LMFDB - Embedded newform 1008.4.h.b.575.16

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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.16
Character	$\chi$	$=$	1008.575
Dual form			1008.4.h.b.575.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.0708i q^{5} -7.00000i q^{7} +O(q^{10})$	$q+13.0708i q^{5} -7.00000i q^{7} +21.9577 q^{11} +44.9771 q^{13} +93.5341i q^{17} +29.0417i q^{19} -63.4509 q^{23} -45.8469 q^{25} +34.8937i q^{29} -234.169i q^{31} +91.4959 q^{35} +260.315 q^{37} -45.1565i q^{41} -36.1248i q^{43} +378.116 q^{47} -49.0000 q^{49} +477.314i q^{53} +287.005i q^{55} +311.748 q^{59} -412.268 q^{61} +587.888i q^{65} +171.786i q^{67} -498.683 q^{71} -479.271 q^{73} -153.704i q^{77} +1253.05i q^{79} +523.525 q^{83} -1222.57 q^{85} -282.522i q^{89} -314.840i q^{91} -379.600 q^{95} -1200.79 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 192 q^{13} - 984 q^{25} + 720 q^{37} - 1176 q^{49} + 2736 q^{61} + 3408 q^{85} + 1152 q^{97}+O(q^{100}) $ 24 * q - 192 * q^13 - 984 * q^25 + 720 * q^37 - 1176 * q^49 + 2736 * q^61 + 3408 * 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
$5$	13.0708i	1.16909i	0.811361	+	0.584546i	$0.198727\pi$
$5$	13.0708i	1.16909i	−0.811361	+	0.584546i	$0.801273\pi$
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	21.9577	0.601863	0.300931	−	0.953646i	$-0.402703\pi$
$11$	21.9577	0.601863	0.300931	+	0.953646i	$0.402703\pi$
$12$	0	0
$13$	44.9771	0.959569	0.479785	−	0.877386i	$-0.340715\pi$
$13$	44.9771	0.959569	0.479785	+	0.877386i	$0.340715\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	93.5341i	1.33443i	0.744864	+	0.667216i	$0.232514\pi$
$17$	93.5341i	1.33443i	−0.744864	+	0.667216i	$0.767486\pi$
$18$	0	0
$19$	29.0417i	0.350665i	0.984509	+	0.175332i	$0.0561000\pi$
$19$	29.0417i	0.350665i	−0.984509	+	0.175332i	$0.943900\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−63.4509	−0.575236	−0.287618	−	0.957745i	$-0.592863\pi$
$23$	−63.4509	−0.575236	−0.287618	+	0.957745i	$0.592863\pi$
$24$	0	0
$25$	−45.8469	−0.366775
$26$	0	0
$27$	0	0
$28$	0	0
$29$	34.8937i	0.223434i	0.993740	+	0.111717i	$0.0356351\pi$
$29$	34.8937i	0.223434i	−0.993740	+	0.111717i	$0.964365\pi$
$30$	0	0
$31$	− 234.169i	− 1.35671i	−0.734733	−	0.678356i	$-0.762693\pi$
$31$	− 234.169i	− 1.35671i	0.734733	−	0.678356i	$-0.237307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	91.4959	0.441875
$36$	0	0
$37$	260.315	1.15663	0.578317	−	0.815812i	$-0.303710\pi$
$37$	260.315	1.15663	0.578317	+	0.815812i	$0.303710\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 45.1565i	− 0.172006i	−0.996295	−	0.0860031i	$-0.972590\pi$
$41$	− 45.1565i	− 0.172006i	0.996295	−	0.0860031i	$-0.0274095\pi$
$42$	0	0
$43$	− 36.1248i	− 0.128116i	−0.997946	−	0.0640580i	$-0.979596\pi$
$43$	− 36.1248i	− 0.128116i	0.997946	−	0.0640580i	$-0.0204043\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	378.116	1.17349	0.586744	−	0.809772i	$-0.300409\pi$
$47$	378.116	1.17349	0.586744	+	0.809772i	$0.300409\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	477.314i	1.23706i	0.785762	+	0.618529i	$0.212271\pi$
$53$	477.314i	1.23706i	−0.785762	+	0.618529i	$0.787729\pi$
$54$	0	0
$55$	287.005i	0.703632i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	311.748	0.687901	0.343951	−	0.938988i	$-0.388235\pi$
$59$	311.748	0.687901	0.343951	+	0.938988i	$0.388235\pi$
$60$	0	0
$61$	−412.268	−0.865337	−0.432669	−	0.901553i	$-0.642428\pi$
$61$	−412.268	−0.865337	−0.432669	+	0.901553i	$0.642428\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	587.888i	1.12182i
$66$	0	0
$67$	171.786i	0.313238i	0.987659	+	0.156619i	$0.0500595\pi$
$67$	171.786i	0.313238i	−0.987659	+	0.156619i	$0.949941\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−498.683	−0.833561	−0.416780	−	0.909007i	$-0.636842\pi$
$71$	−498.683	−0.833561	−0.416780	+	0.909007i	$0.636842\pi$
$72$	0	0
$73$	−479.271	−0.768416	−0.384208	−	0.923247i	$-0.625525\pi$
$73$	−479.271	−0.768416	−0.384208	+	0.923247i	$0.625525\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 153.704i	− 0.227483i
$78$	0	0
$79$	1253.05i	1.78454i	0.451501	+	0.892271i	$0.350889\pi$
$79$	1253.05i	1.78454i	−0.451501	+	0.892271i	$0.649111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	523.525	0.692342	0.346171	−	0.938171i	$-0.387482\pi$
$83$	523.525	0.692342	0.346171	+	0.938171i	$0.387482\pi$
$84$	0	0
$85$	−1222.57	−1.56007
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 282.522i	− 0.336486i	−0.985746	−	0.168243i	$-0.946191\pi$
