LMFDB - Embedded newform 7800.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3732.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 19 $ x^3 - x^2 - 13*x + 19
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.20633$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -4.48688 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.48688 q^{7} +1.00000 q^{9} -1.20633 q^{11} +1.00000 q^{13} -3.00000 q^{17} -7.69321 q^{19} +4.48688 q^{21} -5.69321 q^{23} -1.00000 q^{27} -7.41266 q^{29} -1.20633 q^{31} +1.20633 q^{33} +0.719448 q^{37} -1.00000 q^{39} +1.28055 q^{41} -1.28055 q^{43} -5.92578 q^{47} +13.1321 q^{49} +3.00000 q^{51} -6.69321 q^{53} +7.69321 q^{57} +5.92578 q^{59} +9.41266 q^{61} -4.48688 q^{63} -11.2063 q^{67} +5.69321 q^{69} +5.13211 q^{71} -5.13211 q^{73} +5.41266 q^{77} +2.71945 q^{79} +1.00000 q^{81} -4.89954 q^{83} +7.41266 q^{87} -0.412660 q^{89} -4.48688 q^{91} +1.20633 q^{93} -12.9738 q^{97} -1.20633 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9 + 5 * q^11 + 3 * q^13 - 9 * q^17 - 2 * q^19 + q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 5 * q^31 - 5 * q^33 + 6 * q^37 - 3 * q^39 - 13 * q^47 + 26 * q^49 + 9 * q^51 + q^53 + 2 * q^57 + 13 * q^59 + 11 * q^61 - q^63 - 25 * q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - q^77 + 12 * q^79 + 3 * q^81 + 15 * q^83 + 5 * q^87 + 16 * q^89 - q^91 - 5 * q^93 - 14 * q^97 + 5 * 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
$7$	−4.48688	−1.69588	−0.847941	−	0.530091i	$-0.822158\pi$
$7$	−4.48688	−1.69588	−0.847941	+	0.530091i	$0.822158\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.20633	−0.363722	−0.181861	−	0.983324i	$-0.558212\pi$
$11$	−1.20633	−0.363722	−0.181861	+	0.983324i	$0.558212\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−7.69321	−1.76494	−0.882472	−	0.470365i	$-0.844122\pi$
$19$	−7.69321	−1.76494	−0.882472	+	0.470365i	$0.844122\pi$
$20$	0	0
$21$	4.48688	0.979118
$22$	0	0
$23$	−5.69321	−1.18712	−0.593558	−	0.804791i	$-0.702277\pi$
$23$	−5.69321	−1.18712	−0.593558	+	0.804791i	$0.702277\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.41266	−1.37650	−0.688248	−	0.725475i	$-0.741620\pi$
$29$	−7.41266	−1.37650	−0.688248	+	0.725475i	$0.741620\pi$
$30$	0	0
$31$	−1.20633	−0.216663	−0.108332	−	0.994115i	$-0.534551\pi$
$31$	−1.20633	−0.216663	−0.108332	+	0.994115i	$0.534551\pi$
$32$	0	0
$33$	1.20633	0.209995
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.719448	0.118277	0.0591383	−	0.998250i	$-0.481165\pi$
$37$	0.719448	0.118277	0.0591383	+	0.998250i	$0.481165\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	1.28055	0.199989	0.0999943	−	0.994988i	$-0.468118\pi$
$41$	1.28055	0.199989	0.0999943	+	0.994988i	$0.468118\pi$
$42$	0	0
$43$	−1.28055	−0.195282	−0.0976412	−	0.995222i	$-0.531130\pi$
$43$	−1.28055	−0.195282	−0.0976412	+	0.995222i	$0.531130\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.92578	−0.864364	−0.432182	−	0.901787i	$-0.642256\pi$
$47$	−5.92578	−0.864364	−0.432182	+	0.901787i	$0.642256\pi$
$48$	0	0
$49$	13.1321	1.87602
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−6.69321	−0.919383	−0.459692	−	0.888079i	$-0.652040\pi$
$53$	−6.69321	−0.919383	−0.459692	+	0.888079i	$0.652040\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.69321	1.01899
$58$	0	0
$59$	5.92578	0.771471	0.385735	−	0.922609i	$-0.373948\pi$
$59$	5.92578	0.771471	0.385735	+	0.922609i	$0.373948\pi$
$60$	0	0
$61$	9.41266	1.20517	0.602584	−	0.798056i	$-0.294138\pi$
$61$	9.41266	1.20517	0.602584	+	0.798056i	$0.294138\pi$
$62$	0	0
$63$	−4.48688	−0.565294
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.2063	−1.36907	−0.684536	−	0.728979i	$-0.739995\pi$
$67$	−11.2063	−1.36907	−0.684536	+	0.728979i	$0.739995\pi$
$68$	0	0
$69$	5.69321	0.685382
$70$	0	0
$71$	5.13211	0.609069	0.304535	−	0.952501i	$-0.401499\pi$
$71$	5.13211	0.609069	0.304535	+	0.952501i	$0.401499\pi$
$72$	0	0
$73$	−5.13211	−0.600668	−0.300334	−	0.953834i	$-0.597098\pi$
$73$	−5.13211	−0.600668	−0.300334	+	0.953834i	$0.597098\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.41266	0.616830
$78$	0	0
$79$	2.71945	0.305962	0.152981	−	0.988229i	$-0.451113\pi$
$79$	2.71945	0.305962	0.152981	+	0.988229i	$0.451113\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.89954	−0.537795	−0.268897	−	0.963169i	$-0.586659\pi$
