LMFDB - Embedded newform 1008.4.h.a.575.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(575,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1008.575");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 144x^{10} + 12024x^{8} - 766296x^{6} + 11751192x^{4} + 565147728x^{2} + 9666232489 $ x^12 - 144x^10 + 12024x^8 - 766296x^6 + 11751192x^4 + 565147728*x^2 + 9666232489
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.6
Root			$6.97168 - 5.87875i$ of defining polynomial
Character	$\chi$	$=$	1008.575
Dual form			1008.4.h.a.575.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.60008i q^{5} +7.00000i q^{7} +O(q^{10})$	$q-3.60008i q^{5} +7.00000i q^{7} +47.6725 q^{11} +42.1639 q^{13} +79.4043i q^{17} -8.89010i q^{19} -134.022 q^{23} +112.039 q^{25} -264.848i q^{29} +304.064i q^{31} +25.2005 q^{35} -413.774 q^{37} +68.1112i q^{41} -218.735i q^{43} +457.000 q^{47} -49.0000 q^{49} +210.882i q^{53} -171.625i q^{55} -68.1730 q^{59} +179.160 q^{61} -151.793i q^{65} +665.378i q^{67} +886.237 q^{71} +615.428 q^{73} +333.708i q^{77} +1273.51i q^{79} +952.085 q^{83} +285.862 q^{85} +926.586i q^{89} +295.148i q^{91} -32.0050 q^{95} -1500.95 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) $ 12 * q + 192 * q^13 + 84 * q^25 + 72 * q^37 - 588 * q^49 - 1800 * q^61 + 3144 * q^85 - 1152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 3.60008i	− 0.322001i	−0.986954	−	0.161000i	$-0.948528\pi$
$5$	− 3.60008i	− 0.322001i	0.986954	−	0.161000i	$-0.0514720\pi$
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	47.6725	1.30671	0.653355	−	0.757052i	$-0.273361\pi$
$11$	47.6725	1.30671	0.653355	+	0.757052i	$0.273361\pi$
$12$	0	0
$13$	42.1639	0.899552	0.449776	−	0.893141i	$-0.351504\pi$
$13$	42.1639	0.899552	0.449776	+	0.893141i	$0.351504\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	79.4043i	1.13285i	0.824115	+	0.566423i	$0.191673\pi$
$17$	79.4043i	1.13285i	−0.824115	+	0.566423i	$0.808327\pi$
$18$	0	0
$19$	− 8.89010i	− 0.107344i	−0.998559	−	0.0536718i	$-0.982908\pi$
$19$	− 8.89010i	− 0.107344i	0.998559	−	0.0536718i	$-0.0170925\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−134.022	−1.21502	−0.607510	−	0.794312i	$-0.707831\pi$
$23$	−134.022	−1.21502	−0.607510	+	0.794312i	$0.707831\pi$
$24$	0	0
$25$	112.039	0.896316
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 264.848i	− 1.69590i	−0.530076	−	0.847950i	$-0.677837\pi$
$29$	− 264.848i	− 1.69590i	0.530076	−	0.847950i	$-0.322163\pi$
$30$	0	0
$31$	304.064i	1.76166i	0.473430	+	0.880832i	$0.343016\pi$
$31$	304.064i	1.76166i	−0.473430	+	0.880832i	$0.656984\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	25.2005	0.121705
$36$	0	0
$37$	−413.774	−1.83849	−0.919244	−	0.393689i	$-0.871199\pi$
$37$	−413.774	−1.83849	−0.919244	+	0.393689i	$0.871199\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	68.1112i	0.259444i	0.991550	+	0.129722i	$0.0414084\pi$
$41$	68.1112i	0.259444i	−0.991550	+	0.129722i	$0.958592\pi$
$42$	0	0
$43$	− 218.735i	− 0.775738i	−0.921715	−	0.387869i	$-0.873211\pi$
$43$	− 218.735i	− 0.775738i	0.921715	−	0.387869i	$-0.126789\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	457.000	1.41830	0.709152	−	0.705056i	$-0.249078\pi$
$47$	457.000	1.41830	0.709152	+	0.705056i	$0.249078\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	210.882i	0.546545i	0.961937	+	0.273272i	$0.0881061\pi$
$53$	210.882i	0.546545i	−0.961937	+	0.273272i	$0.911894\pi$
$54$	0	0
$55$	− 171.625i	− 0.420761i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−68.1730	−0.150430	−0.0752150	−	0.997167i	$-0.523964\pi$
$59$	−68.1730	−0.150430	−0.0752150	+	0.997167i	$0.523964\pi$
$60$	0	0
$61$	179.160	0.376050	0.188025	−	0.982164i	$-0.439791\pi$
$61$	179.160	0.376050	0.188025	+	0.982164i	$0.439791\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 151.793i	− 0.289656i
$66$	0	0
$67$	665.378i	1.21327i	0.794982	+	0.606634i	$0.207480\pi$
$67$	665.378i	1.21327i	−0.794982	+	0.606634i	$0.792520\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	886.237	1.48137	0.740683	−	0.671855i	$-0.234502\pi$
$71$	886.237	1.48137	0.740683	+	0.671855i	$0.234502\pi$
$72$	0	0
$73$	615.428	0.986718	0.493359	−	0.869826i	$-0.335769\pi$
$73$	615.428	0.986718	0.493359	+	0.869826i	$0.335769\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	333.708i	0.493890i
$78$	0	0
$79$	1273.51i	1.81368i	0.421475	+	0.906840i	$0.361513\pi$
$79$	1273.51i	1.81368i	−0.421475	+	0.906840i	$0.638487\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	952.085	1.25910	0.629548	−	0.776962i	$-0.283240\pi$
