LMFDB - Embedded newform 456.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,2,Mod(1,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.a (trivial)

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:	yes
Analytic conductor:	$3.64117833217$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{17} +1.00000 q^{19} +3.00000 q^{21} +4.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -10.0000 q^{31} +5.00000 q^{33} -3.00000 q^{35} +2.00000 q^{39} -11.0000 q^{43} +1.00000 q^{45} +9.00000 q^{47} +2.00000 q^{49} +1.00000 q^{51} +10.0000 q^{53} -5.00000 q^{55} -1.00000 q^{57} +4.00000 q^{59} -5.00000 q^{61} -3.00000 q^{63} -2.00000 q^{65} -4.00000 q^{67} -4.00000 q^{69} +8.00000 q^{71} +13.0000 q^{73} +4.00000 q^{75} +15.0000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -1.00000 q^{85} +6.00000 q^{87} -6.00000 q^{89} +6.00000 q^{91} +10.0000 q^{93} +1.00000 q^{95} +2.00000 q^{97} -5.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	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
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−5.00000	−0.674200
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	15.0000	1.70941
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	10.0000	1.03695
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	11.0000	0.968496
$130$	0	0
$131$	−19.0000	−1.66004	−0.830019	−	0.557735i	$-0.811670\pi$
$131$	−19.0000	−1.66004	−0.830019	+	0.557735i	$0.811670\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	10.0000	0.836242
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−10.0000	−0.803219
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	5.00000	0.389249
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.00000	0.365636
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	18.0000	1.26335
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−11.0000	−0.750194
$216$	0	0
$217$	30.0000	2.03653
$218$	0	0
$219$	−13.0000	−0.878459
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	−23.0000	−1.50678	−0.753390	−	0.657574i	$-0.771583\pi$
$233$	−23.0000	−1.50678	−0.753390	+	0.657574i	$0.771583\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−19.0000	−1.19927	−0.599635	−	0.800274i	$-0.704687\pi$
$251$	−19.0000	−1.19927	−0.599635	+	0.800274i	$0.704687\pi$
$252$	0	0
$253$	−20.0000	−1.25739
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−6.00000	−0.363137
$274$	0	0
$275$	20.0000	1.20605
$276$	0	0
$277$	25.0000	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$277$	25.0000	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	−7.00000	−0.416107	−0.208053	−	0.978117i	$-0.566713\pi$
$283$	−7.00000	−0.416107	−0.208053	+	0.978117i	$0.566713\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	−8.00000	−0.462652
$300$	0	0
$301$	33.0000	1.90209
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−5.00000	−0.286299
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	30.0000	1.67968
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	8.00000	0.443760
$326$	0	0
$327$	8.00000	0.442401
$328$	0	0
$329$	−27.0000	−1.48856
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	50.0000	2.70765
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	21.0000	1.12734	0.563670	−	0.826000i	$-0.309389\pi$
$347$	21.0000	1.12734	0.563670	+	0.826000i	$0.309389\pi$
$348$	0	0
$349$	−27.0000	−1.44528	−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000	−1.44528	−0.722638	+	0.691226i	$0.757071\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	13.0000	0.680451
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−30.0000	−1.55752
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	0	0
$387$	−11.0000	−0.559161
$388$	0	0
$389$	25.0000	1.26755	0.633775	−	0.773517i	$-0.281504\pi$
$389$	25.0000	1.26755	0.633775	+	0.773517i	$0.281504\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	19.0000	0.958423
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	0	0
$399$	3.00000	0.150188
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	15.0000	0.734553
$418$	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
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	9.00000	0.437595
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	15.0000	0.725901
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	−30.0000	−1.43182	−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000	−1.43182	−0.715911	+	0.698192i	$0.753988\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−7.00000	−0.332580	−0.166290	−	0.986077i	$-0.553179\pi$
$443$	−7.00000	−0.332580	−0.166290	+	0.986077i	$0.553179\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	1.00000	0.0472984
