LMFDB - Embedded newform 6048.2.j.d.5615.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.2
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.18263 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-4.18263 q^{5} -1.00000i q^{7} -1.86730i q^{11} +3.07604i q^{13} -0.504885i q^{17} -3.05596 q^{19} -5.97229 q^{23} +12.4944 q^{25} -3.52884 q^{29} -10.8980i q^{31} +4.18263i q^{35} +11.3660i q^{37} +7.26960i q^{41} -9.02120 q^{43} +0.327367 q^{47} -1.00000 q^{49} -8.32940 q^{53} +7.81021i q^{55} -5.98008i q^{59} -13.2783i q^{61} -12.8659i q^{65} +9.21499 q^{67} -9.02460 q^{71} +0.416491 q^{73} -1.86730 q^{77} +6.00459i q^{79} +2.62368i q^{83} +2.11175i q^{85} -5.69486i q^{89} +3.07604 q^{91} +12.7819 q^{95} +6.21289 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) $ 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.18263	−1.87053	−0.935264	−	0.353951i	$-0.884838\pi$
$5$	−4.18263	−1.87053	−0.935264	+	0.353951i	$0.884838\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.86730i	− 0.563011i	−0.959560	−	0.281506i	$-0.909166\pi$
$11$	− 1.86730i	− 0.563011i	0.959560	−	0.281506i	$-0.0908338\pi$
$12$	0	0
$13$	3.07604i	0.853139i	0.904455	+	0.426569i	$0.140278\pi$
$13$	3.07604i	0.853139i	−0.904455	+	0.426569i	$0.859722\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.504885i	− 0.122453i	−0.998124	−	0.0612263i	$-0.980499\pi$
$17$	− 0.504885i	− 0.122453i	0.998124	−	0.0612263i	$-0.0195011\pi$
$18$	0	0
$19$	−3.05596	−0.701085	−0.350543	−	0.936547i	$-0.614003\pi$
$19$	−3.05596	−0.701085	−0.350543	+	0.936547i	$0.614003\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.97229	−1.24531	−0.622654	−	0.782497i	$-0.713946\pi$
$23$	−5.97229	−1.24531	−0.622654	+	0.782497i	$0.713946\pi$
$24$	0	0
$25$	12.4944	2.49887
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.52884	−0.655289	−0.327644	−	0.944801i	$-0.606255\pi$
$29$	−3.52884	−0.655289	−0.327644	+	0.944801i	$0.606255\pi$
$30$	0	0
$31$	− 10.8980i	− 1.95734i	−0.205435	−	0.978671i	$-0.565861\pi$
$31$	− 10.8980i	− 1.95734i	0.205435	−	0.978671i	$-0.434139\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.18263i	0.706993i
$36$	0	0
$37$	11.3660i	1.86856i	0.356542	+	0.934279i	$0.383956\pi$
$37$	11.3660i	1.86856i	−0.356542	+	0.934279i	$0.616044\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.26960i	1.13532i	0.823263	+	0.567660i	$0.192151\pi$
$41$	7.26960i	1.13532i	−0.823263	+	0.567660i	$0.807849\pi$
$42$	0	0
$43$	−9.02120	−1.37572	−0.687860	−	0.725844i	$-0.741450\pi$
$43$	−9.02120	−1.37572	−0.687860	+	0.725844i	$0.741450\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.327367	0.0477513	0.0238757	−	0.999715i	$-0.492399\pi$
$47$	0.327367	0.0477513	0.0238757	+	0.999715i	$0.492399\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.32940	−1.14413	−0.572066	−	0.820208i	$-0.693858\pi$
$53$	−8.32940	−1.14413	−0.572066	+	0.820208i	$0.693858\pi$
$54$	0	0
$55$	7.81021i	1.05313i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.98008i	− 0.778540i	−0.921124	−	0.389270i	$-0.872727\pi$
$59$	− 5.98008i	− 0.778540i	0.921124	−	0.389270i	$-0.127273\pi$
$60$	0	0
$61$	− 13.2783i	− 1.70011i	−0.526691	−	0.850057i	$-0.676568\pi$
$61$	− 13.2783i	− 1.70011i	0.526691	−	0.850057i	$-0.323432\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 12.8659i	− 1.59582i
$66$	0	0
$67$	9.21499	1.12579	0.562895	−	0.826528i	$-0.309688\pi$
$67$	9.21499	1.12579	0.562895	+	0.826528i	$0.309688\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.02460	−1.07102	−0.535512	−	0.844528i	$-0.679881\pi$
$71$	−9.02460	−1.07102	−0.535512	+	0.844528i	$0.679881\pi$
$72$	0	0
$73$	0.416491	0.0487466	0.0243733	−	0.999703i	$-0.492241\pi$
$73$	0.416491	0.0487466	0.0243733	+	0.999703i	$0.492241\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.86730	−0.212798
$78$	0	0
$79$	6.00459i	0.675569i	0.941224	+	0.337784i	$0.109677\pi$
$79$	6.00459i	0.675569i	−0.941224	+	0.337784i	$0.890323\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.62368i	0.287987i	0.989579	+	0.143993i	$0.0459944\pi$
$83$	2.62368i	0.287987i	−0.989579	+	0.143993i	$0.954006\pi$
$84$	0	0
$85$	2.11175i	0.229051i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 5.69486i	− 0.603654i	−0.953363	−	0.301827i	$-0.902403\pi$
$89$	− 5.69486i	− 0.603654i	0.953363	−	0.301827i	$-0.0975966\pi$
$90$	0	0
$91$	3.07604	0.322456
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.7819	1.31140
$96$	0	0
$97$	6.21289	0.630824	0.315412	−	0.948955i	$-0.397857\pi$
