LMFDB - Embedded newform 3200.2.d.s.1601.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1601,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3200.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1601.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3200.1601
Dual form			3200.2.d.s.1601.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} +4.24264 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.41421i q^{3} +4.24264 q^{7} +1.00000 q^{9} -5.65685i q^{11} +2.00000i q^{13} -6.00000 q^{17} -2.82843i q^{19} -6.00000i q^{21} -7.07107 q^{23} -5.65685i q^{27} -4.00000i q^{29} -2.82843 q^{31} -8.00000 q^{33} +2.00000i q^{37} +2.82843 q^{39} -8.00000 q^{41} -1.41421i q^{43} +1.41421 q^{47} +11.0000 q^{49} +8.48528i q^{51} -2.00000i q^{53} -4.00000 q^{57} -2.82843i q^{59} -14.0000i q^{61} +4.24264 q^{63} +4.24264i q^{67} +10.0000i q^{69} +2.82843 q^{71} +6.00000 q^{73} -24.0000i q^{77} +16.9706 q^{79} -5.00000 q^{81} +12.7279i q^{83} -5.65685 q^{87} -6.00000 q^{89} +8.48528i q^{91} +4.00000i q^{93} -10.0000 q^{97} -5.65685i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} - 24 q^{17} - 32 q^{33} - 32 q^{41} + 44 q^{49} - 16 q^{57} + 24 q^{73} - 20 q^{81} - 24 q^{89} - 40 q^{97}+O(q^{100}) $ 4 * q + 4 * q^9 - 24 * q^17 - 32 * q^33 - 32 * q^41 + 44 * q^49 - 16 * q^57 + 24 * q^73 - 20 * q^81 - 24 * q^89 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.24264	1.60357	0.801784	−	0.597614i	$-0.203885\pi$
$7$	4.24264	1.60357	0.801784	+	0.597614i	$0.203885\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 5.65685i	− 1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	− 5.65685i	− 1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	− 2.82843i	− 0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$	− 2.82843i	− 0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$	0	0
$21$	− 6.00000i	− 1.30931i
$22$	0	0
$23$	−7.07107	−1.47442	−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−7.07107	−1.47442	−0.737210	+	0.675664i	$0.763857\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	− 4.00000i	− 0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	− 4.00000i	− 0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	−8.00000	−1.39262
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	− 1.41421i	− 0.215666i	−0.994169	−	0.107833i	$-0.965609\pi$
$43$	− 1.41421i	− 0.215666i	0.994169	−	0.107833i	$-0.0343911\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.41421	0.206284	0.103142	−	0.994667i	$-0.467110\pi$
$47$	1.41421	0.206284	0.103142	+	0.994667i	$0.467110\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	8.48528i	1.18818i
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 2.82843i	− 0.368230i	−0.982905	−	0.184115i	$-0.941058\pi$
$59$	− 2.82843i	− 0.368230i	0.982905	−	0.184115i	$-0.0589419\pi$
$60$	0	0
$61$	− 14.0000i	− 1.79252i	−0.443533	−	0.896258i	$-0.646275\pi$
$61$	− 14.0000i	− 1.79252i	0.443533	−	0.896258i	$-0.353725\pi$
$62$	0	0
$63$	4.24264	0.534522
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.24264i	0.518321i	0.965834	+	0.259161i	$0.0834459\pi$
$67$	4.24264i	0.518321i	−0.965834	+	0.259161i	$0.916554\pi$
$68$	0	0
$69$	10.0000i	1.20386i
$70$	0	0
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 24.0000i	− 2.73505i
$78$	0	0
$79$	16.9706	1.90934	0.954669	−	0.297670i	$-0.0962096\pi$
$79$	16.9706	1.90934	0.954669	+	0.297670i	$0.0962096\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	12.7279i	1.39707i	0.715575	+	0.698535i	$0.246165\pi$
$83$	12.7279i	1.39707i	−0.715575	+	0.698535i	$0.753835\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.65685	−0.606478
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	8.48528i	0.889499i
