LMFDB - Embedded newform 4368.2.h.l.337.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1092)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.l.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +3.12311i q^{11} +(-3.56155 - 0.561553i) q^{13} +0.561553i q^{15} -2.00000 q^{17} +2.56155i q^{19} -1.00000i q^{21} +3.68466 q^{23} +4.68466 q^{25} +1.00000 q^{27} +3.43845 q^{29} +2.56155i q^{31} +3.12311i q^{33} +0.561553 q^{35} +(-3.56155 - 0.561553i) q^{39} -10.0000i q^{41} -1.43845 q^{43} +0.561553i q^{45} +4.56155i q^{47} -1.00000 q^{49} -2.00000 q^{51} -0.561553 q^{53} -1.75379 q^{55} +2.56155i q^{57} +15.1231i q^{59} -11.1231 q^{61} -1.00000i q^{63} +(0.315342 - 2.00000i) q^{65} +13.1231i q^{67} +3.68466 q^{69} +9.36932i q^{71} +10.5616i q^{73} +4.68466 q^{75} +3.12311 q^{77} +3.68466 q^{79} +1.00000 q^{81} +5.68466i q^{83} -1.12311i q^{85} +3.43845 q^{87} -5.68466i q^{89} +(-0.561553 + 3.56155i) q^{91} +2.56155i q^{93} -1.43845 q^{95} +15.6847i q^{97} +3.12311i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} - 6 q^{13} - 8 q^{17} - 10 q^{23} - 6 q^{25} + 4 q^{27} + 22 q^{29} - 6 q^{35} - 6 q^{39} - 14 q^{43} - 4 q^{49} - 8 q^{51} + 6 q^{53} - 40 q^{55} - 28 q^{61} + 26 q^{65} - 10 q^{69} - 6 q^{75} - 4 q^{77} - 10 q^{79} + 4 q^{81} + 22 q^{87} + 6 q^{91} - 14 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 - 6 * q^13 - 8 * q^17 - 10 * q^23 - 6 * q^25 + 4 * q^27 + 22 * q^29 - 6 * q^35 - 6 * q^39 - 14 * q^43 - 4 * q^49 - 8 * q^51 + 6 * q^53 - 40 * q^55 - 28 * q^61 + 26 * q^65 - 10 * q^69 - 6 * q^75 - 4 * q^77 - 10 * q^79 + 4 * q^81 + 22 * q^87 + 6 * q^91 - 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			0.561553i			0.251134i	0.992085	+	0.125567i	$0.0400750\pi$
$5$			0.561553i			0.251134i	−0.992085	+	0.125567i	$0.959925\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.12311i			0.941652i	0.882226	+	0.470826i	$0.156044\pi$
$11$			3.12311i			0.941652i	−0.882226	+	0.470826i	$0.843956\pi$
$12$		0			0
$13$	−3.56155	−	0.561553i	−0.987797	−	0.155747i
$13$	−3.56155	−	0.561553i	−0.987797	−	0.155747i
$14$		0			0
$15$			0.561553i			0.144992i
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			2.56155i			0.587661i	0.955858	+	0.293830i	$0.0949300\pi$
$19$			2.56155i			0.587661i	−0.955858	+	0.293830i	$0.905070\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	3.68466			0.768304			0.384152	−	0.923270i	$-0.374494\pi$
$23$	3.68466			0.768304			0.384152	+	0.923270i	$0.374494\pi$
$24$		0			0
$25$	4.68466			0.936932
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	3.43845			0.638504			0.319252	−	0.947670i	$-0.396568\pi$
$29$	3.43845			0.638504			0.319252	+	0.947670i	$0.396568\pi$
$30$		0			0
$31$			2.56155i			0.460068i	0.973183	+	0.230034i	$0.0738838\pi$
$31$			2.56155i			0.460068i	−0.973183	+	0.230034i	$0.926116\pi$
$32$		0			0
$33$			3.12311i			0.543663i
$34$		0			0
$35$	0.561553			0.0949197
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−3.56155	−	0.561553i	−0.570305	−	0.0899204i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$	−1.43845			−0.219361			−0.109681	−	0.993967i	$-0.534983\pi$
$43$	−1.43845			−0.219361			−0.109681	+	0.993967i	$0.534983\pi$
$44$		0			0
$45$			0.561553i			0.0837114i
$46$		0			0
$47$			4.56155i			0.665371i	0.943038	+	0.332685i	$0.107955\pi$
$47$			4.56155i			0.665371i	−0.943038	+	0.332685i	$0.892045\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	−0.561553			−0.0771352			−0.0385676	−	0.999256i	$-0.512279\pi$
$53$	−0.561553			−0.0771352			−0.0385676	+	0.999256i	$0.512279\pi$
$54$		0			0
$55$	−1.75379			−0.236481
$56$		0			0
$57$			2.56155i			0.339286i
$58$		0			0
$59$			15.1231i			1.96886i	0.175775	+	0.984430i	$0.443757\pi$
$59$			15.1231i			1.96886i	−0.175775	+	0.984430i	$0.556243\pi$
$60$		0			0
$61$	−11.1231			−1.42417			−0.712084	−	0.702094i	$-0.752248\pi$
$61$	−11.1231			−1.42417			−0.712084	+	0.702094i	$0.752248\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	0.315342	−	2.00000i	0.0391133	−	0.248069i
$66$		0			0
$67$			13.1231i			1.60324i	0.597832	+	0.801621i	$0.296029\pi$
$67$			13.1231i			1.60324i	−0.597832	+	0.801621i	$0.703971\pi$
$68$		0			0
$69$	3.68466			0.443581
$70$		0			0
$71$			9.36932i			1.11193i	0.831205	+	0.555967i	$0.187652\pi$
$71$			9.36932i			1.11193i	−0.831205	+	0.555967i	$0.812348\pi$
$72$		0			0
$73$			10.5616i			1.23614i	0.786125	+	0.618068i	$0.212084\pi$
$73$			10.5616i			1.23614i	−0.786125	+	0.618068i	$0.787916\pi$
$74$		0			0
$75$	4.68466			0.540938
$76$		0			0
$77$	3.12311			0.355911
$78$		0			0
$79$	3.68466			0.414556			0.207278	−	0.978282i	$-0.433539\pi$
$79$	3.68466			0.414556			0.207278	+	0.978282i	$0.433539\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.68466i			0.623972i	0.950087	+	0.311986i	$0.100994\pi$
$83$			5.68466i			0.623972i	−0.950087	+	0.311986i	$0.899006\pi$
$84$		0			0
$85$		−	1.12311i		−	0.121818i
$86$		0			0
$87$	3.43845			0.368640
$88$		0			0
$89$		−	5.68466i		−	0.602573i	−0.953534	−	0.301286i	$-0.902584\pi$
$89$		−	5.68466i		−	0.602573i	0.953534	−	0.301286i	$-0.0974160\pi$
$90$		0			0
$91$	−0.561553	+	3.56155i	−0.0588667	+	0.373352i
