LMFDB - Embedded newform 1350.2.c.a.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.c (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:	$10.7798042729$
Analytic rank:	$1$
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 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.2.c.a.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} -3.00000 q^{11} +1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -8.00000 q^{19} -3.00000i q^{22} -3.00000i q^{23} -1.00000 q^{26} -2.00000i q^{28} +9.00000 q^{29} -7.00000 q^{31} +1.00000i q^{32} +3.00000 q^{34} +2.00000i q^{37} -8.00000i q^{38} -12.0000 q^{41} +7.00000i q^{43} +3.00000 q^{44} +3.00000 q^{46} -3.00000i q^{47} +3.00000 q^{49} -1.00000i q^{52} -12.0000i q^{53} +2.00000 q^{56} +9.00000i q^{58} -12.0000 q^{59} -10.0000 q^{61} -7.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +3.00000i q^{68} -2.00000i q^{73} -2.00000 q^{74} +8.00000 q^{76} -6.00000i q^{77} +1.00000 q^{79} -12.0000i q^{82} -18.0000i q^{83} -7.00000 q^{86} +3.00000i q^{88} -2.00000 q^{91} +3.00000i q^{92} +3.00000 q^{94} +14.0000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 6 q^{11} - 4 q^{14} + 2 q^{16} - 16 q^{19} - 2 q^{26} + 18 q^{29} - 14 q^{31} + 6 q^{34} - 24 q^{41} + 6 q^{44} + 6 q^{46} + 6 q^{49} + 4 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} - 4 q^{74} + 16 q^{76} + 2 q^{79} - 14 q^{86} - 4 q^{91} + 6 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^11 - 4 * q^14 + 2 * q^16 - 16 * q^19 - 2 * q^26 + 18 * q^29 - 14 * q^31 + 6 * q^34 - 24 * q^41 + 6 * q^44 + 6 * q^46 + 6 * q^49 + 4 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 4 * q^74 + 16 * q^76 + 2 * q^79 - 14 * q^86 - 4 * q^91 + 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	− 3.00000i	− 0.639602i
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	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
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	3.00000	0.442326
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	9.00000i	1.18176i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 7.00000i	− 0.889001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	3.00000i	0.363803i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 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
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	0	0
$82$	− 12.0000i	− 1.32518i
$83$	− 18.0000i	− 1.97576i	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	− 18.0000i	− 1.97576i	0.155230	−	0.987878i	$-0.450388\pi$
$84$	0	0
$85$	0	0
$86$	−7.00000	−0.754829
$87$	0	0
$88$	3.00000i	0.319801i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	3.00000i	0.312772i
$93$	0	0
$94$	3.00000	0.309426
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	12.0000	1.16554
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	2.00000i	0.188982i
$113$	15.0000i	1.41108i	0.708669	+	0.705541i	$0.249296\pi$
$113$	15.0000i	1.41108i	−0.708669	+	0.705541i	$0.750704\pi$
$114$	0	0
$115$	0	0
$116$	−9.00000	−0.835629
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	6.00000	0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 10.0000i	− 0.905357i
$123$	0	0
$124$	7.00000	0.628619
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	4.00000	0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 3.00000i	− 0.250873i
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	8.00000i	0.648886i
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	17.0000i	1.35675i	0.734717	+	0.678374i	$0.237315\pi$
$157$	17.0000i	1.35675i	−0.734717	+	0.678374i	$0.762685\pi$
$158$	1.00000i	0.0795557i
$159$	0	0
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	18.0000	1.39707
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	− 7.00000i	− 0.533745i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	− 2.00000i	− 0.148250i
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	0	0
$187$	9.00000i	0.658145i
$188$	3.00000i	0.218797i
