LMFDB - Embedded newform 4140.2.f.c.829.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (of order $2$, degree $1$, 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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 $ x^14 + 27x^12 + 283x^10 + 1441x^8 + 3596x^6 + 3740x^4 + 772x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.3
Root			$-0.0729221i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.c.829.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.54426 - 1.61718i) q^{5} +3.99468i q^{7} +O(q^{10})$	$q+(-1.54426 - 1.61718i) q^{5} +3.99468i q^{7} -2.31084 q^{11} +5.60596i q^{13} +3.27777i q^{17} -1.49731 q^{19} +1.00000i q^{23} +(-0.230539 + 4.99468i) q^{25} -4.51213 q^{29} -2.38895 q^{31} +(6.46012 - 6.16882i) q^{35} -6.66237i q^{37} -1.07468 q^{41} -12.5260i q^{43} -4.43129i q^{47} -8.95749 q^{49} -2.01735i q^{53} +(3.56853 + 3.73705i) q^{55} +13.1502 q^{59} +11.1266 q^{61} +(9.06585 - 8.65705i) q^{65} +9.91149i q^{67} -4.94360 q^{71} -4.38340i q^{73} -9.23108i q^{77} -9.35481 q^{79} -3.95659i q^{83} +(5.30074 - 5.06172i) q^{85} -11.6996 q^{89} -22.3940 q^{91} +(2.31224 + 2.42142i) q^{95} -16.4888i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q+O(q^{10}) $ 14 * q	$ 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) $ 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.54426	−	1.61718i	−0.690613	−	0.723225i
$5$	−1.54426	−	1.61718i	−0.690613	−	0.723225i
$6$		0			0
$7$			3.99468i			1.50985i	0.655812	+	0.754924i	$0.272326\pi$
$7$			3.99468i			1.50985i	−0.655812	+	0.754924i	$0.727674\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.31084			−0.696745			−0.348372	−	0.937356i	$-0.613266\pi$
$11$	−2.31084			−0.696745			−0.348372	+	0.937356i	$0.613266\pi$
$12$		0			0
$13$			5.60596i			1.55481i	0.628998	+	0.777407i	$0.283465\pi$
$13$			5.60596i			1.55481i	−0.628998	+	0.777407i	$0.716535\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.27777i			0.794976i	0.917607	+	0.397488i	$0.130118\pi$
$17$			3.27777i			0.794976i	−0.917607	+	0.397488i	$0.869882\pi$
$18$		0			0
$19$	−1.49731			−0.343507			−0.171754	−	0.985140i	$-0.554943\pi$
$19$	−1.49731			−0.343507			−0.171754	+	0.985140i	$0.554943\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−0.230539	+	4.99468i	−0.0461077	+	0.998936i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.51213			−0.837882			−0.418941	−	0.908014i	$-0.637599\pi$
$29$	−4.51213			−0.837882			−0.418941	+	0.908014i	$0.637599\pi$
$30$		0			0
$31$	−2.38895			−0.429069			−0.214534	−	0.976716i	$-0.568823\pi$
$31$	−2.38895			−0.429069			−0.214534	+	0.976716i	$0.568823\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.46012	−	6.16882i	1.09196	−	1.04272i
$36$		0			0
$37$		−	6.66237i		−	1.09529i	−0.836712	−	0.547643i	$-0.815525\pi$
$37$		−	6.66237i		−	1.09529i	0.836712	−	0.547643i	$-0.184475\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.07468			−0.167837			−0.0839184	−	0.996473i	$-0.526743\pi$
$41$	−1.07468			−0.167837			−0.0839184	+	0.996473i	$0.526743\pi$
$42$		0			0
$43$		−	12.5260i		−	1.91019i	−0.296297	−	0.955096i	$-0.595752\pi$
$43$		−	12.5260i		−	1.91019i	0.296297	−	0.955096i	$-0.404248\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.43129i		−	0.646370i	−0.946336	−	0.323185i	$-0.895246\pi$
$47$		−	4.43129i		−	0.646370i	0.946336	−	0.323185i	$-0.104754\pi$
$48$		0			0
$49$	−8.95749			−1.27964
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	2.01735i		−	0.277104i	−0.990355	−	0.138552i	$-0.955755\pi$
$53$		−	2.01735i		−	0.277104i	0.990355	−	0.138552i	$-0.0442449\pi$
$54$		0			0
$55$	3.56853	+	3.73705i	0.481181	+	0.503903i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	13.1502			1.71202			0.856008	−	0.516963i	$-0.172938\pi$
$59$	13.1502			1.71202			0.856008	+	0.516963i	$0.172938\pi$
$60$		0			0
$61$	11.1266			1.42462			0.712308	−	0.701867i	$-0.247650\pi$
$61$	11.1266			1.42462			0.712308	+	0.701867i	$0.247650\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.06585	−	8.65705i	1.12448	−	1.07377i
$66$		0			0
$67$			9.91149i			1.21088i	0.795891	+	0.605440i	$0.207003\pi$
$67$			9.91149i			1.21088i	−0.795891	+	0.605440i	$0.792997\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.94360			−0.586697			−0.293348	−	0.956006i	$-0.594770\pi$
$71$	−4.94360			−0.586697			−0.293348	+	0.956006i	$0.594770\pi$
$72$		0			0
$73$		−	4.38340i		−	0.513038i	−0.966539	−	0.256519i	$-0.917424\pi$
$73$		−	4.38340i		−	0.513038i	0.966539	−	0.256519i	$-0.0825757\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	9.23108i		−	1.05198i
$78$		0			0
$79$	−9.35481			−1.05250			−0.526249	−	0.850331i	$-0.676402\pi$
$79$	−9.35481			−1.05250			−0.526249	+	0.850331i	$0.676402\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	3.95659i		−	0.434292i	−0.976139	−	0.217146i	$-0.930325\pi$
$83$		−	3.95659i		−	0.434292i	0.976139	−	0.217146i	$-0.0696748\pi$
$84$		0			0
$85$	5.30074	−	5.06172i	0.574946	−	0.549021i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.6996			−1.24015			−0.620076	−	0.784542i	$-0.712898\pi$
