LMFDB - Embedded newform 1575.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
Analytic rank:	$0$
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:	no (minimal twist has level 315)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{7} +4.41421 q^{8} +O(q^{10})$	$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{7} +4.41421 q^{8} +0.828427 q^{11} +4.82843 q^{13} +2.41421 q^{14} +3.00000 q^{16} -3.65685 q^{17} +2.82843 q^{19} +2.00000 q^{22} +7.65685 q^{23} +11.6569 q^{26} +3.82843 q^{28} -8.00000 q^{29} -8.48528 q^{31} -1.58579 q^{32} -8.82843 q^{34} +6.00000 q^{37} +6.82843 q^{38} -3.65685 q^{41} +9.65685 q^{43} +3.17157 q^{44} +18.4853 q^{46} -4.00000 q^{47} +1.00000 q^{49} +18.4853 q^{52} +10.8284 q^{53} +4.41421 q^{56} -19.3137 q^{58} +4.00000 q^{59} +6.00000 q^{61} -20.4853 q^{62} -9.82843 q^{64} -7.31371 q^{67} -14.0000 q^{68} +6.48528 q^{71} -16.1421 q^{73} +14.4853 q^{74} +10.8284 q^{76} +0.828427 q^{77} -5.65685 q^{79} -8.82843 q^{82} -8.00000 q^{83} +23.3137 q^{86} +3.65685 q^{88} -17.3137 q^{89} +4.82843 q^{91} +29.3137 q^{92} -9.65685 q^{94} +8.82843 q^{97} +2.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 6 q^{8} - 4 q^{11} + 4 q^{13} + 2 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{22} + 4 q^{23} + 12 q^{26} + 2 q^{28} - 16 q^{29} - 6 q^{32} - 12 q^{34} + 12 q^{37} + 8 q^{38} + 4 q^{41} + 8 q^{43} + 12 q^{44} + 20 q^{46} - 8 q^{47} + 2 q^{49} + 20 q^{52} + 16 q^{53} + 6 q^{56} - 16 q^{58} + 8 q^{59} + 12 q^{61} - 24 q^{62} - 14 q^{64} + 8 q^{67} - 28 q^{68} - 4 q^{71} - 4 q^{73} + 12 q^{74} + 16 q^{76} - 4 q^{77} - 12 q^{82} - 16 q^{83} + 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^7 + 6 * q^8 - 4 * q^11 + 4 * q^13 + 2 * q^14 + 6 * q^16 + 4 * q^17 + 4 * q^22 + 4 * q^23 + 12 * q^26 + 2 * q^28 - 16 * q^29 - 6 * q^32 - 12 * q^34 + 12 * q^37 + 8 * q^38 + 4 * q^41 + 8 * q^43 + 12 * q^44 + 20 * q^46 - 8 * q^47 + 2 * q^49 + 20 * q^52 + 16 * q^53 + 6 * q^56 - 16 * q^58 + 8 * q^59 + 12 * q^61 - 24 * q^62 - 14 * q^64 + 8 * q^67 - 28 * q^68 - 4 * q^71 - 4 * q^73 + 12 * q^74 + 16 * q^76 - 4 * q^77 - 12 * q^82 - 16 * q^83 + 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	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	4.41421	1.56066
$9$	0	0
$10$	0	0
$11$	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	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	2.41421	0.645226
$15$	0	0
$16$	3.00000	0.750000
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	7.65685	1.59656	0.798282	−	0.602284i	$-0.205742\pi$
$23$	7.65685	1.59656	0.798282	+	0.602284i	$0.205742\pi$
$24$	0	0
$25$	0	0
$26$	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
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	6.82843	1.10772
$39$	0	0
$40$	0	0
$41$	−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	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	3.17157	0.478133
$45$	0	0
$46$	18.4853	2.72551
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	18.4853	2.56345
$53$	10.8284	1.48740	0.743699	−	0.668514i	$-0.233069\pi$
$53$	10.8284	1.48740	0.743699	+	0.668514i	$0.233069\pi$
$54$	0	0
$55$	0	0
$56$	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
$65$	0	0
$66$	0	0
