LMFDB - Embedded newform 2312.2.b.a.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(577,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-10,0,0,0,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.a.577.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{3} -1.41421i q^{5} -2.82843i q^{7} -5.00000 q^{9} -2.82843i q^{11} +4.00000 q^{13} -4.00000 q^{15} -8.00000 q^{19} -8.00000 q^{21} +2.82843i q^{23} +3.00000 q^{25} +5.65685i q^{27} -4.24264i q^{29} -8.48528i q^{31} -8.00000 q^{33} -4.00000 q^{35} +7.07107i q^{37} -11.3137i q^{39} -1.41421i q^{41} -8.00000 q^{43} +7.07107i q^{45} -1.00000 q^{49} +12.0000 q^{53} -4.00000 q^{55} +22.6274i q^{57} -8.00000 q^{59} +1.41421i q^{61} +14.1421i q^{63} -5.65685i q^{65} +12.0000 q^{67} +8.00000 q^{69} +8.48528i q^{71} -4.24264i q^{73} -8.48528i q^{75} -8.00000 q^{77} -2.82843i q^{79} +1.00000 q^{81} -12.0000 q^{87} +8.00000 q^{89} -11.3137i q^{91} -24.0000 q^{93} +11.3137i q^{95} +4.24264i q^{97} +14.1421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{9} + 8 q^{13} - 8 q^{15} - 16 q^{19} - 16 q^{21} + 6 q^{25} - 16 q^{33} - 8 q^{35} - 16 q^{43} - 2 q^{49} + 24 q^{53} - 8 q^{55} - 16 q^{59} + 24 q^{67} + 16 q^{69} - 16 q^{77} + 2 q^{81} - 24 q^{87}+ \cdots - 48 q^{93}+O(q^{100}) $ 2 * q - 10 * q^9 + 8 * q^13 - 8 * q^15 - 16 * q^19 - 16 * q^21 + 6 * q^25 - 16 * q^33 - 8 * q^35 - 16 * q^43 - 2 * q^49 + 24 * q^53 - 8 * q^55 - 16 * q^59 + 24 * q^67 + 16 * q^69 - 16 * q^77 + 2 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 2.82843i	− 1.63299i	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	− 2.82843i	− 1.63299i	0.577350	−	0.816497i	$-0.304087\pi$
$4$	0	0
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	−5.00000	−1.66667
$10$	0	0
$11$	− 2.82843i	− 0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	− 2.82843i	− 0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	0	0
$17$	0	0
$18$	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
$21$	−8.00000	−1.74574
$22$	0	0
$23$	2.82843i	0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$	2.82843i	0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	5.65685i	1.08866i
$28$	0	0
$29$	− 4.24264i	− 0.787839i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	− 4.24264i	− 0.787839i	0.919145	−	0.393919i	$-0.128881\pi$
$30$	0	0
$31$	− 8.48528i	− 1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$	− 8.48528i	− 1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$	0	0
$33$	−8.00000	−1.39262
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	7.07107i	1.16248i	0.813733	+	0.581238i	$0.197432\pi$
$37$	7.07107i	1.16248i	−0.813733	+	0.581238i	$0.802568\pi$
$38$	0	0
$39$	− 11.3137i	− 1.81164i
$40$	0	0
$41$	− 1.41421i	− 0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$	− 1.41421i	− 0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	7.07107i	1.05409i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	22.6274i	2.99707i
$58$	0	0
$59$	−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
$61$	1.41421i	0.181071i	0.995893	+	0.0905357i	$0.0288579\pi$
$61$	1.41421i	0.181071i	−0.995893	+	0.0905357i	$0.971142\pi$
$62$	0	0
$63$	14.1421i	1.78174i
$64$	0	0
$65$	− 5.65685i	− 0.701646i
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	8.48528i	1.00702i	0.863990	+	0.503509i	$0.167958\pi$
$71$	8.48528i	1.00702i	−0.863990	+	0.503509i	$0.832042\pi$
$72$	0	0
$73$	− 4.24264i	− 0.496564i	−0.968688	−	0.248282i	$-0.920134\pi$
$73$	− 4.24264i	− 0.496564i	0.968688	−	0.248282i	$-0.0798659\pi$
$74$	0	0
$75$	− 8.48528i	− 0.979796i
