LMFDB - Embedded newform 315.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("315.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.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:	$2.51528766367$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +O(q^{10})$	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +2.41421 q^{10} +0.828427 q^{11} -4.82843 q^{13} +2.41421 q^{14} +3.00000 q^{16} +3.65685 q^{17} +2.82843 q^{19} -3.82843 q^{20} -2.00000 q^{22} -7.65685 q^{23} +1.00000 q^{25} +11.6569 q^{26} -3.82843 q^{28} -8.00000 q^{29} -8.48528 q^{31} +1.58579 q^{32} -8.82843 q^{34} +1.00000 q^{35} -6.00000 q^{37} -6.82843 q^{38} +4.41421 q^{40} -3.65685 q^{41} -9.65685 q^{43} +3.17157 q^{44} +18.4853 q^{46} +4.00000 q^{47} +1.00000 q^{49} -2.41421 q^{50} -18.4853 q^{52} -10.8284 q^{53} -0.828427 q^{55} +4.41421 q^{56} +19.3137 q^{58} +4.00000 q^{59} +6.00000 q^{61} +20.4853 q^{62} -9.82843 q^{64} +4.82843 q^{65} +7.31371 q^{67} +14.0000 q^{68} -2.41421 q^{70} +6.48528 q^{71} +16.1421 q^{73} +14.4853 q^{74} +10.8284 q^{76} -0.828427 q^{77} -5.65685 q^{79} -3.00000 q^{80} +8.82843 q^{82} +8.00000 q^{83} -3.65685 q^{85} +23.3137 q^{86} -3.65685 q^{88} -17.3137 q^{89} +4.82843 q^{91} -29.3137 q^{92} -9.65685 q^{94} -2.82843 q^{95} -8.82843 q^{97} -2.41421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{13} + 2 q^{14} + 6 q^{16} - 4 q^{17} - 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{28} - 16 q^{29} + 6 q^{32} - 12 q^{34} + 2 q^{35} - 12 q^{37} - 8 q^{38} + 6 q^{40} + 4 q^{41} - 8 q^{43} + 12 q^{44} + 20 q^{46} + 8 q^{47} + 2 q^{49} - 2 q^{50} - 20 q^{52} - 16 q^{53} + 4 q^{55} + 6 q^{56} + 16 q^{58} + 8 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{67} + 28 q^{68} - 2 q^{70} - 4 q^{71} + 4 q^{73} + 12 q^{74} + 16 q^{76} + 4 q^{77} - 6 q^{80} + 12 q^{82} + 16 q^{83} + 4 q^{85} + 24 q^{86} + 4 q^{88} - 12 q^{89} + 4 q^{91} - 36 q^{92} - 8 q^{94} - 12 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^13 + 2 * q^14 + 6 * q^16 - 4 * q^17 - 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^28 - 16 * q^29 + 6 * q^32 - 12 * q^34 + 2 * q^35 - 12 * q^37 - 8 * q^38 + 6 * q^40 + 4 * q^41 - 8 * q^43 + 12 * q^44 + 20 * q^46 + 8 * q^47 + 2 * q^49 - 2 * q^50 - 20 * q^52 - 16 * q^53 + 4 * q^55 + 6 * q^56 + 16 * q^58 + 8 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^67 + 28 * q^68 - 2 * q^70 - 4 * q^71 + 4 * q^73 + 12 * q^74 + 16 * q^76 + 4 * q^77 - 6 * q^80 + 12 * q^82 + 16 * q^83 + 4 * q^85 + 24 * q^86 + 4 * q^88 - 12 * q^89 + 4 * q^91 - 36 * q^92 - 8 * q^94 - 12 * q^97 - 2 * q^98

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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−4.41421	−1.56066
$9$	0	0
$10$	2.41421	0.763441
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	−4.82843	−1.33916	−0.669582	−	0.742738i	$-0.733527\pi$
$13$	−4.82843	−1.33916	−0.669582	+	0.742738i	$0.733527\pi$
$14$	2.41421	0.645226
$15$	0	0
$16$	3.00000	0.750000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	11.6569	2.28610
$27$	0	0
$28$	−3.82843	−0.723505
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	−8.82843	−1.51406
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−6.82843	−1.10772
$39$	0	0
$40$	4.41421	0.697948
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	3.17157	0.478133
$45$	0	0
$46$	18.4853	2.72551
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.41421	−0.341421
$51$	0	0
$52$	−18.4853	−2.56345
$53$	−10.8284	−1.48740	−0.743699	−	0.668514i	$-0.766931\pi$
$53$	−10.8284	−1.48740	−0.743699	+	0.668514i	$0.766931\pi$
$54$	0	0
$55$	−0.828427	−0.111705
$56$	4.41421	0.589874
$57$	0	0
$58$	19.3137	2.53601
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	20.4853	2.60163
$63$	0	0
$64$	−9.82843	−1.22855
$65$	4.82843	0.598893
$66$	0	0
$67$	7.31371	0.893512	0.446756	−	0.894656i	$-0.352579\pi$
$67$	7.31371	0.893512	0.446756	+	0.894656i	$0.352579\pi$
$68$	14.0000	1.69775
$69$	0	0
$70$	−2.41421	−0.288554
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	0	0
$73$	16.1421	1.88929	0.944647	−	0.328088i	$-0.106404\pi$
$73$	16.1421	1.88929	0.944647	+	0.328088i	$0.106404\pi$
$74$	14.4853	1.68388
$75$	0	0
$76$	10.8284	1.24211
$77$	−0.828427	−0.0944080
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	8.82843	0.974937
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−3.65685	−0.396642
$86$	23.3137	2.51398
$87$	0	0
$88$	−3.65685	−0.389822
$89$	−17.3137	−1.83525	−0.917625	−	0.397448i	$-0.869896\pi$
$89$	−17.3137	−1.83525	−0.917625	+	0.397448i	$0.869896\pi$
$90$	0	0
$91$	4.82843	0.506157
$92$	−29.3137	−3.05617
$93$	0	0
$94$	−9.65685	−0.996028
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−8.82843	−0.896391	−0.448195	−	0.893936i	$-0.647933\pi$
$97$	−8.82843	−0.896391	−0.448195	+	0.893936i	$0.647933\pi$
$98$	−2.41421	−0.243872
$99$	0	0