$67$	−7.31371	−0.893512	−0.446756	−	0.894656i	$-0.647421\pi$
$67$	−7.31371	−0.893512	−0.446756	+	0.894656i	$0.647421\pi$
$68$	−14.0000	−1.69775
$69$	0	0
$70$	0	0
$71$	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	$-0.893596\pi$
$73$	−16.1421	−1.88929	−0.944647	+	0.328088i	$0.893596\pi$
$74$	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	$0.603086\pi$
$80$	0	0
$81$	0	0
$82$	−8.82843	−0.974937
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	8.82843	0.896391	0.448195	−	0.893936i	$-0.352067\pi$
$97$	8.82843	0.896391	0.448195	+	0.893936i	$0.352067\pi$
$98$	2.41421	0.243872
$99$	0	0
$100$	0	0
$101$	−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	$-0.617333\pi$
$103$	−7.31371	−0.720641	−0.360321	+	0.932829i	$0.617333\pi$
$104$	21.3137	2.08998
$105$	0	0
$106$	26.1421	2.53915
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	3.00000	0.283473
$113$	−4.48528	−0.421940	−0.210970	−	0.977493i	$-0.567662\pi$
$113$	−4.48528	−0.421940	−0.210970	+	0.977493i	$0.567662\pi$
$114$	0	0
$115$	0	0
$116$	−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
$125$	0	0
$126$	0	0
$127$	−13.6569	−1.21185	−0.605925	−	0.795522i	$-0.707197\pi$
$127$	−13.6569	−1.21185	−0.605925	+	0.795522i	$0.707197\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	0	0
$131$	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	$-0.584507\pi$
$137$	−6.14214	−0.524758	−0.262379	+	0.964965i	$0.584507\pi$
$138$	0	0
$139$	6.82843	0.579180	0.289590	−	0.957151i	$-0.406481\pi$
$139$	6.82843	0.579180	0.289590	+	0.957151i	$0.406481\pi$
$140$	0	0
$141$	0	0
$142$	15.6569	1.31389
$143$	4.00000	0.334497
$144$	0	0
$145$	0	0
$146$	−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
$155$	0	0
$156$	0	0
$157$	3.17157	0.253119	0.126560	−	0.991959i	$-0.459607\pi$
$157$	3.17157	0.253119	0.126560	+	0.991959i	$0.459607\pi$
$158$	−13.6569	−1.08648
$159$	0	0
$160$	0	0
$161$	7.65685	0.603445
$162$	0	0
$163$	17.6569	1.38299	0.691496	−	0.722380i	$-0.256952\pi$
$163$	17.6569	1.38299	0.691496	+	0.722380i	$0.256952\pi$
$164$	−14.0000	−1.09322
$165$	0	0
$166$	−19.3137	−1.49903
$167$	−23.3137	−1.80407	−0.902034	−	0.431664i	$-0.857927\pi$
$167$	−23.3137	−1.80407	−0.902034	+	0.431664i	$0.857927\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	36.9706	2.81898
$173$	4.34315	0.330203	0.165102	−	0.986277i	$-0.447205\pi$
$173$	4.34315	0.330203	0.165102	+	0.986277i	$0.447205\pi$
$174$	0	0
$175$	0	0
$176$	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
$185$	0	0
$186$	0	0
$187$	−3.02944	−0.221534
$188$	−15.3137	−1.11687
$189$	0	0
$190$	0	0
$191$	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	$-0.260777\pi$
$193$	18.9706	1.36553	0.682765	+	0.730638i	$0.260777\pi$
$194$	21.3137	1.53024
$195$	0	0
$196$	3.82843	0.273459
$197$	−2.82843	−0.201517	−0.100759	−	0.994911i	$-0.532127\pi$
$197$	−2.82843	−0.201517	−0.100759	+	0.994911i	$0.532127\pi$
$198$	0	0
$199$	22.8284	1.61826	0.809132	−	0.587627i	$-0.199938\pi$
$199$	22.8284	1.61826	0.809132	+	0.587627i	$0.199938\pi$
$200$	0	0
$201$	0	0
$202$	−37.7990	−2.65953
$203$	−8.00000	−0.561490
$204$	0	0
$205$	0	0
$206$	−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
$215$	0	0
$216$	0	0
$217$	−8.48528	−0.576018