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	0	0
$202$	− 3.00000i	− 0.211079i
$203$	18.0000i	1.26335i
$204$	0	0
$205$	0	0
$206$	14.0000	0.975426
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	24.0000	1.66011
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	12.0000i	0.824163i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	0	0
$217$	− 14.0000i	− 0.950382i
$218$	4.00000i	0.270914i
$219$	0	0
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−15.0000	−0.997785
$227$	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
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	0	0
$232$	− 9.00000i	− 0.590879i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	6.00000i	0.388922i
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	− 2.00000i	− 0.128565i
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	7.00000i	0.444500i
$249$	0	0
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 15.0000i	− 0.935674i	−0.883815	−	0.467837i	$-0.845033\pi$
$257$	− 15.0000i	− 0.935674i	0.883815	−	0.467837i	$-0.154967\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	3.00000i	0.185341i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	16.0000	0.981023
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	− 14.0000i	− 0.839664i
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	3.00000	0.177394
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	2.00000i	0.117041i
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	− 15.0000i	− 0.868927i
$299$	3.00000	0.173494
$300$	0	0
$301$	−14.0000	−0.806947
$302$	− 1.00000i	− 0.0575435i
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	0	0
$307$	29.0000i	1.65512i	0.561379	+	0.827559i	$0.310271\pi$
$307$	29.0000i	1.65512i	−0.561379	+	0.827559i	$0.689729\pi$
$308$	6.00000i	0.341882i
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	4.00000i	0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$	4.00000i	0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	−17.0000	−0.959366
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	0	0
$321$	0	0
$322$	6.00000i	0.334367i
$323$	24.0000i	1.33540i
$324$	0	0
$325$	0	0
$326$	−19.0000	−1.05231
$327$	0	0
$328$	12.0000i	0.662589i
$329$	6.00000	0.330791
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	18.0000i	0.987878i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 16.0000i	− 0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$	− 16.0000i	− 0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$	12.0000i	0.652714i
$339$	0	0
$340$	0	0
$341$	21.0000	1.13721
$342$	0	0
$343$	20.0000i	1.07990i
$344$	7.00000	0.377415
$345$	0	0
$346$	6.00000	0.322562
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	− 3.00000i	− 0.159901i
$353$	− 3.00000i	− 0.159674i	−0.996808	−	0.0798369i	$-0.974560\pi$
$353$	− 3.00000i	− 0.159674i	0.996808	−	0.0798369i	$-0.0254400\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	12.0000i	0.634220i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	8.00000i	0.420471i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	− 3.00000i	− 0.156386i
$369$	0	0
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	− 29.0000i	− 1.50156i	−0.660551	−	0.750782i	$-0.729677\pi$
$373$	− 29.0000i	− 1.50156i	0.660551	−	0.750782i	$-0.270323\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	−3.00000	−0.154713
$377$	9.00000i	0.463524i
$378$	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
$381$	0	0
$382$	18.0000i	0.920960i
$383$	− 3.00000i	− 0.153293i	−0.997058	−	0.0766464i	$-0.975579\pi$
$383$	− 3.00000i	− 0.153293i	0.997058	−	0.0766464i	$-0.0244213\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	− 14.0000i	− 0.710742i
$389$	−27.0000	−1.36895	−0.684477	−	0.729034i	$-0.739969\pi$
$389$	−27.0000	−1.36895	−0.684477	+	0.729034i	$0.739969\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	−12.0000	−0.604551
$395$	0	0
$396$	0	0
$397$	35.0000i	1.75660i	0.478110	+	0.878300i	$0.341322\pi$