$89$	− 282.522i	− 0.336486i	0.985746	−	0.168243i	$-0.0538094\pi$
$90$	0	0
$91$	− 314.840i	− 0.362683i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−379.600	−0.409959
$96$	0	0
$97$	−1200.79	−1.25693	−0.628464	−	0.777839i	$-0.716316\pi$
$97$	−1200.79	−1.25693	−0.628464	+	0.777839i	$0.716316\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	147.796i	0.145607i	0.997346	+	0.0728034i	$0.0231946\pi$
$101$	147.796i	0.145607i	−0.997346	+	0.0728034i	$0.976805\pi$
$102$	0	0
$103$	1581.41i	1.51283i	0.654094	+	0.756413i	$0.273050\pi$
$103$	1581.41i	1.51283i	−0.654094	+	0.756413i	$0.726950\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	916.913	0.828424	0.414212	−	0.910181i	$-0.364057\pi$
$107$	916.913	0.828424	0.414212	+	0.910181i	$0.364057\pi$
$108$	0	0
$109$	225.114	0.197817	0.0989084	−	0.995097i	$-0.468465\pi$
$109$	225.114	0.197817	0.0989084	+	0.995097i	$0.468465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1449.28i	1.20652i	0.797545	+	0.603259i	$0.206131\pi$
$113$	1449.28i	1.20652i	−0.797545	+	0.603259i	$0.793869\pi$
$114$	0	0
$115$	− 829.356i	− 0.672503i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	654.739	0.504368
$120$	0	0
$121$	−848.860	−0.637761
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1034.60i	0.740298i
$126$	0	0
$127$	− 465.544i	− 0.325278i	−0.986686	−	0.162639i	$-0.947999\pi$
$127$	− 465.544i	− 0.325278i	0.986686	−	0.162639i	$-0.0520007\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−914.559	−0.609965	−0.304982	−	0.952358i	$-0.598651\pi$
$131$	−914.559	−0.609965	−0.304982	+	0.952358i	$0.598651\pi$
$132$	0	0
$133$	203.292	0.132539
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2188.99i	1.36510i	0.730840	+	0.682549i	$0.239128\pi$
$137$	2188.99i	1.36510i	−0.730840	+	0.682549i	$0.760872\pi$
$138$	0	0
$139$	− 1563.65i	− 0.954154i	−0.878861	−	0.477077i	$-0.841696\pi$
$139$	− 1563.65i	− 0.954154i	0.878861	−	0.477077i	$-0.158304\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	987.592	0.577529
$144$	0	0
$145$	−456.090	−0.261215
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1374.33i	0.755633i	0.925881	+	0.377816i	$0.123325\pi$
$149$	1374.33i	0.755633i	−0.925881	+	0.377816i	$0.876675\pi$
$150$	0	0
$151$	3568.18i	1.92301i	0.274785	+	0.961506i	$0.411393\pi$
$151$	3568.18i	1.92301i	−0.274785	+	0.961506i	$0.588607\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3060.79	1.58612
$156$	0	0
$157$	−2092.83	−1.06386	−0.531930	−	0.846789i	$-0.678533\pi$
$157$	−2092.83	−1.06386	−0.531930	+	0.846789i	$0.678533\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	444.156i	0.217419i
$162$	0	0
$163$	− 1357.96i	− 0.652538i	−0.945277	−	0.326269i	$-0.894209\pi$
$163$	− 1357.96i	− 0.652538i	0.945277	−	0.326269i	$-0.105791\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2837.98	1.31503	0.657514	−	0.753442i	$-0.271608\pi$
$167$	2837.98	1.31503	0.657514	+	0.753442i	$0.271608\pi$
$168$	0	0
$169$	−174.062	−0.0792272
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3947.50i	1.73481i	0.497598	+	0.867407i	$0.334215\pi$
$173$	3947.50i	1.73481i	−0.497598	+	0.867407i	$0.665785\pi$
$174$	0	0
$175$	320.928i	0.138628i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2128.74	−0.888880	−0.444440	−	0.895809i	$-0.646597\pi$
$179$	−2128.74	−0.888880	−0.444440	+	0.895809i	$0.646597\pi$
$180$	0	0
$181$	−1890.96	−0.776539	−0.388270	−	0.921546i	$-0.626927\pi$
$181$	−1890.96	−0.776539	−0.388270	+	0.921546i	$0.626927\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3402.53i	1.35221i
$186$	0	0
$187$	2053.79i	0.803145i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1010.49	0.382808	0.191404	−	0.981511i	$-0.438696\pi$
$191$	1010.49	0.382808	0.191404	+	0.981511i	$0.438696\pi$
$192$	0	0
$193$	3852.32	1.43677	0.718383	−	0.695647i	$-0.244882\pi$
$193$	3852.32	1.43677	0.718383	+	0.695647i	$0.244882\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2119.15i	− 0.766411i	−0.923663	−	0.383206i	$-0.874820\pi$
$197$	− 2119.15i	− 0.766411i	0.923663	−	0.383206i	$-0.125180\pi$
$198$	0	0
$199$	1353.42i	0.482119i	0.970510	+	0.241060i	$0.0774949\pi$
$199$	1353.42i	0.482119i	−0.970510	+	0.241060i	$0.922505\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	244.256	0.0844502
$204$	0	0
$205$	590.233	0.201091
$206$	0	0
$207$	0	0
$208$	0	0
$209$	637.689i	0.211052i
$210$	0	0
$211$	− 146.100i	− 0.0476678i	−0.999716	−	0.0238339i	$-0.992413\pi$
$211$	− 146.100i	− 0.0476678i	0.999716	−	0.0238339i	$-0.00758729\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	472.182	0.149779
