LMFDB - Embedded newform 6048.2.j.d.5615.11

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			5615.11
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35025 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-2.35025 q^{5} +1.00000i q^{7} +1.09095i q^{11} +4.10613i q^{13} -4.74991i q^{17} +7.10196 q^{19} -3.99240 q^{23} +0.523653 q^{25} -4.09983 q^{29} -10.8933i q^{31} -2.35025i q^{35} -2.27336i q^{37} -1.29246i q^{41} +8.59200 q^{43} +6.79438 q^{47} -1.00000 q^{49} +0.688525 q^{53} -2.56400i q^{55} +7.72675i q^{59} +9.70563i q^{61} -9.65042i q^{65} -8.23313 q^{67} +6.46122 q^{71} +8.44644 q^{73} -1.09095 q^{77} +5.70061i q^{79} -10.0681i q^{83} +11.1634i q^{85} +2.54950i q^{89} -4.10613 q^{91} -16.6913 q^{95} -15.6447 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.35025	−1.05106	−0.525531	−	0.850775i	$-0.676133\pi$
$5$	−2.35025	−1.05106	−0.525531	+	0.850775i	$0.676133\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.09095i	0.328934i	0.986383	+	0.164467i	$0.0525904\pi$
$11$	1.09095i	0.328934i	−0.986383	+	0.164467i	$0.947410\pi$
$12$	0	0
$13$	4.10613i	1.13884i	0.822048	+	0.569418i	$0.192832\pi$
$13$	4.10613i	1.13884i	−0.822048	+	0.569418i	$0.807168\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.74991i	− 1.15202i	−0.817442	−	0.576011i	$-0.804609\pi$
$17$	− 4.74991i	− 1.15202i	0.817442	−	0.576011i	$-0.195391\pi$
$18$	0	0
$19$	7.10196	1.62930	0.814650	−	0.579953i	$-0.196929\pi$
$19$	7.10196	1.62930	0.814650	+	0.579953i	$0.196929\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.99240	−0.832474	−0.416237	−	0.909256i	$-0.636651\pi$
$23$	−3.99240	−0.832474	−0.416237	+	0.909256i	$0.636651\pi$
$24$	0	0
$25$	0.523653	0.104731
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.09983	−0.761319	−0.380660	−	0.924715i	$-0.624303\pi$
$29$	−4.09983	−0.761319	−0.380660	+	0.924715i	$0.624303\pi$
$30$	0	0
$31$	− 10.8933i	− 1.95649i	−0.207454	−	0.978245i	$-0.566518\pi$
$31$	− 10.8933i	− 1.95649i	0.207454	−	0.978245i	$-0.433482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.35025i	− 0.397264i
$36$	0	0
$37$	− 2.27336i	− 0.373737i	−0.982385	−	0.186869i	$-0.940166\pi$
$37$	− 2.27336i	− 0.373737i	0.982385	−	0.186869i	$-0.0598339\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.29246i	− 0.201848i	−0.994894	−	0.100924i	$-0.967820\pi$
$41$	− 1.29246i	− 0.201848i	0.994894	−	0.100924i	$-0.0321799\pi$
$42$	0	0
$43$	8.59200	1.31027	0.655134	−	0.755513i	$-0.272612\pi$
$43$	8.59200	1.31027	0.655134	+	0.755513i	$0.272612\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.79438	0.991062	0.495531	−	0.868590i	$-0.334974\pi$
$47$	6.79438	0.991062	0.495531	+	0.868590i	$0.334974\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.688525	0.0945762	0.0472881	−	0.998881i	$-0.484942\pi$
$53$	0.688525	0.0945762	0.0472881	+	0.998881i	$0.484942\pi$
$54$	0	0
$55$	− 2.56400i	− 0.345730i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.72675i	1.00594i	0.864305	+	0.502968i	$0.167759\pi$
$59$	7.72675i	1.00594i	−0.864305	+	0.502968i	$0.832241\pi$
$60$	0	0
$61$	9.70563i	1.24268i	0.783542	+	0.621339i	$0.213411\pi$
$61$	9.70563i	1.24268i	−0.783542	+	0.621339i	$0.786589\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 9.65042i	− 1.19699i
$66$	0	0
$67$	−8.23313	−1.00584	−0.502918	−	0.864334i	$-0.667740\pi$
$67$	−8.23313	−1.00584	−0.502918	+	0.864334i	$0.667740\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.46122	0.766806	0.383403	−	0.923581i	$-0.374752\pi$
$71$	6.46122	0.766806	0.383403	+	0.923581i	$0.374752\pi$
$72$	0	0
$73$	8.44644	0.988581	0.494290	−	0.869297i	$-0.335428\pi$
$73$	8.44644	0.988581	0.494290	+	0.869297i	$0.335428\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.09095	−0.124325
$78$	0	0
$79$	5.70061i	0.641369i	0.947186	+	0.320684i	$0.103913\pi$
$79$	5.70061i	0.641369i	−0.947186	+	0.320684i	$0.896087\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 10.0681i	− 1.10512i	−0.833474	−	0.552558i	$-0.813652\pi$
$83$	− 10.0681i	− 1.10512i	0.833474	−	0.552558i	$-0.186348\pi$
$84$	0	0
$85$	11.1634i	1.21085i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.54950i	0.270246i	0.990829	+	0.135123i	$0.0431430\pi$
$89$	2.54950i	0.270246i	−0.990829	+	0.135123i	$0.956857\pi$
$90$	0	0
$91$	−4.10613	−0.430440
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.6913	−1.71250
$96$	0	0
$97$	−15.6447	−1.58847	−0.794237	−	0.607607i	$-0.792129\pi$
