LMFDB - Embedded newform 576.7.h.a.161.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(161,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("576.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.h (of order $2$, degree $1$, 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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 964 x^{14} - 6608 x^{13} + 404616 x^{12} - 2342156 x^{11} + 95125730 x^{10} + \cdots + 39\!\cdots\!00 $ x^16 - 8x^15 + 964x^14 - 6608x^13 + 404616x^12 - 2342156x^11 + 95125730x^10 - 454315424x^9 + 13646460877x^8 - 51885446412x^7 + 1232155738966x^6 - 3516750797184x^5 + 70625158468274x^4 - 135448702737480x^3 + 2383004402377884x^2 - 2315857662932040*x + 39335549337519300
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{36} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.9
Root			$-2.36747 + 7.23051i$ of defining polynomial
Character	$\chi$	$=$	576.161
Dual form			576.7.h.a.161.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.3796 q^{5} -532.635 q^{7} +O(q^{10})$	$q+14.3796 q^{5} -532.635 q^{7} +1840.27 q^{11} -1004.26i q^{13} +3718.73i q^{17} +7863.61i q^{19} -19532.8i q^{23} -15418.2 q^{25} +42502.3 q^{29} -39103.0 q^{31} -7659.11 q^{35} -61799.6i q^{37} -13324.3i q^{41} -57908.9i q^{43} -51926.1i q^{47} +166051. q^{49} +52999.4 q^{53} +26462.5 q^{55} -95850.6 q^{59} +248790. i q^{61} -14440.9i q^{65} +170554. i q^{67} +477668. i q^{71} -481200. q^{73} -980194. q^{77} +358403. q^{79} +85889.4 q^{83} +53474.0i q^{85} +1.09316e6i q^{89} +534906. i q^{91} +113076. i q^{95} +1.01923e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 19888 q^{25} + 388784 q^{49} + 465728 q^{73} + 6555136 q^{97}+O(q^{100}) $ 16 * q - 19888 * q^25 + 388784 * q^49 + 465728 * q^73 + 6555136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	14.3796	0.115037	0.0575186	−	0.998344i	$-0.481681\pi$
$5$	14.3796	0.115037	0.0575186	+	0.998344i	$0.481681\pi$
$6$	0	0
$7$	−532.635	−1.55287	−0.776436	−	0.630196i	$-0.782975\pi$
$7$	−532.635	−1.55287	−0.776436	+	0.630196i	$0.782975\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1840.27	1.38262	0.691312	−	0.722557i	$-0.257033\pi$
$11$	1840.27	1.38262	0.691312	+	0.722557i	$0.257033\pi$
$12$	0	0
$13$	− 1004.26i	− 0.457106i	−0.973531	−	0.228553i	$-0.926601\pi$
$13$	− 1004.26i	− 0.457106i	0.973531	−	0.228553i	$-0.0733995\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3718.73i	0.756916i	0.925618	+	0.378458i	$0.123546\pi$
$17$	3718.73i	0.756916i	−0.925618	+	0.378458i	$0.876454\pi$
$18$	0	0
$19$	7863.61i	1.14647i	0.819392	+	0.573233i	$0.194311\pi$
$19$	7863.61i	1.14647i	−0.819392	+	0.573233i	$0.805689\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 19532.8i	− 1.60539i	−0.596391	−	0.802694i	$-0.703399\pi$
$23$	− 19532.8i	− 1.60539i	0.596391	−	0.802694i	$-0.296601\pi$
$24$	0	0
$25$	−15418.2	−0.986766
$26$	0	0
$27$	0	0
$28$	0	0
$29$	42502.3	1.74268	0.871341	−	0.490679i	$-0.163251\pi$
$29$	42502.3	1.74268	0.871341	+	0.490679i	$0.163251\pi$
$30$	0	0
$31$	−39103.0	−1.31258	−0.656289	−	0.754510i	$-0.727875\pi$
$31$	−39103.0	−1.31258	−0.656289	+	0.754510i	$0.727875\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7659.11	−0.178638
$36$	0	0
$37$	− 61799.6i	− 1.22006i	−0.792379	−	0.610029i	$-0.791158\pi$
$37$	− 61799.6i	− 1.22006i	0.792379	−	0.610029i	$-0.208842\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 13324.3i	− 0.193327i	−0.995317	−	0.0966634i	$-0.969183\pi$
$41$	− 13324.3i	− 0.193327i	0.995317	−	0.0966634i	$-0.0308170\pi$
$42$	0	0
$43$	− 57908.9i	− 0.728350i	−0.931331	−	0.364175i	$-0.881351\pi$
$43$	− 57908.9i	− 0.728350i	0.931331	−	0.364175i	$-0.118649\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 51926.1i	− 0.500141i	−0.968228	−	0.250070i	$-0.919546\pi$
$47$	− 51926.1i	− 0.500141i	0.968228	−	0.250070i	$-0.0804538\pi$
$48$	0	0
$49$	166051.	1.41141
$50$	0	0
$51$	0	0
$52$	0	0
$53$	52999.4	0.355995	0.177997	−	0.984031i	$-0.443038\pi$
$53$	52999.4	0.355995	0.177997	+	0.984031i	$0.443038\pi$
$54$	0	0
$55$	26462.5	0.159053
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−95850.6	−0.466701	−0.233351	−	0.972393i	$-0.574969\pi$
$59$	−95850.6	−0.466701	−0.233351	+	0.972393i	$0.574969\pi$
$60$	0	0
$61$	248790.i	1.09608i	0.836451	+	0.548042i	$0.184627\pi$
$61$	248790.i	1.09608i	−0.836451	+	0.548042i	$0.815373\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 14440.9i	− 0.0525842i
$66$	0	0
$67$	170554.i	0.567072i	0.958962	+	0.283536i	$0.0915075\pi$
$67$	170554.i	0.567072i	−0.958962	+	0.283536i	$0.908493\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	477668.i	1.33460i	0.744790	+	0.667299i	$0.232550\pi$
$71$	477668.i	1.33460i	−0.744790	+	0.667299i	$0.767450\pi$
$72$	0	0
$73$	−481200.	−1.23696	−0.618482	−	0.785799i	$-0.712252\pi$