$216$	0	0
$217$	−1639.19	−0.512789
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4206.89i	1.28048i
$222$	0	0
$223$	− 5505.27i	− 1.65318i	−0.562802	−	0.826592i	$-0.690277\pi$
$223$	− 5505.27i	− 1.65318i	0.562802	−	0.826592i	$-0.309723\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1854.62	0.542271	0.271136	−	0.962541i	$-0.412601\pi$
$227$	1854.62	0.542271	0.271136	+	0.962541i	$0.412601\pi$
$228$	0	0
$229$	2874.89	0.829598	0.414799	−	0.909913i	$-0.363852\pi$
$229$	2874.89	0.829598	0.414799	+	0.909913i	$0.363852\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2979.87i	− 0.837844i	−0.908022	−	0.418922i	$-0.862408\pi$
$233$	− 2979.87i	− 0.837844i	0.908022	−	0.418922i	$-0.137592\pi$
$234$	0	0
$235$	4942.30i	1.37192i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5581.19	1.51053	0.755266	−	0.655418i	$-0.227508\pi$
$239$	5581.19	1.51053	0.755266	+	0.655418i	$0.227508\pi$
$240$	0	0
$241$	28.6887	0.00766806	0.00383403	−	0.999993i	$-0.498780\pi$
$241$	28.6887	0.00766806	0.00383403	+	0.999993i	$0.498780\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 640.471i	− 0.167013i
$246$	0	0
$247$	1306.21i	0.336487i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4822.91	−1.21283	−0.606414	−	0.795149i	$-0.707392\pi$
$251$	−4822.91	−1.21283	−0.606414	+	0.795149i	$0.707392\pi$
$252$	0	0
$253$	−1393.23	−0.346213
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1193.09i	0.289584i	0.989462	+	0.144792i	$0.0462512\pi$
$257$	1193.09i	0.289584i	−0.989462	+	0.144792i	$0.953749\pi$
$258$	0	0
$259$	− 1822.20i	− 0.437167i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3778.06	−0.885798	−0.442899	−	0.896571i	$-0.646050\pi$
$263$	−3778.06	−0.885798	−0.442899	+	0.896571i	$0.646050\pi$
$264$	0	0
$265$	−6238.89	−1.44623
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 5891.11i	− 1.33527i	−0.744489	−	0.667635i	$-0.767307\pi$
$269$	− 5891.11i	− 1.33527i	0.744489	−	0.667635i	$-0.232693\pi$
$270$	0	0
$271$	1968.28i	0.441198i	0.975365	+	0.220599i	$0.0708013\pi$
$271$	1968.28i	0.441198i	−0.975365	+	0.220599i	$0.929199\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1006.69	−0.220748
$276$	0	0
$277$	−6806.25	−1.47635	−0.738173	−	0.674611i	$-0.764311\pi$
$277$	−6806.25	−1.47635	−0.738173	+	0.674611i	$0.764311\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 608.205i	− 0.129119i	−0.997914	−	0.0645595i	$-0.979436\pi$
$281$	− 608.205i	− 0.129119i	0.997914	−	0.0645595i	$-0.0205642\pi$
$282$	0	0
$283$	− 3932.13i	− 0.825940i	−0.910745	−	0.412970i	$-0.864491\pi$
$283$	− 3932.13i	− 0.825940i	0.910745	−	0.412970i	$-0.135509\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−316.095	−0.0650123
$288$	0	0
$289$	−3835.63	−0.780711
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1333.83i	0.265949i	0.991119	+	0.132975i	$0.0424529\pi$
$293$	1333.83i	0.265949i	−0.991119	+	0.132975i	$0.957547\pi$
$294$	0	0
$295$	4074.81i	0.804219i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2853.83	−0.551978
$300$	0	0
$301$	−252.874	−0.0484233
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5388.69i	− 1.01166i
$306$	0	0
$307$	− 1827.17i	− 0.339681i	−0.985472	−	0.169841i	$-0.945675\pi$
$307$	− 1827.17i	− 0.339681i	0.985472	−	0.169841i	$-0.0543253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4257.31	0.776237	0.388118	−	0.921610i	$-0.373125\pi$
$311$	4257.31	0.776237	0.388118	+	0.921610i	$0.373125\pi$
$312$	0	0
$313$	2403.35	0.434011	0.217005	−	0.976170i	$-0.430371\pi$
$313$	2403.35	0.434011	0.217005	+	0.976170i	$0.430371\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5446.58i	0.965016i	0.875891	+	0.482508i	$0.160274\pi$
$317$	5446.58i	0.965016i	−0.875891	+	0.482508i	$0.839726\pi$
$318$	0	0
$319$	766.184i	0.134477i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2716.39	−0.467938
$324$	0	0
$325$	−2062.06	−0.351946
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2646.82i	− 0.443537i
$330$	0	0
$331$	9331.48i	1.54956i	0.632231	+	0.774780i	$0.282139\pi$
$331$	9331.48i	1.54956i	−0.632231	+	0.774780i	$0.717861\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2245.38	−0.366204
$336$	0	0
$337$	5473.41	0.884734	0.442367	−	0.896834i	$-0.354139\pi$
$337$	5473.41	0.884734	0.442367	+	0.896834i	$0.354139\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 5141.82i	− 0.816554i
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8798.11	−1.36112	−0.680558	−	0.732694i	$-0.738262\pi$
$347$	−8798.11	−1.36112	−0.680558	+	0.732694i	$0.738262\pi$
$348$	0	0
