LMFDB - Embedded newform 5200.2.a.ci.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5200,2,Mod(1,5200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.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:	$41.5222090511$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35194 q^{3} +4.17226 q^{7} -1.17226 q^{9} +O(q^{10})$	$q+1.35194 q^{3} +4.17226 q^{7} -1.17226 q^{9} +5.52420 q^{11} -1.00000 q^{13} +0.703878 q^{17} -6.82032 q^{19} +5.64064 q^{21} -2.64806 q^{23} -5.64064 q^{27} +8.17226 q^{29} +9.52420 q^{31} +7.46838 q^{33} +6.87614 q^{37} -1.35194 q^{39} +0.703878 q^{41} -1.35194 q^{43} -8.17226 q^{47} +10.4078 q^{49} +0.951601 q^{51} -5.04840 q^{53} -9.22066 q^{57} +12.2281 q^{59} -0.172260 q^{61} -4.89098 q^{63} +10.8761 q^{67} -3.58002 q^{69} +5.52420 q^{71} +11.2207 q^{73} +23.0484 q^{77} -1.29612 q^{79} -4.10902 q^{81} -9.58002 q^{83} +11.0484 q^{87} +11.7523 q^{89} -4.17226 q^{91} +12.8761 q^{93} -18.3445 q^{97} -6.47580 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 2 q^{7} + 11 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 2 * q^7 + 11 * q^9	$ 3 q + 2 q^{3} - 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{17} - 8 q^{19} - 8 q^{21} - 10 q^{23} + 8 q^{27} + 10 q^{29} + 12 q^{31} + 12 q^{33} + 2 q^{37} - 2 q^{39} - 2 q^{41} - 2 q^{43} - 10 q^{47} + 23 q^{49} + 36 q^{51} + 18 q^{53} + 20 q^{57} + 16 q^{59} + 14 q^{61} - 50 q^{63} + 14 q^{67} + 12 q^{69} - 14 q^{73} + 36 q^{77} - 8 q^{79} + 23 q^{81} - 6 q^{83} - 2 q^{89} + 2 q^{91} + 20 q^{93} - 26 q^{97} - 36 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 2 * q^7 + 11 * q^9 - 3 * q^13 - 2 * q^17 - 8 * q^19 - 8 * q^21 - 10 * q^23 + 8 * q^27 + 10 * q^29 + 12 * q^31 + 12 * q^33 + 2 * q^37 - 2 * q^39 - 2 * q^41 - 2 * q^43 - 10 * q^47 + 23 * q^49 + 36 * q^51 + 18 * q^53 + 20 * q^57 + 16 * q^59 + 14 * q^61 - 50 * q^63 + 14 * q^67 + 12 * q^69 - 14 * q^73 + 36 * q^77 - 8 * q^79 + 23 * q^81 - 6 * q^83 - 2 * q^89 + 2 * q^91 + 20 * q^93 - 26 * q^97 - 36 * 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
$3$	1.35194	0.780542	0.390271	−	0.920700i	$-0.372381\pi$
$3$	1.35194	0.780542	0.390271	+	0.920700i	$0.372381\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.17226	1.57697	0.788483	−	0.615056i	$-0.210867\pi$
$7$	4.17226	1.57697	0.788483	+	0.615056i	$0.210867\pi$
$8$	0	0
$9$	−1.17226	−0.390753
$10$	0	0
$11$	5.52420	1.66561	0.832804	−	0.553567i	$-0.186734\pi$
$11$	5.52420	1.66561	0.832804	+	0.553567i	$0.186734\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.703878	0.170716	0.0853578	−	0.996350i	$-0.472797\pi$
$17$	0.703878	0.170716	0.0853578	+	0.996350i	$0.472797\pi$
$18$	0	0
$19$	−6.82032	−1.56469	−0.782344	−	0.622846i	$-0.785976\pi$
$19$	−6.82032	−1.56469	−0.782344	+	0.622846i	$0.785976\pi$
$20$	0	0
$21$	5.64064	1.23089
$22$	0	0
$23$	−2.64806	−0.552159	−0.276079	−	0.961135i	$-0.589035\pi$
$23$	−2.64806	−0.552159	−0.276079	+	0.961135i	$0.589035\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.64064	−1.08554
$28$	0	0
$29$	8.17226	1.51755	0.758775	−	0.651352i	$-0.225798\pi$
$29$	8.17226	1.51755	0.758775	+	0.651352i	$0.225798\pi$
$30$	0	0
$31$	9.52420	1.71060	0.855298	−	0.518136i	$-0.173374\pi$
$31$	9.52420	1.71060	0.855298	+	0.518136i	$0.173374\pi$
$32$	0	0
$33$	7.46838	1.30008
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.87614	1.13043	0.565215	−	0.824944i	$-0.308793\pi$
$37$	6.87614	1.13043	0.565215	+	0.824944i	$0.308793\pi$
$38$	0	0
$39$	−1.35194	−0.216484
$40$	0	0
$41$	0.703878	0.109927	0.0549637	−	0.998488i	$-0.482496\pi$
$41$	0.703878	0.109927	0.0549637	+	0.998488i	$0.482496\pi$
$42$	0	0
$43$	−1.35194	−0.206169	−0.103084	−	0.994673i	$-0.532871\pi$
$43$	−1.35194	−0.206169	−0.103084	+	0.994673i	$0.532871\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.17226	−1.19205	−0.596023	−	0.802967i	$-0.703253\pi$
$47$	−8.17226	−1.19205	−0.596023	+	0.802967i	$0.703253\pi$
$48$	0	0
$49$	10.4078	1.48682
$50$	0	0
$51$	0.951601	0.133251
$52$	0	0
$53$	−5.04840	−0.693451	−0.346725	−	0.937967i	$-0.612706\pi$
$53$	−5.04840	−0.693451	−0.346725	+	0.937967i	$0.612706\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.22066	−1.22131
$58$	0	0
$59$	12.2281	1.59196	0.795980	−	0.605323i	$-0.206956\pi$
$59$	12.2281	1.59196	0.795980	+	0.605323i	$0.206956\pi$
$60$	0	0
$61$	−0.172260	−0.0220557	−0.0110278	−	0.999939i	$-0.503510\pi$
$61$	−0.172260	−0.0220557	−0.0110278	+	0.999939i	$0.503510\pi$
$62$	0	0
$63$	−4.89098	−0.616205
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8761	1.32873	0.664366	−	0.747407i	$-0.268702\pi$
$67$	10.8761	1.32873	0.664366	+	0.747407i	$0.268702\pi$
$68$	0	0
