LMFDB - Embedded newform 2304.3.b.o.127.4

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			127.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.o.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.92820i q^{5} +10.9282i q^{7} +O(q^{10})$	$q+8.92820i q^{5} +10.9282i q^{7} +1.07180 q^{11} +3.85641i q^{13} -7.85641 q^{17} -17.0718 q^{19} -8.00000i q^{23} -54.7128 q^{25} -3.07180i q^{29} +30.6410i q^{31} -97.5692 q^{35} +45.7128i q^{37} -35.8564 q^{41} +74.6410 q^{43} +42.1436i q^{47} -70.4256 q^{49} -12.9282i q^{53} +9.56922i q^{55} -44.2102 q^{59} +14.0000i q^{61} -34.4308 q^{65} +80.4974 q^{67} -123.138i q^{71} +85.4256 q^{73} +11.7128i q^{77} +55.2154i q^{79} +49.0718 q^{83} -70.1436i q^{85} -105.713 q^{89} -42.1436 q^{91} -152.420i q^{95} +21.1384 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	8.92820i	1.78564i	0.450413	+	0.892820i	$0.351277\pi$
$5$	8.92820i	1.78564i	−0.450413	+	0.892820i	$0.648723\pi$
$6$	0	0
$7$	10.9282i	1.56117i	0.625048	+	0.780586i	$0.285079\pi$
$7$	10.9282i	1.56117i	−0.625048	+	0.780586i	$0.714921\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.07180	0.0974361	0.0487180	−	0.998813i	$-0.484486\pi$
$11$	1.07180	0.0974361	0.0487180	+	0.998813i	$0.484486\pi$
$12$	0	0
$13$	3.85641i	0.296647i	0.988939	+	0.148323i	$0.0473876\pi$
$13$	3.85641i	0.296647i	−0.988939	+	0.148323i	$0.952612\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.85641	−0.462142	−0.231071	−	0.972937i	$-0.574223\pi$
$17$	−7.85641	−0.462142	−0.231071	+	0.972937i	$0.574223\pi$
$18$	0	0
$19$	−17.0718	−0.898516	−0.449258	−	0.893402i	$-0.648312\pi$
$19$	−17.0718	−0.898516	−0.449258	+	0.893402i	$0.648312\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.00000i	− 0.347826i	−0.984761	−	0.173913i	$-0.944359\pi$
$23$	− 8.00000i	− 0.347826i	0.984761	−	0.173913i	$-0.0556412\pi$
$24$	0	0
$25$	−54.7128	−2.18851
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.07180i	− 0.105924i	−0.998597	−	0.0529620i	$-0.983134\pi$
$29$	− 3.07180i	− 0.105924i	0.998597	−	0.0529620i	$-0.0168662\pi$
$30$	0	0
$31$	30.6410i	0.988420i	0.869343	+	0.494210i	$0.164543\pi$
$31$	30.6410i	0.988420i	−0.869343	+	0.494210i	$0.835457\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−97.5692	−2.78769
$36$	0	0
$37$	45.7128i	1.23548i	0.786382	+	0.617741i	$0.211952\pi$
$37$	45.7128i	1.23548i	−0.786382	+	0.617741i	$0.788048\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−35.8564	−0.874546	−0.437273	−	0.899329i	$-0.644056\pi$
$41$	−35.8564	−0.874546	−0.437273	+	0.899329i	$0.644056\pi$
$42$	0	0
$43$	74.6410	1.73584	0.867919	−	0.496706i	$-0.165457\pi$
$43$	74.6410	1.73584	0.867919	+	0.496706i	$0.165457\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	42.1436i	0.896672i	0.893865	+	0.448336i	$0.147983\pi$
$47$	42.1436i	0.896672i	−0.893865	+	0.448336i	$0.852017\pi$
$48$	0	0
$49$	−70.4256	−1.43726
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.9282i	− 0.243928i	−0.992535	−	0.121964i	$-0.961081\pi$
$53$	− 12.9282i	− 0.243928i	0.992535	−	0.121964i	$-0.0389193\pi$
$54$	0	0
$55$	9.56922i	0.173986i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−44.2102	−0.749326	−0.374663	−	0.927161i	$-0.622242\pi$
$59$	−44.2102	−0.749326	−0.374663	+	0.927161i	$0.622242\pi$
$60$	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
$65$	−34.4308	−0.529704
$66$	0	0
$67$	80.4974	1.20145	0.600727	−	0.799454i	$-0.294878\pi$
$67$	80.4974	1.20145	0.600727	+	0.799454i	$0.294878\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 123.138i	− 1.73434i	−0.498009	−	0.867172i	$-0.665935\pi$
$71$	− 123.138i	− 1.73434i	0.498009	−	0.867172i	$-0.334065\pi$
$72$	0	0
$73$	85.4256	1.17021	0.585107	−	0.810956i	$-0.301052\pi$
$73$	85.4256	1.17021	0.585107	+	0.810956i	$0.301052\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.7128i	0.152114i
$78$	0	0
$79$	55.2154i	0.698929i	0.936950	+	0.349464i	$0.113636\pi$
$79$	55.2154i	0.698929i	−0.936950	+	0.349464i	$0.886364\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	49.0718	0.591226	0.295613	−	0.955308i	$-0.404476\pi$
$83$	49.0718	0.591226	0.295613	+	0.955308i	$0.404476\pi$
$84$	0	0
$85$	− 70.1436i	− 0.825219i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−105.713	−1.18778	−0.593892	−	0.804545i	$-0.702409\pi$
$89$	−105.713	−1.18778	−0.593892	+	0.804545i	$0.702409\pi$
$90$	0	0
$91$	−42.1436	−0.463116
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 152.420i	− 1.60443i
$96$	0	0
$97$	21.1384	0.217922	0.108961	−	0.994046i	$-0.465248\pi$