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	− 2.82843i	− 0.318223i	−0.987261	−	0.159111i	$-0.949137\pi$
$79$	− 2.82843i	− 0.318223i	0.987261	−	0.159111i	$-0.0508629\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.0000	−1.28654
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	− 11.3137i	− 1.18600i
$92$	0	0
$93$	−24.0000	−2.48868
$94$	0	0
$95$	11.3137i	1.16076i
$96$	0	0
$97$	4.24264i	0.430775i	0.976529	+	0.215387i	$0.0691014\pi$
$97$	4.24264i	0.430775i	−0.976529	+	0.215387i	$0.930899\pi$
$98$	0	0
$99$	14.1421i	1.42134i
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	11.3137i	1.10410i
$106$	0	0
$107$	2.82843i	0.273434i	0.990610	+	0.136717i	$0.0436552\pi$
$107$	2.82843i	0.273434i	−0.990610	+	0.136717i	$0.956345\pi$
$108$	0	0
$109$	− 4.24264i	− 0.406371i	−0.979140	−	0.203186i	$-0.934871\pi$
$109$	− 4.24264i	− 0.406371i	0.979140	−	0.203186i	$-0.0651295\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	0	0
$113$	9.89949i	0.931266i	0.884978	+	0.465633i	$0.154173\pi$
$113$	9.89949i	0.931266i	−0.884978	+	0.465633i	$0.845827\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−20.0000	−1.84900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	22.6274i	1.99223i
$130$	0	0
$131$	− 2.82843i	− 0.247121i	−0.992337	−	0.123560i	$-0.960569\pi$
$131$	− 2.82843i	− 0.247121i	0.992337	−	0.123560i	$-0.0394313\pi$
$132$	0	0
$133$	22.6274i	1.96205i
$134$	0	0
$135$	8.00000	0.688530
$136$	0	0
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	0	0
$139$	2.82843i	0.239904i	0.992780	+	0.119952i	$0.0382741\pi$
$139$	2.82843i	0.239904i	−0.992780	+	0.119952i	$0.961726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 11.3137i	− 0.946100i
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	2.82843i	0.233285i
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	− 33.9411i	− 2.69171i
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	− 8.48528i	− 0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$	− 8.48528i	− 0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$	0	0
$165$	11.3137i	0.880771i
$166$	0	0
$167$	14.1421i	1.09435i	0.837018	+	0.547176i	$0.184297\pi$
$167$	14.1421i	1.09435i	−0.837018	+	0.547176i	$0.815703\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	40.0000	3.05888
$172$	0	0
$173$	15.5563i	1.18273i	0.806405	+	0.591364i	$0.201410\pi$
$173$	15.5563i	1.18273i	−0.806405	+	0.591364i	$0.798590\pi$
$174$	0	0
$175$	− 8.48528i	− 0.641427i
$176$	0	0
$177$	22.6274i	1.70078i
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	− 7.07107i	− 0.525588i	−0.964852	−	0.262794i	$-0.915356\pi$
$181$	− 7.07107i	− 0.525588i	0.964852	−	0.262794i	$-0.0846440\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	10.0000	0.735215
$186$	0	0
$187$	0	0
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	− 4.24264i	− 0.305392i	−0.988273	−	0.152696i	$-0.951204\pi$
$193$	− 4.24264i	− 0.305392i	0.988273	−	0.152696i	$-0.0487955\pi$
$194$	0	0
$195$	−16.0000	−1.14578
$196$	0	0
$197$	− 18.3848i	− 1.30986i	−0.755689	−	0.654931i	$-0.772698\pi$
$197$	− 18.3848i	− 1.30986i	0.755689	−	0.654931i	$-0.227302\pi$
$198$	0	0
$199$	8.48528i	0.601506i	0.953702	+	0.300753i	$0.0972379\pi$
$199$	8.48528i	0.601506i	−0.953702	+	0.300753i	$0.902762\pi$
$200$	0	0
$201$	− 33.9411i	− 2.39402i
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	− 14.1421i	− 0.982946i
$208$	0	0
$209$	22.6274i	1.56517i
$210$	0	0
$211$	− 8.48528i	− 0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$	− 8.48528i	− 0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$	0	0
$213$	24.0000	1.64445
$214$	0	0
$215$	11.3137i	0.771589i
