LMFDB - Embedded newform 2496.3.k.e.703.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,3,Mod(703,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2496.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2496.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$68.0110739843$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.11
Character	$\chi$	$=$	2496.703
Dual form			2496.3.k.e.703.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +4.98564 q^{5} -3.34189i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +4.98564 q^{5} -3.34189i q^{7} -3.00000 q^{9} +8.65097i q^{11} -3.60555 q^{13} -8.63539i q^{15} -0.936203 q^{17} +29.3196i q^{19} -5.78833 q^{21} -33.3456i q^{23} -0.143370 q^{25} +5.19615i q^{27} -0.223935 q^{29} -35.7681i q^{31} +14.9839 q^{33} -16.6615i q^{35} +22.5963 q^{37} +6.24500i q^{39} -3.58506 q^{41} -68.8231i q^{43} -14.9569 q^{45} +5.87303i q^{47} +37.8317 q^{49} +1.62155i q^{51} +17.0444 q^{53} +43.1306i q^{55} +50.7831 q^{57} -66.0654i q^{59} +114.756 q^{61} +10.0257i q^{63} -17.9760 q^{65} -49.7915i q^{67} -57.7563 q^{69} -78.4873i q^{71} -15.6030 q^{73} +0.248323i q^{75} +28.9106 q^{77} -137.505i q^{79} +9.00000 q^{81} +20.3475i q^{83} -4.66757 q^{85} +0.387867i q^{87} +24.0829 q^{89} +12.0494i q^{91} -61.9521 q^{93} +146.177i q^{95} -136.996 q^{97} -25.9529i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{9}+O(q^{10}) $ 24 * q - 72 * q^9	$ 24 q - 72 q^{9} + 104 q^{25} - 64 q^{29} + 48 q^{33} + 192 q^{37} - 248 q^{49} - 336 q^{53} - 16 q^{61} + 192 q^{69} + 112 q^{73} + 272 q^{77} + 216 q^{81} - 64 q^{85} + 96 q^{93} - 80 q^{97}+O(q^{100}) $ 24 * q - 72 * q^9 + 104 * q^25 - 64 * q^29 + 48 * q^33 + 192 * q^37 - 248 * q^49 - 336 * q^53 - 16 * q^61 + 192 * q^69 + 112 * q^73 + 272 * q^77 + 216 * q^81 - 64 * q^85 + 96 * q^93 - 80 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	4.98564	0.997128	0.498564	−	0.866853i	$-0.333861\pi$
$5$	4.98564	0.997128	0.498564	+	0.866853i	$0.333861\pi$
$6$	0	0
$7$	− 3.34189i	− 0.477413i	−0.971092	−	0.238707i	$-0.923277\pi$
$7$	− 3.34189i	− 0.477413i	0.971092	−	0.238707i	$-0.0767235\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	8.65097i	0.786451i	0.919442	+	0.393226i	$0.128641\pi$
$11$	8.65097i	0.786451i	−0.919442	+	0.393226i	$0.871359\pi$
$12$	0	0
$13$	−3.60555	−0.277350
$13$	−3.60555	−0.277350
$14$	0	0
$15$	− 8.63539i	− 0.575692i
$16$	0	0
$17$	−0.936203	−0.0550707	−0.0275354	−	0.999621i	$-0.508766\pi$
$17$	−0.936203	−0.0550707	−0.0275354	+	0.999621i	$0.508766\pi$
$18$	0	0
$19$	29.3196i	1.54314i	0.636145	+	0.771570i	$0.280528\pi$
$19$	29.3196i	1.54314i	−0.636145	+	0.771570i	$0.719472\pi$
$20$	0	0
$21$	−5.78833	−0.275635
$22$	0	0
$23$	− 33.3456i	− 1.44981i	−0.688849	−	0.724904i	$-0.741884\pi$
$23$	− 33.3456i	− 1.44981i	0.688849	−	0.724904i	$-0.258116\pi$
$24$	0	0
$25$	−0.143370	−0.00573478
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−0.223935	−0.00772190	−0.00386095	−	0.999993i	$-0.501229\pi$
$29$	−0.223935	−0.00772190	−0.00386095	+	0.999993i	$0.501229\pi$
$30$	0	0
$31$	− 35.7681i	− 1.15381i	−0.816812	−	0.576905i	$-0.804260\pi$
$31$	− 35.7681i	− 1.15381i	0.816812	−	0.576905i	$-0.195740\pi$
$32$	0	0
$33$	14.9839	0.454058
$34$	0	0
$35$	− 16.6615i	− 0.476043i
$36$	0	0
$37$	22.5963	0.610710	0.305355	−	0.952239i	$-0.401225\pi$
$37$	22.5963	0.610710	0.305355	+	0.952239i	$0.401225\pi$
$38$	0	0
$39$	6.24500i	0.160128i
$40$	0	0
$41$	−3.58506	−0.0874404	−0.0437202	−	0.999044i	$-0.513921\pi$
$41$	−3.58506	−0.0874404	−0.0437202	+	0.999044i	$0.513921\pi$
$42$	0	0
$43$	− 68.8231i	− 1.60054i	−0.599642	−	0.800268i	$-0.704690\pi$
$43$	− 68.8231i	− 1.60054i	0.599642	−	0.800268i	$-0.295310\pi$
$44$	0	0
$45$	−14.9569	−0.332376
$46$	0	0
$47$	5.87303i	0.124958i	0.998046	+	0.0624791i	$0.0199007\pi$
$47$	5.87303i	0.124958i	−0.998046	+	0.0624791i	$0.980099\pi$
$48$	0	0
$49$	37.8317	0.772076
$50$	0	0
$51$	1.62155i	0.0317951i
$52$	0	0
$53$	17.0444	0.321592	0.160796	−	0.986988i	$-0.448594\pi$
$53$	17.0444	0.321592	0.160796	+	0.986988i	$0.448594\pi$
$54$	0	0
$55$	43.1306i	0.784193i
$56$	0	0
$57$	50.7831	0.890932
$58$	0	0
$59$	− 66.0654i	− 1.11975i	−0.828576	−	0.559876i	$-0.810849\pi$
$59$	− 66.0654i	− 1.11975i	0.828576	−	0.559876i	$-0.189151\pi$
$60$	0	0
$61$	114.756	1.88124	0.940620	−	0.339462i	$-0.110245\pi$
$61$	114.756	1.88124	0.940620	+	0.339462i	$0.110245\pi$
$62$	0	0
$63$	10.0257i	0.159138i
$64$	0	0
$65$	−17.9760	−0.276554
$66$	0	0
$67$	− 49.7915i	− 0.743157i	−0.928402	−	0.371578i	$-0.878817\pi$
$67$	− 49.7915i	− 0.743157i	0.928402	−	0.371578i	$-0.121183\pi$
$68$	0	0
$69$	−57.7563	−0.837047
$70$	0	0
$71$	− 78.4873i	− 1.10545i	−0.833362	−	0.552727i	$-0.813587\pi$
$71$	− 78.4873i	− 1.10545i	0.833362	−	0.552727i	$-0.186413\pi$
$72$	0	0