$73$	−481200.	−1.23696	−0.618482	+	0.785799i	$0.712252\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−980194.	−2.14704
$78$	0	0
$79$	358403.	0.726927	0.363463	−	0.931609i	$-0.381594\pi$
$79$	358403.	0.726927	0.363463	+	0.931609i	$0.381594\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	85889.4	0.150212	0.0751061	−	0.997176i	$-0.476070\pi$
$83$	85889.4	0.150212	0.0751061	+	0.997176i	$0.476070\pi$
$84$	0	0
$85$	53474.0i	0.0870735i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.09316e6i	1.55065i	0.631561	+	0.775326i	$0.282414\pi$
$89$	1.09316e6i	1.55065i	−0.631561	+	0.775326i	$0.717586\pi$
$90$	0	0
$91$	534906.i	0.709828i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	113076.i	0.131886i
$96$	0	0
$97$	1.01923e6	1.11675	0.558377	−	0.829588i	$-0.311424\pi$
$97$	1.01923e6	1.11675	0.558377	+	0.829588i	$0.311424\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.56917e6	1.52302	0.761512	−	0.648150i	$-0.224457\pi$
$101$	1.56917e6	1.52302	0.761512	+	0.648150i	$0.224457\pi$
$102$	0	0
$103$	129905.	0.118881	0.0594407	−	0.998232i	$-0.481068\pi$
$103$	129905.	0.118881	0.0594407	+	0.998232i	$0.481068\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−891258.	−0.727532	−0.363766	−	0.931490i	$-0.618509\pi$
$107$	−891258.	−0.727532	−0.363766	+	0.931490i	$0.618509\pi$
$108$	0	0
$109$	− 26047.2i	− 0.0201132i	−0.999949	−	0.0100566i	$-0.996799\pi$
$109$	− 26047.2i	− 0.0201132i	0.999949	−	0.0100566i	$-0.00320117\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 905823.i	− 0.627781i	−0.949459	−	0.313890i	$-0.898368\pi$
$113$	− 905823.i	− 0.627781i	0.949459	−	0.313890i	$-0.101632\pi$
$114$	0	0
$115$	− 280874.i	− 0.184679i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.98073e6i	− 1.17539i
$120$	0	0
$121$	1.61504e6	0.911648
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−446391.	−0.228552
$126$	0	0
$127$	−2.11555e6	−1.03279	−0.516395	−	0.856350i	$-0.672726\pi$
$127$	−2.11555e6	−1.03279	−0.516395	+	0.856350i	$0.672726\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.08682e6	−1.81791	−0.908953	−	0.416898i	$-0.863117\pi$
$131$	−4.08682e6	−1.81791	−0.908953	+	0.416898i	$0.863117\pi$
$132$	0	0
$133$	− 4.18844e6i	− 1.78032i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.90000e6i	1.12781i	0.825839	+	0.563906i	$0.190702\pi$
$137$	2.90000e6i	1.12781i	−0.825839	+	0.563906i	$0.809298\pi$
$138$	0	0
$139$	1.47985e6i	0.551029i	0.961297	+	0.275515i	$0.0888482\pi$
$139$	1.47985e6i	0.551029i	−0.961297	+	0.275515i	$0.911152\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.84812e6i	− 0.632006i
$144$	0	0
$145$	611167.	0.200473
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.02979e6	1.21822	0.609108	−	0.793087i	$-0.291528\pi$
$149$	4.02979e6	1.21822	0.609108	+	0.793087i	$0.291528\pi$
$150$	0	0
$151$	−1.21957e6	−0.354222	−0.177111	−	0.984191i	$-0.556675\pi$
$151$	−1.21957e6	−0.354222	−0.177111	+	0.984191i	$0.556675\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−562287.	−0.150995
$156$	0	0
$157$	6.97067e6i	1.80126i	0.434591	+	0.900628i	$0.356893\pi$
$157$	6.97067e6i	1.80126i	−0.434591	+	0.900628i	$0.643107\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.04038e7i	2.49296i
$162$	0	0
$163$	6.97866e6i	1.61142i	0.592309	+	0.805711i	$0.298216\pi$
$163$	6.97866e6i	1.61142i	−0.592309	+	0.805711i	$0.701784\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.99941e6i	0.858710i	0.903136	+	0.429355i	$0.141259\pi$
$167$	3.99941e6i	0.858710i	−0.903136	+	0.429355i	$0.858741\pi$
$168$	0	0
$169$	3.81827e6	0.791054
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.10654e6	−1.56566	−0.782830	−	0.622236i	$-0.786224\pi$
$173$	−8.10654e6	−1.56566	−0.782830	+	0.622236i	$0.786224\pi$
$174$	0	0
$175$	8.21229e6	1.53232
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.38256e6	−0.589775	−0.294887	−	0.955532i	$-0.595282\pi$
$179$	−3.38256e6	−0.589775	−0.294887	+	0.955532i	$0.595282\pi$
$180$	0	0
$181$	− 1.56556e6i	− 0.264019i	−0.991248	−	0.132009i	$-0.957857\pi$
$181$	− 1.56556e6i	− 0.264019i	0.991248	−	0.132009i	$-0.0421429\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 888657.i	− 0.140352i
$186$	0	0
$187$	6.84347e6i	1.04653i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.46841e6i	1.35887i	0.733737	+	0.679434i	$0.237774\pi$
$191$	9.46841e6i	1.35887i	−0.733737	+	0.679434i	$0.762226\pi$
$192$	0	0
$193$	3.71838e6	0.517228	0.258614	−	0.965981i	$-0.416734\pi$
$193$	3.71838e6	0.517228	0.258614	+	0.965981i	$0.416734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.13905e6	−0.410582	−0.205291	−	0.978701i	$-0.565814\pi$
$197$	−3.13905e6	−0.410582	−0.205291	+	0.978701i	$0.565814\pi$
$198$	0	0
$199$	−2.62533e6	−0.333139	−0.166569	−	0.986030i	$-0.553269\pi$