$97$	−15.6447	−1.58847	−0.794237	+	0.607607i	$0.792129\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.3962	−1.23347	−0.616736	−	0.787170i	$-0.711546\pi$
$101$	−12.3962	−1.23347	−0.616736	+	0.787170i	$0.711546\pi$
$102$	0	0
$103$	13.2234i	1.30294i	0.758673	+	0.651472i	$0.225848\pi$
$103$	13.2234i	1.30294i	−0.758673	+	0.651472i	$0.774152\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.44530i	0.526417i	0.964739	+	0.263209i	$0.0847808\pi$
$107$	5.44530i	0.526417i	−0.964739	+	0.263209i	$0.915219\pi$
$108$	0	0
$109$	− 0.947776i	− 0.0907805i	−0.998969	−	0.0453902i	$-0.985547\pi$
$109$	− 0.947776i	− 0.0907805i	0.998969	−	0.0453902i	$-0.0144531\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 16.8819i	− 1.58812i	−0.607840	−	0.794059i	$-0.707964\pi$
$113$	− 16.8819i	− 1.58812i	0.607840	−	0.794059i	$-0.292036\pi$
$114$	0	0
$115$	9.38313	0.874981
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.74991	0.435423
$120$	0	0
$121$	9.80982	0.891802
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.5205	0.940983
$126$	0	0
$127$	10.2310i	0.907852i	0.891039	+	0.453926i	$0.149977\pi$
$127$	10.2310i	0.907852i	−0.891039	+	0.453926i	$0.850023\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.9013i	1.38930i	0.719346	+	0.694651i	$0.244441\pi$
$131$	15.9013i	1.38930i	−0.719346	+	0.694651i	$0.755559\pi$
$132$	0	0
$133$	7.10196i	0.615818i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.0619i	− 0.859647i	−0.902913	−	0.429823i	$-0.858576\pi$
$137$	− 10.0619i	− 0.859647i	0.902913	−	0.429823i	$-0.141424\pi$
$138$	0	0
$139$	4.44750	0.377232	0.188616	−	0.982051i	$-0.439600\pi$
$139$	4.44750	0.377232	0.188616	+	0.982051i	$0.439600\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.47959	−0.374602
$144$	0	0
$145$	9.63561	0.800193
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.3264	−1.00981	−0.504907	−	0.863174i	$-0.668473\pi$
$149$	−12.3264	−1.00981	−0.504907	+	0.863174i	$0.668473\pi$
$150$	0	0
$151$	9.25492i	0.753154i	0.926385	+	0.376577i	$0.122899\pi$
$151$	9.25492i	0.753154i	−0.926385	+	0.376577i	$0.877101\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	25.6019i	2.05639i
$156$	0	0
$157$	4.19774i	0.335016i	0.985871	+	0.167508i	$0.0535720\pi$
$157$	4.19774i	0.335016i	−0.985871	+	0.167508i	$0.946428\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.99240i	− 0.314645i
$162$	0	0
$163$	23.6952	1.85595	0.927977	−	0.372639i	$-0.121547\pi$
$163$	23.6952	1.85595	0.927977	+	0.372639i	$0.121547\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.79250	0.370855	0.185427	−	0.982658i	$-0.440633\pi$
$167$	4.79250	0.370855	0.185427	+	0.982658i	$0.440633\pi$
$168$	0	0
$169$	−3.86033	−0.296949
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.2897	1.39054	0.695271	−	0.718747i	$-0.255284\pi$
$173$	18.2897	1.39054	0.695271	+	0.718747i	$0.255284\pi$
$174$	0	0
$175$	0.523653i	0.0395844i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 6.64806i	− 0.496899i	−0.968645	−	0.248450i	$-0.920079\pi$
$179$	− 6.64806i	− 0.496899i	0.968645	−	0.248450i	$-0.0799210\pi$
$180$	0	0
$181$	− 2.46631i	− 0.183320i	−0.995790	−	0.0916598i	$-0.970783\pi$
$181$	− 2.46631i	− 0.183320i	0.995790	−	0.0916598i	$-0.0292172\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.34294i	0.392821i
$186$	0	0
$187$	5.18192	0.378939
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.9048	−0.933760	−0.466880	−	0.884321i	$-0.654622\pi$
$191$	−12.9048	−0.933760	−0.466880	+	0.884321i	$0.654622\pi$
$192$	0	0
$193$	18.3689	1.32222	0.661112	−	0.750287i	$-0.270085\pi$
$193$	18.3689	1.32222	0.661112	+	0.750287i	$0.270085\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.0957	1.07552	0.537761	−	0.843098i	$-0.319270\pi$
$197$	15.0957	1.07552	0.537761	+	0.843098i	$0.319270\pi$
$198$	0	0
$199$	3.51621i	0.249258i	0.992203	+	0.124629i	$0.0397740\pi$
$199$	3.51621i	0.249258i	−0.992203	+	0.124629i	$0.960226\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.09983i	− 0.287752i
$204$	0	0
$205$	3.03759i	0.212155i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.74789i	0.535933i
$210$	0	0
$211$	20.2502	1.39408	0.697041	−	0.717031i	$-0.254500\pi$
$211$	20.2502	1.39408	0.697041	+	0.717031i	$0.254500\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−20.1933	−1.37717
$216$	0	0
$217$	10.8933	0.739484
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.5038	1.31196
$222$	0	0
$223$	12.9963i	0.870295i	0.900359	+	0.435148i	$0.143304\pi$
$223$	12.9963i	0.870295i	−0.900359	+	0.435148i	$0.856696\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.8974i	1.91799i	0.283430	+	0.958993i	$0.408527\pi$