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	− 5.65685i	− 0.568535i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	9.89949	0.975426	0.487713	−	0.873004i	$-0.337831\pi$
$103$	9.89949	0.975426	0.487713	+	0.873004i	$0.337831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.07107i	− 0.683586i	−0.939775	−	0.341793i	$-0.888966\pi$
$107$	− 7.07107i	− 0.683586i	0.939775	−	0.341793i	$-0.111034\pi$
$108$	0	0
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	2.82843	0.268462
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−25.4558	−2.33353
$120$	0	0
$121$	−21.0000	−1.90909
$122$	0	0
$123$	11.3137i	1.02012i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.89949	0.878438	0.439219	−	0.898380i	$-0.355255\pi$
$127$	9.89949	0.878438	0.439219	+	0.898380i	$0.355255\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	− 11.3137i	− 0.988483i	−0.869325	−	0.494242i	$-0.835446\pi$
$131$	− 11.3137i	− 0.988483i	0.869325	−	0.494242i	$-0.164554\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	8.48528i	0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$	8.48528i	0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	− 2.00000i	− 0.168430i
$142$	0	0
$143$	11.3137	0.946100
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 15.5563i	− 1.28307i
$148$	0	0
$149$	22.0000i	1.80231i	0.433497	+	0.901155i	$0.357280\pi$
$149$	22.0000i	1.80231i	−0.433497	+	0.901155i	$0.642720\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	0	0
$159$	−2.82843	−0.224309
$160$	0	0
$161$	−30.0000	−2.36433
$162$	0	0
$163$	− 18.3848i	− 1.44001i	−0.693971	−	0.720003i	$-0.744140\pi$
$163$	− 18.3848i	− 1.44001i	0.693971	−	0.720003i	$-0.255860\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.89949	−0.766046	−0.383023	−	0.923739i	$-0.625117\pi$
$167$	−9.89949	−0.766046	−0.383023	+	0.923739i	$0.625117\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	− 2.82843i	− 0.216295i
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	19.7990i	1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$	19.7990i	1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	− 16.0000i	− 1.18927i	−0.803996	−	0.594635i	$-0.797296\pi$
$181$	− 16.0000i	− 1.18927i	0.803996	−	0.594635i	$-0.202704\pi$
$182$	0	0
$183$	−19.7990	−1.46358
$184$	0	0
$185$	0	0
$186$	0	0
$187$	33.9411i	2.48202i
$188$	0	0
$189$	− 24.0000i	− 1.74574i
$190$	0	0
$191$	2.82843	0.204658	0.102329	−	0.994751i	$-0.467371\pi$
$191$	2.82843	0.204658	0.102329	+	0.994751i	$0.467371\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	− 16.9706i	− 1.19110i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.07107	−0.491473
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	− 16.9706i	− 1.16830i	−0.811645	−	0.584151i	$-0.801428\pi$
$211$	− 16.9706i	− 1.16830i	0.811645	−	0.584151i	$-0.198572\pi$
$212$	0	0
$213$	− 4.00000i	− 0.274075i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	− 8.48528i	− 0.573382i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	−15.5563	−1.04173	−0.520865	−	0.853639i	$-0.674391\pi$
$223$	−15.5563	−1.04173	−0.520865	+	0.853639i	$0.674391\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.07107i	0.469323i	0.972077	+	0.234662i	$0.0753982\pi$
$227$	7.07107i	0.469323i	−0.972077	+	0.234662i	$0.924602\pi$
$228$	0	0
$229$	20.0000i	1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$	20.0000i	1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$	0	0
$231$	−33.9411	−2.23316
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 24.0000i	− 1.55897i
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	− 9.89949i	− 0.635053i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.65685	0.359937
