LMFDB - Embedded newform 7800.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

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:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.73205 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.73205 q^{7} +1.00000 q^{9} +1.73205 q^{11} -1.00000 q^{13} +2.46410 q^{17} -3.46410 q^{19} -1.73205 q^{21} -2.00000 q^{23} +1.00000 q^{27} +3.92820 q^{29} -9.19615 q^{31} +1.73205 q^{33} +0.535898 q^{37} -1.00000 q^{39} -2.53590 q^{41} -8.92820 q^{43} +6.66025 q^{47} -4.00000 q^{49} +2.46410 q^{51} +11.9282 q^{53} -3.46410 q^{57} +1.73205 q^{59} -12.4641 q^{61} -1.73205 q^{63} -4.26795 q^{67} -2.00000 q^{69} -14.3923 q^{71} +3.46410 q^{73} -3.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} -13.7321 q^{83} +3.92820 q^{87} -2.92820 q^{89} +1.73205 q^{91} -9.19615 q^{93} -4.92820 q^{97} +1.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} - 2 q^{13} - 2 q^{17} - 4 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{31} + 8 q^{37} - 2 q^{39} - 12 q^{41} - 4 q^{43} - 4 q^{47} - 8 q^{49} - 2 q^{51} + 10 q^{53} - 18 q^{61} - 12 q^{67} - 4 q^{69} - 8 q^{71} - 6 q^{77} + 28 q^{79} + 2 q^{81} - 24 q^{83} - 6 q^{87} + 8 q^{89} - 8 q^{93} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 2 * q^13 - 2 * q^17 - 4 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^31 + 8 * q^37 - 2 * q^39 - 12 * q^41 - 4 * q^43 - 4 * q^47 - 8 * q^49 - 2 * q^51 + 10 * q^53 - 18 * q^61 - 12 * q^67 - 4 * q^69 - 8 * q^71 - 6 * q^77 + 28 * q^79 + 2 * q^81 - 24 * q^83 - 6 * q^87 + 8 * q^89 - 8 * q^93 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.46410	0.597632	0.298816	−	0.954311i	$-0.403408\pi$
$17$	2.46410	0.597632	0.298816	+	0.954311i	$0.403408\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	−1.73205	−0.377964
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.92820	0.729449	0.364725	−	0.931115i	$-0.381163\pi$
$29$	3.92820	0.729449	0.364725	+	0.931115i	$0.381163\pi$
$30$	0	0
$31$	−9.19615	−1.65168	−0.825839	−	0.563906i	$-0.809298\pi$
$31$	−9.19615	−1.65168	−0.825839	+	0.563906i	$0.809298\pi$
$32$	0	0
$33$	1.73205	0.301511
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.535898	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.53590	−0.396041	−0.198020	−	0.980198i	$-0.563451\pi$
$41$	−2.53590	−0.396041	−0.198020	+	0.980198i	$0.563451\pi$
$42$	0	0
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.66025	0.971498	0.485749	−	0.874098i	$-0.338547\pi$
$47$	6.66025	0.971498	0.485749	+	0.874098i	$0.338547\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	2.46410	0.345043
$52$	0	0
$53$	11.9282	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$53$	11.9282	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	1.73205	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$60$	0	0
$61$	−12.4641	−1.59586	−0.797932	−	0.602747i	$-0.794073\pi$
$61$	−12.4641	−1.59586	−0.797932	+	0.602747i	$0.794073\pi$
$62$	0	0
$63$	−1.73205	−0.218218
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.26795	−0.521413	−0.260706	−	0.965418i	$-0.583955\pi$
$67$	−4.26795	−0.521413	−0.260706	+	0.965418i	$0.583955\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−14.3923	−1.70805	−0.854026	−	0.520230i	$-0.825846\pi$
$71$	−14.3923	−1.70805	−0.854026	+	0.520230i	$0.825846\pi$
$72$	0	0
$73$	3.46410	0.405442	0.202721	−	0.979236i	$-0.435021\pi$
$73$	3.46410	0.405442	0.202721	+	0.979236i	$0.435021\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.7321	−1.50729	−0.753644	−	0.657283i	$-0.771706\pi$
$83$	−13.7321	−1.50729	−0.753644	+	0.657283i	$0.771706\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.92820	0.421148
$88$	0	0
$89$	−2.92820	−0.310389	−0.155194	−	0.987884i	$-0.549600\pi$
$89$	−2.92820	−0.310389	−0.155194	+	0.987884i	$0.549600\pi$
$90$	0	0
$91$	1.73205	0.181568
$92$	0	0
$93$	−9.19615	−0.953597
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	0	0
$99$	1.73205	0.174078
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	15.3205	1.50957	0.754787	−	0.655970i	$-0.227740\pi$
$103$	15.3205	1.50957	0.754787	+	0.655970i	$0.227740\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.46410	0.721582	0.360791	−	0.932647i	$-0.382507\pi$
