LMFDB - Embedded newform 2400.2.k.f.1201.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1201,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2400.1201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2400 = 2^{5} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2400.k (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64$ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$2^{14}$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1201.2
Root			$-0.450129 - 1.34067i$ of defining polynomial
Character	$\chi$	$=$	2400.1201
Dual form			2400.2.k.f.1201.8

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.64265 q^{7} -1.00000 q^{9} +1.51363i q^{11} +3.87086i q^{13} +3.31415 q^{17} -7.08582i q^{19} +2.64265i q^{21} -4.82778 q^{23} +1.00000i q^{27} -2.18513i q^{29} +7.36266 q^{31} +1.51363 q^{33} +7.87086i q^{37} +3.87086 q^{39} +8.72532 q^{41} -1.01641i q^{43} +7.08582 q^{47} -0.0164068 q^{49} -3.31415i q^{51} +4.50820i q^{53} -7.08582 q^{57} +6.79893i q^{59} -3.60104i q^{61} +2.64265 q^{63} -1.01641i q^{67} +4.82778i q^{69} +6.72532 q^{71} +15.5146 q^{73} -4.00000i q^{77} +7.36266 q^{79} +1.00000 q^{81} -7.74173i q^{83} -2.18513 q^{87} +14.7581 q^{89} -10.2293i q^{91} -7.36266i q^{93} +11.1444 q^{97} -1.51363i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100})$ 12 * q - 12 * q^9 + 32 * q^31 - 16 * q^39 - 8 * q^41 + 12 * q^49 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$ .

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.64265	−0.998827	−0.499414	−	0.866364i	$-0.666451\pi$
$7$	−2.64265	−0.998827	−0.499414	+	0.866364i	$0.666451\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.51363i	0.456377i	0.973617	+	0.228189i	$0.0732803\pi$
$11$	1.51363i	0.456377i	−0.973617	+	0.228189i	$0.926720\pi$
$12$	0	0
$13$	3.87086i	1.07358i	0.843714	+	0.536792i	$0.180364\pi$
$13$	3.87086i	1.07358i	−0.843714	+	0.536792i	$0.819636\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.31415	0.803800	0.401900	−	0.915684i	$-0.368350\pi$
$17$	3.31415	0.803800	0.401900	+	0.915684i	$0.368350\pi$
$18$	0	0
$19$	− 7.08582i	− 1.62560i	−0.582545	−	0.812799i	$-0.697943\pi$
$19$	− 7.08582i	− 1.62560i	0.582545	−	0.812799i	$-0.302057\pi$
$20$	0	0
$21$	2.64265i	0.576673i
$22$	0	0
$23$	−4.82778	−1.00666	−0.503331	−	0.864094i	$-0.667892\pi$
$23$	−4.82778	−1.00666	−0.503331	+	0.864094i	$0.667892\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 2.18513i	− 0.405769i	−0.979203	−	0.202885i	$-0.934968\pi$
$29$	− 2.18513i	− 0.405769i	0.979203	−	0.202885i	$-0.0650316\pi$
$30$	0	0
$31$	7.36266	1.32237	0.661187	−	0.750222i	$-0.270053\pi$
$31$	7.36266	1.32237	0.661187	+	0.750222i	$0.270053\pi$
$32$	0	0
$33$	1.51363	0.263490
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.87086i	1.29396i	0.762506	+	0.646981i	$0.223969\pi$
$37$	7.87086i	1.29396i	−0.762506	+	0.646981i	$0.776031\pi$
$38$	0	0
$39$	3.87086	0.619834
$40$	0	0
$41$	8.72532	1.36267	0.681333	−	0.731973i	$-0.261400\pi$
$41$	8.72532	1.36267	0.681333	+	0.731973i	$0.261400\pi$
$42$	0	0
$43$	− 1.01641i	− 0.155001i	−0.996992	−	0.0775003i	$-0.975306\pi$
$43$	− 1.01641i	− 0.155001i	0.996992	−	0.0775003i	$-0.0246939\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.08582	1.03357	0.516786	−	0.856114i	$-0.327128\pi$
$47$	7.08582	1.03357	0.516786	+	0.856114i	$0.327128\pi$
$48$	0	0
$49$	−0.0164068	−0.00234382
$50$	0	0
$51$	− 3.31415i	− 0.464074i
$52$	0	0
$53$	4.50820i	0.619249i	0.950859	+	0.309625i	$0.100203\pi$
$53$	4.50820i	0.619249i	−0.950859	+	0.309625i	$0.899797\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.08582	−0.938539
$58$	0	0
$59$	6.79893i	0.885145i	0.896733	+	0.442573i	$0.145934\pi$
$59$	6.79893i	0.885145i	−0.896733	+	0.442573i	$0.854066\pi$
$60$	0	0
$61$	− 3.60104i	− 0.461065i	−0.973065	−	0.230533i	$-0.925953\pi$
$61$	− 3.60104i	− 0.461065i	0.973065	−	0.230533i	$-0.0740469\pi$
$62$	0	0
$63$	2.64265	0.332942
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.01641i	− 0.124174i	−0.998071	−	0.0620869i	$-0.980224\pi$
$67$	− 1.01641i	− 0.124174i	0.998071	−	0.0620869i	$-0.0197756\pi$
$68$	0	0
$69$	4.82778i	0.581197i
$70$	0	0
$71$	6.72532	0.798149	0.399074	−	0.916919i	$-0.369331\pi$
$71$	6.72532	0.798149	0.399074	+	0.916919i	$0.369331\pi$
$72$	0	0
$73$	15.5146	1.81585	0.907925	−	0.419132i	$-0.137666\pi$
$73$	15.5146	1.81585	0.907925	+	0.419132i	$0.137666\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	7.36266	0.828364	0.414182	−	0.910194i	$-0.364068\pi$
$79$	7.36266	0.828364	0.414182	+	0.910194i	$0.364068\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 7.74173i	− 0.849765i	−0.905248	−	0.424883i	$-0.860315\pi$
$83$	− 7.74173i	− 0.849765i	0.905248	−	0.424883i	$-0.139685\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.18513	−0.234271
