LMFDB - Embedded newform 4704.2.c.c.2353.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(2353,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, 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("4704.2353");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.386672896.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 $ x^8 - x^6 - 2x^5 + 2x^4 - 4x^3 - 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2353.4
Root			$-1.19503 - 0.756243i$ of defining polynomial
Character	$\chi$	$=$	4704.2353
Dual form			4704.2.c.c.2353.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +4.10245i q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +4.10245i q^{5} -1.00000 q^{9} -2.67767i q^{11} -3.02497i q^{13} +4.10245 q^{15} +5.12742 q^{17} +2.78012i q^{19} -7.12742 q^{23} -11.8301 q^{25} +1.00000i q^{27} -8.83006i q^{29} -1.42477 q^{31} -2.67767 q^{33} +1.42477i q^{37} -3.02497 q^{39} -5.12742 q^{41} -2.39980i q^{43} -4.10245i q^{45} -9.56024 q^{47} -5.12742i q^{51} -2.78012i q^{53} +10.9850 q^{55} +2.78012 q^{57} +4.00000i q^{59} +5.17992i q^{61} +12.4098 q^{65} +0.244852i q^{67} +7.12742i q^{69} +4.27787 q^{71} +4.15495 q^{73} +11.8301i q^{75} -6.25484 q^{79} +1.00000 q^{81} -9.35535i q^{83} +21.0350i q^{85} -8.83006 q^{87} -11.2824 q^{89} +1.42477i q^{93} -11.4053 q^{95} -6.69460 q^{97} +2.67767i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 4 q^{15} - 4 q^{17} - 12 q^{23} - 24 q^{25} + 8 q^{31} - 12 q^{33} - 8 q^{39} + 4 q^{41} - 8 q^{55} - 16 q^{57} - 16 q^{65} + 28 q^{71} + 8 q^{73} + 40 q^{79} + 8 q^{81} - 20 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 + 4 * q^15 - 4 * q^17 - 12 * q^23 - 24 * q^25 + 8 * q^31 - 12 * q^33 - 8 * q^39 + 4 * q^41 - 8 * q^55 - 16 * q^57 - 16 * q^65 + 28 * q^71 + 8 * q^73 + 40 * q^79 + 8 * q^81 - 20 * q^89 - 40 * q^95 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	4.10245i	1.83467i	0.398117	+	0.917335i	$0.369664\pi$
$5$	4.10245i	1.83467i	−0.398117	+	0.917335i	$0.630336\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.67767i	− 0.807349i	−0.914903	−	0.403674i	$-0.867733\pi$
$11$	− 2.67767i	− 0.807349i	0.914903	−	0.403674i	$-0.132267\pi$
$12$	0	0
$13$	− 3.02497i	− 0.838976i	−0.907761	−	0.419488i	$-0.862210\pi$
$13$	− 3.02497i	− 0.838976i	0.907761	−	0.419488i	$-0.137790\pi$
$14$	0	0
$15$	4.10245	1.05925
$16$	0	0
$17$	5.12742	1.24358	0.621791	−	0.783183i	$-0.286405\pi$
$17$	5.12742	1.24358	0.621791	+	0.783183i	$0.286405\pi$
$18$	0	0
$19$	2.78012i	0.637803i	0.947788	+	0.318902i	$0.103314\pi$
$19$	2.78012i	0.637803i	−0.947788	+	0.318902i	$0.896686\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.12742	−1.48617	−0.743085	−	0.669197i	$-0.766638\pi$
$23$	−7.12742	−1.48617	−0.743085	+	0.669197i	$0.766638\pi$
$24$	0	0
$25$	−11.8301	−2.36601
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 8.83006i	− 1.63970i	−0.572577	−	0.819851i	$-0.694056\pi$
$29$	− 8.83006i	− 1.63970i	0.572577	−	0.819851i	$-0.305944\pi$
$30$	0	0
$31$	−1.42477	−0.255897	−0.127948	−	0.991781i	$-0.540839\pi$
$31$	−1.42477	−0.255897	−0.127948	+	0.991781i	$0.540839\pi$
$32$	0	0
$33$	−2.67767	−0.466123
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.42477i	0.234231i	0.993118	+	0.117116i	$0.0373648\pi$
$37$	1.42477i	0.234231i	−0.993118	+	0.117116i	$0.962635\pi$
$38$	0	0
$39$	−3.02497	−0.484383
$40$	0	0
$41$	−5.12742	−0.800768	−0.400384	−	0.916347i	$-0.631123\pi$
$41$	−5.12742	−0.800768	−0.400384	+	0.916347i	$0.631123\pi$
$42$	0	0
$43$	− 2.39980i	− 0.365966i	−0.983116	−	0.182983i	$-0.941425\pi$
$43$	− 2.39980i	− 0.365966i	0.983116	−	0.182983i	$-0.0585754\pi$
$44$	0	0
$45$	− 4.10245i	− 0.611557i
$46$	0	0
$47$	−9.56024	−1.39450	−0.697252	−	0.716826i	$-0.745594\pi$
$47$	−9.56024	−1.39450	−0.697252	+	0.716826i	$0.745594\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 5.12742i	− 0.717982i
$52$	0	0
$53$	− 2.78012i	− 0.381879i	−0.981602	−	0.190939i	$-0.938847\pi$
$53$	− 2.78012i	− 0.381879i	0.981602	−	0.190939i	$-0.0611534\pi$
$54$	0	0
$55$	10.9850	1.48122
$56$	0	0
$57$	2.78012	0.368236
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	5.17992i	0.663221i	0.943416	+	0.331610i	$0.107592\pi$
$61$	5.17992i	0.663221i	−0.943416	+	0.331610i	$0.892408\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.4098	1.53924
$66$	0	0
$67$	0.244852i	0.0299135i	0.999888	+	0.0149567i	$0.00476105\pi$
$67$	0.244852i	0.0299135i	−0.999888	+	0.0149567i	$0.995239\pi$
$68$	0	0
$69$	7.12742i	0.858040i
$70$	0	0
$71$	4.27787	0.507690	0.253845	−	0.967245i	$-0.418305\pi$
$71$	4.27787	0.507690	0.253845	+	0.967245i	$0.418305\pi$
$72$	0	0
$73$	4.15495	0.486300	0.243150	−	0.969989i	$-0.421819\pi$
$73$	4.15495	0.486300	0.243150	+	0.969989i	$0.421819\pi$
$74$	0	0
