LMFDB - Embedded newform 6048.2.c.e.3025.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + x^{18} + 4x^{16} + 8x^{12} + 4x^{10} + 32x^{8} + 256x^{4} + 256x^{2} + 1024 $ x^20 + x^18 + 4x^16 + 8x^12 + 4x^10 + 32x^8 + 256x^4 + 256x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.1
Root			$-0.885915 + 1.10234i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.e.3025.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.50133i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.50133i q^{5} +1.00000 q^{7} -3.01331i q^{11} -3.90632i q^{13} +1.38839 q^{17} +4.79460i q^{19} +5.06853 q^{23} -7.25934 q^{25} -4.91070i q^{29} -1.13938 q^{31} -3.50133i q^{35} -9.45610i q^{37} +4.11364 q^{41} -1.51868i q^{43} +10.7641 q^{47} +1.00000 q^{49} -0.431457i q^{53} -10.5506 q^{55} +7.40936i q^{59} -12.9520i q^{61} -13.6773 q^{65} +3.36185i q^{67} +6.26892 q^{71} +10.0859 q^{73} -3.01331i q^{77} -12.9346 q^{79} -17.4139i q^{83} -4.86120i q^{85} -0.818403 q^{89} -3.90632i q^{91} +16.7875 q^{95} -11.4727 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{7}+O(q^{10}) $ 20 * q + 20 * q^7	$ 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) $ 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * q^97

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
$5$	− 3.50133i	− 1.56584i	−0.622120	−	0.782922i	$-0.713728\pi$
$5$	− 3.50133i	− 1.56584i	0.622120	−	0.782922i	$-0.286272\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.01331i	− 0.908547i	−0.890862	−	0.454273i	$-0.849899\pi$
$11$	− 3.01331i	− 0.908547i	0.890862	−	0.454273i	$-0.150101\pi$
$12$	0	0
$13$	− 3.90632i	− 1.08342i	−0.840566	−	0.541709i	$-0.817777\pi$
$13$	− 3.90632i	− 1.08342i	0.840566	−	0.541709i	$-0.182223\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.38839	0.336733	0.168367	−	0.985724i	$-0.446151\pi$
$17$	1.38839	0.336733	0.168367	+	0.985724i	$0.446151\pi$
$18$	0	0
$19$	4.79460i	1.09996i	0.835179	+	0.549978i	$0.185364\pi$
$19$	4.79460i	1.09996i	−0.835179	+	0.549978i	$0.814636\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.06853	1.05686	0.528431	−	0.848976i	$-0.322781\pi$
$23$	5.06853	1.05686	0.528431	+	0.848976i	$0.322781\pi$
$24$	0	0
$25$	−7.25934	−1.45187
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.91070i	− 0.911893i	−0.890007	−	0.455947i	$-0.849301\pi$
$29$	− 4.91070i	− 0.911893i	0.890007	−	0.455947i	$-0.150699\pi$
$30$	0	0
$31$	−1.13938	−0.204639	−0.102320	−	0.994752i	$-0.532626\pi$
$31$	−1.13938	−0.204639	−0.102320	+	0.994752i	$0.532626\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.50133i	− 0.591833i
$36$	0	0
$37$	− 9.45610i	− 1.55457i	−0.629146	−	0.777287i	$-0.716595\pi$
$37$	− 9.45610i	− 1.55457i	0.629146	−	0.777287i	$-0.283405\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.11364	0.642443	0.321222	−	0.947004i	$-0.395907\pi$
$41$	4.11364	0.642443	0.321222	+	0.947004i	$0.395907\pi$
$42$	0	0
$43$	− 1.51868i	− 0.231597i	−0.993273	−	0.115799i	$-0.963057\pi$
$43$	− 1.51868i	− 0.231597i	0.993273	−	0.115799i	$-0.0369427\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.7641	1.57010	0.785051	−	0.619431i	$-0.212637\pi$
$47$	10.7641	1.57010	0.785051	+	0.619431i	$0.212637\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.431457i	− 0.0592651i	−0.999561	−	0.0296326i	$-0.990566\pi$
$53$	− 0.431457i	− 0.0592651i	0.999561	−	0.0296326i	$-0.00943372\pi$
$54$	0	0
$55$	−10.5506	−1.42264
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.40936i	0.964617i	0.876002	+	0.482308i	$0.160202\pi$
$59$	7.40936i	0.964617i	−0.876002	+	0.482308i	$0.839798\pi$
$60$	0	0
$61$	− 12.9520i	− 1.65834i	−0.559000	−	0.829168i	$-0.688815\pi$
$61$	− 12.9520i	− 1.65834i	0.559000	−	0.829168i	$-0.311185\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.6773	−1.69646
$66$	0	0
$67$	3.36185i	0.410715i	0.978687	+	0.205357i	$0.0658357\pi$
$67$	3.36185i	0.410715i	−0.978687	+	0.205357i	$0.934164\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.26892	0.743984	0.371992	−	0.928236i	$-0.378675\pi$
$71$	6.26892	0.743984	0.371992	+	0.928236i	$0.378675\pi$
$72$	0	0
$73$	10.0859	1.18046	0.590230	−	0.807235i	$-0.299037\pi$
$73$	10.0859	1.18046	0.590230	+	0.807235i	$0.299037\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.01331i	− 0.343398i
$78$	0	0
$79$	−12.9346	−1.45526	−0.727631	−	0.685969i	$-0.759378\pi$
$79$	−12.9346	−1.45526	−0.727631	+	0.685969i	$0.759378\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 17.4139i	− 1.91142i	−0.294307	−	0.955711i	$-0.595089\pi$
$83$	− 17.4139i	− 1.91142i	0.294307	−	0.955711i	$-0.404911\pi$
$84$	0	0