$199$	−2.62533e6	−0.333139	−0.166569	+	0.986030i	$0.553269\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.26382e7	−2.70616
$204$	0	0
$205$	− 191598.i	− 0.0222398i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.44712e7i	1.58513i
$210$	0	0
$211$	2.03189e6i	0.216298i	0.994135	+	0.108149i	$0.0344924\pi$
$211$	2.03189e6i	0.216298i	−0.994135	+	0.108149i	$0.965508\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 832710.i	− 0.0837873i
$216$	0	0
$217$	2.08276e7	2.03826
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.73458e6	0.345991
$222$	0	0
$223$	−3.62230e6	−0.326641	−0.163320	−	0.986573i	$-0.552220\pi$
$223$	−3.62230e6	−0.326641	−0.163320	+	0.986573i	$0.552220\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.59003e7	−1.35934	−0.679671	−	0.733517i	$-0.737877\pi$
$227$	−1.59003e7	−1.35934	−0.679671	+	0.733517i	$0.737877\pi$
$228$	0	0
$229$	1.56301e7i	1.30154i	0.759277	+	0.650768i	$0.225553\pi$
$229$	1.56301e7i	1.30154i	−0.759277	+	0.650768i	$0.774447\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.79932e7i	1.42246i	0.702957	+	0.711232i	$0.251863\pi$
$233$	1.79932e7i	1.42246i	−0.702957	+	0.711232i	$0.748137\pi$
$234$	0	0
$235$	− 746679.i	− 0.0575348i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 2.01345e7i	− 1.47485i	−0.675429	−	0.737425i	$-0.736042\pi$
$239$	− 2.01345e7i	− 1.47485i	0.675429	−	0.737425i	$-0.263958\pi$
$240$	0	0
$241$	9.32181e6	0.665962	0.332981	−	0.942934i	$-0.391946\pi$
$241$	9.32181e6	0.665962	0.332981	+	0.942934i	$0.391946\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.38776e6	0.162365
$246$	0	0
$247$	7.89713e6	0.524057
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.90202e6	0.120280	0.0601401	−	0.998190i	$-0.480845\pi$
$251$	1.90202e6	0.120280	0.0601401	+	0.998190i	$0.480845\pi$
$252$	0	0
$253$	− 3.59456e7i	− 2.21965i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.16315e7i	0.685231i	0.939476	+	0.342616i	$0.111313\pi$
$257$	1.16315e7i	0.685231i	−0.939476	+	0.342616i	$0.888687\pi$
$258$	0	0
$259$	3.29167e7i	1.89460i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.05144e7i	− 0.577984i	−0.957332	−	0.288992i	$-0.906680\pi$
$263$	− 1.05144e7i	− 0.577984i	0.957332	−	0.288992i	$-0.0933200\pi$
$264$	0	0
$265$	762113.	0.0409526
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.14970e7	1.61813	0.809064	−	0.587721i	$-0.199975\pi$
$269$	3.14970e7	1.61813	0.809064	+	0.587721i	$0.199975\pi$
$270$	0	0
$271$	−3.21890e7	−1.61733	−0.808667	−	0.588266i	$-0.799811\pi$
$271$	−3.21890e7	−1.61733	−0.808667	+	0.588266i	$0.799811\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.83737e7	−1.36433
$276$	0	0
$277$	− 1.02479e7i	− 0.482167i	−0.970504	−	0.241083i	$-0.922497\pi$
$277$	− 1.02479e7i	− 0.482167i	0.970504	−	0.241083i	$-0.0775027\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.73088e7i	1.68148i	0.541438	+	0.840741i	$0.317880\pi$
$281$	3.73088e7i	1.68148i	−0.541438	+	0.840741i	$0.682120\pi$
$282$	0	0
$283$	3.95134e7i	1.74335i	0.490082	+	0.871676i	$0.336967\pi$
$283$	3.95134e7i	1.74335i	−0.490082	+	0.871676i	$0.663033\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.09698e6i	0.300212i
$288$	0	0
$289$	1.03086e7	0.427078
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.50715e7	−0.996731	−0.498365	−	0.866967i	$-0.666066\pi$
$293$	−2.50715e7	−0.996731	−0.498365	+	0.866967i	$0.666066\pi$
$294$	0	0
$295$	−1.37830e6	−0.0536880
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.96160e7	−0.733833
$300$	0	0
$301$	3.08443e7i	1.13103i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.57752e6i	0.126090i
$306$	0	0
$307$	− 8.64086e6i	− 0.298636i	−0.988789	−	0.149318i	$-0.952292\pi$
$307$	− 8.64086e6i	− 0.298636i	0.988789	−	0.149318i	$-0.0477078\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.01336e7i	0.669329i	0.942337	+	0.334665i	$0.108623\pi$
$311$	2.01336e7i	0.669329i	−0.942337	+	0.334665i	$0.891377\pi$
$312$	0	0
$313$	1.42830e6	0.0465786	0.0232893	−	0.999729i	$-0.492586\pi$
$313$	1.42830e6	0.0465786	0.0232893	+	0.999729i	$0.492586\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.13641e7	0.356745	0.178373	−	0.983963i	$-0.442917\pi$
$317$	1.13641e7	0.356745	0.178373	+	0.983963i	$0.442917\pi$
$318$	0	0
$319$	7.82157e7	2.40947
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.92426e7	−0.867779
$324$	0	0
$325$	1.54839e7i	0.451057i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.76577e7i	0.776655i
$330$	0	0
$331$	− 4.32471e7i	− 1.19254i	−0.802784	−	0.596271i	$-0.796649\pi$
$331$	− 4.32471e7i	− 1.19254i	0.802784	−	0.596271i	$-0.203351\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.45251e6i	0.0652343i
$336$	0	0
$337$	6.62511e7	1.73103	0.865513	−	0.500886i	$-0.166993\pi$
