LMFDB - Embedded newform 1725.2.b.g.1174.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1725,2,Mod(1174,1725)] 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("1725.1174"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1725.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,0,2,0,0,-2,0,8,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$13.7741943487$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 69)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1725.1174
Dual form			1725.2.b.g.1174.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} +4.00000i q^{22} +1.00000i q^{23} +3.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -4.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} -2.00000i q^{38} +6.00000 q^{39} +2.00000 q^{41} -2.00000i q^{42} -10.0000i q^{43} +4.00000 q^{44} -1.00000 q^{46} +1.00000i q^{48} +3.00000 q^{49} +4.00000 q^{51} +6.00000i q^{52} +12.0000i q^{53} -1.00000 q^{54} +6.00000 q^{56} +2.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -6.00000 q^{61} +4.00000i q^{62} +2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} -10.0000i q^{67} +4.00000i q^{68} +1.00000 q^{69} +8.00000 q^{71} -3.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} -2.00000 q^{76} -8.00000i q^{77} +6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +10.0000 q^{86} +2.00000i q^{87} +12.0000i q^{88} +16.0000 q^{89} +12.0000 q^{91} +1.00000i q^{92} -4.00000i q^{93} +5.00000 q^{96} -10.0000i q^{97} +3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 4 q^{14} - 2 q^{16} - 4 q^{19} - 4 q^{21} + 6 q^{24} - 12 q^{26} - 4 q^{29} + 8 q^{31} - 8 q^{34} - 2 q^{36} + 12 q^{39} + 4 q^{41} + 8 q^{44} - 2 q^{46} + 6 q^{49}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 4 * q^14 - 2 * q^16 - 4 * q^19 - 4 * q^21 + 6 * q^24 - 12 * q^26 - 4 * q^29 + 8 * q^31 - 8 * q^34 - 2 * q^36 + 12 * q^39 + 4 * q^41 + 8 * q^44 - 2 * q^46 + 6 * q^49 + 8 * q^51 - 2 * q^54 + 12 * q^56 + 24 * q^59 - 12 * q^61 - 14 * q^64 + 8 * q^66 + 2 * q^69 + 16 * q^71 - 4 * q^74 - 4 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 + 20 * q^86 + 32 * q^89 + 24 * q^91 + 10 * q^96 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	3.00000i	1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 1.00000i	− 0.288675i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	4.00000i	0.852803i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	3.00000	0.612372
$25$	0	0
$26$	−6.00000	−1.17670
$27$	1.00000i	0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000i	0.883883i
$33$	− 4.00000i	− 0.696311i
$34$	−4.00000	−0.685994
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 2.00000i	− 0.324443i
$39$	6.00000	0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	4.00000	0.560112
$52$	6.00000i	0.832050i
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	6.00000	0.801784
$57$	2.00000i	0.264906i
$58$	− 2.00000i	− 0.262613i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	4.00000i	0.508001i
$63$	2.00000i	0.251976i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	4.00000i	0.485071i
$69$	1.00000	0.120386
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	− 3.00000i	− 0.353553i
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−2.00000	−0.229416
$77$	− 8.00000i	− 0.911685i
$78$	6.00000i	0.679366i
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	10.0000	1.07833
$87$	2.00000i	0.214423i
$88$	12.0000i	1.27920i
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	1.00000i	0.104257i
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	5.00000	0.510310
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	3.00000i	0.303046i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	4.00000i	0.396059i
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	−18.0000	−1.76505
$105$	0	0
$106$	−12.0000	−1.16554
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	1.00000i	0.0962250i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	2.00000i	0.188982i
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−2.00000	−0.185695