$218$	24.1421	1.63511
$219$	0	0
$220$	0	0
$221$	−17.6569	−1.18773
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	−1.58579	−0.105955
$225$	0	0
$226$	−10.8284	−0.720296
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	5.31371	0.351140	0.175570	−	0.984467i	$-0.443823\pi$
$229$	5.31371	0.351140	0.175570	+	0.984467i	$0.443823\pi$
$230$	0	0
$231$	0	0
$232$	−35.3137	−2.31846
$233$	6.14214	0.402385	0.201192	−	0.979552i	$-0.435518\pi$
$233$	6.14214	0.402385	0.201192	+	0.979552i	$0.435518\pi$
$234$	0	0
$235$	0	0
$236$	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
$245$	0	0
$246$	0	0
$247$	13.6569	0.868965
$248$	−37.4558	−2.37845
$249$	0	0
$250$	0	0
$251$	−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	$-0.318424\pi$
$257$	17.3137	1.08000	0.540000	+	0.841665i	$0.318424\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	5.65685	0.349482
$263$	−25.3137	−1.56091	−0.780455	−	0.625212i	$-0.785013\pi$
$263$	−25.3137	−1.56091	−0.780455	+	0.625212i	$0.785013\pi$
$264$	0	0
$265$	0	0
$266$	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	0
$276$	0	0
$277$	4.34315	0.260954	0.130477	−	0.991451i	$-0.458349\pi$
$277$	4.34315	0.260954	0.130477	+	0.991451i	$0.458349\pi$
$278$	16.4853	0.988721
$279$	0	0
$280$	0	0
$281$	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	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	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
$290$	0	0
$291$	0	0
$292$	−61.7990	−3.61651
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	0	0
$307$	4.97056	0.283685	0.141843	−	0.989889i	$-0.454697\pi$
$307$	4.97056	0.283685	0.141843	+	0.989889i	$0.454697\pi$
$308$	3.17157	0.180717
$309$	0	0
$310$	0	0
$311$	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	$-0.299661\pi$
$313$	20.8284	1.17729	0.588646	+	0.808391i	$0.299661\pi$
$314$	7.65685	0.432101
$315$	0	0
$316$	−21.6569	−1.21829
$317$	14.1421	0.794301	0.397151	−	0.917753i	$-0.369999\pi$
$317$	14.1421	0.794301	0.397151	+	0.917753i	$0.369999\pi$
$318$	0	0
$319$	−6.62742	−0.371064
$320$	0	0
$321$	0	0
$322$	18.4853	1.03014
$323$	−10.3431	−0.575508
$324$	0	0
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	22.2843	1.21390	0.606951	−	0.794739i	$-0.292392\pi$
$337$	22.2843	1.21390	0.606951	+	0.794739i	$0.292392\pi$
$338$	24.8995	1.35435
$339$	0	0
$340$	0	0
$341$	−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	$-0.502932\pi$
$347$	−0.343146	−0.0184210	−0.00921051	+	0.999958i	$0.502932\pi$
$348$	0	0
$349$	−29.3137	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$349$	−29.3137	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$350$	0	0
$351$	0	0
$352$	−1.31371	−0.0700209
$353$	16.6274	0.884988	0.442494	−	0.896771i	$-0.354094\pi$
$353$	16.6274	0.884988	0.442494	+	0.896771i	$0.354094\pi$
$354$	0	0
$355$	0	0
$356$	−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
$365$	0	0
$366$	0	0
$367$	−24.9706	−1.30345	−0.651726	−	0.758454i	$-0.725955\pi$
$367$	−24.9706	−1.30345	−0.651726	+	0.758454i	$0.725955\pi$
$368$	22.9706	1.19742
$369$	0	0
$370$	0	0
$371$	10.8284	0.562184
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−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	$0.447911\pi$
$380$	0	0
$381$	0	0
