LMFDB - Embedded newform 3450.2.d.i.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.5483886973$
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 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.i.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000i q^{22} +1.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{39} +2.00000 q^{41} -12.0000i q^{43} -4.00000 q^{44} -1.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} -6.00000i q^{58} +6.00000 q^{61} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +2.00000i q^{68} -1.00000 q^{69} +12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -4.00000i q^{83} +12.0000 q^{86} -6.00000i q^{87} -4.00000i q^{88} +6.00000 q^{89} -1.00000i q^{92} +8.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 2 q^{24} + 4 q^{26} - 12 q^{29} + 4 q^{34} + 2 q^{36} + 4 q^{39} + 4 q^{41} - 8 q^{44} - 2 q^{46} + 14 q^{49} + 4 q^{51} + 2 q^{54} + 12 q^{61} - 2 q^{64} - 8 q^{66} - 2 q^{69} + 24 q^{71} + 4 q^{74} - 8 q^{79} + 2 q^{81} + 24 q^{86} + 12 q^{89} + 16 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 2 * q^24 + 4 * q^26 - 12 * q^29 + 4 * q^34 + 2 * q^36 + 4 * q^39 + 4 * q^41 - 8 * q^44 - 2 * q^46 + 14 * q^49 + 4 * q^51 + 2 * q^54 + 12 * q^61 - 2 * q^64 - 8 * q^66 - 2 * q^69 + 24 * q^71 + 4 * q^74 - 8 * q^79 + 2 * q^81 + 24 * q^86 + 12 * q^89 + 16 * q^94 - 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