$73$	−15.6030	−0.213740	−0.106870	−	0.994273i	$-0.534083\pi$
$73$	−15.6030	−0.213740	−0.106870	+	0.994273i	$0.534083\pi$
$74$	0	0
$75$	0.248323i	0.00331098i
$76$	0	0
$77$	28.9106	0.375463
$78$	0	0
$79$	− 137.505i	− 1.74057i	−0.492551	−	0.870283i	$-0.663936\pi$
$79$	− 137.505i	− 1.74057i	0.492551	−	0.870283i	$-0.336064\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	20.3475i	0.245150i	0.992459	+	0.122575i	$0.0391153\pi$
$83$	20.3475i	0.245150i	−0.992459	+	0.122575i	$0.960885\pi$
$84$	0	0
$85$	−4.66757	−0.0549126
$86$	0	0
$87$	0.387867i	0.00445824i
$88$	0	0
$89$	24.0829	0.270594	0.135297	−	0.990805i	$-0.456801\pi$
$89$	24.0829	0.270594	0.135297	+	0.990805i	$0.456801\pi$
$90$	0	0
$91$	12.0494i	0.132411i
$92$	0	0
$93$	−61.9521	−0.666152
$94$	0	0
$95$	146.177i	1.53871i
$96$	0	0
$97$	−136.996	−1.41233	−0.706166	−	0.708046i	$-0.749577\pi$
$97$	−136.996	−1.41233	−0.706166	+	0.708046i	$0.749577\pi$
$98$	0	0
$99$	− 25.9529i	− 0.262150i
$100$	0	0
$101$	−48.5723	−0.480914	−0.240457	−	0.970660i	$-0.577297\pi$
$101$	−48.5723	−0.480914	−0.240457	+	0.970660i	$0.577297\pi$
$102$	0	0
$103$	20.7100i	0.201068i	0.994934	+	0.100534i	$0.0320551\pi$
$103$	20.7100i	0.201068i	−0.994934	+	0.100534i	$0.967945\pi$
$104$	0	0
$105$	−28.8585	−0.274843
$106$	0	0
$107$	− 80.1149i	− 0.748738i	−0.927280	−	0.374369i	$-0.877859\pi$
$107$	− 80.1149i	− 0.748738i	0.927280	−	0.374369i	$-0.122141\pi$
$108$	0	0
$109$	126.510	1.16064	0.580320	−	0.814388i	$-0.302927\pi$
$109$	126.510	1.16064	0.580320	+	0.814388i	$0.302927\pi$
$110$	0	0
$111$	− 39.1379i	− 0.352594i
$112$	0	0
$113$	202.600	1.79292	0.896461	−	0.443123i	$-0.146129\pi$
$113$	202.600	1.79292	0.896461	+	0.443123i	$0.146129\pi$
$114$	0	0
$115$	− 166.249i	− 1.44565i
$116$	0	0
$117$	10.8167	0.0924500
$118$	0	0
$119$	3.12869i	0.0262915i
$120$	0	0
$121$	46.1608	0.381494
$122$	0	0
$123$	6.20950i	0.0504838i
$124$	0	0
$125$	−125.356	−1.00285
$126$	0	0
$127$	− 53.8312i	− 0.423868i	−0.977284	−	0.211934i	$-0.932024\pi$
$127$	− 53.8312i	− 0.423868i	0.977284	−	0.211934i	$-0.0679762\pi$
$128$	0	0
$129$	−119.205	−0.924070
$130$	0	0
$131$	255.421i	1.94978i	0.222695	+	0.974888i	$0.428515\pi$
$131$	255.421i	1.94978i	−0.222695	+	0.974888i	$0.571485\pi$
$132$	0	0
$133$	97.9832	0.736715
$134$	0	0
$135$	25.9062i	0.191897i
$136$	0	0
$137$	47.3297	0.345473	0.172736	−	0.984968i	$-0.444739\pi$
$137$	47.3297	0.345473	0.172736	+	0.984968i	$0.444739\pi$
$138$	0	0
$139$	− 151.671i	− 1.09116i	−0.838059	−	0.545579i	$-0.816310\pi$
$139$	− 151.671i	− 1.09116i	0.838059	−	0.545579i	$-0.183690\pi$
$140$	0	0
$141$	10.1724	0.0721446
$142$	0	0
$143$	− 31.1915i	− 0.218122i
$144$	0	0
$145$	−1.11646	−0.00769972
$146$	0	0
$147$	− 65.5265i	− 0.445759i
$148$	0	0
$149$	−257.046	−1.72514	−0.862569	−	0.505939i	$-0.831146\pi$
$149$	−257.046	−1.72514	−0.862569	+	0.505939i	$0.831146\pi$
$150$	0	0
$151$	− 74.0102i	− 0.490134i	−0.969506	−	0.245067i	$-0.921190\pi$
$151$	− 74.0102i	− 0.490134i	0.969506	−	0.245067i	$-0.0788099\pi$
$152$	0	0
$153$	2.80861	0.0183569
$154$	0	0
$155$	− 178.327i	− 1.15050i
$156$	0	0
$157$	−86.3013	−0.549690	−0.274845	−	0.961489i	$-0.588626\pi$
$157$	−86.3013	−0.549690	−0.274845	+	0.961489i	$0.588626\pi$
$158$	0	0
$159$	− 29.5217i	− 0.185671i
$160$	0	0
$161$	−111.437	−0.692158
$162$	0	0
$163$	124.211i	0.762029i	0.924569	+	0.381015i	$0.124425\pi$
$163$	124.211i	0.762029i	−0.924569	+	0.381015i	$0.875575\pi$
$164$	0	0
$165$	74.7044	0.452754
$166$	0	0
$167$	168.354i	1.00811i	0.863672	+	0.504054i	$0.168159\pi$
$167$	168.354i	1.00811i	−0.863672	+	0.504054i	$0.831841\pi$
$168$	0	0
$169$	13.0000	0.0769231
$170$	0	0
$171$	− 87.9589i	− 0.514380i
$172$	0	0
$173$	323.241	1.86844	0.934222	−	0.356691i	$-0.116095\pi$
$173$	323.241	1.86844	0.934222	+	0.356691i	$0.116095\pi$
$174$	0	0
$175$	0.479126i	0.00273786i
$176$	0	0
$177$	−114.429	−0.646490
$178$	0	0
$179$	106.891i	0.597157i	0.954385	+	0.298579i	$0.0965125\pi$
$179$	106.891i	0.597157i	−0.954385	+	0.298579i	$0.903487\pi$
$180$	0	0
$181$	−49.5800	−0.273923	−0.136961	−	0.990576i	$-0.543734\pi$
$181$	−49.5800	−0.273923	−0.136961	+	0.990576i	$0.543734\pi$
$182$	0	0
$183$	− 198.763i	− 1.08613i
$184$	0	0
$185$	112.657	0.608957
$186$	0	0
$187$	− 8.09906i	− 0.0433105i
$188$	0	0
$189$	17.3650	0.0918783
$190$	0	0
$191$	− 126.760i	− 0.663663i	−0.943339	−	0.331832i	$-0.892333\pi$
$191$	− 126.760i	− 0.663663i	0.943339	−	0.331832i	$-0.107667\pi$
$192$	0	0
$193$	131.610	0.681915	0.340958	−	0.940079i	$-0.389249\pi$
$193$	131.610	0.681915	0.340958	+	0.940079i	$0.389249\pi$
$194$	0	0
$195$	31.1353i	0.159668i
$196$	0	0
$197$	112.332	0.570214	0.285107	−	0.958496i	$-0.407971\pi$
$197$	112.332	0.570214	0.285107	+	0.958496i	$0.407971\pi$
$198$	0	0
$199$	243.277i	1.22250i	0.791439	+	0.611249i	$0.209332\pi$
$199$	243.277i	1.22250i	−0.791439	+	0.611249i	$0.790668\pi$
