LMFDB - Embedded newform 10000.2.a.bo.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + \cdots + 45 $ x^12 - x^11 - 20x^10 + 11x^9 + 144x^8 - 29x^7 - 440x^6 + 4x^5 + 556x^4 + 15x^3 - 285*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.24312$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.602175 q^{3} +3.10675 q^{7} -2.63738 q^{9} +O(q^{10})$	$q-0.602175 q^{3} +3.10675 q^{7} -2.63738 q^{9} +6.35836 q^{11} +1.18880 q^{13} -8.12739 q^{17} +3.07990 q^{19} -1.87081 q^{21} -4.85739 q^{23} +3.39469 q^{27} +1.77689 q^{29} -6.90486 q^{31} -3.82885 q^{33} +6.84968 q^{37} -0.715868 q^{39} -5.93533 q^{41} -4.11000 q^{43} -1.38389 q^{47} +2.65189 q^{49} +4.89411 q^{51} -10.8926 q^{53} -1.85464 q^{57} -10.3333 q^{59} +8.21746 q^{61} -8.19369 q^{63} +0.367587 q^{67} +2.92500 q^{69} -7.47448 q^{71} -5.05096 q^{73} +19.7538 q^{77} -12.6258 q^{79} +5.86795 q^{81} -3.71750 q^{83} -1.07000 q^{87} +8.32720 q^{89} +3.69331 q^{91} +4.15794 q^{93} -4.97243 q^{97} -16.7694 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 26 * q^9	$ 12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 26 * q^9 + q^11 - 4 * q^13 - 8 * q^17 - 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 - 33 * q^31 - 26 * q^33 - 6 * q^37 - 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 + 5 * q^51 - 22 * q^53 - 36 * q^57 - 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 - 19 * q^71 + 5 * q^73 + 13 * q^77 - 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 - 46 * q^91 - 10 * q^93 - 22 * 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
$3$	−0.602175	−0.347666	−0.173833	−	0.984775i	$-0.555615\pi$
$3$	−0.602175	−0.347666	−0.173833	+	0.984775i	$0.555615\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.10675	1.17424	0.587120	−	0.809500i	$-0.300262\pi$
$7$	3.10675	1.17424	0.587120	+	0.809500i	$0.300262\pi$
$8$	0	0
$9$	−2.63738	−0.879128
$10$	0	0
$11$	6.35836	1.91712	0.958558	−	0.284896i	$-0.0919591\pi$
$11$	6.35836	1.91712	0.958558	+	0.284896i	$0.0919591\pi$
$12$	0	0
$13$	1.18880	0.329715	0.164857	−	0.986317i	$-0.447284\pi$
$13$	1.18880	0.329715	0.164857	+	0.986317i	$0.447284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−8.12739	−1.97118	−0.985590	−	0.169150i	$-0.945898\pi$
$17$	−8.12739	−1.97118	−0.985590	+	0.169150i	$0.945898\pi$
$18$	0	0
$19$	3.07990	0.706578	0.353289	−	0.935514i	$-0.385063\pi$
$19$	3.07990	0.706578	0.353289	+	0.935514i	$0.385063\pi$
$20$	0	0
$21$	−1.87081	−0.408244
$22$	0	0
$23$	−4.85739	−1.01284	−0.506418	−	0.862288i	$-0.669031\pi$
$23$	−4.85739	−1.01284	−0.506418	+	0.862288i	$0.669031\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.39469	0.653309
$28$	0	0
$29$	1.77689	0.329961	0.164980	−	0.986297i	$-0.447244\pi$
$29$	1.77689	0.329961	0.164980	+	0.986297i	$0.447244\pi$
$30$	0	0
$31$	−6.90486	−1.24015	−0.620075	−	0.784542i	$-0.712898\pi$
$31$	−6.90486	−1.24015	−0.620075	+	0.784542i	$0.712898\pi$
$32$	0	0
$33$	−3.82885	−0.666516
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.84968	1.12608	0.563040	−	0.826429i	$-0.309632\pi$
$37$	6.84968	1.12608	0.563040	+	0.826429i	$0.309632\pi$
$38$	0	0
$39$	−0.715868	−0.114631
$40$	0	0
$41$	−5.93533	−0.926942	−0.463471	−	0.886112i	$-0.653396\pi$
$41$	−5.93533	−0.926942	−0.463471	+	0.886112i	$0.653396\pi$
$42$	0	0
$43$	−4.11000	−0.626770	−0.313385	−	0.949626i	$-0.601463\pi$
$43$	−4.11000	−0.626770	−0.313385	+	0.949626i	$0.601463\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.38389	−0.201861	−0.100930	−	0.994893i	$-0.532182\pi$
$47$	−1.38389	−0.201861	−0.100930	+	0.994893i	$0.532182\pi$
$48$	0	0
$49$	2.65189	0.378841
$50$	0	0
$51$	4.89411	0.685313
$52$	0	0
$53$	−10.8926	−1.49621	−0.748104	−	0.663582i	$-0.769036\pi$
$53$	−10.8926	−1.49621	−0.748104	+	0.663582i	$0.769036\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.85464	−0.245653
$58$	0	0
$59$	−10.3333	−1.34528	−0.672642	−	0.739968i	$-0.734841\pi$
$59$	−10.3333	−1.34528	−0.672642	+	0.739968i	$0.734841\pi$
$60$	0	0
$61$	8.21746	1.05214	0.526069	−	0.850442i	$-0.323665\pi$
$61$	8.21746	1.05214	0.526069	+	0.850442i	$0.323665\pi$
$62$	0	0
$63$	−8.19369	−1.03231
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.367587	0.0449079	0.0224539	−	0.999748i	$-0.492852\pi$
$67$	0.367587	0.0449079	0.0224539	+	0.999748i	$0.492852\pi$
$68$	0	0
$69$	2.92500	0.352129
$70$	0	0
$71$	−7.47448	−0.887057	−0.443529	−	0.896260i	$-0.646274\pi$
$71$	−7.47448	−0.887057	−0.443529	+	0.896260i	$0.646274\pi$
$72$	0	0
$73$	−5.05096	−0.591170	−0.295585	−	0.955317i	$-0.595514\pi$