$83$	−4.89954	−0.537795	−0.268897	+	0.963169i	$0.586659\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.41266	0.794721
$88$	0	0
$89$	−0.412660	−0.0437419	−0.0218709	−	0.999761i	$-0.506962\pi$
$89$	−0.412660	−0.0437419	−0.0218709	+	0.999761i	$0.506962\pi$
$90$	0	0
$91$	−4.48688	−0.470353
$92$	0	0
$93$	1.20633	0.125091
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.9738	−1.31729	−0.658643	−	0.752456i	$-0.728869\pi$
$97$	−12.9738	−1.31729	−0.658643	+	0.752456i	$0.728869\pi$
$98$	0	0
$99$	−1.20633	−0.121241
$100$	0	0
$101$	13.6670	1.35991	0.679957	−	0.733252i	$-0.261998\pi$
$101$	13.6670	1.35991	0.679957	+	0.733252i	$0.261998\pi$
$102$	0	0
$103$	0.719448	0.0708893	0.0354447	−	0.999372i	$-0.488715\pi$
$103$	0.719448	0.0708893	0.0354447	+	0.999372i	$0.488715\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.7991	1.33401	0.667004	−	0.745054i	$-0.267576\pi$
$107$	13.7991	1.33401	0.667004	+	0.745054i	$0.267576\pi$
$108$	0	0
$109$	0.867892	0.0831290	0.0415645	−	0.999136i	$-0.486766\pi$
$109$	0.867892	0.0831290	0.0415645	+	0.999136i	$0.486766\pi$
$110$	0	0
$111$	−0.719448	−0.0682870
$112$	0	0
$113$	6.41266	0.603252	0.301626	−	0.953426i	$-0.402471\pi$
$113$	6.41266	0.603252	0.301626	+	0.953426i	$0.402471\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	13.4606	1.23394
$120$	0	0
$121$	−9.54477	−0.867706
$122$	0	0
$123$	−1.28055	−0.115463
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.1321	−1.34276	−0.671379	−	0.741114i	$-0.734298\pi$
$127$	−15.1321	−1.34276	−0.671379	+	0.741114i	$0.734298\pi$
$128$	0	0
$129$	1.28055	0.112746
$130$	0	0
$131$	−7.69321	−0.672159	−0.336080	−	0.941834i	$-0.609101\pi$
$131$	−7.69321	−0.672159	−0.336080	+	0.941834i	$0.609101\pi$
$132$	0	0
$133$	34.5185	2.99314
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.7991	−1.69155	−0.845775	−	0.533540i	$-0.820861\pi$
$137$	−19.7991	−1.69155	−0.845775	+	0.533540i	$0.820861\pi$
$138$	0	0
$139$	−19.9475	−1.69193	−0.845964	−	0.533241i	$-0.820974\pi$
$139$	−19.9475	−1.69193	−0.845964	+	0.533241i	$0.820974\pi$
$140$	0	0
$141$	5.92578	0.499041
$142$	0	0
$143$	−1.20633	−0.100878
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.1321	−1.08312
$148$	0	0
$149$	3.85156	0.315532	0.157766	−	0.987477i	$-0.449571\pi$
$149$	3.85156	0.315532	0.157766	+	0.987477i	$0.449571\pi$
$150$	0	0
$151$	−12.0742	−0.982586	−0.491293	−	0.870994i	$-0.663476\pi$
$151$	−12.0742	−0.982586	−0.491293	+	0.870994i	$0.663476\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.69321	−0.534176	−0.267088	−	0.963672i	$-0.586062\pi$
$157$	−6.69321	−0.534176	−0.267088	+	0.963672i	$0.586062\pi$
$158$	0	0
$159$	6.69321	0.530806
$160$	0	0
$161$	25.5448	2.01321
$162$	0	0
$163$	10.2543	0.803180	0.401590	−	0.915820i	$-0.368458\pi$
$163$	10.2543	0.803180	0.401590	+	0.915820i	$0.368458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.71945	−0.365202	−0.182601	−	0.983187i	$-0.558452\pi$
$167$	−4.71945	−0.365202	−0.182601	+	0.983187i	$0.558452\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.69321	−0.588315
$172$	0	0
$173$	−0.280552	−0.0213300	−0.0106650	−	0.999943i	$-0.503395\pi$
$173$	−0.280552	−0.0213300	−0.0106650	+	0.999943i	$0.503395\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.92578	−0.445409
$178$	0	0
$179$	−2.15834	−0.161322	−0.0806611	−	0.996742i	$-0.525703\pi$
$179$	−2.15834	−0.161322	−0.0806611	+	0.996742i	$0.525703\pi$
$180$	0	0
$181$	−5.10587	−0.379516	−0.189758	−	0.981831i	$-0.560770\pi$
$181$	−5.10587	−0.379516	−0.189758	+	0.981831i	$0.560770\pi$
$182$	0	0
$183$	−9.41266	−0.695804
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.61899	0.264647
$188$	0	0
$189$	4.48688	0.326373
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	20.2543	1.45794	0.728969	−	0.684547i	$-0.240000\pi$
$193$	20.2543	1.45794	0.728969	+	0.684547i	$0.240000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.1059	−1.00500	−0.502501	−	0.864577i	$-0.667587\pi$
$197$	−14.1059	−1.00500	−0.502501	+	0.864577i	$0.667587\pi$
$198$	0	0
$199$	13.3864	0.948938	0.474469	−	0.880272i	$-0.342640\pi$
$199$	13.3864	0.948938	0.474469	+	0.880272i	$0.342640\pi$
$200$	0	0
$201$	11.2063	0.790434
$202$	0	0
$203$	33.2597	2.33438
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5.69321	−0.395706
$208$	0	0
$209$	9.28055	0.641949
$210$	0	0
$211$	15.1321	1.04174	0.520869	−	0.853637i	$-0.325608\pi$
