LMFDB - Embedded newform 2646.2.d.e.2645.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(2645,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2646.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.d (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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.16
Root			$0.991445 - 0.130526i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.e.2645.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +4.29725 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +4.29725 q^{5} -1.00000i q^{8} +4.29725i q^{10} +5.20041i q^{11} -2.81444i q^{13} +1.00000 q^{16} -4.02815 q^{17} +1.77998i q^{19} -4.29725 q^{20} -5.20041 q^{22} +7.70319i q^{23} +13.4663 q^{25} +2.81444 q^{26} +5.02884i q^{29} +1.25139i q^{31} +1.00000i q^{32} -4.02815i q^{34} -9.17738 q^{37} -1.77998 q^{38} -4.29725i q^{40} -5.29403 q^{41} -4.53572 q^{43} -5.20041i q^{44} -7.70319 q^{46} +7.59771 q^{47} +13.4663i q^{50} +2.81444i q^{52} -0.167470i q^{53} +22.3475i q^{55} -5.02884 q^{58} +5.99698 q^{59} +7.79367i q^{61} -1.25139 q^{62} -1.00000 q^{64} -12.0944i q^{65} +11.3778 q^{67} +4.02815 q^{68} -0.539455i q^{71} +3.80634i q^{73} -9.17738i q^{74} -1.77998i q^{76} +1.99590 q^{79} +4.29725 q^{80} -5.29403i q^{82} +16.0388 q^{83} -17.3100 q^{85} -4.53572i q^{86} +5.20041 q^{88} -1.29396 q^{89} -7.70319i q^{92} +7.59771i q^{94} +7.64901i q^{95} -17.2596i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	4.29725	1.92179	0.960894	−	0.276916i	$-0.0893125\pi$
$5$	4.29725	1.92179	0.960894	+	0.276916i	$0.0893125\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	4.29725i	1.35891i
$11$	5.20041i	1.56798i	0.620771	+	0.783992i	$0.286820\pi$
$11$	5.20041i	1.56798i	−0.620771	+	0.783992i	$0.713180\pi$
$12$	0	0
$13$	− 2.81444i	− 0.780586i	−0.920691	−	0.390293i	$-0.872374\pi$
$13$	− 2.81444i	− 0.780586i	0.920691	−	0.390293i	$-0.127626\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.02815	−0.976970	−0.488485	−	0.872572i	$-0.662450\pi$
$17$	−4.02815	−0.976970	−0.488485	+	0.872572i	$0.662450\pi$
$18$	0	0
$19$	1.77998i	0.408355i	0.978934	+	0.204178i	$0.0654520\pi$
$19$	1.77998i	0.408355i	−0.978934	+	0.204178i	$0.934548\pi$
$20$	−4.29725	−0.960894
$21$	0	0
$22$	−5.20041	−1.10873
$23$	7.70319i	1.60623i	0.595827	+	0.803113i	$0.296824\pi$
$23$	7.70319i	1.60623i	−0.595827	+	0.803113i	$0.703176\pi$
$24$	0	0
$25$	13.4663	2.69327
$26$	2.81444	0.551958
$27$	0	0
$28$	0	0
$29$	5.02884i	0.933832i	0.884302	+	0.466916i	$0.154635\pi$
$29$	5.02884i	0.933832i	−0.884302	+	0.466916i	$0.845365\pi$
$30$	0	0
$31$	1.25139i	0.224756i	0.993666	+	0.112378i	$0.0358468\pi$
$31$	1.25139i	0.224756i	−0.993666	+	0.112378i	$0.964153\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 4.02815i	− 0.690822i
$35$	0	0
$36$	0	0
$37$	−9.17738	−1.50875	−0.754376	−	0.656443i	$-0.772060\pi$
$37$	−9.17738	−1.50875	−0.754376	+	0.656443i	$0.772060\pi$
$38$	−1.77998	−0.288751
$39$	0	0
$40$	− 4.29725i	− 0.679455i
$41$	−5.29403	−0.826789	−0.413394	−	0.910552i	$-0.635657\pi$
$41$	−5.29403	−0.826789	−0.413394	+	0.910552i	$0.635657\pi$
$42$	0	0
$43$	−4.53572	−0.691690	−0.345845	−	0.938292i	$-0.612408\pi$
$43$	−4.53572	−0.691690	−0.345845	+	0.938292i	$0.612408\pi$
$44$	− 5.20041i	− 0.783992i
$45$	0	0
$46$	−7.70319	−1.13577
$47$	7.59771	1.10824	0.554120	−	0.832437i	$-0.313055\pi$
$47$	7.59771	1.10824	0.554120	+	0.832437i	$0.313055\pi$
$48$	0	0
$49$	0	0
$50$	13.4663i	1.90443i
$51$	0	0
$52$	2.81444i	0.390293i
$53$	− 0.167470i	− 0.0230037i	−0.999934	−	0.0115019i	$-0.996339\pi$
$53$	− 0.167470i	− 0.0230037i	0.999934	−	0.0115019i	$-0.00366124\pi$
$54$	0	0
$55$	22.3475i	3.01333i
$56$	0	0
$57$	0	0
$58$	−5.02884	−0.660319
$59$	5.99698	0.780740	0.390370	−	0.920658i	$-0.372347\pi$
$59$	5.99698	0.780740	0.390370	+	0.920658i	$0.372347\pi$
$60$	0	0
$61$	7.79367i	0.997877i	0.866637	+	0.498939i	$0.166277\pi$
$61$	7.79367i	0.997877i	−0.866637	+	0.498939i	$0.833723\pi$
$62$	−1.25139	−0.158927
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 12.0944i	− 1.50012i
$66$	0	0
$67$	11.3778	1.39002	0.695009	−	0.719001i	$-0.255400\pi$
$67$	11.3778	1.39002	0.695009	+	0.719001i	$0.255400\pi$
$68$	4.02815	0.488485
$69$	0	0
$70$	0	0
$71$	− 0.539455i	− 0.0640216i	−0.999488	−	0.0320108i	$-0.989809\pi$
$71$	− 0.539455i	− 0.0640216i	0.999488	−	0.0320108i	$-0.0101911\pi$
$72$	0	0
$73$	3.80634i	0.445498i	0.974876	+	0.222749i	$0.0715031\pi$
$73$	3.80634i	0.445498i	−0.974876	+	0.222749i	$0.928497\pi$
$74$	− 9.17738i	− 1.06685i
$75$	0	0
$76$	− 1.77998i	− 0.204178i
$77$	0	0
$78$	0	0
$79$	1.99590	0.224556	0.112278	−	0.993677i	$-0.464185\pi$
$79$	1.99590	0.224556	0.112278	+	0.993677i	$0.464185\pi$
$80$	4.29725	0.480447
$81$	0	0
$82$	− 5.29403i	− 0.584628i
$83$	16.0388	1.76049	0.880245	−	0.474520i	$-0.157378\pi$
