LMFDB - Embedded newform 7800.2.a.bn.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
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			1.2
Root			$-0.363328$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.36333 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.36333 q^{7} +1.00000 q^{9} -5.64600 q^{11} -1.00000 q^{13} +2.14134 q^{17} +2.41468 q^{19} -1.36333 q^{21} -2.72666 q^{23} +1.00000 q^{27} -5.28267 q^{29} -1.64600 q^{31} -5.64600 q^{33} +3.86799 q^{37} -1.00000 q^{39} +5.86799 q^{41} -9.00933 q^{43} +3.77801 q^{47} -5.14134 q^{49} +2.14134 q^{51} +12.0093 q^{53} +2.41468 q^{57} +6.19269 q^{59} +11.5946 q^{61} -1.36333 q^{63} -0.324695 q^{67} -2.72666 q^{69} +11.8680 q^{71} +8.28267 q^{73} +7.69735 q^{77} +2.31198 q^{79} +1.00000 q^{81} +10.3727 q^{83} -5.28267 q^{87} +3.73599 q^{89} +1.36333 q^{91} -1.64600 q^{93} -16.8480 q^{97} -5.64600 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * q^99

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
$7$	−1.36333	−0.515290	−0.257645	−	0.966240i	$-0.582946\pi$
$7$	−1.36333	−0.515290	−0.257645	+	0.966240i	$0.582946\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.64600	−1.70233	−0.851167	−	0.524896i	$-0.824104\pi$
$11$	−5.64600	−1.70233	−0.851167	+	0.524896i	$0.824104\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.14134	0.519350	0.259675	−	0.965696i	$-0.416385\pi$
$17$	2.14134	0.519350	0.259675	+	0.965696i	$0.416385\pi$
$18$	0	0
$19$	2.41468	0.553966	0.276983	−	0.960875i	$-0.410666\pi$
$19$	2.41468	0.553966	0.276983	+	0.960875i	$0.410666\pi$
$20$	0	0
$21$	−1.36333	−0.297503
$22$	0	0
$23$	−2.72666	−0.568547	−0.284274	−	0.958743i	$-0.591752\pi$
$23$	−2.72666	−0.568547	−0.284274	+	0.958743i	$0.591752\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.28267	−0.980968	−0.490484	−	0.871450i	$-0.663180\pi$
$29$	−5.28267	−0.980968	−0.490484	+	0.871450i	$0.663180\pi$
$30$	0	0
$31$	−1.64600	−0.295630	−0.147815	−	0.989015i	$-0.547224\pi$
$31$	−1.64600	−0.295630	−0.147815	+	0.989015i	$0.547224\pi$
$32$	0	0
$33$	−5.64600	−0.982843
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.86799	0.635894	0.317947	−	0.948108i	$-0.397007\pi$
$37$	3.86799	0.635894	0.317947	+	0.948108i	$0.397007\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.86799	0.916426	0.458213	−	0.888842i	$-0.348490\pi$
$41$	5.86799	0.916426	0.458213	+	0.888842i	$0.348490\pi$
$42$	0	0
$43$	−9.00933	−1.37391	−0.686955	−	0.726700i	$-0.741053\pi$
$43$	−9.00933	−1.37391	−0.686955	+	0.726700i	$0.741053\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.77801	0.551079	0.275540	−	0.961290i	$-0.411143\pi$
$47$	3.77801	0.551079	0.275540	+	0.961290i	$0.411143\pi$
$48$	0	0
$49$	−5.14134	−0.734477
$50$	0	0
$51$	2.14134	0.299847
$52$	0	0
$53$	12.0093	1.64961	0.824804	−	0.565419i	$-0.191285\pi$
$53$	12.0093	1.64961	0.824804	+	0.565419i	$0.191285\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.41468	0.319832
$58$	0	0
$59$	6.19269	0.806219	0.403110	−	0.915152i	$-0.367929\pi$
$59$	6.19269	0.806219	0.403110	+	0.915152i	$0.367929\pi$
$60$	0	0
$61$	11.5946	1.48454	0.742271	−	0.670099i	$-0.233749\pi$
$61$	11.5946	1.48454	0.742271	+	0.670099i	$0.233749\pi$
$62$	0	0
$63$	−1.36333	−0.171763
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.324695	−0.0396678	−0.0198339	−	0.999803i	$-0.506314\pi$
$67$	−0.324695	−0.0396678	−0.0198339	+	0.999803i	$0.506314\pi$
$68$	0	0
$69$	−2.72666	−0.328251
$70$	0	0
$71$	11.8680	1.40847	0.704236	−	0.709966i	$-0.251290\pi$
$71$	11.8680	1.40847	0.704236	+	0.709966i	$0.251290\pi$
$72$	0	0
$73$	8.28267	0.969413	0.484707	−	0.874677i	$-0.338926\pi$
$73$	8.28267	0.969413	0.484707	+	0.874677i	$0.338926\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.69735	0.877195
$78$	0	0
$79$	2.31198	0.260118	0.130059	−	0.991506i	$-0.458483\pi$
$79$	2.31198	0.260118	0.130059	+	0.991506i	$0.458483\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.3727	1.13855	0.569274	−	0.822148i	$-0.307225\pi$
$83$	10.3727	1.13855	0.569274	+	0.822148i	$0.307225\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.28267	−0.566362
$88$	0	0
$89$	3.73599	0.396014	0.198007	−	0.980201i	$-0.436553\pi$
$89$	3.73599	0.396014	0.198007	+	0.980201i	$0.436553\pi$
$90$	0	0
$91$	1.36333	0.142916
$92$	0	0
$93$	−1.64600	−0.170682
