LMFDB - Embedded newform 10000.2.a.n.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:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+2.61803 q^{3} +3.00000 q^{7} +3.85410 q^{9} +O(q^{10})$	$q+2.61803 q^{3} +3.00000 q^{7} +3.85410 q^{9} +0.236068 q^{11} +1.00000 q^{13} +7.85410 q^{17} +0.854102 q^{19} +7.85410 q^{21} +6.23607 q^{23} +2.23607 q^{27} -0.527864 q^{29} -4.23607 q^{31} +0.618034 q^{33} +7.32624 q^{37} +2.61803 q^{39} -7.47214 q^{41} +1.76393 q^{43} -5.94427 q^{47} +2.00000 q^{49} +20.5623 q^{51} +1.52786 q^{53} +2.23607 q^{57} -4.47214 q^{59} -2.14590 q^{61} +11.5623 q^{63} +5.76393 q^{67} +16.3262 q^{69} +3.00000 q^{71} -5.70820 q^{73} +0.708204 q^{77} +2.76393 q^{79} -5.70820 q^{81} +13.4721 q^{83} -1.38197 q^{87} -4.47214 q^{89} +3.00000 q^{91} -11.0902 q^{93} -10.5623 q^{97} +0.909830 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 6 q^{7} + q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 6 * q^7 + q^9	$ 2 q + 3 q^{3} + 6 q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + 9 q^{17} - 5 q^{19} + 9 q^{21} + 8 q^{23} - 10 q^{29} - 4 q^{31} - q^{33} - q^{37} + 3 q^{39} - 6 q^{41} + 8 q^{43} + 6 q^{47} + 4 q^{49} + 21 q^{51} + 12 q^{53} - 11 q^{61} + 3 q^{63} + 16 q^{67} + 17 q^{69} + 6 q^{71} + 2 q^{73} - 12 q^{77} + 10 q^{79} + 2 q^{81} + 18 q^{83} - 5 q^{87} + 6 q^{91} - 11 q^{93} - q^{97} + 13 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 6 * q^7 + q^9 - 4 * q^11 + 2 * q^13 + 9 * q^17 - 5 * q^19 + 9 * q^21 + 8 * q^23 - 10 * q^29 - 4 * q^31 - q^33 - q^37 + 3 * q^39 - 6 * q^41 + 8 * q^43 + 6 * q^47 + 4 * q^49 + 21 * q^51 + 12 * q^53 - 11 * q^61 + 3 * q^63 + 16 * q^67 + 17 * q^69 + 6 * q^71 + 2 * q^73 - 12 * q^77 + 10 * q^79 + 2 * q^81 + 18 * q^83 - 5 * q^87 + 6 * q^91 - 11 * q^93 - q^97 + 13 * 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
$3$	2.61803	1.51152	0.755761	−	0.654847i	$-0.227267\pi$
$3$	2.61803	1.51152	0.755761	+	0.654847i	$0.227267\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	0.236068	0.0711772	0.0355886	−	0.999367i	$-0.488669\pi$
$11$	0.236068	0.0711772	0.0355886	+	0.999367i	$0.488669\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.85410	1.90490	0.952450	−	0.304696i	$-0.0985548\pi$
$17$	7.85410	1.90490	0.952450	+	0.304696i	$0.0985548\pi$
$18$	0	0
$19$	0.854102	0.195944	0.0979722	−	0.995189i	$-0.468764\pi$
$19$	0.854102	0.195944	0.0979722	+	0.995189i	$0.468764\pi$
$20$	0	0
$21$	7.85410	1.71391
$22$	0	0
$23$	6.23607	1.30031	0.650155	−	0.759802i	$-0.274704\pi$
$23$	6.23607	1.30031	0.650155	+	0.759802i	$0.274704\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−0.527864	−0.0980219	−0.0490109	−	0.998798i	$-0.515607\pi$
$29$	−0.527864	−0.0980219	−0.0490109	+	0.998798i	$0.515607\pi$
$30$	0	0
$31$	−4.23607	−0.760820	−0.380410	−	0.924818i	$-0.624217\pi$
$31$	−4.23607	−0.760820	−0.380410	+	0.924818i	$0.624217\pi$
$32$	0	0
$33$	0.618034	0.107586
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.32624	1.20443	0.602213	−	0.798335i	$-0.294286\pi$
$37$	7.32624	1.20443	0.602213	+	0.798335i	$0.294286\pi$
$38$	0	0
$39$	2.61803	0.419221
$40$	0	0
$41$	−7.47214	−1.16695	−0.583476	−	0.812131i	$-0.698308\pi$
$41$	−7.47214	−1.16695	−0.583476	+	0.812131i	$0.698308\pi$
$42$	0	0
$43$	1.76393	0.268997	0.134499	−	0.990914i	$-0.457058\pi$
$43$	1.76393	0.268997	0.134499	+	0.990914i	$0.457058\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.94427	−0.867061	−0.433531	−	0.901139i	$-0.642732\pi$
$47$	−5.94427	−0.867061	−0.433531	+	0.901139i	$0.642732\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	20.5623	2.87930
$52$	0	0
$53$	1.52786	0.209868	0.104934	−	0.994479i	$-0.466537\pi$
$53$	1.52786	0.209868	0.104934	+	0.994479i	$0.466537\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.23607	0.296174
$58$	0	0
$59$	−4.47214	−0.582223	−0.291111	−	0.956689i	$-0.594025\pi$
$59$	−4.47214	−0.582223	−0.291111	+	0.956689i	$0.594025\pi$
$60$	0	0
$61$	−2.14590	−0.274754	−0.137377	−	0.990519i	$-0.543867\pi$
$61$	−2.14590	−0.274754	−0.137377	+	0.990519i	$0.543867\pi$
$62$	0	0
$63$	11.5623	1.45671
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.76393	0.704176	0.352088	−	0.935967i	$-0.385472\pi$
$67$	5.76393	0.704176	0.352088	+	0.935967i	$0.385472\pi$
$68$	0	0
$69$	16.3262	1.96545
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−5.70820	−0.668095	−0.334047	−	0.942556i	$-0.608415\pi$
$73$	−5.70820	−0.668095	−0.334047	+	0.942556i	$0.608415\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.708204	0.0807073
$78$	0	0
$79$	2.76393	0.310967	0.155483	−	0.987839i	$-0.450307\pi$