$117$	− 6.00000i	− 0.554700i
$118$	12.0000i	1.10469i
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	− 6.00000i	− 0.543214i
$123$	− 2.00000i	− 0.180334i
$124$	4.00000	0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	3.00000i	0.265165i
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	− 4.00000i	− 0.348155i
$133$	4.00000i	0.346844i
$134$	10.0000	0.863868
$135$	0	0
$136$	−12.0000	−1.02899
$137$	8.00000i	0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$	8.00000i	0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$	1.00000i	0.0851257i
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	24.0000i	2.00698i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	− 3.00000i	− 0.247436i
$148$	2.00000i	0.164399i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 6.00000i	− 0.486664i
$153$	− 4.00000i	− 0.323381i
$154$	8.00000	0.644658
$155$	0	0
$156$	6.00000	0.480384
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 10.0000i	− 0.795557i
$159$	12.0000	0.951662
$160$	0	0
$161$	2.00000	0.157622
$162$	1.00000i	0.0785674i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	− 6.00000i	− 0.462910i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	2.00000	0.152944
$172$	− 10.0000i	− 0.762493i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 12.0000i	− 0.901975i
$178$	16.0000i	1.19925i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	12.0000i	0.889499i
$183$	6.00000i	0.443533i
$184$	−3.00000	−0.221163
$185$	0	0
$186$	4.00000	0.293294
$187$	16.0000i	1.17004i
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	7.00000i	0.505181i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	3.00000	0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	− 4.00000i	− 0.284268i
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	14.0000i	0.985037i
$203$	4.00000i	0.280745i
$204$	4.00000	0.280056
$205$	0	0
$206$	−6.00000	−0.418040
$207$	− 1.00000i	− 0.0695048i
$208$	− 6.00000i	− 0.416025i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	12.0000i	0.824163i
$213$	− 8.00000i	− 0.548151i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	− 8.00000i	− 0.543075i
$218$	− 14.0000i	− 0.948200i
$219$	14.0000	0.946032
$220$	0	0
$221$	−24.0000	−1.61441
$222$	2.00000i	0.134231i
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	16.0000	1.06430
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	2.00000i	0.132453i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	− 6.00000i	− 0.393919i
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	10.0000i	0.649570i
$238$	8.00000i	0.518563i
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	5.00000i	0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−6.00000	−0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	− 12.0000i	− 0.763542i
$248$	12.0000i	0.762001i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	2.00000i	0.125988i
$253$	4.00000i	0.251478i
$254$	12.0000	0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	− 10.0000i	− 0.622573i
$259$	4.00000	0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	− 4.00000i	− 0.247121i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	−4.00000	−0.245256
$267$	− 16.0000i	− 0.979184i
$268$	− 10.0000i	− 0.610847i
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	− 4.00000i	− 0.242536i
$273$	− 12.0000i	− 0.726273i
$274$	−8.00000	−0.483298
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 4.00000i	− 0.239904i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	− 4.00000i	− 0.236113i
$288$	− 5.00000i	− 0.294628i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	14.0000i	0.819288i
$293$	− 20.0000i	− 1.16841i	−0.811605	−	0.584206i	$-0.801406\pi$
$293$	− 20.0000i	− 1.16841i	0.811605	−	0.584206i	$-0.198594\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−6.00000	−0.348743
$297$	4.00000i	0.232104i
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−20.0000	−1.15278
$302$	8.00000i	0.460348i
$303$	− 14.0000i	− 0.804279i
$304$	2.00000	0.114708
$305$	0	0
