LMFDB - Embedded newform 2304.3.b.k.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(127,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2304.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.b (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:	$62.7794529086$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.k.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.92820i q^{5} +2.92820i q^{7} +O(q^{10})$	$q-4.92820i q^{5} +2.92820i q^{7} -14.9282 q^{11} -23.8564i q^{13} +19.8564 q^{17} +30.9282 q^{19} +8.00000i q^{23} +0.712813 q^{25} -16.9282i q^{29} +38.6410i q^{31} +14.4308 q^{35} -9.71281i q^{37} -8.14359 q^{41} -5.35898 q^{43} -69.8564i q^{47} +40.4256 q^{49} +0.928203i q^{53} +73.5692i q^{55} -108.210 q^{59} +14.0000i q^{61} -117.569 q^{65} +16.4974 q^{67} -43.1384i q^{71} -25.4256 q^{73} -43.7128i q^{77} -96.7846i q^{79} -62.9282 q^{83} -97.8564i q^{85} -50.2872 q^{89} +69.8564 q^{91} -152.420i q^{95} -145.138 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 32 q^{11} + 24 q^{17} + 96 q^{19} - 108 q^{25} + 224 q^{35} - 88 q^{41} - 160 q^{43} - 60 q^{49} - 128 q^{59} - 304 q^{65} - 128 q^{67} + 120 q^{73} - 224 q^{83} - 312 q^{89} + 224 q^{91} - 248 q^{97}+O(q^{100}) $ 4 * q - 32 * q^11 + 24 * q^17 + 96 * q^19 - 108 * q^25 + 224 * q^35 - 88 * q^41 - 160 * q^43 - 60 * q^49 - 128 * q^59 - 304 * q^65 - 128 * q^67 + 120 * q^73 - 224 * q^83 - 312 * q^89 + 224 * q^91 - 248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 4.92820i	− 0.985641i	−0.870131	−	0.492820i	$-0.835966\pi$
$5$	− 4.92820i	− 0.985641i	0.870131	−	0.492820i	$-0.164034\pi$
$6$	0	0
$7$	2.92820i	0.418315i	0.977882	+	0.209157i	$0.0670721\pi$
$7$	2.92820i	0.418315i	−0.977882	+	0.209157i	$0.932928\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−14.9282	−1.35711	−0.678555	−	0.734550i	$-0.737393\pi$
$11$	−14.9282	−1.35711	−0.678555	+	0.734550i	$0.737393\pi$
$12$	0	0
$13$	− 23.8564i	− 1.83511i	−0.397611	−	0.917554i	$-0.630161\pi$
$13$	− 23.8564i	− 1.83511i	0.397611	−	0.917554i	$-0.369839\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	19.8564	1.16802	0.584012	−	0.811745i	$-0.301482\pi$
$17$	19.8564	1.16802	0.584012	+	0.811745i	$0.301482\pi$
$18$	0	0
$19$	30.9282	1.62780	0.813900	−	0.581005i	$-0.197340\pi$
$19$	30.9282	1.62780	0.813900	+	0.581005i	$0.197340\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000i	0.347826i	0.984761	+	0.173913i	$0.0556412\pi$
$23$	8.00000i	0.347826i	−0.984761	+	0.173913i	$0.944359\pi$
$24$	0	0
$25$	0.712813	0.0285125
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 16.9282i	− 0.583731i	−0.956459	−	0.291866i	$-0.905724\pi$
$29$	− 16.9282i	− 0.583731i	0.956459	−	0.291866i	$-0.0942760\pi$
$30$	0	0
$31$	38.6410i	1.24648i	0.782029	+	0.623242i	$0.214185\pi$
$31$	38.6410i	1.24648i	−0.782029	+	0.623242i	$0.785815\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.4308	0.412308
$36$	0	0
$37$	− 9.71281i	− 0.262508i	−0.991349	−	0.131254i	$-0.958100\pi$
$37$	− 9.71281i	− 0.262508i	0.991349	−	0.131254i	$-0.0419004\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.14359	−0.198624	−0.0993121	−	0.995056i	$-0.531664\pi$
$41$	−8.14359	−0.198624	−0.0993121	+	0.995056i	$0.531664\pi$
$42$	0	0
$43$	−5.35898	−0.124628	−0.0623138	−	0.998057i	$-0.519848\pi$
$43$	−5.35898	−0.124628	−0.0623138	+	0.998057i	$0.519848\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 69.8564i	− 1.48631i	−0.669121	−	0.743153i	$-0.733329\pi$
$47$	− 69.8564i	− 1.48631i	0.669121	−	0.743153i	$-0.266671\pi$
$48$	0	0
$49$	40.4256	0.825013
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.928203i	0.0175133i	0.999962	+	0.00875663i	$0.00278736\pi$
$53$	0.928203i	0.0175133i	−0.999962	+	0.00875663i	$0.997213\pi$
$54$	0	0
$55$	73.5692i	1.33762i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−108.210	−1.83407	−0.917036	−	0.398805i	$-0.869425\pi$
$59$	−108.210	−1.83407	−0.917036	+	0.398805i	$0.869425\pi$
$60$	0	0
$61$	14.0000i	0.229508i	0.993394	+	0.114754i	$0.0366080\pi$
$61$	14.0000i	0.229508i	−0.993394	+	0.114754i	$0.963392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−117.569	−1.80876
$66$	0	0
$67$	16.4974	0.246230	0.123115	−	0.992392i	$-0.460712\pi$
$67$	16.4974	0.246230	0.123115	+	0.992392i	$0.460712\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 43.1384i	− 0.607584i	−0.952738	−	0.303792i	$-0.901747\pi$
$71$	− 43.1384i	− 0.607584i	0.952738	−	0.303792i	$-0.0982527\pi$
$72$	0	0
$73$	−25.4256	−0.348296	−0.174148	−	0.984719i	$-0.555717\pi$
$73$	−25.4256	−0.348296	−0.174148	+	0.984719i	$0.555717\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 43.7128i	− 0.567699i
$78$	0	0
$79$	− 96.7846i	− 1.22512i	−0.790423	−	0.612561i	$-0.790139\pi$
