LMFDB - Embedded newform 6048.2.j.d.5615.46

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			5615.46
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.45

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.69622 q^{5} +1.00000i q^{7} +O(q^{10})$	$q+3.69622 q^{5} +1.00000i q^{7} -2.05843i q^{11} +4.65027i q^{13} +5.73497i q^{17} -3.27215 q^{19} -4.45085 q^{23} +8.66201 q^{25} -10.2594 q^{29} +1.26066i q^{31} +3.69622i q^{35} +3.25539i q^{37} +8.72747i q^{41} +2.56921 q^{43} +2.03048 q^{47} -1.00000 q^{49} -10.0794 q^{53} -7.60840i q^{55} +0.851213i q^{59} -2.31448i q^{61} +17.1884i q^{65} -15.4289 q^{67} -4.44544 q^{71} -6.97349 q^{73} +2.05843 q^{77} +11.6750i q^{79} -12.8349i q^{83} +21.1977i q^{85} +2.35419i q^{89} -4.65027 q^{91} -12.0946 q^{95} +4.02539 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.69622	1.65300	0.826499	−	0.562938i	$-0.190329\pi$
$5$	3.69622	1.65300	0.826499	+	0.562938i	$0.190329\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.05843i	− 0.620640i	−0.950632	−	0.310320i	$-0.899564\pi$
$11$	− 2.05843i	− 0.620640i	0.950632	−	0.310320i	$-0.100436\pi$
$12$	0	0
$13$	4.65027i	1.28975i	0.764287	+	0.644876i	$0.223091\pi$
$13$	4.65027i	1.28975i	−0.764287	+	0.644876i	$0.776909\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.73497i	1.39094i	0.718557	+	0.695468i	$0.244803\pi$
$17$	5.73497i	1.39094i	−0.718557	+	0.695468i	$0.755197\pi$
$18$	0	0
$19$	−3.27215	−0.750682	−0.375341	−	0.926887i	$-0.622474\pi$
$19$	−3.27215	−0.750682	−0.375341	+	0.926887i	$0.622474\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.45085	−0.928067	−0.464033	−	0.885818i	$-0.653598\pi$
$23$	−4.45085	−0.928067	−0.464033	+	0.885818i	$0.653598\pi$
$24$	0	0
$25$	8.66201	1.73240
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.2594	−1.90512	−0.952561	−	0.304346i	$-0.901562\pi$
$29$	−10.2594	−1.90512	−0.952561	+	0.304346i	$0.901562\pi$
$30$	0	0
$31$	1.26066i	0.226422i	0.993571	+	0.113211i	$0.0361136\pi$
$31$	1.26066i	0.226422i	−0.993571	+	0.113211i	$0.963886\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.69622i	0.624775i
$36$	0	0
$37$	3.25539i	0.535183i	0.963532	+	0.267592i	$0.0862278\pi$
$37$	3.25539i	0.535183i	−0.963532	+	0.267592i	$0.913772\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.72747i	1.36300i	0.731817	+	0.681501i	$0.238673\pi$
$41$	8.72747i	1.36300i	−0.731817	+	0.681501i	$0.761327\pi$
$42$	0	0
$43$	2.56921	0.391801	0.195901	−	0.980624i	$-0.437237\pi$
$43$	2.56921	0.391801	0.195901	+	0.980624i	$0.437237\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.03048	0.296176	0.148088	−	0.988974i	$-0.452688\pi$
$47$	2.03048	0.296176	0.148088	+	0.988974i	$0.452688\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.0794	−1.38451	−0.692253	−	0.721655i	$-0.743382\pi$
$53$	−10.0794	−1.38451	−0.692253	+	0.721655i	$0.743382\pi$
$54$	0	0
$55$	− 7.60840i	− 1.02592i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.851213i	0.110818i	0.998464	+	0.0554092i	$0.0176463\pi$
$59$	0.851213i	0.110818i	−0.998464	+	0.0554092i	$0.982354\pi$
$60$	0	0
$61$	− 2.31448i	− 0.296339i	−0.988962	−	0.148170i	$-0.952662\pi$
$61$	− 2.31448i	− 0.296339i	0.988962	−	0.148170i	$-0.0473382\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.1884i	2.13196i
$66$	0	0
$67$	−15.4289	−1.88494	−0.942468	−	0.334296i	$-0.891501\pi$
$67$	−15.4289	−1.88494	−0.942468	+	0.334296i	$0.891501\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.44544	−0.527576	−0.263788	−	0.964581i	$-0.584972\pi$
$71$	−4.44544	−0.527576	−0.263788	+	0.964581i	$0.584972\pi$
$72$	0	0
$73$	−6.97349	−0.816186	−0.408093	−	0.912940i	$-0.633806\pi$
$73$	−6.97349	−0.816186	−0.408093	+	0.912940i	$0.633806\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.05843	0.234580
$78$	0	0
$79$	11.6750i	1.31354i	0.754092	+	0.656769i	$0.228077\pi$
$79$	11.6750i	1.31354i	−0.754092	+	0.656769i	$0.771923\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 12.8349i	− 1.40882i	−0.709796	−	0.704408i	$-0.751213\pi$
$83$	− 12.8349i	− 1.40882i	0.709796	−	0.704408i	$-0.248787\pi$
$84$	0	0
$85$	21.1977i	2.29921i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.35419i	0.249544i	0.992185	+	0.124772i	$0.0398200\pi$
$89$	2.35419i	0.249544i	−0.992185	+	0.124772i	$0.960180\pi$
$90$	0	0
$91$	−4.65027	−0.487480
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.0946	−1.24088
$96$	0	0
$97$	4.02539	0.408717	0.204358	−	0.978896i	$-0.434489\pi$