$100$	3.82843	0.382843
$101$	−15.6569	−1.55792	−0.778958	−	0.627077i	$-0.784251\pi$
$101$	−15.6569	−1.55792	−0.778958	+	0.627077i	$0.784251\pi$
$102$	0	0
$103$	7.31371	0.720641	0.360321	−	0.932829i	$-0.382667\pi$
$103$	7.31371	0.720641	0.360321	+	0.932829i	$0.382667\pi$
$104$	21.3137	2.08998
$105$	0	0
$106$	26.1421	2.53915
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	−3.00000	−0.283473
$113$	4.48528	0.421940	0.210970	−	0.977493i	$-0.432338\pi$
$113$	4.48528	0.421940	0.210970	+	0.977493i	$0.432338\pi$
$114$	0	0
$115$	7.65685	0.714005
$116$	−30.6274	−2.84368
$117$	0	0
$118$	−9.65685	−0.888985
$119$	−3.65685	−0.335223
$120$	0	0
$121$	−10.3137	−0.937610
$122$	−14.4853	−1.31144
$123$	0	0
$124$	−32.4853	−2.91726
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.6569	1.21185	0.605925	−	0.795522i	$-0.292803\pi$
$127$	13.6569	1.21185	0.605925	+	0.795522i	$0.292803\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	−11.6569	−1.02237
$131$	2.34315	0.204722	0.102361	−	0.994747i	$-0.467360\pi$
$131$	2.34315	0.204722	0.102361	+	0.994747i	$0.467360\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	−17.6569	−1.52532
$135$	0	0
$136$	−16.1421	−1.38418
$137$	6.14214	0.524758	0.262379	−	0.964965i	$-0.415493\pi$
$137$	6.14214	0.524758	0.262379	+	0.964965i	$0.415493\pi$
$138$	0	0
$139$	6.82843	0.579180	0.289590	−	0.957151i	$-0.406481\pi$
$139$	6.82843	0.579180	0.289590	+	0.957151i	$0.406481\pi$
$140$	3.82843	0.323561
$141$	0	0
$142$	−15.6569	−1.31389
$143$	−4.00000	−0.334497
$144$	0	0
$145$	8.00000	0.664364
$146$	−38.9706	−3.22523
$147$	0	0
$148$	−22.9706	−1.88817
$149$	1.65685	0.135735	0.0678674	−	0.997694i	$-0.478381\pi$
$149$	1.65685	0.135735	0.0678674	+	0.997694i	$0.478381\pi$
$150$	0	0
$151$	5.65685	0.460348	0.230174	−	0.973149i	$-0.426070\pi$
$151$	5.65685	0.460348	0.230174	+	0.973149i	$0.426070\pi$
$152$	−12.4853	−1.01269
$153$	0	0
$154$	2.00000	0.161165
$155$	8.48528	0.681554
$156$	0	0
$157$	−3.17157	−0.253119	−0.126560	−	0.991959i	$-0.540393\pi$
$157$	−3.17157	−0.253119	−0.126560	+	0.991959i	$0.540393\pi$
$158$	13.6569	1.08648
$159$	0	0
$160$	−1.58579	−0.125367
$161$	7.65685	0.603445
$162$	0	0
$163$	−17.6569	−1.38299	−0.691496	−	0.722380i	$-0.743048\pi$
$163$	−17.6569	−1.38299	−0.691496	+	0.722380i	$0.743048\pi$
$164$	−14.0000	−1.09322
$165$	0	0
$166$	−19.3137	−1.49903
$167$	23.3137	1.80407	0.902034	−	0.431664i	$-0.142073\pi$
$167$	23.3137	1.80407	0.902034	+	0.431664i	$0.142073\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	8.82843	0.677109
$171$	0	0
$172$	−36.9706	−2.81898
$173$	−4.34315	−0.330203	−0.165102	−	0.986277i	$-0.552795\pi$
$173$	−4.34315	−0.330203	−0.165102	+	0.986277i	$0.552795\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	2.48528	0.187335
$177$	0	0
$178$	41.7990	3.13297
$179$	−22.4853	−1.68063	−0.840314	−	0.542099i	$-0.817630\pi$
$179$	−22.4853	−1.68063	−0.840314	+	0.542099i	$0.817630\pi$
$180$	0	0
$181$	−0.343146	−0.0255058	−0.0127529	−	0.999919i	$-0.504059\pi$
$181$	−0.343146	−0.0255058	−0.0127529	+	0.999919i	$0.504059\pi$
$182$	−11.6569	−0.864064
$183$	0	0
$184$	33.7990	2.49169
$185$	6.00000	0.441129
$186$	0	0
$187$	3.02944	0.221534
$188$	15.3137	1.11687
$189$	0	0
$190$	6.82843	0.495386
$191$	0.828427	0.0599429	0.0299714	−	0.999551i	$-0.490458\pi$
$191$	0.828427	0.0599429	0.0299714	+	0.999551i	$0.490458\pi$
$192$	0	0
$193$	−18.9706	−1.36553	−0.682765	−	0.730638i	$-0.739223\pi$
$193$	−18.9706	−1.36553	−0.682765	+	0.730638i	$0.739223\pi$
$194$	21.3137	1.53024
$195$	0	0
$196$	3.82843	0.273459
$197$	2.82843	0.201517	0.100759	−	0.994911i	$-0.467873\pi$
$197$	2.82843	0.201517	0.100759	+	0.994911i	$0.467873\pi$
$198$	0	0
$199$	22.8284	1.61826	0.809132	−	0.587627i	$-0.199938\pi$
$199$	22.8284	1.61826	0.809132	+	0.587627i	$0.199938\pi$
$200$	−4.41421	−0.312132
$201$	0	0
$202$	37.7990	2.65953
$203$	8.00000	0.561490
$204$	0	0
$205$	3.65685	0.255406
$206$	−17.6569	−1.23021
$207$	0	0
$208$	−14.4853	−1.00437
$209$	2.34315	0.162079
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−41.4558	−2.84720
$213$	0	0
$214$	−4.82843	−0.330064
$215$	9.65685	0.658592
$216$	0	0
$217$	8.48528	0.576018
$218$	−24.1421	−1.63511
$219$	0	0
$220$	−3.17157	−0.213827
$221$	−17.6569	−1.18773
$222$	0	0
$223$	−5.65685	−0.378811	−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685	−0.378811	−0.189405	+	0.981899i	$0.560656\pi$
$224$	−1.58579	−0.105955
$225$	0	0
$226$	−10.8284	−0.720296
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	5.31371	0.351140	0.175570	−	0.984467i	$-0.443823\pi$
$229$	5.31371	0.351140	0.175570	+	0.984467i	$0.443823\pi$
$230$	−18.4853	−1.21888
$231$	0	0
$232$	35.3137	2.31846
$233$	−6.14214	−0.402385	−0.201192	−	0.979552i	$-0.564482\pi$
$233$	−6.14214	−0.402385	−0.201192	+	0.979552i	$0.564482\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	15.3137	0.996838