$92$		0			0
$93$			2.56155i			0.265621i
$94$		0			0
$95$	−1.43845			−0.147582
$96$		0			0
$97$			15.6847i			1.59254i	0.604944	+	0.796268i	$0.293195\pi$
$97$			15.6847i			1.59254i	−0.604944	+	0.796268i	$0.706805\pi$
$98$		0			0
$99$			3.12311i			0.313884i
$100$		0			0
$101$	−9.36932			−0.932282			−0.466141	−	0.884710i	$-0.654356\pi$
$101$	−9.36932			−0.932282			−0.466141	+	0.884710i	$0.654356\pi$
$102$		0			0
$103$	1.75379			0.172806			0.0864030	−	0.996260i	$-0.472463\pi$
$103$	1.75379			0.172806			0.0864030	+	0.996260i	$0.472463\pi$
$104$		0			0
$105$	0.561553			0.0548019
$106$		0			0
$107$	16.4924			1.59438			0.797191	−	0.603727i	$-0.206318\pi$
$107$	16.4924			1.59438			0.797191	+	0.603727i	$0.206318\pi$
$108$		0			0
$109$		−	1.12311i		−	0.107574i	−0.998552	−	0.0537870i	$-0.982871\pi$
$109$		−	1.12311i		−	0.107574i	0.998552	−	0.0537870i	$-0.0171292\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.8078			1.39300			0.696499	−	0.717558i	$-0.254740\pi$
$113$	14.8078			1.39300			0.696499	+	0.717558i	$0.254740\pi$
$114$		0			0
$115$			2.06913i			0.192947i
$116$		0			0
$117$	−3.56155	−	0.561553i	−0.329266	−	0.0519156i
$118$		0			0
$119$			2.00000i			0.183340i
$120$		0			0
$121$	1.24621			0.113292
$122$		0			0
$123$		−	10.0000i		−	0.901670i
$124$		0			0
$125$			5.43845i			0.486430i
$126$		0			0
$127$	−12.4924			−1.10852			−0.554262	−	0.832343i	$-0.686999\pi$
$127$	−12.4924			−1.10852			−0.554262	+	0.832343i	$0.686999\pi$
$128$		0			0
$129$	−1.43845			−0.126648
$130$		0			0
$131$	−6.87689			−0.600837			−0.300419	−	0.953807i	$-0.597126\pi$
$131$	−6.87689			−0.600837			−0.300419	+	0.953807i	$0.597126\pi$
$132$		0			0
$133$	2.56155			0.222115
$134$		0			0
$135$			0.561553i			0.0483308i
$136$		0			0
$137$		−	5.36932i		−	0.458732i	−0.973340	−	0.229366i	$-0.926335\pi$
$137$		−	5.36932i		−	0.458732i	0.973340	−	0.229366i	$-0.0736652\pi$
$138$		0			0
$139$	−5.75379			−0.488030			−0.244015	−	0.969771i	$-0.578465\pi$
$139$	−5.75379			−0.488030			−0.244015	+	0.969771i	$0.578465\pi$
$140$		0			0
$141$			4.56155i			0.384152i
$142$		0			0
$143$	1.75379	−	11.1231i	0.146659	−	0.930161i
$144$		0			0
$145$			1.93087i			0.160350i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$			5.36932i			0.439872i	0.975514	+	0.219936i	$0.0705848\pi$
$149$			5.36932i			0.439872i	−0.975514	+	0.219936i	$0.929415\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	−1.43845			−0.115539
$156$		0			0
$157$	−8.24621			−0.658119			−0.329060	−	0.944309i	$-0.606732\pi$
$157$	−8.24621			−0.658119			−0.329060	+	0.944309i	$0.606732\pi$
$158$		0			0
$159$	−0.561553			−0.0445340
$160$		0			0
$161$		−	3.68466i		−	0.290392i
$162$		0			0
$163$			6.87689i			0.538640i	0.963051	+	0.269320i	$0.0867989\pi$
$163$			6.87689i			0.538640i	−0.963051	+	0.269320i	$0.913201\pi$
$164$		0			0
$165$	−1.75379			−0.136532
$166$		0			0
$167$		−	0.561553i		−	0.0434543i	−0.999764	−	0.0217271i	$-0.993083\pi$
$167$		−	0.561553i		−	0.0434543i	0.999764	−	0.0217271i	$-0.00691650\pi$
$168$		0			0
$169$	12.3693	+	4.00000i	0.951486	+	0.307692i
$170$		0			0
$171$			2.56155i			0.195887i
$172$		0			0
$173$	−4.87689			−0.370783			−0.185392	−	0.982665i	$-0.559355\pi$
$173$	−4.87689			−0.370783			−0.185392	+	0.982665i	$0.559355\pi$
$174$		0			0
$175$		−	4.68466i		−	0.354127i
$176$		0			0
$177$			15.1231i			1.13672i
$178$		0			0
$179$	−10.5616			−0.789408			−0.394704	−	0.918808i	$-0.629153\pi$
$179$	−10.5616			−0.789408			−0.394704	+	0.918808i	$0.629153\pi$
$180$		0			0
$181$	17.3693			1.29105			0.645526	−	0.763739i	$-0.276638\pi$
$181$	17.3693			1.29105			0.645526	+	0.763739i	$0.276638\pi$
$182$		0			0
$183$	−11.1231			−0.822244
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	6.24621i		−	0.456768i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$			15.3693i			1.10631i	0.833079	+	0.553154i	$0.186576\pi$
$193$			15.3693i			1.10631i	−0.833079	+	0.553154i	$0.813424\pi$
$194$		0			0
$195$	0.315342	−	2.00000i	0.0225821	−	0.143223i
$196$		0			0
$197$		−	21.3693i		−	1.52250i	−0.648458	−	0.761250i	$-0.724586\pi$
$197$		−	21.3693i		−	1.52250i	0.648458	−	0.761250i	$-0.275414\pi$
$198$		0			0
$199$	25.6155			1.81584			0.907918	−	0.419147i	$-0.137671\pi$
$199$	25.6155			1.81584			0.907918	+	0.419147i	$0.137671\pi$
$200$		0			0
$201$			13.1231i			0.925633i
$202$		0			0
$203$		−	3.43845i		−	0.241332i
$204$		0			0
$205$	5.61553			0.392205
$206$		0			0
$207$	3.68466			0.256101
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−2.56155			−0.176345			−0.0881723	−	0.996105i	$-0.528103\pi$
$211$	−2.56155			−0.176345			−0.0881723	+	0.996105i	$0.528103\pi$
$212$		0			0
$213$			9.36932i			0.641975i
$214$		0			0
$215$		−	0.807764i		−	0.0550891i
$216$		0			0
$217$	2.56155			0.173890
$218$		0			0
$219$			10.5616i			0.713684i
$220$		0			0
$221$	7.12311	+	1.12311i	0.479152	+	0.0755483i
$222$		0			0
$223$		−	10.5616i		−	0.707254i	−0.935387	−	0.353627i	$-0.884948\pi$
$223$		−	10.5616i		−	0.707254i	0.935387	−	0.353627i	$-0.115052\pi$
$224$		0			0
$225$	4.68466			0.312311
$226$		0			0
$227$		−	7.12311i		−	0.472777i	−0.971659	−	0.236389i	$-0.924036\pi$
