LMFDB - Embedded newform 3420.2.bb.d.2773.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(37,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("3420.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.bb (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} + \cdots + 1370 $ x^12 - 4x^11 + 28x^10 - 64x^9 + 236x^8 - 420x^7 + 946x^6 - 1216x^5 + 1896x^4 - 1564x^3 + 2284x^2 - 1088*x + 1370
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2773.5
Root			$1.24060 + 1.01288i$ of defining polynomial
Character	$\chi$	$=$	3420.2773
Dual form			3420.2.bb.d.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.48119 + 1.67513i) q^{5} +(-1.48119 + 1.48119i) q^{7} +O(q^{10})$	$q+(1.48119 + 1.67513i) q^{5} +(-1.48119 + 1.48119i) q^{7} -0.806063 q^{11} +(-0.437032 + 0.437032i) q^{13} +(-3.15633 + 3.15633i) q^{17} +(1.81645 - 3.96239i) q^{19} +(-1.86907 - 1.86907i) q^{23} +(-0.612127 + 4.96239i) q^{25} +4.50696 q^{29} +6.67568i q^{31} +(-4.67513 - 0.287258i) q^{35} +(5.29626 + 5.29626i) q^{37} -11.1826i q^{41} +(-3.86907 - 3.86907i) q^{43} +(-6.83146 + 6.83146i) q^{47} +2.61213i q^{49} +(-8.92916 + 8.92916i) q^{53} +(-1.19394 - 1.35026i) q^{55} -12.9308 q^{59} -2.15633 q^{61} +(-1.37941 - 0.0847564i) q^{65} +(4.94399 + 4.94399i) q^{67} +3.04278i q^{71} +(-6.19394 - 6.19394i) q^{73} +(1.19394 - 1.19394i) q^{77} -1.46417 q^{79} +(-5.32487 - 5.32487i) q^{83} +(-9.96239 - 0.612127i) q^{85} +14.2254 q^{89} -1.29466i q^{91} +(9.32803 - 2.82628i) q^{95} +(7.46498 + 7.46498i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 12 * q - 4 * q^5 + 4 * q^7	$ 12 q - 4 q^{5} + 4 q^{7} - 8 q^{11} + 4 q^{17} - 4 q^{23} - 4 q^{25} - 36 q^{35} - 28 q^{43} - 20 q^{47} - 16 q^{55} + 16 q^{61} - 76 q^{73} + 16 q^{77} - 84 q^{83} - 76 q^{85} + 16 q^{95}+O(q^{100}) $ 12 * q - 4 * q^5 + 4 * q^7 - 8 * q^11 + 4 * q^17 - 4 * q^23 - 4 * q^25 - 36 * q^35 - 28 * q^43 - 20 * q^47 - 16 * q^55 + 16 * q^61 - 76 * q^73 + 16 * q^77 - 84 * q^83 - 76 * q^85 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$6$		0			0
$7$	−1.48119	+	1.48119i	−0.559839	+	0.559839i	−0.929261	−	0.369423i	$-0.879555\pi$
$7$	−1.48119	+	1.48119i	−0.559839	+	0.559839i	0.369423	+	0.929261i	$0.379555\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.806063			−0.243037			−0.121519	−	0.992589i	$-0.538776\pi$
$11$	−0.806063			−0.243037			−0.121519	+	0.992589i	$0.538776\pi$
$12$		0			0
$13$	−0.437032	+	0.437032i	−0.121211	+	0.121211i	−0.765110	−	0.643899i	$-0.777316\pi$
$13$	−0.437032	+	0.437032i	−0.121211	+	0.121211i	0.643899	+	0.765110i	$0.277316\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.15633	+	3.15633i	−0.765521	+	0.765521i	−0.977315	−	0.211793i	$-0.932070\pi$
$17$	−3.15633	+	3.15633i	−0.765521	+	0.765521i	0.211793	+	0.977315i	$0.432070\pi$
$18$		0			0
$19$	1.81645	−	3.96239i	0.416721	−	0.909034i
$19$	1.81645	−	3.96239i	0.416721	−	0.909034i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.86907	−	1.86907i	−0.389728	−	0.389728i	0.484863	−	0.874590i	$-0.338870\pi$
$23$	−1.86907	−	1.86907i	−0.389728	−	0.389728i	−0.874590	+	0.484863i	$0.838870\pi$
$24$		0			0
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.50696			0.836921			0.418460	−	0.908235i	$-0.362570\pi$
$29$	4.50696			0.836921			0.418460	+	0.908235i	$0.362570\pi$
$30$		0			0
$31$			6.67568i			1.19899i	0.800380	+	0.599494i	$0.204631\pi$
$31$			6.67568i			1.19899i	−0.800380	+	0.599494i	$0.795369\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.67513	−	0.287258i	−0.790241	−	0.0485554i
$36$		0			0
$37$	5.29626	+	5.29626i	0.870700	+	0.870700i	0.992549	−	0.121848i	$-0.0388822\pi$
$37$	5.29626	+	5.29626i	0.870700	+	0.870700i	−0.121848	+	0.992549i	$0.538882\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.1826i		−	1.74643i	−0.487332	−	0.873217i	$-0.662030\pi$
$41$		−	11.1826i		−	1.74643i	0.487332	−	0.873217i	$-0.337970\pi$
$42$		0			0
$43$	−3.86907	−	3.86907i	−0.590027	−	0.590027i	0.347611	−	0.937639i	$-0.386993\pi$
$43$	−3.86907	−	3.86907i	−0.590027	−	0.590027i	−0.937639	+	0.347611i	$0.886993\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.83146	+	6.83146i	−0.996470	+	0.996470i	−0.999994	−	0.00352351i	$-0.998878\pi$
$47$	−6.83146	+	6.83146i	−0.996470	+	0.996470i	0.00352351	+	0.999994i	$0.498878\pi$
$48$		0			0
$49$			2.61213i			0.373161i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.92916	+	8.92916i	−1.22651	+	1.22651i	−0.261240	+	0.965274i	$0.584131\pi$
$53$	−8.92916	+	8.92916i	−1.22651	+	1.22651i	−0.965274	+	0.261240i	$0.915869\pi$
$54$		0			0
$55$	−1.19394	−	1.35026i	−0.160990	−	0.182069i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.9308			−1.68344			−0.841721	−	0.539913i	$-0.818457\pi$
$59$	−12.9308			−1.68344			−0.841721	+	0.539913i	$0.818457\pi$
$60$		0			0
$61$	−2.15633			−0.276089			−0.138045	−	0.990426i	$-0.544082\pi$
$61$	−2.15633			−0.276089			−0.138045	+	0.990426i	$0.544082\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.37941	−	0.0847564i	−0.171095	−	0.0105127i
$66$		0			0
$67$	4.94399	+	4.94399i	0.604004	+	0.604004i	0.941373	−	0.337368i	$-0.109537\pi$
$67$	4.94399	+	4.94399i	0.604004	+	0.604004i	−0.337368	+	0.941373i	$0.609537\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.04278i			0.361112i	0.983565	+	0.180556i	$0.0577897\pi$
$71$			3.04278i			0.361112i	−0.983565	+	0.180556i	$0.942210\pi$
$72$		0			0
$73$	−6.19394	−	6.19394i	−0.724945	−	0.724945i	0.244663	−	0.969608i	$-0.421323\pi$
$73$	−6.19394	−	6.19394i	−0.724945	−	0.724945i	−0.969608	+	0.244663i	$0.921323\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.19394	−	1.19394i	0.136062	−	0.136062i
$78$		0			0
$79$	−1.46417			−0.164732			−0.0823660	−	0.996602i	$-0.526248\pi$
