LMFDB - Embedded newform 2475.2.c.n.199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.n.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +O(q^{10})$	$q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +1.00000 q^{11} -5.46410i q^{13} -3.46410 q^{14} -5.00000 q^{16} -5.46410 q^{19} +1.73205i q^{22} +6.92820i q^{23} +9.46410 q^{26} -2.00000i q^{28} -3.46410 q^{29} -10.9282 q^{31} -5.19615i q^{32} -4.92820i q^{37} -9.46410i q^{38} -3.46410 q^{41} +4.92820i q^{43} -1.00000 q^{44} -12.0000 q^{46} +6.92820i q^{47} +3.00000 q^{49} +5.46410i q^{52} +0.928203i q^{53} -3.46410 q^{56} -6.00000i q^{58} -6.92820 q^{59} +2.00000 q^{61} -18.9282i q^{62} -1.00000 q^{64} +8.00000i q^{67} -13.8564 q^{71} +8.39230i q^{73} +8.53590 q^{74} +5.46410 q^{76} +2.00000i q^{77} +6.53590 q^{79} -6.00000i q^{82} +8.53590i q^{83} -8.53590 q^{86} +1.73205i q^{88} +0.928203 q^{89} +10.9282 q^{91} -6.92820i q^{92} -12.0000 q^{94} -10.0000i q^{97} +5.19615i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{11} - 20 q^{16} - 8 q^{19} + 24 q^{26} - 16 q^{31} - 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} + 48 q^{74} + 8 q^{76} + 40 q^{79} - 48 q^{86} - 24 q^{89} + 16 q^{91} - 48 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^11 - 20 * q^16 - 8 * q^19 + 24 * q^26 - 16 * q^31 - 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 + 48 * q^74 + 8 * q^76 + 40 * q^79 - 48 * q^86 - 24 * q^89 + 16 * q^91 - 48 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.73205i	0.612372i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 5.46410i	− 1.51547i	−0.652563	−	0.757735i	$-0.726306\pi$
$13$	− 5.46410i	− 1.51547i	0.652563	−	0.757735i	$-0.273694\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	0	0
$21$	0	0
$22$	1.73205i	0.369274i
$23$	6.92820i	1.44463i	0.691564	+	0.722315i	$0.256922\pi$
$23$	6.92820i	1.44463i	−0.691564	+	0.722315i	$0.743078\pi$
$24$	0	0
$25$	0	0
$26$	9.46410	1.85606
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−10.9282	−1.96276	−0.981382	−	0.192068i	$-0.938481\pi$
$31$	−10.9282	−1.96276	−0.981382	+	0.192068i	$0.938481\pi$
$32$	− 5.19615i	− 0.918559i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.92820i	− 0.810192i	−0.914274	−	0.405096i	$-0.867238\pi$
$37$	− 4.92820i	− 0.810192i	0.914274	−	0.405096i	$-0.132762\pi$
$38$	− 9.46410i	− 1.53528i
$39$	0	0
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	4.92820i	0.751544i	0.926712	+	0.375772i	$0.122622\pi$
$43$	4.92820i	0.751544i	−0.926712	+	0.375772i	$0.877378\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−12.0000	−1.76930
$47$	6.92820i	1.01058i	0.862949	+	0.505291i	$0.168615\pi$
$47$	6.92820i	1.01058i	−0.862949	+	0.505291i	$0.831385\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	5.46410i	0.757735i
$53$	0.928203i	0.127499i	0.997966	+	0.0637493i	$0.0203058\pi$
$53$	0.928203i	0.127499i	−0.997966	+	0.0637493i	$0.979694\pi$
$54$	0	0
$55$	0	0
$56$	−3.46410	−0.462910
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 18.9282i	− 2.40388i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	8.39230i	0.982245i	0.871091	+	0.491122i	$0.163413\pi$
$73$	8.39230i	0.982245i	−0.871091	+	0.491122i	$0.836587\pi$
$74$	8.53590	0.992278
$75$	0	0
$76$	5.46410	0.626775
$77$	2.00000i	0.227921i
$78$	0	0
$79$	6.53590	0.735346	0.367673	−	0.929955i	$-0.380155\pi$
$79$	6.53590	0.735346	0.367673	+	0.929955i	$0.380155\pi$
$80$	0	0
$81$	0	0
$82$	− 6.00000i	− 0.662589i
$83$	8.53590i	0.936937i	0.883480	+	0.468468i	$0.155194\pi$
$83$	8.53590i	0.936937i	−0.883480	+	0.468468i	$0.844806\pi$
$84$	0	0
$85$	0	0
$86$	−8.53590	−0.920450
$87$	0	0
$88$	1.73205i	0.184637i
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	10.9282	1.14559
$92$	− 6.92820i	− 0.722315i
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	5.19615i	0.524891i
$99$	0	0
$100$	0	0
$101$	10.3923	1.03407	0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923	1.03407	0.517036	+	0.855963i	$0.327035\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	9.46410	0.928032
$105$	0	0
$106$	−1.60770	−0.156153
$107$	− 8.53590i	− 0.825196i	−0.910913	−	0.412598i	$-0.864621\pi$
$107$	− 8.53590i	− 0.825196i	0.910913	−	0.412598i	$-0.135379\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	− 10.0000i	− 0.944911i
$113$	− 12.9282i	− 1.21618i	−0.793867	−	0.608092i	$-0.791935\pi$