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.8480	−1.71066	−0.855328	−	0.518086i	$-0.826645\pi$
$97$	−16.8480	−1.71066	−0.855328	+	0.518086i	$0.826645\pi$
$98$	0	0
$99$	−5.64600	−0.567444
$100$	0	0
$101$	−5.87732	−0.584815	−0.292408	−	0.956294i	$-0.594456\pi$
$101$	−5.87732	−0.584815	−0.292408	+	0.956294i	$0.594456\pi$
$102$	0	0
$103$	−14.0187	−1.38130	−0.690650	−	0.723189i	$-0.742675\pi$
$103$	−14.0187	−1.38130	−0.690650	+	0.723189i	$0.742675\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.8773	−1.05155	−0.525775	−	0.850624i	$-0.676225\pi$
$107$	−10.8773	−1.05155	−0.525775	+	0.850624i	$0.676225\pi$
$108$	0	0
$109$	−3.58532	−0.343411	−0.171706	−	0.985148i	$-0.554928\pi$
$109$	−3.58532	−0.343411	−0.171706	+	0.985148i	$0.554928\pi$
$110$	0	0
$111$	3.86799	0.367134
$112$	0	0
$113$	18.5653	1.74648	0.873240	−	0.487290i	$-0.162014\pi$
$113$	18.5653	1.74648	0.873240	+	0.487290i	$0.162014\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−2.91934	−0.267616
$120$	0	0
$121$	20.8773	1.89794
$122$	0	0
$123$	5.86799	0.529099
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.858664	−0.0761941	−0.0380970	−	0.999274i	$-0.512130\pi$
$127$	−0.858664	−0.0761941	−0.0380970	+	0.999274i	$0.512130\pi$
$128$	0	0
$129$	−9.00933	−0.793227
$130$	0	0
$131$	−1.86799	−0.163207	−0.0816036	−	0.996665i	$-0.526004\pi$
$131$	−1.86799	−0.163207	−0.0816036	+	0.996665i	$0.526004\pi$
$132$	0	0
$133$	−3.29200	−0.285453
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.24404	−0.448028	−0.224014	−	0.974586i	$-0.571916\pi$
$137$	−5.24404	−0.448028	−0.224014	+	0.974586i	$0.571916\pi$
$138$	0	0
$139$	8.17997	0.693816	0.346908	−	0.937899i	$-0.387232\pi$
$139$	8.17997	0.693816	0.346908	+	0.937899i	$0.387232\pi$
$140$	0	0
$141$	3.77801	0.318166
$142$	0	0
$143$	5.64600	0.472142
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.14134	−0.424050
$148$	0	0
$149$	9.73599	0.797603	0.398801	−	0.917037i	$-0.369426\pi$
$149$	9.73599	0.797603	0.398801	+	0.917037i	$0.369426\pi$
$150$	0	0
$151$	1.18336	0.0963004	0.0481502	−	0.998840i	$-0.484667\pi$
$151$	1.18336	0.0963004	0.0481502	+	0.998840i	$0.484667\pi$
$152$	0	0
$153$	2.14134	0.173117
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.9800	1.43496	0.717481	−	0.696578i	$-0.245295\pi$
$157$	17.9800	1.43496	0.717481	+	0.696578i	$0.245295\pi$
$158$	0	0
$159$	12.0093	0.952402
$160$	0	0
$161$	3.71733	0.292966
$162$	0	0
$163$	4.28267	0.335445	0.167722	−	0.985834i	$-0.446359\pi$
$163$	4.28267	0.335445	0.167722	+	0.985834i	$0.446359\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.41468	−0.496383	−0.248191	−	0.968711i	$-0.579836\pi$
$167$	−6.41468	−0.496383	−0.248191	+	0.968711i	$0.579836\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.41468	0.184655
$172$	0	0
$173$	17.1800	1.30617	0.653084	−	0.757285i	$-0.273475\pi$
$173$	17.1800	1.30617	0.653084	+	0.757285i	$0.273475\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.19269	0.465471
$178$	0	0
$179$	7.11203	0.531578	0.265789	−	0.964031i	$-0.414368\pi$
$179$	7.11203	0.531578	0.265789	+	0.964031i	$0.414368\pi$
$180$	0	0
$181$	25.4333	1.89045	0.945223	−	0.326427i	$-0.105845\pi$
$181$	25.4333	1.89045	0.945223	+	0.326427i	$0.105845\pi$
$182$	0	0
$183$	11.5946	0.857101
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−12.0900	−0.884107
$188$	0	0
$189$	−1.36333	−0.0991675
$190$	0	0
$191$	16.1214	1.16650	0.583250	−	0.812292i	$-0.301781\pi$
$191$	16.1214	1.16650	0.583250	+	0.812292i	$0.301781\pi$
$192$	0	0
$193$	10.0187	0.721159	0.360579	−	0.932729i	$-0.382579\pi$
$193$	10.0187	0.721159	0.360579	+	0.932729i	$0.382579\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.0187	−0.713800	−0.356900	−	0.934143i	$-0.616166\pi$
$197$	−10.0187	−0.713800	−0.356900	+	0.934143i	$0.616166\pi$
$198$	0	0
$199$	−15.4427	−1.09470	−0.547351	−	0.836903i	$-0.684364\pi$
$199$	−15.4427	−1.09470	−0.547351	+	0.836903i	$0.684364\pi$
$200$	0	0
$201$	−0.324695	−0.0229022
$202$	0	0
$203$	7.20202	0.505482
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.72666	−0.189516
$208$	0	0
$209$	−13.6333	−0.943034
$210$	0	0
$211$	17.8387	1.22807	0.614033	−	0.789280i	$-0.289546\pi$
$211$	17.8387	1.22807	0.614033	+	0.789280i	$0.289546\pi$
$212$	0	0
$213$	11.8680	0.813181
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.24404	0.152335
$218$	0	0
$219$	8.28267	0.559691
$220$	0	0