$97$	4.02539	0.408717	0.204358	+	0.978896i	$0.434489\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.2008	1.41303	0.706517	−	0.707696i	$-0.250265\pi$
$101$	14.2008	1.41303	0.706517	+	0.707696i	$0.250265\pi$
$102$	0	0
$103$	5.30631i	0.522846i	0.965224	+	0.261423i	$0.0841918\pi$
$103$	5.30631i	0.522846i	−0.965224	+	0.261423i	$0.915808\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3669i	1.29223i	0.763240	+	0.646115i	$0.223608\pi$
$107$	13.3669i	1.29223i	−0.763240	+	0.646115i	$0.776392\pi$
$108$	0	0
$109$	− 5.91832i	− 0.566872i	−0.958991	−	0.283436i	$-0.908526\pi$
$109$	− 5.91832i	− 0.566872i	0.958991	−	0.283436i	$-0.0914743\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 18.5535i	− 1.74537i	−0.488287	−	0.872683i	$-0.662378\pi$
$113$	− 18.5535i	− 1.74537i	0.488287	−	0.872683i	$-0.337622\pi$
$114$	0	0
$115$	−16.4513	−1.53409
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.73497	−0.525724
$120$	0	0
$121$	6.76287	0.614806
$122$	0	0
$123$	0	0
$124$	0	0
$125$	13.5356	1.21066
$126$	0	0
$127$	9.76963i	0.866915i	0.901174	+	0.433457i	$0.142707\pi$
$127$	9.76963i	0.866915i	−0.901174	+	0.433457i	$0.857293\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 15.5661i	− 1.36002i	−0.733204	−	0.680009i	$-0.761976\pi$
$131$	− 15.5661i	− 1.36002i	0.733204	−	0.680009i	$-0.238024\pi$
$132$	0	0
$133$	− 3.27215i	− 0.283731i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.54964i	0.730445i	0.930920	+	0.365223i	$0.119007\pi$
$137$	8.54964i	0.730445i	−0.930920	+	0.365223i	$0.880993\pi$
$138$	0	0
$139$	15.4607	1.31136	0.655679	−	0.755040i	$-0.272383\pi$
$139$	15.4607	1.31136	0.655679	+	0.755040i	$0.272383\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.57225	0.800472
$144$	0	0
$145$	−37.9210	−3.14916
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.91181	0.730084	0.365042	−	0.930991i	$-0.381055\pi$
$149$	8.91181	0.730084	0.365042	+	0.930991i	$0.381055\pi$
$150$	0	0
$151$	14.8040i	1.20473i	0.798219	+	0.602367i	$0.205776\pi$
$151$	14.8040i	1.20473i	−0.798219	+	0.602367i	$0.794224\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.65968i	0.374275i
$156$	0	0
$157$	− 14.3716i	− 1.14698i	−0.819214	−	0.573488i	$-0.805590\pi$
$157$	− 14.3716i	− 1.14698i	0.819214	−	0.573488i	$-0.194410\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.45085i	− 0.350776i
$162$	0	0
$163$	−21.7846	−1.70630	−0.853149	−	0.521667i	$-0.825310\pi$
$163$	−21.7846	−1.70630	−0.853149	+	0.521667i	$0.825310\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.4031	−1.03717	−0.518583	−	0.855027i	$-0.673540\pi$
$167$	−13.4031	−1.03717	−0.518583	+	0.855027i	$0.673540\pi$
$168$	0	0
$169$	−8.62498	−0.663460
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.3221	1.01286	0.506429	−	0.862282i	$-0.330965\pi$
$173$	13.3221	1.01286	0.506429	+	0.862282i	$0.330965\pi$
$174$	0	0
$175$	8.66201i	0.654787i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 12.6787i	− 0.947648i	−0.880619	−	0.473824i	$-0.842873\pi$
$179$	− 12.6787i	− 0.947648i	0.880619	−	0.473824i	$-0.157127\pi$
$180$	0	0
$181$	− 11.1255i	− 0.826951i	−0.910515	−	0.413476i	$-0.864315\pi$
$181$	− 11.1255i	− 0.826951i	0.910515	−	0.413476i	$-0.135685\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.0326i	0.884657i
$186$	0	0
$187$	11.8050	0.863270
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.76387	0.561774	0.280887	−	0.959741i	$-0.409371\pi$
$191$	7.76387	0.561774	0.280887	+	0.959741i	$0.409371\pi$
$192$	0	0
$193$	9.09582	0.654731	0.327366	−	0.944898i	$-0.393839\pi$
$193$	9.09582	0.654731	0.327366	+	0.944898i	$0.393839\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.3722	0.738988	0.369494	−	0.929233i	$-0.379531\pi$
$197$	10.3722	0.738988	0.369494	+	0.929233i	$0.379531\pi$
$198$	0	0
$199$	9.92379i	0.703479i	0.936098	+	0.351739i	$0.114410\pi$
$199$	9.92379i	0.703479i	−0.936098	+	0.351739i	$0.885590\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 10.2594i	− 0.720069i
$204$	0	0
$205$	32.2586i	2.25304i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.73549i	0.465903i
$210$	0	0
$211$	10.9624	0.754680	0.377340	−	0.926075i	$-0.376839\pi$
$211$	10.9624	0.754680	0.377340	+	0.926075i	$0.376839\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.49637	0.647647
$216$	0	0
$217$	−1.26066	−0.0855794
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−26.6692	−1.79396
$222$	0	0
$223$	1.18980i	0.0796749i	0.999206	+	0.0398375i	$0.0126840\pi$