$113$	− 12.9282i	− 1.21618i	0.793867	−	0.608092i	$-0.208065\pi$
$114$	0	0
$115$	0	0
$116$	3.46410	0.321634
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.46410i	0.313625i
$123$	0	0
$124$	10.9282	0.981382
$125$	0	0
$126$	0	0
$127$	8.92820i	0.792250i	0.918197	+	0.396125i	$0.129645\pi$
$127$	8.92820i	0.792250i	−0.918197	+	0.396125i	$0.870355\pi$
$128$	− 12.1244i	− 1.07165i
$129$	0	0
$130$	0	0
$131$	−18.9282	−1.65376	−0.826882	−	0.562375i	$-0.809888\pi$
$131$	−18.9282	−1.65376	−0.826882	+	0.562375i	$0.809888\pi$
$132$	0	0
$133$	− 10.9282i	− 0.947595i
$134$	−13.8564	−1.19701
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−12.3923	−1.05110	−0.525551	−	0.850762i	$-0.676141\pi$
$139$	−12.3923	−1.05110	−0.525551	+	0.850762i	$0.676141\pi$
$140$	0	0
$141$	0	0
$142$	− 24.0000i	− 2.01404i
$143$	− 5.46410i	− 0.456931i
$144$	0	0
$145$	0	0
$146$	−14.5359	−1.20300
$147$	0	0
$148$	4.92820i	0.405096i
$149$	−15.4641	−1.26687	−0.633434	−	0.773796i	$-0.718355\pi$
$149$	−15.4641	−1.26687	−0.633434	+	0.773796i	$0.718355\pi$
$150$	0	0
$151$	−20.3923	−1.65950	−0.829751	−	0.558134i	$-0.811518\pi$
$151$	−20.3923	−1.65950	−0.829751	+	0.558134i	$0.811518\pi$
$152$	− 9.46410i	− 0.767640i
$153$	0	0
$154$	−3.46410	−0.279145
$155$	0	0
$156$	0	0
$157$	− 3.07180i	− 0.245156i	−0.992459	−	0.122578i	$-0.960884\pi$
$157$	− 3.07180i	− 0.245156i	0.992459	−	0.122578i	$-0.0391162\pi$
$158$	11.3205i	0.900611i
$159$	0	0
$160$	0	0
$161$	−13.8564	−1.09204
$162$	0	0
$163$	− 9.85641i	− 0.772013i	−0.922496	−	0.386007i	$-0.873854\pi$
$163$	− 9.85641i	− 0.772013i	0.922496	−	0.386007i	$-0.126146\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	−14.7846	−1.14751
$167$	10.3923i	0.804181i	0.915600	+	0.402090i	$0.131716\pi$
$167$	10.3923i	0.804181i	−0.915600	+	0.402090i	$0.868284\pi$
$168$	0	0
$169$	−16.8564	−1.29665
$170$	0	0
$171$	0	0
$172$	− 4.92820i	− 0.375772i
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	0	0
$175$	0	0
$176$	−5.00000	−0.376889
$177$	0	0
$178$	1.60770i	0.120502i
$179$	6.92820	0.517838	0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820	0.517838	0.258919	+	0.965899i	$0.416634\pi$
$180$	0	0
$181$	15.8564	1.17860	0.589299	−	0.807915i	$-0.299404\pi$
$181$	15.8564	1.17860	0.589299	+	0.807915i	$0.299404\pi$
$182$	18.9282i	1.40305i
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 6.92820i	− 0.505291i
$189$	0	0
$190$	0	0
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	0	0
$193$	− 24.3923i	− 1.75580i	−0.478847	−	0.877898i	$-0.658945\pi$
$193$	− 24.3923i	− 1.75580i	0.478847	−	0.877898i	$-0.341055\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	0	0
$201$	0	0
$202$	18.0000i	1.26648i
$203$	− 6.92820i	− 0.486265i
$204$	0	0
$205$	0	0
$206$	13.8564	0.965422
$207$	0	0
$208$	27.3205i	1.89434i
$209$	−5.46410	−0.377960
$210$	0	0
$211$	−8.39230	−0.577750	−0.288875	−	0.957367i	$-0.593281\pi$
$211$	−8.39230	−0.577750	−0.288875	+	0.957367i	$0.593281\pi$
$212$	− 0.928203i	− 0.0637493i
$213$	0	0
$214$	14.7846	1.01066
$215$	0	0
$216$	0	0
$217$	− 21.8564i	− 1.48371i
$218$	17.3205i	1.17309i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 9.85641i	− 0.660034i	−0.943975	−	0.330017i	$-0.892946\pi$
$223$	− 9.85641i	− 0.660034i	0.943975	−	0.330017i	$-0.107054\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	22.3923	1.48951
$227$	− 15.4641i	− 1.02639i	−0.858272	−	0.513194i	$-0.828462\pi$
$227$	− 15.4641i	− 1.02639i	0.858272	−	0.513194i	$-0.171538\pi$
$228$	0	0
$229$	23.8564	1.57648	0.788238	−	0.615371i	$-0.210994\pi$
$229$	23.8564	1.57648	0.788238	+	0.615371i	$0.210994\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	0	0
$235$	0	0
$236$	6.92820	0.450988
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	0.143594	0.00924967	0.00462484	−	0.999989i	$-0.498528\pi$
$241$	0.143594	0.00924967	0.00462484	+	0.999989i	$0.498528\pi$
$242$	1.73205i	0.111340i
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	29.8564i	1.89972i
$248$	− 18.9282i	− 1.20194i
$249$	0	0
$250$	0	0
$251$	1.85641	0.117175	0.0585877	−	0.998282i	$-0.481340\pi$
$251$	1.85641	0.117175	0.0585877	+	0.998282i	$0.481340\pi$
$252$	0	0
$253$	6.92820i	0.435572i
$254$	−15.4641	−0.970304
$255$	0	0
$256$	19.0000	1.18750
$257$	19.8564i	1.23861i	0.785151	+	0.619304i	$0.212585\pi$