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	22.6274i	1.42823i	0.700028	+	0.714115i	$0.253171\pi$
$251$	22.6274i	1.42823i	−0.700028	+	0.714115i	$0.746829\pi$
$252$	0	0
$253$	40.0000i	2.51478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	8.48528i	0.527250i
$260$	0	0
$261$	− 4.00000i	− 0.247594i
$262$	0	0
$263$	7.07107	0.436021	0.218010	−	0.975946i	$-0.430043\pi$
$263$	7.07107	0.436021	0.218010	+	0.975946i	$0.430043\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.48528i	0.519291i
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	−25.4558	−1.54633	−0.773166	−	0.634203i	$-0.781328\pi$
$271$	−25.4558	−1.54633	−0.773166	+	0.634203i	$0.781328\pi$
$272$	0	0
$273$	12.0000	0.726273
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.0000i	1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$	30.0000i	1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$	0	0
$279$	−2.82843	−0.169334
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	− 9.89949i	− 0.588464i	−0.955734	−	0.294232i	$-0.904936\pi$
$283$	− 9.89949i	− 0.588464i	0.955734	−	0.294232i	$-0.0950638\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−33.9411	−2.00348
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	14.1421i	0.829027i
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−32.0000	−1.85683
$298$	0	0
$299$	− 14.1421i	− 0.817861i
$300$	0	0
$301$	− 6.00000i	− 0.345834i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.5563i	0.887848i	0.896065	+	0.443924i	$0.146414\pi$
$307$	15.5563i	0.887848i	−0.896065	+	0.443924i	$0.853586\pi$
$308$	0	0
$309$	− 14.0000i	− 0.796432i
$310$	0	0
$311$	2.82843	0.160385	0.0801927	−	0.996779i	$-0.474446\pi$
$311$	2.82843	0.160385	0.0801927	+	0.996779i	$0.474446\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	−22.6274	−1.26689
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	16.9706i	0.944267i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.48528	0.469237
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	− 22.6274i	− 1.24372i	−0.783130	−	0.621858i	$-0.786378\pi$
$331$	− 22.6274i	− 1.24372i	0.783130	−	0.621858i	$-0.213622\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	− 14.1421i	− 0.768095i
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.24264i	− 0.227757i	−0.993495	−	0.113878i	$-0.963673\pi$
$347$	− 4.24264i	− 0.227757i	0.993495	−	0.113878i	$-0.0363274\pi$
$348$	0	0
$349$	− 12.0000i	− 0.642345i	−0.947021	−	0.321173i	$-0.895923\pi$
$349$	− 12.0000i	− 0.642345i	0.947021	−	0.321173i	$-0.104077\pi$
$350$	0	0
$351$	11.3137	0.603881
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	36.0000i	1.90532i
$358$	0	0
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	11.0000	0.578947
$362$	0	0
$363$	29.6985i	1.55877i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.24264	0.221464	0.110732	−	0.993850i	$-0.464680\pi$
$367$	4.24264	0.221464	0.110732	+	0.993850i	$0.464680\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	0	0
$371$	− 8.48528i	− 0.440534i
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	31.1127i	1.59815i	0.601230	+	0.799076i	$0.294678\pi$
$379$	31.1127i	1.59815i	−0.601230	+	0.799076i	$0.705322\pi$
$380$	0	0
$381$	− 14.0000i	− 0.717242i
$382$	0	0
$383$	7.07107	0.361315	0.180657	−	0.983546i	$-0.442177\pi$
$383$	7.07107	0.361315	0.180657	+	0.983546i	$0.442177\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.41421i	− 0.0718885i
$388$	0	0
$389$	14.0000i	0.709828i	0.934899	+	0.354914i	$0.115490\pi$
$389$	14.0000i	0.709828i	−0.934899	+	0.354914i	$0.884510\pi$
$390$	0	0
$391$	42.4264	2.14560
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	−16.9706	−0.849591