$382$	2.00000	0.102329
$383$	27.3137	1.39567	0.697833	−	0.716261i	$-0.254148\pi$
$383$	27.3137	1.39567	0.697833	+	0.716261i	$0.254148\pi$
$384$	0	0
$385$	0	0
$386$	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
$395$	0	0
$396$	0	0
$397$	−3.85786	−0.193621	−0.0968103	−	0.995303i	$-0.530864\pi$
$397$	−3.85786	−0.193621	−0.0968103	+	0.995303i	$0.530864\pi$
$398$	55.1127	2.76255
$399$	0	0
$400$	0	0
$401$	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	$0.478844\pi$
$410$	0	0
$411$	0	0
$412$	−28.0000	−1.37946
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	−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
$425$	0	0
$426$	0	0
$427$	6.00000	0.290360
$428$	−7.65685	−0.370108
$429$	0	0
$430$	0	0
$431$	−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	$-0.712831\pi$
$433$	−25.7990	−1.23982	−0.619910	+	0.784673i	$0.712831\pi$
$434$	−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	$0.395945\pi$
$440$	0	0
$441$	0	0
$442$	−42.6274	−2.02758
$443$	1.02944	0.0489100	0.0244550	−	0.999701i	$-0.492215\pi$
$443$	1.02944	0.0489100	0.0244550	+	0.999701i	$0.492215\pi$
$444$	0	0
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	−1.31371	−0.0614527	−0.0307263	−	0.999528i	$-0.509782\pi$
$457$	−1.31371	−0.0614527	−0.0307263	+	0.999528i	$0.509782\pi$
$458$	12.8284	0.599433
$459$	0	0
$460$	0	0
$461$	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	$-0.422740\pi$
$463$	10.3431	0.480687	0.240343	+	0.970688i	$0.422740\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	14.8284	0.686914
$467$	8.68629	0.401954	0.200977	−	0.979596i	$-0.435588\pi$
$467$	8.68629	0.401954	0.200977	+	0.979596i	$0.435588\pi$
$468$	0	0
$469$	−7.31371	−0.337716
$470$	0	0
$471$	0	0
$472$	17.6569	0.812723
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	−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
$485$	0	0
$486$	0	0
$487$	13.6569	0.618851	0.309426	−	0.950924i	$-0.399863\pi$
$487$	13.6569	0.618851	0.309426	+	0.950924i	$0.399863\pi$
$488$	26.4853	1.19893
$489$	0	0
$490$	0	0
$491$	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	$0.305282\pi$
$500$	0	0
$501$	0	0
$502$	−44.9706	−2.00713
$503$	−14.6274	−0.652204	−0.326102	−	0.945335i	$-0.605735\pi$
$503$	−14.6274	−0.652204	−0.326102	+	0.945335i	$0.605735\pi$
$504$	0	0
$505$	0	0
$506$	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
$515$	0	0
$516$	0	0
$517$	−3.31371	−0.145737
$518$	14.4853	0.636447
$519$	0	0
$520$	0	0
$521$	−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	$-0.368647\pi$
$523$	18.3431	0.802090	0.401045	+	0.916058i	$0.368647\pi$
$524$	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
$530$	0	0
$531$	0	0
$532$	10.8284	0.469472
$533$	−17.6569	−0.764803
$534$	0	0
$535$	0	0
$536$	−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
$545$	0	0
$546$	0	0
$547$	−0.686292	−0.0293437	−0.0146719	−	0.999892i	$-0.504670\pi$
$547$	−0.686292	−0.0293437	−0.0146719	+	0.999892i	$0.504670\pi$
$548$	−23.5147	−1.00450
$549$	0	0
$550$	0	0
$551$	−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	$-0.307571\pi$
$557$	26.8284	1.13676	0.568378	+	0.822767i	$0.307571\pi$
$558$	0	0
$559$	46.6274	1.97213
$560$	0	0
$561$	0	0
$562$	34.6274	1.46067