$97$	6.21289	0.630824	0.315412	+	0.948955i	$0.397857\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.85216	0.184296	0.0921482	−	0.995745i	$-0.470627\pi$
$101$	1.85216	0.184296	0.0921482	+	0.995745i	$0.470627\pi$
$102$	0	0
$103$	2.74001i	0.269982i	0.990847	+	0.134991i	$0.0431005\pi$
$103$	2.74001i	0.269982i	−0.990847	+	0.134991i	$0.956900\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.22506i	− 0.408452i	−0.978924	−	0.204226i	$-0.934532\pi$
$107$	− 4.22506i	− 0.408452i	0.978924	−	0.204226i	$-0.0654677\pi$
$108$	0	0
$109$	13.9420i	1.33540i	0.744432	+	0.667698i	$0.232720\pi$
$109$	13.9420i	1.33540i	−0.744432	+	0.667698i	$0.767280\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 7.49447i	− 0.705020i	−0.935808	−	0.352510i	$-0.885328\pi$
$113$	− 7.49447i	− 0.705020i	0.935808	−	0.352510i	$-0.114672\pi$
$114$	0	0
$115$	24.9799	2.32938
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.504885	−0.0462827
$120$	0	0
$121$	7.51320	0.683018
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−31.3462	−2.80369
$126$	0	0
$127$	18.0939i	1.60557i	0.596267	+	0.802787i	$0.296650\pi$
$127$	18.0939i	1.60557i	−0.596267	+	0.802787i	$0.703350\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 9.32625i	− 0.814839i	−0.913241	−	0.407419i	$-0.866429\pi$
$131$	− 9.32625i	− 0.814839i	0.913241	−	0.407419i	$-0.133571\pi$
$132$	0	0
$133$	3.05596i	0.264985i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.53069i	0.557955i	0.960298	+	0.278977i	$0.0899954\pi$
$137$	6.53069i	0.557955i	−0.960298	+	0.278977i	$0.910005\pi$
$138$	0	0
$139$	10.1589	0.861664	0.430832	−	0.902432i	$-0.358220\pi$
$139$	10.1589	0.861664	0.430832	+	0.902432i	$0.358220\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.74387	0.480327
$144$	0	0
$145$	14.7598	1.22574
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.72516	−0.796716	−0.398358	−	0.917230i	$-0.630420\pi$
$149$	−9.72516	−0.796716	−0.398358	+	0.917230i	$0.630420\pi$
$150$	0	0
$151$	− 12.4664i	− 1.01450i	−0.861800	−	0.507249i	$-0.830662\pi$
$151$	− 12.4664i	− 1.01450i	0.861800	−	0.507249i	$-0.169338\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	45.5823i	3.66126i
$156$	0	0
$157$	11.3627i	0.906838i	0.891297	+	0.453419i	$0.149796\pi$
$157$	11.3627i	0.906838i	−0.891297	+	0.453419i	$0.850204\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.97229i	0.470682i
$162$	0	0
$163$	3.59449	0.281542	0.140771	−	0.990042i	$-0.455042\pi$
$163$	3.59449	0.281542	0.140771	+	0.990042i	$0.455042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.3801	1.57706	0.788531	−	0.614995i	$-0.210842\pi$
$167$	20.3801	1.57706	0.788531	+	0.614995i	$0.210842\pi$
$168$	0	0
$169$	3.53800	0.272154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.49761	0.341947	0.170974	−	0.985276i	$-0.445309\pi$
$173$	4.49761	0.341947	0.170974	+	0.985276i	$0.445309\pi$
$174$	0	0
$175$	− 12.4944i	− 0.944486i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.63430i	0.196897i	0.995142	+	0.0984485i	$0.0313880\pi$
$179$	2.63430i	0.196897i	−0.995142	+	0.0984485i	$0.968612\pi$
$180$	0	0
$181$	− 10.3748i	− 0.771151i	−0.922676	−	0.385576i	$-0.874003\pi$
$181$	− 10.3748i	− 0.771151i	0.922676	−	0.385576i	$-0.125997\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 47.5397i	− 3.49519i
$186$	0	0
$187$	−0.942770	−0.0689422
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.1707	1.31479	0.657395	−	0.753547i	$-0.271659\pi$
$191$	18.1707	1.31479	0.657395	+	0.753547i	$0.271659\pi$
$192$	0	0
$193$	−16.7704	−1.20716	−0.603578	−	0.797304i	$-0.706259\pi$
$193$	−16.7704	−1.20716	−0.603578	+	0.797304i	$0.706259\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.2240	1.36965	0.684826	−	0.728706i	$-0.259878\pi$
$197$	19.2240	1.36965	0.684826	+	0.728706i	$0.259878\pi$
$198$	0	0
$199$	− 1.68931i	− 0.119752i	−0.998206	−	0.0598761i	$-0.980929\pi$
$199$	− 1.68931i	− 0.119752i	0.998206	−	0.0598761i	$-0.0190706\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.52884i	0.247676i
$204$	0	0
$205$	− 30.4060i	− 2.12365i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.70638i	0.394719i
$210$	0	0
$211$	14.1544	0.974427	0.487213	−	0.873283i	$-0.338013\pi$
$211$	14.1544	0.974427	0.487213	+	0.873283i	$0.338013\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	37.7323	2.57332
$216$	0	0
$217$	−10.8980	−0.739806
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.55304	0.104469
$222$	0	0
$223$	− 1.33032i	− 0.0890851i	−0.999007	−	0.0445425i	$-0.985817\pi$