$223$	1.18980i	0.0796749i	−0.999206	+	0.0398375i	$0.987316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.62724i	0.638983i	0.947589	+	0.319491i	$0.103512\pi$
$227$	9.62724i	0.638983i	−0.947589	+	0.319491i	$0.896488\pi$
$228$	0	0
$229$	21.5143i	1.42170i	0.703343	+	0.710851i	$0.251690\pi$
$229$	21.5143i	1.42170i	−0.703343	+	0.710851i	$0.748310\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.82082i	− 0.446847i	−0.974722	−	0.223423i	$-0.928277\pi$
$233$	− 6.82082i	− 0.446847i	0.974722	−	0.223423i	$-0.0717232\pi$
$234$	0	0
$235$	7.50510	0.489579
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.24506	−0.403959	−0.201980	−	0.979390i	$-0.564737\pi$
$239$	−6.24506	−0.403959	−0.201980	+	0.979390i	$0.564737\pi$
$240$	0	0
$241$	25.5573	1.64629	0.823145	−	0.567831i	$-0.192217\pi$
$241$	25.5573	1.64629	0.823145	+	0.567831i	$0.192217\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.69622	−0.236143
$246$	0	0
$247$	− 15.2164i	− 0.968194i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 0.373698i	− 0.0235876i	−0.999930	−	0.0117938i	$-0.996246\pi$
$251$	− 0.373698i	− 0.0235876i	0.999930	−	0.0117938i	$-0.00375417\pi$
$252$	0	0
$253$	9.16177i	0.575995i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.0514i	− 0.689369i	−0.938719	−	0.344684i	$-0.887986\pi$
$257$	− 11.0514i	− 0.689369i	0.938719	−	0.344684i	$-0.112014\pi$
$258$	0	0
$259$	−3.25539	−0.202280
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.6685	1.70612	0.853058	−	0.521817i	$-0.174745\pi$
$263$	27.6685	1.70612	0.853058	+	0.521817i	$0.174745\pi$
$264$	0	0
$265$	−37.2555	−2.28859
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.3247	1.78796	0.893981	−	0.448105i	$-0.147901\pi$
$269$	29.3247	1.78796	0.893981	+	0.448105i	$0.147901\pi$
$270$	0	0
$271$	5.13414i	0.311877i	0.987767	+	0.155938i	$0.0498401\pi$
$271$	5.13414i	0.311877i	−0.987767	+	0.155938i	$0.950160\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 17.8301i	− 1.07520i
$276$	0	0
$277$	− 5.93089i	− 0.356353i	−0.983999	−	0.178176i	$-0.942980\pi$
$277$	− 5.93089i	− 0.356353i	0.983999	−	0.178176i	$-0.0570197\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 10.7827i	− 0.643242i	−0.946868	−	0.321621i	$-0.895772\pi$
$281$	− 10.7827i	− 0.643242i	0.946868	−	0.321621i	$-0.104228\pi$
$282$	0	0
$283$	18.1556	1.07924	0.539619	−	0.841910i	$-0.318568\pi$
$283$	18.1556	1.07924	0.539619	+	0.841910i	$0.318568\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.72747	−0.515166
$288$	0	0
$289$	−15.8899	−0.934701
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−21.8863	−1.27861	−0.639306	−	0.768953i	$-0.720778\pi$
$293$	−21.8863	−1.27861	−0.639306	+	0.768953i	$0.720778\pi$
$294$	0	0
$295$	3.14627i	0.183183i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 20.6976i	− 1.19698i
$300$	0	0
$301$	2.56921i	0.148087i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.55484i	− 0.489848i
$306$	0	0
$307$	−16.8301	−0.960546	−0.480273	−	0.877119i	$-0.659462\pi$
$307$	−16.8301	−0.960546	−0.480273	+	0.877119i	$0.659462\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.1293	1.87859	0.939294	−	0.343113i	$-0.111481\pi$
$311$	33.1293	1.87859	0.939294	+	0.343113i	$0.111481\pi$
$312$	0	0
$313$	13.4187	0.758469	0.379235	−	0.925301i	$-0.376187\pi$
$313$	13.4187	0.758469	0.379235	+	0.925301i	$0.376187\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.764835	0.0429574	0.0214787	−	0.999769i	$-0.493163\pi$
$317$	0.764835	0.0429574	0.0214787	+	0.999769i	$0.493163\pi$
$318$	0	0
$319$	21.1183i	1.18240i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 18.7657i	− 1.04415i
$324$	0	0
$325$	40.2807i	2.23437i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.03048i	0.111944i
$330$	0	0
$331$	−29.7299	−1.63410	−0.817051	−	0.576566i	$-0.804392\pi$
$331$	−29.7299	−1.63410	−0.817051	+	0.576566i	$0.804392\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−57.0284	−3.11580
$336$	0	0
$337$	−19.0487	−1.03765	−0.518824	−	0.854881i	$-0.673630\pi$
$337$	−19.0487	−1.03765	−0.518824	+	0.854881i	$0.673630\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.59499	0.140526
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 30.0749i	− 1.61451i	−0.590205	−	0.807253i	$-0.700953\pi$
$347$	− 30.0749i	− 1.61451i	0.590205	−	0.807253i	$-0.299047\pi$
$348$	0	0
$349$	28.7995i	1.54160i	0.637078	+	0.770800i	$0.280143\pi$
$349$	28.7995i	1.54160i	−0.637078	+	0.770800i	$0.719857\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 26.2377i	− 1.39649i	−0.715857	−	0.698247i	$-0.753964\pi$