$97$	21.1384	0.217922	0.108961	+	0.994046i	$0.465248\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 172.928i	− 1.71216i	−0.516843	−	0.856080i	$-0.672893\pi$
$101$	− 172.928i	− 1.71216i	0.516843	−	0.856080i	$-0.327107\pi$
$102$	0	0
$103$	16.7846i	0.162957i	0.996675	+	0.0814787i	$0.0259643\pi$
$103$	16.7846i	0.162957i	−0.996675	+	0.0814787i	$0.974036\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	138.067	1.29034	0.645171	−	0.764038i	$-0.276786\pi$
$107$	138.067	1.29034	0.645171	+	0.764038i	$0.276786\pi$
$108$	0	0
$109$	− 7.85641i	− 0.0720771i	−0.999350	−	0.0360386i	$-0.988526\pi$
$109$	− 7.85641i	− 0.0720771i	0.999350	−	0.0360386i	$-0.0114739\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−165.138	−1.46140	−0.730701	−	0.682698i	$-0.760807\pi$
$113$	−165.138	−1.46140	−0.730701	+	0.682698i	$0.760807\pi$
$114$	0	0
$115$	71.4256	0.621092
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 85.8564i	− 0.721482i
$120$	0	0
$121$	−119.851	−0.990506
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 265.282i	− 2.12226i
$126$	0	0
$127$	147.349i	1.16023i	0.814536	+	0.580113i	$0.196992\pi$
$127$	147.349i	1.16023i	−0.814536	+	0.580113i	$0.803008\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	70.9282	0.541437	0.270718	−	0.962659i	$-0.412739\pi$
$131$	70.9282	0.541437	0.270718	+	0.962659i	$0.412739\pi$
$132$	0	0
$133$	− 186.564i	− 1.40274i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	206.133	1.50462	0.752311	−	0.658808i	$-0.228939\pi$
$137$	206.133	1.50462	0.752311	+	0.658808i	$0.228939\pi$
$138$	0	0
$139$	39.9230	0.287216	0.143608	−	0.989635i	$-0.454130\pi$
$139$	39.9230	0.287216	0.143608	+	0.989635i	$0.454130\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.13328i	0.0289041i
$144$	0	0
$145$	27.4256	0.189142
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.353829i	0.00237469i	0.999999	+	0.00118735i	$0.000377944\pi$
$149$	0.353829i	0.00237469i	−0.999999	+	0.00118735i	$0.999622\pi$
$150$	0	0
$151$	− 104.210i	− 0.690134i	−0.938578	−	0.345067i	$-0.887856\pi$
$151$	− 104.210i	− 0.690134i	0.938578	−	0.345067i	$-0.112144\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−273.569	−1.76496
$156$	0	0
$157$	− 221.713i	− 1.41218i	−0.708120	−	0.706092i	$-0.750457\pi$
$157$	− 221.713i	− 1.41218i	0.708120	−	0.706092i	$-0.249543\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	87.4256	0.543016
$162$	0	0
$163$	−175.923	−1.07928	−0.539641	−	0.841895i	$-0.681440\pi$
$163$	−175.923	−1.07928	−0.539641	+	0.841895i	$0.681440\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	237.856i	1.42429i	0.702032	+	0.712145i	$0.252276\pi$
$167$	237.856i	1.42429i	−0.702032	+	0.712145i	$0.747724\pi$
$168$	0	0
$169$	154.128	0.912001
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 71.7795i	− 0.414910i	−0.978245	−	0.207455i	$-0.933482\pi$
$173$	− 71.7795i	− 0.414910i	0.978245	−	0.207455i	$-0.0665181\pi$
$174$	0	0
$175$	− 597.913i	− 3.41664i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.93336	−0.0331473	−0.0165736	−	0.999863i	$-0.505276\pi$
$179$	−5.93336	−0.0331473	−0.0165736	+	0.999863i	$0.505276\pi$
$180$	0	0
$181$	− 150.995i	− 0.834226i	−0.908855	−	0.417113i	$-0.863042\pi$
$181$	− 150.995i	− 0.834226i	0.908855	−	0.417113i	$-0.136958\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−408.133	−2.20613
$186$	0	0
$187$	−8.42047	−0.0450293
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 23.4256i	− 0.122647i	−0.998118	−	0.0613236i	$-0.980468\pi$
$191$	− 23.4256i	− 0.122647i	0.998118	−	0.0613236i	$-0.0195322\pi$
$192$	0	0
$193$	137.426	0.712050	0.356025	−	0.934476i	$-0.384132\pi$
$193$	137.426	0.712050	0.356025	+	0.934476i	$0.384132\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 39.4923i	− 0.200468i	−0.994964	−	0.100234i	$-0.968041\pi$
$197$	− 39.4923i	− 0.200468i	0.994964	−	0.100234i	$-0.0319592\pi$
$198$	0	0
$199$	275.923i	1.38655i	0.720674	+	0.693274i	$0.243832\pi$
$199$	275.923i	1.38655i	−0.720674	+	0.693274i	$0.756168\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	33.5692	0.165366
$204$	0	0
$205$	− 320.133i	− 1.56163i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−18.2975	−0.0875478
$210$	0	0
$211$	189.359	0.897436	0.448718	−	0.893673i	$-0.351881\pi$
$211$	189.359	0.897436	0.448718	+	0.893673i	$0.351881\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	666.410i	3.09958i
$216$	0	0
$217$	−334.851	−1.54309
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 30.2975i	− 0.137093i
$222$	0	0
$223$	144.210i	0.646683i	0.946282	+	0.323341i	$0.104806\pi$
$223$	144.210i	0.646683i	−0.946282	+	0.323341i	$0.895194\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	225.072	0.991506	0.495753	−	0.868464i	$-0.334892\pi$