$107$	7.46410	0.721582	0.360791	+	0.932647i	$0.382507\pi$
$108$	0	0
$109$	−2.53590	−0.242895	−0.121448	−	0.992598i	$-0.538754\pi$
$109$	−2.53590	−0.242895	−0.121448	+	0.992598i	$0.538754\pi$
$110$	0	0
$111$	0.535898	0.0508652
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.26795	−0.391242
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	−2.53590	−0.228654
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.928203	−0.0823647	−0.0411824	−	0.999152i	$-0.513112\pi$
$127$	−0.928203	−0.0823647	−0.0411824	+	0.999152i	$0.513112\pi$
$128$	0	0
$129$	−8.92820	−0.786084
$130$	0	0
$131$	−21.4641	−1.87533	−0.937664	−	0.347544i	$-0.887016\pi$
$131$	−21.4641	−1.87533	−0.937664	+	0.347544i	$0.887016\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.92820	−0.250173	−0.125087	−	0.992146i	$-0.539921\pi$
$137$	−2.92820	−0.250173	−0.125087	+	0.992146i	$0.539921\pi$
$138$	0	0
$139$	−14.3923	−1.22074	−0.610370	−	0.792117i	$-0.708979\pi$
$139$	−14.3923	−1.22074	−0.610370	+	0.792117i	$0.708979\pi$
$140$	0	0
$141$	6.66025	0.560895
$142$	0	0
$143$	−1.73205	−0.144841
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.00000	−0.329914
$148$	0	0
$149$	11.8564	0.971315	0.485657	−	0.874149i	$-0.338580\pi$
$149$	11.8564	0.971315	0.485657	+	0.874149i	$0.338580\pi$
$150$	0	0
$151$	8.12436	0.661151	0.330575	−	0.943780i	$-0.392757\pi$
$151$	8.12436	0.661151	0.330575	+	0.943780i	$0.392757\pi$
$152$	0	0
$153$	2.46410	0.199211
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.39230	0.749588	0.374794	−	0.927108i	$-0.377714\pi$
$157$	9.39230	0.749588	0.374794	+	0.927108i	$0.377714\pi$
$158$	0	0
$159$	11.9282	0.945968
$160$	0	0
$161$	3.46410	0.273009
$162$	0	0
$163$	6.39230	0.500684	0.250342	−	0.968157i	$-0.419457\pi$
$163$	6.39230	0.500684	0.250342	+	0.968157i	$0.419457\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.3205	−1.34030	−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205	−1.34030	−0.670151	+	0.742225i	$0.733770\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.46410	−0.264906
$172$	0	0
$173$	−3.92820	−0.298656	−0.149328	−	0.988788i	$-0.547711\pi$
$173$	−3.92820	−0.298656	−0.149328	+	0.988788i	$0.547711\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.73205	0.130189
$178$	0	0
$179$	−0.392305	−0.0293222	−0.0146611	−	0.999893i	$-0.504667\pi$
$179$	−0.392305	−0.0293222	−0.0146611	+	0.999893i	$0.504667\pi$
$180$	0	0
$181$	−3.53590	−0.262821	−0.131411	−	0.991328i	$-0.541951\pi$
$181$	−3.53590	−0.262821	−0.131411	+	0.991328i	$0.541951\pi$
$182$	0	0
$183$	−12.4641	−0.921373
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.26795	0.312103
$188$	0	0
$189$	−1.73205	−0.125988
$190$	0	0
$191$	14.3923	1.04139	0.520695	−	0.853743i	$-0.325673\pi$
$191$	14.3923	1.04139	0.520695	+	0.853743i	$0.325673\pi$
$192$	0	0
$193$	−1.46410	−0.105388	−0.0526942	−	0.998611i	$-0.516781\pi$
$193$	−1.46410	−0.105388	−0.0526942	+	0.998611i	$0.516781\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.39230	−0.312939	−0.156469	−	0.987683i	$-0.550011\pi$
$197$	−4.39230	−0.312939	−0.156469	+	0.987683i	$0.550011\pi$
$198$	0	0
$199$	21.3205	1.51137	0.755685	−	0.654935i	$-0.227304\pi$
$199$	21.3205	1.51137	0.755685	+	0.654935i	$0.227304\pi$
$200$	0	0
$201$	−4.26795	−0.301038
$202$	0	0
$203$	−6.80385	−0.477536
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	0	0
$213$	−14.3923	−0.986144
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.9282	1.08128
$218$	0	0
$219$	3.46410	0.234082
$220$	0	0
$221$	−2.46410	−0.165753
$222$	0	0
$223$	−28.2487	−1.89167	−0.945837	−	0.324642i	$-0.894756\pi$
$223$	−28.2487	−1.89167	−0.945837	+	0.324642i	$0.894756\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.26795	0.150529	0.0752645	−	0.997164i	$-0.476020\pi$
$227$	2.26795	0.150529	0.0752645	+	0.997164i	$0.476020\pi$
$228$	0	0
$229$	−18.7846	−1.24132	−0.620661	−	0.784079i	$-0.713136\pi$
$229$	−18.7846	−1.24132	−0.620661	+	0.784079i	$0.713136\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.0000	0.909398
$238$	0	0
$239$	3.19615	0.206742	0.103371	−	0.994643i	$-0.467037\pi$
$239$	3.19615	0.206742	0.103371	+	0.994643i	$0.467037\pi$
$240$	0	0
$241$	−20.3923	−1.31358	−0.656792	−	0.754072i	$-0.728087\pi$