$306$	4.00000	0.228665
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	− 8.00000i	− 0.455842i
$309$	6.00000	0.341328
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	18.0000i	1.01905i
$313$	− 2.00000i	− 0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$	− 2.00000i	− 0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	12.0000i	0.672927i
$319$	−8.00000	−0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	2.00000i	0.111456i
$323$	− 8.00000i	− 0.445132i
$324$	1.00000	0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	14.0000i	0.774202i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	16.0000	0.875481
$335$	0	0
$336$	2.00000	0.109109
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	− 23.0000i	− 1.25104i
$339$	−16.0000	−0.869001
$340$	0	0
$341$	16.0000	0.866449
$342$	2.00000i	0.108148i
$343$	− 20.0000i	− 1.07990i
$344$	30.0000	1.61749
$345$	0	0
$346$	2.00000	0.107521
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	2.00000i	0.107211i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	20.0000i	1.06600i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	16.0000	0.847998
$357$	− 8.00000i	− 0.423405i
$358$	− 12.0000i	− 0.634220i
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 2.00000i	− 0.105118i
$363$	− 5.00000i	− 0.262432i
$364$	12.0000	0.628971
$365$	0	0
$366$	−6.00000	−0.313625
$367$	− 38.0000i	− 1.98358i	−0.127862	−	0.991792i	$-0.540812\pi$
$367$	− 38.0000i	− 1.98358i	0.127862	−	0.991792i	$-0.459188\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	−2.00000	−0.104116
$370$	0	0
$371$	24.0000	1.24602
$372$	− 4.00000i	− 0.207390i
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	2.00000i	0.102869i
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	− 12.0000i	− 0.613973i
$383$	28.0000i	1.43073i	0.698749	+	0.715367i	$0.253740\pi$
$383$	28.0000i	1.43073i	−0.698749	+	0.715367i	$0.746260\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−18.0000	−0.916176
$387$	10.0000i	0.508329i
$388$	− 10.0000i	− 0.507673i
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	9.00000i	0.454569i
$393$	4.00000i	0.201773i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	6.00000i	0.300753i
$399$	4.00000	0.200250
$400$	0	0
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	− 10.0000i	− 0.498755i
$403$	24.0000i	1.19553i
$404$	14.0000	0.696526
$405$	0	0
$406$	−4.00000	−0.198517
$407$	8.00000i	0.396545i
$408$	12.0000i	0.594089i
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	6.00000i	0.295599i
$413$	− 24.0000i	− 1.18096i
$414$	1.00000	0.0491473
$415$	0	0
$416$	−30.0000	−1.47087
$417$	4.00000i	0.195881i
$418$	− 8.00000i	− 0.391293i
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	8.00000i	0.389434i
$423$	0	0
$424$	−36.0000	−1.74831
$425$	0	0
$426$	8.00000	0.387601
$427$	12.0000i	0.580721i
$428$	12.0000i	0.580042i
$429$	24.0000	1.15873
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−14.0000	−0.670478
$437$	− 2.00000i	− 0.0956730i
$438$	14.0000i	0.668946i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 24.0000i	− 1.14156i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	0	0
$447$	0	0
$448$	14.0000i	0.661438i
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	− 16.0000i	− 0.752577i
$453$	− 8.00000i	− 0.375873i
$454$	24.0000	1.12638
$455$	0	0
$456$	−6.00000	−0.280976
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 6.00000i	− 0.280362i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	− 8.00000i	− 0.372194i
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	10.0000	0.463241
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	− 6.00000i	− 0.277350i
$469$	−20.0000	−0.923514
$470$	0	0
$471$	10.0000	0.460776
$472$	36.0000i	1.65703i
$473$	− 40.0000i	− 1.83920i
$474$	−10.0000	−0.459315
$475$	0	0
$476$	8.00000	0.366679
$477$	− 12.0000i	− 0.549442i
$478$	− 8.00000i	− 0.365911i
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 2.00000i	− 0.0910975i
$483$	− 2.00000i	− 0.0910032i