$79$	2.76393	0.310967	0.155483	+	0.987839i	$0.450307\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	13.4721	1.47876	0.739380	−	0.673289i	$-0.235119\pi$
$83$	13.4721	1.47876	0.739380	+	0.673289i	$0.235119\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.38197	−0.148162
$88$	0	0
$89$	−4.47214	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	−11.0902	−1.15000
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.5623	−1.07244	−0.536220	−	0.844078i	$-0.680148\pi$
$97$	−10.5623	−1.07244	−0.536220	+	0.844078i	$0.680148\pi$
$98$	0	0
$99$	0.909830	0.0914414
$100$	0	0
$101$	−1.61803	−0.161000	−0.0805002	−	0.996755i	$-0.525652\pi$
$101$	−1.61803	−0.161000	−0.0805002	+	0.996755i	$0.525652\pi$
$102$	0	0
$103$	19.8541	1.95628	0.978141	−	0.207941i	$-0.0666762\pi$
$103$	19.8541	1.95628	0.978141	+	0.207941i	$0.0666762\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.0902	−0.975454	−0.487727	−	0.872996i	$-0.662174\pi$
$107$	−10.0902	−0.975454	−0.487727	+	0.872996i	$0.662174\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	19.1803	1.82052
$112$	0	0
$113$	8.23607	0.774784	0.387392	−	0.921915i	$-0.373376\pi$
$113$	8.23607	0.774784	0.387392	+	0.921915i	$0.373376\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.85410	0.356312
$118$	0	0
$119$	23.5623	2.15995
$120$	0	0
$121$	−10.9443	−0.994934
$122$	0	0
$123$	−19.5623	−1.76387
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.4721	1.10672	0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721	1.10672	0.553362	+	0.832941i	$0.313345\pi$
$128$	0	0
$129$	4.61803	0.406595
$130$	0	0
$131$	−10.4164	−0.910086	−0.455043	−	0.890470i	$-0.650376\pi$
$131$	−10.4164	−0.910086	−0.455043	+	0.890470i	$0.650376\pi$
$132$	0	0
$133$	2.56231	0.222180
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.09017	−0.520318	−0.260159	−	0.965566i	$-0.583775\pi$
$137$	−6.09017	−0.520318	−0.260159	+	0.965566i	$0.583775\pi$
$138$	0	0
$139$	−10.3262	−0.875860	−0.437930	−	0.899009i	$-0.644288\pi$
$139$	−10.3262	−0.875860	−0.437930	+	0.899009i	$0.644288\pi$
$140$	0	0
$141$	−15.5623	−1.31058
$142$	0	0
$143$	0.236068	0.0197410
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.23607	0.431864
$148$	0	0
$149$	−2.23607	−0.183186	−0.0915929	−	0.995797i	$-0.529196\pi$
$149$	−2.23607	−0.183186	−0.0915929	+	0.995797i	$0.529196\pi$
$150$	0	0
$151$	−3.70820	−0.301769	−0.150885	−	0.988551i	$-0.548212\pi$
$151$	−3.70820	−0.301769	−0.150885	+	0.988551i	$0.548212\pi$
$152$	0	0
$153$	30.2705	2.44723
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−15.5623	−1.24201	−0.621004	−	0.783808i	$-0.713275\pi$
$157$	−15.5623	−1.24201	−0.621004	+	0.783808i	$0.713275\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	18.7082	1.47441
$162$	0	0
$163$	4.85410	0.380203	0.190101	−	0.981764i	$-0.439118\pi$
$163$	4.85410	0.380203	0.190101	+	0.981764i	$0.439118\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.6180	−1.20856	−0.604280	−	0.796772i	$-0.706539\pi$
$167$	−15.6180	−1.20856	−0.604280	+	0.796772i	$0.706539\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	3.29180	0.251730
$172$	0	0
$173$	3.23607	0.246034	0.123017	−	0.992405i	$-0.460743\pi$
$173$	3.23607	0.246034	0.123017	+	0.992405i	$0.460743\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.7082	−0.880042
$178$	0	0
$179$	11.1803	0.835658	0.417829	−	0.908526i	$-0.362791\pi$
$179$	11.1803	0.835658	0.417829	+	0.908526i	$0.362791\pi$
$180$	0	0
$181$	−21.4164	−1.59187	−0.795935	−	0.605383i	$-0.793020\pi$
$181$	−21.4164	−1.59187	−0.795935	+	0.605383i	$0.793020\pi$
$182$	0	0
$183$	−5.61803	−0.415297
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.85410	0.135585
$188$	0	0
$189$	6.70820	0.487950
$190$	0	0
$191$	−17.6525	−1.27729	−0.638644	−	0.769502i	$-0.720504\pi$
$191$	−17.6525	−1.27729	−0.638644	+	0.769502i	$0.720504\pi$
$192$	0	0
$193$	16.6525	1.19867	0.599336	−	0.800498i	$-0.295431\pi$
$193$	16.6525	1.19867	0.599336	+	0.800498i	$0.295431\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	17.5623	1.24496	0.622479	−	0.782636i	$-0.286125\pi$
$199$	17.5623	1.24496	0.622479	+	0.782636i	$0.286125\pi$
$200$	0	0
$201$	15.0902	1.06438
$202$	0	0
$203$	−1.58359	−0.111146
$204$	0	0
$205$	0	0
$206$	0	0
$207$	24.0344	1.67051
$208$	0	0
$209$	0.201626	0.0139468
$210$	0	0
$211$	−11.2705	−0.775894	−0.387947	−	0.921682i	$-0.626816\pi$
$211$	−11.2705	−0.775894	−0.387947	+	0.921682i	$0.626816\pi$
$212$	0	0
$213$	7.85410	0.538154
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.7082	−0.862689
$218$	0	0
$219$	−14.9443	−1.00984
$220$	0	0
$221$	7.85410	0.528324
$222$	0	0
