LMFDB - Embedded newform 384.4.j.b.289.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(97,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("384.97");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.j (of order $4$, degree $2$, 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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			289.8
Character	$\chi$	$=$	384.289
Dual form			384.4.j.b.97.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.12132 + 2.12132i) q^{3} +(3.72414 - 3.72414i) q^{5} +20.2675i q^{7} +9.00000i q^{9} +O(q^{10})$	\(q+(2.12132 + 2.12132i) q^{3} +(3.72414 - 3.72414i) q^{5} +20.2675i q^{7} +9.00000i q^{9} +(47.5467 - 47.5467i) q^{11} +(-27.8636 - 27.8636i) q^{13} +15.8002 q^{15} +56.3788 q^{17} +(66.1489 + 66.1489i) q^{19} +(-42.9939 + 42.9939i) q^{21} -3.66579i q^{23} +97.2615i q^{25} +(-19.0919 + 19.0919i) q^{27} +(86.8428 + 86.8428i) q^{29} -102.846 q^{31} +201.724 q^{33} +(75.4791 + 75.4791i) q^{35} +(-66.8267 + 66.8267i) q^{37} -118.215i q^{39} +29.5734i q^{41} +(372.929 - 372.929i) q^{43} +(33.5173 + 33.5173i) q^{45} +539.082 q^{47} -67.7725 q^{49} +(119.598 + 119.598i) q^{51} +(-385.376 + 385.376i) q^{53} -354.142i q^{55} +280.646i q^{57} +(-71.0380 + 71.0380i) q^{59} +(155.133 + 155.133i) q^{61} -182.408 q^{63} -207.536 q^{65} +(178.141 + 178.141i) q^{67} +(7.77631 - 7.77631i) q^{69} +483.585i q^{71} +908.791i q^{73} +(-206.323 + 206.323i) q^{75} +(963.654 + 963.654i) q^{77} +1066.46 q^{79} -81.0000 q^{81} +(-871.541 - 871.541i) q^{83} +(209.963 - 209.963i) q^{85} +368.443i q^{87} +185.044i q^{89} +(564.727 - 564.727i) q^{91} +(-218.168 - 218.168i) q^{93} +492.696 q^{95} -725.140 q^{97} +(427.921 + 427.921i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 40 q^{11} + 120 q^{15} - 24 q^{19} - 400 q^{29} - 744 q^{31} + 456 q^{35} - 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 744 q^{51} - 752 q^{53} + 1376 q^{59} + 912 q^{61} - 504 q^{63} + 976 q^{65} + 2256 q^{67} + 528 q^{69} - 1104 q^{75} - 1904 q^{77} + 5992 q^{79} - 1944 q^{81} - 2680 q^{83} + 240 q^{85} + 3496 q^{91} - 7728 q^{95} + 360 q^{99}+O(q^{100}) $ 24 * q + 40 * q^11 + 120 * q^15 - 24 * q^19 - 400 * q^29 - 744 * q^31 + 456 * q^35 - 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 744 * q^51 - 752 * q^53 + 1376 * q^59 + 912 * q^61 - 504 * q^63 + 976 * q^65 + 2256 * q^67 + 528 * q^69 - 1104 * q^75 - 1904 * q^77 + 5992 * q^79 - 1944 * q^81 - 2680 * q^83 + 240 * q^85 + 3496 * q^91 - 7728 * q^95 + 360 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	2.12132	+	2.12132i	0.408248	+	0.408248i
$3$	2.12132	+	2.12132i	0.408248	+	0.408248i
$4$		0			0
$5$	3.72414	−	3.72414i	0.333097	−	0.333097i	−0.520664	−	0.853762i	$-0.674316\pi$
$5$	3.72414	−	3.72414i	0.333097	−	0.333097i	0.853762	+	0.520664i	$0.174316\pi$
$6$		0			0
$7$			20.2675i			1.09434i	0.837020	+	0.547172i	$0.184296\pi$
$7$			20.2675i			1.09434i	−0.837020	+	0.547172i	$0.815704\pi$
$8$		0			0
$9$			9.00000i			0.333333i
$10$		0			0
$11$	47.5467	−	47.5467i	1.30326	−	1.30326i	0.377081	−	0.926180i	$-0.376928\pi$
$11$	47.5467	−	47.5467i	1.30326	−	1.30326i	0.926180	−	0.377081i	$-0.123072\pi$
$12$		0			0
$13$	−27.8636	−	27.8636i	−0.594460	−	0.594460i	0.344373	−	0.938833i	$-0.388091\pi$
$13$	−27.8636	−	27.8636i	−0.594460	−	0.594460i	−0.938833	+	0.344373i	$0.888091\pi$
$14$		0			0
$15$	15.8002			0.271973
$16$		0			0
$17$	56.3788			0.804345			0.402173	−	0.915564i	$-0.368255\pi$
$17$	56.3788			0.804345			0.402173	+	0.915564i	$0.368255\pi$
$18$		0			0
$19$	66.1489	+	66.1489i	0.798716	+	0.798716i	0.982893	−	0.184177i	$-0.0589621\pi$
$19$	66.1489	+	66.1489i	0.798716	+	0.798716i	−0.184177	+	0.982893i	$0.558962\pi$
$20$		0			0
$21$	−42.9939	+	42.9939i	−0.446764	+	0.446764i
$22$		0			0
$23$		−	3.66579i		−	0.0332335i	−0.999862	−	0.0166167i	$-0.994710\pi$
$23$		−	3.66579i		−	0.0332335i	0.999862	−	0.0166167i	$-0.00528951\pi$
$24$		0			0
$25$			97.2615i			0.778092i
$26$		0			0
$27$	−19.0919	+	19.0919i	−0.136083	+	0.136083i
$28$		0			0
$29$	86.8428	+	86.8428i	0.556079	+	0.556079i	0.928189	−	0.372109i	$-0.121365\pi$
$29$	86.8428	+	86.8428i	0.556079	+	0.556079i	−0.372109	+	0.928189i	$0.621365\pi$
$30$		0			0
$31$	−102.846			−0.595858			−0.297929	−	0.954588i	$-0.596296\pi$
$31$	−102.846			−0.595858			−0.297929	+	0.954588i	$0.596296\pi$
$32$		0			0
$33$	201.724			1.06411
$34$		0			0
$35$	75.4791	+	75.4791i	0.364523	+	0.364523i
$36$		0			0
$37$	−66.8267	+	66.8267i	−0.296925	+	0.296925i	−0.839808	−	0.542883i	$-0.817333\pi$
$37$	−66.8267	+	66.8267i	−0.296925	+	0.296925i	0.542883	+	0.839808i	$0.317333\pi$
$38$		0			0
$39$		−	118.215i		−	0.485374i
$40$		0			0
$41$			29.5734i			0.112649i	0.998413	+	0.0563243i	$0.0179381\pi$
$41$			29.5734i			0.112649i	−0.998413	+	0.0563243i	$0.982062\pi$
$42$		0			0
$43$	372.929	−	372.929i	1.32258	−	1.32258i	0.410905	−	0.911678i	$-0.365213\pi$
$43$	372.929	−	372.929i	1.32258	−	1.32258i	0.911678	−	0.410905i	$-0.134787\pi$
$44$		0			0
$45$	33.5173	+	33.5173i	0.111032	+	0.111032i
$46$		0			0
$47$	539.082			1.67305			0.836524	−	0.547930i	$-0.184584\pi$
$47$	539.082			1.67305			0.836524	+	0.547930i	$0.184584\pi$
$48$		0			0
$49$	−67.7725			−0.197587
$50$		0			0
$51$	119.598	+	119.598i	0.328373	+	0.328373i
$52$		0			0
$53$	−385.376	+	385.376i	−0.998781	+	0.998781i	−0.999999	−	0.00121789i	$-0.999612\pi$
$53$	−385.376	+	385.376i	−0.998781	+	0.998781i	0.00121789	+	0.999999i	$0.499612\pi$
$54$		0			0
$55$		−	354.142i		−	0.868226i
$56$		0			0
$57$			280.646i			0.652149i
$58$		0			0
$59$	−71.0380	+	71.0380i	−0.156752	+	0.156752i	−0.781126	−	0.624374i	$-0.785354\pi$
$59$	−71.0380	+	71.0380i	−0.156752	+	0.156752i	0.624374	+	0.781126i	$0.285354\pi$
$60$		0			0
$61$	155.133	+	155.133i	0.325619	+	0.325619i	0.850918	−	0.525299i	$-0.176046\pi$
$61$	155.133	+	155.133i	0.325619	+	0.325619i	−0.525299	+	0.850918i	$0.676046\pi$
$62$		0			0
$63$	−182.408			−0.364781
$64$		0			0
$65$	−207.536			−0.396026
$66$		0			0
$67$	178.141	+	178.141i	0.324827	+	0.324827i	0.850615	−	0.525788i	$-0.176230\pi$
$67$	178.141	+	178.141i	0.324827	+	0.324827i	−0.525788	+	0.850615i	$0.676230\pi$
$68$		0			0
$69$	7.77631	−	7.77631i	0.0135675	−	0.0135675i
$70$		0			0
$71$			483.585i			0.808323i	0.914688	+	0.404162i	$0.132437\pi$