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	−12.0000	−0.810885
$220$	0	0
$221$	0	0
$222$	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
$225$	−15.0000	−1.00000
$226$	0	0
$227$	19.7990i	1.31411i	0.753845	+	0.657053i	$0.228197\pi$
$227$	19.7990i	1.31411i	−0.753845	+	0.657053i	$0.771803\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	22.6274i	1.48877i
$232$	0	0
$233$	− 7.07107i	− 0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$	− 7.07107i	− 0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	26.8701i	1.73085i	0.501036	+	0.865426i	$0.332952\pi$
$241$	26.8701i	1.73085i	−0.501036	+	0.865426i	$0.667048\pi$
$242$	0	0
$243$	14.1421i	0.907218i
$244$	0	0
$245$	1.41421i	0.0903508i
$246$	0	0
$247$	−32.0000	−2.03611
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	20.0000	1.24274
$260$	0	0
$261$	21.2132i	1.31306i
$262$	0	0
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	− 16.9706i	− 1.04249i
$266$	0	0
$267$	− 22.6274i	− 1.38478i
$268$	0	0
$269$	7.07107i	0.431131i	0.976489	+	0.215565i	$0.0691594\pi$
$269$	7.07107i	0.431131i	−0.976489	+	0.215565i	$0.930841\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	−32.0000	−1.93673
$274$	0	0
$275$	− 8.48528i	− 0.511682i
$276$	0	0
$277$	− 21.2132i	− 1.27458i	−0.770625	−	0.637289i	$-0.780056\pi$
$277$	− 21.2132i	− 1.27458i	0.770625	−	0.637289i	$-0.219944\pi$
$278$	0	0
$279$	42.4264i	2.54000i
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	− 14.1421i	− 0.840663i	−0.907371	−	0.420331i	$-0.861914\pi$
$283$	− 14.1421i	− 0.840663i	0.907371	−	0.420331i	$-0.138086\pi$
$284$	0	0
$285$	32.0000	1.89552
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	0	0
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	11.3137i	0.658710i
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	11.3137i	0.654289i
$300$	0	0
$301$	22.6274i	1.30422i
$302$	0	0
$303$	33.9411i	1.94987i
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	− 22.6274i	− 1.28723i
$310$	0	0
$311$	− 25.4558i	− 1.44347i	−0.692170	−	0.721734i	$-0.743345\pi$
$311$	− 25.4558i	− 1.44347i	0.692170	−	0.721734i	$-0.256655\pi$
$312$	0	0
$313$	− 12.7279i	− 0.719425i	−0.933063	−	0.359712i	$-0.882875\pi$
$313$	− 12.7279i	− 0.719425i	0.933063	−	0.359712i	$-0.117125\pi$
$314$	0	0
$315$	20.0000	1.12687
$316$	0	0
$317$	− 21.2132i	− 1.19145i	−0.803188	−	0.595726i	$-0.796864\pi$
$317$	− 21.2132i	− 1.19145i	0.803188	−	0.595726i	$-0.203136\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	0	0
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	−12.0000	−0.663602
$328$	0	0
$329$	0	0
$330$	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
$333$	− 35.3553i	− 1.93746i
$334$	0	0
$335$	− 16.9706i	− 0.927201i
$336$	0	0
$337$	− 35.3553i	− 1.92593i	−0.269630	−	0.962964i	$-0.586901\pi$
$337$	− 35.3553i	− 1.92593i	0.269630	−	0.962964i	$-0.413099\pi$
$338$	0	0
$339$	28.0000	1.52075
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	− 11.3137i	− 0.609110i
$346$	0	0
$347$	− 25.4558i	− 1.36654i	−0.730165	−	0.683271i	$-0.760557\pi$
$347$	− 25.4558i	− 1.36654i	0.730165	−	0.683271i	$-0.239443\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	22.6274i	1.20776i
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	− 8.48528i	− 0.445362i
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	− 14.1421i	− 0.738213i	−0.929387	−	0.369107i	$-0.879664\pi$
$367$	− 14.1421i	− 0.738213i	0.929387	−	0.369107i	$-0.120336\pi$
$368$	0	0
$369$	7.07107i	0.368105i
$370$	0	0
$371$	− 33.9411i	− 1.76214i
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	−32.0000	−1.65247
$376$	0	0
$377$	− 16.9706i	− 0.874028i
$378$	0	0