$227$	28.8974i	1.91799i	−0.283430	+	0.958993i	$0.591473\pi$
$228$	0	0
$229$	− 3.65438i	− 0.241488i	−0.992684	−	0.120744i	$-0.961472\pi$
$229$	− 3.65438i	− 0.241488i	0.992684	−	0.120744i	$-0.0385281\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.7096i	0.898143i	0.893496	+	0.449071i	$0.148245\pi$
$233$	13.7096i	0.898143i	−0.893496	+	0.449071i	$0.851755\pi$
$234$	0	0
$235$	−15.9685	−1.04167
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.5360	1.32836	0.664181	−	0.747572i	$-0.268780\pi$
$239$	20.5360	1.32836	0.664181	+	0.747572i	$0.268780\pi$
$240$	0	0
$241$	22.1694	1.42806	0.714030	−	0.700116i	$-0.246868\pi$
$241$	22.1694	1.42806	0.714030	+	0.700116i	$0.246868\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.35025	0.150152
$246$	0	0
$247$	29.1616i	1.85551i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.82901i	0.178565i	0.996006	+	0.0892827i	$0.0284575\pi$
$251$	2.82901i	0.178565i	−0.996006	+	0.0892827i	$0.971543\pi$
$252$	0	0
$253$	− 4.35552i	− 0.273829i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2158i	1.19865i	0.800505	+	0.599325i	$0.204564\pi$
$257$	19.2158i	1.19865i	−0.800505	+	0.599325i	$0.795436\pi$
$258$	0	0
$259$	2.27336	0.141259
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.5713	1.63846	0.819230	−	0.573465i	$-0.194401\pi$
$263$	26.5713	1.63846	0.819230	+	0.573465i	$0.194401\pi$
$264$	0	0
$265$	−1.61820	−0.0994054
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.88420	−0.480708	−0.240354	−	0.970685i	$-0.577264\pi$
$269$	−7.88420	−0.480708	−0.240354	+	0.970685i	$0.577264\pi$
$270$	0	0
$271$	− 15.1206i	− 0.918513i	−0.888304	−	0.459256i	$-0.848116\pi$
$271$	− 15.1206i	− 0.918513i	0.888304	−	0.459256i	$-0.151884\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.571280i	0.0344495i
$276$	0	0
$277$	10.1215i	0.608141i	0.952650	+	0.304071i	$0.0983459\pi$
$277$	10.1215i	0.608141i	−0.952650	+	0.304071i	$0.901654\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 23.6645i	− 1.41170i	−0.708359	−	0.705852i	$-0.750564\pi$
$281$	− 23.6645i	− 1.41170i	0.708359	−	0.705852i	$-0.249436\pi$
$282$	0	0
$283$	−18.1838	−1.08091	−0.540457	−	0.841371i	$-0.681749\pi$
$283$	−18.1838	−1.08091	−0.540457	+	0.841371i	$0.681749\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.29246	0.0762914
$288$	0	0
$289$	−5.56163	−0.327155
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.8327	−1.21706	−0.608529	−	0.793532i	$-0.708240\pi$
$293$	−20.8327	−1.21706	−0.608529	+	0.793532i	$0.708240\pi$
$294$	0	0
$295$	− 18.1597i	− 1.05730i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 16.3933i	− 0.948052i
$300$	0	0
$301$	8.59200i	0.495235i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 22.8106i	− 1.30613i
$306$	0	0
$307$	4.57939	0.261360	0.130680	−	0.991425i	$-0.458284\pi$
$307$	4.57939	0.261360	0.130680	+	0.991425i	$0.458284\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.20035	0.351589	0.175795	−	0.984427i	$-0.443751\pi$
$311$	6.20035	0.351589	0.175795	+	0.984427i	$0.443751\pi$
$312$	0	0
$313$	−22.0230	−1.24481	−0.622406	−	0.782695i	$-0.713845\pi$
$313$	−22.0230	−1.24481	−0.622406	+	0.782695i	$0.713845\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.35768	−0.132420	−0.0662102	−	0.997806i	$-0.521091\pi$
$317$	−2.35768	−0.132420	−0.0662102	+	0.997806i	$0.521091\pi$
$318$	0	0
$319$	− 4.47272i	− 0.250424i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 33.7336i	− 1.87699i
$324$	0	0
$325$	2.15019i	0.119271i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.79438i	0.374586i
$330$	0	0
$331$	−2.15331	−0.118357	−0.0591784	−	0.998247i	$-0.518848\pi$
$331$	−2.15331	−0.118357	−0.0591784	+	0.998247i	$0.518848\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.3499	1.05720
$336$	0	0
$337$	23.2469	1.26634	0.633169	−	0.774014i	$-0.281754\pi$
$337$	23.2469	1.26634	0.633169	+	0.774014i	$0.281754\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.8840	0.643556
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.1260i	1.40252i	0.712906	+	0.701260i	$0.247379\pi$
$347$	26.1260i	1.40252i	−0.712906	+	0.701260i	$0.752621\pi$
$348$	0	0
$349$	− 3.68463i	− 0.197234i	−0.995125	−	0.0986169i	$-0.968558\pi$
$349$	− 3.68463i	− 0.197234i	0.995125	−	0.0986169i	$-0.0314418\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.5433i	0.827285i	0.910439	+	0.413642i	$0.135744\pi$
$353$	15.5433i	0.827285i	−0.910439	+	0.413642i	$0.864256\pi$
$354$	0	0
$355$	−15.1855	−0.805960