$79$	−1.46417			−0.164732			−0.0823660	+	0.996602i	$0.526248\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.32487	−	5.32487i	−0.584480	−	0.584480i	0.351651	−	0.936131i	$-0.385620\pi$
$83$	−5.32487	−	5.32487i	−0.584480	−	0.584480i	−0.936131	+	0.351651i	$0.885620\pi$
$84$		0			0
$85$	−9.96239	−	0.612127i	−1.08057	−	0.0663945i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.2254			1.50789			0.753946	−	0.656937i	$-0.228148\pi$
$89$	14.2254			1.50789			0.753946	+	0.656937i	$0.228148\pi$
$90$		0			0
$91$		−	1.29466i		−	0.135717i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.32803	−	2.82628i	0.957036	−	0.289970i
$96$		0			0
$97$	7.46498	+	7.46498i	0.757954	+	0.757954i	0.975950	−	0.217995i	$-0.0699518\pi$
$97$	7.46498	+	7.46498i	0.757954	+	0.757954i	−0.217995	+	0.975950i	$0.569952\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.08110			−0.406085			−0.203042	−	0.979170i	$-0.565083\pi$
$101$	−4.08110			−0.406085			−0.203042	+	0.979170i	$0.565083\pi$
$102$		0			0
$103$	−1.31110	+	1.31110i	−0.129186	+	0.129186i	−0.768743	−	0.639557i	$-0.779118\pi$
$103$	−1.31110	+	1.31110i	−0.129186	+	0.129186i	0.639557	+	0.768743i	$0.279118\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.11271	−	7.11271i	−0.687612	−	0.687612i	0.274092	−	0.961703i	$-0.411623\pi$
$107$	−7.11271	−	7.11271i	−0.687612	−	0.687612i	−0.961703	+	0.274092i	$0.911623\pi$
$108$		0			0
$109$	−19.3225			−1.85076			−0.925379	−	0.379043i	$-0.876253\pi$
$109$	−19.3225			−1.85076			−0.925379	+	0.379043i	$0.876253\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.23865	−	6.23865i	0.586882	−	0.586882i	−0.349903	−	0.936786i	$-0.613786\pi$
$113$	6.23865	−	6.23865i	0.586882	−	0.586882i	0.936786	+	0.349903i	$0.113786\pi$
$114$		0			0
$115$	0.362481	−	5.89938i	0.0338015	−	0.550120i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.35026i		−	0.857137i
$120$		0			0
$121$	−10.3503			−0.940933
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.21933	+	6.32487i	−0.824602	+	0.565713i
$126$		0			0
$127$	−9.80322	−	9.80322i	−0.869895	−	0.869895i	0.122565	−	0.992460i	$-0.460888\pi$
$127$	−9.80322	−	9.80322i	−0.869895	−	0.869895i	−0.992460	+	0.122565i	$0.960888\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.14903			0.799355			0.399677	−	0.916656i	$-0.369122\pi$
$131$	9.14903			0.799355			0.399677	+	0.916656i	$0.369122\pi$
$132$		0			0
$133$	3.17856	+	8.55958i	0.275616	+	0.742209i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.18664	+	3.18664i	−0.272253	+	0.272253i	−0.830007	−	0.557753i	$-0.811663\pi$
$137$	−3.18664	+	3.18664i	−0.272253	+	0.272253i	0.557753	+	0.830007i	$0.311663\pi$
$138$		0			0
$139$		−	15.2447i		−	1.29304i	−0.762897	−	0.646520i	$-0.776224\pi$
$139$		−	15.2447i		−	1.29304i	0.762897	−	0.646520i	$-0.223776\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.352275	−	0.352275i	0.0294587	−	0.0294587i
$144$		0			0
$145$	6.67568	+	7.54974i	0.554385	+	0.626972i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	15.3806i		−	1.26003i	−0.776585	−	0.630013i	$-0.783050\pi$
$149$		−	15.3806i		−	1.26003i	0.776585	−	0.630013i	$-0.216950\pi$
$150$		0			0
$151$			5.21151i			0.424106i	0.977258	+	0.212053i	$0.0680150\pi$
$151$			5.21151i			0.424106i	−0.977258	+	0.212053i	$0.931985\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11.1826	+	9.88798i	−0.898211	+	0.794221i
$156$		0			0
$157$	−10.2447	+	10.2447i	−0.817618	+	0.817618i	−0.985762	−	0.168145i	$-0.946222\pi$
$157$	−10.2447	+	10.2447i	−0.817618	+	0.817618i	0.168145	+	0.985762i	$0.446222\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.53690			0.436369
$162$		0			0
$163$	−10.4133	−	10.4133i	−0.815630	−	0.815630i	0.169841	−	0.985471i	$-0.445674\pi$
$163$	−10.4133	−	10.4133i	−0.815630	−	0.815630i	−0.985471	+	0.169841i	$0.945674\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.42220	+	4.42220i	0.342200	+	0.342200i	0.857194	−	0.514994i	$-0.172206\pi$
$167$	4.42220	+	4.42220i	0.342200	+	0.342200i	−0.514994	+	0.857194i	$0.672206\pi$
$168$		0			0
$169$			12.6180i			0.970616i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.06992	−	4.06992i	0.309431	−	0.309431i	−0.535258	−	0.844689i	$-0.679786\pi$
$173$	4.06992	−	4.06992i	0.309431	−	0.309431i	0.844689	+	0.535258i	$0.179786\pi$
$174$		0			0
$175$	−6.44358	−	8.25694i	−0.487089	−	0.624166i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.87327			0.214758			0.107379	−	0.994218i	$-0.465754\pi$
$179$	2.87327			0.214758			0.107379	+	0.994218i	$0.465754\pi$
$180$		0			0
$181$			18.0278i			1.34000i	0.742362	+	0.669999i	$0.233705\pi$
$181$			18.0278i			1.34000i	−0.742362	+	0.669999i	$0.766295\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.02714	+	16.7167i	−0.0755168	+	1.22904i
$186$		0			0
$187$	2.54420	−	2.54420i	0.186050	−	0.186050i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.43866			0.321170			0.160585	−	0.987022i	$-0.448662\pi$
$191$	4.43866			0.321170			0.160585	+	0.987022i	$0.448662\pi$
$192$		0			0
$193$	−4.42220	+	4.42220i	−0.318317	+	0.318317i	−0.848120	−	0.529804i	$-0.822266\pi$
$193$	−4.42220	+	4.42220i	−0.318317	+	0.318317i	0.529804	+	0.848120i	$0.322266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.3806	+	10.3806i	−0.739586	+	0.739586i	−0.972498	−	0.232912i	$-0.925175\pi$
$197$	−10.3806	+	10.3806i	−0.739586	+	0.739586i	0.232912	+	0.972498i	$0.425175\pi$
$198$		0			0
$199$			8.23743i			0.583936i	0.956428	+	0.291968i	$0.0943100\pi$
$199$			8.23743i			0.583936i	−0.956428	+	0.291968i	$0.905690\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.67568	+	6.67568i	−0.468541	+	0.468541i
$204$		0			0
$205$	18.7324	−	16.5637i	1.30833	−	1.15686i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.46417	+	3.19394i	−0.101279	+	0.220929i
$210$		0			0
