	0.374949i
$839$	29.3951	1.01483	0.507416	−	0.861701i	$-0.330601\pi$
$839$	29.3951	1.01483	0.507416	+	0.861701i	$0.330601\pi$
$840$	0	0
$841$	16.0000	0.551724
$842$	− 14.4164i	− 0.496822i
$843$	0	0
$844$	2.58359	0.0889309
$845$	0	0
$846$	0	0
$847$	− 6.00000i	− 0.206162i
$848$	22.4164i	0.769783i
$849$	0	0
$850$	0	0
$851$	4.58359	0.157124
$852$	0	0
$853$	− 5.83282i	− 0.199712i	−0.995002	−	0.0998559i	$-0.968162\pi$
$853$	− 5.83282i	− 0.199712i	0.995002	−	0.0998559i	$-0.0318382\pi$
$854$	−29.5623	−1.01160
$855$	0	0
$856$	0.201626	0.00689144
$857$	− 34.1803i	− 1.16758i	−0.811905	−	0.583789i	$-0.801569\pi$
$857$	− 34.1803i	− 1.16758i	0.811905	−	0.583789i	$-0.198431\pi$
$858$	0	0
$859$	25.1246	0.857241	0.428620	−	0.903485i	$-0.359000\pi$
$859$	25.1246	0.857241	0.428620	+	0.903485i	$0.359000\pi$
$860$	0	0
$861$	0	0
$862$	− 19.8541i	− 0.676233i
$863$	6.76393i	0.230247i	0.993351	+	0.115123i	$0.0367264\pi$
$863$	6.76393i	0.230247i	−0.993351	+	0.115123i	$0.963274\pi$
$864$	0	0
$865$	0	0
$866$	23.1246	0.785806
$867$	0	0
$868$	− 9.43769i	− 0.320336i
$869$	−1.58359	−0.0537197
$870$	0	0
$871$	67.2492	2.27865
$872$	11.9098i	0.403318i
$873$	0	0
$874$	7.23607	0.244764
$875$	0	0
$876$	0	0
$877$	18.9787i	0.640866i	0.947271	+	0.320433i	$0.103828\pi$
$877$	18.9787i	0.640866i	−0.947271	+	0.320433i	$0.896172\pi$
$878$	48.7426i	1.64498i
$879$	0	0
$880$	0	0
$881$	−45.5410	−1.53432	−0.767158	−	0.641458i	$-0.778330\pi$
$881$	−45.5410	−1.53432	−0.767158	+	0.641458i	$0.778330\pi$
$882$	0	0
$883$	17.8328i	0.600122i	0.953920	+	0.300061i	$0.0970070\pi$
$883$	17.8328i	0.600122i	−0.953920	+	0.300061i	$0.902993\pi$
$884$	−12.7082	−0.427423
$885$	0	0
$886$	−11.6180	−0.390315
$887$	− 42.4721i	− 1.42607i	−0.701126	−	0.713037i	$-0.747319\pi$
$887$	− 42.4721i	− 1.42607i	0.701126	−	0.713037i	$-0.252681\pi$
$888$	0	0
$889$	−29.1246	−0.976808
$890$	0	0
$891$	0	0
$892$	− 1.14590i	− 0.0383675i
$893$	− 30.1246i	− 1.00808i
$894$	0	0
$895$	0	0
$896$	−40.8541	−1.36484
$897$	0	0
$898$	63.5410i	2.12039i
$899$	34.1459	1.13883
$900$	0	0
$901$	19.5623	0.651715
$902$	− 14.5623i	− 0.484872i
$903$	0	0
$904$	−4.59675	−0.152886
$905$	0	0
$906$	0	0
$907$	42.2705i	1.40357i	0.712389	+	0.701785i	$0.247613\pi$
$907$	42.2705i	1.40357i	−0.712389	+	0.701785i	$0.752387\pi$
$908$	4.09017i	0.135737i
$909$	0	0
$910$	0	0
$911$	36.5410	1.21066	0.605329	−	0.795975i	$-0.293042\pi$
$911$	36.5410	1.21066	0.605329	+	0.795975i	$0.293042\pi$
$912$	0	0
$913$	− 1.41641i	− 0.0468763i
$914$	29.1246	0.963357
$915$	0	0
$916$	−0.854102	−0.0282203
$917$	36.8115i	1.21562i
$918$	0	0
$919$	24.0689	0.793959	0.396980	−	0.917827i	$-0.370058\pi$
$919$	24.0689	0.793959	0.396980	+	0.917827i	$0.370058\pi$
$920$	0	0
$921$	0	0
$922$	58.6869i	1.93275i
$923$	− 14.5623i	− 0.479324i
$924$	0	0
$925$	0	0
$926$	−22.8541	−0.751032
$927$	0	0
$928$	22.6869i	0.744735i
$929$	36.7082	1.20436	0.602179	−	0.798361i	$-0.294300\pi$
$929$	36.7082	1.20436	0.602179	+	0.798361i	$0.294300\pi$
$930$	0	0
$931$	−7.23607	−0.237153
$932$	− 16.6180i	− 0.544342i
$933$	0	0
$934$	63.4853	2.07730
$935$	0	0
$936$	0	0