$83$	952.085	1.25910	0.629548	+	0.776962i	$0.283240\pi$
$84$	0	0
$85$	285.862	0.364777
$86$	0	0
$87$	0	0
$88$	0	0
$89$	926.586i	1.10357i	0.833986	+	0.551786i	$0.186054\pi$
$89$	926.586i	1.10357i	−0.833986	+	0.551786i	$0.813946\pi$
$90$	0	0
$91$	295.148i	0.339999i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−32.0050	−0.0345647
$96$	0	0
$97$	−1500.95	−1.57112	−0.785559	−	0.618787i	$-0.787624\pi$
$97$	−1500.95	−1.57112	−0.785559	+	0.618787i	$0.787624\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1095.75i	− 1.07952i	−0.841818	−	0.539761i	$-0.818515\pi$
$101$	− 1095.75i	− 1.07952i	0.841818	−	0.539761i	$-0.181485\pi$
$102$	0	0
$103$	− 1489.30i	− 1.42471i	−0.701821	−	0.712354i	$-0.747629\pi$
$103$	− 1489.30i	− 1.42471i	0.701821	−	0.712354i	$-0.252371\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1505.57	1.36027	0.680137	−	0.733085i	$-0.261920\pi$
$107$	1505.57	1.36027	0.680137	+	0.733085i	$0.261920\pi$
$108$	0	0
$109$	488.435	0.429207	0.214603	−	0.976701i	$-0.431154\pi$
$109$	488.435	0.429207	0.214603	+	0.976701i	$0.431154\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 96.8326i	− 0.0806128i	−0.999187	−	0.0403064i	$-0.987167\pi$
$113$	− 96.8326i	− 0.0806128i	0.999187	−	0.0403064i	$-0.0128334\pi$
$114$	0	0
$115$	482.488i	0.391237i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−555.830	−0.428175
$120$	0	0
$121$	941.670	0.707491
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 853.360i	− 0.610615i
$126$	0	0
$127$	1116.09i	0.779819i	0.920853	+	0.389910i	$0.127494\pi$
$127$	1116.09i	0.779819i	−0.920853	+	0.389910i	$0.872506\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2616.57	1.74512	0.872559	−	0.488509i	$-0.162459\pi$
$131$	2616.57	1.74512	0.872559	+	0.488509i	$0.162459\pi$
$132$	0	0
$133$	62.2307	0.0405721
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1240.80i	− 0.773785i	−0.922125	−	0.386892i	$-0.873548\pi$
$137$	− 1240.80i	− 0.773785i	0.922125	−	0.386892i	$-0.126452\pi$
$138$	0	0
$139$	2295.38i	1.40066i	0.713820	+	0.700329i	$0.246963\pi$
$139$	2295.38i	1.40066i	−0.713820	+	0.700329i	$0.753037\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2010.06	1.17545
$144$	0	0
$145$	−953.474	−0.546081
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1790.49i	0.984448i	0.870468	+	0.492224i	$0.163816\pi$
$149$	1790.49i	0.984448i	−0.870468	+	0.492224i	$0.836184\pi$
$150$	0	0
$151$	2333.70i	1.25771i	0.777523	+	0.628855i	$0.216476\pi$
$151$	2333.70i	1.25771i	−0.777523	+	0.628855i	$0.783524\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1094.65	0.567257
$156$	0	0
$157$	−2415.89	−1.22808	−0.614042	−	0.789274i	$-0.710457\pi$
$157$	−2415.89	−1.22808	−0.614042	+	0.789274i	$0.710457\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 938.152i	− 0.459234i
$162$	0	0
$163$	− 46.1504i	− 0.0221765i	−0.999939	−	0.0110883i	$-0.996470\pi$
$163$	− 46.1504i	− 0.0221765i	0.999939	−	0.0110883i	$-0.00352958\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1421.51	0.658679	0.329340	−	0.944211i	$-0.393174\pi$
$167$	1421.51	0.658679	0.329340	+	0.944211i	$0.393174\pi$
$168$	0	0
$169$	−419.203	−0.190807
$170$	0	0
$171$	0	0
$172$	0	0
$173$	283.147i	0.124435i	0.998063	+	0.0622175i	$0.0198172\pi$
$173$	283.147i	0.124435i	−0.998063	+	0.0622175i	$0.980183\pi$
$174$	0	0
$175$	784.276i	0.338775i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1930.75	0.806208	0.403104	−	0.915154i	$-0.367931\pi$
$179$	1930.75	0.806208	0.403104	+	0.915154i	$0.367931\pi$
$180$	0	0
$181$	4387.12	1.80161	0.900806	−	0.434221i	$-0.142976\pi$
$181$	4387.12	1.80161	0.900806	+	0.434221i	$0.142976\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1489.62i	0.591994i
$186$	0	0
$187$	3785.40i	1.48030i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1173.68	−0.444631	−0.222315	−	0.974975i	$-0.571361\pi$
$191$	−1173.68	−0.444631	−0.222315	+	0.974975i	$0.571361\pi$
$192$	0	0
$193$	1681.36	0.627083	0.313542	−	0.949574i	$-0.398485\pi$
$193$	1681.36	0.627083	0.313542	+	0.949574i	$0.398485\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	918.295i	0.332111i	0.986116	+	0.166055i	$0.0531031\pi$
$197$	918.295i	0.332111i	−0.986116	+	0.166055i	$0.946897\pi$
$198$	0	0
$199$	1191.28i	0.424360i	0.977231	+	0.212180i	$0.0680563\pi$
$199$	1191.28i	0.424360i	−0.977231	+	0.212180i	$0.931944\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1853.94	0.640990
$204$	0	0
$205$	245.206	0.0835410
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 423.814i	− 0.140267i
$210$	0	0