$221$	−2.14134	−0.144042
$222$	0	0
$223$	−6.56534	−0.439648	−0.219824	−	0.975540i	$-0.570548\pi$
$223$	−6.56534	−0.439648	−0.219824	+	0.975540i	$0.570548\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.3820	0.888194	0.444097	−	0.895979i	$-0.353525\pi$
$227$	13.3820	0.888194	0.444097	+	0.895979i	$0.353525\pi$
$228$	0	0
$229$	9.88665	0.653328	0.326664	−	0.945140i	$-0.394075\pi$
$229$	9.88665	0.653328	0.326664	+	0.945140i	$0.394075\pi$
$230$	0	0
$231$	7.69735	0.506449
$232$	0	0
$233$	−26.8667	−1.76009	−0.880047	−	0.474886i	$-0.842489\pi$
$233$	−26.8667	−1.76009	−0.880047	+	0.474886i	$0.842489\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.31198	0.150179
$238$	0	0
$239$	27.4847	1.77784	0.888918	−	0.458066i	$-0.151458\pi$
$239$	27.4847	1.77784	0.888918	+	0.458066i	$0.151458\pi$
$240$	0	0
$241$	−10.0187	−0.645358	−0.322679	−	0.946508i	$-0.604584\pi$
$241$	−10.0187	−0.645358	−0.322679	+	0.946508i	$0.604584\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.41468	−0.153642
$248$	0	0
$249$	10.3727	0.657340
$250$	0	0
$251$	−24.1507	−1.52438	−0.762188	−	0.647355i	$-0.775875\pi$
$251$	−24.1507	−1.52438	−0.762188	+	0.647355i	$0.775875\pi$
$252$	0	0
$253$	15.3947	0.967857
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.15066	0.0717765	0.0358882	−	0.999356i	$-0.488574\pi$
$257$	1.15066	0.0717765	0.0358882	+	0.999356i	$0.488574\pi$
$258$	0	0
$259$	−5.27334	−0.327670
$260$	0	0
$261$	−5.28267	−0.326989
$262$	0	0
$263$	−2.28267	−0.140756	−0.0703778	−	0.997520i	$-0.522420\pi$
$263$	−2.28267	−0.140756	−0.0703778	+	0.997520i	$0.522420\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.73599	0.228639
$268$	0	0
$269$	8.17064	0.498173	0.249086	−	0.968481i	$-0.419870\pi$
$269$	8.17064	0.498173	0.249086	+	0.968481i	$0.419870\pi$
$270$	0	0
$271$	21.5433	1.30866	0.654331	−	0.756208i	$-0.272950\pi$
$271$	21.5433	1.30866	0.654331	+	0.756208i	$0.272950\pi$
$272$	0	0
$273$	1.36333	0.0825124
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.1307	−1.38979	−0.694894	−	0.719112i	$-0.744549\pi$
$277$	−23.1307	−1.38979	−0.694894	+	0.719112i	$0.744549\pi$
$278$	0	0
$279$	−1.64600	−0.0985435
$280$	0	0
$281$	4.96137	0.295970	0.147985	−	0.988990i	$-0.452721\pi$
$281$	4.96137	0.295970	0.147985	+	0.988990i	$0.452721\pi$
$282$	0	0
$283$	4.74531	0.282080	0.141040	−	0.990004i	$-0.454955\pi$
$283$	4.74531	0.282080	0.141040	+	0.990004i	$0.454955\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−12.4147	−0.730275
$290$	0	0
$291$	−16.8480	−0.987648
$292$	0	0
$293$	29.7360	1.73719	0.868597	−	0.495518i	$-0.165022\pi$
$293$	29.7360	1.73719	0.868597	+	0.495518i	$0.165022\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.64600	−0.327614
$298$	0	0
$299$	2.72666	0.157687
$300$	0	0
$301$	12.2827	0.707961
$302$	0	0
$303$	−5.87732	−0.337643
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.4520	1.05311	0.526555	−	0.850141i	$-0.323483\pi$
$307$	18.4520	1.05311	0.526555	+	0.850141i	$0.323483\pi$
$308$	0	0
$309$	−14.0187	−0.797494
$310$	0	0
$311$	15.9160	0.902511	0.451255	−	0.892395i	$-0.350976\pi$
$311$	15.9160	0.902511	0.451255	+	0.892395i	$0.350976\pi$
$312$	0	0
$313$	−1.99067	−0.112519	−0.0562597	−	0.998416i	$-0.517917\pi$
$313$	−1.99067	−0.112519	−0.0562597	+	0.998416i	$0.517917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.26401	−0.127160	−0.0635799	−	0.997977i	$-0.520252\pi$
$317$	−2.26401	−0.127160	−0.0635799	+	0.997977i	$0.520252\pi$
$318$	0	0
$319$	29.8260	1.66993
$320$	0	0
$321$	−10.8773	−0.607113
$322$	0	0
$323$	5.17064	0.287702
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.58532	−0.198269
$328$	0	0
$329$	−5.15066	−0.283965
$330$	0	0
$331$	1.69867	0.0933674	0.0466837	−	0.998910i	$-0.485135\pi$
$331$	1.69867	0.0933674	0.0466837	+	0.998910i	$0.485135\pi$
$332$	0	0
$333$	3.86799	0.211965
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.7067	−1.12796	−0.563982	−	0.825787i	$-0.690731\pi$
$337$	−20.7067	−1.12796	−0.563982	+	0.825787i	$0.690731\pi$
$338$	0	0
$339$	18.5653	1.00833
$340$	0	0
$341$	9.29332	0.503261
$342$	0	0
$343$	16.5526	0.893758
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.97070	0.427889	0.213945	−	0.976846i	$-0.431369\pi$
$347$	7.97070	0.427889	0.213945	+	0.976846i	$0.431369\pi$
$348$	0	0
$349$	6.56534	0.351435	0.175717	−	0.984441i	$-0.443776\pi$