$83$	16.0388	1.76049	0.880245	+	0.474520i	$0.157378\pi$
$84$	0	0
$85$	−17.3100	−1.87753
$86$	− 4.53572i	− 0.489099i
$87$	0	0
$88$	5.20041	0.554366
$89$	−1.29396	−0.137159	−0.0685795	−	0.997646i	$-0.521847\pi$
$89$	−1.29396	−0.137159	−0.0685795	+	0.997646i	$0.521847\pi$
$90$	0	0
$91$	0	0
$92$	− 7.70319i	− 0.803113i
$93$	0	0
$94$	7.59771i	0.783644i
$95$	7.64901i	0.784772i
$96$	0	0
$97$	− 17.2596i	− 1.75245i	−0.481902	−	0.876225i	$-0.660054\pi$
$97$	− 17.2596i	− 1.75245i	0.481902	−	0.876225i	$-0.339946\pi$
$98$	0	0
$99$	0	0
$100$	−13.4663	−1.34663
$101$	14.1659	1.40955	0.704777	−	0.709429i	$-0.251047\pi$
$101$	14.1659	1.40955	0.704777	+	0.709429i	$0.251047\pi$
$102$	0	0
$103$	− 0.817539i	− 0.0805545i	−0.999189	−	0.0402773i	$-0.987176\pi$
$103$	− 0.817539i	− 0.0805545i	0.999189	−	0.0402773i	$-0.0128241\pi$
$104$	−2.81444	−0.275979
$105$	0	0
$106$	0.167470	0.0162661
$107$	− 12.5586i	− 1.21409i	−0.794667	−	0.607045i	$-0.792355\pi$
$107$	− 12.5586i	− 1.21409i	0.794667	−	0.607045i	$-0.207645\pi$
$108$	0	0
$109$	−0.395023	−0.0378363	−0.0189181	−	0.999821i	$-0.506022\pi$
$109$	−0.395023	−0.0378363	−0.0189181	+	0.999821i	$0.506022\pi$
$110$	−22.3475	−2.13075
$111$	0	0
$112$	0	0
$113$	− 16.1520i	− 1.51946i	−0.650241	−	0.759728i	$-0.725332\pi$
$113$	− 16.1520i	− 1.51946i	0.650241	−	0.759728i	$-0.274668\pi$
$114$	0	0
$115$	33.1025i	3.08682i
$116$	− 5.02884i	− 0.466916i
$117$	0	0
$118$	5.99698i	0.552067i
$119$	0	0
$120$	0	0
$121$	−16.0443	−1.45857
$122$	−7.79367	−0.705606
$123$	0	0
$124$	− 1.25139i	− 0.112378i
$125$	36.3820	3.25411
$126$	0	0
$127$	9.04803	0.802883	0.401441	−	0.915885i	$-0.368509\pi$
$127$	9.04803	0.802883	0.401441	+	0.915885i	$0.368509\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	12.0944	1.06075
$131$	−1.75412	−0.153259	−0.0766293	−	0.997060i	$-0.524416\pi$
$131$	−1.75412	−0.153259	−0.0766293	+	0.997060i	$0.524416\pi$
$132$	0	0
$133$	0	0
$134$	11.3778i	0.982891i
$135$	0	0
$136$	4.02815i	0.345411i
$137$	16.4433i	1.40485i	0.711759	+	0.702423i	$0.247899\pi$
$137$	16.4433i	1.40485i	−0.711759	+	0.702423i	$0.752101\pi$
$138$	0	0
$139$	17.6373i	1.49598i	0.663710	+	0.747990i	$0.268981\pi$
$139$	17.6373i	1.49598i	−0.663710	+	0.747990i	$0.731019\pi$
$140$	0	0
$141$	0	0
$142$	0.539455	0.0452701
$143$	14.6363	1.22395
$144$	0	0
$145$	21.6102i	1.79463i
$146$	−3.80634	−0.315015
$147$	0	0
$148$	9.17738	0.754376
$149$	12.2115i	1.00041i	0.865908	+	0.500203i	$0.166741\pi$
$149$	12.2115i	1.00041i	−0.865908	+	0.500203i	$0.833259\pi$
$150$	0	0
$151$	−8.44860	−0.687537	−0.343769	−	0.939054i	$-0.611704\pi$
$151$	−8.44860	−0.687537	−0.343769	+	0.939054i	$0.611704\pi$
$152$	1.77998	0.144375
$153$	0	0
$154$	0	0
$155$	5.37753i	0.431934i
$156$	0	0
$157$	1.97105i	0.157307i	0.996902	+	0.0786534i	$0.0250620\pi$
$157$	1.97105i	0.157307i	−0.996902	+	0.0786534i	$0.974938\pi$
$158$	1.99590i	0.158785i
$159$	0	0
$160$	4.29725i	0.339727i
$161$	0	0
$162$	0	0
$163$	−7.67231	−0.600941	−0.300471	−	0.953791i	$-0.597144\pi$
$163$	−7.67231	−0.600941	−0.300471	+	0.953791i	$0.597144\pi$
$164$	5.29403	0.413394
$165$	0	0
$166$	16.0388i	1.24485i
$167$	−6.19782	−0.479602	−0.239801	−	0.970822i	$-0.577082\pi$
$167$	−6.19782	−0.479602	−0.239801	+	0.970822i	$0.577082\pi$
$168$	0	0
$169$	5.07891	0.390685
$170$	− 17.3100i	− 1.32761i
$171$	0	0
$172$	4.53572	0.345845
$173$	−1.17825	−0.0895805	−0.0447903	−	0.998996i	$-0.514262\pi$
$173$	−1.17825	−0.0895805	−0.0447903	+	0.998996i	$0.514262\pi$
$174$	0	0
$175$	0	0
$176$	5.20041i	0.391996i
$177$	0	0
$178$	− 1.29396i	− 0.0969861i
$179$	− 15.3425i	− 1.14676i	−0.819291	−	0.573378i	$-0.805633\pi$
$179$	− 15.3425i	− 1.14676i	0.819291	−	0.573378i	$-0.194367\pi$
$180$	0	0
$181$	− 5.24186i	− 0.389624i	−0.980841	−	0.194812i	$-0.937590\pi$
$181$	− 5.24186i	− 0.389624i	0.980841	−	0.194812i	$-0.0624097\pi$
$182$	0	0
$183$	0	0
$184$	7.70319	0.567886
$185$	−39.4375	−2.89950
$186$	0	0
$187$	− 20.9480i	− 1.53187i
$188$	−7.59771	−0.554120
$189$	0	0
$190$	−7.64901	−0.554918
$191$	4.63237i	0.335187i	0.985856	+	0.167593i	$0.0535996\pi$
$191$	4.63237i	0.335187i	−0.985856	+	0.167593i	$0.946400\pi$
$192$	0	0
$193$	−7.34315	−0.528571	−0.264286	−	0.964444i	$-0.585136\pi$
$193$	−7.34315	−0.528571	−0.264286	+	0.964444i	$0.585136\pi$
$194$	17.2596	1.23917
$195$	0	0
$196$	0	0
$197$	− 23.5456i	− 1.67756i	−0.544474	−	0.838778i	$-0.683271\pi$
$197$	− 23.5456i	− 1.67756i	0.544474	−	0.838778i	$-0.316729\pi$
$198$	0	0
$199$	7.28291i	0.516272i	0.966109	+	0.258136i	$0.0831082\pi$
$199$	7.28291i	0.516272i	−0.966109	+	0.258136i	$0.916892\pi$
$200$	− 13.4663i	− 0.952215i
$201$	0	0
$202$	14.1659i	0.996706i
$203$	0	0
$204$	0	0
$205$	−22.7498	−1.58891
$206$	0.817539	0.0569607
$207$	0	0
$208$	− 2.81444i	− 0.195147i
$209$	−9.25662	−0.640294
$210$	0	0
$211$	−12.7862	−0.880238	−0.440119	−	0.897939i	$-0.645064\pi$