$211$	954.606i	0.311459i	0.987800	+	0.155729i	$0.0497728\pi$
$211$	954.606i	0.311459i	−0.987800	+	0.155729i	$0.950227\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−787.461	−0.249788
$216$	0	0
$217$	−2128.45	−0.665846
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3348.00i	1.01905i
$222$	0	0
$223$	− 3159.84i	− 0.948872i	−0.880290	−	0.474436i	$-0.842652\pi$
$223$	− 3159.84i	− 0.948872i	0.880290	−	0.474436i	$-0.157348\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2936.85	0.858703	0.429352	−	0.903137i	$-0.358742\pi$
$227$	2936.85	0.858703	0.429352	+	0.903137i	$0.358742\pi$
$228$	0	0
$229$	894.312	0.258069	0.129034	−	0.991640i	$-0.458812\pi$
$229$	894.312	0.258069	0.129034	+	0.991640i	$0.458812\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	307.440i	0.0864422i	0.999066	+	0.0432211i	$0.0137620\pi$
$233$	307.440i	0.0864422i	−0.999066	+	0.0432211i	$0.986238\pi$
$234$	0	0
$235$	− 1645.23i	− 0.456694i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−67.7247	−0.0183295	−0.00916474	−	0.999958i	$-0.502917\pi$
$239$	−67.7247	−0.0183295	−0.00916474	+	0.999958i	$0.502917\pi$
$240$	0	0
$241$	−272.919	−0.0729472	−0.0364736	−	0.999335i	$-0.511612\pi$
$241$	−272.919	−0.0729472	−0.0364736	+	0.999335i	$0.511612\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	176.404i	0.0460001i
$246$	0	0
$247$	− 374.842i	− 0.0965612i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1709.58	−0.429911	−0.214955	−	0.976624i	$-0.568961\pi$
$251$	−1709.58	−0.429911	−0.214955	+	0.976624i	$0.568961\pi$
$252$	0	0
$253$	−6389.15	−1.58768
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6872.06i	− 1.66797i	−0.551790	−	0.833983i	$-0.686055\pi$
$257$	− 6872.06i	− 1.66797i	0.551790	−	0.833983i	$-0.313945\pi$
$258$	0	0
$259$	− 2896.42i	− 0.694883i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2916.14	0.683715	0.341857	−	0.939752i	$-0.388944\pi$
$263$	2916.14	0.683715	0.341857	+	0.939752i	$0.388944\pi$
$264$	0	0
$265$	759.192	0.175988
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4102.37i	− 0.929836i	−0.885354	−	0.464918i	$-0.846084\pi$
$269$	− 4102.37i	− 0.929836i	0.885354	−	0.464918i	$-0.153916\pi$
$270$	0	0
$271$	4732.45i	1.06080i	0.847748	+	0.530399i	$0.177958\pi$
$271$	4732.45i	1.06080i	−0.847748	+	0.530399i	$0.822042\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5341.20	1.17122
$276$	0	0
$277$	−838.409	−0.181860	−0.0909298	−	0.995857i	$-0.528984\pi$
$277$	−838.409	−0.181860	−0.0909298	+	0.995857i	$0.528984\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6781.90i	1.43977i	0.694095	+	0.719883i	$0.255805\pi$
$281$	6781.90i	1.43977i	−0.694095	+	0.719883i	$0.744195\pi$
$282$	0	0
$283$	1303.62i	0.273825i	0.990583	+	0.136912i	$0.0437179\pi$
$283$	1303.62i	0.273825i	−0.990583	+	0.136912i	$0.956282\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−476.779	−0.0980605
$288$	0	0
$289$	−1392.05	−0.283339
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 8332.09i	− 1.66132i	−0.556782	−	0.830658i	$-0.687964\pi$
$293$	− 8332.09i	− 1.66132i	0.556782	−	0.830658i	$-0.312036\pi$
$294$	0	0
$295$	245.428i	0.0484385i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5650.88	−1.09297
$300$	0	0
$301$	1531.14	0.293201
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 644.988i	− 0.121088i
$306$	0	0
$307$	7657.18i	1.42351i	0.702426	+	0.711756i	$0.252100\pi$
$307$	7657.18i	1.42351i	−0.702426	+	0.711756i	$0.747900\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9291.64	−1.69415	−0.847075	−	0.531473i	$-0.821639\pi$
$311$	−9291.64	−1.69415	−0.847075	+	0.531473i	$0.821639\pi$
$312$	0	0
$313$	4108.26	0.741893	0.370947	−	0.928654i	$-0.379033\pi$
$313$	4108.26	0.741893	0.370947	+	0.928654i	$0.379033\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	661.333i	0.117174i	0.998282	+	0.0585870i	$0.0186595\pi$
$317$	661.333i	0.117174i	−0.998282	+	0.0585870i	$0.981340\pi$
$318$	0	0
$319$	− 12626.0i	− 2.21605i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	705.913	0.121604
$324$	0	0
$325$	4724.02	0.806282
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3199.00i	0.536068i
$330$	0	0
$331$	− 2835.98i	− 0.470935i	−0.971882	−	0.235468i	$-0.924338\pi$
$331$	− 2835.98i	− 0.470935i	0.971882	−	0.235468i	$-0.0756622\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2395.41	0.390673
$336$	0	0
$337$	3505.31	0.566607	0.283304	−	0.959030i	$-0.408570\pi$
$337$	3505.31	0.566607	0.283304	+	0.959030i	$0.408570\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14495.5i	2.30198i