$349$	6.56534	0.351435	0.175717	+	0.984441i	$0.443776\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−20.4520	−1.08855	−0.544275	−	0.838907i	$-0.683195\pi$
$353$	−20.4520	−1.08855	−0.544275	+	0.838907i	$0.683195\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.91934	−0.154508
$358$	0	0
$359$	−11.8994	−0.628025	−0.314012	−	0.949419i	$-0.601673\pi$
$359$	−11.8994	−0.628025	−0.314012	+	0.949419i	$0.601673\pi$
$360$	0	0
$361$	−13.1693	−0.693122
$362$	0	0
$363$	20.8773	1.09578
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−23.5853	−1.23114	−0.615572	−	0.788081i	$-0.711075\pi$
$367$	−23.5853	−1.23114	−0.615572	+	0.788081i	$0.711075\pi$
$368$	0	0
$369$	5.86799	0.305475
$370$	0	0
$371$	−16.3727	−0.850026
$372$	0	0
$373$	−25.1693	−1.30322	−0.651609	−	0.758555i	$-0.725906\pi$
$373$	−25.1693	−1.30322	−0.651609	+	0.758555i	$0.725906\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.28267	0.272071
$378$	0	0
$379$	−8.83530	−0.453839	−0.226919	−	0.973914i	$-0.572865\pi$
$379$	−8.83530	−0.453839	−0.226919	+	0.973914i	$0.572865\pi$
$380$	0	0
$381$	−0.858664	−0.0439907
$382$	0	0
$383$	−27.9053	−1.42589	−0.712947	−	0.701218i	$-0.752640\pi$
$383$	−27.9053	−1.42589	−0.712947	+	0.701218i	$0.752640\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.00933	−0.457970
$388$	0	0
$389$	−0.697352	−0.0353571	−0.0176786	−	0.999844i	$-0.505628\pi$
$389$	−0.697352	−0.0353571	−0.0176786	+	0.999844i	$0.505628\pi$
$390$	0	0
$391$	−5.83869	−0.295275
$392$	0	0
$393$	−1.86799	−0.0942278
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.8867	0.897705	0.448853	−	0.893606i	$-0.351833\pi$
$397$	17.8867	0.897705	0.448853	+	0.893606i	$0.351833\pi$
$398$	0	0
$399$	−3.29200	−0.164806
$400$	0	0
$401$	25.1120	1.25404	0.627018	−	0.779005i	$-0.284275\pi$
$401$	25.1120	1.25404	0.627018	+	0.779005i	$0.284275\pi$
$402$	0	0
$403$	1.64600	0.0819931
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.8387	−1.08250
$408$	0	0
$409$	−23.6774	−1.17077	−0.585385	−	0.810755i	$-0.699057\pi$
$409$	−23.6774	−1.17077	−0.585385	+	0.810755i	$0.699057\pi$
$410$	0	0
$411$	−5.24404	−0.258669
$412$	0	0
$413$	−8.44267	−0.415436
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.17997	0.400575
$418$	0	0
$419$	−18.7347	−0.915248	−0.457624	−	0.889146i	$-0.651299\pi$
$419$	−18.7347	−0.915248	−0.457624	+	0.889146i	$0.651299\pi$
$420$	0	0
$421$	4.26401	0.207815	0.103908	−	0.994587i	$-0.466865\pi$
$421$	4.26401	0.207815	0.103908	+	0.994587i	$0.466865\pi$
$422$	0	0
$423$	3.77801	0.183693
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−15.8073	−0.764969
$428$	0	0
$429$	5.64600	0.272591
$430$	0	0
$431$	20.1693	0.971522	0.485761	−	0.874092i	$-0.338543\pi$
$431$	20.1693	0.971522	0.485761	+	0.874092i	$0.338543\pi$
$432$	0	0
$433$	−20.6974	−0.994651	−0.497326	−	0.867564i	$-0.665685\pi$
$433$	−20.6974	−0.994651	−0.497326	+	0.867564i	$0.665685\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.58400	−0.314956
$438$	0	0
$439$	−2.31198	−0.110345	−0.0551723	−	0.998477i	$-0.517571\pi$
$439$	−2.31198	−0.110345	−0.0551723	+	0.998477i	$0.517571\pi$
$440$	0	0
$441$	−5.14134	−0.244826
$442$	0	0
$443$	2.25337	0.107061	0.0535304	−	0.998566i	$-0.482953\pi$
$443$	2.25337	0.107061	0.0535304	+	0.998566i	$0.482953\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.73599	0.460496
$448$	0	0
$449$	22.1693	1.04624	0.523118	−	0.852261i	$-0.324769\pi$
$449$	22.1693	1.04624	0.523118	+	0.852261i	$0.324769\pi$
$450$	0	0
$451$	−33.1307	−1.56006
$452$	0	0
$453$	1.18336	0.0555990
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.4720	1.47220	0.736098	−	0.676875i	$-0.236666\pi$
$457$	31.4720	1.47220	0.736098	+	0.676875i	$0.236666\pi$
$458$	0	0
$459$	2.14134	0.0999490
$460$	0	0
$461$	17.7360	0.826047	0.413024	−	0.910720i	$-0.364473\pi$
$461$	17.7360	0.826047	0.413024	+	0.910720i	$0.364473\pi$
$462$	0	0
$463$	−0.431266	−0.0200426	−0.0100213	−	0.999950i	$-0.503190\pi$
$463$	−0.431266	−0.0200426	−0.0100213	+	0.999950i	$0.503190\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.30265	−0.245377	−0.122689	−	0.992445i	$-0.539152\pi$
$467$	−5.30265	−0.245377	−0.122689	+	0.992445i	$0.539152\pi$
$468$	0	0
$469$	0.442666	0.0204404
$470$	0	0
$471$	17.9800	0.828476
$472$	0	0
$473$	50.8667	2.33885
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0093	0.549869
$478$	0	0