$349$	7885.59	1.20947	0.604737	−	0.796426i	$-0.293278\pi$
$349$	7885.59	1.20947	0.604737	+	0.796426i	$0.293278\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2600.48i	− 0.392095i	−0.980594	−	0.196047i	$-0.937189\pi$
$353$	− 2600.48i	− 0.392095i	0.980594	−	0.196047i	$-0.0628106\pi$
$354$	0	0
$355$	− 6518.21i	− 0.974509i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.6718	0.00392113	0.00196057	−	0.999998i	$-0.499376\pi$
$359$	26.6718	0.00392113	0.00196057	+	0.999998i	$0.499376\pi$
$360$	0	0
$361$	6015.58	0.877034
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6264.47i	− 0.898349i
$366$	0	0
$367$	− 7713.77i	− 1.09715i	−0.836100	−	0.548577i	$-0.815170\pi$
$367$	− 7713.77i	− 1.09715i	0.836100	−	0.548577i	$-0.184830\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3341.20	0.467564
$372$	0	0
$373$	12305.4	1.70818	0.854090	−	0.520125i	$-0.174115\pi$
$373$	12305.4	1.70818	0.854090	+	0.520125i	$0.174115\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1569.42i	0.214401i
$378$	0	0
$379$	− 5565.57i	− 0.754312i	−0.926150	−	0.377156i	$-0.876902\pi$
$379$	− 5565.57i	− 0.754312i	0.926150	−	0.377156i	$-0.123098\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6566.30	0.876037	0.438019	−	0.898966i	$-0.355680\pi$
$383$	6566.30	0.876037	0.438019	+	0.898966i	$0.355680\pi$
$384$	0	0
$385$	2009.04	0.265948
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1887.96i	0.246076i	0.992402	+	0.123038i	$0.0392637\pi$
$389$	1887.96i	0.246076i	−0.992402	+	0.123038i	$0.960736\pi$
$390$	0	0
$391$	− 5934.82i	− 0.767613i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16378.4	−2.08629
$396$	0	0
$397$	219.004	0.0276864	0.0138432	−	0.999904i	$-0.495593\pi$
$397$	219.004	0.0276864	0.0138432	+	0.999904i	$0.495593\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 10548.8i	− 1.31367i	−0.754032	−	0.656837i	$-0.771894\pi$
$401$	− 10548.8i	− 1.31367i	0.754032	−	0.656837i	$-0.228106\pi$
$402$	0	0
$403$	− 10532.3i	− 1.30186i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5715.91	0.696135
$408$	0	0
$409$	3217.81	0.389023	0.194512	−	0.980900i	$-0.437688\pi$
$409$	3217.81	0.389023	0.194512	+	0.980900i	$0.437688\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2182.24i	− 0.260002i
$414$	0	0
$415$	6842.91i	0.809411i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12988.7	−1.51442	−0.757209	−	0.653173i	$-0.773437\pi$
$419$	−12988.7	−1.51442	−0.757209	+	0.653173i	$0.773437\pi$
$420$	0	0
$421$	7399.55	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$421$	7399.55	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4288.25i	− 0.489436i
$426$	0	0
$427$	2885.88i	0.327067i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14086.5	1.57430	0.787152	−	0.616759i	$-0.211555\pi$
$431$	14086.5	1.57430	0.787152	+	0.616759i	$0.211555\pi$
$432$	0	0
$433$	−2743.57	−0.304498	−0.152249	−	0.988342i	$-0.548652\pi$
$433$	−2743.57	−0.304498	−0.152249	+	0.988342i	$0.548652\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1842.72i	− 0.201715i
$438$	0	0
$439$	795.333i	0.0864674i	0.999065	+	0.0432337i	$0.0137660\pi$
$439$	795.333i	0.0864674i	−0.999065	+	0.0432337i	$0.986234\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−934.806	−0.100257	−0.0501287	−	0.998743i	$-0.515963\pi$
$443$	−934.806	−0.100257	−0.0501287	+	0.998743i	$0.515963\pi$
$444$	0	0
$445$	3692.80	0.393383
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 10.1113i	− 0.00106277i	−1.00000		0.000531384i	$-0.999831\pi$
$449$	− 10.1113i	− 0.00106277i	1.00000		0.000531384i	$-0.000169145\pi$
$450$	0	0
$451$	− 991.532i	− 0.103524i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4115.22	0.424010
$456$	0	0
$457$	7299.94	0.747213	0.373607	−	0.927587i	$-0.378121\pi$
$457$	7299.94	0.747213	0.373607	+	0.927587i	$0.378121\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 2153.57i	− 0.217574i	−0.994065	−	0.108787i	$-0.965303\pi$
$461$	− 2153.57i	− 0.217574i	0.994065	−	0.108787i	$-0.0346967\pi$
$462$	0	0
$463$	4719.81i	0.473754i	0.971540	+	0.236877i	$0.0761239\pi$
$463$	4719.81i	0.473754i	−0.971540	+	0.236877i	$0.923876\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6064.78	0.600952	0.300476	−	0.953789i	$-0.402854\pi$
$467$	6064.78	0.600952	0.300476	+	0.953789i	$0.402854\pi$
$468$	0	0
$469$	1202.50	0.118393
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 793.218i	− 0.0771082i
$474$	0	0
$475$	− 1331.47i	− 0.128615i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2552.49	0.243479	0.121739	−	0.992562i	$-0.461153\pi$