$211$	15.1321	1.04174	0.520869	+	0.853637i	$0.325608\pi$
$212$	0	0
$213$	−5.13211	−0.351646
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.41266	0.367435
$218$	0	0
$219$	5.13211	0.346796
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	19.0796	1.27767	0.638833	−	0.769345i	$-0.279417\pi$
$223$	19.0796	1.27767	0.638833	+	0.769345i	$0.279417\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.7674	−1.17927	−0.589633	−	0.807671i	$-0.700728\pi$
$227$	−17.7674	−1.17927	−0.589633	+	0.807671i	$0.700728\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−5.41266	−0.356127
$232$	0	0
$233$	−14.4127	−0.944205	−0.472102	−	0.881544i	$-0.656505\pi$
$233$	−14.4127	−0.944205	−0.472102	+	0.881544i	$0.656505\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.71945	−0.176647
$238$	0	0
$239$	26.0217	1.68321	0.841604	−	0.540096i	$-0.181612\pi$
$239$	26.0217	1.68321	0.841604	+	0.540096i	$0.181612\pi$
$240$	0	0
$241$	3.74568	0.241281	0.120640	−	0.992696i	$-0.461505\pi$
$241$	3.74568	0.241281	0.120640	+	0.992696i	$0.461505\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.69321	−0.489507
$248$	0	0
$249$	4.89954	0.310496
$250$	0	0
$251$	−6.66698	−0.420816	−0.210408	−	0.977614i	$-0.567479\pi$
$251$	−6.66698	−0.420816	−0.210408	+	0.977614i	$0.567479\pi$
$252$	0	0
$253$	6.86789	0.431781
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.1321	1.13105	0.565525	−	0.824731i	$-0.308674\pi$
$257$	18.1321	1.13105	0.565525	+	0.824731i	$0.308674\pi$
$258$	0	0
$259$	−3.22808	−0.200583
$260$	0	0
$261$	−7.41266	−0.458832
$262$	0	0
$263$	3.28055	0.202287	0.101144	−	0.994872i	$-0.467750\pi$
$263$	3.28055	0.202287	0.101144	+	0.994872i	$0.467750\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.412660	0.0252544
$268$	0	0
$269$	−24.9475	−1.52108	−0.760539	−	0.649293i	$-0.775065\pi$
$269$	−24.9475	−1.52108	−0.760539	+	0.649293i	$0.775065\pi$
$270$	0	0
$271$	7.31220	0.444185	0.222092	−	0.975026i	$-0.428711\pi$
$271$	7.31220	0.444185	0.222092	+	0.975026i	$0.428711\pi$
$272$	0	0
$273$	4.48688	0.271558
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.9738	0.779518	0.389759	−	0.920917i	$-0.372558\pi$
$277$	12.9738	0.779518	0.389759	+	0.920917i	$0.372558\pi$
$278$	0	0
$279$	−1.20633	−0.0722211
$280$	0	0
$281$	33.3864	1.99167	0.995834	−	0.0911899i	$-0.0290670\pi$
$281$	33.3864	1.99167	0.995834	+	0.0911899i	$0.0290670\pi$
$282$	0	0
$283$	29.3864	1.74684	0.873421	−	0.486966i	$-0.161897\pi$
$283$	29.3864	1.74684	0.873421	+	0.486966i	$0.161897\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.74568	−0.339157
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	12.9738	0.760535
$292$	0	0
$293$	−10.8253	−0.632422	−0.316211	−	0.948689i	$-0.602411\pi$
$293$	−10.8253	−0.632422	−0.316211	+	0.948689i	$0.602411\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.20633	0.0699984
$298$	0	0
$299$	−5.69321	−0.329247
$300$	0	0
$301$	5.74568	0.331176
$302$	0	0
$303$	−13.6670	−0.785147
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.5448	0.773041	0.386520	−	0.922281i	$-0.373677\pi$
$307$	13.5448	0.773041	0.386520	+	0.922281i	$0.373677\pi$
$308$	0	0
$309$	−0.719448	−0.0409280
$310$	0	0
$311$	−8.26422	−0.468621	−0.234310	−	0.972162i	$-0.575283\pi$
$311$	−8.26422	−0.468621	−0.234310	+	0.972162i	$0.575283\pi$
$312$	0	0
$313$	11.3068	0.639097	0.319549	−	0.947570i	$-0.396469\pi$
$313$	11.3068	0.639097	0.319549	+	0.947570i	$0.396469\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.6407	1.21546	0.607732	−	0.794142i	$-0.292079\pi$
$317$	21.6407	1.21546	0.607732	+	0.794142i	$0.292079\pi$
$318$	0	0
$319$	8.94211	0.500662
$320$	0	0
$321$	−13.7991	−0.770190
$322$	0	0
$323$	23.0796	1.28419
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.867892	−0.0479945
$328$	0	0
$329$	26.5883	1.46586
$330$	0	0
$331$	−28.5185	−1.56752	−0.783760	−	0.621064i	$-0.786701\pi$
$331$	−28.5185	−1.56752	−0.783760	+	0.621064i	$0.786701\pi$
$332$	0	0
$333$	0.719448	0.0394255
$334$	0	0
$335$	0	0
$336$	0	0
$337$	33.6145	1.83110	0.915549	−	0.402206i	$-0.131756\pi$
$337$	33.6145	1.83110	0.915549	+	0.402206i	$0.131756\pi$
$338$	0	0
$339$	−6.41266	−0.348288
$340$	0	0
$341$	1.45523	0.0788052
$342$	0	0
$343$	−27.5140	−1.48562
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−34.8253	−1.86952	−0.934761	−	0.355278i	$-0.884386\pi$
$347$	−34.8253	−1.86952	−0.934761	+	0.355278i	$0.884386\pi$
$348$	0	0