$73$	−5.05096	−0.591170	−0.295585	+	0.955317i	$0.595514\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.7538	2.25116
$78$	0	0
$79$	−12.6258	−1.42052	−0.710258	−	0.703941i	$-0.751422\pi$
$79$	−12.6258	−1.42052	−0.710258	+	0.703941i	$0.751422\pi$
$80$	0	0
$81$	5.86795	0.651995
$82$	0	0
$83$	−3.71750	−0.408048	−0.204024	−	0.978966i	$-0.565402\pi$
$83$	−3.71750	−0.408048	−0.204024	+	0.978966i	$0.565402\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.07000	−0.114716
$88$	0	0
$89$	8.32720	0.882682	0.441341	−	0.897340i	$-0.354503\pi$
$89$	8.32720	0.882682	0.441341	+	0.897340i	$0.354503\pi$
$90$	0	0
$91$	3.69331	0.387164
$92$	0	0
$93$	4.15794	0.431158
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.97243	−0.504874	−0.252437	−	0.967613i	$-0.581232\pi$
$97$	−4.97243	−0.504874	−0.252437	+	0.967613i	$0.581232\pi$
$98$	0	0
$99$	−16.7694	−1.68539
$100$	0	0
$101$	−17.7796	−1.76914	−0.884569	−	0.466409i	$-0.845548\pi$
$101$	−17.7796	−1.76914	−0.884569	+	0.466409i	$0.845548\pi$
$102$	0	0
$103$	−14.4348	−1.42230	−0.711152	−	0.703038i	$-0.751826\pi$
$103$	−14.4348	−1.42230	−0.711152	+	0.703038i	$0.751826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.17066	0.209846	0.104923	−	0.994480i	$-0.466540\pi$
$107$	2.17066	0.209846	0.104923	+	0.994480i	$0.466540\pi$
$108$	0	0
$109$	13.0376	1.24878	0.624388	−	0.781114i	$-0.285348\pi$
$109$	13.0376	1.24878	0.624388	+	0.781114i	$0.285348\pi$
$110$	0	0
$111$	−4.12471	−0.391500
$112$	0	0
$113$	−12.6131	−1.18654	−0.593269	−	0.805005i	$-0.702163\pi$
$113$	−12.6131	−1.18654	−0.593269	+	0.805005i	$0.702163\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.13533	−0.289861
$118$	0	0
$119$	−25.2497	−2.31464
$120$	0	0
$121$	29.4287	2.67534
$122$	0	0
$123$	3.57411	0.322266
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.29220	0.292135	0.146068	−	0.989275i	$-0.453338\pi$
$127$	3.29220	0.292135	0.146068	+	0.989275i	$0.453338\pi$
$128$	0	0
$129$	2.47494	0.217907
$130$	0	0
$131$	1.53983	0.134535	0.0672677	−	0.997735i	$-0.478572\pi$
$131$	1.53983	0.134535	0.0672677	+	0.997735i	$0.478572\pi$
$132$	0	0
$133$	9.56849	0.829693
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.39613	−0.119280	−0.0596399	−	0.998220i	$-0.518995\pi$
$137$	−1.39613	−0.119280	−0.0596399	+	0.998220i	$0.518995\pi$
$138$	0	0
$139$	3.42105	0.290169	0.145085	−	0.989419i	$-0.453655\pi$
$139$	3.42105	0.290169	0.145085	+	0.989419i	$0.453655\pi$
$140$	0	0
$141$	0.833343	0.0701801
$142$	0	0
$143$	7.55883	0.632102
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.59690	−0.131710
$148$	0	0
$149$	−16.3532	−1.33970	−0.669852	−	0.742495i	$-0.733642\pi$
$149$	−16.3532	−1.33970	−0.669852	+	0.742495i	$0.733642\pi$
$150$	0	0
$151$	15.1332	1.23152	0.615759	−	0.787934i	$-0.288849\pi$
$151$	15.1332	1.23152	0.615759	+	0.787934i	$0.288849\pi$
$152$	0	0
$153$	21.4350	1.73292
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.4540	−0.993940	−0.496970	−	0.867768i	$-0.665554\pi$
$157$	−12.4540	−0.993940	−0.496970	+	0.867768i	$0.665554\pi$
$158$	0	0
$159$	6.55923	0.520181
$160$	0	0
$161$	−15.0907	−1.18931
$162$	0	0
$163$	11.3570	0.889547	0.444774	−	0.895643i	$-0.353284\pi$
$163$	11.3570	0.889547	0.444774	+	0.895643i	$0.353284\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.9163	0.922113	0.461057	−	0.887371i	$-0.347470\pi$
$167$	11.9163	0.922113	0.461057	+	0.887371i	$0.347470\pi$
$168$	0	0
$169$	−11.5867	−0.891288
$170$	0	0
$171$	−8.12289	−0.621173
$172$	0	0
$173$	4.22623	0.321314	0.160657	−	0.987010i	$-0.448639\pi$
$173$	4.22623	0.321314	0.160657	+	0.987010i	$0.448639\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.22248	0.467710
$178$	0	0
$179$	−5.21431	−0.389736	−0.194868	−	0.980830i	$-0.562428\pi$
$179$	−5.21431	−0.389736	−0.194868	+	0.980830i	$0.562428\pi$
$180$	0	0
$181$	10.3045	0.765928	0.382964	−	0.923763i	$-0.374903\pi$
$181$	10.3045	0.765928	0.382964	+	0.923763i	$0.374903\pi$
$182$	0	0
$183$	−4.94835	−0.365793
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−51.6768	−3.77898
$188$	0	0
$189$	10.5465	0.767142
$190$	0	0
$191$	0.431679	0.0312352	0.0156176	−	0.999878i	$-0.495029\pi$
$191$	0.431679	0.0312352	0.0156176	+	0.999878i	$0.495029\pi$
$192$	0	0
$193$	−19.7290	−1.42013	−0.710064	−	0.704138i	$-0.751334\pi$
$193$	−19.7290	−1.42013	−0.710064	+	0.704138i	$0.751334\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.4026	1.66736	0.833682	−	0.552245i	$-0.186229\pi$
$197$	23.4026	1.66736	0.833682	+	0.552245i	$0.186229\pi$
$198$	0	0
$199$	−22.1578	−1.57072	−0.785362	−	0.619036i	$-0.787523\pi$