$397$	35.0000i	1.75660i	−0.478110	+	0.878300i	$0.658678\pi$
$398$	− 5.00000i	− 0.250627i
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	− 7.00000i	− 0.348695i
$404$	3.00000	0.149256
$405$	0	0
$406$	−18.0000	−0.893325
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	0	0
$412$	14.0000i	0.689730i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	24.0000i	1.17388i
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	− 10.0000i	− 0.486792i
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	− 6.00000i	− 0.290021i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	14.0000	0.672022
$435$	0	0
$436$	−4.00000	−0.191565
$437$	24.0000i	1.14808i
$438$	0	0
$439$	−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
$441$	0	0
$442$	3.00000i	0.142695i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	− 15.0000i	− 0.705541i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	16.0000i	0.747631i
$459$	0	0
$460$	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
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−18.0000	−0.833834
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	− 21.0000i	− 0.965581i
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	− 18.0000i	− 0.823301i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	5.00000i	0.227744i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	10.0000i	0.452679i
$489$	0	0
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	− 27.0000i	− 1.21602i
$494$	8.00000	0.359937
$495$	0	0
$496$	−7.00000	−0.314309
$497$	0	0
$498$	0	0
$499$	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
$501$	0	0
$502$	− 15.0000i	− 0.669483i
$503$	21.0000i	0.936344i	0.883637	+	0.468172i	$0.155087\pi$
$503$	21.0000i	0.936344i	−0.883637	+	0.468172i	$0.844913\pi$
$504$	0	0
$505$	0	0
$506$	−9.00000	−0.400099
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	15.0000	0.661622
$515$	0	0
$516$	0	0
$517$	9.00000i	0.395820i
$518$	− 4.00000i	− 0.175750i
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	25.0000i	1.09317i	0.837402	+	0.546587i	$0.184073\pi$
$523$	25.0000i	1.09317i	−0.837402	+	0.546587i	$0.815927\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	−12.0000	−0.523225
$527$	21.0000i	0.914774i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	16.0000i	0.693688i
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	− 21.0000i	− 0.905374i
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	− 16.0000i	− 0.687259i
$543$	0	0
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	− 19.0000i	− 0.812381i	−0.913788	−	0.406191i	$-0.866857\pi$
$547$	− 19.0000i	− 0.812381i	0.913788	−	0.406191i	$-0.133143\pi$
$548$	6.00000i	0.256307i
$549$	0	0
$550$	0	0
$551$	−72.0000	−3.06730
$552$	0	0
$553$	2.00000i	0.0850487i
$554$	22.0000	0.934690
$555$	0	0
$556$	14.0000	0.593732
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	0	0
$566$	−16.0000	−0.672530
$567$	0	0
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	3.00000i	0.125436i
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	0	0
$577$	− 16.0000i	− 0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$	− 16.0000i	− 0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$	8.00000i	0.332756i
$579$	0	0
$580$	0	0
$581$	36.0000	1.49353
$582$	0	0
$583$	36.0000i	1.49097i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−30.0000	−1.23929
$587$	− 6.00000i	− 0.247647i	−0.992304	−	0.123823i	$-0.960484\pi$
$587$	− 6.00000i	− 0.247647i	0.992304	−	0.123823i	$-0.0395156\pi$
$588$	0	0
$589$	56.0000	2.30744
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	− 21.0000i	− 0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$	− 21.0000i	− 0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$	0	0
$595$	0	0
$596$	15.0000	0.614424
$597$	0	0
$598$	3.00000i	0.122679i
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	29.0000	1.18293	0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000	1.18293	0.591467	+	0.806329i	$0.298549\pi$