$227$		−	7.12311i		−	0.472777i	0.971659	−	0.236389i	$-0.0759638\pi$
$228$		0			0
$229$			19.3693i			1.27996i	0.768391	+	0.639980i	$0.221057\pi$
$229$			19.3693i			1.27996i	−0.768391	+	0.639980i	$0.778943\pi$
$230$		0			0
$231$	3.12311			0.205485
$232$		0			0
$233$	−17.6847			−1.15856			−0.579280	−	0.815128i	$-0.696666\pi$
$233$	−17.6847			−1.15856			−0.579280	+	0.815128i	$0.696666\pi$
$234$		0			0
$235$	−2.56155			−0.167097
$236$		0			0
$237$	3.68466			0.239344
$238$		0			0
$239$		−	8.24621i		−	0.533403i	−0.963779	−	0.266702i	$-0.914066\pi$
$239$		−	8.24621i		−	0.533403i	0.963779	−	0.266702i	$-0.0859338\pi$
$240$		0			0
$241$			23.0540i			1.48504i	0.669826	+	0.742519i	$0.266369\pi$
$241$			23.0540i			1.48504i	−0.669826	+	0.742519i	$0.733631\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	0.561553i		−	0.0358763i
$246$		0			0
$247$	1.43845	−	9.12311i	0.0915262	−	0.580489i
$248$		0			0
$249$			5.68466i			0.360251i
$250$		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
$253$			11.5076i			0.723475i
$254$		0			0
$255$		−	1.12311i		−	0.0703316i
$256$		0			0
$257$	−5.36932			−0.334929			−0.167464	−	0.985878i	$-0.553558\pi$
$257$	−5.36932			−0.334929			−0.167464	+	0.985878i	$0.553558\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.43845			0.212835
$262$		0			0
$263$	−27.0540			−1.66822			−0.834110	−	0.551598i	$-0.814018\pi$
$263$	−27.0540			−1.66822			−0.834110	+	0.551598i	$0.814018\pi$
$264$		0			0
$265$		−	0.315342i		−	0.0193713i
$266$		0			0
$267$		−	5.68466i		−	0.347895i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$			22.7386i			1.38127i	0.723202	+	0.690637i	$0.242670\pi$
$271$			22.7386i			1.38127i	−0.723202	+	0.690637i	$0.757330\pi$
$272$		0			0
$273$	−0.561553	+	3.56155i	−0.0339867	+	0.215555i
$274$		0			0
$275$			14.6307i			0.882263i
$276$		0			0
$277$	12.5616			0.754751			0.377375	−	0.926060i	$-0.376827\pi$
$277$	12.5616			0.754751			0.377375	+	0.926060i	$0.376827\pi$
$278$		0			0
$279$			2.56155i			0.153356i
$280$		0			0
$281$			18.0000i			1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		0			0
$283$	9.75379			0.579803			0.289901	−	0.957057i	$-0.406378\pi$
$283$	9.75379			0.579803			0.289901	+	0.957057i	$0.406378\pi$
$284$		0			0
$285$	−1.43845			−0.0852063
$286$		0			0
$287$	−10.0000			−0.590281
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$			15.6847i			0.919451i
$292$		0			0
$293$			7.93087i			0.463326i	0.972796	+	0.231663i	$0.0744167\pi$
$293$			7.93087i			0.463326i	−0.972796	+	0.231663i	$0.925583\pi$
$294$		0			0
$295$	−8.49242			−0.494448
$296$		0			0
$297$			3.12311i			0.181221i
$298$		0			0
$299$	−13.1231	−	2.06913i	−0.758929	−	0.119661i
$300$		0			0
$301$			1.43845i			0.0829107i
$302$		0			0
$303$	−9.36932			−0.538253
$304$		0			0
$305$		−	6.24621i		−	0.357657i
$306$		0			0
$307$			8.80776i			0.502686i	0.967898	+	0.251343i	$0.0808722\pi$
$307$			8.80776i			0.502686i	−0.967898	+	0.251343i	$0.919128\pi$
$308$		0			0
$309$	1.75379			0.0997696
$310$		0			0
$311$	25.6155			1.45252			0.726262	−	0.687418i	$-0.241256\pi$
$311$	25.6155			1.45252			0.726262	+	0.687418i	$0.241256\pi$
$312$		0			0
$313$	−16.8769			−0.953938			−0.476969	−	0.878920i	$-0.658265\pi$
$313$	−16.8769			−0.953938			−0.476969	+	0.878920i	$0.658265\pi$
$314$		0			0
$315$	0.561553			0.0316399
$316$		0			0
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		0			0
$319$			10.7386i			0.601248i
$320$		0			0
$321$	16.4924			0.920517
$322$		0			0
$323$		−	5.12311i		−	0.285057i
$324$		0			0
$325$	−16.6847	−	2.63068i	−0.925498	−	0.145924i
$326$		0			0
$327$		−	1.12311i		−	0.0621079i
$328$		0			0
$329$	4.56155			0.251487
$330$		0			0
$331$		−	26.2462i		−	1.44262i	−0.692611	−	0.721311i	$-0.743540\pi$
$331$		−	26.2462i		−	1.44262i	0.692611	−	0.721311i	$-0.256460\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.36932			−0.402629
$336$		0			0
$337$	−1.68466			−0.0917692			−0.0458846	−	0.998947i	$-0.514611\pi$
$337$	−1.68466			−0.0917692			−0.0458846	+	0.998947i	$0.514611\pi$
$338$		0			0
$339$	14.8078			0.804247
$340$		0			0
$341$	−8.00000			−0.433224
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			2.06913i			0.111398i
$346$		0			0
$347$	−8.49242			−0.455897			−0.227949	−	0.973673i	$-0.573202\pi$
$347$	−8.49242			−0.455897			−0.227949	+	0.973673i	$0.573202\pi$
$348$		0			0
$349$		−	26.4233i		−	1.41441i	−0.707010	−	0.707203i	$-0.749957\pi$
$349$		−	26.4233i		−	1.41441i	0.707010	−	0.707203i	$-0.250043\pi$
$350$		0			0
$351$	−3.56155	−	0.561553i	−0.190102	−	0.0299735i
$352$		0			0
$353$		−	28.2462i		−	1.50339i	−0.659509	−	0.751697i	$-0.729236\pi$
$353$		−	28.2462i		−	1.50339i	0.659509	−	0.751697i	$-0.270764\pi$
$354$		0			0
$355$	−5.26137			−0.279244
$356$		0			0
$357$			2.00000i			0.105851i
$358$		0			0
$359$		−	20.2462i		−	1.06855i	−0.845309	−	0.534277i	$-0.820584\pi$
$359$		−	20.2462i		−	1.06855i	0.845309	−	0.534277i	$-0.179416\pi$
$360$		0			0
$361$	12.4384			0.654655
$362$		0			0
$363$	1.24621			0.0654091
$364$		0			0
$365$	−5.93087			−0.310436
$366$		0			0
$367$	11.3693			0.593474			0.296737	−	0.954959i	$-0.404102\pi$
