LMFDB - Embedded newform 10000.2.a.g.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+2.23607 q^{3} -2.61803 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+2.23607 q^{3} -2.61803 q^{7} +2.00000 q^{9} -1.23607 q^{11} +3.38197 q^{13} +5.23607 q^{17} -7.61803 q^{19} -5.85410 q^{21} -5.47214 q^{23} -2.23607 q^{27} +4.38197 q^{29} +4.70820 q^{31} -2.76393 q^{33} -8.23607 q^{37} +7.56231 q^{39} +5.70820 q^{41} -4.38197 q^{43} -0.145898 q^{47} -0.145898 q^{49} +11.7082 q^{51} +7.94427 q^{53} -17.0344 q^{57} -2.38197 q^{59} -13.1803 q^{61} -5.23607 q^{63} -13.7082 q^{67} -12.2361 q^{69} +8.61803 q^{71} +11.0000 q^{73} +3.23607 q^{77} +5.32624 q^{79} -11.0000 q^{81} +4.52786 q^{83} +9.79837 q^{87} +4.00000 q^{89} -8.85410 q^{91} +10.5279 q^{93} -15.2705 q^{97} -2.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q - 3 * q^7 + 4 * q^9	$ 2 q - 3 q^{7} + 4 q^{9} + 2 q^{11} + 9 q^{13} + 6 q^{17} - 13 q^{19} - 5 q^{21} - 2 q^{23} + 11 q^{29} - 4 q^{31} - 10 q^{33} - 12 q^{37} - 5 q^{39} - 2 q^{41} - 11 q^{43} - 7 q^{47} - 7 q^{49} + 10 q^{51} - 2 q^{53} - 5 q^{57} - 7 q^{59} - 4 q^{61} - 6 q^{63} - 14 q^{67} - 20 q^{69} + 15 q^{71} + 22 q^{73} + 2 q^{77} - 5 q^{79} - 22 q^{81} + 18 q^{83} - 5 q^{87} + 8 q^{89} - 11 q^{91} + 30 q^{93} + 3 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 3 * q^7 + 4 * q^9 + 2 * q^11 + 9 * q^13 + 6 * q^17 - 13 * q^19 - 5 * q^21 - 2 * q^23 + 11 * q^29 - 4 * q^31 - 10 * q^33 - 12 * q^37 - 5 * q^39 - 2 * q^41 - 11 * q^43 - 7 * q^47 - 7 * q^49 + 10 * q^51 - 2 * q^53 - 5 * q^57 - 7 * q^59 - 4 * q^61 - 6 * q^63 - 14 * q^67 - 20 * q^69 + 15 * q^71 + 22 * q^73 + 2 * q^77 - 5 * q^79 - 22 * q^81 + 18 * q^83 - 5 * q^87 + 8 * q^89 - 11 * q^91 + 30 * q^93 + 3 * q^97 + 4 * 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
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.61803	−0.989524	−0.494762	−	0.869029i	$-0.664745\pi$
$7$	−2.61803	−0.989524	−0.494762	+	0.869029i	$0.664745\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−1.23607	−0.372689	−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607	−0.372689	−0.186344	+	0.982485i	$0.559664\pi$
$12$	0	0
$13$	3.38197	0.937989	0.468994	−	0.883201i	$-0.344616\pi$
$13$	3.38197	0.937989	0.468994	+	0.883201i	$0.344616\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	−7.61803	−1.74770	−0.873848	−	0.486198i	$-0.838383\pi$
$19$	−7.61803	−1.74770	−0.873848	+	0.486198i	$0.838383\pi$
$20$	0	0
$21$	−5.85410	−1.27747
$22$	0	0
$23$	−5.47214	−1.14102	−0.570510	−	0.821291i	$-0.693254\pi$
$23$	−5.47214	−1.14102	−0.570510	+	0.821291i	$0.693254\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	4.38197	0.813711	0.406855	−	0.913493i	$-0.366625\pi$
$29$	4.38197	0.813711	0.406855	+	0.913493i	$0.366625\pi$
$30$	0	0
$31$	4.70820	0.845618	0.422809	−	0.906219i	$-0.361044\pi$
$31$	4.70820	0.845618	0.422809	+	0.906219i	$0.361044\pi$
$32$	0	0
$33$	−2.76393	−0.481139
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.23607	−1.35400	−0.677001	−	0.735982i	$-0.736721\pi$
$37$	−8.23607	−1.35400	−0.677001	+	0.735982i	$0.736721\pi$
$38$	0	0
$39$	7.56231	1.21094
$40$	0	0
$41$	5.70820	0.891472	0.445736	−	0.895165i	$-0.352942\pi$
$41$	5.70820	0.891472	0.445736	+	0.895165i	$0.352942\pi$
$42$	0	0
$43$	−4.38197	−0.668244	−0.334122	−	0.942530i	$-0.608440\pi$
$43$	−4.38197	−0.668244	−0.334122	+	0.942530i	$0.608440\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.145898	−0.0212814	−0.0106407	−	0.999943i	$-0.503387\pi$
$47$	−0.145898	−0.0212814	−0.0106407	+	0.999943i	$0.503387\pi$
$48$	0	0
$49$	−0.145898	−0.0208426
$50$	0	0
$51$	11.7082	1.63948
$52$	0	0
$53$	7.94427	1.09123	0.545615	−	0.838036i	$-0.316296\pi$
$53$	7.94427	1.09123	0.545615	+	0.838036i	$0.316296\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−17.0344	−2.25627
$58$	0	0
$59$	−2.38197	−0.310106	−0.155053	−	0.987906i	$-0.549555\pi$
$59$	−2.38197	−0.310106	−0.155053	+	0.987906i	$0.549555\pi$
$60$	0	0
$61$	−13.1803	−1.68757	−0.843785	−	0.536682i	$-0.819678\pi$
$61$	−13.1803	−1.68757	−0.843785	+	0.536682i	$0.819678\pi$
$62$	0	0
$63$	−5.23607	−0.659683
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.7082	−1.67472	−0.837362	−	0.546649i	$-0.815903\pi$
$67$	−13.7082	−1.67472	−0.837362	+	0.546649i	$0.815903\pi$
$68$	0	0
$69$	−12.2361	−1.47305
$70$	0	0
$71$	8.61803	1.02277	0.511386	−	0.859351i	$-0.329132\pi$
$71$	8.61803	1.02277	0.511386	+	0.859351i	$0.329132\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.23607	0.368784
$78$	0	0
$79$	5.32624	0.599249	0.299624	−	0.954057i	$-0.403139\pi$