$211$		−	22.5348i		−	1.55136i	−0.631128	−	0.775679i	$-0.717408\pi$
$211$		−	22.5348i		−	1.55136i	0.631128	−	0.775679i	$-0.282592\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.750354	−	12.2120i	0.0511737	−	0.832854i
$216$		0			0
$217$	−9.88798	−	9.88798i	−0.671239	−	0.671239i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.75883i		−	0.185579i
$222$		0			0
$223$	13.4361	−	13.4361i	0.899749	−	0.899749i	−0.0956650	−	0.995414i	$-0.530498\pi$
$223$	13.4361	−	13.4361i	0.899749	−	0.899749i	0.995414	+	0.0956650i	$0.0304978\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.06992	+	4.06992i	0.270130	+	0.270130i	0.829153	−	0.559022i	$-0.188823\pi$
$227$	4.06992	+	4.06992i	0.270130	+	0.270130i	−0.559022	+	0.829153i	$0.688823\pi$
$228$		0			0
$229$		−	4.46898i		−	0.295318i	−0.989038	−	0.147659i	$-0.952826\pi$
$229$		−	4.46898i		−	0.295318i	0.989038	−	0.147659i	$-0.0471738\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.1114	+	11.1114i	0.727933	+	0.727933i	0.970208	−	0.242274i	$-0.0778934\pi$
$233$	11.1114	+	11.1114i	0.727933	+	0.727933i	−0.242274	+	0.970208i	$0.577893\pi$
$234$		0			0
$235$	−21.5623	−	1.32487i	−1.40657	−	0.0864249i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.4387i		−	0.675221i	−0.941286	−	0.337610i	$-0.890381\pi$
$239$		−	10.4387i		−	0.675221i	0.941286	−	0.337610i	$-0.109619\pi$
$240$		0			0
$241$			13.6353i			0.878328i	0.898407	+	0.439164i	$0.144725\pi$
$241$			13.6353i			0.878328i	−0.898407	+	0.439164i	$0.855275\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.37565	+	3.86907i	−0.279550	+	0.247186i
$246$		0			0
$247$	0.937845	+	2.52553i	0.0596737	+	0.160696i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.96239			−0.565701			−0.282850	−	0.959164i	$-0.591280\pi$
$251$	−8.96239			−0.565701			−0.282850	+	0.959164i	$0.591280\pi$
$252$		0			0
$253$	1.50659	+	1.50659i	0.0947183	+	0.0947183i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.76043	−	6.76043i	−0.421704	−	0.421704i	0.464086	−	0.885790i	$-0.346383\pi$
$257$	−6.76043	−	6.76043i	−0.421704	−	0.421704i	−0.885790	+	0.464086i	$0.846383\pi$
$258$		0			0
$259$	−15.6896			−0.974904
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.36248	+	4.36248i	0.269002	+	0.269002i	0.828698	−	0.559696i	$-0.189082\pi$
$263$	4.36248	+	4.36248i	0.269002	+	0.269002i	−0.559696	+	0.828698i	$0.689082\pi$
$264$		0			0
$265$	−28.1833	−	1.73169i	−1.73129	−	0.106377i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.4377			1.06320			0.531598	−	0.846997i	$-0.321592\pi$
$269$	17.4377			1.06320			0.531598	+	0.846997i	$0.321592\pi$
$270$		0			0
$271$	−8.28233			−0.503116			−0.251558	−	0.967842i	$-0.580943\pi$
$271$	−8.28233			−0.503116			−0.251558	+	0.967842i	$0.580943\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.493413	−	4.00000i	0.0297539	−	0.241209i
$276$		0			0
$277$	−9.80606	+	9.80606i	−0.589189	+	0.589189i	−0.937412	−	0.348223i	$-0.886785\pi$
$277$	−9.80606	+	9.80606i	−0.589189	+	0.589189i	0.348223	+	0.937412i	$0.386785\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5.66498i			0.337944i	0.985621	+	0.168972i	$0.0540448\pi$
$281$			5.66498i			0.337944i	−0.985621	+	0.168972i	$0.945955\pi$
$282$		0			0
$283$	4.18172	+	4.18172i	0.248577	+	0.248577i	0.820387	−	0.571809i	$-0.193758\pi$
$283$	4.18172	+	4.18172i	0.248577	+	0.248577i	−0.571809	+	0.820387i	$0.693758\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.5637	+	16.5637i	0.977721	+	0.977721i
$288$		0			0
$289$		−	2.92478i		−	0.172046i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.35054	−	7.35054i	0.429423	−	0.429423i	−0.459009	−	0.888432i	$-0.651795\pi$
$293$	7.35054	−	7.35054i	0.429423	−	0.429423i	0.888432	+	0.459009i	$0.151795\pi$
$294$		0			0
$295$	−19.1530	−	21.6607i	−1.11513	−	1.26114i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.63368			0.0944784
$300$		0			0
$301$	11.4617			0.660640
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.19394	−	3.61213i	−0.182884	−	0.206830i
$306$		0			0
$307$	23.3924	+	23.3924i	1.33508	+	1.33508i	0.900762	+	0.434313i	$0.143009\pi$
$307$	23.3924	+	23.3924i	1.33508	+	1.33508i	0.434313	+	0.900762i	$0.356991\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.2546			−0.638188			−0.319094	−	0.947723i	$-0.603379\pi$
$311$	−11.2546			−0.638188			−0.319094	+	0.947723i	$0.603379\pi$
$312$		0			0
$313$	14.5369	+	14.5369i	0.821674	+	0.821674i	0.986348	−	0.164674i	$-0.0526571\pi$
$313$	14.5369	+	14.5369i	0.821674	+	0.821674i	−0.164674	+	0.986348i	$0.552657\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.87154	+	9.87154i	0.554441	+	0.554441i	0.927719	−	0.373279i	$-0.121766\pi$
$317$	9.87154	+	9.87154i	0.554441	+	0.554441i	−0.373279	+	0.927719i	$0.621766\pi$
$318$		0			0
$319$	−3.63289			−0.203403
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.77329	+	18.2399i	0.376876	+	1.01489i
$324$		0			0
$325$	−1.90120	−	2.43624i	−0.105460	−	0.135138i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	20.2374i		−	1.11573i
$330$		0			0
$331$			20.3110i			1.11639i	0.829709	+	0.558196i	$0.188506\pi$
$331$			20.3110i			1.11639i	−0.829709	+	0.558196i	$0.811494\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.958820	+	15.6048i	−0.0523859	+	0.852583i
$336$		0			0
$337$	11.7892	+	11.7892i	0.642197	+	0.642197i	0.951095	−	0.308898i	$-0.0999601\pi$
$337$	11.7892	+	11.7892i	0.642197	+	0.642197i	−0.308898	+	0.951095i	$0.599960\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	5.38102i		−	0.291399i
$342$		0			0
$343$	−14.2374	−	14.2374i	−0.768749	−	0.768749i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.6302	+	18.6302i	−1.00012	+	1.00012i	−0.000122901		1.00000i	$0.500039\pi$
$347$	−18.6302	+	18.6302i	−1.00012	+	1.00012i	−1.00000		0.000122901i	$0.999961\pi$