$337$	6.62511e7	1.73103	0.865513	+	0.500886i	$0.166993\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.19601e7	−1.81480
$342$	0	0
$343$	−2.57807e7	−0.638871
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.91544e7	0.458437	0.229219	−	0.973375i	$-0.426383\pi$
$347$	1.91544e7	0.458437	0.229219	+	0.973375i	$0.426383\pi$
$348$	0	0
$349$	− 6.86099e6i	− 0.161403i	−0.996738	−	0.0807013i	$-0.974284\pi$
$349$	− 6.86099e6i	− 0.161403i	0.996738	−	0.0807013i	$-0.0257160\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.57294e7i	− 0.584932i	−0.956276	−	0.292466i	$-0.905524\pi$
$353$	− 2.57294e7i	− 0.584932i	0.956276	−	0.292466i	$-0.0944758\pi$
$354$	0	0
$355$	6.86869e6i	0.153528i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.37299e7i	0.729007i	0.931202	+	0.364503i	$0.118761\pi$
$359$	3.37299e7i	0.729007i	−0.931202	+	0.364503i	$0.881239\pi$
$360$	0	0
$361$	−1.47905e7	−0.314385
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.91949e6	−0.142297
$366$	0	0
$367$	−6.34156e7	−1.28291	−0.641457	−	0.767159i	$-0.721670\pi$
$367$	−6.34156e7	−1.28291	−0.641457	+	0.767159i	$0.721670\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.82293e7	−0.552814
$372$	0	0
$373$	6.59689e7i	1.27120i	0.772020	+	0.635598i	$0.219247\pi$
$373$	6.59689e7i	1.27120i	−0.772020	+	0.635598i	$0.780753\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.26834e7i	− 0.796591i
$378$	0	0
$379$	4.89537e7i	0.899225i	0.893224	+	0.449612i	$0.148438\pi$
$379$	4.89537e7i	0.899225i	−0.893224	+	0.449612i	$0.851562\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.14530e7i	0.915830i	0.888996	+	0.457915i	$0.151404\pi$
$383$	5.14530e7i	0.915830i	−0.888996	+	0.457915i	$0.848596\pi$
$384$	0	0
$385$	−1.40948e7	−0.246989
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.56825e7	0.606187	0.303094	−	0.952961i	$-0.401981\pi$
$389$	3.56825e7	0.606187	0.303094	+	0.952961i	$0.401981\pi$
$390$	0	0
$391$	7.26370e7	1.21514
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.15371e6	0.0836236
$396$	0	0
$397$	− 7.15422e7i	− 1.14338i	−0.820469	−	0.571691i	$-0.806288\pi$
$397$	− 7.15422e7i	− 1.14338i	0.820469	−	0.571691i	$-0.193712\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.88082e7i	0.912021i	0.889974	+	0.456011i	$0.150722\pi$
$401$	5.88082e7i	0.912021i	−0.889974	+	0.456011i	$0.849278\pi$
$402$	0	0
$403$	3.92697e7i	0.599987i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.13728e8i	− 1.68688i
$408$	0	0
$409$	−7.97534e7	−1.16568	−0.582840	−	0.812587i	$-0.698058\pi$
$409$	−7.97534e7	−1.16568	−0.582840	+	0.812587i	$0.698058\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.10534e7	0.724727
$414$	0	0
$415$	1.23506e6	0.0172800
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.32343e8	1.79912	0.899558	−	0.436802i	$-0.143889\pi$
$419$	1.32343e8	1.79912	0.899558	+	0.436802i	$0.143889\pi$
$420$	0	0
$421$	− 5.98882e7i	− 0.802592i	−0.915948	−	0.401296i	$-0.868560\pi$
$421$	− 5.98882e7i	− 0.802592i	0.915948	−	0.401296i	$-0.131440\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 5.73362e7i	− 0.746899i
$426$	0	0
$427$	− 1.32514e8i	− 1.70208i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 9.68795e7i	− 1.21004i	−0.796210	−	0.605021i	$-0.793165\pi$
$431$	− 9.68795e7i	− 1.21004i	0.796210	−	0.605021i	$-0.206835\pi$
$432$	0	0
$433$	−1.29433e8	−1.59434	−0.797171	−	0.603754i	$-0.793671\pi$
$433$	−1.29433e8	−1.59434	−0.797171	+	0.603754i	$0.793671\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.53598e8	1.84052
$438$	0	0
$439$	−9.05824e7	−1.07066	−0.535328	−	0.844644i	$-0.679812\pi$
$439$	−9.05824e7	−1.07066	−0.535328	+	0.844644i	$0.679812\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.24978e8	1.43755	0.718773	−	0.695245i	$-0.244704\pi$
$443$	1.24978e8	1.43755	0.718773	+	0.695245i	$0.244704\pi$
$444$	0	0
$445$	1.57193e7i	0.178383i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4.62993e7i	− 0.511488i	−0.966745	−	0.255744i	$-0.917680\pi$
$449$	− 4.62993e7i	− 0.511488i	0.966745	−	0.255744i	$-0.0823205\pi$
$450$	0	0
$451$	− 2.45203e7i	− 0.267298i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.69176e6i	0.0816566i
$456$	0	0
$457$	1.10415e8	1.15686	0.578430	−	0.815732i	$-0.303666\pi$
$457$	1.10415e8	1.15686	0.578430	+	0.815732i	$0.303666\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.38554e8	−1.41421	−0.707107	−	0.707106i	$-0.750000\pi$
$461$	−1.38554e8	−1.41421	−0.707107	+	0.707106i	$0.750000\pi$
$462$	0	0
$463$	−9.36448e7	−0.943498	−0.471749	−	0.881733i	$-0.656377\pi$
$463$	−9.36448e7	−0.943498	−0.471749	+	0.881733i	$0.656377\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.08817e8	−1.06843	−0.534214	−	0.845350i	$-0.679392\pi$
$467$	−1.08817e8	−1.06843	−0.534214	+	0.845350i	$0.679392\pi$
$468$	0	0
$469$	− 9.08432e7i	− 0.880590i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.06568e8i	− 1.00703i