$79$	5.32624	0.599249	0.299624	+	0.954057i	$0.403139\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.52786	0.496998	0.248499	−	0.968632i	$-0.420063\pi$
$83$	4.52786	0.496998	0.248499	+	0.968632i	$0.420063\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.79837	1.05050
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−8.85410	−0.928162
$92$	0	0
$93$	10.5279	1.09169
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.2705	−1.55049	−0.775243	−	0.631664i	$-0.782372\pi$
$97$	−15.2705	−1.55049	−0.775243	+	0.631664i	$0.782372\pi$
$98$	0	0
$99$	−2.47214	−0.248459
$100$	0	0
$101$	−17.9443	−1.78552	−0.892761	−	0.450531i	$-0.851235\pi$
$101$	−17.9443	−1.78552	−0.892761	+	0.450531i	$0.851235\pi$
$102$	0	0
$103$	3.14590	0.309975	0.154987	−	0.987916i	$-0.450466\pi$
$103$	3.14590	0.309975	0.154987	+	0.987916i	$0.450466\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.7082	−1.61524	−0.807622	−	0.589701i	$-0.799246\pi$
$107$	−16.7082	−1.61524	−0.807622	+	0.589701i	$0.799246\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	−18.4164	−1.74801
$112$	0	0
$113$	−10.5623	−0.993618	−0.496809	−	0.867860i	$-0.665495\pi$
$113$	−10.5623	−0.993618	−0.496809	+	0.867860i	$0.665495\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.76393	0.625326
$118$	0	0
$119$	−13.7082	−1.25663
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	12.7639	1.15088
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.52786	−0.667990	−0.333995	−	0.942575i	$-0.608397\pi$
$127$	−7.52786	−0.667990	−0.333995	+	0.942575i	$0.608397\pi$
$128$	0	0
$129$	−9.79837	−0.862699
$130$	0	0
$131$	1.09017	0.0952486	0.0476243	−	0.998865i	$-0.484835\pi$
$131$	1.09017	0.0952486	0.0476243	+	0.998865i	$0.484835\pi$
$132$	0	0
$133$	19.9443	1.72939
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	−9.76393	−0.828166	−0.414083	−	0.910239i	$-0.635898\pi$
$139$	−9.76393	−0.828166	−0.414083	+	0.910239i	$0.635898\pi$
$140$	0	0
$141$	−0.326238	−0.0274742
$142$	0	0
$143$	−4.18034	−0.349578
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.326238	−0.0269077
$148$	0	0
$149$	−18.4164	−1.50873	−0.754365	−	0.656455i	$-0.772055\pi$
$149$	−18.4164	−1.50873	−0.754365	+	0.656455i	$0.772055\pi$
$150$	0	0
$151$	−0.381966	−0.0310840	−0.0155420	−	0.999879i	$-0.504947\pi$
$151$	−0.381966	−0.0310840	−0.0155420	+	0.999879i	$0.504947\pi$
$152$	0	0
$153$	10.4721	0.846622
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.23607	0.338075	0.169038	−	0.985610i	$-0.445934\pi$
$157$	4.23607	0.338075	0.169038	+	0.985610i	$0.445934\pi$
$158$	0	0
$159$	17.7639	1.40877
$160$	0	0
$161$	14.3262	1.12907
$162$	0	0
$163$	−15.1803	−1.18902	−0.594508	−	0.804090i	$-0.702653\pi$
$163$	−15.1803	−1.18902	−0.594508	+	0.804090i	$0.702653\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.0344	1.47293	0.736465	−	0.676476i	$-0.236494\pi$
$167$	19.0344	1.47293	0.736465	+	0.676476i	$0.236494\pi$
$168$	0	0
$169$	−1.56231	−0.120177
$170$	0	0
$171$	−15.2361	−1.16513
$172$	0	0
$173$	6.52786	0.496304	0.248152	−	0.968721i	$-0.420177\pi$
$173$	6.52786	0.496304	0.248152	+	0.968721i	$0.420177\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.32624	−0.400345
$178$	0	0
$179$	−3.29180	−0.246040	−0.123020	−	0.992404i	$-0.539258\pi$
$179$	−3.29180	−0.246040	−0.123020	+	0.992404i	$0.539258\pi$
$180$	0	0
$181$	−8.29180	−0.616324	−0.308162	−	0.951334i	$-0.599714\pi$
$181$	−8.29180	−0.616324	−0.308162	+	0.951334i	$0.599714\pi$
$182$	0	0
$183$	−29.4721	−2.17864
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.47214	−0.473289
$188$	0	0
$189$	5.85410	0.425823
$190$	0	0
$191$	0.291796	0.0211136	0.0105568	−	0.999944i	$-0.496640\pi$
$191$	0.291796	0.0211136	0.0105568	+	0.999944i	$0.496640\pi$
$192$	0	0
$193$	−7.70820	−0.554849	−0.277424	−	0.960747i	$-0.589481\pi$
$193$	−7.70820	−0.554849	−0.277424	+	0.960747i	$0.589481\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.7639	−1.05189	−0.525943	−	0.850520i	$-0.676288\pi$
$197$	−14.7639	−1.05189	−0.525943	+	0.850520i	$0.676288\pi$
$198$	0	0
$199$	−9.38197	−0.665070	−0.332535	−	0.943091i	$-0.607904\pi$
$199$	−9.38197	−0.665070	−0.332535	+	0.943091i	$0.607904\pi$
$200$	0	0
$201$	−30.6525	−2.16206
$202$	0	0
$203$	−11.4721	−0.805186
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.9443	−0.760679
$208$	0	0
$209$	9.41641	0.651347
$210$	0	0
$211$	−19.9443	−1.37302	−0.686510	−	0.727120i	$-0.740858\pi$
$211$	−19.9443	−1.37302	−0.686510	+	0.727120i	$0.740858\pi$
$212$	0	0
$213$	19.2705	1.32039
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.3262	−0.836760
$218$	0	0
$219$	24.5967	1.66209
$220$	0	0
$221$	17.7082	1.19118
$222$	0	0