$89$	−11.6996			−1.24015			−0.620076	+	0.784542i	$0.712898\pi$
$90$		0			0
$91$	−22.3940			−2.34753
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.31224	+	2.42142i	0.237231	+	0.248433i
$96$		0			0
$97$		−	16.4888i		−	1.67418i	−0.547064	−	0.837090i	$-0.684255\pi$
$97$		−	16.4888i		−	1.67418i	0.547064	−	0.837090i	$-0.315745\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.3994			−1.03478			−0.517390	−	0.855750i	$-0.673096\pi$
$101$	−10.3994			−1.03478			−0.517390	+	0.855750i	$0.673096\pi$
$102$		0			0
$103$			12.4044i			1.22225i	0.791536	+	0.611123i	$0.209282\pi$
$103$			12.4044i			1.22225i	−0.791536	+	0.611123i	$0.790718\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.51748i			0.146700i	0.997306	+	0.0733502i	$0.0233691\pi$
$107$			1.51748i			0.146700i	−0.997306	+	0.0733502i	$0.976631\pi$
$108$		0			0
$109$	−4.86276			−0.465767			−0.232884	−	0.972505i	$-0.574816\pi$
$109$	−4.86276			−0.465767			−0.232884	+	0.972505i	$0.574816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.72566i		−	0.162337i	−0.996700	−	0.0811683i	$-0.974135\pi$
$113$		−	1.72566i		−	0.162337i	0.996700	−	0.0811683i	$-0.0258651\pi$
$114$		0			0
$115$	1.61718	−	1.54426i	0.150803	−	0.144003i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13.0937			−1.20029
$120$		0			0
$121$	−5.66001			−0.514546
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.43331	−	7.34025i	0.754298	−	0.656532i
$126$		0			0
$127$			14.1962i			1.25971i	0.776713	+	0.629855i	$0.216886\pi$
$127$			14.1962i			1.25971i	−0.776713	+	0.629855i	$0.783114\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.34301			0.466821			0.233410	−	0.972378i	$-0.425011\pi$
$131$	5.34301			0.466821			0.233410	+	0.972378i	$0.425011\pi$
$132$		0			0
$133$		−	5.98129i		−	0.518644i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	18.1842i		−	1.55358i	−0.629759	−	0.776791i	$-0.716846\pi$
$137$		−	18.1842i		−	1.55358i	0.629759	−	0.776791i	$-0.283154\pi$
$138$		0			0
$139$	−1.86064			−0.157817			−0.0789086	−	0.996882i	$-0.525144\pi$
$139$	−1.86064			−0.157817			−0.0789086	+	0.996882i	$0.525144\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12.9545i		−	1.08331i
$144$		0			0
$145$	6.96789	+	7.29692i	0.578652	+	0.605977i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.33848			0.519268			0.259634	−	0.965707i	$-0.416398\pi$
$149$	6.33848			0.519268			0.259634	+	0.965707i	$0.416398\pi$
$150$		0			0
$151$	7.44809			0.606117			0.303058	−	0.952972i	$-0.401992\pi$
$151$	7.44809			0.606117			0.303058	+	0.952972i	$0.401992\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.68916	+	3.86337i	0.296320	+	0.310313i
$156$		0			0
$157$		−	5.10147i		−	0.407142i	−0.979060	−	0.203571i	$-0.934745\pi$
$157$		−	5.10147i		−	0.407142i	0.979060	−	0.203571i	$-0.0652547\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.99468			−0.314825
$162$		0			0
$163$		−	4.41321i		−	0.345669i	−0.984951	−	0.172835i	$-0.944707\pi$
$163$		−	4.41321i		−	0.345669i	0.984951	−	0.172835i	$-0.0552927\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10.3349i		−	0.799739i	−0.916572	−	0.399869i	$-0.869055\pi$
$167$		−	10.3349i		−	0.799739i	0.916572	−	0.399869i	$-0.130945\pi$
$168$		0			0
$169$	−18.4268			−1.41745
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	9.56659i		−	0.727335i	−0.931529	−	0.363667i	$-0.881525\pi$
$173$		−	9.56659i		−	0.727335i	0.931529	−	0.363667i	$-0.118475\pi$
$174$		0			0
$175$	−19.9522	−	0.920929i	−1.50824	−	0.0696157i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−18.9657			−1.41757			−0.708783	−	0.705427i	$-0.750755\pi$
$179$	−18.9657			−1.41757			−0.708783	+	0.705427i	$0.750755\pi$
$180$		0			0
$181$	24.4964			1.82080			0.910401	−	0.413727i	$-0.135773\pi$
$181$	24.4964			1.82080			0.910401	+	0.413727i	$0.135773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.7742	+	10.2884i	−0.792138	+	0.756419i
$186$		0			0
$187$		−	7.57441i		−	0.553896i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.3403			−1.10998			−0.554992	−	0.831856i	$-0.687279\pi$
$191$	−15.3403			−1.10998			−0.554992	+	0.831856i	$0.687279\pi$
$192$		0			0
$193$		−	11.4740i		−	0.825915i	−0.910750	−	0.412958i	$-0.864496\pi$
$193$		−	11.4740i		−	0.825915i	0.910750	−	0.412958i	$-0.135504\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.8530i			1.34322i	0.740904	+	0.671611i	$0.234397\pi$
$197$			18.8530i			1.34322i	−0.740904	+	0.671611i	$0.765603\pi$
$198$		0			0
$199$	−20.3152			−1.44011			−0.720053	−	0.693919i	$-0.755883\pi$
$199$	−20.3152			−1.44011			−0.720053	+	0.693919i	$0.755883\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	18.0245i		−	1.26507i
$204$		0			0
$205$	1.65958	+	1.73795i	0.115910	+	0.121384i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.46005			0.239337
$210$		0			0
$211$	−0.0136020			−0.000936399			−0.000468199	−	1.00000i	$-0.500149\pi$