$211$	−12.7862	−0.880238	−0.440119	+	0.897939i	$0.645064\pi$
$212$	0.167470i	0.0115019i
$213$	0	0
$214$	12.5586	0.858491
$215$	−19.4911	−1.32928
$216$	0	0
$217$	0	0
$218$	− 0.395023i	− 0.0267543i
$219$	0	0
$220$	− 22.3475i	− 1.50667i
$221$	11.3370i	0.762609i
$222$	0	0
$223$	− 3.38874i	− 0.226927i	−0.993542	−	0.113463i	$-0.963806\pi$
$223$	− 3.38874i	− 0.226927i	0.993542	−	0.113463i	$-0.0361945\pi$
$224$	0	0
$225$	0	0
$226$	16.1520	1.07442
$227$	−1.21149	−0.0804093	−0.0402046	−	0.999191i	$-0.512801\pi$
$227$	−1.21149	−0.0804093	−0.0402046	+	0.999191i	$0.512801\pi$
$228$	0	0
$229$	− 19.6590i	− 1.29910i	−0.760318	−	0.649551i	$-0.774957\pi$
$229$	− 19.6590i	− 1.29910i	0.760318	−	0.649551i	$-0.225043\pi$
$230$	−33.1025	−2.18271
$231$	0	0
$232$	5.02884	0.330160
$233$	15.2758i	1.00075i	0.865808	+	0.500377i	$0.166805\pi$
$233$	15.2758i	1.00075i	−0.865808	+	0.500377i	$0.833195\pi$
$234$	0	0
$235$	32.6493	2.12980
$236$	−5.99698	−0.390370
$237$	0	0
$238$	0	0
$239$	− 0.539712i	− 0.0349111i	−0.999848	−	0.0174555i	$-0.994443\pi$
$239$	− 0.539712i	− 0.0349111i	0.999848	−	0.0174555i	$-0.00555655\pi$
$240$	0	0
$241$	9.37436i	0.603855i	0.953331	+	0.301928i	$0.0976301\pi$
$241$	9.37436i	0.603855i	−0.953331	+	0.301928i	$0.902370\pi$
$242$	− 16.0443i	− 1.03137i
$243$	0	0
$244$	− 7.79367i	− 0.498939i
$245$	0	0
$246$	0	0
$247$	5.00965	0.318756
$248$	1.25139	0.0794633
$249$	0	0
$250$	36.3820i	2.30100i
$251$	−12.2724	−0.774626	−0.387313	−	0.921948i	$-0.626597\pi$
$251$	−12.2724	−0.774626	−0.387313	+	0.921948i	$0.626597\pi$
$252$	0	0
$253$	−40.0597	−2.51853
$254$	9.04803i	0.567724i
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.4783	0.840754	0.420377	−	0.907350i	$-0.361898\pi$
$257$	13.4783	0.840754	0.420377	+	0.907350i	$0.361898\pi$
$258$	0	0
$259$	0	0
$260$	12.0944i	0.750060i
$261$	0	0
$262$	− 1.75412i	− 0.108370i
$263$	− 12.5893i	− 0.776289i	−0.921599	−	0.388144i	$-0.873116\pi$
$263$	− 12.5893i	− 0.776289i	0.921599	−	0.388144i	$-0.126884\pi$
$264$	0	0
$265$	− 0.719659i	− 0.0442083i
$266$	0	0
$267$	0	0
$268$	−11.3778	−0.695009
$269$	6.85395	0.417892	0.208946	−	0.977927i	$-0.432997\pi$
$269$	6.85395	0.417892	0.208946	+	0.977927i	$0.432997\pi$
$270$	0	0
$271$	− 29.8899i	− 1.81568i	−0.419317	−	0.907840i	$-0.637730\pi$
$271$	− 29.8899i	− 1.81568i	0.419317	−	0.907840i	$-0.362270\pi$
$272$	−4.02815	−0.244243
$273$	0	0
$274$	−16.4433	−0.993377
$275$	70.0306i	4.22300i
$276$	0	0
$277$	−30.7992	−1.85055	−0.925274	−	0.379300i	$-0.876165\pi$
$277$	−30.7992	−1.85055	−0.925274	+	0.379300i	$0.876165\pi$
$278$	−17.6373	−1.05782
$279$	0	0
$280$	0	0
$281$	− 6.44161i	− 0.384274i	−0.981368	−	0.192137i	$-0.938458\pi$
$281$	− 6.44161i	− 0.384274i	0.981368	−	0.192137i	$-0.0615418\pi$
$282$	0	0
$283$	13.5392i	0.804821i	0.915459	+	0.402410i	$0.131828\pi$
$283$	13.5392i	0.804821i	−0.915459	+	0.402410i	$0.868172\pi$
$284$	0.539455i	0.0320108i
$285$	0	0
$286$	14.6363i	0.865460i
$287$	0	0
$288$	0	0
$289$	−0.773996	−0.0455292
$290$	−21.6102	−1.26899
$291$	0	0
$292$	− 3.80634i	− 0.222749i
$293$	−26.3170	−1.53745	−0.768727	−	0.639577i	$-0.779109\pi$
$293$	−26.3170	−1.53745	−0.768727	+	0.639577i	$0.779109\pi$
$294$	0	0
$295$	25.7705	1.50042
$296$	9.17738i	0.533424i
$297$	0	0
$298$	−12.2115	−0.707393
$299$	21.6802	1.25380
$300$	0	0
$301$	0	0
$302$	− 8.44860i	− 0.486162i
$303$	0	0
$304$	1.77998i	0.102089i
$305$	33.4913i	1.91771i
$306$	0	0
$307$	− 19.4207i	− 1.10840i	−0.832385	−	0.554198i	$-0.813025\pi$
$307$	− 19.4207i	− 1.10840i	0.832385	−	0.554198i	$-0.186975\pi$
$308$	0	0
$309$	0	0
$310$	−5.37753	−0.305423
$311$	−3.58894	−0.203510	−0.101755	−	0.994809i	$-0.532446\pi$
$311$	−3.58894	−0.203510	−0.101755	+	0.994809i	$0.532446\pi$
$312$	0	0
$313$	12.9198i	0.730272i	0.930954	+	0.365136i	$0.118977\pi$
$313$	12.9198i	0.730272i	−0.930954	+	0.365136i	$0.881023\pi$
$314$	−1.97105	−0.111233
$315$	0	0
$316$	−1.99590	−0.112278
$317$	− 12.0384i	− 0.676143i	−0.941120	−	0.338071i	$-0.890225\pi$
$317$	− 12.0384i	− 0.676143i	0.941120	−	0.338071i	$-0.109775\pi$
$318$	0	0
$319$	−26.1520	−1.46423
$320$	−4.29725	−0.240224
$321$	0	0
$322$	0	0
$323$	− 7.17002i	− 0.398951i
$324$	0	0
$325$	− 37.9003i	− 2.10233i
$326$	− 7.67231i	− 0.424930i
$327$	0	0
$328$	5.29403i	0.292314i
$329$	0	0
$330$	0	0
$331$	25.8226	1.41934	0.709671	−	0.704534i	$-0.248844\pi$
$331$	25.8226	1.41934	0.709671	+	0.704534i	$0.248844\pi$
$332$	−16.0388	−0.880245
$333$	0	0
$334$	− 6.19782i	− 0.339130i
$335$	48.8932	2.67132
$336$	0	0
$337$	34.5378	1.88139	0.940696	−	0.339250i	$-0.110173\pi$
$337$	34.5378	1.88139	0.940696	+	0.339250i	$0.110173\pi$
$338$	5.07891i	0.276256i
$339$	0	0
$340$	17.3100	0.938765
$341$	−6.50774	−0.352414
$342$	0	0
$343$	0	0
$344$	4.53572i	0.244549i
$345$	0	0
$346$	− 1.17825i	− 0.0633430i
