LMFDB - Embedded newform 2800.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{3} -1.00000 q^{7} +7.47214 q^{9} +O(q^{10})$	$q+3.23607 q^{3} -1.00000 q^{7} +7.47214 q^{9} +0.236068 q^{11} +1.23607 q^{13} +2.47214 q^{17} +4.47214 q^{19} -3.23607 q^{21} -6.23607 q^{23} +14.4721 q^{27} +5.00000 q^{29} -3.70820 q^{31} +0.763932 q^{33} +3.00000 q^{37} +4.00000 q^{39} +4.76393 q^{41} -1.76393 q^{43} +2.00000 q^{47} +1.00000 q^{49} +8.00000 q^{51} +8.47214 q^{53} +14.4721 q^{57} -11.7082 q^{59} -9.70820 q^{61} -7.47214 q^{63} +4.23607 q^{67} -20.1803 q^{69} -8.70820 q^{71} -8.76393 q^{73} -0.236068 q^{77} +11.1803 q^{79} +24.4164 q^{81} +7.70820 q^{83} +16.1803 q^{87} +17.2361 q^{89} -1.23607 q^{91} -12.0000 q^{93} +5.23607 q^{97} +1.76393 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{21} - 8 q^{23} + 20 q^{27} + 10 q^{29} + 6 q^{31} + 6 q^{33} + 6 q^{37} + 8 q^{39} + 14 q^{41} - 8 q^{43} + 4 q^{47} + 2 q^{49} + 16 q^{51} + 8 q^{53} + 20 q^{57} - 10 q^{59} - 6 q^{61} - 6 q^{63} + 4 q^{67} - 18 q^{69} - 4 q^{71} - 22 q^{73} + 4 q^{77} + 22 q^{81} + 2 q^{83} + 10 q^{87} + 30 q^{89} + 2 q^{91} - 24 q^{93} + 6 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^21 - 8 * q^23 + 20 * q^27 + 10 * q^29 + 6 * q^31 + 6 * q^33 + 6 * q^37 + 8 * q^39 + 14 * q^41 - 8 * q^43 + 4 * q^47 + 2 * q^49 + 16 * q^51 + 8 * q^53 + 20 * q^57 - 10 * q^59 - 6 * q^61 - 6 * q^63 + 4 * q^67 - 18 * q^69 - 4 * q^71 - 22 * q^73 + 4 * q^77 + 22 * q^81 + 2 * q^83 + 10 * q^87 + 30 * q^89 + 2 * q^91 - 24 * q^93 + 6 * q^97 + 8 * 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
$3$	3.23607	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	3.23607	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.47214	2.49071
$10$	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
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	−3.23607	−0.706168
$22$	0	0
$23$	−6.23607	−1.30031	−0.650155	−	0.759802i	$-0.725296\pi$
$23$	−6.23607	−1.30031	−0.650155	+	0.759802i	$0.725296\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.4721	2.78516
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−3.70820	−0.666013	−0.333007	−	0.942925i	$-0.608063\pi$
$31$	−3.70820	−0.666013	−0.333007	+	0.942925i	$0.608063\pi$
$32$	0	0
$33$	0.763932	0.132983
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	4.76393	0.744001	0.372001	−	0.928232i	$-0.378672\pi$
$41$	4.76393	0.744001	0.372001	+	0.928232i	$0.378672\pi$
$42$	0	0
$43$	−1.76393	−0.268997	−0.134499	−	0.990914i	$-0.542942\pi$
$43$	−1.76393	−0.268997	−0.134499	+	0.990914i	$0.542942\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.4721	1.91688
$58$	0	0
$59$	−11.7082	−1.52428	−0.762139	−	0.647413i	$-0.775851\pi$
$59$	−11.7082	−1.52428	−0.762139	+	0.647413i	$0.775851\pi$
$60$	0	0
$61$	−9.70820	−1.24301	−0.621504	−	0.783411i	$-0.713478\pi$
$61$	−9.70820	−1.24301	−0.621504	+	0.783411i	$0.713478\pi$
$62$	0	0
$63$	−7.47214	−0.941401
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.23607	0.517518	0.258759	−	0.965942i	$-0.416686\pi$
$67$	4.23607	0.517518	0.258759	+	0.965942i	$0.416686\pi$
$68$	0	0
$69$	−20.1803	−2.42943
$70$	0	0
$71$	−8.70820	−1.03347	−0.516737	−	0.856144i	$-0.672853\pi$
$71$	−8.70820	−1.03347	−0.516737	+	0.856144i	$0.672853\pi$
$72$	0	0
$73$	−8.76393	−1.02574	−0.512870	−	0.858466i	$-0.671418\pi$
$73$	−8.76393	−1.02574	−0.512870	+	0.858466i	$0.671418\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.236068	−0.0269024
$78$	0	0
$79$	11.1803	1.25789	0.628943	−	0.777451i	$-0.283488\pi$
$79$	11.1803	1.25789	0.628943	+	0.777451i	$0.283488\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	7.70820	0.846085	0.423043	−	0.906110i	$-0.360962\pi$
$83$	7.70820	0.846085	0.423043	+	0.906110i	$0.360962\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.1803	1.73471
$88$	0	0
$89$	17.2361	1.82702	0.913510	−	0.406817i	$-0.133361\pi$
$89$	17.2361	1.82702	0.913510	+	0.406817i	$0.133361\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	0	0
$93$	−12.0000	−1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.23607	0.531642	0.265821	−	0.964022i	$-0.414357\pi$
$97$	5.23607	0.531642	0.265821	+	0.964022i	$0.414357\pi$
$98$	0	0
$99$	1.76393	0.177282
$100$	0	0
$101$	4.76393	0.474029	0.237014	−	0.971506i	$-0.423831\pi$
$101$	4.76393	0.474029	0.237014	+	0.971506i	$0.423831\pi$
$102$	0	0