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.5496	−0.00286973	−0.00143486	−	0.999999i	$-0.500457\pi$
$347$	−18.5496	−0.00286973	−0.00143486	+	0.999999i	$0.500457\pi$
$348$	0	0
$349$	−4314.42	−0.661735	−0.330867	−	0.943677i	$-0.607341\pi$
$349$	−4314.42	−0.661735	−0.330867	+	0.943677i	$0.607341\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3078.38i	0.464152i	0.972698	+	0.232076i	$0.0745519\pi$
$353$	3078.38i	0.464152i	−0.972698	+	0.232076i	$0.925448\pi$
$354$	0	0
$355$	− 3190.52i	− 0.477001i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2227.68	−0.327501	−0.163750	−	0.986502i	$-0.552359\pi$
$359$	−2227.68	−0.327501	−0.163750	+	0.986502i	$0.552359\pi$
$360$	0	0
$361$	6779.97	0.988477
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2215.59i	− 0.317724i
$366$	0	0
$367$	− 12608.8i	− 1.79339i	−0.442644	−	0.896697i	$-0.645959\pi$
$367$	− 12608.8i	− 1.79339i	0.442644	−	0.896697i	$-0.354041\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1476.17	−0.206575
$372$	0	0
$373$	−4943.48	−0.686230	−0.343115	−	0.939293i	$-0.611482\pi$
$373$	−4943.48	−0.686230	−0.343115	+	0.939293i	$0.611482\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 11167.0i	− 1.52555i
$378$	0	0
$379$	889.338i	0.120534i	0.998182	+	0.0602668i	$0.0191951\pi$
$379$	889.338i	0.120534i	−0.998182	+	0.0602668i	$0.980805\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10237.0	−1.36576	−0.682880	−	0.730530i	$-0.739273\pi$
$383$	−10237.0	−1.36576	−0.682880	+	0.730530i	$0.739273\pi$
$384$	0	0
$385$	1201.37	0.159033
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9962.66i	− 1.29853i	−0.760564	−	0.649263i	$-0.775077\pi$
$389$	− 9962.66i	− 1.29853i	0.760564	−	0.649263i	$-0.224923\pi$
$390$	0	0
$391$	− 10641.9i	− 1.37643i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4584.72	0.584006
$396$	0	0
$397$	−4012.29	−0.507231	−0.253616	−	0.967305i	$-0.581620\pi$
$397$	−4012.29	−0.507231	−0.253616	+	0.967305i	$0.581620\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5952.06i	0.741227i	0.928787	+	0.370613i	$0.120852\pi$
$401$	5952.06i	0.741227i	−0.928787	+	0.370613i	$0.879148\pi$
$402$	0	0
$403$	12820.5i	1.58471i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−19725.7	−2.40237
$408$	0	0
$409$	−5447.34	−0.658567	−0.329283	−	0.944231i	$-0.606807\pi$
$409$	−5447.34	−0.658567	−0.329283	+	0.944231i	$0.606807\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 477.211i	− 0.0568572i
$414$	0	0
$415$	− 3427.58i	− 0.405430i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11545.2	−1.34612	−0.673058	−	0.739590i	$-0.735019\pi$
$419$	−11545.2	−1.34612	−0.673058	+	0.739590i	$0.735019\pi$
$420$	0	0
$421$	−3568.39	−0.413094	−0.206547	−	0.978437i	$-0.566223\pi$
$421$	−3568.39	−0.413094	−0.206547	+	0.978437i	$0.566223\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8896.42i	1.01539i
$426$	0	0
$427$	1254.12i	0.142133i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15894.3	−1.77634	−0.888170	−	0.459514i	$-0.848023\pi$
$431$	−15894.3	−1.77634	−0.888170	+	0.459514i	$0.848023\pi$
$432$	0	0
$433$	8379.60	0.930018	0.465009	−	0.885306i	$-0.346051\pi$
$433$	8379.60	0.930018	0.465009	+	0.885306i	$0.346051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1191.47i	0.130425i
$438$	0	0
$439$	− 3502.44i	− 0.380779i	−0.981709	−	0.190390i	$-0.939025\pi$
$439$	− 3502.44i	− 0.380779i	0.981709	−	0.190390i	$-0.0609752\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8582.21	−0.920436	−0.460218	−	0.887806i	$-0.652229\pi$
$443$	−8582.21	−0.920436	−0.460218	+	0.887806i	$0.652229\pi$
$444$	0	0
$445$	3335.78	0.355351
$446$	0	0
$447$	0	0
$448$	0	0
$449$	814.202i	0.0855781i	0.999084	+	0.0427890i	$0.0136243\pi$
$449$	814.202i	0.0855781i	−0.999084	+	0.0427890i	$0.986376\pi$
$450$	0	0
$451$	3247.03i	0.339017i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1062.55	0.109480
$456$	0	0
$457$	−14759.3	−1.51075	−0.755375	−	0.655293i	$-0.772545\pi$
$457$	−14759.3	−1.51075	−0.755375	+	0.655293i	$0.772545\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	887.622i	0.0896761i	0.998994	+	0.0448381i	$0.0142772\pi$
$461$	887.622i	0.0896761i	−0.998994	+	0.0448381i	$0.985723\pi$
$462$	0	0
$463$	− 7447.20i	− 0.747518i	−0.927526	−	0.373759i	$-0.878069\pi$
$463$	− 7447.20i	− 0.747518i	0.927526	−	0.373759i	$-0.121931\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14375.6	1.42446	0.712230	−	0.701946i	$-0.247685\pi$
$467$	14375.6	1.42446	0.712230	+	0.701946i	$0.247685\pi$
$468$	0	0
$469$	−4657.65	−0.458572
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 10427.6i	− 1.01366i