$88$	0	0
$89$	14.7581	1.56436	0.782180	−	0.623053i	$-0.214108\pi$
$89$	14.7581	1.56436	0.782180	+	0.623053i	$0.214108\pi$
$90$	0	0
$91$	− 10.2293i	− 1.07233i
$92$	0	0
$93$	− 7.36266i	− 0.763472i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.1444	1.13154	0.565769	−	0.824563i	$-0.308579\pi$
$97$	11.1444	1.13154	0.565769	+	0.824563i	$0.308579\pi$
$98$	0	0
$99$	− 1.51363i	− 0.152126i
$100$	0	0
$101$	− 13.3295i	− 1.32633i	−0.748471	−	0.663167i	$-0.769212\pi$
$101$	− 13.3295i	− 1.32633i	0.748471	−	0.663167i	$-0.230788\pi$
$102$	0	0
$103$	0.958386	0.0944326	0.0472163	−	0.998885i	$-0.484965\pi$
$103$	0.958386	0.0944326	0.0472163	+	0.998885i	$0.484965\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	0.769233i	0.0736792i	0.999321	+	0.0368396i	$0.0117291\pi$
$109$	0.769233i	0.0736792i	−0.999321	+	0.0368396i	$0.988271\pi$
$110$	0	0
$111$	7.87086	0.747069
$112$	0	0
$113$	−14.4585	−1.36014	−0.680071	−	0.733146i	$-0.738051\pi$
$113$	−14.4585	−1.36014	−0.680071	+	0.733146i	$0.738051\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.87086i	− 0.357862i
$118$	0	0
$119$	−8.75814	−0.802857
$120$	0	0
$121$	8.70892	0.791720
$122$	0	0
$123$	− 8.72532i	− 0.786736i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.5290	−1.02303	−0.511516	−	0.859274i	$-0.670916\pi$
$127$	−11.5290	−1.02303	−0.511516	+	0.859274i	$0.670916\pi$
$128$	0	0
$129$	−1.01641	−0.0894896
$130$	0	0
$131$	− 7.37270i	− 0.644156i	−0.946713	−	0.322078i	$-0.895619\pi$
$131$	− 7.37270i	− 0.644156i	0.946713	−	0.322078i	$-0.104381\pi$
$132$	0	0
$133$	18.7253i	1.62369i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.88792	−0.332167	−0.166084	−	0.986112i	$-0.553112\pi$
$137$	−3.88792	−0.332167	−0.166084	+	0.986112i	$0.553112\pi$
$138$	0	0
$139$	14.6291i	1.24083i	0.784275	+	0.620414i	$0.213035\pi$
$139$	14.6291i	1.24083i	−0.784275	+	0.620414i	$0.786965\pi$
$140$	0	0
$141$	− 7.08582i	− 0.596733i
$142$	0	0
$143$	−5.85907	−0.489960
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.0164068i	0.00135321i
$148$	0	0
$149$	− 11.0715i	− 0.907010i	−0.891254	−	0.453505i	$-0.850173\pi$
$149$	− 11.0715i	− 0.907010i	0.891254	−	0.453505i	$-0.149827\pi$
$150$	0	0
$151$	−0.637339	−0.0518659	−0.0259329	−	0.999664i	$-0.508256\pi$
$151$	−0.637339	−0.0518659	−0.0259329	+	0.999664i	$0.508256\pi$
$152$	0	0
$153$	−3.31415	−0.267933
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 0.129135i	− 0.0103061i	−0.999987	−	0.00515306i	$-0.998360\pi$
$157$	− 0.129135i	− 0.0103061i	0.999987	−	0.00515306i	$-0.00164028\pi$
$158$	0	0
$159$	4.50820	0.357524
$160$	0	0
$161$	12.7581	1.00548
$162$	0	0
$163$	19.4835i	1.52606i	0.646362	+	0.763031i	$0.276290\pi$
$163$	19.4835i	1.52606i	−0.646362	+	0.763031i	$0.723710\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.80052	−0.139328	−0.0696641	−	0.997571i	$-0.522193\pi$
$167$	−1.80052	−0.139328	−0.0696641	+	0.997571i	$0.522193\pi$
$168$	0	0
$169$	−1.98359	−0.152584
$170$	0	0
$171$	7.08582i	0.541866i
$172$	0	0
$173$	− 23.2335i	− 1.76641i	−0.468985	−	0.883206i	$-0.655380\pi$
$173$	− 23.2335i	− 1.76641i	0.468985	−	0.883206i	$-0.344620\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.79893	0.511039
$178$	0	0
$179$	2.85664i	0.213515i	0.994285	+	0.106757i	$0.0340468\pi$
$179$	2.85664i	0.213515i	−0.994285	+	0.106757i	$0.965953\pi$
$180$	0	0
$181$	5.28530i	0.392853i	0.980519	+	0.196427i	$0.0629337\pi$
$181$	5.28530i	0.392853i	−0.980519	+	0.196427i	$0.937066\pi$
$182$	0	0
$183$	−3.60104	−0.266196
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.01641i	0.366836i
$188$	0	0
$189$	− 2.64265i	− 0.192224i
$190$	0	0
$191$	5.96719	0.431770	0.215885	−	0.976419i	$-0.430736\pi$
$191$	5.96719	0.431770	0.215885	+	0.976419i	$0.430736\pi$
$192$	0	0
$193$	−14.9409	−1.07547	−0.537733	−	0.843115i	$-0.680719\pi$
$193$	−14.9409	−1.07547	−0.537733	+	0.843115i	$0.680719\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.23353i	− 0.230379i	−0.993344	−	0.115190i	$-0.963252\pi$
$197$	− 3.23353i	− 0.230379i	0.993344	−	0.115190i	$-0.0367476\pi$
$198$	0	0
$199$	8.12080	0.575668	0.287834	−	0.957680i	$-0.407065\pi$
$199$	8.12080	0.575668	0.287834	+	0.957680i	$0.407065\pi$
$200$	0	0
$201$	−1.01641	−0.0716918
$202$	0	0
$203$	5.77454i	0.405293i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.82778	0.335554
$208$	0	0
$209$	10.7253	0.741886
$210$	0	0
$211$	13.7141i	0.944119i	0.881567	+	0.472059i	$0.156489\pi$
$211$	13.7141i	0.944119i	−0.881567	+	0.472059i	$0.843511\pi$
$212$	0	0
$213$	− 6.72532i	− 0.460812i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−19.4569	−1.32082
$218$	0	0
$219$	− 15.5146i	− 1.04838i
$220$	0	0
$221$	12.8286i	0.862947i
$222$	0	0
$223$	9.84472	0.659251	0.329626	−	0.944112i	$-0.393077\pi$