$103$	−8.47214	−0.834784	−0.417392	−	0.908726i	$-0.637056\pi$
$103$	−8.47214	−0.834784	−0.417392	+	0.908726i	$0.637056\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	8.41641	0.806146	0.403073	−	0.915168i	$-0.367942\pi$
$109$	8.41641	0.806146	0.403073	+	0.915168i	$0.367942\pi$
$110$	0	0
$111$	9.70820	0.921462
$112$	0	0
$113$	−14.4164	−1.35618	−0.678091	−	0.734978i	$-0.737192\pi$
$113$	−14.4164	−1.35618	−0.678091	+	0.734978i	$0.737192\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.23607	0.853875
$118$	0	0
$119$	−2.47214	−0.226620
$120$	0	0
$121$	−10.9443	−0.994934
$122$	0	0
$123$	15.4164	1.39005
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.6525	−1.21146	−0.605731	−	0.795670i	$-0.707119\pi$
$127$	−13.6525	−1.21146	−0.605731	+	0.795670i	$0.707119\pi$
$128$	0	0
$129$	−5.70820	−0.502579
$130$	0	0
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	0	0
$133$	−4.47214	−0.387783
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.9443	−0.935032	−0.467516	−	0.883985i	$-0.654851\pi$
$137$	−10.9443	−0.935032	−0.467516	+	0.883985i	$0.654851\pi$
$138$	0	0
$139$	−10.6525	−0.903531	−0.451766	−	0.892137i	$-0.649206\pi$
$139$	−10.6525	−0.903531	−0.451766	+	0.892137i	$0.649206\pi$
$140$	0	0
$141$	6.47214	0.545052
$142$	0	0
$143$	0.291796	0.0244012
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.23607	0.266906
$148$	0	0
$149$	3.94427	0.323127	0.161564	−	0.986862i	$-0.448346\pi$
$149$	3.94427	0.323127	0.161564	+	0.986862i	$0.448346\pi$
$150$	0	0
$151$	20.2361	1.64679	0.823394	−	0.567470i	$-0.192078\pi$
$151$	20.2361	1.64679	0.823394	+	0.567470i	$0.192078\pi$
$152$	0	0
$153$	18.4721	1.49338
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.763932	0.0609684	0.0304842	−	0.999535i	$-0.490295\pi$
$157$	0.763932	0.0609684	0.0304842	+	0.999535i	$0.490295\pi$
$158$	0	0
$159$	27.4164	2.17426
$160$	0	0
$161$	6.23607	0.491471
$162$	0	0
$163$	1.52786	0.119672	0.0598358	−	0.998208i	$-0.480942\pi$
$163$	1.52786	0.119672	0.0598358	+	0.998208i	$0.480942\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.23607	−0.405179	−0.202590	−	0.979264i	$-0.564936\pi$
$167$	−5.23607	−0.405179	−0.202590	+	0.979264i	$0.564936\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	33.4164	2.55542
$172$	0	0
$173$	−11.5279	−0.876447	−0.438224	−	0.898866i	$-0.644392\pi$
$173$	−11.5279	−0.876447	−0.438224	+	0.898866i	$0.644392\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−37.8885	−2.84788
$178$	0	0
$179$	23.4164	1.75022	0.875112	−	0.483920i	$-0.160787\pi$
$179$	23.4164	1.75022	0.875112	+	0.483920i	$0.160787\pi$
$180$	0	0
$181$	8.18034	0.608040	0.304020	−	0.952666i	$-0.401671\pi$
$181$	8.18034	0.608040	0.304020	+	0.952666i	$0.401671\pi$
$182$	0	0
$183$	−31.4164	−2.32237
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.583592	0.0426765
$188$	0	0
$189$	−14.4721	−1.05269
$190$	0	0
$191$	−6.47214	−0.468307	−0.234154	−	0.972200i	$-0.575232\pi$
$191$	−6.47214	−0.468307	−0.234154	+	0.972200i	$0.575232\pi$
$192$	0	0
$193$	12.4164	0.893753	0.446876	−	0.894596i	$-0.352536\pi$
$193$	12.4164	0.893753	0.446876	+	0.894596i	$0.352536\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.47214	−0.104885	−0.0524427	−	0.998624i	$-0.516701\pi$
$197$	−1.47214	−0.104885	−0.0524427	+	0.998624i	$0.516701\pi$
$198$	0	0
$199$	−7.23607	−0.512951	−0.256476	−	0.966551i	$-0.582561\pi$
$199$	−7.23607	−0.512951	−0.256476	+	0.966551i	$0.582561\pi$
$200$	0	0
$201$	13.7082	0.966902
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−46.5967	−3.23870
$208$	0	0
$209$	1.05573	0.0730262
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−28.1803	−1.93089
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.70820	0.251729
$218$	0	0
$219$	−28.3607	−1.91644
$220$	0	0
$221$	3.05573	0.205551
$222$	0	0
$223$	−20.1803	−1.35138	−0.675688	−	0.737188i	$-0.736153\pi$
$223$	−20.1803	−1.35138	−0.675688	+	0.737188i	$0.736153\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.4164	−1.42146	−0.710728	−	0.703466i	$-0.751635\pi$
$227$	−21.4164	−1.42146	−0.710728	+	0.703466i	$0.751635\pi$
$228$	0	0
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	−0.763932	−0.0502630
$232$	0	0
$233$	7.94427	0.520447	0.260223	−	0.965548i	$-0.416204\pi$
$233$	7.94427	0.520447	0.260223	+	0.965548i	$0.416204\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	36.1803	2.35017
$238$	0	0