$479$	−2.07934	−0.0950074	−0.0475037	−	0.998871i	$-0.515127\pi$
$479$	−2.07934	−0.0950074	−0.0475037	+	0.998871i	$0.515127\pi$
$480$	0	0
$481$	−3.86799	−0.176365
$482$	0	0
$483$	3.71733	0.169144
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15.3633	−0.696179	−0.348089	−	0.937461i	$-0.613169\pi$
$487$	−15.3633	−0.696179	−0.348089	+	0.937461i	$0.613169\pi$
$488$	0	0
$489$	4.28267	0.193669
$490$	0	0
$491$	28.2173	1.27343	0.636714	−	0.771100i	$-0.280293\pi$
$491$	28.2173	1.27343	0.636714	+	0.771100i	$0.280293\pi$
$492$	0	0
$493$	−11.3120	−0.509466
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.1800	−0.725771
$498$	0	0
$499$	7.15405	0.320259	0.160130	−	0.987096i	$-0.448809\pi$
$499$	7.15405	0.320259	0.160130	+	0.987096i	$0.448809\pi$
$500$	0	0
$501$	−6.41468	−0.286587
$502$	0	0
$503$	5.09337	0.227102	0.113551	−	0.993532i	$-0.463777\pi$
$503$	5.09337	0.227102	0.113551	+	0.993532i	$0.463777\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−21.4906	−0.952555	−0.476278	−	0.879295i	$-0.658014\pi$
$509$	−21.4906	−0.952555	−0.476278	+	0.879295i	$0.658014\pi$
$510$	0	0
$511$	−11.2920	−0.499529
$512$	0	0
$513$	2.41468	0.106611
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−21.3306	−0.938120
$518$	0	0
$519$	17.1800	0.754117
$520$	0	0
$521$	25.7360	1.12751	0.563757	−	0.825941i	$-0.309355\pi$
$521$	25.7360	1.12751	0.563757	+	0.825941i	$0.309355\pi$
$522$	0	0
$523$	−0.972014	−0.0425032	−0.0212516	−	0.999774i	$-0.506765\pi$
$523$	−0.972014	−0.0425032	−0.0212516	+	0.999774i	$0.506765\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.52464	−0.153536
$528$	0	0
$529$	−15.5653	−0.676754
$530$	0	0
$531$	6.19269	0.268740
$532$	0	0
$533$	−5.86799	−0.254171
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.11203	0.306907
$538$	0	0
$539$	29.0280	1.25032
$540$	0	0
$541$	13.6774	0.588036	0.294018	−	0.955800i	$-0.405007\pi$
$541$	13.6774	0.588036	0.294018	+	0.955800i	$0.405007\pi$
$542$	0	0
$543$	25.4333	1.09145
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−33.7546	−1.44324	−0.721622	−	0.692287i	$-0.756603\pi$
$547$	−33.7546	−1.44324	−0.721622	+	0.692287i	$0.756603\pi$
$548$	0	0
$549$	11.5946	0.494848
$550$	0	0
$551$	−12.7560	−0.543422
$552$	0	0
$553$	−3.15198	−0.134036
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.2827	0.435691	0.217845	−	0.975983i	$-0.430097\pi$
$557$	10.2827	0.435691	0.217845	+	0.975983i	$0.430097\pi$
$558$	0	0
$559$	9.00933	0.381054
$560$	0	0
$561$	−12.0900	−0.510440
$562$	0	0
$563$	−35.3841	−1.49126	−0.745630	−	0.666360i	$-0.767851\pi$
$563$	−35.3841	−1.49126	−0.745630	+	0.666360i	$0.767851\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.36333	−0.0572544
$568$	0	0
$569$	21.7160	0.910382	0.455191	−	0.890394i	$-0.349571\pi$
$569$	21.7160	0.910382	0.455191	+	0.890394i	$0.349571\pi$
$570$	0	0
$571$	24.9507	1.04416	0.522078	−	0.852898i	$-0.325157\pi$
$571$	24.9507	1.04416	0.522078	+	0.852898i	$0.325157\pi$
$572$	0	0
$573$	16.1214	0.673479
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27.1893	−1.13191	−0.565953	−	0.824438i	$-0.691492\pi$
$577$	−27.1893	−1.13191	−0.565953	+	0.824438i	$0.691492\pi$
$578$	0	0
$579$	10.0187	0.416361
$580$	0	0
$581$	−14.1413	−0.586681
$582$	0	0
$583$	−67.8047	−2.80818
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−47.5220	−1.96144	−0.980721	−	0.195411i	$-0.937396\pi$
$587$	−47.5220	−1.96144	−0.980721	+	0.195411i	$0.937396\pi$
$588$	0	0
$589$	−3.97456	−0.163769
$590$	0	0
$591$	−10.0187	−0.412112
$592$	0	0
$593$	17.9453	0.736923	0.368462	−	0.929643i	$-0.379885\pi$
$593$	17.9453	0.736923	0.368462	+	0.929643i	$0.379885\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.4427	−0.632026
$598$	0	0
$599$	15.4720	0.632168	0.316084	−	0.948731i	$-0.397632\pi$
$599$	15.4720	0.632168	0.316084	+	0.948731i	$0.397632\pi$
$600$	0	0
$601$	45.7453	1.86599	0.932995	−	0.359889i	$-0.117185\pi$
$601$	45.7453	1.86599	0.932995	+	0.359889i	$0.117185\pi$
$602$	0	0
$603$	−0.324695	−0.0132226
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.3467	−0.622905	−0.311453	−	0.950262i	$-0.600815\pi$
$607$	−15.3467	−0.622905	−0.311453	+	0.950262i	$0.600815\pi$
$608$	0	0
$609$	7.20202	0.291840
$610$	0	0
$611$	−3.77801	−0.152842
$612$	0	0
$613$	−5.67738	−0.229307	−0.114654	−	0.993406i	$-0.536576\pi$