$79$	− 96.7846i	− 1.22512i	0.790423	−	0.612561i	$-0.209861\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−62.9282	−0.758171	−0.379086	−	0.925362i	$-0.623761\pi$
$83$	−62.9282	−0.758171	−0.379086	+	0.925362i	$0.623761\pi$
$84$	0	0
$85$	− 97.8564i	− 1.15125i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−50.2872	−0.565025	−0.282512	−	0.959264i	$-0.591168\pi$
$89$	−50.2872	−0.565025	−0.282512	+	0.959264i	$0.591168\pi$
$90$	0	0
$91$	69.8564	0.767653
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 152.420i	− 1.60443i
$96$	0	0
$97$	−145.138	−1.49627	−0.748136	−	0.663545i	$-0.769051\pi$
$97$	−145.138	−1.49627	−0.748136	+	0.663545i	$0.769051\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 159.072i	− 1.57497i	−0.616335	−	0.787484i	$-0.711383\pi$
$101$	− 159.072i	− 1.57497i	0.616335	−	0.787484i	$-0.288617\pi$
$102$	0	0
$103$	24.7846i	0.240627i	0.992736	+	0.120314i	$0.0383900\pi$
$103$	24.7846i	0.240627i	−0.992736	+	0.120314i	$0.961610\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	42.0666	0.393146	0.196573	−	0.980489i	$-0.437019\pi$
$107$	42.0666	0.393146	0.196573	+	0.980489i	$0.437019\pi$
$108$	0	0
$109$	19.8564i	0.182169i	0.995843	+	0.0910844i	$0.0290333\pi$
$109$	19.8564i	0.182169i	−0.995843	+	0.0910844i	$0.970967\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.13844	0.0100747	0.00503734	−	0.999987i	$-0.498397\pi$
$113$	1.13844	0.0100747	0.00503734	+	0.999987i	$0.498397\pi$
$114$	0	0
$115$	39.4256	0.342832
$116$	0	0
$117$	0	0
$118$	0	0
$119$	58.1436i	0.488602i
$120$	0	0
$121$	101.851	0.841746
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 126.718i	− 1.01374i
$126$	0	0
$127$	171.349i	1.34920i	0.738182	+	0.674601i	$0.235684\pi$
$127$	171.349i	1.34920i	−0.738182	+	0.674601i	$0.764316\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−57.0718	−0.435663	−0.217831	−	0.975986i	$-0.569898\pi$
$131$	−57.0718	−0.435663	−0.217831	+	0.975986i	$0.569898\pi$
$132$	0	0
$133$	90.5641i	0.680933i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−154.133	−1.12506	−0.562530	−	0.826777i	$-0.690172\pi$
$137$	−154.133	−1.12506	−0.562530	+	0.826777i	$0.690172\pi$
$138$	0	0
$139$	167.923	1.20808	0.604040	−	0.796954i	$-0.293557\pi$
$139$	167.923	1.20808	0.604040	+	0.796954i	$0.293557\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	356.133i	2.49044i
$144$	0	0
$145$	−83.4256	−0.575349
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 124.354i	− 0.834589i	−0.908771	−	0.417295i	$-0.862978\pi$
$149$	− 124.354i	− 0.834589i	0.908771	−	0.417295i	$-0.137022\pi$
$150$	0	0
$151$	− 48.2102i	− 0.319273i	−0.987176	−	0.159637i	$-0.948968\pi$
$151$	− 48.2102i	− 0.319273i	0.987176	−	0.159637i	$-0.0510322\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	190.431	1.22859
$156$	0	0
$157$	− 166.287i	− 1.05915i	−0.848262	−	0.529577i	$-0.822351\pi$
$157$	− 166.287i	− 1.05915i	0.848262	−	0.529577i	$-0.177649\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−23.4256	−0.145501
$162$	0	0
$163$	−31.9230	−0.195847	−0.0979235	−	0.995194i	$-0.531220\pi$
$163$	−31.9230	−0.195847	−0.0979235	+	0.995194i	$0.531220\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 210.144i	− 1.25834i	−0.777266	−	0.629172i	$-0.783394\pi$
$167$	− 210.144i	− 1.25834i	0.777266	−	0.629172i	$-0.216606\pi$
$168$	0	0
$169$	−400.128	−2.36762
$170$	0	0
$171$	0	0
$172$	0	0
$173$	163.779i	0.946702i	0.880874	+	0.473351i	$0.156956\pi$
$173$	163.779i	0.946702i	−0.880874	+	0.473351i	$0.843044\pi$
$174$	0	0
$175$	2.08726i	0.0119272i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	186.067	1.03948	0.519739	−	0.854325i	$-0.326029\pi$
$179$	186.067	1.03948	0.519739	+	0.854325i	$0.326029\pi$
$180$	0	0
$181$	42.9948i	0.237541i	0.992922	+	0.118770i	$0.0378952\pi$
$181$	42.9948i	0.237541i	−0.992922	+	0.118770i	$0.962105\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−47.8667	−0.258739
$186$	0	0
$187$	−296.420	−1.58514
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 87.4256i	− 0.457726i	−0.973459	−	0.228863i	$-0.926499\pi$
$191$	− 87.4256i	− 0.457726i	0.973459	−	0.228863i	$-0.0735007\pi$
$192$	0	0
$193$	26.5744	0.137691	0.0688455	−	0.997627i	$-0.478068\pi$
$193$	26.5744	0.137691	0.0688455	+	0.997627i	$0.478068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	251.492i	1.27661i	0.769783	+	0.638305i	$0.220364\pi$
$197$	251.492i	1.27661i	−0.769783	+	0.638305i	$0.779636\pi$
$198$	0	0
$199$	− 68.0770i	− 0.342095i	−0.985263	−	0.171048i	$-0.945285\pi$
$199$	− 68.0770i	− 0.342095i	0.985263	−	0.171048i	$-0.0547152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	49.5692	0.244183
$204$	0	0
$205$	40.1333i	0.195772i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−461.703	−2.20910
$210$	0	0
$211$	−258.641	−1.22579	−0.612893	−	0.790166i	$-0.709995\pi$