$257$	19.8564i	1.23861i	−0.785151	+	0.619304i	$0.787415\pi$
$258$	0	0
$259$	9.85641	0.612447
$260$	0	0
$261$	0	0
$262$	− 32.7846i	− 2.02544i
$263$	20.5359i	1.26630i	0.774030	+	0.633149i	$0.218238\pi$
$263$	20.5359i	1.26630i	−0.774030	+	0.633149i	$0.781762\pi$
$264$	0	0
$265$	0	0
$266$	18.9282	1.16056
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	19.8564	1.21067	0.605333	−	0.795972i	$-0.293040\pi$
$269$	19.8564	1.21067	0.605333	+	0.795972i	$0.293040\pi$
$270$	0	0
$271$	−11.6077	−0.705117	−0.352559	−	0.935790i	$-0.614688\pi$
$271$	−11.6077	−0.705117	−0.352559	+	0.935790i	$0.614688\pi$
$272$	0	0
$273$	0	0
$274$	−31.1769	−1.88347
$275$	0	0
$276$	0	0
$277$	29.4641i	1.77033i	0.465281	+	0.885163i	$0.345953\pi$
$277$	29.4641i	1.77033i	−0.465281	+	0.885163i	$0.654047\pi$
$278$	− 21.4641i	− 1.28733i
$279$	0	0
$280$	0	0
$281$	−3.46410	−0.206651	−0.103325	−	0.994648i	$-0.532948\pi$
$281$	−3.46410	−0.206651	−0.103325	+	0.994648i	$0.532948\pi$
$282$	0	0
$283$	4.92820i	0.292951i	0.989214	+	0.146476i	$0.0467930\pi$
$283$	4.92820i	0.292951i	−0.989214	+	0.146476i	$0.953207\pi$
$284$	13.8564	0.822226
$285$	0	0
$286$	9.46410	0.559624
$287$	− 6.92820i	− 0.408959i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	− 8.39230i	− 0.491122i
$293$	− 13.8564i	− 0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$	− 13.8564i	− 0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$	0	0
$295$	0	0
$296$	8.53590	0.496139
$297$	0	0
$298$	− 26.7846i	− 1.55159i
$299$	37.8564	2.18929
$300$	0	0
$301$	−9.85641	−0.568114
$302$	− 35.3205i	− 2.03247i
$303$	0	0
$304$	27.3205	1.56694
$305$	0	0
$306$	0	0
$307$	14.0000i	0.799022i	0.916728	+	0.399511i	$0.130820\pi$
$307$	14.0000i	0.799022i	−0.916728	+	0.399511i	$0.869180\pi$
$308$	− 2.00000i	− 0.113961i
$309$	0	0
$310$	0	0
$311$	−5.07180	−0.287595	−0.143798	−	0.989607i	$-0.545931\pi$
$311$	−5.07180	−0.287595	−0.143798	+	0.989607i	$0.545931\pi$
$312$	0	0
$313$	− 20.9282i	− 1.18293i	−0.806330	−	0.591466i	$-0.798549\pi$
$313$	− 20.9282i	− 1.18293i	0.806330	−	0.591466i	$-0.201451\pi$
$314$	5.32051	0.300254
$315$	0	0
$316$	−6.53590	−0.367673
$317$	24.9282i	1.40011i	0.714090	+	0.700054i	$0.246841\pi$
$317$	24.9282i	1.40011i	−0.714090	+	0.700054i	$0.753159\pi$
$318$	0	0
$319$	−3.46410	−0.193952
$320$	0	0
$321$	0	0
$322$	− 24.0000i	− 1.33747i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	17.0718	0.945519
$327$	0	0
$328$	− 6.00000i	− 0.331295i
$329$	−13.8564	−0.763928
$330$	0	0
$331$	9.85641	0.541757	0.270879	−	0.962614i	$-0.412686\pi$
$331$	9.85641	0.541757	0.270879	+	0.962614i	$0.412686\pi$
$332$	− 8.53590i	− 0.468468i
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	33.1769i	1.80726i	0.428312	+	0.903631i	$0.359108\pi$
$337$	33.1769i	1.80726i	−0.428312	+	0.903631i	$0.640892\pi$
$338$	− 29.1962i	− 1.58806i
$339$	0	0
$340$	0	0
$341$	−10.9282	−0.591795
$342$	0	0
$343$	20.0000i	1.07990i
$344$	−8.53590	−0.460225
$345$	0	0
$346$	20.7846	1.11739
$347$	− 22.3923i	− 1.20208i	−0.799218	−	0.601041i	$-0.794753\pi$
$347$	− 22.3923i	− 1.20208i	0.799218	−	0.601041i	$-0.205247\pi$
$348$	0	0
$349$	8.14359	0.435917	0.217958	−	0.975958i	$-0.430060\pi$
$349$	8.14359	0.435917	0.217958	+	0.975958i	$0.430060\pi$
$350$	0	0
$351$	0	0
$352$	− 5.19615i	− 0.276956i
$353$	12.9282i	0.688099i	0.938952	+	0.344049i	$0.111799\pi$
$353$	12.9282i	0.688099i	−0.938952	+	0.344049i	$0.888201\pi$
$354$	0	0
$355$	0	0
$356$	−0.928203	−0.0491947
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−20.7846	−1.09697	−0.548485	−	0.836160i	$-0.684795\pi$
$359$	−20.7846	−1.09697	−0.548485	+	0.836160i	$0.684795\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	27.4641i	1.44348i
$363$	0	0
$364$	−10.9282	−0.572793
$365$	0	0
$366$	0	0
$367$	20.0000i	1.04399i	0.852948	+	0.521996i	$0.174812\pi$
$367$	20.0000i	1.04399i	−0.852948	+	0.521996i	$0.825188\pi$
$368$	− 34.6410i	− 1.80579i
$369$	0	0
$370$	0	0
$371$	−1.85641	−0.0963798
$372$	0	0
$373$	20.3923i	1.05587i	0.849284	+	0.527937i	$0.177034\pi$
$373$	20.3923i	1.05587i	−0.849284	+	0.527937i	$0.822966\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	18.9282i	0.974852i
$378$	0	0
$379$	17.8564	0.917222	0.458611	−	0.888637i	$-0.348347\pi$
$379$	17.8564	0.917222	0.458611	+	0.888637i	$0.348347\pi$
$380$	0	0
$381$	0	0
$382$	32.7846i	1.67741i