$367$	11.3693			0.593474			0.296737	+	0.954959i	$0.404102\pi$
$368$		0			0
$369$		−	10.0000i		−	0.520579i
$370$		0			0
$371$			0.561553i			0.0291544i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$			5.43845i			0.280840i
$376$		0			0
$377$	−12.2462	−	1.93087i	−0.630712	−	0.0994448i
$378$		0			0
$379$			36.9848i			1.89978i	0.312578	+	0.949892i	$0.398807\pi$
$379$			36.9848i			1.89978i	−0.312578	+	0.949892i	$0.601193\pi$
$380$		0			0
$381$	−12.4924			−0.640006
$382$		0			0
$383$		−	20.8769i		−	1.06676i	−0.845876	−	0.533380i	$-0.820922\pi$
$383$		−	20.8769i		−	1.06676i	0.845876	−	0.533380i	$-0.179078\pi$
$384$		0			0
$385$			1.75379i			0.0893814i
$386$		0			0
$387$	−1.43845			−0.0731204
$388$		0			0
$389$	4.24621			0.215291			0.107646	−	0.994189i	$-0.465669\pi$
$389$	4.24621			0.215291			0.107646	+	0.994189i	$0.465669\pi$
$390$		0			0
$391$	−7.36932			−0.372682
$392$		0			0
$393$	−6.87689			−0.346893
$394$		0			0
$395$			2.06913i			0.104109i
$396$		0			0
$397$			11.0540i			0.554783i	0.960757	+	0.277392i	$0.0894699\pi$
$397$			11.0540i			0.554783i	−0.960757	+	0.277392i	$0.910530\pi$
$398$		0			0
$399$	2.56155			0.128238
$400$		0			0
$401$			2.49242i			0.124466i	0.998062	+	0.0622328i	$0.0198221\pi$
$401$			2.49242i			0.124466i	−0.998062	+	0.0622328i	$0.980178\pi$
$402$		0			0
$403$	1.43845	−	9.12311i	0.0716542	−	0.454454i
$404$		0			0
$405$			0.561553i			0.0279038i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	2.56155i		−	0.126661i	−0.997993	−	0.0633303i	$-0.979828\pi$
$409$		−	2.56155i		−	0.126661i	0.997993	−	0.0633303i	$-0.0201722\pi$
$410$		0			0
$411$		−	5.36932i		−	0.264849i
$412$		0			0
$413$	15.1231			0.744159
$414$		0			0
$415$	−3.19224			−0.156701
$416$		0			0
$417$	−5.75379			−0.281764
$418$		0			0
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$		−	4.63068i		−	0.225686i	−0.993613	−	0.112843i	$-0.964004\pi$
$421$		−	4.63068i		−	0.225686i	0.993613	−	0.112843i	$-0.0359957\pi$
$422$		0			0
$423$			4.56155i			0.221790i
$424$		0			0
$425$	−9.36932			−0.454479
$426$		0			0
$427$			11.1231i			0.538285i
$428$		0			0
$429$	1.75379	−	11.1231i	0.0846737	−	0.537029i
$430$		0			0
$431$			8.87689i			0.427585i	0.976879	+	0.213792i	$0.0685816\pi$
$431$			8.87689i			0.427585i	−0.976879	+	0.213792i	$0.931418\pi$
$432$		0			0
$433$	20.7386			0.996635			0.498318	−	0.866995i	$-0.333951\pi$
$433$	20.7386			0.996635			0.498318	+	0.866995i	$0.333951\pi$
$434$		0			0
$435$			1.93087i			0.0925781i
$436$		0			0
$437$			9.43845i			0.451502i
$438$		0			0
$439$	−6.24621			−0.298115			−0.149058	−	0.988828i	$-0.547624\pi$
$439$	−6.24621			−0.298115			−0.149058	+	0.988828i	$0.547624\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	24.3153			1.15526			0.577628	−	0.816300i	$-0.303978\pi$
$443$	24.3153			1.15526			0.577628	+	0.816300i	$0.303978\pi$
$444$		0			0
$445$	3.19224			0.151326
$446$		0			0
$447$			5.36932i			0.253960i
$448$		0			0
$449$		−	10.0000i		−	0.471929i	−0.971762	−	0.235965i	$-0.924175\pi$
$449$		−	10.0000i		−	0.471929i	0.971762	−	0.235965i	$-0.0758249\pi$
$450$		0			0
$451$	31.2311			1.47061
$452$		0			0
$453$		−	12.0000i		−	0.563809i
$454$		0			0
$455$	−2.00000	−	0.315342i	−0.0937614	−	0.0147834i
$456$		0			0
$457$		−	4.63068i		−	0.216614i	−0.994117	−	0.108307i	$-0.965457\pi$
$457$		−	4.63068i		−	0.216614i	0.994117	−	0.108307i	$-0.0345430\pi$
$458$		0			0
$459$	−2.00000			−0.0933520
$460$		0			0
$461$		−	3.75379i		−	0.174831i	−0.996172	−	0.0874157i	$-0.972139\pi$
$461$		−	3.75379i		−	0.174831i	0.996172	−	0.0874157i	$-0.0278608\pi$
$462$		0			0
$463$		−	17.6155i		−	0.818663i	−0.912386	−	0.409332i	$-0.865762\pi$
$463$		−	17.6155i		−	0.818663i	0.912386	−	0.409332i	$-0.134238\pi$
$464$		0			0
$465$	−1.43845			−0.0667064
$466$		0			0
$467$	−17.1231			−0.792363			−0.396181	−	0.918172i	$-0.629665\pi$
$467$	−17.1231			−0.792363			−0.396181	+	0.918172i	$0.629665\pi$
$468$		0			0
$469$	13.1231			0.605969
$470$		0			0
$471$	−8.24621			−0.379965
$472$		0			0
$473$		−	4.49242i		−	0.206562i
$474$		0			0
$475$			12.0000i			0.550598i
$476$		0			0
$477$	−0.561553			−0.0257117
$478$		0			0
$479$		−	35.3002i		−	1.61291i	−0.591298	−	0.806453i	$-0.701384\pi$
$479$		−	35.3002i		−	1.61291i	0.591298	−	0.806453i	$-0.298616\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		−	3.68466i		−	0.167658i
$484$		0			0
$485$	−8.80776			−0.399940
$486$		0			0
$487$		−	6.24621i		−	0.283043i	−0.989935	−	0.141521i	$-0.954801\pi$
$487$		−	6.24621i		−	0.283043i	0.989935	−	0.141521i	$-0.0451994\pi$
$488$		0			0
$489$			6.87689i			0.310984i
$490$		0			0
$491$	−6.24621			−0.281888			−0.140944	−	0.990018i	$-0.545014\pi$
$491$	−6.24621			−0.281888			−0.140944	+	0.990018i	$0.545014\pi$
$492$		0			0
$493$	−6.87689			−0.309720
$494$		0			0
$495$	−1.75379			−0.0788269
$496$		0			0
$497$	9.36932			0.420271
$498$		0			0
$499$			41.6155i			1.86297i	0.363784	+	0.931483i	$0.381485\pi$
$499$			41.6155i			1.86297i	−0.363784	+	0.931483i	$0.618515\pi$
$500$		0			0
$501$		−	0.561553i		−	0.0250883i
$502$		0			0
$503$	−28.4924			−1.27041			−0.635207	−	0.772342i	$-0.719085\pi$
