LMFDB - Embedded newform 2800.2.g.s.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 175)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.s.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +O(q^{10})$	$q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +0.236068 q^{11} -1.23607i q^{13} +2.47214i q^{17} -4.47214 q^{19} -3.23607 q^{21} +6.23607i q^{23} +14.4721i q^{27} -5.00000 q^{29} -3.70820 q^{31} -0.763932i q^{33} +3.00000i q^{37} -4.00000 q^{39} +4.76393 q^{41} +1.76393i q^{43} +2.00000i q^{47} -1.00000 q^{49} +8.00000 q^{51} -8.47214i q^{53} +14.4721i q^{57} +11.7082 q^{59} -9.70820 q^{61} +7.47214i q^{63} +4.23607i q^{67} +20.1803 q^{69} -8.70820 q^{71} +8.76393i q^{73} -0.236068i q^{77} -11.1803 q^{79} +24.4164 q^{81} -7.70820i q^{83} +16.1803i q^{87} -17.2361 q^{89} -1.23607 q^{91} +12.0000i q^{93} +5.23607i q^{97} -1.76393 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 8 q^{11} - 4 q^{21} - 20 q^{29} + 12 q^{31} - 16 q^{39} + 28 q^{41} - 4 q^{49} + 32 q^{51} + 20 q^{59} - 12 q^{61} + 36 q^{69} - 8 q^{71} + 44 q^{81} - 60 q^{89} + 4 q^{91} - 16 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 8 * q^11 - 4 * q^21 - 20 * q^29 + 12 * q^31 - 16 * q^39 + 28 * q^41 - 4 * q^49 + 32 * q^51 + 20 * q^59 - 12 * q^61 + 36 * q^69 - 8 * q^71 + 44 * q^81 - 60 * q^89 + 4 * q^91 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.23607i	− 1.86834i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	− 3.23607i	− 1.86834i	0.356822	−	0.934172i	$-0.383860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	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
$13$	− 1.23607i	− 0.342824i	−0.985199	−	0.171412i	$-0.945167\pi$
$13$	− 1.23607i	− 0.342824i	0.985199	−	0.171412i	$-0.0548329\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214i	0.599581i	0.954005	+	0.299791i	$0.0969168\pi$
$17$	2.47214i	0.599581i	−0.954005	+	0.299791i	$0.903083\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	−3.23607	−0.706168
$22$	0	0
$23$	6.23607i	1.30031i	0.759802	+	0.650155i	$0.225296\pi$
$23$	6.23607i	1.30031i	−0.759802	+	0.650155i	$0.774704\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.4721i	2.78516i
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	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
$33$	− 0.763932i	− 0.132983i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	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
$43$	1.76393i	0.268997i	0.990914	+	0.134499i	$0.0429424\pi$
$43$	1.76393i	0.268997i	−0.990914	+	0.134499i	$0.957058\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	− 8.47214i	− 1.16374i	−0.813283	−	0.581869i	$-0.802322\pi$
$53$	− 8.47214i	− 1.16374i	0.813283	−	0.581869i	$-0.197678\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.4721i	1.91688i
$58$	0	0
$59$	11.7082	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$59$	11.7082	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$60$	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
$63$	7.47214i	0.941401i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.23607i	0.517518i	0.965942	+	0.258759i	$0.0833136\pi$
$67$	4.23607i	0.517518i	−0.965942	+	0.258759i	$0.916686\pi$
$68$	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
$73$	8.76393i	1.02574i	0.858466	+	0.512870i	$0.171418\pi$
$73$	8.76393i	1.02574i	−0.858466	+	0.512870i	$0.828582\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.236068i	− 0.0269024i
$78$	0	0
$79$	−11.1803	−1.25789	−0.628943	−	0.777451i	$-0.716512\pi$
$79$	−11.1803	−1.25789	−0.628943	+	0.777451i	$0.716512\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	− 7.70820i	− 0.846085i	−0.906110	−	0.423043i	$-0.860962\pi$
$83$	− 7.70820i	− 0.846085i	0.906110	−	0.423043i	$-0.139038\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.1803i	1.73471i
$88$	0	0
$89$	−17.2361	−1.82702	−0.913510	−	0.406817i	$-0.866639\pi$
$89$	−17.2361	−1.82702	−0.913510	+	0.406817i	$0.866639\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	0	0
$93$	12.0000i	1.24434i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.23607i	0.531642i	0.964022	+	0.265821i	$0.0856430\pi$
$97$	5.23607i	0.531642i	−0.964022	+	0.265821i	$0.914357\pi$
$98$	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