$383$	− 13.8564i	− 0.708029i	−0.935240	−	0.354015i	$-0.884816\pi$
$383$	− 13.8564i	− 0.708029i	0.935240	−	0.354015i	$-0.115184\pi$
$384$	0	0
$385$	0	0
$386$	42.2487	2.15040
$387$	0	0
$388$	10.0000i	0.507673i
$389$	11.0718	0.561362	0.280681	−	0.959801i	$-0.409440\pi$
$389$	11.0718	0.561362	0.280681	+	0.959801i	$0.409440\pi$
$390$	0	0
$391$	0	0
$392$	5.19615i	0.262445i
$393$	0	0
$394$	−20.7846	−1.04711
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	42.9282i	2.15180i
$399$	0	0
$400$	0	0
$401$	7.85641	0.392330	0.196165	−	0.980571i	$-0.437151\pi$
$401$	7.85641	0.392330	0.196165	+	0.980571i	$0.437151\pi$
$402$	0	0
$403$	59.7128i	2.97451i
$404$	−10.3923	−0.517036
$405$	0	0
$406$	12.0000	0.595550
$407$	− 4.92820i	− 0.244282i
$408$	0	0
$409$	6.78461	0.335477	0.167739	−	0.985831i	$-0.446354\pi$
$409$	6.78461	0.335477	0.167739	+	0.985831i	$0.446354\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	− 13.8564i	− 0.681829i
$414$	0	0
$415$	0	0
$416$	−28.3923	−1.39205
$417$	0	0
$418$	− 9.46410i	− 0.462904i
$419$	30.9282	1.51094	0.755471	−	0.655182i	$-0.227408\pi$
$419$	30.9282	1.51094	0.755471	+	0.655182i	$0.227408\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 14.5359i	− 0.707596i
$423$	0	0
$424$	−1.60770	−0.0780766
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	8.53590i	0.412598i
$429$	0	0
$430$	0	0
$431$	−8.78461	−0.423140	−0.211570	−	0.977363i	$-0.567858\pi$
$431$	−8.78461	−0.423140	−0.211570	+	0.977363i	$0.567858\pi$
$432$	0	0
$433$	− 0.143594i	− 0.00690067i	−0.999994	−	0.00345033i	$-0.998902\pi$
$433$	− 0.143594i	− 0.00690067i	0.999994	−	0.00345033i	$-0.00109828\pi$
$434$	37.8564	1.81717
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 37.8564i	− 1.81092i
$438$	0	0
$439$	−33.1769	−1.58345	−0.791724	−	0.610879i	$-0.790816\pi$
$439$	−33.1769	−1.58345	−0.791724	+	0.610879i	$0.790816\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	17.0718	0.808373
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	−26.7846	−1.26404	−0.632022	−	0.774950i	$-0.717775\pi$
$449$	−26.7846	−1.26404	−0.632022	+	0.774950i	$0.717775\pi$
$450$	0	0
$451$	−3.46410	−0.163118
$452$	12.9282i	0.608092i
$453$	0	0
$454$	26.7846	1.25706
$455$	0	0
$456$	0	0
$457$	12.3923i	0.579688i	0.957074	+	0.289844i	$0.0936034\pi$
$457$	12.3923i	0.579688i	−0.957074	+	0.289844i	$0.906397\pi$
$458$	41.3205i	1.93078i
$459$	0	0
$460$	0	0
$461$	−36.2487	−1.68827	−0.844135	−	0.536130i	$-0.819886\pi$
$461$	−36.2487	−1.68827	−0.844135	+	0.536130i	$0.819886\pi$
$462$	0	0
$463$	28.0000i	1.30127i	0.759390	+	0.650635i	$0.225497\pi$
$463$	28.0000i	1.30127i	−0.759390	+	0.650635i	$0.774503\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	−20.7846	−0.962828
$467$	− 5.07180i	− 0.234695i	−0.993091	−	0.117347i	$-0.962561\pi$
$467$	− 5.07180i	− 0.234695i	0.993091	−	0.117347i	$-0.0374391\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	− 12.0000i	− 0.552345i
$473$	4.92820i	0.226599i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	− 20.7846i	− 0.950666i
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−26.9282	−1.22782
$482$	0.248711i	0.0113285i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 31.7128i	− 1.43704i	−0.695504	−	0.718522i	$-0.744819\pi$
$487$	− 31.7128i	− 1.43704i	0.695504	−	0.718522i	$-0.255181\pi$
$488$	3.46410i	0.156813i
$489$	0	0
$490$	0	0
$491$	30.9282	1.39577	0.697885	−	0.716210i	$-0.254125\pi$
$491$	30.9282	1.39577	0.697885	+	0.716210i	$0.254125\pi$
$492$	0	0
$493$	0	0
$494$	−51.7128	−2.32667
$495$	0	0
$496$	54.6410	2.45345
$497$	− 27.7128i	− 1.24309i
$498$	0	0
$499$	−28.7846	−1.28858	−0.644288	−	0.764783i	$-0.722846\pi$
$499$	−28.7846	−1.28858	−0.644288	+	0.764783i	$0.722846\pi$
$500$	0	0
$501$	0	0
$502$	3.21539i	0.143510i
$503$	31.1769i	1.39011i	0.718957	+	0.695055i	$0.244620\pi$
$503$	31.1769i	1.39011i	−0.718957	+	0.695055i	$0.755380\pi$
$504$	0	0
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	− 8.92820i	− 0.396125i
$509$	19.8564	0.880120	0.440060	−	0.897968i	$-0.354957\pi$
$509$	19.8564	0.880120	0.440060	+	0.897968i	$0.354957\pi$
$510$	0	0
$511$	−16.7846	−0.742507
$512$	8.66025i	0.382733i
$513$	0	0
$514$	−34.3923	−1.51698
$515$	0	0
$516$	0	0
$517$	6.92820i	0.304702i
$518$	17.0718i	0.750092i