$474$	0	0
$475$	− 996.042i	− 0.0962138i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1168.44	0.111456	0.0557280	−	0.998446i	$-0.482252\pi$
$479$	1168.44	0.111456	0.0557280	+	0.998446i	$0.482252\pi$
$480$	0	0
$481$	−17446.3	−1.65381
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5403.54i	0.505901i
$486$	0	0
$487$	− 2400.43i	− 0.223355i	−0.993745	−	0.111677i	$-0.964378\pi$
$487$	− 2400.43i	− 0.223355i	0.993745	−	0.111677i	$-0.0356224\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17207.0	1.58155	0.790774	−	0.612108i	$-0.209678\pi$
$491$	17207.0	1.58155	0.790774	+	0.612108i	$0.209678\pi$
$492$	0	0
$493$	21030.1	1.92119
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6203.66i	0.559904i
$498$	0	0
$499$	− 9304.20i	− 0.834696i	−0.908747	−	0.417348i	$-0.862960\pi$
$499$	− 9304.20i	− 0.834696i	0.908747	−	0.417348i	$-0.137040\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12560.6	−1.11342	−0.556708	−	0.830708i	$-0.687936\pi$
$503$	−12560.6	−1.11342	−0.556708	+	0.830708i	$0.687936\pi$
$504$	0	0
$505$	−3944.80	−0.347607
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7580.38i	− 0.660107i	−0.943962	−	0.330053i	$-0.892933\pi$
$509$	− 7580.38i	− 0.660107i	0.943962	−	0.330053i	$-0.107067\pi$
$510$	0	0
$511$	4308.00i	0.372944i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5361.59	−0.458757
$516$	0	0
$517$	21786.3	1.85331
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 20449.0i	− 1.71955i	−0.510671	−	0.859776i	$-0.670603\pi$
$521$	− 20449.0i	− 1.71955i	0.510671	−	0.859776i	$-0.329397\pi$
$522$	0	0
$523$	− 16372.1i	− 1.36884i	−0.729090	−	0.684418i	$-0.760056\pi$
$523$	− 16372.1i	− 1.36884i	0.729090	−	0.684418i	$-0.239944\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24144.0	−1.99569
$528$	0	0
$529$	5794.81	0.476273
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2871.84i	0.233383i
$534$	0	0
$535$	− 5420.18i	− 0.438009i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2335.95	−0.186673
$540$	0	0
$541$	1343.15	0.106740	0.0533702	−	0.998575i	$-0.483004\pi$
$541$	1343.15	0.106740	0.0533702	+	0.998575i	$0.483004\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1758.40i	− 0.138205i
$546$	0	0
$547$	− 8325.07i	− 0.650739i	−0.945587	−	0.325369i	$-0.894511\pi$
$547$	− 8325.07i	− 0.650739i	0.945587	−	0.325369i	$-0.105489\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2354.53	−0.182044
$552$	0	0
$553$	−8914.55	−0.685507
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11194.4i	0.851568i	0.904825	+	0.425784i	$0.140002\pi$
$557$	11194.4i	0.851568i	−0.904825	+	0.425784i	$0.859998\pi$
$558$	0	0
$559$	− 9222.71i	− 0.697816i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21109.7	−1.58022	−0.790112	−	0.612962i	$-0.789978\pi$
$563$	−21109.7	−1.58022	−0.790112	+	0.612962i	$0.789978\pi$
$564$	0	0
$565$	−348.605	−0.0259574
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17201.7i	1.26737i	0.773592	+	0.633684i	$0.218458\pi$
$569$	17201.7i	1.26737i	−0.773592	+	0.633684i	$0.781542\pi$
$570$	0	0
$571$	− 749.269i	− 0.0549141i	−0.999623	−	0.0274570i	$-0.991259\pi$
$571$	− 749.269i	− 0.0549141i	0.999623	−	0.0274570i	$-0.00874095\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−15015.7	−1.08904
$576$	0	0
$577$	−3430.59	−0.247517	−0.123759	−	0.992312i	$-0.539495\pi$
$577$	−3430.59	−0.247517	−0.123759	+	0.992312i	$0.539495\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6664.60i	0.475893i
$582$	0	0
$583$	10053.3i	0.714176i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19117.2	1.34421	0.672106	−	0.740455i	$-0.265390\pi$
$587$	19117.2	1.34421	0.672106	+	0.740455i	$0.265390\pi$
$588$	0	0
$589$	2703.16	0.189103
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2501.41i	0.173222i	0.996242	+	0.0866111i	$0.0276037\pi$
$593$	2501.41i	0.173222i	−0.996242	+	0.0866111i	$0.972396\pi$
$594$	0	0
$595$	2001.03i	0.137873i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5306.86	0.361991	0.180995	−	0.983484i	$-0.442068\pi$
$599$	5306.86	0.361991	0.180995	+	0.983484i	$0.442068\pi$
$600$	0	0
$601$	25702.9	1.74450	0.872250	−	0.489060i	$-0.162660\pi$
$601$	25702.9	1.74450	0.872250	+	0.489060i	$0.162660\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3390.08i	− 0.227812i
$606$	0	0
$607$	− 18644.3i	− 1.24670i	−0.781942	−	0.623351i	$-0.785771\pi$
$607$	− 18644.3i	− 1.24670i	0.781942	−	0.623351i	$-0.214229\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	19268.9	1.27584
$612$	0	0
$613$	−16222.0	−1.06884	−0.534421	−	0.845219i	$-0.679470\pi$