$241$	−20.3923	−1.31358	−0.656792	+	0.754072i	$0.728087\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.46410	0.220416
$248$	0	0
$249$	−13.7321	−0.870233
$250$	0	0
$251$	4.39230	0.277240	0.138620	−	0.990346i	$-0.455733\pi$
$251$	4.39230	0.277240	0.138620	+	0.990346i	$0.455733\pi$
$252$	0	0
$253$	−3.46410	−0.217786
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2487	1.20070	0.600351	−	0.799737i	$-0.295028\pi$
$257$	19.2487	1.20070	0.600351	+	0.799737i	$0.295028\pi$
$258$	0	0
$259$	−0.928203	−0.0576757
$260$	0	0
$261$	3.92820	0.243150
$262$	0	0
$263$	−23.3205	−1.43800	−0.719002	−	0.695008i	$-0.755401\pi$
$263$	−23.3205	−1.43800	−0.719002	+	0.695008i	$0.755401\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.92820	−0.179203
$268$	0	0
$269$	−19.0000	−1.15845	−0.579225	−	0.815168i	$-0.696645\pi$
$269$	−19.0000	−1.15845	−0.579225	+	0.815168i	$0.696645\pi$
$270$	0	0
$271$	−21.7321	−1.32013	−0.660064	−	0.751209i	$-0.729471\pi$
$271$	−21.7321	−1.32013	−0.660064	+	0.751209i	$0.729471\pi$
$272$	0	0
$273$	1.73205	0.104828
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.9282	1.25745	0.628727	−	0.777626i	$-0.283576\pi$
$277$	20.9282	1.25745	0.628727	+	0.777626i	$0.283576\pi$
$278$	0	0
$279$	−9.19615	−0.550559
$280$	0	0
$281$	12.9282	0.771232	0.385616	−	0.922659i	$-0.373989\pi$
$281$	12.9282	0.771232	0.385616	+	0.922659i	$0.373989\pi$
$282$	0	0
$283$	−9.32051	−0.554047	−0.277023	−	0.960863i	$-0.589348\pi$
$283$	−9.32051	−0.554047	−0.277023	+	0.960863i	$0.589348\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.39230	0.259270
$288$	0	0
$289$	−10.9282	−0.642835
$290$	0	0
$291$	−4.92820	−0.288896
$292$	0	0
$293$	15.8564	0.926341	0.463171	−	0.886269i	$-0.346712\pi$
$293$	15.8564	0.926341	0.463171	+	0.886269i	$0.346712\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.73205	0.100504
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	15.4641	0.891336
$302$	0	0
$303$	3.00000	0.172345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4.53590	−0.258877	−0.129439	−	0.991587i	$-0.541318\pi$
$307$	−4.53590	−0.258877	−0.129439	+	0.991587i	$0.541318\pi$
$308$	0	0
$309$	15.3205	0.871553
$310$	0	0
$311$	−7.46410	−0.423250	−0.211625	−	0.977351i	$-0.567876\pi$
$311$	−7.46410	−0.423250	−0.211625	+	0.977351i	$0.567876\pi$
$312$	0	0
$313$	1.92820	0.108988	0.0544942	−	0.998514i	$-0.482645\pi$
$313$	1.92820	0.108988	0.0544942	+	0.998514i	$0.482645\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.5359	0.704086	0.352043	−	0.935984i	$-0.385487\pi$
$317$	12.5359	0.704086	0.352043	+	0.935984i	$0.385487\pi$
$318$	0	0
$319$	6.80385	0.380942
$320$	0	0
$321$	7.46410	0.416606
$322$	0	0
$323$	−8.53590	−0.474950
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.53590	−0.140236
$328$	0	0
$329$	−11.5359	−0.635995
$330$	0	0
$331$	−4.53590	−0.249316	−0.124658	−	0.992200i	$-0.539783\pi$
$331$	−4.53590	−0.249316	−0.124658	+	0.992200i	$0.539783\pi$
$332$	0	0
$333$	0.535898	0.0293671
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.14359	0.171242	0.0856212	−	0.996328i	$-0.472713\pi$
$337$	3.14359	0.171242	0.0856212	+	0.996328i	$0.472713\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−15.9282	−0.862561
$342$	0	0
$343$	19.0526	1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.2487	−0.872277	−0.436138	−	0.899880i	$-0.643654\pi$
$347$	−16.2487	−0.872277	−0.436138	+	0.899880i	$0.643654\pi$
$348$	0	0
$349$	−14.3923	−0.770402	−0.385201	−	0.922833i	$-0.625868\pi$
$349$	−14.3923	−0.770402	−0.385201	+	0.922833i	$0.625868\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	7.32051	0.389631	0.194816	−	0.980840i	$-0.437589\pi$
$353$	7.32051	0.389631	0.194816	+	0.980840i	$0.437589\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.26795	−0.225884
$358$	0	0
$359$	3.73205	0.196970	0.0984851	−	0.995139i	$-0.468600\pi$
$359$	3.73205	0.196970	0.0984851	+	0.995139i	$0.468600\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	−8.00000	−0.419891
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.92820	−0.361649	−0.180825	−	0.983515i	$-0.557877\pi$
$367$	−6.92820	−0.361649	−0.180825	+	0.983515i	$0.557877\pi$
$368$	0	0
$369$	−2.53590	−0.132014
$370$	0	0