$223$	− 1.33032i	− 0.0890851i	0.999007	−	0.0445425i	$-0.0141830\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.34646i	− 0.421229i	−0.977569	−	0.210615i	$-0.932453\pi$
$227$	− 6.34646i	− 0.421229i	0.977569	−	0.210615i	$-0.0675465\pi$
$228$	0	0
$229$	− 13.9920i	− 0.924619i	−0.886719	−	0.462309i	$-0.847021\pi$
$229$	− 13.9920i	− 0.924619i	0.886719	−	0.462309i	$-0.152979\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0797i	0.922392i	0.887298	+	0.461196i	$0.152580\pi$
$233$	14.0797i	0.922392i	−0.887298	+	0.461196i	$0.847420\pi$
$234$	0	0
$235$	−1.36925	−0.0893202
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.1737	−0.852134	−0.426067	−	0.904691i	$-0.640101\pi$
$239$	−13.1737	−0.852134	−0.426067	+	0.904691i	$0.640101\pi$
$240$	0	0
$241$	7.46846	0.481086	0.240543	−	0.970639i	$-0.422675\pi$
$241$	7.46846	0.481086	0.240543	+	0.970639i	$0.422675\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.18263	0.267218
$246$	0	0
$247$	− 9.40024i	− 0.598123i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.3203i	1.03013i	0.857151	+	0.515065i	$0.172232\pi$
$251$	16.3203i	1.03013i	−0.857151	+	0.515065i	$0.827768\pi$
$252$	0	0
$253$	11.1520i	0.701123i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.6939i	0.854205i	0.904203	+	0.427102i	$0.140466\pi$
$257$	13.6939i	0.854205i	−0.904203	+	0.427102i	$0.859534\pi$
$258$	0	0
$259$	11.3660	0.706249
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.63480	−0.285794	−0.142897	−	0.989738i	$-0.545642\pi$
$263$	−4.63480	−0.285794	−0.142897	+	0.989738i	$0.545642\pi$
$264$	0	0
$265$	34.8388	2.14013
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.6544	1.74709	0.873545	−	0.486743i	$-0.161815\pi$
$269$	28.6544	1.74709	0.873545	+	0.486743i	$0.161815\pi$
$270$	0	0
$271$	8.27458i	0.502645i	0.967903	+	0.251322i	$0.0808654\pi$
$271$	8.27458i	0.502645i	−0.967903	+	0.251322i	$0.919135\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 23.3307i	− 1.40689i
$276$	0	0
$277$	12.5278i	0.752724i	0.926473	+	0.376362i	$0.122825\pi$
$277$	12.5278i	0.752724i	−0.926473	+	0.376362i	$0.877175\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.63112i	0.514889i	0.966293	+	0.257445i	$0.0828805\pi$
$281$	8.63112i	0.514889i	−0.966293	+	0.257445i	$0.917119\pi$
$282$	0	0
$283$	−8.73498	−0.519241	−0.259620	−	0.965711i	$-0.583597\pi$
$283$	−8.73498	−0.519241	−0.259620	+	0.965711i	$0.583597\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.26960	0.429111
$288$	0	0
$289$	16.7451	0.985005
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.3572	1.77348	0.886742	−	0.462265i	$-0.152963\pi$
$293$	30.3572	1.77348	0.886742	+	0.462265i	$0.152963\pi$
$294$	0	0
$295$	25.0124i	1.45628i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 18.3710i	− 1.06242i
$300$	0	0
$301$	9.02120i	0.519973i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	55.5382i	3.18011i
$306$	0	0
$307$	29.1705	1.66485	0.832425	−	0.554138i	$-0.186952\pi$
$307$	29.1705	1.66485	0.832425	+	0.554138i	$0.186952\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.4778	0.764253	0.382127	−	0.924110i	$-0.375192\pi$
$311$	13.4778	0.764253	0.382127	+	0.924110i	$0.375192\pi$
$312$	0	0
$313$	20.7291	1.17168	0.585838	−	0.810428i	$-0.300765\pi$
$313$	20.7291	1.17168	0.585838	+	0.810428i	$0.300765\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.453146	0.0254512	0.0127256	−	0.999919i	$-0.495949\pi$
$317$	0.453146	0.0254512	0.0127256	+	0.999919i	$0.495949\pi$
$318$	0	0
$319$	6.58939i	0.368935i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.54291i	0.0858497i
$324$	0	0
$325$	38.4331i	2.13189i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 0.327367i	− 0.0180483i
$330$	0	0
$331$	−24.8798	−1.36752	−0.683759	−	0.729708i	$-0.739656\pi$
$331$	−24.8798	−1.36752	−0.683759	+	0.729708i	$0.739656\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−38.5429	−2.10582
$336$	0	0
$337$	21.8036	1.18772	0.593860	−	0.804568i	$-0.297603\pi$
$337$	21.8036	1.18772	0.593860	+	0.804568i	$0.297603\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.3498	−1.10201
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.7794i	1.11550i	0.830010	+	0.557748i	$0.188335\pi$
$347$	20.7794i	1.11550i	−0.830010	+	0.557748i	$0.811665\pi$
$348$	0	0
$349$	− 30.5928i	− 1.63760i	−0.574081	−	0.818799i	$-0.694640\pi$
$349$	− 30.5928i	− 1.63760i	0.574081	−	0.818799i	$-0.305360\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 19.5174i	− 1.03881i	−0.854529	−	0.519403i	$-0.826154\pi$
$353$	− 19.5174i	− 1.03881i	0.854529	−	0.519403i	$-0.173846\pi$