$227$	225.072	0.991506	0.495753	+	0.868464i	$0.334892\pi$
$228$	0	0
$229$	233.846i	1.02116i	0.859830	+	0.510581i	$0.170570\pi$
$229$	233.846i	1.02116i	−0.859830	+	0.510581i	$0.829430\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−218.862	−0.939320	−0.469660	−	0.882847i	$-0.655623\pi$
$233$	−218.862	−0.939320	−0.469660	+	0.882847i	$0.655623\pi$
$234$	0	0
$235$	−376.267	−1.60113
$236$	0	0
$237$	0	0
$238$	0	0
$239$	371.979i	1.55640i	0.628017	+	0.778200i	$0.283867\pi$
$239$	371.979i	1.55640i	−0.628017	+	0.778200i	$0.716133\pi$
$240$	0	0
$241$	−328.564	−1.36334	−0.681668	−	0.731661i	$-0.738745\pi$
$241$	−328.564	−1.36334	−0.681668	+	0.731661i	$0.738745\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 628.774i	− 2.56643i
$246$	0	0
$247$	− 65.8358i	− 0.266542i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	148.210	0.590479	0.295240	−	0.955423i	$-0.404601\pi$
$251$	148.210	0.590479	0.295240	+	0.955423i	$0.404601\pi$
$252$	0	0
$253$	− 8.57437i	− 0.0338908i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−351.703	−1.36849	−0.684246	−	0.729251i	$-0.739869\pi$
$257$	−351.703	−1.36849	−0.684246	+	0.729251i	$0.739869\pi$
$258$	0	0
$259$	−499.559	−1.92880
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 242.144i	− 0.920698i	−0.887738	−	0.460349i	$-0.847724\pi$
$263$	− 242.144i	− 0.920698i	0.887738	−	0.460349i	$-0.152276\pi$
$264$	0	0
$265$	115.426	0.435568
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 75.6462i	− 0.281213i	−0.990066	−	0.140606i	$-0.955095\pi$
$269$	− 75.6462i	− 0.281213i	0.990066	−	0.140606i	$-0.0449052\pi$
$270$	0	0
$271$	150.795i	0.556439i	0.960518	+	0.278219i	$0.0897442\pi$
$271$	150.795i	0.556439i	−0.960518	+	0.278219i	$0.910256\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−58.6410	−0.213240
$276$	0	0
$277$	213.559i	0.770971i	0.922714	+	0.385485i	$0.125966\pi$
$277$	213.559i	0.770971i	−0.922714	+	0.385485i	$0.874034\pi$
$278$	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
$283$	−179.790	−0.635300	−0.317650	−	0.948208i	$-0.602894\pi$
$283$	−179.790	−0.635300	−0.317650	+	0.948208i	$0.602894\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 391.846i	− 1.36532i
$288$	0	0
$289$	−227.277	−0.786425
$290$	0	0
$291$	0	0
$292$	0	0
$293$	517.902i	1.76759i	0.467879	+	0.883793i	$0.345018\pi$
$293$	517.902i	1.76759i	−0.467879	+	0.883793i	$0.654982\pi$
$294$	0	0
$295$	− 394.718i	− 1.33803i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	30.8513	0.103181
$300$	0	0
$301$	815.692i	2.70994i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−124.995	−0.409819
$306$	0	0
$307$	−407.769	−1.32824	−0.664119	−	0.747627i	$-0.731193\pi$
$307$	−407.769	−1.32824	−0.664119	+	0.747627i	$0.731193\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 368.995i	− 1.18648i	−0.805026	−	0.593239i	$-0.797849\pi$
$311$	− 368.995i	− 1.18648i	0.805026	−	0.593239i	$-0.202151\pi$
$312$	0	0
$313$	−341.979	−1.09259	−0.546293	−	0.837594i	$-0.683961\pi$
$313$	−341.979	−1.09259	−0.546293	+	0.837594i	$0.683961\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 396.487i	− 1.25075i	−0.780325	−	0.625374i	$-0.784946\pi$
$317$	− 396.487i	− 1.25075i	0.780325	−	0.625374i	$-0.215054\pi$
$318$	0	0
$319$	− 3.29234i	− 0.0103208i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	134.123	0.415241
$324$	0	0
$325$	− 210.995i	− 0.649215i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−460.554	−1.39986
$330$	0	0
$331$	267.061	0.806832	0.403416	−	0.915017i	$-0.367823\pi$
$331$	267.061	0.806832	0.403416	+	0.915017i	$0.367823\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	718.697i	2.14537i
$336$	0	0
$337$	−149.426	−0.443399	−0.221700	−	0.975115i	$-0.571160\pi$
$337$	−149.426	−0.443399	−0.221700	+	0.975115i	$0.571160\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.8409i	0.0963077i
$342$	0	0
$343$	− 234.144i	− 0.682634i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	529.913	1.52713	0.763563	−	0.645733i	$-0.223448\pi$
$347$	529.913	1.52713	0.763563	+	0.645733i	$0.223448\pi$
$348$	0	0
$349$	75.7025i	0.216913i	0.994101	+	0.108456i	$0.0345908\pi$
$349$	75.7025i	0.216913i	−0.994101	+	0.108456i	$0.965409\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−325.138	−0.921072	−0.460536	−	0.887641i	$-0.652343\pi$
$353$	−325.138	−0.921072	−0.460536	+	0.887641i	$0.652343\pi$
$354$	0	0
$355$	1099.41	3.09692
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 195.713i	− 0.545161i	−0.962133	−	0.272581i	$-0.912123\pi$
