LMFDB - Embedded newform 3450.2.d.m.2899.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} +2.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -4.00000 q^{29} -1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -8.00000i q^{38} -4.00000 q^{39} -2.00000 q^{41} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +8.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +2.00000 q^{61} -1.00000 q^{64} -2.00000 q^{66} +6.00000i q^{67} +6.00000i q^{68} +1.00000 q^{69} +10.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} -8.00000 q^{76} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +8.00000i q^{83} +2.00000 q^{86} -4.00000i q^{87} -2.00000i q^{88} +12.0000 q^{89} +1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} +16.0000i q^{97} -7.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{11} + 2 q^{16} + 16 q^{19} - 2 q^{24} + 8 q^{26} - 8 q^{29} - 12 q^{34} + 2 q^{36} - 8 q^{39} - 4 q^{41} + 4 q^{44} - 2 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} - 16 q^{59} + 4 q^{61} - 2 q^{64} - 4 q^{66} + 2 q^{69} + 20 q^{71} + 4 q^{74} - 16 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{86} + 24 q^{89} + 24 q^{94} + 2 q^{96} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^11 + 2 * q^16 + 16 * q^19 - 2 * q^24 + 8 * q^26 - 8 * q^29 - 12 * q^34 + 2 * q^36 - 8 * q^39 - 4 * q^41 + 4 * q^44 - 2 * q^46 + 14 * q^49 + 12 * q^51 - 2 * q^54 - 16 * q^59 + 4 * q^61 - 2 * q^64 - 4 * q^66 + 2 * q^69 + 20 * q^71 + 4 * q^74 - 16 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^86 + 24 * q^89 + 24 * q^94 + 2 * q^96 + 4 * 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
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	− 1.00000i	− 0.288675i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.00000i	0.235702i
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	2.00000i	0.426401i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 2.00000i	− 0.348155i
$34$	−6.00000	−1.02899
$35$	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	$0.947432\pi$
$38$	− 8.00000i	− 1.29777i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	6.00000	0.840168
$52$	− 4.00000i	− 0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	8.00000i	1.05963i
$58$	4.00000i	0.525226i
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	6.00000i	0.727607i
$69$	1.00000	0.120386
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	4.00000i	0.452911i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	− 4.00000i	− 0.428845i
$88$	− 2.00000i	− 0.213201i
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	1.00000i	0.104257i
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	− 7.00000i	− 0.707107i
$99$	2.00000	0.201008
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	− 6.00000i	− 0.594089i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	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
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	4.00000	0.371391
$117$	− 4.00000i	− 0.369800i
$118$	8.00000i	0.736460i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 2.00000i	− 0.181071i
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	1.00000i	0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	2.00000i	0.174078i
$133$	0	0
$134$	6.00000	0.518321
$135$	0	0
$136$	6.00000	0.514496
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	− 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
$141$	−12.0000	−1.01058
$142$	− 10.0000i	− 0.839181i
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	7.00000i	0.577350i
$148$	− 2.00000i	− 0.164399i
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	8.00000i	0.648886i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 8.00000i	− 0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	8.00000	0.620920
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−8.00000	−0.611775
$172$	− 2.00000i	− 0.152499i
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−2.00000	−0.150756
$177$	− 8.00000i	− 0.601317i
$178$	− 12.0000i	− 0.899438i
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	12.0000i	0.877527i
$188$	− 12.0000i	− 0.875190i
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	26.0000i	1.87152i	0.352636	+	0.935760i	$0.385285\pi$
$193$	26.0000i	1.87152i	−0.352636	+	0.935760i	$0.614715\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−7.00000	−0.500000
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	− 2.00000i	− 0.142134i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	− 12.0000i	− 0.844317i
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	4.00000i	0.277350i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000i	0.412082i