$379$	− 2.82843i	− 0.145287i	−0.997358	−	0.0726433i	$-0.976857\pi$
$379$	− 2.82843i	− 0.145287i	0.997358	−	0.0726433i	$-0.0231435\pi$
$380$	0	0
$381$	− 11.3137i	− 0.579619i
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	11.3137i	0.576600i
$386$	0	0
$387$	40.0000	2.03331
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	− 12.7279i	− 0.638796i	−0.947621	−	0.319398i	$-0.896519\pi$
$397$	− 12.7279i	− 0.638796i	0.947621	−	0.319398i	$-0.103481\pi$
$398$	0	0
$399$	64.0000	3.20401
$400$	0	0
$401$	15.5563i	0.776847i	0.921481	+	0.388424i	$0.126980\pi$
$401$	15.5563i	0.776847i	−0.921481	+	0.388424i	$0.873020\pi$
$402$	0	0
$403$	− 33.9411i	− 1.69073i
$404$	0	0
$405$	− 1.41421i	− 0.0702728i
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	− 45.2548i	− 2.23226i
$412$	0	0
$413$	22.6274i	1.11342i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	− 8.48528i	− 0.414533i	−0.978285	−	0.207267i	$-0.933543\pi$
$419$	− 8.48528i	− 0.414533i	0.978285	−	0.207267i	$-0.0664567\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000	0.193574
$428$	0	0
$429$	−32.0000	−1.54497
$430$	0	0
$431$	8.48528i	0.408722i	0.978896	+	0.204361i	$0.0655116\pi$
$431$	8.48528i	0.408722i	−0.978896	+	0.204361i	$0.934488\pi$
$432$	0	0
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	0	0
$435$	16.9706i	0.813676i
$436$	0	0
$437$	− 22.6274i	− 1.08242i
$438$	0	0
$439$	− 25.4558i	− 1.21494i	−0.794342	−	0.607471i	$-0.792184\pi$
$439$	− 25.4558i	− 1.21494i	0.794342	−	0.607471i	$-0.207816\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	− 11.3137i	− 0.536321i
$446$	0	0
$447$	62.2254i	2.94316i
$448$	0	0
$449$	18.3848i	0.867631i	0.901002	+	0.433816i	$0.142833\pi$
$449$	18.3848i	0.867631i	−0.901002	+	0.433816i	$0.857167\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	− 33.9411i	− 1.59469i
$454$	0	0
$455$	−16.0000	−0.750092
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	33.9411i	1.57398i
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	− 33.9411i	− 1.56726i
$470$	0	0
$471$	− 5.65685i	− 0.260654i
$472$	0	0
$473$	22.6274i	1.04041i
$474$	0	0
$475$	−24.0000	−1.10120
$476$	0	0
$477$	−60.0000	−2.74721
$478$	0	0
$479$	31.1127i	1.42158i	0.703407	+	0.710788i	$0.251661\pi$
$479$	31.1127i	1.42158i	−0.703407	+	0.710788i	$0.748339\pi$
$480$	0	0
$481$	28.2843i	1.28965i
$482$	0	0
$483$	− 22.6274i	− 1.02958i
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	19.7990i	0.897178i	0.893738	+	0.448589i	$0.148073\pi$
$487$	19.7990i	0.897178i	−0.893738	+	0.448589i	$0.851927\pi$
$488$	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
$493$	0	0
$494$	0	0
$495$	20.0000	0.898933
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	8.48528i	0.379853i	0.981798	+	0.189927i	$0.0608250\pi$
$499$	8.48528i	0.379853i	−0.981798	+	0.189927i	$0.939175\pi$
$500$	0	0
$501$	40.0000	1.78707
$502$	0	0
$503$	− 19.7990i	− 0.882793i	−0.897312	−	0.441397i	$-0.854483\pi$
$503$	− 19.7990i	− 0.882793i	0.897312	−	0.441397i	$-0.145517\pi$
$504$	0	0
$505$	16.9706i	0.755180i
$506$	0	0
$507$	− 8.48528i	− 0.376845i
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	− 45.2548i	− 1.99805i
$514$	0	0
$515$	− 11.3137i	− 0.498542i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	44.0000	1.93139
$520$	0	0
$521$	− 43.8406i	− 1.92069i	−0.278810	−	0.960346i	$-0.589940\pi$
$521$	− 43.8406i	− 1.92069i	0.278810	−	0.960346i	$-0.410060\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	−24.0000	−1.04745
$526$	0	0
$527$	0	0
$528$	0	0
$529$	15.0000	0.652174
$530$	0	0
$531$	40.0000	1.73585
$532$	0	0
$533$	− 5.65685i	− 0.245026i