$503$	−28.4924			−1.27041			−0.635207	+	0.772342i	$0.719085\pi$
$504$		0			0
$505$		−	5.26137i		−	0.234128i
$506$		0			0
$507$	12.3693	+	4.00000i	0.549341	+	0.177646i
$508$		0			0
$509$		−	41.6847i		−	1.84764i	−0.382827	−	0.923820i	$-0.625050\pi$
$509$		−	41.6847i		−	1.84764i	0.382827	−	0.923820i	$-0.374950\pi$
$510$		0			0
$511$	10.5616			0.467216
$512$		0			0
$513$			2.56155i			0.113095i
$514$		0			0
$515$			0.984845i			0.0433975i
$516$		0			0
$517$	−14.2462			−0.626548
$518$		0			0
$519$	−4.87689			−0.214072
$520$		0			0
$521$	21.3693			0.936207			0.468103	−	0.883674i	$-0.344937\pi$
$521$	21.3693			0.936207			0.468103	+	0.883674i	$0.344937\pi$
$522$		0			0
$523$	−5.75379			−0.251596			−0.125798	−	0.992056i	$-0.540149\pi$
$523$	−5.75379			−0.251596			−0.125798	+	0.992056i	$0.540149\pi$
$524$		0			0
$525$		−	4.68466i		−	0.204455i
$526$		0			0
$527$		−	5.12311i		−	0.223166i
$528$		0			0
$529$	−9.42329			−0.409708
$530$		0			0
$531$			15.1231i			0.656287i
$532$		0			0
$533$	−5.61553	+	35.6155i	−0.243236	+	1.54268i
$534$		0			0
$535$			9.26137i			0.400404i
$536$		0			0
$537$	−10.5616			−0.455765
$538$		0			0
$539$		−	3.12311i		−	0.134522i
$540$		0			0
$541$		−	20.9848i		−	0.902209i	−0.892471	−	0.451104i	$-0.851030\pi$
$541$		−	20.9848i		−	0.902209i	0.892471	−	0.451104i	$-0.148970\pi$
$542$		0			0
$543$	17.3693			0.745389
$544$		0			0
$545$	0.630683			0.0270155
$546$		0			0
$547$	−32.1771			−1.37579			−0.687896	−	0.725809i	$-0.741466\pi$
$547$	−32.1771			−1.37579			−0.687896	+	0.725809i	$0.741466\pi$
$548$		0			0
$549$	−11.1231			−0.474723
$550$		0			0
$551$			8.80776i			0.375223i
$552$		0			0
$553$		−	3.68466i		−	0.156688i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	3.12311i		−	0.132330i	−0.997809	−	0.0661651i	$-0.978924\pi$
$557$		−	3.12311i		−	0.132330i	0.997809	−	0.0661651i	$-0.0210764\pi$
$558$		0			0
$559$	5.12311	+	0.807764i	0.216684	+	0.0341648i
$560$		0			0
$561$		−	6.24621i		−	0.263715i
$562$		0			0
$563$	−3.36932			−0.142000			−0.0709999	−	0.997476i	$-0.522619\pi$
$563$	−3.36932			−0.142000			−0.0709999	+	0.997476i	$0.522619\pi$
$564$		0			0
$565$			8.31534i			0.349829i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−32.4233			−1.35926			−0.679628	−	0.733557i	$-0.737859\pi$
$569$	−32.4233			−1.35926			−0.679628	+	0.733557i	$0.737859\pi$
$570$		0			0
$571$	12.8078			0.535988			0.267994	−	0.963421i	$-0.413639\pi$
$571$	12.8078			0.535988			0.267994	+	0.963421i	$0.413639\pi$
$572$		0			0
$573$	16.0000			0.668410
$574$		0			0
$575$	17.2614			0.719849
$576$		0			0
$577$		−	33.6155i		−	1.39943i	−0.714421	−	0.699716i	$-0.753310\pi$
$577$		−	33.6155i		−	1.39943i	0.714421	−	0.699716i	$-0.246690\pi$
$578$		0			0
$579$			15.3693i			0.638727i
$580$		0			0
$581$	5.68466			0.235839
$582$		0			0
$583$		−	1.75379i		−	0.0726345i
$584$		0			0
$585$	0.315342	−	2.00000i	0.0130378	−	0.0826898i
$586$		0			0
$587$			11.4384i			0.472115i	0.971739	+	0.236058i	$0.0758554\pi$
$587$			11.4384i			0.472115i	−0.971739	+	0.236058i	$0.924145\pi$
$588$		0			0
$589$	−6.56155			−0.270364
$590$		0			0
$591$		−	21.3693i		−	0.879016i
$592$		0			0
$593$			24.4233i			1.00294i	0.865174	+	0.501472i	$0.167208\pi$
$593$			24.4233i			1.00294i	−0.865174	+	0.501472i	$0.832792\pi$
$594$		0			0
$595$	−1.12311			−0.0460428
$596$		0			0
$597$	25.6155			1.04837
$598$		0			0
$599$	22.5616			0.921840			0.460920	−	0.887442i	$-0.347519\pi$
$599$	22.5616			0.921840			0.460920	+	0.887442i	$0.347519\pi$
$600$		0			0
$601$	22.4924			0.917485			0.458743	−	0.888569i	$-0.348300\pi$
$601$	22.4924			0.917485			0.458743	+	0.888569i	$0.348300\pi$
$602$		0			0
$603$			13.1231i			0.534414i
$604$		0			0
$605$			0.699813i			0.0284515i
$606$		0			0
$607$	−0.492423			−0.0199868			−0.00999341	−	0.999950i	$-0.503181\pi$
$607$	−0.492423			−0.0199868			−0.00999341	+	0.999950i	$0.503181\pi$
$608$		0			0
$609$		−	3.43845i		−	0.139333i
$610$		0			0
$611$	2.56155	−	16.2462i	0.103629	−	0.657251i
$612$		0			0
$613$		−	5.12311i		−	0.206920i	−0.994634	−	0.103460i	$-0.967009\pi$
$613$		−	5.12311i		−	0.206920i	0.994634	−	0.103460i	$-0.0329914\pi$
$614$		0			0
$615$	5.61553			0.226440
$616$		0			0
$617$		−	37.8617i		−	1.52426i	−0.647426	−	0.762128i	$-0.724155\pi$
$617$		−	37.8617i		−	1.52426i	0.647426	−	0.762128i	$-0.275845\pi$
$618$		0			0
$619$			2.24621i			0.0902829i	0.998981	+	0.0451414i	$0.0143738\pi$
$619$			2.24621i			0.0902829i	−0.998981	+	0.0451414i	$0.985626\pi$
$620$		0			0
$621$	3.68466			0.147860
$622$		0			0
$623$	−5.68466			−0.227751
$624$		0			0
$625$	20.3693			0.814773
$626$		0			0
$627$	−8.00000			−0.319489
$628$		0			0
$629$		0			0
$630$		0			0
$631$			26.7386i			1.06445i	0.846604	+	0.532224i	$0.178644\pi$
$631$			26.7386i			1.06445i	−0.846604	+	0.532224i	$0.821356\pi$
$632$		0			0
$633$	−2.56155			−0.101813
$634$		0			0
$635$		−	7.01515i		−	0.278388i
$636$		0			0
$637$	3.56155	+	0.561553i	0.141114	+	0.0222495i
$638$		0			0
$639$			9.36932i			0.370644i
$640$		0			0
$641$	33.6847			1.33046			0.665232	−	0.746637i	$-0.268333\pi$
$641$	33.6847			1.33046			0.665232	+	0.746637i	$0.268333\pi$
$642$		0			0