$237$	0	0
$238$	8.82843	0.572262
$239$	−4.14214	−0.267932	−0.133966	−	0.990986i	$-0.542771\pi$
$239$	−4.14214	−0.267932	−0.133966	+	0.990986i	$0.542771\pi$
$240$	0	0
$241$	3.65685	0.235559	0.117779	−	0.993040i	$-0.462422\pi$
$241$	3.65685	0.235559	0.117779	+	0.993040i	$0.462422\pi$
$242$	24.8995	1.60060
$243$	0	0
$244$	22.9706	1.47054
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−13.6569	−0.868965
$248$	37.4558	2.37845
$249$	0	0
$250$	2.41421	0.152688
$251$	−18.6274	−1.17575	−0.587876	−	0.808951i	$-0.700036\pi$
$251$	−18.6274	−1.17575	−0.587876	+	0.808951i	$0.700036\pi$
$252$	0	0
$253$	−6.34315	−0.398790
$254$	−32.9706	−2.06876
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−17.3137	−1.08000	−0.540000	−	0.841665i	$-0.681576\pi$
$257$	−17.3137	−1.08000	−0.540000	+	0.841665i	$0.681576\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	18.4853	1.14641
$261$	0	0
$262$	−5.65685	−0.349482
$263$	25.3137	1.56091	0.780455	−	0.625212i	$-0.214987\pi$
$263$	25.3137	1.56091	0.780455	+	0.625212i	$0.214987\pi$
$264$	0	0
$265$	10.8284	0.665185
$266$	6.82843	0.418678
$267$	0	0
$268$	28.0000	1.71037
$269$	18.9706	1.15666	0.578328	−	0.815804i	$-0.303705\pi$
$269$	18.9706	1.15666	0.578328	+	0.815804i	$0.303705\pi$
$270$	0	0
$271$	−7.51472	−0.456487	−0.228243	−	0.973604i	$-0.573298\pi$
$271$	−7.51472	−0.456487	−0.228243	+	0.973604i	$0.573298\pi$
$272$	10.9706	0.665188
$273$	0	0
$274$	−14.8284	−0.895818
$275$	0.828427	0.0499560
$276$	0	0
$277$	−4.34315	−0.260954	−0.130477	−	0.991451i	$-0.541651\pi$
$277$	−4.34315	−0.260954	−0.130477	+	0.991451i	$0.541651\pi$
$278$	−16.4853	−0.988721
$279$	0	0
$280$	−4.41421	−0.263800
$281$	14.3431	0.855640	0.427820	−	0.903864i	$-0.359282\pi$
$281$	14.3431	0.855640	0.427820	+	0.903864i	$0.359282\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	24.8284	1.47330
$285$	0	0
$286$	9.65685	0.571022
$287$	3.65685	0.215857
$288$	0	0
$289$	−3.62742	−0.213377
$290$	−19.3137	−1.13414
$291$	0	0
$292$	61.7990	3.61651
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	26.4853	1.53943
$297$	0	0
$298$	−4.00000	−0.231714
$299$	36.9706	2.13806
$300$	0	0
$301$	9.65685	0.556612
$302$	−13.6569	−0.785864
$303$	0	0
$304$	8.48528	0.486664
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−4.97056	−0.283685	−0.141843	−	0.989889i	$-0.545303\pi$
$307$	−4.97056	−0.283685	−0.141843	+	0.989889i	$0.545303\pi$
$308$	−3.17157	−0.180717
$309$	0	0
$310$	−20.4853	−1.16349
$311$	33.6569	1.90851	0.954253	−	0.299002i	$-0.0966537\pi$
$311$	33.6569	1.90851	0.954253	+	0.299002i	$0.0966537\pi$
$312$	0	0
$313$	−20.8284	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$313$	−20.8284	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$314$	7.65685	0.432101
$315$	0	0
$316$	−21.6569	−1.21829
$317$	−14.1421	−0.794301	−0.397151	−	0.917753i	$-0.630001\pi$
$317$	−14.1421	−0.794301	−0.397151	+	0.917753i	$0.630001\pi$
$318$	0	0
$319$	−6.62742	−0.371064
$320$	9.82843	0.549426
$321$	0	0
$322$	−18.4853	−1.03014
$323$	10.3431	0.575508
$324$	0	0
$325$	−4.82843	−0.267833
$326$	42.6274	2.36091
$327$	0	0
$328$	16.1421	0.891300
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	30.6274	1.68090
$333$	0	0
$334$	−56.2843	−3.07974
$335$	−7.31371	−0.399591
$336$	0	0
$337$	−22.2843	−1.21390	−0.606951	−	0.794739i	$-0.707608\pi$
$337$	−22.2843	−1.21390	−0.606951	+	0.794739i	$0.707608\pi$
$338$	−24.8995	−1.35435
$339$	0	0
$340$	−14.0000	−0.759257
$341$	−7.02944	−0.380665
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	42.6274	2.29832
$345$	0	0
$346$	10.4853	0.563692
$347$	0.343146	0.0184210	0.00921051	−	0.999958i	$-0.497068\pi$
$347$	0.343146	0.0184210	0.00921051	+	0.999958i	$0.497068\pi$
$348$	0	0
$349$	−29.3137	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$349$	−29.3137	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$350$	2.41421	0.129045
$351$	0	0
$352$	1.31371	0.0700209
$353$	−16.6274	−0.884988	−0.442494	−	0.896771i	$-0.645906\pi$
$353$	−16.6274	−0.884988	−0.442494	+	0.896771i	$0.645906\pi$
$354$	0	0
$355$	−6.48528	−0.344203
$356$	−66.2843	−3.51306
$357$	0	0
$358$	54.2843	2.86901
$359$	−10.4853	−0.553392	−0.276696	−	0.960958i	$-0.589239\pi$
$359$	−10.4853	−0.553392	−0.276696	+	0.960958i	$0.589239\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0.828427	0.0435412
$363$	0	0
$364$	18.4853	0.968892
$365$	−16.1421	−0.844918
$366$	0	0
$367$	24.9706	1.30345	0.651726	−	0.758454i	$-0.274045\pi$
$367$	24.9706	1.30345	0.651726	+	0.758454i	$0.274045\pi$
$368$	−22.9706	−1.19742
$369$	0	0
$370$	−14.4853	−0.753054
$371$	10.8284	0.562184
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−7.31371	−0.378183
$375$	0	0
$376$	−17.6569	−0.910583
$377$	38.6274	1.98941
$378$	0	0
$379$	6.34315	0.325826	0.162913	−	0.986640i	$-0.447911\pi$
$379$	6.34315	0.325826	0.162913	+	0.986640i	$0.447911\pi$
$380$	−10.8284	−0.555487