$239$	5.52786	0.357568	0.178784	−	0.983888i	$-0.442784\pi$
$239$	5.52786	0.357568	0.178784	+	0.983888i	$0.442784\pi$
$240$	0	0
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	0	0
$243$	35.5967	2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.52786	0.351730
$248$	0	0
$249$	24.9443	1.58078
$250$	0	0
$251$	−6.47214	−0.408518	−0.204259	−	0.978917i	$-0.565478\pi$
$251$	−6.47214	−0.408518	−0.204259	+	0.978917i	$0.565478\pi$
$252$	0	0
$253$	−1.47214	−0.0925524
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.6525	−0.789240	−0.394620	−	0.918844i	$-0.629124\pi$
$257$	−12.6525	−0.789240	−0.394620	+	0.918844i	$0.629124\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	37.3607	2.31257
$262$	0	0
$263$	−16.2361	−1.00116	−0.500579	−	0.865691i	$-0.666880\pi$
$263$	−16.2361	−1.00116	−0.500579	+	0.865691i	$0.666880\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	55.7771	3.41350
$268$	0	0
$269$	−11.7082	−0.713862	−0.356931	−	0.934131i	$-0.616177\pi$
$269$	−11.7082	−0.713862	−0.356931	+	0.934131i	$0.616177\pi$
$270$	0	0
$271$	−23.7082	−1.44017	−0.720085	−	0.693885i	$-0.755897\pi$
$271$	−23.7082	−1.44017	−0.720085	+	0.693885i	$0.755897\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	0	0
$279$	−27.7082	−1.65885
$280$	0	0
$281$	−15.3607	−0.916341	−0.458171	−	0.888864i	$-0.651495\pi$
$281$	−15.3607	−0.916341	−0.458171	+	0.888864i	$0.651495\pi$
$282$	0	0
$283$	−17.4164	−1.03530	−0.517649	−	0.855593i	$-0.673193\pi$
$283$	−17.4164	−1.03530	−0.517649	+	0.855593i	$0.673193\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.76393	−0.281206
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	16.9443	0.993291
$292$	0	0
$293$	−31.1246	−1.81832	−0.909160	−	0.416448i	$-0.863275\pi$
$293$	−31.1246	−1.81832	−0.909160	+	0.416448i	$0.863275\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.41641	0.198240
$298$	0	0
$299$	−7.70820	−0.445777
$300$	0	0
$301$	1.76393	0.101671
$302$	0	0
$303$	15.4164	0.885649
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4.58359	−0.261599	−0.130800	−	0.991409i	$-0.541754\pi$
$307$	−4.58359	−0.261599	−0.130800	+	0.991409i	$0.541754\pi$
$308$	0	0
$309$	−27.4164	−1.55966
$310$	0	0
$311$	−24.3607	−1.38137	−0.690684	−	0.723157i	$-0.742690\pi$
$311$	−24.3607	−1.38137	−0.690684	+	0.723157i	$0.742690\pi$
$312$	0	0
$313$	19.5279	1.10378	0.551890	−	0.833917i	$-0.313907\pi$
$313$	19.5279	1.10378	0.551890	+	0.833917i	$0.313907\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.3607	1.42440	0.712199	−	0.701978i	$-0.247699\pi$
$317$	25.3607	1.42440	0.712199	+	0.701978i	$0.247699\pi$
$318$	0	0
$319$	1.18034	0.0660863
$320$	0	0
$321$	−25.8885	−1.44496
$322$	0	0
$323$	11.0557	0.615157
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.2361	1.50616
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	24.7082	1.35809	0.679043	−	0.734099i	$-0.262395\pi$
$331$	24.7082	1.35809	0.679043	+	0.734099i	$0.262395\pi$
$332$	0	0
$333$	22.4164	1.22841
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.4721	−0.897294	−0.448647	−	0.893709i	$-0.648094\pi$
$337$	−16.4721	−0.897294	−0.448647	+	0.893709i	$0.648094\pi$
$338$	0	0
$339$	−46.6525	−2.53381
$340$	0	0
$341$	−0.875388	−0.0474049
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.2361	−1.08633	−0.543165	−	0.839626i	$-0.682774\pi$
$347$	−20.2361	−1.08633	−0.543165	+	0.839626i	$0.682774\pi$
$348$	0	0
$349$	4.47214	0.239388	0.119694	−	0.992811i	$-0.461809\pi$
$349$	4.47214	0.239388	0.119694	+	0.992811i	$0.461809\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	0	0
$353$	−2.18034	−0.116048	−0.0580239	−	0.998315i	$-0.518480\pi$
$353$	−2.18034	−0.116048	−0.0580239	+	0.998315i	$0.518480\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	30.1246	1.58992	0.794958	−	0.606664i	$-0.207493\pi$
$359$	30.1246	1.58992	0.794958	+	0.606664i	$0.207493\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−35.4164	−1.85888
$364$	0	0
$365$	0	0
$366$	0	0
$367$	37.1246	1.93789	0.968944	−	0.247278i	$-0.0795362\pi$
$367$	37.1246	1.93789	0.968944	+	0.247278i	$0.0795362\pi$
$368$	0	0
$369$	35.5967	1.85309
$370$	0	0
$371$	−8.47214	−0.439851
$372$	0	0
$373$	−37.8328	−1.95891	−0.979454	−	0.201665i	$-0.935365\pi$
$373$	−37.8328	−1.95891	−0.979454	+	0.201665i	$0.935365\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.18034	0.318304
$378$	0	0