$479$	2552.49	0.243479	0.121739	+	0.992562i	$0.461153\pi$
$480$	0	0
$481$	11708.2	1.10987
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 15695.4i	− 1.46946i
$486$	0	0
$487$	− 15818.2i	− 1.47185i	−0.677065	−	0.735923i	$-0.736748\pi$
$487$	− 15818.2i	− 1.47185i	0.677065	−	0.735923i	$-0.263252\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16680.2	−1.53313	−0.766565	−	0.642167i	$-0.778036\pi$
$491$	−16680.2	−1.53313	−0.766565	+	0.642167i	$0.778036\pi$
$492$	0	0
$493$	−3263.75	−0.298158
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3490.78i	0.315056i
$498$	0	0
$499$	− 20469.1i	− 1.83632i	−0.396207	−	0.918161i	$-0.629674\pi$
$499$	− 20469.1i	− 1.83632i	0.396207	−	0.918161i	$-0.370326\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13067.6	−1.15836	−0.579178	−	0.815201i	$-0.696626\pi$
$503$	−13067.6	−1.15836	−0.579178	+	0.815201i	$0.696626\pi$
$504$	0	0
$505$	−1931.82	−0.170228
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 15409.6i	− 1.34188i	−0.741510	−	0.670942i	$-0.765890\pi$
$509$	− 15409.6i	− 1.34188i	0.741510	−	0.670942i	$-0.234110\pi$
$510$	0	0
$511$	3354.89i	0.290434i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−20670.4	−1.76863
$516$	0	0
$517$	8302.56	0.706279
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14845.8i	1.24838i	0.781272	+	0.624191i	$0.214571\pi$
$521$	14845.8i	1.24838i	−0.781272	+	0.624191i	$0.785429\pi$
$522$	0	0
$523$	8784.15i	0.734424i	0.930137	+	0.367212i	$0.119688\pi$
$523$	8784.15i	0.734424i	−0.930137	+	0.367212i	$0.880312\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21902.8	1.81044
$528$	0	0
$529$	−8140.99	−0.669104
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2031.01i	− 0.165052i
$534$	0	0
$535$	11984.8i	0.968503i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1075.93	−0.0859804
$540$	0	0
$541$	4184.75	0.332563	0.166282	−	0.986078i	$-0.446824\pi$
$541$	4184.75	0.332563	0.166282	+	0.986078i	$0.446824\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2942.43i	0.231266i
$546$	0	0
$547$	− 746.339i	− 0.0583385i	−0.999574	−	0.0291692i	$-0.990714\pi$
$547$	− 746.339i	− 0.0583385i	0.999574	−	0.0291692i	$-0.00928617\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1013.37	−0.0783505
$552$	0	0
$553$	8771.33	0.674493
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7866.24i	− 0.598390i	−0.954192	−	0.299195i	$-0.903282\pi$
$557$	− 7866.24i	− 0.598390i	0.954192	−	0.299195i	$-0.0967182\pi$
$558$	0	0
$559$	− 1624.79i	− 0.122936i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14918.7	−1.11678	−0.558391	−	0.829578i	$-0.688581\pi$
$563$	−14918.7	−1.11678	−0.558391	+	0.829578i	$0.688581\pi$
$564$	0	0
$565$	−18943.3	−1.41053
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 16898.6i	− 1.24504i	−0.782605	−	0.622518i	$-0.786110\pi$
$569$	− 16898.6i	− 1.24504i	0.782605	−	0.622518i	$-0.213890\pi$
$570$	0	0
$571$	− 17855.6i	− 1.30864i	−0.756217	−	0.654321i	$-0.772954\pi$
$571$	− 17855.6i	− 1.30864i	0.756217	−	0.654321i	$-0.227046\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2909.02	0.210982
$576$	0	0
$577$	14413.3	1.03992	0.519960	−	0.854191i	$-0.325947\pi$
$577$	14413.3	1.03992	0.519960	+	0.854191i	$0.325947\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3664.68i	− 0.261681i
$582$	0	0
$583$	10480.7i	0.744539i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27666.5	−1.94535	−0.972675	−	0.232172i	$-0.925417\pi$
$587$	−27666.5	−1.94535	−0.972675	+	0.232172i	$0.925417\pi$
$588$	0	0
$589$	6800.68	0.475751
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15104.9i	− 1.04601i	−0.852330	−	0.523005i	$-0.824811\pi$
$593$	− 15104.9i	− 1.04601i	0.852330	−	0.523005i	$-0.175189\pi$
$594$	0	0
$595$	8557.99i	0.589653i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10189.8	−0.695062	−0.347531	−	0.937669i	$-0.612980\pi$
$599$	−10189.8	−0.695062	−0.347531	+	0.937669i	$0.612980\pi$
$600$	0	0
$601$	−13809.2	−0.937251	−0.468626	−	0.883397i	$-0.655251\pi$
$601$	−13809.2	−0.937251	−0.468626	+	0.883397i	$0.655251\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 11095.3i	− 0.745601i
$606$	0	0
$607$	14411.4i	0.963660i	0.876265	+	0.481830i	$0.160028\pi$
$607$	14411.4i	0.963660i	−0.876265	+	0.481830i	$0.839972\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17006.6	1.12604
$612$	0	0
$613$	−19462.0	−1.28232	−0.641162	−	0.767406i	$-0.721547\pi$
$613$	−19462.0	−1.28232	−0.641162	+	0.767406i	$0.721547\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26668.9i	− 1.74011i	−0.492952	−	0.870056i	$-0.664082\pi$