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.82843i	0.121829i
$540$	0	0
$541$	− 9.89949i	− 0.425613i	−0.977094	−	0.212806i	$-0.931740\pi$
$541$	− 9.89949i	− 0.425613i	0.977094	−	0.212806i	$-0.0682603\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	2.82843i	0.120935i	0.998170	+	0.0604674i	$0.0192591\pi$
$547$	2.82843i	0.120935i	−0.998170	+	0.0604674i	$0.980741\pi$
$548$	0	0
$549$	− 7.07107i	− 0.301786i
$550$	0	0
$551$	33.9411i	1.44594i
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	− 28.2843i	− 1.20060i
$556$	0	0
$557$	−20.0000	−0.847427	−0.423714	−	0.905796i	$-0.639274\pi$
$557$	−20.0000	−0.847427	−0.423714	+	0.905796i	$0.639274\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	− 2.82843i	− 0.118783i
$568$	0	0
$569$	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
$571$	− 42.4264i	− 1.77549i	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	− 42.4264i	− 1.77549i	0.460336	−	0.887745i	$-0.347729\pi$
$572$	0	0
$573$	− 22.6274i	− 0.945274i
$574$	0	0
$575$	8.48528i	0.353861i
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	0	0
$579$	−12.0000	−0.498703
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 33.9411i	− 1.40570i
$584$	0	0
$585$	28.2843i	1.16941i
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	67.8823i	2.79704i
$590$	0	0
$591$	−52.0000	−2.13899
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24.0000	0.982255
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	− 38.1838i	− 1.55755i	−0.627304	−	0.778774i	$-0.715842\pi$
$601$	− 38.1838i	− 1.55755i	0.627304	−	0.778774i	$-0.284158\pi$
$602$	0	0
$603$	−60.0000	−2.44339
$604$	0	0
$605$	− 4.24264i	− 0.172488i
$606$	0	0
$607$	19.7990i	0.803616i	0.915724	+	0.401808i	$0.131618\pi$
$607$	19.7990i	0.803616i	−0.915724	+	0.401808i	$0.868382\pi$
$608$	0	0
$609$	33.9411i	1.37536i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	0	0
$615$	5.65685i	0.228106i
$616$	0	0
$617$	4.24264i	0.170802i	0.996347	+	0.0854011i	$0.0272172\pi$
$617$	4.24264i	0.170802i	−0.996347	+	0.0854011i	$0.972783\pi$
$618$	0	0
$619$	19.7990i	0.795789i	0.917431	+	0.397894i	$0.130259\pi$
$619$	19.7990i	0.795789i	−0.917431	+	0.397894i	$0.869741\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	− 22.6274i	− 0.906548i
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	64.0000	2.55591
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	− 5.65685i	− 0.224485i
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	− 42.4264i	− 1.67836i
$640$	0	0
$641$	− 7.07107i	− 0.279290i	−0.990202	−	0.139645i	$-0.955404\pi$
$641$	− 7.07107i	− 0.279290i	0.990202	−	0.139645i	$-0.0445962\pi$
$642$	0	0
$643$	8.48528i	0.334627i	0.985904	+	0.167313i	$0.0535092\pi$
$643$	8.48528i	0.334627i	−0.985904	+	0.167313i	$0.946491\pi$
$644$	0	0
$645$	32.0000	1.26000
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	22.6274i	0.888204i
$650$	0	0
$651$	67.8823i	2.66052i
$652$	0	0
$653$	− 7.07107i	− 0.276712i	−0.990383	−	0.138356i	$-0.955818\pi$
$653$	− 7.07107i	− 0.276712i	0.990383	−	0.138356i	$-0.0441819\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	21.2132i	0.827606i
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	28.0000	1.08907	0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000	1.08907	0.544537	+	0.838737i	$0.316705\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	32.0000	1.24091
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	− 11.3137i	− 0.437413i
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	− 4.24264i	− 0.163542i	−0.996651	−	0.0817709i	$-0.973942\pi$
$673$	− 4.24264i	− 0.163542i	0.996651	−	0.0817709i	$-0.0260576\pi$