$359$	− 195.713i	− 0.545161i	0.962133	−	0.272581i	$-0.0878771\pi$
$360$	0	0
$361$	−69.5538	−0.192670
$362$	0	0
$363$	0	0
$364$	0	0
$365$	762.697i	2.08958i
$366$	0	0
$367$	− 257.051i	− 0.700412i	−0.936673	−	0.350206i	$-0.886112\pi$
$367$	− 257.051i	− 0.700412i	0.936673	−	0.350206i	$-0.113888\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	141.282	0.380814
$372$	0	0
$373$	− 339.128i	− 0.909191i	−0.890698	−	0.454595i	$-0.849784\pi$
$373$	− 339.128i	− 0.909191i	0.890698	−	0.454595i	$-0.150216\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.8461	0.0314220
$378$	0	0
$379$	163.215	0.430647	0.215324	−	0.976543i	$-0.430919\pi$
$379$	163.215	0.430647	0.215324	+	0.976543i	$0.430919\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 76.5538i	− 0.199879i	−0.994994	−	0.0999396i	$-0.968135\pi$
$383$	− 76.5538i	− 0.199879i	0.994994	−	0.0999396i	$-0.0318650\pi$
$384$	0	0
$385$	−104.574	−0.271622
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 338.477i	− 0.870120i	−0.900401	−	0.435060i	$-0.856727\pi$
$389$	− 338.477i	− 0.870120i	0.900401	−	0.435060i	$-0.143273\pi$
$390$	0	0
$391$	62.8513i	0.160745i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−492.974	−1.24804
$396$	0	0
$397$	7.72312i	0.0194537i	0.999953	+	0.00972685i	$0.00309620\pi$
$397$	7.72312i	0.0194537i	−0.999953	+	0.00972685i	$0.996904\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−448.123	−1.11751	−0.558757	−	0.829332i	$-0.688721\pi$
$401$	−448.123	−1.11751	−0.558757	+	0.829332i	$0.688721\pi$
$402$	0	0
$403$	−118.164	−0.293211
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.9948i	0.120380i
$408$	0	0
$409$	317.692	0.776754	0.388377	−	0.921501i	$-0.373036\pi$
$409$	317.692	0.776754	0.388377	+	0.921501i	$0.373036\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 483.138i	− 1.16983i
$414$	0	0
$415$	438.123i	1.05572i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−660.056	−1.57531	−0.787657	−	0.616114i	$-0.788706\pi$
$419$	−660.056	−1.57531	−0.787657	+	0.616114i	$0.788706\pi$
$420$	0	0
$421$	− 42.4411i	− 0.100810i	−0.998729	−	0.0504051i	$-0.983949\pi$
$421$	− 42.4411i	− 0.100810i	0.998729	−	0.0504051i	$-0.0160512\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	429.846	1.01140
$426$	0	0
$427$	−152.995	−0.358302
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 302.123i	− 0.700981i	−0.936566	−	0.350491i	$-0.886015\pi$
$431$	− 302.123i	− 0.700981i	0.936566	−	0.350491i	$-0.113985\pi$
$432$	0	0
$433$	376.277	0.869000	0.434500	−	0.900672i	$-0.356925\pi$
$433$	376.277	0.869000	0.434500	+	0.900672i	$0.356925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	136.574i	0.312527i
$438$	0	0
$439$	− 394.928i	− 0.899609i	−0.893127	−	0.449804i	$-0.851494\pi$
$439$	− 394.928i	− 0.899609i	0.893127	−	0.449804i	$-0.148506\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	666.487	1.50449	0.752243	−	0.658886i	$-0.228972\pi$
$443$	666.487	1.50449	0.752243	+	0.658886i	$0.228972\pi$
$444$	0	0
$445$	− 943.825i	− 2.12096i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	793.825	1.76799	0.883993	−	0.467501i	$-0.154845\pi$
$449$	793.825	1.76799	0.883993	+	0.467501i	$0.154845\pi$
$450$	0	0
$451$	−38.4308	−0.0852124
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 376.267i	− 0.826959i
$456$	0	0
$457$	−693.138	−1.51671	−0.758357	−	0.651839i	$-0.773998\pi$
$457$	−693.138	−1.51671	−0.758357	+	0.651839i	$0.773998\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	99.0924i	0.214951i	0.994208	+	0.107476i	$0.0342767\pi$
$461$	99.0924i	0.214951i	−0.994208	+	0.107476i	$0.965723\pi$
$462$	0	0
$463$	− 308.077i	− 0.665393i	−0.943034	−	0.332696i	$-0.892042\pi$
$463$	− 308.077i	− 0.665393i	0.943034	−	0.332696i	$-0.107958\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−295.195	−0.632109	−0.316054	−	0.948741i	$-0.602358\pi$
$467$	−295.195	−0.632109	−0.316054	+	0.948741i	$0.602358\pi$
$468$	0	0
$469$	879.692i	1.87568i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	80.0000	0.169133
$474$	0	0
$475$	934.046	1.96641
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 291.559i	− 0.608682i	−0.952563	−	0.304341i	$-0.901564\pi$
$479$	− 291.559i	− 0.608682i	0.952563	−	0.304341i	$-0.0984363\pi$
$480$	0	0
$481$	−176.287	−0.366501
$482$	0	0
$483$	0	0
$484$	0	0
$485$	188.728i	0.389130i
$486$	0	0
$487$	729.779i	1.49852i	0.662276	+	0.749260i	$0.269591\pi$
$487$	729.779i	1.49852i	−0.662276	+	0.749260i	$0.730409\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−258.487	−0.526450	−0.263225	−	0.964734i	$-0.584786\pi$
$491$	−258.487	−0.526450	−0.263225	+	0.964734i	$0.584786\pi$