$213$	10.0000i	0.685189i
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	2.00000	0.135147
$220$	0	0
$221$	24.0000	1.61441
$222$	2.00000i	0.134231i
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	0	0
$225$	0	0
$226$	18.0000	1.19734
$227$	− 28.0000i	− 1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$	− 28.0000i	− 1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$	− 8.00000i	− 0.529813i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	− 4.00000i	− 0.262613i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	8.00000	0.520756
$237$	8.00000i	0.519656i
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	7.00000i	0.449977i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−2.00000	−0.127515
$247$	32.0000i	2.03611i
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	−2.00000	−0.125491
$255$	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	$-0.240706\pi$
$258$	2.00000i	0.124515i
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	4.00000i	0.247121i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	0	0
$267$	12.0000i	0.734388i
$268$	− 6.00000i	− 0.366508i
$269$	−8.00000	−0.487769	−0.243884	−	0.969804i	$-0.578422\pi$
$269$	−8.00000	−0.487769	−0.243884	+	0.969804i	$0.578422\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	12.0000i	0.714590i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−16.0000	−0.937937
$292$	2.00000i	0.117041i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−2.00000	−0.116248
$297$	2.00000i	0.116052i
$298$	14.0000i	0.810998i
$299$	4.00000	0.231326
$300$	0	0
$301$	0	0
$302$	4.00000i	0.230174i
$303$	12.0000i	0.689382i
$304$	8.00000	0.458831
$305$	0	0
$306$	6.00000	0.342997
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	− 4.00000i	− 0.226455i
$313$	28.0000i	1.58265i	0.611393	+	0.791327i	$0.290609\pi$
$313$	28.0000i	1.58265i	−0.611393	+	0.791327i	$0.709391\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	− 6.00000i	− 0.336463i
$319$	8.00000	0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 48.0000i	− 2.67079i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	10.0000i	0.553001i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	− 8.00000i	− 0.439057i
$333$	− 2.00000i	− 0.109599i
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	28.0000i	1.52526i	0.646837	+	0.762629i	$0.276092\pi$
$337$	28.0000i	1.52526i	−0.646837	+	0.762629i	$0.723908\pi$
$338$	3.00000i	0.163178i
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	8.00000i	0.432590i
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−10.0000	−0.537603
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	4.00000i	0.214423i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	2.00000i	0.106600i
$353$	10.0000i	0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$	10.0000i	0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	−12.0000	−0.635999
$357$	0	0
$358$	16.0000i	0.845626i
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 10.0000i	− 0.525588i
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	− 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
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	−12.0000	−0.618853
$377$	− 16.0000i	− 0.824042i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	− 24.0000i	− 1.22795i
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	26.0000	1.32337
$387$	− 2.00000i	− 0.101666i
$388$	− 16.0000i	− 0.812277i
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	7.00000i	0.353553i
$393$	− 4.00000i	− 0.201773i
$394$	18.0000	0.906827
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	6.00000i	0.299253i
$403$	0	0
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	− 4.00000i	− 0.198273i
$408$	6.00000i	0.297044i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	4.00000	0.196116
$417$	− 4.00000i	− 0.195881i
$418$	16.0000i	0.782586i
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	− 4.00000i	− 0.194717i
$423$	− 12.0000i	− 0.583460i
$424$	6.00000	0.291386
$425$	0	0
$426$	10.0000	0.484502
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	8.00000	0.386244
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	16.0000i	0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$	16.0000i	0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 8.00000i	− 0.382692i
$438$	− 2.00000i	− 0.0955637i
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 24.0000i	− 1.14156i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	− 14.0000i	− 0.662177i
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	− 18.0000i	− 0.846649i
$453$	− 4.00000i	− 0.187936i