$85$	− 4.86120i	− 0.527272i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.818403	−0.0867506	−0.0433753	−	0.999059i	$-0.513811\pi$
$89$	−0.818403	−0.0867506	−0.0433753	+	0.999059i	$0.513811\pi$
$90$	0	0
$91$	− 3.90632i	− 0.409494i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.7875	1.72236
$96$	0	0
$97$	−11.4727	−1.16488	−0.582441	−	0.812873i	$-0.697902\pi$
$97$	−11.4727	−1.16488	−0.582441	+	0.812873i	$0.697902\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.38541i	− 0.535868i	−0.963437	−	0.267934i	$-0.913659\pi$
$101$	− 5.38541i	− 0.535868i	0.963437	−	0.267934i	$-0.0863410\pi$
$102$	0	0
$103$	4.82652	0.475571	0.237785	−	0.971318i	$-0.423579\pi$
$103$	4.82652	0.475571	0.237785	+	0.971318i	$0.423579\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.92215i	0.475842i	0.971284	+	0.237921i	$0.0764660\pi$
$107$	4.92215i	0.475842i	−0.971284	+	0.237921i	$0.923534\pi$
$108$	0	0
$109$	20.1342i	1.92851i	0.264974	+	0.964255i	$0.414637\pi$
$109$	20.1342i	1.92851i	−0.264974	+	0.964255i	$0.585363\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.2438	−1.24587	−0.622937	−	0.782272i	$-0.714061\pi$
$113$	−13.2438	−1.24587	−0.622937	+	0.782272i	$0.714061\pi$
$114$	0	0
$115$	− 17.7466i	− 1.65488i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.38839	0.127273
$120$	0	0
$121$	1.91997	0.174543
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.91070i	0.707554i
$126$	0	0
$127$	−10.2252	−0.907343	−0.453671	−	0.891169i	$-0.649886\pi$
$127$	−10.2252	−0.907343	−0.453671	+	0.891169i	$0.649886\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.54726i	− 0.135185i	−0.997713	−	0.0675924i	$-0.978468\pi$
$131$	− 1.54726i	− 0.135185i	0.997713	−	0.0675924i	$-0.0215317\pi$
$132$	0	0
$133$	4.79460i	0.415744i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.46379	0.637674	0.318837	−	0.947809i	$-0.396708\pi$
$137$	7.46379	0.637674	0.318837	+	0.947809i	$0.396708\pi$
$138$	0	0
$139$	16.6383i	1.41125i	0.708588	+	0.705623i	$0.249333\pi$
$139$	16.6383i	1.41125i	−0.708588	+	0.705623i	$0.750667\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−11.7709	−0.984336
$144$	0	0
$145$	−17.1940	−1.42788
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.3846i	1.75190i	0.482405	+	0.875948i	$0.339763\pi$
$149$	21.3846i	1.75190i	−0.482405	+	0.875948i	$0.660237\pi$
$150$	0	0
$151$	−1.21341	−0.0987459	−0.0493730	−	0.998780i	$-0.515722\pi$
$151$	−1.21341	−0.0987459	−0.0493730	+	0.998780i	$0.515722\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.98936i	0.320433i
$156$	0	0
$157$	8.15645i	0.650955i	0.945550	+	0.325478i	$0.105525\pi$
$157$	8.15645i	0.650955i	−0.945550	+	0.325478i	$0.894475\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.06853	0.399456
$162$	0	0
$163$	− 22.8278i	− 1.78801i	−0.448055	−	0.894006i	$-0.647883\pi$
$163$	− 22.8278i	− 1.78801i	0.448055	−	0.894006i	$-0.352117\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.3575	−1.34316	−0.671581	−	0.740931i	$-0.734384\pi$
$167$	−17.3575	−1.34316	−0.671581	+	0.740931i	$0.734384\pi$
$168$	0	0
$169$	−2.25934	−0.173795
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.5784i	1.79263i	0.443415	+	0.896316i	$0.353767\pi$
$173$	23.5784i	1.79263i	−0.443415	+	0.896316i	$0.646233\pi$
$174$	0	0
$175$	−7.25934	−0.548754
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 12.0965i	− 0.904134i	−0.891984	−	0.452067i	$-0.850687\pi$
$179$	− 12.0965i	− 0.904134i	0.891984	−	0.452067i	$-0.149313\pi$
$180$	0	0
$181$	− 3.70662i	− 0.275511i	−0.990466	−	0.137755i	$-0.956011\pi$
$181$	− 3.70662i	− 0.275511i	0.990466	−	0.137755i	$-0.0439888\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−33.1090	−2.43422
$186$	0	0
$187$	− 4.18364i	− 0.305938i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.2770	−0.815977	−0.407988	−	0.912987i	$-0.633770\pi$
$191$	−11.2770	−0.815977	−0.407988	+	0.912987i	$0.633770\pi$
$192$	0	0
$193$	−3.15188	−0.226877	−0.113439	−	0.993545i	$-0.536187\pi$
$193$	−3.15188	−0.226877	−0.113439	+	0.993545i	$0.536187\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.4819i	− 0.818052i	−0.912523	−	0.409026i	$-0.865869\pi$
$197$	− 11.4819i	− 0.818052i	0.912523	−	0.409026i	$-0.134131\pi$
$198$	0	0
$199$	3.38405	0.239889	0.119944	−	0.992781i	$-0.461728\pi$
$199$	3.38405	0.239889	0.119944	+	0.992781i	$0.461728\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.91070i	− 0.344663i
$204$	0	0
$205$	− 14.4032i	− 1.00597i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	14.4476	0.999362
$210$	0	0
$211$	17.0774i	1.17565i	0.808987	+	0.587827i	$0.200016\pi$