$379$	11.1803	0.574295	0.287148	−	0.957886i	$-0.407293\pi$
$379$	11.1803	0.574295	0.287148	+	0.957886i	$0.407293\pi$
$380$	0	0
$381$	−44.1803	−2.26343
$382$	0	0
$383$	33.2361	1.69828	0.849142	−	0.528165i	$-0.177120\pi$
$383$	33.2361	1.69828	0.849142	+	0.528165i	$0.177120\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.1803	−0.669994
$388$	0	0
$389$	2.88854	0.146455	0.0732275	−	0.997315i	$-0.476670\pi$
$389$	2.88854	0.146455	0.0732275	+	0.997315i	$0.476670\pi$
$390$	0	0
$391$	−15.4164	−0.779641
$392$	0	0
$393$	54.8328	2.76595
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.05573	0.454494	0.227247	−	0.973837i	$-0.427028\pi$
$397$	9.05573	0.454494	0.227247	+	0.973837i	$0.427028\pi$
$398$	0	0
$399$	−14.4721	−0.724513
$400$	0	0
$401$	2.52786	0.126236	0.0631178	−	0.998006i	$-0.479896\pi$
$401$	2.52786	0.126236	0.0631178	+	0.998006i	$0.479896\pi$
$402$	0	0
$403$	−4.58359	−0.228325
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.708204	0.0351044
$408$	0	0
$409$	24.4721	1.21007	0.605035	−	0.796199i	$-0.293159\pi$
$409$	24.4721	1.21007	0.605035	+	0.796199i	$0.293159\pi$
$410$	0	0
$411$	−35.4164	−1.74696
$412$	0	0
$413$	11.7082	0.576123
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−34.4721	−1.68811
$418$	0	0
$419$	26.1803	1.27899	0.639497	−	0.768794i	$-0.279143\pi$
$419$	26.1803	1.27899	0.639497	+	0.768794i	$0.279143\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	0	0
$423$	14.9443	0.726615
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.70820	0.469813
$428$	0	0
$429$	0.944272	0.0455899
$430$	0	0
$431$	−17.5279	−0.844288	−0.422144	−	0.906529i	$-0.638722\pi$
$431$	−17.5279	−0.844288	−0.422144	+	0.906529i	$0.638722\pi$
$432$	0	0
$433$	−28.3607	−1.36293	−0.681464	−	0.731852i	$-0.738656\pi$
$433$	−28.3607	−1.36293	−0.681464	+	0.731852i	$0.738656\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.8885	−1.33409
$438$	0	0
$439$	8.29180	0.395746	0.197873	−	0.980228i	$-0.436597\pi$
$439$	8.29180	0.395746	0.197873	+	0.980228i	$0.436597\pi$
$440$	0	0
$441$	7.47214	0.355816
$442$	0	0
$443$	19.4164	0.922501	0.461251	−	0.887270i	$-0.347401\pi$
$443$	19.4164	0.922501	0.461251	+	0.887270i	$0.347401\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.7639	0.603713
$448$	0	0
$449$	20.5279	0.968770	0.484385	−	0.874855i	$-0.339043\pi$
$449$	20.5279	0.968770	0.484385	+	0.874855i	$0.339043\pi$
$450$	0	0
$451$	1.12461	0.0529559
$452$	0	0
$453$	65.4853	3.07677
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.5279	−0.586029	−0.293014	−	0.956108i	$-0.594658\pi$
$457$	−12.5279	−0.586029	−0.293014	+	0.956108i	$0.594658\pi$
$458$	0	0
$459$	35.7771	1.66993
$460$	0	0
$461$	−14.1803	−0.660444	−0.330222	−	0.943903i	$-0.607124\pi$
$461$	−14.1803	−0.660444	−0.330222	+	0.943903i	$0.607124\pi$
$462$	0	0
$463$	13.8885	0.645455	0.322728	−	0.946492i	$-0.395400\pi$
$463$	13.8885	0.645455	0.322728	+	0.946492i	$0.395400\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.94427	−0.321343	−0.160671	−	0.987008i	$-0.551366\pi$
$467$	−6.94427	−0.321343	−0.160671	+	0.987008i	$0.551366\pi$
$468$	0	0
$469$	−4.23607	−0.195603
$470$	0	0
$471$	2.47214	0.113910
$472$	0	0
$473$	−0.416408	−0.0191465
$474$	0	0
$475$	0	0
$476$	0	0
$477$	63.3050	2.89853
$478$	0	0
$479$	26.1803	1.19621	0.598105	−	0.801418i	$-0.295921\pi$
$479$	26.1803	1.19621	0.598105	+	0.801418i	$0.295921\pi$
$480$	0	0
$481$	3.70820	0.169080
$482$	0	0
$483$	20.1803	0.918237
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.76393	−0.261189	−0.130594	−	0.991436i	$-0.541689\pi$
$487$	−5.76393	−0.261189	−0.130594	+	0.991436i	$0.541689\pi$
$488$	0	0
$489$	4.94427	0.223588
$490$	0	0
$491$	5.76393	0.260123	0.130061	−	0.991506i	$-0.458483\pi$
$491$	5.76393	0.260123	0.130061	+	0.991506i	$0.458483\pi$
$492$	0	0
$493$	12.3607	0.556697
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.70820	0.390616
$498$	0	0
$499$	−11.0557	−0.494922	−0.247461	−	0.968898i	$-0.579596\pi$
$499$	−11.0557	−0.494922	−0.247461	+	0.968898i	$0.579596\pi$
$500$	0	0
$501$	−16.9443	−0.757014
$502$	0	0
$503$	8.11146	0.361672	0.180836	−	0.983513i	$-0.442120\pi$
$503$	8.11146	0.361672	0.180836	+	0.983513i	$0.442120\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−37.1246	−1.64876
$508$	0	0
$509$	40.6525	1.80189	0.900945	−	0.433934i	$-0.142875\pi$
$509$	40.6525	1.80189	0.900945	+	0.433934i	$0.142875\pi$