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−8.00000	−0.374634
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	− 10.0000i	− 0.467269i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	4.00000i	0.184900i
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	− 8.00000i	− 0.368230i
$473$	− 4.00000i	− 0.183920i
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	− 14.0000i	− 0.640345i
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	6.00000i	0.273293i
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	2.00000i	0.0905357i
$489$	16.0000	0.723545
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	2.00000i	0.0901670i
$493$	24.0000i	1.08091i
$494$	32.0000	1.43975
$495$	0	0
$496$	0	0
$497$	0	0
$498$	8.00000i	0.358489i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	− 2.00000i	− 0.0892644i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	0	0
$506$	2.00000	0.0889108
$507$	− 3.00000i	− 0.133235i
$508$	2.00000i	0.0887357i
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 8.00000i	− 0.353209i
$514$	−22.0000	−0.970378
$515$	0	0
$516$	2.00000	0.0880451
$517$	− 24.0000i	− 1.05552i
$518$	0	0
$519$	10.0000	0.438951
$520$	0	0
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	− 4.00000i	− 0.175075i
$523$	30.0000i	1.31181i	0.754844	+	0.655904i	$0.227712\pi$
$523$	30.0000i	1.31181i	−0.754844	+	0.655904i	$0.772288\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 2.00000i	− 0.0870388i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	− 8.00000i	− 0.346518i
$534$	12.0000	0.519291
$535$	0	0
$536$	−6.00000	−0.259161
$537$	− 16.0000i	− 0.690451i
$538$	8.00000i	0.344904i
$539$	−14.0000	−0.603023
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	− 24.0000i	− 1.03089i
$543$	10.0000i	0.429141i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	2.00000i	0.0854358i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−32.0000	−1.36325
$552$	1.00000i	0.0425628i
$553$	0	0
$554$	28.0000	1.18961
$555$	0	0
$556$	4.00000	0.169638
$557$	− 38.0000i	− 1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	− 38.0000i	− 1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−12.0000	−0.506640
$562$	− 12.0000i	− 0.506189i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	14.0000	0.588464
$567$	0	0
$568$	10.0000i	0.419591i
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	8.00000i	0.334497i
$573$	24.0000i	1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	19.0000i	0.790296i
$579$	−26.0000	−1.08052
$580$	0	0
$581$	0	0
$582$	16.0000i	0.663221i
$583$	12.0000i	0.496989i
$584$	2.00000	0.0827606
$585$	0	0
$586$	−14.0000	−0.578335
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	− 7.00000i	− 0.288675i
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	2.00000i	0.0821995i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	14.0000	0.573462
$597$	16.0000i	0.654836i
$598$	− 4.00000i	− 0.163572i
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	− 6.00000i	− 0.244339i
$604$	4.00000	0.162758
$605$	0	0
$606$	12.0000	0.487467
$607$	10.0000i	0.405887i	0.979190	+	0.202944i	$0.0650509\pi$
$607$	10.0000i	0.405887i	−0.979190	+	0.202944i	$0.934949\pi$
$608$	− 8.00000i	− 0.324443i
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	− 6.00000i	− 0.242536i
$613$	− 30.0000i	− 1.21169i	−0.795583	−	0.605844i	$-0.792835\pi$
$613$	− 30.0000i	− 1.21169i	0.795583	−	0.605844i	$-0.207165\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	− 26.0000i	− 1.04251i
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	28.0000	1.11911
$627$	− 16.0000i	− 0.638978i
$628$	− 10.0000i	− 0.399043i
$629$	12.0000	0.478471
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	8.00000i	0.318223i
$633$	4.00000i	0.158986i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	28.0000i	1.10940i
$638$	− 8.00000i	− 0.316723i
$639$	−10.0000	−0.395594
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	12.0000i	0.473602i
$643$	− 10.0000i	− 0.394362i	−0.980367	−	0.197181i	$-0.936821\pi$
$643$	− 10.0000i	− 0.394362i	0.980367	−	0.197181i	$-0.0631786\pi$
$644$	0	0
$645$	0	0
$646$	−48.0000	−1.88853
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	1.00000i	0.0392837i
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	2.00000i	0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$	2.00000i	0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	− 28.0000i	− 1.08825i
$663$	24.0000i	0.932083i
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	4.00000i	0.154881i
$668$	− 16.0000i	− 0.619059i
$669$	2.00000	0.0773245
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	28.0000	1.07852
$675$	0	0
$676$	3.00000	0.115385