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.3607	1.01952	0.509762	−	0.860315i	$-0.329733\pi$
$227$	15.3607	1.01952	0.509762	+	0.860315i	$0.329733\pi$
$228$	0	0
$229$	17.5623	1.16055	0.580275	−	0.814421i	$-0.302945\pi$
$229$	17.5623	1.16055	0.580275	+	0.814421i	$0.302945\pi$
$230$	0	0
$231$	1.85410	0.121991
$232$	0	0
$233$	27.7082	1.81522	0.907612	−	0.419809i	$-0.137903\pi$
$233$	27.7082	1.81522	0.907612	+	0.419809i	$0.137903\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.23607	0.470033
$238$	0	0
$239$	18.0902	1.17016	0.585078	−	0.810977i	$-0.301064\pi$
$239$	18.0902	1.17016	0.585078	+	0.810977i	$0.301064\pi$
$240$	0	0
$241$	13.3820	0.862008	0.431004	−	0.902350i	$-0.358159\pi$
$241$	13.3820	0.862008	0.431004	+	0.902350i	$0.358159\pi$
$242$	0	0
$243$	−21.6525	−1.38901
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.854102	0.0543452
$248$	0	0
$249$	35.2705	2.23518
$250$	0	0
$251$	−23.1803	−1.46313	−0.731565	−	0.681772i	$-0.761210\pi$
$251$	−23.1803	−1.46313	−0.731565	+	0.681772i	$0.761210\pi$
$252$	0	0
$253$	1.47214	0.0925524
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.7426	−1.04438	−0.522189	−	0.852830i	$-0.674884\pi$
$257$	−16.7426	−1.04438	−0.522189	+	0.852830i	$0.674884\pi$
$258$	0	0
$259$	21.9787	1.36569
$260$	0	0
$261$	−2.03444	−0.125929
$262$	0	0
$263$	−2.38197	−0.146878	−0.0734392	−	0.997300i	$-0.523397\pi$
$263$	−2.38197	−0.146878	−0.0734392	+	0.997300i	$0.523397\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.7082	−0.716530
$268$	0	0
$269$	13.0902	0.798122	0.399061	−	0.916924i	$-0.369336\pi$
$269$	13.0902	0.798122	0.399061	+	0.916924i	$0.369336\pi$
$270$	0	0
$271$	9.50658	0.577483	0.288742	−	0.957407i	$-0.406763\pi$
$271$	9.50658	0.577483	0.288742	+	0.957407i	$0.406763\pi$
$272$	0	0
$273$	7.85410	0.475352
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.7082	0.823646	0.411823	−	0.911264i	$-0.364892\pi$
$277$	13.7082	0.823646	0.411823	+	0.911264i	$0.364892\pi$
$278$	0	0
$279$	−16.3262	−0.977426
$280$	0	0
$281$	−19.1803	−1.14420	−0.572102	−	0.820183i	$-0.693872\pi$
$281$	−19.1803	−1.14420	−0.572102	+	0.820183i	$0.693872\pi$
$282$	0	0
$283$	25.7082	1.52819	0.764097	−	0.645101i	$-0.223185\pi$
$283$	25.7082	1.52819	0.764097	+	0.645101i	$0.223185\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−22.4164	−1.32320
$288$	0	0
$289$	44.6869	2.62864
$290$	0	0
$291$	−27.6525	−1.62102
$292$	0	0
$293$	−11.5623	−0.675477	−0.337739	−	0.941240i	$-0.609662\pi$
$293$	−11.5623	−0.675477	−0.337739	+	0.941240i	$0.609662\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.527864	0.0306298
$298$	0	0
$299$	6.23607	0.360641
$300$	0	0
$301$	5.29180	0.305014
$302$	0	0
$303$	−4.23607	−0.243356
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−17.1246	−0.977353	−0.488677	−	0.872465i	$-0.662520\pi$
$307$	−17.1246	−0.977353	−0.488677	+	0.872465i	$0.662520\pi$
$308$	0	0
$309$	51.9787	2.95697
$310$	0	0
$311$	21.0902	1.19591	0.597957	−	0.801528i	$-0.295979\pi$
$311$	21.0902	1.19591	0.597957	+	0.801528i	$0.295979\pi$
$312$	0	0
$313$	3.56231	0.201353	0.100677	−	0.994919i	$-0.467899\pi$
$313$	3.56231	0.201353	0.100677	+	0.994919i	$0.467899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.6180	0.596368	0.298184	−	0.954508i	$-0.403619\pi$
$317$	10.6180	0.596368	0.298184	+	0.954508i	$0.403619\pi$
$318$	0	0
$319$	−0.124612	−0.00697692
$320$	0	0
$321$	−26.4164	−1.47442
$322$	0	0
$323$	6.70820	0.373254
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−39.2705	−2.17166
$328$	0	0
$329$	−17.8328	−0.983155
$330$	0	0
$331$	−4.56231	−0.250767	−0.125384	−	0.992108i	$-0.540016\pi$
$331$	−4.56231	−0.250767	−0.125384	+	0.992108i	$0.540016\pi$
$332$	0	0
$333$	28.2361	1.54733
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−31.4164	−1.71136	−0.855680	−	0.517505i	$-0.826861\pi$
$337$	−31.4164	−1.71136	−0.855680	+	0.517505i	$0.826861\pi$
$338$	0	0
$339$	21.5623	1.17110
$340$	0	0
$341$	−1.00000	−0.0541530
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.90983	0.531988	0.265994	−	0.963975i	$-0.414300\pi$
$347$	9.90983	0.531988	0.265994	+	0.963975i	$0.414300\pi$
$348$	0	0
$349$	−22.2361	−1.19027	−0.595135	−	0.803626i	$-0.702901\pi$
$349$	−22.2361	−1.19027	−0.595135	+	0.803626i	$0.702901\pi$
$350$	0	0
$351$	2.23607	0.119352
$352$	0	0
$353$	−16.7639	−0.892254	−0.446127	−	0.894970i	$-0.647197\pi$
$353$	−16.7639	−0.892254	−0.446127	+	0.894970i	$0.647197\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	61.6869	3.26482
$358$	0	0
$359$	−3.09017	−0.163093	−0.0815465	−	0.996670i	$-0.525986\pi$