$617$	− 26668.9i	− 1.74011i	0.492952	−	0.870056i	$-0.335918\pi$
$618$	0	0
$619$	2154.61i	0.139905i	0.997550	+	0.0699524i	$0.0222847\pi$
$619$	2154.61i	0.139905i	−0.997550	+	0.0699524i	$0.977715\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1977.65	−0.127180
$624$	0	0
$625$	−19253.9	−1.23225
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24348.3i	1.54345i
$630$	0	0
$631$	26162.2i	1.65056i	0.564725	+	0.825279i	$0.308982\pi$
$631$	26162.2i	1.65056i	−0.564725	+	0.825279i	$0.691018\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6085.05	0.380280
$636$	0	0
$637$	−2203.88	−0.137081
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2612.69i	0.160991i	0.996755	+	0.0804954i	$0.0256502\pi$
$641$	2612.69i	0.160991i	−0.996755	+	0.0804954i	$0.974350\pi$
$642$	0	0
$643$	− 22177.5i	− 1.36018i	−0.733130	−	0.680089i	$-0.761941\pi$
$643$	− 22177.5i	− 1.36018i	0.733130	−	0.680089i	$-0.238059\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21862.2	−1.32842	−0.664212	−	0.747544i	$-0.731233\pi$
$647$	−21862.2	−1.32842	−0.664212	+	0.747544i	$0.731233\pi$
$648$	0	0
$649$	6845.27	0.414022
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16524.9i	0.990306i	0.868806	+	0.495153i	$0.164888\pi$
$653$	16524.9i	0.990306i	−0.868806	+	0.495153i	$0.835112\pi$
$654$	0	0
$655$	− 11954.1i	− 0.713105i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33324.8	1.96988	0.984938	−	0.172909i	$-0.0553167\pi$
$659$	33324.8	1.96988	0.984938	+	0.172909i	$0.0553167\pi$
$660$	0	0
$661$	20681.6	1.21698	0.608488	−	0.793563i	$-0.291776\pi$
$661$	20681.6	1.21698	0.608488	+	0.793563i	$0.291776\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2657.20i	0.154950i
$666$	0	0
$667$	− 2214.03i	− 0.128527i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9052.46	−0.520814
$672$	0	0
$673$	−12205.9	−0.699114	−0.349557	−	0.936915i	$-0.613668\pi$
$673$	−12205.9	−0.699114	−0.349557	+	0.936915i	$0.613668\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4737.38i	− 0.268940i	−0.990918	−	0.134470i	$-0.957067\pi$
$677$	− 4737.38i	− 0.268940i	0.990918	−	0.134470i	$-0.0429332\pi$
$678$	0	0
$679$	8405.54i	0.475074i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25228.5	1.41339	0.706693	−	0.707520i	$-0.250186\pi$
$683$	25228.5	1.41339	0.706693	+	0.707520i	$0.250186\pi$
$684$	0	0
$685$	−28612.0	−1.59592
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21468.2i	1.18704i
$690$	0	0
$691$	− 28292.5i	− 1.55760i	−0.627275	−	0.778798i	$-0.715830\pi$
$691$	− 28292.5i	− 1.55760i	0.627275	−	0.778798i	$-0.284170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20438.3	1.11549
$696$	0	0
$697$	4223.67	0.229531
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21867.6i	1.17821i	0.808056	+	0.589106i	$0.200520\pi$
$701$	21867.6i	1.17821i	−0.808056	+	0.589106i	$0.799480\pi$
$702$	0	0
$703$	7559.98i	0.405591i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1034.57	0.0550342
$708$	0	0
$709$	20888.6	1.10647	0.553234	−	0.833026i	$-0.313393\pi$
$709$	20888.6	1.10647	0.553234	+	0.833026i	$0.313393\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14858.3i	0.780429i
$714$	0	0
$715$	12908.7i	0.675184i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	27118.8	1.40662	0.703312	−	0.710882i	$-0.251704\pi$
$719$	27118.8	1.40662	0.703312	+	0.710882i	$0.251704\pi$
$720$	0	0
$721$	11069.9	0.571794
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1599.77i	− 0.0819501i
$726$	0	0
$727$	14943.1i	0.762322i	0.924509	+	0.381161i	$0.124476\pi$
$727$	14943.1i	0.762322i	−0.924509	+	0.381161i	$0.875524\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3378.91	0.170962
$732$	0	0
$733$	3150.28	0.158743	0.0793713	−	0.996845i	$-0.474709\pi$
$733$	3150.28	0.158743	0.0793713	+	0.996845i	$0.474709\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3772.01i	0.188526i
$738$	0	0
$739$	− 19675.6i	− 0.979401i	−0.871891	−	0.489701i	$-0.837106\pi$
$739$	− 19675.6i	− 0.979401i	0.871891	−	0.489701i	$-0.162894\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13513.3	0.667236	0.333618	−	0.942708i	$-0.391730\pi$
$743$	13513.3	0.667236	0.333618	+	0.942708i	$0.391730\pi$
$744$	0	0
$745$	−17963.6	−0.883404
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 6418.39i	− 0.313115i
$750$	0	0
$751$	6634.93i	0.322386i	0.986923	+	0.161193i	$0.0515342\pi$
$751$	6634.93i	0.322386i	−0.986923	+	0.161193i	$0.948466\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−46639.2	−2.24818
$756$	0	0
$757$	31608.8	1.51763	0.758813	−	0.651309i	$-0.225780\pi$