$200$	0	0
$201$	−86.2414	−0.429062
$202$	0	0
$203$	0.748367i	0.00368654i
$204$	0	0
$205$	−17.8738	−0.0871893
$206$	0	0
$207$	100.037i	0.483270i
$208$	0	0
$209$	−253.643	−1.21360
$210$	0	0
$211$	322.242i	1.52721i	0.645681	+	0.763607i	$0.276574\pi$
$211$	322.242i	1.52721i	−0.645681	+	0.763607i	$0.723426\pi$
$212$	0	0
$213$	−135.944	−0.638234
$214$	0	0
$215$	− 343.127i	− 1.59594i
$216$	0	0
$217$	−119.533	−0.550844
$218$	0	0
$219$	27.0252i	0.123403i
$220$	0	0
$221$	3.37553	0.0152739
$222$	0	0
$223$	− 197.953i	− 0.887681i	−0.896106	−	0.443840i	$-0.853616\pi$
$223$	− 197.953i	− 0.887681i	0.896106	−	0.443840i	$-0.146384\pi$
$224$	0	0
$225$	0.430109	0.00191159
$226$	0	0
$227$	− 324.837i	− 1.43100i	−0.698613	−	0.715499i	$-0.746199\pi$
$227$	− 324.837i	− 1.43100i	0.698613	−	0.715499i	$-0.253801\pi$
$228$	0	0
$229$	−371.743	−1.62333	−0.811666	−	0.584122i	$-0.801439\pi$
$229$	−371.743	−1.62333	−0.811666	+	0.584122i	$0.801439\pi$
$230$	0	0
$231$	− 50.0747i	− 0.216773i
$232$	0	0
$233$	210.436	0.903158	0.451579	−	0.892231i	$-0.350861\pi$
$233$	210.436	0.903158	0.451579	+	0.892231i	$0.350861\pi$
$234$	0	0
$235$	29.2808i	0.124599i
$236$	0	0
$237$	−238.165	−1.00492
$238$	0	0
$239$	− 317.305i	− 1.32763i	−0.747895	−	0.663817i	$-0.768935\pi$
$239$	− 317.305i	− 1.32763i	0.747895	−	0.663817i	$-0.231065\pi$
$240$	0	0
$241$	101.711	0.422038	0.211019	−	0.977482i	$-0.432322\pi$
$241$	101.711	0.422038	0.211019	+	0.977482i	$0.432322\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	188.616	0.769859
$246$	0	0
$247$	− 105.713i	− 0.427990i
$248$	0	0
$249$	35.2429	0.141538
$250$	0	0
$251$	− 401.077i	− 1.59792i	−0.601387	−	0.798958i	$-0.705385\pi$
$251$	− 401.077i	− 1.59792i	0.601387	−	0.798958i	$-0.294615\pi$
$252$	0	0
$253$	288.472	1.14020
$254$	0	0
$255$	8.08447i	0.0317038i
$256$	0	0
$257$	−1.84794	−0.00719042	−0.00359521	−	0.999994i	$-0.501144\pi$
$257$	−1.84794	−0.00719042	−0.00359521	+	0.999994i	$0.501144\pi$
$258$	0	0
$259$	− 75.5144i	− 0.291561i
$260$	0	0
$261$	0.671805	0.00257397
$262$	0	0
$263$	− 366.786i	− 1.39463i	−0.716767	−	0.697313i	$-0.754379\pi$
$263$	− 366.786i	− 1.39463i	0.716767	−	0.697313i	$-0.245621\pi$
$264$	0	0
$265$	84.9771	0.320668
$266$	0	0
$267$	− 41.7128i	− 0.156228i
$268$	0	0
$269$	−121.585	−0.451990	−0.225995	−	0.974128i	$-0.572563\pi$
$269$	−121.585	−0.451990	−0.225995	+	0.974128i	$0.572563\pi$
$270$	0	0
$271$	83.5517i	0.308309i	0.988047	+	0.154154i	$0.0492653\pi$
$271$	83.5517i	0.308309i	−0.988047	+	0.154154i	$0.950735\pi$
$272$	0	0
$273$	20.8701	0.0764473
$274$	0	0
$275$	− 1.24029i	− 0.00451013i
$276$	0	0
$277$	−295.425	−1.06652	−0.533258	−	0.845953i	$-0.679033\pi$
$277$	−295.425	−1.06652	−0.533258	+	0.845953i	$0.679033\pi$
$278$	0	0
$279$	107.304i	0.384603i
$280$	0	0
$281$	−388.170	−1.38139	−0.690694	−	0.723147i	$-0.742695\pi$
$281$	−388.170	−1.38139	−0.690694	+	0.723147i	$0.742695\pi$
$282$	0	0
$283$	214.538i	0.758083i	0.925380	+	0.379042i	$0.123746\pi$
$283$	214.538i	0.758083i	−0.925380	+	0.379042i	$0.876254\pi$
$284$	0	0
$285$	253.186	0.888374
$286$	0	0
$287$	11.9809i	0.0417452i
$288$	0	0
$289$	−288.124	−0.996967
$290$	0	0
$291$	237.285i	0.815411i
$292$	0	0
$293$	173.110	0.590819	0.295409	−	0.955371i	$-0.404544\pi$
$293$	173.110	0.590819	0.295409	+	0.955371i	$0.404544\pi$
$294$	0	0
$295$	− 329.379i	− 1.11654i
$296$	0	0
$297$	−44.9517	−0.151353
$298$	0	0
$299$	120.229i	0.402105i
$300$	0	0
$301$	−229.999	−0.764118
$302$	0	0
$303$	84.1297i	0.277656i
$304$	0	0
$305$	572.130	1.87584
$306$	0	0
$307$	− 353.625i	− 1.15187i	−0.817494	−	0.575937i	$-0.804637\pi$
$307$	− 353.625i	− 1.15187i	0.817494	−	0.575937i	$-0.195363\pi$
$308$	0	0
$309$	35.8707	0.116086
$310$	0	0
$311$	0.0737574i	0 0.000237162i	1.00000		0.000118581i	$3.77455e-5\pi$
$311$	0.0737574i	0 0.000237162i	−1.00000		0.000118581i	$0.999962\pi$
$312$	0	0
$313$	262.790	0.839585	0.419793	−	0.907620i	$-0.362103\pi$
$313$	262.790	0.839585	0.419793	+	0.907620i	$0.362103\pi$
$314$	0	0
$315$	49.9845i	0.158681i
$316$	0	0
$317$	363.488	1.14665	0.573326	−	0.819328i	$-0.305653\pi$
$317$	363.488	1.14665	0.573326	+	0.819328i	$0.305653\pi$
$318$	0	0
$319$	− 1.93725i	− 0.00607290i
$320$	0	0
$321$	−138.763	−0.432284
$322$	0	0
$323$	− 27.4491i	− 0.0849818i
$324$	0	0
$325$	0.516926	0.00159054
$326$	0	0
$327$	− 219.121i	− 0.670096i
$328$	0	0
$329$	19.6271	0.0596567
$330$	0	0
$331$	− 534.064i	− 1.61349i	−0.590902	−	0.806743i	$-0.701228\pi$
$331$	− 534.064i	− 1.61349i	0.590902	−	0.806743i	$-0.298772\pi$
$332$	0	0
$333$	−67.7889	−0.203570
$334$	0	0
$335$	− 248.243i	− 0.741023i
$336$	0	0
$337$	537.056	1.59364	0.796819	−	0.604218i	$-0.206514\pi$
$337$	537.056	1.59364	0.796819	+	0.604218i	$0.206514\pi$
$338$	0	0
$339$	− 350.914i	− 1.03514i
$340$	0	0
$341$	309.428	0.907415