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.0367	1.21583	0.607914	−	0.794003i	$-0.292006\pi$
$359$	23.0367	1.21583	0.607914	+	0.794003i	$0.292006\pi$
$360$	0	0
$361$	31.4378	1.65462
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.8512	−1.03906
$366$	0	0
$367$	16.1135i	0.841116i	0.907266	+	0.420558i	$0.138166\pi$
$367$	16.1135i	0.841116i	−0.907266	+	0.420558i	$0.861834\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.688525i	0.0357464i
$372$	0	0
$373$	8.37018i	0.433392i	0.976239	+	0.216696i	$0.0695280\pi$
$373$	8.37018i	0.433392i	−0.976239	+	0.216696i	$0.930472\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 16.8344i	− 0.867018i
$378$	0	0
$379$	4.66739	0.239748	0.119874	−	0.992789i	$-0.461751\pi$
$379$	4.66739	0.239748	0.119874	+	0.992789i	$0.461751\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.1113	−0.567761	−0.283880	−	0.958860i	$-0.591622\pi$
$383$	−11.1113	−0.567761	−0.283880	+	0.958860i	$0.591622\pi$
$384$	0	0
$385$	2.56400	0.130674
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.8483	1.71618	0.858090	−	0.513500i	$-0.171651\pi$
$389$	33.8483	1.71618	0.858090	+	0.513500i	$0.171651\pi$
$390$	0	0
$391$	18.9636i	0.959028i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 13.3978i	− 0.674118i
$396$	0	0
$397$	30.2554i	1.51848i	0.650813	+	0.759238i	$0.274428\pi$
$397$	30.2554i	1.51848i	−0.650813	+	0.759238i	$0.725572\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.7603i	1.78578i	0.450272	+	0.892891i	$0.351327\pi$
$401$	35.7603i	1.78578i	−0.450272	+	0.892891i	$0.648673\pi$
$402$	0	0
$403$	44.7292	2.22812
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.48012	0.122935
$408$	0	0
$409$	25.9283	1.28207	0.641036	−	0.767511i	$-0.278505\pi$
$409$	25.9283	1.28207	0.641036	+	0.767511i	$0.278505\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.72675	−0.380208
$414$	0	0
$415$	23.6625i	1.16155i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.32586i	0.309038i	0.987990	+	0.154519i	$0.0493829\pi$
$419$	6.32586i	0.309038i	−0.987990	+	0.154519i	$0.950617\pi$
$420$	0	0
$421$	− 27.3586i	− 1.33338i	−0.745336	−	0.666689i	$-0.767711\pi$
$421$	− 27.3586i	− 1.33338i	0.745336	−	0.666689i	$-0.232289\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 2.48730i	− 0.120652i
$426$	0	0
$427$	−9.70563	−0.469688
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.5526	−1.32716	−0.663582	−	0.748104i	$-0.730965\pi$
$431$	−27.5526	−1.32716	−0.663582	+	0.748104i	$0.730965\pi$
$432$	0	0
$433$	13.0479	0.627040	0.313520	−	0.949582i	$-0.398492\pi$
$433$	13.0479	0.627040	0.313520	+	0.949582i	$0.398492\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−28.3539	−1.35635
$438$	0	0
$439$	6.42338i	0.306571i	0.988182	+	0.153286i	$0.0489854\pi$
$439$	6.42338i	0.306571i	−0.988182	+	0.153286i	$0.951015\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 22.1824i	− 1.05392i	−0.849891	−	0.526959i	$-0.823332\pi$
$443$	− 22.1824i	− 1.05392i	0.849891	−	0.526959i	$-0.176668\pi$
$444$	0	0
$445$	− 5.99195i	− 0.284046i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.86593i	− 0.0880586i	−0.999030	−	0.0440293i	$-0.985981\pi$
$449$	− 1.86593i	− 0.0880586i	0.999030	−	0.0440293i	$-0.0140195\pi$
$450$	0	0
$451$	1.41001	0.0663947
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.65042	0.452419
$456$	0	0
$457$	−28.3835	−1.32773	−0.663863	−	0.747854i	$-0.731084\pi$
$457$	−28.3835	−1.32773	−0.663863	+	0.747854i	$0.731084\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−33.2364	−1.54797	−0.773986	−	0.633202i	$-0.781740\pi$
$461$	−33.2364	−1.54797	−0.773986	+	0.633202i	$0.781740\pi$
$462$	0	0
$463$	2.11779i	0.0984219i	0.998788	+	0.0492110i	$0.0156707\pi$
$463$	2.11779i	0.0984219i	−0.998788	+	0.0492110i	$0.984329\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 13.7416i	− 0.635884i	−0.948110	−	0.317942i	$-0.897008\pi$
$467$	− 13.7416i	− 0.635884i	0.948110	−	0.317942i	$-0.102992\pi$
$468$	0	0
$469$	− 8.23313i	− 0.380171i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.37345i	0.430992i
$474$	0	0
$475$	3.71896	0.170638
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.78287	−0.401299	−0.200650	−	0.979663i	$-0.564305\pi$
$479$	−8.78287	−0.401299	−0.200650	+	0.979663i	$0.564305\pi$
$480$	0	0
$481$	9.33470	0.425626
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.7688	1.66959
$486$	0	0
$487$	− 4.42586i	− 0.200555i	−0.994960	−	0.100277i	$-0.968027\pi$