$474$	0	0
$475$	− 1.21243e8i	− 1.13129i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.14845e8i	1.04497i	0.852647	+	0.522487i	$0.174996\pi$
$479$	1.14845e8i	1.04497i	−0.852647	+	0.522487i	$0.825004\pi$
$480$	0	0
$481$	−6.20631e7	−0.557697
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.46562e7	0.128468
$486$	0	0
$487$	4.88400e7	0.422852	0.211426	−	0.977394i	$-0.432189\pi$
$487$	4.88400e7	0.422852	0.211426	+	0.977394i	$0.432189\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.74342e8	1.47285	0.736425	−	0.676519i	$-0.236513\pi$
$491$	1.74342e8	1.47285	0.736425	+	0.676519i	$0.236513\pi$
$492$	0	0
$493$	1.58054e8i	1.31906i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.54423e8i	− 2.07246i
$498$	0	0
$499$	− 1.75384e7i	− 0.141152i	−0.997506	−	0.0705762i	$-0.977516\pi$
$499$	− 1.75384e7i	− 0.141152i	0.997506	−	0.0705762i	$-0.0224838\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 2.19431e6i	− 0.0172423i	−0.999963	−	0.00862114i	$-0.997256\pi$
$503$	− 2.19431e6i	− 0.0172423i	0.999963	−	0.00862114i	$-0.00274423\pi$
$504$	0	0
$505$	2.25642e7	0.175204
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.55819e8	−1.18159	−0.590794	−	0.806823i	$-0.701185\pi$
$509$	−1.55819e8	−1.18159	−0.590794	+	0.806823i	$0.701185\pi$
$510$	0	0
$511$	2.56304e8	1.92085
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.86799e6	0.0136758
$516$	0	0
$517$	− 9.55582e7i	− 0.691506i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.84025e8i	− 1.30126i	−0.759394	−	0.650631i	$-0.774505\pi$
$521$	− 1.84025e8i	− 1.30126i	0.759394	−	0.650631i	$-0.225495\pi$
$522$	0	0
$523$	− 1.66067e8i	− 1.16086i	−0.814311	−	0.580428i	$-0.802885\pi$
$523$	− 1.66067e8i	− 1.16086i	0.814311	−	0.580428i	$-0.197115\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.45413e8i	− 0.993511i
$528$	0	0
$529$	−2.33493e8	−1.57727
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.33811e7	−0.0883709
$534$	0	0
$535$	−1.28160e7	−0.0836932
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.05579e8	1.95145
$540$	0	0
$541$	2.97979e8i	1.88189i	0.338563	+	0.940944i	$0.390059\pi$
$541$	2.97979e8i	1.88189i	−0.338563	+	0.940944i	$0.609941\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 374549.i	− 0.00231376i
$546$	0	0
$547$	− 6.85340e7i	− 0.418740i	−0.977836	−	0.209370i	$-0.932859\pi$
$547$	− 6.85340e7i	− 0.418740i	0.977836	−	0.209370i	$-0.0671413\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.34221e8i	1.99793i
$552$	0	0
$553$	−1.90898e8	−1.12882
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.92323e6	−0.0516365	−0.0258182	−	0.999667i	$-0.508219\pi$
$557$	−8.92323e6	−0.0516365	−0.0258182	+	0.999667i	$0.508219\pi$
$558$	0	0
$559$	−5.81558e7	−0.332933
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.27610e8	0.715088	0.357544	−	0.933896i	$-0.383614\pi$
$563$	1.27610e8	0.715088	0.357544	+	0.933896i	$0.383614\pi$
$564$	0	0
$565$	− 1.30254e7i	− 0.0722181i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.40278e7i	0.510410i	0.966887	+	0.255205i	$0.0821430\pi$
$569$	9.40278e7i	0.510410i	−0.966887	+	0.255205i	$0.917857\pi$
$570$	0	0
$571$	− 5.09226e7i	− 0.273528i	−0.990604	−	0.136764i	$-0.956330\pi$
$571$	− 5.09226e7i	− 0.273528i	0.990604	−	0.136764i	$-0.0436703\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.01160e8i	1.58414i
$576$	0	0
$577$	−5.59555e7	−0.291283	−0.145641	−	0.989337i	$-0.546525\pi$
$577$	−5.59555e7	−0.291283	−0.145641	+	0.989337i	$0.546525\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.57477e7	−0.233260
$582$	0	0
$583$	9.75333e7	0.492206
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.98205e8	1.47435	0.737175	−	0.675702i	$-0.236160\pi$
$587$	2.98205e8	1.47435	0.737175	+	0.675702i	$0.236160\pi$
$588$	0	0
$589$	− 3.07491e8i	− 1.50483i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 3.66078e7i	− 0.175554i	−0.996140	−	0.0877769i	$-0.972024\pi$
$593$	− 3.66078e7i	− 0.175554i	0.996140	−	0.0877769i	$-0.0279763\pi$
$594$	0	0
$595$	− 2.84821e7i	− 0.135214i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 2.08514e8i	− 0.970186i	−0.874463	−	0.485093i	$-0.838786\pi$
$599$	− 2.08514e8i	− 0.970186i	0.874463	−	0.485093i	$-0.161214\pi$
$600$	0	0
$601$	−2.59296e8	−1.19446	−0.597232	−	0.802069i	$-0.703733\pi$
$601$	−2.59296e8	−1.19446	−0.597232	+	0.802069i	$0.703733\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.32237e7	0.104873
$606$	0	0
$607$	−3.21946e8	−1.43952	−0.719759	−	0.694224i	$-0.755748\pi$
$607$	−3.21946e8	−1.43952	−0.719759	+	0.694224i	$0.755748\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.21475e7	−0.228618
$612$	0	0
$613$	1.19987e8i	0.520899i	0.965487	+	0.260450i	$0.0838708\pi$