$484$	5.00000	0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	24.0000i	1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$	24.0000i	1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	− 18.0000i	− 0.814822i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	− 8.00000i	− 0.360302i
$494$	12.0000	0.539906
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 16.0000i	− 0.717698i
$498$	− 12.0000i	− 0.537733i
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	24.0000i	1.07117i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	−4.00000	−0.177822
$507$	23.0000i	1.02147i
$508$	− 12.0000i	− 0.532414i
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	− 11.0000i	− 0.486136i
$513$	− 2.00000i	− 0.0883022i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	4.00000i	0.175750i
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	2.00000i	0.0875376i
$523$	− 38.0000i	− 1.66162i	−0.556553	−	0.830812i	$-0.687876\pi$
$523$	− 38.0000i	− 1.66162i	0.556553	−	0.830812i	$-0.312124\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−12.0000	−0.523225
$527$	16.0000i	0.696971i
$528$	4.00000i	0.174078i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−12.0000	−0.520756
$532$	4.00000i	0.173422i
$533$	12.0000i	0.519778i
$534$	16.0000	0.692388
$535$	0	0
$536$	30.0000	1.29580
$537$	12.0000i	0.517838i
$538$	− 10.0000i	− 0.431131i
$539$	12.0000	0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 20.0000i	− 0.859074i
$543$	2.00000i	0.0858282i
$544$	−20.0000	−0.857493
$545$	0	0
$546$	12.0000	0.513553
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	8.00000i	0.341743i
$549$	6.00000	0.256074
$550$	0	0
$551$	4.00000	0.170406
$552$	3.00000i	0.127688i
$553$	20.0000i	0.850487i
$554$	10.0000	0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 4.00000i	− 0.169485i	−0.996403	−	0.0847427i	$-0.972993\pi$
$557$	− 4.00000i	− 0.169485i	0.996403	−	0.0847427i	$-0.0270068\pi$
$558$	− 4.00000i	− 0.169334i
$559$	60.0000	2.53773
$560$	0	0
$561$	16.0000	0.675521
$562$	− 24.0000i	− 1.01238i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	6.00000	0.252199
$567$	− 2.00000i	− 0.0839921i
$568$	24.0000i	1.00702i
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	24.0000i	1.00349i
$573$	12.0000i	0.501307i
$574$	4.00000	0.166957
$575$	0	0
$576$	7.00000	0.291667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	1.00000i	0.0415945i
$579$	18.0000	0.748054
$580$	0	0
$581$	−24.0000	−0.995688
$582$	− 10.0000i	− 0.414513i
$583$	48.0000i	1.98796i
$584$	−42.0000	−1.73797
$585$	0	0
$586$	20.0000	0.826192
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 3.00000i	− 0.123718i
$589$	−8.00000	−0.329634
$590$	0	0
$591$	18.0000	0.740421
$592$	− 2.00000i	− 0.0821995i
$593$	− 34.0000i	− 1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	− 34.0000i	− 1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	0	0
$597$	− 6.00000i	− 0.245564i
$598$	− 6.00000i	− 0.245358i
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	− 20.0000i	− 0.815139i
$603$	10.0000i	0.407231i
$604$	8.00000	0.325515
$605$	0	0
$606$	14.0000	0.568711
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	− 10.0000i	− 0.405554i
$609$	4.00000	0.162088
$610$	0	0
$611$	0	0
$612$	− 4.00000i	− 0.161690i
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	24.0000	0.966988
$617$	− 24.0000i	− 0.966204i	−0.875564	−	0.483102i	$-0.839510\pi$
$617$	− 24.0000i	− 0.966204i	0.875564	−	0.483102i	$-0.160490\pi$
$618$	6.00000i	0.241355i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	− 32.0000i	− 1.28205i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	2.00000	0.0799361
$627$	8.00000i	0.319489i
$628$	10.0000i	0.399043i
$629$	−8.00000	−0.318981
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	− 30.0000i	− 1.19334i
$633$	− 8.00000i	− 0.317971i
$634$	30.0000	1.19145
$635$	0	0
$636$	12.0000	0.475831
$637$	18.0000i	0.713186i
$638$	− 8.00000i	− 0.316723i
$639$	−8.00000	−0.316475
$640$	0	0
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	12.0000i	0.473602i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	8.00000	0.314756