$563$	29.9411	1.26187	0.630934	−	0.775837i	$-0.282672\pi$
$563$	29.9411	1.26187	0.630934	+	0.775837i	$0.282672\pi$
$564$	0	0
$565$	0	0
$566$	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
$575$	0	0
$576$	0	0
$577$	25.5147	1.06219	0.531096	−	0.847312i	$-0.321780\pi$
$577$	25.5147	1.06219	0.531096	+	0.847312i	$0.321780\pi$
$578$	−8.75736	−0.364258
$579$	0	0
$580$	0	0
$581$	−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	$-0.246603\pi$
$587$	34.6274	1.42923	0.714613	+	0.699520i	$0.246603\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	0	0
$592$	18.0000	0.739795
$593$	−17.3137	−0.710989	−0.355494	−	0.934678i	$-0.615687\pi$
$593$	−17.3137	−0.710989	−0.355494	+	0.934678i	$0.615687\pi$
$594$	0	0
$595$	0	0
$596$	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
$605$	0	0
$606$	0	0
$607$	−15.0294	−0.610026	−0.305013	−	0.952348i	$-0.598661\pi$
$607$	−15.0294	−0.610026	−0.305013	+	0.952348i	$0.598661\pi$
$608$	−4.48528	−0.181902
$609$	0	0
$610$	0	0
$611$	−19.3137	−0.781349
$612$	0	0
$613$	−14.9706	−0.604655	−0.302328	−	0.953204i	$-0.597764\pi$
$613$	−14.9706	−0.604655	−0.302328	+	0.953204i	$0.597764\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	3.65685	0.147339
$617$	−21.1716	−0.852335	−0.426168	−	0.904644i	$-0.640137\pi$
$617$	−21.1716	−0.852335	−0.426168	+	0.904644i	$0.640137\pi$
$618$	0	0
$619$	−28.4853	−1.14492	−0.572460	−	0.819933i	$-0.694011\pi$
$619$	−28.4853	−1.14492	−0.572460	+	0.819933i	$0.694011\pi$
$620$	0	0
$621$	0	0
$622$	81.2548	3.25802
$623$	−17.3137	−0.693659
$624$	0	0
$625$	0	0
$626$	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
$635$	0	0
$636$	0	0
$637$	4.82843	0.191309
$638$	−16.0000	−0.633446
$639$	0	0
$640$	0	0
$641$	−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	$-0.645221\pi$
$643$	−22.3431	−0.881128	−0.440564	+	0.897721i	$0.645221\pi$
$644$	29.3137	1.15512
$645$	0	0
$646$	−24.9706	−0.982454
$647$	18.6274	0.732319	0.366160	−	0.930552i	$-0.380672\pi$
$647$	18.6274	0.732319	0.366160	+	0.930552i	$0.380672\pi$
$648$	0	0
$649$	3.31371	0.130074
$650$	0	0
$651$	0	0
$652$	67.5980	2.64734
$653$	5.85786	0.229236	0.114618	−	0.993410i	$-0.463436\pi$
$653$	5.85786	0.229236	0.114618	+	0.993410i	$0.463436\pi$
$654$	0	0
$655$	0	0
$656$	−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
$665$	0	0
$666$	0	0
$667$	−61.2548	−2.37180
$668$	−89.2548	−3.45337
$669$	0	0
$670$	0	0
$671$	4.97056	0.191886
$672$	0	0
$673$	−4.62742	−0.178374	−0.0891869	−	0.996015i	$-0.528427\pi$
$673$	−4.62742	−0.178374	−0.0891869	+	0.996015i	$0.528427\pi$
$674$	53.7990	2.07226
$675$	0	0
$676$	39.4853	1.51866
$677$	21.3137	0.819152	0.409576	−	0.912276i	$-0.365677\pi$
$677$	21.3137	0.819152	0.409576	+	0.912276i	$0.365677\pi$
$678$	0	0
$679$	8.82843	0.338804
$680$	0	0
$681$	0	0
$682$	−16.9706	−0.649836
$683$	1.02944	0.0393903	0.0196952	−	0.999806i	$-0.493730\pi$
$683$	1.02944	0.0393903	0.0196952	+	0.999806i	$0.493730\pi$
$684$	0	0
$685$	0	0
$686$	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
$695$	0	0
$696$	0	0
$697$	13.3726	0.506523
$698$	−70.7696	−2.67867
$699$	0	0
$700$	0	0
$701$	−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	$0.398925\pi$
$710$	0	0
$711$	0	0
$712$	−76.4264	−2.86420