$371$	−20.6603	−1.07263
$372$	0	0
$373$	21.3923	1.10765	0.553826	−	0.832633i	$-0.313167\pi$
$373$	21.3923	1.10765	0.553826	+	0.832633i	$0.313167\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.92820	−0.202313
$378$	0	0
$379$	4.80385	0.246757	0.123379	−	0.992360i	$-0.460627\pi$
$379$	4.80385	0.246757	0.123379	+	0.992360i	$0.460627\pi$
$380$	0	0
$381$	−0.928203	−0.0475533
$382$	0	0
$383$	−8.53590	−0.436164	−0.218082	−	0.975930i	$-0.569980\pi$
$383$	−8.53590	−0.436164	−0.218082	+	0.975930i	$0.569980\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.92820	−0.453846
$388$	0	0
$389$	−7.85641	−0.398336	−0.199168	−	0.979965i	$-0.563824\pi$
$389$	−7.85641	−0.398336	−0.199168	+	0.979965i	$0.563824\pi$
$390$	0	0
$391$	−4.92820	−0.249230
$392$	0	0
$393$	−21.4641	−1.08272
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	−38.1051	−1.90288	−0.951439	−	0.307836i	$-0.900395\pi$
$401$	−38.1051	−1.90288	−0.951439	+	0.307836i	$0.900395\pi$
$402$	0	0
$403$	9.19615	0.458093
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.928203	0.0460093
$408$	0	0
$409$	11.3205	0.559763	0.279882	−	0.960035i	$-0.409705\pi$
$409$	11.3205	0.559763	0.279882	+	0.960035i	$0.409705\pi$
$410$	0	0
$411$	−2.92820	−0.144438
$412$	0	0
$413$	−3.00000	−0.147620
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−14.3923	−0.704794
$418$	0	0
$419$	−23.3205	−1.13928	−0.569641	−	0.821894i	$-0.692918\pi$
$419$	−23.3205	−1.13928	−0.569641	+	0.821894i	$0.692918\pi$
$420$	0	0
$421$	−5.07180	−0.247184	−0.123592	−	0.992333i	$-0.539441\pi$
$421$	−5.07180	−0.247184	−0.123592	+	0.992333i	$0.539441\pi$
$422$	0	0
$423$	6.66025	0.323833
$424$	0	0
$425$	0	0
$426$	0	0
$427$	21.5885	1.04474
$428$	0	0
$429$	−1.73205	−0.0836242
$430$	0	0
$431$	2.39230	0.115233	0.0576166	−	0.998339i	$-0.481650\pi$
$431$	2.39230	0.115233	0.0576166	+	0.998339i	$0.481650\pi$
$432$	0	0
$433$	−0.928203	−0.0446066	−0.0223033	−	0.999751i	$-0.507100\pi$
$433$	−0.928203	−0.0446066	−0.0223033	+	0.999751i	$0.507100\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.92820	0.331421
$438$	0	0
$439$	10.7846	0.514721	0.257361	−	0.966315i	$-0.417147\pi$
$439$	10.7846	0.514721	0.257361	+	0.966315i	$0.417147\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−8.92820	−0.424192	−0.212096	−	0.977249i	$-0.568029\pi$
$443$	−8.92820	−0.424192	−0.212096	+	0.977249i	$0.568029\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.8564	0.560789
$448$	0	0
$449$	−34.9282	−1.64836	−0.824182	−	0.566325i	$-0.808365\pi$
$449$	−34.9282	−1.64836	−0.824182	+	0.566325i	$0.808365\pi$
$450$	0	0
$451$	−4.39230	−0.206826
$452$	0	0
$453$	8.12436	0.381716
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.3205	−0.529551	−0.264776	−	0.964310i	$-0.585298\pi$
$457$	−11.3205	−0.529551	−0.264776	+	0.964310i	$0.585298\pi$
$458$	0	0
$459$	2.46410	0.115014
$460$	0	0
$461$	−7.07180	−0.329366	−0.164683	−	0.986347i	$-0.552660\pi$
$461$	−7.07180	−0.329366	−0.164683	+	0.986347i	$0.552660\pi$
$462$	0	0
$463$	6.26795	0.291296	0.145648	−	0.989336i	$-0.453473\pi$
$463$	6.26795	0.291296	0.145648	+	0.989336i	$0.453473\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.8564	1.19649	0.598246	−	0.801313i	$-0.295865\pi$
$467$	25.8564	1.19649	0.598246	+	0.801313i	$0.295865\pi$
$468$	0	0
$469$	7.39230	0.341345
$470$	0	0
$471$	9.39230	0.432775
$472$	0	0
$473$	−15.4641	−0.711040
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.9282	0.546155
$478$	0	0
$479$	−25.0526	−1.14468	−0.572340	−	0.820016i	$-0.693964\pi$
$479$	−25.0526	−1.14468	−0.572340	+	0.820016i	$0.693964\pi$
$480$	0	0
$481$	−0.535898	−0.0244349
$482$	0	0
$483$	3.46410	0.157622
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.80385	0.308312	0.154156	−	0.988047i	$-0.450734\pi$
$487$	6.80385	0.308312	0.154156	+	0.988047i	$0.450734\pi$
$488$	0	0
$489$	6.39230	0.289070
$490$	0	0
$491$	−2.67949	−0.120924	−0.0604619	−	0.998171i	$-0.519257\pi$
$491$	−2.67949	−0.120924	−0.0604619	+	0.998171i	$0.519257\pi$
$492$	0	0
$493$	9.67949	0.435942
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.9282	1.11818
$498$	0	0
$499$	9.58846	0.429238	0.214619	−	0.976698i	$-0.431149\pi$
$499$	9.58846	0.429238	0.214619	+	0.976698i	$0.431149\pi$