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	3.00000i	0.117851i
$649$	48.0000	1.88416
$650$	0	0
$651$	−8.00000	−0.313545
$652$	− 8.00000i	− 0.313304i
$653$	42.0000i	1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	− 14.0000i	− 0.546192i
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	24.0000i	0.932083i
$664$	36.0000	1.39707
$665$	0	0
$666$	2.00000	0.0774984
$667$	− 2.00000i	− 0.0774403i
$668$	− 16.0000i	− 0.619059i
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	− 10.0000i	− 0.385758i
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−23.0000	−0.884615
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	− 16.0000i	− 0.614476i
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−24.0000	−0.919682
$682$	16.0000i	0.612672i
$683$	52.0000i	1.98972i	0.101237	+	0.994862i	$0.467720\pi$
$683$	52.0000i	1.98972i	−0.101237	+	0.994862i	$0.532280\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	20.0000	0.763604
$687$	6.00000i	0.228914i
$688$	10.0000i	0.381246i
$689$	−72.0000	−2.74298
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	− 2.00000i	− 0.0760286i
$693$	8.00000i	0.303895i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	8.00000i	0.303022i
$698$	− 2.00000i	− 0.0757011i
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	− 6.00000i	− 0.226455i
$703$	− 4.00000i	− 0.150863i
$704$	−28.0000	−1.05529
$705$	0	0
$706$	6.00000	0.225813
$707$	− 28.0000i	− 1.05305i
$708$	− 12.0000i	− 0.450988i
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	48.0000i	1.79888i
$713$	4.00000i	0.149801i
$714$	8.00000	0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	8.00000i	0.298765i
$718$	− 20.0000i	− 0.746393i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	− 15.0000i	− 0.558242i
$723$	2.00000i	0.0743808i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	5.00000	0.185567
$727$	6.00000i	0.222528i	0.993791	+	0.111264i	$0.0354899\pi$
$727$	6.00000i	0.222528i	−0.993791	+	0.111264i	$0.964510\pi$
$728$	36.0000i	1.33425i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	6.00000i	0.221766i
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	38.0000	1.40261
$735$	0	0
$736$	−5.00000	−0.184302
$737$	− 40.0000i	− 1.47342i
$738$	− 2.00000i	− 0.0736210i
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	24.0000i	0.881068i
$743$	20.0000i	0.733729i	0.930274	+	0.366864i	$0.119569\pi$
$743$	20.0000i	0.733729i	−0.930274	+	0.366864i	$0.880431\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	34.0000	1.24483
$747$	12.0000i	0.439057i
$748$	16.0000i	0.585018i
$749$	24.0000	0.876941
$750$	0	0
$751$	30.0000	1.09472	0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000	1.09472	0.547358	+	0.836899i	$0.315634\pi$
$752$	0	0
$753$	− 24.0000i	− 0.874609i
$754$	12.0000	0.437014
$755$	0	0
$756$	2.00000	0.0727393
$757$	46.0000i	1.67190i	0.548807	+	0.835949i	$0.315082\pi$
$757$	46.0000i	1.67190i	−0.548807	+	0.835949i	$0.684918\pi$
$758$	− 10.0000i	− 0.363216i
$759$	4.00000	0.145191
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	− 12.0000i	− 0.434714i
$763$	28.0000i	1.01367i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−28.0000	−1.01168
$767$	72.0000i	2.59977i
$768$	17.0000i	0.613435i
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	18.0000i	0.647834i
$773$	32.0000i	1.15096i	0.817816	+	0.575480i	$0.195185\pi$
$773$	32.0000i	1.15096i	−0.817816	+	0.575480i	$0.804815\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	30.0000	1.07694
$777$	− 4.00000i	− 0.143499i
$778$	20.0000i	0.717035i
$779$	−4.00000	−0.143315
$780$	0	0
$781$	32.0000	1.14505
$782$	− 4.00000i	− 0.143040i
$783$	− 2.00000i	− 0.0714742i
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−4.00000	−0.142675
$787$	2.00000i	0.0712923i	0.999364	+	0.0356462i	$0.0113489\pi$
$787$	2.00000i	0.0712923i	−0.999364	+	0.0356462i	$0.988651\pi$
$788$	18.0000i	0.641223i
$789$	12.0000	0.427211
$790$	0	0
$791$	−32.0000	−1.13779
$792$	− 12.0000i	− 0.426401i
$793$	− 36.0000i	− 1.27840i
$794$	18.0000	0.638796