$353$	− 26.2377i	− 1.39649i	0.715857	−	0.698247i	$-0.246036\pi$
$354$	0	0
$355$	−16.4313	−0.872083
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.4178	0.972053	0.486027	−	0.873944i	$-0.338446\pi$
$359$	18.4178	0.972053	0.486027	+	0.873944i	$0.338446\pi$
$360$	0	0
$361$	−8.29305	−0.436476
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−25.7755	−1.34915
$366$	0	0
$367$	37.2862i	1.94632i	0.230127	+	0.973161i	$0.426086\pi$
$367$	37.2862i	1.94632i	−0.230127	+	0.973161i	$0.573914\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 10.0794i	− 0.523294i
$372$	0	0
$373$	− 4.59095i	− 0.237711i	−0.992912	−	0.118855i	$-0.962078\pi$
$373$	− 4.59095i	− 0.237711i	0.992912	−	0.118855i	$-0.0379224\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 47.7090i	− 2.45714i
$378$	0	0
$379$	21.0149	1.07947	0.539733	−	0.841836i	$-0.318525\pi$
$379$	21.0149	1.07947	0.539733	+	0.841836i	$0.318525\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.09047	−0.464501	−0.232251	−	0.972656i	$-0.574609\pi$
$383$	−9.09047	−0.464501	−0.232251	+	0.972656i	$0.574609\pi$
$384$	0	0
$385$	7.60840	0.387760
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.06330	−0.104614	−0.0523068	−	0.998631i	$-0.516657\pi$
$389$	−2.06330	−0.104614	−0.0523068	+	0.998631i	$0.516657\pi$
$390$	0	0
$391$	− 25.5255i	− 1.29088i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	43.1533i	2.17128i
$396$	0	0
$397$	0.0369436i	0.00185415i	1.00000		0.000927073i	$0.000295096\pi$
$397$	0.0369436i	0.00185415i	−1.00000		0.000927073i	$0.999705\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.6814i	0.932906i	0.884546	+	0.466453i	$0.154468\pi$
$401$	18.6814i	0.932906i	−0.884546	+	0.466453i	$0.845532\pi$
$402$	0	0
$403$	−5.86242	−0.292028
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.70100	0.332156
$408$	0	0
$409$	−11.5630	−0.571755	−0.285878	−	0.958266i	$-0.592285\pi$
$409$	−11.5630	−0.571755	−0.285878	+	0.958266i	$0.592285\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.851213	−0.0418854
$414$	0	0
$415$	− 47.4406i	− 2.32877i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.5911i	0.712822i	0.934329	+	0.356411i	$0.116000\pi$
$419$	14.5911i	0.712822i	−0.934329	+	0.356411i	$0.884000\pi$
$420$	0	0
$421$	17.6204i	0.858767i	0.903122	+	0.429384i	$0.141269\pi$
$421$	17.6204i	0.858767i	−0.903122	+	0.429384i	$0.858731\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	49.6764i	2.40966i
$426$	0	0
$427$	2.31448	0.112006
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.6500	0.850170	0.425085	−	0.905153i	$-0.360244\pi$
$431$	17.6500	0.850170	0.425085	+	0.905153i	$0.360244\pi$
$432$	0	0
$433$	−6.31965	−0.303703	−0.151852	−	0.988403i	$-0.548524\pi$
$433$	−6.31965	−0.303703	−0.151852	+	0.988403i	$0.548524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.5638	0.696683
$438$	0	0
$439$	10.1993i	0.486786i	0.969928	+	0.243393i	$0.0782605\pi$
$439$	10.1993i	0.486786i	−0.969928	+	0.243393i	$0.921739\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.46579i	0.449733i	0.974390	+	0.224867i	$0.0721947\pi$
$443$	9.46579i	0.449733i	−0.974390	+	0.224867i	$0.927805\pi$
$444$	0	0
$445$	8.70161i	0.412496i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	28.1482i	1.32840i	0.747556	+	0.664198i	$0.231227\pi$
$449$	28.1482i	1.32840i	−0.747556	+	0.664198i	$0.768773\pi$
$450$	0	0
$451$	17.9649	0.845934
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.1884	−0.805804
$456$	0	0
$457$	−34.9936	−1.63693	−0.818465	−	0.574556i	$-0.805175\pi$
$457$	−34.9936	−1.63693	−0.818465	+	0.574556i	$0.805175\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.2558	1.40915	0.704577	−	0.709628i	$-0.251137\pi$
$461$	30.2558	1.40915	0.704577	+	0.709628i	$0.251137\pi$
$462$	0	0
$463$	2.87077i	0.133416i	0.997773	+	0.0667080i	$0.0212496\pi$
$463$	2.87077i	0.133416i	−0.997773	+	0.0667080i	$0.978750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.1957i	1.67494i	0.546484	+	0.837469i	$0.315966\pi$
$467$	36.1957i	1.67494i	−0.546484	+	0.837469i	$0.684034\pi$
$468$	0	0
$469$	− 15.4289i	− 0.712439i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5.28855i	− 0.243168i
$474$	0	0
$475$	−28.3434	−1.30048
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17.1156	0.782030	0.391015	−	0.920384i	$-0.372124\pi$
$479$	17.1156	0.782030	0.391015	+	0.920384i	$0.372124\pi$
$480$	0	0
$481$	−15.1384	−0.690253
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.8787	0.675608
$486$	0	0