$211$	−0.0136020			−0.000936399			−0.000468199		1.00000i	$0.500149\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−20.2567	+	19.3433i	−1.38150	+	1.31920i
$216$		0			0
$217$		−	9.54311i		−	0.647828i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.3751			−1.23604
$222$		0			0
$223$		−	8.61622i		−	0.576985i	−0.957482	−	0.288493i	$-0.906846\pi$
$223$		−	8.61622i		−	0.576985i	0.957482	−	0.288493i	$-0.0931540\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			16.2173i			1.07638i	0.842824	+	0.538190i	$0.180892\pi$
$227$			16.2173i			1.07638i	−0.842824	+	0.538190i	$0.819108\pi$
$228$		0			0
$229$	−11.2202			−0.741450			−0.370725	−	0.928743i	$-0.620891\pi$
$229$	−11.2202			−0.741450			−0.370725	+	0.928743i	$0.620891\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.24723i		−	0.0817090i	−0.999165	−	0.0408545i	$-0.986992\pi$
$233$		−	1.24723i		−	0.0817090i	0.999165	−	0.0408545i	$-0.0130080\pi$
$234$		0			0
$235$	−7.16619	+	6.84305i	−0.467471	+	0.446391i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.8930			0.898663			0.449332	−	0.893365i	$-0.351662\pi$
$239$	13.8930			0.898663			0.449332	+	0.893365i	$0.351662\pi$
$240$		0			0
$241$	16.1962			1.04329			0.521645	−	0.853163i	$-0.325319\pi$
$241$	16.1962			1.04329			0.521645	+	0.853163i	$0.325319\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.8327	+	14.4859i	0.883737	+	0.925468i
$246$		0			0
$247$		−	8.39388i		−	0.534090i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.451519			0.0284996			0.0142498	−	0.999898i	$-0.495464\pi$
$251$	0.451519			0.0284996			0.0142498	+	0.999898i	$0.495464\pi$
$252$		0			0
$253$		−	2.31084i		−	0.145281i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.94554i			0.246116i	0.992399	+	0.123058i	$0.0392701\pi$
$257$			3.94554i			0.246116i	−0.992399	+	0.123058i	$0.960730\pi$
$258$		0			0
$259$	26.6140			1.65372
$260$		0			0
$261$		0			0
$262$		0			0
$263$			21.7884i			1.34353i	0.740765	+	0.671764i	$0.234463\pi$
$263$			21.7884i			1.34353i	−0.740765	+	0.671764i	$0.765537\pi$
$264$		0			0
$265$	−3.26242	+	3.11531i	−0.200409	+	0.191372i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−29.3071			−1.78688			−0.893442	−	0.449179i	$-0.851717\pi$
$269$	−29.3071			−1.78688			−0.893442	+	0.449179i	$0.851717\pi$
$270$		0			0
$271$	15.6009			0.947685			0.473842	−	0.880610i	$-0.342867\pi$
$271$	15.6009			0.947685			0.473842	+	0.880610i	$0.342867\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.532739	−	11.5419i	0.0321253	−	0.696004i
$276$		0			0
$277$		−	1.76382i		−	0.105977i	−0.998595	−	0.0529887i	$-0.983125\pi$
$277$		−	1.76382i		−	0.105977i	0.998595	−	0.0529887i	$-0.0168747\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−28.6750			−1.71061			−0.855305	−	0.518125i	$-0.826630\pi$
$281$	−28.6750			−1.71061			−0.855305	+	0.518125i	$0.826630\pi$
$282$		0			0
$283$			23.6822i			1.40776i	0.710319	+	0.703880i	$0.248551\pi$
$283$			23.6822i			1.40776i	−0.710319	+	0.703880i	$0.751449\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4.29300i		−	0.253408i
$288$		0			0
$289$	6.25621			0.368013
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	32.7897i		−	1.91559i	−0.287444	−	0.957797i	$-0.592805\pi$
$293$		−	32.7897i		−	1.91559i	0.287444	−	0.957797i	$-0.407195\pi$
$294$		0			0
$295$	−20.3073	−	21.2663i	−1.18234	−	1.23817i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.60596			−0.324201
$300$		0			0
$301$	50.0373			2.88410
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−17.1823	−	17.9937i	−0.983858	−	1.03032i
$306$		0			0
$307$		−	2.50214i		−	0.142804i	−0.997448	−	0.0714022i	$-0.977253\pi$
$307$		−	2.50214i		−	0.142804i	0.997448	−	0.0714022i	$-0.0227474\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−33.4641			−1.89757			−0.948787	−	0.315918i	$-0.897688\pi$
$311$	−33.4641			−1.89757			−0.948787	+	0.315918i	$0.897688\pi$
$312$		0			0
$313$		−	24.5389i		−	1.38702i	−0.720446	−	0.693511i	$-0.756063\pi$
$313$		−	24.5389i		−	1.38702i	0.720446	−	0.693511i	$-0.243937\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			24.5232i			1.37736i	0.725066	+	0.688680i	$0.241809\pi$
$317$			24.5232i			1.37736i	−0.725066	+	0.688680i	$0.758191\pi$
$318$		0			0
$319$	10.4268			0.583790
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.90785i		−	0.273080i
$324$		0			0
$325$	−28.0000	−	1.29239i	−1.55316	−	0.0716890i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.7016			0.975920
$330$		0			0
$331$	−3.00518			−0.165179			−0.0825897	−	0.996584i	$-0.526319\pi$
$331$	−3.00518			−0.165179			−0.0825897	+	0.996584i	$0.526319\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.0287	−	15.3059i	0.875739	−	0.836250i
$336$		0			0
$337$			1.83422i			0.0999162i	0.998751	+	0.0499581i	$0.0159088\pi$
$337$			1.83422i			0.0999162i	−0.998751	+	0.0499581i	$0.984091\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.52049			0.298951
$342$		0			0
$343$		−	7.81954i		−	0.422215i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			19.7001i			1.05756i	0.848759	+	0.528779i	$0.177350\pi$