$211$	17.0774i	1.17565i	−0.808987	+	0.587827i	$0.799984\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.31742	−0.362645
$216$	0	0
$217$	−1.13938	−0.0773463
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5.42348i	− 0.364823i
$222$	0	0
$223$	−24.1161	−1.61494	−0.807468	−	0.589912i	$-0.799163\pi$
$223$	−24.1161	−1.61494	−0.807468	+	0.589912i	$0.799163\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.60050i	0.504463i	0.967667	+	0.252231i	$0.0811644\pi$
$227$	7.60050i	0.504463i	−0.967667	+	0.252231i	$0.918836\pi$
$228$	0	0
$229$	− 24.0017i	− 1.58608i	−0.609171	−	0.793039i	$-0.708498\pi$
$229$	− 24.0017i	− 1.58608i	0.609171	−	0.793039i	$-0.291502\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.6276	−0.696237	−0.348118	−	0.937451i	$-0.613179\pi$
$233$	−10.6276	−0.696237	−0.348118	+	0.937451i	$0.613179\pi$
$234$	0	0
$235$	− 37.6886i	− 2.45853i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.0645	1.81534	0.907671	−	0.419682i	$-0.137858\pi$
$239$	28.0645	1.81534	0.907671	+	0.419682i	$0.137858\pi$
$240$	0	0
$241$	−20.7612	−1.33734	−0.668672	−	0.743558i	$-0.733137\pi$
$241$	−20.7612	−1.33734	−0.668672	+	0.743558i	$0.733137\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.50133i	− 0.223692i
$246$	0	0
$247$	18.7292	1.19171
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.4352i	1.35298i	0.736454	+	0.676488i	$0.236499\pi$
$251$	21.4352i	1.35298i	−0.736454	+	0.676488i	$0.763501\pi$
$252$	0	0
$253$	− 15.2730i	− 0.960208i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.6807	−0.853380	−0.426690	−	0.904398i	$-0.640320\pi$
$257$	−13.6807	−0.853380	−0.426690	+	0.904398i	$0.640320\pi$
$258$	0	0
$259$	− 9.45610i	− 0.587574i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.8063	1.15965	0.579823	−	0.814743i	$-0.303122\pi$
$263$	18.8063	1.15965	0.579823	+	0.814743i	$0.303122\pi$
$264$	0	0
$265$	−1.51067	−0.0927999
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.1654i	1.65630i	0.560504	+	0.828152i	$0.310607\pi$
$269$	27.1654i	1.65630i	−0.560504	+	0.828152i	$0.689393\pi$
$270$	0	0
$271$	−0.618132	−0.0375488	−0.0187744	−	0.999824i	$-0.505976\pi$
$271$	−0.618132	−0.0375488	−0.0187744	+	0.999824i	$0.505976\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.8746i	1.31909i
$276$	0	0
$277$	− 16.4479i	− 0.988260i	−0.869388	−	0.494130i	$-0.835487\pi$
$277$	− 16.4479i	− 0.988260i	0.869388	−	0.494130i	$-0.164513\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.6482	1.58970	0.794849	−	0.606807i	$-0.207550\pi$
$281$	26.6482	1.58970	0.794849	+	0.606807i	$0.207550\pi$
$282$	0	0
$283$	− 5.96019i	− 0.354297i	−0.984184	−	0.177148i	$-0.943313\pi$
$283$	− 5.96019i	− 0.354297i	0.984184	−	0.177148i	$-0.0566872\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.11364	0.242821
$288$	0	0
$289$	−15.0724	−0.886611
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.52262i	− 0.0889522i	−0.999010	−	0.0444761i	$-0.985838\pi$
$293$	− 1.52262i	− 0.0889522i	0.999010	−	0.0444761i	$-0.0141619\pi$
$294$	0	0
$295$	25.9426	1.51044
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 19.7993i	− 1.14502i
$300$	0	0
$301$	− 1.51868i	− 0.0875355i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−45.3493	−2.59669
$306$	0	0
$307$	− 5.29191i	− 0.302025i	−0.988532	−	0.151013i	$-0.951747\pi$
$307$	− 5.29191i	− 0.302025i	0.988532	−	0.151013i	$-0.0482534\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.95423	−0.394338	−0.197169	−	0.980370i	$-0.563175\pi$
$311$	−6.95423	−0.394338	−0.197169	+	0.980370i	$0.563175\pi$
$312$	0	0
$313$	−11.7834	−0.666039	−0.333020	−	0.942920i	$-0.608068\pi$
$313$	−11.7834	−0.666039	−0.333020	+	0.942920i	$0.608068\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.74196i	0.0978384i	0.998803	+	0.0489192i	$0.0155777\pi$
$317$	1.74196i	0.0978384i	−0.998803	+	0.0489192i	$0.984422\pi$
$318$	0	0
$319$	−14.7974	−0.828498
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.65676i	0.370392i
$324$	0	0
$325$	28.3573i	1.57298i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.7641	0.593443
$330$	0	0
$331$	22.6068i	1.24258i	0.783579	+	0.621292i	$0.213392\pi$
$331$	22.6068i	1.24258i	−0.783579	+	0.621292i	$0.786608\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.7709	0.643116
$336$	0	0
$337$	24.7468	1.34804	0.674021	−	0.738712i	$-0.264566\pi$
$337$	24.7468	1.34804	0.674021	+	0.738712i	$0.264566\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.43331i	0.185924i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.17968i	− 0.170694i	−0.996351	−	0.0853471i	$-0.972800\pi$