$487$	26.8336i	1.21595i	0.793957	+	0.607974i	$0.208018\pi$
$487$	26.8336i	1.21595i	−0.793957	+	0.607974i	$0.791982\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.17336i	0.323729i	0.986813	+	0.161865i	$0.0517508\pi$
$491$	7.17336i	0.323729i	−0.986813	+	0.161865i	$0.948249\pi$
$492$	0	0
$493$	− 58.8374i	− 2.64990i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.44544i	− 0.199405i
$498$	0	0
$499$	36.8731	1.65067	0.825334	−	0.564646i	$-0.190987\pi$
$499$	36.8731	1.65067	0.825334	+	0.564646i	$0.190987\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.4855	0.779642	0.389821	−	0.920891i	$-0.372537\pi$
$503$	17.4855	0.779642	0.389821	+	0.920891i	$0.372537\pi$
$504$	0	0
$505$	52.4893	2.33574
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.6034	−1.71106	−0.855532	−	0.517750i	$-0.826770\pi$
$509$	−38.6034	−1.71106	−0.855532	+	0.517750i	$0.826770\pi$
$510$	0	0
$511$	− 6.97349i	− 0.308489i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.6133i	0.864264i
$516$	0	0
$517$	− 4.17960i	− 0.183819i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.3406i	1.28543i	0.766104	+	0.642716i	$0.222193\pi$
$521$	29.3406i	1.28543i	−0.766104	+	0.642716i	$0.777807\pi$
$522$	0	0
$523$	7.90612	0.345711	0.172855	−	0.984947i	$-0.444701\pi$
$523$	7.90612	0.345711	0.172855	+	0.984947i	$0.444701\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.22987	−0.314938
$528$	0	0
$529$	−3.18992	−0.138692
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−40.5851	−1.75793
$534$	0	0
$535$	49.4071i	2.13606i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.05843i	0.0886629i
$540$	0	0
$541$	6.25573i	0.268955i	0.990917	+	0.134477i	$0.0429356\pi$
$541$	6.25573i	0.268955i	−0.990917	+	0.134477i	$0.957064\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 21.8754i	− 0.937038i
$546$	0	0
$547$	−18.4597	−0.789279	−0.394640	−	0.918836i	$-0.629131\pi$
$547$	−18.4597	−0.789279	−0.394640	+	0.918836i	$0.629131\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	33.5703	1.43014
$552$	0	0
$553$	−11.6750	−0.496471
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.6497	−0.451243	−0.225622	−	0.974215i	$-0.572441\pi$
$557$	−10.6497	−0.451243	−0.225622	+	0.974215i	$0.572441\pi$
$558$	0	0
$559$	11.9475i	0.505327i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.9117i	0.459874i	0.973206	+	0.229937i	$0.0738520\pi$
$563$	10.9117i	0.459874i	−0.973206	+	0.229937i	$0.926148\pi$
$564$	0	0
$565$	− 68.5778i	− 2.88509i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 10.7145i	− 0.449175i	−0.974454	−	0.224588i	$-0.927896\pi$
$569$	− 10.7145i	− 0.449175i	0.974454	−	0.224588i	$-0.0721035\pi$
$570$	0	0
$571$	21.6134	0.904493	0.452247	−	0.891893i	$-0.350623\pi$
$571$	21.6134	0.904493	0.452247	+	0.891893i	$0.350623\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−38.5533	−1.60779
$576$	0	0
$577$	17.2449	0.717914	0.358957	−	0.933354i	$-0.383133\pi$
$577$	17.2449	0.717914	0.358957	+	0.933354i	$0.383133\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.8349	0.532482
$582$	0	0
$583$	20.7476i	0.859279i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.83868i	0.282263i	0.989991	+	0.141131i	$0.0450740\pi$
$587$	6.83868i	0.282263i	−0.989991	+	0.141131i	$0.954926\pi$
$588$	0	0
$589$	− 4.12508i	− 0.169971i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.2262i	0.953787i	0.878961	+	0.476893i	$0.158237\pi$
$593$	23.2262i	0.953787i	−0.878961	+	0.476893i	$0.841763\pi$
$594$	0	0
$595$	−21.1977	−0.869021
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.71919	−0.192821	−0.0964105	−	0.995342i	$-0.530736\pi$
$599$	−4.71919	−0.192821	−0.0964105	+	0.995342i	$0.530736\pi$
$600$	0	0
$601$	−0.230392	−0.00939787	−0.00469893	−	0.999989i	$-0.501496\pi$
$601$	−0.230392	−0.00939787	−0.00469893	+	0.999989i	$0.501496\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.9970	1.01627
$606$	0	0
$607$	− 5.43638i	− 0.220656i	−0.993895	−	0.110328i	$-0.964810\pi$
$607$	− 5.43638i	− 0.220656i	0.993895	−	0.110328i	$-0.0351901\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.44228i	0.381994i
$612$	0	0
$613$	24.7976i	1.00156i	0.865573	+	0.500782i	$0.166954\pi$
$613$	24.7976i	1.00156i	−0.865573	+	0.500782i	$0.833046\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13.4024i	− 0.539561i	−0.962922	−	0.269781i	$-0.913049\pi$
$617$	− 13.4024i	− 0.539561i	0.962922	−	0.269781i	$-0.0869512\pi$
$618$	0	0
$619$	12.8992	0.518465	0.259232	−	0.965815i	$-0.416530\pi$