$381$	0	0
$382$	−2.00000	−0.102329
$383$	−27.3137	−1.39567	−0.697833	−	0.716261i	$-0.745852\pi$
$383$	−27.3137	−1.39567	−0.697833	+	0.716261i	$0.745852\pi$
$384$	0	0
$385$	0.828427	0.0422206
$386$	45.7990	2.33111
$387$	0	0
$388$	−33.7990	−1.71588
$389$	−29.6569	−1.50366	−0.751831	−	0.659356i	$-0.770829\pi$
$389$	−29.6569	−1.50366	−0.751831	+	0.659356i	$0.770829\pi$
$390$	0	0
$391$	−28.0000	−1.41602
$392$	−4.41421	−0.222951
$393$	0	0
$394$	−6.82843	−0.344011
$395$	5.65685	0.284627
$396$	0	0
$397$	3.85786	0.193621	0.0968103	−	0.995303i	$-0.469136\pi$
$397$	3.85786	0.193621	0.0968103	+	0.995303i	$0.469136\pi$
$398$	−55.1127	−2.76255
$399$	0	0
$400$	3.00000	0.150000
$401$	18.6274	0.930209	0.465104	−	0.885256i	$-0.346017\pi$
$401$	18.6274	0.930209	0.465104	+	0.885256i	$0.346017\pi$
$402$	0	0
$403$	40.9706	2.04089
$404$	−59.9411	−2.98218
$405$	0	0
$406$	−19.3137	−0.958523
$407$	−4.97056	−0.246382
$408$	0	0
$409$	2.68629	0.132829	0.0664143	−	0.997792i	$-0.478844\pi$
$409$	2.68629	0.132829	0.0664143	+	0.997792i	$0.478844\pi$
$410$	−8.82843	−0.436005
$411$	0	0
$412$	28.0000	1.37946
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−7.65685	−0.375408
$417$	0	0
$418$	−5.65685	−0.276686
$419$	10.6274	0.519183	0.259592	−	0.965718i	$-0.416412\pi$
$419$	10.6274	0.519183	0.259592	+	0.965718i	$0.416412\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	28.9706	1.41026
$423$	0	0
$424$	47.7990	2.32132
$425$	3.65685	0.177383
$426$	0	0
$427$	−6.00000	−0.290360
$428$	7.65685	0.370108
$429$	0	0
$430$	−23.3137	−1.12429
$431$	−12.8284	−0.617924	−0.308962	−	0.951074i	$-0.599982\pi$
$431$	−12.8284	−0.617924	−0.308962	+	0.951074i	$0.599982\pi$
$432$	0	0
$433$	25.7990	1.23982	0.619910	−	0.784673i	$-0.287169\pi$
$433$	25.7990	1.23982	0.619910	+	0.784673i	$0.287169\pi$
$434$	−20.4853	−0.983325
$435$	0	0
$436$	38.2843	1.83348
$437$	−21.6569	−1.03599
$438$	0	0
$439$	13.4558	0.642212	0.321106	−	0.947043i	$-0.395945\pi$
$439$	13.4558	0.642212	0.321106	+	0.947043i	$0.395945\pi$
$440$	3.65685	0.174334
$441$	0	0
$442$	42.6274	2.02758
$443$	−1.02944	−0.0489100	−0.0244550	−	0.999701i	$-0.507785\pi$
$443$	−1.02944	−0.0489100	−0.0244550	+	0.999701i	$0.507785\pi$
$444$	0	0
$445$	17.3137	0.820748
$446$	13.6569	0.646671
$447$	0	0
$448$	9.82843	0.464350
$449$	−3.31371	−0.156384	−0.0781918	−	0.996938i	$-0.524915\pi$
$449$	−3.31371	−0.156384	−0.0781918	+	0.996938i	$0.524915\pi$
$450$	0	0
$451$	−3.02944	−0.142651
$452$	17.1716	0.807683
$453$	0	0
$454$	−48.2843	−2.26609
$455$	−4.82843	−0.226360
$456$	0	0
$457$	1.31371	0.0614527	0.0307263	−	0.999528i	$-0.490218\pi$
$457$	1.31371	0.0614527	0.0307263	+	0.999528i	$0.490218\pi$
$458$	−12.8284	−0.599433
$459$	0	0
$460$	29.3137	1.36676
$461$	28.6274	1.33331	0.666656	−	0.745366i	$-0.267725\pi$
$461$	28.6274	1.33331	0.666656	+	0.745366i	$0.267725\pi$
$462$	0	0
$463$	−10.3431	−0.480687	−0.240343	−	0.970688i	$-0.577260\pi$
$463$	−10.3431	−0.480687	−0.240343	+	0.970688i	$0.577260\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	14.8284	0.686914
$467$	−8.68629	−0.401954	−0.200977	−	0.979596i	$-0.564412\pi$
$467$	−8.68629	−0.401954	−0.200977	+	0.979596i	$0.564412\pi$
$468$	0	0
$469$	−7.31371	−0.337716
$470$	9.65685	0.445437
$471$	0	0
$472$	−17.6569	−0.812723
$473$	−8.00000	−0.367840
$474$	0	0
$475$	2.82843	0.129777
$476$	−14.0000	−0.641689
$477$	0	0
$478$	10.0000	0.457389
$479$	41.6569	1.90335	0.951675	−	0.307107i	$-0.0993608\pi$
$479$	41.6569	1.90335	0.951675	+	0.307107i	$0.0993608\pi$
$480$	0	0
$481$	28.9706	1.32094
$482$	−8.82843	−0.402124
$483$	0	0
$484$	−39.4853	−1.79479
$485$	8.82843	0.400878
$486$	0	0
$487$	−13.6569	−0.618851	−0.309426	−	0.950924i	$-0.600137\pi$
$487$	−13.6569	−0.618851	−0.309426	+	0.950924i	$0.600137\pi$
$488$	−26.4853	−1.19893
$489$	0	0
$490$	2.41421	0.109063
$491$	16.1421	0.728484	0.364242	−	0.931304i	$-0.381328\pi$
$491$	16.1421	0.728484	0.364242	+	0.931304i	$0.381328\pi$
$492$	0	0
$493$	−29.2548	−1.31757
$494$	32.9706	1.48342
$495$	0	0
$496$	−25.4558	−1.14300
$497$	−6.48528	−0.290905
$498$	0	0
$499$	25.6569	1.14856	0.574279	−	0.818659i	$-0.305282\pi$
$499$	25.6569	1.14856	0.574279	+	0.818659i	$0.305282\pi$
$500$	−3.82843	−0.171212
$501$	0	0
$502$	44.9706	2.00713
$503$	14.6274	0.652204	0.326102	−	0.945335i	$-0.394265\pi$
$503$	14.6274	0.652204	0.326102	+	0.945335i	$0.394265\pi$
$504$	0	0
$505$	15.6569	0.696721
$506$	15.3137	0.680777
$507$	0	0
$508$	52.2843	2.31974
$509$	−16.3431	−0.724397	−0.362199	−	0.932101i	$-0.617974\pi$
$509$	−16.3431	−0.724397	−0.362199	+	0.932101i	$0.617974\pi$
$510$	0	0
$511$	−16.1421	−0.714086
$512$	31.2426	1.38074
$513$	0	0
$514$	41.7990	1.84367
$515$	−7.31371	−0.322281
$516$	0	0
$517$	3.31371	0.145737
$518$	−14.4853	−0.636447
$519$	0	0
$520$	−21.3137	−0.934668
$521$	−26.9706	−1.18160	−0.590801	−	0.806817i	$-0.701188\pi$