$492$	0	0
$493$	24.1333i	0.0489519i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1345.68	2.70761
$498$	0	0
$499$	−495.195	−0.992374	−0.496187	−	0.868216i	$-0.665267\pi$
$499$	−495.195	−0.992374	−0.496187	+	0.868216i	$0.665267\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 698.831i	− 1.38933i	−0.719336	−	0.694663i	$-0.755554\pi$
$503$	− 698.831i	− 1.38933i	0.719336	−	0.694663i	$-0.244446\pi$
$504$	0	0
$505$	1543.94	3.05730
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.1948i	0.0416400i	0.999783	+	0.0208200i	$0.00662769\pi$
$509$	21.1948i	0.0416400i	−0.999783	+	0.0208200i	$0.993372\pi$
$510$	0	0
$511$	933.549i	1.82691i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−149.856	−0.290983
$516$	0	0
$517$	45.1694i	0.0873682i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	167.015	0.320567	0.160284	−	0.987071i	$-0.448759\pi$
$521$	167.015	0.320567	0.160284	+	0.987071i	$0.448759\pi$
$522$	0	0
$523$	450.908	0.862156	0.431078	−	0.902315i	$-0.358133\pi$
$523$	450.908	0.862156	0.431078	+	0.902315i	$0.358133\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 240.728i	− 0.456790i
$528$	0	0
$529$	465.000	0.879017
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 138.277i	− 0.259431i
$534$	0	0
$535$	1232.69i	2.30409i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−75.4820	−0.140041
$540$	0	0
$541$	79.5692i	0.147078i	0.997292	+	0.0735390i	$0.0234293\pi$
$541$	79.5692i	0.147078i	−0.997292	+	0.0735390i	$0.976571\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	70.1436	0.128704
$546$	0	0
$547$	−349.933	−0.639732	−0.319866	−	0.947463i	$-0.603638\pi$
$547$	−349.933	−0.639732	−0.319866	+	0.947463i	$0.603638\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	52.4411i	0.0951744i
$552$	0	0
$553$	−603.405	−1.09115
$554$	0	0
$555$	0	0
$556$	0	0
$557$	874.210i	1.56950i	0.619814	+	0.784749i	$0.287208\pi$
$557$	874.210i	1.56950i	−0.619814	+	0.784749i	$0.712792\pi$
$558$	0	0
$559$	287.846i	0.514930i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	104.497	0.185608	0.0928041	−	0.995684i	$-0.470417\pi$
$563$	104.497	0.185608	0.0928041	+	0.995684i	$0.470417\pi$
$564$	0	0
$565$	− 1474.39i	− 2.60954i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	136.697	0.240241	0.120121	−	0.992759i	$-0.461672\pi$
$569$	136.697	0.240241	0.120121	+	0.992759i	$0.461672\pi$
$570$	0	0
$571$	−566.928	−0.992869	−0.496435	−	0.868074i	$-0.665358\pi$
$571$	−566.928	−0.992869	−0.496435	+	0.868074i	$0.665358\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	437.703i	0.761222i
$576$	0	0
$577$	778.574	1.34935	0.674675	−	0.738115i	$-0.264284\pi$
$577$	778.574	1.34935	0.674675	+	0.738115i	$0.264284\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	536.267i	0.923006i
$582$	0	0
$583$	− 13.8564i	− 0.0237674i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1053.89	−1.79539	−0.897693	−	0.440621i	$-0.854758\pi$
$587$	−1053.89	−1.79539	−0.897693	+	0.440621i	$0.854758\pi$
$588$	0	0
$589$	− 523.097i	− 0.888111i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−544.543	−0.918286	−0.459143	−	0.888362i	$-0.651843\pi$
$593$	−544.543	−0.918286	−0.459143	+	0.888362i	$0.651843\pi$
$594$	0	0
$595$	766.543	1.28831
$596$	0	0
$597$	0	0
$598$	0	0
$599$	948.246i	1.58305i	0.611138	+	0.791524i	$0.290712\pi$
$599$	948.246i	1.58305i	−0.611138	+	0.791524i	$0.709288\pi$
$600$	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
$605$	− 1070.06i	− 1.76869i
$606$	0	0
$607$	− 1096.06i	− 1.80569i	−0.429962	−	0.902847i	$-0.641473\pi$
$607$	− 1096.06i	− 1.80569i	0.429962	−	0.902847i	$-0.358527\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−162.523	−0.265995
$612$	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
$617$	−385.979	−0.625574	−0.312787	−	0.949823i	$-0.601263\pi$
$617$	−385.979	−0.625574	−0.312787	+	0.949823i	$0.601263\pi$
$618$	0	0
$619$	−558.887	−0.902887	−0.451443	−	0.892300i	$-0.649091\pi$
$619$	−558.887	−0.902887	−0.451443	+	0.892300i	$0.649091\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1155.25i	− 1.85434i
$624$	0	0
$625$	1000.67	1.60107
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 359.138i	− 0.570967i
$630$	0	0
$631$	− 420.231i	− 0.665976i	−0.942931	−	0.332988i	$-0.891943\pi$
$631$	− 420.231i	− 0.665976i	0.942931	−	0.332988i	$-0.108057\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1315.56	−2.07175
$636$	0	0
$637$	− 271.590i	− 0.426358i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−256.964	−0.400880	−0.200440	−	0.979706i	$-0.564237\pi$
$641$	−256.964	−0.400880	−0.200440	+	0.979706i	$0.564237\pi$