$71$			483.585i			0.808323i	−0.914688	+	0.404162i	$0.867563\pi$
$72$		0			0
$73$			908.791i			1.45707i	0.685010	+	0.728534i	$0.259798\pi$
$73$			908.791i			1.45707i	−0.685010	+	0.728534i	$0.740202\pi$
$74$		0			0
$75$	−206.323	+	206.323i	−0.317655	+	0.317655i
$76$		0			0
$77$	963.654	+	963.654i	1.42622	+	1.42622i
$78$		0			0
$79$	1066.46			1.51882			0.759409	−	0.650613i	$-0.225488\pi$
$79$	1066.46			1.51882			0.759409	+	0.650613i	$0.225488\pi$
$80$		0			0
$81$	−81.0000			−0.111111
$82$		0			0
$83$	−871.541	−	871.541i	−1.15258	−	1.15258i	−0.986034	−	0.166544i	$-0.946739\pi$
$83$	−871.541	−	871.541i	−1.15258	−	1.15258i	−0.166544	−	0.986034i	$-0.553261\pi$
$84$		0			0
$85$	209.963	−	209.963i	0.267925	−	0.267925i
$86$		0			0
$87$			368.443i			0.454037i
$88$		0			0
$89$			185.044i			0.220388i	0.993910	+	0.110194i	$0.0351473\pi$
$89$			185.044i			0.220388i	−0.993910	+	0.110194i	$0.964853\pi$
$90$		0			0
$91$	564.727	−	564.727i	0.650543	−	0.650543i
$92$		0			0
$93$	−218.168	−	218.168i	−0.243258	−	0.243258i
$94$		0			0
$95$	492.696			0.532100
$96$		0			0
$97$	−725.140			−0.759039			−0.379519	−	0.925184i	$-0.623911\pi$
$97$	−725.140			−0.759039			−0.379519	+	0.925184i	$0.623911\pi$
$98$		0			0
$99$	427.921	+	427.921i	0.434420	+	0.434420i
$100$		0			0
$101$	1141.34	−	1141.34i	1.12443	−	1.12443i	0.133359	−	0.991068i	$-0.457424\pi$
$101$	1141.34	−	1141.34i	1.12443	−	1.12443i	0.991068	−	0.133359i	$-0.0425765\pi$
$102$		0			0
$103$		−	1099.46i		−	1.05177i	−0.850555	−	0.525886i	$-0.823734\pi$
$103$		−	1099.46i		−	1.05177i	0.850555	−	0.525886i	$-0.176266\pi$
$104$		0			0
$105$			320.231i			0.297632i
$106$		0			0
$107$	−361.438	+	361.438i	−0.326556	+	0.326556i	−0.851275	−	0.524719i	$-0.824170\pi$
$107$	−361.438	+	361.438i	−0.326556	+	0.326556i	0.524719	+	0.851275i	$0.324170\pi$
$108$		0			0
$109$	−1405.61	−	1405.61i	−1.23517	−	1.23517i	−0.961953	−	0.273215i	$-0.911913\pi$
$109$	−1405.61	−	1405.61i	−1.23517	−	1.23517i	−0.273215	−	0.961953i	$-0.588087\pi$
$110$		0			0
$111$	−283.522			−0.242439
$112$		0			0
$113$	−290.830			−0.242115			−0.121058	−	0.992645i	$-0.538629\pi$
$113$	−290.830			−0.242115			−0.121058	+	0.992645i	$0.538629\pi$
$114$		0			0
$115$	−13.6519	−	13.6519i	−0.0110700	−	0.0110700i
$116$		0			0
$117$	250.773	−	250.773i	0.198153	−	0.198153i
$118$		0			0
$119$			1142.66i			0.880230i
$120$		0			0
$121$		−	3190.38i		−	2.39698i
$122$		0			0
$123$	−62.7347	+	62.7347i	−0.0459886	+	0.0459886i
$124$		0			0
$125$	827.734	+	827.734i	0.592278	+	0.592278i
$126$		0			0
$127$	278.588			0.194651			0.0973256	−	0.995253i	$-0.468971\pi$
$127$	278.588			0.194651			0.0973256	+	0.995253i	$0.468971\pi$
$128$		0			0
$129$	1582.20			1.07988
$130$		0			0
$131$	666.234	+	666.234i	0.444345	+	0.444345i	0.893469	−	0.449124i	$-0.148264\pi$
$131$	666.234	+	666.234i	0.444345	+	0.444345i	−0.449124	+	0.893469i	$0.648264\pi$
$132$		0			0
$133$	−1340.67	+	1340.67i	−0.874069	+	0.874069i
$134$		0			0
$135$			142.202i			0.0906576i
$136$		0			0
$137$		−	632.580i		−	0.394489i	−0.980354	−	0.197244i	$-0.936801\pi$
$137$		−	632.580i		−	0.394489i	0.980354	−	0.197244i	$-0.0631992\pi$
$138$		0			0
$139$	−1097.95	+	1097.95i	−0.669976	+	0.669976i	−0.957710	−	0.287734i	$-0.907098\pi$
$139$	−1097.95	+	1097.95i	−0.669976	+	0.669976i	0.287734	+	0.957710i	$0.407098\pi$
$140$		0			0
$141$	1143.57	+	1143.57i	0.683019	+	0.683019i
$142$		0			0
$143$	−2649.65			−1.54947
$144$		0			0
$145$	646.830			0.370457
$146$		0			0
$147$	−143.767	−	143.767i	−0.0806647	−	0.0806647i
$148$		0			0
$149$	−963.349	+	963.349i	−0.529669	+	0.529669i	−0.920474	−	0.390805i	$-0.872197\pi$
$149$	−963.349	+	963.349i	−0.529669	+	0.529669i	0.390805	+	0.920474i	$0.372197\pi$
$150$		0			0
$151$		−	2047.36i		−	1.10339i	−0.834046	−	0.551694i	$-0.813981\pi$
$151$		−	2047.36i		−	1.10339i	0.834046	−	0.551694i	$-0.186019\pi$
$152$		0			0
$153$			507.409i			0.268115i
$154$		0			0
$155$	−383.011	+	383.011i	−0.198479	+	0.198479i
$156$		0			0
$157$	−2663.57	−	2663.57i	−1.35399	−	1.35399i	−0.881147	−	0.472843i	$-0.843228\pi$
$157$	−2663.57	−	2663.57i	−1.35399	−	1.35399i	−0.472843	−	0.881147i	$-0.656772\pi$
$158$		0			0
$159$	−1635.01			−0.815502
$160$		0			0
$161$	74.2964			0.0363688
$162$		0			0
$163$	−1003.92	−	1003.92i	−0.482410	−	0.482410i	0.423491	−	0.905900i	$-0.360805\pi$
$163$	−1003.92	−	1003.92i	−0.482410	−	0.482410i	−0.905900	+	0.423491i	$0.860805\pi$
$164$		0			0
$165$	751.248	−	751.248i	0.354452	−	0.354452i
$166$		0			0
$167$			1282.05i			0.594061i	0.954868	+	0.297031i	$0.0959964\pi$
$167$			1282.05i			0.594061i	−0.954868	+	0.297031i	$0.904004\pi$
$168$		0			0
$169$		−	644.238i		−	0.293235i
$170$		0			0
$171$	−595.340	+	595.340i	−0.266239	+	0.266239i
$172$		0			0
$173$	−1974.96	−	1974.96i	−0.867937	−	0.867937i	0.124306	−	0.992244i	$-0.460329\pi$
$173$	−1974.96	−	1974.96i	−0.867937	−	0.867937i	−0.992244	+	0.124306i	$0.960329\pi$
$174$		0			0
$175$	−1971.25			−0.851500
$176$		0			0
$177$	−301.389			−0.127987
$178$		0			0
$179$	−66.3425	−	66.3425i	−0.0277021	−	0.0277021i	0.693120	−	0.720822i	$-0.256236\pi$
$179$	−66.3425	−	66.3425i	−0.0277021	−	0.0277021i	−0.720822	+	0.693120i	$0.756236\pi$
$180$		0			0
$181$	371.784	−	371.784i	0.152677	−	0.152677i	−0.626636	−	0.779312i	$-0.715569\pi$
$181$	371.784	−	371.784i	0.152677	−	0.152677i	0.779312	+	0.626636i	$0.215569\pi$
$182$		0			0
$183$			658.174i			0.265867i
$184$		0			0
$185$			497.744i			0.197810i
$186$		0			0
$187$	2680.63	−	2680.63i	1.04827	−	1.04827i
$188$		0			0
$189$	−386.945	−	386.945i	−0.148921	−	0.148921i
$190$		0			0
$191$	−2857.35			−1.08246			−0.541232	−	0.840873i	$-0.682042\pi$
$191$	−2857.35			−1.08246			−0.541232	+	0.840873i	$0.682042\pi$
$192$		0			0
$193$	−4318.58			−1.61066			−0.805332	−	0.592824i	$-0.798013\pi$
$193$	−4318.58			−1.61066			−0.805332	+	0.592824i	$0.798013\pi$
$194$		0			0
$195$	−440.251	−	440.251i	−0.161677	−	0.161677i
$196$		0			0
$197$	1400.78	−	1400.78i	0.506605	−	0.506605i	−0.406878	−	0.913483i	$-0.633382\pi$
$197$	1400.78	−	1400.78i	0.506605	−	0.506605i	0.913483	+	0.406878i	$0.133382\pi$
$198$		0			0
$199$			2154.43i			0.767454i	0.923447	+	0.383727i	$0.125360\pi$
$199$			2154.43i			0.767454i	−0.923447	+	0.383727i	$0.874640\pi$