$613$	−16222.0	−1.06884	−0.534421	+	0.845219i	$0.679470\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18380.6i	1.19931i	0.800259	+	0.599654i	$0.204695\pi$
$617$	18380.6i	1.19931i	−0.800259	+	0.599654i	$0.795305\pi$
$618$	0	0
$619$	23174.4i	1.50478i	0.658719	+	0.752389i	$0.271099\pi$
$619$	23174.4i	1.50478i	−0.658719	+	0.752389i	$0.728901\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6486.10	−0.417111
$624$	0	0
$625$	10932.8	0.699697
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 32855.4i	− 2.08272i
$630$	0	0
$631$	− 6609.12i	− 0.416965i	−0.978026	−	0.208483i	$-0.933148\pi$
$631$	− 6609.12i	− 0.416965i	0.978026	−	0.208483i	$-0.0668525\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4018.01	0.251102
$636$	0	0
$637$	−2066.03	−0.128507
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 32175.5i	− 1.98261i	−0.131576	−	0.991306i	$-0.542004\pi$
$641$	− 32175.5i	− 1.98261i	0.131576	−	0.991306i	$-0.457996\pi$
$642$	0	0
$643$	2045.07i	0.125427i	0.998032	+	0.0627137i	$0.0199755\pi$
$643$	2045.07i	0.125427i	−0.998032	+	0.0627137i	$0.980024\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6627.83	−0.402731	−0.201365	−	0.979516i	$-0.564538\pi$
$647$	−6627.83	−0.402731	−0.201365	+	0.979516i	$0.564538\pi$
$648$	0	0
$649$	−3249.98	−0.196568
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18808.9i	1.12718i	0.826054	+	0.563591i	$0.190580\pi$
$653$	18808.9i	1.12718i	−0.826054	+	0.563591i	$0.809420\pi$
$654$	0	0
$655$	− 9419.84i	− 0.561929i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11555.9	0.683085	0.341542	−	0.939866i	$-0.389051\pi$
$659$	11555.9	0.683085	0.341542	+	0.939866i	$0.389051\pi$
$660$	0	0
$661$	−13791.5	−0.811536	−0.405768	−	0.913976i	$-0.632996\pi$
$661$	−13791.5	−0.811536	−0.405768	+	0.913976i	$0.632996\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 224.035i	− 0.0130642i
$666$	0	0
$667$	35495.4i	2.06055i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8540.99	0.491388
$672$	0	0
$673$	−17234.4	−0.987127	−0.493563	−	0.869710i	$-0.664306\pi$
$673$	−17234.4	−0.987127	−0.493563	+	0.869710i	$0.664306\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4978.57i	− 0.282632i	−0.989965	−	0.141316i	$-0.954867\pi$
$677$	− 4978.57i	− 0.282632i	0.989965	−	0.141316i	$-0.0451334\pi$
$678$	0	0
$679$	− 10506.7i	− 0.593827i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3865.49	−0.216558	−0.108279	−	0.994121i	$-0.534534\pi$
$683$	−3865.49	−0.216558	−0.108279	+	0.994121i	$0.534534\pi$
$684$	0	0
$685$	−4466.97	−0.249159
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8891.62i	0.491645i
$690$	0	0
$691$	7070.72i	0.389266i	0.980876	+	0.194633i	$0.0623516\pi$
$691$	7070.72i	0.389266i	−0.980876	+	0.194633i	$0.937648\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8263.54	0.451013
$696$	0	0
$697$	−5408.33	−0.293910
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30926.5i	1.66630i	0.553047	+	0.833150i	$0.313465\pi$
$701$	30926.5i	1.66630i	−0.553047	+	0.833150i	$0.686535\pi$
$702$	0	0
$703$	3678.49i	0.197350i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7670.28	0.408021
$708$	0	0
$709$	−33606.0	−1.78011	−0.890056	−	0.455851i	$-0.849335\pi$
$709$	−33606.0	−1.78011	−0.890056	+	0.455851i	$0.849335\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 40751.2i	− 2.14046i
$714$	0	0
$715$	− 7236.37i	− 0.378497i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13025.9	−0.675641	−0.337821	−	0.941211i	$-0.609690\pi$
$719$	−13025.9	−0.675641	−0.337821	+	0.941211i	$0.609690\pi$
$720$	0	0
$721$	10425.1	0.538489
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 29673.5i	− 1.52006i
$726$	0	0
$727$	8072.97i	0.411843i	0.978569	+	0.205922i	$0.0660192\pi$
$727$	8072.97i	0.411843i	−0.978569	+	0.205922i	$0.933981\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17368.5	0.878791
$732$	0	0
$733$	−6967.76	−0.351105	−0.175553	−	0.984470i	$-0.556171\pi$
$733$	−6967.76	−0.351105	−0.175553	+	0.984470i	$0.556171\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31720.3i	1.58539i
$738$	0	0
$739$	26105.3i	1.29946i	0.760165	+	0.649730i	$0.225118\pi$
$739$	26105.3i	1.29946i	−0.760165	+	0.649730i	$0.774882\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35531.4	1.75440	0.877202	−	0.480121i	$-0.159407\pi$
$743$	35531.4	1.75440	0.877202	+	0.480121i	$0.159407\pi$
$744$	0	0
$745$	6445.91	0.316993
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10539.0i	0.514135i
$750$	0	0
$751$	− 22942.8i	− 1.11477i	−0.830253	−	0.557387i	$-0.811804\pi$