$347$	− 3.17968i	− 0.170694i	0.996351	−	0.0853471i	$-0.0271999\pi$
$348$	0	0
$349$	− 12.0444i	− 0.644722i	−0.946617	−	0.322361i	$-0.895523\pi$
$349$	− 12.0444i	− 0.644722i	0.946617	−	0.322361i	$-0.104477\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.3700	1.50998	0.754991	−	0.655736i	$-0.227641\pi$
$353$	28.3700	1.50998	0.754991	+	0.655736i	$0.227641\pi$
$354$	0	0
$355$	− 21.9496i	− 1.16496i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−26.2701	−1.38648	−0.693241	−	0.720706i	$-0.743818\pi$
$359$	−26.2701	−1.38648	−0.693241	+	0.720706i	$0.743818\pi$
$360$	0	0
$361$	−3.98817	−0.209904
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 35.3139i	− 1.84842i
$366$	0	0
$367$	−15.6261	−0.815678	−0.407839	−	0.913054i	$-0.633717\pi$
$367$	−15.6261	−0.815678	−0.407839	+	0.913054i	$0.633717\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 0.431457i	− 0.0224001i
$372$	0	0
$373$	3.63905i	0.188423i	0.995552	+	0.0942113i	$0.0300329\pi$
$373$	3.63905i	0.188423i	−0.995552	+	0.0942113i	$0.969967\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.1827	−0.987962
$378$	0	0
$379$	4.66150i	0.239445i	0.992807	+	0.119723i	$0.0382005\pi$
$379$	4.66150i	0.239445i	−0.992807	+	0.119723i	$0.961799\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.9040	−0.608268	−0.304134	−	0.952629i	$-0.598367\pi$
$383$	−11.9040	−0.608268	−0.304134	+	0.952629i	$0.598367\pi$
$384$	0	0
$385$	−10.5506	−0.537708
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 35.6321i	− 1.80662i	−0.428987	−	0.903311i	$-0.641129\pi$
$389$	− 35.6321i	− 1.80662i	0.428987	−	0.903311i	$-0.358871\pi$
$390$	0	0
$391$	7.03708	0.355880
$392$	0	0
$393$	0	0
$394$	0	0
$395$	45.2885i	2.27871i
$396$	0	0
$397$	10.9738i	0.550760i	0.961335	+	0.275380i	$0.0888037\pi$
$397$	10.9738i	0.550760i	−0.961335	+	0.275380i	$0.911196\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.65382	−0.332276	−0.166138	−	0.986103i	$-0.553130\pi$
$401$	−6.65382	−0.332276	−0.166138	+	0.986103i	$0.553130\pi$
$402$	0	0
$403$	4.45079i	0.221710i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.4942	−1.41240
$408$	0	0
$409$	−33.8304	−1.67281	−0.836404	−	0.548114i	$-0.815346\pi$
$409$	−33.8304	−1.67281	−0.836404	+	0.548114i	$0.815346\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.40936i	0.364591i
$414$	0	0
$415$	−60.9718	−2.99299
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 10.6763i	− 0.521571i	−0.965397	−	0.260786i	$-0.916018\pi$
$419$	− 10.6763i	− 0.521571i	0.965397	−	0.260786i	$-0.0839816\pi$
$420$	0	0
$421$	15.4063i	0.750855i	0.926852	+	0.375427i	$0.122504\pi$
$421$	15.4063i	0.750855i	−0.926852	+	0.375427i	$0.877496\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.0788	−0.488892
$426$	0	0
$427$	− 12.9520i	− 0.626792i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.7179	1.23879	0.619393	−	0.785081i	$-0.287379\pi$
$431$	25.7179	1.23879	0.619393	+	0.785081i	$0.287379\pi$
$432$	0	0
$433$	−7.26219	−0.348998	−0.174499	−	0.984657i	$-0.555831\pi$
$433$	−7.26219	−0.348998	−0.174499	+	0.984657i	$0.555831\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.3016i	1.16250i
$438$	0	0
$439$	−28.7252	−1.37098	−0.685488	−	0.728084i	$-0.740411\pi$
$439$	−28.7252	−1.37098	−0.685488	+	0.728084i	$0.740411\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.70732i	− 0.271163i	−0.990766	−	0.135582i	$-0.956710\pi$
$443$	− 5.70732i	− 0.271163i	0.990766	−	0.135582i	$-0.0432903\pi$
$444$	0	0
$445$	2.86550i	0.135838i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.9514	1.60227	0.801134	−	0.598485i	$-0.204230\pi$
$449$	33.9514	1.60227	0.801134	+	0.598485i	$0.204230\pi$
$450$	0	0
$451$	− 12.3957i	− 0.583690i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13.6773	−0.641203
$456$	0	0
$457$	35.4661	1.65903	0.829517	−	0.558482i	$-0.188616\pi$
$457$	35.4661	1.65903	0.829517	+	0.558482i	$0.188616\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 26.3291i	− 1.22627i	−0.789979	−	0.613134i	$-0.789908\pi$
$461$	− 26.3291i	− 1.22627i	0.789979	−	0.613134i	$-0.210092\pi$
$462$	0	0
$463$	5.45714	0.253615	0.126807	−	0.991927i	$-0.459527\pi$
$463$	5.45714	0.253615	0.126807	+	0.991927i	$0.459527\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.77349i	0.267165i	0.991038	+	0.133583i	$0.0426482\pi$
$467$	5.77349i	0.267165i	−0.991038	+	0.133583i	$0.957352\pi$
$468$	0	0
$469$	3.36185i	0.155236i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.57626	−0.210417
$474$	0	0
$475$	− 34.8056i	− 1.59699i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−31.3044	−1.43033	−0.715167	−	0.698954i	$-0.753649\pi$