$602$	− 14.0000i	− 0.570597i
$603$	0	0
$604$	1.00000	0.0406894
$605$	0	0
$606$	0	0
$607$	− 22.0000i	− 0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	− 22.0000i	− 0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$	− 8.00000i	− 0.324443i
$609$	0	0
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	25.0000i	1.00974i	0.863195	+	0.504870i	$0.168460\pi$
$613$	25.0000i	1.00974i	−0.863195	+	0.504870i	$0.831540\pi$
$614$	−29.0000	−1.17034
$615$	0	0
$616$	−6.00000	−0.241747
$617$	− 21.0000i	− 0.845428i	−0.906263	−	0.422714i	$-0.861077\pi$
$617$	− 21.0000i	− 0.845428i	0.906263	−	0.422714i	$-0.138923\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	12.0000i	0.481156i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−4.00000	−0.159872
$627$	0	0
$628$	− 17.0000i	− 0.678374i
$629$	6.00000	0.239236
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 1.00000i	− 0.0397779i
$633$	0	0
$634$	−24.0000	−0.953162
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	− 27.0000i	− 1.06894i
$639$	0	0
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	− 41.0000i	− 1.61688i	−0.588577	−	0.808441i	$-0.700312\pi$
$643$	− 41.0000i	− 1.61688i	0.588577	−	0.808441i	$-0.299688\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	−24.0000	−0.944267
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	− 19.0000i	− 0.744097i
$653$	− 48.0000i	− 1.87839i	−0.343391	−	0.939193i	$-0.611576\pi$
$653$	− 48.0000i	− 1.87839i	0.343391	−	0.939193i	$-0.388424\pi$
$654$	0	0
$655$	0	0
$656$	−12.0000	−0.468521
$657$	0	0
$658$	6.00000i	0.233904i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	− 10.0000i	− 0.388661i
$663$	0	0
$664$	−18.0000	−0.698535
$665$	0	0
$666$	0	0
$667$	− 27.0000i	− 1.04544i
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	16.0000	0.616297
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 36.0000i	− 1.38359i	−0.722093	−	0.691796i	$-0.756820\pi$
$677$	− 36.0000i	− 1.38359i	0.722093	−	0.691796i	$-0.243180\pi$
$678$	0	0
$679$	−28.0000	−1.07454
$680$	0	0
$681$	0	0
$682$	21.0000i	0.804132i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	−20.0000	−0.763604
$687$	0	0
$688$	7.00000i	0.266872i
$689$	12.0000	0.457164
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	− 26.0000i	− 0.984115i
$699$	0	0
$700$	0	0
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	0	0
$703$	− 16.0000i	− 0.603451i
$704$	3.00000	0.113067
$705$	0	0
$706$	3.00000	0.112906
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21.0000i	0.786456i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	24.0000i	0.895672i
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	45.0000i	1.67473i
$723$	0	0
$724$	−8.00000	−0.297318
$725$	0	0
$726$	0	0
$727$	44.0000i	1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$	44.0000i	1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$	2.00000i	0.0741249i
$729$	0	0
$730$	0	0
$731$	21.0000	0.776713
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	3.00000	0.110581
$737$	12.0000i	0.442026i
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	24.0000i	0.881068i
$743$	− 33.0000i	− 1.21065i	−0.795977	−	0.605326i	$-0.793043\pi$
$743$	− 33.0000i	− 1.21065i	0.795977	−	0.605326i	$-0.206957\pi$
$744$	0	0
$745$	0	0
$746$	29.0000	1.06177
$747$	0	0
$748$	− 9.00000i	− 0.329073i
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	− 3.00000i	− 0.109399i
$753$	0	0
$754$	−9.00000	−0.327761
$755$	0	0
$756$	0	0
$757$	− 7.00000i	− 0.254419i	−0.991876	−	0.127210i	$-0.959398\pi$
$757$	− 7.00000i	− 0.254419i	0.991876	−	0.127210i	$-0.0406021\pi$
$758$	− 8.00000i	− 0.290573i
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	3.00000	0.108394
$767$	− 12.0000i	− 0.433295i
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	0	0
$772$	14.0000i	0.503871i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	− 27.0000i	− 0.967997i
$779$	96.0000	3.43956
$780$	0	0
$781$	0	0