$677$	10.0000i	0.384331i	0.981363	+	0.192166i	$0.0615511\pi$
$677$	10.0000i	0.384331i	−0.981363	+	0.192166i	$0.938449\pi$
$678$	18.0000i	0.691286i
$679$	0	0
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	0	0
$687$	10.0000i	0.381524i
$688$	2.00000i	0.0762493i
$689$	24.0000	0.914327
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	10.0000i	0.380143i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	4.00000	0.151620
$697$	12.0000i	0.454532i
$698$	26.0000i	0.984115i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	− 4.00000i	− 0.150970i
$703$	16.0000i	0.603451i
$704$	2.00000	0.0753778
$705$	0	0
$706$	10.0000	0.376355
$707$	0	0
$708$	8.00000i	0.300658i
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	12.0000i	0.449719i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	16.0000	0.597948
$717$	14.0000i	0.522840i
$718$	− 16.0000i	− 0.597115i
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	0	0
$722$	− 45.0000i	− 1.67473i
$723$	− 6.00000i	− 0.223142i
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−7.00000	−0.259794
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	− 2.00000i	− 0.0739221i
$733$	− 42.0000i	− 1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$	− 42.0000i	− 1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	− 12.0000i	− 0.442026i
$738$	− 2.00000i	− 0.0736210i
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	− 8.00000i	− 0.292705i
$748$	− 12.0000i	− 0.438763i
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	12.0000i	0.437595i
$753$	2.00000i	0.0728841i
$754$	−16.0000	−0.582686
$755$	0	0
$756$	0	0
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	12.0000i	0.435860i
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	− 2.00000i	− 0.0724524i
$763$	0	0
$764$	−24.0000	−0.868290
$765$	0	0
$766$	−32.0000	−1.15621
$767$	− 32.0000i	− 1.15545i
$768$	1.00000i	0.0360844i
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	− 26.0000i	− 0.935760i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−16.0000	−0.574367
$777$	0	0
$778$	26.0000i	0.932145i
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−20.0000	−0.715656
$782$	6.00000i	0.214560i
$783$	4.00000i	0.142948i
$784$	7.00000	0.250000
$785$	0	0
$786$	−4.00000	−0.142675
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	2.00000i	0.0710669i
$793$	8.00000i	0.284088i
$794$	−28.0000	−0.993683
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	−12.0000	−0.423999
$802$	8.00000i	0.282490i
$803$	4.00000i	0.141157i
$804$	6.00000	0.211604
$805$	0	0
$806$	0	0
$807$	− 8.00000i	− 0.281613i
$808$	12.0000i	0.422159i
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	24.0000i	0.841717i
$814$	−4.00000	−0.140200
$815$	0	0
$816$	6.00000	0.210042
$817$	16.0000i	0.559769i
$818$	− 10.0000i	− 0.349642i
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	− 2.00000i	− 0.0697580i
$823$	− 22.0000i	− 0.766872i	−0.923567	−	0.383436i	$-0.874741\pi$
$823$	− 22.0000i	− 0.766872i	0.923567	−	0.383436i	$-0.125259\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	− 4.00000i	− 0.138675i
$833$	− 42.0000i	− 1.45521i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	30.0000i	1.03633i
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	14.0000i	0.482472i
$843$	12.0000i	0.413302i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−12.0000	−0.412568
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	−14.0000	−0.480479
$850$	0	0
$851$	2.00000	0.0685591
$852$	− 10.0000i	− 0.342594i
$853$	36.0000i	1.23262i	0.787505	+	0.616308i	$0.211372\pi$
$853$	36.0000i	1.23262i	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	− 8.00000i	− 0.273115i
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	0	0
$862$	8.00000i	0.272481i
$863$	− 20.0000i	− 0.680808i	−0.940279	−	0.340404i	$-0.889436\pi$
$863$	− 20.0000i	− 0.680808i	0.940279	−	0.340404i	$-0.110564\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	16.0000	0.543702
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−24.0000	−0.813209
$872$	10.0000i	0.338643i
$873$	− 16.0000i	− 0.541518i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	20.0000i	0.675352i	0.941262	+	0.337676i	$0.109641\pi$
$877$	20.0000i	0.675352i	−0.941262	+	0.337676i	$0.890359\pi$
$878$	28.0000i	0.944954i
$879$	14.0000	0.472208
$880$	0	0
$881$	4.00000	0.134763	0.0673817	−	0.997727i	$-0.478535\pi$
$881$	4.00000	0.134763	0.0673817	+	0.997727i	$0.478535\pi$
$882$	7.00000i	0.235702i
$883$	32.0000i	1.07689i	0.842662	+	0.538443i	$0.180987\pi$
$883$	32.0000i	1.07689i	−0.842662	+	0.538443i	$0.819013\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−4.00000	−0.134383