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	35.0000	1.63723	0.818615	−	0.574342i	$-0.194742\pi$
$457$	35.0000	1.63723	0.818615	+	0.574342i	$0.194742\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	7.00000	0.326023	0.163011	−	0.986624i	$-0.447879\pi$
$461$	7.00000	0.326023	0.163011	+	0.986624i	$0.447879\pi$
$462$	0	0
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	0	0
$465$	10.0000	0.463739
$466$	0	0
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	55.0000	2.52890
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	−5.00000	−0.224733
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	39.0000	1.74588	0.872940	−	0.487828i	$-0.162211\pi$
$499$	39.0000	1.74588	0.872940	+	0.487828i	$0.162211\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−39.0000	−1.72526
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	10.0000	0.440653
$516$	0	0
$517$	−45.0000	−1.97910
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	8.00000	0.350486	0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000	0.350486	0.175243	+	0.984525i	$0.443929\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	10.0000	0.435607
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	0	0
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	−10.0000	−0.430730
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	0	0
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.00000	−0.211857	−0.105928	−	0.994374i	$-0.533781\pi$
$557$	−5.00000	−0.211857	−0.105928	+	0.994374i	$0.533781\pi$
$558$	0	0
$559$	22.0000	0.930501
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	0	0
$575$	−16.0000	−0.667246
$576$	0	0
$577$	35.0000	1.45707	0.728535	−	0.685009i	$-0.240202\pi$
$577$	35.0000	1.45707	0.728535	+	0.685009i	$0.240202\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−50.0000	−2.07079
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−35.0000	−1.44460	−0.722302	−	0.691577i	$-0.756916\pi$
$587$	−35.0000	−1.44460	−0.722302	+	0.691577i	$0.756916\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	11.0000	0.450200
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	−18.0000	−0.729397
$610$	0	0
$611$	−18.0000	−0.728202
$612$	0	0
$613$	5.00000	0.201948	0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000	0.201948	0.100974	+	0.994889i	$0.467804\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	5.00000	0.199681
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−47.0000	−1.87104	−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000	−1.87104	−0.935520	+	0.353273i	$0.885069\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	0	0
$645$	11.0000	0.433125
$646$	0	0
$647$	−25.0000	−0.982851	−0.491426	−	0.870919i	$-0.663524\pi$
$647$	−25.0000	−0.982851	−0.491426	+	0.870919i	$0.663524\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	−30.0000	−1.17579
$652$	0	0
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	−19.0000	−0.742391
$656$	0	0
$657$	13.0000	0.507178
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	25.0000	0.965114
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	6.00000	0.229920
$682$	0	0
$683$	34.0000	1.30097	0.650487	−	0.759517i	$-0.274565\pi$
$683$	34.0000	1.30097	0.650487	+	0.759517i	$0.274565\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	0	0
$687$	11.0000	0.419676
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	0	0
$693$	15.0000	0.569803
$694$	0	0
$695$	−15.0000	−0.568982
$696$	0	0
$697$	0	0
$698$	0	0
$699$	23.0000	0.869940
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−9.00000	−0.338960
$706$	0	0
$707$	18.0000	0.676960
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−40.0000	−1.49801
$714$	0	0
$715$	10.0000	0.373979
$716$	0	0
$717$	−11.0000	−0.410803
$718$	0	0
$719$	−17.0000	−0.633993	−0.316997	−	0.948427i	$-0.602674\pi$
$719$	−17.0000	−0.633993	−0.316997	+	0.948427i	$0.602674\pi$
$720$	0	0
$721$	−30.0000	−1.11726
$722$	0	0
$723$	12.0000	0.446285
$724$	0	0
$725$	24.0000	0.891338
$726$	0	0
$727$	39.0000	1.44643	0.723215	−	0.690623i	$-0.242664\pi$
$727$	39.0000	1.44643	0.723215	+	0.690623i	$0.242664\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.0000	0.406850
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	20.0000	0.736709
$738$	0	0
$739$	−7.00000	−0.257499	−0.128750	−	0.991677i	$-0.541096\pi$
$739$	−7.00000	−0.257499	−0.128750	+	0.991677i	$0.541096\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	−1.00000	−0.0366372
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−54.0000	−1.97312
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	19.0000	0.692398
$754$	0	0
$755$	20.0000	0.727875