$782$	− 9.00000i	− 0.321839i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 31.0000i	− 1.10503i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	− 31.0000i	− 1.10503i	0.833503	−	0.552515i	$-0.186332\pi$
$788$	− 12.0000i	− 0.427482i
$789$	0	0
$790$	0	0
$791$	−30.0000	−1.06668
$792$	0	0
$793$	− 10.0000i	− 0.355110i
$794$	−35.0000	−1.24210
$795$	0	0
$796$	5.00000	0.177220
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	0	0
$802$	− 12.0000i	− 0.423735i
$803$	6.00000i	0.211735i
$804$	0	0
$805$	0	0
$806$	7.00000	0.246564
$807$	0	0
$808$	3.00000i	0.105540i
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−10.0000	−0.351147	−0.175574	−	0.984466i	$-0.556178\pi$
$811$	−10.0000	−0.351147	−0.175574	+	0.984466i	$0.556178\pi$
$812$	− 18.0000i	− 0.631676i
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	− 56.0000i	− 1.95919i
$818$	25.0000i	0.874105i
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	24.0000	0.835067
$827$	− 42.0000i	− 1.46048i	−0.683189	−	0.730242i	$-0.739408\pi$
$827$	− 42.0000i	− 1.46048i	0.683189	−	0.730242i	$-0.260592\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	− 9.00000i	− 0.311832i
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	− 3.00000i	− 0.103633i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	8.00000i	0.275698i
$843$	0	0
$844$	10.0000	0.344214
$845$	0	0
$846$	0	0
$847$	− 4.00000i	− 0.137442i
$848$	− 12.0000i	− 0.412082i
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	1.00000i	0.0342393i	0.999853	+	0.0171197i	$0.00544963\pi$
$853$	1.00000i	0.0342393i	−0.999853	+	0.0171197i	$0.994550\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	6.00000	0.205076
$857$	− 30.0000i	− 1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$	− 30.0000i	− 1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 33.0000i	− 1.12333i	−0.827364	−	0.561667i	$-0.810160\pi$
$863$	− 33.0000i	− 1.12333i	0.827364	−	0.561667i	$-0.189840\pi$
$864$	0	0
$865$	0	0
$866$	14.0000	0.475739
$867$	0	0
$868$	14.0000i	0.475191i
$869$	−3.00000	−0.101768
$870$	0	0
$871$	4.00000	0.135535
$872$	− 4.00000i	− 0.135457i
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	53.0000i	1.78968i	0.446384	+	0.894841i	$0.352711\pi$
$877$	53.0000i	1.78968i	−0.446384	+	0.894841i	$0.647289\pi$
$878$	− 8.00000i	− 0.269987i
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	0	0
$887$	− 15.0000i	− 0.503651i	−0.967773	−	0.251825i	$-0.918969\pi$
$887$	− 15.0000i	− 0.503651i	0.967773	−	0.251825i	$-0.0810309\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	− 4.00000i	− 0.133930i
$893$	24.0000i	0.803129i
$894$	0	0
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	6.00000i	0.200223i
$899$	−63.0000	−2.10117
$900$	0	0
$901$	−36.0000	−1.19933
$902$	36.0000i	1.19867i
$903$	0	0
$904$	15.0000	0.498893
$905$	0	0
$906$	0	0
$907$	53.0000i	1.75984i	0.475125	+	0.879918i	$0.342403\pi$
$907$	53.0000i	1.75984i	−0.475125	+	0.879918i	$0.657597\pi$
$908$	− 12.0000i	− 0.398234i
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	54.0000i	1.78714i
$914$	28.0000	0.926158
$915$	0	0
$916$	−16.0000	−0.528655
$917$	6.00000i	0.198137i
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	0	0
$922$	30.0000i	0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−22.0000	−0.722965
$927$	0	0
$928$	9.00000i	0.295439i
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−24.0000	−0.786568
$932$	− 18.0000i	− 0.589610i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	0	0
$941$	3.00000	0.0977972	0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000	0.0977972	0.0488986	+	0.998804i	$0.484429\pi$
$942$	0	0
$943$	36.0000i	1.17232i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	21.0000	0.682769
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	0	0
$952$	− 6.00000i	− 0.194461i
$953$	57.0000i	1.84641i	0.384307	+	0.923206i	$0.374441\pi$
$953$	57.0000i	1.84641i	−0.384307	+	0.923206i	$0.625559\pi$
$954$	0	0
$955$	0	0
$956$	18.0000	0.582162