$211$	−258.641	−1.22579	−0.612893	+	0.790166i	$0.709995\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	26.4102i	0.122838i
$216$	0	0
$217$	−113.149	−0.521423
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 473.703i	− 2.14345i
$222$	0	0
$223$	8.21024i	0.0368172i	0.999831	+	0.0184086i	$0.00585997\pi$
$223$	8.21024i	0.0368172i	−0.999831	+	0.0184086i	$0.994140\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−238.928	−1.05255	−0.526274	−	0.850315i	$-0.676411\pi$
$227$	−238.928	−1.05255	−0.526274	+	0.850315i	$0.676411\pi$
$228$	0	0
$229$	− 181.846i	− 0.794088i	−0.917800	−	0.397044i	$-0.870036\pi$
$229$	− 181.846i	− 0.794088i	0.917800	−	0.397044i	$-0.129964\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−385.138	−1.65295	−0.826477	−	0.562970i	$-0.809659\pi$
$233$	−385.138	−1.65295	−0.826477	+	0.562970i	$0.809659\pi$
$234$	0	0
$235$	−344.267	−1.46496
$236$	0	0
$237$	0	0
$238$	0	0
$239$	403.979i	1.69029i	0.534537	+	0.845145i	$0.320486\pi$
$239$	403.979i	1.69029i	−0.534537	+	0.845145i	$0.679514\pi$
$240$	0	0
$241$	−51.4359	−0.213427	−0.106714	−	0.994290i	$-0.534033\pi$
$241$	−51.4359	−0.213427	−0.106714	+	0.994290i	$0.534033\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 199.226i	− 0.813166i
$246$	0	0
$247$	− 737.836i	− 2.98719i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.21024	0.0167738	0.00838692	−	0.999965i	$-0.497330\pi$
$251$	4.21024	0.0167738	0.00838692	+	0.999965i	$0.497330\pi$
$252$	0	0
$253$	− 119.426i	− 0.472038i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	91.7025	0.356819	0.178410	−	0.983956i	$-0.442905\pi$
$257$	91.7025	0.356819	0.178410	+	0.983956i	$0.442905\pi$
$258$	0	0
$259$	28.4411	0.109811
$260$	0	0
$261$	0	0
$262$	0	0
$263$	269.856i	1.02607i	0.858368	+	0.513035i	$0.171479\pi$
$263$	269.856i	1.02607i	−0.858368	+	0.513035i	$0.828521\pi$
$264$	0	0
$265$	4.57437	0.0172618
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 200.354i	− 0.744810i	−0.928070	−	0.372405i	$-0.878533\pi$
$269$	− 200.354i	− 0.744810i	0.928070	−	0.372405i	$-0.121467\pi$
$270$	0	0
$271$	− 497.205i	− 1.83471i	−0.398076	−	0.917353i	$-0.630322\pi$
$271$	− 497.205i	− 1.83471i	0.398076	−	0.917353i	$-0.369678\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.6410	−0.0386946
$276$	0	0
$277$	− 257.559i	− 0.929816i	−0.885359	−	0.464908i	$-0.846087\pi$
$277$	− 257.559i	− 0.929816i	0.885359	−	0.464908i	$-0.153913\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	194.000	0.690391	0.345196	−	0.938531i	$-0.387813\pi$
$281$	194.000	0.690391	0.345196	+	0.938531i	$0.387813\pi$
$282$	0	0
$283$	332.210	1.17389	0.586944	−	0.809628i	$-0.300331\pi$
$283$	332.210	1.17389	0.586944	+	0.809628i	$0.300331\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 23.8461i	− 0.0830874i
$288$	0	0
$289$	105.277	0.364280
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 465.902i	− 1.59011i	−0.606537	−	0.795055i	$-0.707442\pi$
$293$	− 465.902i	− 1.59011i	0.606537	−	0.795055i	$-0.292558\pi$
$294$	0	0
$295$	533.282i	1.80774i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	190.851	0.638299
$300$	0	0
$301$	− 15.6922i	− 0.0521335i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	68.9948	0.226213
$306$	0	0
$307$	−215.769	−0.702831	−0.351416	−	0.936220i	$-0.614300\pi$
$307$	−215.769	−0.702831	−0.351416	+	0.936220i	$0.614300\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	175.005i	0.562718i	0.959603	+	0.281359i	$0.0907851\pi$
$311$	175.005i	0.562718i	−0.959603	+	0.281359i	$0.909215\pi$
$312$	0	0
$313$	433.979	1.38652	0.693258	−	0.720690i	$-0.256175\pi$
$313$	433.979	1.38652	0.693258	+	0.720690i	$0.256175\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	88.4871i	0.279139i	0.990212	+	0.139570i	$0.0445719\pi$
$317$	88.4871i	0.279139i	−0.990212	+	0.139570i	$0.955428\pi$
$318$	0	0
$319$	252.708i	0.792187i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	614.123	1.90131
$324$	0	0
$325$	− 17.0052i	− 0.0523236i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	204.554	0.621744
$330$	0	0
$331$	107.061	0.323449	0.161724	−	0.986836i	$-0.448294\pi$
$331$	107.061	0.323449	0.161724	+	0.986836i	$0.448294\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 81.3027i	− 0.242694i
$336$	0	0
$337$	−38.5744	−0.114464	−0.0572320	−	0.998361i	$-0.518227\pi$
$337$	−38.5744	−0.114464	−0.0572320	+	0.998361i	$0.518227\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 576.841i	− 1.69162i
$342$	0	0
$343$	261.856i	0.763430i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	65.9127	0.189950	0.0949751	−	0.995480i	$-0.469723\pi$
$347$	65.9127	0.189950	0.0949751	+	0.995480i	$0.469723\pi$
$348$	0	0
$349$	− 367.703i	− 1.05359i	−0.849993	−	0.526794i	$-0.823394\pi$