$349$	−2.66698	−0.142760	−0.0713800	−	0.997449i	$-0.522740\pi$
$349$	−2.66698	−0.142760	−0.0713800	+	0.997449i	$0.522740\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−4.45523	−0.237128	−0.118564	−	0.992946i	$-0.537829\pi$
$353$	−4.45523	−0.237128	−0.118564	+	0.992946i	$0.537829\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−13.4606	−0.712413
$358$	0	0
$359$	7.92578	0.418307	0.209153	−	0.977883i	$-0.432929\pi$
$359$	7.92578	0.418307	0.209153	+	0.977883i	$0.432929\pi$
$360$	0	0
$361$	40.1855	2.11503
$362$	0	0
$363$	9.54477	0.500970
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.7728	0.979935	0.489967	−	0.871741i	$-0.337009\pi$
$367$	18.7728	0.979935	0.489967	+	0.871741i	$0.337009\pi$
$368$	0	0
$369$	1.28055	0.0666629
$370$	0	0
$371$	30.0316	1.55917
$372$	0	0
$373$	30.2281	1.56515	0.782575	−	0.622556i	$-0.213906\pi$
$373$	30.2281	1.56515	0.782575	+	0.622556i	$0.213906\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.41266	−0.381771
$378$	0	0
$379$	11.0579	0.568005	0.284003	−	0.958823i	$-0.408338\pi$
$379$	11.0579	0.568005	0.284003	+	0.958823i	$0.408338\pi$
$380$	0	0
$381$	15.1321	0.775241
$382$	0	0
$383$	22.2543	1.13714	0.568571	−	0.822634i	$-0.307496\pi$
$383$	22.2543	1.13714	0.568571	+	0.822634i	$0.307496\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.28055	−0.0650941
$388$	0	0
$389$	23.2380	1.17821	0.589106	−	0.808056i	$-0.299480\pi$
$389$	23.2380	1.17821	0.589106	+	0.808056i	$0.299480\pi$
$390$	0	0
$391$	17.0796	0.863754
$392$	0	0
$393$	7.69321	0.388071
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.36019	−0.218832	−0.109416	−	0.993996i	$-0.534898\pi$
$397$	−4.36019	−0.218832	−0.109416	+	0.993996i	$0.534898\pi$
$398$	0	0
$399$	−34.5185	−1.72809
$400$	0	0
$401$	−30.8787	−1.54201	−0.771005	−	0.636829i	$-0.780246\pi$
$401$	−30.8787	−1.54201	−0.771005	+	0.636829i	$0.780246\pi$
$402$	0	0
$403$	−1.20633	−0.0600916
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.867892	−0.0430198
$408$	0	0
$409$	−32.8787	−1.62575	−0.812874	−	0.582440i	$-0.802098\pi$
$409$	−32.8787	−1.62575	−0.812874	+	0.582440i	$0.802098\pi$
$410$	0	0
$411$	19.7991	0.976617
$412$	0	0
$413$	−26.5883	−1.30832
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.9475	0.976835
$418$	0	0
$419$	20.7194	1.01221	0.506106	−	0.862471i	$-0.331085\pi$
$419$	20.7194	1.01221	0.506106	+	0.862471i	$0.331085\pi$
$420$	0	0
$421$	−24.2117	−1.18001	−0.590004	−	0.807400i	$-0.700874\pi$
$421$	−24.2117	−1.18001	−0.590004	+	0.807400i	$0.700874\pi$
$422$	0	0
$423$	−5.92578	−0.288121
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−42.2335	−2.04382
$428$	0	0
$429$	1.20633	0.0582422
$430$	0	0
$431$	0.920365	0.0443324	0.0221662	−	0.999754i	$-0.492944\pi$
$431$	0.920365	0.0443324	0.0221662	+	0.999754i	$0.492944\pi$
$432$	0	0
$433$	−13.7991	−0.663142	−0.331571	−	0.943430i	$-0.607579\pi$
$433$	−13.7991	−0.663142	−0.331571	+	0.943430i	$0.607579\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	43.7991	2.09519
$438$	0	0
$439$	21.3765	1.02025	0.510123	−	0.860102i	$-0.329600\pi$
$439$	21.3765	1.02025	0.510123	+	0.860102i	$0.329600\pi$
$440$	0	0
$441$	13.1321	0.625338
$442$	0	0
$443$	−10.1059	−0.480144	−0.240072	−	0.970755i	$-0.577171\pi$
$443$	−10.1059	−0.480144	−0.240072	+	0.970755i	$0.577171\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.85156	−0.182172
$448$	0	0
$449$	−28.5086	−1.34541	−0.672703	−	0.739913i	$-0.734867\pi$
$449$	−28.5086	−1.34541	−0.672703	+	0.739913i	$0.734867\pi$
$450$	0	0
$451$	−1.54477	−0.0727403
$452$	0	0
$453$	12.0742	0.567296
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.4923	1.37959	0.689796	−	0.724004i	$-0.257700\pi$
$457$	29.4923	1.37959	0.689796	+	0.724004i	$0.257700\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	3.73578	0.173993	0.0869964	−	0.996209i	$-0.472273\pi$
$461$	3.73578	0.173993	0.0869964	+	0.996209i	$0.472273\pi$
$462$	0	0
$463$	−7.00541	−0.325569	−0.162785	−	0.986662i	$-0.552048\pi$
$463$	−7.00541	−0.325569	−0.162785	+	0.986662i	$0.552048\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.5349	−1.18161	−0.590806	−	0.806813i	$-0.701190\pi$
$467$	−25.5349	−1.18161	−0.590806	+	0.806813i	$0.701190\pi$
$468$	0	0
$469$	50.2815	2.32178
$470$	0	0
$471$	6.69321	0.308407
$472$	0	0
$473$	1.54477	0.0710285
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.69321	−0.306461
$478$	0	0