$713$	−64.9706	−2.43317
$714$	0	0
$715$	0	0
$716$	−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
$725$	0	0
$726$	0	0
$727$	6.34315	0.235254	0.117627	−	0.993058i	$-0.462471\pi$
$727$	6.34315	0.235254	0.117627	+	0.993058i	$0.462471\pi$
$728$	21.3137	0.789939
$729$	0	0
$730$	0	0
$731$	−35.3137	−1.30612
$732$	0	0
$733$	1.51472	0.0559474	0.0279737	−	0.999609i	$-0.491095\pi$
$733$	1.51472	0.0559474	0.0279737	+	0.999609i	$0.491095\pi$
$734$	−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	$0.611311\pi$
$740$	0	0
$741$	0	0
$742$	26.1421	0.959708
$743$	−29.3137	−1.07542	−0.537708	−	0.843131i	$-0.680710\pi$
$743$	−29.3137	−1.07542	−0.537708	+	0.843131i	$0.680710\pi$
$744$	0	0
$745$	0	0
$746$	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
$755$	0	0
$756$	0	0
$757$	−34.9706	−1.27103	−0.635513	−	0.772090i	$-0.719211\pi$
$757$	−34.9706	−1.27103	−0.635513	+	0.772090i	$0.719211\pi$
$758$	15.3137	0.556219
$759$	0	0
$760$	0	0
$761$	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	$0.615322\pi$
$770$	0	0
$771$	0	0
$772$	72.6274	2.61392
$773$	36.3431	1.30717	0.653586	−	0.756852i	$-0.273264\pi$
$773$	36.3431	1.30717	0.653586	+	0.756852i	$0.273264\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	0	0
$787$	−2.34315	−0.0835241	−0.0417621	−	0.999128i	$-0.513297\pi$
$787$	−2.34315	−0.0835241	−0.0417621	+	0.999128i	$0.513297\pi$
$788$	−10.8284	−0.385747
$789$	0	0
$790$	0	0
$791$	−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	$-0.141556\pi$
$797$	50.9706	1.80547	0.902735	+	0.430197i	$0.141556\pi$
$798$	0	0
$799$	14.6274	0.517481
$800$	0	0
$801$	0	0
$802$	44.9706	1.58797
$803$	−13.3726	−0.471908
$804$	0	0
$805$	0	0
$806$	−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
$815$	0	0
$816$	0	0
$817$	27.3137	0.955586
$818$	6.48528	0.226753
$819$	0	0
$820$	0	0
$821$	−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	$-0.557697\pi$
$823$	−10.3431	−0.360539	−0.180270	+	0.983617i	$0.557697\pi$
$824$	−32.2843	−1.12468
$825$	0	0
$826$	9.65685	0.336005
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	0	0
$829$	−11.6569	−0.404859	−0.202430	−	0.979297i	$-0.564884\pi$
$829$	−11.6569	−0.404859	−0.202430	+	0.979297i	$0.564884\pi$
$830$	0	0
$831$	0	0
$832$	−47.4558	−1.64524
$833$	−3.65685	−0.126702
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	−10.3137	−0.354383
$848$	32.4853	1.11555
$849$	0	0
$850$	0	0
$851$	45.9411	1.57484
$852$	0	0
$853$	−42.4853	−1.45467	−0.727334	−	0.686283i	$-0.759241\pi$
$853$	−42.4853	−1.45467	−0.727334	+	0.686283i	$0.759241\pi$
$854$	14.4853	0.495676
$855$	0	0
$856$	−8.82843	−0.301749
$857$	0.343146	0.0117216	0.00586082	−	0.999983i	$-0.498134\pi$
$857$	0.343146	0.0117216	0.00586082	+	0.999983i	$0.498134\pi$
$858$	0	0
$859$	−38.1421	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$859$	−38.1421	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$860$	0	0
$861$	0	0
$862$	−30.9706	−1.05486
$863$	54.2843	1.84786	0.923929	−	0.382564i	$-0.124959\pi$
$863$	54.2843	1.84786	0.923929	+	0.382564i	$0.124959\pi$
$864$	0	0
$865$	0	0
$866$	−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
$875$	0	0
$876$	0	0
$877$	20.3431	0.686939	0.343470	−	0.939164i	$-0.388398\pi$