$347$			19.7001i			1.05756i	−0.848759	+	0.528779i	$0.822650\pi$
$348$		0			0
$349$	−19.1625			−1.02575			−0.512874	−	0.858464i	$-0.671419\pi$
$349$	−19.1625			−1.02575			−0.512874	+	0.858464i	$0.671419\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	21.9329i		−	1.16737i	−0.811979	−	0.583686i	$-0.801610\pi$
$353$		−	21.9329i		−	1.16737i	0.811979	−	0.583686i	$-0.198390\pi$
$354$		0			0
$355$	7.63418	+	7.99468i	0.405180	+	0.424314i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0257			−0.634693			−0.317347	−	0.948310i	$-0.602792\pi$
$359$	−12.0257			−0.634693			−0.317347	+	0.948310i	$0.602792\pi$
$360$		0			0
$361$	−16.7581			−0.882003
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.08875	+	6.76910i	−0.371042	+	0.354311i
$366$		0			0
$367$		−	3.39198i		−	0.177060i	−0.996074	−	0.0885298i	$-0.971783\pi$
$367$		−	3.39198i		−	0.177060i	0.996074	−	0.0885298i	$-0.0282169\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.05867			0.418385
$372$		0			0
$373$		−	14.4907i		−	0.750299i	−0.926964	−	0.375150i	$-0.877591\pi$
$373$		−	14.4907i		−	0.750299i	0.926964	−	0.375150i	$-0.122409\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	25.2948i		−	1.30275i
$378$		0			0
$379$	10.2473			0.526366			0.263183	−	0.964746i	$-0.415228\pi$
$379$	10.2473			0.526366			0.263183	+	0.964746i	$0.415228\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			21.0326i			1.07471i	0.843355	+	0.537357i	$0.180577\pi$
$383$			21.0326i			1.07471i	−0.843355	+	0.537357i	$0.819423\pi$
$384$		0			0
$385$	−14.9283	+	14.2552i	−0.760817	+	0.726510i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.4421			0.985754			0.492877	−	0.870099i	$-0.335945\pi$
$389$	19.4421			0.985754			0.492877	+	0.870099i	$0.335945\pi$
$390$		0			0
$391$	−3.27777			−0.165764
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.4462	+	15.1284i	0.726869	+	0.761192i
$396$		0			0
$397$		−	4.53486i		−	0.227598i	−0.993504	−	0.113799i	$-0.963698\pi$
$397$		−	4.53486i		−	0.227598i	0.993504	−	0.113799i	$-0.0363020\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−26.8076			−1.33871			−0.669353	−	0.742944i	$-0.733429\pi$
$401$	−26.8076			−1.33871			−0.669353	+	0.742944i	$0.733429\pi$
$402$		0			0
$403$		−	13.3924i		−	0.667122i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			15.3957i			0.763135i
$408$		0			0
$409$	25.5992			1.26580			0.632899	−	0.774234i	$-0.281865\pi$
$409$	25.5992			1.26580			0.632899	+	0.774234i	$0.281865\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			52.5310i			2.58488i
$414$		0			0
$415$	−6.39851	+	6.10999i	−0.314091	+	0.299928i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	36.9334			1.80431			0.902157	−	0.431407i	$-0.141983\pi$
$419$	36.9334			1.80431			0.902157	+	0.431407i	$0.141983\pi$
$420$		0			0
$421$	18.4129			0.897391			0.448696	−	0.893685i	$-0.351889\pi$
$421$	18.4129			0.897391			0.448696	+	0.893685i	$0.351889\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.3714	−	0.755653i	−0.794131	−	0.0366546i
$426$		0			0
$427$			44.4473i			2.15095i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	32.2149			1.55174			0.775869	−	0.630894i	$-0.217312\pi$
$431$	32.2149			1.55174			0.775869	+	0.630894i	$0.217312\pi$
$432$		0			0
$433$		−	38.7313i		−	1.86131i	−0.365900	−	0.930654i	$-0.619239\pi$
$433$		−	38.7313i		−	1.86131i	0.365900	−	0.930654i	$-0.380761\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.49731i		−	0.0716262i
$438$		0			0
$439$	−12.4945			−0.596329			−0.298165	−	0.954514i	$-0.596374\pi$
$439$	−12.4945			−0.596329			−0.298165	+	0.954514i	$0.596374\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	24.8438i		−	1.18036i	−0.807270	−	0.590182i	$-0.799056\pi$
$443$		−	24.8438i		−	1.18036i	0.807270	−	0.590182i	$-0.200944\pi$
$444$		0			0
$445$	18.0671	+	18.9203i	0.856464	+	0.896908i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.3084			−0.675254			−0.337627	−	0.941280i	$-0.609624\pi$
$449$	−14.3084			−0.675254			−0.337627	+	0.941280i	$0.609624\pi$
$450$		0			0
$451$	2.48341			0.116939
$452$		0			0
$453$		0			0
$454$		0			0
$455$	34.5822	+	36.2152i	1.62124	+	1.69779i
$456$		0			0
$457$			30.0544i			1.40589i	0.711247	+	0.702943i	$0.248131\pi$
$457$			30.0544i			1.40589i	−0.711247	+	0.702943i	$0.751869\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.27280			0.199004			0.0995021	−	0.995037i	$-0.468275\pi$
$461$	4.27280			0.199004			0.0995021	+	0.995037i	$0.468275\pi$
$462$		0			0
$463$		−	36.5765i		−	1.69986i	−0.526899	−	0.849928i	$-0.676646\pi$
$463$		−	36.5765i		−	1.69986i	0.526899	−	0.849928i	$-0.323354\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	34.4252i		−	1.59301i	−0.604634	−	0.796503i	$-0.706681\pi$
$467$		−	34.4252i		−	1.59301i	0.604634	−	0.796503i	$-0.293319\pi$
$468$		0			0
$469$	−39.5932			−1.82825
$470$		0			0
$471$		0			0
$472$		0			0
$473$			28.9455i			1.33092i
$474$		0			0