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

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
$2$	1.00000i	0.707107i
$3$	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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−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
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	1.00000	0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	4.00000i	0.696311i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	2.00000	0.280056
$52$	2.00000i	0.277350i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	2.00000i	0.242536i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000i	0.117851i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	0	0
$78$	2.00000i	0.226455i
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	− 6.00000i	− 0.643268i
$88$	− 4.00000i	− 0.426401i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	− 1.00000i	− 0.104257i
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	7.00000i	0.707107i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	2.00000i	0.198030i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	1.00000i	0.0962250i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	6.00000i	0.543214i
$123$	2.00000i	0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	12.0000	1.05654
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 4.00000i	− 0.348155i
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	−2.00000	−0.171499
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	− 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
$141$	8.00000	0.673722
$142$	12.0000i	1.00702i
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	7.00000i	0.577350i
$148$	2.00000i	0.164399i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	− 4.00000i	− 0.318223i
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	12.0000i	0.914991i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	6.00000i	0.449719i
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	6.00000i	0.443533i
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−7.00000	−0.500000
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	− 4.00000i	− 0.284268i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	6.00000i	0.422159i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−8.00000	−0.557386
$207$	− 1.00000i	− 0.0695048i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	2.00000i	0.137361i
$213$	12.0000i	0.822226i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	2.00000i	0.135457i
$219$	6.00000	0.405442
$220$	0	0
$221$	−4.00000	−0.269069
$222$	2.00000i	0.134231i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	6.00000	0.399114
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 26.0000i	− 1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	− 26.0000i	− 1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000i	0.321412i
$243$	1.00000i	0.0641500i
$244$	−6.00000	−0.384111
$245$	0	0
$246$	−2.00000	−0.127515
$247$	0	0
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	4.00000i	0.251478i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	12.0000i	0.747087i
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	12.0000i	0.733017i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−14.0000	−0.845771
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	8.00000i	0.476393i
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	6.00000i	0.351123i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	−2.00000	−0.116248
$297$	− 4.00000i	− 0.232104i
$298$	10.0000i	0.579284i
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	− 16.0000i	− 0.920697i
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	− 2.00000i	− 0.113228i
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	4.00000	0.225018
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	2.00000i	0.112154i
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	2.00000i	0.110600i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	4.00000i	0.219529i
$333$	2.00000i	0.109599i
$334$	8.00000	0.437741
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	9.00000i	0.489535i
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	14.0000	0.752645
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	6.00000i	0.321634i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	4.00000i	0.213201i
$353$	− 10.0000i	− 0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$	− 10.0000i	− 0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	8.00000i	0.422813i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	6.00000i	0.315353i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	−6.00000	−0.313625
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	1.00000i	0.0521286i
$369$	−2.00000	−0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−8.00000	−0.412568
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	16.0000i	0.818631i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	12.0000i	0.609994i
$388$	− 10.0000i	− 0.507673i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	− 7.00000i	− 0.353553i
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	4.00000	0.201008
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	4.00000i	0.200502i
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	12.0000i	0.598506i
$403$	0	0
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	− 2.00000i	− 0.0990148i
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	− 8.00000i	− 0.394132i
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	2.00000	0.0980581
$417$	− 4.00000i	− 0.195881i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	4.00000i	0.194717i
$423$	8.00000i	0.388973i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	−12.0000	−0.581402
$427$	0	0
$428$	− 4.00000i	− 0.193347i
$429$	8.00000	0.386244
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	6.00000i	0.286691i
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 4.00000i	− 0.190261i
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−16.0000	−0.757622
$447$	10.0000i	0.472984i
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	6.00000i	0.282216i
$453$	− 16.0000i	− 0.751746i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	− 14.0000i	− 0.654177i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	26.0000	1.20443
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	− 48.0000i	− 2.20704i
$474$	4.00000	0.183726
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	12.0000i	0.548867i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	− 22.0000i	− 1.00207i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	− 6.00000i	− 0.271607i
$489$	20.0000	0.904431
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	4.00000i	0.179244i
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	12.0000i	0.535586i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	9.00000i	0.399704i
$508$	− 16.0000i	− 0.709885i
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	22.0000	0.970378
$515$	0	0
$516$	−12.0000	−0.528271
$517$	− 32.0000i	− 1.40736i
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	6.00000i	0.262613i
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	0	0
$526$	−8.00000	−0.348817
$527$	0	0
$528$	4.00000i	0.174078i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−12.0000	−0.518321
$537$	8.00000i	0.345225i
$538$	18.0000i	0.776035i
$539$	28.0000	1.20605
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	− 8.00000i	− 0.343629i
$543$	6.00000i	0.257485i
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	− 14.0000i	− 0.598050i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	1.00000i	0.0425628i
$553$	0	0
$554$	14.0000	0.594803
$555$	0	0
$556$	4.00000	0.169638
$557$	26.0000i	1.10166i	0.834619	+	0.550828i	$0.185688\pi$
$557$	26.0000i	1.10166i	−0.834619	+	0.550828i	$0.814312\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	10.0000i	0.421825i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−28.0000	−1.17693
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	8.00000i	0.334497i
$573$	16.0000i	0.668410i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	13.0000i	0.540729i
$579$	6.00000	0.249351
$580$	0	0
$581$	0	0
$582$	− 10.0000i	− 0.414513i
$583$	− 8.00000i	− 0.331326i
$584$	−6.00000	−0.248282
$585$	0	0
$586$	18.0000	0.743573
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	− 7.00000i	− 0.288675i
$589$	0	0
$590$	0	0