$69$	−3.58002	−0.430983
$70$	0	0
$71$	5.52420	0.655602	0.327801	−	0.944747i	$-0.393692\pi$
$71$	5.52420	0.655602	0.327801	+	0.944747i	$0.393692\pi$
$72$	0	0
$73$	11.2207	1.31328	0.656639	−	0.754205i	$-0.271977\pi$
$73$	11.2207	1.31328	0.656639	+	0.754205i	$0.271977\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23.0484	2.62661
$78$	0	0
$79$	−1.29612	−0.145825	−0.0729125	−	0.997338i	$-0.523229\pi$
$79$	−1.29612	−0.145825	−0.0729125	+	0.997338i	$0.523229\pi$
$80$	0	0
$81$	−4.10902	−0.456558
$82$	0	0
$83$	−9.58002	−1.05154	−0.525772	−	0.850626i	$-0.676223\pi$
$83$	−9.58002	−1.05154	−0.525772	+	0.850626i	$0.676223\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.0484	1.18451
$88$	0	0
$89$	11.7523	1.24574	0.622869	−	0.782326i	$-0.285967\pi$
$89$	11.7523	1.24574	0.622869	+	0.782326i	$0.285967\pi$
$90$	0	0
$91$	−4.17226	−0.437372
$92$	0	0
$93$	12.8761	1.33519
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.3445	−1.86260	−0.931302	−	0.364248i	$-0.881326\pi$
$97$	−18.3445	−1.86260	−0.931302	+	0.364248i	$0.881326\pi$
$98$	0	0
$99$	−6.47580	−0.650842
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−10.9926	−1.08313	−0.541566	−	0.840658i	$-0.682168\pi$
$103$	−10.9926	−1.08313	−0.541566	+	0.840658i	$0.682168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.99258	0.675998	0.337999	−	0.941146i	$-0.390250\pi$
$107$	6.99258	0.675998	0.337999	+	0.941146i	$0.390250\pi$
$108$	0	0
$109$	3.40776	0.326404	0.163202	−	0.986593i	$-0.447818\pi$
$109$	3.40776	0.326404	0.163202	+	0.986593i	$0.447818\pi$
$110$	0	0
$111$	9.29612	0.882349
$112$	0	0
$113$	−3.64064	−0.342483	−0.171241	−	0.985229i	$-0.554778\pi$
$113$	−3.64064	−0.342483	−0.171241	+	0.985229i	$0.554778\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.17226	0.108376
$118$	0	0
$119$	2.93676	0.269213
$120$	0	0
$121$	19.5168	1.77425
$122$	0	0
$123$	0.951601	0.0858030
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.69646	−0.505479	−0.252740	−	0.967534i	$-0.581332\pi$
$127$	−5.69646	−0.505479	−0.252740	+	0.967534i	$0.581332\pi$
$128$	0	0
$129$	−1.82774	−0.160923
$130$	0	0
$131$	−17.7523	−1.55102	−0.775512	−	0.631333i	$-0.782508\pi$
$131$	−17.7523	−1.55102	−0.775512	+	0.631333i	$0.782508\pi$
$132$	0	0
$133$	−28.4562	−2.46746
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.93676	0.763519	0.381760	−	0.924262i	$-0.375318\pi$
$137$	8.93676	0.763519	0.381760	+	0.924262i	$0.375318\pi$
$138$	0	0
$139$	−13.2961	−1.12776	−0.563881	−	0.825856i	$-0.690692\pi$
$139$	−13.2961	−1.12776	−0.563881	+	0.825856i	$0.690692\pi$
$140$	0	0
$141$	−11.0484	−0.930443
$142$	0	0
$143$	−5.52420	−0.461957
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.0707	1.16053
$148$	0	0
$149$	−8.93676	−0.732128	−0.366064	−	0.930590i	$-0.619295\pi$
$149$	−8.93676	−0.732128	−0.366064	+	0.930590i	$0.619295\pi$
$150$	0	0
$151$	−2.93196	−0.238599	−0.119300	−	0.992858i	$-0.538065\pi$
$151$	−2.93196	−0.238599	−0.119300	+	0.992858i	$0.538065\pi$
$152$	0	0
$153$	−0.825129	−0.0667077
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.592243	−0.0472662	−0.0236331	−	0.999721i	$-0.507523\pi$
$157$	−0.592243	−0.0472662	−0.0236331	+	0.999721i	$0.507523\pi$
$158$	0	0
$159$	−6.82513	−0.541268
$160$	0	0
$161$	−11.0484	−0.870736
$162$	0	0
$163$	16.6284	1.30244	0.651219	−	0.758890i	$-0.274258\pi$
$163$	16.6284	1.30244	0.651219	+	0.758890i	$0.274258\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.82774	−0.296199	−0.148100	−	0.988972i	$-0.547316\pi$
$167$	−3.82774	−0.296199	−0.148100	+	0.988972i	$0.547316\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.99519	0.611408
$172$	0	0
$173$	−14.1116	−1.07289	−0.536444	−	0.843936i	$-0.680233\pi$
$173$	−14.1116	−1.07289	−0.536444	+	0.843936i	$0.680233\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.5316	1.24259
$178$	0	0
$179$	−5.75228	−0.429945	−0.214973	−	0.976620i	$-0.568966\pi$
$179$	−5.75228	−0.429945	−0.214973	+	0.976620i	$0.568966\pi$
$180$	0	0
$181$	2.76450	0.205484	0.102742	−	0.994708i	$-0.467238\pi$
$181$	2.76450	0.205484	0.102742	+	0.994708i	$0.467238\pi$
$182$	0	0
$183$	−0.232886	−0.0172154
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.88836	0.284345
$188$	0	0
$189$	−23.5342	−1.71186
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	14.3445	1.03254	0.516271	−	0.856426i	$-0.327320\pi$
$193$	14.3445	1.03254	0.516271	+	0.856426i	$0.327320\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.40776	0.527781	0.263890	−	0.964553i	$-0.414994\pi$
$197$	7.40776	0.527781	0.263890	+	0.964553i	$0.414994\pi$
$198$	0	0
$199$	−1.75228	−0.124216	−0.0621078	−	0.998069i	$-0.519782\pi$