$347$	11.4390i	0.614075i	0.951697	+	0.307038i	$0.0993377\pi$
$347$	11.4390i	0.614075i	−0.951697	+	0.307038i	$0.900662\pi$
$348$	0	0
$349$	9.93436i	0.531774i	0.964004	+	0.265887i	$0.0856648\pi$
$349$	9.93436i	0.531774i	−0.964004	+	0.265887i	$0.914335\pi$
$350$	0	0
$351$	0	0
$352$	−5.20041	−0.277183
$353$	−2.80734	−0.149419	−0.0747097	−	0.997205i	$-0.523803\pi$
$353$	−2.80734	−0.149419	−0.0747097	+	0.997205i	$0.523803\pi$
$354$	0	0
$355$	− 2.31817i	− 0.123036i
$356$	1.29396	0.0685795
$357$	0	0
$358$	15.3425	0.810879
$359$	− 3.70282i	− 0.195427i	−0.995215	−	0.0977137i	$-0.968847\pi$
$359$	− 3.70282i	− 0.195427i	0.995215	−	0.0977137i	$-0.0311530\pi$
$360$	0	0
$361$	15.8317	0.833246
$362$	5.24186	0.275506
$363$	0	0
$364$	0	0
$365$	16.3568i	0.856154i
$366$	0	0
$367$	− 13.3176i	− 0.695171i	−0.937648	−	0.347586i	$-0.887002\pi$
$367$	− 13.3176i	− 0.695171i	0.937648	−	0.347586i	$-0.112998\pi$
$368$	7.70319i	0.401556i
$369$	0	0
$370$	− 39.4375i	− 2.05026i
$371$	0	0
$372$	0	0
$373$	−2.09640	−0.108548	−0.0542738	−	0.998526i	$-0.517284\pi$
$373$	−2.09640	−0.108548	−0.0542738	+	0.998526i	$0.517284\pi$
$374$	20.9480	1.08320
$375$	0	0
$376$	− 7.59771i	− 0.391822i
$377$	14.1534	0.728936
$378$	0	0
$379$	−10.5916	−0.544053	−0.272027	−	0.962290i	$-0.587694\pi$
$379$	−10.5916	−0.544053	−0.272027	+	0.962290i	$0.587694\pi$
$380$	− 7.64901i	− 0.392386i
$381$	0	0
$382$	−4.63237	−0.237013
$383$	29.3742	1.50095	0.750476	−	0.660898i	$-0.229825\pi$
$383$	29.3742	1.50095	0.750476	+	0.660898i	$0.229825\pi$
$384$	0	0
$385$	0	0
$386$	− 7.34315i	− 0.373756i
$387$	0	0
$388$	17.2596i	0.876225i
$389$	0.139250i	0.00706028i	0.999994	+	0.00353014i	$0.00112368\pi$
$389$	0.139250i	0.00706028i	−0.999994	+	0.00353014i	$0.998876\pi$
$390$	0	0
$391$	− 31.0296i	− 1.56923i
$392$	0	0
$393$	0	0
$394$	23.5456	1.18621
$395$	8.57687	0.431549
$396$	0	0
$397$	5.13731i	0.257834i	0.991655	+	0.128917i	$0.0411501\pi$
$397$	5.13731i	0.257834i	−0.991655	+	0.128917i	$0.958850\pi$
$398$	−7.28291	−0.365059
$399$	0	0
$400$	13.4663	0.673317
$401$	10.0437i	0.501557i	0.968044	+	0.250778i	$0.0806866\pi$
$401$	10.0437i	0.501557i	−0.968044	+	0.250778i	$0.919313\pi$
$402$	0	0
$403$	3.52196	0.175442
$404$	−14.1659	−0.704777
$405$	0	0
$406$	0	0
$407$	− 47.7261i	− 2.36570i
$408$	0	0
$409$	9.66617i	0.477961i	0.971024	+	0.238981i	$0.0768132\pi$
$409$	9.66617i	0.477961i	−0.971024	+	0.238981i	$0.923187\pi$
$410$	− 22.7498i	− 1.12353i
$411$	0	0
$412$	0.817539i	0.0402773i
$413$	0	0
$414$	0	0
$415$	68.9228	3.38329
$416$	2.81444	0.137989
$417$	0	0
$418$	− 9.25662i	− 0.452756i
$419$	17.8462	0.871846	0.435923	−	0.899984i	$-0.356422\pi$
$419$	17.8462	0.871846	0.435923	+	0.899984i	$0.356422\pi$
$420$	0	0
$421$	5.87102	0.286136	0.143068	−	0.989713i	$-0.454303\pi$
$421$	5.87102	0.286136	0.143068	+	0.989713i	$0.454303\pi$
$422$	− 12.7862i	− 0.622422i
$423$	0	0
$424$	−0.167470	−0.00813305
$425$	−54.2445	−2.63124
$426$	0	0
$427$	0	0
$428$	12.5586i	0.607045i
$429$	0	0
$430$	− 19.4911i	− 0.939944i
$431$	− 26.1698i	− 1.26055i	−0.776370	−	0.630277i	$-0.782941\pi$
$431$	− 26.1698i	− 1.26055i	0.776370	−	0.630277i	$-0.217059\pi$
$432$	0	0
$433$	− 16.2277i	− 0.779852i	−0.920846	−	0.389926i	$-0.872501\pi$
$433$	− 16.2277i	− 0.779852i	0.920846	−	0.389926i	$-0.127499\pi$
$434$	0	0
$435$	0	0
$436$	0.395023	0.0189181
$437$	−13.7115	−0.655910
$438$	0	0
$439$	33.8953i	1.61773i	0.587991	+	0.808867i	$0.299919\pi$
$439$	33.8953i	1.61773i	−0.587991	+	0.808867i	$0.700081\pi$
$440$	22.3475	1.06537
$441$	0	0
$442$	−11.3370	−0.539246
$443$	− 6.11944i	− 0.290743i	−0.989377	−	0.145372i	$-0.953562\pi$
$443$	− 6.11944i	− 0.290743i	0.989377	−	0.145372i	$-0.0464378\pi$
$444$	0	0
$445$	−5.56045	−0.263591
$446$	3.38874	0.160461
$447$	0	0
$448$	0	0
$449$	− 39.0982i	− 1.84516i	−0.385806	−	0.922580i	$-0.626077\pi$
$449$	− 39.0982i	− 1.84516i	0.385806	−	0.922580i	$-0.373923\pi$
$450$	0	0
$451$	− 27.5312i	− 1.29639i
$452$	16.1520i	0.759728i
$453$	0	0
$454$	− 1.21149i	− 0.0568579i
$455$	0	0
$456$	0	0
$457$	−5.24264	−0.245240	−0.122620	−	0.992454i	$-0.539130\pi$
$457$	−5.24264	−0.245240	−0.122620	+	0.992454i	$0.539130\pi$
$458$	19.6590	0.918604
$459$	0	0
$460$	− 33.1025i	− 1.54341i
$461$	−22.4583	−1.04599	−0.522993	−	0.852337i	$-0.675185\pi$
$461$	−22.4583	−1.04599	−0.522993	+	0.852337i	$0.675185\pi$
$462$	0	0
$463$	−16.5278	−0.768110	−0.384055	−	0.923310i	$-0.625473\pi$
$463$	−16.5278	−0.768110	−0.384055	+	0.923310i	$0.625473\pi$
$464$	5.02884i	0.233458i
$465$	0	0
$466$	−15.2758	−0.707640
$467$	−30.1761	−1.39638	−0.698192	−	0.715910i	$-0.746012\pi$
$467$	−30.1761	−1.39638	−0.698192	+	0.715910i	$0.746012\pi$
$468$	0	0
$469$	0	0
$470$	32.6493i	1.50600i
$471$	0	0
$472$	− 5.99698i	− 0.276033i
$473$	− 23.5876i	− 1.08456i
$474$	0	0
$475$	23.9698i	1.09981i
$476$	0	0
$477$	0	0
$478$	0.539712	0.0246859