$199$	−22.1578	−1.57072	−0.785362	+	0.619036i	$0.787523\pi$
$200$	0	0
$201$	−0.221352	−0.0156129
$202$	0	0
$203$	5.52036	0.387453
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.8108	0.890413
$208$	0	0
$209$	19.5831	1.35459
$210$	0	0
$211$	7.10573	0.489178	0.244589	−	0.969627i	$-0.421347\pi$
$211$	7.10573	0.489178	0.244589	+	0.969627i	$0.421347\pi$
$212$	0	0
$213$	4.50095	0.308400
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−21.4517	−1.45623
$218$	0	0
$219$	3.04156	0.205530
$220$	0	0
$221$	−9.66186	−0.649927
$222$	0	0
$223$	3.78945	0.253760	0.126880	−	0.991918i	$-0.459504\pi$
$223$	3.78945	0.253760	0.126880	+	0.991918i	$0.459504\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.66253	−0.110346	−0.0551730	−	0.998477i	$-0.517571\pi$
$227$	−1.66253	−0.110346	−0.0551730	+	0.998477i	$0.517571\pi$
$228$	0	0
$229$	10.2238	0.675608	0.337804	−	0.941217i	$-0.390316\pi$
$229$	10.2238	0.675608	0.337804	+	0.941217i	$0.390316\pi$
$230$	0	0
$231$	−11.8953	−0.782651
$232$	0	0
$233$	−2.26497	−0.148383	−0.0741916	−	0.997244i	$-0.523638\pi$
$233$	−2.26497	−0.148383	−0.0741916	+	0.997244i	$0.523638\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.60296	0.493865
$238$	0	0
$239$	14.6133	0.945254	0.472627	−	0.881262i	$-0.343306\pi$
$239$	14.6133	0.945254	0.472627	+	0.881262i	$0.343306\pi$
$240$	0	0
$241$	22.2188	1.43124	0.715618	−	0.698492i	$-0.246145\pi$
$241$	22.2188	1.43124	0.715618	+	0.698492i	$0.246145\pi$
$242$	0	0
$243$	−13.7176	−0.879986
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.66140	0.232969
$248$	0	0
$249$	2.23858	0.141864
$250$	0	0
$251$	22.7251	1.43440	0.717198	−	0.696869i	$-0.245424\pi$
$251$	22.7251	1.43440	0.717198	+	0.696869i	$0.245424\pi$
$252$	0	0
$253$	−30.8850	−1.94173
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.6188	−1.03666	−0.518328	−	0.855182i	$-0.673445\pi$
$257$	−16.6188	−1.03666	−0.518328	+	0.855182i	$0.673445\pi$
$258$	0	0
$259$	21.2802	1.32229
$260$	0	0
$261$	−4.68635	−0.290078
$262$	0	0
$263$	−26.5978	−1.64009	−0.820045	−	0.572299i	$-0.806052\pi$
$263$	−26.5978	−1.64009	−0.820045	+	0.572299i	$0.806052\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.01444	−0.306878
$268$	0	0
$269$	17.5220	1.06834	0.534169	−	0.845378i	$-0.320625\pi$
$269$	17.5220	1.06834	0.534169	+	0.845378i	$0.320625\pi$
$270$	0	0
$271$	8.48232	0.515265	0.257632	−	0.966243i	$-0.417058\pi$
$271$	8.48232	0.515265	0.257632	+	0.966243i	$0.417058\pi$
$272$	0	0
$273$	−2.22402	−0.134604
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.21456	−0.193144	−0.0965722	−	0.995326i	$-0.530788\pi$
$277$	−3.21456	−0.193144	−0.0965722	+	0.995326i	$0.530788\pi$
$278$	0	0
$279$	18.2108	1.09025
$280$	0	0
$281$	17.8256	1.06338	0.531692	−	0.846938i	$-0.321556\pi$
$281$	17.8256	1.06338	0.531692	+	0.846938i	$0.321556\pi$
$282$	0	0
$283$	18.1479	1.07878	0.539391	−	0.842055i	$-0.318654\pi$
$283$	18.1479	1.07878	0.539391	+	0.842055i	$0.318654\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.4396	−1.08845
$288$	0	0
$289$	49.0544	2.88555
$290$	0	0
$291$	2.99427	0.175527
$292$	0	0
$293$	0.197873	0.0115599	0.00577993	−	0.999983i	$-0.498160\pi$
$293$	0.197873	0.0115599	0.00577993	+	0.999983i	$0.498160\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	21.5847	1.25247
$298$	0	0
$299$	−5.77449	−0.333947
$300$	0	0
$301$	−12.7688	−0.735979
$302$	0	0
$303$	10.7064	0.615069
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.9718	1.02571	0.512853	−	0.858476i	$-0.328589\pi$
$307$	17.9718	1.02571	0.512853	+	0.858476i	$0.328589\pi$
$308$	0	0
$309$	8.69229	0.494487
$310$	0	0
$311$	−4.42515	−0.250928	−0.125464	−	0.992098i	$-0.540042\pi$
$311$	−4.42515	−0.250928	−0.125464	+	0.992098i	$0.540042\pi$
$312$	0	0
$313$	−5.99889	−0.339078	−0.169539	−	0.985524i	$-0.554228\pi$
$313$	−5.99889	−0.339078	−0.169539	+	0.985524i	$0.554228\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.9957	−1.57240	−0.786198	−	0.617974i	$-0.787954\pi$
$317$	−27.9957	−1.57240	−0.786198	+	0.617974i	$0.787954\pi$
$318$	0	0
$319$	11.2981	0.632573
$320$	0	0
$321$	−1.30712	−0.0729563
$322$	0	0
$323$	−25.0316	−1.39279
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.85093	−0.434157
$328$	0	0
$329$	−4.29939	−0.237033
$330$	0	0
$331$	−11.5695	−0.635917	−0.317958	−	0.948105i	$-0.602997\pi$
$331$	−11.5695	−0.635917	−0.317958	+	0.948105i	$0.602997\pi$
$332$	0	0
$333$	−18.0652	−0.989970
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.6448	−0.852228	−0.426114	−	0.904669i	$-0.640118\pi$
$337$	−15.6448	−0.852228	−0.426114	+	0.904669i	$0.640118\pi$