$619$	12.8992	0.518465	0.259232	+	0.965815i	$0.416530\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.35419	−0.0943188
$624$	0	0
$625$	6.72041	0.268817
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.6696	−0.744405
$630$	0	0
$631$	24.0217i	0.956288i	0.878281	+	0.478144i	$0.158690\pi$
$631$	24.0217i	0.956288i	−0.878281	+	0.478144i	$0.841310\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	36.1107i	1.43301i
$636$	0	0
$637$	− 4.65027i	− 0.184250i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 10.4176i	− 0.411469i	−0.978608	−	0.205735i	$-0.934042\pi$
$641$	− 10.4176i	− 0.411469i	0.978608	−	0.205735i	$-0.0659583\pi$
$642$	0	0
$643$	−42.2723	−1.66706	−0.833528	−	0.552477i	$-0.813683\pi$
$643$	−42.2723	−1.66706	−0.833528	+	0.552477i	$0.813683\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.75700	−0.304959	−0.152480	−	0.988307i	$-0.548726\pi$
$647$	−7.75700	−0.304959	−0.152480	+	0.988307i	$0.548726\pi$
$648$	0	0
$649$	1.75216	0.0687784
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.9159	−0.779370	−0.389685	−	0.920948i	$-0.627416\pi$
$653$	−19.9159	−0.779370	−0.389685	+	0.920948i	$0.627416\pi$
$654$	0	0
$655$	− 57.5357i	− 2.24811i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	39.4456i	1.53658i	0.640101	+	0.768291i	$0.278892\pi$
$659$	39.4456i	1.53658i	−0.640101	+	0.768291i	$0.721108\pi$
$660$	0	0
$661$	35.8650i	1.39499i	0.716591	+	0.697494i	$0.245702\pi$
$661$	35.8650i	1.39499i	−0.716591	+	0.697494i	$0.754298\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 12.0946i	− 0.469007i
$666$	0	0
$667$	45.6631	1.76808
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.76420	−0.183920
$672$	0	0
$673$	30.5668	1.17827	0.589133	−	0.808036i	$-0.299470\pi$
$673$	30.5668	1.17827	0.589133	+	0.808036i	$0.299470\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.14804	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$677$	−7.14804	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$678$	0	0
$679$	4.02539i	0.154480i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.07293i	0.0793186i	0.999213	+	0.0396593i	$0.0126273\pi$
$683$	2.07293i	0.0793186i	−0.999213	+	0.0396593i	$0.987373\pi$
$684$	0	0
$685$	31.6013i	1.20742i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 46.8717i	− 1.78567i
$690$	0	0
$691$	−34.7325	−1.32129	−0.660644	−	0.750700i	$-0.729717\pi$
$691$	−34.7325	−1.32129	−0.660644	+	0.750700i	$0.729717\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	57.1460	2.16767
$696$	0	0
$697$	−50.0518	−1.89585
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.1113	1.13729	0.568643	−	0.822584i	$-0.307469\pi$
$701$	30.1113	1.13729	0.568643	+	0.822584i	$0.307469\pi$
$702$	0	0
$703$	− 10.6521i	− 0.401752i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.2008i	0.534077i
$708$	0	0
$709$	30.7194i	1.15369i	0.816853	+	0.576846i	$0.195717\pi$
$709$	30.7194i	1.15369i	−0.816853	+	0.576846i	$0.804283\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 5.61102i	− 0.210135i
$714$	0	0
$715$	35.3811	1.32318
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.93357	0.0721099	0.0360549	−	0.999350i	$-0.488521\pi$
$719$	1.93357	0.0721099	0.0360549	+	0.999350i	$0.488521\pi$
$720$	0	0
$721$	−5.30631	−0.197617
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−88.8671	−3.30044
$726$	0	0
$727$	21.4593i	0.795881i	0.917411	+	0.397940i	$0.130275\pi$
$727$	21.4593i	0.795881i	−0.917411	+	0.397940i	$0.869725\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.7344i	0.544970i
$732$	0	0
$733$	7.43101i	0.274471i	0.990538	+	0.137235i	$0.0438217\pi$
$733$	7.43101i	0.274471i	−0.990538	+	0.137235i	$0.956178\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.7592i	1.16987i
$738$	0	0
$739$	6.48141	0.238423	0.119211	−	0.992869i	$-0.461963\pi$
$739$	6.48141	0.238423	0.119211	+	0.992869i	$0.461963\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−50.4752	−1.85176	−0.925879	−	0.377821i	$-0.876673\pi$
$743$	−50.4752	−1.85176	−0.925879	+	0.377821i	$0.876673\pi$
$744$	0	0
$745$	32.9400	1.20683
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13.3669	−0.488417
$750$	0	0
$751$	− 34.8254i	− 1.27080i	−0.772185	−	0.635398i	$-0.780836\pi$
$751$	− 34.8254i	− 1.27080i	0.772185	−	0.635398i	$-0.219164\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	54.7189i	1.99142i
$756$	0	0
$757$	− 15.6422i	− 0.568524i	−0.958747	−	0.284262i	$-0.908251\pi$
$757$	− 15.6422i	− 0.568524i	0.958747	−	0.284262i	$-0.0917485\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 8.77808i	− 0.318205i	−0.987262	−	0.159103i	$-0.949140\pi$