$510$	0	0
$511$	8.76393	0.387694
$512$	0	0
$513$	64.7214	2.85752
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.472136	0.0207645
$518$	0	0
$519$	−37.3050	−1.63751
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	−16.3607	−0.715403	−0.357701	−	0.933836i	$-0.616439\pi$
$523$	−16.3607	−0.715403	−0.357701	+	0.933836i	$0.616439\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.16718	−0.399329
$528$	0	0
$529$	15.8885	0.690806
$530$	0	0
$531$	−87.4853	−3.79654
$532$	0	0
$533$	5.88854	0.255061
$534$	0	0
$535$	0	0
$536$	0	0
$537$	75.7771	3.27002
$538$	0	0
$539$	0.236068	0.0101682
$540$	0	0
$541$	15.9443	0.685498	0.342749	−	0.939427i	$-0.388642\pi$
$541$	15.9443	0.685498	0.342749	+	0.939427i	$0.388642\pi$
$542$	0	0
$543$	26.4721	1.13603
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.76393	0.417476	0.208738	−	0.977972i	$-0.433064\pi$
$547$	9.76393	0.417476	0.208738	+	0.977972i	$0.433064\pi$
$548$	0	0
$549$	−72.5410	−3.09598
$550$	0	0
$551$	22.3607	0.952597
$552$	0	0
$553$	−11.1803	−0.475436
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.11146	−0.386065	−0.193032	−	0.981192i	$-0.561832\pi$
$557$	−9.11146	−0.386065	−0.193032	+	0.981192i	$0.561832\pi$
$558$	0	0
$559$	−2.18034	−0.0922186
$560$	0	0
$561$	1.88854	0.0797344
$562$	0	0
$563$	−17.4164	−0.734014	−0.367007	−	0.930218i	$-0.619618\pi$
$563$	−17.4164	−0.734014	−0.367007	+	0.930218i	$0.619618\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−24.4164	−1.02539
$568$	0	0
$569$	−3.94427	−0.165352	−0.0826762	−	0.996576i	$-0.526347\pi$
$569$	−3.94427	−0.165352	−0.0826762	+	0.996576i	$0.526347\pi$
$570$	0	0
$571$	−36.5967	−1.53153	−0.765763	−	0.643123i	$-0.777638\pi$
$571$	−36.5967	−1.53153	−0.765763	+	0.643123i	$0.777638\pi$
$572$	0	0
$573$	−20.9443	−0.874960
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	40.1803	1.66984
$580$	0	0
$581$	−7.70820	−0.319790
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.7639	1.02212	0.511058	−	0.859546i	$-0.329254\pi$
$587$	24.7639	1.02212	0.511058	+	0.859546i	$0.329254\pi$
$588$	0	0
$589$	−16.5836	−0.683315
$590$	0	0
$591$	−4.76393	−0.195962
$592$	0	0
$593$	−37.3050	−1.53193	−0.765965	−	0.642882i	$-0.777739\pi$
$593$	−37.3050	−1.53193	−0.765965	+	0.642882i	$0.777739\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−23.4164	−0.958370
$598$	0	0
$599$	11.1803	0.456816	0.228408	−	0.973565i	$-0.426648\pi$
$599$	11.1803	0.456816	0.228408	+	0.973565i	$0.426648\pi$
$600$	0	0
$601$	−36.9443	−1.50699	−0.753494	−	0.657455i	$-0.771633\pi$
$601$	−36.9443	−1.50699	−0.753494	+	0.657455i	$0.771633\pi$
$602$	0	0
$603$	31.6525	1.28899
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.12461	0.289179	0.144590	−	0.989492i	$-0.453814\pi$
$607$	7.12461	0.289179	0.144590	+	0.989492i	$0.453814\pi$
$608$	0	0
$609$	−16.1803	−0.655660
$610$	0	0
$611$	2.47214	0.100012
$612$	0	0
$613$	−44.4164	−1.79396	−0.896981	−	0.442069i	$-0.854245\pi$
$613$	−44.4164	−1.79396	−0.896981	+	0.442069i	$0.854245\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.94427	−0.239307	−0.119654	−	0.992816i	$-0.538178\pi$
$617$	−5.94427	−0.239307	−0.119654	+	0.992816i	$0.538178\pi$
$618$	0	0
$619$	11.7082	0.470592	0.235296	−	0.971924i	$-0.424394\pi$
$619$	11.7082	0.470592	0.235296	+	0.971924i	$0.424394\pi$
$620$	0	0
$621$	−90.2492	−3.62158
$622$	0	0
$623$	−17.2361	−0.690548
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.41641	0.136438
$628$	0	0
$629$	7.41641	0.295712
$630$	0	0
$631$	−27.6525	−1.10083	−0.550414	−	0.834892i	$-0.685530\pi$
$631$	−27.6525	−1.10083	−0.550414	+	0.834892i	$0.685530\pi$
$632$	0	0
$633$	−38.8328	−1.54347
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.23607	0.0489748
$638$	0	0
$639$	−65.0689	−2.57409
$640$	0	0
$641$	43.8328	1.73129	0.865646	−	0.500656i	$-0.166908\pi$
$641$	43.8328	1.73129	0.865646	+	0.500656i	$0.166908\pi$
$642$	0	0
$643$	−18.4721	−0.728470	−0.364235	−	0.931307i	$-0.618669\pi$
$643$	−18.4721	−0.728470	−0.364235	+	0.931307i	$0.618669\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.8885	0.781899	0.390950	−	0.920412i	$-0.372147\pi$
$647$	19.8885	0.781899	0.390950	+	0.920412i	$0.372147\pi$
$648$	0	0
$649$	−2.76393	−0.108494
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	25.0557	0.980506	0.490253	−	0.871580i	$-0.336904\pi$