$75$	11.8301i	1.36602i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.25484	−0.703724	−0.351862	−	0.936052i	$-0.614451\pi$
$79$	−6.25484	−0.703724	−0.351862	+	0.936052i	$0.614451\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 9.35535i	− 1.02688i	−0.858125	−	0.513441i	$-0.828370\pi$
$83$	− 9.35535i	− 1.02688i	0.858125	−	0.513441i	$-0.171630\pi$
$84$	0	0
$85$	21.0350i	2.28156i
$86$	0	0
$87$	−8.83006	−0.946682
$88$	0	0
$89$	−11.2824	−1.19593	−0.597964	−	0.801523i	$-0.704024\pi$
$89$	−11.2824	−1.19593	−0.597964	+	0.801523i	$0.704024\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.42477i	0.147742i
$94$	0	0
$95$	−11.4053	−1.17016
$96$	0	0
$97$	−6.69460	−0.679733	−0.339867	−	0.940474i	$-0.610382\pi$
$97$	−6.69460	−0.679733	−0.339867	+	0.940474i	$0.610382\pi$
$98$	0	0
$99$	2.67767i	0.269116i
$100$	0	0
$101$	− 1.45779i	− 0.145056i	−0.997366	−	0.0725279i	$-0.976893\pi$
$101$	− 1.45779i	− 0.145056i	0.997366	−	0.0725279i	$-0.0231066\pi$
$102$	0	0
$103$	8.13547	0.801611	0.400806	−	0.916163i	$-0.368730\pi$
$103$	8.13547	0.801611	0.400806	+	0.916163i	$0.368730\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.32233i	− 0.514529i	−0.966341	−	0.257264i	$-0.917179\pi$
$107$	− 5.32233i	− 0.514529i	0.966341	−	0.257264i	$-0.0828211\pi$
$108$	0	0
$109$	− 13.5602i	− 1.29884i	−0.760432	−	0.649418i	$-0.775013\pi$
$109$	− 13.5602i	− 1.29884i	0.760432	−	0.649418i	$-0.224987\pi$
$110$	0	0
$111$	1.42477	0.135233
$112$	0	0
$113$	−4.15495	−0.390865	−0.195432	−	0.980717i	$-0.562611\pi$
$113$	−4.15495	−0.390865	−0.195432	+	0.980717i	$0.562611\pi$
$114$	0	0
$115$	− 29.2398i	− 2.72663i
$116$	0	0
$117$	3.02497i	0.279659i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.83006	0.348188
$122$	0	0
$123$	5.12742i	0.462324i
$124$	0	0
$125$	− 28.0200i	− 2.50618i
$126$	0	0
$127$	0.694597	0.0616355	0.0308177	−	0.999525i	$-0.490189\pi$
$127$	0.694597	0.0616355	0.0308177	+	0.999525i	$0.490189\pi$
$128$	0	0
$129$	−2.39980	−0.211291
$130$	0	0
$131$	− 12.2049i	− 1.06635i	−0.846006	−	0.533173i	$-0.820999\pi$
$131$	− 12.2049i	− 1.06635i	0.846006	−	0.533173i	$-0.179001\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.10245	−0.353082
$136$	0	0
$137$	−15.1044	−1.29045	−0.645227	−	0.763991i	$-0.723237\pi$
$137$	−15.1044	−1.29045	−0.645227	+	0.763991i	$0.723237\pi$
$138$	0	0
$139$	− 7.61018i	− 0.645487i	−0.946486	−	0.322744i	$-0.895395\pi$
$139$	− 7.61018i	− 0.645487i	0.946486	−	0.322744i	$-0.104605\pi$
$140$	0	0
$141$	9.56024i	0.805117i
$142$	0	0
$143$	−8.09989	−0.677347
$144$	0	0
$145$	36.2249	3.00831
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.21988i	0.427629i	0.976874	+	0.213815i	$0.0685889\pi$
$149$	5.21988i	0.427629i	−0.976874	+	0.213815i	$0.931411\pi$
$150$	0	0
$151$	5.56024	0.452486	0.226243	−	0.974071i	$-0.427356\pi$
$151$	5.56024	0.452486	0.226243	+	0.974071i	$0.427356\pi$
$152$	0	0
$153$	−5.12742	−0.414527
$154$	0	0
$155$	− 5.84505i	− 0.469486i
$156$	0	0
$157$	0.519169i	0.0414342i	0.999785	+	0.0207171i	$0.00659493\pi$
$157$	0.519169i	0.0414342i	−0.999785	+	0.0207171i	$0.993405\pi$
$158$	0	0
$159$	−2.78012	−0.220478
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 8.65464i	− 0.677883i	−0.940807	−	0.338942i	$-0.889931\pi$
$163$	− 8.65464i	− 0.677883i	0.940807	−	0.338942i	$-0.110069\pi$
$164$	0	0
$165$	− 10.9850i	− 0.855182i
$166$	0	0
$167$	−25.1044	−1.94264	−0.971318	−	0.237786i	$-0.923578\pi$
$167$	−25.1044	−1.94264	−0.971318	+	0.237786i	$0.923578\pi$
$168$	0	0
$169$	3.84954	0.296119
$170$	0	0
$171$	− 2.78012i	− 0.212601i
$172$	0	0
$173$	1.94750i	0.148066i	0.997256	+	0.0740328i	$0.0235869\pi$
$173$	1.94750i	0.148066i	−0.997256	+	0.0740328i	$0.976413\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	− 22.6976i	− 1.69650i	−0.529595	−	0.848251i	$-0.677656\pi$
$179$	− 22.6976i	− 1.69650i	0.529595	−	0.848251i	$-0.322344\pi$
$180$	0	0
$181$	− 10.5353i	− 0.783080i	−0.920161	−	0.391540i	$-0.871942\pi$
$181$	− 10.5353i	− 0.783080i	0.920161	−	0.391540i	$-0.128058\pi$
$182$	0	0
$183$	5.17992	0.382911
$184$	0	0
$185$	−5.84505	−0.429737
$186$	0	0
$187$	− 13.7296i	− 1.00400i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.27787	0.309536	0.154768	−	0.987951i	$-0.450537\pi$
$191$	4.27787	0.309536	0.154768	+	0.987951i	$0.450537\pi$
$192$	0	0
$193$	23.1400	1.66565	0.832825	−	0.553536i	$-0.186722\pi$
$193$	23.1400	1.66565	0.832825	+	0.553536i	$0.186722\pi$
$194$	0	0
$195$	− 12.4098i	− 0.888683i
$196$	0	0
$197$	− 19.4747i	− 1.38752i	−0.720208	−	0.693758i	$-0.755954\pi$
$197$	− 19.4747i	− 1.38752i	0.720208	−	0.693758i	$-0.244046\pi$
$198$	0	0
$199$	−17.5602	−1.24481	−0.622406	−	0.782694i	$-0.713845\pi$
$199$	−17.5602	−1.24481	−0.622406	+	0.782694i	$0.713845\pi$
$200$	0	0
$201$	0.244852	0.0172705
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 21.0350i	− 1.46915i