$348$		0			0
$349$			21.0640i			1.12753i	0.825936	+	0.563764i	$0.190647\pi$
$349$			21.0640i			1.12753i	−0.825936	+	0.563764i	$0.809353\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.99271	+	1.99271i	0.106061	+	0.106061i	0.758146	−	0.652085i	$-0.226105\pi$
$353$	1.99271	+	1.99271i	0.106061	+	0.106061i	−0.652085	+	0.758146i	$0.726105\pi$
$354$		0			0
$355$	−5.09706	+	4.50696i	−0.270524	+	0.239204i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			28.6458i			1.51187i	0.654649	+	0.755933i	$0.272816\pi$
$359$			28.6458i			1.51187i	−0.654649	+	0.755933i	$0.727184\pi$
$360$		0			0
$361$	−12.4010	−	14.3949i	−0.652687	−	0.757628i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.20123	−	19.5501i	0.0628753	−	1.02330i
$366$		0			0
$367$	−12.1368	+	12.1368i	−0.633536	+	0.633536i	−0.948953	−	0.315417i	$-0.897856\pi$
$367$	−12.1368	+	12.1368i	−0.633536	+	0.633536i	0.315417	+	0.948953i	$0.397856\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	26.4516i		−	1.37330i
$372$		0			0
$373$	16.4789	−	16.4789i	0.853245	−	0.853245i	−0.137286	−	0.990531i	$-0.543838\pi$
$373$	16.4789	−	16.4789i	0.853245	−	0.853245i	0.990531	+	0.137286i	$0.0438381\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.96968	+	1.96968i	−0.101444	+	0.101444i
$378$		0			0
$379$	2.92834			0.150419			0.0752094	−	0.997168i	$-0.476037\pi$
$379$	2.92834			0.150419			0.0752094	+	0.997168i	$0.476037\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−21.3381	+	21.3381i	−1.09033	+	1.09033i	−0.0948343	+	0.995493i	$0.530232\pi$
$383$	−21.3381	+	21.3381i	−1.09033	+	1.09033i	−0.995493	+	0.0948343i	$0.969768\pi$
$384$		0			0
$385$	3.76845	+	0.231548i	0.192058	+	0.0118008i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			24.5647i			1.24548i	0.782430	+	0.622739i	$0.213980\pi$
$389$			24.5647i			1.24548i	−0.782430	+	0.622739i	$0.786020\pi$
$390$		0			0
$391$	11.7988			0.596689
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.16872	−	2.45268i	−0.109120	−	0.123408i
$396$		0			0
$397$	−5.23155	+	5.23155i	−0.262564	+	0.262564i	−0.826095	−	0.563531i	$-0.809443\pi$
$397$	−5.23155	+	5.23155i	−0.262564	+	0.262564i	0.563531	+	0.826095i	$0.309443\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.7324i			0.935450i	0.883874	+	0.467725i	$0.154926\pi$
$401$			18.7324i			0.935450i	−0.883874	+	0.467725i	$0.845074\pi$
$402$		0			0
$403$	−2.91748	−	2.91748i	−0.145330	−	0.145330i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.26912	−	4.26912i	−0.211613	−	0.211613i
$408$		0			0
$409$	−14.9850			−0.740962			−0.370481	−	0.928840i	$-0.620807\pi$
$409$	−14.9850			−0.740962			−0.370481	+	0.928840i	$0.620807\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	19.1530	−	19.1530i	0.942456	−	0.942456i
$414$		0			0
$415$	1.03269	−	16.8070i	0.0506926	−	0.825024i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	34.2882i		−	1.67509i	−0.546369	−	0.837544i	$-0.683990\pi$
$419$		−	34.2882i		−	1.67509i	0.546369	−	0.837544i	$-0.316010\pi$
$420$		0			0
$421$		−	21.6056i		−	1.05299i	−0.850177	−	0.526497i	$-0.823505\pi$
$421$		−	21.6056i		−	1.05299i	0.850177	−	0.526497i	$-0.176495\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.7308	−	17.5950i	−0.666044	−	0.853482i
$426$		0			0
$427$	3.19394	−	3.19394i	0.154565	−	0.154565i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	19.7431i		−	0.950990i	−0.879718	−	0.475495i	$-0.842269\pi$
$431$		−	19.7431i		−	0.950990i	0.879718	−	0.475495i	$-0.157731\pi$
$432$		0			0
$433$	11.0296	−	11.0296i	0.530047	−	0.530047i	−0.390539	−	0.920586i	$-0.627712\pi$
$433$	11.0296	−	11.0296i	0.530047	−	0.530047i	0.920586	+	0.390539i	$0.127712\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.8010	+	4.01091i	−0.516683	+	0.191868i
$438$		0			0
$439$	12.3628			0.590047			0.295023	−	0.955490i	$-0.404673\pi$
$439$	12.3628			0.590047			0.295023	+	0.955490i	$0.404673\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.1744	−	11.1744i	−0.530913	−	0.530913i	0.389931	−	0.920844i	$-0.372499\pi$
$443$	−11.1744	−	11.1744i	−0.530913	−	0.530913i	−0.920844	+	0.389931i	$0.872499\pi$
$444$		0			0
$445$	21.0706	+	23.8294i	0.998843	+	1.12962i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.90615			0.137150			0.0685748	−	0.997646i	$-0.478155\pi$
$449$	2.90615			0.137150			0.0685748	+	0.997646i	$0.478155\pi$
$450$		0			0
$451$			9.01391i			0.424449i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.16872	−	1.91764i	0.101671	−	0.0899003i
$456$		0			0
$457$	14.1490	−	14.1490i	0.661864	−	0.661864i	−0.293955	−	0.955819i	$-0.594972\pi$
$457$	14.1490	−	14.1490i	0.661864	−	0.661864i	0.955819	+	0.293955i	$0.0949717\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.6859			0.870291			0.435145	−	0.900360i	$-0.356697\pi$
$461$	18.6859			0.870291			0.435145	+	0.900360i	$0.356697\pi$
$462$		0			0
$463$	0.906679	+	0.906679i	0.0421369	+	0.0421369i	0.727861	−	0.685724i	$-0.240514\pi$
$463$	0.906679	+	0.906679i	0.0421369	+	0.0421369i	−0.685724	+	0.727861i	$0.740514\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.49437	+	8.49437i	−0.393072	+	0.393072i	−0.875781	−	0.482709i	$-0.839653\pi$
$467$	−8.49437	+	8.49437i	−0.393072	+	0.393072i	0.482709	+	0.875781i	$0.339653\pi$
$468$		0			0
$469$	−14.6460			−0.676290
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.11871	+	3.11871i	0.143399	+	0.143399i
$474$		0			0
$475$	18.5510	+	11.4394i	0.851179	+	0.524875i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	15.9307i		−	0.727890i	−0.931420	−	0.363945i	$-0.881430\pi$
$479$		−	15.9307i		−	0.727890i	0.931420	−	0.363945i	$-0.118570\pi$
$480$		0			0
$481$	−4.62927			−0.211077
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.44773	+	23.5619i	−0.0657382	+	1.06989i