$653$	25.0557	0.980506	0.490253	+	0.871580i	$0.336904\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−65.4853	−2.55482
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−42.7214	−1.66167	−0.830834	−	0.556520i	$-0.812136\pi$
$661$	−42.7214	−1.66167	−0.830834	+	0.556520i	$0.812136\pi$
$662$	0	0
$663$	9.88854	0.384039
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−31.1803	−1.20731
$668$	0	0
$669$	−65.3050	−2.52484
$670$	0	0
$671$	−2.29180	−0.0884738
$672$	0	0
$673$	19.5279	0.752744	0.376372	−	0.926469i	$-0.377171\pi$
$673$	19.5279	0.752744	0.376372	+	0.926469i	$0.377171\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.3607	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$677$	−14.3607	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$678$	0	0
$679$	−5.23607	−0.200942
$680$	0	0
$681$	−69.3050	−2.65577
$682$	0	0
$683$	−14.1246	−0.540463	−0.270232	−	0.962795i	$-0.587100\pi$
$683$	−14.1246	−0.540463	−0.270232	+	0.962795i	$0.587100\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14.4721	−0.552146
$688$	0	0
$689$	10.4721	0.398957
$690$	0	0
$691$	4.18034	0.159028	0.0795138	−	0.996834i	$-0.474663\pi$
$691$	4.18034	0.159028	0.0795138	+	0.996834i	$0.474663\pi$
$692$	0	0
$693$	−1.76393	−0.0670062
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.7771	0.446089
$698$	0	0
$699$	25.7082	0.972374
$700$	0	0
$701$	−29.0557	−1.09742	−0.548710	−	0.836013i	$-0.684881\pi$
$701$	−29.0557	−1.09742	−0.548710	+	0.836013i	$0.684881\pi$
$702$	0	0
$703$	13.4164	0.506009
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.76393	−0.179166
$708$	0	0
$709$	−12.1115	−0.454855	−0.227428	−	0.973795i	$-0.573032\pi$
$709$	−12.1115	−0.454855	−0.227428	+	0.973795i	$0.573032\pi$
$710$	0	0
$711$	83.5410	3.13303
$712$	0	0
$713$	23.1246	0.866024
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.8885	0.668060
$718$	0	0
$719$	−16.1803	−0.603425	−0.301712	−	0.953399i	$-0.597558\pi$
$719$	−16.1803	−0.603425	−0.301712	+	0.953399i	$0.597558\pi$
$720$	0	0
$721$	8.47214	0.315519
$722$	0	0
$723$	−11.4164	−0.424581
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.05573	0.113331	0.0566653	−	0.998393i	$-0.481953\pi$
$727$	3.05573	0.113331	0.0566653	+	0.998393i	$0.481953\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	−4.36068	−0.161286
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.00000	0.0368355
$738$	0	0
$739$	−25.6525	−0.943642	−0.471821	−	0.881694i	$-0.656403\pi$
$739$	−25.6525	−0.943642	−0.471821	+	0.881694i	$0.656403\pi$
$740$	0	0
$741$	17.8885	0.657152
$742$	0	0
$743$	10.4721	0.384185	0.192093	−	0.981377i	$-0.438473\pi$
$743$	10.4721	0.384185	0.192093	+	0.981377i	$0.438473\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	57.5967	2.10735
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	−3.05573	−0.111505	−0.0557526	−	0.998445i	$-0.517756\pi$
$751$	−3.05573	−0.111505	−0.0557526	+	0.998445i	$0.517756\pi$
$752$	0	0
$753$	−20.9443	−0.763252
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19.5836	0.711778	0.355889	−	0.934528i	$-0.384178\pi$
$757$	19.5836	0.711778	0.355889	+	0.934528i	$0.384178\pi$
$758$	0	0
$759$	−4.76393	−0.172920
$760$	0	0
$761$	27.7771	1.00692	0.503459	−	0.864019i	$-0.332060\pi$
$761$	27.7771	1.00692	0.503459	+	0.864019i	$0.332060\pi$
$762$	0	0
$763$	−8.41641	−0.304694
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.4721	−0.522559
$768$	0	0
$769$	−43.0132	−1.55109	−0.775547	−	0.631290i	$-0.782526\pi$
$769$	−43.0132	−1.55109	−0.775547	+	0.631290i	$0.782526\pi$
$770$	0	0
$771$	−40.9443	−1.47457
$772$	0	0
$773$	50.1803	1.80486	0.902431	−	0.430835i	$-0.141781\pi$
$773$	50.1803	1.80486	0.902431	+	0.430835i	$0.141781\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.70820	−0.348280
$778$	0	0
$779$	21.3050	0.763329
$780$	0	0
$781$	−2.05573	−0.0735597
$782$	0	0
$783$	72.3607	2.58596
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−40.7639	−1.45308	−0.726539	−	0.687126i	$-0.758872\pi$
$787$	−40.7639	−1.45308	−0.726539	+	0.687126i	$0.758872\pi$
$788$	0	0
$789$	−52.5410	−1.87051
$790$	0	0
$791$	14.4164	0.512588
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.4164	−1.25451	−0.627257	−	0.778813i	$-0.715822\pi$
$797$	−35.4164	−1.25451	−0.627257	+	0.778813i	$0.715822\pi$
$798$	0	0
$799$	4.94427	0.174916
$800$	0	0
$801$	128.790	4.55058
$802$	0	0
$803$	−2.06888	−0.0730093
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−37.8885	−1.33374