$338$	0	0
$339$	7.59527	0.412519
$340$	0	0
$341$	−43.9036	−2.37751
$342$	0	0
$343$	−13.5085	−0.729390
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.8151	0.634267	0.317134	−	0.948381i	$-0.397280\pi$
$347$	11.8151	0.634267	0.317134	+	0.948381i	$0.397280\pi$
$348$	0	0
$349$	−10.4189	−0.557709	−0.278855	−	0.960333i	$-0.589955\pi$
$349$	−10.4189	−0.557709	−0.278855	+	0.960333i	$0.589955\pi$
$350$	0	0
$351$	4.03562	0.215406
$352$	0	0
$353$	2.26845	0.120738	0.0603688	−	0.998176i	$-0.480772\pi$
$353$	2.26845	0.120738	0.0603688	+	0.998176i	$0.480772\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	15.2048	0.804722
$358$	0	0
$359$	9.12455	0.481575	0.240788	−	0.970578i	$-0.422594\pi$
$359$	9.12455	0.481575	0.240788	+	0.970578i	$0.422594\pi$
$360$	0	0
$361$	−9.51419	−0.500747
$362$	0	0
$363$	−17.7212	−0.930124
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−19.0423	−0.993998	−0.496999	−	0.867751i	$-0.665565\pi$
$367$	−19.0423	−0.993998	−0.496999	+	0.867751i	$0.665565\pi$
$368$	0	0
$369$	15.6537	0.814901
$370$	0	0
$371$	−33.8404	−1.75691
$372$	0	0
$373$	−14.1434	−0.732318	−0.366159	−	0.930552i	$-0.619327\pi$
$373$	−14.1434	−0.732318	−0.366159	+	0.930552i	$0.619327\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.11238	0.108793
$378$	0	0
$379$	−30.5371	−1.56859	−0.784294	−	0.620390i	$-0.786974\pi$
$379$	−30.5371	−1.56859	−0.784294	+	0.620390i	$0.786974\pi$
$380$	0	0
$381$	−1.98248	−0.101565
$382$	0	0
$383$	−25.2695	−1.29121	−0.645606	−	0.763670i	$-0.723395\pi$
$383$	−25.2695	−1.29121	−0.645606	+	0.763670i	$0.723395\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.8397	0.551011
$388$	0	0
$389$	5.24243	0.265802	0.132901	−	0.991129i	$-0.457571\pi$
$389$	5.24243	0.265802	0.132901	+	0.991129i	$0.457571\pi$
$390$	0	0
$391$	39.4779	1.99648
$392$	0	0
$393$	−0.927246	−0.0467734
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−26.2225	−1.31607	−0.658034	−	0.752988i	$-0.728612\pi$
$397$	−26.2225	−1.31607	−0.658034	+	0.752988i	$0.728612\pi$
$398$	0	0
$399$	−5.76191	−0.288456
$400$	0	0
$401$	2.36232	0.117969	0.0589844	−	0.998259i	$-0.481214\pi$
$401$	2.36232	0.117969	0.0589844	+	0.998259i	$0.481214\pi$
$402$	0	0
$403$	−8.20852	−0.408896
$404$	0	0
$405$	0	0
$406$	0	0
$407$	43.5527	2.15883
$408$	0	0
$409$	16.7447	0.827971	0.413985	−	0.910284i	$-0.364136\pi$
$409$	16.7447	0.827971	0.413985	+	0.910284i	$0.364136\pi$
$410$	0	0
$411$	0.840717	0.0414695
$412$	0	0
$413$	−32.1031	−1.57969
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.06007	−0.100882
$418$	0	0
$419$	−27.5497	−1.34589	−0.672946	−	0.739692i	$-0.734971\pi$
$419$	−27.5497	−1.34589	−0.672946	+	0.739692i	$0.734971\pi$
$420$	0	0
$421$	30.1029	1.46713	0.733564	−	0.679620i	$-0.237855\pi$
$421$	30.1029	1.46713	0.733564	+	0.679620i	$0.237855\pi$
$422$	0	0
$423$	3.64984	0.177461
$424$	0	0
$425$	0	0
$426$	0	0
$427$	25.5296	1.23546
$428$	0	0
$429$	−4.55174	−0.219760
$430$	0	0
$431$	−2.51170	−0.120985	−0.0604923	−	0.998169i	$-0.519267\pi$
$431$	−2.51170	−0.120985	−0.0604923	+	0.998169i	$0.519267\pi$
$432$	0	0
$433$	−23.2843	−1.11897	−0.559487	−	0.828839i	$-0.689002\pi$
$433$	−23.2843	−1.11897	−0.559487	+	0.828839i	$0.689002\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14.9603	−0.715649
$438$	0	0
$439$	−17.7176	−0.845614	−0.422807	−	0.906220i	$-0.638955\pi$
$439$	−17.7176	−0.845614	−0.422807	+	0.906220i	$0.638955\pi$
$440$	0	0
$441$	−6.99405	−0.333050
$442$	0	0
$443$	2.05642	0.0977035	0.0488517	−	0.998806i	$-0.484444\pi$
$443$	2.05642	0.0977035	0.0488517	+	0.998806i	$0.484444\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.84747	0.465770
$448$	0	0
$449$	22.9654	1.08381	0.541903	−	0.840441i	$-0.317704\pi$
$449$	22.9654	1.08381	0.541903	+	0.840441i	$0.317704\pi$
$450$	0	0
$451$	−37.7389	−1.77706
$452$	0	0
$453$	−9.11281	−0.428157
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.8819	1.11715	0.558573	−	0.829455i	$-0.311349\pi$
$457$	23.8819	1.11715	0.558573	+	0.829455i	$0.311349\pi$
$458$	0	0
$459$	−27.5900	−1.28779
$460$	0	0
$461$	−9.65079	−0.449482	−0.224741	−	0.974419i	$-0.572154\pi$
$461$	−9.65079	−0.449482	−0.224741	+	0.974419i	$0.572154\pi$
$462$	0	0
$463$	35.6053	1.65472	0.827359	−	0.561674i	$-0.189842\pi$
$463$	35.6053	1.65472	0.827359	+	0.561674i	$0.189842\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.9822	−1.38741	−0.693705	−	0.720259i	$-0.744023\pi$
$467$	−29.9822	−1.38741	−0.693705	+	0.720259i	$0.744023\pi$
$468$	0	0
$469$	1.14200	0.0527326
$470$	0	0
$471$	7.49951	0.345559
$472$	0	0