$486$		0			0
$487$	1.02714	+	1.02714i	0.0465441	+	0.0465441i	0.729996	−	0.683452i	$-0.239522\pi$
$487$	1.02714	+	1.02714i	0.0465441	+	0.0465441i	−0.683452	+	0.729996i	$0.739522\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	16.1866			0.730493			0.365246	−	0.930911i	$-0.380985\pi$
$491$	16.1866			0.730493			0.365246	+	0.930911i	$0.380985\pi$
$492$		0			0
$493$	−14.2254	+	14.2254i	−0.640681	+	0.640681i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.50696	−	4.50696i	−0.202165	−	0.202165i
$498$		0			0
$499$		−	15.0943i		−	0.675713i	−0.941198	−	0.337856i	$-0.890298\pi$
$499$		−	15.0943i		−	0.675713i	0.941198	−	0.337856i	$-0.109702\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	26.9429	+	26.9429i	1.20132	+	1.20132i	0.973764	+	0.227559i	$0.0730745\pi$
$503$	26.9429	+	26.9429i	1.20132	+	1.20132i	0.227559	+	0.973764i	$0.426925\pi$
$504$		0			0
$505$	−6.04491	−	6.83638i	−0.268995	−	0.304215i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.6484			−1.09252			−0.546261	−	0.837615i	$-0.683949\pi$
$509$	−24.6484			−1.09252			−0.546261	+	0.837615i	$0.683949\pi$
$510$		0			0
$511$	18.3488			0.811705
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.13824	−	0.254269i	−0.182353	−	0.0112044i
$516$		0			0
$517$	5.50659	−	5.50659i	0.242179	−	0.242179i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			7.60481i			0.333173i	0.986027	+	0.166586i	$0.0532745\pi$
$521$			7.60481i			0.333173i	−0.986027	+	0.166586i	$0.946726\pi$
$522$		0			0
$523$	−22.5645	+	22.5645i	−0.986675	+	0.986675i	−0.999912	−	0.0132372i	$-0.995786\pi$
$523$	−22.5645	+	22.5645i	−0.986675	+	0.986675i	0.0132372	+	0.999912i	$0.495786\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.0706	−	21.0706i	−0.917850	−	0.917850i
$528$		0			0
$529$		−	16.0132i		−	0.696225i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.88717	+	4.88717i	0.211687	+	0.211687i
$534$		0			0
$535$	1.37941	−	22.4500i	0.0596373	−	0.970599i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	2.10554i		−	0.0906920i
$540$		0			0
$541$	36.3331			1.56208			0.781041	−	0.624479i	$-0.214689\pi$
$541$	36.3331			1.56208			0.781041	+	0.624479i	$0.214689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−28.6203	−	32.3677i	−1.22596	−	1.38648i
$546$		0			0
$547$	7.18103	+	7.18103i	0.307039	+	0.307039i	0.843760	−	0.536721i	$-0.180337\pi$
$547$	7.18103	+	7.18103i	0.307039	+	0.307039i	−0.536721	+	0.843760i	$0.680337\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.18664	−	17.8583i	0.348763	−	0.760790i
$552$		0			0
$553$	2.16872	−	2.16872i	0.0922234	−	0.0922234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.3054	+	24.3054i	−1.02985	+	1.02985i	−0.0303105	+	0.999541i	$0.509650\pi$
$557$	−24.3054	+	24.3054i	−1.02985	+	1.02985i	−0.999541	+	0.0303105i	$0.990350\pi$
$558$		0			0
$559$	3.38181			0.143035
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.7743	−	15.7743i	0.664809	−	0.664809i	−0.291700	−	0.956510i	$-0.594221\pi$
$563$	15.7743	−	15.7743i	0.664809	−	0.664809i	0.956510	+	0.291700i	$0.0942210\pi$
$564$		0			0
$565$	19.6912	+	1.20990i	0.828415	+	0.0509009i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.45189			0.186633			0.0933164	−	0.995637i	$-0.470253\pi$
$569$	4.45189			0.186633			0.0933164	+	0.995637i	$0.470253\pi$
$570$		0			0
$571$	29.6834			1.24221			0.621105	−	0.783727i	$-0.286684\pi$
$571$	29.6834			1.24221			0.621105	+	0.783727i	$0.286684\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10.4191	−	8.13093i	0.434508	−	0.339083i
$576$		0			0
$577$	−12.7235	+	12.7235i	−0.529688	+	0.529688i	−0.920479	−	0.390791i	$-0.872201\pi$
$577$	−12.7235	+	12.7235i	−0.529688	+	0.529688i	0.390791	+	0.920479i	$0.372201\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.7743			0.654430
$582$		0			0
$583$	7.19747	−	7.19747i	0.298089	−	0.298089i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.48849	+	2.48849i	−0.102711	+	0.102711i	−0.756595	−	0.653884i	$-0.773138\pi$
$587$	−2.48849	+	2.48849i	−0.102711	+	0.102711i	0.653884	+	0.756595i	$0.273138\pi$
$588$		0			0
$589$	26.4516	+	12.1260i	1.08992	+	0.499643i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.78892	+	2.78892i	0.114527	+	0.114527i	0.762048	−	0.647521i	$-0.224194\pi$
$593$	2.78892	+	2.78892i	0.114527	+	0.114527i	−0.647521	+	0.762048i	$0.724194\pi$
$594$		0			0
$595$	15.6629	−	13.8496i	0.642117	−	0.567776i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	46.1947			1.88746			0.943732	−	0.330711i	$-0.107288\pi$
$599$	46.1947			1.88746			0.943732	+	0.330711i	$0.107288\pi$
$600$		0			0
$601$		−	28.8714i		−	1.17769i	−0.808246	−	0.588845i	$-0.799583\pi$
$601$		−	28.8714i		−	1.17769i	0.808246	−	0.588845i	$-0.200417\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.3307	−	17.3380i	−0.623284	−	0.704892i
$606$		0			0
$607$	−15.5365	−	15.5365i	−0.630608	−	0.630608i	0.317613	−	0.948220i	$-0.397119\pi$
$607$	−15.5365	−	15.5365i	−0.630608	−	0.630608i	−0.948220	+	0.317613i	$0.897119\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	5.97113i		−	0.241566i
$612$		0			0
$613$	23.4241	+	23.4241i	0.946089	+	0.946089i	0.998619	−	0.0525301i	$-0.0167285\pi$
$613$	23.4241	+	23.4241i	0.946089	+	0.946089i	−0.0525301	+	0.998619i	$0.516729\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.4558	−	16.4558i	0.662486	−	0.662486i	−0.293480	−	0.955965i	$-0.594813\pi$
$617$	16.4558	−	16.4558i	0.662486	−	0.662486i	0.955965	+	0.293480i	$0.0948133\pi$
$618$		0			0
$619$		−	41.2711i		−	1.65882i	−0.558637	−	0.829412i	$-0.688676\pi$
$619$		−	41.2711i		−	1.65882i	0.558637	−	0.829412i	$-0.311324\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−21.0706	+	21.0706i	−0.844176	+	0.844176i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.4335			−1.33308