$613$	1.19987e8i	0.520899i	−0.965487	+	0.260450i	$0.916129\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.56694e8i	− 1.51859i	−0.650748	−	0.759294i	$-0.725544\pi$
$617$	− 3.56694e8i	− 1.51859i	0.650748	−	0.759294i	$-0.274456\pi$
$618$	0	0
$619$	5.20499e7i	0.219456i	0.993962	+	0.109728i	$0.0349980\pi$
$619$	5.20499e7i	0.219456i	−0.993962	+	0.109728i	$0.965002\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 5.82256e8i	− 2.40796i
$624$	0	0
$625$	2.34491e8	0.960474
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.29816e8	0.923482
$630$	0	0
$631$	1.25904e8	0.501131	0.250565	−	0.968100i	$-0.419384\pi$
$631$	1.25904e8	0.501131	0.250565	+	0.968100i	$0.419384\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.04209e7	−0.118809
$636$	0	0
$637$	− 1.66759e8i	− 0.645166i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3.04089e8i	− 1.15459i	−0.816537	−	0.577294i	$-0.804109\pi$
$641$	− 3.04089e8i	− 1.15459i	0.816537	−	0.577294i	$-0.195891\pi$
$642$	0	0
$643$	1.05787e8i	0.397921i	0.980007	+	0.198961i	$0.0637566\pi$
$643$	1.05787e8i	0.397921i	−0.980007	+	0.198961i	$0.936243\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.17062e8i	− 1.17066i	−0.810795	−	0.585330i	$-0.800965\pi$
$647$	− 3.17062e8i	− 1.17066i	0.810795	−	0.585330i	$-0.199035\pi$
$648$	0	0
$649$	−1.76391e8	−0.645272
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.23336e8	1.16122	0.580610	−	0.814182i	$-0.302814\pi$
$653$	3.23336e8	1.16122	0.580610	+	0.814182i	$0.302814\pi$
$654$	0	0
$655$	−5.87670e7	−0.209127
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.55905e8	−0.544757	−0.272379	−	0.962190i	$-0.587810\pi$
$659$	−1.55905e8	−0.544757	−0.272379	+	0.962190i	$0.587810\pi$
$660$	0	0
$661$	6.52039e7i	0.225771i	0.993608	+	0.112886i	$0.0360094\pi$
$661$	6.52039e7i	0.225771i	−0.993608	+	0.112886i	$0.963991\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 6.02283e7i	− 0.204803i
$666$	0	0
$667$	− 8.30186e8i	− 2.79768i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.57842e8i	1.51547i
$672$	0	0
$673$	3.26382e8	1.07073	0.535365	−	0.844621i	$-0.320174\pi$
$673$	3.26382e8	1.07073	0.535365	+	0.844621i	$0.320174\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.44774e8	0.466580	0.233290	−	0.972407i	$-0.425051\pi$
$677$	1.44774e8	0.466580	0.233290	+	0.972407i	$0.425051\pi$
$678$	0	0
$679$	−5.42878e8	−1.73418
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.29560e8	1.66208	0.831042	−	0.556210i	$-0.187745\pi$
$683$	5.29560e8	1.66208	0.831042	+	0.556210i	$0.187745\pi$
$684$	0	0
$685$	4.17010e7i	0.129740i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 5.32253e7i	− 0.162727i
$690$	0	0
$691$	2.84591e8i	0.862555i	0.902219	+	0.431277i	$0.141937\pi$
$691$	2.84591e8i	0.862555i	−0.902219	+	0.431277i	$0.858063\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.12798e7i	0.0633888i
$696$	0	0
$697$	4.95494e7	0.146332
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.68763e8	−0.780216	−0.390108	−	0.920769i	$-0.627562\pi$
$701$	−2.68763e8	−0.780216	−0.390108	+	0.920769i	$0.627562\pi$
$702$	0	0
$703$	4.85968e8	1.39876
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.35797e8	−2.36506
$708$	0	0
$709$	− 3.60170e7i	− 0.101057i	−0.998723	−	0.0505287i	$-0.983909\pi$
$709$	− 3.60170e7i	− 0.101057i	0.998723	−	0.0505287i	$-0.0160906\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.63789e8i	2.10720i
$714$	0	0
$715$	− 2.65753e7i	− 0.0727042i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1.01849e8i	− 0.274012i	−0.990570	−	0.137006i	$-0.956252\pi$
$719$	− 1.01849e8i	− 0.274012i	0.990570	−	0.137006i	$-0.0437479\pi$
$720$	0	0
$721$	−6.91919e7	−0.184608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.55309e8	−1.71962
$726$	0	0
$727$	1.74848e7	0.0455049	0.0227524	−	0.999741i	$-0.492757\pi$
$727$	1.74848e7	0.0455049	0.0227524	+	0.999741i	$0.492757\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.15347e8	0.551299
$732$	0	0
$733$	− 5.93038e8i	− 1.50581i	−0.658128	−	0.752906i	$-0.728652\pi$
$733$	− 5.93038e8i	− 1.50581i	0.658128	−	0.752906i	$-0.271348\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.13866e8i	0.784047i
$738$	0	0
$739$	6.40859e8i	1.58792i	0.607969	+	0.793961i	$0.291985\pi$
$739$	6.40859e8i	1.58792i	−0.607969	+	0.793961i	$0.708015\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.54358e8i	0.863925i	0.901892	+	0.431963i	$0.142179\pi$
$743$	3.54358e8i	0.863925i	−0.901892	+	0.431963i	$0.857821\pi$
$744$	0	0
$745$	5.79470e7	0.140140
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.74715e8	1.12976
$750$	0	0
$751$	1.91251e8	0.451527	0.225763	−	0.974182i	$-0.427512\pi$
$751$	1.91251e8	0.451527	0.225763	+	0.974182i	$0.427512\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.75370e7	−0.0407487
$756$	0	0
$757$	5.16142e8i	1.18982i	0.803792	+	0.594910i	$0.202812\pi$