$643$		−	21.7538i		−	0.857886i	−0.903332	−	0.428943i	$-0.858886\pi$
$643$		−	21.7538i		−	0.857886i	0.903332	−	0.428943i	$-0.141114\pi$
$644$		0			0
$645$		−	0.807764i		−	0.0318057i
$646$		0			0
$647$	7.36932			0.289718			0.144859	−	0.989452i	$-0.453727\pi$
$647$	7.36932			0.289718			0.144859	+	0.989452i	$0.453727\pi$
$648$		0			0
$649$	−47.2311			−1.85398
$650$		0			0
$651$	2.56155			0.100395
$652$		0			0
$653$	−11.7538			−0.459961			−0.229981	−	0.973195i	$-0.573866\pi$
$653$	−11.7538			−0.459961			−0.229981	+	0.973195i	$0.573866\pi$
$654$		0			0
$655$		−	3.86174i		−	0.150891i
$656$		0			0
$657$			10.5616i			0.412045i
$658$		0			0
$659$	32.6695			1.27262			0.636312	−	0.771432i	$-0.280459\pi$
$659$	32.6695			1.27262			0.636312	+	0.771432i	$0.280459\pi$
$660$		0			0
$661$		−	4.94602i		−	0.192378i	−0.995363	−	0.0961890i	$-0.969335\pi$
$661$		−	4.94602i		−	0.192378i	0.995363	−	0.0961890i	$-0.0306653\pi$
$662$		0			0
$663$	7.12311	+	1.12311i	0.276638	+	0.0436178i
$664$		0			0
$665$			1.43845i			0.0557806i
$666$		0			0
$667$	12.6695			0.490565
$668$		0			0
$669$		−	10.5616i		−	0.408333i
$670$		0			0
$671$		−	34.7386i		−	1.34107i
$672$		0			0
$673$	−10.1771			−0.392298			−0.196149	−	0.980574i	$-0.562844\pi$
$673$	−10.1771			−0.392298			−0.196149	+	0.980574i	$0.562844\pi$
$674$		0			0
$675$	4.68466			0.180313
$676$		0			0
$677$	4.73863			0.182120			0.0910602	−	0.995845i	$-0.470974\pi$
$677$	4.73863			0.182120			0.0910602	+	0.995845i	$0.470974\pi$
$678$		0			0
$679$	15.6847			0.601922
$680$		0			0
$681$		−	7.12311i		−	0.272958i
$682$		0			0
$683$		−	1.36932i		−	0.0523955i	−0.999657	−	0.0261977i	$-0.991660\pi$
$683$		−	1.36932i		−	0.0523955i	0.999657	−	0.0261977i	$-0.00833995\pi$
$684$		0			0
$685$	3.01515			0.115203
$686$		0			0
$687$			19.3693i			0.738986i
$688$		0			0
$689$	2.00000	+	0.315342i	0.0761939	+	0.0120136i
$690$		0			0
$691$			5.93087i			0.225621i	0.993617	+	0.112810i	$0.0359853\pi$
$691$			5.93087i			0.225621i	−0.993617	+	0.112810i	$0.964015\pi$
$692$		0			0
$693$	3.12311			0.118637
$694$		0			0
$695$		−	3.23106i		−	0.122561i
$696$		0			0
$697$			20.0000i			0.757554i
$698$		0			0
$699$	−17.6847			−0.668895
$700$		0			0
$701$	30.6695			1.15837			0.579186	−	0.815196i	$-0.303371\pi$
$701$	30.6695			1.15837			0.579186	+	0.815196i	$0.303371\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−2.56155			−0.0964737
$706$		0			0
$707$			9.36932i			0.352369i
$708$		0			0
$709$			17.7538i			0.666758i	0.942793	+	0.333379i	$0.108189\pi$
$709$			17.7538i			0.666758i	−0.942793	+	0.333379i	$0.891811\pi$
$710$		0			0
$711$	3.68466			0.138185
$712$		0			0
$713$			9.43845i			0.353473i
$714$		0			0
$715$	6.24621	+	0.984845i	0.233595	+	0.0368311i
$716$		0			0
$717$		−	8.24621i		−	0.307960i
$718$		0			0
$719$	−40.3542			−1.50496			−0.752478	−	0.658617i	$-0.771142\pi$
$719$	−40.3542			−1.50496			−0.752478	+	0.658617i	$0.771142\pi$
$720$		0			0
$721$		−	1.75379i		−	0.0653145i
$722$		0			0
$723$			23.0540i			0.857387i
$724$		0			0
$725$	16.1080			0.598234
$726$		0			0
$727$	−17.6155			−0.653324			−0.326662	−	0.945141i	$-0.605924\pi$
$727$	−17.6155			−0.653324			−0.326662	+	0.945141i	$0.605924\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	2.87689			0.106406
$732$		0			0
$733$		−	42.5616i		−	1.57205i	−0.618197	−	0.786023i	$-0.712136\pi$
$733$		−	42.5616i		−	1.57205i	0.618197	−	0.786023i	$-0.287864\pi$
$734$		0			0
$735$		−	0.561553i		−	0.0207132i
$736$		0			0
$737$	−40.9848			−1.50970
$738$		0			0
$739$			50.2462i			1.84834i	0.381985	+	0.924168i	$0.375240\pi$
$739$			50.2462i			1.84834i	−0.381985	+	0.924168i	$0.624760\pi$
$740$		0			0
$741$	1.43845	−	9.12311i	0.0528427	−	0.335146i
$742$		0			0
$743$		−	28.7386i		−	1.05432i	−0.849767	−	0.527159i	$-0.823257\pi$
$743$		−	28.7386i		−	1.05432i	0.849767	−	0.527159i	$-0.176743\pi$
$744$		0			0
$745$	−3.01515			−0.110467
$746$		0			0
$747$			5.68466i			0.207991i
$748$		0			0
$749$		−	16.4924i		−	0.602620i
$750$		0			0
$751$	−29.4384			−1.07422			−0.537112	−	0.843511i	$-0.680485\pi$
$751$	−29.4384			−1.07422			−0.537112	+	0.843511i	$0.680485\pi$
$752$		0			0
$753$	4.00000			0.145768
$754$		0			0
$755$	6.73863			0.245244
$756$		0			0
$757$	−9.05398			−0.329072			−0.164536	−	0.986371i	$-0.552613\pi$
$757$	−9.05398			−0.329072			−0.164536	+	0.986371i	$0.552613\pi$
$758$		0			0
$759$			11.5076i			0.417699i
$760$		0			0
$761$			23.9309i			0.867493i	0.901035	+	0.433747i	$0.142809\pi$
$761$			23.9309i			0.867493i	−0.901035	+	0.433747i	$0.857191\pi$
$762$		0			0
$763$	−1.12311			−0.0406592
$764$		0			0
$765$		−	1.12311i		−	0.0406060i
$766$		0			0
$767$	8.49242	−	53.8617i	0.306644	−	1.94483i
$768$		0			0
$769$		−	48.1771i		−	1.73731i	−0.495418	−	0.868655i	$-0.664985\pi$
$769$		−	48.1771i		−	1.73731i	0.495418	−	0.868655i	$-0.335015\pi$
$770$		0			0
$771$	−5.36932			−0.193371
$772$		0			0
$773$		−	15.7538i		−	0.566624i	−0.959028	−	0.283312i	$-0.908567\pi$
$773$		−	15.7538i		−	0.566624i	0.959028	−	0.283312i	$-0.0914333\pi$
$774$		0			0
$775$			12.0000i			0.431053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	25.6155			0.917772