$200$		0			0
$201$			755.789i			0.265220i
$202$		0			0
$203$	−1760.09	+	1760.09i	−0.608542	+	0.608542i
$204$		0			0
$205$	110.136	+	110.136i	0.0375230	+	0.0375230i
$206$		0			0
$207$	32.9921			0.0110778
$208$		0			0
$209$	6290.33			2.08187
$210$		0			0
$211$	−663.531	−	663.531i	−0.216490	−	0.216490i	0.590528	−	0.807017i	$-0.298920\pi$
$211$	−663.531	−	663.531i	−0.216490	−	0.216490i	−0.807017	+	0.590528i	$0.798920\pi$
$212$		0			0
$213$	−1025.84	+	1025.84i	−0.329997	+	0.329997i
$214$		0			0
$215$		−	2777.68i		−	0.881098i
$216$		0			0
$217$		−	2084.42i		−	0.652073i
$218$		0			0
$219$	−1927.84	+	1927.84i	−0.594846	+	0.594846i
$220$		0			0
$221$	−1570.92	−	1570.92i	−0.478151	−	0.478151i
$222$		0			0
$223$	432.791			0.129963			0.0649816	−	0.997886i	$-0.479301\pi$
$223$	432.791			0.129963			0.0649816	+	0.997886i	$0.479301\pi$
$224$		0			0
$225$	−875.354			−0.259364
$226$		0			0
$227$	−482.544	−	482.544i	−0.141091	−	0.141091i	0.633034	−	0.774124i	$-0.281809\pi$
$227$	−482.544	−	482.544i	−0.141091	−	0.141091i	−0.774124	+	0.633034i	$0.781809\pi$
$228$		0			0
$229$	2717.39	−	2717.39i	0.784150	−	0.784150i	−0.196378	−	0.980528i	$-0.562918\pi$
$229$	2717.39	−	2717.39i	0.784150	−	0.784150i	0.980528	+	0.196378i	$0.0629180\pi$
$230$		0			0
$231$			4088.44i			1.16450i
$232$		0			0
$233$		−	1214.38i		−	0.341445i	−0.985319	−	0.170722i	$-0.945390\pi$
$233$		−	1214.38i		−	0.341445i	0.985319	−	0.170722i	$-0.0546102\pi$
$234$		0			0
$235$	2007.62	−	2007.62i	0.557288	−	0.557288i
$236$		0			0
$237$	2262.31	+	2262.31i	0.620055	+	0.620055i
$238$		0			0
$239$	2676.38			0.724353			0.362177	−	0.932110i	$-0.382034\pi$
$239$	2676.38			0.724353			0.362177	+	0.932110i	$0.382034\pi$
$240$		0			0
$241$	−2415.81			−0.645708			−0.322854	−	0.946449i	$-0.604642\pi$
$241$	−2415.81			−0.645708			−0.322854	+	0.946449i	$0.604642\pi$
$242$		0			0
$243$	−171.827	−	171.827i	−0.0453609	−	0.0453609i
$244$		0			0
$245$	−252.394	+	252.394i	−0.0658158	+	0.0658158i
$246$		0			0
$247$		−	3686.29i		−	0.949609i
$248$		0			0
$249$		−	3697.63i		−	0.941076i
$250$		0			0
$251$	−1492.39	+	1492.39i	−0.375295	+	0.375295i	−0.869402	−	0.494106i	$-0.835495\pi$
$251$	−1492.39	+	1492.39i	−0.375295	+	0.375295i	0.494106	+	0.869402i	$0.335495\pi$
$252$		0			0
$253$	−174.296	−	174.296i	−0.0433119	−	0.0433119i
$254$		0			0
$255$	890.797			0.218760
$256$		0			0
$257$	−3266.32			−0.792791			−0.396395	−	0.918080i	$-0.629739\pi$
$257$	−3266.32			−0.792791			−0.396395	+	0.918080i	$0.629739\pi$
$258$		0			0
$259$	−1354.41	−	1354.41i	−0.324938	−	0.324938i
$260$		0			0
$261$	−781.585	+	781.585i	−0.185360	+	0.185360i
$262$		0			0
$263$		−	893.279i		−	0.209437i	−0.994502	−	0.104719i	$-0.966606\pi$
$263$		−	893.279i		−	0.209437i	0.994502	−	0.104719i	$-0.0333942\pi$
$264$		0			0
$265$			2870.39i			0.665383i
$266$		0			0
$267$	−392.537	+	392.537i	−0.0899732	+	0.0899732i
$268$		0			0
$269$	2619.13	+	2619.13i	0.593647	+	0.593647i	0.938615	−	0.344967i	$-0.112110\pi$
$269$	2619.13	+	2619.13i	0.593647	+	0.593647i	−0.344967	+	0.938615i	$0.612110\pi$
$270$		0			0
$271$	5424.67			1.21596			0.607980	−	0.793952i	$-0.291980\pi$
$271$	5424.67			1.21596			0.607980	+	0.793952i	$0.291980\pi$
$272$		0			0
$273$	2395.93			0.531166
$274$		0			0
$275$	4624.47	+	4624.47i	1.01406	+	1.01406i
$276$		0			0
$277$	−2242.55	+	2242.55i	−0.486432	+	0.486432i	−0.907178	−	0.420746i	$-0.861768\pi$
$277$	−2242.55	+	2242.55i	−0.486432	+	0.486432i	0.420746	+	0.907178i	$0.361768\pi$
$278$		0			0
$279$		−	925.610i		−	0.198619i
$280$		0			0
$281$			3622.87i			0.769119i	0.923100	+	0.384559i	$0.125647\pi$
$281$			3622.87i			0.769119i	−0.923100	+	0.384559i	$0.874353\pi$
$282$		0			0
$283$	0.00634733	−	0.00634733i	1.33325e−6	−	1.33325e-6i	−0.707106	−	0.707107i	$-0.750000\pi$
$283$	0.00634733	−	0.00634733i	1.33325e−6	−	1.33325e-6i	0.707107	+	0.707106i	$0.250000\pi$
$284$		0			0
$285$	1045.17	+	1045.17i	0.217229	+	0.217229i
$286$		0			0
$287$	−599.380			−0.123276
$288$		0			0
$289$	−1734.43			−0.353028
$290$		0			0
$291$	−1538.25	−	1538.25i	−0.309876	−	0.309876i
$292$		0			0
$293$	24.9384	−	24.9384i	0.00497241	−	0.00497241i	−0.704616	−	0.709589i	$-0.748881\pi$
$293$	24.9384	−	24.9384i	0.00497241	−	0.00497241i	0.709589	+	0.704616i	$0.248881\pi$
$294$		0			0
$295$			529.112i			0.104427i
$296$		0			0
$297$			1815.51i			0.354703i
$298$		0			0
$299$	−102.142	+	102.142i	−0.0197560	+	0.0197560i
$300$		0			0
$301$	7558.34	+	7558.34i	1.44736	+	1.44736i
$302$		0			0
$303$	4842.28			0.918091
$304$		0			0
$305$	1155.48			0.216926
$306$		0			0
$307$	618.757	+	618.757i	0.115030	+	0.115030i	0.762279	−	0.647249i	$-0.224080\pi$
$307$	618.757	+	618.757i	0.115030	+	0.115030i	−0.647249	+	0.762279i	$0.724080\pi$
$308$		0			0
$309$	2332.30	−	2332.30i	0.429384	−	0.429384i
$310$		0			0
$311$			4183.92i			0.762856i	0.924399	+	0.381428i	$0.124568\pi$
$311$			4183.92i			0.762856i	−0.924399	+	0.381428i	$0.875432\pi$
$312$		0			0
$313$		−	2124.18i		−	0.383597i	−0.981434	−	0.191799i	$-0.938568\pi$
$313$		−	2124.18i		−	0.383597i	0.981434	−	0.191799i	$-0.0614321\pi$
$314$		0			0
$315$	−679.312	+	679.312i	−0.121508	+	0.121508i
$316$		0			0
$317$	−1064.95	−	1064.95i	−0.188687	−	0.188687i	0.606442	−	0.795128i	$-0.292596\pi$
$317$	−1064.95	−	1064.95i	−0.188687	−	0.188687i	−0.795128	+	0.606442i	$0.792596\pi$
$318$		0			0
$319$	8258.18			1.44943
$320$		0			0
$321$	−1533.45			−0.266632
$322$		0			0
$323$	3729.40	+	3729.40i	0.642443	+	0.642443i
$324$		0			0
$325$	2710.06	−	2710.06i	0.462545	−	0.462545i
$326$		0			0
$327$		−	5963.51i		−	1.00851i
$328$		0			0
$329$			10925.9i			1.83089i
$330$		0			0
$331$	1785.55	−	1785.55i	0.296504	−	0.296504i	−0.543139	−	0.839643i	$-0.682764\pi$
$331$	1785.55	−	1785.55i	0.296504	−	0.296504i	0.839643	+	0.543139i	$0.182764\pi$
$332$		0			0
$333$	−601.440	−	601.440i	−0.0989751	−	0.0989751i
$334$		0			0
$335$	1326.85			0.216398
$336$		0			0
$337$	549.365			0.0888006			0.0444003	−	0.999014i	$-0.485862\pi$
$337$	549.365			0.0888006			0.0444003	+	0.999014i	$0.485862\pi$
$338$		0			0
$339$	−616.944	−	616.944i	−0.0988430	−	0.0988430i
$340$		0			0
$341$	−4889.97	+	4889.97i	−0.776559	+	0.776559i
$342$		0			0
$343$			5578.18i			0.878115i
$344$		0			0