$206$	0	0
$207$	7.12742	0.495390
$208$	0	0
$209$	7.44425	0.514930
$210$	0	0
$211$	23.9600i	1.64948i	0.565514	+	0.824739i	$0.308678\pi$
$211$	23.9600i	1.64948i	−0.565514	+	0.824739i	$0.691322\pi$
$212$	0	0
$213$	− 4.27787i	− 0.293115i
$214$	0	0
$215$	9.84505	0.671427
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 4.15495i	− 0.280765i
$220$	0	0
$221$	− 15.5103i	− 1.04334i
$222$	0	0
$223$	2.74966	0.184131	0.0920653	−	0.995753i	$-0.470653\pi$
$223$	2.74966	0.184131	0.0920653	+	0.995753i	$0.470653\pi$
$224$	0	0
$225$	11.8301	0.788671
$226$	0	0
$227$	8.30478i	0.551208i	0.961271	+	0.275604i	$0.0888778\pi$
$227$	8.30478i	0.551208i	−0.961271	+	0.275604i	$0.911122\pi$
$228$	0	0
$229$	− 11.9245i	− 0.787991i	−0.919112	−	0.393995i	$-0.871093\pi$
$229$	− 11.9245i	− 0.787991i	0.919112	−	0.393995i	$-0.128907\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.56024	−0.233239	−0.116620	−	0.993177i	$-0.537206\pi$
$233$	−3.56024	−0.233239	−0.116620	+	0.993177i	$0.537206\pi$
$234$	0	0
$235$	− 39.2204i	− 2.55845i
$236$	0	0
$237$	6.25484i	0.406295i
$238$	0	0
$239$	−26.9425	−1.74277	−0.871383	−	0.490604i	$-0.836776\pi$
$239$	−26.9425	−1.74277	−0.871383	+	0.490604i	$0.836776\pi$
$240$	0	0
$241$	−25.8151	−1.66290	−0.831448	−	0.555603i	$-0.812487\pi$
$241$	−25.8151	−1.66290	−0.831448	+	0.555603i	$0.812487\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.40978	0.535102
$248$	0	0
$249$	−9.35535	−0.592871
$250$	0	0
$251$	27.1543i	1.71397i	0.515345	+	0.856983i	$0.327664\pi$
$251$	27.1543i	1.71397i	−0.515345	+	0.856983i	$0.672336\pi$
$252$	0	0
$253$	19.0849i	1.19986i
$254$	0	0
$255$	21.0350	1.31726
$256$	0	0
$257$	31.3823	1.95757	0.978786	−	0.204887i	$-0.0656827\pi$
$257$	31.3823	1.95757	0.978786	+	0.204887i	$0.0656827\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.83006i	0.546567i
$262$	0	0
$263$	11.0275	0.679987	0.339993	−	0.940428i	$-0.389575\pi$
$263$	11.0275	0.679987	0.339993	+	0.940428i	$0.389575\pi$
$264$	0	0
$265$	11.4053	0.700621
$266$	0	0
$267$	11.2824i	0.690470i
$268$	0	0
$269$	− 21.0019i	− 1.28051i	−0.768162	−	0.640255i	$-0.778829\pi$
$269$	− 21.0019i	− 1.28051i	0.768162	−	0.640255i	$-0.221171\pi$
$270$	0	0
$271$	28.6451	1.74007	0.870034	−	0.492991i	$-0.164097\pi$
$271$	28.6451	1.74007	0.870034	+	0.492991i	$0.164097\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	31.6771i	1.91020i
$276$	0	0
$277$	2.96504i	0.178152i	0.996025	+	0.0890761i	$0.0283914\pi$
$277$	2.96504i	0.178152i	−0.996025	+	0.0890761i	$0.971609\pi$
$278$	0	0
$279$	1.42477	0.0852989
$280$	0	0
$281$	26.4098	1.57548	0.787738	−	0.616011i	$-0.211252\pi$
$281$	26.4098	1.57548	0.787738	+	0.616011i	$0.211252\pi$
$282$	0	0
$283$	11.2698i	0.669922i	0.942232	+	0.334961i	$0.108723\pi$
$283$	11.2698i	0.669922i	−0.942232	+	0.334961i	$0.891277\pi$
$284$	0	0
$285$	11.4053i	0.675591i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.29042	0.546495
$290$	0	0
$291$	6.69460i	0.392444i
$292$	0	0
$293$	− 10.5622i	− 0.617049i	−0.951216	−	0.308524i	$-0.900165\pi$
$293$	− 10.5622i	− 0.617049i	0.951216	−	0.308524i	$-0.0998351\pi$
$294$	0	0
$295$	−16.4098	−0.955415
$296$	0	0
$297$	2.67767	0.155374
$298$	0	0
$299$	21.5602i	1.24686i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.45779	−0.0837480
$304$	0	0
$305$	−21.2503	−1.21679
$306$	0	0
$307$	− 10.7801i	− 0.615254i	−0.951507	−	0.307627i	$-0.900465\pi$
$307$	− 10.7801i	− 0.615254i	0.951507	−	0.307627i	$-0.0995349\pi$
$308$	0	0
$309$	− 8.13547i	− 0.462811i
$310$	0	0
$311$	9.56024	0.542111	0.271056	−	0.962564i	$-0.412627\pi$
$311$	9.56024	0.542111	0.271056	+	0.962564i	$0.412627\pi$
$312$	0	0
$313$	−7.69909	−0.435178	−0.217589	−	0.976040i	$-0.569819\pi$
$313$	−7.69909	−0.435178	−0.217589	+	0.976040i	$0.569819\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.7801i	− 0.605472i	−0.953074	−	0.302736i	$-0.902100\pi$
$317$	− 10.7801i	− 0.605472i	0.953074	−	0.302736i	$-0.0979000\pi$
$318$	0	0
$319$	−23.6440	−1.32381
$320$	0	0
$321$	−5.32233	−0.297063
$322$	0	0
$323$	14.2548i	0.793160i
$324$	0	0
$325$	35.7856i	1.98503i
$326$	0	0
$327$	−13.5602	−0.749883
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 25.2104i	− 1.38569i	−0.721088	−	0.692844i	$-0.756357\pi$
$331$	− 25.2104i	− 1.38569i	0.721088	−	0.692844i	$-0.243643\pi$
$332$	0	0
$333$	− 1.42477i	− 0.0780770i
$334$	0	0
$335$	−1.00449	−0.0548813
$336$	0	0
$337$	−23.9700	−1.30573	−0.652865	−	0.757474i	$-0.726433\pi$
$337$	−23.9700	−1.30573	−0.652865	+	0.757474i	$0.726433\pi$
$338$	0	0
$339$	4.15495i	0.225666i
$340$	0	0
$341$	3.81508i	0.206598i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−29.2398	−1.57422
$346$	0	0
$347$	− 6.98757i	− 0.375112i	−0.982254	−	0.187556i	$-0.939943\pi$
$347$	− 6.98757i	− 0.375112i	0.982254	−	0.187556i	$-0.0600567\pi$