$751$	− 22942.8i	− 1.11477i	0.830253	−	0.557387i	$-0.188196\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8401.51	0.404983
$756$	0	0
$757$	6203.28	0.297836	0.148918	−	0.988850i	$-0.452421\pi$
$757$	6203.28	0.297836	0.148918	+	0.988850i	$0.452421\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38979.8i	1.85679i	0.371593	+	0.928396i	$0.378812\pi$
$761$	38979.8i	1.85679i	−0.371593	+	0.928396i	$0.621188\pi$
$762$	0	0
$763$	3419.04i	0.162225i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2874.44	−0.135320
$768$	0	0
$769$	32100.5	1.50530	0.752648	−	0.658423i	$-0.228776\pi$
$769$	32100.5	1.50530	0.752648	+	0.658423i	$0.228776\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8832.52i	0.410975i	0.978660	+	0.205488i	$0.0658780\pi$
$773$	8832.52i	0.410975i	−0.978660	+	0.205488i	$0.934122\pi$
$774$	0	0
$775$	34067.2i	1.57901i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	605.516	0.0278496
$780$	0	0
$781$	42249.1	1.93571
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8697.39i	0.395443i
$786$	0	0
$787$	− 29962.1i	− 1.35709i	−0.734558	−	0.678546i	$-0.762610\pi$
$787$	− 29962.1i	− 1.35709i	0.734558	−	0.678546i	$-0.237390\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	677.828	0.0304688
$792$	0	0
$793$	7554.07	0.338276
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 39903.2i	− 1.77345i	−0.462293	−	0.886727i	$-0.652973\pi$
$797$	− 39903.2i	− 1.77345i	0.462293	−	0.886727i	$-0.347027\pi$
$798$	0	0
$799$	36287.7i	1.60672i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29339.0	1.28935
$804$	0	0
$805$	−3377.42	−0.147874
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5172.61i	0.224795i	0.993663	+	0.112398i	$0.0358530\pi$
$809$	5172.61i	0.224795i	−0.993663	+	0.112398i	$0.964147\pi$
$810$	0	0
$811$	13639.6i	0.590570i	0.955409	+	0.295285i	$0.0954145\pi$
$811$	13639.6i	0.590570i	−0.955409	+	0.295285i	$0.904585\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−166.145	−0.00714086
$816$	0	0
$817$	−1944.57	−0.0832705
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 19617.8i	− 0.833942i	−0.908920	−	0.416971i	$-0.863092\pi$
$821$	− 19617.8i	− 0.833942i	0.908920	−	0.416971i	$-0.136908\pi$
$822$	0	0
$823$	5228.10i	0.221434i	0.993852	+	0.110717i	$0.0353147\pi$
$823$	5228.10i	0.221434i	−0.993852	+	0.110717i	$0.964685\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33001.7	−1.38765	−0.693823	−	0.720146i	$-0.744075\pi$
$827$	−33001.7	−1.38765	−0.693823	+	0.720146i	$0.744075\pi$
$828$	0	0
$829$	14852.8	0.622265	0.311132	−	0.950367i	$-0.399292\pi$
$829$	14852.8	0.622265	0.311132	+	0.950367i	$0.399292\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3890.81i	− 0.161835i
$834$	0	0
$835$	− 5117.53i	− 0.212095i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32978.4	−1.35702	−0.678510	−	0.734591i	$-0.737374\pi$
$839$	−32978.4	−1.35702	−0.678510	+	0.734591i	$0.737374\pi$
$840$	0	0
$841$	−45755.6	−1.87608
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1509.16i	0.0614400i
$846$	0	0
$847$	6591.69i	0.267406i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	55454.7	2.23380
$852$	0	0
$853$	28614.7	1.14859	0.574297	−	0.818647i	$-0.305276\pi$
$853$	28614.7	1.14859	0.574297	+	0.818647i	$0.305276\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4665.12i	− 0.185948i	−0.995669	−	0.0929740i	$-0.970363\pi$
$857$	− 4665.12i	− 0.185948i	0.995669	−	0.0929740i	$-0.0296373\pi$
$858$	0	0
$859$	36024.7i	1.43090i	0.698662	+	0.715452i	$0.253779\pi$
$859$	36024.7i	1.43090i	−0.698662	+	0.715452i	$0.746221\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11837.5	0.466920	0.233460	−	0.972366i	$-0.424995\pi$
$863$	11837.5	0.466920	0.233460	+	0.972366i	$0.424995\pi$
$864$	0	0
$865$	1019.35	0.0400681
$866$	0	0
$867$	0	0
$868$	0	0
$869$	60711.3i	2.36995i
$870$	0	0
$871$	28055.0i	1.09140i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5973.52	0.230791
$876$	0	0
$877$	−37061.2	−1.42699	−0.713494	−	0.700662i	$-0.752888\pi$
$877$	−37061.2	−1.42699	−0.713494	+	0.700662i	$0.752888\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 34593.5i	− 1.32291i	−0.749984	−	0.661456i	$-0.769939\pi$
$881$	− 34593.5i	− 1.32291i	0.749984	−	0.661456i	$-0.230061\pi$
$882$	0	0
$883$	46369.3i	1.76722i	0.468226	+	0.883609i	$0.344893\pi$
$883$	46369.3i	1.76722i	−0.468226	+	0.883609i	$0.655107\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	859.625	0.0325404	0.0162702	−	0.999868i	$-0.494821\pi$
$887$	859.625	0.0325404	0.0162702	+	0.999868i	$0.494821\pi$
$888$	0	0
$889$	−7812.64	−0.294744