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	22.0000i	0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$	22.0000i	0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$	18.9282	0.826882
$525$	0	0
$526$	−35.5692	−1.55089
$527$	0	0
$528$	0	0
$529$	−25.0000	−1.08696
$530$	0	0
$531$	0	0
$532$	10.9282i	0.473798i
$533$	18.9282i	0.819871i
$534$	0	0
$535$	0	0
$536$	−13.8564	−0.598506
$537$	0	0
$538$	34.3923i	1.48276i
$539$	3.00000	0.129219
$540$	0	0
$541$	27.8564	1.19764	0.598820	−	0.800883i	$-0.295636\pi$
$541$	27.8564	1.19764	0.598820	+	0.800883i	$0.295636\pi$
$542$	− 20.1051i	− 0.863589i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	− 18.0000i	− 0.768922i
$549$	0	0
$550$	0	0
$551$	18.9282	0.806369
$552$	0	0
$553$	13.0718i	0.555869i
$554$	−51.0333	−2.16820
$555$	0	0
$556$	12.3923	0.525551
$557$	− 3.21539i	− 0.136240i	−0.997677	−	0.0681202i	$-0.978300\pi$
$557$	− 3.21539i	− 0.136240i	0.997677	−	0.0681202i	$-0.0217001\pi$
$558$	0	0
$559$	26.9282	1.13894
$560$	0	0
$561$	0	0
$562$	− 6.00000i	− 0.253095i
$563$	10.3923i	0.437983i	0.975727	+	0.218992i	$0.0702768\pi$
$563$	10.3923i	0.437983i	−0.975727	+	0.218992i	$0.929723\pi$
$564$	0	0
$565$	0	0
$566$	−8.53590	−0.358791
$567$	0	0
$568$	− 24.0000i	− 1.00702i
$569$	5.32051	0.223047	0.111524	−	0.993762i	$-0.464427\pi$
$569$	5.32051	0.223047	0.111524	+	0.993762i	$0.464427\pi$
$570$	0	0
$571$	3.60770	0.150977	0.0754887	−	0.997147i	$-0.475948\pi$
$571$	3.60770	0.150977	0.0754887	+	0.997147i	$0.475948\pi$
$572$	5.46410i	0.228466i
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	− 18.7846i	− 0.782014i	−0.920388	−	0.391007i	$-0.872127\pi$
$577$	− 18.7846i	− 0.782014i	0.920388	−	0.391007i	$-0.127873\pi$
$578$	29.4449i	1.22474i
$579$	0	0
$580$	0	0
$581$	−17.0718	−0.708257
$582$	0	0
$583$	0.928203i	0.0384422i
$584$	−14.5359	−0.601500
$585$	0	0
$586$	24.0000	0.991431
$587$	18.9282i	0.781251i	0.920550	+	0.390625i	$0.127741\pi$
$587$	18.9282i	0.781251i	−0.920550	+	0.390625i	$0.872259\pi$
$588$	0	0
$589$	59.7128	2.46042
$590$	0	0
$591$	0	0
$592$	24.6410i	1.01274i
$593$	8.78461i	0.360741i	0.983599	+	0.180370i	$0.0577296\pi$
$593$	8.78461i	0.360741i	−0.983599	+	0.180370i	$0.942270\pi$
$594$	0	0
$595$	0	0
$596$	15.4641	0.633434
$597$	0	0
$598$	65.5692i	2.68132i
$599$	−37.8564	−1.54677	−0.773385	−	0.633936i	$-0.781438\pi$
$599$	−37.8564	−1.54677	−0.773385	+	0.633936i	$0.781438\pi$
$600$	0	0
$601$	−32.6410	−1.33145	−0.665727	−	0.746195i	$-0.731879\pi$
$601$	−32.6410	−1.33145	−0.665727	+	0.746195i	$0.731879\pi$
$602$	− 17.0718i	− 0.695794i
$603$	0	0
$604$	20.3923	0.829751
$605$	0	0
$606$	0	0
$607$	− 18.7846i	− 0.762444i	−0.924484	−	0.381222i	$-0.875503\pi$
$607$	− 18.7846i	− 0.762444i	0.924484	−	0.381222i	$-0.124497\pi$
$608$	28.3923i	1.15146i
$609$	0	0
$610$	0	0
$611$	37.8564	1.53151
$612$	0	0
$613$	20.3923i	0.823637i	0.911266	+	0.411819i	$0.135106\pi$
$613$	20.3923i	0.823637i	−0.911266	+	0.411819i	$0.864894\pi$
$614$	−24.2487	−0.978598
$615$	0	0
$616$	−3.46410	−0.139573
$617$	36.9282i	1.48667i	0.668917	+	0.743337i	$0.266758\pi$
$617$	36.9282i	1.48667i	−0.668917	+	0.743337i	$0.733242\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	− 8.78461i	− 0.352231i
$623$	1.85641i	0.0743754i
$624$	0	0
$625$	0	0
$626$	36.2487	1.44879
$627$	0	0
$628$	3.07180i	0.122578i
$629$	0	0
$630$	0	0
$631$	−21.0718	−0.838855	−0.419427	−	0.907789i	$-0.637769\pi$
$631$	−21.0718	−0.838855	−0.419427	+	0.907789i	$0.637769\pi$
$632$	11.3205i	0.450306i
$633$	0	0
$634$	−43.1769	−1.71477
$635$	0	0
$636$	0	0
$637$	− 16.3923i	− 0.649487i
$638$	− 6.00000i	− 0.237542i
$639$	0	0
$640$	0	0
$641$	12.9282	0.510633	0.255317	−	0.966857i	$-0.417820\pi$
$641$	12.9282	0.510633	0.255317	+	0.966857i	$0.417820\pi$
$642$	0	0
$643$	− 37.5692i	− 1.48159i	−0.671734	−	0.740793i	$-0.734450\pi$
$643$	− 37.5692i	− 1.48159i	0.671734	−	0.740793i	$-0.265550\pi$
$644$	13.8564	0.546019
$645$	0	0
$646$	0	0
$647$	− 27.7128i	− 1.08950i	−0.838597	−	0.544752i	$-0.816624\pi$
$647$	− 27.7128i	− 1.08950i	0.838597	−	0.544752i	$-0.183376\pi$
$648$	0	0
$649$	−6.92820	−0.271956
$650$	0	0
$651$	0	0
$652$	9.85641i	0.386007i
$653$	19.8564i	0.777041i	0.921440	+	0.388521i	$0.127014\pi$
$653$	19.8564i	0.777041i	−0.921440	+	0.388521i	$0.872986\pi$
$654$	0	0
$655$	0	0
$656$	17.3205	0.676252
$657$	0	0
$658$	− 24.0000i	− 0.935617i
$659$	−15.7128	−0.612084	−0.306042	−	0.952018i	$-0.599005\pi$