$613$	−5.67738	−0.229307	−0.114654	+	0.993406i	$0.536576\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.8867	0.881123	0.440562	−	0.897722i	$-0.354779\pi$
$617$	21.8867	0.881123	0.440562	+	0.897722i	$0.354779\pi$
$618$	0	0
$619$	21.4134	0.860676	0.430338	−	0.902668i	$-0.358394\pi$
$619$	21.4134	0.860676	0.430338	+	0.902668i	$0.358394\pi$
$620$	0	0
$621$	−2.72666	−0.109417
$622$	0	0
$623$	−5.09337	−0.204062
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13.6333	−0.544461
$628$	0	0
$629$	8.28267	0.330252
$630$	0	0
$631$	35.1893	1.40086	0.700432	−	0.713719i	$-0.252991\pi$
$631$	35.1893	1.40086	0.700432	+	0.713719i	$0.252991\pi$
$632$	0	0
$633$	17.8387	0.709024
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.14134	0.203707
$638$	0	0
$639$	11.8680	0.469491
$640$	0	0
$641$	33.5547	1.32533	0.662665	−	0.748916i	$-0.269425\pi$
$641$	33.5547	1.32533	0.662665	+	0.748916i	$0.269425\pi$
$642$	0	0
$643$	18.7160	0.738087	0.369044	−	0.929412i	$-0.379685\pi$
$643$	18.7160	0.738087	0.369044	+	0.929412i	$0.379685\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.5653	−0.651251	−0.325625	−	0.945499i	$-0.605575\pi$
$647$	−16.5653	−0.651251	−0.325625	+	0.945499i	$0.605575\pi$
$648$	0	0
$649$	−34.9639	−1.37245
$650$	0	0
$651$	2.24404	0.0879508
$652$	0	0
$653$	26.5454	1.03880	0.519400	−	0.854531i	$-0.326155\pi$
$653$	26.5454	1.03880	0.519400	+	0.854531i	$0.326155\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.28267	0.323138
$658$	0	0
$659$	36.3306	1.41524	0.707620	−	0.706593i	$-0.249769\pi$
$659$	36.3306	1.41524	0.707620	+	0.706593i	$0.249769\pi$
$660$	0	0
$661$	22.2279	0.864566	0.432283	−	0.901738i	$-0.357708\pi$
$661$	22.2279	0.864566	0.432283	+	0.901738i	$0.357708\pi$
$662$	0	0
$663$	−2.14134	−0.0831626
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.4040	0.557726
$668$	0	0
$669$	−6.56534	−0.253831
$670$	0	0
$671$	−65.4634	−2.52719
$672$	0	0
$673$	1.46264	0.0563807	0.0281903	−	0.999603i	$-0.491026\pi$
$673$	1.46264	0.0563807	0.0281903	+	0.999603i	$0.491026\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.5653	0.713524	0.356762	−	0.934195i	$-0.383881\pi$
$677$	18.5653	0.713524	0.356762	+	0.934195i	$0.383881\pi$
$678$	0	0
$679$	22.9694	0.881484
$680$	0	0
$681$	13.3820	0.512799
$682$	0	0
$683$	6.03138	0.230784	0.115392	−	0.993320i	$-0.463188\pi$
$683$	6.03138	0.230784	0.115392	+	0.993320i	$0.463188\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.88665	0.377199
$688$	0	0
$689$	−12.0093	−0.457519
$690$	0	0
$691$	14.1566	0.538543	0.269271	−	0.963064i	$-0.413217\pi$
$691$	14.1566	0.538543	0.269271	+	0.963064i	$0.413217\pi$
$692$	0	0
$693$	7.69735	0.292398
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.5653	0.475946
$698$	0	0
$699$	−26.8667	−1.01619
$700$	0	0
$701$	6.26270	0.236539	0.118269	−	0.992982i	$-0.462265\pi$
$701$	6.26270	0.236539	0.118269	+	0.992982i	$0.462265\pi$
$702$	0	0
$703$	9.33996	0.352263
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.01272	0.301349
$708$	0	0
$709$	6.28267	0.235951	0.117975	−	0.993017i	$-0.462360\pi$
$709$	6.28267	0.235951	0.117975	+	0.993017i	$0.462360\pi$
$710$	0	0
$711$	2.31198	0.0867059
$712$	0	0
$713$	4.48808	0.168080
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.4847	1.02643
$718$	0	0
$719$	−28.9253	−1.07873	−0.539366	−	0.842072i	$-0.681336\pi$
$719$	−28.9253	−1.07873	−0.539366	+	0.842072i	$0.681336\pi$
$720$	0	0
$721$	19.1120	0.711769
$722$	0	0
$723$	−10.0187	−0.372598
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−40.2427	−1.49252	−0.746260	−	0.665655i	$-0.768152\pi$
$727$	−40.2427	−1.49252	−0.746260	+	0.665655i	$0.768152\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.2920	−0.713540
$732$	0	0
$733$	19.7907	0.730987	0.365494	−	0.930814i	$-0.380900\pi$
$733$	19.7907	0.730987	0.365494	+	0.930814i	$0.380900\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.83323	0.0675278
$738$	0	0
$739$	−18.1634	−0.668151	−0.334075	−	0.942546i	$-0.608424\pi$
$739$	−18.1634	−0.668151	−0.334075	+	0.942546i	$0.608424\pi$
$740$	0	0
$741$	−2.41468	−0.0887055
$742$	0	0
$743$	24.2661	0.890236	0.445118	−	0.895472i	$-0.353162\pi$
$743$	24.2661	0.890236	0.445118	+	0.895472i	$0.353162\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.3727	0.379516
$748$	0	0
$749$	14.8294	0.541853
$750$	0	0
$751$	28.2720	1.03166	0.515830	−	0.856691i	$-0.327483\pi$