$780$		0			0
$781$	−29.2614			−1.04705
$782$		0			0
$783$	3.43845			0.122880
$784$		0			0
$785$		−	4.63068i		−	0.165276i
$786$		0			0
$787$		−	8.80776i		−	0.313963i	−0.987602	−	0.156981i	$-0.949824\pi$
$787$		−	8.80776i		−	0.313963i	0.987602	−	0.156981i	$-0.0501763\pi$
$788$		0			0
$789$	−27.0540			−0.963147
$790$		0			0
$791$		−	14.8078i		−	0.526503i
$792$		0			0
$793$	39.6155	+	6.24621i	1.40679	+	0.221809i
$794$		0			0
$795$		−	0.315342i		−	0.0111840i
$796$		0			0
$797$	−41.2311			−1.46048			−0.730239	−	0.683191i	$-0.760592\pi$
$797$	−41.2311			−1.46048			−0.730239	+	0.683191i	$0.760592\pi$
$798$		0			0
$799$		−	9.12311i		−	0.322752i
$800$		0			0
$801$		−	5.68466i		−	0.200858i
$802$		0			0
$803$	−32.9848			−1.16401
$804$		0			0
$805$	2.06913			0.0729273
$806$		0			0
$807$	10.0000			0.352017
$808$		0			0
$809$	42.3153			1.48773			0.743864	−	0.668331i	$-0.232991\pi$
$809$	42.3153			1.48773			0.743864	+	0.668331i	$0.232991\pi$
$810$		0			0
$811$			36.0000i			1.26413i	0.774915	+	0.632065i	$0.217793\pi$
$811$			36.0000i			1.26413i	−0.774915	+	0.632065i	$0.782207\pi$
$812$		0			0
$813$			22.7386i			0.797479i
$814$		0			0
$815$	−3.86174			−0.135271
$816$		0			0
$817$		−	3.68466i		−	0.128910i
$818$		0			0
$819$	−0.561553	+	3.56155i	−0.0196222	+	0.124451i
$820$		0			0
$821$		−	5.50758i		−	0.192216i	−0.995371	−	0.0961079i	$-0.969361\pi$
$821$		−	5.50758i		−	0.192216i	0.995371	−	0.0961079i	$-0.0306394\pi$
$822$		0			0
$823$	−42.7386			−1.48978			−0.744888	−	0.667190i	$-0.767497\pi$
$823$	−42.7386			−1.48978			−0.744888	+	0.667190i	$0.767497\pi$
$824$		0			0
$825$			14.6307i			0.509375i
$826$		0			0
$827$		−	16.7386i		−	0.582059i	−0.956714	−	0.291030i	$-0.906002\pi$
$827$		−	16.7386i		−	0.582059i	0.956714	−	0.291030i	$-0.0939978\pi$
$828$		0			0
$829$	−20.7386			−0.720283			−0.360141	−	0.932898i	$-0.617272\pi$
$829$	−20.7386			−0.720283			−0.360141	+	0.932898i	$0.617272\pi$
$830$		0			0
$831$	12.5616			0.435755
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$	0.315342			0.0109128
$836$		0			0
$837$			2.56155i			0.0885402i
$838$		0			0
$839$		−	3.12311i		−	0.107822i	−0.998546	−	0.0539108i	$-0.982831\pi$
$839$		−	3.12311i		−	0.107822i	0.998546	−	0.0539108i	$-0.0171687\pi$
$840$		0			0
$841$	−17.1771			−0.592313
$842$		0			0
$843$			18.0000i			0.619953i
$844$		0			0
$845$	−2.24621	+	6.94602i	−0.0772720	+	0.238951i
$846$		0			0
$847$		−	1.24621i		−	0.0428203i
$848$		0			0
$849$	9.75379			0.334749
$850$		0			0
$851$		0			0
$852$		0			0
$853$			45.9309i			1.57264i	0.617818	+	0.786322i	$0.288017\pi$
$853$			45.9309i			1.57264i	−0.617818	+	0.786322i	$0.711983\pi$
$854$		0			0
$855$	−1.43845			−0.0491939
$856$		0			0
$857$	25.3693			0.866599			0.433300	−	0.901250i	$-0.357349\pi$
$857$	25.3693			0.866599			0.433300	+	0.901250i	$0.357349\pi$
$858$		0			0
$859$	1.12311			0.0383199			0.0191599	−	0.999816i	$-0.493901\pi$
$859$	1.12311			0.0383199			0.0191599	+	0.999816i	$0.493901\pi$
$860$		0			0
$861$	−10.0000			−0.340799
$862$		0			0
$863$		−	40.1080i		−	1.36529i	−0.730750	−	0.682645i	$-0.760829\pi$
$863$		−	40.1080i		−	1.36529i	0.730750	−	0.682645i	$-0.239171\pi$
$864$		0			0
$865$		−	2.73863i		−	0.0931163i
$866$		0			0
$867$	−13.0000			−0.441503
$868$		0			0
$869$			11.5076i			0.390368i
$870$		0			0
$871$	7.36932	−	46.7386i	0.249700	−	1.58368i
$872$		0			0
$873$			15.6847i			0.530845i
$874$		0			0
$875$	5.43845			0.183853
$876$		0			0
$877$		−	9.61553i		−	0.324693i	−0.986734	−	0.162347i	$-0.948094\pi$
$877$		−	9.61553i		−	0.324693i	0.986734	−	0.162347i	$-0.0519063\pi$
$878$		0			0
$879$			7.93087i			0.267502i
$880$		0			0
$881$	−40.2462			−1.35593			−0.677965	−	0.735095i	$-0.737138\pi$
$881$	−40.2462			−1.35593			−0.677965	+	0.735095i	$0.737138\pi$
$882$		0			0
$883$	−9.75379			−0.328241			−0.164121	−	0.986440i	$-0.552479\pi$
$883$	−9.75379			−0.328241			−0.164121	+	0.986440i	$0.552479\pi$
$884$		0			0
$885$	−8.49242			−0.285470
$886$		0			0
$887$	−43.8617			−1.47273			−0.736367	−	0.676583i	$-0.763460\pi$
$887$	−43.8617			−1.47273			−0.736367	+	0.676583i	$0.763460\pi$
$888$		0			0
$889$			12.4924i			0.418982i
$890$		0			0
$891$			3.12311i			0.104628i
$892$		0			0
$893$	−11.6847			−0.391012
$894$		0			0
$895$		−	5.93087i		−	0.198247i
$896$		0			0
$897$	−13.1231	−	2.06913i	−0.438168	−	0.0690863i
$898$		0			0
$899$			8.80776i			0.293755i
$900$		0			0
$901$	1.12311			0.0374161
$902$		0			0
$903$			1.43845i			0.0478685i
$904$		0			0
$905$			9.75379i			0.324227i
$906$		0			0
$907$	50.5616			1.67887			0.839434	−	0.543461i	$-0.182886\pi$
$907$	50.5616			1.67887			0.839434	+	0.543461i	$0.182886\pi$
$908$		0			0
$909$	−9.36932			−0.310761
$910$		0			0
$911$	−24.8078			−0.821918			−0.410959	−	0.911654i	$-0.634806\pi$
$911$	−24.8078			−0.821918			−0.410959	+	0.911654i	$0.634806\pi$
$912$		0			0
$913$	−17.7538			−0.587565
$914$		0			0
$915$		−	6.24621i		−	0.206493i
$916$		0			0
$917$			6.87689i			0.227095i
$918$		0			0
$919$	−12.4924			−0.412087			−0.206043	−	0.978543i	$-0.566059\pi$
$919$	−12.4924			−0.412087			−0.206043	+	0.978543i	$0.566059\pi$
$920$		0			0