$642$	0	0
$643$	467.215	0.726618	0.363309	−	0.931669i	$-0.381647\pi$
$643$	467.215	0.726618	0.363309	+	0.931669i	$0.381647\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	971.138i	1.50099i	0.660878	+	0.750493i	$0.270184\pi$
$647$	971.138i	1.50099i	−0.660878	+	0.750493i	$0.729816\pi$
$648$	0	0
$649$	−47.3844	−0.0730114
$650$	0	0
$651$	0	0
$652$	0	0
$653$	439.380i	0.672863i	0.941708	+	0.336432i	$0.109220\pi$
$653$	439.380i	0.672863i	−0.941708	+	0.336432i	$0.890780\pi$
$654$	0	0
$655$	633.261i	0.966811i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1092.48	−1.65778	−0.828890	−	0.559412i	$-0.811027\pi$
$659$	−1092.48	−1.65778	−0.828890	+	0.559412i	$0.811027\pi$
$660$	0	0
$661$	122.267i	0.184972i	0.995714	+	0.0924861i	$0.0294814\pi$
$661$	122.267i	0.184972i	−0.995714	+	0.0924861i	$0.970519\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1665.68	2.50478
$666$	0	0
$667$	−24.5744	−0.0368431
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15.0052i	0.0223624i
$672$	0	0
$673$	−796.851	−1.18403	−0.592014	−	0.805927i	$-0.701667\pi$
$673$	−796.851	−1.18403	−0.592014	+	0.805927i	$0.701667\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1185.19i	1.75066i	0.483529	+	0.875328i	$0.339355\pi$
$677$	1185.19i	1.75066i	−0.483529	+	0.875328i	$0.660645\pi$
$678$	0	0
$679$	231.005i	0.340214i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	850.446	1.24516	0.622581	−	0.782555i	$-0.286084\pi$
$683$	850.446	1.24516	0.622581	+	0.782555i	$0.286084\pi$
$684$	0	0
$685$	1840.40i	2.68672i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	49.8564	0.0723605
$690$	0	0
$691$	1161.49	1.68089	0.840443	−	0.541900i	$-0.182295\pi$
$691$	1161.49	1.68089	0.840443	+	0.541900i	$0.182295\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	356.441i	0.512865i
$696$	0	0
$697$	281.703	0.404164
$698$	0	0
$699$	0	0
$700$	0	0
$701$	177.215i	0.252804i	0.991979	+	0.126402i	$0.0403429\pi$
$701$	177.215i	0.252804i	−0.991979	+	0.126402i	$0.959657\pi$
$702$	0	0
$703$	− 780.400i	− 1.11010i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1889.79	2.67298
$708$	0	0
$709$	329.846i	0.465227i	0.972569	+	0.232614i	$0.0747277\pi$
$709$	329.846i	0.465227i	−0.972569	+	0.232614i	$0.925272\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	245.128	0.343798
$714$	0	0
$715$	−36.9028	−0.0516123
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1343.27i	1.86825i	0.356946	+	0.934125i	$0.383818\pi$
$719$	1343.27i	1.86825i	−0.356946	+	0.934125i	$0.616182\pi$
$720$	0	0
$721$	−183.426	−0.254404
$722$	0	0
$723$	0	0
$724$	0	0
$725$	168.067i	0.231816i
$726$	0	0
$727$	6.64102i	0.00913482i	0.999990	+	0.00456741i	$0.00145386\pi$
$727$	6.64102i	0.00913482i	−0.999990	+	0.00456741i	$0.998546\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−586.410	−0.802203
$732$	0	0
$733$	1396.39i	1.90503i	0.304486	+	0.952517i	$0.401515\pi$
$733$	1396.39i	1.90503i	−0.304486	+	0.952517i	$0.598485\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	86.2769	0.117065
$738$	0	0
$739$	−276.939	−0.374748	−0.187374	−	0.982289i	$-0.559998\pi$
$739$	−276.939	−0.374748	−0.187374	+	0.982289i	$0.559998\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 424.728i	− 0.571640i	−0.958283	−	0.285820i	$-0.907734\pi$
$743$	− 424.728i	− 0.571640i	0.958283	−	0.285820i	$-0.0922659\pi$
$744$	0	0
$745$	−3.15906	−0.00424035
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1508.82i	2.01445i
$750$	0	0
$751$	1008.32i	1.34264i	0.741167	+	0.671320i	$0.234272\pi$
$751$	1008.32i	1.34264i	−0.741167	+	0.671320i	$0.765728\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	930.410	1.23233
$756$	0	0
$757$	− 2.70766i	− 0.00357683i	−0.999998	−	0.00178841i	$-0.999431\pi$
$757$	− 2.70766i	− 0.00357683i	0.999998	−	0.00178841i	$-0.000569270\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	656.431	0.862590	0.431295	−	0.902211i	$-0.358057\pi$
$761$	656.431	0.862590	0.431295	+	0.902211i	$0.358057\pi$
$762$	0	0
$763$	85.8564	0.112525
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 170.493i	− 0.222285i
$768$	0	0
$769$	−533.959	−0.694355	−0.347177	−	0.937799i	$-0.612860\pi$
$769$	−533.959	−0.694355	−0.347177	+	0.937799i	$0.612860\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 350.918i	− 0.453969i	−0.973898	−	0.226984i	$-0.927113\pi$
$773$	− 350.918i	− 0.453969i	0.973898	−	0.226984i	$-0.0728866\pi$
$774$	0	0
$775$	− 1676.46i	− 2.16317i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	612.133	0.785794
$780$	0	0
$781$	− 131.979i	− 0.168988i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1979.50	2.52165