$199$	−1.75228	−0.124216	−0.0621078	+	0.998069i	$0.519782\pi$
$200$	0	0
$201$	14.7039	1.03713
$202$	0	0
$203$	34.0968	2.39313
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.10422	0.215758
$208$	0	0
$209$	−37.6768	−2.60616
$210$	0	0
$211$	9.75228	0.671374	0.335687	−	0.941974i	$-0.391031\pi$
$211$	9.75228	0.671374	0.335687	+	0.941974i	$0.391031\pi$
$212$	0	0
$213$	7.46838	0.511725
$214$	0	0
$215$	0	0
$216$	0	0
$217$	39.7374	2.69755
$218$	0	0
$219$	15.1696	1.02507
$220$	0	0
$221$	−0.703878	−0.0473480
$222$	0	0
$223$	2.76450	0.185125	0.0925624	−	0.995707i	$-0.470494\pi$
$223$	2.76450	0.185125	0.0925624	+	0.995707i	$0.470494\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.8129	1.18228	0.591142	−	0.806568i	$-0.298677\pi$
$227$	17.8129	1.18228	0.591142	+	0.806568i	$0.298677\pi$
$228$	0	0
$229$	10.1116	0.668196	0.334098	−	0.942538i	$-0.391568\pi$
$229$	10.1116	0.668196	0.334098	+	0.942538i	$0.391568\pi$
$230$	0	0
$231$	31.1600	2.05018
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.75228	−0.113823
$238$	0	0
$239$	−9.86872	−0.638354	−0.319177	−	0.947695i	$-0.603407\pi$
$239$	−9.86872	−0.638354	−0.319177	+	0.947695i	$0.603407\pi$
$240$	0	0
$241$	−1.88836	−0.121640	−0.0608201	−	0.998149i	$-0.519372\pi$
$241$	−1.88836	−0.121640	−0.0608201	+	0.998149i	$0.519372\pi$
$242$	0	0
$243$	11.3668	0.729179
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.82032	0.433967
$248$	0	0
$249$	−12.9516	−0.820774
$250$	0	0
$251$	−2.35936	−0.148921	−0.0744607	−	0.997224i	$-0.523724\pi$
$251$	−2.35936	−0.148921	−0.0744607	+	0.997224i	$0.523724\pi$
$252$	0	0
$253$	−14.6284	−0.919681
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.65548	0.103266	0.0516330	−	0.998666i	$-0.483557\pi$
$257$	1.65548	0.103266	0.0516330	+	0.998666i	$0.483557\pi$
$258$	0	0
$259$	28.6890	1.78265
$260$	0	0
$261$	−9.58002	−0.592988
$262$	0	0
$263$	−16.6332	−1.02565	−0.512824	−	0.858494i	$-0.671401\pi$
$263$	−16.6332	−1.02565	−0.512824	+	0.858494i	$0.671401\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.8884	0.972352
$268$	0	0
$269$	16.0968	0.981439	0.490720	−	0.871318i	$-0.336734\pi$
$269$	16.0968	0.981439	0.490720	+	0.871318i	$0.336734\pi$
$270$	0	0
$271$	10.4758	0.636360	0.318180	−	0.948030i	$-0.396928\pi$
$271$	10.4758	0.636360	0.318180	+	0.948030i	$0.396928\pi$
$272$	0	0
$273$	−5.64064	−0.341387
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.3929	−1.04504	−0.522520	−	0.852627i	$-0.675008\pi$
$277$	−17.3929	−1.04504	−0.522520	+	0.852627i	$0.675008\pi$
$278$	0	0
$279$	−11.1648	−0.668422
$280$	0	0
$281$	11.2961	0.673870	0.336935	−	0.941528i	$-0.390610\pi$
$281$	11.2961	0.673870	0.336935	+	0.941528i	$0.390610\pi$
$282$	0	0
$283$	9.24030	0.549279	0.274640	−	0.961547i	$-0.411441\pi$
$283$	9.24030	0.549279	0.274640	+	0.961547i	$0.411441\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.93676	0.173352
$288$	0	0
$289$	−16.5046	−0.970856
$290$	0	0
$291$	−24.8007	−1.45384
$292$	0	0
$293$	16.4051	0.958399	0.479199	−	0.877706i	$-0.340927\pi$
$293$	16.4051	0.958399	0.479199	+	0.877706i	$0.340927\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−31.1600	−1.80809
$298$	0	0
$299$	2.64806	0.153141
$300$	0	0
$301$	−5.64064	−0.325121
$302$	0	0
$303$	8.11164	0.466001
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.1090	−0.862318	−0.431159	−	0.902276i	$-0.641895\pi$
$307$	−15.1090	−0.862318	−0.431159	+	0.902276i	$0.641895\pi$
$308$	0	0
$309$	−14.8613	−0.845430
$310$	0	0
$311$	8.11164	0.459969	0.229984	−	0.973194i	$-0.426133\pi$
$311$	8.11164	0.459969	0.229984	+	0.973194i	$0.426133\pi$
$312$	0	0
$313$	1.39292	0.0787325	0.0393662	−	0.999225i	$-0.487466\pi$
$313$	1.39292	0.0787325	0.0393662	+	0.999225i	$0.487466\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.8129	1.67446	0.837230	−	0.546851i	$-0.184174\pi$
$317$	29.8129	1.67446	0.837230	+	0.546851i	$0.184174\pi$
$318$	0	0
$319$	45.1452	2.52765
$320$	0	0
$321$	9.45355	0.527645
$322$	0	0
$323$	−4.80068	−0.267117
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.60708	0.254772
$328$	0	0
$329$	−34.0968	−1.87982
$330$	0	0
$331$	19.1648	1.05339	0.526697	−	0.850053i	$-0.323430\pi$
$331$	19.1648	1.05339	0.526697	+	0.850053i	$0.323430\pi$
$332$	0	0
$333$	−8.06063	−0.441720
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.6406	−0.634106	−0.317053	−	0.948408i	$-0.602693\pi$
$337$	−11.6406	−0.634106	−0.317053	+	0.948408i	$0.602693\pi$
$338$	0	0
$339$	−4.92193	−0.267322
$340$	0	0
$341$	52.6136	2.84919
$342$	0	0
$343$	14.2180	0.767702
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.7597	−1.22180	−0.610902	−	0.791706i	$-0.709193\pi$
$347$	−22.7597	−1.22180	−0.610902	+	0.791706i	$0.709193\pi$