$479$	−31.3044	−1.43033	−0.715167	+	0.698954i	$0.753649\pi$
$480$	0	0
$481$	−36.9386	−1.68425
$482$	0	0
$483$	0	0
$484$	0	0
$485$	40.1699i	1.82402i
$486$	0	0
$487$	25.7136	1.16519	0.582597	−	0.812761i	$-0.302037\pi$
$487$	25.7136	1.16519	0.582597	+	0.812761i	$0.302037\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.7216i	− 0.574116i	−0.957913	−	0.287058i	$-0.907323\pi$
$491$	− 12.7216i	− 0.574116i	0.957913	−	0.287058i	$-0.0926773\pi$
$492$	0	0
$493$	− 6.81794i	− 0.307065i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.26892	0.281199
$498$	0	0
$499$	37.9382i	1.69835i	0.528115	+	0.849173i	$0.322899\pi$
$499$	37.9382i	1.69835i	−0.528115	+	0.849173i	$0.677101\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.9704	0.756674	0.378337	−	0.925668i	$-0.376496\pi$
$503$	16.9704	0.756674	0.378337	+	0.925668i	$0.376496\pi$
$504$	0	0
$505$	−18.8561	−0.839086
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.6789i	0.517656i	0.965923	+	0.258828i	$0.0833363\pi$
$509$	11.6789i	0.517656i	−0.965923	+	0.258828i	$0.916664\pi$
$510$	0	0
$511$	10.0859	0.446172
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 16.8992i	− 0.744670i
$516$	0	0
$517$	− 32.4355i	− 1.42651i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.82516	0.211394	0.105697	−	0.994398i	$-0.466293\pi$
$521$	4.82516	0.211394	0.105697	+	0.994398i	$0.466293\pi$
$522$	0	0
$523$	− 24.7958i	− 1.08424i	−0.840300	−	0.542121i	$-0.817621\pi$
$523$	− 24.7958i	− 1.08424i	0.840300	−	0.542121i	$-0.182379\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.58190	−0.0689088
$528$	0	0
$529$	2.68998	0.116956
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 16.0692i	− 0.696035i
$534$	0	0
$535$	17.2341	0.745095
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.01331i	− 0.129792i
$540$	0	0
$541$	− 14.0510i	− 0.604100i	−0.953292	−	0.302050i	$-0.902329\pi$
$541$	− 14.0510i	− 0.604100i	0.953292	−	0.302050i	$-0.0976709\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	70.4967	3.01975
$546$	0	0
$547$	8.56879i	0.366375i	0.983078	+	0.183188i	$0.0586416\pi$
$547$	8.56879i	0.366375i	−0.983078	+	0.183188i	$0.941358\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	23.5448	1.00304
$552$	0	0
$553$	−12.9346	−0.550037
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.47669i	− 0.147312i	−0.997284	−	0.0736560i	$-0.976533\pi$
$557$	− 3.47669i	− 0.147312i	0.997284	−	0.0736560i	$-0.0234667\pi$
$558$	0	0
$559$	−5.93247	−0.250917
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.82325i	0.287566i	0.989609	+	0.143783i	$0.0459267\pi$
$563$	6.82325i	0.287566i	−0.989609	+	0.143783i	$0.954073\pi$
$564$	0	0
$565$	46.3711i	1.95085i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.1322	1.05360	0.526799	−	0.849990i	$-0.323392\pi$
$569$	25.1322	1.05360	0.526799	+	0.849990i	$0.323392\pi$
$570$	0	0
$571$	− 14.1758i	− 0.593238i	−0.954996	−	0.296619i	$-0.904141\pi$
$571$	− 14.1758i	− 0.593238i	0.954996	−	0.296619i	$-0.0958591\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−36.7942	−1.53442
$576$	0	0
$577$	18.3021	0.761927	0.380963	−	0.924590i	$-0.375592\pi$
$577$	18.3021	0.761927	0.380963	+	0.924590i	$0.375592\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 17.4139i	− 0.722450i
$582$	0	0
$583$	−1.30011	−0.0538451
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.70640i	− 0.276803i	−0.990376	−	0.138401i	$-0.955804\pi$
$587$	− 6.70640i	− 0.276803i	0.990376	−	0.138401i	$-0.0441964\pi$
$588$	0	0
$589$	− 5.46288i	− 0.225094i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17.9084	−0.735411	−0.367706	−	0.929942i	$-0.619857\pi$
$593$	−17.9084	−0.735411	−0.367706	+	0.929942i	$0.619857\pi$
$594$	0	0
$595$	− 4.86120i	− 0.199290i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.9838	0.979953	0.489976	−	0.871736i	$-0.337005\pi$
$599$	23.9838	0.979953	0.489976	+	0.871736i	$0.337005\pi$
$600$	0	0
$601$	−17.2548	−0.703840	−0.351920	−	0.936030i	$-0.614471\pi$
$601$	−17.2548	−0.703840	−0.351920	+	0.936030i	$0.614471\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.72246i	− 0.273307i
$606$	0	0
$607$	−38.0189	−1.54314	−0.771569	−	0.636146i	$-0.780528\pi$
$607$	−38.0189	−1.54314	−0.771569	+	0.636146i	$0.780528\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 42.0479i	− 1.70108i
$612$	0	0
$613$	33.1990i	1.34089i	0.741957	+	0.670447i	$0.233898\pi$
$613$	33.1990i	1.34089i	−0.741957	+	0.670447i	$0.766102\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.827310	0.0333062	0.0166531	−	0.999861i	$-0.494699\pi$
$617$	0.827310	0.0333062	0.0166531	+	0.999861i	$0.494699\pi$