$761$	− 8.77808i	− 0.318205i	0.987262	−	0.159103i	$-0.0508600\pi$
$762$	0	0
$763$	5.91832	0.214257
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.95837	−0.142928
$768$	0	0
$769$	20.4092	0.735975	0.367988	−	0.929831i	$-0.380047\pi$
$769$	20.4092	0.735975	0.367988	+	0.929831i	$0.380047\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.204759	−0.00736468	−0.00368234	−	0.999993i	$-0.501172\pi$
$773$	−0.204759	−0.00736468	−0.00368234	+	0.999993i	$0.501172\pi$
$774$	0	0
$775$	10.9199i	0.392254i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 28.5576i	− 1.02318i
$780$	0	0
$781$	9.15062i	0.327435i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 53.1205i	− 1.89595i
$786$	0	0
$787$	−51.9705	−1.85255	−0.926274	−	0.376850i	$-0.877007\pi$
$787$	−51.9705	−1.85255	−0.926274	+	0.376850i	$0.877007\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.5535	0.659687
$792$	0	0
$793$	10.7630	0.382204
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.9252	0.599522	0.299761	−	0.954014i	$-0.403093\pi$
$797$	16.9252	0.599522	0.299761	+	0.954014i	$0.403093\pi$
$798$	0	0
$799$	11.6448i	0.411962i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	14.3544i	0.506557i
$804$	0	0
$805$	− 16.4513i	− 0.579832i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.4440i	0.367191i	0.983002	+	0.183596i	$0.0587737\pi$
$809$	10.4440i	0.367191i	−0.983002	+	0.183596i	$0.941226\pi$
$810$	0	0
$811$	−15.2976	−0.537170	−0.268585	−	0.963256i	$-0.586556\pi$
$811$	−15.2976	−0.537170	−0.268585	+	0.963256i	$0.586556\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−80.5204	−2.82051
$816$	0	0
$817$	−8.40685	−0.294118
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.7719	0.724946	0.362473	−	0.931994i	$-0.381933\pi$
$821$	20.7719	0.724946	0.362473	+	0.931994i	$0.381933\pi$
$822$	0	0
$823$	6.01697i	0.209738i	0.994486	+	0.104869i	$0.0334424\pi$
$823$	6.01697i	0.209738i	−0.994486	+	0.104869i	$0.966558\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 53.8903i	− 1.87395i	−0.349399	−	0.936974i	$-0.613614\pi$
$827$	− 53.8903i	− 1.87395i	0.349399	−	0.936974i	$-0.386386\pi$
$828$	0	0
$829$	5.69863i	0.197922i	0.995091	+	0.0989608i	$0.0315519\pi$
$829$	5.69863i	0.197922i	−0.995091	+	0.0989608i	$0.968448\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.73497i	− 0.198705i
$834$	0	0
$835$	−49.5409	−1.71443
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.43315	−0.325668	−0.162834	−	0.986653i	$-0.552064\pi$
$839$	−9.43315	−0.325668	−0.162834	+	0.986653i	$0.552064\pi$
$840$	0	0
$841$	76.2553	2.62949
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−31.8798	−1.09670
$846$	0	0
$847$	6.76287i	0.232375i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 14.4893i	− 0.496686i
$852$	0	0
$853$	− 3.49254i	− 0.119582i	−0.998211	−	0.0597912i	$-0.980957\pi$
$853$	− 3.49254i	− 0.119582i	0.998211	−	0.0597912i	$-0.0190435\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.2047i	0.382745i	0.981517	+	0.191372i	$0.0612938\pi$
$857$	11.2047i	0.382745i	−0.981517	+	0.191372i	$0.938706\pi$
$858$	0	0
$859$	−43.4564	−1.48271	−0.741357	−	0.671111i	$-0.765817\pi$
$859$	−43.4564	−1.48271	−0.741357	+	0.671111i	$0.765817\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.8312	0.675063	0.337532	−	0.941314i	$-0.390408\pi$
$863$	19.8312	0.675063	0.337532	+	0.941314i	$0.390408\pi$
$864$	0	0
$865$	49.2412	1.67425
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.0321	0.815235
$870$	0	0
$871$	− 71.7483i	− 2.43110i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.5356i	0.457587i
$876$	0	0
$877$	− 40.6652i	− 1.37317i	−0.727051	−	0.686584i	$-0.759110\pi$
$877$	− 40.6652i	− 1.37317i	0.727051	−	0.686584i	$-0.240890\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	46.8298i	1.57774i	0.614562	+	0.788868i	$0.289333\pi$
$881$	46.8298i	1.57774i	−0.614562	+	0.788868i	$0.710667\pi$
$882$	0	0
$883$	18.8388	0.633975	0.316988	−	0.948430i	$-0.397329\pi$
$883$	18.8388	0.633975	0.316988	+	0.948430i	$0.397329\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.52873	0.0513297	0.0256648	−	0.999671i	$-0.491830\pi$
$887$	1.52873	0.0513297	0.0256648	+	0.999671i	$0.491830\pi$
$888$	0	0
$889$	−9.76963	−0.327663
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.64403	−0.222334
$894$	0	0
$895$	− 46.8631i	− 1.56646i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 12.9336i	− 0.431361i
$900$	0	0