$521$	−26.9706	−1.18160	−0.590801	+	0.806817i	$0.701188\pi$
$522$	0	0
$523$	−18.3431	−0.802090	−0.401045	−	0.916058i	$-0.631353\pi$
$523$	−18.3431	−0.802090	−0.401045	+	0.916058i	$0.631353\pi$
$524$	8.97056	0.391881
$525$	0	0
$526$	−61.1127	−2.66464
$527$	−31.0294	−1.35166
$528$	0	0
$529$	35.6274	1.54902
$530$	−26.1421	−1.13554
$531$	0	0
$532$	−10.8284	−0.469472
$533$	17.6569	0.764803
$534$	0	0
$535$	−2.00000	−0.0864675
$536$	−32.2843	−1.39447
$537$	0	0
$538$	−45.7990	−1.97453
$539$	0.828427	0.0356829
$540$	0	0
$541$	33.3137	1.43227	0.716134	−	0.697963i	$-0.245910\pi$
$541$	33.3137	1.43227	0.716134	+	0.697963i	$0.245910\pi$
$542$	18.1421	0.779271
$543$	0	0
$544$	5.79899	0.248630
$545$	−10.0000	−0.428353
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	23.5147	1.00450
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	−22.6274	−0.963960
$552$	0	0
$553$	5.65685	0.240554
$554$	10.4853	0.445477
$555$	0	0
$556$	26.1421	1.10867
$557$	−26.8284	−1.13676	−0.568378	−	0.822767i	$-0.692429\pi$
$557$	−26.8284	−1.13676	−0.568378	+	0.822767i	$0.692429\pi$
$558$	0	0
$559$	46.6274	1.97213
$560$	3.00000	0.126773
$561$	0	0
$562$	−34.6274	−1.46067
$563$	−29.9411	−1.26187	−0.630934	−	0.775837i	$-0.717328\pi$
$563$	−29.9411	−1.26187	−0.630934	+	0.775837i	$0.717328\pi$
$564$	0	0
$565$	−4.48528	−0.188697
$566$	9.65685	0.405908
$567$	0	0
$568$	−28.6274	−1.20118
$569$	39.3137	1.64812	0.824058	−	0.566505i	$-0.191705\pi$
$569$	39.3137	1.64812	0.824058	+	0.566505i	$0.191705\pi$
$570$	0	0
$571$	−23.3137	−0.975648	−0.487824	−	0.872942i	$-0.662209\pi$
$571$	−23.3137	−0.975648	−0.487824	+	0.872942i	$0.662209\pi$
$572$	−15.3137	−0.640298
$573$	0	0
$574$	−8.82843	−0.368491
$575$	−7.65685	−0.319313
$576$	0	0
$577$	−25.5147	−1.06219	−0.531096	−	0.847312i	$-0.678220\pi$
$577$	−25.5147	−1.06219	−0.531096	+	0.847312i	$0.678220\pi$
$578$	8.75736	0.364258
$579$	0	0
$580$	30.6274	1.27173
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−8.97056	−0.371523
$584$	−71.2548	−2.94855
$585$	0	0
$586$	4.82843	0.199460
$587$	−34.6274	−1.42923	−0.714613	−	0.699520i	$-0.753397\pi$
$587$	−34.6274	−1.42923	−0.714613	+	0.699520i	$0.753397\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	9.65685	0.397566
$591$	0	0
$592$	−18.0000	−0.739795
$593$	17.3137	0.710989	0.355494	−	0.934678i	$-0.384313\pi$
$593$	17.3137	0.710989	0.355494	+	0.934678i	$0.384313\pi$
$594$	0	0
$595$	3.65685	0.149916
$596$	6.34315	0.259825
$597$	0	0
$598$	−89.2548	−3.64990
$599$	20.1421	0.822985	0.411493	−	0.911413i	$-0.365008\pi$
$599$	20.1421	0.822985	0.411493	+	0.911413i	$0.365008\pi$
$600$	0	0
$601$	−26.2843	−1.07216	−0.536079	−	0.844168i	$-0.680095\pi$
$601$	−26.2843	−1.07216	−0.536079	+	0.844168i	$0.680095\pi$
$602$	−23.3137	−0.950196
$603$	0	0
$604$	21.6569	0.881205
$605$	10.3137	0.419312
$606$	0	0
$607$	15.0294	0.610026	0.305013	−	0.952348i	$-0.401339\pi$
$607$	15.0294	0.610026	0.305013	+	0.952348i	$0.401339\pi$
$608$	4.48528	0.181902
$609$	0	0
$610$	14.4853	0.586492
$611$	−19.3137	−0.781349
$612$	0	0
$613$	14.9706	0.604655	0.302328	−	0.953204i	$-0.402236\pi$
$613$	14.9706	0.604655	0.302328	+	0.953204i	$0.402236\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	3.65685	0.147339
$617$	21.1716	0.852335	0.426168	−	0.904644i	$-0.359863\pi$
$617$	21.1716	0.852335	0.426168	+	0.904644i	$0.359863\pi$
$618$	0	0
$619$	−28.4853	−1.14492	−0.572460	−	0.819933i	$-0.694011\pi$
$619$	−28.4853	−1.14492	−0.572460	+	0.819933i	$0.694011\pi$
$620$	32.4853	1.30464
$621$	0	0
$622$	−81.2548	−3.25802
$623$	17.3137	0.693659
$624$	0	0
$625$	1.00000	0.0400000
$626$	50.2843	2.00976
$627$	0	0
$628$	−12.1421	−0.484524
$629$	−21.9411	−0.874850
$630$	0	0
$631$	−5.65685	−0.225196	−0.112598	−	0.993641i	$-0.535917\pi$
$631$	−5.65685	−0.225196	−0.112598	+	0.993641i	$0.535917\pi$
$632$	24.9706	0.993276
$633$	0	0
$634$	34.1421	1.35596
$635$	−13.6569	−0.541956
$636$	0	0
$637$	−4.82843	−0.191309
$638$	16.0000	0.633446
$639$	0	0
$640$	−20.5563	−0.812561
$641$	−12.9706	−0.512306	−0.256153	−	0.966636i	$-0.582455\pi$
$641$	−12.9706	−0.512306	−0.256153	+	0.966636i	$0.582455\pi$
$642$	0	0
$643$	22.3431	0.881128	0.440564	−	0.897721i	$-0.354779\pi$
$643$	22.3431	0.881128	0.440564	+	0.897721i	$0.354779\pi$
$644$	29.3137	1.15512
$645$	0	0
$646$	−24.9706	−0.982454
$647$	−18.6274	−0.732319	−0.366160	−	0.930552i	$-0.619328\pi$
$647$	−18.6274	−0.732319	−0.366160	+	0.930552i	$0.619328\pi$
$648$	0	0
$649$	3.31371	0.130074
$650$	11.6569	0.457219
$651$	0	0
$652$	−67.5980	−2.64734
$653$	−5.85786	−0.229236	−0.114618	−	0.993410i	$-0.536564\pi$
$653$	−5.85786	−0.229236	−0.114618	+	0.993410i	$0.536564\pi$
$654$	0	0
$655$	−2.34315	−0.0915543
$656$	−10.9706	−0.428329
$657$	0	0
$658$	9.65685	0.376463
$659$	−32.1421	−1.25208	−0.626040	−	0.779791i	$-0.715325\pi$
$659$	−32.1421	−1.25208	−0.626040	+	0.779791i	$0.715325\pi$