$348$	0	0
$349$	20.8900i	1.11822i	0.829095	+	0.559108i	$0.188856\pi$
$349$	20.8900i	1.11822i	−0.829095	+	0.559108i	$0.811144\pi$
$350$	0	0
$351$	3.02497	0.161461
$352$	0	0
$353$	−3.38225	−0.180019	−0.0900096	−	0.995941i	$-0.528690\pi$
$353$	−3.38225	−0.180019	−0.0900096	+	0.995941i	$0.528690\pi$
$354$	0	0
$355$	17.5497i	0.931444i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.6877	−1.09185	−0.545926	−	0.837833i	$-0.683822\pi$
$359$	−20.6877	−1.09185	−0.545926	+	0.837833i	$0.683822\pi$
$360$	0	0
$361$	11.2709	0.593207
$362$	0	0
$363$	− 3.83006i	− 0.201026i
$364$	0	0
$365$	17.0455i	0.892200i
$366$	0	0
$367$	−9.56024	−0.499040	−0.249520	−	0.968370i	$-0.580273\pi$
$367$	−9.56024	−0.499040	−0.249520	+	0.968370i	$0.580273\pi$
$368$	0	0
$369$	5.12742	0.266923
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.95006i	0.100970i	0.998725	+	0.0504850i	$0.0160767\pi$
$373$	1.95006i	0.100970i	−0.998725	+	0.0504850i	$0.983923\pi$
$374$	0	0
$375$	−28.0200	−1.44694
$376$	0	0
$377$	−26.7107	−1.37567
$378$	0	0
$379$	− 32.3698i	− 1.66273i	−0.555730	−	0.831363i	$-0.687561\pi$
$379$	− 32.3698i	− 1.66273i	0.555730	−	0.831363i	$-0.312439\pi$
$380$	0	0
$381$	− 0.694597i	− 0.0355853i
$382$	0	0
$383$	−17.8451	−0.911840	−0.455920	−	0.890021i	$-0.650690\pi$
$383$	−17.8451	−0.911840	−0.455920	+	0.890021i	$0.650690\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.39980i	0.121989i
$388$	0	0
$389$	6.11937i	0.310264i	0.987894	+	0.155132i	$0.0495803\pi$
$389$	6.11937i	0.310264i	−0.987894	+	0.155132i	$0.950420\pi$
$390$	0	0
$391$	−36.5453	−1.84817
$392$	0	0
$393$	−12.2049	−0.615655
$394$	0	0
$395$	− 25.6601i	− 1.29110i
$396$	0	0
$397$	− 3.71957i	− 0.186680i	−0.995634	−	0.0933399i	$-0.970246\pi$
$397$	− 3.71957i	− 0.186680i	0.995634	−	0.0933399i	$-0.0297543\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.5258	−1.62426	−0.812130	−	0.583477i	$-0.801692\pi$
$401$	−32.5258	−1.62426	−0.812130	+	0.583477i	$0.801692\pi$
$402$	0	0
$403$	4.30990i	0.214691i
$404$	0	0
$405$	4.10245i	0.203852i
$406$	0	0
$407$	3.81508	0.189106
$408$	0	0
$409$	1.13436	0.0560903	0.0280452	−	0.999607i	$-0.491072\pi$
$409$	1.13436	0.0560903	0.0280452	+	0.999607i	$0.491072\pi$
$410$	0	0
$411$	15.1044i	0.745044i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	38.3798	1.88399
$416$	0	0
$417$	−7.61018	−0.372672
$418$	0	0
$419$	1.56024i	0.0762227i	0.999273	+	0.0381113i	$0.0121342\pi$
$419$	1.56024i	0.0762227i	−0.999273	+	0.0381113i	$0.987866\pi$
$420$	0	0
$421$	− 31.8845i	− 1.55396i	−0.629528	−	0.776978i	$-0.716752\pi$
$421$	− 31.8845i	− 1.55396i	0.629528	−	0.776978i	$-0.283248\pi$
$422$	0	0
$423$	9.56024	0.464835
$424$	0	0
$425$	−60.6577	−2.94233
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.09989i	0.391066i
$430$	0	0
$431$	−8.87258	−0.427377	−0.213689	−	0.976902i	$-0.568548\pi$
$431$	−8.87258	−0.427377	−0.213689	+	0.976902i	$0.568548\pi$
$432$	0	0
$433$	−11.5602	−0.555550	−0.277775	−	0.960646i	$-0.589597\pi$
$433$	−11.5602	−0.555550	−0.277775	+	0.960646i	$0.589597\pi$
$434$	0	0
$435$	− 36.2249i	− 1.73685i
$436$	0	0
$437$	− 19.8151i	− 0.947884i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.87807i	− 0.279276i	−0.990203	−	0.139638i	$-0.955406\pi$
$443$	− 5.87807i	− 0.279276i	0.990203	−	0.139638i	$-0.0445938\pi$
$444$	0	0
$445$	− 46.2853i	− 2.19413i
$446$	0	0
$447$	5.21988	0.246892
$448$	0	0
$449$	−11.8451	−0.559003	−0.279501	−	0.960145i	$-0.590169\pi$
$449$	−11.8451	−0.559003	−0.279501	+	0.960145i	$0.590169\pi$
$450$	0	0
$451$	13.7296i	0.646499i
$452$	0	0
$453$	− 5.56024i	− 0.261243i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.2088	−1.31955	−0.659775	−	0.751463i	$-0.729348\pi$
$457$	−28.2088	−1.31955	−0.659775	+	0.751463i	$0.729348\pi$
$458$	0	0
$459$	5.12742i	0.239327i
$460$	0	0
$461$	11.0519i	0.514737i	0.966313	+	0.257369i	$0.0828555\pi$
$461$	11.0519i	0.514737i	−0.966313	+	0.257369i	$0.917145\pi$
$462$	0	0
$463$	22.2548	1.03427	0.517135	−	0.855904i	$-0.326999\pi$
$463$	22.2548	1.03427	0.517135	+	0.855904i	$0.326999\pi$
$464$	0	0
$465$	−5.84505	−0.271058
$466$	0	0
$467$	21.4552i	0.992830i	0.868085	+	0.496415i	$0.165351\pi$
$467$	21.4552i	0.992830i	−0.868085	+	0.496415i	$0.834649\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.519169	0.0239221
$472$	0	0
$473$	−6.42588	−0.295462
$474$	0	0
$475$	− 32.8890i	− 1.50905i
$476$	0	0
$477$	2.78012i	0.127293i
$478$	0	0
$479$	8.09989	0.370093	0.185047	−	0.982730i	$-0.440756\pi$
$479$	8.09989	0.370093	0.185047	+	0.982730i	$0.440756\pi$
$480$	0	0
$481$	4.30990	0.196514
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 27.4642i	− 1.24709i
$486$	0	0
$487$	16.4098	0.743598	0.371799	−	0.928313i	$-0.378741\pi$
$487$	16.4098	0.743598	0.371799	+	0.928313i	$0.378741\pi$