$921$			8.80776i			0.290226i
$922$		0			0
$923$	5.26137	−	33.3693i	0.173180	−	1.09836i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.75379			0.0576020
$928$		0			0
$929$		−	39.9309i		−	1.31009i	−0.755590	−	0.655045i	$-0.772650\pi$
$929$		−	39.9309i		−	1.31009i	0.755590	−	0.655045i	$-0.227350\pi$
$930$		0			0
$931$		−	2.56155i		−	0.0839515i
$932$		0			0
$933$	25.6155			0.838615
$934$		0			0
$935$	3.50758			0.114710
$936$		0			0
$937$	25.8617			0.844866			0.422433	−	0.906394i	$-0.361176\pi$
$937$	25.8617			0.844866			0.422433	+	0.906394i	$0.361176\pi$
$938$		0			0
$939$	−16.8769			−0.550757
$940$		0			0
$941$			43.9309i			1.43211i	0.698046	+	0.716053i	$0.254053\pi$
$941$			43.9309i			1.43211i	−0.698046	+	0.716053i	$0.745947\pi$
$942$		0			0
$943$		−	36.8466i		−	1.19989i
$944$		0			0
$945$	0.561553			0.0182673
$946$		0			0
$947$		−	18.4924i		−	0.600923i	−0.953794	−	0.300461i	$-0.902859\pi$
$947$		−	18.4924i		−	0.600923i	0.953794	−	0.300461i	$-0.0971407\pi$
$948$		0			0
$949$	5.93087	−	37.6155i	0.192524	−	1.22105i
$950$		0			0
$951$		−	6.00000i		−	0.194563i
$952$		0			0
$953$	−34.3153			−1.11158			−0.555791	−	0.831322i	$-0.687585\pi$
$953$	−34.3153			−1.11158			−0.555791	+	0.831322i	$0.687585\pi$
$954$		0			0
$955$			8.98485i			0.290743i
$956$		0			0
$957$			10.7386i			0.347131i
$958$		0			0
$959$	−5.36932			−0.173384
$960$		0			0
$961$	24.4384			0.788337
$962$		0			0
$963$	16.4924			0.531461
$964$		0			0
$965$	−8.63068			−0.277832
$966$		0			0
$967$			0.492423i			0.0158352i	0.999969	+	0.00791762i	$0.00252028\pi$
$967$			0.492423i			0.0158352i	−0.999969	+	0.00791762i	$0.997480\pi$
$968$		0			0
$969$		−	5.12311i		−	0.164578i
$970$		0			0
$971$	−41.4773			−1.33107			−0.665534	−	0.746367i	$-0.731796\pi$
$971$	−41.4773			−1.33107			−0.665534	+	0.746367i	$0.731796\pi$
$972$		0			0
$973$			5.75379i			0.184458i
$974$		0			0
$975$	−16.6847	−	2.63068i	−0.534337	−	0.0842493i
$976$		0			0
$977$		−	61.3693i		−	1.96338i	−0.190490	−	0.981689i	$-0.561008\pi$
$977$		−	61.3693i		−	1.96338i	0.190490	−	0.981689i	$-0.438992\pi$
$978$		0			0
$979$	17.7538			0.567414
$980$		0			0
$981$		−	1.12311i		−	0.0358580i
$982$		0			0
$983$		−	36.5616i		−	1.16613i	−0.812425	−	0.583066i	$-0.801853\pi$
$983$		−	36.5616i		−	1.16613i	0.812425	−	0.583066i	$-0.198147\pi$
$984$		0			0
$985$	12.0000			0.382352
$986$		0			0
$987$	4.56155			0.145196
$988$		0			0
$989$	−5.30019			−0.168536
$990$		0			0
$991$	1.75379			0.0557109			0.0278555	−	0.999612i	$-0.491132\pi$
$991$	1.75379			0.0557109			0.0278555	+	0.999612i	$0.491132\pi$
$992$		0			0
$993$		−	26.2462i		−	0.832898i
$994$		0			0
$995$			14.3845i			0.456018i
$996$		0			0
$997$	44.1080			1.39691			0.698456	−	0.715653i	$-0.253871\pi$
$997$	44.1080			1.39691			0.698456	+	0.715653i	$0.253871\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.l.337.3		4
4.3	odd	2		1092.2.e.e.337.3	yes	4
12.11	even	2		3276.2.e.e.2521.2		4
13.12	even	2	inner	4368.2.h.l.337.2		4
28.27	even	2		7644.2.e.i.4705.2		4
52.51	odd	2		1092.2.e.e.337.2	✓	4
156.155	even	2		3276.2.e.e.2521.3		4
364.363	even	2		7644.2.e.i.4705.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.e.e.337.2	✓	4	52.51	odd	2
1092.2.e.e.337.3	yes	4	4.3	odd	2
3276.2.e.e.2521.2		4	12.11	even	2
3276.2.e.e.2521.3		4	156.155	even	2
4368.2.h.l.337.2		4	13.12	even	2	inner
4368.2.h.l.337.3		4	1.1	even	1	trivial
7644.2.e.i.4705.2		4	28.27	even	2
7644.2.e.i.4705.3		4	364.363	even	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.561553i q^{5} -1.00000i q^{7} +1.00000 q^{9} +3.12311i q^{11} +(-3.56155 - 0.561553i) q^{13} +0.561553i q^{15} -2.00000 q^{17} +2.56155i q^{19} -1.00000i q^{21} +3.68466 q^{23} +4.68466 q^{25} +1.00000 q^{27} +3.43845 q^{29} +2.56155i q^{31} +3.12311i q^{33} +0.561553 q^{35} +(-3.56155 - 0.561553i) q^{39} -10.0000i q^{41} -1.43845 q^{43} +0.561553i q^{45} +4.56155i q^{47} -1.00000 q^{49} -2.00000 q^{51} -0.561553 q^{53} -1.75379 q^{55} +2.56155i q^{57} +15.1231i q^{59} -11.1231 q^{61} -1.00000i q^{63} +(0.315342 - 2.00000i) q^{65} +13.1231i q^{67} +3.68466 q^{69} +9.36932i q^{71} +10.5616i q^{73} +4.68466 q^{75} +3.12311 q^{77} +3.68466 q^{79} +1.00000 q^{81} +5.68466i q^{83} -1.12311i q^{85} +3.43845 q^{87} -5.68466i q^{89} +(-0.561553 + 3.56155i) q^{91} +2.56155i q^{93} -1.43845 q^{95} +15.6847i q^{97} +3.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} - 6 q^{13} - 8 q^{17} - 10 q^{23} - 6 q^{25} + 4 q^{27} + 22 q^{29} - 6 q^{35} - 6 q^{39} - 14 q^{43} - 4 q^{49} - 8 q^{51} + 6 q^{53} - 40 q^{55} - 28 q^{61} + 26 q^{65} - 10 q^{69} - 6 q^{75} - 4 q^{77} - 10 q^{79} + 4 q^{81} + 22 q^{87} + 6 q^{91} - 14 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 - 6 * q^13 - 8 * q^17 - 10 * q^23 - 6 * q^25 + 4 * q^27 + 22 * q^29 - 6 * q^35 - 6 * q^39 - 14 * q^43 - 4 * q^49 - 8 * q^51 + 6 * q^53 - 40 * q^55 - 28 * q^61 + 26 * q^65 - 10 * q^69 - 6 * q^75 - 4 * q^77 - 10 * q^79 + 4 * q^81 + 22 * q^87 + 6 * q^91 - 14 * q^95

Introduction

L-functions

Modular forms

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Database

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