$660$	0	0
$661$	49.3137	1.91808	0.959040	−	0.283269i	$-0.0914189\pi$
$661$	49.3137	1.91808	0.959040	+	0.283269i	$0.0914189\pi$
$662$	28.9706	1.12597
$663$	0	0
$664$	−35.3137	−1.37044
$665$	2.82843	0.109682
$666$	0	0
$667$	61.2548	2.37180
$668$	89.2548	3.45337
$669$	0	0
$670$	17.6569	0.682144
$671$	4.97056	0.191886
$672$	0	0
$673$	4.62742	0.178374	0.0891869	−	0.996015i	$-0.471573\pi$
$673$	4.62742	0.178374	0.0891869	+	0.996015i	$0.471573\pi$
$674$	53.7990	2.07226
$675$	0	0
$676$	39.4853	1.51866
$677$	−21.3137	−0.819152	−0.409576	−	0.912276i	$-0.634323\pi$
$677$	−21.3137	−0.819152	−0.409576	+	0.912276i	$0.634323\pi$
$678$	0	0
$679$	8.82843	0.338804
$680$	16.1421	0.619023
$681$	0	0
$682$	16.9706	0.649836
$683$	−1.02944	−0.0393903	−0.0196952	−	0.999806i	$-0.506270\pi$
$683$	−1.02944	−0.0393903	−0.0196952	+	0.999806i	$0.506270\pi$
$684$	0	0
$685$	−6.14214	−0.234679
$686$	2.41421	0.0921751
$687$	0	0
$688$	−28.9706	−1.10449
$689$	52.2843	1.99187
$690$	0	0
$691$	−10.1421	−0.385825	−0.192913	−	0.981216i	$-0.561793\pi$
$691$	−10.1421	−0.385825	−0.192913	+	0.981216i	$0.561793\pi$
$692$	−16.6274	−0.632080
$693$	0	0
$694$	−0.828427	−0.0314467
$695$	−6.82843	−0.259017
$696$	0	0
$697$	−13.3726	−0.506523
$698$	70.7696	2.67867
$699$	0	0
$700$	−3.82843	−0.144701
$701$	−3.31371	−0.125157	−0.0625785	−	0.998040i	$-0.519932\pi$
$701$	−3.31371	−0.125157	−0.0625785	+	0.998040i	$0.519932\pi$
$702$	0	0
$703$	−16.9706	−0.640057
$704$	−8.14214	−0.306868
$705$	0	0
$706$	40.1421	1.51077
$707$	15.6569	0.588837
$708$	0	0
$709$	16.6274	0.624456	0.312228	−	0.950007i	$-0.398925\pi$
$709$	16.6274	0.624456	0.312228	+	0.950007i	$0.398925\pi$
$710$	15.6569	0.587591
$711$	0	0
$712$	76.4264	2.86420
$713$	64.9706	2.43317
$714$	0	0
$715$	4.00000	0.149592
$716$	−86.0833	−3.21708
$717$	0	0
$718$	25.3137	0.944699
$719$	−20.6863	−0.771468	−0.385734	−	0.922610i	$-0.626052\pi$
$719$	−20.6863	−0.771468	−0.385734	+	0.922610i	$0.626052\pi$
$720$	0	0
$721$	−7.31371	−0.272377
$722$	26.5563	0.988325
$723$	0	0
$724$	−1.31371	−0.0488236
$725$	−8.00000	−0.297113
$726$	0	0
$727$	−6.34315	−0.235254	−0.117627	−	0.993058i	$-0.537529\pi$
$727$	−6.34315	−0.235254	−0.117627	+	0.993058i	$0.537529\pi$
$728$	−21.3137	−0.789939
$729$	0	0
$730$	38.9706	1.44237
$731$	−35.3137	−1.30612
$732$	0	0
$733$	−1.51472	−0.0559474	−0.0279737	−	0.999609i	$-0.508905\pi$
$733$	−1.51472	−0.0559474	−0.0279737	+	0.999609i	$0.508905\pi$
$734$	−60.2843	−2.22513
$735$	0	0
$736$	−12.1421	−0.447565
$737$	6.05887	0.223182
$738$	0	0
$739$	−18.6274	−0.685221	−0.342610	−	0.939478i	$-0.611311\pi$
$739$	−18.6274	−0.685221	−0.342610	+	0.939478i	$0.611311\pi$
$740$	22.9706	0.844415
$741$	0	0
$742$	−26.1421	−0.959708
$743$	29.3137	1.07542	0.537708	−	0.843131i	$-0.319290\pi$
$743$	29.3137	1.07542	0.537708	+	0.843131i	$0.319290\pi$
$744$	0	0
$745$	−1.65685	−0.0607024
$746$	24.1421	0.883906
$747$	0	0
$748$	11.5980	0.424064
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	46.6274	1.70146	0.850729	−	0.525604i	$-0.176161\pi$
$751$	46.6274	1.70146	0.850729	+	0.525604i	$0.176161\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	−93.2548	−3.39614
$755$	−5.65685	−0.205874
$756$	0	0
$757$	34.9706	1.27103	0.635513	−	0.772090i	$-0.280789\pi$
$757$	34.9706	1.27103	0.635513	+	0.772090i	$0.280789\pi$
$758$	−15.3137	−0.556219
$759$	0	0
$760$	12.4853	0.452889
$761$	24.6274	0.892743	0.446372	−	0.894848i	$-0.352716\pi$
$761$	24.6274	0.892743	0.446372	+	0.894848i	$0.352716\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	3.17157	0.114743
$765$	0	0
$766$	65.9411	2.38255
$767$	−19.3137	−0.697378
$768$	0	0
$769$	−19.6569	−0.708844	−0.354422	−	0.935086i	$-0.615322\pi$
$769$	−19.6569	−0.708844	−0.354422	+	0.935086i	$0.615322\pi$
$770$	−2.00000	−0.0720750
$771$	0	0
$772$	−72.6274	−2.61392
$773$	−36.3431	−1.30717	−0.653586	−	0.756852i	$-0.726736\pi$
$773$	−36.3431	−1.30717	−0.653586	+	0.756852i	$0.726736\pi$
$774$	0	0
$775$	−8.48528	−0.304800
$776$	38.9706	1.39896
$777$	0	0
$778$	71.5980	2.56691
$779$	−10.3431	−0.370582
$780$	0	0
$781$	5.37258	0.192246
$782$	67.5980	2.41730
$783$	0	0
$784$	3.00000	0.107143
$785$	3.17157	0.113198
$786$	0	0
$787$	2.34315	0.0835241	0.0417621	−	0.999128i	$-0.486703\pi$
$787$	2.34315	0.0835241	0.0417621	+	0.999128i	$0.486703\pi$
$788$	10.8284	0.385747
$789$	0	0
$790$	−13.6569	−0.485889
$791$	−4.48528	−0.159478
$792$	0	0
$793$	−28.9706	−1.02877
$794$	−9.31371	−0.330531
$795$	0	0
$796$	87.3970	3.09770
$797$	−50.9706	−1.80547	−0.902735	−	0.430197i	$-0.858444\pi$
$797$	−50.9706	−1.80547	−0.902735	+	0.430197i	$0.858444\pi$
$798$	0	0
$799$	14.6274	0.517481
$800$	1.58579	0.0560660
$801$	0	0
$802$	−44.9706	−1.58797
$803$	13.3726	0.471908
$804$	0	0
$805$	−7.65685	−0.269869
$806$	−98.9117	−3.48402
$807$	0	0
$808$	69.1127	2.43138
$809$	2.34315	0.0823806	0.0411903	−	0.999151i	$-0.486885\pi$