$937$	− 9.43769i	− 0.308316i	−0.988046	−	0.154158i	$-0.950734\pi$
$937$	− 9.43769i	− 0.308316i	0.988046	−	0.154158i	$-0.0492665\pi$
$938$	67.2492i	2.19576i
$939$	0	0
$940$	0	0
$941$	27.2705	0.888993	0.444497	−	0.895781i	$-0.353383\pi$
$941$	27.2705	0.888993	0.444497	+	0.895781i	$0.353383\pi$
$942$	0	0
$943$	3.70820i	0.120756i
$944$	−20.1246	−0.655000
$945$	0	0
$946$	−43.6869	−1.42038
$947$	− 22.5967i	− 0.734296i	−0.930163	−	0.367148i	$-0.880334\pi$
$947$	− 22.5967i	− 0.734296i	0.930163	−	0.367148i	$-0.119666\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	0	0
$951$	0	0
$952$	28.4164i	0.920981i
$953$	− 13.1591i	− 0.426264i	−0.977023	−	0.213132i	$-0.931634\pi$
$953$	− 13.1591i	− 0.426264i	0.977023	−	0.213132i	$-0.0683664\pi$
$954$	0	0
$955$	0	0
$956$	−15.9787	−0.516789
$957$	0	0
$958$	52.6869i	1.70224i
$959$	−16.8541	−0.544247
$960$	0	0
$961$	−5.09017	−0.164199
$962$	29.1246i	0.939015i
$963$	0	0
$964$	11.8541	0.381795
$965$	0	0
$966$	0	0
$967$	− 16.1459i	− 0.519217i	−0.965714	−	0.259609i	$-0.916406\pi$
$967$	− 16.1459i	− 0.519217i	0.965714	−	0.259609i	$-0.0835935\pi$
$968$	4.47214i	0.143740i
$969$	0	0
$970$	0	0
$971$	51.5410	1.65403	0.827015	−	0.562180i	$-0.190037\pi$
$971$	51.5410	1.65403	0.827015	+	0.562180i	$0.190037\pi$
$972$	0	0
$973$	36.7082i	1.17681i
$974$	−6.00000	−0.192252
$975$	0	0
$976$	29.5623	0.946266
$977$	23.3820i	0.748055i	0.927418	+	0.374028i	$0.122024\pi$
$977$	23.3820i	0.748055i	−0.927418	+	0.374028i	$0.877976\pi$
$978$	0	0
$979$	−40.2492	−1.28637
$980$	0	0
$981$	0	0
$982$	58.6869i	1.87277i
$983$	5.25735i	0.167684i	0.996479	+	0.0838418i	$0.0267190\pi$
$983$	5.25735i	0.167684i	−0.996479	+	0.0838418i	$0.973281\pi$
$984$	0	0
$985$	0	0
$986$	45.9787	1.46426
$987$	0	0
$988$	10.8541i	0.345315i
$989$	11.1246	0.353742
$990$	0	0
$991$	−26.8197	−0.851955	−0.425977	−	0.904734i	$-0.640070\pi$
$991$	−26.8197	−0.851955	−0.425977	+	0.904734i	$0.640070\pi$
$992$	17.2148i	0.546570i
$993$	0	0
$994$	14.5623	0.461888
$995$	0	0
$996$	0	0
$997$	− 52.8541i	− 1.67391i	−0.547275	−	0.836953i	$-0.684335\pi$
$997$	− 52.8541i	− 1.67391i	0.547275	−	0.836953i	$-0.315665\pi$
$998$	68.5410i	2.16963i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1125.2.b.f.874.1		4
3.2	odd	2		125.2.b.b.124.4		4
5.2	odd	4		1125.2.a.d.1.2		2
5.3	odd	4		1125.2.a.c.1.1		2
5.4	even	2	inner	1125.2.b.f.874.4		4
12.11	even	2		2000.2.c.e.1249.3		4
15.2	even	4		125.2.a.a.1.1	✓	2
15.8	even	4		125.2.a.b.1.2	yes	2
15.14	odd	2		125.2.b.b.124.1		4
60.23	odd	4		2000.2.a.a.1.2		2
60.47	odd	4		2000.2.a.l.1.1		2
60.59	even	2		2000.2.c.e.1249.2		4
75.2	even	20		625.2.d.d.501.1		4
75.8	even	20		625.2.d.a.251.1		4
75.11	odd	10		625.2.e.g.124.2		8
75.14	odd	10		625.2.e.g.124.1		8
75.17	even	20		625.2.d.j.251.1		4
75.23	even	20		625.2.d.g.501.1		4
75.29	odd	10		625.2.e.d.249.1		8
75.38	even	20		625.2.d.g.126.1		4
75.41	odd	10		625.2.e.g.499.1		8
75.44	odd	10		625.2.e.d.374.2		8
75.47	even	20		625.2.d.j.376.1		4
75.53	even	20		625.2.d.a.376.1		4
75.56	odd	10		625.2.e.d.374.1		8
75.59	odd	10		625.2.e.g.499.2		8