$757$	5.16142e8i	1.18982i	−0.803792	+	0.594910i	$0.797188\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 2.06817e8i	− 0.469281i	−0.972082	−	0.234640i	$-0.924609\pi$
$761$	− 2.06817e8i	− 0.469281i	0.972082	−	0.234640i	$-0.0753912\pi$
$762$	0	0
$763$	1.38736e7i	0.0312332i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.62592e7i	0.213332i
$768$	0	0
$769$	−3.45353e8	−0.759423	−0.379711	−	0.925105i	$-0.623977\pi$
$769$	−3.45353e8	−0.759423	−0.379711	+	0.925105i	$0.623977\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.05512e8	−1.31094	−0.655472	−	0.755219i	$-0.727530\pi$
$773$	−6.05512e8	−1.31094	−0.655472	+	0.755219i	$0.727530\pi$
$774$	0	0
$775$	6.02899e8	1.29521
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.04777e8	0.221643
$780$	0	0
$781$	8.79038e8i	1.84525i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.00236e8i	0.207211i
$786$	0	0
$787$	7.52622e8i	1.54402i	0.635611	+	0.772009i	$0.280748\pi$
$787$	7.52622e8i	1.54402i	−0.635611	+	0.772009i	$0.719252\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.82473e8i	0.974863i
$792$	0	0
$793$	2.49851e8	0.501027
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.45541e8	0.880062	0.440031	−	0.897983i	$-0.354967\pi$
$797$	4.45541e8	0.880062	0.440031	+	0.897983i	$0.354967\pi$
$798$	0	0
$799$	1.93099e8	0.378565
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.85539e8	−1.71026
$804$	0	0
$805$	1.49603e8i	0.286783i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.08582e8i	0.960540i	0.877121	+	0.480270i	$0.159461\pi$
$809$	5.08582e8i	0.960540i	−0.877121	+	0.480270i	$0.840539\pi$
$810$	0	0
$811$	4.00397e8i	0.750633i	0.926897	+	0.375317i	$0.122466\pi$
$811$	4.00397e8i	0.750633i	−0.926897	+	0.375317i	$0.877534\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.00351e8i	0.185373i
$816$	0	0
$817$	4.55373e8	0.835028
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.91805e8	−1.61154	−0.805769	−	0.592230i	$-0.798248\pi$
$821$	−8.91805e8	−1.61154	−0.805769	+	0.592230i	$0.798248\pi$
$822$	0	0
$823$	−4.37666e8	−0.785134	−0.392567	−	0.919723i	$-0.628413\pi$
$823$	−4.37666e8	−0.785134	−0.392567	+	0.919723i	$0.628413\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.07859e8	−0.367497	−0.183748	−	0.982973i	$-0.558823\pi$
$827$	−2.07859e8	−0.367497	−0.183748	+	0.982973i	$0.558823\pi$
$828$	0	0
$829$	− 3.17032e8i	− 0.556467i	−0.960513	−	0.278234i	$-0.910251\pi$
$829$	− 3.17032e8i	− 0.556467i	0.960513	−	0.278234i	$-0.0897490\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.17499e8i	1.06832i
$834$	0	0
$835$	5.75101e7i	0.0987836i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 3.78582e8i	− 0.641024i	−0.947245	−	0.320512i	$-0.896145\pi$
$839$	− 3.78582e8i	− 0.641024i	0.947245	−	0.320512i	$-0.103855\pi$
$840$	0	0
$841$	1.21162e9	2.03694
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.49053e7	0.0910006
$846$	0	0
$847$	−8.60227e8	−1.41567
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.20712e9	−1.95867
$852$	0	0
$853$	− 5.37740e8i	− 0.866413i	−0.901295	−	0.433206i	$-0.857382\pi$
$853$	− 5.37740e8i	− 0.866413i	0.901295	−	0.433206i	$-0.142618\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.49442e8i	0.555179i	0.960700	+	0.277589i	$0.0895355\pi$
$857$	3.49442e8i	0.555179i	−0.960700	+	0.277589i	$0.910465\pi$
$858$	0	0
$859$	− 8.75469e8i	− 1.38121i	−0.723230	−	0.690607i	$-0.757343\pi$
$859$	− 8.75469e8i	− 1.38121i	0.723230	−	0.690607i	$-0.242657\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 5.65174e8i	− 0.879325i	−0.898163	−	0.439663i	$-0.855098\pi$
$863$	− 5.65174e8i	− 0.879325i	0.898163	−	0.439663i	$-0.144902\pi$
$864$	0	0
$865$	−1.16569e8	−0.180109
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.59559e8	1.00507
$870$	0	0
$871$	1.71281e8	0.259212
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.37763e8	0.354912
$876$	0	0
$877$	− 1.08310e9i	− 1.60573i	−0.596164	−	0.802863i	$-0.703309\pi$
$877$	− 1.08310e9i	− 1.60573i	0.596164	−	0.802863i	$-0.296691\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.68321e8i	1.41609i	0.706166	+	0.708047i	$0.250423\pi$
$881$	9.68321e8i	1.41609i	−0.706166	+	0.708047i	$0.749577\pi$
$882$	0	0
$883$	1.60909e8i	0.233721i	0.993148	+	0.116860i	$0.0372830\pi$
$883$	1.60909e8i	0.233721i	−0.993148	+	0.116860i	$0.962717\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.06540e9i	− 1.52666i	−0.646011	−	0.763328i	$-0.723564\pi$
$887$	− 1.06540e9i	− 1.52666i	0.646011	−	0.763328i	$-0.276436\pi$
$888$	0	0
$889$	1.12682e9	1.60379
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.08327e8	0.573395
$894$	0	0
$895$	−4.86400e7	−0.0678460
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.66197e9	−2.28740
$900$	0	0
$901$	1.97090e8i	0.269458i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 2.25122e7i	− 0.0303720i