$877$	20.3431	0.686939	0.343470	+	0.939164i	$0.388398\pi$
$878$	32.4853	1.09633
$879$	0	0
$880$	0	0
$881$	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	$-0.539272\pi$
$883$	−7.31371	−0.246126	−0.123063	+	0.992399i	$0.539272\pi$
$884$	−67.5980	−2.27357
$885$	0	0
$886$	2.48528	0.0834947
$887$	−31.3137	−1.05141	−0.525706	−	0.850667i	$-0.676199\pi$
$887$	−31.3137	−1.05141	−0.525706	+	0.850667i	$0.676199\pi$
$888$	0	0
$889$	−13.6569	−0.458036
$890$	0	0
$891$	0	0
$892$	21.6569	0.725125
$893$	−11.3137	−0.378599
$894$	0	0
$895$	0	0
$896$	−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	0
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−76.5685	−2.54102
$909$	0	0
$910$	0	0
$911$	−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	$0.621741\pi$
$920$	0	0
$921$	0	0
$922$	69.1127	2.27611
$923$	31.3137	1.03070
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	13.7990	0.450793	0.225397	−	0.974267i	$-0.427632\pi$
$937$	13.7990	0.450793	0.225397	+	0.974267i	$0.427632\pi$
$938$	−17.6569	−0.576517
$939$	0	0
$940$	0	0
$941$	−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	$-0.599737\pi$
$947$	−18.9706	−0.616460	−0.308230	+	0.951312i	$0.599737\pi$
$948$	0	0
$949$	−77.9411	−2.53008
$950$	0	0
$951$	0	0
$952$	−16.1421	−0.523170
$953$	−51.1127	−1.65570	−0.827851	−	0.560948i	$-0.810437\pi$
$953$	−51.1127	−1.65570	−0.827851	+	0.560948i	$0.810437\pi$
$954$	0	0
$955$	0	0
$956$	−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
$965$	0	0
$966$	0	0
$967$	38.6274	1.24217	0.621087	−	0.783742i	$-0.286691\pi$
$967$	38.6274	1.24217	0.621087	+	0.783742i	$0.286691\pi$
$968$	−45.5269	−1.46329
$969$	0	0
$970$	0	0
$971$	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	$-0.468675\pi$
$977$	6.14214	0.196504	0.0982522	+	0.995162i	$0.468675\pi$
$978$	0	0
$979$	−14.3431	−0.458409
$980$	0	0
$981$	0	0
$982$	38.9706	1.24360
$983$	−41.9411	−1.33771	−0.668857	−	0.743391i	$-0.733216\pi$
$983$	−41.9411	−1.33771	−0.668857	+	0.743391i	$0.733216\pi$
$984$	0	0
$985$	0	0
$986$	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
$995$	0	0
$996$	0	0
$997$	37.5147	1.18810	0.594052	−	0.804427i	$-0.297528\pi$
$997$	37.5147	1.18810	0.594052	+	0.804427i	$0.297528\pi$
$998$	61.9411	1.96071
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.u.1.2		2
3.2	odd	2		1575.2.a.m.1.1		2
5.2	odd	4		1575.2.d.h.1324.4		4
5.3	odd	4		1575.2.d.h.1324.1		4
5.4	even	2		315.2.a.c.1.1	✓	2
15.2	even	4		1575.2.d.j.1324.1		4
15.8	even	4		1575.2.d.j.1324.4		4
15.14	odd	2		315.2.a.f.1.2	yes	2
20.19	odd	2		5040.2.a.bu.1.1		2
35.34	odd	2		2205.2.a.p.1.1		2
60.59	even	2		5040.2.a.bx.1.2		2
105.104	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	5.4	even	2
315.2.a.f.1.2	yes	2	15.14	odd	2
1575.2.a.m.1.1		2	3.2	odd	2
1575.2.a.u.1.2		2	1.1	even	1	trivial
1575.2.d.h.1324.1		4	5.3	odd	4
1575.2.d.h.1324.4		4	5.2	odd	4
1575.2.d.j.1324.1		4	15.2	even	4
1575.2.d.j.1324.4		4	15.8	even	4
2205.2.a.p.1.1		2	35.34	odd	2
2205.2.a.y.1.2		2	105.104	even	2
5040.2.a.bu.1.1		2	20.19	odd	2
5040.2.a.bx.1.2		2	60.59	even	2

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

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

Introduction

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