$479$	35.4561	1.62003	0.810016	−	0.586408i	$-0.199458\pi$
$479$	35.4561	1.62003	0.810016	+	0.586408i	$0.199458\pi$
$480$	0	0
$481$	25.8292i	1.17771i
$482$	−9.37436	−0.426990
$483$	0	0
$484$	16.0443	0.729286
$485$	− 74.1690i	− 3.36784i
$486$	0	0
$487$	19.6313	0.889577	0.444789	−	0.895636i	$-0.353279\pi$
$487$	19.6313	0.889577	0.444789	+	0.895636i	$0.353279\pi$
$488$	7.79367	0.352803
$489$	0	0
$490$	0	0
$491$	13.8081i	0.623149i	0.950222	+	0.311574i	$0.100856\pi$
$491$	13.8081i	0.623149i	−0.950222	+	0.311574i	$0.899144\pi$
$492$	0	0
$493$	− 20.2569i	− 0.912326i
$494$	5.00965i	0.225395i
$495$	0	0
$496$	1.25139i	0.0561890i
$497$	0	0
$498$	0	0
$499$	−15.7029	−0.702960	−0.351480	−	0.936195i	$-0.614321\pi$
$499$	−15.7029	−0.702960	−0.351480	+	0.936195i	$0.614321\pi$
$500$	−36.3820	−1.62705
$501$	0	0
$502$	− 12.2724i	− 0.547743i
$503$	9.82270	0.437972	0.218986	−	0.975728i	$-0.429725\pi$
$503$	9.82270	0.437972	0.218986	+	0.975728i	$0.429725\pi$
$504$	0	0
$505$	60.8742	2.70887
$506$	− 40.0597i	− 1.78087i
$507$	0	0
$508$	−9.04803	−0.401441
$509$	7.79995	0.345727	0.172863	−	0.984946i	$-0.444698\pi$
$509$	7.79995	0.345727	0.172863	+	0.984946i	$0.444698\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	13.4783i	0.594503i
$515$	− 3.51317i	− 0.154809i
$516$	0	0
$517$	39.5112i	1.73770i
$518$	0	0
$519$	0	0
$520$	−12.0944	−0.530373
$521$	13.3093	0.583092	0.291546	−	0.956557i	$-0.405830\pi$
$521$	13.3093	0.583092	0.291546	+	0.956557i	$0.405830\pi$
$522$	0	0
$523$	43.5992i	1.90646i	0.302249	+	0.953229i	$0.402263\pi$
$523$	43.5992i	1.90646i	−0.302249	+	0.953229i	$0.597737\pi$
$524$	1.75412	0.0766293
$525$	0	0
$526$	12.5893	0.548919
$527$	− 5.04079i	− 0.219580i
$528$	0	0
$529$	−36.3391	−1.57996
$530$	0.719659	0.0312600
$531$	0	0
$532$	0	0
$533$	14.8998i	0.645380i
$534$	0	0
$535$	− 53.9676i	− 2.33322i
$536$	− 11.3778i	− 0.491446i
$537$	0	0
$538$	6.85395i	0.295495i
$539$	0	0
$540$	0	0
$541$	3.54936	0.152599	0.0762995	−	0.997085i	$-0.475689\pi$
$541$	3.54936	0.152599	0.0762995	+	0.997085i	$0.475689\pi$
$542$	29.8899	1.28388
$543$	0	0
$544$	− 4.02815i	− 0.172706i
$545$	−1.69751	−0.0727134
$546$	0	0
$547$	−23.3055	−0.996471	−0.498236	−	0.867042i	$-0.666019\pi$
$547$	−23.3055	−0.996471	−0.498236	+	0.867042i	$0.666019\pi$
$548$	− 16.4433i	− 0.702423i
$549$	0	0
$550$	−70.0306	−2.98611
$551$	−8.95123	−0.381335
$552$	0	0
$553$	0	0
$554$	− 30.7992i	− 1.30853i
$555$	0	0
$556$	− 17.6373i	− 0.747990i
$557$	25.4845i	1.07981i	0.841726	+	0.539906i	$0.181540\pi$
$557$	25.4845i	1.07981i	−0.841726	+	0.539906i	$0.818460\pi$
$558$	0	0
$559$	12.7655i	0.539924i
$560$	0	0
$561$	0	0
$562$	6.44161	0.271723
$563$	27.9806	1.17924	0.589621	−	0.807680i	$-0.299277\pi$
$563$	27.9806	1.17924	0.589621	+	0.807680i	$0.299277\pi$
$564$	0	0
$565$	− 69.4093i	− 2.92007i
$566$	−13.5392	−0.569094
$567$	0	0
$568$	−0.539455	−0.0226350
$569$	− 12.7865i	− 0.536036i	−0.963414	−	0.268018i	$-0.913631\pi$
$569$	− 12.7865i	− 0.536036i	0.963414	−	0.268018i	$-0.0863687\pi$
$570$	0	0
$571$	−24.9288	−1.04324	−0.521620	−	0.853178i	$-0.674672\pi$
$571$	−24.9288	−1.04324	−0.521620	+	0.853178i	$0.674672\pi$
$572$	−14.6363	−0.611973
$573$	0	0
$574$	0	0
$575$	103.734i	4.32600i
$576$	0	0
$577$	− 16.1318i	− 0.671576i	−0.941938	−	0.335788i	$-0.890997\pi$
$577$	− 16.1318i	− 0.671576i	0.941938	−	0.335788i	$-0.109003\pi$
$578$	− 0.773996i	− 0.0321940i
$579$	0	0
$580$	− 21.6102i	− 0.897314i
$581$	0	0
$582$	0	0
$583$	0.870912	0.0360695
$584$	3.80634	0.157507
$585$	0	0
$586$	− 26.3170i	− 1.08714i
$587$	22.9149	0.945797	0.472899	−	0.881117i	$-0.343208\pi$
$587$	22.9149	0.945797	0.472899	+	0.881117i	$0.343208\pi$
$588$	0	0
$589$	−2.22745	−0.0917803
$590$	25.7705i	1.06096i
$591$	0	0
$592$	−9.17738	−0.377188
$593$	14.0878	0.578518	0.289259	−	0.957251i	$-0.406591\pi$
$593$	14.0878	0.578518	0.289259	+	0.957251i	$0.406591\pi$
$594$	0	0
$595$	0	0
$596$	− 12.2115i	− 0.500203i
$597$	0	0
$598$	21.6802i	0.886568i
$599$	33.5681i	1.37155i	0.727812	+	0.685777i	$0.240537\pi$
$599$	33.5681i	1.37155i	−0.727812	+	0.685777i	$0.759463\pi$
$600$	0	0
$601$	− 25.9839i	− 1.05990i	−0.848027	−	0.529952i	$-0.822210\pi$
$601$	− 25.9839i	− 1.05990i	0.848027	−	0.529952i	$-0.177790\pi$
$602$	0	0
$603$	0	0
$604$	8.44860	0.343769
$605$	−68.9463	−2.80307
$606$	0	0
$607$	− 17.8701i	− 0.725324i	−0.931921	−	0.362662i	$-0.881868\pi$
$607$	− 17.8701i	− 0.725324i	0.931921	−	0.362662i	$-0.118132\pi$
$608$	−1.77998	−0.0721877
$609$	0	0
$610$	−33.4913	−1.35602
$611$	− 21.3833i	− 0.865077i
$612$	0	0
$613$	10.0387	0.405460	0.202730	−	0.979235i	$-0.435019\pi$
$613$	10.0387	0.405460	0.202730	+	0.979235i	$0.435019\pi$
$614$	19.4207	0.783754
$615$	0	0
$616$	0	0
$617$	5.79319i	0.233225i	0.993177	+	0.116612i	$0.0372035\pi$
$617$	5.79319i	0.233225i	−0.993177	+	0.116612i	$0.962796\pi$
$618$	0	0