$223$	−5.70820	−0.382250	−0.191125	−	0.981566i	$-0.561214\pi$
$223$	−5.70820	−0.382250	−0.191125	+	0.981566i	$0.561214\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.5967	1.69892	0.849458	−	0.527656i	$-0.176929\pi$
$227$	25.5967	1.69892	0.849458	+	0.527656i	$0.176929\pi$
$228$	0	0
$229$	20.1803	1.33355	0.666777	−	0.745257i	$-0.267673\pi$
$229$	20.1803	1.33355	0.666777	+	0.745257i	$0.267673\pi$
$230$	0	0
$231$	7.23607	0.476098
$232$	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
$237$	11.9098	0.773627
$238$	0	0
$239$	22.2361	1.43833	0.719166	−	0.694838i	$-0.244524\pi$
$239$	22.2361	1.43833	0.719166	+	0.694838i	$0.244524\pi$
$240$	0	0
$241$	11.4721	0.738985	0.369493	−	0.929234i	$-0.379532\pi$
$241$	11.4721	0.738985	0.369493	+	0.929234i	$0.379532\pi$
$242$	0	0
$243$	−17.8885	−1.14755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−25.7639	−1.63932
$248$	0	0
$249$	10.1246	0.641621
$250$	0	0
$251$	8.05573	0.508473	0.254237	−	0.967142i	$-0.418176\pi$
$251$	8.05573	0.508473	0.254237	+	0.967142i	$0.418176\pi$
$252$	0	0
$253$	6.76393	0.425245
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.61803	−0.225687	−0.112843	−	0.993613i	$-0.535996\pi$
$257$	−3.61803	−0.225687	−0.112843	+	0.993613i	$0.535996\pi$
$258$	0	0
$259$	21.5623	1.33982
$260$	0	0
$261$	8.76393	0.542474
$262$	0	0
$263$	13.8541	0.854281	0.427140	−	0.904185i	$-0.359521\pi$
$263$	13.8541	0.854281	0.427140	+	0.904185i	$0.359521\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.94427	0.547381
$268$	0	0
$269$	−14.2918	−0.871386	−0.435693	−	0.900095i	$-0.643497\pi$
$269$	−14.2918	−0.871386	−0.435693	+	0.900095i	$0.643497\pi$
$270$	0	0
$271$	−12.9443	−0.786309	−0.393154	−	0.919473i	$-0.628616\pi$
$271$	−12.9443	−0.786309	−0.393154	+	0.919473i	$0.628616\pi$
$272$	0	0
$273$	−19.7984	−1.19825
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.1803	1.27260	0.636302	−	0.771440i	$-0.280463\pi$
$277$	21.1803	1.27260	0.636302	+	0.771440i	$0.280463\pi$
$278$	0	0
$279$	9.41641	0.563746
$280$	0	0
$281$	9.14590	0.545599	0.272799	−	0.962071i	$-0.412051\pi$
$281$	9.14590	0.545599	0.272799	+	0.962071i	$0.412051\pi$
$282$	0	0
$283$	7.27051	0.432187	0.216093	−	0.976373i	$-0.430668\pi$
$283$	7.27051	0.432187	0.216093	+	0.976373i	$0.430668\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.9443	−0.882132
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	−34.1459	−2.00167
$292$	0	0
$293$	0.472136	0.0275825	0.0137912	−	0.999905i	$-0.495610\pi$
$293$	0.472136	0.0275825	0.0137912	+	0.999905i	$0.495610\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.76393	0.160380
$298$	0	0
$299$	−18.5066	−1.07026
$300$	0	0
$301$	11.4721	0.661243
$302$	0	0
$303$	−40.1246	−2.30510
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.7639	−0.614330	−0.307165	−	0.951656i	$-0.599380\pi$
$307$	−10.7639	−0.614330	−0.307165	+	0.951656i	$0.599380\pi$
$308$	0	0
$309$	7.03444	0.400175
$310$	0	0
$311$	4.90983	0.278411	0.139205	−	0.990264i	$-0.455545\pi$
$311$	4.90983	0.278411	0.139205	+	0.990264i	$0.455545\pi$
$312$	0	0
$313$	5.70820	0.322647	0.161323	−	0.986902i	$-0.448424\pi$
$313$	5.70820	0.322647	0.161323	+	0.986902i	$0.448424\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.2361	−0.799577	−0.399789	−	0.916607i	$-0.630916\pi$
$317$	−14.2361	−0.799577	−0.399789	+	0.916607i	$0.630916\pi$
$318$	0	0
$319$	−5.41641	−0.303261
$320$	0	0
$321$	−37.3607	−2.08527
$322$	0	0
$323$	−39.8885	−2.21946
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.58359	0.364073
$328$	0	0
$329$	0.381966	0.0210585
$330$	0	0
$331$	−32.0689	−1.76267	−0.881333	−	0.472496i	$-0.843353\pi$
$331$	−32.0689	−1.76267	−0.881333	+	0.472496i	$0.843353\pi$
$332$	0	0
$333$	−16.4721	−0.902667
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.09017	−0.495173	−0.247587	−	0.968866i	$-0.579637\pi$
$337$	−9.09017	−0.495173	−0.247587	+	0.968866i	$0.579637\pi$
$338$	0	0
$339$	−23.6180	−1.28276
$340$	0	0
$341$	−5.81966	−0.315152
$342$	0	0
$343$	18.7082	1.01015
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.85410	−0.367947	−0.183974	−	0.982931i	$-0.558896\pi$
$347$	−6.85410	−0.367947	−0.183974	+	0.982931i	$0.558896\pi$
$348$	0	0
$349$	25.1246	1.34489	0.672445	−	0.740147i	$-0.265244\pi$
$349$	25.1246	1.34489	0.672445	+	0.740147i	$0.265244\pi$
$350$	0	0
$351$	−7.56231	−0.403646
$352$	0	0
$353$	−29.5623	−1.57344	−0.786721	−	0.617308i	$-0.788223\pi$
$353$	−29.5623	−1.57344	−0.786721	+	0.617308i	$0.788223\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−30.6525	−1.62230
$358$	0	0
$359$	−25.2705	−1.33373	−0.666863	−	0.745180i	$-0.732363\pi$