$345$		−	57.9202i		−	0.00903860i
$346$		0			0
$347$	6594.28	−	6594.28i	1.02017	−	1.02017i	0.0203788	−	0.999792i	$-0.493513\pi$
$347$	6594.28	−	6594.28i	1.02017	−	1.02017i	0.999792	−	0.0203788i	$-0.00648721\pi$
$348$		0			0
$349$	−6550.56	−	6550.56i	−1.00471	−	1.00471i	−0.999989	−	0.00471950i	$-0.998498\pi$
$349$	−6550.56	−	6550.56i	−1.00471	−	1.00471i	−0.00471950	−	0.999989i	$-0.501502\pi$
$350$		0			0
$351$	1063.94			0.161791
$352$		0			0
$353$	−939.592			−0.141670			−0.0708349	−	0.997488i	$-0.522566\pi$
$353$	−939.592			−0.141670			−0.0708349	+	0.997488i	$0.522566\pi$
$354$		0			0
$355$	1800.94	+	1800.94i	0.269250	+	0.269250i
$356$		0			0
$357$	−2423.95	+	2423.95i	−0.359352	+	0.359352i
$358$		0			0
$359$		−	11175.4i		−	1.64294i	−0.570254	−	0.821469i	$-0.693155\pi$
$359$		−	11175.4i		−	1.64294i	0.570254	−	0.821469i	$-0.306845\pi$
$360$		0			0
$361$			1892.35i			0.275893i
$362$		0			0
$363$	6767.82	−	6767.82i	0.978563	−	0.978563i
$364$		0			0
$365$	3384.47	+	3384.47i	0.485346	+	0.485346i
$366$		0			0
$367$	−10474.8			−1.48986			−0.744930	−	0.667143i	$-0.767517\pi$
$367$	−10474.8			−1.48986			−0.744930	+	0.667143i	$0.767517\pi$
$368$		0			0
$369$	−266.161			−0.0375496
$370$		0			0
$371$	−7810.61	−	7810.61i	−1.09301	−	1.09301i
$372$		0			0
$373$	−7160.10	+	7160.10i	−0.993930	+	0.993930i	−0.999982	−	0.00605119i	$-0.998074\pi$
$373$	−7160.10	+	7160.10i	−0.993930	+	0.993930i	0.00605119	+	0.999982i	$0.498074\pi$
$374$		0			0
$375$			3511.78i			0.483593i
$376$		0			0
$377$		−	4839.51i		−	0.661134i
$378$		0			0
$379$	−3707.91	+	3707.91i	−0.502540	+	0.502540i	−0.912226	−	0.409687i	$-0.865638\pi$
$379$	−3707.91	+	3707.91i	−0.502540	+	0.502540i	0.409687	+	0.912226i	$0.365638\pi$
$380$		0			0
$381$	590.975	+	590.975i	0.0794660	+	0.0794660i
$382$		0			0
$383$	1619.09			0.216010			0.108005	−	0.994150i	$-0.465554\pi$
$383$	1619.09			0.216010			0.108005	+	0.994150i	$0.465554\pi$
$384$		0			0
$385$	7177.57			0.950137
$386$		0			0
$387$	3356.36	+	3356.36i	0.440861	+	0.440861i
$388$		0			0
$389$	3130.25	−	3130.25i	0.407995	−	0.407995i	−0.473044	−	0.881039i	$-0.656845\pi$
$389$	3130.25	−	3130.25i	0.407995	−	0.407995i	0.881039	+	0.473044i	$0.156845\pi$
$390$		0			0
$391$		−	206.673i		−	0.0267312i
$392$		0			0
$393$			2826.59i			0.362806i
$394$		0			0
$395$	3971.67	−	3971.67i	0.505914	−	0.505914i
$396$		0			0
$397$	4797.49	+	4797.49i	0.606497	+	0.606497i	0.942029	−	0.335532i	$-0.108916\pi$
$397$	4797.49	+	4797.49i	0.606497	+	0.606497i	−0.335532	+	0.942029i	$0.608916\pi$
$398$		0			0
$399$	−5688.00			−0.713674
$400$		0			0
$401$	10424.9			1.29825			0.649123	−	0.760684i	$-0.275136\pi$
$401$	10424.9			1.29825			0.649123	+	0.760684i	$0.275136\pi$
$402$		0			0
$403$	2865.65	+	2865.65i	0.354214	+	0.354214i
$404$		0			0
$405$	−301.656	+	301.656i	−0.0370108	+	0.0370108i
$406$		0			0
$407$			6354.78i			0.773943i
$408$		0			0
$409$			12366.0i			1.49501i	0.664255	+	0.747506i	$0.268749\pi$
$409$			12366.0i			1.49501i	−0.664255	+	0.747506i	$0.731251\pi$
$410$		0			0
$411$	1341.90	−	1341.90i	0.161049	−	0.161049i
$412$		0			0
$413$	−1439.77	−	1439.77i	−0.171540	−	0.171540i
$414$		0			0
$415$	−6491.48			−0.767842
$416$		0			0
$417$	−4658.20			−0.547034
$418$		0			0
$419$	1809.65	+	1809.65i	0.210996	+	0.210996i	0.804691	−	0.593694i	$-0.202331\pi$
$419$	1809.65	+	1809.65i	0.210996	+	0.210996i	−0.593694	+	0.804691i	$0.702331\pi$
$420$		0			0
$421$	−2481.06	+	2481.06i	−0.287219	+	0.287219i	−0.835980	−	0.548760i	$-0.815100\pi$
$421$	−2481.06	+	2481.06i	−0.287219	+	0.287219i	0.548760	+	0.835980i	$0.315100\pi$
$422$		0			0
$423$			4851.74i			0.557683i
$424$		0			0
$425$			5483.49i			0.625855i
$426$		0			0
$427$	−3144.17	+	3144.17i	−0.356339	+	0.356339i
$428$		0			0
$429$	−5620.75	−	5620.75i	−0.632570	−	0.632570i
$430$		0			0
$431$	−4925.25			−0.550443			−0.275221	−	0.961381i	$-0.588751\pi$
$431$	−4925.25			−0.550443			−0.275221	+	0.961381i	$0.588751\pi$
$432$		0			0
$433$	13472.5			1.49526			0.747630	−	0.664115i	$-0.231192\pi$
$433$	13472.5			1.49526			0.747630	+	0.664115i	$0.231192\pi$
$434$		0			0
$435$	1372.13	+	1372.13i	0.151239	+	0.151239i
$436$		0			0
$437$	242.488	−	242.488i	0.0265441	−	0.0265441i
$438$		0			0
$439$		−	13.7683i		−	0.00149687i	−1.00000		0.000748434i	$-0.999762\pi$
$439$		−	13.7683i		−	0.00149687i	1.00000		0.000748434i	$-0.000238234\pi$
$440$		0			0
$441$		−	609.952i		−	0.0658624i
$442$		0			0
$443$	112.762	−	112.762i	0.0120936	−	0.0120936i	−0.701034	−	0.713128i	$-0.747278\pi$
$443$	112.762	−	112.762i	0.0120936	−	0.0120936i	0.713128	+	0.701034i	$0.247278\pi$
$444$		0			0
$445$	689.129	+	689.129i	0.0734108	+	0.0734108i
$446$		0			0
$447$	−4087.14			−0.432473
$448$		0			0
$449$	−17094.6			−1.79676			−0.898380	−	0.439218i	$-0.855255\pi$
$449$	−17094.6			−1.79676			−0.898380	+	0.439218i	$0.855255\pi$
$450$		0			0
$451$	1406.12	+	1406.12i	0.146811	+	0.146811i
$452$		0			0
$453$	4343.10	−	4343.10i	0.450457	−	0.450457i
$454$		0			0
$455$		−	4206.24i		−	0.433389i
$456$		0			0
$457$		−	4074.04i		−	0.417015i	−0.978021	−	0.208507i	$-0.933140\pi$
$457$		−	4074.04i		−	0.417015i	0.978021	−	0.208507i	$-0.0668605\pi$
$458$		0			0
$459$	−1076.38	+	1076.38i	−0.109458	+	0.109458i
$460$		0			0
$461$	−2329.41	−	2329.41i	−0.235339	−	0.235339i	0.579578	−	0.814917i	$-0.303217\pi$
$461$	−2329.41	−	2329.41i	−0.235339	−	0.235339i	−0.814917	+	0.579578i	$0.803217\pi$
$462$		0			0
$463$	12448.0			1.24948			0.624738	−	0.780834i	$-0.285206\pi$
$463$	12448.0			1.24948			0.624738	+	0.780834i	$0.285206\pi$
$464$		0			0
$465$	−1624.98			−0.162057
$466$		0			0
$467$	−366.937	−	366.937i	−0.0363594	−	0.0363594i	0.688693	−	0.725053i	$-0.258185\pi$
$467$	−366.937	−	366.937i	−0.0363594	−	0.0363594i	−0.725053	+	0.688693i	$0.758185\pi$
$468$		0			0
$469$	−3610.48	+	3610.48i	−0.355472	+	0.355472i
$470$		0			0
$471$		−	11300.6i		−	1.10553i
$472$		0			0
$473$		−	35463.1i		−	3.44734i
$474$		0			0
$475$	−6433.74	+	6433.74i	−0.621474	+	0.621474i
$476$		0			0
$477$	−3468.38	−	3468.38i	−0.332927	−	0.332927i
$478$		0			0
$479$	18458.4			1.76072			0.880361	−	0.474304i	$-0.157300\pi$
$479$	18458.4			1.76072			0.880361	+	0.474304i	$0.157300\pi$
$480$		0			0
$481$	3724.07			0.353020
$482$		0			0