$500$	0	0
$501$	−17.3205	−0.773823
$502$	0	0
$503$	−14.5359	−0.648124	−0.324062	−	0.946036i	$-0.605049\pi$
$503$	−14.5359	−0.648124	−0.324062	+	0.946036i	$0.605049\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−14.2487	−0.631563	−0.315782	−	0.948832i	$-0.602267\pi$
$509$	−14.2487	−0.631563	−0.315782	+	0.948832i	$0.602267\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	−3.46410	−0.152944
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.5359	0.507348
$518$	0	0
$519$	−3.92820	−0.172429
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	21.3205	0.932281	0.466140	−	0.884711i	$-0.345644\pi$
$523$	21.3205	0.932281	0.466140	+	0.884711i	$0.345644\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22.6603	−0.987096
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	1.73205	0.0751646
$532$	0	0
$533$	2.53590	0.109842
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.392305	−0.0169292
$538$	0	0
$539$	−6.92820	−0.298419
$540$	0	0
$541$	−18.3923	−0.790747	−0.395373	−	0.918520i	$-0.629385\pi$
$541$	−18.3923	−0.790747	−0.395373	+	0.918520i	$0.629385\pi$
$542$	0	0
$543$	−3.53590	−0.151740
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.46410	0.404656	0.202328	−	0.979318i	$-0.435149\pi$
$547$	9.46410	0.404656	0.202328	+	0.979318i	$0.435149\pi$
$548$	0	0
$549$	−12.4641	−0.531955
$550$	0	0
$551$	−13.6077	−0.579707
$552$	0	0
$553$	−24.2487	−1.03116
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.3923	1.37251	0.686253	−	0.727363i	$-0.259254\pi$
$557$	32.3923	1.37251	0.686253	+	0.727363i	$0.259254\pi$
$558$	0	0
$559$	8.92820	0.377623
$560$	0	0
$561$	4.26795	0.180193
$562$	0	0
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.73205	−0.0727393
$568$	0	0
$569$	19.2487	0.806948	0.403474	−	0.914991i	$-0.367803\pi$
$569$	19.2487	0.806948	0.403474	+	0.914991i	$0.367803\pi$
$570$	0	0
$571$	−27.8564	−1.16575	−0.582877	−	0.812560i	$-0.698073\pi$
$571$	−27.8564	−1.16575	−0.582877	+	0.812560i	$0.698073\pi$
$572$	0	0
$573$	14.3923	0.601247
$574$	0	0
$575$	0	0
$576$	0	0
$577$	32.6410	1.35886	0.679432	−	0.733739i	$-0.262226\pi$
$577$	32.6410	1.35886	0.679432	+	0.733739i	$0.262226\pi$
$578$	0	0
$579$	−1.46410	−0.0608460
$580$	0	0
$581$	23.7846	0.986752
$582$	0	0
$583$	20.6603	0.855660
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.5885	−0.973600	−0.486800	−	0.873514i	$-0.661836\pi$
$587$	−23.5885	−0.973600	−0.486800	+	0.873514i	$0.661836\pi$
$588$	0	0
$589$	31.8564	1.31262
$590$	0	0
$591$	−4.39230	−0.180675
$592$	0	0
$593$	12.9282	0.530898	0.265449	−	0.964125i	$-0.414480\pi$
$593$	12.9282	0.530898	0.265449	+	0.964125i	$0.414480\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.3205	0.872590
$598$	0	0
$599$	29.1769	1.19214	0.596068	−	0.802934i	$-0.296729\pi$
$599$	29.1769	1.19214	0.596068	+	0.802934i	$0.296729\pi$
$600$	0	0
$601$	22.7128	0.926475	0.463237	−	0.886234i	$-0.346688\pi$
$601$	22.7128	0.926475	0.463237	+	0.886234i	$0.346688\pi$
$602$	0	0
$603$	−4.26795	−0.173804
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.6795	−0.433467	−0.216734	−	0.976231i	$-0.569540\pi$
$607$	−10.6795	−0.433467	−0.216734	+	0.976231i	$0.569540\pi$
$608$	0	0
$609$	−6.80385	−0.275706
$610$	0	0
$611$	−6.66025	−0.269445
$612$	0	0
$613$	−11.0718	−0.447186	−0.223593	−	0.974683i	$-0.571779\pi$
$613$	−11.0718	−0.447186	−0.223593	+	0.974683i	$0.571779\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.85641	0.316287	0.158144	−	0.987416i	$-0.449449\pi$
$617$	7.85641	0.316287	0.158144	+	0.987416i	$0.449449\pi$
$618$	0	0
$619$	25.3205	1.01772	0.508859	−	0.860850i	$-0.330068\pi$
$619$	25.3205	1.01772	0.508859	+	0.860850i	$0.330068\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	0	0
$623$	5.07180	0.203197
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	1.32051	0.0526521
$630$	0	0
$631$	40.2487	1.60228	0.801138	−	0.598480i	$-0.204228\pi$
$631$	40.2487	1.60228	0.801138	+	0.598480i	$0.204228\pi$
$632$	0	0
$633$	−26.0000	−1.03341
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	−14.3923	−0.569351
$640$	0	0
$641$	49.3923	1.95088	0.975439	−	0.220268i	$-0.0706932\pi$
$641$	49.3923	1.95088	0.975439	+	0.220268i	$0.0706932\pi$
$642$	0	0