$223$	9.84472	0.659251	0.329626	+	0.944112i	$0.393077\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.70892i	− 0.378914i	−0.981889	−	0.189457i	$-0.939327\pi$
$227$	− 5.70892i	− 0.378914i	0.981889	−	0.189457i	$-0.0606728\pi$
$228$	0	0
$229$	− 0.769233i	− 0.0508324i	−0.999677	−	0.0254162i	$-0.991909\pi$
$229$	− 0.769233i	− 0.0508324i	0.999677	−	0.0254162i	$-0.00809109\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	18.4008	1.20548	0.602739	−	0.797939i	$-0.294076\pi$
$233$	18.4008	1.20548	0.602739	+	0.797939i	$0.294076\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 7.36266i	− 0.478256i
$238$	0	0
$239$	−10.0328	−0.648969	−0.324484	−	0.945891i	$-0.605191\pi$
$239$	−10.0328	−0.648969	−0.324484	+	0.945891i	$0.605191\pi$
$240$	0	0
$241$	10.7581	0.692992	0.346496	−	0.938051i	$-0.387371\pi$
$241$	10.7581	0.692992	0.346496	+	0.938051i	$0.387371\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	27.4282	1.74522
$248$	0	0
$249$	−7.74173	−0.490612
$250$	0	0
$251$	12.6580i	0.798966i	0.916741	+	0.399483i	$0.130810\pi$
$251$	12.6580i	0.798966i	−0.916741	+	0.399483i	$0.869190\pi$
$252$	0	0
$253$	− 7.30749i	− 0.459418i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.3110	−0.830316	−0.415158	−	0.909749i	$-0.636274\pi$
$257$	−13.3110	−0.830316	−0.415158	+	0.909749i	$0.636274\pi$
$258$	0	0
$259$	− 20.7999i	− 1.29244i
$260$	0	0
$261$	2.18513i	0.135256i
$262$	0	0
$263$	18.4256	1.13617	0.568087	−	0.822969i	$-0.307684\pi$
$263$	18.4256	1.13617	0.568087	+	0.822969i	$0.307684\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 14.7581i	− 0.903183i
$268$	0	0
$269$	− 3.86940i	− 0.235921i	−0.993018	−	0.117961i	$-0.962364\pi$
$269$	− 3.86940i	− 0.235921i	0.993018	−	0.117961i	$-0.0376357\pi$
$270$	0	0
$271$	17.3955	1.05670	0.528350	−	0.849027i	$-0.322811\pi$
$271$	17.3955	1.05670	0.528350	+	0.849027i	$0.322811\pi$
$272$	0	0
$273$	−10.2293	−0.619108
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.887271i	− 0.0533110i	−0.999645	−	0.0266555i	$-0.991514\pi$
$277$	− 0.887271i	− 0.0533110i	0.999645	−	0.0266555i	$-0.00848571\pi$
$278$	0	0
$279$	−7.36266	−0.440791
$280$	0	0
$281$	−13.4835	−0.804356	−0.402178	−	0.915562i	$-0.631747\pi$
$281$	−13.4835	−0.804356	−0.402178	+	0.915562i	$0.631747\pi$
$282$	0	0
$283$	28.4342i	1.69024i	0.534577	+	0.845120i	$0.320471\pi$
$283$	28.4342i	1.69024i	−0.534577	+	0.845120i	$0.679529\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.0580	−1.36107
$288$	0	0
$289$	−6.01641	−0.353906
$290$	0	0
$291$	− 11.1444i	− 0.653294i
$292$	0	0
$293$	− 7.99166i	− 0.466878i	−0.972371	−	0.233439i	$-0.925002\pi$
$293$	− 7.99166i	− 0.466878i	0.972371	−	0.233439i	$-0.0749979\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.51363	−0.0878299
$298$	0	0
$299$	− 18.6877i	− 1.08074i
$300$	0	0
$301$	2.68601i	0.154819i
$302$	0	0
$303$	−13.3295	−0.765760
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.4506i	0.995961i	0.867188	+	0.497980i	$0.165925\pi$
$307$	17.4506i	0.995961i	−0.867188	+	0.497980i	$0.834075\pi$
$308$	0	0
$309$	− 0.958386i	− 0.0545207i
$310$	0	0
$311$	−21.4506	−1.21635	−0.608177	−	0.793801i	$-0.708099\pi$
$311$	−21.4506	−1.21635	−0.608177	+	0.793801i	$0.708099\pi$
$312$	0	0
$313$	−7.73879	−0.437422	−0.218711	−	0.975790i	$-0.570185\pi$
$313$	−7.73879	−0.437422	−0.218711	+	0.975790i	$0.570185\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.2335i	− 0.630938i	−0.948936	−	0.315469i	$-0.897838\pi$
$317$	− 11.2335i	− 0.630938i	0.948936	−	0.315469i	$-0.102162\pi$
$318$	0	0
$319$	3.30749	0.185184
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 23.4835i	− 1.30665i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.769233	0.0425387
$328$	0	0
$329$	−18.7253	−1.03236
$330$	0	0
$331$	− 8.00084i	− 0.439766i	−0.975526	−	0.219883i	$-0.929432\pi$
$331$	− 8.00084i	− 0.439766i	0.975526	−	0.219883i	$-0.0705676\pi$
$332$	0	0
$333$	− 7.87086i	− 0.431321i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.5692	1.17495	0.587474	−	0.809243i	$-0.300123\pi$
$337$	21.5692	1.17495	0.587474	+	0.809243i	$0.300123\pi$
$338$	0	0
$339$	14.4585i	0.785279i
$340$	0	0
$341$	11.1444i	0.603501i
$342$	0	0
$343$	18.5419	1.00117
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.7089i	1.16540i	0.812689	+	0.582698i	$0.198003\pi$
$347$	21.7089i	1.16540i	−0.812689	+	0.582698i	$0.801997\pi$
$348$	0	0
$349$	24.7422i	1.32442i	0.749318	+	0.662211i	$0.230382\pi$
$349$	24.7422i	1.32442i	−0.749318	+	0.662211i	$0.769618\pi$
$350$	0	0
$351$	−3.87086	−0.206611
$352$	0	0
$353$	−3.31415	−0.176394	−0.0881972	−	0.996103i	$-0.528111\pi$
$353$	−3.31415	−0.176394	−0.0881972	+	0.996103i	$0.528111\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.75814i	0.463530i
$358$	0	0
$359$	16.7581	0.884461	0.442230	−	0.896902i	$-0.354187\pi$