$359$	−3.09017	−0.163093	−0.0815465	+	0.996670i	$0.525986\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	0	0
$363$	−28.6525	−1.50386
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.5410	1.64643	0.823214	−	0.567731i	$-0.192179\pi$
$367$	31.5410	1.64643	0.823214	+	0.567731i	$0.192179\pi$
$368$	0	0
$369$	−28.7984	−1.49918
$370$	0	0
$371$	4.58359	0.237968
$372$	0	0
$373$	−16.2361	−0.840672	−0.420336	−	0.907369i	$-0.638088\pi$
$373$	−16.2361	−0.840672	−0.420336	+	0.907369i	$0.638088\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.527864	−0.0271864
$378$	0	0
$379$	−0.527864	−0.0271146	−0.0135573	−	0.999908i	$-0.504316\pi$
$379$	−0.527864	−0.0271146	−0.0135573	+	0.999908i	$0.504316\pi$
$380$	0	0
$381$	32.6525	1.67284
$382$	0	0
$383$	4.32624	0.221060	0.110530	−	0.993873i	$-0.464745\pi$
$383$	4.32624	0.221060	0.110530	+	0.993873i	$0.464745\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.79837	0.345581
$388$	0	0
$389$	−33.2148	−1.68406	−0.842028	−	0.539434i	$-0.818638\pi$
$389$	−33.2148	−1.68406	−0.842028	+	0.539434i	$0.818638\pi$
$390$	0	0
$391$	48.9787	2.47696
$392$	0	0
$393$	−27.2705	−1.37562
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.76393	−0.289283	−0.144642	−	0.989484i	$-0.546203\pi$
$397$	−5.76393	−0.289283	−0.144642	+	0.989484i	$0.546203\pi$
$398$	0	0
$399$	6.70820	0.335830
$400$	0	0
$401$	8.18034	0.408507	0.204253	−	0.978918i	$-0.434523\pi$
$401$	8.18034	0.408507	0.204253	+	0.978918i	$0.434523\pi$
$402$	0	0
$403$	−4.23607	−0.211014
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.72949	0.0857276
$408$	0	0
$409$	6.70820	0.331699	0.165850	−	0.986151i	$-0.446963\pi$
$409$	6.70820	0.331699	0.165850	+	0.986151i	$0.446963\pi$
$410$	0	0
$411$	−15.9443	−0.786473
$412$	0	0
$413$	−13.4164	−0.660178
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−27.0344	−1.32388
$418$	0	0
$419$	−27.2361	−1.33057	−0.665284	−	0.746590i	$-0.731690\pi$
$419$	−27.2361	−1.33057	−0.665284	+	0.746590i	$0.731690\pi$
$420$	0	0
$421$	−12.2705	−0.598028	−0.299014	−	0.954249i	$-0.596658\pi$
$421$	−12.2705	−0.598028	−0.299014	+	0.954249i	$0.596658\pi$
$422$	0	0
$423$	−22.9098	−1.11391
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.43769	−0.311542
$428$	0	0
$429$	0.618034	0.0298390
$430$	0	0
$431$	32.0689	1.54470	0.772352	−	0.635195i	$-0.219080\pi$
$431$	32.0689	1.54470	0.772352	+	0.635195i	$0.219080\pi$
$432$	0	0
$433$	23.3607	1.12264	0.561321	−	0.827598i	$-0.310293\pi$
$433$	23.3607	1.12264	0.561321	+	0.827598i	$0.310293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.32624	0.254789
$438$	0	0
$439$	5.65248	0.269778	0.134889	−	0.990861i	$-0.456932\pi$
$439$	5.65248	0.269778	0.134889	+	0.990861i	$0.456932\pi$
$440$	0	0
$441$	7.70820	0.367057
$442$	0	0
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.85410	−0.276890
$448$	0	0
$449$	13.9443	0.658071	0.329035	−	0.944318i	$-0.393276\pi$
$449$	13.9443	0.658071	0.329035	+	0.944318i	$0.393276\pi$
$450$	0	0
$451$	−1.76393	−0.0830603
$452$	0	0
$453$	−9.70820	−0.456131
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.2148	−0.524605	−0.262303	−	0.964986i	$-0.584482\pi$
$457$	−11.2148	−0.524605	−0.262303	+	0.964986i	$0.584482\pi$
$458$	0	0
$459$	17.5623	0.819738
$460$	0	0
$461$	28.1803	1.31249	0.656245	−	0.754548i	$-0.272144\pi$
$461$	28.1803	1.31249	0.656245	+	0.754548i	$0.272144\pi$
$462$	0	0
$463$	−11.9787	−0.556698	−0.278349	−	0.960480i	$-0.589787\pi$
$463$	−11.9787	−0.556698	−0.278349	+	0.960480i	$0.589787\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.18034	−0.147169	−0.0735843	−	0.997289i	$-0.523444\pi$
$467$	−3.18034	−0.147169	−0.0735843	+	0.997289i	$0.523444\pi$
$468$	0	0
$469$	17.2918	0.798461
$470$	0	0
$471$	−40.7426	−1.87732
$472$	0	0
$473$	0.416408	0.0191465
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.88854	0.269618
$478$	0	0
$479$	39.7984	1.81843	0.909217	−	0.416322i	$-0.136681\pi$
$479$	39.7984	1.81843	0.909217	+	0.416322i	$0.136681\pi$
$480$	0	0
$481$	7.32624	0.334048
$482$	0	0
$483$	48.9787	2.22861
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.1246	0.821305	0.410652	−	0.911792i	$-0.365301\pi$
$487$	18.1246	0.821305	0.410652	+	0.911792i	$0.365301\pi$
$488$	0	0
$489$	12.7082	0.574685
$490$	0	0
$491$	−14.7639	−0.666287	−0.333143	−	0.942876i	$-0.608109\pi$
$491$	−14.7639	−0.666287	−0.333143	+	0.942876i	$0.608109\pi$
$492$	0	0
$493$	−4.14590	−0.186722
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.00000	0.403705
$498$	0	0
$499$	−40.8541	−1.82888	−0.914440	−	0.404721i	$-0.867369\pi$
$499$	−40.8541	−1.82888	−0.914440	+	0.404721i	$0.867369\pi$
$500$	0	0
$501$	−40.8885	−1.82677
$502$	0	0