$349$	− 367.703i	− 1.05359i	0.849993	−	0.526794i	$-0.176606\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−158.862	−0.450033	−0.225016	−	0.974355i	$-0.572244\pi$
$353$	−158.862	−0.450033	−0.225016	+	0.974355i	$0.572244\pi$
$354$	0	0
$355$	−212.595	−0.598859
$356$	0	0
$357$	0	0
$358$	0	0
$359$	140.287i	0.390772i	0.980726	+	0.195386i	$0.0625960\pi$
$359$	140.287i	0.390772i	−0.980726	+	0.195386i	$0.937404\pi$
$360$	0	0
$361$	595.554	1.64973
$362$	0	0
$363$	0	0
$364$	0	0
$365$	125.303i	0.343295i
$366$	0	0
$367$	− 505.051i	− 1.37616i	−0.725634	−	0.688081i	$-0.758454\pi$
$367$	− 505.051i	− 1.37616i	0.725634	−	0.688081i	$-0.241546\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.71797	−0.00732606
$372$	0	0
$373$	215.128i	0.576751i	0.957517	+	0.288376i	$0.0931151\pi$
$373$	215.128i	0.576751i	−0.957517	+	0.288376i	$0.906885\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−403.846	−1.07121
$378$	0	0
$379$	−204.785	−0.540329	−0.270164	−	0.962814i	$-0.587078\pi$
$379$	−204.785	−0.540329	−0.270164	+	0.962814i	$0.587078\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 588.554i	− 1.53669i	−0.640034	−	0.768347i	$-0.721080\pi$
$383$	− 588.554i	− 1.53669i	0.640034	−	0.768347i	$-0.278920\pi$
$384$	0	0
$385$	−215.426	−0.559547
$386$	0	0
$387$	0	0
$388$	0	0
$389$	534.477i	1.37398i	0.726669	+	0.686988i	$0.241068\pi$
$389$	534.477i	1.37398i	−0.726669	+	0.686988i	$0.758932\pi$
$390$	0	0
$391$	158.851i	0.406269i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−476.974	−1.20753
$396$	0	0
$397$	340.277i	0.857121i	0.903513	+	0.428560i	$0.140979\pi$
$397$	340.277i	0.857121i	−0.903513	+	0.428560i	$0.859021\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	300.123	0.748436	0.374218	−	0.927341i	$-0.377911\pi$
$401$	300.123	0.748436	0.374218	+	0.927341i	$0.377911\pi$
$402$	0	0
$403$	921.836	2.28743
$404$	0	0
$405$	0	0
$406$	0	0
$407$	144.995i	0.356253i
$408$	0	0
$409$	−513.692	−1.25597	−0.627986	−	0.778225i	$-0.716120\pi$
$409$	−513.692	−1.25597	−0.627986	+	0.778225i	$0.716120\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 316.862i	− 0.767219i
$414$	0	0
$415$	310.123i	0.747284i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	91.9437	0.219436	0.109718	−	0.993963i	$-0.465005\pi$
$419$	91.9437	0.219436	0.109718	+	0.993963i	$0.465005\pi$
$420$	0	0
$421$	− 513.559i	− 1.21985i	−0.792457	−	0.609927i	$-0.791199\pi$
$421$	− 513.559i	− 1.21985i	0.792457	−	0.609927i	$-0.208801\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	14.1539	0.0333033
$426$	0	0
$427$	−40.9948	−0.0960067
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 446.123i	− 1.03509i	−0.855657	−	0.517544i	$-0.826846\pi$
$431$	− 446.123i	− 1.03509i	0.855657	−	0.517544i	$-0.173154\pi$
$432$	0	0
$433$	43.7231	0.100977	0.0504886	−	0.998725i	$-0.483922\pi$
$433$	43.7231	0.100977	0.0504886	+	0.998725i	$0.483922\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	247.426i	0.566191i
$438$	0	0
$439$	381.072i	0.868045i	0.900902	+	0.434023i	$0.142906\pi$
$439$	381.072i	0.868045i	−0.900902	+	0.434023i	$0.857094\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−181.513	−0.409736	−0.204868	−	0.978790i	$-0.565676\pi$
$443$	−181.513	−0.409736	−0.204868	+	0.978790i	$0.565676\pi$
$444$	0	0
$445$	247.825i	0.556911i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−397.825	−0.886026	−0.443013	−	0.896515i	$-0.646090\pi$
$449$	−397.825	−0.886026	−0.443013	+	0.896515i	$0.646090\pi$
$450$	0	0
$451$	121.569	0.269555
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 344.267i	− 0.756630i
$456$	0	0
$457$	−526.862	−1.15287	−0.576435	−	0.817143i	$-0.695557\pi$
$457$	−526.862	−1.15287	−0.576435	+	0.817143i	$0.695557\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	888.908i	1.92822i	0.265510	+	0.964108i	$0.414460\pi$
$461$	888.908i	1.92822i	−0.265510	+	0.964108i	$0.585540\pi$
$462$	0	0
$463$	515.923i	1.11430i	0.830410	+	0.557152i	$0.188106\pi$
$463$	515.923i	1.11430i	−0.830410	+	0.557152i	$0.811894\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−439.195	−0.940460	−0.470230	−	0.882544i	$-0.655829\pi$
$467$	−439.195	−0.940460	−0.470230	+	0.882544i	$0.655829\pi$
$468$	0	0
$469$	48.3078i	0.103002i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	80.0000	0.169133
$474$	0	0
$475$	22.0460	0.0464127
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 179.559i	− 0.374862i	−0.982278	−	0.187431i	$-0.939984\pi$
$479$	− 179.559i	− 0.374862i	0.982278	−	0.187431i	$-0.0600161\pi$
$480$	0	0
$481$	−231.713	−0.481731
$482$	0	0
$483$	0	0
$484$	0	0
$485$	715.272i	1.47479i
$486$	0	0
$487$	− 494.221i	− 1.01483i	−0.861703	−	0.507413i	$-0.830602\pi$
$487$	− 494.221i	− 1.01483i	0.861703	−	0.507413i	$-0.169398\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−226.487	−0.461277	−0.230639	−	0.973039i	$-0.574082\pi$