$591$	−22.0000	−0.904959
$592$	− 2.00000i	− 0.0821995i
$593$	− 42.0000i	− 1.72473i	−0.506284	−	0.862367i	$-0.668981\pi$
$593$	− 42.0000i	− 1.72473i	0.506284	−	0.862367i	$-0.331019\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	4.00000i	0.163709i
$598$	2.00000i	0.0817861i
$599$	−44.0000	−1.79779	−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000	−1.79779	−0.898896	+	0.438163i	$0.855629\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	16.0000	0.651031
$605$	0	0
$606$	−6.00000	−0.243733
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	− 2.00000i	− 0.0808452i
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	− 8.00000i	− 0.321807i
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	− 12.0000i	− 0.481156i
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	4.00000i	0.159111i
$633$	4.00000i	0.158986i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	− 14.0000i	− 0.554700i
$638$	− 24.0000i	− 0.950169i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	− 4.00000i	− 0.157867i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16.0000i	− 0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$	− 16.0000i	− 0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	2.00000	0.0780869
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	− 12.0000i	− 0.466393i
$663$	− 4.00000i	− 0.155347i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	− 6.00000i	− 0.232321i
$668$	8.00000i	0.309529i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	6.00000i	0.230429i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 14.0000i	− 0.534133i
$688$	− 12.0000i	− 0.457496i
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	14.0000i	0.532200i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	− 4.00000i	− 0.151511i
$698$	2.00000i	0.0757011i
$699$	26.0000	0.983410
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	0	0
$704$	−4.00000	−0.150756
$705$	0	0
$706$	10.0000	0.376355
$707$	0	0
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−8.00000	−0.298974
$717$	12.0000i	0.448148i
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	0	0
$722$	− 19.0000i	− 0.707107i
$723$	− 22.0000i	− 0.818189i
$724$	−6.00000	−0.222988
$725$	0	0
$726$	−5.00000	−0.185567
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	− 6.00000i	− 0.221766i
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	− 48.0000i	− 1.76810i
$738$	− 2.00000i	− 0.0736210i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	4.00000i	0.146352i
$748$	8.00000i	0.292509i
$749$	0	0
$750$	0	0
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	− 8.00000i	− 0.291730i
$753$	12.0000i	0.437304i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	− 50.0000i	− 1.81728i	−0.417579	−	0.908640i	$-0.637121\pi$
$757$	− 50.0000i	− 1.81728i	0.417579	−	0.908640i	$-0.362879\pi$
$758$	− 8.00000i	− 0.290573i
$759$	−4.00000	−0.145191
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	− 16.0000i	− 0.579619i
$763$	0	0
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	6.00000i	0.215945i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	0	0
$780$	0	0
$781$	48.0000	1.71758
$782$	2.00000i	0.0715199i
$783$	6.00000i	0.214423i
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	− 22.0000i	− 0.783718i
$789$	−8.00000	−0.284808
$790$	0	0
$791$	0	0
$792$	4.00000i	0.142134i
$793$	− 12.0000i	− 0.426132i
$794$	−10.0000	−0.354887
$795$	0	0
$796$	−4.00000	−0.141776
$797$	− 38.0000i	− 1.34603i	−0.739629	−	0.673015i	$-0.764999\pi$
$797$	− 38.0000i	− 1.34603i	0.739629	−	0.673015i	$-0.235001\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	18.0000i	0.635602i
$803$	− 24.0000i	− 0.846942i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	0	0
$807$	18.0000i	0.633630i
$808$	− 6.00000i	− 0.211079i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	8.00000	0.280400
$815$	0	0
$816$	2.00000	0.0700140
$817$	0	0
$818$	− 10.0000i	− 0.349642i
$819$	0	0
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	− 14.0000i	− 0.488306i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	1.00000i	0.0347524i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	2.00000i	0.0693375i
$833$	− 14.0000i	− 0.485071i
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	0	0
$838$	12.0000i	0.414533i
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 2.00000i	− 0.0689246i
$843$	10.0000i	0.344418i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−8.00000	−0.275046
$847$	0	0
$848$	− 2.00000i	− 0.0686803i
$849$	−28.0000	−0.960958
$850$	0	0
$851$	2.00000	0.0685591
$852$	− 12.0000i	− 0.411113i
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	0	0
$855$	0	0
$856$	4.00000	0.136717
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	8.00000i	0.273115i
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	32.0000i	1.08992i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	26.0000	0.883516
$867$	13.0000i	0.441503i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−24.0000	−0.813209
$872$	− 2.00000i	− 0.0677285i
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	− 7.00000i	− 0.235702i
$883$	12.0000i	0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$	12.0000i	0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−36.0000	−1.20944
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	2.00000i	0.0667781i
$898$	14.0000i	0.467186i
$899$	0	0
$900$	0	0
$901$	−4.00000	−0.133259
$902$	8.00000i	0.266371i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	16.0000	0.531564
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	− 2.00000i	− 0.0660098i
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	30.0000i	0.987997i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	− 8.00000i	− 0.262754i
$928$	− 6.00000i	− 0.196960i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	26.0000i	0.851658i
$933$	− 12.0000i	− 0.392862i
$934$	36.0000	1.17796
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	− 14.0000i	− 0.456145i
$943$	2.00000i	0.0651290i
$944$	0	0
$945$	0	0
$946$	48.0000	1.56061
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	4.00000i	0.129914i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−12.0000	−0.388108
$957$	− 24.0000i	− 0.775810i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 4.00000i	− 0.128965i
$963$	− 4.00000i	− 0.128898i
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	6.00000	0.192055
$977$	− 26.0000i	− 0.831814i	−0.909407	−	0.415907i	$-0.863464\pi$
$977$	− 26.0000i	− 0.831814i	0.909407	−	0.415907i	$-0.136536\pi$
$978$	20.0000i	0.639529i
$979$	24.0000	0.767043
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	− 32.0000i	− 1.02116i
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	− 12.0000i	− 0.379853i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.i.2899.2		2
5.2	odd	4		690.2.a.f.1.1	✓	1
5.3	odd	4		3450.2.a.q.1.1		1
5.4	even	2	inner	3450.2.d.i.2899.1		2
15.2	even	4		2070.2.a.m.1.1		1
20.7	even	4		5520.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.f.1.1	✓	1	5.2	odd	4
2070.2.a.m.1.1		1	15.2	even	4
3450.2.a.q.1.1		1	5.3	odd	4
3450.2.d.i.2899.1		2	5.4	even	2	inner
3450.2.d.i.2899.2		2	1.1	even	1	trivial
5520.2.a.j.1.1		1	20.7	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000i q^{22} +1.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{39} +2.00000 q^{41} -12.0000i q^{43} -4.00000 q^{44} -1.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} -6.00000i q^{58} +6.00000 q^{61} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +2.00000i q^{68} -1.00000 q^{69} +12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -4.00000i q^{83} +12.0000 q^{86} -6.00000i q^{87} -4.00000i q^{88} +6.00000 q^{89} -1.00000i q^{92} +8.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 2 q^{24} + 4 q^{26} - 12 q^{29} + 4 q^{34} + 2 q^{36} + 4 q^{39} + 4 q^{41} - 8 q^{44} - 2 q^{46} + 14 q^{49} + 4 q^{51} + 2 q^{54} + 12 q^{61} - 2 q^{64} - 8 q^{66} - 2 q^{69} + 24 q^{71} + 4 q^{74} - 8 q^{79} + 2 q^{81} + 24 q^{86} + 12 q^{89} + 16 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 2 * q^24 + 4 * q^26 - 12 * q^29 + 4 * q^34 + 2 * q^36 + 4 * q^39 + 4 * q^41 - 8 * q^44 - 2 * q^46 + 14 * q^49 + 4 * q^51 + 2 * q^54 + 12 * q^61 - 2 * q^64 - 8 * q^66 - 2 * q^69 + 24 * q^71 + 4 * q^74 - 8 * q^79 + 2 * q^81 + 24 * q^86 + 12 * q^89 + 16 * q^94 - 2 * q^96 - 8 * q^99

Introduction

L-functions

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