$503$	−7.50658	−0.334702	−0.167351	−	0.985897i	$-0.553521\pi$
$503$	−7.50658	−0.334702	−0.167351	+	0.985897i	$0.553521\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.4164	−1.39525
$508$	0	0
$509$	−13.0902	−0.580212	−0.290106	−	0.956995i	$-0.593690\pi$
$509$	−13.0902	−0.580212	−0.290106	+	0.956995i	$0.593690\pi$
$510$	0	0
$511$	−17.1246	−0.757548
$512$	0	0
$513$	1.90983	0.0843211
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.40325	−0.0617150
$518$	0	0
$519$	8.47214	0.371885
$520$	0	0
$521$	32.4508	1.42170	0.710849	−	0.703345i	$-0.248311\pi$
$521$	32.4508	1.42170	0.710849	+	0.703345i	$0.248311\pi$
$522$	0	0
$523$	−26.6525	−1.16543	−0.582716	−	0.812676i	$-0.698010\pi$
$523$	−26.6525	−1.16543	−0.582716	+	0.812676i	$0.698010\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.2705	−1.44929
$528$	0	0
$529$	15.8885	0.690806
$530$	0	0
$531$	−17.2361	−0.747982
$532$	0	0
$533$	−7.47214	−0.323654
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.2705	1.26312
$538$	0	0
$539$	0.472136	0.0203363
$540$	0	0
$541$	28.7082	1.23426	0.617131	−	0.786860i	$-0.288295\pi$
$541$	28.7082	1.23426	0.617131	+	0.786860i	$0.288295\pi$
$542$	0	0
$543$	−56.0689	−2.40615
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−10.6180	−0.453994	−0.226997	−	0.973895i	$-0.572891\pi$
$547$	−10.6180	−0.453994	−0.226997	+	0.973895i	$0.572891\pi$
$548$	0	0
$549$	−8.27051	−0.352977
$550$	0	0
$551$	−0.450850	−0.0192068
$552$	0	0
$553$	8.29180	0.352603
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.8885	−0.461362	−0.230681	−	0.973029i	$-0.574095\pi$
$557$	−10.8885	−0.461362	−0.230681	+	0.973029i	$0.574095\pi$
$558$	0	0
$559$	1.76393	0.0746064
$560$	0	0
$561$	4.85410	0.204940
$562$	0	0
$563$	34.8541	1.46893	0.734463	−	0.678649i	$-0.237434\pi$
$563$	34.8541	1.46893	0.734463	+	0.678649i	$0.237434\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−17.1246	−0.719166
$568$	0	0
$569$	22.2361	0.932184	0.466092	−	0.884736i	$-0.345661\pi$
$569$	22.2361	0.932184	0.466092	+	0.884736i	$0.345661\pi$
$570$	0	0
$571$	−1.67376	−0.0700448	−0.0350224	−	0.999387i	$-0.511150\pi$
$571$	−1.67376	−0.0700448	−0.0350224	+	0.999387i	$0.511150\pi$
$572$	0	0
$573$	−46.2148	−1.93065
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	43.5967	1.81182
$580$	0	0
$581$	40.4164	1.67676
$582$	0	0
$583$	0.360680	0.0149378
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.8197	0.487850	0.243925	−	0.969794i	$-0.421565\pi$
$587$	11.8197	0.487850	0.243925	+	0.969794i	$0.421565\pi$
$588$	0	0
$589$	−3.61803	−0.149078
$590$	0	0
$591$	31.4164	1.29230
$592$	0	0
$593$	−47.0132	−1.93060	−0.965299	−	0.261145i	$-0.915900\pi$
$593$	−47.0132	−1.93060	−0.965299	+	0.261145i	$0.915900\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	45.9787	1.88178
$598$	0	0
$599$	8.94427	0.365453	0.182727	−	0.983164i	$-0.441508\pi$
$599$	8.94427	0.365453	0.182727	+	0.983164i	$0.441508\pi$
$600$	0	0
$601$	−14.8328	−0.605043	−0.302522	−	0.953143i	$-0.597828\pi$
$601$	−14.8328	−0.605043	−0.302522	+	0.953143i	$0.597828\pi$
$602$	0	0
$603$	22.2148	0.904656
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.1459	1.10182	0.550909	−	0.834565i	$-0.314281\pi$
$607$	27.1459	1.10182	0.550909	+	0.834565i	$0.314281\pi$
$608$	0	0
$609$	−4.14590	−0.168000
$610$	0	0
$611$	−5.94427	−0.240480
$612$	0	0
$613$	−41.5623	−1.67869	−0.839343	−	0.543602i	$-0.817060\pi$
$613$	−41.5623	−1.67869	−0.839343	+	0.543602i	$0.817060\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.3607	−1.22227	−0.611137	−	0.791524i	$-0.709288\pi$
$617$	−30.3607	−1.22227	−0.611137	+	0.791524i	$0.709288\pi$
$618$	0	0
$619$	46.9574	1.88738	0.943689	−	0.330833i	$-0.107330\pi$
$619$	46.9574	1.88738	0.943689	+	0.330833i	$0.107330\pi$
$620$	0	0
$621$	13.9443	0.559564
$622$	0	0
$623$	−13.4164	−0.537517
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.527864	0.0210809
$628$	0	0
$629$	57.5410	2.29431
$630$	0	0
$631$	−8.83282	−0.351629	−0.175814	−	0.984423i	$-0.556256\pi$
$631$	−8.83282	−0.351629	−0.175814	+	0.984423i	$0.556256\pi$
$632$	0	0
$633$	−29.5066	−1.17278
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	11.5623	0.457398
$640$	0	0
$641$	14.3607	0.567213	0.283606	−	0.958941i	$-0.408469\pi$
$641$	14.3607	0.567213	0.283606	+	0.958941i	$0.408469\pi$
$642$	0	0
$643$	36.2361	1.42901	0.714506	−	0.699630i	$-0.246652\pi$
$643$	36.2361	1.42901	0.714506	+	0.699630i	$0.246652\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.65248	−0.300850	−0.150425	−	0.988621i	$-0.548064\pi$
$647$	−7.65248	−0.300850	−0.150425	+	0.988621i	$0.548064\pi$
$648$	0	0
$649$	−1.05573	−0.0414410