$901$	− 57.8048i	− 1.92576i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 41.1222i	− 1.36695i
$906$	0	0
$907$	24.5682	0.815773	0.407886	−	0.913033i	$-0.366266\pi$
$907$	24.5682	0.815773	0.407886	+	0.913033i	$0.366266\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−14.8864	−0.493209	−0.246604	−	0.969116i	$-0.579315\pi$
$911$	−14.8864	−0.493209	−0.246604	+	0.969116i	$0.579315\pi$
$912$	0	0
$913$	−26.4198	−0.874367
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.5661	0.514038
$918$	0	0
$919$	− 55.6464i	− 1.83560i	−0.397038	−	0.917802i	$-0.629962\pi$
$919$	− 55.6464i	− 1.83560i	0.397038	−	0.917802i	$-0.370038\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 20.6725i	− 0.680443i
$924$	0	0
$925$	28.1982i	0.927153i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 29.3098i	− 0.961623i	−0.876824	−	0.480812i	$-0.840342\pi$
$929$	− 29.3098i	− 0.961623i	0.876824	−	0.480812i	$-0.159658\pi$
$930$	0	0
$931$	3.27215	0.107240
$932$	0	0
$933$	0	0
$934$	0	0
$935$	43.6340	1.42698
$936$	0	0
$937$	12.6626	0.413669	0.206834	−	0.978376i	$-0.433684\pi$
$937$	12.6626	0.413669	0.206834	+	0.978376i	$0.433684\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.6135	0.867575	0.433788	−	0.901015i	$-0.357177\pi$
$941$	26.6135	0.867575	0.433788	+	0.901015i	$0.357177\pi$
$942$	0	0
$943$	− 38.8447i	− 1.26496i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.05029i	− 0.131617i	−0.997832	−	0.0658084i	$-0.979037\pi$
$947$	− 4.05029i	− 0.131617i	0.997832	−	0.0658084i	$-0.0209626\pi$
$948$	0	0
$949$	− 32.4286i	− 1.05268i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 7.19155i	− 0.232957i	−0.993193	−	0.116479i	$-0.962839\pi$
$953$	− 7.19155i	− 0.232957i	0.993193	−	0.116479i	$-0.0371606\pi$
$954$	0	0
$955$	28.6969	0.928611
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.54964	−0.276082
$960$	0	0
$961$	29.4107	0.948733
$962$	0	0
$963$	0	0
$964$	0	0
$965$	33.6201	1.08227
$966$	0	0
$967$	− 46.3071i	− 1.48914i	−0.667547	−	0.744568i	$-0.732656\pi$
$967$	− 46.3071i	− 1.48914i	0.667547	−	0.744568i	$-0.267344\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 22.1689i	− 0.711432i	−0.934594	−	0.355716i	$-0.884237\pi$
$971$	− 22.1689i	− 0.711432i	0.934594	−	0.355716i	$-0.115763\pi$
$972$	0	0
$973$	15.4607i	0.495647i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 13.0854i	− 0.418638i	−0.977847	−	0.209319i	$-0.932875\pi$
$977$	− 13.0854i	− 0.418638i	0.977847	−	0.209319i	$-0.0671248\pi$
$978$	0	0
$979$	4.84594	0.154877
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14.9454	−0.476684	−0.238342	−	0.971181i	$-0.576604\pi$
$983$	−14.9454	−0.476684	−0.238342	+	0.971181i	$0.576604\pi$
$984$	0	0
$985$	38.3379	1.22155
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.4352	−0.363618
$990$	0	0
$991$	25.6587i	0.815075i	0.913188	+	0.407537i	$0.133612\pi$
$991$	25.6587i	0.815075i	−0.913188	+	0.407537i	$0.866388\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	36.6805i	1.16285i
$996$	0	0
$997$	10.1507i	0.321476i	0.986997	+	0.160738i	$0.0513874\pi$
$997$	10.1507i	0.321476i	−0.986997	+	0.160738i	$0.948613\pi$
$998$	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.5615.46		48
3.2	odd	2	inner	6048.2.j.d.5615.3		48
4.3	odd	2		1512.2.j.d.323.15	✓	48
8.3	odd	2	inner	6048.2.j.d.5615.4		48
8.5	even	2		1512.2.j.d.323.33	yes	48
12.11	even	2		1512.2.j.d.323.34	yes	48
24.5	odd	2		1512.2.j.d.323.16	yes	48
24.11	even	2	inner	6048.2.j.d.5615.45		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.15	✓	48	4.3	odd	2
1512.2.j.d.323.16	yes	48	24.5	odd	2
1512.2.j.d.323.33	yes	48	8.5	even	2
1512.2.j.d.323.34	yes	48	12.11	even	2
6048.2.j.d.5615.3		48	3.2	odd	2	inner
6048.2.j.d.5615.4		48	8.3	odd	2	inner
6048.2.j.d.5615.45		48	24.11	even	2	inner
6048.2.j.d.5615.46		48	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.69622 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+3.69622 q^{5} +1.00000i q^{7} -2.05843i q^{11} +4.65027i q^{13} +5.73497i q^{17} -3.27215 q^{19} -4.45085 q^{23} +8.66201 q^{25} -10.2594 q^{29} +1.26066i q^{31} +3.69622i q^{35} +3.25539i q^{37} +8.72747i q^{41} +2.56921 q^{43} +2.03048 q^{47} -1.00000 q^{49} -10.0794 q^{53} -7.60840i q^{55} +0.851213i q^{59} -2.31448i q^{61} +17.1884i q^{65} -15.4289 q^{67} -4.44544 q^{71} -6.97349 q^{73} +2.05843 q^{77} +11.6750i q^{79} -12.8349i q^{83} +21.1977i q^{85} +2.35419i q^{89} -4.65027 q^{91} -12.0946 q^{95} +4.02539 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more