$618$	0	0
$619$	42.3695i	1.70297i	0.524377	+	0.851486i	$0.324298\pi$
$619$	42.3695i	1.70297i	−0.524377	+	0.851486i	$0.675702\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.818403	−0.0327886
$624$	0	0
$625$	−8.59870	−0.343948
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 13.1287i	− 0.523477i
$630$	0	0
$631$	18.5333	0.737801	0.368901	−	0.929469i	$-0.379734\pi$
$631$	18.5333	0.737801	0.368901	+	0.929469i	$0.379734\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	35.8020i	1.42076i
$636$	0	0
$637$	− 3.90632i	− 0.154774i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.3013	0.525371	0.262686	−	0.964881i	$-0.415392\pi$
$641$	13.3013	0.525371	0.262686	+	0.964881i	$0.415392\pi$
$642$	0	0
$643$	− 12.1885i	− 0.480668i	−0.970690	−	0.240334i	$-0.922743\pi$
$643$	− 12.1885i	− 0.480668i	0.970690	−	0.240334i	$-0.0772570\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.4813	−0.726576	−0.363288	−	0.931677i	$-0.618346\pi$
$647$	−18.4813	−0.726576	−0.363288	+	0.931677i	$0.618346\pi$
$648$	0	0
$649$	22.3267	0.876399
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 25.0787i	− 0.981405i	−0.871327	−	0.490703i	$-0.836740\pi$
$653$	− 25.0787i	− 0.981405i	0.871327	−	0.490703i	$-0.163260\pi$
$654$	0	0
$655$	−5.41748	−0.211678
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 40.3331i	− 1.57115i	−0.618764	−	0.785577i	$-0.712366\pi$
$659$	− 40.3331i	− 1.57115i	0.618764	−	0.785577i	$-0.287634\pi$
$660$	0	0
$661$	− 21.1566i	− 0.822896i	−0.911433	−	0.411448i	$-0.865023\pi$
$661$	− 21.1566i	− 0.822896i	0.911433	−	0.411448i	$-0.134977\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.7875	0.650991
$666$	0	0
$667$	− 24.8900i	− 0.963745i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−39.0284	−1.50667
$672$	0	0
$673$	45.8540	1.76754	0.883772	−	0.467918i	$-0.154996\pi$
$673$	45.8540	1.76754	0.883772	+	0.467918i	$0.154996\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 41.0495i	− 1.57766i	−0.614612	−	0.788830i	$-0.710687\pi$
$677$	− 41.0495i	− 1.57766i	0.614612	−	0.788830i	$-0.289313\pi$
$678$	0	0
$679$	−11.4727	−0.440284
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.87089i	0.301171i	0.988597	+	0.150586i	$0.0481159\pi$
$683$	7.87089i	0.301171i	−0.988597	+	0.150586i	$0.951884\pi$
$684$	0	0
$685$	− 26.1332i	− 0.998499i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.68541	−0.0642089
$690$	0	0
$691$	− 33.4478i	− 1.27241i	−0.771519	−	0.636207i	$-0.780503\pi$
$691$	− 33.4478i	− 1.27241i	0.771519	−	0.636207i	$-0.219497\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	58.2564	2.20979
$696$	0	0
$697$	5.71133	0.216332
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.9716i	0.754317i	0.926149	+	0.377158i	$0.123099\pi$
$701$	19.9716i	0.754317i	−0.926149	+	0.377158i	$0.876901\pi$
$702$	0	0
$703$	45.3382	1.70996
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.38541i	− 0.202539i
$708$	0	0
$709$	− 38.1379i	− 1.43230i	−0.697947	−	0.716149i	$-0.745903\pi$
$709$	− 38.1379i	− 1.43230i	0.697947	−	0.716149i	$-0.254097\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.77499	−0.216275
$714$	0	0
$715$	41.2140i	1.54132i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33.8150	−1.26109	−0.630543	−	0.776155i	$-0.717168\pi$
$719$	−33.8150	−1.26109	−0.630543	+	0.776155i	$0.717168\pi$
$720$	0	0
$721$	4.82652	0.179749
$722$	0	0
$723$	0	0
$724$	0	0
$725$	35.6484i	1.32395i
$726$	0	0
$727$	38.9720	1.44539	0.722696	−	0.691166i	$-0.242903\pi$
$727$	38.9720	1.44539	0.722696	+	0.691166i	$0.242903\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.10852i	− 0.0779865i
$732$	0	0
$733$	− 13.2978i	− 0.491164i	−0.969376	−	0.245582i	$-0.921021\pi$
$733$	− 13.2978i	− 0.491164i	0.969376	−	0.245582i	$-0.0789790\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.1303	0.373154
$738$	0	0
$739$	37.4030i	1.37589i	0.725762	+	0.687946i	$0.241487\pi$
$739$	37.4030i	1.37589i	−0.725762	+	0.687946i	$0.758513\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.1138	1.14145	0.570727	−	0.821140i	$-0.306662\pi$
$743$	31.1138	1.14145	0.570727	+	0.821140i	$0.306662\pi$
$744$	0	0
$745$	74.8747	2.74320
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.92215i	0.179851i
$750$	0	0
$751$	−18.6412	−0.680227	−0.340114	−	0.940384i	$-0.610466\pi$
$751$	−18.6412	−0.680227	−0.340114	+	0.940384i	$0.610466\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.24855i	0.154621i
$756$	0	0
$757$	29.8972i	1.08663i	0.839528	+	0.543317i	$0.182832\pi$