$475$	0.345189	−	7.47860i	0.0158383	−	0.343142i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.31790			−0.425746			−0.212873	−	0.977080i	$-0.568282\pi$
$479$	−9.31790			−0.425746			−0.212873	+	0.977080i	$0.568282\pi$
$480$		0			0
$481$	37.3490			1.70297
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−26.6653	+	25.4629i	−1.21081	+	1.15621i
$486$		0			0
$487$			5.49269i			0.248897i	0.992226	+	0.124449i	$0.0397162\pi$
$487$			5.49269i			0.248897i	−0.992226	+	0.124449i	$0.960284\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.97963			−0.134469			−0.0672344	−	0.997737i	$-0.521418\pi$
$491$	−2.97963			−0.134469			−0.0672344	+	0.997737i	$0.521418\pi$
$492$		0			0
$493$		−	14.7897i		−	0.666096i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	19.7481i		−	0.885823i
$498$		0			0
$499$	−2.38895			−0.106944			−0.0534721	−	0.998569i	$-0.517029\pi$
$499$	−2.38895			−0.106944			−0.0534721	+	0.998569i	$0.517029\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	25.4790i		−	1.13605i	−0.823010	−	0.568027i	$-0.807707\pi$
$503$		−	25.4790i		−	1.13605i	0.823010	−	0.568027i	$-0.192293\pi$
$504$		0			0
$505$	16.0594	+	16.8177i	0.714632	+	0.748378i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.3695			−0.503942			−0.251971	−	0.967735i	$-0.581079\pi$
$509$	−11.3695			−0.503942			−0.251971	+	0.967735i	$0.581079\pi$
$510$		0			0
$511$	17.5103			0.774610
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.0602	−	19.1556i	0.883958	−	0.844099i
$516$		0			0
$517$			10.2400i			0.450355i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.6336			−0.772543			−0.386271	−	0.922385i	$-0.626237\pi$
$521$	−17.6336			−0.772543			−0.386271	+	0.922385i	$0.626237\pi$
$522$		0			0
$523$		−	3.47209i		−	0.151824i	−0.997115	−	0.0759120i	$-0.975813\pi$
$523$		−	3.47209i		−	0.151824i	0.997115	−	0.0759120i	$-0.0241868\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	7.83044i		−	0.341099i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	6.02461i		−	0.260955i
$534$		0			0
$535$	2.45404	−	2.34338i	0.106097	−	0.101313i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.6993			0.891583
$540$		0			0
$541$	−43.0785			−1.85209			−0.926046	−	0.377411i	$-0.876814\pi$
$541$	−43.0785			−1.85209			−0.926046	+	0.377411i	$0.876814\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.50935	+	7.86395i	0.321665	+	0.336855i
$546$		0			0
$547$		−	3.12643i		−	0.133677i	−0.997764	−	0.0668383i	$-0.978709\pi$
$547$		−	3.12643i		−	0.133677i	0.997764	−	0.0668383i	$-0.0212912\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.75607			0.287818
$552$		0			0
$553$		−	37.3695i		−	1.58911i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	32.9391i		−	1.39567i	−0.716257	−	0.697837i	$-0.754146\pi$
$557$		−	32.9391i		−	1.39567i	0.716257	−	0.697837i	$-0.245854\pi$
$558$		0			0
$559$	70.2201			2.96999
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	24.4548i		−	1.03065i	−0.856996	−	0.515324i	$-0.827672\pi$
$563$		−	24.4548i		−	1.03065i	0.856996	−	0.515324i	$-0.172328\pi$
$564$		0			0
$565$	−2.79070	+	2.66486i	−0.117406	+	0.112112i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.16894			−0.216693			−0.108347	−	0.994113i	$-0.534556\pi$
$569$	−5.16894			−0.216693			−0.108347	+	0.994113i	$0.534556\pi$
$570$		0			0
$571$	−17.0392			−0.713070			−0.356535	−	0.934282i	$-0.616042\pi$
$571$	−17.0392			−0.713070			−0.356535	+	0.934282i	$0.616042\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.99468	−	0.230539i	−0.208293	−	0.00961413i
$576$		0			0
$577$			38.2584i			1.59272i	0.604824	+	0.796359i	$0.293243\pi$
$577$			38.2584i			1.59272i	−0.604824	+	0.796359i	$0.706757\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.8053			0.655715
$582$		0			0
$583$			4.66178i			0.193071i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			25.6094i			1.05701i	0.848929	+	0.528506i	$0.177248\pi$
$587$			25.6094i			1.05701i	−0.848929	+	0.528506i	$0.822752\pi$
$588$		0			0
$589$	3.57701			0.147388
$590$		0			0
$591$		0			0
$592$		0			0
$593$			5.83920i			0.239787i	0.992787	+	0.119894i	$0.0382553\pi$
$593$			5.83920i			0.239787i	−0.992787	+	0.119894i	$0.961745\pi$
$594$		0			0
$595$	20.2200	+	21.1748i	0.828938	+	0.868082i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	43.4013			1.77333			0.886665	−	0.462413i	$-0.153016\pi$
$599$	43.4013			1.77333			0.886665	+	0.462413i	$0.153016\pi$
$600$		0			0
$601$	−35.3983			−1.44393			−0.721964	−	0.691931i	$-0.756760\pi$
$601$	−35.3983			−1.44393			−0.721964	+	0.691931i	$0.756760\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.74051	+	9.15325i	0.355352	+	0.372133i
$606$		0			0
$607$			35.9548i			1.45936i	0.683788	+	0.729681i	$0.260331\pi$
$607$			35.9548i			1.45936i	−0.683788	+	0.729681i	$0.739669\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.8416			1.00499
$612$		0			0
$613$			6.70382i			0.270765i	0.990793	+	0.135382i	$0.0432263\pi$
$613$			6.70382i			0.270765i	−0.990793	+	0.135382i	$0.956774\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.0722i			0.486010i	0.970025	+	0.243005i	$0.0781332\pi$