$342$	0	0
$343$	− 290.182i	− 0.846013i
$344$	0	0
$345$	−287.952	−0.834644
$346$	0	0
$347$	− 349.707i	− 1.00780i	−0.863762	−	0.503900i	$-0.831898\pi$
$347$	− 349.707i	− 1.00780i	0.863762	−	0.503900i	$-0.168102\pi$
$348$	0	0
$349$	−139.999	−0.401144	−0.200572	−	0.979679i	$-0.564280\pi$
$349$	−139.999	−0.401144	−0.200572	+	0.979679i	$0.564280\pi$
$350$	0	0
$351$	− 18.7350i	− 0.0533761i
$352$	0	0
$353$	−341.896	−0.968543	−0.484272	−	0.874918i	$-0.660915\pi$
$353$	−341.896	−0.968543	−0.484272	+	0.874918i	$0.660915\pi$
$354$	0	0
$355$	− 391.309i	− 1.10228i
$356$	0	0
$357$	5.41905	0.0151794
$358$	0	0
$359$	534.175i	1.48795i	0.668206	+	0.743976i	$0.267062\pi$
$359$	534.175i	1.48795i	−0.668206	+	0.743976i	$0.732938\pi$
$360$	0	0
$361$	−498.642	−1.38128
$362$	0	0
$363$	− 79.9528i	− 0.220256i
$364$	0	0
$365$	−77.7911	−0.213126
$366$	0	0
$367$	453.584i	1.23592i	0.786208	+	0.617962i	$0.212042\pi$
$367$	453.584i	1.23592i	−0.786208	+	0.617962i	$0.787958\pi$
$368$	0	0
$369$	10.7552	0.0291468
$370$	0	0
$371$	− 56.9605i	− 0.153532i
$372$	0	0
$373$	−341.544	−0.915668	−0.457834	−	0.889038i	$-0.651375\pi$
$373$	−341.544	−0.915668	−0.457834	+	0.889038i	$0.651375\pi$
$374$	0	0
$375$	217.123i	0.578994i
$376$	0	0
$377$	0.807409	0.00214167
$378$	0	0
$379$	36.6412i	0.0966787i	0.998831	+	0.0483393i	$0.0153929\pi$
$379$	36.6412i	0.0966787i	−0.998831	+	0.0483393i	$0.984607\pi$
$380$	0	0
$381$	−93.2384	−0.244720
$382$	0	0
$383$	166.098i	0.433675i	0.976208	+	0.216838i	$0.0695742\pi$
$383$	166.098i	0.433675i	−0.976208	+	0.216838i	$0.930426\pi$
$384$	0	0
$385$	144.138	0.374384
$386$	0	0
$387$	206.469i	0.533512i
$388$	0	0
$389$	408.616	1.05043	0.525213	−	0.850971i	$-0.323986\pi$
$389$	408.616	1.05043	0.525213	+	0.850971i	$0.323986\pi$
$390$	0	0
$391$	31.2182i	0.0798420i
$392$	0	0
$393$	442.402	1.12570
$394$	0	0
$395$	− 685.550i	− 1.73557i
$396$	0	0
$397$	−476.592	−1.20048	−0.600242	−	0.799819i	$-0.704929\pi$
$397$	−476.592	−1.20048	−0.600242	+	0.799819i	$0.704929\pi$
$398$	0	0
$399$	− 169.712i	− 0.425343i
$400$	0	0
$401$	134.833	0.336243	0.168121	−	0.985766i	$-0.446230\pi$
$401$	134.833	0.336243	0.168121	+	0.985766i	$0.446230\pi$
$402$	0	0
$403$	128.964i	0.320009i
$404$	0	0
$405$	44.8708	0.110792
$406$	0	0
$407$	195.480i	0.480294i
$408$	0	0
$409$	−640.526	−1.56608	−0.783039	−	0.621972i	$-0.786332\pi$
$409$	−640.526	−1.56608	−0.783039	+	0.621972i	$0.786332\pi$
$410$	0	0
$411$	− 81.9775i	− 0.199459i
$412$	0	0
$413$	−220.784	−0.534585
$414$	0	0
$415$	101.445i	0.244446i
$416$	0	0
$417$	−262.702	−0.629981
$418$	0	0
$419$	128.427i	0.306509i	0.988187	+	0.153255i	$0.0489755\pi$
$419$	128.427i	0.306509i	−0.988187	+	0.153255i	$0.951025\pi$
$420$	0	0
$421$	−579.481	−1.37644	−0.688220	−	0.725502i	$-0.741608\pi$
$421$	−579.481	−1.37644	−0.688220	+	0.725502i	$0.741608\pi$
$422$	0	0
$423$	− 17.6191i	− 0.0416527i
$424$	0	0
$425$	0.134223	0.000315819 0
$426$	0	0
$427$	− 383.501i	− 0.898129i
$428$	0	0
$429$	−54.0253	−0.125933
$430$	0	0
$431$	− 358.772i	− 0.832417i	−0.909269	−	0.416208i	$-0.863359\pi$
$431$	− 358.772i	− 0.832417i	0.909269	−	0.416208i	$-0.136641\pi$
$432$	0	0
$433$	−406.338	−0.938424	−0.469212	−	0.883085i	$-0.655462\pi$
$433$	−406.338	−0.938424	−0.469212	+	0.883085i	$0.655462\pi$
$434$	0	0
$435$	1.93376i	0.00444544i
$436$	0	0
$437$	977.681	2.23726
$438$	0	0
$439$	− 573.804i	− 1.30707i	−0.756896	−	0.653535i	$-0.773285\pi$
$439$	− 573.804i	− 1.30707i	0.756896	−	0.653535i	$-0.226715\pi$
$440$	0	0
$441$	−113.495	−0.257359
$442$	0	0
$443$	96.6497i	0.218171i	0.994032	+	0.109085i	$0.0347922\pi$
$443$	96.6497i	0.218171i	−0.994032	+	0.109085i	$0.965208\pi$
$444$	0	0
$445$	120.069	0.269817
$446$	0	0
$447$	445.216i	0.996009i
$448$	0	0
$449$	−11.9281	−0.0265660	−0.0132830	−	0.999912i	$-0.504228\pi$
$449$	−11.9281	−0.0265660	−0.0132830	+	0.999912i	$0.504228\pi$
$450$	0	0
$451$	− 31.0142i	− 0.0687677i
$452$	0	0
$453$	−128.189	−0.282979
$454$	0	0
$455$	60.0739i	0.132030i
$456$	0	0
$457$	620.161	1.35703	0.678514	−	0.734588i	$-0.262624\pi$
$457$	620.161	1.35703	0.678514	+	0.734588i	$0.262624\pi$
$458$	0	0
$459$	− 4.86465i	− 0.0105984i
$460$	0	0
$461$	−478.720	−1.03844	−0.519220	−	0.854641i	$-0.673777\pi$
$461$	−478.720	−1.03844	−0.519220	+	0.854641i	$0.673777\pi$
$462$	0	0
$463$	59.7601i	0.129072i	0.997915	+	0.0645358i	$0.0205567\pi$
$463$	59.7601i	0.129072i	−0.997915	+	0.0645358i	$0.979443\pi$
$464$	0	0
$465$	−308.871	−0.664239
$466$	0	0
$467$	658.094i	1.40919i	0.709608	+	0.704597i	$0.248872\pi$
$467$	658.094i	1.40919i	−0.709608	+	0.704597i	$0.751128\pi$
$468$	0	0
$469$	−166.398	−0.354793
$470$	0	0
$471$	149.478i	0.317364i
$472$	0	0
$473$	595.386	1.25874
$474$	0	0
$475$	− 4.20354i	− 0.00884957i
$476$	0	0
$477$	−51.1331	−0.107197
$478$	0	0
$479$	875.844i	1.82848i	0.405169	+	0.914242i	$0.367213\pi$