$354$	0	0
$355$	37.7466	2.00338
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.3969	1.07651	0.538254	−	0.842783i	$-0.319084\pi$
$359$	20.3969	1.07651	0.538254	+	0.842783i	$0.319084\pi$
$360$	0	0
$361$	−9.66111	−0.508479
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.74203	−0.0911818
$366$	0	0
$367$	25.6785i	1.34041i	0.742177	+	0.670203i	$0.233793\pi$
$367$	25.6785i	1.34041i	−0.742177	+	0.670203i	$0.766207\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.32940i	0.432441i
$372$	0	0
$373$	− 12.4313i	− 0.643669i	−0.946796	−	0.321835i	$-0.895700\pi$
$373$	− 12.4313i	− 0.643669i	0.946796	−	0.321835i	$-0.104300\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.8548i	− 0.559053i
$378$	0	0
$379$	9.90048	0.508553	0.254277	−	0.967132i	$-0.418163\pi$
$379$	9.90048	0.508553	0.254277	+	0.967132i	$0.418163\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−23.6536	−1.20864	−0.604320	−	0.796741i	$-0.706555\pi$
$383$	−23.6536	−1.20864	−0.604320	+	0.796741i	$0.706555\pi$
$384$	0	0
$385$	7.81021	0.398045
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.14513	−0.159465	−0.0797323	−	0.996816i	$-0.525407\pi$
$389$	−3.14513	−0.159465	−0.0797323	+	0.996816i	$0.525407\pi$
$390$	0	0
$391$	3.01532i	0.152491i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 25.1150i	− 1.26367i
$396$	0	0
$397$	12.2176i	0.613183i	0.951841	+	0.306592i	$0.0991886\pi$
$397$	12.2176i	0.613183i	−0.951841	+	0.306592i	$0.900811\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 35.0797i	− 1.75180i	−0.482494	−	0.875899i	$-0.660269\pi$
$401$	− 35.0797i	− 1.75180i	0.482494	−	0.875899i	$-0.339731\pi$
$402$	0	0
$403$	33.5227	1.66988
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.2237	1.05202
$408$	0	0
$409$	−27.3489	−1.35231	−0.676157	−	0.736757i	$-0.736356\pi$
$409$	−27.3489	−1.35231	−0.676157	+	0.736757i	$0.736356\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.98008	−0.294260
$414$	0	0
$415$	− 10.9739i	− 0.538687i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.6442i	0.715417i	0.933833	+	0.357709i	$0.116442\pi$
$419$	14.6442i	0.715417i	−0.933833	+	0.357709i	$0.883558\pi$
$420$	0	0
$421$	− 20.2168i	− 0.985305i	−0.870226	−	0.492653i	$-0.836027\pi$
$421$	− 20.2168i	− 0.985305i	0.870226	−	0.492653i	$-0.163973\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 6.30822i	− 0.305994i
$426$	0	0
$427$	−13.2783	−0.642583
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.6656	−0.947260	−0.473630	−	0.880724i	$-0.657057\pi$
$431$	−19.6656	−0.947260	−0.473630	+	0.880724i	$0.657057\pi$
$432$	0	0
$433$	11.9996	0.576666	0.288333	−	0.957530i	$-0.406899\pi$
$433$	11.9996	0.576666	0.288333	+	0.957530i	$0.406899\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.2511	0.873068
$438$	0	0
$439$	7.82113i	0.373282i	0.982428	+	0.186641i	$0.0597601\pi$
$439$	7.82113i	0.373282i	−0.982428	+	0.186641i	$0.940240\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 29.9854i	− 1.42465i	−0.701850	−	0.712325i	$-0.747642\pi$
$443$	− 29.9854i	− 1.42465i	0.701850	−	0.712325i	$-0.252358\pi$
$444$	0	0
$445$	23.8195i	1.12915i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	42.0892i	1.98631i	0.116797	+	0.993156i	$0.462737\pi$
$449$	42.0892i	1.98631i	−0.116797	+	0.993156i	$0.537263\pi$
$450$	0	0
$451$	13.5745	0.639198
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.8659	−0.603163
$456$	0	0
$457$	−12.1655	−0.569079	−0.284540	−	0.958664i	$-0.591841\pi$
$457$	−12.1655	−0.569079	−0.284540	+	0.958664i	$0.591841\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.0054	1.35091	0.675457	−	0.737399i	$-0.263946\pi$
$461$	29.0054	1.35091	0.675457	+	0.737399i	$0.263946\pi$
$462$	0	0
$463$	5.21317i	0.242277i	0.992636	+	0.121138i	$0.0386545\pi$
$463$	5.21317i	0.242277i	−0.992636	+	0.121138i	$0.961346\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.3061i	0.569460i	0.958608	+	0.284730i	$0.0919039\pi$
$467$	12.3061i	0.569460i	−0.958608	+	0.284730i	$0.908096\pi$
$468$	0	0
$469$	− 9.21499i	− 0.425509i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.8452i	0.774545i
$474$	0	0
$475$	−38.1823	−1.75192
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.9609	−0.912038	−0.456019	−	0.889970i	$-0.650725\pi$
$479$	−19.9609	−0.912038	−0.456019	+	0.889970i	$0.650725\pi$
$480$	0	0
$481$	−34.9622	−1.59414
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−25.9862	−1.17997
$486$	0	0
$487$	− 5.61420i	− 0.254404i	−0.991877	−	0.127202i	$-0.959400\pi$