$479$	2.23257	0.102009	0.0510043	−	0.998698i	$-0.483758\pi$
$479$	2.23257	0.102009	0.0510043	+	0.998698i	$0.483758\pi$
$480$	0	0
$481$	0.719448	0.0328040
$482$	0	0
$483$	−25.5448	−1.16233
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−26.4443	−1.19831	−0.599153	−	0.800635i	$-0.704496\pi$
$487$	−26.4443	−1.19831	−0.599153	+	0.800635i	$0.704496\pi$
$488$	0	0
$489$	−10.2543	−0.463716
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	22.2380	1.00155
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23.0272	−1.03291
$498$	0	0
$499$	41.2597	1.84704	0.923520	−	0.383551i	$-0.125299\pi$
$499$	41.2597	1.84704	0.923520	+	0.383551i	$0.125299\pi$
$500$	0	0
$501$	4.71945	0.210849
$502$	0	0
$503$	−25.1321	−1.12059	−0.560293	−	0.828295i	$-0.689311\pi$
$503$	−25.1321	−1.12059	−0.560293	+	0.828295i	$0.689311\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−34.2018	−1.51597	−0.757985	−	0.652272i	$-0.773816\pi$
$509$	−34.2018	−1.51597	−0.757985	+	0.652272i	$0.773816\pi$
$510$	0	0
$511$	23.0272	1.01866
$512$	0	0
$513$	7.69321	0.339664
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.14844	0.314388
$518$	0	0
$519$	0.280552	0.0123149
$520$	0	0
$521$	−1.17468	−0.0514637	−0.0257318	−	0.999669i	$-0.508192\pi$
$521$	−1.17468	−0.0514637	−0.0257318	+	0.999669i	$0.508192\pi$
$522$	0	0
$523$	2.20092	0.0962394	0.0481197	−	0.998842i	$-0.484677\pi$
$523$	2.20092	0.0962394	0.0481197	+	0.998842i	$0.484677\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.61899	0.157646
$528$	0	0
$529$	9.41266	0.409246
$530$	0	0
$531$	5.92578	0.257157
$532$	0	0
$533$	1.28055	0.0554669
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.15834	0.0931394
$538$	0	0
$539$	−15.8417	−0.682348
$540$	0	0
$541$	−9.13211	−0.392620	−0.196310	−	0.980542i	$-0.562896\pi$
$541$	−9.13211	−0.392620	−0.196310	+	0.980542i	$0.562896\pi$
$542$	0	0
$543$	5.10587	0.219114
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−38.0534	−1.62705	−0.813523	−	0.581533i	$-0.802453\pi$
$547$	−38.0534	−1.62705	−0.813523	+	0.581533i	$0.802453\pi$
$548$	0	0
$549$	9.41266	0.401723
$550$	0	0
$551$	57.0272	2.42944
$552$	0	0
$553$	−12.2018	−0.518875
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.2543	0.688717	0.344359	−	0.938838i	$-0.388096\pi$
$557$	16.2543	0.688717	0.344359	+	0.938838i	$0.388096\pi$
$558$	0	0
$559$	−1.28055	−0.0541616
$560$	0	0
$561$	−3.61899	−0.152794
$562$	0	0
$563$	7.95743	0.335366	0.167683	−	0.985841i	$-0.446372\pi$
$563$	7.95743	0.335366	0.167683	+	0.985841i	$0.446372\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.48688	−0.188431
$568$	0	0
$569$	−30.2806	−1.26943	−0.634713	−	0.772748i	$-0.718882\pi$
$569$	−30.2806	−1.26943	−0.634713	+	0.772748i	$0.718882\pi$
$570$	0	0
$571$	15.6407	0.654545	0.327272	−	0.944930i	$-0.393871\pi$
$571$	15.6407	0.654545	0.327272	+	0.944930i	$0.393871\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.67688	−0.361223	−0.180612	−	0.983555i	$-0.557808\pi$
$577$	−8.67688	−0.361223	−0.180612	+	0.983555i	$0.557808\pi$
$578$	0	0
$579$	−20.2543	−0.841741
$580$	0	0
$581$	21.9837	0.912036
$582$	0	0
$583$	8.07422	0.334400
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.7150	−1.47411	−0.737057	−	0.675830i	$-0.763785\pi$
$587$	−35.7150	−1.47411	−0.737057	+	0.675830i	$0.763785\pi$
$588$	0	0
$589$	9.28055	0.382398
$590$	0	0
$591$	14.1059	0.580238
$592$	0	0
$593$	−26.5086	−1.08858	−0.544289	−	0.838897i	$-0.683201\pi$
$593$	−26.5086	−1.08858	−0.544289	+	0.838897i	$0.683201\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.3864	−0.547870
$598$	0	0
$599$	−35.9050	−1.46704	−0.733518	−	0.679670i	$-0.762123\pi$
$599$	−35.9050	−1.46704	−0.733518	+	0.679670i	$0.762123\pi$
$600$	0	0
$601$	27.5185	1.12250	0.561252	−	0.827645i	$-0.310320\pi$
$601$	27.5185	1.12250	0.561252	+	0.827645i	$0.310320\pi$
$602$	0	0
$603$	−11.2063	−0.456357
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.53487	−0.305831	−0.152915	−	0.988239i	$-0.548866\pi$
$607$	−7.53487	−0.305831	−0.152915	+	0.988239i	$0.548866\pi$
$608$	0	0
$609$	−33.2597	−1.34775
$610$	0	0
$611$	−5.92578	−0.239731
$612$	0	0
$613$	22.8253	0.921906	0.460953	−	0.887425i	$-0.347508\pi$
$613$	22.8253	0.921906	0.460953	+	0.887425i	$0.347508\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.5349	−1.59161	−0.795807	−	0.605550i	$-0.792953\pi$
$617$	−39.5349	−1.59161	−0.795807	+	0.605550i	$0.792953\pi$