$359$	16.7581	0.884461	0.442230	+	0.896902i	$0.354187\pi$
$360$	0	0
$361$	−31.2088	−1.64257
$362$	0	0
$363$	− 8.70892i	− 0.457100i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.5324	1.48938	0.744690	−	0.667411i	$-0.232597\pi$
$367$	28.5324	1.48938	0.744690	+	0.667411i	$0.232597\pi$
$368$	0	0
$369$	−8.72532	−0.454222
$370$	0	0
$371$	− 11.9136i	− 0.618523i
$372$	0	0
$373$	37.5798i	1.94581i	0.231211	+	0.972904i	$0.425731\pi$
$373$	37.5798i	1.94581i	−0.231211	+	0.972904i	$0.574269\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.45836	0.435628
$378$	0	0
$379$	6.74456i	0.346445i	0.984883	+	0.173222i	$0.0554179\pi$
$379$	6.74456i	0.346445i	−0.984883	+	0.173222i	$0.944582\pi$
$380$	0	0
$381$	11.5290i	0.590648i
$382$	0	0
$383$	21.8312	1.11552	0.557762	−	0.830001i	$-0.311660\pi$
$383$	21.8312	1.11552	0.557762	+	0.830001i	$0.311660\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.01641i	0.0516669i
$388$	0	0
$389$	8.81344i	0.446859i	0.974720	+	0.223429i	$0.0717252\pi$
$389$	8.81344i	0.446859i	−0.974720	+	0.223429i	$0.928275\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−7.37270	−0.371904
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.821644i	0.0412372i	0.999787	+	0.0206186i	$0.00656356\pi$
$397$	0.821644i	0.0412372i	−0.999787	+	0.0206186i	$0.993436\pi$
$398$	0	0
$399$	18.7253	0.937439
$400$	0	0
$401$	−12.7253	−0.635472	−0.317736	−	0.948179i	$-0.602923\pi$
$401$	−12.7253	−0.635472	−0.317736	+	0.948179i	$0.602923\pi$
$402$	0	0
$403$	28.4999i	1.41968i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.9136	−0.590535
$408$	0	0
$409$	2.25827	0.111664	0.0558321	−	0.998440i	$-0.482219\pi$
$409$	2.25827	0.111664	0.0558321	+	0.998440i	$0.482219\pi$
$410$	0	0
$411$	3.88792i	0.191777i
$412$	0	0
$413$	− 17.9672i	− 0.884107i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.6291	0.716392
$418$	0	0
$419$	33.4579i	1.63453i	0.576264	+	0.817263i	$0.304510\pi$
$419$	33.4579i	1.63453i	−0.576264	+	0.817263i	$0.695490\pi$
$420$	0	0
$421$	− 11.3398i	− 0.552669i	−0.961061	−	0.276335i	$-0.910880\pi$
$421$	− 11.3398i	− 0.552669i	0.961061	−	0.276335i	$-0.0891198\pi$
$422$	0	0
$423$	−7.08582	−0.344524
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.51627i	0.460525i
$428$	0	0
$429$	5.85907i	0.282878i
$430$	0	0
$431$	−10.6597	−0.513459	−0.256730	−	0.966483i	$-0.582645\pi$
$431$	−10.6597	−0.513459	−0.256730	+	0.966483i	$0.582645\pi$
$432$	0	0
$433$	26.5132	1.27414	0.637072	−	0.770805i	$-0.280146\pi$
$433$	26.5132	1.27414	0.637072	+	0.770805i	$0.280146\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.2088i	1.63643i
$438$	0	0
$439$	−32.8789	−1.56923	−0.784613	−	0.619986i	$-0.787138\pi$
$439$	−32.8789	−1.56923	−0.784613	+	0.619986i	$0.787138\pi$
$440$	0	0
$441$	0.0164068	0.000781274 0
$442$	0	0
$443$	− 5.70892i	− 0.271239i	−0.990761	−	0.135619i	$-0.956698\pi$
$443$	− 5.70892i	− 0.271239i	0.990761	−	0.135619i	$-0.0433024\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−11.0715	−0.523662
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	13.2069i	0.621890i
$452$	0	0
$453$	0.637339i	0.0299448i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.94229	−0.184413	−0.0922064	−	0.995740i	$-0.529392\pi$
$457$	−3.94229	−0.184413	−0.0922064	+	0.995740i	$0.529392\pi$
$458$	0	0
$459$	3.31415i	0.154691i
$460$	0	0
$461$	− 33.8969i	− 1.57874i	−0.613920	−	0.789369i	$-0.710408\pi$
$461$	− 33.8969i	− 1.57874i	0.613920	−	0.789369i	$-0.289592\pi$
$462$	0	0
$463$	−22.8688	−1.06280	−0.531402	−	0.847120i	$-0.678335\pi$
$463$	−22.8688	−1.06280	−0.531402	+	0.847120i	$0.678335\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 15.7417i	− 0.728440i	−0.931313	−	0.364220i	$-0.881336\pi$
$467$	− 15.7417i	− 0.728440i	0.931313	−	0.364220i	$-0.118664\pi$
$468$	0	0
$469$	2.68601i	0.124028i
$470$	0	0
$471$	−0.129135	−0.00595024
$472$	0	0
$473$	1.53847	0.0707388
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.50820i	− 0.206416i
$478$	0	0
$479$	−20.6925	−0.945465	−0.472732	−	0.881206i	$-0.656732\pi$
$479$	−20.6925	−0.945465	−0.472732	+	0.881206i	$0.656732\pi$
$480$	0	0
$481$	−30.4671	−1.38918
$482$	0	0
$483$	− 12.7581i	− 0.580515i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−30.8401	−1.39750	−0.698750	−	0.715366i	$-0.746260\pi$
$487$	−30.8401	−1.39750	−0.698750	+	0.715366i	$0.746260\pi$
$488$	0	0
$489$	19.4835	0.881072
$490$	0	0
$491$	10.9737i	0.495238i	0.968858	+	0.247619i	$0.0796481\pi$
$491$	10.9737i	0.495238i	−0.968858	+	0.247619i	$0.920352\pi$
$492$	0	0
$493$	− 7.24186i	− 0.326157i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.7727	−0.797213
$498$	0	0
$499$	− 3.71729i	− 0.166409i	−0.996533	−	0.0832044i	$-0.973485\pi$
$499$	− 3.71729i	− 0.166409i	0.996533	−	0.0832044i	$-0.0265154\pi$
$500$	0	0