$487$	− 5.61420i	− 0.254404i	0.991877	−	0.127202i	$-0.0405996\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 9.07131i	− 0.409382i	−0.978827	−	0.204691i	$-0.934381\pi$
$491$	− 9.07131i	− 0.409382i	0.978827	−	0.204691i	$-0.0656190\pi$
$492$	0	0
$493$	1.78166i	0.0802418i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.02460i	0.404809i
$498$	0	0
$499$	36.4346	1.63103	0.815517	−	0.578732i	$-0.196452\pi$
$499$	36.4346	1.63103	0.815517	+	0.578732i	$0.196452\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.10995	−0.183253	−0.0916267	−	0.995793i	$-0.529207\pi$
$503$	−4.10995	−0.183253	−0.0916267	+	0.995793i	$0.529207\pi$
$504$	0	0
$505$	−7.74688	−0.344732
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.5175	0.776450	0.388225	−	0.921565i	$-0.373088\pi$
$509$	17.5175	0.776450	0.388225	+	0.921565i	$0.373088\pi$
$510$	0	0
$511$	− 0.416491i	− 0.0184245i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 11.4605i	− 0.505008i
$516$	0	0
$517$	− 0.611291i	− 0.0268845i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 40.7567i	− 1.78558i	−0.450472	−	0.892791i	$-0.648744\pi$
$521$	− 40.7567i	− 1.78558i	0.450472	−	0.892791i	$-0.351256\pi$
$522$	0	0
$523$	−7.03412	−0.307581	−0.153790	−	0.988103i	$-0.549148\pi$
$523$	−7.03412	−0.307581	−0.153790	+	0.988103i	$0.549148\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.50225	−0.239682
$528$	0	0
$529$	12.6683	0.550794
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−22.3615	−0.968586
$534$	0	0
$535$	17.6718i	0.764020i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.86730i	0.0804302i
$540$	0	0
$541$	− 19.6764i	− 0.845953i	−0.906141	−	0.422976i	$-0.860985\pi$
$541$	− 19.6764i	− 0.845953i	0.906141	−	0.422976i	$-0.139015\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 58.3140i	− 2.49790i
$546$	0	0
$547$	−17.8078	−0.761408	−0.380704	−	0.924697i	$-0.624318\pi$
$547$	−17.8078	−0.761408	−0.380704	+	0.924697i	$0.624318\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.7840	0.459413
$552$	0	0
$553$	6.00459	0.255341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.99646	−0.254078	−0.127039	−	0.991898i	$-0.540547\pi$
$557$	−5.99646	−0.254078	−0.127039	+	0.991898i	$0.540547\pi$
$558$	0	0
$559$	− 27.7495i	− 1.17368i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 20.4848i	− 0.863333i	−0.902033	−	0.431667i	$-0.857926\pi$
$563$	− 20.4848i	− 0.863333i	0.902033	−	0.431667i	$-0.142074\pi$
$564$	0	0
$565$	31.3466i	1.31876i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.1889i	1.60096i	0.599359	+	0.800480i	$0.295422\pi$
$569$	38.1889i	1.60096i	−0.599359	+	0.800480i	$0.704578\pi$
$570$	0	0
$571$	−11.3551	−0.475198	−0.237599	−	0.971363i	$-0.576360\pi$
$571$	−11.3551	−0.475198	−0.237599	+	0.971363i	$0.576360\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−74.6200	−3.11187
$576$	0	0
$577$	−20.7869	−0.865371	−0.432685	−	0.901545i	$-0.642434\pi$
$577$	−20.7869	−0.865371	−0.432685	+	0.901545i	$0.642434\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.62368	0.108849
$582$	0	0
$583$	15.5535i	0.644159i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.3849i	1.00647i	0.864148	+	0.503237i	$0.167858\pi$
$587$	24.3849i	1.00647i	−0.864148	+	0.503237i	$0.832142\pi$
$588$	0	0
$589$	33.3039i	1.37226i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.5053i	1.33483i	0.744684	+	0.667417i	$0.232600\pi$
$593$	32.5053i	1.33483i	−0.744684	+	0.667417i	$0.767400\pi$
$594$	0	0
$595$	2.11175	0.0865731
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.4800	0.428200	0.214100	−	0.976812i	$-0.431318\pi$
$599$	10.4800	0.428200	0.214100	+	0.976812i	$0.431318\pi$
$600$	0	0
$601$	−22.4946	−0.917575	−0.458787	−	0.888546i	$-0.651716\pi$
$601$	−22.4946	−0.917575	−0.458787	+	0.888546i	$0.651716\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−31.4249	−1.27761
$606$	0	0
$607$	− 32.3672i	− 1.31374i	−0.754002	−	0.656872i	$-0.771879\pi$
$607$	− 32.3672i	− 1.31374i	0.754002	−	0.656872i	$-0.228121\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.00699i	0.0407385i
$612$	0	0
$613$	− 34.8714i	− 1.40844i	−0.709981	−	0.704220i	$-0.751297\pi$
$613$	− 34.8714i	− 1.40844i	0.709981	−	0.704220i	$-0.248703\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.47324i	− 0.0995688i	−0.998760	−	0.0497844i	$-0.984147\pi$
$617$	− 2.47324i	− 0.0995688i	0.998760	−	0.0497844i	$-0.0158534\pi$
$618$	0	0
$619$	15.2375	0.612447	0.306223	−	0.951960i	$-0.400935\pi$