$619$	− 28.3424i	− 1.13918i	−0.821930	−	0.569589i	$-0.807102\pi$
$619$	− 28.3424i	− 1.13918i	0.821930	−	0.569589i	$-0.192898\pi$
$620$	− 5.37753i	− 0.215967i
$621$	0	0
$622$	− 3.58894i	− 0.143903i
$623$	0	0
$624$	0	0
$625$	89.0108	3.56043
$626$	−12.9198	−0.516380
$627$	0	0
$628$	− 1.97105i	− 0.0786534i
$629$	36.9679	1.47401
$630$	0	0
$631$	22.3316	0.889005	0.444503	−	0.895777i	$-0.353380\pi$
$631$	22.3316	0.889005	0.444503	+	0.895777i	$0.353380\pi$
$632$	− 1.99590i	− 0.0793925i
$633$	0	0
$634$	12.0384	0.478105
$635$	38.8816	1.54297
$636$	0	0
$637$	0	0
$638$	− 26.1520i	− 1.03537i
$639$	0	0
$640$	− 4.29725i	− 0.169864i
$641$	− 30.2138i	− 1.19337i	−0.802474	−	0.596687i	$-0.796483\pi$
$641$	− 30.2138i	− 1.19337i	0.802474	−	0.596687i	$-0.203517\pi$
$642$	0	0
$643$	− 28.0613i	− 1.10663i	−0.832972	−	0.553315i	$-0.813363\pi$
$643$	− 28.0613i	− 1.10663i	0.832972	−	0.553315i	$-0.186637\pi$
$644$	0	0
$645$	0	0
$646$	7.17002	0.282101
$647$	24.9682	0.981600	0.490800	−	0.871272i	$-0.336705\pi$
$647$	24.9682	0.981600	0.490800	+	0.871272i	$0.336705\pi$
$648$	0	0
$649$	31.1868i	1.22419i
$650$	37.9003	1.48657
$651$	0	0
$652$	7.67231	0.300471
$653$	− 35.2971i	− 1.38128i	−0.723198	−	0.690641i	$-0.757328\pi$
$653$	− 35.2971i	− 1.38128i	0.723198	−	0.690641i	$-0.242672\pi$
$654$	0	0
$655$	−7.53791	−0.294530
$656$	−5.29403	−0.206697
$657$	0	0
$658$	0	0
$659$	− 8.73576i	− 0.340297i	−0.985418	−	0.170149i	$-0.945575\pi$
$659$	− 8.73576i	− 0.340297i	0.985418	−	0.170149i	$-0.0544248\pi$
$660$	0	0
$661$	− 35.5348i	− 1.38214i	−0.722785	−	0.691072i	$-0.757139\pi$
$661$	− 35.5348i	− 1.38214i	0.722785	−	0.691072i	$-0.242861\pi$
$662$	25.8226i	1.00363i
$663$	0	0
$664$	− 16.0388i	− 0.622427i
$665$	0	0
$666$	0	0
$667$	−38.7381	−1.49994
$668$	6.19782	0.239801
$669$	0	0
$670$	48.8932i	1.88891i
$671$	−40.5303	−1.56465
$672$	0	0
$673$	7.14709	0.275500	0.137750	−	0.990467i	$-0.456013\pi$
$673$	7.14709	0.275500	0.137750	+	0.990467i	$0.456013\pi$
$674$	34.5378i	1.33035i
$675$	0	0
$676$	−5.07891	−0.195343
$677$	20.1163	0.773132	0.386566	−	0.922262i	$-0.373661\pi$
$677$	20.1163	0.773132	0.386566	+	0.922262i	$0.373661\pi$
$678$	0	0
$679$	0	0
$680$	17.3100i	0.663807i
$681$	0	0
$682$	− 6.50774i	− 0.249194i
$683$	− 18.7325i	− 0.716781i	−0.933572	−	0.358390i	$-0.883326\pi$
$683$	− 18.7325i	− 0.716781i	0.933572	−	0.358390i	$-0.116674\pi$
$684$	0	0
$685$	70.6610i	2.69982i
$686$	0	0
$687$	0	0
$688$	−4.53572	−0.172923
$689$	−0.471334	−0.0179564
$690$	0	0
$691$	− 31.9637i	− 1.21596i	−0.793954	−	0.607978i	$-0.791981\pi$
$691$	− 31.9637i	− 1.21596i	0.793954	−	0.607978i	$-0.208019\pi$
$692$	1.17825	0.0447903
$693$	0	0
$694$	−11.4390	−0.434217
$695$	75.7920i	2.87496i
$696$	0	0
$697$	21.3252	0.807748
$698$	−9.93436	−0.376021
$699$	0	0
$700$	0	0
$701$	− 24.1148i	− 0.910805i	−0.890286	−	0.455403i	$-0.849495\pi$
$701$	− 24.1148i	− 0.910805i	0.890286	−	0.455403i	$-0.150505\pi$
$702$	0	0
$703$	− 16.3355i	− 0.616106i
$704$	− 5.20041i	− 0.195998i
$705$	0	0
$706$	− 2.80734i	− 0.105655i
$707$	0	0
$708$	0	0
$709$	28.0545	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$709$	28.0545	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$710$	2.31817	0.0869995
$711$	0	0
$712$	1.29396i	0.0484931i
$713$	−9.63968	−0.361009
$714$	0	0
$715$	62.8957	2.35216
$716$	15.3425i	0.573378i
$717$	0	0
$718$	3.70282	0.138188
$719$	31.4581	1.17319	0.586594	−	0.809881i	$-0.300468\pi$
$719$	31.4581	1.17319	0.586594	+	0.809881i	$0.300468\pi$
$720$	0	0
$721$	0	0
$722$	15.8317i	0.589194i
$723$	0	0
$724$	5.24186i	0.194812i
$725$	67.7201i	2.51506i
$726$	0	0
$727$	− 35.9687i	− 1.33400i	−0.745056	−	0.667002i	$-0.767577\pi$
$727$	− 35.9687i	− 1.33400i	0.745056	−	0.667002i	$-0.232423\pi$
$728$	0	0
$729$	0	0
$730$	−16.3568	−0.605392
$731$	18.2705	0.675761
$732$	0	0
$733$	− 13.2830i	− 0.490619i	−0.969445	−	0.245309i	$-0.921110\pi$
$733$	− 13.2830i	− 0.490619i	0.969445	−	0.245309i	$-0.0788895\pi$
$734$	13.3176	0.491560
$735$	0	0
$736$	−7.70319	−0.283943
$737$	59.1692i	2.17953i
$738$	0	0
$739$	−43.1049	−1.58564	−0.792819	−	0.609457i	$-0.791388\pi$
$739$	−43.1049	−1.58564	−0.792819	+	0.609457i	$0.791388\pi$
$740$	39.4375	1.44975
$741$	0	0
$742$	0	0
$743$	− 11.1826i	− 0.410248i	−0.978736	−	0.205124i	$-0.934240\pi$
$743$	− 11.1826i	− 0.410248i	0.978736	−	0.205124i	$-0.0657598\pi$
$744$	0	0
$745$	52.4759i	1.92257i
$746$	− 2.09640i	− 0.0767547i
$747$	0	0
$748$	20.9480i	0.765937i
$749$	0	0
$750$	0	0
$751$	−38.3361	−1.39891	−0.699453	−	0.714679i	$-0.746573\pi$
$751$	−38.3361	−1.39891	−0.699453	+	0.714679i	$0.746573\pi$
$752$	7.59771	0.277060
$753$	0	0
$754$	14.1534i	0.515436i
$755$	−36.3057	−1.32130
$756$	0	0
$757$	47.1806	1.71481	0.857405	−	0.514642i	$-0.172075\pi$
$757$	47.1806	1.71481	0.857405	+	0.514642i	$0.172075\pi$
$758$	− 10.5916i	− 0.384704i