$501$	1.80052i	0.0804412i
$502$	0	0
$503$	−39.9451	−1.78107	−0.890533	−	0.454919i	$-0.849668\pi$
$503$	−39.9451	−1.78107	−0.890533	+	0.454919i	$0.849668\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.98359i	0.0880945i
$508$	0	0
$509$	− 0.0728979i	− 0.00323114i	−0.999999	−	0.00161557i	$-0.999486\pi$
$509$	− 0.0728979i	− 0.00323114i	0.999999	−	0.00161557i	$-0.000514253\pi$
$510$	0	0
$511$	−40.9997	−1.81372
$512$	0	0
$513$	7.08582	0.312846
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.7253i	0.471699i
$518$	0	0
$519$	−23.2335	−1.01984
$520$	0	0
$521$	−11.9672	−0.524292	−0.262146	−	0.965028i	$-0.584430\pi$
$521$	−11.9672	−0.524292	−0.262146	+	0.965028i	$0.584430\pi$
$522$	0	0
$523$	− 16.0656i	− 0.702501i	−0.936282	−	0.351250i	$-0.885757\pi$
$523$	− 16.0656i	− 0.702501i	0.936282	−	0.351250i	$-0.114243\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.4010	1.06292
$528$	0	0
$529$	0.307491	0.0133692
$530$	0	0
$531$	− 6.79893i	− 0.295048i
$532$	0	0
$533$	33.7745i	1.46294i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.85664	0.123273
$538$	0	0
$539$	− 0.0248338i	− 0.00106967i
$540$	0	0
$541$	15.8559i	0.681698i	0.940118	+	0.340849i	$0.110715\pi$
$541$	15.8559i	0.681698i	−0.940118	+	0.340849i	$0.889285\pi$
$542$	0	0
$543$	5.28530	0.226814
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.95078i	0.211680i	0.994383	+	0.105840i	$0.0337531\pi$
$547$	4.95078i	0.211680i	−0.994383	+	0.105840i	$0.966247\pi$
$548$	0	0
$549$	3.60104i	0.153688i
$550$	0	0
$551$	−15.4835	−0.659618
$552$	0	0
$553$	−19.4569	−0.827393
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.26634i	0.0536565i	0.999640	+	0.0268283i	$0.00854073\pi$
$557$	1.26634i	0.0536565i	−0.999640	+	0.0268283i	$0.991459\pi$
$558$	0	0
$559$	3.93437	0.166406
$560$	0	0
$561$	5.01641	0.211793
$562$	0	0
$563$	5.70892i	0.240602i	0.992737	+	0.120301i	$0.0383860\pi$
$563$	5.70892i	0.240602i	−0.992737	+	0.120301i	$0.961614\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.64265	−0.110981
$568$	0	0
$569$	−2.75814	−0.115627	−0.0578135	−	0.998327i	$-0.518413\pi$
$569$	−2.75814	−0.115627	−0.0578135	+	0.998327i	$0.518413\pi$
$570$	0	0
$571$	− 25.7735i	− 1.07859i	−0.842118	−	0.539294i	$-0.818691\pi$
$571$	− 25.7735i	− 1.07859i	0.842118	−	0.539294i	$-0.181309\pi$
$572$	0	0
$573$	− 5.96719i	− 0.249283i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−32.7135	−1.36188	−0.680941	−	0.732338i	$-0.738429\pi$
$577$	−32.7135	−1.36188	−0.680941	+	0.732338i	$0.738429\pi$
$578$	0	0
$579$	14.9409i	0.620921i
$580$	0	0
$581$	20.4587i	0.848769i
$582$	0	0
$583$	−6.82376	−0.282611
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 43.4835i	− 1.79475i	−0.441264	−	0.897377i	$-0.645470\pi$
$587$	− 43.4835i	− 1.79475i	0.441264	−	0.897377i	$-0.354530\pi$
$588$	0	0
$589$	− 52.1705i	− 2.14965i
$590$	0	0
$591$	−3.23353	−0.133009
$592$	0	0
$593$	7.83021	0.321548	0.160774	−	0.986991i	$-0.448601\pi$
$593$	7.83021	0.321548	0.160774	+	0.986991i	$0.448601\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.12080i	− 0.332362i
$598$	0	0
$599$	32.7581	1.33846	0.669231	−	0.743055i	$-0.266624\pi$
$599$	32.7581	1.33846	0.669231	+	0.743055i	$0.266624\pi$
$600$	0	0
$601$	17.8074	0.726377	0.363189	−	0.931716i	$-0.381688\pi$
$601$	17.8074	0.726377	0.363189	+	0.931716i	$0.381688\pi$
$602$	0	0
$603$	1.01641i	0.0413913i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.41188	0.138484	0.0692420	−	0.997600i	$-0.477942\pi$
$607$	3.41188	0.138484	0.0692420	+	0.997600i	$0.477942\pi$
$608$	0	0
$609$	5.77454	0.233996
$610$	0	0
$611$	27.4282i	1.10963i
$612$	0	0
$613$	36.6290i	1.47943i	0.672920	+	0.739716i	$0.265040\pi$
$613$	36.6290i	1.47943i	−0.672920	+	0.739716i	$0.734960\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.3979	1.62636	0.813180	−	0.582012i	$-0.197734\pi$
$617$	40.3979	1.62636	0.813180	+	0.582012i	$0.197734\pi$
$618$	0	0
$619$	24.5172i	0.985430i	0.870191	+	0.492715i	$0.163996\pi$
$619$	24.5172i	0.985430i	−0.870191	+	0.492715i	$0.836004\pi$
$620$	0	0
$621$	− 4.82778i	− 0.193732i
$622$	0	0
$623$	−39.0006	−1.56252
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 10.7253i	− 0.428328i
$628$	0	0
$629$	26.0852i	1.04009i
$630$	0	0
$631$	−18.7805	−0.747640	−0.373820	−	0.927501i	$-0.621952\pi$
$631$	−18.7805	−0.747640	−0.373820	+	0.927501i	$0.621952\pi$
$632$	0	0
$633$	13.7141	0.545087
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 0.0635083i	− 0.00251629i
$638$	0	0
$639$	−6.72532	−0.266050
$640$	0	0
$641$	−15.5163	−0.612856	−0.306428	−	0.951894i	$-0.599134\pi$
$641$	−15.5163	−0.612856	−0.306428	+	0.951894i	$0.599134\pi$
$642$	0	0
$643$	− 17.4506i	− 0.688186i	−0.938936	−	0.344093i	$-0.888186\pi$