$479$	875.844i	1.82848i	−0.405169	+	0.914242i	$0.632787\pi$
$480$	0	0
$481$	−81.4721	−0.169381
$482$	0	0
$483$	193.015i	0.399618i
$484$	0	0
$485$	−683.014	−1.40828
$486$	0	0
$487$	− 138.229i	− 0.283838i	−0.989878	−	0.141919i	$-0.954673\pi$
$487$	− 138.229i	− 0.283838i	0.989878	−	0.141919i	$-0.0453273\pi$
$488$	0	0
$489$	215.139	0.439958
$490$	0	0
$491$	515.656i	1.05022i	0.851035	+	0.525108i	$0.175975\pi$
$491$	515.656i	1.05022i	−0.851035	+	0.525108i	$0.824025\pi$
$492$	0	0
$493$	0.209648	0.000425250 0
$494$	0	0
$495$	− 129.392i	− 0.261398i
$496$	0	0
$497$	−262.296	−0.527759
$498$	0	0
$499$	− 238.135i	− 0.477224i	−0.971115	−	0.238612i	$-0.923308\pi$
$499$	− 238.135i	− 0.477224i	0.971115	−	0.238612i	$-0.0766924\pi$
$500$	0	0
$501$	291.598	0.582032
$502$	0	0
$503$	− 702.621i	− 1.39686i	−0.715678	−	0.698430i	$-0.753882\pi$
$503$	− 702.621i	− 1.39686i	0.715678	−	0.698430i	$-0.246118\pi$
$504$	0	0
$505$	−242.164	−0.479533
$506$	0	0
$507$	− 22.5167i	− 0.0444116i
$508$	0	0
$509$	587.488	1.15420	0.577100	−	0.816673i	$-0.304184\pi$
$509$	587.488	1.15420	0.577100	+	0.816673i	$0.304184\pi$
$510$	0	0
$511$	52.1437i	0.102042i
$512$	0	0
$513$	−152.349	−0.296977
$514$	0	0
$515$	103.253i	0.200490i
$516$	0	0
$517$	−50.8074	−0.0982735
$518$	0	0
$519$	− 559.870i	− 1.07875i
$520$	0	0
$521$	724.631	1.39085	0.695423	−	0.718600i	$-0.255217\pi$
$521$	724.631	1.39085	0.695423	+	0.718600i	$0.255217\pi$
$522$	0	0
$523$	892.068i	1.70568i	0.522176	+	0.852838i	$0.325121\pi$
$523$	892.068i	1.70568i	−0.522176	+	0.852838i	$0.674879\pi$
$524$	0	0
$525$	0.829870	0.00158071
$526$	0	0
$527$	33.4862i	0.0635411i
$528$	0	0
$529$	−582.929	−1.10195
$530$	0	0
$531$	198.196i	0.373251i
$532$	0	0
$533$	12.9261	0.0242516
$534$	0	0
$535$	− 399.424i	− 0.746588i
$536$	0	0
$537$	185.141	0.344769
$538$	0	0
$539$	327.281i	0.607201i
$540$	0	0
$541$	237.144	0.438343	0.219172	−	0.975686i	$-0.429665\pi$
$541$	237.144	0.438343	0.219172	+	0.975686i	$0.429665\pi$
$542$	0	0
$543$	85.8751i	0.158149i
$544$	0	0
$545$	630.733	1.15731
$546$	0	0
$547$	855.557i	1.56409i	0.623222	+	0.782045i	$0.285823\pi$
$547$	855.557i	1.56409i	−0.623222	+	0.782045i	$0.714177\pi$
$548$	0	0
$549$	−344.267	−0.627080
$550$	0	0
$551$	− 6.56569i	− 0.0119160i
$552$	0	0
$553$	−459.526	−0.830970
$554$	0	0
$555$	− 195.128i	− 0.351581i
$556$	0	0
$557$	417.936	0.750334	0.375167	−	0.926957i	$-0.377585\pi$
$557$	417.936	0.750334	0.375167	+	0.926957i	$0.377585\pi$
$558$	0	0
$559$	248.145i	0.443909i
$560$	0	0
$561$	−14.0280	−0.0250053
$562$	0	0
$563$	623.197i	1.10692i	0.832875	+	0.553461i	$0.186693\pi$
$563$	623.197i	1.10692i	−0.832875	+	0.553461i	$0.813307\pi$
$564$	0	0
$565$	1010.09	1.78777
$566$	0	0
$567$	− 30.0770i	− 0.0530459i
$568$	0	0
$569$	928.386	1.63161	0.815805	−	0.578328i	$-0.196294\pi$
$569$	928.386	1.63161	0.815805	+	0.578328i	$0.196294\pi$
$570$	0	0
$571$	668.041i	1.16995i	0.811052	+	0.584974i	$0.198895\pi$
$571$	668.041i	1.16995i	−0.811052	+	0.584974i	$0.801105\pi$
$572$	0	0
$573$	−219.554	−0.383166
$574$	0	0
$575$	4.78074i	0.00831434i
$576$	0	0
$577$	−396.211	−0.686675	−0.343337	−	0.939212i	$-0.611557\pi$
$577$	−396.211	−0.686675	−0.343337	+	0.939212i	$0.611557\pi$
$578$	0	0
$579$	− 227.955i	− 0.393704i
$580$	0	0
$581$	67.9991	0.117038
$582$	0	0
$583$	147.450i	0.252916i
$584$	0	0
$585$	53.9280	0.0921846
$586$	0	0
$587$	557.230i	0.949285i	0.880179	+	0.474642i	$0.157423\pi$
$587$	557.230i	0.949285i	−0.880179	+	0.474642i	$0.842577\pi$
$588$	0	0
$589$	1048.71	1.78049
$590$	0	0
$591$	− 194.565i	− 0.329213i
$592$	0	0
$593$	439.125	0.740515	0.370257	−	0.928929i	$-0.379269\pi$
$593$	439.125	0.740515	0.370257	+	0.928929i	$0.379269\pi$
$594$	0	0
$595$	15.5985i	0.0262160i
$596$	0	0
$597$	421.368	0.705809
$598$	0	0
$599$	26.3814i	0.0440424i	0.999758	+	0.0220212i	$0.00701013\pi$
$599$	26.3814i	0.0440424i	−0.999758	+	0.0220212i	$0.992990\pi$
$600$	0	0
$601$	−34.6958	−0.0577300	−0.0288650	−	0.999583i	$-0.509189\pi$
$601$	−34.6958	−0.0577300	−0.0288650	+	0.999583i	$0.509189\pi$
$602$	0	0
$603$	149.375i	0.247719i
$604$	0	0
$605$	230.141	0.380399
$606$	0	0
$607$	− 2.23322i	− 0.00367912i	−0.999998	−	0.00183956i	$-0.999414\pi$
$607$	− 2.23322i	− 0.00367912i	0.999998	−	0.00183956i	$-0.000585550\pi$
$608$	0	0
$609$	1.29621	0.00212842
$610$	0	0
$611$	− 21.1755i	− 0.0346572i
$612$	0	0
$613$	784.667	1.28004	0.640022	−	0.768357i	$-0.278925\pi$
$613$	784.667	1.28004	0.640022	+	0.768357i	$0.278925\pi$
$614$	0	0
$615$	30.9584i	0.0503388i
$616$	0	0
$617$	720.280	1.16739	0.583695	−	0.811973i	$-0.301606\pi$
$617$	720.280	1.16739	0.583695	+	0.811973i	$0.301606\pi$
$618$	0	0
$619$	− 856.400i	− 1.38352i	−0.722127	−	0.691761i	$-0.756835\pi$
$619$	− 856.400i	− 1.38352i	0.722127	−	0.691761i	$-0.243165\pi$
$620$	0	0
$621$	173.269	0.279016