$809$	2.34315	0.0823806	0.0411903	+	0.999151i	$0.486885\pi$
$810$	0	0
$811$	−25.1716	−0.883893	−0.441947	−	0.897041i	$-0.645712\pi$
$811$	−25.1716	−0.883893	−0.441947	+	0.897041i	$0.645712\pi$
$812$	30.6274	1.07481
$813$	0	0
$814$	12.0000	0.420600
$815$	17.6569	0.618493
$816$	0	0
$817$	−27.3137	−0.955586
$818$	−6.48528	−0.226753
$819$	0	0
$820$	14.0000	0.488901
$821$	−29.9411	−1.04495	−0.522476	−	0.852654i	$-0.674992\pi$
$821$	−29.9411	−1.04495	−0.522476	+	0.852654i	$0.674992\pi$
$822$	0	0
$823$	10.3431	0.360539	0.180270	−	0.983617i	$-0.442303\pi$
$823$	10.3431	0.360539	0.180270	+	0.983617i	$0.442303\pi$
$824$	−32.2843	−1.12468
$825$	0	0
$826$	9.65685	0.336005
$827$	−46.0000	−1.59958	−0.799788	−	0.600282i	$-0.795055\pi$
$827$	−46.0000	−1.59958	−0.799788	+	0.600282i	$0.795055\pi$
$828$	0	0
$829$	−11.6569	−0.404859	−0.202430	−	0.979297i	$-0.564884\pi$
$829$	−11.6569	−0.404859	−0.202430	+	0.979297i	$0.564884\pi$
$830$	19.3137	0.670389
$831$	0	0
$832$	47.4558	1.64524
$833$	3.65685	0.126702
$834$	0	0
$835$	−23.3137	−0.806804
$836$	8.97056	0.310253
$837$	0	0
$838$	−25.6569	−0.886301
$839$	−17.6569	−0.609582	−0.304791	−	0.952419i	$-0.598587\pi$
$839$	−17.6569	−0.609582	−0.304791	+	0.952419i	$0.598587\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−33.7990	−1.16479
$843$	0	0
$844$	−45.9411	−1.58136
$845$	−10.3137	−0.354802
$846$	0	0
$847$	10.3137	0.354383
$848$	−32.4853	−1.11555
$849$	0	0
$850$	−8.82843	−0.302813
$851$	45.9411	1.57484
$852$	0	0
$853$	42.4853	1.45467	0.727334	−	0.686283i	$-0.240759\pi$
$853$	42.4853	1.45467	0.727334	+	0.686283i	$0.240759\pi$
$854$	14.4853	0.495676
$855$	0	0
$856$	−8.82843	−0.301749
$857$	−0.343146	−0.0117216	−0.00586082	−	0.999983i	$-0.501866\pi$
$857$	−0.343146	−0.0117216	−0.00586082	+	0.999983i	$0.501866\pi$
$858$	0	0
$859$	−38.1421	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$859$	−38.1421	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$860$	36.9706	1.26069
$861$	0	0
$862$	30.9706	1.05486
$863$	−54.2843	−1.84786	−0.923929	−	0.382564i	$-0.875041\pi$
$863$	−54.2843	−1.84786	−0.923929	+	0.382564i	$0.875041\pi$
$864$	0	0
$865$	4.34315	0.147671
$866$	−62.2843	−2.11651
$867$	0	0
$868$	32.4853	1.10262
$869$	−4.68629	−0.158972
$870$	0	0
$871$	−35.3137	−1.19656
$872$	−44.1421	−1.49484
$873$	0	0
$874$	52.2843	1.76854
$875$	1.00000	0.0338062
$876$	0	0
$877$	−20.3431	−0.686939	−0.343470	−	0.939164i	$-0.611602\pi$
$877$	−20.3431	−0.686939	−0.343470	+	0.939164i	$0.611602\pi$
$878$	−32.4853	−1.09633
$879$	0	0
$880$	−2.48528	−0.0837788
$881$	10.6863	0.360030	0.180015	−	0.983664i	$-0.442385\pi$
$881$	10.6863	0.360030	0.180015	+	0.983664i	$0.442385\pi$
$882$	0	0
$883$	7.31371	0.246126	0.123063	−	0.992399i	$-0.460728\pi$
$883$	7.31371	0.246126	0.123063	+	0.992399i	$0.460728\pi$
$884$	−67.5980	−2.27357
$885$	0	0
$886$	2.48528	0.0834947
$887$	31.3137	1.05141	0.525706	−	0.850667i	$-0.323801\pi$
$887$	31.3137	1.05141	0.525706	+	0.850667i	$0.323801\pi$
$888$	0	0
$889$	−13.6569	−0.458036
$890$	−41.7990	−1.40111
$891$	0	0
$892$	−21.6569	−0.725125
$893$	11.3137	0.378599
$894$	0	0
$895$	22.4853	0.751600
$896$	−20.5563	−0.686739
$897$	0	0
$898$	8.00000	0.266963
$899$	67.8823	2.26400
$900$	0	0
$901$	−39.5980	−1.31920
$902$	7.31371	0.243520
$903$	0	0
$904$	−19.7990	−0.658505
$905$	0.343146	0.0114066
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	76.5685	2.54102
$909$	0	0
$910$	11.6569	0.386421
$911$	−3.85786	−0.127817	−0.0639084	−	0.997956i	$-0.520357\pi$
$911$	−3.85786	−0.127817	−0.0639084	+	0.997956i	$0.520357\pi$
$912$	0	0
$913$	6.62742	0.219335
$914$	−3.17157	−0.104906
$915$	0	0
$916$	20.3431	0.672156
$917$	−2.34315	−0.0773775
$918$	0	0
$919$	−22.6274	−0.746410	−0.373205	−	0.927749i	$-0.621741\pi$
$919$	−22.6274	−0.746410	−0.373205	+	0.927749i	$0.621741\pi$
$920$	−33.7990	−1.11432
$921$	0	0
$922$	−69.1127	−2.27611
$923$	−31.3137	−1.03070
$924$	0	0
$925$	−6.00000	−0.197279
$926$	24.9706	0.820584
$927$	0	0
$928$	−12.6863	−0.416448
$929$	51.9411	1.70413	0.852067	−	0.523434i	$-0.175349\pi$
$929$	51.9411	1.70413	0.852067	+	0.523434i	$0.175349\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	−23.5147	−0.770250
$933$	0	0
$934$	20.9706	0.686178
$935$	−3.02944	−0.0990732
$936$	0	0
$937$	−13.7990	−0.450793	−0.225397	−	0.974267i	$-0.572368\pi$
$937$	−13.7990	−0.450793	−0.225397	+	0.974267i	$0.572368\pi$
$938$	17.6569	0.576517
$939$	0	0
$940$	−15.3137	−0.499478
$941$	−4.62742	−0.150849	−0.0754247	−	0.997151i	$-0.524031\pi$
$941$	−4.62742	−0.150849	−0.0754247	+	0.997151i	$0.524031\pi$
$942$	0	0
$943$	28.0000	0.911805
$944$	12.0000	0.390567
$945$	0	0
$946$	19.3137	0.627943
$947$	18.9706	0.616460	0.308230	−	0.951312i	$-0.400263\pi$
$947$	18.9706	0.616460	0.308230	+	0.951312i	$0.400263\pi$
$948$	0	0
$949$	−77.9411	−2.53008
$950$	−6.82843	−0.221543
$951$	0	0
$952$	16.1421	0.523170