$473$	−26.1329	−1.20159
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.7279	1.31536
$478$	0	0
$479$	−7.71442	−0.352481	−0.176240	−	0.984347i	$-0.556394\pi$
$479$	−7.71442	−0.352481	−0.176240	+	0.984347i	$0.556394\pi$
$480$	0	0
$481$	8.14292	0.371285
$482$	0	0
$483$	9.08725	0.413484
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.35315	0.106631	0.0533157	−	0.998578i	$-0.483021\pi$
$487$	2.35315	0.106631	0.0533157	+	0.998578i	$0.483021\pi$
$488$	0	0
$489$	−6.83889	−0.309265
$490$	0	0
$491$	−16.8083	−0.758547	−0.379274	−	0.925285i	$-0.623826\pi$
$491$	−16.8083	−0.758547	−0.379274	+	0.925285i	$0.623826\pi$
$492$	0	0
$493$	−14.4415	−0.650412
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23.2213	−1.04162
$498$	0	0
$499$	5.98815	0.268066	0.134033	−	0.990977i	$-0.457207\pi$
$499$	5.98815	0.268066	0.134033	+	0.990977i	$0.457207\pi$
$500$	0	0
$501$	−7.17572	−0.320587
$502$	0	0
$503$	−40.8272	−1.82040	−0.910199	−	0.414172i	$-0.864071\pi$
$503$	−40.8272	−1.82040	−0.910199	+	0.414172i	$0.864071\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.97725	0.309871
$508$	0	0
$509$	−5.00315	−0.221761	−0.110880	−	0.993834i	$-0.535367\pi$
$509$	−5.00315	−0.221761	−0.110880	+	0.993834i	$0.535367\pi$
$510$	0	0
$511$	−15.6920	−0.694175
$512$	0	0
$513$	10.4553	0.461614
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8.79925	−0.386991
$518$	0	0
$519$	−2.54493	−0.111710
$520$	0	0
$521$	−12.1683	−0.533103	−0.266552	−	0.963821i	$-0.585884\pi$
$521$	−12.1683	−0.533103	−0.266552	+	0.963821i	$0.585884\pi$
$522$	0	0
$523$	−27.4118	−1.19863	−0.599317	−	0.800512i	$-0.704561\pi$
$523$	−27.4118	−1.19863	−0.599317	+	0.800512i	$0.704561\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	56.1185	2.44456
$528$	0	0
$529$	0.594283	0.0258384
$530$	0	0
$531$	27.2530	1.18268
$532$	0	0
$533$	−7.05594	−0.305626
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.13993	0.135498
$538$	0	0
$539$	16.8617	0.726283
$540$	0	0
$541$	37.0226	1.59173	0.795864	−	0.605476i	$-0.207017\pi$
$541$	37.0226	1.59173	0.795864	+	0.605476i	$0.207017\pi$
$542$	0	0
$543$	−6.20512	−0.266287
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.6039	−1.35129	−0.675643	−	0.737229i	$-0.736134\pi$
$547$	−31.6039	−1.35129	−0.675643	+	0.737229i	$0.736134\pi$
$548$	0	0
$549$	−21.6726	−0.924964
$550$	0	0
$551$	5.47266	0.233143
$552$	0	0
$553$	−39.2253	−1.66803
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.49371	0.0632907	0.0316454	−	0.999499i	$-0.489925\pi$
$557$	1.49371	0.0632907	0.0316454	+	0.999499i	$0.489925\pi$
$558$	0	0
$559$	−4.88599	−0.206655
$560$	0	0
$561$	31.1185	1.31382
$562$	0	0
$563$	−33.1859	−1.39862	−0.699310	−	0.714819i	$-0.746509\pi$
$563$	−33.1859	−1.39862	−0.699310	+	0.714819i	$0.746509\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.2303	0.765599
$568$	0	0
$569$	−15.3354	−0.642894	−0.321447	−	0.946928i	$-0.604169\pi$
$569$	−15.3354	−0.642894	−0.321447	+	0.946928i	$0.604169\pi$
$570$	0	0
$571$	−17.4822	−0.731607	−0.365804	−	0.930692i	$-0.619206\pi$
$571$	−17.4822	−0.731607	−0.365804	+	0.930692i	$0.619206\pi$
$572$	0	0
$573$	−0.259947	−0.0108594
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.9776	1.58103	0.790514	−	0.612444i	$-0.209814\pi$
$577$	37.9776	1.58103	0.790514	+	0.612444i	$0.209814\pi$
$578$	0	0
$579$	11.8803	0.493730
$580$	0	0
$581$	−11.5493	−0.479147
$582$	0	0
$583$	−69.2588	−2.86840
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.8357	−1.14890	−0.574451	−	0.818539i	$-0.694785\pi$
$587$	−27.8357	−1.14890	−0.574451	+	0.818539i	$0.694785\pi$
$588$	0	0
$589$	−21.2663	−0.876263
$590$	0	0
$591$	−14.0924	−0.579686
$592$	0	0
$593$	23.1501	0.950661	0.475330	−	0.879807i	$-0.342329\pi$
$593$	23.1501	0.950661	0.475330	+	0.879807i	$0.342329\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.3429	0.546088
$598$	0	0
$599$	−38.4067	−1.56926	−0.784628	−	0.619967i	$-0.787146\pi$
$599$	−38.4067	−1.56926	−0.784628	+	0.619967i	$0.787146\pi$
$600$	0	0
$601$	−15.9611	−0.651068	−0.325534	−	0.945530i	$-0.605544\pi$
$601$	−15.9611	−0.651068	−0.325534	+	0.945530i	$0.605544\pi$
$602$	0	0
$603$	−0.969467	−0.0394798
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.6038	−0.958047	−0.479024	−	0.877802i	$-0.659009\pi$
$607$	−23.6038	−0.958047	−0.479024	+	0.877802i	$0.659009\pi$
$608$	0	0
$609$	−3.32422	−0.134704
$610$	0	0
$611$	−1.64517	−0.0665564
$612$	0	0
$613$	−4.03568	−0.163000	−0.0814998	−	0.996673i	$-0.525971\pi$
$613$	−4.03568	−0.163000	−0.0814998	+	0.996673i	$0.525971\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.7044	0.793268	0.396634	−	0.917977i	$-0.370178\pi$