$617$			12.0722i			0.486010i	−0.970025	+	0.243005i	$0.921867\pi$
$618$		0			0
$619$	13.3418			0.536254			0.268127	−	0.963384i	$-0.413595\pi$
$619$	13.3418			0.536254			0.268127	+	0.963384i	$0.413595\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	46.7360i		−	1.87244i
$624$		0			0
$625$	−24.8937	−	2.30294i	−0.995748	−	0.0921174i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	21.8377			0.870727
$630$		0			0
$631$	30.3182			1.20695			0.603474	−	0.797383i	$-0.293783\pi$
$631$	30.3182			1.20695			0.603474	+	0.797383i	$0.293783\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	22.9578	−	21.9226i	0.911053	−	0.869972i
$636$		0			0
$637$		−	50.2153i		−	1.98960i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	29.3840			1.16060			0.580299	−	0.814403i	$-0.302936\pi$
$641$	29.3840			1.16060			0.580299	+	0.814403i	$0.302936\pi$
$642$		0			0
$643$			33.3568i			1.31546i	0.753252	+	0.657732i	$0.228484\pi$
$643$			33.3568i			1.31546i	−0.753252	+	0.657732i	$0.771516\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			29.5633i			1.16225i	0.813814	+	0.581126i	$0.197388\pi$
$647$			29.5633i			1.16225i	−0.813814	+	0.581126i	$0.802612\pi$
$648$		0			0
$649$	−30.3881			−1.19284
$650$		0			0
$651$		0			0
$652$		0			0
$653$			26.4562i			1.03531i	0.855589	+	0.517655i	$0.173195\pi$
$653$			26.4562i			1.03531i	−0.855589	+	0.517655i	$0.826805\pi$
$654$		0			0
$655$	−8.25098	−	8.64060i	−0.322392	−	0.337616i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	31.9043			1.24282			0.621408	−	0.783487i	$-0.286561\pi$
$659$	31.9043			1.24282			0.621408	+	0.783487i	$0.286561\pi$
$660$		0			0
$661$	11.3266			0.440553			0.220277	−	0.975437i	$-0.429304\pi$
$661$	11.3266			0.440553			0.220277	+	0.975437i	$0.429304\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.67282	+	9.23665i	−0.375096	+	0.358182i
$666$		0			0
$667$		−	4.51213i		−	0.174710i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−25.7118			−0.992594
$672$		0			0
$673$		−	15.1178i		−	0.582749i	−0.956609	−	0.291375i	$-0.905887\pi$
$673$		−	15.1178i		−	0.582749i	0.956609	−	0.291375i	$-0.0941126\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	9.77610i		−	0.375726i	−0.982195	−	0.187863i	$-0.939844\pi$
$677$		−	9.77610i		−	0.375726i	0.982195	−	0.187863i	$-0.0601561\pi$
$678$		0			0
$679$	65.8674			2.52776
$680$		0			0
$681$		0			0
$682$		0			0
$683$			2.52943i			0.0967861i	0.998828	+	0.0483931i	$0.0154100\pi$
$683$			2.52943i			0.0967861i	−0.998828	+	0.0483931i	$0.984590\pi$
$684$		0			0
$685$	−29.4071	+	28.0811i	−1.12359	+	1.07292i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11.3092			0.430846
$690$		0			0
$691$	−5.95336			−0.226476			−0.113238	−	0.993568i	$-0.536122\pi$
$691$	−5.95336			−0.226476			−0.113238	+	0.993568i	$0.536122\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.87330	+	3.00898i	0.108991	+	0.114137i
$696$		0			0
$697$		−	3.52255i		−	0.133426i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.0260			0.454215			0.227108	−	0.973870i	$-0.427073\pi$
$701$	12.0260			0.454215			0.227108	+	0.973870i	$0.427073\pi$
$702$		0			0
$703$			9.97565i			0.376239i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	41.5423i		−	1.56236i
$708$		0			0
$709$	−7.49518			−0.281488			−0.140744	−	0.990046i	$-0.544949\pi$
$709$	−7.49518			−0.281488			−0.140744	+	0.990046i	$0.544949\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2.38895i		−	0.0894670i
$714$		0			0
$715$	−20.9497	+	20.0051i	−0.783476	+	0.748147i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−37.2523			−1.38928			−0.694639	−	0.719359i	$-0.744436\pi$
$719$	−37.2523			−1.38928			−0.694639	+	0.719359i	$0.744436\pi$
$720$		0			0
$721$	−49.5518			−1.84541
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.04022	−	22.5367i	0.0386328	−	0.836990i
$726$		0			0
$727$			31.5588i			1.17045i	0.810870	+	0.585226i	$0.198994\pi$
$727$			31.5588i			1.17045i	−0.810870	+	0.585226i	$0.801006\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	41.0573			1.51856
$732$		0			0
$733$		−	3.54807i		−	0.131051i	−0.997851	−	0.0655254i	$-0.979128\pi$
$733$		−	3.54807i		−	0.131051i	0.997851	−	0.0655254i	$-0.0208723\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	22.9039i		−	0.843675i
$738$		0			0
$739$	−4.75173			−0.174795			−0.0873976	−	0.996174i	$-0.527855\pi$
$739$	−4.75173			−0.174795			−0.0873976	+	0.996174i	$0.527855\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31.7958i			1.16647i	0.812302	+	0.583237i	$0.198214\pi$
$743$			31.7958i			1.16647i	−0.812302	+	0.583237i	$0.801786\pi$
$744$		0			0
$745$	−9.78824	−	10.2505i	−0.358613	−	0.375548i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.06186			−0.221495
$750$		0			0
$751$	−15.7363			−0.574225			−0.287113	−	0.957897i	$-0.592695\pi$
$751$	−15.7363			−0.574225			−0.287113	+	0.957897i	$0.592695\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.5018	−	12.0449i	−0.418592	−	0.438359i