$483$	157.607	+	157.607i	0.0148475	+	0.0148475i
$484$		0			0
$485$	−2700.52	+	2700.52i	−0.252834	+	0.252834i
$486$		0			0
$487$		−	3880.55i		−	0.361077i	−0.983568	−	0.180538i	$-0.942216\pi$
$487$		−	3880.55i		−	0.361077i	0.983568	−	0.180538i	$-0.0577840\pi$
$488$		0			0
$489$		−	4259.26i		−	0.393886i
$490$		0			0
$491$	−3893.20	+	3893.20i	−0.357836	+	0.357836i	−0.863015	−	0.505179i	$-0.831427\pi$
$491$	−3893.20	+	3893.20i	−0.357836	+	0.357836i	0.505179	+	0.863015i	$0.331427\pi$
$492$		0			0
$493$	4896.10	+	4896.10i	0.447280	+	0.447280i
$494$		0			0
$495$	3187.27			0.289409
$496$		0			0
$497$	−9801.06			−0.884583
$498$		0			0
$499$	5061.87	+	5061.87i	0.454110	+	0.454110i	0.896716	−	0.442606i	$-0.145946\pi$
$499$	5061.87	+	5061.87i	0.454110	+	0.454110i	−0.442606	+	0.896716i	$0.645946\pi$
$500$		0			0
$501$	−2719.65	+	2719.65i	−0.242525	+	0.242525i
$502$		0			0
$503$			11814.7i			1.04730i	0.851933	+	0.523651i	$0.175430\pi$
$503$			11814.7i			1.04730i	−0.851933	+	0.523651i	$0.824570\pi$
$504$		0			0
$505$		−	8500.99i		−	0.749088i
$506$		0			0
$507$	1366.63	−	1366.63i	0.119713	−	0.119713i
$508$		0			0
$509$	−7553.16	−	7553.16i	−0.657737	−	0.657737i	0.297107	−	0.954844i	$-0.403978\pi$
$509$	−7553.16	−	7553.16i	−0.657737	−	0.657737i	−0.954844	+	0.297107i	$0.903978\pi$
$510$		0			0
$511$	−18418.9			−1.59453
$512$		0			0
$513$	−2525.81			−0.217383
$514$		0			0
$515$	−4094.53	−	4094.53i	−0.350343	−	0.350343i
$516$		0			0
$517$	25631.6	−	25631.6i	2.18042	−	2.18042i
$518$		0			0
$519$		−	8379.03i		−	0.708668i
$520$		0			0
$521$			1696.39i			0.142649i	0.997453	+	0.0713245i	$0.0227226\pi$
$521$			1696.39i			0.142649i	−0.997453	+	0.0713245i	$0.977277\pi$
$522$		0			0
$523$	−6684.89	+	6684.89i	−0.558910	+	0.558910i	−0.928997	−	0.370087i	$-0.879328\pi$
$523$	−6684.89	+	6684.89i	−0.558910	+	0.558910i	0.370087	+	0.928997i	$0.379328\pi$
$524$		0			0
$525$	−4181.65	−	4181.65i	−0.347623	−	0.347623i
$526$		0			0
$527$	−5798.31			−0.479276
$528$		0			0
$529$	12153.6			0.998896
$530$		0			0
$531$	−639.342	−	639.342i	−0.0522506	−	0.0522506i
$532$		0			0
$533$	824.023	−	824.023i	0.0669651	−	0.0669651i
$534$		0			0
$535$			2692.09i			0.217550i
$536$		0			0
$537$		−	281.468i		−	0.0226187i
$538$		0			0
$539$	−3222.36	+	3222.36i	−0.257508	+	0.257508i
$540$		0			0
$541$	5108.56	+	5108.56i	0.405978	+	0.405978i	0.880333	−	0.474355i	$-0.157319\pi$
$541$	5108.56	+	5108.56i	0.405978	+	0.405978i	−0.474355	+	0.880333i	$0.657319\pi$
$542$		0			0
$543$	1577.34			0.124660
$544$		0			0
$545$	−10469.4			−0.822863
$546$		0			0
$547$	−13162.0	−	13162.0i	−1.02883	−	1.02883i	−0.999572	−	0.0292549i	$-0.990687\pi$
$547$	−13162.0	−	13162.0i	−1.02883	−	1.02883i	−0.0292549	−	0.999572i	$-0.509313\pi$
$548$		0			0
$549$	−1396.20	+	1396.20i	−0.108540	+	0.108540i
$550$		0			0
$551$			11489.1i			0.888299i
$552$		0			0
$553$			21614.6i			1.66211i
$554$		0			0
$555$	−1055.87	+	1055.87i	−0.0807557	+	0.0807557i
$556$		0			0
$557$	10462.8	+	10462.8i	0.795911	+	0.795911i	0.982448	−	0.186537i	$-0.0597263\pi$
$557$	10462.8	+	10462.8i	0.795911	+	0.795911i	−0.186537	+	0.982448i	$0.559726\pi$
$558$		0			0
$559$	−20782.3			−1.57245
$560$		0			0
$561$	11372.9			0.855911
$562$		0			0
$563$	−11673.6	−	11673.6i	−0.873857	−	0.873857i	0.119033	−	0.992890i	$-0.462020\pi$
$563$	−11673.6	−	11673.6i	−0.873857	−	0.873857i	−0.992890	+	0.119033i	$0.962020\pi$
$564$		0			0
$565$	−1083.09	+	1083.09i	−0.0806479	+	0.0806479i
$566$		0			0
$567$		−	1641.67i		−	0.121594i
$568$		0			0
$569$		−	238.804i		−	0.0175943i	−0.999961	−	0.00879717i	$-0.997200\pi$
$569$		−	238.804i		−	0.0175943i	0.999961	−	0.00879717i	$-0.00280026\pi$
$570$		0			0
$571$	−14395.8	+	14395.8i	−1.05507	+	1.05507i	−0.0566747	+	0.998393i	$0.518050\pi$
$571$	−14395.8	+	14395.8i	−1.05507	+	1.05507i	−0.998393	+	0.0566747i	$0.981950\pi$
$572$		0			0
$573$	−6061.36	−	6061.36i	−0.441914	−	0.441914i
$574$		0			0
$575$	356.540			0.0258587
$576$		0			0
$577$	−14732.2			−1.06293			−0.531463	−	0.847081i	$-0.678358\pi$
$577$	−14732.2			−1.06293			−0.531463	+	0.847081i	$0.678358\pi$
$578$		0			0
$579$	−9161.09	−	9161.09i	−0.657551	−	0.657551i
$580$		0			0
$581$	17664.0	−	17664.0i	1.26132	−	1.26132i
$582$		0			0
$583$			36646.7i			2.60335i
$584$		0			0
$585$		−	1867.83i		−	0.132009i
$586$		0			0
$587$	4120.47	−	4120.47i	0.289727	−	0.289727i	−0.547245	−	0.836972i	$-0.684323\pi$
$587$	4120.47	−	4120.47i	0.289727	−	0.289727i	0.836972	+	0.547245i	$0.184323\pi$
$588$		0			0
$589$	−6803.12	−	6803.12i	−0.475921	−	0.475921i
$590$		0			0
$591$	5942.99			0.413641
$592$		0			0
$593$	−18995.5			−1.31543			−0.657716	−	0.753266i	$-0.728477\pi$
$593$	−18995.5			−1.31543			−0.657716	+	0.753266i	$0.728477\pi$
$594$		0			0
$595$	4255.43	+	4255.43i	0.293202	+	0.293202i
$596$		0			0
$597$	−4570.23	+	4570.23i	−0.313312	+	0.313312i
$598$		0			0
$599$		−	19402.7i		−	1.32349i	−0.749728	−	0.661747i	$-0.769815\pi$
$599$		−	19402.7i		−	1.32349i	0.749728	−	0.661747i	$-0.230185\pi$
$600$		0			0
$601$		−	14134.5i		−	0.959329i	−0.877452	−	0.479665i	$-0.840758\pi$
$601$		−	14134.5i		−	0.959329i	0.877452	−	0.479665i	$-0.159242\pi$
$602$		0			0
$603$	−1603.27	+	1603.27i	−0.108276	+	0.108276i
$604$		0			0
$605$	−11881.4	−	11881.4i	−0.798428	−	0.798428i
$606$		0			0
$607$	−2098.86			−0.140346			−0.0701731	−	0.997535i	$-0.522355\pi$
$607$	−2098.86			−0.140346			−0.0701731	+	0.997535i	$0.522355\pi$
$608$		0			0
$609$	−7467.42			−0.496872
$610$		0			0
$611$	−15020.8	−	15020.8i	−0.994560	−	0.994560i
$612$		0			0
$613$	−6577.63	+	6577.63i	−0.433390	+	0.433390i	−0.889780	−	0.456390i	$-0.849142\pi$
$613$	−6577.63	+	6577.63i	−0.433390	+	0.433390i	0.456390	+	0.889780i	$0.349142\pi$
$614$		0			0
$615$			467.266i			0.0306374i
$616$		0			0
$617$		−	8011.43i		−	0.522736i	−0.965239	−	0.261368i	$-0.915826\pi$
$617$		−	8011.43i		−	0.522736i	0.965239	−	0.261368i	$-0.0841736\pi$
$618$		0			0
$619$	7930.76	−	7930.76i	0.514966	−	0.514966i	−0.401078	−	0.916044i	$-0.631364\pi$
$619$	7930.76	−	7930.76i	0.514966	−	0.514966i	0.916044	+	0.401078i	$0.131364\pi$
$620$		0			0
$621$	69.9868	+	69.9868i	0.00452250	+	0.00452250i
$622$		0			0
$623$	−3750.37			−0.241181
$624$		0			0
$625$	−5992.49			−0.383520
$626$		0			0
$627$	13343.8	+	13343.8i	0.849920	+	0.849920i