$348$	0	0
$349$	1.04840	0.0561195	0.0280598	−	0.999606i	$-0.491067\pi$
$349$	1.04840	0.0561195	0.0280598	+	0.999606i	$0.491067\pi$
$350$	0	0
$351$	5.64064	0.301075
$352$	0	0
$353$	−4.40515	−0.234462	−0.117231	−	0.993105i	$-0.537402\pi$
$353$	−4.40515	−0.234462	−0.117231	+	0.993105i	$0.537402\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.97033	0.210132
$358$	0	0
$359$	−24.2281	−1.27871	−0.639355	−	0.768912i	$-0.720798\pi$
$359$	−24.2281	−1.27871	−0.639355	+	0.768912i	$0.720798\pi$
$360$	0	0
$361$	27.5168	1.44825
$362$	0	0
$363$	26.3855	1.38488
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−27.3371	−1.42699	−0.713493	−	0.700663i	$-0.752888\pi$
$367$	−27.3371	−1.42699	−0.713493	+	0.700663i	$0.752888\pi$
$368$	0	0
$369$	−0.825129	−0.0429545
$370$	0	0
$371$	−21.0632	−1.09355
$372$	0	0
$373$	−22.6890	−1.17479	−0.587397	−	0.809299i	$-0.699847\pi$
$373$	−22.6890	−1.17479	−0.587397	+	0.809299i	$0.699847\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.17226	−0.420893
$378$	0	0
$379$	−30.3249	−1.55768	−0.778842	−	0.627220i	$-0.784193\pi$
$379$	−30.3249	−1.55768	−0.778842	+	0.627220i	$0.784193\pi$
$380$	0	0
$381$	−7.70127	−0.394548
$382$	0	0
$383$	−5.23550	−0.267521	−0.133761	−	0.991014i	$-0.542705\pi$
$383$	−5.23550	−0.267521	−0.133761	+	0.991014i	$0.542705\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.58482	0.0805612
$388$	0	0
$389$	23.8735	1.21044	0.605218	−	0.796060i	$-0.293086\pi$
$389$	23.8735	1.21044	0.605218	+	0.796060i	$0.293086\pi$
$390$	0	0
$391$	−1.86391	−0.0942621
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−39.2207	−1.96843	−0.984214	−	0.176981i	$-0.943367\pi$
$397$	−39.2207	−1.96843	−0.984214	+	0.176981i	$0.943367\pi$
$398$	0	0
$399$	−38.4710	−1.92596
$400$	0	0
$401$	31.0336	1.54974	0.774871	−	0.632119i	$-0.217815\pi$
$401$	31.0336	1.54974	0.774871	+	0.632119i	$0.217815\pi$
$402$	0	0
$403$	−9.52420	−0.474434
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.9852	1.88285
$408$	0	0
$409$	−18.1116	−0.895563	−0.447781	−	0.894143i	$-0.647786\pi$
$409$	−18.1116	−0.895563	−0.447781	+	0.894143i	$0.647786\pi$
$410$	0	0
$411$	12.0820	0.595959
$412$	0	0
$413$	51.0187	2.51047
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.9755	−0.880266
$418$	0	0
$419$	23.0484	1.12599	0.562994	−	0.826461i	$-0.309649\pi$
$419$	23.0484	1.12599	0.562994	+	0.826461i	$0.309649\pi$
$420$	0	0
$421$	−34.9123	−1.70152	−0.850761	−	0.525553i	$-0.823859\pi$
$421$	−34.9123	−1.70152	−0.850761	+	0.525553i	$0.823859\pi$
$422$	0	0
$423$	9.58002	0.465796
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.718715	−0.0347811
$428$	0	0
$429$	−7.46838	−0.360577
$430$	0	0
$431$	−6.47580	−0.311928	−0.155964	−	0.987763i	$-0.549848\pi$
$431$	−6.47580	−0.311928	−0.155964	+	0.987763i	$0.549848\pi$
$432$	0	0
$433$	24.8155	1.19256	0.596279	−	0.802777i	$-0.296645\pi$
$433$	24.8155	1.19256	0.596279	+	0.802777i	$0.296645\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.0606	0.863957
$438$	0	0
$439$	−10.8155	−0.516196	−0.258098	−	0.966119i	$-0.583096\pi$
$439$	−10.8155	−0.516196	−0.258098	+	0.966119i	$0.583096\pi$
$440$	0	0
$441$	−12.2006	−0.580981
$442$	0	0
$443$	18.0410	0.857153	0.428576	−	0.903506i	$-0.359015\pi$
$443$	18.0410	0.857153	0.428576	+	0.903506i	$0.359015\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.0820	−0.571457
$448$	0	0
$449$	27.1452	1.28106	0.640531	−	0.767933i	$-0.278714\pi$
$449$	27.1452	1.28106	0.640531	+	0.767933i	$0.278714\pi$
$450$	0	0
$451$	3.88836	0.183096
$452$	0	0
$453$	−3.96383	−0.186237
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.75228	0.175524	0.0877621	−	0.996141i	$-0.472028\pi$
$457$	3.75228	0.175524	0.0877621	+	0.996141i	$0.472028\pi$
$458$	0	0
$459$	−3.97033	−0.185319
$460$	0	0
$461$	36.3297	1.69204	0.846021	−	0.533150i	$-0.178992\pi$
$461$	36.3297	1.69204	0.846021	+	0.533150i	$0.178992\pi$
$462$	0	0
$463$	−29.4684	−1.36951	−0.684756	−	0.728772i	$-0.740091\pi$
$463$	−29.4684	−1.36951	−0.684756	+	0.728772i	$0.740091\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.5848	−0.813729	−0.406864	−	0.913489i	$-0.633378\pi$
$467$	−17.5848	−0.813729	−0.406864	+	0.913489i	$0.633378\pi$
$468$	0	0
$469$	45.3781	2.09537
$470$	0	0
$471$	−0.800677	−0.0368932
$472$	0	0
$473$	−7.46838	−0.343397
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.91804	0.270968
$478$	0	0
$479$	11.2765	0.515235	0.257618	−	0.966247i	$-0.417062\pi$
$479$	11.2765	0.515235	0.257618	+	0.966247i	$0.417062\pi$
$480$	0	0
$481$	−6.87614	−0.313525
$482$	0	0
$483$	−14.9368	−0.679646
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−26.1574	−1.18531	−0.592653	−	0.805458i	$-0.701919\pi$