$622$	0	0
$623$	− 80.4825i	− 0.129185i
$624$	0	0
$625$	−621.395	−0.994232
$626$	0	0
$627$	439.323i	0.700675i
$628$	0	0
$629$	−21.1547	−0.0336323
$630$	0	0
$631$	315.588i	0.500139i	0.968228	+	0.250070i	$0.0804535\pi$
$631$	315.588i	0.500139i	−0.968228	+	0.250070i	$0.919547\pi$
$632$	0	0
$633$	558.140	0.881738
$634$	0	0
$635$	− 268.383i	− 0.422651i
$636$	0	0
$637$	−136.404	−0.214135
$638$	0	0
$639$	235.462i	0.368485i
$640$	0	0
$641$	1033.39	1.61215	0.806073	−	0.591816i	$-0.201589\pi$
$641$	1033.39	1.61215	0.806073	+	0.591816i	$0.201589\pi$
$642$	0	0
$643$	316.889i	0.492828i	0.969165	+	0.246414i	$0.0792523\pi$
$643$	316.889i	0.492828i	−0.969165	+	0.246414i	$0.920748\pi$
$644$	0	0
$645$	−594.314	−0.921417
$646$	0	0
$647$	458.255i	0.708277i	0.935193	+	0.354138i	$0.115226\pi$
$647$	458.255i	0.708277i	−0.935193	+	0.354138i	$0.884774\pi$
$648$	0	0
$649$	571.530	0.880631
$650$	0	0
$651$	207.037i	0.318030i
$652$	0	0
$653$	−12.9221	−0.0197889	−0.00989444	−	0.999951i	$-0.503150\pi$
$653$	−12.9221	−0.0197889	−0.00989444	+	0.999951i	$0.503150\pi$
$654$	0	0
$655$	1273.44i	1.94418i
$656$	0	0
$657$	46.8091	0.0712467
$658$	0	0
$659$	− 345.890i	− 0.524871i	−0.964949	−	0.262436i	$-0.915474\pi$
$659$	− 345.890i	− 0.524871i	0.964949	−	0.262436i	$-0.0845257\pi$
$660$	0	0
$661$	−1018.12	−1.54026	−0.770132	−	0.637884i	$-0.779810\pi$
$661$	−1018.12	−1.54026	−0.770132	+	0.637884i	$0.779810\pi$
$662$	0	0
$663$	− 5.84658i	− 0.00881838i
$664$	0	0
$665$	488.509	0.734600
$666$	0	0
$667$	7.46725i	0.0111953i
$668$	0	0
$669$	−342.864	−0.512503
$670$	0	0
$671$	992.747i	1.47950i
$672$	0	0
$673$	−970.343	−1.44182	−0.720909	−	0.693030i	$-0.756275\pi$
$673$	−970.343	−1.44182	−0.720909	+	0.693030i	$0.756275\pi$
$674$	0	0
$675$	− 0.744970i	− 0.00110366i
$676$	0	0
$677$	647.080	0.955804	0.477902	−	0.878413i	$-0.341397\pi$
$677$	647.080	0.955804	0.477902	+	0.878413i	$0.341397\pi$
$678$	0	0
$679$	457.827i	0.674267i
$680$	0	0
$681$	−562.634	−0.826187
$682$	0	0
$683$	− 596.337i	− 0.873114i	−0.899677	−	0.436557i	$-0.856198\pi$
$683$	− 596.337i	− 0.873114i	0.899677	−	0.436557i	$-0.143802\pi$
$684$	0	0
$685$	235.969	0.344481
$686$	0	0
$687$	643.878i	0.937231i
$688$	0	0
$689$	−61.4543	−0.0891935
$690$	0	0
$691$	876.399i	1.26831i	0.773208	+	0.634153i	$0.218651\pi$
$691$	876.399i	1.26831i	−0.773208	+	0.634153i	$0.781349\pi$
$692$	0	0
$693$	−86.7318	−0.125154
$694$	0	0
$695$	− 756.178i	− 1.08803i
$696$	0	0
$697$	3.35634	0.00481541
$698$	0	0
$699$	− 364.485i	− 0.521438i
$700$	0	0
$701$	−56.3928	−0.0804462	−0.0402231	−	0.999191i	$-0.512807\pi$
$701$	−56.3928	−0.0804462	−0.0402231	+	0.999191i	$0.512807\pi$
$702$	0	0
$703$	662.515i	0.942411i
$704$	0	0
$705$	50.7159	0.0719375
$706$	0	0
$707$	162.323i	0.229595i
$708$	0	0
$709$	−1151.81	−1.62456	−0.812280	−	0.583267i	$-0.801774\pi$
$709$	−1151.81	−1.62456	−0.812280	+	0.583267i	$0.801774\pi$
$710$	0	0
$711$	412.514i	0.580189i
$712$	0	0
$713$	−1192.71	−1.67280
$714$	0	0
$715$	− 155.510i	− 0.217496i
$716$	0	0
$717$	−549.588	−0.766510
$718$	0	0
$719$	876.648i	1.21926i	0.792686	+	0.609630i	$0.208682\pi$
$719$	876.648i	1.21926i	−0.792686	+	0.609630i	$0.791318\pi$
$720$	0	0
$721$	69.2105	0.0959924
$722$	0	0
$723$	− 176.169i	− 0.243664i
$724$	0	0
$725$	0.0321055	4.42834e−5 0
$726$	0	0
$727$	− 484.847i	− 0.666915i	−0.942765	−	0.333457i	$-0.891785\pi$
$727$	− 484.847i	− 0.666915i	0.942765	−	0.333457i	$-0.108215\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	64.4323i	0.0881427i
$732$	0	0
$733$	−243.342	−0.331980	−0.165990	−	0.986127i	$-0.553082\pi$
$733$	−243.342	−0.331980	−0.165990	+	0.986127i	$0.553082\pi$
$734$	0	0
$735$	− 326.692i	− 0.444479i
$736$	0	0
$737$	430.745	0.584457
$738$	0	0
$739$	697.193i	0.943427i	0.881752	+	0.471714i	$0.156364\pi$
$739$	697.193i	0.943427i	−0.881752	+	0.471714i	$0.843636\pi$
$740$	0	0
$741$	−183.101	−0.247100
$742$	0	0
$743$	− 637.384i	− 0.857852i	−0.903340	−	0.428926i	$-0.858892\pi$
$743$	− 637.384i	− 0.857852i	0.903340	−	0.428926i	$-0.141108\pi$
$744$	0	0
$745$	−1281.54	−1.72019
$746$	0	0
$747$	− 61.0424i	− 0.0817168i
$748$	0	0
$749$	−267.736	−0.357457
$750$	0	0
$751$	576.281i	0.767352i	0.923468	+	0.383676i	$0.125342\pi$
$751$	576.281i	0.767352i	−0.923468	+	0.383676i	$0.874658\pi$
$752$	0	0
$753$	−694.685	−0.922557
$754$	0	0
$755$	− 368.988i	− 0.488726i
$756$	0	0
$757$	−457.887	−0.604870	−0.302435	−	0.953170i	$-0.597800\pi$
$757$	−457.887	−0.604870	−0.302435	+	0.953170i	$0.597800\pi$
$758$	0	0
$759$	− 499.648i	− 0.658297i
$760$	0	0
$761$	−531.702	−0.698688	−0.349344	−	0.936995i	$-0.613596\pi$
$761$	−531.702	−0.698688	−0.349344	+	0.936995i	$0.613596\pi$
$762$	0	0
$763$	− 422.782i	− 0.554105i
$764$	0	0
$765$	14.0027	0.0183042
$766$	0	0
$767$	238.202i	0.310564i
$768$	0	0