$491$	−226.487	−0.461277	−0.230639	+	0.973039i	$0.574082\pi$
$492$	0	0
$493$	− 336.133i	− 0.681812i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	126.318	0.254161
$498$	0	0
$499$	−239.195	−0.479348	−0.239674	−	0.970853i	$-0.577041\pi$
$499$	−239.195	−0.479348	−0.239674	+	0.970853i	$0.577041\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 298.831i	− 0.594097i	−0.954862	−	0.297048i	$-0.903998\pi$
$503$	− 298.831i	− 0.594097i	0.954862	−	0.297048i	$-0.0960023\pi$
$504$	0	0
$505$	−783.938	−1.55235
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 713.195i	− 1.40117i	−0.713570	−	0.700584i	$-0.752923\pi$
$509$	− 713.195i	− 1.40117i	0.713570	−	0.700584i	$-0.247077\pi$
$510$	0	0
$511$	− 74.4514i	− 0.145697i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	122.144	0.237172
$516$	0	0
$517$	1042.83i	2.01708i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	748.985	1.43759	0.718795	−	0.695222i	$-0.244694\pi$
$521$	748.985	1.43759	0.718795	+	0.695222i	$0.244694\pi$
$522$	0	0
$523$	338.908	0.648007	0.324003	−	0.946056i	$-0.394971\pi$
$523$	338.908	0.648007	0.324003	+	0.946056i	$0.394971\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	767.272i	1.45592i
$528$	0	0
$529$	465.000	0.879017
$530$	0	0
$531$	0	0
$532$	0	0
$533$	194.277i	0.364497i
$534$	0	0
$535$	− 207.313i	− 0.387501i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−603.482	−1.11963
$540$	0	0
$541$	− 3.56922i	− 0.00659745i	−0.999995	−	0.00329872i	$-0.998950\pi$
$541$	− 3.56922i	− 0.00659745i	0.999995	−	0.00329872i	$-0.00105002\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	97.8564	0.179553
$546$	0	0
$547$	530.067	0.969043	0.484522	−	0.874779i	$-0.338994\pi$
$547$	530.067	0.969043	0.484522	+	0.874779i	$0.338994\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 523.559i	− 0.950198i
$552$	0	0
$553$	283.405	0.512486
$554$	0	0
$555$	0	0
$556$	0	0
$557$	721.790i	1.29585i	0.761703	+	0.647926i	$0.224363\pi$
$557$	721.790i	1.29585i	−0.761703	+	0.647926i	$0.775637\pi$
$558$	0	0
$559$	127.846i	0.228705i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.50258	−0.0133261	−0.00666303	−	0.999978i	$-0.502121\pi$
$563$	−7.50258	−0.0133261	−0.00666303	+	0.999978i	$0.502121\pi$
$564$	0	0
$565$	− 5.61046i	− 0.00993001i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−500.697	−0.879960	−0.439980	−	0.898008i	$-0.645015\pi$
$569$	−500.697	−0.879960	−0.439980	+	0.898008i	$0.645015\pi$
$570$	0	0
$571$	553.072	0.968602	0.484301	−	0.874901i	$-0.339074\pi$
$571$	553.072	0.968602	0.484301	+	0.874901i	$0.339074\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.70250i	0.00991740i
$576$	0	0
$577$	889.426	1.54147	0.770733	−	0.637159i	$-0.219890\pi$
$577$	889.426	1.54147	0.770733	+	0.637159i	$0.219890\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 184.267i	− 0.317154i
$582$	0	0
$583$	− 13.8564i	− 0.0237674i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−317.892	−0.541554	−0.270777	−	0.962642i	$-0.587281\pi$
$587$	−317.892	−0.541554	−0.270777	+	0.962642i	$0.587281\pi$
$588$	0	0
$589$	1195.10i	2.02903i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	508.543	0.857577	0.428789	−	0.903405i	$-0.358940\pi$
$593$	508.543	0.857577	0.428789	+	0.903405i	$0.358940\pi$
$594$	0	0
$595$	286.543	0.481586
$596$	0	0
$597$	0	0
$598$	0	0
$599$	548.246i	0.915269i	0.889140	+	0.457634i	$0.151303\pi$
$599$	548.246i	0.915269i	−0.889140	+	0.457634i	$0.848697\pi$
$600$	0	0
$601$	542.000	0.901830	0.450915	−	0.892567i	$-0.351098\pi$
$601$	542.000	0.901830	0.450915	+	0.892567i	$0.351098\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 501.944i	− 0.829659i
$606$	0	0
$607$	527.944i	0.869759i	0.900489	+	0.434879i	$0.143209\pi$
$607$	527.944i	0.869759i	−0.900489	+	0.434879i	$0.856791\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1666.52	−2.72753
$612$	0	0
$613$	738.000i	1.20392i	0.798528	+	0.601958i	$0.205612\pi$
$613$	738.000i	1.20392i	−0.798528	+	0.601958i	$0.794388\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	389.979	0.632057	0.316029	−	0.948750i	$-0.397650\pi$
$617$	389.979	0.632057	0.316029	+	0.948750i	$0.397650\pi$
$618$	0	0
$619$	−1006.89	−1.62663	−0.813317	−	0.581820i	$-0.802341\pi$
$619$	−1006.89	−1.62663	−0.813317	+	0.581820i	$0.802341\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 147.251i	− 0.236358i
$624$	0	0
$625$	−606.672	−0.970675
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 192.862i	− 0.306616i
$630$	0	0
$631$	1043.77i	1.65415i	0.562091	+	0.827075i	$0.309997\pi$
$631$	1043.77i	1.65415i	−0.562091	+	0.827075i	$0.690003\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	844.441	1.32983
$636$	0	0
$637$	− 964.410i	− 1.51399i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1100.96	1.71757	0.858786	−	0.512334i	$-0.171219\pi$