$619$	15.2375	0.612447	0.306223	+	0.951960i	$0.400935\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.69486	−0.228160
$624$	0	0
$625$	68.6375	2.74550
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.73852	0.228810
$630$	0	0
$631$	23.2064i	0.923832i	0.886924	+	0.461916i	$0.152838\pi$
$631$	23.2064i	0.923832i	−0.886924	+	0.461916i	$0.847162\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 75.6800i	− 3.00327i
$636$	0	0
$637$	− 3.07604i	− 0.121877i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	29.5988i	1.16908i	0.811364	+	0.584541i	$0.198725\pi$
$641$	29.5988i	1.16908i	−0.811364	+	0.584541i	$0.801275\pi$
$642$	0	0
$643$	−28.8060	−1.13600	−0.567999	−	0.823029i	$-0.692282\pi$
$643$	−28.8060	−1.13600	−0.567999	+	0.823029i	$0.692282\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.3622	−1.07572	−0.537860	−	0.843034i	$-0.680767\pi$
$647$	−27.3622	−1.07572	−0.537860	+	0.843034i	$0.680767\pi$
$648$	0	0
$649$	−11.1666	−0.438327
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.9292	1.56255	0.781275	−	0.624187i	$-0.214570\pi$
$653$	39.9292	1.56255	0.781275	+	0.624187i	$0.214570\pi$
$654$	0	0
$655$	39.0082i	1.52418i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.9809i	0.583572i	0.956484	+	0.291786i	$0.0942495\pi$
$659$	14.9809i	0.583572i	−0.956484	+	0.291786i	$0.905751\pi$
$660$	0	0
$661$	33.0963i	1.28730i	0.765322	+	0.643648i	$0.222580\pi$
$661$	33.0963i	1.28730i	−0.765322	+	0.643648i	$0.777420\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 12.7819i	− 0.495662i
$666$	0	0
$667$	21.0753	0.816037
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.7945	−0.957183
$672$	0	0
$673$	−12.1189	−0.467150	−0.233575	−	0.972339i	$-0.575043\pi$
$673$	−12.1189	−0.467150	−0.233575	+	0.972339i	$0.575043\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.7231	−1.06549	−0.532743	−	0.846277i	$-0.678839\pi$
$677$	−27.7231	−1.06549	−0.532743	+	0.846277i	$0.678839\pi$
$678$	0	0
$679$	− 6.21289i	− 0.238429i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 9.79813i	− 0.374915i	−0.982273	−	0.187457i	$-0.939975\pi$
$683$	− 9.79813i	− 0.374915i	0.982273	−	0.187457i	$-0.0600247\pi$
$684$	0	0
$685$	− 27.3155i	− 1.04367i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 25.6215i	− 0.976103i
$690$	0	0
$691$	37.4765	1.42567	0.712837	−	0.701330i	$-0.247410\pi$
$691$	37.4765	1.42567	0.712837	+	0.701330i	$0.247410\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−42.4908	−1.61177
$696$	0	0
$697$	3.67031	0.139023
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.6525	0.855575	0.427787	−	0.903879i	$-0.359293\pi$
$701$	22.6525	0.855575	0.427787	+	0.903879i	$0.359293\pi$
$702$	0	0
$703$	− 34.7340i	− 1.31002i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.85216i	− 0.0696575i
$708$	0	0
$709$	− 1.58535i	− 0.0595393i	−0.999557	−	0.0297696i	$-0.990523\pi$
$709$	− 1.58535i	− 0.0595393i	0.999557	−	0.0297696i	$-0.00947737\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	65.0861i	2.43749i
$714$	0	0
$715$	−24.0245	−0.898465
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.7228	−0.660948	−0.330474	−	0.943815i	$-0.607209\pi$
$719$	−17.7228	−0.660948	−0.330474	+	0.943815i	$0.607209\pi$
$720$	0	0
$721$	2.74001	0.102043
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−44.0906	−1.63749
$726$	0	0
$727$	− 3.31812i	− 0.123062i	−0.998105	−	0.0615312i	$-0.980402\pi$
$727$	− 3.31812i	− 0.123062i	0.998105	−	0.0615312i	$-0.0195984\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.55467i	0.168460i
$732$	0	0
$733$	− 18.8547i	− 0.696414i	−0.937418	−	0.348207i	$-0.886791\pi$
$733$	− 18.8547i	− 0.696414i	0.937418	−	0.348207i	$-0.113209\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 17.2071i	− 0.633832i
$738$	0	0
$739$	33.3560	1.22702	0.613510	−	0.789687i	$-0.289757\pi$
$739$	33.3560	1.22702	0.613510	+	0.789687i	$0.289757\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.0826	0.993565	0.496783	−	0.867875i	$-0.334515\pi$
$743$	27.0826	0.993565	0.496783	+	0.867875i	$0.334515\pi$
$744$	0	0
$745$	40.6767	1.49028
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.22506	−0.154380
$750$	0	0
$751$	44.3284i	1.61757i	0.588107	+	0.808783i	$0.299874\pi$
$751$	44.3284i	1.61757i	−0.588107	+	0.808783i	$0.700126\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	52.1421i	1.89765i
$756$	0	0
$757$	12.3093i	0.447391i	0.974659	+	0.223695i	$0.0718121\pi$
$757$	12.3093i	0.447391i	−0.974659	+	0.223695i	$0.928188\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.85054i	− 0.0670820i	−0.999437	−	0.0335410i	$-0.989322\pi$