$887$	− 36.0000i	− 1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$	− 36.0000i	− 1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	2.00000i	0.0669650i
$893$	96.0000i	3.21252i
$894$	−14.0000	−0.468230
$895$	0	0
$896$	0	0
$897$	4.00000i	0.133556i
$898$	30.0000i	1.00111i
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	− 4.00000i	− 0.133185i
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−4.00000	−0.132891
$907$	22.0000i	0.730498i	0.930910	+	0.365249i	$0.119016\pi$
$907$	22.0000i	0.730498i	−0.930910	+	0.365249i	$0.880984\pi$
$908$	28.0000i	0.929213i
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	8.00000i	0.264906i
$913$	− 16.0000i	− 0.529523i
$914$	8.00000	0.264616
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	6.00000i	0.198030i
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	12.0000i	0.395199i
$923$	40.0000i	1.31662i
$924$	0	0
$925$	0	0
$926$	−26.0000	−0.854413
$927$	0	0
$928$	4.00000i	0.131306i
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	56.0000	1.83533
$932$	− 6.00000i	− 0.196537i
$933$	26.0000i	0.851202i
$934$	24.0000	0.785304
$935$	0	0
$936$	4.00000	0.130744
$937$	− 32.0000i	− 1.04539i	−0.852518	−	0.522697i	$-0.824926\pi$
$937$	− 32.0000i	− 1.04539i	0.852518	−	0.522697i	$-0.175074\pi$
$938$	0	0
$939$	−28.0000	−0.913745
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	10.0000i	0.325818i
$943$	2.00000i	0.0651290i
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−4.00000	−0.130051
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	− 8.00000i	− 0.259828i
$949$	8.00000	0.259691
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−14.0000	−0.452792
$957$	8.00000i	0.258603i
$958$	− 16.0000i	− 0.516937i
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	8.00000i	0.257930i
$963$	− 12.0000i	− 0.386695i
$964$	6.00000	0.193247
$965$	0	0
$966$	0	0
$967$	42.0000i	1.35063i	0.737530	+	0.675314i	$0.235992\pi$
$967$	42.0000i	1.35063i	−0.737530	+	0.675314i	$0.764008\pi$
$968$	− 7.00000i	− 0.224989i
$969$	48.0000	1.54198
$970$	0	0
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	26.0000	0.833094
$975$	0	0
$976$	2.00000	0.0640184
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	− 16.0000i	− 0.511624i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−10.0000	−0.319275
$982$	− 20.0000i	− 0.638226i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	24.0000	0.764316
$987$	0	0
$988$	− 32.0000i	− 1.01806i
$989$	2.00000	0.0635963
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	28.0000i	0.888553i
$994$	0	0
$995$	0	0
$996$	8.00000	0.253490
$997$	4.00000i	0.126681i	0.997992	+	0.0633406i	$0.0201755\pi$
$997$	4.00000i	0.126681i	−0.997992	+	0.0633406i	$0.979825\pi$
$998$	− 4.00000i	− 0.126618i
$999$	2.00000	0.0632772

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.j.1.1	✓	1	5.2	odd	4
2070.2.a.c.1.1		1	15.2	even	4
3450.2.a.b.1.1		1	5.3	odd	4
3450.2.d.m.2899.1		2	1.1	even	1	trivial
3450.2.d.m.2899.2		2	5.4	even	2	inner
5520.2.a.l.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} +2.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -4.00000 q^{29} -1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -8.00000i q^{38} -4.00000 q^{39} -2.00000 q^{41} +2.00000i q^{43} +2.00000 q^{44} -1.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +8.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} +2.00000 q^{61} -1.00000 q^{64} -2.00000 q^{66} +6.00000i q^{67} +6.00000i q^{68} +1.00000 q^{69} +10.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} -8.00000 q^{76} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +8.00000i q^{83} +2.00000 q^{86} -4.00000i q^{87} -2.00000i q^{88} +12.0000 q^{89} +1.00000i q^{92} +12.0000 q^{94} +1.00000 q^{96} +16.0000i q^{97} -7.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{11} + 2 q^{16} + 16 q^{19} - 2 q^{24} + 8 q^{26} - 8 q^{29} - 12 q^{34} + 2 q^{36} - 8 q^{39} - 4 q^{41} + 4 q^{44} - 2 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} - 16 q^{59} + 4 q^{61} - 2 q^{64} - 4 q^{66} + 2 q^{69} + 20 q^{71} + 4 q^{74} - 16 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{86} + 24 q^{89} + 24 q^{94} + 2 q^{96} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^11 + 2 * q^16 + 16 * q^19 - 2 * q^24 + 8 * q^26 - 8 * q^29 - 12 * q^34 + 2 * q^36 - 8 * q^39 - 4 * q^41 + 4 * q^44 - 2 * q^46 + 14 * q^49 + 12 * q^51 - 2 * q^54 - 16 * q^59 + 4 * q^61 - 2 * q^64 - 4 * q^66 + 2 * q^69 + 20 * q^71 + 4 * q^74 - 16 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^86 + 24 * q^89 + 24 * q^94 + 2 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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