$756$	0	0
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	0	0
$759$	20.0000	0.725954
$760$	0	0
$761$	−1.00000	−0.0362500	−0.0181250	−	0.999836i	$-0.505770\pi$
$761$	−1.00000	−0.0362500	−0.0181250	+	0.999836i	$0.505770\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	0	0
$765$	−1.00000	−0.0361551
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	0	0
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−40.0000	−1.43131
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	0	0
$789$	23.0000	0.818822
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	−10.0000	−0.354663
$796$	0	0
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−65.0000	−2.29380
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	22.0000	0.774437
$808$	0	0
$809$	17.0000	0.597688	0.298844	−	0.954302i	$-0.403399\pi$
$809$	17.0000	0.597688	0.298844	+	0.954302i	$0.403399\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	−11.0000	−0.384841
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	−9.00000	−0.314102	−0.157051	−	0.987590i	$-0.550199\pi$
$821$	−9.00000	−0.314102	−0.157051	+	0.987590i	$0.550199\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	−20.0000	−0.696311
$826$	0	0
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	−25.0000	−0.867240
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	22.0000	0.757720
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−42.0000	−1.44314
$848$	0	0
$849$	7.00000	0.240239
$850$	0	0
$851$	0	0
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	−15.0000	−0.511793	−0.255897	−	0.966704i	$-0.582371\pi$
$859$	−15.0000	−0.511793	−0.255897	+	0.966704i	$0.582371\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.0000	0.953131	0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000	0.953131	0.476566	+	0.879139i	$0.341881\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	−20.0000	−0.678454
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	27.0000	0.912767
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−37.0000	−1.24656	−0.623281	−	0.781998i	$-0.714201\pi$
$881$	−37.0000	−1.24656	−0.623281	+	0.781998i	$0.714201\pi$
$882$	0	0
$883$	−31.0000	−1.04323	−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000	−1.04323	−0.521617	+	0.853180i	$0.674671\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	56.0000	1.88030	0.940148	−	0.340766i	$-0.110687\pi$
$887$	56.0000	1.88030	0.940148	+	0.340766i	$0.110687\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	9.00000	0.301174
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	60.0000	2.00111
$900$	0	0
$901$	−10.0000	−0.333148
$902$	0	0
$903$	−33.0000	−1.09817
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	5.00000	0.165295
$916$	0	0
$917$	57.0000	1.88231
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.0000	0.328443
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	27.0000	0.883940
$934$	0	0
$935$	5.00000	0.163517
$936$	0	0
$937$	−3.00000	−0.0980057	−0.0490029	−	0.998799i	$-0.515604\pi$
$937$	−3.00000	−0.0980057	−0.0490029	+	0.998799i	$0.515604\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	−26.0000	−0.843996
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	0	0
$955$	15.0000	0.485389
$956$	0	0
$957$	−30.0000	−0.969762
$958$	0	0
$959$	27.0000	0.871875
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	18.0000	0.580042
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	1.00000	0.0321246
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	45.0000	1.44263
$974$	0	0
$975$	−8.00000	−0.256205
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	0	0
$981$	−8.00000	−0.255420
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	27.0000	0.859419
$988$	0	0
$989$	−44.0000	−1.39912
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	−32.0000	−1.01549
$994$	0	0
$995$	−11.0000	−0.348723
$996$	0	0
$997$	−35.0000	−1.10846	−0.554231	−	0.832363i	$-0.686987\pi$
$997$	−35.0000	−1.10846	−0.554231	+	0.832363i	$0.686987\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.a.a.1.1	✓	1
3.2	odd	2		1368.2.a.d.1.1		1
4.3	odd	2		912.2.a.i.1.1		1
8.3	odd	2		3648.2.a.g.1.1		1
8.5	even	2		3648.2.a.z.1.1		1
12.11	even	2		2736.2.a.i.1.1		1
19.18	odd	2		8664.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.a.1.1	✓	1	1.1	even	1	trivial
912.2.a.i.1.1		1	4.3	odd	2
1368.2.a.d.1.1		1	3.2	odd	2
2736.2.a.i.1.1		1	12.11	even	2
3648.2.a.g.1.1		1	8.3	odd	2
3648.2.a.z.1.1		1	8.5	even	2
8664.2.a.l.1.1		1	19.18	odd	2

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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)

Introduction

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