$618$	0	0
$619$	−10.9837	−0.441471	−0.220735	−	0.975334i	$-0.570846\pi$
$619$	−10.9837	−0.441471	−0.220735	+	0.975334i	$0.570846\pi$
$620$	0	0
$621$	5.69321	0.228461
$622$	0	0
$623$	1.85156	0.0741810
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.28055	−0.370630
$628$	0	0
$629$	−2.15834	−0.0860588
$630$	0	0
$631$	−34.6670	−1.38007	−0.690035	−	0.723776i	$-0.742405\pi$
$631$	−34.6670	−1.38007	−0.690035	+	0.723776i	$0.742405\pi$
$632$	0	0
$633$	−15.1321	−0.601447
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.1321	0.520313
$638$	0	0
$639$	5.13211	0.203023
$640$	0	0
$641$	10.1222	0.399803	0.199902	−	0.979816i	$-0.435938\pi$
$641$	10.1222	0.399803	0.199902	+	0.979816i	$0.435938\pi$
$642$	0	0
$643$	14.5710	0.574624	0.287312	−	0.957837i	$-0.407238\pi$
$643$	14.5710	0.574624	0.287312	+	0.957837i	$0.407238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.1855	1.69780	0.848899	−	0.528556i	$-0.177266\pi$
$647$	43.1855	1.69780	0.848899	+	0.528556i	$0.177266\pi$
$648$	0	0
$649$	−7.14844	−0.280601
$650$	0	0
$651$	−5.41266	−0.212139
$652$	0	0
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.13211	−0.200223
$658$	0	0
$659$	−25.7892	−1.00460	−0.502302	−	0.864692i	$-0.667513\pi$
$659$	−25.7892	−1.00460	−0.502302	+	0.864692i	$0.667513\pi$
$660$	0	0
$661$	28.6670	1.11502	0.557508	−	0.830172i	$-0.311758\pi$
$661$	28.6670	1.11502	0.557508	+	0.830172i	$0.311758\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	42.2018	1.63406
$668$	0	0
$669$	−19.0796	−0.737661
$670$	0	0
$671$	−11.3548	−0.438346
$672$	0	0
$673$	−7.40276	−0.285355	−0.142678	−	0.989769i	$-0.545571\pi$
$673$	−7.40276	−0.285355	−0.142678	+	0.989769i	$0.545571\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.3864	1.74434	0.872171	−	0.489201i	$-0.162712\pi$
$677$	45.3864	1.74434	0.872171	+	0.489201i	$0.162712\pi$
$678$	0	0
$679$	58.2117	2.23396
$680$	0	0
$681$	17.7674	0.680850
$682$	0	0
$683$	48.5403	1.85734	0.928671	−	0.370904i	$-0.120952\pi$
$683$	48.5403	1.85734	0.928671	+	0.370904i	$0.120952\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	−6.69321	−0.254991
$690$	0	0
$691$	−0.803571	−0.0305693	−0.0152846	−	0.999883i	$-0.504865\pi$
$691$	−0.803571	−0.0305693	−0.0152846	+	0.999883i	$0.504865\pi$
$692$	0	0
$693$	5.41266	0.205610
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.84166	−0.145513
$698$	0	0
$699$	14.4127	0.545137
$700$	0	0
$701$	−31.6571	−1.19567	−0.597836	−	0.801619i	$-0.703972\pi$
$701$	−31.6571	−1.19567	−0.597836	+	0.801619i	$0.703972\pi$
$702$	0	0
$703$	−5.53487	−0.208751
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−61.3221	−2.30626
$708$	0	0
$709$	−8.66698	−0.325495	−0.162748	−	0.986668i	$-0.552036\pi$
$709$	−8.66698	−0.325495	−0.162748	+	0.986668i	$0.552036\pi$
$710$	0	0
$711$	2.71945	0.101987
$712$	0	0
$713$	6.86789	0.257205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−26.0217	−0.971800
$718$	0	0
$719$	20.9312	0.780602	0.390301	−	0.920687i	$-0.372371\pi$
$719$	20.9312	0.780602	0.390301	+	0.920687i	$0.372371\pi$
$720$	0	0
$721$	−3.22808	−0.120220
$722$	0	0
$723$	−3.74568	−0.139304
$724$	0	0
$725$	0	0
$726$	0	0
$727$	42.6670	1.58243	0.791215	−	0.611538i	$-0.209449\pi$
$727$	42.6670	1.58243	0.791215	+	0.611538i	$0.209449\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.84166	0.142089
$732$	0	0
$733$	−20.2117	−0.746538	−0.373269	−	0.927723i	$-0.621763\pi$
$733$	−20.2117	−0.746538	−0.373269	+	0.927723i	$0.621763\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.5185	0.497962
$738$	0	0
$739$	39.0054	1.43484	0.717419	−	0.696642i	$-0.245324\pi$
$739$	39.0054	1.43484	0.717419	+	0.696642i	$0.245324\pi$
$740$	0	0
$741$	7.69321	0.282617
$742$	0	0
$743$	−35.3548	−1.29704	−0.648520	−	0.761197i	$-0.724612\pi$
$743$	−35.3548	−1.29704	−0.648520	+	0.761197i	$0.724612\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4.89954	−0.179265
$748$	0	0
$749$	−61.9149	−2.26232
$750$	0	0
$751$	−36.4661	−1.33067	−0.665333	−	0.746547i	$-0.731710\pi$
$751$	−36.4661	−1.33067	−0.665333	+	0.746547i	$0.731710\pi$
$752$	0	0
$753$	6.66698	0.242958
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.82532	−0.139034	−0.0695168	−	0.997581i	$-0.522146\pi$
$757$	−3.82532	−0.139034	−0.0695168	+	0.997581i	$0.522146\pi$
$758$	0	0
$759$	−6.86789	−0.249289
$760$	0	0
$761$	30.2543	1.09672	0.548359	−	0.836243i	$-0.315253\pi$