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 4062.77i	− 0.152246i
$894$	0	0
$895$	− 6950.86i	− 0.259599i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	80530.9	2.98760
$900$	0	0
$901$	−16744.9	−0.619151
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 15794.0i	− 0.580120i
$906$	0	0
$907$	− 37379.7i	− 1.36844i	−0.729277	−	0.684219i	$-0.760143\pi$
$907$	− 37379.7i	− 1.36844i	0.729277	−	0.684219i	$-0.239857\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6094.16	−0.221634	−0.110817	−	0.993841i	$-0.535347\pi$
$911$	−6094.16	−0.221634	−0.110817	+	0.993841i	$0.535347\pi$
$912$	0	0
$913$	45388.3	1.64527
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18316.0i	0.659592i
$918$	0	0
$919$	− 24074.7i	− 0.864146i	−0.901839	−	0.432073i	$-0.857782\pi$
$919$	− 24074.7i	− 0.864146i	0.901839	−	0.432073i	$-0.142218\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37367.2	1.33256
$924$	0	0
$925$	−46359.0	−1.64787
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 711.813i	− 0.0251387i	−0.999921	−	0.0125693i	$-0.995999\pi$
$929$	− 711.813i	− 0.0251387i	0.999921	−	0.0125693i	$-0.00400105\pi$
$930$	0	0
$931$	435.615i	0.0153348i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13627.7	0.476658
$936$	0	0
$937$	33822.8	1.17924	0.589618	−	0.807682i	$-0.299278\pi$
$937$	33822.8	1.17924	0.589618	+	0.807682i	$0.299278\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 49771.8i	− 1.72424i	−0.506701	−	0.862122i	$-0.669135\pi$
$941$	− 49771.8i	− 1.72424i	0.506701	−	0.862122i	$-0.330865\pi$
$942$	0	0
$943$	− 9128.38i	− 0.315229i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35457.5	1.21670	0.608349	−	0.793669i	$-0.291832\pi$
$947$	35457.5	1.21670	0.608349	+	0.793669i	$0.291832\pi$
$948$	0	0
$949$	25948.9	0.887604
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24581.1i	− 0.835528i	−0.908556	−	0.417764i	$-0.862814\pi$
$953$	− 24581.1i	− 0.835528i	0.908556	−	0.417764i	$-0.137186\pi$
$954$	0	0
$955$	4225.33i	0.143171i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8685.59	0.292463
$960$	0	0
$961$	−62664.1	−2.10346
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6053.03i	− 0.201921i
$966$	0	0
$967$	− 33117.9i	− 1.10134i	−0.834722	−	0.550672i	$-0.814371\pi$
$967$	− 33117.9i	− 1.10134i	0.834722	−	0.550672i	$-0.185629\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10702.9	−0.353731	−0.176865	−	0.984235i	$-0.556596\pi$
$971$	−10702.9	−0.353731	−0.176865	+	0.984235i	$0.556596\pi$
$972$	0	0
$973$	−16067.6	−0.529399
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11861.1i	− 0.388403i	−0.980962	−	0.194202i	$-0.937788\pi$
$977$	− 11861.1i	− 0.388403i	0.980962	−	0.194202i	$-0.0622116\pi$
$978$	0	0
$979$	44172.7i	1.44205i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12494.0	0.405390	0.202695	−	0.979242i	$-0.435030\pi$
$983$	12494.0	0.405390	0.202695	+	0.979242i	$0.435030\pi$
$984$	0	0
$985$	3305.93	0.106940
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29315.2i	0.942537i
$990$	0	0
$991$	38449.7i	1.23249i	0.787556	+	0.616244i	$0.211346\pi$
$991$	38449.7i	1.23249i	−0.787556	+	0.616244i	$0.788654\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4288.70	0.136644
$996$	0	0
$997$	17139.7	0.544454	0.272227	−	0.962233i	$-0.412240\pi$
$997$	17139.7	0.544454	0.272227	+	0.962233i	$0.412240\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.h.a.575.6	yes	12
3.2	odd	2	inner	1008.4.h.a.575.8	yes	12
4.3	odd	2	inner	1008.4.h.a.575.5	✓	12
12.11	even	2	inner	1008.4.h.a.575.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.4.h.a.575.5	✓	12	4.3	odd	2	inner
1008.4.h.a.575.6	yes	12	1.1	even	1	trivial
1008.4.h.a.575.7	yes	12	12.11	even	2	inner
1008.4.h.a.575.8	yes	12	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.60008i q^{5} +7.00000i q^{7} +O(q^{10})\)	\(q-3.60008i q^{5} +7.00000i q^{7} +47.6725 q^{11} +42.1639 q^{13} +79.4043i q^{17} -8.89010i q^{19} -134.022 q^{23} +112.039 q^{25} -264.848i q^{29} +304.064i q^{31} +25.2005 q^{35} -413.774 q^{37} +68.1112i q^{41} -218.735i q^{43} +457.000 q^{47} -49.0000 q^{49} +210.882i q^{53} -171.625i q^{55} -68.1730 q^{59} +179.160 q^{61} -151.793i q^{65} +665.378i q^{67} +886.237 q^{71} +615.428 q^{73} +333.708i q^{77} +1273.51i q^{79} +952.085 q^{83} +285.862 q^{85} +926.586i q^{89} +295.148i q^{91} -32.0050 q^{95} -1500.95 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) \) 12 * q + 192 * q^13 + 84 * q^25 + 72 * q^37 - 588 * q^49 - 1800 * q^61 + 3144 * q^85 - 1152 * q^97

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