$786$	0	0
$787$	−391.041	−0.496875	−0.248438	−	0.968648i	$-0.579917\pi$
$787$	−391.041	−0.496875	−0.248438	+	0.968648i	$0.579917\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1804.67i	− 2.28150i
$792$	0	0
$793$	−53.9897	−0.0680828
$794$	0	0
$795$	0	0
$796$	0	0
$797$	138.210i	0.173413i	0.996234	+	0.0867065i	$0.0276343\pi$
$797$	138.210i	0.173413i	−0.996234	+	0.0867065i	$0.972366\pi$
$798$	0	0
$799$	− 331.097i	− 0.414389i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	91.5589	0.114021
$804$	0	0
$805$	780.554i	0.969632i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−919.528	−1.13662	−0.568311	−	0.822814i	$-0.692403\pi$
$809$	−919.528	−1.13662	−0.568311	+	0.822814i	$0.692403\pi$
$810$	0	0
$811$	−181.359	−0.223624	−0.111812	−	0.993729i	$-0.535665\pi$
$811$	−181.359	−0.223624	−0.111812	+	0.993729i	$0.535665\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1570.68i	− 1.92721i
$816$	0	0
$817$	−1274.26	−1.55968
$818$	0	0
$819$	0	0
$820$	0	0
$821$	839.359i	1.02236i	0.859473	+	0.511181i	$0.170792\pi$
$821$	839.359i	1.02236i	−0.859473	+	0.511181i	$0.829208\pi$
$822$	0	0
$823$	− 26.1999i	− 0.0318347i	−0.999873	−	0.0159173i	$-0.994933\pi$
$823$	− 26.1999i	− 0.0318347i	0.999873	−	0.0159173i	$-0.00506686\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.38991	−0.00893581	−0.00446790	−	0.999990i	$-0.501422\pi$
$827$	−7.38991	−0.00893581	−0.00446790	+	0.999990i	$0.501422\pi$
$828$	0	0
$829$	− 340.410i	− 0.410627i	−0.978696	−	0.205314i	$-0.934179\pi$
$829$	− 340.410i	− 0.410627i	0.978696	−	0.205314i	$-0.0658215\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	553.292	0.664216
$834$	0	0
$835$	−2123.63	−2.54327
$836$	0	0
$837$	0	0
$838$	0	0
$839$	734.810i	0.875816i	0.899020	+	0.437908i	$0.144281\pi$
$839$	734.810i	0.875816i	−0.899020	+	0.437908i	$0.855719\pi$
$840$	0	0
$841$	831.564	0.988780
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1376.09i	1.62851i
$846$	0	0
$847$	− 1309.76i	− 1.54635i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	365.703	0.429733
$852$	0	0
$853$	778.800i	0.913013i	0.889720	+	0.456506i	$0.150899\pi$
$853$	778.800i	0.913013i	−0.889720	+	0.456506i	$0.849101\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1129.52	−1.31799	−0.658995	−	0.752147i	$-0.729018\pi$
$857$	−1129.52	−1.31799	−0.658995	+	0.752147i	$0.729018\pi$
$858$	0	0
$859$	−1373.85	−1.59936	−0.799680	−	0.600426i	$-0.794998\pi$
$859$	−1373.85	−1.59936	−0.799680	+	0.600426i	$0.794998\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	539.405i	0.625035i	0.949912	+	0.312517i	$0.101172\pi$
$863$	539.405i	0.625035i	−0.949912	+	0.312517i	$0.898828\pi$
$864$	0	0
$865$	640.862	0.740880
$866$	0	0
$867$	0	0
$868$	0	0
$869$	59.1797i	0.0681009i
$870$	0	0
$871$	310.431i	0.356407i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2899.06	3.31321
$876$	0	0
$877$	668.236i	0.761956i	0.924584	+	0.380978i	$0.124413\pi$
$877$	668.236i	0.761956i	−0.924584	+	0.380978i	$0.875587\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1170.53	−1.32864	−0.664321	−	0.747448i	$-0.731279\pi$
$881$	−1170.53	−1.32864	−0.664321	+	0.747448i	$0.731279\pi$
$882$	0	0
$883$	32.9179	0.0372796	0.0186398	−	0.999826i	$-0.494066\pi$
$883$	32.9179	0.0372796	0.0186398	+	0.999826i	$0.494066\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	655.846i	0.739398i	0.929152	+	0.369699i	$0.120539\pi$
$887$	655.846i	0.739398i	−0.929152	+	0.369699i	$0.879461\pi$
$888$	0	0
$889$	−1610.26	−1.81131
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 719.467i	− 0.805674i
$894$	0	0
$895$	− 52.9742i	− 0.0591891i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	94.1230	0.104697
$900$	0	0
$901$	101.569i	0.112729i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1348.11	1.48963
$906$	0	0
$907$	378.949	0.417805	0.208902	−	0.977937i	$-0.433011\pi$
$907$	378.949	0.417805	0.208902	+	0.977937i	$0.433011\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 571.938i	− 0.627814i	−0.949454	−	0.313907i	$-0.898362\pi$
$911$	− 571.938i	− 0.627814i	0.949454	−	0.313907i	$-0.101638\pi$
$912$	0	0
$913$	52.5950	0.0576068
$914$	0	0
$915$	0	0
$916$	0	0
$917$	775.118i	0.845276i
$918$	0	0
$919$	97.8207i	0.106443i	0.998583	+	0.0532213i	$0.0169489\pi$
$919$	97.8207i	0.106443i	−0.998583	+	0.0532213i	$0.983051\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	474.872	0.514487
$924$	0	0
$925$	− 2501.08i	− 2.70387i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1228.06	−1.32192	−0.660959	−	0.750422i	$-0.729850\pi$
$929$	−1228.06	−1.32192	−0.660959	+	0.750422i	$0.729850\pi$