$359$	−25.2705	−1.33373	−0.666863	+	0.745180i	$0.732363\pi$
$360$	0	0
$361$	39.0344	2.05444
$362$	0	0
$363$	−21.1803	−1.11168
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−19.2705	−1.00591	−0.502956	−	0.864312i	$-0.667754\pi$
$367$	−19.2705	−1.00591	−0.502956	+	0.864312i	$0.667754\pi$
$368$	0	0
$369$	11.4164	0.594314
$370$	0	0
$371$	−20.7984	−1.07980
$372$	0	0
$373$	16.0902	0.833117	0.416559	−	0.909109i	$-0.363236\pi$
$373$	16.0902	0.833117	0.416559	+	0.909109i	$0.363236\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.8197	0.763251
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	−16.8328	−0.862371
$382$	0	0
$383$	−13.7639	−0.703304	−0.351652	−	0.936131i	$-0.614380\pi$
$383$	−13.7639	−0.703304	−0.351652	+	0.936131i	$0.614380\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.76393	−0.445496
$388$	0	0
$389$	−12.4164	−0.629537	−0.314768	−	0.949168i	$-0.601927\pi$
$389$	−12.4164	−0.629537	−0.314768	+	0.949168i	$0.601927\pi$
$390$	0	0
$391$	−28.6525	−1.44902
$392$	0	0
$393$	2.43769	0.122965
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.0902	1.00830	0.504148	−	0.863617i	$-0.331806\pi$
$397$	20.0902	1.00830	0.504148	+	0.863617i	$0.331806\pi$
$398$	0	0
$399$	44.5967	2.23263
$400$	0	0
$401$	−36.5967	−1.82755	−0.913777	−	0.406216i	$-0.866848\pi$
$401$	−36.5967	−1.82755	−0.913777	+	0.406216i	$0.866848\pi$
$402$	0	0
$403$	15.9230	0.793180
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.1803	0.504621
$408$	0	0
$409$	−20.5279	−1.01504	−0.507519	−	0.861641i	$-0.669437\pi$
$409$	−20.5279	−1.01504	−0.507519	+	0.861641i	$0.669437\pi$
$410$	0	0
$411$	11.1803	0.551485
$412$	0	0
$413$	6.23607	0.306857
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−21.8328	−1.06916
$418$	0	0
$419$	−26.1246	−1.27627	−0.638135	−	0.769924i	$-0.720294\pi$
$419$	−26.1246	−1.27627	−0.638135	+	0.769924i	$0.720294\pi$
$420$	0	0
$421$	4.94427	0.240969	0.120485	−	0.992715i	$-0.461555\pi$
$421$	4.94427	0.240969	0.120485	+	0.992715i	$0.461555\pi$
$422$	0	0
$423$	−0.291796	−0.0141876
$424$	0	0
$425$	0	0
$426$	0	0
$427$	34.5066	1.66989
$428$	0	0
$429$	−9.34752	−0.451303
$430$	0	0
$431$	−3.29180	−0.158560	−0.0792801	−	0.996852i	$-0.525262\pi$
$431$	−3.29180	−0.158560	−0.0792801	+	0.996852i	$0.525262\pi$
$432$	0	0
$433$	2.27051	0.109114	0.0545569	−	0.998511i	$-0.482625\pi$
$433$	2.27051	0.109114	0.0545569	+	0.998511i	$0.482625\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.6869	1.99416
$438$	0	0
$439$	−16.7984	−0.801743	−0.400871	−	0.916134i	$-0.631293\pi$
$439$	−16.7984	−0.801743	−0.400871	+	0.916134i	$0.631293\pi$
$440$	0	0
$441$	−0.291796	−0.0138951
$442$	0	0
$443$	8.34752	0.396603	0.198301	−	0.980141i	$-0.436458\pi$
$443$	8.34752	0.396603	0.198301	+	0.980141i	$0.436458\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−41.1803	−1.94776
$448$	0	0
$449$	−23.9098	−1.12837	−0.564187	−	0.825647i	$-0.690810\pi$
$449$	−23.9098	−1.12837	−0.564187	+	0.825647i	$0.690810\pi$
$450$	0	0
$451$	−7.05573	−0.332241
$452$	0	0
$453$	−0.854102	−0.0401292
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.47214	−0.209198	−0.104599	−	0.994514i	$-0.533356\pi$
$457$	−4.47214	−0.209198	−0.104599	+	0.994514i	$0.533356\pi$
$458$	0	0
$459$	−11.7082	−0.546492
$460$	0	0
$461$	−21.2918	−0.991658	−0.495829	−	0.868420i	$-0.665136\pi$
$461$	−21.2918	−0.991658	−0.495829	+	0.868420i	$0.665136\pi$
$462$	0	0
$463$	15.9443	0.740993	0.370497	−	0.928834i	$-0.379188\pi$
$463$	15.9443	0.740993	0.370497	+	0.928834i	$0.379188\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.27051	0.336439	0.168220	−	0.985750i	$-0.446198\pi$
$467$	7.27051	0.336439	0.168220	+	0.985750i	$0.446198\pi$
$468$	0	0
$469$	35.8885	1.65718
$470$	0	0
$471$	9.47214	0.436453
$472$	0	0
$473$	5.41641	0.249047
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15.8885	0.727486
$478$	0	0
$479$	−22.3820	−1.02266	−0.511329	−	0.859385i	$-0.670847\pi$
$479$	−22.3820	−1.02266	−0.511329	+	0.859385i	$0.670847\pi$
$480$	0	0
$481$	−27.8541	−1.27004
$482$	0	0
$483$	32.0344	1.45762
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.7082	1.21026	0.605132	−	0.796125i	$-0.293120\pi$
$487$	26.7082	1.21026	0.605132	+	0.796125i	$0.293120\pi$
$488$	0	0
$489$	−33.9443	−1.53501
$490$	0	0
$491$	−3.76393	−0.169864	−0.0849319	−	0.996387i	$-0.527067\pi$
$491$	−3.76393	−0.169864	−0.0849319	+	0.996387i	$0.527067\pi$
$492$	0	0
$493$	22.9443	1.03336
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.5623	−1.01206
$498$	0	0
$499$	−29.5623	−1.32339	−0.661695	−	0.749773i	$-0.730163\pi$
$499$	−29.5623	−1.32339	−0.661695	+	0.749773i	$0.730163\pi$
$500$	0	0
$501$	42.5623	1.90154