$643$	− 17.4506i	− 0.688186i	0.938936	−	0.344093i	$-0.111814\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.1403	0.516600	0.258300	−	0.966065i	$-0.416838\pi$
$647$	13.1403	0.516600	0.258300	+	0.966065i	$0.416838\pi$
$648$	0	0
$649$	−10.2911	−0.403960
$650$	0	0
$651$	19.4569i	0.762577i
$652$	0	0
$653$	− 14.7993i	− 0.579141i	−0.957157	−	0.289570i	$-0.906488\pi$
$653$	− 14.7993i	− 0.579141i	0.957157	−	0.289570i	$-0.0935124\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−15.5146	−0.605284
$658$	0	0
$659$	− 7.99614i	− 0.311485i	−0.987798	−	0.155743i	$-0.950223\pi$
$659$	− 7.99614i	− 0.311485i	0.987798	−	0.155743i	$-0.0497771\pi$
$660$	0	0
$661$	− 0.915029i	− 0.0355905i	−0.999842	−	0.0177953i	$-0.994335\pi$
$661$	− 0.915029i	− 0.0355905i	0.999842	−	0.0177953i	$-0.00566470\pi$
$662$	0	0
$663$	12.8286	0.498223
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.5494i	0.408473i
$668$	0	0
$669$	− 9.84472i	− 0.380619i
$670$	0	0
$671$	5.45065	0.210420
$672$	0	0
$673$	34.3978	1.32594	0.662969	−	0.748647i	$-0.269296\pi$
$673$	34.3978	1.32594	0.662969	+	0.748647i	$0.269296\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 40.1676i	− 1.54377i	−0.635764	−	0.771884i	$-0.719315\pi$
$677$	− 40.1676i	− 1.54377i	0.635764	−	0.771884i	$-0.280685\pi$
$678$	0	0
$679$	−29.4506	−1.13021
$680$	0	0
$681$	−5.70892	−0.218766
$682$	0	0
$683$	− 33.2580i	− 1.27258i	−0.771449	−	0.636291i	$-0.780468\pi$
$683$	− 33.2580i	− 1.27258i	0.771449	−	0.636291i	$-0.219532\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−0.769233	−0.0293481
$688$	0	0
$689$	−17.4506	−0.664817
$690$	0	0
$691$	50.2241i	1.91062i	0.295611	+	0.955308i	$0.404477\pi$
$691$	50.2241i	1.91062i	−0.295611	+	0.955308i	$0.595523\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	28.9170	1.09531
$698$	0	0
$699$	− 18.4008i	− 0.695983i
$700$	0	0
$701$	23.7543i	0.897188i	0.893736	+	0.448594i	$0.148075\pi$
$701$	23.7543i	0.897188i	−0.893736	+	0.448594i	$0.851925\pi$
$702$	0	0
$703$	55.7715	2.10346
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.2252i	1.32478i
$708$	0	0
$709$	− 36.3146i	− 1.36382i	−0.731435	−	0.681911i	$-0.761149\pi$
$709$	− 36.3146i	− 1.36382i	0.731435	−	0.681911i	$-0.238851\pi$
$710$	0	0
$711$	−7.36266	−0.276121
$712$	0	0
$713$	−35.5453	−1.33118
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.0328i	0.374682i
$718$	0	0
$719$	−30.7253	−1.14586	−0.572931	−	0.819604i	$-0.694194\pi$
$719$	−30.7253	−1.14586	−0.572931	+	0.819604i	$0.694194\pi$
$720$	0	0
$721$	−2.53268	−0.0943219
$722$	0	0
$723$	− 10.7581i	− 0.400099i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5.47445	−0.203036	−0.101518	−	0.994834i	$-0.532370\pi$
$727$	−5.47445	−0.203036	−0.101518	+	0.994834i	$0.532370\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 3.36852i	− 0.124589i
$732$	0	0
$733$	− 17.1455i	− 0.633285i	−0.948545	−	0.316643i	$-0.897444\pi$
$733$	− 17.1455i	− 0.633285i	0.948545	−	0.316643i	$-0.102556\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.53847	0.0566701
$738$	0	0
$739$	− 11.6019i	− 0.426782i	−0.976967	−	0.213391i	$-0.931549\pi$
$739$	− 11.6019i	− 0.426782i	0.976967	−	0.213391i	$-0.0684508\pi$
$740$	0	0
$741$	− 27.4282i	− 1.00760i
$742$	0	0
$743$	23.6613	0.868048	0.434024	−	0.900901i	$-0.357093\pi$
$743$	23.6613	0.868048	0.434024	+	0.900901i	$0.357093\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.74173i	0.283255i
$748$	0	0
$749$	10.5706i	0.386241i
$750$	0	0
$751$	11.4283	0.417024	0.208512	−	0.978020i	$-0.433138\pi$
$751$	11.4283	0.417024	0.208512	+	0.978020i	$0.433138\pi$
$752$	0	0
$753$	12.6580	0.461283
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19.1784i	0.697049i	0.937300	+	0.348525i	$0.113317\pi$
$757$	19.1784i	0.697049i	−0.937300	+	0.348525i	$0.886683\pi$
$758$	0	0
$759$	−7.30749	−0.265245
$760$	0	0
$761$	4.03281	0.146189	0.0730947	−	0.997325i	$-0.476712\pi$
$761$	4.03281	0.146189	0.0730947	+	0.997325i	$0.476712\pi$
$762$	0	0
$763$	− 2.03281i	− 0.0735928i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.3177	−0.950279
$768$	0	0
$769$	−2.95078	−0.106408	−0.0532039	−	0.998584i	$-0.516943\pi$
$769$	−2.95078	−0.106408	−0.0532039	+	0.998584i	$0.516943\pi$
$770$	0	0
$771$	13.3110i	0.479383i
$772$	0	0
$773$	45.2663i	1.62812i	0.580783	+	0.814059i	$0.302747\pi$
$773$	45.2663i	1.62812i	−0.580783	+	0.814059i	$0.697253\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−20.7999	−0.746193
$778$	0	0
$779$	− 61.8260i	− 2.21515i
$780$	0	0
$781$	10.1797i	0.364257i
$782$	0	0
$783$	2.18513	0.0780903
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 52.9997i	− 1.88924i	−0.328171	−	0.944618i	$-0.606432\pi$
$787$	− 52.9997i	− 1.88924i	0.328171	−	0.944618i	$-0.393568\pi$
$788$	0	0
$789$	− 18.4256i	− 0.655970i
$790$	0	0
$791$	38.2088	1.35855
$792$	0	0
$793$	13.9391	0.494993