$808$	0	0
$809$	29.4721	1.03619	0.518093	−	0.855325i	$-0.326642\pi$
$809$	29.4721	1.03619	0.518093	+	0.855325i	$0.326642\pi$
$810$	0	0
$811$	42.7214	1.50015	0.750075	−	0.661353i	$-0.230017\pi$
$811$	42.7214	1.50015	0.750075	+	0.661353i	$0.230017\pi$
$812$	0	0
$813$	−76.7214	−2.69074
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.88854	−0.275985
$818$	0	0
$819$	−9.23607	−0.322734
$820$	0	0
$821$	28.8328	1.00627	0.503136	−	0.864207i	$-0.332179\pi$
$821$	28.8328	1.00627	0.503136	+	0.864207i	$0.332179\pi$
$822$	0	0
$823$	31.6525	1.10334	0.551668	−	0.834064i	$-0.313992\pi$
$823$	31.6525	1.10334	0.551668	+	0.834064i	$0.313992\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.5410	−1.44452	−0.722261	−	0.691620i	$-0.756897\pi$
$827$	−41.5410	−1.44452	−0.722261	+	0.691620i	$0.756897\pi$
$828$	0	0
$829$	−7.63932	−0.265325	−0.132662	−	0.991161i	$-0.542353\pi$
$829$	−7.63932	−0.265325	−0.132662	+	0.991161i	$0.542353\pi$
$830$	0	0
$831$	−64.3607	−2.23265
$832$	0	0
$833$	2.47214	0.0856544
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−53.6656	−1.85496
$838$	0	0
$839$	30.6525	1.05824	0.529120	−	0.848547i	$-0.322522\pi$
$839$	30.6525	1.05824	0.529120	+	0.848547i	$0.322522\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−49.7082	−1.71204
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.9443	0.376050
$848$	0	0
$849$	−56.3607	−1.93429
$850$	0	0
$851$	−18.7082	−0.641309
$852$	0	0
$853$	27.4164	0.938720	0.469360	−	0.883007i	$-0.344485\pi$
$853$	27.4164	0.938720	0.469360	+	0.883007i	$0.344485\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.8197	−0.540389	−0.270195	−	0.962806i	$-0.587088\pi$
$857$	−15.8197	−0.540389	−0.270195	+	0.962806i	$0.587088\pi$
$858$	0	0
$859$	−22.3607	−0.762937	−0.381468	−	0.924382i	$-0.624581\pi$
$859$	−22.3607	−0.762937	−0.381468	+	0.924382i	$0.624581\pi$
$860$	0	0
$861$	−15.4164	−0.525390
$862$	0	0
$863$	−18.3475	−0.624557	−0.312278	−	0.949991i	$-0.601092\pi$
$863$	−18.3475	−0.624557	−0.312278	+	0.949991i	$0.601092\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−35.2361	−1.19668
$868$	0	0
$869$	2.63932	0.0895328
$870$	0	0
$871$	5.23607	0.177417
$872$	0	0
$873$	39.1246	1.32417
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.3607	1.02521	0.512604	−	0.858625i	$-0.328681\pi$
$877$	30.3607	1.02521	0.512604	+	0.858625i	$0.328681\pi$
$878$	0	0
$879$	−100.721	−3.39725
$880$	0	0
$881$	5.81966	0.196069	0.0980347	−	0.995183i	$-0.468744\pi$
$881$	5.81966	0.196069	0.0980347	+	0.995183i	$0.468744\pi$
$882$	0	0
$883$	1.40325	0.0472232	0.0236116	−	0.999721i	$-0.492483\pi$
$883$	1.40325	0.0472232	0.0236116	+	0.999721i	$0.492483\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.3475	0.716780	0.358390	−	0.933572i	$-0.383326\pi$
$887$	21.3475	0.716780	0.358390	+	0.933572i	$0.383326\pi$
$888$	0	0
$889$	13.6525	0.457889
$890$	0	0
$891$	5.76393	0.193099
$892$	0	0
$893$	8.94427	0.299309
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−24.9443	−0.832865
$898$	0	0
$899$	−18.5410	−0.618378
$900$	0	0
$901$	20.9443	0.697755
$902$	0	0
$903$	5.70820	0.189957
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−34.8328	−1.15660	−0.578302	−	0.815823i	$-0.696285\pi$
$907$	−34.8328	−1.15660	−0.578302	+	0.815823i	$0.696285\pi$
$908$	0	0
$909$	35.5967	1.18067
$910$	0	0
$911$	−0.819660	−0.0271566	−0.0135783	−	0.999908i	$-0.504322\pi$
$911$	−0.819660	−0.0271566	−0.0135783	+	0.999908i	$0.504322\pi$
$912$	0	0
$913$	1.81966	0.0602220
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.9443	−0.559549
$918$	0	0
$919$	27.7639	0.915848	0.457924	−	0.888991i	$-0.348593\pi$
$919$	27.7639	0.915848	0.457924	+	0.888991i	$0.348593\pi$
$920$	0	0
$921$	−14.8328	−0.488758
$922$	0	0
$923$	−10.7639	−0.354299
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−63.3050	−2.07921
$928$	0	0
$929$	38.2918	1.25631	0.628157	−	0.778087i	$-0.283810\pi$
$929$	38.2918	1.25631	0.628157	+	0.778087i	$0.283810\pi$
$930$	0	0
$931$	4.47214	0.146568
$932$	0	0
$933$	−78.8328	−2.58087
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.2361	1.15111	0.575556	−	0.817762i	$-0.304786\pi$
$937$	35.2361	1.15111	0.575556	+	0.817762i	$0.304786\pi$
$938$	0	0
$939$	63.1935	2.06224
$940$	0	0
$941$	−5.23607	−0.170691	−0.0853455	−	0.996351i	$-0.527199\pi$
$941$	−5.23607	−0.170691	−0.0853455	+	0.996351i	$0.527199\pi$