$502$	0	0
$503$	15.8885	0.708435	0.354218	−	0.935163i	$-0.384747\pi$
$503$	15.8885	0.708435	0.354218	+	0.935163i	$0.384747\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.49342	−0.155148
$508$	0	0
$509$	31.0902	1.37805	0.689024	−	0.724739i	$-0.258040\pi$
$509$	31.0902	1.37805	0.689024	+	0.724739i	$0.258040\pi$
$510$	0	0
$511$	−28.7984	−1.27397
$512$	0	0
$513$	17.0344	0.752089
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.180340	0.00793134
$518$	0	0
$519$	14.5967	0.640726
$520$	0	0
$521$	8.41641	0.368730	0.184365	−	0.982858i	$-0.440977\pi$
$521$	8.41641	0.368730	0.184365	+	0.982858i	$0.440977\pi$
$522$	0	0
$523$	18.3820	0.803787	0.401894	−	0.915686i	$-0.368352\pi$
$523$	18.3820	0.803787	0.401894	+	0.915686i	$0.368352\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.6525	1.07388
$528$	0	0
$529$	6.94427	0.301925
$530$	0	0
$531$	−4.76393	−0.206737
$532$	0	0
$533$	19.3050	0.836190
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.36068	−0.317637
$538$	0	0
$539$	0.180340	0.00776779
$540$	0	0
$541$	8.65248	0.371999	0.185999	−	0.982550i	$-0.440448\pi$
$541$	8.65248	0.371999	0.185999	+	0.982550i	$0.440448\pi$
$542$	0	0
$543$	−18.5410	−0.795671
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.36068	−0.400234	−0.200117	−	0.979772i	$-0.564132\pi$
$547$	−9.36068	−0.400234	−0.200117	+	0.979772i	$0.564132\pi$
$548$	0	0
$549$	−26.3607	−1.12505
$550$	0	0
$551$	−33.3820	−1.42212
$552$	0	0
$553$	−13.9443	−0.592971
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.81966	−0.331330	−0.165665	−	0.986182i	$-0.552977\pi$
$557$	−7.81966	−0.331330	−0.165665	+	0.986182i	$0.552977\pi$
$558$	0	0
$559$	−14.8197	−0.626805
$560$	0	0
$561$	−14.4721	−0.611014
$562$	0	0
$563$	−1.96556	−0.0828384	−0.0414192	−	0.999142i	$-0.513188\pi$
$563$	−1.96556	−0.0828384	−0.0414192	+	0.999142i	$0.513188\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	28.7984	1.20942
$568$	0	0
$569$	17.4721	0.732470	0.366235	−	0.930522i	$-0.380647\pi$
$569$	17.4721	0.732470	0.366235	+	0.930522i	$0.380647\pi$
$570$	0	0
$571$	32.3050	1.35192	0.675960	−	0.736938i	$-0.263729\pi$
$571$	32.3050	1.35192	0.675960	+	0.736938i	$0.263729\pi$
$572$	0	0
$573$	0.652476	0.0272576
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.8885	0.661449	0.330724	−	0.943727i	$-0.392707\pi$
$577$	15.8885	0.661449	0.330724	+	0.943727i	$0.392707\pi$
$578$	0	0
$579$	−17.2361	−0.716307
$580$	0	0
$581$	−11.8541	−0.491791
$582$	0	0
$583$	−9.81966	−0.406689
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.4721	1.29899	0.649497	−	0.760364i	$-0.274979\pi$
$587$	31.4721	1.29899	0.649497	+	0.760364i	$0.274979\pi$
$588$	0	0
$589$	−35.8673	−1.47788
$590$	0	0
$591$	−33.0132	−1.35798
$592$	0	0
$593$	19.5623	0.803328	0.401664	−	0.915787i	$-0.368432\pi$
$593$	19.5623	0.803328	0.401664	+	0.915787i	$0.368432\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−20.9787	−0.858602
$598$	0	0
$599$	15.1803	0.620252	0.310126	−	0.950695i	$-0.399629\pi$
$599$	15.1803	0.620252	0.310126	+	0.950695i	$0.399629\pi$
$600$	0	0
$601$	−22.4377	−0.915253	−0.457626	−	0.889145i	$-0.651300\pi$
$601$	−22.4377	−0.915253	−0.457626	+	0.889145i	$0.651300\pi$
$602$	0	0
$603$	−27.4164	−1.11648
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.27051	−0.0515684	−0.0257842	−	0.999668i	$-0.508208\pi$
$607$	−1.27051	−0.0515684	−0.0257842	+	0.999668i	$0.508208\pi$
$608$	0	0
$609$	−25.6525	−1.03949
$610$	0	0
$611$	−0.493422	−0.0199617
$612$	0	0
$613$	−10.0344	−0.405287	−0.202644	−	0.979253i	$-0.564953\pi$
$613$	−10.0344	−0.405287	−0.202644	+	0.979253i	$0.564953\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.5967	1.23178	0.615889	−	0.787833i	$-0.288797\pi$
$617$	30.5967	1.23178	0.615889	+	0.787833i	$0.288797\pi$
$618$	0	0
$619$	40.7082	1.63620	0.818100	−	0.575075i	$-0.195027\pi$
$619$	40.7082	1.63620	0.818100	+	0.575075i	$0.195027\pi$
$620$	0	0
$621$	12.2361	0.491016
$622$	0	0
$623$	−10.4721	−0.419557
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.0557	0.840885
$628$	0	0
$629$	−43.1246	−1.71949
$630$	0	0
$631$	−7.83282	−0.311819	−0.155910	−	0.987771i	$-0.549831\pi$
$631$	−7.83282	−0.311819	−0.155910	+	0.987771i	$0.549831\pi$
$632$	0	0
$633$	−44.5967	−1.77256
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.493422	−0.0195501
$638$	0	0
$639$	17.2361	0.681848
$640$	0	0
$641$	−17.4508	−0.689267	−0.344634	−	0.938737i	$-0.611997\pi$
$641$	−17.4508	−0.689267	−0.344634	+	0.938737i	$0.611997\pi$
$642$	0	0
$643$	−19.0557	−0.751485	−0.375742	−	0.926724i	$-0.622612\pi$