$630$		0			0
$631$	27.1939			1.08257			0.541287	−	0.840838i	$-0.317937\pi$
$631$	27.1939			1.08257			0.541287	+	0.840838i	$0.317937\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.90120	−	30.9421i	0.0754469	−	1.22790i
$636$		0			0
$637$	−1.14158	−	1.14158i	−0.0452311	−	0.0452311i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	4.96042i		−	0.195925i	−0.995190	−	0.0979625i	$-0.968767\pi$
$641$		−	4.96042i		−	0.195925i	0.995190	−	0.0979625i	$-0.0312325\pi$
$642$		0			0
$643$	20.3176	+	20.3176i	0.801247	+	0.801247i	0.983290	−	0.182044i	$-0.0582712\pi$
$643$	20.3176	+	20.3176i	0.801247	+	0.801247i	−0.182044	+	0.983290i	$0.558271\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.16125	+	8.16125i	−0.320852	+	0.320852i	−0.849094	−	0.528242i	$-0.822851\pi$
$647$	−8.16125	+	8.16125i	−0.320852	+	0.320852i	0.528242	+	0.849094i	$0.322851\pi$
$648$		0			0
$649$	10.4230			0.409139
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.4387	+	23.4387i	0.917226	+	0.917226i	0.996827	−	0.0796012i	$-0.0253647\pi$
$653$	23.4387	+	23.4387i	0.917226	+	0.917226i	−0.0796012	+	0.996827i	$0.525365\pi$
$654$		0			0
$655$	13.5515	+	15.3258i	0.529501	+	0.598830i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.84519			−0.266651			−0.133325	−	0.991072i	$-0.542566\pi$
$659$	−6.84519			−0.266651			−0.133325	+	0.991072i	$0.542566\pi$
$660$		0			0
$661$			13.0674i			0.508263i	0.967170	+	0.254131i	$0.0817896\pi$
$661$			13.0674i			0.508263i	−0.967170	+	0.254131i	$0.918210\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.63035	+	18.0029i	−0.373449	+	0.698122i
$666$		0			0
$667$	−8.42380	−	8.42380i	−0.326171	−	0.326171i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.73813			0.0671000
$672$		0			0
$673$	−9.16699	+	9.16699i	−0.353361	+	0.353361i	−0.861359	−	0.507997i	$-0.830386\pi$
$673$	−9.16699	+	9.16699i	−0.353361	+	0.353361i	0.507997	+	0.861359i	$0.330386\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.13824	+	4.13824i	0.159046	+	0.159046i	0.782144	−	0.623098i	$-0.214126\pi$
$677$	4.13824	+	4.13824i	0.159046	+	0.159046i	−0.623098	+	0.782144i	$0.714126\pi$
$678$		0			0
$679$	−22.1142			−0.848665
$680$		0			0
$681$		0			0
$682$		0			0
$683$	34.7907	−	34.7907i	1.33123	−	1.33123i	0.426956	−	0.904273i	$-0.359586\pi$
$683$	34.7907	−	34.7907i	1.33123	−	1.33123i	0.904273	−	0.426956i	$-0.140414\pi$
$684$		0			0
$685$	−10.0581	−	0.618006i	−0.384299	−	0.0236128i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	7.80465i		−	0.297333i
$690$		0			0
$691$	11.1939			0.425837			0.212919	−	0.977070i	$-0.431703\pi$
$691$	11.1939			0.425837			0.212919	+	0.977070i	$0.431703\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	25.5369	−	22.5804i	0.968670	−	0.856523i
$696$		0			0
$697$	35.2960	+	35.2960i	1.33693	+	1.33693i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.75386			0.0662425			0.0331213	−	0.999451i	$-0.489455\pi$
$701$	1.75386			0.0662425			0.0331213	+	0.999451i	$0.489455\pi$
$702$		0			0
$703$	30.6062	−	11.3655i	1.15434	−	0.428657i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.04491	−	6.04491i	0.227342	−	0.227342i
$708$		0			0
$709$			27.4109i			1.02944i	0.857359	+	0.514719i	$0.172104\pi$
$709$			27.4109i			1.02944i	−0.857359	+	0.514719i	$0.827896\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.4773	−	12.4773i	0.467278	−	0.467278i
$714$		0			0
$715$	1.11190	+	0.0683191i	0.0415825	+	0.00255499i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			18.2170i			0.679378i	0.940538	+	0.339689i	$0.110322\pi$
$719$			18.2170i			0.679378i	−0.940538	+	0.339689i	$0.889678\pi$
$720$		0			0
$721$		−	3.88397i		−	0.144647i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.75883	+	22.3653i	−0.102460	+	0.830625i
$726$		0			0
$727$	2.28726	−	2.28726i	0.0848297	−	0.0848297i	−0.663419	−	0.748248i	$-0.730895\pi$
$727$	2.28726	−	2.28726i	0.0848297	−	0.0848297i	0.748248	+	0.663419i	$0.230895\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	24.4241			0.903357
$732$		0			0
$733$	25.8872	+	25.8872i	0.956164	+	0.956164i	0.999079	−	0.0429145i	$-0.0136643\pi$
$733$	25.8872	+	25.8872i	0.956164	+	0.956164i	−0.0429145	+	0.999079i	$0.513664\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.98517	−	3.98517i	−0.146796	−	0.146796i
$738$		0			0
$739$			48.8021i			1.79521i	0.440797	+	0.897607i	$0.354696\pi$
$739$			48.8021i			1.79521i	−0.440797	+	0.897607i	$0.645304\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.674864	−	0.674864i	0.0247583	−	0.0247583i	−0.694619	−	0.719378i	$-0.744427\pi$
$743$	0.674864	−	0.674864i	0.0247583	−	0.0247583i	0.719378	+	0.694619i	$0.244427\pi$
$744$		0			0
$745$	25.7645	−	22.7816i	0.943938	−	0.834654i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21.0706			0.769903
$750$		0			0
$751$			11.7727i			0.429593i	0.976659	+	0.214797i	$0.0689089\pi$
$751$			11.7727i			0.429593i	−0.976659	+	0.214797i	$0.931091\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.72996	+	7.71925i	−0.317716	+	0.280932i
$756$		0			0
$757$	36.0943	−	36.0943i	1.31187	−	1.31187i	0.391832	−	0.920037i	$-0.371841\pi$
$757$	36.0943	−	36.0943i	1.31187	−	1.31187i	0.920037	−	0.391832i	$-0.128159\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.6194			0.674953			0.337477	−	0.941334i	$-0.390427\pi$
$761$	18.6194			0.674953			0.337477	+	0.941334i	$0.390427\pi$
$762$		0			0
$763$	28.6203	−	28.6203i	1.03613	−	1.03613i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.65115	−	5.65115i	0.204051	−	0.204051i
$768$		0			0
$769$			23.5720i			0.850027i	0.905187	+	0.425013i	$0.139731\pi$
$769$			23.5720i			0.850027i	−0.905187	+	0.425013i	$0.860269\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	27.7627	−	27.7627i	0.998556	−	0.998556i	−0.00144323	−	0.999999i	$-0.500459\pi$