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	− 5.65685i	− 0.281788i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.3137	0.560800
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	2.82843i	0.139516i
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	− 8.48528i	− 0.414533i	−0.978285	−	0.207267i	$-0.933543\pi$
$419$	− 8.48528i	− 0.414533i	0.978285	−	0.207267i	$-0.0664567\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	1.41421	0.0687614
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 59.3970i	− 2.87442i
$428$	0	0
$429$	− 16.0000i	− 0.772487i
$430$	0	0
$431$	31.1127	1.49865	0.749323	−	0.662205i	$-0.230379\pi$
$431$	31.1127	1.49865	0.749323	+	0.662205i	$0.230379\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.0000i	0.956730i
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	11.0000	0.523810
$442$	0	0
$443$	− 15.5563i	− 0.739104i	−0.929210	−	0.369552i	$-0.879511\pi$
$443$	− 15.5563i	− 0.739104i	0.929210	−	0.369552i	$-0.120489\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	31.1127	1.47158
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	45.2548i	2.13097i
$452$	0	0
$453$	− 20.0000i	− 0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	33.9411i	1.58424i
$460$	0	0
$461$	24.0000i	1.11779i	0.829238	+	0.558896i	$0.188775\pi$
$461$	24.0000i	1.11779i	−0.829238	+	0.558896i	$0.811225\pi$
$462$	0	0
$463$	−7.07107	−0.328620	−0.164310	−	0.986409i	$-0.552540\pi$
$463$	−7.07107	−0.328620	−0.164310	+	0.986409i	$0.552540\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.89949i	− 0.458094i	−0.973415	−	0.229047i	$-0.926439\pi$
$467$	− 9.89949i	− 0.458094i	0.973415	−	0.229047i	$-0.0735609\pi$
$468$	0	0
$469$	18.0000i	0.831163i
$470$	0	0
$471$	−25.4558	−1.17294
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	33.9411	1.55081	0.775405	−	0.631464i	$-0.217546\pi$
$479$	33.9411	1.55081	0.775405	+	0.631464i	$0.217546\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	42.4264i	1.93047i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.7279	−0.576757	−0.288379	−	0.957516i	$-0.593116\pi$
$487$	−12.7279	−0.576757	−0.288379	+	0.957516i	$0.593116\pi$
$488$	0	0
$489$	−26.0000	−1.17576
$490$	0	0
$491$	− 16.9706i	− 0.765871i	−0.923775	−	0.382935i	$-0.874913\pi$
$491$	− 16.9706i	− 0.765871i	0.923775	−	0.382935i	$-0.125087\pi$
$492$	0	0
$493$	24.0000i	1.08091i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	− 25.4558i	− 1.13956i	−0.821797	−	0.569780i	$-0.807028\pi$
$499$	− 25.4558i	− 1.13956i	0.821797	−	0.569780i	$-0.192972\pi$
$500$	0	0
$501$	14.0000i	0.625474i
$502$	0	0
$503$	32.5269	1.45030	0.725152	−	0.688589i	$-0.241770\pi$
$503$	32.5269	1.45030	0.725152	+	0.688589i	$0.241770\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.7279i	− 0.565267i
$508$	0	0
$509$	− 20.0000i	− 0.886484i	−0.896402	−	0.443242i	$-0.853828\pi$
$509$	− 20.0000i	− 0.886484i	0.896402	−	0.443242i	$-0.146172\pi$
$510$	0	0
$511$	25.4558	1.12610
$512$	0	0
$513$	−16.0000	−0.706417
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	−25.4558	−1.11739
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	12.7279i	0.556553i	0.960501	+	0.278277i	$0.0897632\pi$
$523$	12.7279i	0.556553i	−0.960501	+	0.278277i	$0.910237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.9706	0.739249
$528$	0	0
$529$	27.0000	1.17391
$530$	0	0
$531$	− 2.82843i	− 0.122743i
$532$	0	0
$533$	− 16.0000i	− 0.693037i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	28.0000	1.20829
$538$	0	0
$539$	− 62.2254i	− 2.68024i
$540$	0	0
$541$	40.0000i	1.71973i	0.510518	+	0.859867i	$0.329454\pi$
$541$	40.0000i	1.71973i	−0.510518	+	0.859867i	$0.670546\pi$
$542$	0	0