$757$	29.8972i	1.08663i	−0.839528	+	0.543317i	$0.817168\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.6967	1.54776	0.773878	−	0.633335i	$-0.218315\pi$
$761$	42.6967	1.54776	0.773878	+	0.633335i	$0.218315\pi$
$762$	0	0
$763$	20.1342i	0.728909i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.9433	1.04508
$768$	0	0
$769$	40.5127	1.46092	0.730462	−	0.682953i	$-0.239305\pi$
$769$	40.5127	1.46092	0.730462	+	0.682953i	$0.239305\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 14.6650i	− 0.527465i	−0.964596	−	0.263732i	$-0.915046\pi$
$773$	− 14.6650i	− 0.527465i	0.964596	−	0.263732i	$-0.0849536\pi$
$774$	0	0
$775$	8.27116	0.297109
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.7233i	0.706660i
$780$	0	0
$781$	− 18.8902i	− 0.675944i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	28.5584	1.01929
$786$	0	0
$787$	− 17.3453i	− 0.618292i	−0.951015	−	0.309146i	$-0.899957\pi$
$787$	− 17.3453i	− 0.618292i	0.951015	−	0.309146i	$-0.100043\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.2438	−0.470896
$792$	0	0
$793$	−50.5947	−1.79667
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.9239i	− 0.776585i	−0.921536	−	0.388292i	$-0.873065\pi$
$797$	− 21.9239i	− 0.776585i	0.921536	−	0.388292i	$-0.126935\pi$
$798$	0	0
$799$	14.9447	0.528705
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 30.3918i	− 1.07250i
$804$	0	0
$805$	− 17.7466i	− 0.625486i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−19.2482	−0.676731	−0.338366	−	0.941015i	$-0.609874\pi$
$809$	−19.2482	−0.676731	−0.338366	+	0.941015i	$0.609874\pi$
$810$	0	0
$811$	12.4354i	0.436665i	0.975874	+	0.218333i	$0.0700617\pi$
$811$	12.4354i	0.436665i	−0.975874	+	0.218333i	$0.929938\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−79.9278	−2.79975
$816$	0	0
$817$	7.28148	0.254747
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 55.1514i	− 1.92480i	−0.271640	−	0.962399i	$-0.587566\pi$
$821$	− 55.1514i	− 1.92480i	0.271640	−	0.962399i	$-0.412434\pi$
$822$	0	0
$823$	−15.6639	−0.546009	−0.273004	−	0.962013i	$-0.588017\pi$
$823$	−15.6639	−0.546009	−0.273004	+	0.962013i	$0.588017\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.1178i	0.560470i	0.959931	+	0.280235i	$0.0904124\pi$
$827$	16.1178i	0.560470i	−0.959931	+	0.280235i	$0.909588\pi$
$828$	0	0
$829$	42.5904i	1.47923i	0.673032	+	0.739613i	$0.264992\pi$
$829$	42.5904i	1.47923i	−0.673032	+	0.739613i	$0.735008\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.38839	0.0481047
$834$	0	0
$835$	60.7743i	2.10318i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21.1741	−0.731011	−0.365505	−	0.930809i	$-0.619104\pi$
$839$	−21.1741	−0.731011	−0.365505	+	0.930809i	$0.619104\pi$
$840$	0	0
$841$	4.88507	0.168451
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.91070i	0.272136i
$846$	0	0
$847$	1.91997	0.0659710
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 47.9285i	− 1.64297i
$852$	0	0
$853$	− 13.2958i	− 0.455240i	−0.973750	−	0.227620i	$-0.926906\pi$
$853$	− 13.2958i	− 0.455240i	0.973750	−	0.227620i	$-0.0730944\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.1940	0.894770	0.447385	−	0.894341i	$-0.352355\pi$
$857$	26.1940	0.894770	0.447385	+	0.894341i	$0.352355\pi$
$858$	0	0
$859$	− 5.90287i	− 0.201403i	−0.994917	−	0.100702i	$-0.967891\pi$
$859$	− 5.90287i	− 0.201403i	0.994917	−	0.100702i	$-0.0321088\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.1022	−0.411964	−0.205982	−	0.978556i	$-0.566039\pi$
$863$	−12.1022	−0.411964	−0.205982	+	0.978556i	$0.566039\pi$
$864$	0	0
$865$	82.5558	2.80698
$866$	0	0
$867$	0	0
$868$	0	0
$869$	38.9761i	1.32217i
$870$	0	0
$871$	13.1325	0.444976
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.91070i	0.267430i
$876$	0	0
$877$	− 27.9450i	− 0.943634i	−0.881697	−	0.471817i	$-0.843598\pi$
$877$	− 27.9450i	− 0.943634i	0.881697	−	0.471817i	$-0.156402\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.6503	0.897872	0.448936	−	0.893564i	$-0.351803\pi$
$881$	26.6503	0.897872	0.448936	+	0.893564i	$0.351803\pi$
$882$	0	0
$883$	4.47024i	0.150435i	0.997167	+	0.0752177i	$0.0239652\pi$
$883$	4.47024i	0.150435i	−0.997167	+	0.0752177i	$0.976035\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.53323	0.286518	0.143259	−	0.989685i	$-0.454242\pi$
$887$	8.53323	0.286518	0.143259	+	0.989685i	$0.454242\pi$
$888$	0	0
$889$	−10.2252	−0.342943
$890$	0	0
$891$	0	0
$892$	0	0
$893$	51.6094i	1.72704i
$894$	0	0
$895$	−42.3539	−1.41573
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.59516i	0.186609i
$900$	0	0
$901$	− 0.599028i	− 0.0199565i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.9781	−0.431407