$769$	458.481	0.596205	0.298102	−	0.954534i	$-0.403646\pi$
$769$	458.481	0.596205	0.298102	+	0.954534i	$0.403646\pi$
$770$	0	0
$771$	3.20072i	0.00415139i
$772$	0	0
$773$	−481.016	−0.622272	−0.311136	−	0.950365i	$-0.600709\pi$
$773$	−481.016	−0.622272	−0.311136	+	0.950365i	$0.600709\pi$
$774$	0	0
$775$	5.12805i	0.00661684i
$776$	0	0
$777$	−130.795	−0.168333
$778$	0	0
$779$	− 105.113i	− 0.134933i
$780$	0	0
$781$	678.991	0.869386
$782$	0	0
$783$	− 1.16360i	− 0.00148608i
$784$	0	0
$785$	−430.267	−0.548111
$786$	0	0
$787$	− 375.007i	− 0.476502i	−0.971204	−	0.238251i	$-0.923426\pi$
$787$	− 375.007i	− 0.476502i	0.971204	−	0.238251i	$-0.0765740\pi$
$788$	0	0
$789$	−635.293	−0.805187
$790$	0	0
$791$	− 677.068i	− 0.855965i
$792$	0	0
$793$	−413.757	−0.521762
$794$	0	0
$795$	− 147.185i	− 0.185138i
$796$	0	0
$797$	609.620	0.764893	0.382447	−	0.923978i	$-0.375082\pi$
$797$	609.620	0.764893	0.382447	+	0.923978i	$0.375082\pi$
$798$	0	0
$799$	− 5.49835i	− 0.00688154i
$800$	0	0
$801$	−72.2487	−0.0901982
$802$	0	0
$803$	− 134.981i	− 0.168096i
$804$	0	0
$805$	−555.587	−0.690171
$806$	0	0
$807$	210.592i	0.260957i
$808$	0	0
$809$	−422.519	−0.522273	−0.261136	−	0.965302i	$-0.584097\pi$
$809$	−422.519	−0.522273	−0.261136	+	0.965302i	$0.584097\pi$
$810$	0	0
$811$	− 266.225i	− 0.328267i	−0.986438	−	0.164134i	$-0.947517\pi$
$811$	− 266.225i	− 0.328267i	0.986438	−	0.164134i	$-0.0524829\pi$
$812$	0	0
$813$	144.716	0.178002
$814$	0	0
$815$	619.270i	0.759841i
$816$	0	0
$817$	2017.87	2.46985
$818$	0	0
$819$	− 36.1481i	− 0.0441369i
$820$	0	0
$821$	1138.40	1.38660	0.693300	−	0.720649i	$-0.256156\pi$
$821$	1138.40	1.38660	0.693300	+	0.720649i	$0.256156\pi$
$822$	0	0
$823$	− 708.475i	− 0.860845i	−0.902628	−	0.430422i	$-0.858365\pi$
$823$	− 708.475i	− 0.860845i	0.902628	−	0.430422i	$-0.141635\pi$
$824$	0	0
$825$	−2.14824	−0.00260392
$826$	0	0
$827$	1378.55i	1.66693i	0.552573	+	0.833464i	$0.313646\pi$
$827$	1378.55i	1.66693i	−0.552573	+	0.833464i	$0.686354\pi$
$828$	0	0
$829$	−46.8483	−0.0565119	−0.0282559	−	0.999601i	$-0.508995\pi$
$829$	−46.8483	−0.0565119	−0.0282559	+	0.999601i	$0.508995\pi$
$830$	0	0
$831$	511.691i	0.615754i
$832$	0	0
$833$	−35.4182	−0.0425188
$834$	0	0
$835$	839.354i	1.00521i
$836$	0	0
$837$	185.856	0.222051
$838$	0	0
$839$	1278.61i	1.52396i	0.647598	+	0.761982i	$0.275774\pi$
$839$	1278.61i	1.52396i	−0.647598	+	0.761982i	$0.724226\pi$
$840$	0	0
$841$	−840.950	−0.999940
$842$	0	0
$843$	672.331i	0.797545i
$844$	0	0
$845$	64.8134	0.0767022
$846$	0	0
$847$	− 154.264i	− 0.182130i
$848$	0	0
$849$	371.590	0.437680
$850$	0	0
$851$	− 753.487i	− 0.885413i
$852$	0	0
$853$	−57.6539	−0.0675895	−0.0337948	−	0.999429i	$-0.510759\pi$
$853$	−57.6539	−0.0675895	−0.0337948	+	0.999429i	$0.510759\pi$
$854$	0	0
$855$	− 438.532i	− 0.512903i
$856$	0	0
$857$	−831.023	−0.969689	−0.484844	−	0.874601i	$-0.661124\pi$
$857$	−831.023	−0.969689	−0.484844	+	0.874601i	$0.661124\pi$
$858$	0	0
$859$	− 909.734i	− 1.05906i	−0.848291	−	0.529531i	$-0.822368\pi$
$859$	− 909.734i	− 1.05906i	0.848291	−	0.529531i	$-0.177632\pi$
$860$	0	0
$861$	20.7515	0.0241016
$862$	0	0
$863$	− 26.8875i	− 0.0311558i	−0.999879	−	0.0155779i	$-0.995041\pi$
$863$	− 26.8875i	− 0.0311558i	0.999879	−	0.0155779i	$-0.00495880\pi$
$864$	0	0
$865$	1611.56	1.86308
$866$	0	0
$867$	499.045i	0.575599i
$868$	0	0
$869$	1189.55	1.36887
$870$	0	0
$871$	179.526i	0.206115i
$872$	0	0
$873$	410.989	0.470778
$874$	0	0
$875$	418.926i	0.478773i
$876$	0	0
$877$	−33.1871	−0.0378416	−0.0189208	−	0.999821i	$-0.506023\pi$
$877$	−33.1871	−0.0378416	−0.0189208	+	0.999821i	$0.506023\pi$
$878$	0	0
$879$	− 299.835i	− 0.341109i
$880$	0	0
$881$	1372.98	1.55843	0.779217	−	0.626754i	$-0.215617\pi$
$881$	1372.98	1.55843	0.779217	+	0.626754i	$0.215617\pi$
$882$	0	0
$883$	− 341.308i	− 0.386532i	−0.981146	−	0.193266i	$-0.938092\pi$
$883$	− 341.308i	− 0.386532i	0.981146	−	0.193266i	$-0.0619081\pi$
$884$	0	0
$885$	−570.500	−0.644633
$886$	0	0
$887$	816.519i	0.920540i	0.887779	+	0.460270i	$0.152247\pi$
$887$	816.519i	0.920540i	−0.887779	+	0.460270i	$0.847753\pi$
$888$	0	0
$889$	−179.898	−0.202360
$890$	0	0
$891$	77.8587i	0.0873835i
$892$	0	0
$893$	−172.195	−0.192828
$894$	0	0
$895$	532.921i	0.595443i
$896$	0	0
$897$	208.243	0.232155
$898$	0	0
$899$	8.00972i	0.00890959i
$900$	0	0
$901$	−15.9570	−0.0177103
$902$	0	0
$903$	398.371i	0.441164i
$904$	0	0
$905$	−247.188	−0.273136
$906$	0	0
$907$	− 410.472i	− 0.452560i	−0.974062	−	0.226280i	$-0.927343\pi$
$907$	− 410.472i	− 0.452560i	0.974062	−	0.226280i	$-0.0726565\pi$
$908$	0	0
$909$	145.717	0.160305
$910$	0	0
$911$	− 50.0265i	− 0.0549138i	−0.999623	−	0.0274569i	$-0.991259\pi$
$911$	− 50.0265i	− 0.0549138i	0.999623	−	0.0274569i	$-0.00874090\pi$
$912$	0	0
$913$	−176.025	−0.192799
$914$	0	0
$915$	− 990.959i	− 1.08302i
$916$	0	0