$543$	−22.6274	−0.971035
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.24264i	− 0.181402i	−0.995878	−	0.0907011i	$-0.971089\pi$
$547$	− 4.24264i	− 0.181402i	0.995878	−	0.0907011i	$-0.0289108\pi$
$548$	0	0
$549$	− 14.0000i	− 0.597505i
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	72.0000	3.06175
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	2.82843	0.119630
$560$	0	0
$561$	48.0000	2.02656
$562$	0	0
$563$	− 35.3553i	− 1.49005i	−0.667037	−	0.745025i	$-0.732438\pi$
$563$	− 35.3553i	− 1.49005i	0.667037	−	0.745025i	$-0.267562\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.2132	−0.890871
$568$	0	0
$569$	−28.0000	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$570$	0	0
$571$	− 22.6274i	− 0.946928i	−0.880813	−	0.473464i	$-0.843003\pi$
$571$	− 22.6274i	− 0.946928i	0.880813	−	0.473464i	$-0.156997\pi$
$572$	0	0
$573$	− 4.00000i	− 0.167102i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	8.48528i	0.352636i
$580$	0	0
$581$	54.0000i	2.24030i
$582$	0	0
$583$	−11.3137	−0.468566
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.7279i	0.525338i	0.964886	+	0.262669i	$0.0846027\pi$
$587$	12.7279i	0.525338i	−0.964886	+	0.262669i	$0.915397\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	−8.48528	−0.349038
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	39.5980	1.61793	0.808965	−	0.587857i	$-0.200028\pi$
$599$	39.5980	1.61793	0.808965	+	0.587857i	$0.200028\pi$
$600$	0	0
$601$	−32.0000	−1.30531	−0.652654	−	0.757656i	$-0.726344\pi$
$601$	−32.0000	−1.30531	−0.652654	+	0.757656i	$0.726344\pi$
$602$	0	0
$603$	4.24264i	0.172774i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.89949	−0.401808	−0.200904	−	0.979611i	$-0.564388\pi$
$607$	−9.89949	−0.401808	−0.200904	+	0.979611i	$0.564388\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	2.82843i	0.114426i
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	− 19.7990i	− 0.795789i	−0.917431	−	0.397894i	$-0.869741\pi$
$619$	− 19.7990i	− 0.795789i	0.917431	−	0.397894i	$-0.130259\pi$
$620$	0	0
$621$	40.0000i	1.60514i
$622$	0	0
$623$	−25.4558	−1.01987
$624$	0	0
$625$	0	0
$626$	0	0
$627$	22.6274i	0.903652i
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	−36.7696	−1.46377	−0.731886	−	0.681427i	$-0.761360\pi$
$631$	−36.7696	−1.46377	−0.731886	+	0.681427i	$0.761360\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.0000i	0.871672i
$638$	0	0
$639$	2.82843	0.111891
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	7.07107i	0.278856i	0.990232	+	0.139428i	$0.0445263\pi$
$643$	7.07107i	0.278856i	−0.990232	+	0.139428i	$0.955474\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.5269	1.27876	0.639382	−	0.768889i	$-0.279190\pi$
$647$	32.5269	1.27876	0.639382	+	0.768889i	$0.279190\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	16.9706i	0.665129i
$652$	0	0
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	25.4558i	0.991619i	0.868431	+	0.495809i	$0.165129\pi$
$659$	25.4558i	0.991619i	−0.868431	+	0.495809i	$0.834871\pi$
$660$	0	0
$661$	10.0000i	0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$	10.0000i	0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	0	0
$663$	−16.9706	−0.659082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.2843i	1.09517i
$668$	0	0
$669$	22.0000i	0.850569i
$670$	0	0
$671$	−79.1960	−3.05733
$672$	0	0
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.0000i	0.384331i	0.981363	+	0.192166i	$0.0615511\pi$
$677$	10.0000i	0.384331i	−0.981363	+	0.192166i	$0.938449\pi$
$678$	0	0
$679$	−42.4264	−1.62818
$680$	0	0
$681$	10.0000	0.383201
$682$	0	0
$683$	− 35.3553i	− 1.35283i	−0.736519	−	0.676417i	$-0.763532\pi$