$674$	0	0
$675$	16.9706i	0.653197i
$676$	0	0
$677$	− 18.3848i	− 0.706584i	−0.935513	−	0.353292i	$-0.885062\pi$
$677$	− 18.3848i	− 0.706584i	0.935513	−	0.353292i	$-0.114938\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	56.0000	2.14592
$682$	0	0
$683$	2.82843i	0.108227i	0.998535	+	0.0541134i	$0.0172332\pi$
$683$	2.82843i	0.108227i	−0.998535	+	0.0541134i	$0.982767\pi$
$684$	0	0
$685$	− 22.6274i	− 0.864549i
$686$	0	0
$687$	50.9117i	1.94240i
$688$	0	0
$689$	48.0000	1.82865
$690$	0	0
$691$	25.4558i	0.968386i	0.874961	+	0.484193i	$0.160887\pi$
$691$	25.4558i	0.968386i	−0.874961	+	0.484193i	$0.839113\pi$
$692$	0	0
$693$	40.0000	1.51947
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−20.0000	−0.756469
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	− 56.5685i	− 2.13352i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	33.9411i	1.27649i
$708$	0	0
$709$	− 21.2132i	− 0.796679i	−0.917238	−	0.398339i	$-0.869587\pi$
$709$	− 21.2132i	− 0.796679i	0.917238	−	0.398339i	$-0.130413\pi$
$710$	0	0
$711$	14.1421i	0.530372i
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	−16.0000	−0.598366
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.7990i	0.738378i	0.929354	+	0.369189i	$0.120364\pi$
$719$	19.7990i	0.738378i	−0.929354	+	0.369189i	$0.879636\pi$
$720$	0	0
$721$	− 22.6274i	− 0.842689i
$722$	0	0
$723$	76.0000	2.82647
$724$	0	0
$725$	− 12.7279i	− 0.472703i
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	43.0000	1.59259
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	− 33.9411i	− 1.25024i
$738$	0	0
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	0	0
$741$	90.5097i	3.32496i
$742$	0	0
$743$	36.7696i	1.34894i	0.738300	+	0.674472i	$0.235629\pi$
$743$	36.7696i	1.34894i	−0.738300	+	0.674472i	$0.764371\pi$
$744$	0	0
$745$	31.1127i	1.13988i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	− 42.4264i	− 1.54816i	−0.633087	−	0.774081i	$-0.718212\pi$
$751$	− 42.4264i	− 1.54816i	0.633087	−	0.774081i	$-0.281788\pi$
$752$	0	0
$753$	− 11.3137i	− 0.412294i
$754$	0	0
$755$	− 16.9706i	− 0.617622i
$756$	0	0
$757$	14.0000	0.508839	0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000	0.508839	0.254419	+	0.967094i	$0.418116\pi$
$758$	0	0
$759$	− 22.6274i	− 0.821323i
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.0000	−1.15545
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	62.2254i	2.24099i
$772$	0	0
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	0	0
$775$	− 25.4558i	− 0.914401i
$776$	0	0
$777$	− 56.5685i	− 2.02939i
$778$	0	0
$779$	11.3137i	0.405356i
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	− 2.82843i	− 0.100951i
$786$	0	0
$787$	− 42.4264i	− 1.51234i	−0.654376	−	0.756169i	$-0.727069\pi$
$787$	− 42.4264i	− 1.51234i	0.654376	−	0.756169i	$-0.272931\pi$
$788$	0	0
$789$	11.3137i	0.402779i
$790$	0	0
$791$	28.0000	0.995565
$792$	0	0
$793$	5.65685i	0.200881i
$794$	0	0
$795$	−48.0000	−1.70238
$796$	0	0
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−40.0000	−1.41333
$802$	0	0
$803$	−12.0000	−0.423471
$804$	0	0
$805$	− 11.3137i	− 0.398756i
$806$	0	0
$807$	20.0000	0.704033
$808$	0	0
$809$	1.41421i	0.0497211i	0.999691	+	0.0248606i	$0.00791417\pi$
$809$	1.41421i	0.0497211i	−0.999691	+	0.0248606i	$0.992086\pi$
$810$	0	0
$811$	− 8.48528i	− 0.297959i	−0.988840	−	0.148979i	$-0.952401\pi$
$811$	− 8.48528i	− 0.297959i	0.988840	−	0.148979i	$-0.0475988\pi$
$812$	0	0
$813$	22.6274i	0.793578i
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	64.0000	2.23908
$818$	0	0
$819$	56.5685i	1.97666i
$820$	0	0
$821$	41.0122i	1.43134i	0.698441	+	0.715668i	$0.253877\pi$