$650$	0	0
$651$	−33.2705	−1.30397
$652$	0	0
$653$	40.5967	1.58867	0.794337	−	0.607478i	$-0.207819\pi$
$653$	40.5967	1.58867	0.794337	+	0.607478i	$0.207819\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	−5.65248	−0.220189	−0.110095	−	0.993921i	$-0.535115\pi$
$659$	−5.65248	−0.220189	−0.110095	+	0.993921i	$0.535115\pi$
$660$	0	0
$661$	6.27051	0.243895	0.121947	−	0.992537i	$-0.461086\pi$
$661$	6.27051	0.243895	0.121947	+	0.992537i	$0.461086\pi$
$662$	0	0
$663$	20.5623	0.798574
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.29180	−0.127459
$668$	0	0
$669$	23.5623	0.910971
$670$	0	0
$671$	−0.506578	−0.0195562
$672$	0	0
$673$	−1.23607	−0.0476469	−0.0238235	−	0.999716i	$-0.507584\pi$
$673$	−1.23607	−0.0476469	−0.0238235	+	0.999716i	$0.507584\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.6180	0.408084	0.204042	−	0.978962i	$-0.434592\pi$
$677$	10.6180	0.408084	0.204042	+	0.978962i	$0.434592\pi$
$678$	0	0
$679$	−31.6869	−1.21603
$680$	0	0
$681$	40.2148	1.54103
$682$	0	0
$683$	−33.3607	−1.27651	−0.638255	−	0.769825i	$-0.720344\pi$
$683$	−33.3607	−1.27651	−0.638255	+	0.769825i	$0.720344\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	45.9787	1.75420
$688$	0	0
$689$	1.52786	0.0582070
$690$	0	0
$691$	−1.27051	−0.0483325	−0.0241662	−	0.999708i	$-0.507693\pi$
$691$	−1.27051	−0.0483325	−0.0241662	+	0.999708i	$0.507693\pi$
$692$	0	0
$693$	2.72949	0.103685
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−58.6869	−2.22293
$698$	0	0
$699$	72.5410	2.74375
$700$	0	0
$701$	−4.18034	−0.157889	−0.0789446	−	0.996879i	$-0.525155\pi$
$701$	−4.18034	−0.157889	−0.0789446	+	0.996879i	$0.525155\pi$
$702$	0	0
$703$	6.25735	0.236001
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.85410	−0.182557
$708$	0	0
$709$	7.76393	0.291581	0.145790	−	0.989316i	$-0.453428\pi$
$709$	7.76393	0.291581	0.145790	+	0.989316i	$0.453428\pi$
$710$	0	0
$711$	10.6525	0.399499
$712$	0	0
$713$	−26.4164	−0.989302
$714$	0	0
$715$	0	0
$716$	0	0
$717$	47.3607	1.76872
$718$	0	0
$719$	26.5066	0.988529	0.494264	−	0.869312i	$-0.335438\pi$
$719$	26.5066	0.988529	0.494264	+	0.869312i	$0.335438\pi$
$720$	0	0
$721$	59.5623	2.21822
$722$	0	0
$723$	35.0344	1.30294
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.6525	1.06266	0.531331	−	0.847164i	$-0.321692\pi$
$727$	28.6525	1.06266	0.531331	+	0.847164i	$0.321692\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	13.8541	0.512412
$732$	0	0
$733$	−39.7771	−1.46920	−0.734600	−	0.678500i	$-0.762630\pi$
$733$	−39.7771	−1.46920	−0.734600	+	0.678500i	$0.762630\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.36068	0.0501213
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	2.23607	0.0821440
$742$	0	0
$743$	18.2705	0.670280	0.335140	−	0.942168i	$-0.391216\pi$
$743$	18.2705	0.670280	0.335140	+	0.942168i	$0.391216\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	51.9230	1.89976
$748$	0	0
$749$	−30.2705	−1.10606
$750$	0	0
$751$	−1.14590	−0.0418144	−0.0209072	−	0.999781i	$-0.506655\pi$
$751$	−1.14590	−0.0418144	−0.0209072	+	0.999781i	$0.506655\pi$
$752$	0	0
$753$	−60.6869	−2.21155
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.4164	0.378591	0.189295	−	0.981920i	$-0.439380\pi$
$757$	10.4164	0.378591	0.189295	+	0.981920i	$0.439380\pi$
$758$	0	0
$759$	3.85410	0.139895
$760$	0	0
$761$	−37.9230	−1.37471	−0.687354	−	0.726323i	$-0.741228\pi$
$761$	−37.9230	−1.37471	−0.687354	+	0.726323i	$0.741228\pi$
$762$	0	0
$763$	−45.0000	−1.62911
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.47214	−0.161479
$768$	0	0
$769$	−23.4164	−0.844417	−0.422209	−	0.906499i	$-0.638745\pi$
$769$	−23.4164	−0.844417	−0.422209	+	0.906499i	$0.638745\pi$
$770$	0	0
$771$	−43.8328	−1.57860
$772$	0	0
$773$	6.20163	0.223057	0.111528	−	0.993761i	$-0.464425\pi$
$773$	6.20163	0.223057	0.111528	+	0.993761i	$0.464425\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	57.5410	2.06427
$778$	0	0
$779$	−6.38197	−0.228658
$780$	0	0
$781$	0.708204	0.0253415
$782$	0	0
$783$	−1.18034	−0.0421819
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15.4164	−0.549536	−0.274768	−	0.961511i	$-0.588601\pi$
$787$	−15.4164	−0.549536	−0.274768	+	0.961511i	$0.588601\pi$
$788$	0	0
$789$	−6.23607	−0.222010
$790$	0	0
$791$	24.7082	0.878523
$792$	0	0
$793$	−2.14590	−0.0762031
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.38197	0.119795	0.0598977	−	0.998205i	$-0.480923\pi$
$797$	3.38197	0.119795	0.0598977	+	0.998205i	$0.480923\pi$
$798$	0	0
$799$	−46.6869	−1.65166
$800$	0	0
$801$	−17.2361	−0.609007
$802$	0	0
$803$	−1.34752	−0.0475531
$804$	0	0
$805$	0	0