$659$	−15.7128	−0.612084	−0.306042	+	0.952018i	$0.599005\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	17.0718i	0.663514i
$663$	0	0
$664$	−14.7846	−0.573754
$665$	0	0
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	− 10.3923i	− 0.402090i
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	3.32051i	0.127996i	0.997950	+	0.0639981i	$0.0203852\pi$
$673$	3.32051i	0.127996i	−0.997950	+	0.0639981i	$0.979615\pi$
$674$	−57.4641	−2.21343
$675$	0	0
$676$	16.8564	0.648323
$677$	− 8.78461i	− 0.337620i	−0.985649	−	0.168810i	$-0.946008\pi$
$677$	− 8.78461i	− 0.337620i	0.985649	−	0.168810i	$-0.0539924\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	− 18.9282i	− 0.724798i
$683$	− 32.7846i	− 1.25447i	−0.778831	−	0.627234i	$-0.784187\pi$
$683$	− 32.7846i	− 1.25447i	0.778831	−	0.627234i	$-0.215813\pi$
$684$	0	0
$685$	0	0
$686$	−34.6410	−1.32260
$687$	0	0
$688$	− 24.6410i	− 0.939430i
$689$	5.07180	0.193220
$690$	0	0
$691$	47.7128	1.81508	0.907540	−	0.419965i	$-0.137958\pi$
$691$	47.7128	1.81508	0.907540	+	0.419965i	$0.137958\pi$
$692$	12.0000i	0.456172i
$693$	0	0
$694$	38.7846	1.47224
$695$	0	0
$696$	0	0
$697$	0	0
$698$	14.1051i	0.533887i
$699$	0	0
$700$	0	0
$701$	−39.4641	−1.49054	−0.745269	−	0.666764i	$-0.767679\pi$
$701$	−39.4641	−1.49054	−0.745269	+	0.666764i	$0.767679\pi$
$702$	0	0
$703$	26.9282i	1.01562i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−22.3923	−0.842746
$707$	20.7846i	0.781686i
$708$	0	0
$709$	11.8564	0.445277	0.222638	−	0.974901i	$-0.428533\pi$
$709$	11.8564	0.445277	0.222638	+	0.974901i	$0.428533\pi$
$710$	0	0
$711$	0	0
$712$	1.60770i	0.0602509i
$713$	− 75.7128i	− 2.83547i
$714$	0	0
$715$	0	0
$716$	−6.92820	−0.258919
$717$	0	0
$718$	− 36.0000i	− 1.34351i
$719$	−5.07180	−0.189146	−0.0945731	−	0.995518i	$-0.530149\pi$
$719$	−5.07180	−0.189146	−0.0945731	+	0.995518i	$0.530149\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	18.8038i	0.699807i
$723$	0	0
$724$	−15.8564	−0.589299
$725$	0	0
$726$	0	0
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	18.9282i	0.701526i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 53.9615i	− 1.99311i	−0.0829082	−	0.996557i	$-0.526421\pi$
$733$	− 53.9615i	− 1.99311i	0.0829082	−	0.996557i	$-0.473579\pi$
$734$	−34.6410	−1.27862
$735$	0	0
$736$	36.0000	1.32698
$737$	8.00000i	0.294684i
$738$	0	0
$739$	−17.4641	−0.642427	−0.321214	−	0.947007i	$-0.604091\pi$
$739$	−17.4641	−0.642427	−0.321214	+	0.947007i	$0.604091\pi$
$740$	0	0
$741$	0	0
$742$	− 3.21539i	− 0.118041i
$743$	25.6077i	0.939455i	0.882811	+	0.469728i	$0.155648\pi$
$743$	25.6077i	0.939455i	−0.882811	+	0.469728i	$0.844352\pi$
$744$	0	0
$745$	0	0
$746$	−35.3205	−1.29318
$747$	0	0
$748$	0	0
$749$	17.0718	0.623790
$750$	0	0
$751$	26.9282	0.982624	0.491312	−	0.870984i	$-0.336517\pi$
$751$	26.9282	0.982624	0.491312	+	0.870984i	$0.336517\pi$
$752$	− 34.6410i	− 1.26323i
$753$	0	0
$754$	−32.7846	−1.19395
$755$	0	0
$756$	0	0
$757$	34.7846i	1.26427i	0.774859	+	0.632134i	$0.217821\pi$
$757$	34.7846i	1.26427i	−0.774859	+	0.632134i	$0.782179\pi$
$758$	30.9282i	1.12336i
$759$	0	0
$760$	0	0
$761$	32.5359	1.17943	0.589713	−	0.807613i	$-0.299241\pi$
$761$	32.5359	1.17943	0.589713	+	0.807613i	$0.299241\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	−18.9282	−0.684798
$765$	0	0
$766$	24.0000	0.867155
$767$	37.8564i	1.36692i
$768$	0	0
$769$	−50.4974	−1.82098	−0.910492	−	0.413527i	$-0.864297\pi$
$769$	−50.4974	−1.82098	−0.910492	+	0.413527i	$0.864297\pi$
$770$	0	0
$771$	0	0
$772$	24.3923i	0.877898i
$773$	− 4.14359i	− 0.149035i	−0.997220	−	0.0745174i	$-0.976258\pi$
$773$	− 4.14359i	− 0.149035i	0.997220	−	0.0745174i	$-0.0237416\pi$
$774$	0	0
$775$	0	0
$776$	17.3205	0.621770
$777$	0	0
$778$	19.1769i	0.687526i
$779$	18.9282	0.678173
$780$	0	0
$781$	−13.8564	−0.495821
$782$	0	0
$783$	0	0
$784$	−15.0000	−0.535714
$785$	0	0
$786$	0	0
$787$	22.7846i	0.812184i	0.913832	+	0.406092i	$0.133109\pi$
$787$	22.7846i	0.812184i	−0.913832	+	0.406092i	$0.866891\pi$
$788$	− 12.0000i	− 0.427482i
$789$	0	0
$790$	0	0
$791$	25.8564	0.919348
$792$	0	0
$793$	− 10.9282i	− 0.388072i
$794$	−3.46410	−0.122936
$795$	0	0
$796$	−24.7846	−0.878467
$797$	52.6410i	1.86464i	0.361634	+	0.932320i	$0.382219\pi$
$797$	52.6410i	1.86464i	−0.361634	+	0.932320i	$0.617781\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	13.6077i	0.480504i