$821$	41.0122i	1.43134i	−0.698441	+	0.715668i	$0.746123\pi$
$822$	0	0
$823$	− 14.1421i	− 0.492964i	−0.969147	−	0.246482i	$-0.920725\pi$
$823$	− 14.1421i	− 0.492964i	0.969147	−	0.246482i	$-0.0792746\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	− 19.7990i	− 0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$827$	− 19.7990i	− 0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	−60.0000	−2.08138
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.0000	0.692129
$836$	0	0
$837$	48.0000	1.65912
$838$	0	0
$839$	− 19.7990i	− 0.683537i	−0.939784	−	0.341769i	$-0.888974\pi$
$839$	− 19.7990i	− 0.683537i	0.939784	−	0.341769i	$-0.111026\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	− 45.2548i	− 1.55866i
$844$	0	0
$845$	− 4.24264i	− 0.145951i
$846$	0	0
$847$	− 8.48528i	− 0.291558i
$848$	0	0
$849$	−40.0000	−1.37280
$850$	0	0
$851$	−20.0000	−0.685591
$852$	0	0
$853$	− 32.5269i	− 1.11370i	−0.830613	−	0.556850i	$-0.812010\pi$
$853$	− 32.5269i	− 1.11370i	0.830613	−	0.556850i	$-0.187990\pi$
$854$	0	0
$855$	− 56.5685i	− 1.93460i
$856$	0	0
$857$	21.2132i	0.724629i	0.932056	+	0.362315i	$0.118013\pi$
$857$	21.2132i	0.724629i	−0.932056	+	0.362315i	$0.881987\pi$
$858$	0	0
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	0	0
$861$	11.3137i	0.385570i
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	− 21.2132i	− 0.717958i
$874$	0	0
$875$	−32.0000	−1.08180
$876$	0	0
$877$	41.0122i	1.38488i	0.721474	+	0.692442i	$0.243465\pi$
$877$	41.0122i	1.38488i	−0.721474	+	0.692442i	$0.756535\pi$
$878$	0	0
$879$	73.5391i	2.48041i
$880$	0	0
$881$	32.5269i	1.09586i	0.836524	+	0.547930i	$0.184584\pi$
$881$	32.5269i	1.09586i	−0.836524	+	0.547930i	$0.815416\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	32.0000	1.07567
$886$	0	0
$887$	48.0833i	1.61448i	0.590225	+	0.807239i	$0.299039\pi$
$887$	48.0833i	1.61448i	−0.590225	+	0.807239i	$0.700961\pi$
$888$	0	0
$889$	− 11.3137i	− 0.379450i
$890$	0	0
$891$	− 2.82843i	− 0.0947559i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	−36.0000	−1.20067
$900$	0	0
$901$	0	0
$902$	0	0
$903$	64.0000	2.12979
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	19.7990i	0.657415i	0.944432	+	0.328707i	$0.106613\pi$
$907$	19.7990i	0.657415i	−0.944432	+	0.328707i	$0.893387\pi$
$908$	0	0
$909$	60.0000	1.99007
$910$	0	0
$911$	48.0833i	1.59307i	0.604593	+	0.796535i	$0.293336\pi$
$911$	48.0833i	1.59307i	−0.604593	+	0.796535i	$0.706664\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 5.65685i	− 0.187010i
$916$	0	0
$917$	−8.00000	−0.264183
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	33.9411i	1.11840i
$922$	0	0
$923$	33.9411i	1.11719i
$924$	0	0
$925$	21.2132i	0.697486i
$926$	0	0
$927$	−40.0000	−1.31377
$928$	0	0
$929$	52.3259i	1.71676i	0.513017	+	0.858379i	$0.328528\pi$
$929$	52.3259i	1.71676i	−0.513017	+	0.858379i	$0.671472\pi$
$930$	0	0
$931$	8.00000	0.262189
$932$	0	0
$933$	−72.0000	−2.35717
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	0	0
$939$	−36.0000	−1.17482
$940$	0	0
$941$	46.6690i	1.52137i	0.649123	+	0.760684i	$0.275136\pi$
$941$	46.6690i	1.52137i	−0.649123	+	0.760684i	$0.724864\pi$
$942$	0	0
$943$	4.00000	0.130258
$944$	0	0
$945$	− 22.6274i	− 0.736070i
$946$	0	0
$947$	25.4558i	0.827204i	0.910458	+	0.413602i	$0.135729\pi$
$947$	25.4558i	0.827204i	−0.910458	+	0.413602i	$0.864271\pi$
$948$	0	0
$949$	− 16.9706i	− 0.550888i
$950$	0	0
$951$	−60.0000	−1.94563
$952$	0	0
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	0	0
$955$	− 11.3137i	− 0.366103i
$956$	0	0
$957$	33.9411i	1.09716i
$958$	0	0
$959$	− 45.2548i	− 1.46135i