$751$	28.2720	1.03166	0.515830	+	0.856691i	$0.327483\pi$
$752$	0	0
$753$	−24.1507	−0.880099
$754$	0	0
$755$	0	0
$756$	0	0
$757$	37.0853	1.34789	0.673944	−	0.738783i	$-0.264599\pi$
$757$	37.0853	1.34789	0.673944	+	0.738783i	$0.264599\pi$
$758$	0	0
$759$	15.3947	0.558792
$760$	0	0
$761$	28.4919	1.03283	0.516416	−	0.856338i	$-0.327266\pi$
$761$	28.4919	1.03283	0.516416	+	0.856338i	$0.327266\pi$
$762$	0	0
$763$	4.88797	0.176956
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.19269	−0.223605
$768$	0	0
$769$	−27.6774	−0.998072	−0.499036	−	0.866581i	$-0.666312\pi$
$769$	−27.6774	−0.998072	−0.499036	+	0.866581i	$0.666312\pi$
$770$	0	0
$771$	1.15066	0.0414402
$772$	0	0
$773$	32.6426	1.17407	0.587037	−	0.809560i	$-0.300294\pi$
$773$	32.6426	1.17407	0.587037	+	0.809560i	$0.300294\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.27334	−0.189180
$778$	0	0
$779$	14.1693	0.507669
$780$	0	0
$781$	−67.0067	−2.39769
$782$	0	0
$783$	−5.28267	−0.188787
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−50.6740	−1.80633	−0.903166	−	0.429291i	$-0.858764\pi$
$787$	−50.6740	−1.80633	−0.903166	+	0.429291i	$0.858764\pi$
$788$	0	0
$789$	−2.28267	−0.0812653
$790$	0	0
$791$	−25.3107	−0.899943
$792$	0	0
$793$	−11.5946	−0.411738
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.3586	−0.862827	−0.431413	−	0.902154i	$-0.641985\pi$
$797$	−24.3586	−0.862827	−0.431413	+	0.902154i	$0.641985\pi$
$798$	0	0
$799$	8.08998	0.286203
$800$	0	0
$801$	3.73599	0.132005
$802$	0	0
$803$	−46.7640	−1.65026
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.17064	0.287620
$808$	0	0
$809$	−50.6027	−1.77909	−0.889547	−	0.456843i	$-0.848980\pi$
$809$	−50.6027	−1.77909	−0.889547	+	0.456843i	$0.848980\pi$
$810$	0	0
$811$	18.4168	0.646700	0.323350	−	0.946280i	$-0.395191\pi$
$811$	18.4168	0.646700	0.323350	+	0.946280i	$0.395191\pi$
$812$	0	0
$813$	21.5433	0.755556
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−21.7546	−0.761099
$818$	0	0
$819$	1.36333	0.0476385
$820$	0	0
$821$	−31.4906	−1.09903	−0.549515	−	0.835484i	$-0.685188\pi$
$821$	−31.4906	−1.09903	−0.549515	+	0.835484i	$0.685188\pi$
$822$	0	0
$823$	18.7347	0.653049	0.326525	−	0.945189i	$-0.394122\pi$
$823$	18.7347	0.653049	0.326525	+	0.945189i	$0.394122\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.7580	−0.582734	−0.291367	−	0.956611i	$-0.594110\pi$
$827$	−16.7580	−0.582734	−0.291367	+	0.956611i	$0.594110\pi$
$828$	0	0
$829$	−20.1014	−0.698150	−0.349075	−	0.937095i	$-0.613504\pi$
$829$	−20.1014	−0.698150	−0.349075	+	0.937095i	$0.613504\pi$
$830$	0	0
$831$	−23.1307	−0.802395
$832$	0	0
$833$	−11.0093	−0.381451
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.64600	−0.0568941
$838$	0	0
$839$	−6.82936	−0.235776	−0.117888	−	0.993027i	$-0.537612\pi$
$839$	−6.82936	−0.235776	−0.117888	+	0.993027i	$0.537612\pi$
$840$	0	0
$841$	−1.09337	−0.0377026
$842$	0	0
$843$	4.96137	0.170879
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−28.4626	−0.977988
$848$	0	0
$849$	4.74531	0.162859
$850$	0	0
$851$	−10.5467	−0.361536
$852$	0	0
$853$	10.2279	0.350198	0.175099	−	0.984551i	$-0.443975\pi$
$853$	10.2279	0.350198	0.175099	+	0.984551i	$0.443975\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.1307	1.06340	0.531702	−	0.846931i	$-0.321553\pi$
$857$	31.1307	1.06340	0.531702	+	0.846931i	$0.321553\pi$
$858$	0	0
$859$	−33.0093	−1.12626	−0.563132	−	0.826367i	$-0.690404\pi$
$859$	−33.0093	−1.12626	−0.563132	+	0.826367i	$0.690404\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	44.4206	1.51210	0.756048	−	0.654516i	$-0.227128\pi$
$863$	44.4206	1.51210	0.756048	+	0.654516i	$0.227128\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−12.4147	−0.421625
$868$	0	0
$869$	−13.0534	−0.442807
$870$	0	0
$871$	0.324695	0.0110019
$872$	0	0
$873$	−16.8480	−0.570219
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.8867	0.536454	0.268227	−	0.963356i	$-0.413562\pi$
$877$	15.8867	0.536454	0.268227	+	0.963356i	$0.413562\pi$
$878$	0	0
$879$	29.7360	1.00297
$880$	0	0
$881$	3.13201	0.105520	0.0527600	−	0.998607i	$-0.483198\pi$
$881$	3.13201	0.105520	0.0527600	+	0.998607i	$0.483198\pi$
$882$	0	0
$883$	−33.1307	−1.11494	−0.557468	−	0.830198i	$-0.688227\pi$
$883$	−33.1307	−1.11494	−0.557468	+	0.830198i	$0.688227\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.8200	−0.665491	−0.332746	−	0.943017i	$-0.607975\pi$