$641$	1100.96	1.71757	0.858786	+	0.512334i	$0.171219\pi$
$642$	0	0
$643$	−508.785	−0.791267	−0.395633	−	0.918409i	$-0.629475\pi$
$643$	−508.785	−0.791267	−0.395633	+	0.918409i	$0.629475\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 804.862i	− 1.24399i	−0.783021	−	0.621995i	$-0.786322\pi$
$647$	− 804.862i	− 1.24399i	0.783021	−	0.621995i	$-0.213678\pi$
$648$	0	0
$649$	1615.38	2.48904
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1284.62i	1.96726i	0.180202	+	0.983630i	$0.442325\pi$
$653$	1284.62i	1.96726i	−0.180202	+	0.983630i	$0.557675\pi$
$654$	0	0
$655$	281.261i	0.429407i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	219.523	0.333116	0.166558	−	0.986032i	$-0.446735\pi$
$659$	219.523	0.333116	0.166558	+	0.986032i	$0.446735\pi$
$660$	0	0
$661$	− 598.267i	− 0.905093i	−0.891741	−	0.452547i	$-0.850516\pi$
$661$	− 598.267i	− 0.905093i	0.891741	−	0.452547i	$-0.149484\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	446.318	0.671155
$666$	0	0
$667$	135.426	0.203037
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 208.995i	− 0.311468i
$672$	0	0
$673$	−575.149	−0.854604	−0.427302	−	0.904109i	$-0.640536\pi$
$673$	−575.149	−0.854604	−0.427302	+	0.904109i	$0.640536\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	450.805i	0.665887i	0.942947	+	0.332943i	$0.108042\pi$
$677$	450.805i	0.665887i	−0.942947	+	0.332943i	$0.891958\pi$
$678$	0	0
$679$	− 424.995i	− 0.625913i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1186.45	1.73711	0.868555	−	0.495593i	$-0.165049\pi$
$683$	1186.45	1.73711	0.868555	+	0.495593i	$0.165049\pi$
$684$	0	0
$685$	759.600i	1.10891i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	22.1436	0.0321387
$690$	0	0
$691$	−870.508	−1.25978	−0.629890	−	0.776685i	$-0.716900\pi$
$691$	−870.508	−1.25978	−0.629890	+	0.776685i	$0.716900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 827.559i	− 1.19073i
$696$	0	0
$697$	−161.703	−0.231998
$698$	0	0
$699$	0	0
$700$	0	0
$701$	218.785i	0.312104i	0.987749	+	0.156052i	$0.0498767\pi$
$701$	218.785i	0.312104i	−0.987749	+	0.156052i	$0.950123\pi$
$702$	0	0
$703$	− 300.400i	− 0.427311i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	465.795	0.658832
$708$	0	0
$709$	− 85.8461i	− 0.121081i	−0.998166	−	0.0605403i	$-0.980718\pi$
$709$	− 85.8461i	− 0.121081i	0.998166	−	0.0605403i	$-0.0192824\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−309.128	−0.433560
$714$	0	0
$715$	1755.10	2.45468
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 816.728i	− 1.13592i	−0.823055	−	0.567961i	$-0.807732\pi$
$719$	− 816.728i	− 1.13592i	0.823055	−	0.567961i	$-0.192268\pi$
$720$	0	0
$721$	−72.5744	−0.100658
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12.0666i	− 0.0166436i
$726$	0	0
$727$	62.6410i	0.0861637i	0.999072	+	0.0430819i	$0.0137176\pi$
$727$	62.6410i	0.0861637i	−0.999072	+	0.0430819i	$0.986282\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−106.410	−0.145568
$732$	0	0
$733$	− 72.3895i	− 0.0987579i	−0.998780	−	0.0493790i	$-0.984276\pi$
$733$	− 72.3895i	− 0.0987579i	0.998780	−	0.0493790i	$-0.0157242\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−246.277	−0.334161
$738$	0	0
$739$	651.061	0.881003	0.440502	−	0.897752i	$-0.354801\pi$
$739$	651.061	0.881003	0.440502	+	0.897752i	$0.354801\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	951.272i	1.28031i	0.768245	+	0.640156i	$0.221130\pi$
$743$	951.272i	1.28031i	−0.768245	+	0.640156i	$0.778870\pi$
$744$	0	0
$745$	−612.841	−0.822605
$746$	0	0
$747$	0	0
$748$	0	0
$749$	123.180i	0.164459i
$750$	0	0
$751$	280.323i	0.373266i	0.982430	+	0.186633i	$0.0597576\pi$
$751$	280.323i	0.373266i	−0.982430	+	0.186633i	$0.940242\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−237.590	−0.314689
$756$	0	0
$757$	246.708i	0.325902i	0.986634	+	0.162951i	$0.0521012\pi$
$757$	246.708i	0.325902i	−0.986634	+	0.162951i	$0.947899\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	739.569	0.971839	0.485919	−	0.874004i	$-0.338485\pi$
$761$	739.569	0.971839	0.485919	+	0.874004i	$0.338485\pi$
$762$	0	0
$763$	−58.1436	−0.0762039
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2581.51i	3.36572i
$768$	0	0
$769$	1017.96	1.32374	0.661872	−	0.749617i	$-0.269762\pi$
$769$	1017.96	1.32374	0.661872	+	0.749617i	$0.269762\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	50.9179i	0.0658705i	0.999457	+	0.0329352i	$0.0104855\pi$
$773$	50.9179i	0.0658705i	−0.999457	+	0.0329352i	$0.989514\pi$
$774$	0	0
$775$	27.5438i	0.0355404i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−251.867	−0.323321
$780$	0	0
$781$	643.979i	0.824557i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−819.497	−1.04395
$786$	0	0
$787$	−759.041	−0.964474	−0.482237	−	0.876041i	$-0.660176\pi$