$759$	0	0
$760$	7.64901	0.277459
$761$	28.0668	1.01742	0.508711	−	0.860937i	$-0.330122\pi$
$761$	28.0668	1.01742	0.508711	+	0.860937i	$0.330122\pi$
$762$	0	0
$763$	0	0
$764$	− 4.63237i	− 0.167593i
$765$	0	0
$766$	29.3742i	1.06133i
$767$	− 16.8782i	− 0.609435i
$768$	0	0
$769$	− 25.9545i	− 0.935944i	−0.883743	−	0.467972i	$-0.844985\pi$
$769$	− 25.9545i	− 0.935944i	0.883743	−	0.467972i	$-0.155015\pi$
$770$	0	0
$771$	0	0
$772$	7.34315	0.264286
$773$	−21.0644	−0.757634	−0.378817	−	0.925472i	$-0.623669\pi$
$773$	−21.0644	−0.757634	−0.378817	+	0.925472i	$0.623669\pi$
$774$	0	0
$775$	16.8516i	0.605329i
$776$	−17.2596	−0.619585
$777$	0	0
$778$	−0.139250	−0.00499237
$779$	− 9.42326i	− 0.337623i
$780$	0	0
$781$	2.80539	0.100385
$782$	31.0296	1.10962
$783$	0	0
$784$	0	0
$785$	8.47009i	0.302310i
$786$	0	0
$787$	19.2179i	0.685046i	0.939510	+	0.342523i	$0.111281\pi$
$787$	19.2179i	0.685046i	−0.939510	+	0.342523i	$0.888719\pi$
$788$	23.5456i	0.838778i
$789$	0	0
$790$	8.57687i	0.305151i
$791$	0	0
$792$	0	0
$793$	21.9348	0.778929
$794$	−5.13731	−0.182316
$795$	0	0
$796$	− 7.28291i	− 0.258136i
$797$	−32.2444	−1.14216	−0.571078	−	0.820896i	$-0.693475\pi$
$797$	−32.2444	−1.14216	−0.571078	+	0.820896i	$0.693475\pi$
$798$	0	0
$799$	−30.6047	−1.08272
$800$	13.4663i	0.476107i
$801$	0	0
$802$	−10.0437	−0.354654
$803$	−19.7945	−0.698534
$804$	0	0
$805$	0	0
$806$	3.52196i	0.124056i
$807$	0	0
$808$	− 14.1659i	− 0.498353i
$809$	45.0695i	1.58456i	0.610157	+	0.792280i	$0.291106\pi$
$809$	45.0695i	1.58456i	−0.610157	+	0.792280i	$0.708894\pi$
$810$	0	0
$811$	30.5466i	1.07264i	0.844016	+	0.536319i	$0.180185\pi$
$811$	30.5466i	1.07264i	−0.844016	+	0.536319i	$0.819815\pi$
$812$	0	0
$813$	0	0
$814$	47.7261	1.67280
$815$	−32.9698	−1.15488
$816$	0	0
$817$	− 8.07348i	− 0.282455i
$818$	−9.66617	−0.337970
$819$	0	0
$820$	22.7498	0.794457
$821$	− 26.9642i	− 0.941057i	−0.882385	−	0.470528i	$-0.844063\pi$
$821$	− 26.9642i	− 0.941057i	0.882385	−	0.470528i	$-0.155937\pi$
$822$	0	0
$823$	8.86392	0.308977	0.154488	−	0.987995i	$-0.450627\pi$
$823$	8.86392	0.308977	0.154488	+	0.987995i	$0.450627\pi$
$824$	−0.817539	−0.0284803
$825$	0	0
$826$	0	0
$827$	37.8611i	1.31656i	0.752773	+	0.658280i	$0.228716\pi$
$827$	37.8611i	1.31656i	−0.752773	+	0.658280i	$0.771284\pi$
$828$	0	0
$829$	− 17.9864i	− 0.624694i	−0.949968	−	0.312347i	$-0.898885\pi$
$829$	− 17.9864i	− 0.624694i	0.949968	−	0.312347i	$-0.101115\pi$
$830$	68.9228i	2.39235i
$831$	0	0
$832$	2.81444i	0.0975733i
$833$	0	0
$834$	0	0
$835$	−26.6336	−0.921693
$836$	9.25662	0.320147
$837$	0	0
$838$	17.8462i	0.616488i
$839$	−46.9701	−1.62159	−0.810793	−	0.585332i	$-0.800964\pi$
$839$	−46.9701	−1.62159	−0.810793	+	0.585332i	$0.800964\pi$
$840$	0	0
$841$	3.71077	0.127958
$842$	5.87102i	0.202329i
$843$	0	0
$844$	12.7862	0.440119
$845$	21.8253	0.750815
$846$	0	0
$847$	0	0
$848$	− 0.167470i	− 0.00575094i
$849$	0	0
$850$	− 54.2445i	− 1.86057i
$851$	− 70.6950i	− 2.42339i
$852$	0	0
$853$	− 15.5809i	− 0.533479i	−0.963769	−	0.266740i	$-0.914054\pi$
$853$	− 15.5809i	− 0.533479i	0.963769	−	0.266740i	$-0.0859463\pi$
$854$	0	0
$855$	0	0
$856$	−12.5586	−0.429246
$857$	9.43539	0.322307	0.161153	−	0.986929i	$-0.448479\pi$
$857$	9.43539	0.322307	0.161153	+	0.986929i	$0.448479\pi$
$858$	0	0
$859$	47.2724i	1.61291i	0.591293	+	0.806457i	$0.298618\pi$
$859$	47.2724i	1.61291i	−0.591293	+	0.806457i	$0.701382\pi$
$860$	19.4911	0.664641
$861$	0	0
$862$	26.1698	0.891347
$863$	20.3117i	0.691417i	0.938342	+	0.345709i	$0.112361\pi$
$863$	20.3117i	0.691417i	−0.938342	+	0.345709i	$0.887639\pi$
$864$	0	0
$865$	−5.06323	−0.172155
$866$	16.2277	0.551439
$867$	0	0
$868$	0	0
$869$	10.3795i	0.352100i
$870$	0	0
$871$	− 32.0221i	− 1.08503i
$872$	0.395023i	0.0133772i
$873$	0	0
$874$	− 13.7115i	− 0.463799i
$875$	0	0
$876$	0	0
$877$	27.2321	0.919564	0.459782	−	0.888032i	$-0.347928\pi$
$877$	27.2321	0.919564	0.459782	+	0.888032i	$0.347928\pi$
$878$	−33.8953	−1.14391
$879$	0	0
$880$	22.3475i	0.753333i
$881$	−13.3664	−0.450326	−0.225163	−	0.974321i	$-0.572291\pi$
$881$	−13.3664	−0.450326	−0.225163	+	0.974321i	$0.572291\pi$
$882$	0	0
$883$	18.6560	0.627824	0.313912	−	0.949452i	$-0.398360\pi$
$883$	18.6560	0.627824	0.313912	+	0.949452i	$0.398360\pi$
$884$	− 11.3370i	− 0.381305i
$885$	0	0
$886$	6.11944	0.205587
$887$	−3.12571	−0.104951	−0.0524756	−	0.998622i	$-0.516711\pi$
$887$	−3.12571	−0.104951	−0.0524756	+	0.998622i	$0.516711\pi$
$888$	0	0
$889$	0	0
$890$	− 5.56045i	− 0.186387i
$891$	0	0
$892$	3.38874i	0.113463i
$893$	13.5238i	0.452556i
$894$	0	0
$895$	− 65.9307i	− 2.20382i
$896$	0	0
$897$	0	0
$898$	39.0982	1.30472
$899$	−6.29304	−0.209885
$900$	0	0
$901$	0.674594i	0.0224740i
$902$	27.5312	0.916687
$903$	0	0
$904$	−16.1520	−0.537209
$905$	− 22.5256i	− 0.748775i
$906$	0	0