$803$	8.39230i	0.296158i
$804$	0	0
$805$	0	0
$806$	−103.426	−3.64301
$807$	0	0
$808$	18.0000i	0.633238i
$809$	−15.4641	−0.543689	−0.271844	−	0.962341i	$-0.587634\pi$
$809$	−15.4641	−0.543689	−0.271844	+	0.962341i	$0.587634\pi$
$810$	0	0
$811$	12.3923	0.435153	0.217576	−	0.976043i	$-0.430185\pi$
$811$	12.3923	0.435153	0.217576	+	0.976043i	$0.430185\pi$
$812$	6.92820i	0.243132i
$813$	0	0
$814$	8.53590	0.299183
$815$	0	0
$816$	0	0
$817$	− 26.9282i	− 0.942099i
$818$	11.7513i	0.410874i
$819$	0	0
$820$	0	0
$821$	−20.5359	−0.716708	−0.358354	−	0.933586i	$-0.616662\pi$
$821$	−20.5359	−0.716708	−0.358354	+	0.933586i	$0.616662\pi$
$822$	0	0
$823$	33.5692i	1.17015i	0.810979	+	0.585075i	$0.198935\pi$
$823$	33.5692i	1.17015i	−0.810979	+	0.585075i	$0.801065\pi$
$824$	13.8564	0.482711
$825$	0	0
$826$	24.0000	0.835067
$827$	− 22.3923i	− 0.778657i	−0.921099	−	0.389328i	$-0.872707\pi$
$827$	− 22.3923i	− 0.778657i	0.921099	−	0.389328i	$-0.127293\pi$
$828$	0	0
$829$	−29.7128	−1.03197	−0.515984	−	0.856598i	$-0.672574\pi$
$829$	−29.7128	−1.03197	−0.515984	+	0.856598i	$0.672574\pi$
$830$	0	0
$831$	0	0
$832$	5.46410i	0.189434i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	5.46410	0.188980
$837$	0	0
$838$	53.5692i	1.85052i
$839$	56.7846	1.96042	0.980211	−	0.197954i	$-0.0634298\pi$
$839$	56.7846	1.96042	0.980211	+	0.197954i	$0.0634298\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	3.46410i	0.119381i
$843$	0	0
$844$	8.39230	0.288875
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	− 4.64102i	− 0.159373i
$849$	0	0
$850$	0	0
$851$	34.1436	1.17043
$852$	0	0
$853$	− 3.60770i	− 0.123525i	−0.998091	−	0.0617626i	$-0.980328\pi$
$853$	− 3.60770i	− 0.123525i	0.998091	−	0.0617626i	$-0.0196722\pi$
$854$	−6.92820	−0.237078
$855$	0	0
$856$	14.7846	0.505328
$857$	37.8564i	1.29315i	0.762850	+	0.646575i	$0.223799\pi$
$857$	37.8564i	1.29315i	−0.762850	+	0.646575i	$0.776201\pi$
$858$	0	0
$859$	7.71281	0.263158	0.131579	−	0.991306i	$-0.457995\pi$
$859$	7.71281	0.263158	0.131579	+	0.991306i	$0.457995\pi$
$860$	0	0
$861$	0	0
$862$	− 15.2154i	− 0.518238i
$863$	− 37.8564i	− 1.28865i	−0.764753	−	0.644324i	$-0.777139\pi$
$863$	− 37.8564i	− 1.28865i	0.764753	−	0.644324i	$-0.222861\pi$
$864$	0	0
$865$	0	0
$866$	0.248711	0.00845155
$867$	0	0
$868$	21.8564i	0.741855i
$869$	6.53590	0.221715
$870$	0	0
$871$	43.7128	1.48115
$872$	17.3205i	0.586546i
$873$	0	0
$874$	65.5692	2.21791
$875$	0	0
$876$	0	0
$877$	− 34.2487i	− 1.15650i	−0.815861	−	0.578248i	$-0.803736\pi$
$877$	− 34.2487i	− 1.15650i	0.815861	−	0.578248i	$-0.196264\pi$
$878$	− 57.4641i	− 1.93932i
$879$	0	0
$880$	0	0
$881$	0.928203	0.0312720	0.0156360	−	0.999878i	$-0.495023\pi$
$881$	0.928203	0.0312720	0.0156360	+	0.999878i	$0.495023\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	0	0
$886$	−20.7846	−0.698273
$887$	− 12.2487i	− 0.411271i	−0.978629	−	0.205636i	$-0.934074\pi$
$887$	− 12.2487i	− 0.411271i	0.978629	−	0.205636i	$-0.0659262\pi$
$888$	0	0
$889$	−17.8564	−0.598885
$890$	0	0
$891$	0	0
$892$	9.85641i	0.330017i
$893$	− 37.8564i	− 1.26682i
$894$	0	0
$895$	0	0
$896$	24.2487	0.810093
$897$	0	0
$898$	− 46.3923i	− 1.54813i
$899$	37.8564	1.26258
$900$	0	0
$901$	0	0
$902$	− 6.00000i	− 0.199778i
$903$	0	0
$904$	22.3923	0.744757
$905$	0	0
$906$	0	0
$907$	18.1436i	0.602448i	0.953553	+	0.301224i	$0.0973952\pi$
$907$	18.1436i	0.602448i	−0.953553	+	0.301224i	$0.902605\pi$
$908$	15.4641i	0.513194i
$909$	0	0
$910$	0	0
$911$	−18.9282	−0.627119	−0.313560	−	0.949568i	$-0.601522\pi$
$911$	−18.9282	−0.627119	−0.313560	+	0.949568i	$0.601522\pi$
$912$	0	0
$913$	8.53590i	0.282497i
$914$	−21.4641	−0.709969
$915$	0	0
$916$	−23.8564	−0.788238
$917$	− 37.8564i	− 1.25013i
$918$	0	0
$919$	32.3923	1.06852	0.534262	−	0.845319i	$-0.320590\pi$
$919$	32.3923	1.06852	0.534262	+	0.845319i	$0.320590\pi$
$920$	0	0
$921$	0	0
$922$	− 62.7846i	− 2.06770i
$923$	75.7128i	2.49212i
$924$	0	0
$925$	0	0
$926$	−48.4974	−1.59372
$927$	0	0
$928$	18.0000i	0.590879i
$929$	2.78461	0.0913601	0.0456800	−	0.998956i	$-0.485455\pi$
$929$	2.78461	0.0913601	0.0456800	+	0.998956i	$0.485455\pi$
$930$	0	0
$931$	−16.3923	−0.537236
$932$	− 12.0000i	− 0.393073i
$933$	0	0
$934$	8.78461	0.287441
$935$	0	0
$936$	0	0
$937$	− 20.3923i	− 0.666188i	−0.942894	−	0.333094i	$-0.891907\pi$