$795$	0	0
$796$	6.00000	0.212664
$797$	44.0000i	1.55856i	0.626676	+	0.779280i	$0.284415\pi$
$797$	44.0000i	1.55856i	−0.626676	+	0.779280i	$0.715585\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	0	0
$801$	−16.0000	−0.565332
$802$	− 4.00000i	− 0.141245i
$803$	56.0000i	1.97620i
$804$	−10.0000	−0.352673
$805$	0	0
$806$	−24.0000	−0.845364
$807$	10.0000i	0.352017i
$808$	42.0000i	1.47755i
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	4.00000i	0.140372i
$813$	20.0000i	0.701431i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	−4.00000	−0.140028
$817$	20.0000i	0.699711i
$818$	26.0000i	0.909069i
$819$	−12.0000	−0.419314
$820$	0	0
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	8.00000i	0.279032i
$823$	− 52.0000i	− 1.81261i	−0.422628	−	0.906303i	$-0.638892\pi$
$823$	− 52.0000i	− 1.81261i	0.422628	−	0.906303i	$-0.361108\pi$
$824$	−18.0000	−0.627060
$825$	0	0
$826$	24.0000	0.835067
$827$	− 16.0000i	− 0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$	− 16.0000i	− 0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	− 42.0000i	− 1.45609i
$833$	12.0000i	0.415775i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−8.00000	−0.276686
$837$	4.00000i	0.138260i
$838$	4.00000i	0.138178i
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	38.0000i	1.30957i
$843$	24.0000i	0.826604i
$844$	8.00000	0.275371
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	− 12.0000i	− 0.412082i
$849$	−6.00000	−0.205919
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	− 8.00000i	− 0.274075i
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	−36.0000	−1.23045
$857$	2.00000i	0.0683187i	0.999416	+	0.0341593i	$0.0108754\pi$
$857$	2.00000i	0.0683187i	−0.999416	+	0.0341593i	$0.989125\pi$
$858$	24.0000i	0.819346i
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	2.00000	0.0679628
$867$	− 1.00000i	− 0.0339618i
$868$	− 8.00000i	− 0.271538i
$869$	−40.0000	−1.35691
$870$	0	0
$871$	60.0000	2.03302
$872$	− 42.0000i	− 1.42230i
$873$	10.0000i	0.338449i
$874$	2.00000	0.0676510
$875$	0	0
$876$	14.0000	0.473016
$877$	− 42.0000i	− 1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$	− 42.0000i	− 1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$	− 8.00000i	− 0.269987i
$879$	−20.0000	−0.674583
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	− 3.00000i	− 0.101015i
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−12.0000	−0.403148
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	6.00000i	0.201347i
$889$	−24.0000	−0.804934
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	6.00000	0.200446
$897$	6.00000i	0.200334i
$898$	− 10.0000i	− 0.333704i
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−48.0000	−1.59911
$902$	8.00000i	0.266371i
$903$	20.0000i	0.665558i
$904$	48.0000	1.59646
$905$	0	0
$906$	8.00000	0.265782
$907$	− 34.0000i	− 1.12895i	−0.825450	−	0.564476i	$-0.809078\pi$
$907$	− 34.0000i	− 1.12895i	0.825450	−	0.564476i	$-0.190922\pi$
$908$	− 24.0000i	− 0.796468i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	− 2.00000i	− 0.0662266i
$913$	− 48.0000i	− 1.58857i
$914$	18.0000	0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	8.00000i	0.264183i
$918$	− 4.00000i	− 0.132020i
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	2.00000i	0.0658665i
$923$	48.0000i	1.57994i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	−8.00000	−0.262896
$927$	− 6.00000i	− 0.197066i
$928$	− 10.0000i	− 0.328266i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	− 10.0000i	− 0.327561i
$933$	0	0
$934$	−8.00000	−0.261768
$935$	0	0
$936$	18.0000	0.588348
$937$	34.0000i	1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$	34.0000i	1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$	− 20.0000i	− 0.653023i
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	10.0000i	0.325818i
$943$	2.00000i	0.0651290i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	40.0000	1.30051
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	10.0000i	0.324785i
$949$	−84.0000	−2.72676
$950$	0	0