$643$	−19.0557	−0.751485	−0.375742	+	0.926724i	$0.622612\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.6525	0.890561	0.445280	−	0.895391i	$-0.353104\pi$
$647$	22.6525	0.890561	0.445280	+	0.895391i	$0.353104\pi$
$648$	0	0
$649$	2.94427	0.115573
$650$	0	0
$651$	−27.5623	−1.08025
$652$	0	0
$653$	30.5066	1.19381	0.596907	−	0.802310i	$-0.296396\pi$
$653$	30.5066	1.19381	0.596907	+	0.802310i	$0.296396\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	22.0000	0.858302
$658$	0	0
$659$	−0.111456	−0.00434172	−0.00217086	−	0.999998i	$-0.500691\pi$
$659$	−0.111456	−0.00434172	−0.00217086	+	0.999998i	$0.500691\pi$
$660$	0	0
$661$	−34.5623	−1.34432	−0.672159	−	0.740407i	$-0.734633\pi$
$661$	−34.5623	−1.34432	−0.672159	+	0.740407i	$0.734633\pi$
$662$	0	0
$663$	39.5967	1.53781
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.9787	−0.928460
$668$	0	0
$669$	−12.7639	−0.493482
$670$	0	0
$671$	16.2918	0.628938
$672$	0	0
$673$	−33.1246	−1.27686	−0.638430	−	0.769680i	$-0.720416\pi$
$673$	−33.1246	−1.27686	−0.638430	+	0.769680i	$0.720416\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.85410	−0.301858	−0.150929	−	0.988545i	$-0.548226\pi$
$677$	−7.85410	−0.301858	−0.150929	+	0.988545i	$0.548226\pi$
$678$	0	0
$679$	39.9787	1.53424
$680$	0	0
$681$	57.2361	2.19329
$682$	0	0
$683$	20.7082	0.792377	0.396189	−	0.918169i	$-0.370333\pi$
$683$	20.7082	0.792377	0.396189	+	0.918169i	$0.370333\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	45.1246	1.72161
$688$	0	0
$689$	26.8673	1.02356
$690$	0	0
$691$	46.8541	1.78241	0.891207	−	0.453597i	$-0.149859\pi$
$691$	46.8541	1.78241	0.891207	+	0.453597i	$0.149859\pi$
$692$	0	0
$693$	6.47214	0.245856
$694$	0	0
$695$	0	0
$696$	0	0
$697$	29.8885	1.13211
$698$	0	0
$699$	13.4164	0.507455
$700$	0	0
$701$	2.18034	0.0823503	0.0411752	−	0.999152i	$-0.486890\pi$
$701$	2.18034	0.0823503	0.0411752	+	0.999152i	$0.486890\pi$
$702$	0	0
$703$	62.7426	2.36638
$704$	0	0
$705$	0	0
$706$	0	0
$707$	46.9787	1.76682
$708$	0	0
$709$	−36.5967	−1.37442	−0.687210	−	0.726459i	$-0.741165\pi$
$709$	−36.5967	−1.37442	−0.687210	+	0.726459i	$0.741165\pi$
$710$	0	0
$711$	10.6525	0.399499
$712$	0	0
$713$	−25.7639	−0.964867
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.7214	1.85688
$718$	0	0
$719$	38.3050	1.42853	0.714267	−	0.699873i	$-0.246760\pi$
$719$	38.3050	1.42853	0.714267	+	0.699873i	$0.246760\pi$
$720$	0	0
$721$	−8.23607	−0.306727
$722$	0	0
$723$	25.6525	0.954026
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.3262	−0.420067	−0.210033	−	0.977694i	$-0.567357\pi$
$727$	−11.3262	−0.420067	−0.210033	+	0.977694i	$0.567357\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	−22.9443	−0.848625
$732$	0	0
$733$	10.8541	0.400905	0.200453	−	0.979703i	$-0.435759\pi$
$733$	10.8541	0.400905	0.200453	+	0.979703i	$0.435759\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.9443	0.624150
$738$	0	0
$739$	−11.3262	−0.416642	−0.208321	−	0.978060i	$-0.566800\pi$
$739$	−11.3262	−0.416642	−0.208321	+	0.978060i	$0.566800\pi$
$740$	0	0
$741$	−57.6099	−2.11635
$742$	0	0
$743$	46.2492	1.69672	0.848360	−	0.529420i	$-0.177591\pi$
$743$	46.2492	1.69672	0.848360	+	0.529420i	$0.177591\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.05573	0.331332
$748$	0	0
$749$	43.7426	1.59832
$750$	0	0
$751$	31.7639	1.15908	0.579541	−	0.814943i	$-0.303232\pi$
$751$	31.7639	1.15908	0.579541	+	0.814943i	$0.303232\pi$
$752$	0	0
$753$	18.0132	0.656436
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.0000	0.545184	0.272592	−	0.962130i	$-0.412119\pi$
$757$	15.0000	0.545184	0.272592	+	0.962130i	$0.412119\pi$
$758$	0	0
$759$	15.1246	0.548989
$760$	0	0
$761$	−25.1459	−0.911538	−0.455769	−	0.890098i	$-0.650636\pi$
$761$	−25.1459	−0.911538	−0.455769	+	0.890098i	$0.650636\pi$
$762$	0	0
$763$	−7.70820	−0.279056
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.05573	−0.290875
$768$	0	0
$769$	39.5279	1.42541	0.712706	−	0.701463i	$-0.247470\pi$
$769$	39.5279	1.42541	0.712706	+	0.701463i	$0.247470\pi$
$770$	0	0
$771$	−8.09017	−0.291360
$772$	0	0
$773$	26.6738	0.959389	0.479694	−	0.877436i	$-0.340748\pi$
$773$	26.6738	0.959389	0.479694	+	0.877436i	$0.340748\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	48.2148	1.72970
$778$	0	0
$779$	−43.4853	−1.55802
$780$	0	0
$781$	−10.6525	−0.381176
$782$	0	0
$783$	−9.79837	−0.350165
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−30.2918	−1.07979	−0.539893	−	0.841734i	$-0.681535\pi$
$787$	−30.2918	−1.07979	−0.539893	+	0.841734i	$0.681535\pi$
$788$	0	0
$789$	30.9787	1.10287
$790$	0	0
$791$	27.6525	0.983209
$792$	0	0
$793$	−44.5755	−1.58292
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.29180	−0.187445	−0.0937225	−	0.995598i	$-0.529877\pi$