$487$	− 4.42586i	− 0.200555i	0.994960	−	0.100277i	$-0.0319730\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.5511i	0.837199i	0.908171	+	0.418599i	$0.137479\pi$
$491$	18.5511i	0.837199i	−0.908171	+	0.418599i	$0.862521\pi$
$492$	0	0
$493$	19.4738i	0.877056i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.46122i	0.289825i
$498$	0	0
$499$	−5.23350	−0.234284	−0.117142	−	0.993115i	$-0.537373\pi$
$499$	−5.23350	−0.234284	−0.117142	+	0.993115i	$0.537373\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.2800	1.21635	0.608177	−	0.793802i	$-0.291901\pi$
$503$	27.2800	1.21635	0.608177	+	0.793802i	$0.291901\pi$
$504$	0	0
$505$	29.1342	1.29646
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.0131	−1.50761	−0.753803	−	0.657101i	$-0.771783\pi$
$509$	−34.0131	−1.50761	−0.753803	+	0.657101i	$0.771783\pi$
$510$	0	0
$511$	8.44644i	0.373648i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 31.0783i	− 1.36947i
$516$	0	0
$517$	7.41234i	0.325994i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.1052i	1.49417i	0.664726	+	0.747087i	$0.268548\pi$
$521$	34.1052i	1.49417i	−0.664726	+	0.747087i	$0.731452\pi$
$522$	0	0
$523$	−24.7371	−1.08168	−0.540840	−	0.841126i	$-0.681894\pi$
$523$	−24.7371	−1.08168	−0.540840	+	0.841126i	$0.681894\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−51.7420	−2.25392
$528$	0	0
$529$	−7.06071	−0.306987
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.30700	0.229872
$534$	0	0
$535$	− 12.7978i	− 0.553297i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.09095i	− 0.0469906i
$540$	0	0
$541$	1.11043i	0.0477412i	0.999715	+	0.0238706i	$0.00759897\pi$
$541$	1.11043i	0.0477412i	−0.999715	+	0.0238706i	$0.992401\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.22751i	0.0954159i
$546$	0	0
$547$	43.6930	1.86818	0.934089	−	0.357040i	$-0.116214\pi$
$547$	43.6930	1.86818	0.934089	+	0.357040i	$0.116214\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29.1168	−1.24042
$552$	0	0
$553$	−5.70061	−0.242415
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.0536	1.40053	0.700263	−	0.713884i	$-0.253066\pi$
$557$	33.0536	1.40053	0.700263	+	0.713884i	$0.253066\pi$
$558$	0	0
$559$	35.2799i	1.49218i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.6097i	0.742159i	0.928601	+	0.371080i	$0.121012\pi$
$563$	17.6097i	0.742159i	−0.928601	+	0.371080i	$0.878988\pi$
$564$	0	0
$565$	39.6767i	1.66921i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12.0355i	− 0.504555i	−0.967655	−	0.252277i	$-0.918820\pi$
$569$	− 12.0355i	− 0.504555i	0.967655	−	0.252277i	$-0.0811795\pi$
$570$	0	0
$571$	6.04104	0.252810	0.126405	−	0.991979i	$-0.459656\pi$
$571$	6.04104	0.252810	0.126405	+	0.991979i	$0.459656\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.09063	−0.0871854
$576$	0	0
$577$	−31.2605	−1.30139	−0.650696	−	0.759339i	$-0.725523\pi$
$577$	−31.2605	−1.30139	−0.650696	+	0.759339i	$0.725523\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.0681	0.417695
$582$	0	0
$583$	0.751147i	0.0311093i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 29.7168i	− 1.22654i	−0.789872	−	0.613272i	$-0.789853\pi$
$587$	− 29.7168i	− 1.22654i	0.789872	−	0.613272i	$-0.210147\pi$
$588$	0	0
$589$	− 77.3635i	− 3.18771i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 19.9618i	− 0.819732i	−0.912146	−	0.409866i	$-0.865575\pi$
$593$	− 19.9618i	− 0.819732i	0.912146	−	0.409866i	$-0.134425\pi$
$594$	0	0
$595$	−11.1634	−0.457657
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.5569	−1.08509	−0.542543	−	0.840028i	$-0.682539\pi$
$599$	−26.5569	−1.08509	−0.542543	+	0.840028i	$0.682539\pi$
$600$	0	0
$601$	−19.6439	−0.801289	−0.400645	−	0.916233i	$-0.631214\pi$
$601$	−19.6439	−0.801289	−0.400645	+	0.916233i	$0.631214\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−23.0555	−0.937339
$606$	0	0
$607$	29.0877i	1.18063i	0.807172	+	0.590316i	$0.200997\pi$
$607$	29.0877i	1.18063i	−0.807172	+	0.590316i	$0.799003\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.8986i	1.12866i
$612$	0	0
$613$	9.44223i	0.381368i	0.981651	+	0.190684i	$0.0610706\pi$
$613$	9.44223i	0.381368i	−0.981651	+	0.190684i	$0.938929\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.4177i	1.18431i	0.805823	+	0.592157i	$0.201723\pi$
$617$	29.4177i	1.18431i	−0.805823	+	0.592157i	$0.798277\pi$
$618$	0	0
$619$	−4.45956	−0.179245	−0.0896225	−	0.995976i	$-0.528566\pi$
$619$	−4.45956	−0.179245	−0.0896225	+	0.995976i	$0.528566\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.54950	−0.102144
$624$	0	0
$625$	−27.3441	−1.09376