$942$	0	0
$943$	−29.7082	−0.967432
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.8328	−1.13191	−0.565957	−	0.824435i	$-0.691493\pi$
$947$	−34.8328	−1.13191	−0.565957	+	0.824435i	$0.691493\pi$
$948$	0	0
$949$	−10.8328	−0.351648
$950$	0	0
$951$	82.0689	2.66127
$952$	0	0
$953$	3.47214	0.112474	0.0562368	−	0.998417i	$-0.482090\pi$
$953$	3.47214	0.112474	0.0562368	+	0.998417i	$0.482090\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.81966	0.123472
$958$	0	0
$959$	10.9443	0.353409
$960$	0	0
$961$	−17.2492	−0.556427
$962$	0	0
$963$	−59.7771	−1.92629
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.1115	0.453794	0.226897	−	0.973919i	$-0.427142\pi$
$967$	14.1115	0.453794	0.226897	+	0.973919i	$0.427142\pi$
$968$	0	0
$969$	35.7771	1.14933
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	10.6525	0.341503
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.4721	−0.367026	−0.183513	−	0.983017i	$-0.558747\pi$
$977$	−11.4721	−0.367026	−0.183513	+	0.983017i	$0.558747\pi$
$978$	0	0
$979$	4.06888	0.130042
$980$	0	0
$981$	62.8885	2.00788
$982$	0	0
$983$	34.5410	1.10169	0.550844	−	0.834608i	$-0.314306\pi$
$983$	34.5410	1.10169	0.550844	+	0.834608i	$0.314306\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.47214	−0.206010
$988$	0	0
$989$	11.0000	0.349780
$990$	0	0
$991$	−13.1803	−0.418687	−0.209344	−	0.977842i	$-0.567133\pi$
$991$	−13.1803	−0.418687	−0.209344	+	0.977842i	$0.567133\pi$
$992$	0	0
$993$	79.9574	2.53737
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−45.4164	−1.43835	−0.719176	−	0.694828i	$-0.755481\pi$
$997$	−45.4164	−1.43835	−0.719176	+	0.694828i	$0.755481\pi$
$998$	0	0
$999$	43.4164	1.37363

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bp.1.2		2
4.3	odd	2		175.2.a.e.1.1	yes	2
5.2	odd	4		2800.2.g.s.449.1		4
5.3	odd	4		2800.2.g.s.449.4		4
5.4	even	2		2800.2.a.bh.1.1		2
12.11	even	2		1575.2.a.n.1.2		2
20.3	even	4		175.2.b.c.99.3		4
20.7	even	4		175.2.b.c.99.2		4
20.19	odd	2		175.2.a.d.1.2	✓	2
28.27	even	2		1225.2.a.u.1.1		2
60.23	odd	4		1575.2.d.k.1324.2		4
60.47	odd	4		1575.2.d.k.1324.3		4
60.59	even	2		1575.2.a.s.1.1		2
140.27	odd	4		1225.2.b.k.99.2		4
140.83	odd	4		1225.2.b.k.99.3		4
140.139	even	2		1225.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.a.d.1.2	✓	2	20.19	odd	2
175.2.a.e.1.1	yes	2	4.3	odd	2
175.2.b.c.99.2		4	20.7	even	4
175.2.b.c.99.3		4	20.3	even	4
1225.2.a.n.1.2		2	140.139	even	2
1225.2.a.u.1.1		2	28.27	even	2
1225.2.b.k.99.2		4	140.27	odd	4
1225.2.b.k.99.3		4	140.83	odd	4
1575.2.a.n.1.2		2	12.11	even	2
1575.2.a.s.1.1		2	60.59	even	2
1575.2.d.k.1324.2		4	60.23	odd	4
1575.2.d.k.1324.3		4	60.47	odd	4
2800.2.a.bh.1.1		2	5.4	even	2
2800.2.a.bp.1.2		2	1.1	even	1	trivial
2800.2.g.s.449.1		4	5.2	odd	4
2800.2.g.s.449.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{3} -1.00000 q^{7} +7.47214 q^{9} +O(q^{10})\)	\(q+3.23607 q^{3} -1.00000 q^{7} +7.47214 q^{9} +0.236068 q^{11} +1.23607 q^{13} +2.47214 q^{17} +4.47214 q^{19} -3.23607 q^{21} -6.23607 q^{23} +14.4721 q^{27} +5.00000 q^{29} -3.70820 q^{31} +0.763932 q^{33} +3.00000 q^{37} +4.00000 q^{39} +4.76393 q^{41} -1.76393 q^{43} +2.00000 q^{47} +1.00000 q^{49} +8.00000 q^{51} +8.47214 q^{53} +14.4721 q^{57} -11.7082 q^{59} -9.70820 q^{61} -7.47214 q^{63} +4.23607 q^{67} -20.1803 q^{69} -8.70820 q^{71} -8.76393 q^{73} -0.236068 q^{77} +11.1803 q^{79} +24.4164 q^{81} +7.70820 q^{83} +16.1803 q^{87} +17.2361 q^{89} -1.23607 q^{91} -12.0000 q^{93} +5.23607 q^{97} +1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{21} - 8 q^{23} + 20 q^{27} + 10 q^{29} + 6 q^{31} + 6 q^{33} + 6 q^{37} + 8 q^{39} + 14 q^{41} - 8 q^{43} + 4 q^{47} + 2 q^{49} + 16 q^{51} + 8 q^{53} + 20 q^{57} - 10 q^{59} - 6 q^{61} - 6 q^{63} + 4 q^{67} - 18 q^{69} - 4 q^{71} - 22 q^{73} + 4 q^{77} + 22 q^{81} + 2 q^{83} + 10 q^{87} + 30 q^{89} + 2 q^{91} - 24 q^{93} + 6 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^21 - 8 * q^23 + 20 * q^27 + 10 * q^29 + 6 * q^31 + 6 * q^33 + 6 * q^37 + 8 * q^39 + 14 * q^41 - 8 * q^43 + 4 * q^47 + 2 * q^49 + 16 * q^51 + 8 * q^53 + 20 * q^57 - 10 * q^59 - 6 * q^61 - 6 * q^63 + 4 * q^67 - 18 * q^69 - 4 * q^71 - 22 * q^73 + 4 * q^77 + 22 * q^81 + 2 * q^83 + 10 * q^87 + 30 * q^89 + 2 * q^91 - 24 * q^93 + 6 * q^97 + 8 * q^99

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