$103$	8.47214i	0.834784i	0.908726	+	0.417392i	$0.137056\pi$
$103$	8.47214i	0.834784i	−0.908726	+	0.417392i	$0.862944\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	0	0
$109$	−8.41641	−0.806146	−0.403073	−	0.915168i	$-0.632058\pi$
$109$	−8.41641	−0.806146	−0.403073	+	0.915168i	$0.632058\pi$
$110$	0	0
$111$	9.70820	0.921462
$112$	0	0
$113$	14.4164i	1.35618i	0.734978	+	0.678091i	$0.237192\pi$
$113$	14.4164i	1.35618i	−0.734978	+	0.678091i	$0.762808\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.23607i	0.853875i
$118$	0	0
$119$	2.47214	0.226620
$120$	0	0
$121$	−10.9443	−0.994934
$122$	0	0
$123$	− 15.4164i	− 1.39005i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.6525i	− 1.21146i	−0.795670	−	0.605731i	$-0.792881\pi$
$127$	− 13.6525i	− 1.21146i	0.795670	−	0.605731i	$-0.207119\pi$
$128$	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
$133$	4.47214i	0.387783i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.9443i	− 0.935032i	−0.883985	−	0.467516i	$-0.845149\pi$
$137$	− 10.9443i	− 0.935032i	0.883985	−	0.467516i	$-0.154851\pi$
$138$	0	0
$139$	10.6525	0.903531	0.451766	−	0.892137i	$-0.350794\pi$
$139$	10.6525	0.903531	0.451766	+	0.892137i	$0.350794\pi$
$140$	0	0
$141$	6.47214	0.545052
$142$	0	0
$143$	− 0.291796i	− 0.0244012i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.23607i	0.266906i
$148$	0	0
$149$	−3.94427	−0.323127	−0.161564	−	0.986862i	$-0.551654\pi$
$149$	−3.94427	−0.323127	−0.161564	+	0.986862i	$0.551654\pi$
$150$	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
$153$	− 18.4721i	− 1.49338i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.763932i	0.0609684i	0.999535	+	0.0304842i	$0.00970493\pi$
$157$	0.763932i	0.0609684i	−0.999535	+	0.0304842i	$0.990295\pi$
$158$	0	0
$159$	−27.4164	−2.17426
$160$	0	0
$161$	6.23607	0.491471
$162$	0	0
$163$	− 1.52786i	− 0.119672i	−0.998208	−	0.0598358i	$-0.980942\pi$
$163$	− 1.52786i	− 0.119672i	0.998208	−	0.0598358i	$-0.0190577\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.23607i	− 0.405179i	−0.979264	−	0.202590i	$-0.935064\pi$
$167$	− 5.23607i	− 0.405179i	0.979264	−	0.202590i	$-0.0649357\pi$
$168$	0	0
$169$	11.4721	0.882472
$170$	0	0
$171$	33.4164	2.55542
$172$	0	0
$173$	11.5279i	0.876447i	0.898866	+	0.438224i	$0.144392\pi$
$173$	11.5279i	0.876447i	−0.898866	+	0.438224i	$0.855608\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 37.8885i	− 2.84788i
$178$	0	0
$179$	−23.4164	−1.75022	−0.875112	−	0.483920i	$-0.839213\pi$
$179$	−23.4164	−1.75022	−0.875112	+	0.483920i	$0.839213\pi$
$180$	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
$183$	31.4164i	2.32237i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.583592i	0.0426765i
$188$	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
$193$	− 12.4164i	− 0.893753i	−0.894596	−	0.446876i	$-0.852536\pi$
$193$	− 12.4164i	− 0.893753i	0.894596	−	0.446876i	$-0.147464\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.47214i	− 0.104885i	−0.998624	−	0.0524427i	$-0.983299\pi$
$197$	− 1.47214i	− 0.104885i	0.998624	−	0.0524427i	$-0.0167007\pi$
$198$	0	0
$199$	7.23607	0.512951	0.256476	−	0.966551i	$-0.417439\pi$
$199$	7.23607	0.512951	0.256476	+	0.966551i	$0.417439\pi$
$200$	0	0
$201$	13.7082	0.966902
$202$	0	0
$203$	5.00000i	0.350931i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 46.5967i	− 3.23870i
$208$	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
$213$	28.1803i	1.93089i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.70820i	0.251729i
$218$	0	0
$219$	28.3607	1.91644
$220$	0	0
$221$	3.05573	0.205551
$222$	0	0
$223$	20.1803i	1.35138i	0.737188	+	0.675688i	$0.236153\pi$
$223$	20.1803i	1.35138i	−0.737188	+	0.675688i	$0.763847\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.4164i	− 1.42146i	−0.703466	−	0.710728i	$-0.748365\pi$
$227$	− 21.4164i	− 1.42146i	0.703466	−	0.710728i	$-0.251635\pi$
$228$	0	0
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	0	0
$231$	−0.763932	−0.0502630
$232$	0	0
$233$	− 7.94427i	− 0.520447i	−0.965548	−	0.260223i	$-0.916204\pi$
$233$	− 7.94427i	− 0.520447i	0.965548	−	0.260223i	$-0.0837962\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	36.1803i	2.35017i
$238$	0	0
$239$	−5.52786	−0.357568	−0.178784	−	0.983888i	$-0.557216\pi$