$683$	− 35.3553i	− 1.35283i	0.736519	−	0.676417i	$-0.236468\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.2843	1.07911
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	− 22.6274i	− 0.860788i	−0.902641	−	0.430394i	$-0.858375\pi$
$691$	− 22.6274i	− 0.860788i	0.902641	−	0.430394i	$-0.141625\pi$
$692$	0	0
$693$	− 24.0000i	− 0.911685i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	48.0000	1.81813
$698$	0	0
$699$	8.48528i	0.320943i
$700$	0	0
$701$	− 18.0000i	− 0.679851i	−0.940452	−	0.339925i	$-0.889598\pi$
$701$	− 18.0000i	− 0.679851i	0.940452	−	0.339925i	$-0.110402\pi$
$702$	0	0
$703$	5.65685	0.213352
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.0000i	1.05156i	0.850620	+	0.525781i	$0.176227\pi$
$709$	28.0000i	1.05156i	−0.850620	+	0.525781i	$0.823773\pi$
$710$	0	0
$711$	16.9706	0.636446
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0000i	0.597531i
$718$	0	0
$719$	−11.3137	−0.421930	−0.210965	−	0.977494i	$-0.567661\pi$
$719$	−11.3137	−0.421930	−0.210965	+	0.977494i	$0.567661\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	0	0
$723$	− 11.3137i	− 0.420761i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.24264	0.157351	0.0786754	−	0.996900i	$-0.474931\pi$
$727$	4.24264	0.157351	0.0786754	+	0.996900i	$0.474931\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	8.48528i	0.313839i
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	− 25.4558i	− 0.936408i	−0.883620	−	0.468204i	$-0.844901\pi$
$739$	− 25.4558i	− 0.936408i	0.883620	−	0.468204i	$-0.155099\pi$
$740$	0	0
$741$	− 8.00000i	− 0.293887i
$742$	0	0
$743$	−32.5269	−1.19330	−0.596648	−	0.802503i	$-0.703501\pi$
$743$	−32.5269	−1.19330	−0.596648	+	0.802503i	$0.703501\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.7279i	0.465690i
$748$	0	0
$749$	− 30.0000i	− 1.09618i
$750$	0	0
$751$	8.48528	0.309632	0.154816	−	0.987943i	$-0.450521\pi$
$751$	8.48528	0.309632	0.154816	+	0.987943i	$0.450521\pi$
$752$	0	0
$753$	32.0000	1.16614
$754$	0	0
$755$	0	0
$756$	0	0
$757$	46.0000i	1.67190i	0.548807	+	0.835949i	$0.315082\pi$
$757$	46.0000i	1.67190i	−0.548807	+	0.835949i	$0.684918\pi$
$758$	0	0
$759$	56.5685	2.05331
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	25.4558i	0.921563i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.65685	0.204257
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	− 25.4558i	− 0.916770i
$772$	0	0
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	22.6274i	0.810711i
$780$	0	0
$781$	− 16.0000i	− 0.572525i
$782$	0	0
$783$	−22.6274	−0.808638
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 46.6690i	− 1.66357i	−0.555097	−	0.831786i	$-0.687319\pi$
$787$	− 46.6690i	− 1.66357i	0.555097	−	0.831786i	$-0.312681\pi$
$788$	0	0
$789$	− 10.0000i	− 0.356009i
$790$	0	0
$791$	42.4264	1.50851
$792$	0	0
$793$	28.0000	0.994309
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10.0000i	− 0.354218i	−0.984191	−	0.177109i	$-0.943325\pi$
$797$	− 10.0000i	− 0.354218i	0.984191	−	0.177109i	$-0.0566745\pi$
$798$	0	0
$799$	−8.48528	−0.300188
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	− 33.9411i	− 1.19776i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.48528	−0.298696
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	36.0000i	1.26258i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	8.48528i	0.296500i
$820$	0	0
$821$	26.0000i	0.907406i	0.891153	+	0.453703i	$0.149897\pi$
$821$	26.0000i	0.907406i	−0.891153	+	0.453703i	$0.850103\pi$
$822$	0	0
$823$	26.8701	0.936631	0.468316	−	0.883561i	$-0.344861\pi$
$823$	26.8701	0.936631	0.468316	+	0.883561i	$0.344861\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.07107i	0.245885i	0.992414	+	0.122943i	$0.0392331\pi$