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.5738i	− 0.587075i	−0.955948	−	0.293538i	$-0.905167\pi$
$797$	− 16.5738i	− 0.587075i	0.955948	−	0.293538i	$-0.0948326\pi$
$798$	0	0
$799$	23.4835	0.830785
$800$	0	0
$801$	−14.7581	−0.521453
$802$	0	0
$803$	23.4835i	0.828713i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.86940	−0.136209
$808$	0	0
$809$	−37.5491	−1.32016	−0.660078	−	0.751197i	$-0.729477\pi$
$809$	−37.5491	−1.32016	−0.660078	+	0.751197i	$0.729477\pi$
$810$	0	0
$811$	− 32.1102i	− 1.12754i	−0.825931	−	0.563771i	$-0.809350\pi$
$811$	− 32.1102i	− 1.12754i	0.825931	−	0.563771i	$-0.190650\pi$
$812$	0	0
$813$	− 17.3955i	− 0.610086i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.20207	−0.251969
$818$	0	0
$819$	10.2293i	0.357442i
$820$	0	0
$821$	29.3809i	1.02540i	0.858568	+	0.512699i	$0.171354\pi$
$821$	29.3809i	1.02540i	−0.858568	+	0.512699i	$0.828646\pi$
$822$	0	0
$823$	−28.3866	−0.989495	−0.494748	−	0.869037i	$-0.664740\pi$
$823$	−28.3866	−0.989495	−0.494748	+	0.869037i	$0.664740\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.45065i	− 0.0504439i	−0.999682	−	0.0252219i	$-0.991971\pi$
$827$	− 1.45065i	− 0.0504439i	0.999682	−	0.0252219i	$-0.00802924\pi$
$828$	0	0
$829$	37.4621i	1.30111i	0.759458	+	0.650556i	$0.225464\pi$
$829$	37.4621i	1.30111i	−0.759458	+	0.650556i	$0.774536\pi$
$830$	0	0
$831$	−0.887271	−0.0307791
$832$	0	0
$833$	−0.0543744	−0.00188396
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.36266i	0.254491i
$838$	0	0
$839$	48.7581	1.68332	0.841659	−	0.540010i	$-0.181579\pi$
$839$	48.7581	1.68332	0.841659	+	0.540010i	$0.181579\pi$
$840$	0	0
$841$	24.2252	0.835351
$842$	0	0
$843$	13.4835i	0.464395i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−23.0146	−0.790791
$848$	0	0
$849$	28.4342	0.975861
$850$	0	0
$851$	− 37.9988i	− 1.30258i
$852$	0	0
$853$	− 4.37073i	− 0.149651i	−0.997197	−	0.0748255i	$-0.976160\pi$
$853$	− 4.37073i	− 0.149651i	0.997197	−	0.0748255i	$-0.0238400\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.5130	0.700712	0.350356	−	0.936617i	$-0.386061\pi$
$857$	20.5130	0.700712	0.350356	+	0.936617i	$0.386061\pi$
$858$	0	0
$859$	10.1131i	0.345054i	0.985005	+	0.172527i	$0.0551932\pi$
$859$	10.1131i	0.345054i	−0.985005	+	0.172527i	$0.944807\pi$
$860$	0	0
$861$	23.0580i	0.785813i
$862$	0	0
$863$	−13.2861	−0.452266	−0.226133	−	0.974096i	$-0.572608\pi$
$863$	−13.2861	−0.452266	−0.226133	+	0.974096i	$0.572608\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.01641i	0.204328i
$868$	0	0
$869$	11.1444i	0.378047i
$870$	0	0
$871$	3.93437	0.133311
$872$	0	0
$873$	−11.1444	−0.377180
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 33.6454i	− 1.13612i	−0.822986	−	0.568062i	$-0.807693\pi$
$877$	− 33.6454i	− 1.13612i	0.822986	−	0.568062i	$-0.192307\pi$
$878$	0	0
$879$	−7.99166	−0.269552
$880$	0	0
$881$	−32.7909	−1.10476	−0.552378	−	0.833594i	$-0.686279\pi$
$881$	−32.7909	−1.10476	−0.552378	+	0.833594i	$0.686279\pi$
$882$	0	0
$883$	− 33.4506i	− 1.12570i	−0.826558	−	0.562852i	$-0.809704\pi$
$883$	− 33.4506i	− 1.12570i	0.826558	−	0.562852i	$-0.190296\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.8924	1.17157	0.585785	−	0.810466i	$-0.300786\pi$
$887$	34.8924	1.17157	0.585785	+	0.810466i	$0.300786\pi$
$888$	0	0
$889$	30.4671	1.02183
$890$	0	0
$891$	1.51363i	0.0507086i
$892$	0	0
$893$	− 50.2088i	− 1.68017i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−18.6877	−0.623964
$898$	0	0
$899$	− 16.0884i	− 0.536578i
$900$	0	0
$901$	14.9409i	0.497752i
$902$	0	0
$903$	2.68601	0.0893847
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 30.9836i	− 1.02879i	−0.857552	−	0.514397i	$-0.828016\pi$
$907$	− 30.9836i	− 1.02879i	0.857552	−	0.514397i	$-0.171984\pi$
$908$	0	0
$909$	13.3295i	0.442112i
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	11.7181	0.387814
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.4835i	0.643400i
$918$	0	0
$919$	15.6043	0.514737	0.257368	−	0.966313i	$-0.417145\pi$
$919$	15.6043	0.514737	0.257368	+	0.966313i	$0.417145\pi$
$920$	0	0
$921$	17.4506	0.575018
$922$	0	0
$923$	26.0328i	0.856880i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.958386	−0.0314775
$928$	0	0
$929$	−38.9341	−1.27739	−0.638693	−	0.769461i	$-0.720525\pi$
$929$	−38.9341	−1.27739	−0.638693	+	0.769461i	$0.720525\pi$
$930$	0	0
$931$	0.116255i	0.00381011i
$932$	0	0
$933$	21.4506i	0.702263i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.6027	0.640393	0.320197	−	0.947351i	$-0.396251\pi$
$937$	19.6027	0.640393	0.320197	+	0.947351i	$0.396251\pi$
$938$	0	0
$939$	7.73879i	0.252546i
$940$	0	0
$941$	− 25.0476i	− 0.816530i	−0.912864	−	0.408265i	$-0.866134\pi$
$941$	− 25.0476i	− 0.816530i	0.912864	−	0.408265i	$-0.133866\pi$
$942$	0	0
$943$	−42.1240	−1.37175