$487$	−26.1574	−1.18531	−0.592653	+	0.805458i	$0.701919\pi$
$488$	0	0
$489$	22.4806	1.01661
$490$	0	0
$491$	−0.456156	−0.0205860	−0.0102930	−	0.999947i	$-0.503276\pi$
$491$	−0.456156	−0.0205860	−0.0102930	+	0.999947i	$0.503276\pi$
$492$	0	0
$493$	5.75228	0.259070
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23.0484	1.03386
$498$	0	0
$499$	12.4610	0.557829	0.278915	−	0.960316i	$-0.410025\pi$
$499$	12.4610	0.557829	0.278915	+	0.960316i	$0.410025\pi$
$500$	0	0
$501$	−5.17487	−0.231196
$502$	0	0
$503$	−11.7933	−0.525835	−0.262918	−	0.964818i	$-0.584685\pi$
$503$	−11.7933	−0.525835	−0.262918	+	0.964818i	$0.584685\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.35194	0.0600417
$508$	0	0
$509$	−29.0484	−1.28755	−0.643774	−	0.765216i	$-0.722632\pi$
$509$	−29.0484	−1.28755	−0.643774	+	0.765216i	$0.722632\pi$
$510$	0	0
$511$	46.8155	2.07100
$512$	0	0
$513$	38.4710	1.69854
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−45.1452	−1.98548
$518$	0	0
$519$	−19.0781	−0.837434
$520$	0	0
$521$	−15.3323	−0.671720	−0.335860	−	0.941912i	$-0.609027\pi$
$521$	−15.3323	−0.671720	−0.335860	+	0.941912i	$0.609027\pi$
$522$	0	0
$523$	15.4487	0.675526	0.337763	−	0.941231i	$-0.390330\pi$
$523$	15.4487	0.675526	0.337763	+	0.941231i	$0.390330\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.70388	0.292026
$528$	0	0
$529$	−15.9878	−0.695121
$530$	0	0
$531$	−14.3345	−0.622064
$532$	0	0
$533$	−0.703878	−0.0304884
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.77673	−0.335591
$538$	0	0
$539$	57.4945	2.47646
$540$	0	0
$541$	−38.8007	−1.66817	−0.834086	−	0.551635i	$-0.814004\pi$
$541$	−38.8007	−1.66817	−0.834086	+	0.551635i	$0.814004\pi$
$542$	0	0
$543$	3.73744	0.160389
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.9926	0.641036	0.320518	−	0.947242i	$-0.396143\pi$
$547$	14.9926	0.641036	0.320518	+	0.947242i	$0.396143\pi$
$548$	0	0
$549$	0.201934	0.00861833
$550$	0	0
$551$	−55.7374	−2.37449
$552$	0	0
$553$	−5.40776	−0.229961
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.3807	1.11779	0.558893	−	0.829240i	$-0.311226\pi$
$557$	26.3807	1.11779	0.558893	+	0.829240i	$0.311226\pi$
$558$	0	0
$559$	1.35194	0.0571809
$560$	0	0
$561$	5.25683	0.221944
$562$	0	0
$563$	−42.0410	−1.77182	−0.885908	−	0.463861i	$-0.846464\pi$
$563$	−42.0410	−1.77182	−0.885908	+	0.463861i	$0.846464\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−17.1439	−0.719977
$568$	0	0
$569$	30.2691	1.26894	0.634472	−	0.772945i	$-0.281217\pi$
$569$	30.2691	1.26894	0.634472	+	0.772945i	$0.281217\pi$
$570$	0	0
$571$	12.1116	0.506856	0.253428	−	0.967354i	$-0.418442\pi$
$571$	12.1116	0.506856	0.253428	+	0.967354i	$0.418442\pi$
$572$	0	0
$573$	−16.2233	−0.677737
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.40515	0.349911	0.174955	−	0.984576i	$-0.444022\pi$
$577$	8.40515	0.349911	0.174955	+	0.984576i	$0.444022\pi$
$578$	0	0
$579$	19.3929	0.805942
$580$	0	0
$581$	−39.9703	−1.65825
$582$	0	0
$583$	−27.8884	−1.15502
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.4439	1.95822	0.979110	−	0.203330i	$-0.0651764\pi$
$587$	47.4439	1.95822	0.979110	+	0.203330i	$0.0651764\pi$
$588$	0	0
$589$	−64.9581	−2.67655
$590$	0	0
$591$	10.0148	0.411955
$592$	0	0
$593$	−32.4413	−1.33221	−0.666103	−	0.745860i	$-0.732039\pi$
$593$	−32.4413	−1.33221	−0.666103	+	0.745860i	$0.732039\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.36897	−0.0969556
$598$	0	0
$599$	34.6742	1.41675	0.708375	−	0.705836i	$-0.249429\pi$
$599$	34.6742	1.41675	0.708375	+	0.705836i	$0.249429\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−12.7497	−0.519207
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.30354	−0.0934978	−0.0467489	−	0.998907i	$-0.514886\pi$
$607$	−2.30354	−0.0934978	−0.0467489	+	0.998907i	$0.514886\pi$
$608$	0	0
$609$	46.0968	1.86794
$610$	0	0
$611$	8.17226	0.330614
$612$	0	0
$613$	15.8735	0.641126	0.320563	−	0.947227i	$-0.396128\pi$
$613$	15.8735	0.641126	0.320563	+	0.947227i	$0.396128\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.3445	−0.416455	−0.208227	−	0.978080i	$-0.566769\pi$
$617$	−10.3445	−0.416455	−0.208227	+	0.978080i	$0.566769\pi$
$618$	0	0
$619$	−37.1500	−1.49318	−0.746592	−	0.665282i	$-0.768311\pi$
$619$	−37.1500	−1.49318	−0.746592	+	0.665282i	$0.768311\pi$
$620$	0	0
$621$	14.9368	0.599392
$622$	0	0
$623$	49.0336	1.96449
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−50.9368	−2.03422
$628$	0	0
$629$	4.83997	0.192982
$630$	0	0
$631$	−10.2132	−0.406583	−0.203291	−	0.979118i	$-0.565164\pi$
$631$	−10.2132	−0.406583	−0.203291	+	0.979118i	$0.565164\pi$
$632$	0	0
$633$	13.1845	0.524036