$806$	0	0
$807$	34.2705	1.20638
$808$	0	0
$809$	2.23607	0.0786160	0.0393080	−	0.999227i	$-0.487485\pi$
$809$	2.23607	0.0786160	0.0393080	+	0.999227i	$0.487485\pi$
$810$	0	0
$811$	−23.0557	−0.809596	−0.404798	−	0.914406i	$-0.632658\pi$
$811$	−23.0557	−0.809596	−0.404798	+	0.914406i	$0.632658\pi$
$812$	0	0
$813$	24.8885	0.872879
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.50658	0.0527085
$818$	0	0
$819$	11.5623	0.404020
$820$	0	0
$821$	−30.1591	−1.05256	−0.526279	−	0.850312i	$-0.676413\pi$
$821$	−30.1591	−1.05256	−0.526279	+	0.850312i	$0.676413\pi$
$822$	0	0
$823$	−32.8328	−1.14448	−0.572240	−	0.820086i	$-0.693925\pi$
$823$	−32.8328	−1.14448	−0.572240	+	0.820086i	$0.693925\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.0344	−0.835759	−0.417880	−	0.908502i	$-0.637227\pi$
$827$	−24.0344	−0.835759	−0.417880	+	0.908502i	$0.637227\pi$
$828$	0	0
$829$	26.5836	0.923286	0.461643	−	0.887066i	$-0.347260\pi$
$829$	26.5836	0.923286	0.461643	+	0.887066i	$0.347260\pi$
$830$	0	0
$831$	35.8885	1.24496
$832$	0	0
$833$	15.7082	0.544257
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.47214	−0.327405
$838$	0	0
$839$	−25.2492	−0.871700	−0.435850	−	0.900019i	$-0.643552\pi$
$839$	−25.2492	−0.871700	−0.435850	+	0.900019i	$0.643552\pi$
$840$	0	0
$841$	−28.7214	−0.990392
$842$	0	0
$843$	−50.2148	−1.72949
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−32.8328	−1.12815
$848$	0	0
$849$	67.3050	2.30990
$850$	0	0
$851$	45.6869	1.56613
$852$	0	0
$853$	−24.8541	−0.850988	−0.425494	−	0.904961i	$-0.639900\pi$
$853$	−24.8541	−0.850988	−0.425494	+	0.904961i	$0.639900\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.3394	1.20717	0.603585	−	0.797298i	$-0.293738\pi$
$857$	35.3394	1.20717	0.603585	+	0.797298i	$0.293738\pi$
$858$	0	0
$859$	39.3951	1.34414	0.672072	−	0.740486i	$-0.265404\pi$
$859$	39.3951	1.34414	0.672072	+	0.740486i	$0.265404\pi$
$860$	0	0
$861$	−58.6869	−2.00004
$862$	0	0
$863$	6.76393	0.230247	0.115123	−	0.993351i	$-0.463274\pi$
$863$	6.76393	0.230247	0.115123	+	0.993351i	$0.463274\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	116.992	3.97325
$868$	0	0
$869$	0.652476	0.0221337
$870$	0	0
$871$	5.76393	0.195303
$872$	0	0
$873$	−40.7082	−1.37776
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.5623	0.829410	0.414705	−	0.909956i	$-0.363885\pi$
$877$	24.5623	0.829410	0.414705	+	0.909956i	$0.363885\pi$
$878$	0	0
$879$	−30.2705	−1.02100
$880$	0	0
$881$	15.8197	0.532978	0.266489	−	0.963838i	$-0.414136\pi$
$881$	15.8197	0.532978	0.266489	+	0.963838i	$0.414136\pi$
$882$	0	0
$883$	−18.6869	−0.628865	−0.314432	−	0.949280i	$-0.601814\pi$
$883$	−18.6869	−0.628865	−0.314432	+	0.949280i	$0.601814\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.8197	−0.363289	−0.181644	−	0.983364i	$-0.558142\pi$
$887$	−10.8197	−0.363289	−0.181644	+	0.983364i	$0.558142\pi$
$888$	0	0
$889$	37.4164	1.25491
$890$	0	0
$891$	−1.34752	−0.0451438
$892$	0	0
$893$	−5.07701	−0.169896
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.3262	0.545117
$898$	0	0
$899$	2.23607	0.0745770
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	13.8541	0.461036
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.3820	−0.776385	−0.388193	−	0.921578i	$-0.626901\pi$
$907$	−23.3820	−0.776385	−0.388193	+	0.921578i	$0.626901\pi$
$908$	0	0
$909$	−6.23607	−0.206837
$910$	0	0
$911$	−10.4164	−0.345111	−0.172555	−	0.985000i	$-0.555202\pi$
$911$	−10.4164	−0.345111	−0.172555	+	0.985000i	$0.555202\pi$
$912$	0	0
$913$	3.18034	0.105254
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−31.2492	−1.03194
$918$	0	0
$919$	−25.6525	−0.846197	−0.423099	−	0.906084i	$-0.639058\pi$
$919$	−25.6525	−0.846197	−0.423099	+	0.906084i	$0.639058\pi$
$920$	0	0
$921$	−44.8328	−1.47729
$922$	0	0
$923$	3.00000	0.0987462
$924$	0	0
$925$	0	0
$926$	0	0
$927$	76.5197	2.51324
$928$	0	0
$929$	−17.1115	−0.561409	−0.280704	−	0.959794i	$-0.590568\pi$
$929$	−17.1115	−0.561409	−0.280704	+	0.959794i	$0.590568\pi$
$930$	0	0
$931$	1.70820	0.0559841
$932$	0	0
$933$	55.2148	1.80765
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.9230	−0.748861	−0.374431	−	0.927255i	$-0.622162\pi$
$937$	−22.9230	−0.748861	−0.374431	+	0.927255i	$0.622162\pi$
$938$	0	0
$939$	9.32624	0.304350
$940$	0	0
$941$	−11.6180	−0.378737	−0.189369	−	0.981906i	$-0.560644\pi$
$941$	−11.6180	−0.378737	−0.189369	+	0.981906i	$0.560644\pi$
$942$	0	0
$943$	−46.5967	−1.51740
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.6180	1.35240	0.676202	−	0.736716i	$-0.263625\pi$
$947$	41.6180	1.35240	0.676202	+	0.736716i	$0.263625\pi$
$948$	0	0