$787$	−759.041	−0.964474	−0.482237	+	0.876041i	$0.660176\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.33358i	0.00421439i
$792$	0	0
$793$	333.990	0.421172
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14.2102i	− 0.0178297i	−0.999960	−	0.00891483i	$-0.997162\pi$
$797$	− 14.2102i	− 0.0178297i	0.999960	−	0.00891483i	$-0.00283772\pi$
$798$	0	0
$799$	− 1387.10i	− 1.73604i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	379.559	0.472676
$804$	0	0
$805$	115.446i	0.143411i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	715.528	0.884460	0.442230	−	0.896902i	$-0.354188\pi$
$809$	715.528	0.884460	0.442230	+	0.896902i	$0.354188\pi$
$810$	0	0
$811$	250.641	0.309052	0.154526	−	0.987989i	$-0.450615\pi$
$811$	250.641	0.309052	0.154526	+	0.987989i	$0.450615\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	157.323i	0.193035i
$816$	0	0
$817$	−165.744	−0.202869
$818$	0	0
$819$	0	0
$820$	0	0
$821$	908.641i	1.10675i	0.832933	+	0.553375i	$0.186660\pi$
$821$	908.641i	1.10675i	−0.832933	+	0.553375i	$0.813340\pi$
$822$	0	0
$823$	− 514.200i	− 0.624787i	−0.949953	−	0.312394i	$-0.898869\pi$
$823$	− 514.200i	− 0.624787i	0.949953	−	0.312394i	$-0.101131\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1240.61	1.50013	0.750067	−	0.661362i	$-0.230021\pi$
$827$	1240.61	1.50013	0.750067	+	0.661362i	$0.230021\pi$
$828$	0	0
$829$	352.410i	0.425103i	0.977150	+	0.212551i	$0.0681773\pi$
$829$	352.410i	0.425103i	−0.977150	+	0.212551i	$0.931823\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	802.708	0.963635
$834$	0	0
$835$	−1035.63	−1.24028
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1038.81i	1.23815i	0.785331	+	0.619076i	$0.212493\pi$
$839$	1038.81i	1.23815i	−0.785331	+	0.619076i	$0.787507\pi$
$840$	0	0
$841$	554.436	0.659258
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1971.91i	2.33362i
$846$	0	0
$847$	298.241i	0.352115i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	77.7025	0.0913073
$852$	0	0
$853$	− 1382.80i	− 1.62110i	−0.585668	−	0.810551i	$-0.699168\pi$
$853$	− 1382.80i	− 1.62110i	0.585668	−	0.810551i	$-0.300832\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	893.518	1.04261	0.521306	−	0.853370i	$-0.325445\pi$
$857$	893.518	1.04261	0.521306	+	0.853370i	$0.325445\pi$
$858$	0	0
$859$	−1549.85	−1.80425	−0.902125	−	0.431475i	$-0.857993\pi$
$859$	−1549.85	−1.80425	−0.902125	+	0.431475i	$0.857993\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	347.405i	0.402555i	0.979534	+	0.201278i	$0.0645093\pi$
$863$	347.405i	0.402555i	−0.979534	+	0.201278i	$0.935491\pi$
$864$	0	0
$865$	807.138	0.933108
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1444.82i	1.66262i
$870$	0	0
$871$	− 393.569i	− 0.451859i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	371.056	0.424064
$876$	0	0
$877$	− 1216.24i	− 1.38681i	−0.720546	−	0.693407i	$-0.756109\pi$
$877$	− 1216.24i	− 1.38681i	0.720546	−	0.693407i	$-0.243891\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	270.533	0.307075	0.153538	−	0.988143i	$-0.450933\pi$
$881$	270.533	0.307075	0.153538	+	0.988143i	$0.450933\pi$
$882$	0	0
$883$	368.918	0.417801	0.208900	−	0.977937i	$-0.433012\pi$
$883$	368.918	0.417801	0.208900	+	0.977937i	$0.433012\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 240.154i	− 0.270748i	−0.990795	−	0.135374i	$-0.956776\pi$
$887$	− 240.154i	− 0.270748i	0.990795	−	0.135374i	$-0.0432237\pi$
$888$	0	0
$889$	−501.744	−0.564391
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2160.53i	− 2.41941i
$894$	0	0
$895$	− 916.974i	− 1.02455i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	654.123	0.727612
$900$	0	0
$901$	18.4308i	0.0204559i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	211.887	0.234130
$906$	0	0
$907$	−1141.05	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$907$	−1141.05	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1755.94i	− 1.92748i	−0.266835	−	0.963742i	$-0.585978\pi$
$911$	− 1755.94i	− 1.92748i	0.266835	−	0.963742i	$-0.414022\pi$
$912$	0	0
$913$	939.405	1.02892
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 167.118i	− 0.182244i
$918$	0	0
$919$	− 1414.18i	− 1.53882i	−0.638753	−	0.769412i	$-0.720549\pi$
$919$	− 1414.18i	− 1.53882i	0.638753	−	0.769412i	$-0.279451\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1029.13	−1.11498
$924$	0	0
$925$	− 6.92342i	− 0.00748478i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1848.06	1.98930	0.994651	−	0.103295i	$-0.0329386\pi$
$929$	1848.06	1.98930	0.994651	+	0.103295i	$0.0329386\pi$
$930$	0	0
$931$	1250.29	1.34296
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1460.82i	1.56237i
$936$	0	0
$937$	170.554	0.182021	0.0910105	−	0.995850i	$-0.470990\pi$