$628$		0			0
$629$	−3767.61	+	3767.61i	−0.238831	+	0.238831i
$630$		0			0
$631$			205.500i			0.0129649i	0.999979	+	0.00648243i	$0.00206344\pi$
$631$			205.500i			0.0129649i	−0.999979	+	0.00648243i	$0.997937\pi$
$632$		0			0
$633$		−	2815.12i		−	0.176763i
$634$		0			0
$635$	1037.50	−	1037.50i	0.0648378	−	0.0648378i
$636$		0			0
$637$	1888.39	+	1888.39i	0.117458	+	0.117458i
$638$		0			0
$639$	−4352.26			−0.269441
$640$		0			0
$641$	24945.2			1.53709			0.768545	−	0.639795i	$-0.220981\pi$
$641$	24945.2			1.53709			0.768545	+	0.639795i	$0.220981\pi$
$642$		0			0
$643$	−2568.72	−	2568.72i	−0.157543	−	0.157543i	0.623934	−	0.781477i	$-0.285533\pi$
$643$	−2568.72	−	2568.72i	−0.157543	−	0.157543i	−0.781477	+	0.623934i	$0.785533\pi$
$644$		0			0
$645$	5892.35	−	5892.35i	0.359707	−	0.359707i
$646$		0			0
$647$		−	24443.7i		−	1.48529i	−0.669687	−	0.742643i	$-0.733572\pi$
$647$		−	24443.7i		−	1.48529i	0.669687	−	0.742643i	$-0.266428\pi$
$648$		0			0
$649$			6755.25i			0.408578i
$650$		0			0
$651$	4421.73	−	4421.73i	0.266208	−	0.266208i
$652$		0			0
$653$	9074.73	+	9074.73i	0.543831	+	0.543831i	0.924650	−	0.380819i	$-0.124358\pi$
$653$	9074.73	+	9074.73i	0.543831	+	0.543831i	−0.380819	+	0.924650i	$0.624358\pi$
$654$		0			0
$655$	4962.30			0.296020
$656$		0			0
$657$	−8179.12			−0.485689
$658$		0			0
$659$	7745.96	+	7745.96i	0.457875	+	0.457875i	0.897957	−	0.440082i	$-0.145051\pi$
$659$	7745.96	+	7745.96i	0.457875	+	0.457875i	−0.440082	+	0.897957i	$0.645051\pi$
$660$		0			0
$661$	6346.77	−	6346.77i	0.373465	−	0.373465i	−0.495272	−	0.868738i	$-0.664932\pi$
$661$	6346.77	−	6346.77i	0.373465	−	0.373465i	0.868738	+	0.495272i	$0.164932\pi$
$662$		0			0
$663$		−	6664.84i		−	0.390409i
$664$		0			0
$665$			9985.72i			0.582300i
$666$		0			0
$667$	318.347	−	318.347i	0.0184804	−	0.0184804i
$668$		0			0
$669$	918.088	+	918.088i	0.0530573	+	0.0530573i
$670$		0			0
$671$	14752.1			0.848734
$672$		0			0
$673$	−10029.4			−0.574450			−0.287225	−	0.957863i	$-0.592733\pi$
$673$	−10029.4			−0.574450			−0.287225	+	0.957863i	$0.592733\pi$
$674$		0			0
$675$	−1856.91	−	1856.91i	−0.105885	−	0.105885i
$676$		0			0
$677$	−16757.3	+	16757.3i	−0.951309	+	0.951309i	−0.998868	−	0.0475592i	$-0.984856\pi$
$677$	−16757.3	+	16757.3i	−0.951309	+	0.951309i	0.0475592	+	0.998868i	$0.484856\pi$
$678$		0			0
$679$		−	14696.8i		−	0.830649i
$680$		0			0
$681$		−	2047.26i		−	0.115200i
$682$		0			0
$683$	−15221.0	+	15221.0i	−0.852731	+	0.852731i	−0.990469	−	0.137738i	$-0.956017\pi$
$683$	−15221.0	+	15221.0i	−0.852731	+	0.852731i	0.137738	+	0.990469i	$0.456017\pi$
$684$		0			0
$685$	−2355.82	−	2355.82i	−0.131403	−	0.131403i
$686$		0			0
$687$	11528.9			0.640256
$688$		0			0
$689$	21475.9			1.18747
$690$		0			0
$691$	12184.0	+	12184.0i	0.670769	+	0.670769i	0.957893	−	0.287125i	$-0.0926994\pi$
$691$	12184.0	+	12184.0i	0.670769	+	0.670769i	−0.287125	+	0.957893i	$0.592699\pi$
$692$		0			0
$693$	−8672.89	+	8672.89i	−0.475405	+	0.475405i
$694$		0			0
$695$			8177.83i			0.446335i
$696$		0			0
$697$			1667.32i			0.0906084i
$698$		0			0
$699$	2576.09	−	2576.09i	0.139394	−	0.139394i
$700$		0			0
$701$	420.818	+	420.818i	0.0226734	+	0.0226734i	0.718353	−	0.695679i	$-0.244896\pi$
$701$	420.818	+	420.818i	0.0226734	+	0.0226734i	−0.695679	+	0.718353i	$0.744896\pi$
$702$		0			0
$703$	−8841.02			−0.474318
$704$		0			0
$705$	8517.61			0.455024
$706$		0			0
$707$	23132.0	+	23132.0i	1.23051	+	1.23051i
$708$		0			0
$709$	−9956.34	+	9956.34i	−0.527388	+	0.527388i	−0.919793	−	0.392405i	$-0.871643\pi$
$709$	−9956.34	+	9956.34i	−0.527388	+	0.527388i	0.392405	+	0.919793i	$0.371643\pi$
$710$		0			0
$711$			9598.18i			0.506273i
$712$		0			0
$713$			377.010i			0.0198024i
$714$		0			0
$715$	−9867.67	+	9867.67i	−0.516126	+	0.516126i
$716$		0			0
$717$	5677.45	+	5677.45i	0.295716	+	0.295716i
$718$		0			0
$719$	−20878.5			−1.08295			−0.541473	−	0.840718i	$-0.682133\pi$
$719$	−20878.5			−1.08295			−0.541473	+	0.840718i	$0.682133\pi$
$720$		0			0
$721$	22283.2			1.15100
$722$		0			0
$723$	−5124.70	−	5124.70i	−0.263609	−	0.263609i
$724$		0			0
$725$	−8446.46	+	8446.46i	−0.432681	+	0.432681i
$726$		0			0
$727$		−	2726.84i		−	0.139110i	−0.997578	−	0.0695549i	$-0.977842\pi$
$727$		−	2726.84i		−	0.139110i	0.997578	−	0.0695549i	$-0.0221579\pi$
$728$		0			0
$729$		−	729.000i		−	0.0370370i
$730$		0			0
$731$	21025.3	−	21025.3i	1.06381	−	1.06381i
$732$		0			0
$733$	1081.50	+	1081.50i	0.0544967	+	0.0544967i	0.733830	−	0.679333i	$-0.237731\pi$
$733$	1081.50	+	1081.50i	0.0544967	+	0.0544967i	−0.679333	+	0.733830i	$0.737731\pi$
$734$		0			0
$735$	−1070.82			−0.0537384
$736$		0			0
$737$	16940.1			0.846669
$738$		0			0
$739$	15055.3	+	15055.3i	0.749416	+	0.749416i	0.974369	−	0.224954i	$-0.0722232\pi$
$739$	15055.3	+	15055.3i	0.749416	+	0.749416i	−0.224954	+	0.974369i	$0.572223\pi$
$740$		0			0
$741$	7819.81	−	7819.81i	0.387676	−	0.387676i
$742$		0			0
$743$			21909.2i			1.08179i	0.841090	+	0.540896i	$0.181915\pi$
$743$			21909.2i			1.08179i	−0.841090	+	0.540896i	$0.818085\pi$
$744$		0			0
$745$			7175.30i			0.352863i
$746$		0			0
$747$	7843.86	−	7843.86i	0.384193	−	0.384193i
$748$		0			0
$749$	−7325.45	−	7325.45i	−0.357365	−	0.357365i
$750$		0			0
$751$	4501.15			0.218707			0.109354	−	0.994003i	$-0.465122\pi$
$751$	4501.15			0.218707			0.109354	+	0.994003i	$0.465122\pi$
$752$		0			0
$753$	−6331.69			−0.306427
$754$		0			0
$755$	−7624.66	−	7624.66i	−0.367536	−	0.367536i
$756$		0			0
$757$	3638.27	−	3638.27i	0.174683	−	0.174683i	−0.614350	−	0.789034i	$-0.710582\pi$
$757$	3638.27	−	3638.27i	0.174683	−	0.174683i	0.789034	+	0.614350i	$0.210582\pi$
$758$		0			0
$759$		−	739.476i		−	0.0353640i
$760$		0			0
$761$			7208.65i			0.343381i	0.985151	+	0.171691i	$0.0549230\pi$
$761$			7208.65i			0.343381i	−0.985151	+	0.171691i	$0.945077\pi$
$762$		0			0
$763$	28488.3	−	28488.3i	1.35170	−	1.35170i
$764$		0			0
$765$	1889.66	+	1889.66i	0.0893085	+	0.0893085i
$766$		0			0
$767$	3958.75			0.186365
$768$		0			0
$769$	−6428.86			−0.301470			−0.150735	−	0.988574i	$-0.548164\pi$
$769$	−6428.86			−0.301470			−0.150735	+	0.988574i	$0.548164\pi$
$770$		0			0
$771$	−6928.90	−	6928.90i	−0.323655	−	0.323655i
$772$		0			0