$756$		0			0
$757$			26.2259i			0.953198i	0.879121	+	0.476599i	$0.158131\pi$
$757$			26.2259i			0.953198i	−0.879121	+	0.476599i	$0.841869\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−37.1436			−1.34646			−0.673228	−	0.739435i	$-0.735093\pi$
$761$	−37.1436			−1.34646			−0.673228	+	0.739435i	$0.735093\pi$
$762$		0			0
$763$		−	19.4252i		−	0.703238i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			73.7197i			2.66187i
$768$		0			0
$769$	21.4226			0.772519			0.386259	−	0.922390i	$-0.373767\pi$
$769$	21.4226			0.772519			0.386259	+	0.922390i	$0.373767\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	52.0313i		−	1.87143i	−0.352751	−	0.935717i	$-0.614754\pi$
$773$		−	52.0313i		−	1.87143i	0.352751	−	0.935717i	$-0.385246\pi$
$774$		0			0
$775$	0.550746	−	11.9321i	0.0197834	−	0.428612i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.60913			0.0576531
$780$		0			0
$781$	11.4239			0.408778
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.25000	+	7.87799i	−0.294455	+	0.281177i
$786$		0			0
$787$			29.3028i			1.04453i	0.852782	+	0.522266i	$0.174913\pi$
$787$			29.3028i			1.04453i	−0.852782	+	0.522266i	$0.825087\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.89347			0.245103
$792$		0			0
$793$			62.3754i			2.21501i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			25.3855i			0.899200i	0.893230	+	0.449600i	$0.148434\pi$
$797$			25.3855i			0.899200i	−0.893230	+	0.449600i	$0.851566\pi$
$798$		0			0
$799$	14.5248			0.513849
$800$		0			0
$801$		0			0
$802$		0			0
$803$			10.1293i			0.357457i
$804$		0			0
$805$	6.16882	+	6.46012i	0.217422	+	0.227689i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	43.0003			1.51181			0.755905	−	0.654681i	$-0.227197\pi$
$809$	43.0003			1.51181			0.755905	+	0.654681i	$0.227197\pi$
$810$		0			0
$811$	−16.0750			−0.564470			−0.282235	−	0.959345i	$-0.591076\pi$
$811$	−16.0750			−0.564470			−0.282235	+	0.959345i	$0.591076\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.13696	+	6.81514i	−0.249997	+	0.238724i
$816$		0			0
$817$			18.7553i			0.656165i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.9668			0.696846			0.348423	−	0.937337i	$-0.386717\pi$
$821$	19.9668			0.696846			0.348423	+	0.937337i	$0.386717\pi$
$822$		0			0
$823$		−	22.9199i		−	0.798939i	−0.916747	−	0.399469i	$-0.869194\pi$
$823$		−	22.9199i		−	0.798939i	0.916747	−	0.399469i	$-0.130806\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			20.2103i			0.702779i	0.936229	+	0.351390i	$0.114291\pi$
$827$			20.2103i			0.702779i	−0.936229	+	0.351390i	$0.885709\pi$
$828$		0			0
$829$	−38.7839			−1.34702			−0.673510	−	0.739178i	$-0.735214\pi$
$829$	−38.7839			−1.34702			−0.673510	+	0.739178i	$0.735214\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	29.3606i		−	1.01728i
$834$		0			0
$835$	−16.7134	+	15.9598i	−0.578391	+	0.552310i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−46.6947			−1.61208			−0.806040	−	0.591861i	$-0.798393\pi$
$839$	−46.6947			−1.61208			−0.806040	+	0.591861i	$0.798393\pi$
$840$		0			0
$841$	−8.64068			−0.297954
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.4557	+	29.7995i	0.978907	+	1.02513i
$846$		0			0
$847$		−	22.6099i		−	0.776887i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.66237			0.228383
$852$		0			0
$853$			27.3238i			0.935549i	0.883848	+	0.467775i	$0.154944\pi$
$853$			27.3238i			0.935549i	−0.883848	+	0.467775i	$0.845056\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	24.8111i		−	0.847530i	−0.905772	−	0.423765i	$-0.860708\pi$
$857$		−	24.8111i		−	0.847530i	0.905772	−	0.423765i	$-0.139292\pi$
$858$		0			0
$859$	−34.1139			−1.16395			−0.581975	−	0.813207i	$-0.697720\pi$
$859$	−34.1139			−1.16395			−0.581975	+	0.813207i	$0.697720\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			25.8989i			0.881609i	0.897603	+	0.440805i	$0.145307\pi$
$863$			25.8989i			0.881609i	−0.897603	+	0.440805i	$0.854693\pi$
$864$		0			0
$865$	−15.4709	+	14.7733i	−0.526026	+	0.502307i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21.6175			0.733323
$870$		0			0
$871$	−55.5634			−1.88269
$872$		0			0
$873$		0			0
$874$		0			0
$875$	29.3220	+	33.6884i	0.991264	+	1.13888i
$876$		0			0
$877$		−	1.59319i		−	0.0537980i	−0.999638	−	0.0268990i	$-0.991437\pi$
$877$		−	1.59319i		−	0.0537980i	0.999638	−	0.0268990i	$-0.00856326\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−45.7424			−1.54110			−0.770551	−	0.637379i	$-0.780019\pi$
$881$	−45.7424			−1.54110			−0.770551	+	0.637379i	$0.780019\pi$
$882$		0			0
$883$		−	38.8682i		−	1.30802i	−0.756487	−	0.654009i	$-0.773086\pi$
$883$		−	38.8682i		−	1.30802i	0.756487	−	0.654009i	$-0.226914\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			56.7934i			1.90694i	0.301495	+	0.953468i	$0.402514\pi$
$887$			56.7934i			1.90694i	−0.301495	+	0.953468i	$0.597486\pi$
$888$		0			0
$889$	−56.7093			−1.90197
$890$		0			0
$891$		0			0
$892$		0			0
$893$			6.63503i			0.222033i
$894$		0			0