$827$	7.07107i	0.245885i	−0.992414	+	0.122943i	$0.960767\pi$
$828$	0	0
$829$	26.0000i	0.903017i	0.892267	+	0.451509i	$0.149114\pi$
$829$	26.0000i	0.903017i	−0.892267	+	0.451509i	$0.850886\pi$
$830$	0	0
$831$	42.4264	1.47176
$832$	0	0
$833$	−66.0000	−2.28676
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	−28.2843	−0.976481	−0.488241	−	0.872709i	$-0.662361\pi$
$839$	−28.2843	−0.976481	−0.488241	+	0.872709i	$0.662361\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	− 11.3137i	− 0.389665i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−89.0955	−3.06136
$848$	0	0
$849$	−14.0000	−0.480479
$850$	0	0
$851$	− 14.1421i	− 0.484786i
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	− 2.82843i	− 0.0965047i	−0.998835	−	0.0482523i	$-0.984635\pi$
$859$	− 2.82843i	− 0.0965047i	0.998835	−	0.0482523i	$-0.0153652\pi$
$860$	0	0
$861$	48.0000i	1.63584i
$862$	0	0
$863$	55.1543	1.87748	0.938738	−	0.344633i	$-0.111997\pi$
$863$	55.1543	1.87748	0.938738	+	0.344633i	$0.111997\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 26.8701i	− 0.912555i
$868$	0	0
$869$	− 96.0000i	− 3.25658i
$870$	0	0
$871$	−8.48528	−0.287513
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	0	0
$879$	−8.48528	−0.286201
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	21.2132i	0.713881i	0.934127	+	0.356941i	$0.116180\pi$
$883$	21.2132i	0.713881i	−0.934127	+	0.356941i	$0.883820\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.8701	0.902208	0.451104	−	0.892471i	$-0.351030\pi$
$887$	26.8701	0.902208	0.451104	+	0.892471i	$0.351030\pi$
$888$	0	0
$889$	42.0000	1.40863
$890$	0	0
$891$	28.2843i	0.947559i
$892$	0	0
$893$	− 4.00000i	− 0.133855i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−20.0000	−0.667781
$898$	0	0
$899$	11.3137i	0.377333i
$900$	0	0
$901$	12.0000i	0.399778i
$902$	0	0
$903$	−8.48528	−0.282372
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.41421i	− 0.0469582i	−0.999724	−	0.0234791i	$-0.992526\pi$
$907$	− 1.41421i	− 0.0469582i	0.999724	−	0.0234791i	$-0.00747431\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.1127	1.03081	0.515405	−	0.856947i	$-0.327642\pi$
$911$	31.1127	1.03081	0.515405	+	0.856947i	$0.327642\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	−39.5980	−1.30622	−0.653108	−	0.757264i	$-0.726535\pi$
$919$	−39.5980	−1.30622	−0.653108	+	0.757264i	$0.726535\pi$
$920$	0	0
$921$	22.0000	0.724925
$922$	0	0
$923$	5.65685i	0.186198i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.89949	0.325142
$928$	0	0
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	0	0
$931$	− 31.1127i	− 1.01968i
$932$	0	0
$933$	− 4.00000i	− 0.130954i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	8.48528i	0.276907i
$940$	0	0
$941$	− 48.0000i	− 1.56476i	−0.622804	−	0.782378i	$-0.714007\pi$
$941$	− 48.0000i	− 1.56476i	0.622804	−	0.782378i	$-0.285993\pi$
$942$	0	0
$943$	56.5685	1.84213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.5563i	0.505513i	0.967530	+	0.252757i	$0.0813372\pi$
$947$	15.5563i	0.505513i	−0.967530	+	0.252757i	$0.918663\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	−2.82843	−0.0917180
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.0000i	1.03441i
$958$	0	0
$959$	−8.48528	−0.274004
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	− 7.07107i	− 0.227862i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.24264	−0.136434	−0.0682171	−	0.997671i	$-0.521731\pi$
$967$	−4.24264	−0.136434	−0.0682171	+	0.997671i	$0.521731\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	28.2843i	0.907685i	0.891082	+	0.453843i	$0.149947\pi$
$971$	28.2843i	0.907685i	−0.891082	+	0.453843i	$0.850053\pi$
$972$	0	0