$887$	−19.8200	−0.665491	−0.332746	+	0.943017i	$0.607975\pi$
$888$	0	0
$889$	1.17064	0.0392620
$890$	0	0
$891$	−5.64600	−0.189148
$892$	0	0
$893$	9.12268	0.305279
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.72666	0.0910404
$898$	0	0
$899$	8.69528	0.290004
$900$	0	0
$901$	25.7160	0.856724
$902$	0	0
$903$	12.2827	0.408742
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.6426	−0.486200	−0.243100	−	0.970001i	$-0.578164\pi$
$907$	−14.6426	−0.486200	−0.243100	+	0.970001i	$0.578164\pi$
$908$	0	0
$909$	−5.87732	−0.194938
$910$	0	0
$911$	−37.8973	−1.25559	−0.627797	−	0.778377i	$-0.716043\pi$
$911$	−37.8973	−1.25559	−0.627797	+	0.778377i	$0.716043\pi$
$912$	0	0
$913$	−58.5640	−1.93819
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.54669	0.0840990
$918$	0	0
$919$	26.2534	0.866019	0.433009	−	0.901389i	$-0.357452\pi$
$919$	26.2534	0.866019	0.433009	+	0.901389i	$0.357452\pi$
$920$	0	0
$921$	18.4520	0.608014
$922$	0	0
$923$	−11.8680	−0.390640
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14.0187	−0.460433
$928$	0	0
$929$	52.2066	1.71284	0.856422	−	0.516276i	$-0.172682\pi$
$929$	52.2066	1.71284	0.856422	+	0.516276i	$0.172682\pi$
$930$	0	0
$931$	−12.4147	−0.406875
$932$	0	0
$933$	15.9160	0.521065
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.3213	−0.533194	−0.266597	−	0.963808i	$-0.585899\pi$
$937$	−16.3213	−0.533194	−0.266597	+	0.963808i	$0.585899\pi$
$938$	0	0
$939$	−1.99067	−0.0649631
$940$	0	0
$941$	35.7546	1.16557	0.582784	−	0.812627i	$-0.301963\pi$
$941$	35.7546	1.16557	0.582784	+	0.812627i	$0.301963\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.73260	0.283771	0.141886	−	0.989883i	$-0.454683\pi$
$947$	8.73260	0.283771	0.141886	+	0.989883i	$0.454683\pi$
$948$	0	0
$949$	−8.28267	−0.268867
$950$	0	0
$951$	−2.26401	−0.0734157
$952$	0	0
$953$	28.8280	0.933832	0.466916	−	0.884302i	$-0.345365\pi$
$953$	28.8280	0.933832	0.466916	+	0.884302i	$0.345365\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	29.8260	0.964137
$958$	0	0
$959$	7.14935	0.230864
$960$	0	0
$961$	−28.2907	−0.912603
$962$	0	0
$963$	−10.8773	−0.350517
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.0687	1.22421	0.612103	−	0.790778i	$-0.290324\pi$
$967$	38.0687	1.22421	0.612103	+	0.790778i	$0.290324\pi$
$968$	0	0
$969$	5.17064	0.166105
$970$	0	0
$971$	1.72534	0.0553687	0.0276844	−	0.999617i	$-0.491187\pi$
$971$	1.72534	0.0553687	0.0276844	+	0.999617i	$0.491187\pi$
$972$	0	0
$973$	−11.1520	−0.357516
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.7173	0.694799	0.347399	−	0.937717i	$-0.387065\pi$
$977$	21.7173	0.694799	0.347399	+	0.937717i	$0.387065\pi$
$978$	0	0
$979$	−21.0934	−0.674147
$980$	0	0
$981$	−3.58532	−0.114470
$982$	0	0
$983$	13.0153	0.415123	0.207561	−	0.978222i	$-0.433447\pi$
$983$	13.0153	0.415123	0.207561	+	0.978222i	$0.433447\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.15066	−0.163947
$988$	0	0
$989$	24.5653	0.781133
$990$	0	0
$991$	23.1413	0.735109	0.367554	−	0.930002i	$-0.380195\pi$
$991$	23.1413	0.735109	0.367554	+	0.930002i	$0.380195\pi$
$992$	0	0
$993$	1.69867	0.0539057
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34.9707	1.10753	0.553767	−	0.832672i	$-0.313190\pi$
$997$	34.9707	1.10753	0.553767	+	0.832672i	$0.313190\pi$
$998$	0	0
$999$	3.86799	0.122378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bn.1.2	yes	3
5.4	even	2		7800.2.a.bm.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bm.1.2	✓	3	5.4	even	2
7800.2.a.bn.1.2	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.36333 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.36333 q^{7} +1.00000 q^{9} -5.64600 q^{11} -1.00000 q^{13} +2.14134 q^{17} +2.41468 q^{19} -1.36333 q^{21} -2.72666 q^{23} +1.00000 q^{27} -5.28267 q^{29} -1.64600 q^{31} -5.64600 q^{33} +3.86799 q^{37} -1.00000 q^{39} +5.86799 q^{41} -9.00933 q^{43} +3.77801 q^{47} -5.14134 q^{49} +2.14134 q^{51} +12.0093 q^{53} +2.41468 q^{57} +6.19269 q^{59} +11.5946 q^{61} -1.36333 q^{63} -0.324695 q^{67} -2.72666 q^{69} +11.8680 q^{71} +8.28267 q^{73} +7.69735 q^{77} +2.31198 q^{79} +1.00000 q^{81} +10.3727 q^{83} -5.28267 q^{87} +3.73599 q^{89} +1.36333 q^{91} -1.64600 q^{93} -16.8480 q^{97} -5.64600 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * q^99

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