$488$	0	0
$489$	−8.65464	−0.391376
$490$	0	0
$491$	39.5971i	1.78699i	0.449070	+	0.893497i	$0.351755\pi$
$491$	39.5971i	1.78699i	−0.449070	+	0.893497i	$0.648245\pi$
$492$	0	0
$493$	− 45.2754i	− 2.03910i
$494$	0	0
$495$	−10.9850	−0.493740
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.14946i	0.0514568i	0.999669	+	0.0257284i	$0.00819050\pi$
$499$	1.14946i	0.0514568i	−0.999669	+	0.0257284i	$0.991809\pi$
$500$	0	0
$501$	25.1044i	1.12158i
$502$	0	0
$503$	9.00449	0.401490	0.200745	−	0.979643i	$-0.435664\pi$
$503$	9.00449	0.401490	0.200745	+	0.979643i	$0.435664\pi$
$504$	0	0
$505$	5.98052	0.266130
$506$	0	0
$507$	− 3.84954i	− 0.170964i
$508$	0	0
$509$	36.5122i	1.61838i	0.587550	+	0.809188i	$0.300093\pi$
$509$	36.5122i	1.61838i	−0.587550	+	0.809188i	$0.699907\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.78012	−0.122745
$514$	0	0
$515$	33.3753i	1.47069i
$516$	0	0
$517$	25.5992i	1.12585i
$518$	0	0
$519$	1.94750	0.0854857
$520$	0	0
$521$	18.0929	0.792666	0.396333	−	0.918107i	$-0.370282\pi$
$521$	18.0929	0.792666	0.396333	+	0.918107i	$0.370282\pi$
$522$	0	0
$523$	− 14.7107i	− 0.643254i	−0.946866	−	0.321627i	$-0.895770\pi$
$523$	− 14.7107i	− 0.643254i	0.946866	−	0.321627i	$-0.104230\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.30540	−0.318228
$528$	0	0
$529$	27.8001	1.20870
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	15.5103i	0.671825i
$534$	0	0
$535$	21.8346	0.943990
$536$	0	0
$537$	−22.6976	−0.979476
$538$	0	0
$539$	0	0
$540$	0	0
$541$	37.7157i	1.62152i	0.585376	+	0.810762i	$0.300947\pi$
$541$	37.7157i	1.62152i	−0.585376	+	0.810762i	$0.699053\pi$
$542$	0	0
$543$	−10.5353	−0.452112
$544$	0	0
$545$	55.6302	2.38293
$546$	0	0
$547$	23.1144i	0.988299i	0.869377	+	0.494149i	$0.164520\pi$
$547$	23.1144i	0.988299i	−0.869377	+	0.494149i	$0.835480\pi$
$548$	0	0
$549$	− 5.17992i	− 0.221074i
$550$	0	0
$551$	24.5486	1.04581
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.84505i	0.248109i
$556$	0	0
$557$	11.1899i	0.474131i	0.971494	+	0.237066i	$0.0761857\pi$
$557$	11.1899i	0.474131i	−0.971494	+	0.237066i	$0.923814\pi$
$558$	0	0
$559$	−7.25933	−0.307037
$560$	0	0
$561$	−13.7296	−0.579662
$562$	0	0
$563$	− 2.81058i	− 0.118452i	−0.998245	−	0.0592260i	$-0.981137\pi$
$563$	− 2.81058i	− 0.118452i	0.998245	−	0.0592260i	$-0.0188632\pi$
$564$	0	0
$565$	− 17.0455i	− 0.717108i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0699	−0.841374	−0.420687	−	0.907206i	$-0.638211\pi$
$569$	−20.0699	−0.841374	−0.420687	+	0.907206i	$0.638211\pi$
$570$	0	0
$571$	2.46584i	0.103192i	0.998668	+	0.0515961i	$0.0164309\pi$
$571$	2.46584i	0.103192i	−0.998668	+	0.0515961i	$0.983569\pi$
$572$	0	0
$573$	− 4.27787i	− 0.178711i
$574$	0	0
$575$	84.3178	3.51630
$576$	0	0
$577$	−4.84954	−0.201889	−0.100945	−	0.994892i	$-0.532186\pi$
$577$	−4.84954	−0.201889	−0.100945	+	0.994892i	$0.532186\pi$
$578$	0	0
$579$	− 23.1400i	− 0.961664i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.44425	−0.308309
$584$	0	0
$585$	−12.4098	−0.513081
$586$	0	0
$587$	19.0155i	0.784853i	0.919783	+	0.392426i	$0.128364\pi$
$587$	19.0155i	0.784853i	−0.919783	+	0.392426i	$0.871636\pi$
$588$	0	0
$589$	− 3.96104i	− 0.163212i
$590$	0	0
$591$	−19.4747	−0.801083
$592$	0	0
$593$	−24.0379	−0.987118	−0.493559	−	0.869712i	$-0.664304\pi$
$593$	−24.0379	−0.987118	−0.493559	+	0.869712i	$0.664304\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.5602i	0.718693i
$598$	0	0
$599$	−33.9631	−1.38769	−0.693847	−	0.720122i	$-0.744086\pi$
$599$	−33.9631	−1.38769	−0.693847	+	0.720122i	$0.744086\pi$
$600$	0	0
$601$	−10.0999	−0.411983	−0.205992	−	0.978554i	$-0.566042\pi$
$601$	−10.0999	−0.411983	−0.205992	+	0.978554i	$0.566042\pi$
$602$	0	0
$603$	− 0.244852i	− 0.00997115i
$604$	0	0
$605$	15.7126i	0.638809i
$606$	0	0
$607$	8.17105	0.331653	0.165826	−	0.986155i	$-0.446971\pi$
$607$	8.17105	0.331653	0.165826	+	0.986155i	$0.446971\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	28.9195i	1.16996i
$612$	0	0
$613$	23.0206i	0.929793i	0.885365	+	0.464896i	$0.153908\pi$
$613$	23.0206i	0.929793i	−0.885365	+	0.464896i	$0.846092\pi$
$614$	0	0
$615$	−21.0350	−0.848211
$616$	0	0
$617$	31.1044	1.25222	0.626108	−	0.779737i	$-0.284647\pi$
$617$	31.1044	1.25222	0.626108	+	0.779737i	$0.284647\pi$
$618$	0	0
$619$	40.9195i	1.64469i	0.568988	+	0.822346i	$0.307335\pi$
$619$	40.9195i	1.64469i	−0.568988	+	0.822346i	$0.692665\pi$
$620$	0	0
$621$	− 7.12742i	− 0.286013i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	55.8001	2.23200
$626$	0	0
$627$	− 7.44425i	− 0.297295i
$628$	0	0
$629$	7.30540i	0.291286i
$630$	0	0
$631$	2.29380	0.0913147	0.0456573	−	0.998957i	$-0.485462\pi$
$631$	2.29380	0.0913147	0.0456573	+	0.998957i	$0.485462\pi$
$632$	0	0
$633$	23.9600	0.952326
$634$	0	0