$973$	36.0000i	1.15411i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	33.9411i	1.08476i
$980$	0	0
$981$	6.00000i	0.191565i
$982$	0	0
$983$	18.3848	0.586383	0.293192	−	0.956054i	$-0.405283\pi$
$983$	18.3848	0.586383	0.293192	+	0.956054i	$0.405283\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 8.48528i	− 0.270089i
$988$	0	0
$989$	10.0000i	0.317982i
$990$	0	0
$991$	−8.48528	−0.269544	−0.134772	−	0.990877i	$-0.543030\pi$
$991$	−8.48528	−0.269544	−0.134772	+	0.990877i	$0.543030\pi$
$992$	0	0
$993$	−32.0000	−1.01549
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	11.3137	0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.d.s.1601.2		4
4.3	odd	2	inner	3200.2.d.s.1601.3		4
5.2	odd	4		3200.2.f.j.449.2		4
5.3	odd	4		3200.2.f.i.449.3		4
5.4	even	2		640.2.d.d.321.3	yes	4
8.3	odd	2	inner	3200.2.d.s.1601.1		4
8.5	even	2	inner	3200.2.d.s.1601.4		4
15.14	odd	2		5760.2.k.o.2881.3		4
16.3	odd	4		6400.2.a.bl.1.1		2
16.5	even	4		6400.2.a.bn.1.1		2
16.11	odd	4		6400.2.a.bn.1.2		2
16.13	even	4		6400.2.a.bl.1.2		2
20.3	even	4		3200.2.f.i.449.2		4
20.7	even	4		3200.2.f.j.449.3		4
20.19	odd	2		640.2.d.d.321.1	✓	4
40.3	even	4		3200.2.f.j.449.4		4
40.13	odd	4		3200.2.f.j.449.1		4
40.19	odd	2		640.2.d.d.321.4	yes	4
40.27	even	4		3200.2.f.i.449.1		4
40.29	even	2		640.2.d.d.321.2	yes	4
40.37	odd	4		3200.2.f.i.449.4		4
60.59	even	2		5760.2.k.o.2881.4		4
80.19	odd	4		1280.2.a.f.1.2		2
80.29	even	4		1280.2.a.f.1.1		2
80.59	odd	4		1280.2.a.j.1.1		2
80.69	even	4		1280.2.a.j.1.2		2
120.29	odd	2		5760.2.k.o.2881.1		4
120.59	even	2		5760.2.k.o.2881.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.d.321.1	✓	4	20.19	odd	2
640.2.d.d.321.2	yes	4	40.29	even	2
640.2.d.d.321.3	yes	4	5.4	even	2
640.2.d.d.321.4	yes	4	40.19	odd	2
1280.2.a.f.1.1		2	80.29	even	4
1280.2.a.f.1.2		2	80.19	odd	4
1280.2.a.j.1.1		2	80.59	odd	4
1280.2.a.j.1.2		2	80.69	even	4
3200.2.d.s.1601.1		4	8.3	odd	2	inner
3200.2.d.s.1601.2		4	1.1	even	1	trivial
3200.2.d.s.1601.3		4	4.3	odd	2	inner
3200.2.d.s.1601.4		4	8.5	even	2	inner
3200.2.f.i.449.1		4	40.27	even	4
3200.2.f.i.449.2		4	20.3	even	4
3200.2.f.i.449.3		4	5.3	odd	4
3200.2.f.i.449.4		4	40.37	odd	4
3200.2.f.j.449.1		4	40.13	odd	4
3200.2.f.j.449.2		4	5.2	odd	4
3200.2.f.j.449.3		4	20.7	even	4
3200.2.f.j.449.4		4	40.3	even	4
5760.2.k.o.2881.1		4	120.29	odd	2
5760.2.k.o.2881.2		4	120.59	even	2
5760.2.k.o.2881.3		4	15.14	odd	2
5760.2.k.o.2881.4		4	60.59	even	2
6400.2.a.bl.1.1		2	16.3	odd	4
6400.2.a.bl.1.2		2	16.13	even	4
6400.2.a.bn.1.1		2	16.5	even	4
6400.2.a.bn.1.2		2	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} +4.24264 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{3} +4.24264 q^{7} +1.00000 q^{9} -5.65685i q^{11} +2.00000i q^{13} -6.00000 q^{17} -2.82843i q^{19} -6.00000i q^{21} -7.07107 q^{23} -5.65685i q^{27} -4.00000i q^{29} -2.82843 q^{31} -8.00000 q^{33} +2.00000i q^{37} +2.82843 q^{39} -8.00000 q^{41} -1.41421i q^{43} +1.41421 q^{47} +11.0000 q^{49} +8.48528i q^{51} -2.00000i q^{53} -4.00000 q^{57} -2.82843i q^{59} -14.0000i q^{61} +4.24264 q^{63} +4.24264i q^{67} +10.0000i q^{69} +2.82843 q^{71} +6.00000 q^{73} -24.0000i q^{77} +16.9706 q^{79} -5.00000 q^{81} +12.7279i q^{83} -5.65685 q^{87} -6.00000 q^{89} +8.48528i q^{91} +4.00000i q^{93} -10.0000 q^{97} -5.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} - 24 q^{17} - 32 q^{33} - 32 q^{41} + 44 q^{49} - 16 q^{57} + 24 q^{73} - 20 q^{81} - 24 q^{89} - 40 q^{97}+O(q^{100}) \) 4 * q + 4 * q^9 - 24 * q^17 - 32 * q^33 - 32 * q^41 + 44 * q^49 - 16 * q^57 + 24 * q^73 - 20 * q^81 - 24 * q^89 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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