$773$	27.7627	−	27.7627i	0.998556	−	0.998556i	0.999999	+	0.00144323i	$0.000459394\pi$
$774$		0			0
$775$	−33.1273	−	4.08636i	−1.18997	−	0.146786i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−44.3099	−	20.3127i	−1.58757	−	0.727776i
$780$		0			0
$781$		−	2.45268i		−	0.0877637i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−32.3357	−	1.98683i	−1.15411	−	0.0709129i
$786$		0			0
$787$	−20.7019	−	20.7019i	−0.737943	−	0.737943i	0.234237	−	0.972180i	$-0.424741\pi$
$787$	−20.7019	−	20.7019i	−0.737943	−	0.737943i	−0.972180	+	0.234237i	$0.924741\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			18.4813i			0.657119i
$792$		0			0
$793$	0.942383	−	0.942383i	0.0334650	−	0.0334650i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.81471	+	8.81471i	0.312233	+	0.312233i	0.845774	−	0.533541i	$-0.179139\pi$
$797$	8.81471	+	8.81471i	0.312233	+	0.312233i	−0.533541	+	0.845774i	$0.679139\pi$
$798$		0			0
$799$		−	43.1246i		−	1.52564i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.99271	+	4.99271i	0.176189	+	0.176189i
$804$		0			0
$805$	8.20123	+	9.27504i	0.289055	+	0.326902i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.73672i		−	0.236850i	−0.992963	−	0.118425i	$-0.962215\pi$
$809$		−	6.73672i		−	0.236850i	0.992963	−	0.118425i	$-0.0377846\pi$
$810$		0			0
$811$		−	1.29466i		−	0.0454616i	−0.999742	−	0.0227308i	$-0.992764\pi$
$811$		−	1.29466i		−	0.0454616i	0.999742	−	0.0227308i	$-0.00723606\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.01951	−	32.8677i	0.0707405	−	1.15130i
$816$		0			0
$817$	−22.3587	+	8.30280i	−0.782232	+	0.290478i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23.7137			0.827614			0.413807	−	0.910365i	$-0.364199\pi$
$821$	23.7137			0.827614			0.413807	+	0.910365i	$0.364199\pi$
$822$		0			0
$823$	2.15140	+	2.15140i	0.0749931	+	0.0749931i	0.743608	−	0.668615i	$-0.233113\pi$
$823$	2.15140	+	2.15140i	0.0749931	+	0.0749931i	−0.668615	+	0.743608i	$0.733113\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17.4896	+	17.4896i	0.608173	+	0.608173i	0.942468	−	0.334295i	$-0.108498\pi$
$827$	17.4896	+	17.4896i	0.608173	+	0.608173i	−0.334295	+	0.942468i	$0.608498\pi$
$828$		0			0
$829$	21.7752			0.756283			0.378141	−	0.925748i	$-0.376563\pi$
$829$	21.7752			0.756283			0.378141	+	0.925748i	$0.376563\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.24472	−	8.24472i	−0.285663	−	0.285663i
$834$		0			0
$835$	−0.857626	+	13.9579i	−0.0296794	+	0.483033i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−30.6196			−1.05710			−0.528552	−	0.848901i	$-0.677265\pi$
$839$	−30.6196			−1.05710			−0.528552	+	0.848901i	$0.677265\pi$
$840$		0			0
$841$	−8.68735			−0.299564
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.1368	+	18.6897i	−0.727128	+	0.642946i
$846$		0			0
$847$	15.3307	−	15.3307i	0.526771	−	0.526771i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	19.7981i		−	0.678672i
$852$		0			0
$853$	−7.18664	−	7.18664i	−0.246066	−	0.246066i	0.573288	−	0.819354i	$-0.305668\pi$
$853$	−7.18664	−	7.18664i	−0.246066	−	0.246066i	−0.819354	+	0.573288i	$0.805668\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.3522	−	19.3522i	−0.661057	−	0.661057i	0.294572	−	0.955629i	$-0.404823\pi$
$857$	−19.3522	−	19.3522i	−0.661057	−	0.661057i	−0.955629	+	0.294572i	$0.904823\pi$
$858$		0			0
$859$		−	30.5599i		−	1.04269i	−0.853346	−	0.521346i	$-0.825430\pi$
$859$		−	30.5599i		−	1.04269i	0.853346	−	0.521346i	$-0.174570\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.199200	−	0.199200i	0.00678084	−	0.00678084i	−0.703708	−	0.710489i	$-0.748474\pi$
$863$	0.199200	−	0.199200i	0.00678084	−	0.00678084i	0.710489	+	0.703708i	$0.248474\pi$
$864$		0			0
$865$	12.8460	+	0.789307i	0.436777	+	0.0268372i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.18021			0.0400360
$870$		0			0
$871$	−4.32136			−0.146424
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.28726	−	23.0240i	0.144936	−	0.778353i
$876$		0			0
$877$	−23.1546	−	23.1546i	−0.781874	−	0.781874i	0.198273	−	0.980147i	$-0.436467\pi$
$877$	−23.1546	−	23.1546i	−0.781874	−	0.781874i	−0.980147	+	0.198273i	$0.936467\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.84367			−0.331642			−0.165821	−	0.986156i	$-0.553027\pi$
$881$	−9.84367			−0.331642			−0.165821	+	0.986156i	$0.553027\pi$
$882$		0			0
$883$	3.71274	+	3.71274i	0.124944	+	0.124944i	0.766814	−	0.641870i	$-0.221841\pi$
$883$	3.71274	+	3.71274i	0.124944	+	0.124944i	−0.641870	+	0.766814i	$0.721841\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	35.8697	+	35.8697i	1.20439	+	1.20439i	0.972819	+	0.231568i	$0.0743855\pi$
$887$	35.8697	+	35.8697i	1.20439	+	1.20439i	0.231568	+	0.972819i	$0.425615\pi$
$888$		0			0
$889$	29.0409			0.974002
$890$		0			0
$891$		0			0
$892$		0			0
$893$	14.6599	+	39.4779i	0.490575	+	1.32108i
$894$		0			0
$895$	4.25587	+	4.81311i	0.142258	+	0.160884i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			30.0870i			1.00346i
$900$		0			0
$901$		−	56.3666i		−	1.87784i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−30.1990	+	26.7027i	−1.00385	+	0.887628i
$906$		0			0
$907$	−6.52260	−	6.52260i	−0.216579	−	0.216579i	0.590476	−	0.807055i	$-0.298940\pi$
$907$	−6.52260	−	6.52260i	−0.216579	−	0.216579i	−0.807055	+	0.590476i	$0.798940\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	20.4476i		−	0.677460i	−0.940884	−	0.338730i	$-0.890003\pi$
$911$		−	20.4476i		−	0.677460i	0.940884	−	0.338730i	$-0.109997\pi$
$912$		0			0
$913$	4.29218	+	4.29218i	0.142051	+	0.142051i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−13.5515	+	13.5515i	−0.447510	+	0.447510i
$918$		0			0