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−10.4078	−0.412370
$638$	0	0
$639$	−6.47580	−0.256179
$640$	0	0
$641$	−16.0968	−0.635785	−0.317893	−	0.948127i	$-0.602975\pi$
$641$	−16.0968	−0.635785	−0.317893	+	0.948127i	$0.602975\pi$
$642$	0	0
$643$	−18.0458	−0.711656	−0.355828	−	0.934551i	$-0.615801\pi$
$643$	−18.0458	−0.711656	−0.355828	+	0.934551i	$0.615801\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−43.5700	−1.71291	−0.856456	−	0.516219i	$-0.827339\pi$
$647$	−43.5700	−1.71291	−0.856456	+	0.516219i	$0.827339\pi$
$648$	0	0
$649$	67.5503	2.65158
$650$	0	0
$651$	53.7226	2.10555
$652$	0	0
$653$	22.3445	0.874409	0.437204	−	0.899362i	$-0.355969\pi$
$653$	22.3445	0.874409	0.437204	+	0.899362i	$0.355969\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.1535	−0.513168
$658$	0	0
$659$	−28.9219	−1.12664	−0.563319	−	0.826239i	$-0.690476\pi$
$659$	−28.9219	−1.12664	−0.563319	+	0.826239i	$0.690476\pi$
$660$	0	0
$661$	−18.6890	−0.726919	−0.363460	−	0.931610i	$-0.618405\pi$
$661$	−18.6890	−0.726919	−0.363460	+	0.931610i	$0.618405\pi$
$662$	0	0
$663$	−0.951601	−0.0369571
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−21.6406	−0.837929
$668$	0	0
$669$	3.73744	0.144498
$670$	0	0
$671$	−0.951601	−0.0367361
$672$	0	0
$673$	−23.5194	−0.906606	−0.453303	−	0.891357i	$-0.649754\pi$
$673$	−23.5194	−0.906606	−0.453303	+	0.891357i	$0.649754\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.35936	0.321276	0.160638	−	0.987013i	$-0.448645\pi$
$677$	8.35936	0.321276	0.160638	+	0.987013i	$0.448645\pi$
$678$	0	0
$679$	−76.5381	−2.93726
$680$	0	0
$681$	24.0820	0.922823
$682$	0	0
$683$	−11.9394	−0.456847	−0.228424	−	0.973562i	$-0.573357\pi$
$683$	−11.9394	−0.456847	−0.228424	+	0.973562i	$0.573357\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.6703	0.521555
$688$	0	0
$689$	5.04840	0.192329
$690$	0	0
$691$	−15.4274	−0.586886	−0.293443	−	0.955977i	$-0.594801\pi$
$691$	−15.4274	−0.586886	−0.293443	+	0.955977i	$0.594801\pi$
$692$	0	0
$693$	−27.0187	−1.02636
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.495445	0.0187663
$698$	0	0
$699$	8.11164	0.306810
$700$	0	0
$701$	37.2813	1.40809	0.704047	−	0.710153i	$-0.251374\pi$
$701$	37.2813	1.40809	0.704047	+	0.710153i	$0.251374\pi$
$702$	0	0
$703$	−46.8975	−1.76877
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.0336	0.941484
$708$	0	0
$709$	−11.9852	−0.450112	−0.225056	−	0.974346i	$-0.572257\pi$
$709$	−11.9852	−0.450112	−0.225056	+	0.974346i	$0.572257\pi$
$710$	0	0
$711$	1.51939	0.0569817
$712$	0	0
$713$	−25.2207	−0.944521
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−13.3419	−0.498263
$718$	0	0
$719$	−23.6258	−0.881094	−0.440547	−	0.897730i	$-0.645215\pi$
$719$	−23.6258	−0.881094	−0.440547	+	0.897730i	$0.645215\pi$
$720$	0	0
$721$	−45.8639	−1.70806
$722$	0	0
$723$	−2.55295	−0.0949454
$724$	0	0
$725$	0	0
$726$	0	0
$727$	22.1526	0.821595	0.410798	−	0.911727i	$-0.365250\pi$
$727$	22.1526	0.821595	0.410798	+	0.911727i	$0.365250\pi$
$728$	0	0
$729$	27.6943	1.02571
$730$	0	0
$731$	−0.951601	−0.0351962
$732$	0	0
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.0820	2.21315
$738$	0	0
$739$	19.0436	0.700530	0.350265	−	0.936651i	$-0.386092\pi$
$739$	19.0436	0.700530	0.350265	+	0.936651i	$0.386092\pi$
$740$	0	0
$741$	9.22066	0.338729
$742$	0	0
$743$	6.26906	0.229989	0.114995	−	0.993366i	$-0.463315\pi$
$743$	6.26906	0.229989	0.114995	+	0.993366i	$0.463315\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.2303	0.410894
$748$	0	0
$749$	29.1749	1.06603
$750$	0	0
$751$	20.9219	0.763452	0.381726	−	0.924276i	$-0.375330\pi$
$751$	20.9219	0.763452	0.381726	+	0.924276i	$0.375330\pi$
$752$	0	0
$753$	−3.18971	−0.116239
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.8975	1.77721	0.888604	−	0.458674i	$-0.151676\pi$
$757$	48.8975	1.77721	0.888604	+	0.458674i	$0.151676\pi$
$758$	0	0
$759$	−19.7767	−0.717850
$760$	0	0
$761$	−4.09680	−0.148509	−0.0742544	−	0.997239i	$-0.523658\pi$
$761$	−4.09680	−0.148509	−0.0742544	+	0.997239i	$0.523658\pi$
$762$	0	0
$763$	14.2180	0.514728
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.2281	−0.441530
$768$	0	0
$769$	−28.7858	−1.03804	−0.519022	−	0.854761i	$-0.673704\pi$
$769$	−28.7858	−1.03804	−0.519022	+	0.854761i	$0.673704\pi$
$770$	0	0
$771$	2.23811	0.0806035
$772$	0	0
$773$	−7.22066	−0.259709	−0.129855	−	0.991533i	$-0.541451\pi$
$773$	−7.22066	−0.259709	−0.129855	+	0.991533i	$0.541451\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	38.7858	1.39143
$778$	0	0
$779$	−4.80068	−0.172002
$780$	0	0
$781$	30.5168	1.09198
$782$	0	0
$783$	−46.0968	−1.64737