$757$	31608.8	1.51763	0.758813	+	0.651309i	$0.225780\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32761.5i	1.56058i	0.625417	+	0.780291i	$0.284929\pi$
$761$	32761.5i	1.56058i	−0.625417	+	0.780291i	$0.715071\pi$
$762$	0	0
$763$	− 1575.80i	− 0.0747678i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14021.5	0.660089
$768$	0	0
$769$	19982.9	0.937061	0.468531	−	0.883447i	$-0.344784\pi$
$769$	19982.9	0.937061	0.468531	+	0.883447i	$0.344784\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14740.5i	0.685873i	0.939359	+	0.342936i	$0.111422\pi$
$773$	14740.5i	0.685873i	−0.939359	+	0.342936i	$0.888578\pi$
$774$	0	0
$775$	10735.9i	0.497608i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1311.42	0.0603165
$780$	0	0
$781$	−10949.9	−0.501689
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 27355.0i	− 1.24375i
$786$	0	0
$787$	− 14775.5i	− 0.669239i	−0.942353	−	0.334619i	$-0.891392\pi$
$787$	− 14775.5i	− 0.669239i	0.942353	−	0.334619i	$-0.108608\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10144.9	0.456021
$792$	0	0
$793$	−18542.6	−0.830351
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32796.2i	1.45759i	0.684731	+	0.728796i	$0.259920\pi$
$797$	32796.2i	1.45759i	−0.684731	+	0.728796i	$0.740080\pi$
$798$	0	0
$799$	35366.8i	1.56594i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10523.7	−0.462481
$804$	0	0
$805$	−5805.49	−0.254182
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38663.8i	1.68028i	0.542370	+	0.840139i	$0.317527\pi$
$809$	38663.8i	1.68028i	−0.542370	+	0.840139i	$0.682473\pi$
$810$	0	0
$811$	− 33801.6i	− 1.46354i	−0.681549	−	0.731772i	$-0.738694\pi$
$811$	− 33801.6i	− 1.46354i	0.681549	−	0.731772i	$-0.261306\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17749.7	0.762876
$816$	0	0
$817$	1049.13	0.0449257
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 7615.89i	− 0.323747i	−0.986812	−	0.161874i	$-0.948246\pi$
$821$	− 7615.89i	− 0.323747i	0.986812	−	0.161874i	$-0.0517537\pi$
$822$	0	0
$823$	− 6527.22i	− 0.276458i	−0.990400	−	0.138229i	$-0.955859\pi$
$823$	− 6527.22i	− 0.276458i	0.990400	−	0.138229i	$-0.0441409\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19935.2	0.838229	0.419114	−	0.907933i	$-0.362341\pi$
$827$	19935.2	0.838229	0.419114	+	0.907933i	$0.362341\pi$
$828$	0	0
$829$	−37669.8	−1.57820	−0.789098	−	0.614267i	$-0.789452\pi$
$829$	−37669.8	−1.57820	−0.789098	+	0.614267i	$0.789452\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 4583.17i	− 0.190633i
$834$	0	0
$835$	37094.8i	1.53739i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37529.0	−1.54427	−0.772136	−	0.635457i	$-0.780812\pi$
$839$	−37529.0	−1.54427	−0.772136	+	0.635457i	$0.780812\pi$
$840$	0	0
$841$	23171.4	0.950077
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2275.14i	− 0.0926238i
$846$	0	0
$847$	5942.02i	0.241051i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16517.2	−0.665337
$852$	0	0
$853$	−11145.2	−0.447369	−0.223684	−	0.974662i	$-0.571809\pi$
$853$	−11145.2	−0.447369	−0.223684	+	0.974662i	$0.571809\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6101.33i	− 0.243194i	−0.992580	−	0.121597i	$-0.961198\pi$
$857$	− 6101.33i	− 0.243194i	0.992580	−	0.121597i	$-0.0388016\pi$
$858$	0	0
$859$	− 5205.70i	− 0.206771i	−0.994641	−	0.103385i	$-0.967032\pi$
$859$	− 5205.70i	− 0.206771i	0.994641	−	0.103385i	$-0.0329675\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26139.7	1.03106	0.515530	−	0.856872i	$-0.327595\pi$
$863$	26139.7	1.03106	0.515530	+	0.856872i	$0.327595\pi$
$864$	0	0
$865$	−51597.2	−2.02816
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27514.0i	1.07405i
$870$	0	0
$871$	7726.42i	0.300574i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7242.19	0.279806
$876$	0	0
$877$	41049.7	1.58056	0.790278	−	0.612748i	$-0.209936\pi$
$877$	41049.7	1.58056	0.790278	+	0.612748i	$0.209936\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	40215.1i	1.53789i	0.639315	+	0.768945i	$0.279218\pi$
$881$	40215.1i	1.53789i	−0.639315	+	0.768945i	$0.720782\pi$
$882$	0	0
$883$	23012.9i	0.877061i	0.898716	+	0.438530i	$0.144501\pi$
$883$	23012.9i	0.877061i	−0.898716	+	0.438530i	$0.855499\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39681.3	1.50211	0.751053	−	0.660242i	$-0.229546\pi$
$887$	39681.3	1.50211	0.751053	+	0.660242i	$0.229546\pi$
$888$	0	0
$889$	−3258.81	−0.122944
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10981.1i	0.411501i
$894$	0	0