$951$	−30.0000	−0.972817
$952$	24.0000i	0.777844i
$953$	− 12.0000i	− 0.388718i	−0.980930	−	0.194359i	$-0.937737\pi$
$953$	− 12.0000i	− 0.388718i	0.980930	−	0.194359i	$-0.0622627\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	−8.00000	−0.258738
$957$	8.00000i	0.258603i
$958$	− 16.0000i	− 0.516937i
$959$	16.0000	0.516667
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 12.0000i	− 0.386896i
$963$	− 12.0000i	− 0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	2.00000	0.0643489
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	15.0000i	0.482118i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	8.00000i	0.256468i
$974$	−24.0000	−0.769010
$975$	0	0
$976$	6.00000	0.192055
$977$	16.0000i	0.511885i	0.966692	+	0.255943i	$0.0823858\pi$
$977$	16.0000i	0.511885i	−0.966692	+	0.255943i	$0.917614\pi$
$978$	− 8.00000i	− 0.255812i
$979$	64.0000	2.04545
$980$	0	0
$981$	14.0000	0.446986
$982$	− 20.0000i	− 0.638226i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	8.00000	0.254772
$987$	0	0
$988$	− 12.0000i	− 0.381771i
$989$	10.0000	0.317982
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	20.0000i	0.635001i
$993$	0	0
$994$	16.0000	0.507489
$995$	0	0
$996$	−12.0000	−0.380235
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	8.00000i	0.253236i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.b.g.1174.2		2
5.2	odd	4		1725.2.a.e.1.1		1
5.3	odd	4		69.2.a.a.1.1	✓	1
5.4	even	2	inner	1725.2.b.g.1174.1		2
15.2	even	4		5175.2.a.v.1.1		1
15.8	even	4		207.2.a.a.1.1		1
20.3	even	4		1104.2.a.c.1.1		1
35.13	even	4		3381.2.a.k.1.1		1
40.3	even	4		4416.2.a.x.1.1		1
40.13	odd	4		4416.2.a.f.1.1		1
55.43	even	4		8349.2.a.a.1.1		1
60.23	odd	4		3312.2.a.k.1.1		1
115.68	even	4		1587.2.a.e.1.1		1
345.68	odd	4		4761.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.a.1.1	✓	1	5.3	odd	4
207.2.a.a.1.1		1	15.8	even	4
1104.2.a.c.1.1		1	20.3	even	4
1587.2.a.e.1.1		1	115.68	even	4
1725.2.a.e.1.1		1	5.2	odd	4
1725.2.b.g.1174.1		2	5.4	even	2	inner
1725.2.b.g.1174.2		2	1.1	even	1	trivial
3312.2.a.k.1.1		1	60.23	odd	4
3381.2.a.k.1.1		1	35.13	even	4
4416.2.a.f.1.1		1	40.13	odd	4
4416.2.a.x.1.1		1	40.3	even	4
4761.2.a.b.1.1		1	345.68	odd	4
5175.2.a.v.1.1		1	15.2	even	4
8349.2.a.a.1.1		1	55.43	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:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} +4.00000i q^{22} +1.00000i q^{23} +3.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -4.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} -2.00000i q^{38} +6.00000 q^{39} +2.00000 q^{41} -2.00000i q^{42} -10.0000i q^{43} +4.00000 q^{44} -1.00000 q^{46} +1.00000i q^{48} +3.00000 q^{49} +4.00000 q^{51} +6.00000i q^{52} +12.0000i q^{53} -1.00000 q^{54} +6.00000 q^{56} +2.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -6.00000 q^{61} +4.00000i q^{62} +2.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} -10.0000i q^{67} +4.00000i q^{68} +1.00000 q^{69} +8.00000 q^{71} -3.00000i q^{72} +14.0000i q^{73} -2.00000 q^{74} -2.00000 q^{76} -8.00000i q^{77} +6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +10.0000 q^{86} +2.00000i q^{87} +12.0000i q^{88} +16.0000 q^{89} +12.0000 q^{91} +1.00000i q^{92} -4.00000i q^{93} +5.00000 q^{96} -10.0000i q^{97} +3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 4 q^{14} - 2 q^{16} - 4 q^{19} - 4 q^{21} + 6 q^{24} - 12 q^{26} - 4 q^{29} + 8 q^{31} - 8 q^{34} - 2 q^{36} + 12 q^{39} + 4 q^{41} + 8 q^{44} - 2 q^{46} + 6 q^{49}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 4 * q^14 - 2 * q^16 - 4 * q^19 - 4 * q^21 + 6 * q^24 - 12 * q^26 - 4 * q^29 + 8 * q^31 - 8 * q^34 - 2 * q^36 + 12 * q^39 + 4 * q^41 + 8 * q^44 - 2 * q^46 + 6 * q^49 + 8 * q^51 - 2 * q^54 + 12 * q^56 + 24 * q^59 - 12 * q^61 - 14 * q^64 + 8 * q^66 + 2 * q^69 + 16 * q^71 - 4 * q^74 - 4 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 + 20 * q^86 + 32 * q^89 + 24 * q^91 + 10 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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