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.5800	1.48217	0.741084	−	0.671413i	$-0.234312\pi$
$787$	41.5800	1.48217	0.741084	+	0.671413i	$0.234312\pi$
$788$	0	0
$789$	−22.4871	−0.800562
$790$	0	0
$791$	−15.1897	−0.540084
$792$	0	0
$793$	0.172260	0.00611715
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.4078	−0.687458	−0.343729	−	0.939069i	$-0.611690\pi$
$797$	−19.4078	−0.687458	−0.343729	+	0.939069i	$0.611690\pi$
$798$	0	0
$799$	−5.75228	−0.203501
$800$	0	0
$801$	−13.7767	−0.486777
$802$	0	0
$803$	61.9852	2.18741
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.7619	0.766055
$808$	0	0
$809$	43.7981	1.53986	0.769929	−	0.638130i	$-0.220292\pi$
$809$	43.7981	1.53986	0.769929	+	0.638130i	$0.220292\pi$
$810$	0	0
$811$	5.75709	0.202159	0.101079	−	0.994878i	$-0.467770\pi$
$811$	5.75709	0.202159	0.101079	+	0.994878i	$0.467770\pi$
$812$	0	0
$813$	14.1626	0.496706
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.22066	0.322590
$818$	0	0
$819$	4.89098	0.170905
$820$	0	0
$821$	−32.9368	−1.14950	−0.574750	−	0.818329i	$-0.694901\pi$
$821$	−32.9368	−1.14950	−0.574750	+	0.818329i	$0.694901\pi$
$822$	0	0
$823$	−57.0894	−1.99001	−0.995005	−	0.0998217i	$-0.968173\pi$
$823$	−57.0894	−1.99001	−0.995005	+	0.0998217i	$0.968173\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.4535	−0.954653	−0.477327	−	0.878726i	$-0.658394\pi$
$827$	−27.4535	−0.954653	−0.477327	+	0.878726i	$0.658394\pi$
$828$	0	0
$829$	10.4200	0.361901	0.180950	−	0.983492i	$-0.442083\pi$
$829$	10.4200	0.361901	0.180950	+	0.983492i	$0.442083\pi$
$830$	0	0
$831$	−23.5142	−0.815698
$832$	0	0
$833$	7.32580	0.253824
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−53.7226	−1.85692
$838$	0	0
$839$	−40.1984	−1.38780	−0.693902	−	0.720070i	$-0.744110\pi$
$839$	−40.1984	−1.38780	−0.693902	+	0.720070i	$0.744110\pi$
$840$	0	0
$841$	37.7858	1.30296
$842$	0	0
$843$	15.2717	0.525984
$844$	0	0
$845$	0	0
$846$	0	0
$847$	81.4291	2.79794
$848$	0	0
$849$	12.4923	0.428736
$850$	0	0
$851$	−18.2084	−0.624177
$852$	0	0
$853$	−34.8761	−1.19414	−0.597068	−	0.802191i	$-0.703668\pi$
$853$	−34.8761	−1.19414	−0.597068	+	0.802191i	$0.703668\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.1600	−0.859450	−0.429725	−	0.902960i	$-0.641390\pi$
$857$	−25.1600	−0.859450	−0.429725	+	0.902960i	$0.641390\pi$
$858$	0	0
$859$	15.0484	0.513445	0.256722	−	0.966485i	$-0.417357\pi$
$859$	15.0484	0.513445	0.256722	+	0.966485i	$0.417357\pi$
$860$	0	0
$861$	3.97033	0.135308
$862$	0	0
$863$	2.29873	0.0782498	0.0391249	−	0.999234i	$-0.487543\pi$
$863$	2.29873	0.0782498	0.0391249	+	0.999234i	$0.487543\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−22.3132	−0.757794
$868$	0	0
$869$	−7.16003	−0.242888
$870$	0	0
$871$	−10.8761	−0.368524
$872$	0	0
$873$	21.5046	0.727819
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	0	0
$879$	22.1788	0.748071
$880$	0	0
$881$	−48.1426	−1.62196	−0.810982	−	0.585070i	$-0.801067\pi$
$881$	−48.1426	−1.62196	−0.810982	+	0.585070i	$0.801067\pi$
$882$	0	0
$883$	−12.8565	−0.432655	−0.216328	−	0.976321i	$-0.569408\pi$
$883$	−12.8565	−0.432655	−0.216328	+	0.976321i	$0.569408\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.5216	−1.09197	−0.545984	−	0.837796i	$-0.683844\pi$
$887$	−32.5216	−1.09197	−0.545984	+	0.837796i	$0.683844\pi$
$888$	0	0
$889$	−23.7671	−0.797123
$890$	0	0
$891$	−22.6991	−0.760447
$892$	0	0
$893$	55.7374	1.86518
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.58002	0.119533
$898$	0	0
$899$	77.8342	2.59592
$900$	0	0
$901$	−3.55346	−0.118383
$902$	0	0
$903$	−7.62581	−0.253771
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−21.0894	−0.700261	−0.350131	−	0.936701i	$-0.613863\pi$
$907$	−21.0894	−0.700261	−0.350131	+	0.936701i	$0.613863\pi$
$908$	0	0
$909$	−7.03356	−0.233289
$910$	0	0
$911$	−43.1600	−1.42996	−0.714978	−	0.699147i	$-0.753563\pi$
$911$	−43.1600	−1.42996	−0.714978	+	0.699147i	$0.753563\pi$
$912$	0	0
$913$	−52.9219	−1.75146
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−74.0671	−2.44591
$918$	0	0
$919$	−11.8884	−0.392161	−0.196080	−	0.980588i	$-0.562821\pi$
$919$	−11.8884	−0.392161	−0.196080	+	0.980588i	$0.562821\pi$
$920$	0	0
$921$	−20.4265	−0.673075
$922$	0	0
$923$	−5.52420	−0.181831
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.8862	0.423237
$928$	0	0
$929$	−34.4265	−1.12950	−0.564748	−	0.825263i	$-0.691027\pi$
$929$	−34.4265	−1.12950	−0.564748	+	0.825263i	$0.691027\pi$
$930$	0	0
$931$	−70.9842	−2.32641
$932$	0	0
$933$	10.9664	0.359025
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	1.88314	0.0614541
$940$	0	0
$941$	−38.1116	−1.24240	−0.621202	−	0.783651i	$-0.713355\pi$