$906$	0	0
$907$	30.0969i	0.999353i	0.866212	+	0.499676i	$0.166548\pi$
$907$	30.0969i	0.999353i	−0.866212	+	0.499676i	$0.833452\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.93001	0.196470	0.0982350	−	0.995163i	$-0.468680\pi$
$911$	5.93001	0.196470	0.0982350	+	0.995163i	$0.468680\pi$
$912$	0	0
$913$	−52.4734	−1.73662
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.54726i	− 0.0510951i
$918$	0	0
$919$	−39.3264	−1.29726	−0.648630	−	0.761104i	$-0.724658\pi$
$919$	−39.3264	−1.29726	−0.648630	+	0.761104i	$0.724658\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 24.4884i	− 0.806046i
$924$	0	0
$925$	68.6450i	2.25704i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.9339	−1.17895	−0.589476	−	0.807786i	$-0.700666\pi$
$929$	−35.9339	−1.17895	−0.589476	+	0.807786i	$0.700666\pi$
$930$	0	0
$931$	4.79460i	0.157137i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.6483	−0.479051
$936$	0	0
$937$	−24.6365	−0.804838	−0.402419	−	0.915456i	$-0.631830\pi$
$937$	−24.6365	−0.804838	−0.402419	+	0.915456i	$0.631830\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23.2069i	0.756522i	0.925699	+	0.378261i	$0.123478\pi$
$941$	23.2069i	0.756522i	−0.925699	+	0.378261i	$0.876522\pi$
$942$	0	0
$943$	20.8501	0.678973
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 19.6763i	− 0.639394i	−0.947520	−	0.319697i	$-0.896419\pi$
$947$	− 19.6763i	− 0.639394i	0.947520	−	0.319697i	$-0.103581\pi$
$948$	0	0
$949$	− 39.3986i	− 1.27893i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.96974	0.322952	0.161476	−	0.986877i	$-0.448375\pi$
$953$	9.96974	0.322952	0.161476	+	0.986877i	$0.448375\pi$
$954$	0	0
$955$	39.4846i	1.27769i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.46379	0.241018
$960$	0	0
$961$	−29.7018	−0.958123
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.0358i	0.355254i
$966$	0	0
$967$	7.20443	0.231679	0.115839	−	0.993268i	$-0.463044\pi$
$967$	7.20443	0.231679	0.115839	+	0.993268i	$0.463044\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.0495i	1.02852i	0.857635	+	0.514258i	$0.171933\pi$
$971$	32.0495i	1.02852i	−0.857635	+	0.514258i	$0.828067\pi$
$972$	0	0
$973$	16.6383i	0.533401i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.17655	0.261591	0.130796	−	0.991409i	$-0.458247\pi$
$977$	8.17655	0.261591	0.130796	+	0.991409i	$0.458247\pi$
$978$	0	0
$979$	2.46610i	0.0788169i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.2990	1.03018	0.515089	−	0.857137i	$-0.327759\pi$
$983$	32.2990	1.03018	0.515089	+	0.857137i	$0.327759\pi$
$984$	0	0
$985$	−40.2020	−1.28094
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 7.69749i	− 0.244766i
$990$	0	0
$991$	43.7283	1.38907	0.694537	−	0.719457i	$-0.255609\pi$
$991$	43.7283	1.38907	0.694537	+	0.719457i	$0.255609\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 11.8487i	− 0.375628i
$996$	0	0
$997$	4.64199i	0.147013i	0.997295	+	0.0735066i	$0.0234190\pi$
$997$	4.64199i	0.147013i	−0.997295	+	0.0735066i	$0.976581\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.e.3025.1		20
3.2	odd	2	inner	6048.2.c.e.3025.19		20
4.3	odd	2		1512.2.c.e.757.16	yes	20
8.3	odd	2		1512.2.c.e.757.15	yes	20
8.5	even	2	inner	6048.2.c.e.3025.20		20
12.11	even	2		1512.2.c.e.757.5	✓	20
24.5	odd	2	inner	6048.2.c.e.3025.2		20
24.11	even	2		1512.2.c.e.757.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.e.757.5	✓	20	12.11	even	2
1512.2.c.e.757.6	yes	20	24.11	even	2
1512.2.c.e.757.15	yes	20	8.3	odd	2
1512.2.c.e.757.16	yes	20	4.3	odd	2
6048.2.c.e.3025.1		20	1.1	even	1	trivial
6048.2.c.e.3025.2		20	24.5	odd	2	inner
6048.2.c.e.3025.19		20	3.2	odd	2	inner
6048.2.c.e.3025.20		20	8.5	even	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.50133i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.50133i q^{5} +1.00000 q^{7} -3.01331i q^{11} -3.90632i q^{13} +1.38839 q^{17} +4.79460i q^{19} +5.06853 q^{23} -7.25934 q^{25} -4.91070i q^{29} -1.13938 q^{31} -3.50133i q^{35} -9.45610i q^{37} +4.11364 q^{41} -1.51868i q^{43} +10.7641 q^{47} +1.00000 q^{49} -0.431457i q^{53} -10.5506 q^{55} +7.40936i q^{59} -12.9520i q^{61} -13.6773 q^{65} +3.36185i q^{67} +6.26892 q^{71} +10.0859 q^{73} -3.01331i q^{77} -12.9346 q^{79} -17.4139i q^{83} -4.86120i q^{85} -0.818403 q^{89} -3.90632i q^{91} +16.7875 q^{95} -11.4727 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{7}+O(q^{10}) \) 20 * q + 20 * q^7	\( 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) \) 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * q^97

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