$930$	0	0
$931$	1202.29	1.29140
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 75.1797i	− 0.0804061i
$936$	0	0
$937$	−494.554	−0.527806	−0.263903	−	0.964549i	$-0.585010\pi$
$937$	−494.554	−0.527806	−0.263903	+	0.964549i	$0.585010\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1154.48i	1.22686i	0.789748	+	0.613431i	$0.210211\pi$
$941$	1154.48i	1.22686i	−0.789748	+	0.613431i	$0.789789\pi$
$942$	0	0
$943$	286.851i	0.304190i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1085.43	1.14618	0.573089	−	0.819493i	$-0.305745\pi$
$947$	1085.43	1.14618	0.573089	+	0.819493i	$0.305745\pi$
$948$	0	0
$949$	329.436i	0.347140i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53.8667	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$953$	53.8667	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$954$	0	0
$955$	209.149	0.219004
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2252.67i	2.34897i
$960$	0	0
$961$	22.1281	0.0230261
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1226.96i	1.27147i
$966$	0	0
$967$	− 172.918i	− 0.178819i	−0.995995	−	0.0894095i	$-0.971502\pi$
$967$	− 172.918i	− 0.178819i	0.995995	−	0.0894095i	$-0.0284980\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−984.877	−1.01429	−0.507146	−	0.861860i	$-0.669299\pi$
$971$	−984.877	−1.01429	−0.507146	+	0.861860i	$0.669299\pi$
$972$	0	0
$973$	436.287i	0.448394i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	675.856	0.691767	0.345884	−	0.938277i	$-0.387579\pi$
$977$	675.856	0.691767	0.345884	+	0.938277i	$0.387579\pi$
$978$	0	0
$979$	−113.303	−0.115733
$980$	0	0
$981$	0	0
$982$	0	0
$983$	564.441i	0.574203i	0.957900	+	0.287101i	$0.0926916\pi$
$983$	564.441i	0.574203i	−0.957900	+	0.287101i	$0.907308\pi$
$984$	0	0
$985$	352.595	0.357964
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 597.128i	− 0.603770i
$990$	0	0
$991$	− 1392.44i	− 1.40508i	−0.711644	−	0.702541i	$-0.752049\pi$
$991$	− 1392.44i	− 1.40508i	0.711644	−	0.702541i	$-0.247951\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2463.50	−2.47588
$996$	0	0
$997$	123.723i	0.124095i	0.998073	+	0.0620477i	$0.0197631\pi$
$997$	123.723i	0.124095i	−0.998073	+	0.0620477i	$0.980237\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.b.o.127.4		4
3.2	odd	2		768.3.b.a.127.1		4
4.3	odd	2		2304.3.b.k.127.4		4
8.3	odd	2	inner	2304.3.b.o.127.1		4
8.5	even	2		2304.3.b.k.127.1		4
12.11	even	2		768.3.b.d.127.3		4
16.3	odd	4		288.3.g.d.127.4		4
16.5	even	4		576.3.g.j.127.1		4
16.11	odd	4		576.3.g.j.127.2		4
16.13	even	4		288.3.g.d.127.3		4
24.5	odd	2		768.3.b.d.127.4		4
24.11	even	2		768.3.b.a.127.2		4
48.5	odd	4		192.3.g.c.127.4		4
48.11	even	4		192.3.g.c.127.2		4
48.29	odd	4		96.3.g.a.31.1	✓	4
48.35	even	4		96.3.g.a.31.3	yes	4
240.29	odd	4		2400.3.e.a.1951.4		4
240.77	even	4		2400.3.j.b.799.2		4
240.83	odd	4		2400.3.j.b.799.1		4
240.173	even	4		2400.3.j.a.799.4		4
240.179	even	4		2400.3.e.a.1951.1		4
240.227	odd	4		2400.3.j.a.799.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.g.a.31.1	✓	4	48.29	odd	4
96.3.g.a.31.3	yes	4	48.35	even	4
192.3.g.c.127.2		4	48.11	even	4
192.3.g.c.127.4		4	48.5	odd	4
288.3.g.d.127.3		4	16.13	even	4
288.3.g.d.127.4		4	16.3	odd	4
576.3.g.j.127.1		4	16.5	even	4
576.3.g.j.127.2		4	16.11	odd	4
768.3.b.a.127.1		4	3.2	odd	2
768.3.b.a.127.2		4	24.11	even	2
768.3.b.d.127.3		4	12.11	even	2
768.3.b.d.127.4		4	24.5	odd	2
2304.3.b.k.127.1		4	8.5	even	2
2304.3.b.k.127.4		4	4.3	odd	2
2304.3.b.o.127.1		4	8.3	odd	2	inner
2304.3.b.o.127.4		4	1.1	even	1	trivial
2400.3.e.a.1951.1		4	240.179	even	4
2400.3.e.a.1951.4		4	240.29	odd	4
2400.3.j.a.799.3		4	240.227	odd	4
2400.3.j.a.799.4		4	240.173	even	4
2400.3.j.b.799.1		4	240.83	odd	4
2400.3.j.b.799.2		4	240.77	even	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)

\(f(q)\)	\(=\)	\(q+8.92820i q^{5} +10.9282i q^{7} +O(q^{10})\)	\(q+8.92820i q^{5} +10.9282i q^{7} +1.07180 q^{11} +3.85641i q^{13} -7.85641 q^{17} -17.0718 q^{19} -8.00000i q^{23} -54.7128 q^{25} -3.07180i q^{29} +30.6410i q^{31} -97.5692 q^{35} +45.7128i q^{37} -35.8564 q^{41} +74.6410 q^{43} +42.1436i q^{47} -70.4256 q^{49} -12.9282i q^{53} +9.56922i q^{55} -44.2102 q^{59} +14.0000i q^{61} -34.4308 q^{65} +80.4974 q^{67} -123.138i q^{71} +85.4256 q^{73} +11.7128i q^{77} +55.2154i q^{79} +49.0718 q^{83} -70.1436i q^{85} -105.713 q^{89} -42.1436 q^{91} -152.420i q^{95} +21.1384 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 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

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