$960$	0	0
$961$	−41.0000	−1.32258
$962$	0	0
$963$	− 14.1421i	− 0.455724i
$964$	0	0
$965$	−6.00000	−0.193147
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	0	0
$975$	− 33.9411i	− 1.08699i
$976$	0	0
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	0	0
$979$	− 22.6274i	− 0.723175i
$980$	0	0
$981$	21.2132i	0.677285i
$982$	0	0
$983$	− 48.0833i	− 1.53362i	−0.641875	−	0.766809i	$-0.721843\pi$
$983$	− 48.0833i	− 1.53362i	0.641875	−	0.766809i	$-0.278157\pi$
$984$	0	0
$985$	−26.0000	−0.828429
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 22.6274i	− 0.719510i
$990$	0	0
$991$	8.48528i	0.269544i	0.990877	+	0.134772i	$0.0430302\pi$
$991$	8.48528i	0.269544i	−0.990877	+	0.134772i	$0.956970\pi$
$992$	0	0
$993$	− 90.5097i	− 2.87224i
$994$	0	0
$995$	12.0000	0.380426
$996$	0	0
$997$	7.07107i	0.223943i	0.993711	+	0.111971i	$0.0357165\pi$
$997$	7.07107i	0.223943i	−0.993711	+	0.111971i	$0.964283\pi$
$998$	0	0
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.a.577.1		2
17.4	even	4		2312.2.a.k.1.1		2
17.9	even	8		136.2.k.d.89.1	yes	2
17.13	even	4		2312.2.a.k.1.2		2
17.15	even	8		136.2.k.d.81.1	✓	2
17.16	even	2	inner	2312.2.b.a.577.2		2
51.26	odd	8		1224.2.w.e.361.1		2
51.32	odd	8		1224.2.w.e.217.1		2
68.15	odd	8		272.2.o.a.81.1		2
68.43	odd	8		272.2.o.a.225.1		2
68.47	odd	4		4624.2.a.t.1.1		2
68.55	odd	4		4624.2.a.t.1.2		2
136.43	odd	8		1088.2.o.q.769.1		2
136.77	even	8		1088.2.o.b.769.1		2
136.83	odd	8		1088.2.o.q.897.1		2
136.117	even	8		1088.2.o.b.897.1		2
204.83	even	8		2448.2.be.i.1441.1		2
204.179	even	8		2448.2.be.i.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.k.d.81.1	✓	2	17.15	even	8
136.2.k.d.89.1	yes	2	17.9	even	8
272.2.o.a.81.1		2	68.15	odd	8
272.2.o.a.225.1		2	68.43	odd	8
1088.2.o.b.769.1		2	136.77	even	8
1088.2.o.b.897.1		2	136.117	even	8
1088.2.o.q.769.1		2	136.43	odd	8
1088.2.o.q.897.1		2	136.83	odd	8
1224.2.w.e.217.1		2	51.32	odd	8
1224.2.w.e.361.1		2	51.26	odd	8
2312.2.a.k.1.1		2	17.4	even	4
2312.2.a.k.1.2		2	17.13	even	4
2312.2.b.a.577.1		2	1.1	even	1	trivial
2312.2.b.a.577.2		2	17.16	even	2	inner
2448.2.be.i.1441.1		2	204.83	even	8
2448.2.be.i.1585.1		2	204.179	even	8
4624.2.a.t.1.1		2	68.47	odd	4
4624.2.a.t.1.2		2	68.55	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{3} -1.41421i q^{5} -2.82843i q^{7} -5.00000 q^{9} -2.82843i q^{11} +4.00000 q^{13} -4.00000 q^{15} -8.00000 q^{19} -8.00000 q^{21} +2.82843i q^{23} +3.00000 q^{25} +5.65685i q^{27} -4.24264i q^{29} -8.48528i q^{31} -8.00000 q^{33} -4.00000 q^{35} +7.07107i q^{37} -11.3137i q^{39} -1.41421i q^{41} -8.00000 q^{43} +7.07107i q^{45} -1.00000 q^{49} +12.0000 q^{53} -4.00000 q^{55} +22.6274i q^{57} -8.00000 q^{59} +1.41421i q^{61} +14.1421i q^{63} -5.65685i q^{65} +12.0000 q^{67} +8.00000 q^{69} +8.48528i q^{71} -4.24264i q^{73} -8.48528i q^{75} -8.00000 q^{77} -2.82843i q^{79} +1.00000 q^{81} -12.0000 q^{87} +8.00000 q^{89} -11.3137i q^{91} -24.0000 q^{93} +11.3137i q^{95} +4.24264i q^{97} +14.1421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{9} + 8 q^{13} - 8 q^{15} - 16 q^{19} - 16 q^{21} + 6 q^{25} - 16 q^{33} - 8 q^{35} - 16 q^{43} - 2 q^{49} + 24 q^{53} - 8 q^{55} - 16 q^{59} + 24 q^{67} + 16 q^{69} - 16 q^{77} + 2 q^{81} - 24 q^{87}+ \cdots - 48 q^{93}+O(q^{100}) \) 2 * q - 10 * q^9 + 8 * q^13 - 8 * q^15 - 16 * q^19 - 16 * q^21 + 6 * q^25 - 16 * q^33 - 8 * q^35 - 16 * q^43 - 2 * q^49 + 24 * q^53 - 8 * q^55 - 16 * q^59 + 24 * q^67 + 16 * q^69 - 16 * q^77 + 2 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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