$906$	0	0
$907$	5.33348e8i	0.714807i	0.933950	+	0.357404i	$0.116338\pi$
$907$	5.33348e8i	0.714807i	−0.933950	+	0.357404i	$0.883662\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.01975e9i	1.34878i	0.738377	+	0.674388i	$0.235593\pi$
$911$	1.01975e9i	1.34878i	−0.738377	+	0.674388i	$0.764407\pi$
$912$	0	0
$913$	1.58060e8	0.207687
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.17678e9	2.82298
$918$	0	0
$919$	−1.70464e8	−0.219627	−0.109814	−	0.993952i	$-0.535025\pi$
$919$	−1.70464e8	−0.219627	−0.109814	+	0.993952i	$0.535025\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.79704e8	0.610054
$924$	0	0
$925$	9.52841e8i	1.20391i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.76995e8i	0.220756i	0.993890	+	0.110378i	$0.0352062\pi$
$929$	1.76995e8i	0.220756i	−0.993890	+	0.110378i	$0.964794\pi$
$930$	0	0
$931$	1.30576e9i	1.61814i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.84067e7i	0.120390i
$936$	0	0
$937$	−9.47406e8	−1.15164	−0.575821	−	0.817576i	$-0.695318\pi$
$937$	−9.47406e8	−1.15164	−0.575821	+	0.817576i	$0.695318\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.18896e9	−1.42691	−0.713457	−	0.700699i	$-0.752871\pi$
$941$	−1.18896e9	−1.42691	−0.713457	+	0.700699i	$0.752871\pi$
$942$	0	0
$943$	−2.60260e8	−0.310365
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.32630e8	−1.09814	−0.549072	−	0.835775i	$-0.685019\pi$
$947$	−9.32630e8	−1.09814	−0.549072	+	0.835775i	$0.685019\pi$
$948$	0	0
$949$	4.83251e8i	0.565424i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.90222e8i	− 0.219777i	−0.993944	−	0.109888i	$-0.964951\pi$
$953$	− 1.90222e8i	− 0.219777i	0.993944	−	0.109888i	$-0.0350494\pi$
$954$	0	0
$955$	1.36152e8i	0.156320i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.54464e9i	− 1.75135i
$960$	0	0
$961$	6.41540e8	0.722859
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.34691e7	0.0595005
$966$	0	0
$967$	3.01606e8	0.333549	0.166775	−	0.985995i	$-0.446665\pi$
$967$	3.01606e8	0.333549	0.166775	+	0.985995i	$0.446665\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.86164e8	0.312577	0.156289	−	0.987711i	$-0.450047\pi$
$971$	2.86164e8	0.312577	0.156289	+	0.987711i	$0.450047\pi$
$972$	0	0
$973$	− 7.88223e8i	− 0.855678i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.63284e9i	1.75090i	0.483313	+	0.875448i	$0.339433\pi$
$977$	1.63284e9i	1.75090i	−0.483313	+	0.875448i	$0.660567\pi$
$978$	0	0
$979$	2.01171e9i	2.14397i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.09869e9i	1.15668i	0.815794	+	0.578342i	$0.196300\pi$
$983$	1.09869e9i	1.15668i	−0.815794	+	0.578342i	$0.803700\pi$
$984$	0	0
$985$	−4.51385e7	−0.0472322
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.13112e9	−1.16928
$990$	0	0
$991$	2.53484e6	0.00260453	0.00130227	−	0.999999i	$-0.499585\pi$
$991$	2.53484e6	0.00260453	0.00130227	+	0.999999i	$0.499585\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.77514e7	−0.0383233
$996$	0	0
$997$	− 1.83353e9i	− 1.85013i	−0.379810	−	0.925065i	$-0.624011\pi$
$997$	− 1.83353e9i	− 1.85013i	0.379810	−	0.925065i	$-0.375989\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.h.a.161.9	yes	16
3.2	odd	2	inner	576.7.h.a.161.5	✓	16
4.3	odd	2	inner	576.7.h.a.161.11	yes	16
8.3	odd	2	inner	576.7.h.a.161.8	yes	16
8.5	even	2	inner	576.7.h.a.161.6	yes	16
12.11	even	2	inner	576.7.h.a.161.7	yes	16
24.5	odd	2	inner	576.7.h.a.161.10	yes	16
24.11	even	2	inner	576.7.h.a.161.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.7.h.a.161.5	✓	16	3.2	odd	2	inner
576.7.h.a.161.6	yes	16	8.5	even	2	inner
576.7.h.a.161.7	yes	16	12.11	even	2	inner
576.7.h.a.161.8	yes	16	8.3	odd	2	inner
576.7.h.a.161.9	yes	16	1.1	even	1	trivial
576.7.h.a.161.10	yes	16	24.5	odd	2	inner
576.7.h.a.161.11	yes	16	4.3	odd	2	inner
576.7.h.a.161.12	yes	16	24.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+14.3796 q^{5} -532.635 q^{7} +O(q^{10})\)	\(q+14.3796 q^{5} -532.635 q^{7} +1840.27 q^{11} -1004.26i q^{13} +3718.73i q^{17} +7863.61i q^{19} -19532.8i q^{23} -15418.2 q^{25} +42502.3 q^{29} -39103.0 q^{31} -7659.11 q^{35} -61799.6i q^{37} -13324.3i q^{41} -57908.9i q^{43} -51926.1i q^{47} +166051. q^{49} +52999.4 q^{53} +26462.5 q^{55} -95850.6 q^{59} +248790. i q^{61} -14440.9i q^{65} +170554. i q^{67} +477668. i q^{71} -481200. q^{73} -980194. q^{77} +358403. q^{79} +85889.4 q^{83} +53474.0i q^{85} +1.09316e6i q^{89} +534906. i q^{91} +113076. i q^{95} +1.01923e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 19888 q^{25} + 388784 q^{49} + 465728 q^{73} + 6555136 q^{97}+O(q^{100}) \) 16 * q - 19888 * q^25 + 388784 * q^49 + 465728 * q^73 + 6555136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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