$907$	−3.81672	−0.126732	−0.0633660	−	0.997990i	$-0.520184\pi$
$907$	−3.81672	−0.126732	−0.0633660	+	0.997990i	$0.520184\pi$
$908$	1.21149	0.0402046
$909$	0	0
$910$	0	0
$911$	48.5006i	1.60690i	0.595373	+	0.803449i	$0.297004\pi$
$911$	48.5006i	1.60690i	−0.595373	+	0.803449i	$0.702996\pi$
$912$	0	0
$913$	83.4085i	2.76042i
$914$	− 5.24264i	− 0.173411i
$915$	0	0
$916$	19.6590i	0.649551i
$917$	0	0
$918$	0	0
$919$	−21.8847	−0.721908	−0.360954	−	0.932584i	$-0.617549\pi$
$919$	−21.8847	−0.721908	−0.360954	+	0.932584i	$0.617549\pi$
$920$	33.1025	1.09136
$921$	0	0
$922$	− 22.4583i	− 0.739624i
$923$	−1.51827	−0.0499743
$924$	0	0
$925$	−123.586	−4.06347
$926$	− 16.5278i	− 0.543136i
$927$	0	0
$928$	−5.02884	−0.165080
$929$	−23.6339	−0.775402	−0.387701	−	0.921785i	$-0.626731\pi$
$929$	−23.6339	−0.775402	−0.387701	+	0.921785i	$0.626731\pi$
$930$	0	0
$931$	0	0
$932$	− 15.2758i	− 0.500377i
$933$	0	0
$934$	− 30.1761i	− 0.987393i
$935$	− 90.0190i	− 2.94394i
$936$	0	0
$937$	− 34.9674i	− 1.14234i	−0.820833	−	0.571168i	$-0.806491\pi$
$937$	− 34.9674i	− 1.14234i	0.820833	−	0.571168i	$-0.193509\pi$
$938$	0	0
$939$	0	0
$940$	−32.6493	−1.06490
$941$	−12.7640	−0.416093	−0.208047	−	0.978119i	$-0.566711\pi$
$941$	−12.7640	−0.416093	−0.208047	+	0.978119i	$0.566711\pi$
$942$	0	0
$943$	− 40.7809i	− 1.32801i
$944$	5.99698	0.195185
$945$	0	0
$946$	23.5876	0.766899
$947$	4.45356i	0.144721i	0.997379	+	0.0723606i	$0.0230532\pi$
$947$	4.45356i	0.144721i	−0.997379	+	0.0723606i	$0.976947\pi$
$948$	0	0
$949$	10.7127	0.347750
$950$	−23.9698	−0.777683
$951$	0	0
$952$	0	0
$953$	− 20.5722i	− 0.666398i	−0.942857	−	0.333199i	$-0.891872\pi$
$953$	− 20.5722i	− 0.666398i	0.942857	−	0.333199i	$-0.108128\pi$
$954$	0	0
$955$	19.9065i	0.644158i
$956$	0.539712i	0.0174555i
$957$	0	0
$958$	35.4561i	1.14554i
$959$	0	0
$960$	0	0
$961$	29.4340	0.949485
$962$	−25.8292	−0.832767
$963$	0	0
$964$	− 9.37436i	− 0.301928i
$965$	−31.5553	−1.01580
$966$	0	0
$967$	−31.6779	−1.01869	−0.509346	−	0.860562i	$-0.670112\pi$
$967$	−31.6779	−1.01869	−0.509346	+	0.860562i	$0.670112\pi$
$968$	16.0443i	0.515683i
$969$	0	0
$970$	74.1690	2.38142
$971$	−9.60450	−0.308223	−0.154112	−	0.988053i	$-0.549252\pi$
$971$	−9.60450	−0.308223	−0.154112	+	0.988053i	$0.549252\pi$
$972$	0	0
$973$	0	0
$974$	19.6313i	0.629026i
$975$	0	0
$976$	7.79367i	0.249469i
$977$	− 40.5256i	− 1.29653i	−0.761416	−	0.648264i	$-0.775495\pi$
$977$	− 40.5256i	− 1.29653i	0.761416	−	0.648264i	$-0.224505\pi$
$978$	0	0
$979$	− 6.72911i	− 0.215063i
$980$	0	0
$981$	0	0
$982$	−13.8081	−0.440633
$983$	25.2959	0.806815	0.403407	−	0.915021i	$-0.367826\pi$
$983$	25.2959	0.806815	0.403407	+	0.915021i	$0.367826\pi$
$984$	0	0
$985$	− 101.181i	− 3.22391i
$986$	20.2569	0.645112
$987$	0	0
$988$	−5.00965	−0.159378
$989$	− 34.9395i	− 1.11101i
$990$	0	0
$991$	−22.6706	−0.720156	−0.360078	−	0.932922i	$-0.617250\pi$
$991$	−22.6706	−0.720156	−0.360078	+	0.932922i	$0.617250\pi$
$992$	−1.25139	−0.0397316
$993$	0	0
$994$	0	0
$995$	31.2965i	0.992165i
$996$	0	0
$997$	− 48.0079i	− 1.52042i	−0.649675	−	0.760212i	$-0.725095\pi$
$997$	− 48.0079i	− 1.52042i	0.649675	−	0.760212i	$-0.274905\pi$
$998$	− 15.7029i	− 0.497068i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.e.2645.16	yes	16
3.2	odd	2	inner	2646.2.d.e.2645.1	✓	16
7.6	odd	2	inner	2646.2.d.e.2645.9	yes	16
21.20	even	2	inner	2646.2.d.e.2645.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.d.e.2645.1	✓	16	3.2	odd	2	inner
2646.2.d.e.2645.8	yes	16	21.20	even	2	inner
2646.2.d.e.2645.9	yes	16	7.6	odd	2	inner
2646.2.d.e.2645.16	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.29725 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.29725 q^{5} -1.00000i q^{8} +4.29725i q^{10} +5.20041i q^{11} -2.81444i q^{13} +1.00000 q^{16} -4.02815 q^{17} +1.77998i q^{19} -4.29725 q^{20} -5.20041 q^{22} +7.70319i q^{23} +13.4663 q^{25} +2.81444 q^{26} +5.02884i q^{29} +1.25139i q^{31} +1.00000i q^{32} -4.02815i q^{34} -9.17738 q^{37} -1.77998 q^{38} -4.29725i q^{40} -5.29403 q^{41} -4.53572 q^{43} -5.20041i q^{44} -7.70319 q^{46} +7.59771 q^{47} +13.4663i q^{50} +2.81444i q^{52} -0.167470i q^{53} +22.3475i q^{55} -5.02884 q^{58} +5.99698 q^{59} +7.79367i q^{61} -1.25139 q^{62} -1.00000 q^{64} -12.0944i q^{65} +11.3778 q^{67} +4.02815 q^{68} -0.539455i q^{71} +3.80634i q^{73} -9.17738i q^{74} -1.77998i q^{76} +1.99590 q^{79} +4.29725 q^{80} -5.29403i q^{82} +16.0388 q^{83} -17.3100 q^{85} -4.53572i q^{86} +5.20041 q^{88} -1.29396 q^{89} -7.70319i q^{92} +7.59771i q^{94} +7.64901i q^{95} -17.2596i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 32 * q^22 + 48 * q^25 - 32 * q^37 + 16 * q^43 - 48 * q^46 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 32 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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