$797$	−5.29180	−0.187445	−0.0937225	+	0.995598i	$0.529877\pi$
$798$	0	0
$799$	−0.763932	−0.0270260
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	−13.5967	−0.479819
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.9574	−1.12495
$808$	0	0
$809$	7.20163	0.253196	0.126598	−	0.991954i	$-0.459594\pi$
$809$	7.20163	0.253196	0.126598	+	0.991954i	$0.459594\pi$
$810$	0	0
$811$	45.3607	1.59283	0.796414	−	0.604751i	$-0.206727\pi$
$811$	45.3607	1.59283	0.796414	+	0.604751i	$0.206727\pi$
$812$	0	0
$813$	−28.9443	−1.01512
$814$	0	0
$815$	0	0
$816$	0	0
$817$	33.3820	1.16789
$818$	0	0
$819$	−17.7082	−0.618775
$820$	0	0
$821$	−13.8541	−0.483511	−0.241756	−	0.970337i	$-0.577723\pi$
$821$	−13.8541	−0.483511	−0.241756	+	0.970337i	$0.577723\pi$
$822$	0	0
$823$	13.5967	0.473953	0.236976	−	0.971515i	$-0.423844\pi$
$823$	13.5967	0.473953	0.236976	+	0.971515i	$0.423844\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.8541	−1.14245	−0.571225	−	0.820794i	$-0.693531\pi$
$827$	−32.8541	−1.14245	−0.571225	+	0.820794i	$0.693531\pi$
$828$	0	0
$829$	−4.43769	−0.154127	−0.0770637	−	0.997026i	$-0.524554\pi$
$829$	−4.43769	−0.154127	−0.0770637	+	0.997026i	$0.524554\pi$
$830$	0	0
$831$	47.3607	1.64292
$832$	0	0
$833$	−0.763932	−0.0264687
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.5279	−0.363896
$838$	0	0
$839$	9.79837	0.338277	0.169139	−	0.985592i	$-0.445901\pi$
$839$	9.79837	0.338277	0.169139	+	0.985592i	$0.445901\pi$
$840$	0	0
$841$	−9.79837	−0.337875
$842$	0	0
$843$	20.4508	0.704365
$844$	0	0
$845$	0	0
$846$	0	0
$847$	24.7984	0.852082
$848$	0	0
$849$	16.2574	0.557951
$850$	0	0
$851$	45.0689	1.54494
$852$	0	0
$853$	5.41641	0.185454	0.0927271	−	0.995692i	$-0.470442\pi$
$853$	5.41641	0.185454	0.0927271	+	0.995692i	$0.470442\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.14590	−0.107462	−0.0537309	−	0.998555i	$-0.517111\pi$
$857$	−3.14590	−0.107462	−0.0537309	+	0.998555i	$0.517111\pi$
$858$	0	0
$859$	−30.5967	−1.04395	−0.521974	−	0.852962i	$-0.674804\pi$
$859$	−30.5967	−1.04395	−0.521974	+	0.852962i	$0.674804\pi$
$860$	0	0
$861$	−33.4164	−1.13883
$862$	0	0
$863$	13.5066	0.459769	0.229885	−	0.973218i	$-0.426165\pi$
$863$	13.5066	0.459769	0.229885	+	0.973218i	$0.426165\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	23.2918	0.791031
$868$	0	0
$869$	−6.58359	−0.223333
$870$	0	0
$871$	−46.3607	−1.57087
$872$	0	0
$873$	−30.5410	−1.03366
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.6525	1.17013	0.585065	−	0.810986i	$-0.301069\pi$
$877$	34.6525	1.17013	0.585065	+	0.810986i	$0.301069\pi$
$878$	0	0
$879$	1.05573	0.0356088
$880$	0	0
$881$	−43.4164	−1.46274	−0.731368	−	0.681983i	$-0.761118\pi$
$881$	−43.4164	−1.46274	−0.731368	+	0.681983i	$0.761118\pi$
$882$	0	0
$883$	−30.4721	−1.02547	−0.512735	−	0.858547i	$-0.671367\pi$
$883$	−30.4721	−1.02547	−0.512735	+	0.858547i	$0.671367\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.11146	0.0708958	0.0354479	−	0.999372i	$-0.488714\pi$
$887$	2.11146	0.0708958	0.0354479	+	0.999372i	$0.488714\pi$
$888$	0	0
$889$	19.7082	0.660992
$890$	0	0
$891$	13.5967	0.455508
$892$	0	0
$893$	1.11146	0.0371935
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−41.3820	−1.38170
$898$	0	0
$899$	20.6312	0.688089
$900$	0	0
$901$	41.5967	1.38579
$902$	0	0
$903$	25.6525	0.853661
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−33.1803	−1.10174	−0.550868	−	0.834593i	$-0.685703\pi$
$907$	−33.1803	−1.10174	−0.550868	+	0.834593i	$0.685703\pi$
$908$	0	0
$909$	−35.8885	−1.19035
$910$	0	0
$911$	−27.3607	−0.906500	−0.453250	−	0.891384i	$-0.649735\pi$
$911$	−27.3607	−0.906500	−0.453250	+	0.891384i	$0.649735\pi$
$912$	0	0
$913$	−5.59675	−0.185225
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.85410	−0.0942508
$918$	0	0
$919$	4.56231	0.150497	0.0752483	−	0.997165i	$-0.476025\pi$
$919$	4.56231	0.150497	0.0752483	+	0.997165i	$0.476025\pi$
$920$	0	0
$921$	−24.0689	−0.793097
$922$	0	0
$923$	29.1459	0.959349
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.29180	0.206650
$928$	0	0
$929$	38.4377	1.26110	0.630550	−	0.776149i	$-0.282829\pi$
$929$	38.4377	1.26110	0.630550	+	0.776149i	$0.282829\pi$
$930$	0	0
$931$	1.11146	0.0364265
$932$	0	0
$933$	10.9787	0.359427
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.5623	0.671741	0.335871	−	0.941908i	$-0.390970\pi$
$937$	20.5623	0.671741	0.335871	+	0.941908i	$0.390970\pi$
$938$	0	0
$939$	12.7639	0.416535
$940$	0	0
$941$	−47.9443	−1.56294	−0.781469	−	0.623944i	$-0.785529\pi$
$941$	−47.9443	−1.56294	−0.781469	+	0.623944i	$0.785529\pi$