$761$	− 1.85054i	− 0.0670820i	0.999437	−	0.0335410i	$-0.0106784\pi$
$762$	0	0
$763$	13.9420	0.504733
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.3949	0.664203
$768$	0	0
$769$	−31.2463	−1.12677	−0.563386	−	0.826194i	$-0.690502\pi$
$769$	−31.2463	−1.12677	−0.563386	+	0.826194i	$0.690502\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.4521	1.59883	0.799415	−	0.600779i	$-0.205143\pi$
$773$	44.4521	1.59883	0.799415	+	0.600779i	$0.205143\pi$
$774$	0	0
$775$	− 136.164i	− 4.89115i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 22.2156i	− 0.795956i
$780$	0	0
$781$	16.8516i	0.602998i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 47.5257i	− 1.69627i
$786$	0	0
$787$	−28.0995	−1.00164	−0.500820	−	0.865551i	$-0.666968\pi$
$787$	−28.0995	−1.00164	−0.500820	+	0.865551i	$0.666968\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.49447	−0.266473
$792$	0	0
$793$	40.8446	1.45043
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.9492	−1.52134	−0.760670	−	0.649139i	$-0.775129\pi$
$797$	−42.9492	−1.52134	−0.760670	+	0.649139i	$0.775129\pi$
$798$	0	0
$799$	− 0.165282i	− 0.00584727i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 0.777712i	− 0.0274449i
$804$	0	0
$805$	− 24.9799i	− 0.880425i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8.54301i	− 0.300356i	−0.988659	−	0.150178i	$-0.952015\pi$
$809$	− 8.54301i	− 0.300356i	0.988659	−	0.150178i	$-0.0479847\pi$
$810$	0	0
$811$	−34.6880	−1.21806	−0.609030	−	0.793147i	$-0.708441\pi$
$811$	−34.6880	−1.21806	−0.609030	+	0.793147i	$0.708441\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.0344	−0.526632
$816$	0	0
$817$	27.5684	0.964497
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.46951	0.295588	0.147794	−	0.989018i	$-0.452783\pi$
$821$	8.46951	0.295588	0.147794	+	0.989018i	$0.452783\pi$
$822$	0	0
$823$	− 44.5109i	− 1.55155i	−0.631008	−	0.775777i	$-0.717358\pi$
$823$	− 44.5109i	− 1.55155i	0.631008	−	0.775777i	$-0.282642\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.3895i	1.26539i	0.774403	+	0.632693i	$0.218051\pi$
$827$	36.3895i	1.26539i	−0.774403	+	0.632693i	$0.781949\pi$
$828$	0	0
$829$	− 39.2607i	− 1.36358i	−0.731548	−	0.681790i	$-0.761202\pi$
$829$	− 39.2607i	− 1.36358i	0.731548	−	0.681790i	$-0.238798\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.504885i	0.0174932i
$834$	0	0
$835$	−85.2425	−2.94994
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15.1101	−0.521659	−0.260830	−	0.965385i	$-0.583996\pi$
$839$	−15.1101	−0.521659	−0.260830	+	0.965385i	$0.583996\pi$
$840$	0	0
$841$	−16.5473	−0.570596
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14.7981	−0.509072
$846$	0	0
$847$	− 7.51320i	− 0.258157i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 67.8810i	− 2.32693i
$852$	0	0
$853$	− 23.4153i	− 0.801724i	−0.916138	−	0.400862i	$-0.868711\pi$
$853$	− 23.4153i	− 0.801724i	0.916138	−	0.400862i	$-0.131289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 20.7800i	− 0.709833i	−0.934898	−	0.354916i	$-0.884509\pi$
$857$	− 20.7800i	− 0.709833i	0.934898	−	0.354916i	$-0.115491\pi$
$858$	0	0
$859$	−12.0088	−0.409735	−0.204868	−	0.978790i	$-0.565676\pi$
$859$	−12.0088	−0.409735	−0.204868	+	0.978790i	$0.565676\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.5254	−0.392328	−0.196164	−	0.980571i	$-0.562849\pi$
$863$	−11.5254	−0.392328	−0.196164	+	0.980571i	$0.562849\pi$
$864$	0	0
$865$	−18.8118	−0.639622
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.2123	0.380353
$870$	0	0
$871$	28.3456i	0.960455i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	31.3462i	1.05969i
$876$	0	0
$877$	− 7.25986i	− 0.245148i	−0.992459	−	0.122574i	$-0.960885\pi$
$877$	− 7.25986i	− 0.245148i	0.992459	−	0.122574i	$-0.0391149\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 10.7411i	− 0.361876i	−0.983495	−	0.180938i	$-0.942087\pi$
$881$	− 10.7411i	− 0.361876i	0.983495	−	0.180938i	$-0.0579133\pi$
$882$	0	0
$883$	−9.09009	−0.305906	−0.152953	−	0.988233i	$-0.548878\pi$
$883$	−9.09009	−0.305906	−0.152953	+	0.988233i	$0.548878\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.0988	0.977041	0.488521	−	0.872552i	$-0.337537\pi$
$887$	29.0988	0.977041	0.488521	+	0.872552i	$0.337537\pi$
$888$	0	0
$889$	18.0939	0.606850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.00042	−0.0334778
$894$	0	0
$895$	− 11.0183i	− 0.368301i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	38.4574i	1.28262i
$900$	0	0