$635$	2.84954i	0.113081i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.27787	−0.169230
$640$	0	0
$641$	27.0654	1.06902	0.534510	−	0.845162i	$-0.320496\pi$
$641$	27.0654	1.06902	0.534510	+	0.845162i	$0.320496\pi$
$642$	0	0
$643$	31.9895i	1.26154i	0.775969	+	0.630771i	$0.217261\pi$
$643$	31.9895i	1.26154i	−0.775969	+	0.630771i	$0.782739\pi$
$644$	0	0
$645$	− 9.84505i	− 0.387649i
$646$	0	0
$647$	−12.5947	−0.495149	−0.247575	−	0.968869i	$-0.579634\pi$
$647$	−12.5947	−0.495149	−0.247575	+	0.968869i	$0.579634\pi$
$648$	0	0
$649$	10.7107	0.420432
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.8346i	− 0.854452i	−0.904145	−	0.427226i	$-0.859491\pi$
$653$	− 21.8346i	− 0.854452i	0.904145	−	0.427226i	$-0.140509\pi$
$654$	0	0
$655$	50.0699	1.95639
$656$	0	0
$657$	−4.15495	−0.162100
$658$	0	0
$659$	− 24.7865i	− 0.965547i	−0.875745	−	0.482773i	$-0.839629\pi$
$659$	− 24.7865i	− 0.965547i	0.875745	−	0.482773i	$-0.160371\pi$
$660$	0	0
$661$	3.09101i	0.120227i	0.998192	+	0.0601133i	$0.0191462\pi$
$661$	3.09101i	0.120227i	−0.998192	+	0.0601133i	$0.980854\pi$
$662$	0	0
$663$	−15.5103	−0.602370
$664$	0	0
$665$	0	0
$666$	0	0
$667$	62.9356i	2.43687i
$668$	0	0
$669$	− 2.74966i	− 0.106308i
$670$	0	0
$671$	13.8701	0.535451
$672$	0	0
$673$	−27.6496	−1.06581	−0.532907	−	0.846174i	$-0.678901\pi$
$673$	−27.6496	−1.06581	−0.532907	+	0.846174i	$0.678901\pi$
$674$	0	0
$675$	− 11.8301i	− 0.455339i
$676$	0	0
$677$	5.84249i	0.224545i	0.993677	+	0.112273i	$0.0358130\pi$
$677$	5.84249i	0.224545i	−0.993677	+	0.112273i	$0.964187\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.30478	0.318240
$682$	0	0
$683$	15.5433i	0.594748i	0.954761	+	0.297374i	$0.0961109\pi$
$683$	15.5433i	0.594748i	−0.954761	+	0.297374i	$0.903889\pi$
$684$	0	0
$685$	− 61.9649i	− 2.36756i
$686$	0	0
$687$	−11.9245	−0.454947
$688$	0	0
$689$	−8.40978	−0.320387
$690$	0	0
$691$	− 42.9304i	− 1.63315i	−0.577239	−	0.816575i	$-0.695870\pi$
$691$	− 42.9304i	− 1.63315i	0.577239	−	0.816575i	$-0.304130\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	31.2204	1.18426
$696$	0	0
$697$	−26.2904	−0.995820
$698$	0	0
$699$	3.56024i	0.134661i
$700$	0	0
$701$	− 5.21476i	− 0.196959i	−0.995139	−	0.0984795i	$-0.968602\pi$
$701$	− 5.21476i	− 0.196959i	0.995139	−	0.0984795i	$-0.0313979\pi$
$702$	0	0
$703$	−3.96104	−0.149393
$704$	0	0
$705$	−39.2204	−1.47712
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 41.6601i	− 1.56458i	−0.622915	−	0.782289i	$-0.714052\pi$
$709$	− 41.6601i	− 1.56458i	0.622915	−	0.782289i	$-0.285948\pi$
$710$	0	0
$711$	6.25484	0.234575
$712$	0	0
$713$	10.1549	0.380306
$714$	0	0
$715$	− 33.2294i	− 1.24271i
$716$	0	0
$717$	26.9425i	1.00619i
$718$	0	0
$719$	14.3547	0.535341	0.267670	−	0.963511i	$-0.413746\pi$
$719$	14.3547	0.535341	0.267670	+	0.963511i	$0.413746\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	25.8151i	0.960073i
$724$	0	0
$725$	104.460i	3.87955i
$726$	0	0
$727$	53.4647	1.98290	0.991448	−	0.130501i	$-0.0416587\pi$
$727$	53.4647	1.98290	0.991448	+	0.130501i	$0.0416587\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 12.3048i	− 0.455109i
$732$	0	0
$733$	− 49.1149i	− 1.81410i	−0.421025	−	0.907049i	$-0.638329\pi$
$733$	− 49.1149i	− 1.81410i	0.421025	−	0.907049i	$-0.361671\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.655634	0.0241506
$738$	0	0
$739$	7.75515i	0.285278i	0.989775	+	0.142639i	$0.0455587\pi$
$739$	7.75515i	0.285278i	−0.989775	+	0.142639i	$0.954441\pi$
$740$	0	0
$741$	− 8.40978i	− 0.308941i
$742$	0	0
$743$	−30.9886	−1.13686	−0.568430	−	0.822732i	$-0.692449\pi$
$743$	−30.9886	−1.13686	−0.568430	+	0.822732i	$0.692449\pi$
$744$	0	0
$745$	−21.4143	−0.784558
$746$	0	0
$747$	9.35535i	0.342294i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	34.9105	1.27390	0.636951	−	0.770905i	$-0.280196\pi$
$751$	34.9105	1.27390	0.636951	+	0.770905i	$0.280196\pi$
$752$	0	0
$753$	27.1543	0.989559
$754$	0	0
$755$	22.8106i	0.830162i
$756$	0	0
$757$	9.25034i	0.336209i	0.985769	+	0.168105i	$0.0537647\pi$
$757$	9.25034i	0.336209i	−0.985769	+	0.168105i	$0.946235\pi$
$758$	0	0
$759$	19.0849	0.692738
$760$	0	0
$761$	11.8381	0.429131	0.214566	−	0.976710i	$-0.431166\pi$
$761$	11.8381	0.429131	0.214566	+	0.976710i	$0.431166\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 21.0350i	− 0.760520i
$766$	0	0
$767$	12.0999	0.436902
$768$	0	0
$769$	20.0699	0.723740	0.361870	−	0.932229i	$-0.382139\pi$
$769$	20.0699	0.723740	0.361870	+	0.932229i	$0.382139\pi$
$770$	0	0
$771$	− 31.3823i	− 1.13020i
$772$	0	0
$773$	18.8521i	0.678063i	0.940775	+	0.339032i	$0.110099\pi$
$773$	18.8521i	0.678063i	−0.940775	+	0.339032i	$0.889901\pi$
$774$	0	0
$775$	16.8551	0.605455
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 14.2548i	− 0.510733i
$780$	0	0
$781$	− 11.4547i	− 0.409883i