$617$	19.7044	0.793268	0.396634	+	0.917977i	$0.370178\pi$
$618$	0	0
$619$	2.03406	0.0817556	0.0408778	−	0.999164i	$-0.486985\pi$
$619$	2.03406	0.0817556	0.0408778	+	0.999164i	$0.486985\pi$
$620$	0	0
$621$	−16.4894	−0.661696
$622$	0	0
$623$	25.8705	1.03648
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−11.7925	−0.470946
$628$	0	0
$629$	−55.6700	−2.21971
$630$	0	0
$631$	−27.6586	−1.10107	−0.550535	−	0.834812i	$-0.685576\pi$
$631$	−27.6586	−1.10107	−0.550535	+	0.834812i	$0.685576\pi$
$632$	0	0
$633$	−4.27889	−0.170071
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.15257	0.124910
$638$	0	0
$639$	19.7131	0.779837
$640$	0	0
$641$	15.2931	0.604040	0.302020	−	0.953302i	$-0.402339\pi$
$641$	15.2931	0.604040	0.302020	+	0.953302i	$0.402339\pi$
$642$	0	0
$643$	−24.9770	−0.984995	−0.492498	−	0.870314i	$-0.663916\pi$
$643$	−24.9770	−0.984995	−0.492498	+	0.870314i	$0.663916\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.3973	−0.801901	−0.400951	−	0.916100i	$-0.631320\pi$
$647$	−20.3973	−0.801901	−0.400951	+	0.916100i	$0.631320\pi$
$648$	0	0
$649$	−65.7030	−2.57907
$650$	0	0
$651$	12.9177	0.506283
$652$	0	0
$653$	39.9371	1.56286	0.781429	−	0.623994i	$-0.214491\pi$
$653$	39.9371	1.56286	0.781429	+	0.623994i	$0.214491\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.3213	0.519714
$658$	0	0
$659$	49.5534	1.93033	0.965164	−	0.261647i	$-0.0842655\pi$
$659$	49.5534	1.93033	0.965164	+	0.261647i	$0.0842655\pi$
$660$	0	0
$661$	6.85042	0.266451	0.133225	−	0.991086i	$-0.457467\pi$
$661$	6.85042	0.266451	0.133225	+	0.991086i	$0.457467\pi$
$662$	0	0
$663$	5.81813	0.225958
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.63107	−0.334196
$668$	0	0
$669$	−2.28191	−0.0882238
$670$	0	0
$671$	52.2495	2.01707
$672$	0	0
$673$	−2.54761	−0.0982031	−0.0491015	−	0.998794i	$-0.515636\pi$
$673$	−2.54761	−0.0982031	−0.0491015	+	0.998794i	$0.515636\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.0184	0.615639	0.307819	−	0.951445i	$-0.400401\pi$
$677$	16.0184	0.615639	0.307819	+	0.951445i	$0.400401\pi$
$678$	0	0
$679$	−15.4481	−0.592843
$680$	0	0
$681$	1.00113	0.0383635
$682$	0	0
$683$	27.3521	1.04660	0.523299	−	0.852149i	$-0.324701\pi$
$683$	27.3521	1.04660	0.523299	+	0.852149i	$0.324701\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.15652	−0.234886
$688$	0	0
$689$	−12.9491	−0.493322
$690$	0	0
$691$	−41.1677	−1.56609	−0.783047	−	0.621962i	$-0.786336\pi$
$691$	−41.1677	−1.56609	−0.783047	+	0.621962i	$0.786336\pi$
$692$	0	0
$693$	−52.0984	−1.97906
$694$	0	0
$695$	0	0
$696$	0	0
$697$	48.2387	1.82717
$698$	0	0
$699$	1.36391	0.0515878
$700$	0	0
$701$	30.2268	1.14165	0.570825	−	0.821072i	$-0.306623\pi$
$701$	30.2268	1.14165	0.570825	+	0.821072i	$0.306623\pi$
$702$	0	0
$703$	21.0964	0.795665
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−55.2368	−2.07739
$708$	0	0
$709$	−15.1066	−0.567340	−0.283670	−	0.958922i	$-0.591552\pi$
$709$	−15.1066	−0.567340	−0.283670	+	0.958922i	$0.591552\pi$
$710$	0	0
$711$	33.2992	1.24882
$712$	0	0
$713$	33.5396	1.25607
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.79975	−0.328633
$718$	0	0
$719$	43.7520	1.63167	0.815837	−	0.578282i	$-0.196277\pi$
$719$	43.7520	1.63167	0.815837	+	0.578282i	$0.196277\pi$
$720$	0	0
$721$	−44.8454	−1.67013
$722$	0	0
$723$	−13.3796	−0.497592
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.4399	0.535546	0.267773	−	0.963482i	$-0.413712\pi$
$727$	14.4399	0.535546	0.267773	+	0.963482i	$0.413712\pi$
$728$	0	0
$729$	−9.34345	−0.346054
$730$	0	0
$731$	33.4036	1.23548
$732$	0	0
$733$	29.7425	1.09856	0.549281	−	0.835637i	$-0.314902\pi$
$733$	29.7425	1.09856	0.549281	+	0.835637i	$0.314902\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.33725	0.0860936
$738$	0	0
$739$	−1.19411	−0.0439261	−0.0219631	−	0.999759i	$-0.506992\pi$
$739$	−1.19411	−0.0439261	−0.0219631	+	0.999759i	$0.506992\pi$
$740$	0	0
$741$	−2.20480	−0.0809955
$742$	0	0
$743$	37.0610	1.35964	0.679818	−	0.733381i	$-0.262059\pi$
$743$	37.0610	1.35964	0.679818	+	0.733381i	$0.262059\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.80447	0.358727
$748$	0	0
$749$	6.74371	0.246410
$750$	0	0
$751$	−14.0317	−0.512025	−0.256013	−	0.966673i	$-0.582409\pi$
$751$	−14.0317	−0.512025	−0.256013	+	0.966673i	$0.582409\pi$
$752$	0	0
$753$	−13.6845	−0.498691
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.3279	−1.06594	−0.532970	−	0.846134i	$-0.678924\pi$
$757$	−29.3279	−1.06594	−0.532970	+	0.846134i	$0.678924\pi$