$901$	4.20539i	0.140102i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	43.3938i	1.44246i
$906$	0	0
$907$	−29.9962	−0.996008	−0.498004	−	0.867175i	$-0.665933\pi$
$907$	−29.9962	−0.996008	−0.498004	+	0.867175i	$0.665933\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.3253	1.43543	0.717716	−	0.696336i	$-0.245188\pi$
$911$	43.3253	1.43543	0.717716	+	0.696336i	$0.245188\pi$
$912$	0	0
$913$	4.89919	0.162140
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.32625	−0.307980
$918$	0	0
$919$	44.6079i	1.47148i	0.677264	+	0.735740i	$0.263165\pi$
$919$	44.6079i	1.47148i	−0.677264	+	0.735740i	$0.736835\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 27.7600i	− 0.913732i
$924$	0	0
$925$	142.011i	4.66929i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.73251i	0.253695i	0.991922	+	0.126848i	$0.0404860\pi$
$929$	7.73251i	0.253695i	−0.991922	+	0.126848i	$0.959514\pi$
$930$	0	0
$931$	3.05596	0.100155
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.94326	0.128958
$936$	0	0
$937$	25.6210	0.837002	0.418501	−	0.908216i	$-0.362556\pi$
$937$	25.6210	0.837002	0.418501	+	0.908216i	$0.362556\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.8529	−1.00578	−0.502888	−	0.864352i	$-0.667729\pi$
$941$	−30.8529	−1.00578	−0.502888	+	0.864352i	$0.667729\pi$
$942$	0	0
$943$	− 43.4161i	− 1.41382i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.5682i	1.28579i	0.765953	+	0.642897i	$0.222268\pi$
$947$	39.5682i	1.28579i	−0.765953	+	0.642897i	$0.777732\pi$
$948$	0	0
$949$	1.28114i	0.0415876i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 4.36408i	− 0.141366i	−0.997499	−	0.0706832i	$-0.977482\pi$
$953$	− 4.36408i	− 0.141366i	0.997499	−	0.0706832i	$-0.0225179\pi$
$954$	0	0
$955$	−76.0015	−2.45935
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.53069	0.210887
$960$	0	0
$961$	−87.7668	−2.83119
$962$	0	0
$963$	0	0
$964$	0	0
$965$	70.1442	2.25802
$966$	0	0
$967$	51.1938i	1.64628i	0.567837	+	0.823141i	$0.307780\pi$
$967$	51.1938i	1.64628i	−0.567837	+	0.823141i	$0.692220\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	47.6773i	1.53004i	0.644008	+	0.765019i	$0.277270\pi$
$971$	47.6773i	1.53004i	−0.644008	+	0.765019i	$0.722730\pi$
$972$	0	0
$973$	− 10.1589i	− 0.325678i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.1656i	0.325225i	0.986690	+	0.162613i	$0.0519921\pi$
$977$	10.1656i	0.325225i	−0.986690	+	0.162613i	$0.948008\pi$
$978$	0	0
$979$	−10.6340	−0.339864
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.35716	0.234657	0.117328	−	0.993093i	$-0.462567\pi$
$983$	7.35716	0.234657	0.117328	+	0.993093i	$0.462567\pi$
$984$	0	0
$985$	−80.4068	−2.56197
$986$	0	0
$987$	0	0
$988$	0	0
$989$	53.8772	1.71320
$990$	0	0
$991$	18.5907i	0.590552i	0.955412	+	0.295276i	$0.0954116\pi$
$991$	18.5907i	0.590552i	−0.955412	+	0.295276i	$0.904588\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.06577i	0.224000i
$996$	0	0
$997$	4.19288i	0.132790i	0.997793	+	0.0663950i	$0.0211497\pi$
$997$	4.19288i	0.132790i	−0.997793	+	0.0663950i	$0.978850\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.d.5615.2		48
3.2	odd	2	inner	6048.2.j.d.5615.47		48
4.3	odd	2		1512.2.j.d.323.45	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.48		48
8.5	even	2		1512.2.j.d.323.3	✓	48
12.11	even	2		1512.2.j.d.323.4	yes	48
24.5	odd	2		1512.2.j.d.323.46	yes	48
24.11	even	2	inner	6048.2.j.d.5615.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.3	✓	48	8.5	even	2
1512.2.j.d.323.4	yes	48	12.11	even	2
1512.2.j.d.323.45	yes	48	4.3	odd	2
1512.2.j.d.323.46	yes	48	24.5	odd	2
6048.2.j.d.5615.1		48	24.11	even	2	inner
6048.2.j.d.5615.2		48	1.1	even	1	trivial
6048.2.j.d.5615.47		48	3.2	odd	2	inner
6048.2.j.d.5615.48		48	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.18263 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-4.18263 q^{5} -1.00000i q^{7} -1.86730i q^{11} +3.07604i q^{13} -0.504885i q^{17} -3.05596 q^{19} -5.97229 q^{23} +12.4944 q^{25} -3.52884 q^{29} -10.8980i q^{31} +4.18263i q^{35} +11.3660i q^{37} +7.26960i q^{41} -9.02120 q^{43} +0.327367 q^{47} -1.00000 q^{49} -8.32940 q^{53} +7.81021i q^{55} -5.98008i q^{59} -13.2783i q^{61} -12.8659i q^{65} +9.21499 q^{67} -9.02460 q^{71} +0.416491 q^{73} -1.86730 q^{77} +6.00459i q^{79} +2.62368i q^{83} +2.11175i q^{85} -5.69486i q^{89} +3.07604 q^{91} +12.7819 q^{95} +6.21289 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) \) 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

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