$643$	21.3205	0.840799	0.420399	−	0.907339i	$-0.361890\pi$
$643$	21.3205	0.840799	0.420399	+	0.907339i	$0.361890\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.7128	1.87578	0.937892	−	0.346927i	$-0.112775\pi$
$647$	47.7128	1.87578	0.937892	+	0.346927i	$0.112775\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	0	0
$651$	15.9282	0.624276
$652$	0	0
$653$	35.9282	1.40598	0.702990	−	0.711200i	$-0.251848\pi$
$653$	35.9282	1.40598	0.702990	+	0.711200i	$0.251848\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.46410	0.135147
$658$	0	0
$659$	29.7128	1.15745	0.578723	−	0.815524i	$-0.303551\pi$
$659$	29.7128	1.15745	0.578723	+	0.815524i	$0.303551\pi$
$660$	0	0
$661$	−36.3923	−1.41550	−0.707748	−	0.706465i	$-0.750289\pi$
$661$	−36.3923	−1.41550	−0.707748	+	0.706465i	$0.750289\pi$
$662$	0	0
$663$	−2.46410	−0.0956978
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.85641	−0.304201
$668$	0	0
$669$	−28.2487	−1.09216
$670$	0	0
$671$	−21.5885	−0.833413
$672$	0	0
$673$	38.8564	1.49780	0.748902	−	0.662681i	$-0.230581\pi$
$673$	38.8564	1.49780	0.748902	+	0.662681i	$0.230581\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−40.6410	−1.56196	−0.780981	−	0.624555i	$-0.785280\pi$
$677$	−40.6410	−1.56196	−0.780981	+	0.624555i	$0.785280\pi$
$678$	0	0
$679$	8.53590	0.327578
$680$	0	0
$681$	2.26795	0.0869080
$682$	0	0
$683$	−43.0526	−1.64736	−0.823680	−	0.567055i	$-0.808083\pi$
$683$	−43.0526	−1.64736	−0.823680	+	0.567055i	$0.808083\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18.7846	−0.716678
$688$	0	0
$689$	−11.9282	−0.454428
$690$	0	0
$691$	−38.1244	−1.45032	−0.725159	−	0.688581i	$-0.758234\pi$
$691$	−38.1244	−1.45032	−0.725159	+	0.688581i	$0.758234\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.24871	−0.236687
$698$	0	0
$699$	−19.8564	−0.751038
$700$	0	0
$701$	1.14359	0.0431929	0.0215965	−	0.999767i	$-0.493125\pi$
$701$	1.14359	0.0431929	0.0215965	+	0.999767i	$0.493125\pi$
$702$	0	0
$703$	−1.85641	−0.0700157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.19615	−0.195421
$708$	0	0
$709$	−17.4641	−0.655878	−0.327939	−	0.944699i	$-0.606354\pi$
$709$	−17.4641	−0.655878	−0.327939	+	0.944699i	$0.606354\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	18.3923	0.688797
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.19615	0.119362
$718$	0	0
$719$	−13.4641	−0.502126	−0.251063	−	0.967971i	$-0.580780\pi$
$719$	−13.4641	−0.502126	−0.251063	+	0.967971i	$0.580780\pi$
$720$	0	0
$721$	−26.5359	−0.988248
$722$	0	0
$723$	−20.3923	−0.758398
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22.2487	−0.825159	−0.412580	−	0.910922i	$-0.635372\pi$
$727$	−22.2487	−0.825159	−0.412580	+	0.910922i	$0.635372\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.0000	−0.813699
$732$	0	0
$733$	−16.7846	−0.619954	−0.309977	−	0.950744i	$-0.600321\pi$
$733$	−16.7846	−0.619954	−0.309977	+	0.950744i	$0.600321\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.39230	−0.272299
$738$	0	0
$739$	36.2679	1.33414	0.667069	−	0.744996i	$-0.267549\pi$
$739$	36.2679	1.33414	0.667069	+	0.744996i	$0.267549\pi$
$740$	0	0
$741$	3.46410	0.127257
$742$	0	0
$743$	−32.8038	−1.20346	−0.601728	−	0.798701i	$-0.705521\pi$
$743$	−32.8038	−1.20346	−0.601728	+	0.798701i	$0.705521\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−13.7321	−0.502429
$748$	0	0
$749$	−12.9282	−0.472386
$750$	0	0
$751$	−15.8564	−0.578608	−0.289304	−	0.957237i	$-0.593424\pi$
$751$	−15.8564	−0.578608	−0.289304	+	0.957237i	$0.593424\pi$
$752$	0	0
$753$	4.39230	0.160064
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.4641	1.25262	0.626310	−	0.779574i	$-0.284565\pi$
$757$	34.4641	1.25262	0.626310	+	0.779574i	$0.284565\pi$
$758$	0	0
$759$	−3.46410	−0.125739
$760$	0	0
$761$	38.3923	1.39172	0.695860	−	0.718177i	$-0.255023\pi$
$761$	38.3923	1.39172	0.695860	+	0.718177i	$0.255023\pi$
$762$	0	0
$763$	4.39230	0.159012
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.73205	−0.0625407
$768$	0	0
$769$	20.7846	0.749512	0.374756	−	0.927123i	$-0.377726\pi$
$769$	20.7846	0.749512	0.374756	+	0.927123i	$0.377726\pi$
$770$	0	0
$771$	19.2487	0.693225
$772$	0	0
$773$	−11.8564	−0.426445	−0.213223	−	0.977004i	$-0.568396\pi$