$917$	853.589	0.930850
$918$	0	0
$919$	− 219.907i	− 0.239289i	−0.992817	−	0.119645i	$-0.961824\pi$
$919$	− 219.907i	− 0.239289i	0.992817	−	0.119645i	$-0.0381755\pi$
$920$	0	0
$921$	−612.497	−0.665035
$922$	0	0
$923$	282.990i	0.306598i
$924$	0	0
$925$	−3.23962	−0.00350229
$926$	0	0
$927$	− 62.1299i	− 0.0670226i
$928$	0	0
$929$	−1432.81	−1.54231	−0.771157	−	0.636645i	$-0.780322\pi$
$929$	−1432.81	−1.54231	−0.771157	+	0.636645i	$0.780322\pi$
$930$	0	0
$931$	1109.21i	1.19142i
$932$	0	0
$933$	0.127751	0.000136925 0
$934$	0	0
$935$	− 40.3790i	− 0.0431861i
$936$	0	0
$937$	208.757	0.222793	0.111397	−	0.993776i	$-0.464468\pi$
$937$	208.757	0.222793	0.111397	+	0.993776i	$0.464468\pi$
$938$	0	0
$939$	− 455.166i	− 0.484735i
$940$	0	0
$941$	−431.307	−0.458350	−0.229175	−	0.973385i	$-0.573603\pi$
$941$	−431.307	−0.458350	−0.229175	+	0.973385i	$0.573603\pi$
$942$	0	0
$943$	119.546i	0.126772i
$944$	0	0
$945$	86.5756	0.0916144
$946$	0	0
$947$	1737.46i	1.83470i	0.398077	+	0.917352i	$0.369678\pi$
$947$	1737.46i	1.83470i	−0.398077	+	0.917352i	$0.630322\pi$
$948$	0	0
$949$	56.2575	0.0592808
$950$	0	0
$951$	− 629.580i	− 0.662019i
$952$	0	0
$953$	883.649	0.927229	0.463615	−	0.886037i	$-0.346552\pi$
$953$	883.649	0.927229	0.463615	+	0.886037i	$0.346552\pi$
$954$	0	0
$955$	− 631.978i	− 0.661757i
$956$	0	0
$957$	−3.35542	−0.00350619
$958$	0	0
$959$	− 158.171i	− 0.164933i
$960$	0	0
$961$	−318.356	−0.331275
$962$	0	0
$963$	240.345i	0.249579i
$964$	0	0
$965$	656.158	0.679957
$966$	0	0
$967$	− 1605.15i	− 1.65992i	−0.557820	−	0.829962i	$-0.688362\pi$
$967$	− 1605.15i	− 1.65992i	0.557820	−	0.829962i	$-0.311638\pi$
$968$	0	0
$969$	−47.5433	−0.0490643
$970$	0	0
$971$	− 229.236i	− 0.236083i	−0.993009	−	0.118041i	$-0.962338\pi$
$971$	− 229.236i	− 0.236083i	0.993009	−	0.118041i	$-0.0376615\pi$
$972$	0	0
$973$	−506.869	−0.520934
$974$	0	0
$975$	− 0.895343i	0 0.000918300i
$976$	0	0
$977$	311.466	0.318798	0.159399	−	0.987214i	$-0.449044\pi$
$977$	311.466	0.318798	0.159399	+	0.987214i	$0.449044\pi$
$978$	0	0
$979$	208.340i	0.212809i
$980$	0	0
$981$	−379.529	−0.386880
$982$	0	0
$983$	− 379.569i	− 0.386133i	−0.981186	−	0.193067i	$-0.938157\pi$
$983$	− 379.569i	− 0.386133i	0.981186	−	0.193067i	$-0.0618433\pi$
$984$	0	0
$985$	560.048	0.568577
$986$	0	0
$987$	− 33.9951i	− 0.0344428i
$988$	0	0
$989$	−2294.95	−2.32047
$990$	0	0
$991$	900.695i	0.908875i	0.890779	+	0.454438i	$0.150160\pi$
$991$	900.695i	0.908875i	−0.890779	+	0.454438i	$0.849840\pi$
$992$	0	0
$993$	−925.026	−0.931547
$994$	0	0
$995$	1212.89i	1.21899i
$996$	0	0
$997$	875.161	0.877795	0.438897	−	0.898537i	$-0.355369\pi$
$997$	875.161	0.877795	0.438897	+	0.898537i	$0.355369\pi$
$998$	0	0
$999$	117.414i	0.117531i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.3.k.e.703.11		24
4.3	odd	2	inner	2496.3.k.e.703.12		24
8.3	odd	2		156.3.f.a.79.2	yes	24
8.5	even	2		156.3.f.a.79.1	✓	24
24.5	odd	2		468.3.f.b.235.24		24
24.11	even	2		468.3.f.b.235.23		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.3.f.a.79.1	✓	24	8.5	even	2
156.3.f.a.79.2	yes	24	8.3	odd	2
468.3.f.b.235.23		24	24.11	even	2
468.3.f.b.235.24		24	24.5	odd	2
2496.3.k.e.703.11		24	1.1	even	1	trivial
2496.3.k.e.703.12		24	4.3	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2496.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +4.98564 q^{5} -3.34189i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +4.98564 q^{5} -3.34189i q^{7} -3.00000 q^{9} +8.65097i q^{11} -3.60555 q^{13} -8.63539i q^{15} -0.936203 q^{17} +29.3196i q^{19} -5.78833 q^{21} -33.3456i q^{23} -0.143370 q^{25} +5.19615i q^{27} -0.223935 q^{29} -35.7681i q^{31} +14.9839 q^{33} -16.6615i q^{35} +22.5963 q^{37} +6.24500i q^{39} -3.58506 q^{41} -68.8231i q^{43} -14.9569 q^{45} +5.87303i q^{47} +37.8317 q^{49} +1.62155i q^{51} +17.0444 q^{53} +43.1306i q^{55} +50.7831 q^{57} -66.0654i q^{59} +114.756 q^{61} +10.0257i q^{63} -17.9760 q^{65} -49.7915i q^{67} -57.7563 q^{69} -78.4873i q^{71} -15.6030 q^{73} +0.248323i q^{75} +28.9106 q^{77} -137.505i q^{79} +9.00000 q^{81} +20.3475i q^{83} -4.66757 q^{85} +0.387867i q^{87} +24.0829 q^{89} +12.0494i q^{91} -61.9521 q^{93} +146.177i q^{95} -136.996 q^{97} -25.9529i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{9}+O(q^{10}) \) 24 * q - 72 * q^9	\( 24 q - 72 q^{9} + 104 q^{25} - 64 q^{29} + 48 q^{33} + 192 q^{37} - 248 q^{49} - 336 q^{53} - 16 q^{61} + 192 q^{69} + 112 q^{73} + 272 q^{77} + 216 q^{81} - 64 q^{85} + 96 q^{93} - 80 q^{97}+O(q^{100}) \) 24 * q - 72 * q^9 + 104 * q^25 - 64 * q^29 + 48 * q^33 + 192 * q^37 - 248 * q^49 - 336 * q^53 - 16 * q^61 + 192 * q^69 + 112 * q^73 + 272 * q^77 + 216 * q^81 - 64 * q^85 + 96 * q^93 - 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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