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10.7982	−0.430554
$630$	0	0
$631$	− 23.6013i	− 0.939553i	−0.882786	−	0.469776i	$-0.844335\pi$
$631$	− 23.6013i	− 0.939553i	0.882786	−	0.469776i	$-0.155665\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 24.0453i	− 0.954208i
$636$	0	0
$637$	− 4.10613i	− 0.162691i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 6.81316i	− 0.269104i	−0.990907	−	0.134552i	$-0.957041\pi$
$641$	− 6.81316i	− 0.269104i	0.990907	−	0.134552i	$-0.0429595\pi$
$642$	0	0
$643$	15.8069	0.623361	0.311681	−	0.950187i	$-0.399108\pi$
$643$	15.8069	0.623361	0.311681	+	0.950187i	$0.399108\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.5205	1.67165	0.835827	−	0.548992i	$-0.184988\pi$
$647$	42.5205	1.67165	0.835827	+	0.548992i	$0.184988\pi$
$648$	0	0
$649$	−8.42950	−0.330887
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.40359	−0.0940597	−0.0470298	−	0.998893i	$-0.514976\pi$
$653$	−2.40359	−0.0940597	−0.0470298	+	0.998893i	$0.514976\pi$
$654$	0	0
$655$	− 37.3720i	− 1.46024i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 26.3292i	− 1.02564i	−0.858497	−	0.512819i	$-0.828601\pi$
$659$	− 26.3292i	− 1.02564i	0.858497	−	0.512819i	$-0.171399\pi$
$660$	0	0
$661$	− 8.09353i	− 0.314802i	−0.987535	−	0.157401i	$-0.949689\pi$
$661$	− 8.09353i	− 0.314802i	0.987535	−	0.157401i	$-0.0503115\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 16.6913i	− 0.647262i
$666$	0	0
$667$	16.3682	0.633778
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.5884	−0.408759
$672$	0	0
$673$	3.41009	0.131449	0.0657246	−	0.997838i	$-0.479064\pi$
$673$	3.41009	0.131449	0.0657246	+	0.997838i	$0.479064\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.4265	0.477589	0.238794	−	0.971070i	$-0.423248\pi$
$677$	12.4265	0.477589	0.238794	+	0.971070i	$0.423248\pi$
$678$	0	0
$679$	− 15.6447i	− 0.600387i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 19.4879i	− 0.745682i	−0.927895	−	0.372841i	$-0.878384\pi$
$683$	− 19.4879i	− 0.745682i	0.927895	−	0.372841i	$-0.121616\pi$
$684$	0	0
$685$	23.6480i	0.903542i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.82717i	0.107707i
$690$	0	0
$691$	−2.44853	−0.0931464	−0.0465732	−	0.998915i	$-0.514830\pi$
$691$	−2.44853	−0.0931464	−0.0465732	+	0.998915i	$0.514830\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.4527	−0.396494
$696$	0	0
$697$	−6.13906	−0.232533
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−21.8472	−0.825157	−0.412578	−	0.910922i	$-0.635372\pi$
$701$	−21.8472	−0.825157	−0.412578	+	0.910922i	$0.635372\pi$
$702$	0	0
$703$	− 16.1453i	− 0.608930i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.3962i	− 0.466209i
$708$	0	0
$709$	− 50.4998i	− 1.89656i	−0.317437	−	0.948279i	$-0.602822\pi$
$709$	− 50.4998i	− 1.89656i	0.317437	−	0.948279i	$-0.397178\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	43.4903i	1.62873i
$714$	0	0
$715$	10.5281	0.393730
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13.7799	0.513904	0.256952	−	0.966424i	$-0.417282\pi$
$719$	13.7799	0.513904	0.256952	+	0.966424i	$0.417282\pi$
$720$	0	0
$721$	−13.2234	−0.492467
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.14689	−0.0797334
$726$	0	0
$727$	− 12.5617i	− 0.465888i	−0.972490	−	0.232944i	$-0.925164\pi$
$727$	− 12.5617i	− 0.465888i	0.972490	−	0.232944i	$-0.0748358\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 40.8112i	− 1.50946i
$732$	0	0
$733$	1.04203i	0.0384883i	0.999815	+	0.0192442i	$0.00612599\pi$
$733$	1.04203i	0.0384883i	−0.999815	+	0.0192442i	$0.993874\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.98195i	− 0.330854i
$738$	0	0
$739$	−3.73631	−0.137442	−0.0687212	−	0.997636i	$-0.521892\pi$
$739$	−3.73631	−0.137442	−0.0687212	+	0.997636i	$0.521892\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.44956	−0.163239	−0.0816193	−	0.996664i	$-0.526009\pi$
$743$	−4.44956	−0.163239	−0.0816193	+	0.996664i	$0.526009\pi$
$744$	0	0
$745$	28.9700	1.06138
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.44530	−0.198967
$750$	0	0
$751$	− 23.5520i	− 0.859426i	−0.902965	−	0.429713i	$-0.858615\pi$
$751$	− 23.5520i	− 0.859426i	0.902965	−	0.429713i	$-0.141385\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 21.7513i	− 0.791612i
$756$	0	0
$757$	16.5240i	0.600576i	0.953849	+	0.300288i	$0.0970829\pi$
$757$	16.5240i	0.600576i	−0.953849	+	0.300288i	$0.902917\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.27488i	0.336214i	0.985769	+	0.168107i	$0.0537655\pi$