$773$	−21187.9	+	21187.9i	−0.985867	+	0.985867i	−0.999902	−	0.0140348i	$-0.995532\pi$
$773$	−21187.9	+	21187.9i	−0.985867	+	0.985867i	0.0140348	+	0.999902i	$0.495532\pi$
$774$		0			0
$775$		−	10002.9i		−	0.463633i
$776$		0			0
$777$		−	5746.28i		−	0.265311i
$778$		0			0
$779$	−1956.25	+	1956.25i	−0.0899742	+	0.0899742i
$780$		0			0
$781$	22992.9	+	22992.9i	1.05346	+	1.05346i
$782$		0			0
$783$	−3315.99			−0.151346
$784$		0			0
$785$	−19839.1			−0.902021
$786$		0			0
$787$	−25780.3	−	25780.3i	−1.16769	−	1.16769i	−0.982751	−	0.184935i	$-0.940793\pi$
$787$	−25780.3	−	25780.3i	−1.16769	−	1.16769i	−0.184935	−	0.982751i	$-0.559207\pi$
$788$		0			0
$789$	1894.93	−	1894.93i	0.0855024	−	0.0855024i
$790$		0			0
$791$		−	5894.41i		−	0.264957i
$792$		0			0
$793$		−	8645.14i		−	0.387135i
$794$		0			0
$795$	−6089.01	+	6089.01i	−0.271641	+	0.271641i
$796$		0			0
$797$	−19825.4	−	19825.4i	−0.881121	−	0.881121i	0.112528	−	0.993649i	$-0.464105\pi$
$797$	−19825.4	−	19825.4i	−0.881121	−	0.881121i	−0.993649	+	0.112528i	$0.964105\pi$
$798$		0			0
$799$	30392.8			1.34571
$800$		0			0
$801$	−1665.39			−0.0734628
$802$		0			0
$803$	43210.0	+	43210.0i	1.89894	+	1.89894i
$804$		0			0
$805$	276.691	−	276.691i	0.0121144	−	0.0121144i
$806$		0			0
$807$			11112.0i			0.484711i
$808$		0			0
$809$			40019.8i			1.73921i	0.493749	+	0.869605i	$0.335626\pi$
$809$			40019.8i			1.73921i	−0.493749	+	0.869605i	$0.664374\pi$
$810$		0			0
$811$	24193.5	−	24193.5i	1.04753	−	1.04753i	0.0487212	−	0.998812i	$-0.484485\pi$
$811$	24193.5	−	24193.5i	1.04753	−	1.04753i	0.998812	−	0.0487212i	$-0.0155146\pi$
$812$		0			0
$813$	11507.5	+	11507.5i	0.496414	+	0.496414i
$814$		0			0
$815$	−7477.45			−0.321379
$816$		0			0
$817$	49337.6			2.11274
$818$		0			0
$819$	5082.54	+	5082.54i	0.216848	+	0.216848i
$820$		0			0
$821$	18924.0	−	18924.0i	0.804450	−	0.804450i	−0.179338	−	0.983788i	$-0.557395\pi$
$821$	18924.0	−	18924.0i	0.804450	−	0.804450i	0.983788	+	0.179338i	$0.0573954\pi$
$822$		0			0
$823$			4894.21i			0.207292i	0.994614	+	0.103646i	$0.0330509\pi$
$823$			4894.21i			0.207292i	−0.994614	+	0.103646i	$0.966949\pi$
$824$		0			0
$825$			19620.0i			0.827975i
$826$		0			0
$827$	19015.3	−	19015.3i	0.799549	−	0.799549i	−0.183475	−	0.983024i	$-0.558735\pi$
$827$	19015.3	−	19015.3i	0.799549	−	0.799549i	0.983024	+	0.183475i	$0.0587348\pi$
$828$		0			0
$829$	7608.33	+	7608.33i	0.318755	+	0.318755i	0.848289	−	0.529534i	$-0.177633\pi$
$829$	7608.33	+	7608.33i	0.318755	+	0.318755i	−0.529534	+	0.848289i	$0.677633\pi$
$830$		0			0
$831$	−9514.33			−0.397170
$832$		0			0
$833$	−3820.93			−0.158928
$834$		0			0
$835$	4774.55	+	4774.55i	0.197880	+	0.197880i
$836$		0			0
$837$	1963.51	−	1963.51i	0.0810860	−	0.0810860i
$838$		0			0
$839$		−	31098.4i		−	1.27966i	−0.768517	−	0.639830i	$-0.779005\pi$
$839$		−	31098.4i		−	1.27966i	0.768517	−	0.639830i	$-0.220995\pi$
$840$		0			0
$841$		−	9305.65i		−	0.381551i
$842$		0			0
$843$	−7685.27	+	7685.27i	−0.313991	+	0.313991i
$844$		0			0
$845$	−2399.23	−	2399.23i	−0.0976759	−	0.0976759i
$846$		0			0
$847$	64661.1			2.62312
$848$		0			0
$849$	0.0269294			1.08859e−6
$850$		0			0
$851$	244.972	+	244.972i	0.00986786	+	0.00986786i
$852$		0			0
$853$	−28324.9	+	28324.9i	−1.13696	+	1.13696i	−0.147968	+	0.988992i	$0.547273\pi$
$853$	−28324.9	+	28324.9i	−1.13696	+	1.13696i	−0.988992	+	0.147968i	$0.952727\pi$
$854$		0			0
$855$			4434.26i			0.177367i
$856$		0			0
$857$			14990.8i			0.597520i	0.954328	+	0.298760i	$0.0965730\pi$
$857$			14990.8i			0.597520i	−0.954328	+	0.298760i	$0.903427\pi$
$858$		0			0
$859$	−19751.5	+	19751.5i	−0.784533	+	0.784533i	−0.980592	−	0.196059i	$-0.937185\pi$
$859$	−19751.5	+	19751.5i	−0.784533	+	0.784533i	0.196059	+	0.980592i	$0.437185\pi$
$860$		0			0
$861$	−1271.48	−	1271.48i	−0.0503273	−	0.0503273i
$862$		0			0
$863$	−9084.96			−0.358349			−0.179175	−	0.983817i	$-0.557343\pi$
$863$	−9084.96			−0.358349			−0.179175	+	0.983817i	$0.557343\pi$
$864$		0			0
$865$	−14710.0			−0.578215
$866$		0			0
$867$	−3679.28	−	3679.28i	−0.144123	−	0.144123i
$868$		0			0
$869$	50706.9	−	50706.9i	1.97942	−	1.97942i
$870$		0			0
$871$		−	9927.32i		−	0.386193i
$872$		0			0
$873$		−	6526.26i		−	0.253013i
$874$		0			0
$875$	−16776.1	+	16776.1i	−0.648155	+	0.648155i
$876$		0			0
$877$	−18336.8	−	18336.8i	−0.706031	−	0.706031i	0.259668	−	0.965698i	$-0.416387\pi$
$877$	−18336.8	−	18336.8i	−0.706031	−	0.706031i	−0.965698	+	0.259668i	$0.916387\pi$
$878$		0			0
$879$	105.805			0.00405995
$880$		0			0
$881$	13402.7			0.512543			0.256271	−	0.966605i	$-0.417506\pi$
$881$	13402.7			0.512543			0.256271	+	0.966605i	$0.417506\pi$
$882$		0			0
$883$	−16285.1	−	16285.1i	−0.620656	−	0.620656i	0.325044	−	0.945699i	$-0.394621\pi$
$883$	−16285.1	−	16285.1i	−0.620656	−	0.620656i	−0.945699	+	0.325044i	$0.894621\pi$
$884$		0			0
$885$	−1122.42	+	1122.42i	−0.0426323	+	0.0426323i
$886$		0			0
$887$			35406.7i			1.34029i	0.742229	+	0.670146i	$0.233769\pi$
$887$			35406.7i			1.34029i	−0.742229	+	0.670146i	$0.766231\pi$
$888$		0			0
$889$			5646.29i			0.213015i
$890$		0			0
$891$	−3851.28	+	3851.28i	−0.144807	+	0.144807i
$892$		0			0
$893$	35659.7	+	35659.7i	1.33629	+	1.33629i
$894$		0			0
$895$	−494.138			−0.0184550
$896$		0			0
$897$	−433.352			−0.0161307
$898$		0			0
$899$	−8931.39	−	8931.39i	−0.331344	−	0.331344i
$900$		0			0
$901$	−21727.0	+	21727.0i	−0.803365	+	0.803365i
$902$		0			0
$903$			32067.3i			1.18176i
$904$		0			0
$905$		−	2769.15i		−	0.101712i
$906$		0			0
$907$	8884.93	−	8884.93i	0.325269	−	0.325269i	−0.525515	−	0.850784i	$-0.676127\pi$
$907$	8884.93	−	8884.93i	0.325269	−	0.325269i	0.850784	+	0.525515i	$0.176127\pi$
$908$		0			0
$909$	10272.0	+	10272.0i	0.374809	+	0.374809i
$910$		0			0
$911$	16668.9			0.606219			0.303110	−	0.952956i	$-0.401975\pi$
$911$	16668.9			0.606219			0.303110	+	0.952956i	$0.401975\pi$
$912$		0			0
$913$	−82877.8			−3.00422
$914$		0			0
$915$	2451.14	+	2451.14i	0.0885596	+	0.0885596i
$916$		0			0
$917$	−13502.9	+	13502.9i	−0.486266	+	0.486266i
$918$		0			0
$919$		−	26028.5i		−	0.934276i	−0.884184	−	0.467138i	$-0.845285\pi$
$919$		−	26028.5i		−	0.934276i	0.884184	−	0.467138i	$-0.154715\pi$
$920$		0			0
$921$			2625.16i			0.0939218i
$922$		0			0
$923$	13474.4	−	13474.4i	0.480516	−	0.480516i