$937$	− 20.3923i	− 0.666188i	0.942894	−	0.333094i	$-0.108093\pi$
$938$	− 27.7128i	− 0.904855i
$939$	0	0
$940$	0	0
$941$	−27.4641	−0.895304	−0.447652	−	0.894208i	$-0.647740\pi$
$941$	−27.4641	−0.895304	−0.447652	+	0.894208i	$0.647740\pi$
$942$	0	0
$943$	− 24.0000i	− 0.781548i
$944$	34.6410	1.12747
$945$	0	0
$946$	−8.53590	−0.277526
$947$	18.9282i	0.615084i	0.951535	+	0.307542i	$0.0995065\pi$
$947$	18.9282i	0.615084i	−0.951535	+	0.307542i	$0.900494\pi$
$948$	0	0
$949$	45.8564	1.48856
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 3.21539i	− 0.104157i	−0.998643	−	0.0520784i	$-0.983415\pi$
$953$	− 3.21539i	− 0.104157i	0.998643	−	0.0520784i	$-0.0165846\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	20.7846i	0.671520i
$959$	−36.0000	−1.16250
$960$	0	0
$961$	88.4256	2.85244
$962$	− 46.6410i	− 1.50377i
$963$	0	0
$964$	−0.143594	−0.00462484
$965$	0	0
$966$	0	0
$967$	22.7846i	0.732704i	0.930476	+	0.366352i	$0.119393\pi$
$967$	22.7846i	0.732704i	−0.930476	+	0.366352i	$0.880607\pi$
$968$	1.73205i	0.0556702i
$969$	0	0
$970$	0	0
$971$	−25.8564	−0.829772	−0.414886	−	0.909873i	$-0.636178\pi$
$971$	−25.8564	−0.829772	−0.414886	+	0.909873i	$0.636178\pi$
$972$	0	0
$973$	− 24.7846i	− 0.794558i
$974$	54.9282	1.76001
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 47.5692i	− 1.52187i	−0.648826	−	0.760937i	$-0.724740\pi$
$977$	− 47.5692i	− 1.52187i	0.648826	−	0.760937i	$-0.275260\pi$
$978$	0	0
$979$	0.928203	0.0296655
$980$	0	0
$981$	0	0
$982$	53.5692i	1.70946i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 29.8564i	− 0.949859i
$989$	−34.1436	−1.08570
$990$	0	0
$991$	−7.21539	−0.229204	−0.114602	−	0.993411i	$-0.536559\pi$
$991$	−7.21539	−0.229204	−0.114602	+	0.993411i	$0.536559\pi$
$992$	56.7846i	1.80291i
$993$	0	0
$994$	48.0000	1.52247
$995$	0	0
$996$	0	0
$997$	27.6077i	0.874344i	0.899378	+	0.437172i	$0.144020\pi$
$997$	27.6077i	0.874344i	−0.899378	+	0.437172i	$0.855980\pi$
$998$	− 49.8564i	− 1.57818i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.n.199.4		4
3.2	odd	2		825.2.c.c.199.1		4
5.2	odd	4		2475.2.a.r.1.1		2
5.3	odd	4		495.2.a.c.1.2		2
5.4	even	2	inner	2475.2.c.n.199.1		4
15.2	even	4		825.2.a.e.1.2		2
15.8	even	4		165.2.a.b.1.1	✓	2
15.14	odd	2		825.2.c.c.199.4		4
20.3	even	4		7920.2.a.bz.1.2		2
55.43	even	4		5445.2.a.s.1.1		2
60.23	odd	4		2640.2.a.x.1.2		2
105.83	odd	4		8085.2.a.bd.1.1		2
165.32	odd	4		9075.2.a.bh.1.1		2
165.98	odd	4		1815.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.1	✓	2	15.8	even	4
495.2.a.c.1.2		2	5.3	odd	4
825.2.a.e.1.2		2	15.2	even	4
825.2.c.c.199.1		4	3.2	odd	2
825.2.c.c.199.4		4	15.14	odd	2
1815.2.a.i.1.2		2	165.98	odd	4
2475.2.a.r.1.1		2	5.2	odd	4
2475.2.c.n.199.1		4	5.4	even	2	inner
2475.2.c.n.199.4		4	1.1	even	1	trivial
2640.2.a.x.1.2		2	60.23	odd	4
5445.2.a.s.1.1		2	55.43	even	4
7920.2.a.bz.1.2		2	20.3	even	4
8085.2.a.bd.1.1		2	105.83	odd	4
9075.2.a.bh.1.1		2	165.32	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +O(q^{10})\)	\(q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +1.00000 q^{11} -5.46410i q^{13} -3.46410 q^{14} -5.00000 q^{16} -5.46410 q^{19} +1.73205i q^{22} +6.92820i q^{23} +9.46410 q^{26} -2.00000i q^{28} -3.46410 q^{29} -10.9282 q^{31} -5.19615i q^{32} -4.92820i q^{37} -9.46410i q^{38} -3.46410 q^{41} +4.92820i q^{43} -1.00000 q^{44} -12.0000 q^{46} +6.92820i q^{47} +3.00000 q^{49} +5.46410i q^{52} +0.928203i q^{53} -3.46410 q^{56} -6.00000i q^{58} -6.92820 q^{59} +2.00000 q^{61} -18.9282i q^{62} -1.00000 q^{64} +8.00000i q^{67} -13.8564 q^{71} +8.39230i q^{73} +8.53590 q^{74} +5.46410 q^{76} +2.00000i q^{77} +6.53590 q^{79} -6.00000i q^{82} +8.53590i q^{83} -8.53590 q^{86} +1.73205i q^{88} +0.928203 q^{89} +10.9282 q^{91} -6.92820i q^{92} -12.0000 q^{94} -10.0000i q^{97} +5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{11} - 20 q^{16} - 8 q^{19} + 24 q^{26} - 16 q^{31} - 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} + 48 q^{74} + 8 q^{76} + 40 q^{79} - 48 q^{86} - 24 q^{89} + 16 q^{91} - 48 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^11 - 20 * q^16 - 8 * q^19 + 24 * q^26 - 16 * q^31 - 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 + 48 * q^74 + 8 * q^76 + 40 * q^79 - 48 * q^86 - 24 * q^89 + 16 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more