$937$	170.554	0.182021	0.0910105	+	0.995850i	$0.470990\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	281.523i	0.299174i	0.988749	+	0.149587i	$0.0477945\pi$
$941$	281.523i	0.299174i	−0.988749	+	0.149587i	$0.952206\pi$
$942$	0	0
$943$	− 65.1487i	− 0.0690867i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1533.43	1.61925	0.809625	−	0.586947i	$-0.199670\pi$
$947$	1533.43	1.61925	0.809625	+	0.586947i	$0.199670\pi$
$948$	0	0
$949$	606.564i	0.639161i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	414.133	0.434557	0.217279	−	0.976110i	$-0.430282\pi$
$953$	414.133	0.434557	0.217279	+	0.976110i	$0.430282\pi$
$954$	0	0
$955$	−430.851	−0.451153
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 451.334i	− 0.470629i
$960$	0	0
$961$	−532.128	−0.553723
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 130.964i	− 0.135714i
$966$	0	0
$967$	− 228.918i	− 0.236730i	−0.992970	−	0.118365i	$-0.962235\pi$
$967$	− 228.918i	− 0.236730i	0.992970	−	0.118365i	$-0.0377653\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−968.877	−0.997813	−0.498907	−	0.866656i	$-0.666265\pi$
$971$	−968.877	−0.997813	−0.498907	+	0.866656i	$0.666265\pi$
$972$	0	0
$973$	491.713i	0.505357i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	648.144	0.663402	0.331701	−	0.943385i	$-0.392378\pi$
$977$	648.144	0.663402	0.331701	+	0.943385i	$0.392378\pi$
$978$	0	0
$979$	750.697	0.766800
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1035.56i	− 1.05347i	−0.850030	−	0.526734i	$-0.823416\pi$
$983$	− 1035.56i	− 1.05347i	0.850030	−	0.526734i	$-0.176584\pi$
$984$	0	0
$985$	1239.41	1.25828
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 42.8719i	− 0.0433487i
$990$	0	0
$991$	− 1032.44i	− 1.04181i	−0.853614	−	0.520906i	$-0.825594\pi$
$991$	− 1032.44i	− 1.04181i	0.853614	−	0.520906i	$-0.174406\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−335.497	−0.337183
$996$	0	0
$997$	456.277i	0.457650i	0.973468	+	0.228825i	$0.0734883\pi$
$997$	456.277i	0.457650i	−0.973468	+	0.228825i	$0.926512\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.b.k.127.2		4
3.2	odd	2		768.3.b.d.127.2		4
4.3	odd	2		2304.3.b.o.127.2		4
8.3	odd	2	inner	2304.3.b.k.127.3		4
8.5	even	2		2304.3.b.o.127.3		4
12.11	even	2		768.3.b.a.127.4		4
16.3	odd	4		288.3.g.d.127.2		4
16.5	even	4		576.3.g.j.127.3		4
16.11	odd	4		576.3.g.j.127.4		4
16.13	even	4		288.3.g.d.127.1		4
24.5	odd	2		768.3.b.a.127.3		4
24.11	even	2		768.3.b.d.127.1		4
48.5	odd	4		192.3.g.c.127.3		4
48.11	even	4		192.3.g.c.127.1		4
48.29	odd	4		96.3.g.a.31.2	✓	4
48.35	even	4		96.3.g.a.31.4	yes	4
240.29	odd	4		2400.3.e.a.1951.3		4
240.77	even	4		2400.3.j.a.799.1		4
240.83	odd	4		2400.3.j.a.799.2		4
240.173	even	4		2400.3.j.b.799.3		4
240.179	even	4		2400.3.e.a.1951.2		4
240.227	odd	4		2400.3.j.b.799.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.g.a.31.2	✓	4	48.29	odd	4
96.3.g.a.31.4	yes	4	48.35	even	4
192.3.g.c.127.1		4	48.11	even	4
192.3.g.c.127.3		4	48.5	odd	4
288.3.g.d.127.1		4	16.13	even	4
288.3.g.d.127.2		4	16.3	odd	4
576.3.g.j.127.3		4	16.5	even	4
576.3.g.j.127.4		4	16.11	odd	4
768.3.b.a.127.3		4	24.5	odd	2
768.3.b.a.127.4		4	12.11	even	2
768.3.b.d.127.1		4	24.11	even	2
768.3.b.d.127.2		4	3.2	odd	2
2304.3.b.k.127.2		4	1.1	even	1	trivial
2304.3.b.k.127.3		4	8.3	odd	2	inner
2304.3.b.o.127.2		4	4.3	odd	2
2304.3.b.o.127.3		4	8.5	even	2
2400.3.e.a.1951.2		4	240.179	even	4
2400.3.e.a.1951.3		4	240.29	odd	4
2400.3.j.a.799.1		4	240.77	even	4
2400.3.j.a.799.2		4	240.83	odd	4
2400.3.j.b.799.3		4	240.173	even	4
2400.3.j.b.799.4		4	240.227	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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.92820i q^{5} +2.92820i q^{7} +O(q^{10})\)	\(q-4.92820i q^{5} +2.92820i q^{7} -14.9282 q^{11} -23.8564i q^{13} +19.8564 q^{17} +30.9282 q^{19} +8.00000i q^{23} +0.712813 q^{25} -16.9282i q^{29} +38.6410i q^{31} +14.4308 q^{35} -9.71281i q^{37} -8.14359 q^{41} -5.35898 q^{43} -69.8564i q^{47} +40.4256 q^{49} +0.928203i q^{53} +73.5692i q^{55} -108.210 q^{59} +14.0000i q^{61} -117.569 q^{65} +16.4974 q^{67} -43.1384i q^{71} -25.4256 q^{73} -43.7128i q^{77} -96.7846i q^{79} -62.9282 q^{83} -97.8564i q^{85} -50.2872 q^{89} +69.8564 q^{91} -152.420i q^{95} -145.138 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 32 q^{11} + 24 q^{17} + 96 q^{19} - 108 q^{25} + 224 q^{35} - 88 q^{41} - 160 q^{43} - 60 q^{49} - 128 q^{59} - 304 q^{65} - 128 q^{67} + 120 q^{73} - 224 q^{83} - 312 q^{89} + 224 q^{91} - 248 q^{97}+O(q^{100}) \) 4 * q - 32 * q^11 + 24 * q^17 + 96 * q^19 - 108 * q^25 + 224 * q^35 - 88 * q^41 - 160 * q^43 - 60 * q^49 - 128 * q^59 - 304 * q^65 - 128 * q^67 + 120 * q^73 - 224 * q^83 - 312 * q^89 + 224 * q^91 - 248 * q^97

Introduction

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