$953$	51.1127	1.65570	0.827851	−	0.560948i	$-0.189563\pi$
$953$	51.1127	1.65570	0.827851	+	0.560948i	$0.189563\pi$
$954$	0	0
$955$	−0.828427	−0.0268073
$956$	−15.8579	−0.512880
$957$	0	0
$958$	−100.569	−3.24922
$959$	−6.14214	−0.198340
$960$	0	0
$961$	41.0000	1.32258
$962$	−69.9411	−2.25499
$963$	0	0
$964$	14.0000	0.450910
$965$	18.9706	0.610684
$966$	0	0
$967$	−38.6274	−1.24217	−0.621087	−	0.783742i	$-0.713309\pi$
$967$	−38.6274	−1.24217	−0.621087	+	0.783742i	$0.713309\pi$
$968$	45.5269	1.46329
$969$	0	0
$970$	−21.3137	−0.684342
$971$	16.9706	0.544611	0.272306	−	0.962211i	$-0.412214\pi$
$971$	16.9706	0.544611	0.272306	+	0.962211i	$0.412214\pi$
$972$	0	0
$973$	−6.82843	−0.218909
$974$	32.9706	1.05644
$975$	0	0
$976$	18.0000	0.576166
$977$	−6.14214	−0.196504	−0.0982522	−	0.995162i	$-0.531325\pi$
$977$	−6.14214	−0.196504	−0.0982522	+	0.995162i	$0.531325\pi$
$978$	0	0
$979$	−14.3431	−0.458409
$980$	−3.82843	−0.122295
$981$	0	0
$982$	−38.9706	−1.24360
$983$	41.9411	1.33771	0.668857	−	0.743391i	$-0.266784\pi$
$983$	41.9411	1.33771	0.668857	+	0.743391i	$0.266784\pi$
$984$	0	0
$985$	−2.82843	−0.0901212
$986$	70.6274	2.24924
$987$	0	0
$988$	−52.2843	−1.66338
$989$	73.9411	2.35119
$990$	0	0
$991$	−51.3137	−1.63003	−0.815017	−	0.579437i	$-0.803272\pi$
$991$	−51.3137	−1.63003	−0.815017	+	0.579437i	$0.803272\pi$
$992$	−13.4558	−0.427223
$993$	0	0
$994$	15.6569	0.496605
$995$	−22.8284	−0.723710
$996$	0	0
$997$	−37.5147	−1.18810	−0.594052	−	0.804427i	$-0.702472\pi$
$997$	−37.5147	−1.18810	−0.594052	+	0.804427i	$0.702472\pi$
$998$	−61.9411	−1.96071
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.a.c.1.1	✓	2
3.2	odd	2		315.2.a.f.1.2	yes	2
4.3	odd	2		5040.2.a.bu.1.1		2
5.2	odd	4		1575.2.d.h.1324.1		4
5.3	odd	4		1575.2.d.h.1324.4		4
5.4	even	2		1575.2.a.u.1.2		2
7.6	odd	2		2205.2.a.p.1.1		2
12.11	even	2		5040.2.a.bx.1.2		2
15.2	even	4		1575.2.d.j.1324.4		4
15.8	even	4		1575.2.d.j.1324.1		4
15.14	odd	2		1575.2.a.m.1.1		2
21.20	even	2		2205.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.a.c.1.1	✓	2	1.1	even	1	trivial
315.2.a.f.1.2	yes	2	3.2	odd	2
1575.2.a.m.1.1		2	15.14	odd	2
1575.2.a.u.1.2		2	5.4	even	2
1575.2.d.h.1324.1		4	5.2	odd	4
1575.2.d.h.1324.4		4	5.3	odd	4
1575.2.d.j.1324.1		4	15.8	even	4
1575.2.d.j.1324.4		4	15.2	even	4
2205.2.a.p.1.1		2	7.6	odd	2
2205.2.a.y.1.2		2	21.20	even	2
5040.2.a.bu.1.1		2	4.3	odd	2
5040.2.a.bx.1.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -1.00000 q^{7} -4.41421 q^{8} +2.41421 q^{10} +0.828427 q^{11} -4.82843 q^{13} +2.41421 q^{14} +3.00000 q^{16} +3.65685 q^{17} +2.82843 q^{19} -3.82843 q^{20} -2.00000 q^{22} -7.65685 q^{23} +1.00000 q^{25} +11.6569 q^{26} -3.82843 q^{28} -8.00000 q^{29} -8.48528 q^{31} +1.58579 q^{32} -8.82843 q^{34} +1.00000 q^{35} -6.00000 q^{37} -6.82843 q^{38} +4.41421 q^{40} -3.65685 q^{41} -9.65685 q^{43} +3.17157 q^{44} +18.4853 q^{46} +4.00000 q^{47} +1.00000 q^{49} -2.41421 q^{50} -18.4853 q^{52} -10.8284 q^{53} -0.828427 q^{55} +4.41421 q^{56} +19.3137 q^{58} +4.00000 q^{59} +6.00000 q^{61} +20.4853 q^{62} -9.82843 q^{64} +4.82843 q^{65} +7.31371 q^{67} +14.0000 q^{68} -2.41421 q^{70} +6.48528 q^{71} +16.1421 q^{73} +14.4853 q^{74} +10.8284 q^{76} -0.828427 q^{77} -5.65685 q^{79} -3.00000 q^{80} +8.82843 q^{82} +8.00000 q^{83} -3.65685 q^{85} +23.3137 q^{86} -3.65685 q^{88} -17.3137 q^{89} +4.82843 q^{91} -29.3137 q^{92} -9.65685 q^{94} -2.82843 q^{95} -8.82843 q^{97} -2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{13} + 2 q^{14} + 6 q^{16} - 4 q^{17} - 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{28} - 16 q^{29} + 6 q^{32} - 12 q^{34} + 2 q^{35} - 12 q^{37} - 8 q^{38} + 6 q^{40} + 4 q^{41} - 8 q^{43} + 12 q^{44} + 20 q^{46} + 8 q^{47} + 2 q^{49} - 2 q^{50} - 20 q^{52} - 16 q^{53} + 4 q^{55} + 6 q^{56} + 16 q^{58} + 8 q^{59} + 12 q^{61} + 24 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{67} + 28 q^{68} - 2 q^{70} - 4 q^{71} + 4 q^{73} + 12 q^{74} + 16 q^{76} + 4 q^{77} - 6 q^{80} + 12 q^{82} + 16 q^{83} + 4 q^{85} + 24 q^{86} + 4 q^{88} - 12 q^{89} + 4 q^{91} - 36 q^{92} - 8 q^{94} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^13 + 2 * q^14 + 6 * q^16 - 4 * q^17 - 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^28 - 16 * q^29 + 6 * q^32 - 12 * q^34 + 2 * q^35 - 12 * q^37 - 8 * q^38 + 6 * q^40 + 4 * q^41 - 8 * q^43 + 12 * q^44 + 20 * q^46 + 8 * q^47 + 2 * q^49 - 2 * q^50 - 20 * q^52 - 16 * q^53 + 4 * q^55 + 6 * q^56 + 16 * q^58 + 8 * q^59 + 12 * q^61 + 24 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^67 + 28 * q^68 - 2 * q^70 - 4 * q^71 + 4 * q^73 + 12 * q^74 + 16 * q^76 + 4 * q^77 - 6 * q^80 + 12 * q^82 + 16 * q^83 + 4 * q^85 + 24 * q^86 + 4 * q^88 - 12 * q^89 + 4 * q^91 - 36 * q^92 - 8 * q^94 - 12 * q^97 - 2 * q^98

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