$761$	9.27488i	0.336214i	−0.985769	+	0.168107i	$0.946235\pi$
$762$	0	0
$763$	0.947776	0.0343118
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.7270	−1.14560
$768$	0	0
$769$	18.2106	0.656690	0.328345	−	0.944558i	$-0.393509\pi$
$769$	18.2106	0.656690	0.328345	+	0.944558i	$0.393509\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.51105	0.0543486	0.0271743	−	0.999631i	$-0.491349\pi$
$773$	1.51105	0.0543486	0.0271743	+	0.999631i	$0.491349\pi$
$774$	0	0
$775$	− 5.70429i	− 0.204904i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 9.17898i	− 0.328871i
$780$	0	0
$781$	7.04888i	0.252229i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 9.86571i	− 0.352122i
$786$	0	0
$787$	3.25985	0.116201	0.0581005	−	0.998311i	$-0.481496\pi$
$787$	3.25985	0.116201	0.0581005	+	0.998311i	$0.481496\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.8819	0.600252
$792$	0	0
$793$	−39.8526	−1.41521
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.5236	1.78964	0.894819	−	0.446428i	$-0.147304\pi$
$797$	50.5236	1.78964	0.894819	+	0.446428i	$0.147304\pi$
$798$	0	0
$799$	− 32.2727i	− 1.14172i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.21465i	0.325178i
$804$	0	0
$805$	9.38313i	0.330712i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17.9908i	0.632523i	0.948672	+	0.316261i	$0.102428\pi$
$809$	17.9908i	0.632523i	−0.948672	+	0.316261i	$0.897572\pi$
$810$	0	0
$811$	−22.2314	−0.780650	−0.390325	−	0.920677i	$-0.627637\pi$
$811$	−22.2314	−0.780650	−0.390325	+	0.920677i	$0.627637\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55.6896	−1.95072
$816$	0	0
$817$	61.0200	2.13482
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.7163	1.00221	0.501103	−	0.865388i	$-0.332928\pi$
$821$	28.7163	1.00221	0.501103	+	0.865388i	$0.332928\pi$
$822$	0	0
$823$	− 46.1721i	− 1.60946i	−0.593641	−	0.804730i	$-0.702310\pi$
$823$	− 46.1721i	− 1.60946i	0.593641	−	0.804730i	$-0.297690\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 55.5900i	− 1.93305i	−0.256568	−	0.966526i	$-0.582592\pi$
$827$	− 55.5900i	− 1.93305i	0.256568	−	0.966526i	$-0.417408\pi$
$828$	0	0
$829$	7.13019i	0.247642i	0.992305	+	0.123821i	$0.0395148\pi$
$829$	7.13019i	0.247642i	−0.992305	+	0.123821i	$0.960485\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.74991i	0.164575i
$834$	0	0
$835$	−11.2636	−0.389791
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33.3725	1.15215	0.576073	−	0.817399i	$-0.304585\pi$
$839$	33.3725	1.15215	0.576073	+	0.817399i	$0.304585\pi$
$840$	0	0
$841$	−12.1914	−0.420393
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.07273	0.312111
$846$	0	0
$847$	9.80982i	0.337070i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.07615i	0.311127i
$852$	0	0
$853$	34.4031i	1.17794i	0.808155	+	0.588969i	$0.200466\pi$
$853$	34.4031i	1.17794i	−0.808155	+	0.588969i	$0.799534\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.5369i	− 0.838164i	−0.907948	−	0.419082i	$-0.862352\pi$
$857$	− 24.5369i	− 0.838164i	0.907948	−	0.419082i	$-0.137648\pi$
$858$	0	0
$859$	−49.8496	−1.70085	−0.850424	−	0.526098i	$-0.823655\pi$
$859$	−49.8496	−1.70085	−0.850424	+	0.526098i	$0.823655\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.18405	0.0403055	0.0201528	−	0.999797i	$-0.493585\pi$
$863$	1.18405	0.0403055	0.0201528	+	0.999797i	$0.493585\pi$
$864$	0	0
$865$	−42.9854	−1.46155
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.21909	−0.210968
$870$	0	0
$871$	− 33.8063i	− 1.14548i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.5205i	0.355658i
$876$	0	0
$877$	52.6756i	1.77873i	0.457200	+	0.889364i	$0.348853\pi$
$877$	52.6756i	1.77873i	−0.457200	+	0.889364i	$0.651147\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 48.5193i	− 1.63466i	−0.576172	−	0.817329i	$-0.695454\pi$
$881$	− 48.5193i	− 1.63466i	0.576172	−	0.817329i	$-0.304546\pi$
$882$	0	0
$883$	−36.3496	−1.22326	−0.611630	−	0.791144i	$-0.709486\pi$
$883$	−36.3496	−1.22326	−0.611630	+	0.791144i	$0.709486\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.5979	−0.490149	−0.245075	−	0.969504i	$-0.578812\pi$
$887$	−14.5979	−0.490149	−0.245075	+	0.969504i	$0.578812\pi$
$888$	0	0
$889$	−10.2310	−0.343136
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.2534	1.61474
$894$	0	0
$895$	15.6246i	0.522272i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	44.6606i	1.48951i
$900$	0	0
$901$	− 3.27043i	− 0.108954i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.79644i	0.192680i
$906$	0	0