$949$	−5.70820	−0.185296
$950$	0	0
$951$	27.7984	0.901424
$952$	0	0
$953$	42.3820	1.37289	0.686443	−	0.727183i	$-0.259171\pi$
$953$	42.3820	1.37289	0.686443	+	0.727183i	$0.259171\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.326238	−0.0105458
$958$	0	0
$959$	−18.2705	−0.589986
$960$	0	0
$961$	−13.0557	−0.421153
$962$	0	0
$963$	−38.8885	−1.25317
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−27.1246	−0.872269	−0.436134	−	0.899882i	$-0.643653\pi$
$967$	−27.1246	−0.872269	−0.436134	+	0.899882i	$0.643653\pi$
$968$	0	0
$969$	17.5623	0.564183
$970$	0	0
$971$	52.0689	1.67097	0.835485	−	0.549513i	$-0.185187\pi$
$971$	52.0689	1.67097	0.835485	+	0.549513i	$0.185187\pi$
$972$	0	0
$973$	−30.9787	−0.993132
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26.0132	−0.832235	−0.416117	−	0.909311i	$-0.636609\pi$
$977$	−26.0132	−0.832235	−0.416117	+	0.909311i	$0.636609\pi$
$978$	0	0
$979$	−1.05573	−0.0337412
$980$	0	0
$981$	−57.8115	−1.84578
$982$	0	0
$983$	46.4853	1.48265	0.741325	−	0.671146i	$-0.234198\pi$
$983$	46.4853	1.48265	0.741325	+	0.671146i	$0.234198\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−46.6869	−1.48606
$988$	0	0
$989$	11.0000	0.349780
$990$	0	0
$991$	−52.4508	−1.66616	−0.833078	−	0.553155i	$-0.813424\pi$
$991$	−52.4508	−1.66616	−0.833078	+	0.553155i	$0.813424\pi$
$992$	0	0
$993$	−11.9443	−0.379040
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−53.2492	−1.68642	−0.843210	−	0.537584i	$-0.819337\pi$
$997$	−53.2492	−1.68642	−0.843210	+	0.537584i	$0.819337\pi$
$998$	0	0
$999$	16.3820	0.518302

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.n.1.2		2
4.3	odd	2		1250.2.a.a.1.1		2
5.4	even	2		10000.2.a.a.1.1		2
20.3	even	4		1250.2.b.b.1249.3		4
20.7	even	4		1250.2.b.b.1249.2		4
20.19	odd	2		1250.2.a.d.1.2		2
25.6	even	5		400.2.u.c.161.1		4
25.21	even	5		400.2.u.c.241.1		4
100.3	even	20		250.2.e.b.49.2		8
100.19	odd	10		250.2.d.a.51.1		4
100.31	odd	10		50.2.d.a.11.1	✓	4
100.47	even	20		250.2.e.b.49.1		8
100.67	even	20		250.2.e.b.199.2		8
100.71	odd	10		50.2.d.a.41.1	yes	4
100.79	odd	10		250.2.d.a.201.1		4
100.83	even	20		250.2.e.b.199.1		8
300.71	even	10		450.2.h.a.91.1		4
300.131	even	10		450.2.h.a.361.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.d.a.11.1	✓	4	100.31	odd	10
50.2.d.a.41.1	yes	4	100.71	odd	10
250.2.d.a.51.1		4	100.19	odd	10
250.2.d.a.201.1		4	100.79	odd	10
250.2.e.b.49.1		8	100.47	even	20
250.2.e.b.49.2		8	100.3	even	20
250.2.e.b.199.1		8	100.83	even	20
250.2.e.b.199.2		8	100.67	even	20
400.2.u.c.161.1		4	25.6	even	5
400.2.u.c.241.1		4	25.21	even	5
450.2.h.a.91.1		4	300.71	even	10
450.2.h.a.361.1		4	300.131	even	10
1250.2.a.a.1.1		2	4.3	odd	2
1250.2.a.d.1.2		2	20.19	odd	2
1250.2.b.b.1249.2		4	20.7	even	4
1250.2.b.b.1249.3		4	20.3	even	4
10000.2.a.a.1.1		2	5.4	even	2
10000.2.a.n.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{3} +3.00000 q^{7} +3.85410 q^{9} +O(q^{10})\)	\(q+2.61803 q^{3} +3.00000 q^{7} +3.85410 q^{9} +0.236068 q^{11} +1.00000 q^{13} +7.85410 q^{17} +0.854102 q^{19} +7.85410 q^{21} +6.23607 q^{23} +2.23607 q^{27} -0.527864 q^{29} -4.23607 q^{31} +0.618034 q^{33} +7.32624 q^{37} +2.61803 q^{39} -7.47214 q^{41} +1.76393 q^{43} -5.94427 q^{47} +2.00000 q^{49} +20.5623 q^{51} +1.52786 q^{53} +2.23607 q^{57} -4.47214 q^{59} -2.14590 q^{61} +11.5623 q^{63} +5.76393 q^{67} +16.3262 q^{69} +3.00000 q^{71} -5.70820 q^{73} +0.708204 q^{77} +2.76393 q^{79} -5.70820 q^{81} +13.4721 q^{83} -1.38197 q^{87} -4.47214 q^{89} +3.00000 q^{91} -11.0902 q^{93} -10.5623 q^{97} +0.909830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 6 q^{7} + q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 6 * q^7 + q^9	\( 2 q + 3 q^{3} + 6 q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + 9 q^{17} - 5 q^{19} + 9 q^{21} + 8 q^{23} - 10 q^{29} - 4 q^{31} - q^{33} - q^{37} + 3 q^{39} - 6 q^{41} + 8 q^{43} + 6 q^{47} + 4 q^{49} + 21 q^{51} + 12 q^{53} - 11 q^{61} + 3 q^{63} + 16 q^{67} + 17 q^{69} + 6 q^{71} + 2 q^{73} - 12 q^{77} + 10 q^{79} + 2 q^{81} + 18 q^{83} - 5 q^{87} + 6 q^{91} - 11 q^{93} - q^{97} + 13 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 6 * q^7 + q^9 - 4 * q^11 + 2 * q^13 + 9 * q^17 - 5 * q^19 + 9 * q^21 + 8 * q^23 - 10 * q^29 - 4 * q^31 - q^33 - q^37 + 3 * q^39 - 6 * q^41 + 8 * q^43 + 6 * q^47 + 4 * q^49 + 21 * q^51 + 12 * q^53 - 11 * q^61 + 3 * q^63 + 16 * q^67 + 17 * q^69 + 6 * q^71 + 2 * q^73 - 12 * q^77 + 10 * q^79 + 2 * q^81 + 18 * q^83 - 5 * q^87 + 6 * q^91 - 11 * q^93 - q^97 + 13 * q^99

Introduction

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