$895$	29.2880	+	30.6710i	0.978989	+	1.02522i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10.7793			0.359509
$900$		0			0
$901$	6.61241			0.220291
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−37.8287	−	39.6151i	−1.25747	−	1.31685i
$906$		0			0
$907$		−	38.7967i		−	1.28822i	−0.764931	−	0.644112i	$-0.777227\pi$
$907$		−	38.7967i		−	1.28822i	0.764931	−	0.644112i	$-0.222773\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−31.2655			−1.03587			−0.517936	−	0.855419i	$-0.673300\pi$
$911$	−31.2655			−1.03587			−0.517936	+	0.855419i	$0.673300\pi$
$912$		0			0
$913$			9.14305i			0.302591i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			21.3436i			0.704828i
$918$		0			0
$919$	29.1623			0.961976			0.480988	−	0.876727i	$-0.340278\pi$
$919$	29.1623			0.961976			0.480988	+	0.876727i	$0.340278\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	27.7136i		−	0.912205i
$924$		0			0
$925$	33.2764	+	1.53593i	1.09412	+	0.0505012i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−32.4054			−1.06319			−0.531593	−	0.847000i	$-0.678406\pi$
$929$	−32.4054			−1.06319			−0.531593	+	0.847000i	$0.678406\pi$
$930$		0			0
$931$	13.4122			0.439566
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.2492	+	11.6968i	−0.400591	+	0.382528i
$936$		0			0
$937$		−	35.1464i		−	1.14818i	−0.818792	−	0.574091i	$-0.805356\pi$
$937$		−	35.1464i		−	1.14818i	0.818792	−	0.574091i	$-0.194644\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.1574			−0.331121			−0.165561	−	0.986200i	$-0.552943\pi$
$941$	−10.1574			−0.331121			−0.165561	+	0.986200i	$0.552943\pi$
$942$		0			0
$943$		−	1.07468i		−	0.0349964i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	14.8674i		−	0.483126i	−0.970385	−	0.241563i	$-0.922340\pi$
$947$		−	14.8674i		−	0.483126i	0.970385	−	0.241563i	$-0.0776601\pi$
$948$		0			0
$949$	24.5732			0.797679
$950$		0			0
$951$		0			0
$952$		0			0
$953$			3.40621i			0.110338i	0.998477	+	0.0551690i	$0.0175698\pi$
$953$			3.40621i			0.110338i	−0.998477	+	0.0551690i	$0.982430\pi$
$954$		0			0
$955$	23.6893	+	24.8080i	0.766569	+	0.802767i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	72.6401			2.34567
$960$		0			0
$961$	−25.2929			−0.815900
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.5555	+	17.7188i	−0.597322	+	0.570388i
$966$		0			0
$967$			26.5021i			0.852251i	0.904664	+	0.426126i	$0.140122\pi$
$967$			26.5021i			0.852251i	−0.904664	+	0.426126i	$0.859878\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15.8259			0.507878			0.253939	−	0.967220i	$-0.418274\pi$
$971$	15.8259			0.507878			0.253939	+	0.967220i	$0.418274\pi$
$972$		0			0
$973$		−	7.43265i		−	0.238280i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	18.0343i		−	0.576968i	−0.957485	−	0.288484i	$-0.906849\pi$
$977$		−	18.0343i		−	0.576968i	0.957485	−	0.288484i	$-0.0931512\pi$
$978$		0			0
$979$	27.0358			0.864069
$980$		0			0
$981$		0			0
$982$		0			0
$983$			42.4751i			1.35475i	0.735640	+	0.677373i	$0.236882\pi$
$983$			42.4751i			1.35475i	−0.735640	+	0.677373i	$0.763118\pi$
$984$		0			0
$985$	30.4887	−	29.1139i	0.971451	−	0.927646i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.5260			0.398303
$990$		0			0
$991$	−32.8613			−1.04387			−0.521936	−	0.852985i	$-0.674790\pi$
$991$	−32.8613			−1.04387			−0.521936	+	0.852985i	$0.674790\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	31.3719	+	32.8533i	0.994556	+	1.04152i
$996$		0			0
$997$			17.7690i			0.562750i	0.959598	+	0.281375i	$0.0907904\pi$
$997$			17.7690i			0.562750i	−0.959598	+	0.281375i	$0.909210\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.c.829.3		14
3.2	odd	2		1380.2.f.b.829.13	yes	14
5.4	even	2	inner	4140.2.f.c.829.4		14
15.2	even	4		6900.2.a.bd.1.1		7
15.8	even	4		6900.2.a.bc.1.7		7
15.14	odd	2		1380.2.f.b.829.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.6	✓	14	15.14	odd	2
1380.2.f.b.829.13	yes	14	3.2	odd	2
4140.2.f.c.829.3		14	1.1	even	1	trivial
4140.2.f.c.829.4		14	5.4	even	2	inner
6900.2.a.bc.1.7		7	15.8	even	4
6900.2.a.bd.1.1		7	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.54426 - 1.61718i) q^{5} +3.99468i q^{7} +O(q^{10})\)	\(q+(-1.54426 - 1.61718i) q^{5} +3.99468i q^{7} -2.31084 q^{11} +5.60596i q^{13} +3.27777i q^{17} -1.49731 q^{19} +1.00000i q^{23} +(-0.230539 + 4.99468i) q^{25} -4.51213 q^{29} -2.38895 q^{31} +(6.46012 - 6.16882i) q^{35} -6.66237i q^{37} -1.07468 q^{41} -12.5260i q^{43} -4.43129i q^{47} -8.95749 q^{49} -2.01735i q^{53} +(3.56853 + 3.73705i) q^{55} +13.1502 q^{59} +11.1266 q^{61} +(9.06585 - 8.65705i) q^{65} +9.91149i q^{67} -4.94360 q^{71} -4.38340i q^{73} -9.23108i q^{77} -9.35481 q^{79} -3.95659i q^{83} +(5.30074 - 5.06172i) q^{85} -11.6996 q^{89} -22.3940 q^{91} +(2.31224 + 2.42142i) q^{95} -16.4888i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 14 * q	\( 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) \) 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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