$761$	30.2543	1.09672	0.548359	+	0.836243i	$0.315253\pi$
$762$	0	0
$763$	−3.89413	−0.140977
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.92578	0.213967
$768$	0	0
$769$	−38.9738	−1.40543	−0.702715	−	0.711472i	$-0.748029\pi$
$769$	−38.9738	−1.40543	−0.702715	+	0.711472i	$0.748029\pi$
$770$	0	0
$771$	−18.1321	−0.653012
$772$	0	0
$773$	26.9213	0.968292	0.484146	−	0.874987i	$-0.339130\pi$
$773$	26.9213	0.968292	0.484146	+	0.874987i	$0.339130\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.22808	0.115807
$778$	0	0
$779$	−9.85156	−0.352969
$780$	0	0
$781$	−6.19102	−0.221532
$782$	0	0
$783$	7.41266	0.264907
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−22.0316	−0.785344	−0.392672	−	0.919679i	$-0.628449\pi$
$787$	−22.0316	−0.785344	−0.392672	+	0.919679i	$0.628449\pi$
$788$	0	0
$789$	−3.28055	−0.116791
$790$	0	0
$791$	−28.7728	−1.02304
$792$	0	0
$793$	9.41266	0.334253
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.7466	1.72669	0.863347	−	0.504611i	$-0.168364\pi$
$797$	48.7466	1.72669	0.863347	+	0.504611i	$0.168364\pi$
$798$	0	0
$799$	17.7773	0.628917
$800$	0	0
$801$	−0.412660	−0.0145806
$802$	0	0
$803$	6.19102	0.218476
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.9475	0.878195
$808$	0	0
$809$	−25.7991	−0.907047	−0.453524	−	0.891244i	$-0.649833\pi$
$809$	−25.7991	−0.907047	−0.453524	+	0.891244i	$0.649833\pi$
$810$	0	0
$811$	40.1375	1.40942	0.704710	−	0.709496i	$-0.251077\pi$
$811$	40.1375	1.40942	0.704710	+	0.709496i	$0.251077\pi$
$812$	0	0
$813$	−7.31220	−0.256450
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.85156	0.344662
$818$	0	0
$819$	−4.48688	−0.156784
$820$	0	0
$821$	48.3176	1.68630	0.843148	−	0.537681i	$-0.180700\pi$
$821$	48.3176	1.68630	0.843148	+	0.537681i	$0.180700\pi$
$822$	0	0
$823$	18.0633	0.629647	0.314824	−	0.949150i	$-0.398055\pi$
$823$	18.0633	0.629647	0.314824	+	0.949150i	$0.398055\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.2696	−1.36554	−0.682769	−	0.730634i	$-0.739225\pi$
$827$	−39.2696	−1.36554	−0.682769	+	0.730634i	$0.739225\pi$
$828$	0	0
$829$	55.9213	1.94223	0.971113	−	0.238619i	$-0.0766946\pi$
$829$	55.9213	1.94223	0.971113	+	0.238619i	$0.0766946\pi$
$830$	0	0
$831$	−12.9738	−0.450055
$832$	0	0
$833$	−39.3963	−1.36500
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.20633	0.0416969
$838$	0	0
$839$	−45.9574	−1.58663	−0.793313	−	0.608814i	$-0.791646\pi$
$839$	−45.9574	−1.58663	−0.793313	+	0.608814i	$0.791646\pi$
$840$	0	0
$841$	25.9475	0.894742
$842$	0	0
$843$	−33.3864	−1.14989
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.8262	1.47153
$848$	0	0
$849$	−29.3864	−1.00854
$850$	0	0
$851$	−4.09597	−0.140408
$852$	0	0
$853$	−32.7204	−1.12032	−0.560162	−	0.828383i	$-0.689261\pi$
$853$	−32.7204	−1.12032	−0.560162	+	0.828383i	$0.689261\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.8951	−0.884558	−0.442279	−	0.896877i	$-0.645830\pi$
$857$	−25.8951	−0.884558	−0.442279	+	0.896877i	$0.645830\pi$
$858$	0	0
$859$	1.57744	0.0538215	0.0269108	−	0.999638i	$-0.491433\pi$
$859$	1.57744	0.0538215	0.0269108	+	0.999638i	$0.491433\pi$
$860$	0	0
$861$	5.74568	0.195812
$862$	0	0
$863$	−32.2761	−1.09869	−0.549345	−	0.835596i	$-0.685123\pi$
$863$	−32.2761	−1.09869	−0.549345	+	0.835596i	$0.685123\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−3.28055	−0.111285
$870$	0	0
$871$	−11.2063	−0.379712
$872$	0	0
$873$	−12.9738	−0.439095
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.7991	0.398427	0.199213	−	0.979956i	$-0.436161\pi$
$877$	11.7991	0.398427	0.199213	+	0.979956i	$0.436161\pi$
$878$	0	0
$879$	10.8253	0.365129
$880$	0	0
$881$	−46.0796	−1.55246	−0.776231	−	0.630448i	$-0.782871\pi$
$881$	−46.0796	−1.55246	−0.776231	+	0.630448i	$0.782871\pi$
$882$	0	0
$883$	9.32312	0.313748	0.156874	−	0.987619i	$-0.449858\pi$
$883$	9.32312	0.313748	0.156874	+	0.987619i	$0.449858\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−53.1954	−1.78613	−0.893063	−	0.449931i	$-0.851449\pi$
$887$	−53.1954	−1.78613	−0.893063	+	0.449931i	$0.851449\pi$
$888$	0	0
$889$	67.8960	2.27716
$890$	0	0
$891$	−1.20633	−0.0404136
$892$	0	0
$893$	45.5883	1.52555
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.69321	0.190091
$898$	0	0
$899$	8.94211	0.298236
$900$	0	0
$901$	20.0796	0.668950
$902$	0	0
$903$	−5.74568	−0.191204
$904$	0	0
$905$	0	0
$906$	0	0