$924$		0			0
$925$	−6499.66	−	6499.66i	−0.231035	−	0.231035i
$926$		0			0
$927$	9895.10			0.350591
$928$		0			0
$929$	−13806.2			−0.487584			−0.243792	−	0.969828i	$-0.578391\pi$
$929$	−13806.2			−0.487584			−0.243792	+	0.969828i	$0.578391\pi$
$930$		0			0
$931$	−4483.07	−	4483.07i	−0.157816	−	0.157816i
$932$		0			0
$933$	−8875.43	+	8875.43i	−0.311435	+	0.311435i
$934$		0			0
$935$		−	19966.1i		−	0.698354i
$936$		0			0
$937$		−	1798.08i		−	0.0626901i	−0.999509	−	0.0313451i	$-0.990021\pi$
$937$		−	1798.08i		−	0.0626901i	0.999509	−	0.0313451i	$-0.00997908\pi$
$938$		0			0
$939$	4506.08	−	4506.08i	0.156603	−	0.156603i
$940$		0			0
$941$	8722.11	+	8722.11i	0.302160	+	0.302160i	0.841858	−	0.539698i	$-0.181462\pi$
$941$	8722.11	+	8722.11i	0.302160	+	0.302160i	−0.539698	+	0.841858i	$0.681462\pi$
$942$		0			0
$943$	108.410			0.00374371
$944$		0			0
$945$	−2882.08			−0.0992106
$946$		0			0
$947$	−15872.2	−	15872.2i	−0.544644	−	0.544644i	0.380243	−	0.924887i	$-0.375840\pi$
$947$	−15872.2	−	15872.2i	−0.544644	−	0.544644i	−0.924887	+	0.380243i	$0.875840\pi$
$948$		0			0
$949$	25322.2	−	25322.2i	0.866168	−	0.866168i
$950$		0			0
$951$		−	4518.21i		−	0.154062i
$952$		0			0
$953$			34643.6i			1.17756i	0.808292	+	0.588782i	$0.200392\pi$
$953$			34643.6i			1.17756i	−0.808292	+	0.588782i	$0.799608\pi$
$954$		0			0
$955$	−10641.2	+	10641.2i	−0.360566	+	0.360566i
$956$		0			0
$957$	17518.2	+	17518.2i	0.591729	+	0.591729i
$958$		0			0
$959$	12820.8			0.431706
$960$		0			0
$961$	−19213.8			−0.644953
$962$		0			0
$963$	−3252.94	−	3252.94i	−0.108852	−	0.108852i
$964$		0			0
$965$	−16083.0	+	16083.0i	−0.536508	+	0.536508i
$966$		0			0
$967$		−	22531.8i		−	0.749301i	−0.927166	−	0.374650i	$-0.877763\pi$
$967$		−	22531.8i		−	0.749301i	0.927166	−	0.374650i	$-0.122237\pi$
$968$		0			0
$969$			15822.5i			0.524553i
$970$		0			0
$971$	−5854.80	+	5854.80i	−0.193501	+	0.193501i	−0.797207	−	0.603706i	$-0.793690\pi$
$971$	−5854.80	+	5854.80i	−0.193501	+	0.193501i	0.603706	+	0.797207i	$0.293690\pi$
$972$		0			0
$973$	−22252.7	−	22252.7i	−0.733184	−	0.733184i
$974$		0			0
$975$	11497.8			0.377666
$976$		0			0
$977$	50312.5			1.64753			0.823766	−	0.566930i	$-0.191869\pi$
$977$	50312.5			1.64753			0.823766	+	0.566930i	$0.191869\pi$
$978$		0			0
$979$	8798.21	+	8798.21i	0.287224	+	0.287224i
$980$		0			0
$981$	12650.5	−	12650.5i	0.411723	−	0.411723i
$982$		0			0
$983$		−	1796.13i		−	0.0582783i	−0.999575	−	0.0291391i	$-0.990723\pi$
$983$		−	1796.13i		−	0.0582783i	0.999575	−	0.0291391i	$-0.00927659\pi$
$984$		0			0
$985$		−	10433.4i		−	0.337498i
$986$		0			0
$987$	−23177.3	+	23177.3i	−0.747457	+	0.747457i
$988$		0			0
$989$	−1367.08	−	1367.08i	−0.0439540	−	0.0439540i
$990$		0			0
$991$	−32363.5			−1.03740			−0.518698	−	0.854958i	$-0.673583\pi$
$991$	−32363.5			−1.03740			−0.518698	+	0.854958i	$0.673583\pi$
$992$		0			0
$993$	7575.45			0.242094
$994$		0			0
$995$	8023.40	+	8023.40i	0.255637	+	0.255637i
$996$		0			0
$997$	−680.434	+	680.434i	−0.0216144	+	0.0216144i	−0.717831	−	0.696217i	$-0.754865\pi$
$997$	−680.434	+	680.434i	−0.0216144	+	0.0216144i	0.696217	+	0.717831i	$0.254865\pi$
$998$		0			0
$999$		−	2551.69i		−	0.0808128i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.j.b.289.8		24
4.3	odd	2		384.4.j.a.289.5		24
8.3	odd	2		192.4.j.a.145.9		24
8.5	even	2		48.4.j.a.13.1	✓	24
16.3	odd	4		192.4.j.a.49.9		24
16.5	even	4	inner	384.4.j.b.97.8		24
16.11	odd	4		384.4.j.a.97.5		24
16.13	even	4		48.4.j.a.37.1	yes	24
24.5	odd	2		144.4.k.b.109.12		24
24.11	even	2		576.4.k.b.145.8		24
48.29	odd	4		144.4.k.b.37.12		24
48.35	even	4		576.4.k.b.433.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.1	✓	24	8.5	even	2
48.4.j.a.37.1	yes	24	16.13	even	4
144.4.k.b.37.12		24	48.29	odd	4
144.4.k.b.109.12		24	24.5	odd	2
192.4.j.a.49.9		24	16.3	odd	4
192.4.j.a.145.9		24	8.3	odd	2
384.4.j.a.97.5		24	16.11	odd	4
384.4.j.a.289.5		24	4.3	odd	2
384.4.j.b.97.8		24	16.5	even	4	inner
384.4.j.b.289.8		24	1.1	even	1	trivial
576.4.k.b.145.8		24	24.11	even	2
576.4.k.b.433.8		24	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.12132 + 2.12132i) q^{3} +(3.72414 - 3.72414i) q^{5} +20.2675i q^{7} +9.00000i q^{9} +O(q^{10})\)	\(q+(2.12132 + 2.12132i) q^{3} +(3.72414 - 3.72414i) q^{5} +20.2675i q^{7} +9.00000i q^{9} +(47.5467 - 47.5467i) q^{11} +(-27.8636 - 27.8636i) q^{13} +15.8002 q^{15} +56.3788 q^{17} +(66.1489 + 66.1489i) q^{19} +(-42.9939 + 42.9939i) q^{21} -3.66579i q^{23} +97.2615i q^{25} +(-19.0919 + 19.0919i) q^{27} +(86.8428 + 86.8428i) q^{29} -102.846 q^{31} +201.724 q^{33} +(75.4791 + 75.4791i) q^{35} +(-66.8267 + 66.8267i) q^{37} -118.215i q^{39} +29.5734i q^{41} +(372.929 - 372.929i) q^{43} +(33.5173 + 33.5173i) q^{45} +539.082 q^{47} -67.7725 q^{49} +(119.598 + 119.598i) q^{51} +(-385.376 + 385.376i) q^{53} -354.142i q^{55} +280.646i q^{57} +(-71.0380 + 71.0380i) q^{59} +(155.133 + 155.133i) q^{61} -182.408 q^{63} -207.536 q^{65} +(178.141 + 178.141i) q^{67} +(7.77631 - 7.77631i) q^{69} +483.585i q^{71} +908.791i q^{73} +(-206.323 + 206.323i) q^{75} +(963.654 + 963.654i) q^{77} +1066.46 q^{79} -81.0000 q^{81} +(-871.541 - 871.541i) q^{83} +(209.963 - 209.963i) q^{85} +368.443i q^{87} +185.044i q^{89} +(564.727 - 564.727i) q^{91} +(-218.168 - 218.168i) q^{93} +492.696 q^{95} -725.140 q^{97} +(427.921 + 427.921i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 40 q^{11} + 120 q^{15} - 24 q^{19} - 400 q^{29} - 744 q^{31} + 456 q^{35} - 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 744 q^{51} - 752 q^{53} + 1376 q^{59} + 912 q^{61} - 504 q^{63} + 976 q^{65} + 2256 q^{67} + 528 q^{69} - 1104 q^{75} - 1904 q^{77} + 5992 q^{79} - 1944 q^{81} - 2680 q^{83} + 240 q^{85} + 3496 q^{91} - 7728 q^{95} + 360 q^{99}+O(q^{100}) \) 24 * q + 40 * q^11 + 120 * q^15 - 24 * q^19 - 400 * q^29 - 744 * q^31 + 456 * q^35 - 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 744 * q^51 - 752 * q^53 + 1376 * q^59 + 912 * q^61 - 504 * q^63 + 976 * q^65 + 2256 * q^67 + 528 * q^69 - 1104 * q^75 - 1904 * q^77 + 5992 * q^79 - 1944 * q^81 - 2680 * q^83 + 240 * q^85 + 3496 * q^91 - 7728 * q^95 + 360 * q^99

Introduction

L-functions

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