$758$	0	0
$759$	18.5982	0.675072
$760$	0	0
$761$	4.30489	0.156052	0.0780260	−	0.996951i	$-0.475138\pi$
$761$	4.30489	0.156052	0.0780260	+	0.996951i	$0.475138\pi$
$762$	0	0
$763$	40.5046	1.46636
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.2843	−0.443560
$768$	0	0
$769$	−15.9671	−0.575787	−0.287893	−	0.957662i	$-0.592955\pi$
$769$	−15.9671	−0.575787	−0.287893	+	0.957662i	$0.592955\pi$
$770$	0	0
$771$	10.0075	0.360410
$772$	0	0
$773$	8.23894	0.296334	0.148167	−	0.988962i	$-0.452663\pi$
$773$	8.23894	0.296334	0.148167	+	0.988962i	$0.452663\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12.8144	−0.459715
$778$	0	0
$779$	−18.2802	−0.654958
$780$	0	0
$781$	−47.5254	−1.70059
$782$	0	0
$783$	6.03201	0.215566
$784$	0	0
$785$	0	0
$786$	0	0
$787$	45.6963	1.62890	0.814449	−	0.580236i	$-0.197040\pi$
$787$	45.6963	1.62890	0.814449	+	0.580236i	$0.197040\pi$
$788$	0	0
$789$	16.0165	0.570204
$790$	0	0
$791$	−39.1856	−1.39328
$792$	0	0
$793$	9.76894	0.346905
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20.1740	−0.714600	−0.357300	−	0.933990i	$-0.616303\pi$
$797$	−20.1740	−0.714600	−0.357300	+	0.933990i	$0.616303\pi$
$798$	0	0
$799$	11.2474	0.397904
$800$	0	0
$801$	−21.9620	−0.775990
$802$	0	0
$803$	−32.1158	−1.13334
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.5513	−0.371425
$808$	0	0
$809$	29.4759	1.03632	0.518160	−	0.855284i	$-0.326617\pi$
$809$	29.4759	1.03632	0.518160	+	0.855284i	$0.326617\pi$
$810$	0	0
$811$	35.5397	1.24797	0.623984	−	0.781437i	$-0.285513\pi$
$811$	35.5397	1.24797	0.623984	+	0.781437i	$0.285513\pi$
$812$	0	0
$813$	−5.10785	−0.179140
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.6584	−0.442862
$818$	0	0
$819$	−9.74069	−0.340367
$820$	0	0
$821$	17.4941	0.610550	0.305275	−	0.952264i	$-0.401252\pi$
$821$	17.4941	0.610550	0.305275	+	0.952264i	$0.401252\pi$
$822$	0	0
$823$	−5.90349	−0.205783	−0.102891	−	0.994693i	$-0.532809\pi$
$823$	−5.90349	−0.205783	−0.102891	+	0.994693i	$0.532809\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.6684	−0.823031	−0.411516	−	0.911403i	$-0.635000\pi$
$827$	−23.6684	−0.823031	−0.411516	+	0.911403i	$0.635000\pi$
$828$	0	0
$829$	29.5025	1.02466	0.512332	−	0.858787i	$-0.328782\pi$
$829$	29.5025	1.02466	0.512332	+	0.858787i	$0.328782\pi$
$830$	0	0
$831$	1.93573	0.0671497
$832$	0	0
$833$	−21.5529	−0.746765
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−23.4399	−0.810201
$838$	0	0
$839$	19.0605	0.658042	0.329021	−	0.944323i	$-0.393281\pi$
$839$	19.0605	0.658042	0.329021	+	0.944323i	$0.393281\pi$
$840$	0	0
$841$	−25.8427	−0.891126
$842$	0	0
$843$	−10.7341	−0.369703
$844$	0	0
$845$	0	0
$846$	0	0
$847$	91.4276	3.14149
$848$	0	0
$849$	−10.9282	−0.375056
$850$	0	0
$851$	−33.2716	−1.14054
$852$	0	0
$853$	47.6597	1.63184	0.815918	−	0.578167i	$-0.196232\pi$
$853$	47.6597	1.63184	0.815918	+	0.578167i	$0.196232\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.5879	0.737429	0.368714	−	0.929543i	$-0.379798\pi$
$857$	21.5879	0.737429	0.368714	+	0.929543i	$0.379798\pi$
$858$	0	0
$859$	39.3103	1.34125	0.670625	−	0.741797i	$-0.266026\pi$
$859$	39.3103	1.34125	0.670625	+	0.741797i	$0.266026\pi$
$860$	0	0
$861$	11.1039	0.378418
$862$	0	0
$863$	−24.3801	−0.829907	−0.414953	−	0.909843i	$-0.636202\pi$
$863$	−24.3801	−0.829907	−0.414953	+	0.909843i	$0.636202\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−29.5394	−1.00321
$868$	0	0
$869$	−80.2795	−2.72330
$870$	0	0
$871$	0.436988	0.0148068
$872$	0	0
$873$	13.1142	0.443849
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.6016	−0.763200	−0.381600	−	0.924327i	$-0.624627\pi$
$877$	−22.6016	−0.763200	−0.381600	+	0.924327i	$0.624627\pi$
$878$	0	0
$879$	−0.119154	−0.00401897
$880$	0	0
$881$	−37.8725	−1.27596	−0.637979	−	0.770054i	$-0.720229\pi$
$881$	−37.8725	−1.27596	−0.637979	+	0.770054i	$0.720229\pi$
$882$	0	0
$883$	−12.1707	−0.409576	−0.204788	−	0.978806i	$-0.565651\pi$
$883$	−12.1707	−0.409576	−0.204788	+	0.978806i	$0.565651\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.2456	1.04912	0.524562	−	0.851372i	$-0.324229\pi$
$887$	31.2456	1.04912	0.524562	+	0.851372i	$0.324229\pi$
$888$	0	0
$889$	10.2280	0.343037
$890$	0	0
$891$	37.3105	1.24995
$892$	0	0
$893$	−4.26224	−0.142630
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.47725	0.116102
$898$	0	0
$899$	−12.2692	−0.409201
$900$	0	0
$901$	88.5280	2.94930
$902$	0	0
$903$	7.68903	0.255875
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.5178	−1.01333	−0.506664	−	0.862144i	$-0.669121\pi$