75.62	even	20		625.2.d.d.126.1		4
75.71	odd	10		625.2.e.d.249.2		8
105.62	odd	4		6125.2.a.d.1.1		2
105.83	odd	4		6125.2.a.g.1.2		2
120.53	even	4		8000.2.a.d.1.2		2
120.77	even	4		8000.2.a.v.1.1		2
120.83	odd	4		8000.2.a.u.1.1		2
120.107	odd	4		8000.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.a.a.1.1	✓	2	15.2	even	4
125.2.a.b.1.2	yes	2	15.8	even	4
125.2.b.b.124.1		4	15.14	odd	2
125.2.b.b.124.4		4	3.2	odd	2
625.2.d.a.251.1		4	75.8	even	20
625.2.d.a.376.1		4	75.53	even	20
625.2.d.d.126.1		4	75.62	even	20
625.2.d.d.501.1		4	75.2	even	20
625.2.d.g.126.1		4	75.38	even	20
625.2.d.g.501.1		4	75.23	even	20
625.2.d.j.251.1		4	75.17	even	20
625.2.d.j.376.1		4	75.47	even	20
625.2.e.d.249.1		8	75.29	odd	10
625.2.e.d.249.2		8	75.71	odd	10
625.2.e.d.374.1		8	75.56	odd	10
625.2.e.d.374.2		8	75.44	odd	10
625.2.e.g.124.1		8	75.14	odd	10
625.2.e.g.124.2		8	75.11	odd	10
625.2.e.g.499.1		8	75.41	odd	10
625.2.e.g.499.2		8	75.59	odd	10
1125.2.a.c.1.1		2	5.3	odd	4
1125.2.a.d.1.2		2	5.2	odd	4
1125.2.b.f.874.1		4	1.1	even	1	trivial
1125.2.b.f.874.4		4	5.4	even	2	inner
2000.2.a.a.1.2		2	60.23	odd	4
2000.2.a.l.1.1		2	60.47	odd	4
2000.2.c.e.1249.2		4	60.59	even	2
2000.2.c.e.1249.3		4	12.11	even	2
6125.2.a.d.1.1		2	105.62	odd	4
6125.2.a.g.1.2		2	105.83	odd	4
8000.2.a.c.1.2		2	120.107	odd	4
8000.2.a.d.1.2		2	120.53	even	4
8000.2.a.u.1.1		2	120.83	odd	4
8000.2.a.v.1.1		2	120.77	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:	\( N \)	\(=\)	\( 1125 = 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1125.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} -0.618034 q^{4} +3.00000i q^{7} -2.23607i q^{8} +3.00000 q^{11} -4.85410i q^{13} +4.85410 q^{14} -4.85410 q^{16} +4.23607i q^{17} +3.61803 q^{19} -4.85410i q^{22} +1.23607i q^{23} -7.85410 q^{26} -1.85410i q^{28} +6.70820 q^{29} +5.09017 q^{31} +3.38197i q^{32} +6.85410 q^{34} -3.70820i q^{37} -5.85410i q^{38} +3.00000 q^{41} -9.00000i q^{43} -1.85410 q^{44} +2.00000 q^{46} -8.32624i q^{47} -2.00000 q^{49} +3.00000i q^{52} -4.61803i q^{53} +6.70820 q^{56} -10.8541i q^{58} +4.14590 q^{59} -6.09017 q^{61} -8.23607i q^{62} -4.23607 q^{64} +13.8541i q^{67} -2.61803i q^{68} +3.00000 q^{71} +1.85410i q^{73} -6.00000 q^{74} -2.23607 q^{76} +9.00000i q^{77} -0.527864 q^{79} -4.85410i q^{82} -0.472136i q^{83} -14.5623 q^{86} -6.70820i q^{88} -13.4164 q^{89} +14.5623 q^{91} -0.763932i q^{92} -13.4721 q^{94} -7.85410i q^{97} +3.23607i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 12 q^{11} + 6 q^{14} - 6 q^{16} + 10 q^{19} - 18 q^{26} - 2 q^{31} + 14 q^{34} + 12 q^{41} + 6 q^{44} + 8 q^{46} - 8 q^{49} + 30 q^{59} - 2 q^{61} - 8 q^{64} + 12 q^{71} - 24 q^{74} - 20 q^{79}+ \cdots - 36 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 + 12 * q^11 + 6 * q^14 - 6 * q^16 + 10 * q^19 - 18 * q^26 - 2 * q^31 + 14 * q^34 + 12 * q^41 + 6 * q^44 + 8 * q^46 - 8 * q^49 + 30 * q^59 - 2 * q^61 - 8 * q^64 + 12 * q^71 - 24 * q^74 - 20 * q^79 - 18 * q^86 + 18 * q^91 - 36 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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