$907$	−3.77723	−0.125421	−0.0627104	−	0.998032i	$-0.519974\pi$
$907$	−3.77723	−0.125421	−0.0627104	+	0.998032i	$0.519974\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.5255	0.680040	0.340020	−	0.940418i	$-0.389566\pi$
$911$	20.5255	0.680040	0.340020	+	0.940418i	$0.389566\pi$
$912$	0	0
$913$	10.9838	0.363511
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.9013	−0.525107
$918$	0	0
$919$	19.8082i	0.653413i	0.945126	+	0.326706i	$0.105939\pi$
$919$	19.8082i	0.653413i	−0.945126	+	0.326706i	$0.894061\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	26.5306i	0.873267i
$924$	0	0
$925$	− 1.19045i	− 0.0391417i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 2.90178i	− 0.0952044i	−0.998866	−	0.0476022i	$-0.984842\pi$
$929$	− 2.90178i	− 0.0952044i	0.998866	−	0.0476022i	$-0.0151580\pi$
$930$	0	0
$931$	−7.10196	−0.232757
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.1788	−0.398289
$936$	0	0
$937$	−9.02197	−0.294735	−0.147368	−	0.989082i	$-0.547080\pi$
$937$	−9.02197	−0.294735	−0.147368	+	0.989082i	$0.547080\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.0744	−0.850000	−0.425000	−	0.905193i	$-0.639726\pi$
$941$	−26.0744	−0.850000	−0.425000	+	0.905193i	$0.639726\pi$
$942$	0	0
$943$	5.16001i	0.168033i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.0960i	1.49792i	0.662616	+	0.748960i	$0.269446\pi$
$947$	46.0960i	1.49792i	−0.662616	+	0.748960i	$0.730554\pi$
$948$	0	0
$949$	34.6822i	1.12583i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43.0803i	1.39551i	0.716338	+	0.697754i	$0.245817\pi$
$953$	43.0803i	1.39551i	−0.716338	+	0.697754i	$0.754183\pi$
$954$	0	0
$955$	30.3295	0.981439
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.0619	0.324916
$960$	0	0
$961$	−87.6634	−2.82785
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−43.1715	−1.38974
$966$	0	0
$967$	0.389484i	0.0125249i	0.999980	+	0.00626247i	$0.00199342\pi$
$967$	0.389484i	0.0125249i	−0.999980	+	0.00626247i	$0.998007\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.1601i	1.32089i	0.750875	+	0.660445i	$0.229632\pi$
$971$	41.1601i	1.32089i	−0.750875	+	0.660445i	$0.770368\pi$
$972$	0	0
$973$	4.44750i	0.142580i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.9292i	1.50140i	0.660644	+	0.750699i	$0.270283\pi$
$977$	46.9292i	1.50140i	−0.660644	+	0.750699i	$0.729717\pi$
$978$	0	0
$979$	−2.78138	−0.0888933
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.0802	1.24646	0.623232	−	0.782037i	$-0.285819\pi$
$983$	39.0802	1.24646	0.623232	+	0.782037i	$0.285819\pi$
$984$	0	0
$985$	−35.4785	−1.13044
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34.3027	−1.09076
$990$	0	0
$991$	23.1043i	0.733932i	0.930234	+	0.366966i	$0.119603\pi$
$991$	23.1043i	0.733932i	−0.930234	+	0.366966i	$0.880397\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 8.26396i	− 0.261985i
$996$	0	0
$997$	− 47.9056i	− 1.51719i	−0.651565	−	0.758593i	$-0.725887\pi$
$997$	− 47.9056i	− 1.51719i	0.651565	−	0.758593i	$-0.274113\pi$
$998$	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.5615.11		48
3.2	odd	2	inner	6048.2.j.d.5615.38		48
4.3	odd	2		1512.2.j.d.323.37	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.37		48
8.5	even	2		1512.2.j.d.323.11	✓	48
12.11	even	2		1512.2.j.d.323.12	yes	48
24.5	odd	2		1512.2.j.d.323.38	yes	48
24.11	even	2	inner	6048.2.j.d.5615.12		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.11	✓	48	8.5	even	2
1512.2.j.d.323.12	yes	48	12.11	even	2
1512.2.j.d.323.37	yes	48	4.3	odd	2
1512.2.j.d.323.38	yes	48	24.5	odd	2
6048.2.j.d.5615.11		48	1.1	even	1	trivial
6048.2.j.d.5615.12		48	24.11	even	2	inner
6048.2.j.d.5615.37		48	8.3	odd	2	inner
6048.2.j.d.5615.38		48	3.2	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)

\(f(q)\)	\(=\)	\(q-2.35025 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-2.35025 q^{5} +1.00000i q^{7} +1.09095i q^{11} +4.10613i q^{13} -4.74991i q^{17} +7.10196 q^{19} -3.99240 q^{23} +0.523653 q^{25} -4.09983 q^{29} -10.8933i q^{31} -2.35025i q^{35} -2.27336i q^{37} -1.29246i q^{41} +8.59200 q^{43} +6.79438 q^{47} -1.00000 q^{49} +0.688525 q^{53} -2.56400i q^{55} +7.72675i q^{59} +9.70563i q^{61} -9.65042i q^{65} -8.23313 q^{67} +6.46122 q^{71} +8.44644 q^{73} -1.09095 q^{77} +5.70061i q^{79} -10.0681i q^{83} +11.1634i q^{85} +2.54950i q^{89} -4.10613 q^{91} -16.6913 q^{95} -15.6447 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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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