$782$	0	0
$783$	8.83006	0.315561
$784$	0	0
$785$	−2.12986	−0.0760181
$786$	0	0
$787$	25.9111i	0.923631i	0.886976	+	0.461815i	$0.152802\pi$
$787$	25.9111i	0.923631i	−0.886976	+	0.461815i	$0.847198\pi$
$788$	0	0
$789$	− 11.0275i	− 0.392590i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	15.6691	0.556427
$794$	0	0
$795$	− 11.4053i	− 0.404504i
$796$	0	0
$797$	− 11.1318i	− 0.394309i	−0.980372	−	0.197154i	$-0.936830\pi$
$797$	− 11.1318i	− 0.394309i	0.980372	−	0.197154i	$-0.0631700\pi$
$798$	0	0
$799$	−49.0193	−1.73418
$800$	0	0
$801$	11.2824	0.398643
$802$	0	0
$803$	− 11.1256i	− 0.392614i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21.0019	−0.739303
$808$	0	0
$809$	33.8151	1.18887	0.594437	−	0.804142i	$-0.297375\pi$
$809$	33.8151	1.18887	0.594437	+	0.804142i	$0.297375\pi$
$810$	0	0
$811$	− 32.5097i	− 1.14157i	−0.821100	−	0.570784i	$-0.806639\pi$
$811$	− 32.5097i	− 1.14157i	0.821100	−	0.570784i	$-0.193361\pi$
$812$	0	0
$813$	− 28.6451i	− 1.00463i
$814$	0	0
$815$	35.5052	1.24369
$816$	0	0
$817$	6.67173	0.233414
$818$	0	0
$819$	0	0
$820$	0	0
$821$	40.7501i	1.42219i	0.703096	+	0.711095i	$0.251800\pi$
$821$	40.7501i	1.42219i	−0.703096	+	0.711095i	$0.748200\pi$
$822$	0	0
$823$	34.2159	1.19269	0.596345	−	0.802728i	$-0.296619\pi$
$823$	34.2159	1.19269	0.596345	+	0.802728i	$0.296619\pi$
$824$	0	0
$825$	31.6771	1.10285
$826$	0	0
$827$	37.2424i	1.29505i	0.762046	+	0.647523i	$0.224195\pi$
$827$	37.2424i	1.29505i	−0.762046	+	0.647523i	$0.775805\pi$
$828$	0	0
$829$	11.2959i	0.392323i	0.980572	+	0.196162i	$0.0628477\pi$
$829$	11.2959i	0.392323i	−0.980572	+	0.196162i	$0.937152\pi$
$830$	0	0
$831$	2.96504	0.102856
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 102.989i	− 3.56409i
$836$	0	0
$837$	− 1.42477i	− 0.0492473i
$838$	0	0
$839$	−38.8746	−1.34210	−0.671051	−	0.741412i	$-0.734157\pi$
$839$	−38.8746	−1.34210	−0.671051	+	0.741412i	$0.734157\pi$
$840$	0	0
$841$	−48.9700	−1.68862
$842$	0	0
$843$	− 26.4098i	− 0.909601i
$844$	0	0
$845$	15.7925i	0.543280i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.2698	0.386779
$850$	0	0
$851$	− 10.1549i	− 0.348107i
$852$	0	0
$853$	25.5507i	0.874841i	0.899257	+	0.437420i	$0.144108\pi$
$853$	25.5507i	0.874841i	−0.899257	+	0.437420i	$0.855892\pi$
$854$	0	0
$855$	11.4053	0.390053
$856$	0	0
$857$	−20.7038	−0.707227	−0.353613	−	0.935392i	$-0.615047\pi$
$857$	−20.7038	−0.707227	−0.353613	+	0.935392i	$0.615047\pi$
$858$	0	0
$859$	− 9.66863i	− 0.329889i	−0.986303	−	0.164945i	$-0.947255\pi$
$859$	− 9.66863i	− 0.329889i	0.986303	−	0.164945i	$-0.0527445\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.5981	−1.07561	−0.537806	−	0.843068i	$-0.680747\pi$
$863$	−31.5981	−1.07561	−0.537806	+	0.843068i	$0.680747\pi$
$864$	0	0
$865$	−7.98950	−0.271651
$866$	0	0
$867$	− 9.29042i	− 0.315519i
$868$	0	0
$869$	16.7484i	0.568151i
$870$	0	0
$871$	0.740671	0.0250967
$872$	0	0
$873$	6.69460	0.226578
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 4.50967i	− 0.152281i	−0.997097	−	0.0761404i	$-0.975740\pi$
$877$	− 4.50967i	− 0.152281i	0.997097	−	0.0761404i	$-0.0242597\pi$
$878$	0	0
$879$	−10.5622	−0.356253
$880$	0	0
$881$	−25.8471	−0.870811	−0.435405	−	0.900234i	$-0.643395\pi$
$881$	−25.8471	−0.870811	−0.435405	+	0.900234i	$0.643395\pi$
$882$	0	0
$883$	58.5786i	1.97133i	0.168725	+	0.985663i	$0.446035\pi$
$883$	58.5786i	1.97133i	−0.168725	+	0.985663i	$0.553965\pi$
$884$	0	0
$885$	16.4098i	0.551609i
$886$	0	0
$887$	−20.2249	−0.679084	−0.339542	−	0.940591i	$-0.610272\pi$
$887$	−20.2249	−0.679084	−0.339542	+	0.940591i	$0.610272\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 2.67767i	− 0.0897054i
$892$	0	0
$893$	− 26.5786i	− 0.889419i
$894$	0	0
$895$	93.1158	3.11252
$896$	0	0
$897$	21.5602	0.719875
$898$	0	0
$899$	12.5808i	0.419594i
$900$	0	0
$901$	− 14.2548i	− 0.474897i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	43.2204	1.43669
$906$	0	0
$907$	45.3103i	1.50450i	0.658876	+	0.752251i	$0.271032\pi$
$907$	45.3103i	1.50450i	−0.658876	+	0.752251i	$0.728968\pi$
$908$	0	0
$909$	1.45779i	0.0483520i
$910$	0	0
$911$	12.5488	0.415761	0.207880	−	0.978154i	$-0.433344\pi$
$911$	12.5488	0.415761	0.207880	+	0.978154i	$0.433344\pi$
$912$	0	0
$913$	−25.0506	−0.829053
$914$	0	0
$915$	21.2503i	0.702515i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	43.9400	1.44945	0.724724	−	0.689039i	$-0.241967\pi$
$919$	43.9400	1.44945	0.724724	+	0.689039i	$0.241967\pi$
$920$	0	0
$921$	−10.7801	−0.355217
$922$	0	0
$923$	− 12.9404i	− 0.425940i
$924$	0	0
$925$	− 16.8551i	− 0.554194i
$926$	0	0
$927$	−8.13547	−0.267204
$928$	0	0
$929$	53.7370	1.76305	0.881527	−	0.472134i	$-0.156516\pi$
$929$	53.7370	1.76305	0.881527	+	0.472134i	$0.156516\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 9.56024i	− 0.312988i