$942$	0	0
$943$	−31.2361	−1.01719
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.0132	1.78769	0.893844	−	0.448379i	$-0.147998\pi$
$947$	55.0132	1.78769	0.893844	+	0.448379i	$0.147998\pi$
$948$	0	0
$949$	37.2016	1.20762
$950$	0	0
$951$	−31.8328	−1.03225
$952$	0	0
$953$	−42.2148	−1.36747	−0.683735	−	0.729730i	$-0.739646\pi$
$953$	−42.2148	−1.36747	−0.683735	+	0.729730i	$0.739646\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12.1115	−0.391508
$958$	0	0
$959$	−13.0902	−0.422704
$960$	0	0
$961$	−8.83282	−0.284930
$962$	0	0
$963$	−33.4164	−1.07683
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.472136	−0.0151829	−0.00759143	−	0.999971i	$-0.502416\pi$
$967$	−0.472136	−0.0151829	−0.00759143	+	0.999971i	$0.502416\pi$
$968$	0	0
$969$	−89.1935	−2.86531
$970$	0	0
$971$	−27.4377	−0.880518	−0.440259	−	0.897871i	$-0.645113\pi$
$971$	−27.4377	−0.880518	−0.440259	+	0.897871i	$0.645113\pi$
$972$	0	0
$973$	25.5623	0.819490
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.7082	−0.598528	−0.299264	−	0.954170i	$-0.596741\pi$
$977$	−18.7082	−0.598528	−0.299264	+	0.954170i	$0.596741\pi$
$978$	0	0
$979$	−4.94427	−0.158020
$980$	0	0
$981$	5.88854	0.188007
$982$	0	0
$983$	−3.79837	−0.121149	−0.0605747	−	0.998164i	$-0.519293\pi$
$983$	−3.79837	−0.121149	−0.0605747	+	0.998164i	$0.519293\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.854102	0.0271864
$988$	0	0
$989$	23.9787	0.762479
$990$	0	0
$991$	51.1803	1.62580	0.812899	−	0.582405i	$-0.197888\pi$
$991$	51.1803	1.62580	0.812899	+	0.582405i	$0.197888\pi$
$992$	0	0
$993$	−71.7082	−2.27559
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−38.5279	−1.22019	−0.610095	−	0.792328i	$-0.708869\pi$
$997$	−38.5279	−1.22019	−0.610095	+	0.792328i	$0.708869\pi$
$998$	0	0
$999$	18.4164	0.582669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.g.1.2		2
4.3	odd	2		5000.2.a.c.1.1		2
5.4	even	2		10000.2.a.i.1.1		2
20.19	odd	2		5000.2.a.a.1.2		2
25.11	even	5		400.2.u.a.321.1		4
25.16	even	5		400.2.u.a.81.1		4
100.11	odd	10		200.2.m.a.121.1	yes	4
100.23	even	20		1000.2.q.a.649.1		8
100.27	even	20		1000.2.q.a.649.2		8
100.39	odd	10		1000.2.m.a.601.1		4
100.59	odd	10		1000.2.m.a.401.1		4
100.63	even	20		1000.2.q.a.849.2		8
100.87	even	20		1000.2.q.a.849.1		8
100.91	odd	10		200.2.m.a.81.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.m.a.81.1	✓	4	100.91	odd	10
200.2.m.a.121.1	yes	4	100.11	odd	10
400.2.u.a.81.1		4	25.16	even	5
400.2.u.a.321.1		4	25.11	even	5
1000.2.m.a.401.1		4	100.59	odd	10
1000.2.m.a.601.1		4	100.39	odd	10
1000.2.q.a.649.1		8	100.23	even	20
1000.2.q.a.649.2		8	100.27	even	20
1000.2.q.a.849.1		8	100.87	even	20
1000.2.q.a.849.2		8	100.63	even	20
5000.2.a.a.1.2		2	20.19	odd	2
5000.2.a.c.1.1		2	4.3	odd	2
10000.2.a.g.1.2		2	1.1	even	1	trivial
10000.2.a.i.1.1		2	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)

\(f(q)\)	\(=\)	\(q+2.23607 q^{3} -2.61803 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{3} -2.61803 q^{7} +2.00000 q^{9} -1.23607 q^{11} +3.38197 q^{13} +5.23607 q^{17} -7.61803 q^{19} -5.85410 q^{21} -5.47214 q^{23} -2.23607 q^{27} +4.38197 q^{29} +4.70820 q^{31} -2.76393 q^{33} -8.23607 q^{37} +7.56231 q^{39} +5.70820 q^{41} -4.38197 q^{43} -0.145898 q^{47} -0.145898 q^{49} +11.7082 q^{51} +7.94427 q^{53} -17.0344 q^{57} -2.38197 q^{59} -13.1803 q^{61} -5.23607 q^{63} -13.7082 q^{67} -12.2361 q^{69} +8.61803 q^{71} +11.0000 q^{73} +3.23607 q^{77} +5.32624 q^{79} -11.0000 q^{81} +4.52786 q^{83} +9.79837 q^{87} +4.00000 q^{89} -8.85410 q^{91} +10.5279 q^{93} -15.2705 q^{97} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q - 3 * q^7 + 4 * q^9	\( 2 q - 3 q^{7} + 4 q^{9} + 2 q^{11} + 9 q^{13} + 6 q^{17} - 13 q^{19} - 5 q^{21} - 2 q^{23} + 11 q^{29} - 4 q^{31} - 10 q^{33} - 12 q^{37} - 5 q^{39} - 2 q^{41} - 11 q^{43} - 7 q^{47} - 7 q^{49} + 10 q^{51} - 2 q^{53} - 5 q^{57} - 7 q^{59} - 4 q^{61} - 6 q^{63} - 14 q^{67} - 20 q^{69} + 15 q^{71} + 22 q^{73} + 2 q^{77} - 5 q^{79} - 22 q^{81} + 18 q^{83} - 5 q^{87} + 8 q^{89} - 11 q^{91} + 30 q^{93} + 3 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 3 * q^7 + 4 * q^9 + 2 * q^11 + 9 * q^13 + 6 * q^17 - 13 * q^19 - 5 * q^21 - 2 * q^23 + 11 * q^29 - 4 * q^31 - 10 * q^33 - 12 * q^37 - 5 * q^39 - 2 * q^41 - 11 * q^43 - 7 * q^47 - 7 * q^49 + 10 * q^51 - 2 * q^53 - 5 * q^57 - 7 * q^59 - 4 * q^61 - 6 * q^63 - 14 * q^67 - 20 * q^69 + 15 * q^71 + 22 * q^73 + 2 * q^77 - 5 * q^79 - 22 * q^81 + 18 * q^83 - 5 * q^87 + 8 * q^89 - 11 * q^91 + 30 * q^93 + 3 * q^97 + 4 * q^99

Introduction

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