$895$	− 27824.4i	− 1.03918i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8171.04	0.303136
$900$	0	0
$901$	−44645.1	−1.65077
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 24716.4i	− 0.907845i
$906$	0	0
$907$	− 13739.3i	− 0.502983i	−0.967859	−	0.251492i	$-0.919079\pi$
$907$	− 13739.3i	− 0.502983i	0.967859	−	0.251492i	$-0.0809211\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47938.0	1.74342	0.871712	−	0.490019i	$-0.163010\pi$
$911$	47938.0	1.74342	0.871712	+	0.490019i	$0.163010\pi$
$912$	0	0
$913$	11495.4	0.416695
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6401.91i	0.230545i
$918$	0	0
$919$	42399.0i	1.52189i	0.648819	+	0.760943i	$0.275263\pi$
$919$	42399.0i	1.52189i	−0.648819	+	0.760943i	$0.724737\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−22429.3	−0.799859
$924$	0	0
$925$	−11934.6	−0.424224
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25846.9i	0.912821i	0.889769	+	0.456410i	$0.150865\pi$
$929$	25846.9i	0.912821i	−0.889769	+	0.456410i	$0.849135\pi$
$930$	0	0
$931$	− 1423.04i	− 0.0500949i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−26844.8	−0.938950
$936$	0	0
$937$	−35213.3	−1.22772	−0.613858	−	0.789417i	$-0.710383\pi$
$937$	−35213.3	−1.22772	−0.613858	+	0.789417i	$0.710383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10725.9i	0.371577i	0.982590	+	0.185789i	$0.0594840\pi$
$941$	10725.9i	0.371577i	−0.982590	+	0.185789i	$0.940516\pi$
$942$	0	0
$943$	2865.22i	0.0989441i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17634.8	−0.605127	−0.302564	−	0.953129i	$-0.597842\pi$
$947$	−17634.8	−0.605127	−0.302564	+	0.953129i	$0.597842\pi$
$948$	0	0
$949$	−21556.2	−0.737348
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6498.81i	0.220899i	0.993882	+	0.110450i	$0.0352291\pi$
$953$	6498.81i	0.220899i	−0.993882	+	0.110450i	$0.964771\pi$
$954$	0	0
$955$	13207.9i	0.447538i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15323.0	0.515958
$960$	0	0
$961$	−25044.4	−0.840668
$962$	0	0
$963$	0	0
$964$	0	0
$965$	50353.0i	1.67971i
$966$	0	0
$967$	− 49539.1i	− 1.64743i	−0.567001	−	0.823717i	$-0.691896\pi$
$967$	− 49539.1i	− 1.64743i	0.567001	−	0.823717i	$-0.308104\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41207.2	1.36190	0.680949	−	0.732331i	$-0.261568\pi$
$971$	41207.2	1.36190	0.680949	+	0.732331i	$0.261568\pi$
$972$	0	0
$973$	−10945.6	−0.360636
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 437.078i	− 0.0143126i	−0.999974	−	0.00715628i	$-0.997722\pi$
$977$	− 437.078i	− 0.0143126i	0.999974	−	0.00715628i	$-0.00227793\pi$
$978$	0	0
$979$	− 6203.53i	− 0.202518i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6784.87	−0.220146	−0.110073	−	0.993923i	$-0.535109\pi$
$983$	−6784.87	−0.220146	−0.110073	+	0.993923i	$0.535109\pi$
$984$	0	0
$985$	27699.0	0.896005
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2292.15i	0.0736969i
$990$	0	0
$991$	38962.5i	1.24893i	0.781054	+	0.624463i	$0.214682\pi$
$991$	38962.5i	1.24893i	−0.781054	+	0.624463i	$0.785318\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17690.4	−0.563641
$996$	0	0
$997$	−45276.6	−1.43824	−0.719119	−	0.694887i	$-0.755454\pi$
$997$	−45276.6	−1.43824	−0.719119	+	0.694887i	$0.755454\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.h.b.575.16	yes	24
3.2	odd	2	inner	1008.4.h.b.575.9	✓	24
4.3	odd	2	inner	1008.4.h.b.575.10	yes	24
12.11	even	2	inner	1008.4.h.b.575.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.4.h.b.575.9	✓	24	3.2	odd	2	inner
1008.4.h.b.575.10	yes	24	4.3	odd	2	inner
1008.4.h.b.575.15	yes	24	12.11	even	2	inner
1008.4.h.b.575.16	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+13.0708i q^{5} -7.00000i q^{7} +O(q^{10})\)	\(q+13.0708i q^{5} -7.00000i q^{7} +21.9577 q^{11} +44.9771 q^{13} +93.5341i q^{17} +29.0417i q^{19} -63.4509 q^{23} -45.8469 q^{25} +34.8937i q^{29} -234.169i q^{31} +91.4959 q^{35} +260.315 q^{37} -45.1565i q^{41} -36.1248i q^{43} +378.116 q^{47} -49.0000 q^{49} +477.314i q^{53} +287.005i q^{55} +311.748 q^{59} -412.268 q^{61} +587.888i q^{65} +171.786i q^{67} -498.683 q^{71} -479.271 q^{73} -153.704i q^{77} +1253.05i q^{79} +523.525 q^{83} -1222.57 q^{85} -282.522i q^{89} -314.840i q^{91} -379.600 q^{95} -1200.79 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 192 q^{13} - 984 q^{25} + 720 q^{37} - 1176 q^{49} + 2736 q^{61} + 3408 q^{85} + 1152 q^{97}+O(q^{100}) \) 24 * q - 192 * q^13 - 984 * q^25 + 720 * q^37 - 1176 * q^49 + 2736 * q^61 + 3408 * q^85 + 1152 * q^97

Introduction

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