$934$	0	0
$935$	56.3247	1.84202
$936$	0	0
$937$	12.7107	0.415240	0.207620	−	0.978210i	$-0.433428\pi$
$937$	12.7107	0.415240	0.207620	+	0.978210i	$0.433428\pi$
$938$	0	0
$939$	7.69909i	0.251250i
$940$	0	0
$941$	4.02253i	0.131131i	0.997848	+	0.0655653i	$0.0208851\pi$
$941$	4.02253i	0.131131i	−0.997848	+	0.0655653i	$0.979115\pi$
$942$	0	0
$943$	36.5453	1.19008
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 43.1413i	− 1.40190i	−0.713209	−	0.700951i	$-0.752759\pi$
$947$	− 43.1413i	− 1.40190i	0.713209	−	0.700951i	$-0.247241\pi$
$948$	0	0
$949$	− 12.5686i	− 0.407994i
$950$	0	0
$951$	−10.7801	−0.349569
$952$	0	0
$953$	−10.2387	−0.331665	−0.165833	−	0.986154i	$-0.553031\pi$
$953$	−10.2387	−0.331665	−0.165833	+	0.986154i	$0.553031\pi$
$954$	0	0
$955$	17.5497i	0.567896i
$956$	0	0
$957$	23.6440i	0.764303i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.9700	−0.934517
$962$	0	0
$963$	5.32233i	0.171510i
$964$	0	0
$965$	94.9304i	3.05592i
$966$	0	0
$967$	−46.9356	−1.50935	−0.754673	−	0.656101i	$-0.772204\pi$
$967$	−46.9356	−1.50935	−0.754673	+	0.656101i	$0.772204\pi$
$968$	0	0
$969$	14.2548	0.457931
$970$	0	0
$971$	− 17.5602i	− 0.563535i	−0.959483	−	0.281767i	$-0.909079\pi$
$971$	− 17.5602i	− 0.563535i	0.959483	−	0.281767i	$-0.0909207\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	35.7856	1.14606
$976$	0	0
$977$	−41.5303	−1.32867	−0.664335	−	0.747435i	$-0.731285\pi$
$977$	−41.5303	−1.32867	−0.664335	+	0.747435i	$0.731285\pi$
$978$	0	0
$979$	30.2105i	0.965532i
$980$	0	0
$981$	13.5602i	0.432945i
$982$	0	0
$983$	−50.4797	−1.61005	−0.805026	−	0.593239i	$-0.797849\pi$
$983$	−50.4797	−1.61005	−0.805026	+	0.593239i	$0.797849\pi$
$984$	0	0
$985$	79.8940	2.54563
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.1044i	0.543888i
$990$	0	0
$991$	−1.46035	−0.0463896	−0.0231948	−	0.999731i	$-0.507384\pi$
$991$	−1.46035	−0.0463896	−0.0231948	+	0.999731i	$0.507384\pi$
$992$	0	0
$993$	−25.2104	−0.800027
$994$	0	0
$995$	− 72.0399i	− 2.28382i
$996$	0	0
$997$	21.8796i	0.692935i	0.938062	+	0.346467i	$0.112619\pi$
$997$	21.8796i	0.692935i	−0.938062	+	0.346467i	$0.887381\pi$
$998$	0	0
$999$	−1.42477	−0.0450778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.c.c.2353.4		8
4.3	odd	2		1176.2.c.c.589.8		8
7.6	odd	2		672.2.c.b.337.5		8
8.3	odd	2		1176.2.c.c.589.7		8
8.5	even	2	inner	4704.2.c.c.2353.5		8
21.20	even	2		2016.2.c.e.1009.8		8
28.27	even	2		168.2.c.b.85.8	yes	8
56.13	odd	2		672.2.c.b.337.4		8
56.27	even	2		168.2.c.b.85.7	✓	8
84.83	odd	2		504.2.c.f.253.1		8
112.13	odd	4		5376.2.a.bl.1.1		4
112.27	even	4		5376.2.a.bm.1.4		4
112.69	odd	4		5376.2.a.bq.1.4		4
112.83	even	4		5376.2.a.bp.1.1		4
168.83	odd	2		504.2.c.f.253.2		8
168.125	even	2		2016.2.c.e.1009.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.7	✓	8	56.27	even	2
168.2.c.b.85.8	yes	8	28.27	even	2
504.2.c.f.253.1		8	84.83	odd	2
504.2.c.f.253.2		8	168.83	odd	2
672.2.c.b.337.4		8	56.13	odd	2
672.2.c.b.337.5		8	7.6	odd	2
1176.2.c.c.589.7		8	8.3	odd	2
1176.2.c.c.589.8		8	4.3	odd	2
2016.2.c.e.1009.1		8	168.125	even	2
2016.2.c.e.1009.8		8	21.20	even	2
4704.2.c.c.2353.4		8	1.1	even	1	trivial
4704.2.c.c.2353.5		8	8.5	even	2	inner
5376.2.a.bl.1.1		4	112.13	odd	4
5376.2.a.bm.1.4		4	112.27	even	4
5376.2.a.bp.1.1		4	112.83	even	4
5376.2.a.bq.1.4		4	112.69	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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +4.10245i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +4.10245i q^{5} -1.00000 q^{9} -2.67767i q^{11} -3.02497i q^{13} +4.10245 q^{15} +5.12742 q^{17} +2.78012i q^{19} -7.12742 q^{23} -11.8301 q^{25} +1.00000i q^{27} -8.83006i q^{29} -1.42477 q^{31} -2.67767 q^{33} +1.42477i q^{37} -3.02497 q^{39} -5.12742 q^{41} -2.39980i q^{43} -4.10245i q^{45} -9.56024 q^{47} -5.12742i q^{51} -2.78012i q^{53} +10.9850 q^{55} +2.78012 q^{57} +4.00000i q^{59} +5.17992i q^{61} +12.4098 q^{65} +0.244852i q^{67} +7.12742i q^{69} +4.27787 q^{71} +4.15495 q^{73} +11.8301i q^{75} -6.25484 q^{79} +1.00000 q^{81} -9.35535i q^{83} +21.0350i q^{85} -8.83006 q^{87} -11.2824 q^{89} +1.42477i q^{93} -11.4053 q^{95} -6.69460 q^{97} +2.67767i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 4 q^{15} - 4 q^{17} - 12 q^{23} - 24 q^{25} + 8 q^{31} - 12 q^{33} - 8 q^{39} + 4 q^{41} - 8 q^{55} - 16 q^{57} - 16 q^{65} + 28 q^{71} + 8 q^{73} + 40 q^{79} + 8 q^{81} - 20 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 + 4 * q^15 - 4 * q^17 - 12 * q^23 - 24 * q^25 + 8 * q^31 - 12 * q^33 - 8 * q^39 + 4 * q^41 - 8 * q^55 - 16 * q^57 - 16 * q^65 + 28 * q^71 + 8 * q^73 + 40 * q^79 + 8 * q^81 - 20 * q^89 - 40 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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