$907$	56.7204	1.88337	0.941685	−	0.336497i	$-0.109242\pi$
$907$	56.7204	1.88337	0.941685	+	0.336497i	$0.109242\pi$
$908$	0	0
$909$	13.6670	0.453305
$910$	0	0
$911$	33.5883	1.11283	0.556414	−	0.830905i	$-0.312177\pi$
$911$	33.5883	1.11283	0.556414	+	0.830905i	$0.312177\pi$
$912$	0	0
$913$	5.91046	0.195608
$914$	0	0
$915$	0	0
$916$	0	0
$917$	34.5185	1.13990
$918$	0	0
$919$	−5.58734	−0.184309	−0.0921547	−	0.995745i	$-0.529375\pi$
$919$	−5.58734	−0.184309	−0.0921547	+	0.995745i	$0.529375\pi$
$920$	0	0
$921$	−13.5448	−0.446315
$922$	0	0
$923$	5.13211	0.168925
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.719448	0.0236298
$928$	0	0
$929$	22.2117	0.728744	0.364372	−	0.931254i	$-0.381284\pi$
$929$	22.2117	0.728744	0.364372	+	0.931254i	$0.381284\pi$
$930$	0	0
$931$	−101.028	−3.31106
$932$	0	0
$933$	8.26422	0.270558
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.23798	−0.203786	−0.101893	−	0.994795i	$-0.532490\pi$
$937$	−6.23798	−0.203786	−0.101893	+	0.994795i	$0.532490\pi$
$938$	0	0
$939$	−11.3068	−0.368983
$940$	0	0
$941$	−11.1846	−0.364607	−0.182303	−	0.983242i	$-0.558355\pi$
$941$	−11.1846	−0.364607	−0.182303	+	0.983242i	$0.558355\pi$
$942$	0	0
$943$	−7.29045	−0.237410
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.9094	0.354509	0.177255	−	0.984165i	$-0.443278\pi$
$947$	10.9094	0.354509	0.177255	+	0.984165i	$0.443278\pi$
$948$	0	0
$949$	−5.13211	−0.166595
$950$	0	0
$951$	−21.6407	−0.701749
$952$	0	0
$953$	−26.2806	−0.851311	−0.425655	−	0.904885i	$-0.639956\pi$
$953$	−26.2806	−0.851311	−0.425655	+	0.904885i	$0.639956\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.94211	−0.289057
$958$	0	0
$959$	88.8361	2.86867
$960$	0	0
$961$	−29.5448	−0.953057
$962$	0	0
$963$	13.7991	0.444669
$964$	0	0
$965$	0	0
$966$	0	0
$967$	13.4705	0.433184	0.216592	−	0.976262i	$-0.430506\pi$
$967$	13.4705	0.433184	0.216592	+	0.976262i	$0.430506\pi$
$968$	0	0
$969$	−23.0796	−0.741425
$970$	0	0
$971$	−11.5349	−0.370172	−0.185086	−	0.982722i	$-0.559256\pi$
$971$	−11.5349	−0.370172	−0.185086	+	0.982722i	$0.559256\pi$
$972$	0	0
$973$	89.5022	2.86931
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.2117	−0.518660	−0.259330	−	0.965789i	$-0.583502\pi$
$977$	−16.2117	−0.518660	−0.259330	+	0.965789i	$0.583502\pi$
$978$	0	0
$979$	0.497804	0.0159099
$980$	0	0
$981$	0.867892	0.0277097
$982$	0	0
$983$	17.1014	0.545449	0.272725	−	0.962092i	$-0.412075\pi$
$983$	17.1014	0.545449	0.272725	+	0.962092i	$0.412075\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−26.5883	−0.846314
$988$	0	0
$989$	7.29045	0.231823
$990$	0	0
$991$	−59.0470	−1.87569	−0.937844	−	0.347056i	$-0.887181\pi$
$991$	−59.0470	−1.87569	−0.937844	+	0.347056i	$0.887181\pi$
$992$	0	0
$993$	28.5185	0.905008
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−47.5195	−1.50496	−0.752478	−	0.658617i	$-0.771142\pi$
$997$	−47.5195	−1.50496	−0.752478	+	0.658617i	$0.771142\pi$
$998$	0	0
$999$	−0.719448	−0.0227623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bh.1.1	✓	3
5.4	even	2		7800.2.a.bs.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bh.1.1	✓	3	1.1	even	1	trivial
7800.2.a.bs.1.3	yes	3	5.4	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.48688 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.48688 q^{7} +1.00000 q^{9} -1.20633 q^{11} +1.00000 q^{13} -3.00000 q^{17} -7.69321 q^{19} +4.48688 q^{21} -5.69321 q^{23} -1.00000 q^{27} -7.41266 q^{29} -1.20633 q^{31} +1.20633 q^{33} +0.719448 q^{37} -1.00000 q^{39} +1.28055 q^{41} -1.28055 q^{43} -5.92578 q^{47} +13.1321 q^{49} +3.00000 q^{51} -6.69321 q^{53} +7.69321 q^{57} +5.92578 q^{59} +9.41266 q^{61} -4.48688 q^{63} -11.2063 q^{67} +5.69321 q^{69} +5.13211 q^{71} -5.13211 q^{73} +5.41266 q^{77} +2.71945 q^{79} +1.00000 q^{81} -4.89954 q^{83} +7.41266 q^{87} -0.412660 q^{89} -4.48688 q^{91} +1.20633 q^{93} -12.9738 q^{97} -1.20633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9 + 5 * q^11 + 3 * q^13 - 9 * q^17 - 2 * q^19 + q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 5 * q^31 - 5 * q^33 + 6 * q^37 - 3 * q^39 - 13 * q^47 + 26 * q^49 + 9 * q^51 + q^53 + 2 * q^57 + 13 * q^59 + 11 * q^61 - q^63 - 25 * q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - q^77 + 12 * q^79 + 3 * q^81 + 15 * q^83 + 5 * q^87 + 16 * q^89 - q^91 - 5 * q^93 - 14 * q^97 + 5 * q^99

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