$941$	−38.1116	−1.24240	−0.621202	+	0.783651i	$0.713355\pi$
$942$	0	0
$943$	−1.86391	−0.0606974
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.9245	0.452487	0.226243	−	0.974071i	$-0.427356\pi$
$947$	13.9245	0.452487	0.226243	+	0.974071i	$0.427356\pi$
$948$	0	0
$949$	−11.2207	−0.364238
$950$	0	0
$951$	40.3052	1.30699
$952$	0	0
$953$	23.7523	0.769412	0.384706	−	0.923039i	$-0.374303\pi$
$953$	23.7523	0.769412	0.384706	+	0.923039i	$0.374303\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	61.0336	1.97293
$958$	0	0
$959$	37.2865	1.20404
$960$	0	0
$961$	59.7104	1.92614
$962$	0	0
$963$	−8.19713	−0.264149
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−6.87614	−0.221122	−0.110561	−	0.993869i	$-0.535265\pi$
$967$	−6.87614	−0.221122	−0.110561	+	0.993869i	$0.535265\pi$
$968$	0	0
$969$	−6.49022	−0.208496
$970$	0	0
$971$	−6.24772	−0.200499	−0.100249	−	0.994962i	$-0.531964\pi$
$971$	−6.24772	−0.200499	−0.100249	+	0.994962i	$0.531964\pi$
$972$	0	0
$973$	−55.4749	−1.77844
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.5316	1.10476	0.552382	−	0.833591i	$-0.313719\pi$
$977$	34.5316	1.10476	0.552382	+	0.833591i	$0.313719\pi$
$978$	0	0
$979$	64.9219	2.07491
$980$	0	0
$981$	−3.99478	−0.127543
$982$	0	0
$983$	−11.5652	−0.368872	−0.184436	−	0.982845i	$-0.559046\pi$
$983$	−11.5652	−0.368872	−0.184436	+	0.982845i	$0.559046\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−46.0968	−1.46728
$988$	0	0
$989$	3.58002	0.113838
$990$	0	0
$991$	−3.77673	−0.119972	−0.0599859	−	0.998199i	$-0.519106\pi$
$991$	−3.77673	−0.119972	−0.0599859	+	0.998199i	$0.519106\pi$
$992$	0	0
$993$	25.9097	0.822220
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−54.4265	−1.72370	−0.861852	−	0.507160i	$-0.830695\pi$
$997$	−54.4265	−1.72370	−0.861852	+	0.507160i	$0.830695\pi$
$998$	0	0
$999$	−38.7858	−1.22713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.ci.1.2		3
4.3	odd	2		1300.2.a.i.1.2		3
5.4	even	2		1040.2.a.o.1.2		3
15.14	odd	2		9360.2.a.da.1.1		3
20.3	even	4		1300.2.c.f.1249.3		6
20.7	even	4		1300.2.c.f.1249.4		6
20.19	odd	2		260.2.a.b.1.2	✓	3
40.19	odd	2		4160.2.a.bo.1.2		3
40.29	even	2		4160.2.a.br.1.2		3
60.59	even	2		2340.2.a.n.1.3		3
260.99	even	4		3380.2.f.h.3041.4		6
260.239	even	4		3380.2.f.h.3041.3		6
260.259	odd	2		3380.2.a.o.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.2	✓	3	20.19	odd	2
1040.2.a.o.1.2		3	5.4	even	2
1300.2.a.i.1.2		3	4.3	odd	2
1300.2.c.f.1249.3		6	20.3	even	4
1300.2.c.f.1249.4		6	20.7	even	4
2340.2.a.n.1.3		3	60.59	even	2
3380.2.a.o.1.2		3	260.259	odd	2
3380.2.f.h.3041.3		6	260.239	even	4
3380.2.f.h.3041.4		6	260.99	even	4
4160.2.a.bo.1.2		3	40.19	odd	2
4160.2.a.br.1.2		3	40.29	even	2
5200.2.a.ci.1.2		3	1.1	even	1	trivial
9360.2.a.da.1.1		3	15.14	odd	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:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35194 q^{3} +4.17226 q^{7} -1.17226 q^{9} +O(q^{10})\)	\(q+1.35194 q^{3} +4.17226 q^{7} -1.17226 q^{9} +5.52420 q^{11} -1.00000 q^{13} +0.703878 q^{17} -6.82032 q^{19} +5.64064 q^{21} -2.64806 q^{23} -5.64064 q^{27} +8.17226 q^{29} +9.52420 q^{31} +7.46838 q^{33} +6.87614 q^{37} -1.35194 q^{39} +0.703878 q^{41} -1.35194 q^{43} -8.17226 q^{47} +10.4078 q^{49} +0.951601 q^{51} -5.04840 q^{53} -9.22066 q^{57} +12.2281 q^{59} -0.172260 q^{61} -4.89098 q^{63} +10.8761 q^{67} -3.58002 q^{69} +5.52420 q^{71} +11.2207 q^{73} +23.0484 q^{77} -1.29612 q^{79} -4.10902 q^{81} -9.58002 q^{83} +11.0484 q^{87} +11.7523 q^{89} -4.17226 q^{91} +12.8761 q^{93} -18.3445 q^{97} -6.47580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 2 q^{7} + 11 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 2 * q^7 + 11 * q^9	\( 3 q + 2 q^{3} - 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{17} - 8 q^{19} - 8 q^{21} - 10 q^{23} + 8 q^{27} + 10 q^{29} + 12 q^{31} + 12 q^{33} + 2 q^{37} - 2 q^{39} - 2 q^{41} - 2 q^{43} - 10 q^{47} + 23 q^{49} + 36 q^{51} + 18 q^{53} + 20 q^{57} + 16 q^{59} + 14 q^{61} - 50 q^{63} + 14 q^{67} + 12 q^{69} - 14 q^{73} + 36 q^{77} - 8 q^{79} + 23 q^{81} - 6 q^{83} - 2 q^{89} + 2 q^{91} + 20 q^{93} - 26 q^{97} - 36 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 2 * q^7 + 11 * q^9 - 3 * q^13 - 2 * q^17 - 8 * q^19 - 8 * q^21 - 10 * q^23 + 8 * q^27 + 10 * q^29 + 12 * q^31 + 12 * q^33 + 2 * q^37 - 2 * q^39 - 2 * q^41 - 2 * q^43 - 10 * q^47 + 23 * q^49 + 36 * q^51 + 18 * q^53 + 20 * q^57 + 16 * q^59 + 14 * q^61 - 50 * q^63 + 14 * q^67 + 12 * q^69 - 14 * q^73 + 36 * q^77 - 8 * q^79 + 23 * q^81 - 6 * q^83 - 2 * q^89 + 2 * q^91 + 20 * q^93 - 26 * q^97 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more