$773$	−11.8564	−0.426445	−0.213223	+	0.977004i	$0.568396\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.928203	−0.0332991
$778$	0	0
$779$	8.78461	0.314741
$780$	0	0
$781$	−24.9282	−0.892001
$782$	0	0
$783$	3.92820	0.140383
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12.8038	−0.456408	−0.228204	−	0.973613i	$-0.573285\pi$
$787$	−12.8038	−0.456408	−0.228204	+	0.973613i	$0.573285\pi$
$788$	0	0
$789$	−23.3205	−0.830232
$790$	0	0
$791$	17.3205	0.615846
$792$	0	0
$793$	12.4641	0.442613
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.8564	−0.809615	−0.404808	−	0.914402i	$-0.632662\pi$
$797$	−22.8564	−0.809615	−0.404808	+	0.914402i	$0.632662\pi$
$798$	0	0
$799$	16.4115	0.580599
$800$	0	0
$801$	−2.92820	−0.103463
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.0000	−0.668832
$808$	0	0
$809$	11.0718	0.389264	0.194632	−	0.980876i	$-0.437649\pi$
$809$	11.0718	0.389264	0.194632	+	0.980876i	$0.437649\pi$
$810$	0	0
$811$	1.58846	0.0557783	0.0278891	−	0.999611i	$-0.491121\pi$
$811$	1.58846	0.0557783	0.0278891	+	0.999611i	$0.491121\pi$
$812$	0	0
$813$	−21.7321	−0.762176
$814$	0	0
$815$	0	0
$816$	0	0
$817$	30.9282	1.08204
$818$	0	0
$819$	1.73205	0.0605228
$820$	0	0
$821$	27.4641	0.958504	0.479252	−	0.877677i	$-0.340908\pi$
$821$	27.4641	0.958504	0.479252	+	0.877677i	$0.340908\pi$
$822$	0	0
$823$	10.1436	0.353583	0.176792	−	0.984248i	$-0.443428\pi$
$823$	10.1436	0.353583	0.176792	+	0.984248i	$0.443428\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.1244	−0.699792	−0.349896	−	0.936788i	$-0.613783\pi$
$827$	−20.1244	−0.699792	−0.349896	+	0.936788i	$0.613783\pi$
$828$	0	0
$829$	14.6077	0.507346	0.253673	−	0.967290i	$-0.418361\pi$
$829$	14.6077	0.507346	0.253673	+	0.967290i	$0.418361\pi$
$830$	0	0
$831$	20.9282	0.725991
$832$	0	0
$833$	−9.85641	−0.341504
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.19615	−0.317866
$838$	0	0
$839$	−29.3205	−1.01226	−0.506128	−	0.862458i	$-0.668924\pi$
$839$	−29.3205	−1.01226	−0.506128	+	0.862458i	$0.668924\pi$
$840$	0	0
$841$	−13.5692	−0.467904
$842$	0	0
$843$	12.9282	0.445271
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13.8564	0.476112
$848$	0	0
$849$	−9.32051	−0.319879
$850$	0	0
$851$	−1.07180	−0.0367407
$852$	0	0
$853$	48.7846	1.67035	0.835177	−	0.549982i	$-0.185365\pi$
$853$	48.7846	1.67035	0.835177	+	0.549982i	$0.185365\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.7128	−0.741696	−0.370848	−	0.928694i	$-0.620933\pi$
$857$	−21.7128	−0.741696	−0.370848	+	0.928694i	$0.620933\pi$
$858$	0	0
$859$	−9.71281	−0.331397	−0.165698	−	0.986176i	$-0.552988\pi$
$859$	−9.71281	−0.331397	−0.165698	+	0.986176i	$0.552988\pi$
$860$	0	0
$861$	4.39230	0.149689
$862$	0	0
$863$	−10.9474	−0.372655	−0.186328	−	0.982488i	$-0.559659\pi$
$863$	−10.9474	−0.372655	−0.186328	+	0.982488i	$0.559659\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.9282	−0.371141
$868$	0	0
$869$	24.2487	0.822581
$870$	0	0
$871$	4.26795	0.144614
$872$	0	0
$873$	−4.92820	−0.166794
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.9282	1.58465	0.792326	−	0.610097i	$-0.208870\pi$
$877$	46.9282	1.58465	0.792326	+	0.610097i	$0.208870\pi$
$878$	0	0
$879$	15.8564	0.534823
$880$	0	0
$881$	−0.464102	−0.0156360	−0.00781799	−	0.999969i	$-0.502489\pi$
$881$	−0.464102	−0.0156360	−0.00781799	+	0.999969i	$0.502489\pi$
$882$	0	0
$883$	21.0718	0.709122	0.354561	−	0.935033i	$-0.384630\pi$
$883$	21.0718	0.709122	0.354561	+	0.935033i	$0.384630\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.78461	0.0934980	0.0467490	−	0.998907i	$-0.485114\pi$
$887$	2.78461	0.0934980	0.0467490	+	0.998907i	$0.485114\pi$
$888$	0	0
$889$	1.60770	0.0539204
$890$	0	0
$891$	1.73205	0.0580259
$892$	0	0
$893$	−23.0718	−0.772068
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	−36.1244	−1.20481
$900$	0	0
$901$	29.3923	0.979200
$902$	0	0
$903$	15.4641	0.514613
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	−0.928203	−0.0307527	−0.0153764	−	0.999882i	$-0.504895\pi$
$911$	−0.928203	−0.0307527	−0.0153764	+	0.999882i	$0.504895\pi$
$912$	0	0
$913$	−23.7846	−0.787156
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.1769	1.22769
$918$	0	0
$919$	−57.0333	−1.88136	−0.940678	−	0.339301i	$-0.889809\pi$