$907$	−30.5178	−1.01333	−0.506664	+	0.862144i	$0.669121\pi$
$908$	0	0
$909$	46.8917	1.55530
$910$	0	0
$911$	−39.6004	−1.31202	−0.656011	−	0.754752i	$-0.727757\pi$
$911$	−39.6004	−1.31202	−0.656011	+	0.754752i	$0.727757\pi$
$912$	0	0
$913$	−23.6372	−0.782276
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.78386	0.157977
$918$	0	0
$919$	−22.7476	−0.750373	−0.375186	−	0.926949i	$-0.622421\pi$
$919$	−22.7476	−0.750373	−0.375186	+	0.926949i	$0.622421\pi$
$920$	0	0
$921$	−10.8222	−0.356603
$922$	0	0
$923$	−8.88568	−0.292476
$924$	0	0
$925$	0	0
$926$	0	0
$927$	38.0702	1.25039
$928$	0	0
$929$	−8.39101	−0.275300	−0.137650	−	0.990481i	$-0.543955\pi$
$929$	−8.39101	−0.275300	−0.137650	+	0.990481i	$0.543955\pi$
$930$	0	0
$931$	8.16756	0.267681
$932$	0	0
$933$	2.66472	0.0872390
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−59.1357	−1.93188	−0.965939	−	0.258769i	$-0.916683\pi$
$937$	−59.1357	−1.93188	−0.965939	+	0.258769i	$0.916683\pi$
$938$	0	0
$939$	3.61239	0.117886
$940$	0	0
$941$	−32.8002	−1.06926	−0.534629	−	0.845087i	$-0.679549\pi$
$941$	−32.8002	−1.06926	−0.534629	+	0.845087i	$0.679549\pi$
$942$	0	0
$943$	28.8302	0.938841
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.70905	−0.120528	−0.0602639	−	0.998182i	$-0.519194\pi$
$947$	−3.70905	−0.120528	−0.0602639	+	0.998182i	$0.519194\pi$
$948$	0	0
$949$	−6.00459	−0.194917
$950$	0	0
$951$	16.8583	0.546669
$952$	0	0
$953$	−19.8814	−0.644021	−0.322011	−	0.946736i	$-0.604359\pi$
$953$	−19.8814	−0.644021	−0.322011	+	0.946736i	$0.604359\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.80345	−0.219924
$958$	0	0
$959$	−4.33744	−0.140063
$960$	0	0
$961$	16.6771	0.537972
$962$	0	0
$963$	−5.72488	−0.184482
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.63562	−0.116914	−0.0584569	−	0.998290i	$-0.518618\pi$
$967$	−3.63562	−0.116914	−0.0584569	+	0.998290i	$0.518618\pi$
$968$	0	0
$969$	15.0734	0.484227
$970$	0	0
$971$	−35.7500	−1.14727	−0.573636	−	0.819110i	$-0.694468\pi$
$971$	−35.7500	−1.14727	−0.573636	+	0.819110i	$0.694468\pi$
$972$	0	0
$973$	10.6283	0.340729
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.3516	−0.651105	−0.325552	−	0.945524i	$-0.605550\pi$
$977$	−20.3516	−0.651105	−0.325552	+	0.945524i	$0.605550\pi$
$978$	0	0
$979$	52.9473	1.69220
$980$	0	0
$981$	−34.3852	−1.09783
$982$	0	0
$983$	34.2562	1.09260	0.546302	−	0.837588i	$-0.316035\pi$
$983$	34.2562	1.09260	0.546302	+	0.837588i	$0.316035\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.58899	0.0824084
$988$	0	0
$989$	19.9639	0.634815
$990$	0	0
$991$	28.8906	0.917739	0.458870	−	0.888504i	$-0.348255\pi$
$991$	28.8906	0.917739	0.458870	+	0.888504i	$0.348255\pi$
$992$	0	0
$993$	6.96686	0.221087
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−15.3014	−0.484599	−0.242299	−	0.970202i	$-0.577902\pi$
$997$	−15.3014	−0.484599	−0.242299	+	0.970202i	$0.577902\pi$
$998$	0	0
$999$	23.2526	0.735679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bo.1.5		12
4.3	odd	2		5000.2.a.p.1.8	yes	12
5.4	even	2		10000.2.a.bp.1.8		12
20.19	odd	2		5000.2.a.o.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.o.1.5	✓	12	20.19	odd	2
5000.2.a.p.1.8	yes	12	4.3	odd	2
10000.2.a.bo.1.5		12	1.1	even	1	trivial
10000.2.a.bp.1.8		12	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-0.602175 q^{3} +3.10675 q^{7} -2.63738 q^{9} +O(q^{10})\)	\(q-0.602175 q^{3} +3.10675 q^{7} -2.63738 q^{9} +6.35836 q^{11} +1.18880 q^{13} -8.12739 q^{17} +3.07990 q^{19} -1.87081 q^{21} -4.85739 q^{23} +3.39469 q^{27} +1.77689 q^{29} -6.90486 q^{31} -3.82885 q^{33} +6.84968 q^{37} -0.715868 q^{39} -5.93533 q^{41} -4.11000 q^{43} -1.38389 q^{47} +2.65189 q^{49} +4.89411 q^{51} -10.8926 q^{53} -1.85464 q^{57} -10.3333 q^{59} +8.21746 q^{61} -8.19369 q^{63} +0.367587 q^{67} +2.92500 q^{69} -7.47448 q^{71} -5.05096 q^{73} +19.7538 q^{77} -12.6258 q^{79} +5.86795 q^{81} -3.71750 q^{83} -1.07000 q^{87} +8.32720 q^{89} +3.69331 q^{91} +4.15794 q^{93} -4.97243 q^{97} -16.7694 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 26 * q^9	\( 12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 26 * q^9 + q^11 - 4 * q^13 - 8 * q^17 - 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 - 33 * q^31 - 26 * q^33 - 6 * q^37 - 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 + 5 * q^51 - 22 * q^53 - 36 * q^57 - 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 - 19 * q^71 + 5 * q^73 + 13 * q^77 - 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 - 46 * q^91 - 10 * q^93 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more