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.93437i	− 0.257832i	−0.991655	−	0.128916i	$-0.958850\pi$
$947$	− 7.93437i	− 0.257832i	0.991655	−	0.128916i	$-0.0411498\pi$
$948$	0	0
$949$	60.0550i	1.94947i
$950$	0	0
$951$	−11.2335	−0.364272
$952$	0	0
$953$	−11.4809	−0.371903	−0.185952	−	0.982559i	$-0.559537\pi$
$953$	−11.4809	−0.371903	−0.185952	+	0.982559i	$0.559537\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.30749i	− 0.106916i
$958$	0	0
$959$	10.2744	0.331778
$960$	0	0
$961$	23.2088	0.748670
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.8993	−0.511285	−0.255643	−	0.966771i	$-0.582287\pi$
$967$	−15.8993	−0.511285	−0.255643	+	0.966771i	$0.582287\pi$
$968$	0	0
$969$	−23.4835	−0.754397
$970$	0	0
$971$	− 40.6600i	− 1.30484i	−0.757857	−	0.652421i	$-0.773754\pi$
$971$	− 40.6600i	− 1.30484i	0.757857	−	0.652421i	$-0.226246\pi$
$972$	0	0
$973$	− 38.6597i	− 1.23937i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26.5676	−0.849972	−0.424986	−	0.905200i	$-0.639721\pi$
$977$	−26.5676	−0.849972	−0.424986	+	0.905200i	$0.639721\pi$
$978$	0	0
$979$	22.3384i	0.713938i
$980$	0	0
$981$	− 0.769233i	− 0.0245597i
$982$	0	0
$983$	−9.88057	−0.315141	−0.157571	−	0.987508i	$-0.550366\pi$
$983$	−9.88057	−0.315141	−0.157571	+	0.987508i	$0.550366\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	18.7253i	0.596034i
$988$	0	0
$989$	4.90699i	0.156033i
$990$	0	0
$991$	53.0549	1.68534	0.842672	−	0.538427i	$-0.180981\pi$
$991$	53.0549	1.68534	0.842672	+	0.538427i	$0.180981\pi$
$992$	0	0
$993$	−8.00084	−0.253899
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32.3051i	1.02311i	0.859250	+	0.511556i	$0.170931\pi$
$997$	32.3051i	1.02311i	−0.859250	+	0.511556i	$0.829069\pi$
$998$	0	0
$999$	−7.87086	−0.249023

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.k.f.1201.2		12
3.2	odd	2		7200.2.k.u.3601.3		12
4.3	odd	2		600.2.k.f.301.6		12
5.2	odd	4		480.2.d.a.49.4		6
5.3	odd	4		480.2.d.b.49.4		6
5.4	even	2	inner	2400.2.k.f.1201.11		12
8.3	odd	2		600.2.k.f.301.5		12
8.5	even	2	inner	2400.2.k.f.1201.8		12
12.11	even	2		1800.2.k.u.901.7		12
15.2	even	4		1440.2.d.e.1009.3		6
15.8	even	4		1440.2.d.f.1009.3		6
15.14	odd	2		7200.2.k.u.3601.9		12
20.3	even	4		120.2.d.b.109.6	yes	6
20.7	even	4		120.2.d.a.109.1	✓	6
20.19	odd	2		600.2.k.f.301.7		12
24.5	odd	2		7200.2.k.u.3601.4		12
24.11	even	2		1800.2.k.u.901.8		12
40.3	even	4		120.2.d.a.109.2	yes	6
40.13	odd	4		480.2.d.a.49.3		6
40.19	odd	2		600.2.k.f.301.8		12
40.27	even	4		120.2.d.b.109.5	yes	6
40.29	even	2	inner	2400.2.k.f.1201.5		12
40.37	odd	4		480.2.d.b.49.3		6
60.23	odd	4		360.2.d.e.109.1		6
60.47	odd	4		360.2.d.f.109.6		6
60.59	even	2		1800.2.k.u.901.6		12
80.3	even	4		3840.2.f.l.769.6		12
80.13	odd	4		3840.2.f.m.769.12		12
80.27	even	4		3840.2.f.l.769.1		12
80.37	odd	4		3840.2.f.m.769.7		12
80.43	even	4		3840.2.f.l.769.7		12
80.53	odd	4		3840.2.f.m.769.1		12
80.67	even	4		3840.2.f.l.769.12		12
80.77	odd	4		3840.2.f.m.769.6		12
120.29	odd	2		7200.2.k.u.3601.10		12
120.53	even	4		1440.2.d.e.1009.4		6
120.59	even	2		1800.2.k.u.901.5		12
120.77	even	4		1440.2.d.f.1009.4		6
120.83	odd	4		360.2.d.f.109.5		6
120.107	odd	4		360.2.d.e.109.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.1	✓	6	20.7	even	4
120.2.d.a.109.2	yes	6	40.3	even	4
120.2.d.b.109.5	yes	6	40.27	even	4
120.2.d.b.109.6	yes	6	20.3	even	4
360.2.d.e.109.1		6	60.23	odd	4
360.2.d.e.109.2		6	120.107	odd	4
360.2.d.f.109.5		6	120.83	odd	4
360.2.d.f.109.6		6	60.47	odd	4
480.2.d.a.49.3		6	40.13	odd	4
480.2.d.a.49.4		6	5.2	odd	4
480.2.d.b.49.3		6	40.37	odd	4
480.2.d.b.49.4		6	5.3	odd	4
600.2.k.f.301.5		12	8.3	odd	2
600.2.k.f.301.6		12	4.3	odd	2
600.2.k.f.301.7		12	20.19	odd	2
600.2.k.f.301.8		12	40.19	odd	2
1440.2.d.e.1009.3		6	15.2	even	4
1440.2.d.e.1009.4		6	120.53	even	4
1440.2.d.f.1009.3		6	15.8	even	4
1440.2.d.f.1009.4		6	120.77	even	4
1800.2.k.u.901.5		12	120.59	even	2
1800.2.k.u.901.6		12	60.59	even	2
1800.2.k.u.901.7		12	12.11	even	2
1800.2.k.u.901.8		12	24.11	even	2
2400.2.k.f.1201.2		12	1.1	even	1	trivial
2400.2.k.f.1201.5		12	40.29	even	2	inner
2400.2.k.f.1201.8		12	8.5	even	2	inner
2400.2.k.f.1201.11		12	5.4	even	2	inner
3840.2.f.l.769.1		12	80.27	even	4
3840.2.f.l.769.6		12	80.3	even	4
3840.2.f.l.769.7		12	80.43	even	4
3840.2.f.l.769.12		12	80.67	even	4
3840.2.f.m.769.1		12	80.53	odd	4
3840.2.f.m.769.6		12	80.77	odd	4
3840.2.f.m.769.7		12	80.37	odd	4
3840.2.f.m.769.12		12	80.13	odd	4
7200.2.k.u.3601.3		12	3.2	odd	2
7200.2.k.u.3601.4		12	24.5	odd	2
7200.2.k.u.3601.9		12	15.14	odd	2
7200.2.k.u.3601.10		12	120.29	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2400.2.k.f.1201.2

Level	$2400$
Weight	$2$
Character	2400.1201
Analytic conductor	$19.164$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$