$919$			13.9756i			0.461011i	0.973071	+	0.230506i	$0.0740380\pi$
$919$			13.9756i			0.461011i	−0.973071	+	0.230506i	$0.925962\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.32979	−	1.32979i	−0.0437707	−	0.0437707i
$924$		0			0
$925$	−29.5241	+	23.0401i	−0.970746	+	0.757555i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			22.9478i			0.752893i	0.926438	+	0.376446i	$0.122854\pi$
$929$			22.9478i			0.752893i	−0.926438	+	0.376446i	$0.877146\pi$
$930$		0			0
$931$	10.3503	+	4.74479i	0.339216	+	0.155504i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.03032	+	0.493413i	0.262619	+	0.0161363i
$936$		0			0
$937$	−11.9076	+	11.9076i	−0.389005	+	0.389005i	−0.874333	−	0.485327i	$-0.838700\pi$
$937$	−11.9076	+	11.9076i	−0.389005	+	0.389005i	0.485327	+	0.874333i	$0.338700\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	34.1709i		−	1.11394i	−0.830533	−	0.556969i	$-0.811964\pi$
$941$		−	34.1709i		−	1.11394i	0.830533	−	0.556969i	$-0.188036\pi$
$942$		0			0
$943$	−20.9011	+	20.9011i	−0.680633	+	0.680633i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.00096	−	6.00096i	0.195005	−	0.195005i	−0.602850	−	0.797855i	$-0.705968\pi$
$947$	6.00096	−	6.00096i	0.195005	−	0.195005i	0.797855	+	0.602850i	$0.205968\pi$
$948$		0			0
$949$	5.41389			0.175742
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−16.0583	+	16.0583i	−0.520179	+	0.520179i	−0.917626	−	0.397446i	$-0.869897\pi$
$953$	−16.0583	+	16.0583i	−0.520179	+	0.520179i	0.397446	+	0.917626i	$0.369897\pi$
$954$		0			0
$955$	6.57452	+	7.43533i	0.212746	+	0.240602i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	9.44007i		−	0.304836i
$960$		0			0
$961$	−13.5647			−0.437570
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−13.9579	−	0.857626i	−0.449321	−	0.0276080i
$966$		0			0
$967$	16.6956	−	16.6956i	0.536894	−	0.536894i	−0.385721	−	0.922615i	$-0.626047\pi$
$967$	16.6956	−	16.6956i	0.536894	−	0.536894i	0.922615	+	0.385721i	$0.126047\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			10.6476i			0.341698i	0.985297	+	0.170849i	$0.0546510\pi$
$971$			10.6476i			0.341698i	−0.985297	+	0.170849i	$0.945349\pi$
$972$		0			0
$973$	22.5804	+	22.5804i	0.723894	+	0.723894i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	29.3085	+	29.3085i	0.937661	+	0.937661i	0.998168	−	0.0605070i	$-0.0192717\pi$
$977$	29.3085	+	29.3085i	0.937661	+	0.937661i	−0.0605070	+	0.998168i	$0.519272\pi$
$978$		0			0
$979$	−11.4666			−0.366474
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.77527	−	2.77527i	0.0885172	−	0.0885172i	−0.661462	−	0.749979i	$-0.730064\pi$
$983$	2.77527	−	2.77527i	0.0885172	−	0.0885172i	0.749979	+	0.661462i	$0.230064\pi$
$984$		0			0
$985$	−32.7645	−	2.01317i	−1.04396	−	0.0641451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			14.4631i			0.459900i
$990$		0			0
$991$		−	9.18342i		−	0.291721i	−0.989305	−	0.145861i	$-0.953405\pi$
$991$		−	9.18342i		−	0.291721i	0.989305	−	0.145861i	$-0.0465951\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.7988	+	12.2012i	−0.437450	+	0.386805i
$996$		0			0
$997$	−37.8773	+	37.8773i	−1.19959	+	1.19959i	−0.225296	+	0.974290i	$0.572335\pi$
$997$	−37.8773	+	37.8773i	−1.19959	+	1.19959i	−0.974290	+	0.225296i	$0.927665\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.bb.d.2773.5		12
3.2	odd	2		380.2.l.b.113.6	yes	12
5.2	odd	4	inner	3420.2.bb.d.37.6		12
15.2	even	4		380.2.l.b.37.1	✓	12
15.8	even	4		1900.2.l.b.1557.6		12
15.14	odd	2		1900.2.l.b.493.1		12
19.18	odd	2	inner	3420.2.bb.d.2773.6		12
57.56	even	2		380.2.l.b.113.1	yes	12
95.37	even	4	inner	3420.2.bb.d.37.5		12
285.113	odd	4		1900.2.l.b.1557.1		12
285.227	odd	4		380.2.l.b.37.6	yes	12
285.284	even	2		1900.2.l.b.493.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.l.b.37.1	✓	12	15.2	even	4
380.2.l.b.37.6	yes	12	285.227	odd	4
380.2.l.b.113.1	yes	12	57.56	even	2
380.2.l.b.113.6	yes	12	3.2	odd	2
1900.2.l.b.493.1		12	15.14	odd	2
1900.2.l.b.493.6		12	285.284	even	2
1900.2.l.b.1557.1		12	285.113	odd	4
1900.2.l.b.1557.6		12	15.8	even	4
3420.2.bb.d.37.5		12	95.37	even	4	inner
3420.2.bb.d.37.6		12	5.2	odd	4	inner
3420.2.bb.d.2773.5		12	1.1	even	1	trivial
3420.2.bb.d.2773.6		12	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.bb (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.48119 + 1.67513i) q^{5} +(-1.48119 + 1.48119i) q^{7} +O(q^{10})\)	\(q+(1.48119 + 1.67513i) q^{5} +(-1.48119 + 1.48119i) q^{7} -0.806063 q^{11} +(-0.437032 + 0.437032i) q^{13} +(-3.15633 + 3.15633i) q^{17} +(1.81645 - 3.96239i) q^{19} +(-1.86907 - 1.86907i) q^{23} +(-0.612127 + 4.96239i) q^{25} +4.50696 q^{29} +6.67568i q^{31} +(-4.67513 - 0.287258i) q^{35} +(5.29626 + 5.29626i) q^{37} -11.1826i q^{41} +(-3.86907 - 3.86907i) q^{43} +(-6.83146 + 6.83146i) q^{47} +2.61213i q^{49} +(-8.92916 + 8.92916i) q^{53} +(-1.19394 - 1.35026i) q^{55} -12.9308 q^{59} -2.15633 q^{61} +(-1.37941 - 0.0847564i) q^{65} +(4.94399 + 4.94399i) q^{67} +3.04278i q^{71} +(-6.19394 - 6.19394i) q^{73} +(1.19394 - 1.19394i) q^{77} -1.46417 q^{79} +(-5.32487 - 5.32487i) q^{83} +(-9.96239 - 0.612127i) q^{85} +14.2254 q^{89} -1.29466i q^{91} +(9.32803 - 2.82628i) q^{95} +(7.46498 + 7.46498i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 12 * q - 4 * q^5 + 4 * q^7	\( 12 q - 4 q^{5} + 4 q^{7} - 8 q^{11} + 4 q^{17} - 4 q^{23} - 4 q^{25} - 36 q^{35} - 28 q^{43} - 20 q^{47} - 16 q^{55} + 16 q^{61} - 76 q^{73} + 16 q^{77} - 84 q^{83} - 76 q^{85} + 16 q^{95}+O(q^{100}) \) 12 * q - 4 * q^5 + 4 * q^7 - 8 * q^11 + 4 * q^17 - 4 * q^23 - 4 * q^25 - 36 * q^35 - 28 * q^43 - 20 * q^47 - 16 * q^55 + 16 * q^61 - 76 * q^73 + 16 * q^77 - 84 * q^83 - 76 * q^85 + 16 * q^95

Introduction

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