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	18.0000	0.580645
$962$	− 2.00000i	− 0.0644826i
$963$	0	0
$964$	−5.00000	−0.161039
$965$	0	0
$966$	0	0
$967$	− 58.0000i	− 1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$	− 58.0000i	− 1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$	2.00000i	0.0642824i
$969$	0	0
$970$	0	0
$971$	33.0000	1.05902	0.529510	−	0.848304i	$-0.322376\pi$
$971$	33.0000	1.05902	0.529510	+	0.848304i	$0.322376\pi$
$972$	0	0
$973$	− 28.0000i	− 0.897639i
$974$	4.00000	0.128168
$975$	0	0
$976$	−10.0000	−0.320092
$977$	33.0000i	1.05576i	0.849318	+	0.527882i	$0.177014\pi$
$977$	33.0000i	1.05576i	−0.849318	+	0.527882i	$0.822986\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 36.0000i	− 1.14881i
$983$	45.0000i	1.43528i	0.696416	+	0.717639i	$0.254777\pi$
$983$	45.0000i	1.43528i	−0.696416	+	0.717639i	$0.745223\pi$
$984$	0	0
$985$	0	0
$986$	27.0000	0.859855
$987$	0	0
$988$	8.00000i	0.254514i
$989$	21.0000	0.667761
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	− 7.00000i	− 0.222250i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	17.0000i	0.538395i	0.963085	+	0.269198i	$0.0867585\pi$
$997$	17.0000i	0.538395i	−0.963085	+	0.269198i	$0.913241\pi$
$998$	4.00000i	0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.c.a.649.2		2
3.2	odd	2		1350.2.c.l.649.1		2
5.2	odd	4		1350.2.a.c.1.1		1
5.3	odd	4		270.2.a.d.1.1	yes	1
5.4	even	2	inner	1350.2.c.a.649.1		2
15.2	even	4		1350.2.a.p.1.1		1
15.8	even	4		270.2.a.a.1.1	✓	1
15.14	odd	2		1350.2.c.l.649.2		2
20.3	even	4		2160.2.a.p.1.1		1
40.3	even	4		8640.2.a.f.1.1		1
40.13	odd	4		8640.2.a.z.1.1		1
45.13	odd	12		810.2.e.a.541.1		2
45.23	even	12		810.2.e.k.541.1		2
45.38	even	12		810.2.e.k.271.1		2
45.43	odd	12		810.2.e.a.271.1		2
60.23	odd	4		2160.2.a.a.1.1		1
120.53	even	4		8640.2.a.by.1.1		1
120.83	odd	4		8640.2.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.a.a.1.1	✓	1	15.8	even	4
270.2.a.d.1.1	yes	1	5.3	odd	4
810.2.e.a.271.1		2	45.43	odd	12
810.2.e.a.541.1		2	45.13	odd	12
810.2.e.k.271.1		2	45.38	even	12
810.2.e.k.541.1		2	45.23	even	12
1350.2.a.c.1.1		1	5.2	odd	4
1350.2.a.p.1.1		1	15.2	even	4
1350.2.c.a.649.1		2	5.4	even	2	inner
1350.2.c.a.649.2		2	1.1	even	1	trivial
1350.2.c.l.649.1		2	3.2	odd	2
1350.2.c.l.649.2		2	15.14	odd	2
2160.2.a.a.1.1		1	60.23	odd	4
2160.2.a.p.1.1		1	20.3	even	4
8640.2.a.f.1.1		1	40.3	even	4
8640.2.a.z.1.1		1	40.13	odd	4
8640.2.a.bo.1.1		1	120.83	odd	4
8640.2.a.by.1.1		1	120.53	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} -3.00000 q^{11} +1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -8.00000 q^{19} -3.00000i q^{22} -3.00000i q^{23} -1.00000 q^{26} -2.00000i q^{28} +9.00000 q^{29} -7.00000 q^{31} +1.00000i q^{32} +3.00000 q^{34} +2.00000i q^{37} -8.00000i q^{38} -12.0000 q^{41} +7.00000i q^{43} +3.00000 q^{44} +3.00000 q^{46} -3.00000i q^{47} +3.00000 q^{49} -1.00000i q^{52} -12.0000i q^{53} +2.00000 q^{56} +9.00000i q^{58} -12.0000 q^{59} -10.0000 q^{61} -7.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +3.00000i q^{68} -2.00000i q^{73} -2.00000 q^{74} +8.00000 q^{76} -6.00000i q^{77} +1.00000 q^{79} -12.0000i q^{82} -18.0000i q^{83} -7.00000 q^{86} +3.00000i q^{88} -2.00000 q^{91} +3.00000i q^{92} +3.00000 q^{94} +14.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 6 q^{11} - 4 q^{14} + 2 q^{16} - 16 q^{19} - 2 q^{26} + 18 q^{29} - 14 q^{31} + 6 q^{34} - 24 q^{41} + 6 q^{44} + 6 q^{46} + 6 q^{49} + 4 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} - 4 q^{74} + 16 q^{76} + 2 q^{79} - 14 q^{86} - 4 q^{91} + 6 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^11 - 4 * q^14 + 2 * q^16 - 16 * q^19 - 2 * q^26 + 18 * q^29 - 14 * q^31 + 6 * q^34 - 24 * q^41 + 6 * q^44 + 6 * q^46 + 6 * q^49 + 4 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 4 * q^74 + 16 * q^76 + 2 * q^79 - 14 * q^86 - 4 * q^91 + 6 * q^94

Introduction

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