$919$	−57.0333	−1.88136	−0.940678	+	0.339301i	$0.889809\pi$
$920$	0	0
$921$	−4.53590	−0.149463
$922$	0	0
$923$	14.3923	0.473728
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.3205	0.503192
$928$	0	0
$929$	−52.6410	−1.72710	−0.863548	−	0.504267i	$-0.831763\pi$
$929$	−52.6410	−1.72710	−0.863548	+	0.504267i	$0.831763\pi$
$930$	0	0
$931$	13.8564	0.454125
$932$	0	0
$933$	−7.46410	−0.244364
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.6410	1.49103	0.745514	−	0.666491i	$-0.232204\pi$
$937$	45.6410	1.49103	0.745514	+	0.666491i	$0.232204\pi$
$938$	0	0
$939$	1.92820	0.0629245
$940$	0	0
$941$	3.75129	0.122289	0.0611443	−	0.998129i	$-0.480525\pi$
$941$	3.75129	0.122289	0.0611443	+	0.998129i	$0.480525\pi$
$942$	0	0
$943$	5.07180	0.165160
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.33975	−0.238510	−0.119255	−	0.992864i	$-0.538051\pi$
$947$	−7.33975	−0.238510	−0.119255	+	0.992864i	$0.538051\pi$
$948$	0	0
$949$	−3.46410	−0.112449
$950$	0	0
$951$	12.5359	0.406504
$952$	0	0
$953$	−29.2487	−0.947459	−0.473729	−	0.880670i	$-0.657093\pi$
$953$	−29.2487	−0.947459	−0.473729	+	0.880670i	$0.657093\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.80385	0.219937
$958$	0	0
$959$	5.07180	0.163777
$960$	0	0
$961$	53.5692	1.72804
$962$	0	0
$963$	7.46410	0.240527
$964$	0	0
$965$	0	0
$966$	0	0
$967$	33.4449	1.07551	0.537757	−	0.843100i	$-0.319272\pi$
$967$	33.4449	1.07551	0.537757	+	0.843100i	$0.319272\pi$
$968$	0	0
$969$	−8.53590	−0.274213
$970$	0	0
$971$	−27.1769	−0.872149	−0.436074	−	0.899911i	$-0.643632\pi$
$971$	−27.1769	−0.872149	−0.436074	+	0.899911i	$0.643632\pi$
$972$	0	0
$973$	24.9282	0.799162
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41.5692	1.32992	0.664959	−	0.746880i	$-0.268449\pi$
$977$	41.5692	1.32992	0.664959	+	0.746880i	$0.268449\pi$
$978$	0	0
$979$	−5.07180	−0.162095
$980$	0	0
$981$	−2.53590	−0.0809650
$982$	0	0
$983$	15.7321	0.501774	0.250887	−	0.968016i	$-0.419278\pi$
$983$	15.7321	0.501774	0.250887	+	0.968016i	$0.419278\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−11.5359	−0.367192
$988$	0	0
$989$	17.8564	0.567801
$990$	0	0
$991$	−49.7128	−1.57918	−0.789590	−	0.613635i	$-0.789707\pi$
$991$	−49.7128	−1.57918	−0.789590	+	0.613635i	$0.789707\pi$
$992$	0	0
$993$	−4.53590	−0.143942
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−46.0333	−1.45789	−0.728945	−	0.684572i	$-0.759989\pi$
$997$	−46.0333	−1.45789	−0.728945	+	0.684572i	$0.759989\pi$
$998$	0	0
$999$	0.535898	0.0169551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bc.1.1	yes	2
5.4	even	2		7800.2.a.y.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.y.1.2	✓	2	5.4	even	2
7800.2.a.bc.1.1	yes	2	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.73205 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.73205 q^{7} +1.00000 q^{9} +1.73205 q^{11} -1.00000 q^{13} +2.46410 q^{17} -3.46410 q^{19} -1.73205 q^{21} -2.00000 q^{23} +1.00000 q^{27} +3.92820 q^{29} -9.19615 q^{31} +1.73205 q^{33} +0.535898 q^{37} -1.00000 q^{39} -2.53590 q^{41} -8.92820 q^{43} +6.66025 q^{47} -4.00000 q^{49} +2.46410 q^{51} +11.9282 q^{53} -3.46410 q^{57} +1.73205 q^{59} -12.4641 q^{61} -1.73205 q^{63} -4.26795 q^{67} -2.00000 q^{69} -14.3923 q^{71} +3.46410 q^{73} -3.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} -13.7321 q^{83} +3.92820 q^{87} -2.92820 q^{89} +1.73205 q^{91} -9.19615 q^{93} -4.92820 q^{97} +1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} - 2 q^{13} - 2 q^{17} - 4 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{31} + 8 q^{37} - 2 q^{39} - 12 q^{41} - 4 q^{43} - 4 q^{47} - 8 q^{49} - 2 q^{51} + 10 q^{53} - 18 q^{61} - 12 q^{67} - 4 q^{69} - 8 q^{71} - 6 q^{77} + 28 q^{79} + 2 q^{81} - 24 q^{83} - 6 q^{87} + 8 q^{89} - 8 q^{93} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 2 * q^13 - 2 * q^17 - 4 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^31 + 8 * q^37 - 2 * q^39 - 12 * q^41 - 4 * q^43 - 4 * q^47 - 8 * q^49 - 2 * q^51 + 10 * q^53 - 18 * q^61 - 12 * q^67 - 4 * q^69 - 8 * q^71 - 6 * q^77 + 28 * q^79 + 2 * q^81 - 24 * q^83 - 6 * q^87 + 8 * q^89 - 8 * q^93 + 4 * q^97

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