$239$	−5.52786	−0.357568	−0.178784	+	0.983888i	$0.557216\pi$
$240$	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
$243$	− 35.5967i	− 2.28353i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.52786i	0.351730i
$248$	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
$253$	1.47214i	0.0925524i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.6525i	− 0.789240i	−0.918844	−	0.394620i	$-0.870876\pi$
$257$	− 12.6525i	− 0.789240i	0.918844	−	0.394620i	$-0.129124\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	37.3607	2.31257
$262$	0	0
$263$	16.2361i	1.00116i	0.865691	+	0.500579i	$0.166880\pi$
$263$	16.2361i	1.00116i	−0.865691	+	0.500579i	$0.833120\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	55.7771i	3.41350i
$268$	0	0
$269$	11.7082	0.713862	0.356931	−	0.934131i	$-0.383823\pi$
$269$	11.7082	0.713862	0.356931	+	0.934131i	$0.383823\pi$
$270$	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
$273$	4.00000i	0.242091i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 19.8885i	− 1.19499i	−0.801874	−	0.597493i	$-0.796163\pi$
$277$	− 19.8885i	− 1.19499i	0.801874	−	0.597493i	$-0.203837\pi$
$278$	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
$283$	17.4164i	1.03530i	0.855593	+	0.517649i	$0.173193\pi$
$283$	17.4164i	1.03530i	−0.855593	+	0.517649i	$0.826807\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.76393i	− 0.281206i
$288$	0	0
$289$	10.8885	0.640503
$290$	0	0
$291$	16.9443	0.993291
$292$	0	0
$293$	31.1246i	1.81832i	0.416448	+	0.909160i	$0.363275\pi$
$293$	31.1246i	1.81832i	−0.416448	+	0.909160i	$0.636725\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.41641i	0.198240i
$298$	0	0
$299$	7.70820	0.445777
$300$	0	0
$301$	1.76393	0.101671
$302$	0	0
$303$	− 15.4164i	− 0.885649i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.58359i	− 0.261599i	−0.991409	−	0.130800i	$-0.958246\pi$
$307$	− 4.58359i	− 0.261599i	0.991409	−	0.130800i	$-0.0417545\pi$
$308$	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
$313$	− 19.5279i	− 1.10378i	−0.833917	−	0.551890i	$-0.813907\pi$
$313$	− 19.5279i	− 1.10378i	0.833917	−	0.551890i	$-0.186093\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.3607i	1.42440i	0.701978	+	0.712199i	$0.252301\pi$
$317$	25.3607i	1.42440i	−0.701978	+	0.712199i	$0.747699\pi$
$318$	0	0
$319$	−1.18034	−0.0660863
$320$	0	0
$321$	−25.8885	−1.44496
$322$	0	0
$323$	− 11.0557i	− 0.615157i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	27.2361i	1.50616i
$328$	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
$333$	− 22.4164i	− 1.22841i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 16.4721i	− 0.897294i	−0.893709	−	0.448647i	$-0.851906\pi$
$337$	− 16.4721i	− 0.897294i	0.893709	−	0.448647i	$-0.148094\pi$
$338$	0	0
$339$	46.6525	2.53381
$340$	0	0
$341$	−0.875388	−0.0474049
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.2361i	− 1.08633i	−0.839626	−	0.543165i	$-0.817226\pi$
$347$	− 20.2361i	− 1.08633i	0.839626	−	0.543165i	$-0.182774\pi$
$348$	0	0
$349$	−4.47214	−0.239388	−0.119694	−	0.992811i	$-0.538191\pi$
$349$	−4.47214	−0.239388	−0.119694	+	0.992811i	$0.538191\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	0	0
$353$	2.18034i	0.116048i	0.998315	+	0.0580239i	$0.0184800\pi$
$353$	2.18034i	0.116048i	−0.998315	+	0.0580239i	$0.981520\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.00000i	− 0.423405i
$358$	0	0
$359$	−30.1246	−1.58992	−0.794958	−	0.606664i	$-0.792507\pi$
$359$	−30.1246	−1.58992	−0.794958	+	0.606664i	$0.792507\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	35.4164i	1.85888i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	37.1246i	1.93789i	0.247278	+	0.968944i	$0.420464\pi$
$367$	37.1246i	1.93789i	−0.247278	+	0.968944i	$0.579536\pi$
$368$	0	0
$369$	−35.5967	−1.85309
$370$	0	0
$371$	−8.47214	−0.439851
$372$	0	0
$373$	37.8328i	1.95891i	0.201665	+	0.979454i	$0.435365\pi$
$373$	37.8328i	1.95891i	−0.201665	+	0.979454i	$0.564635\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.18034i	0.318304i
$378$	0	0
$379$	−11.1803	−0.574295	−0.287148	−	0.957886i	$-0.592707\pi$
$379$	−11.1803	−0.574295	−0.287148	+	0.957886i	$0.592707\pi$
$380$	0	0
$381$	−44.1803	−2.26343
$382$	0	0
$383$	− 33.2361i	− 1.69828i	−0.528165	−	0.849142i	$-0.677120\pi$
$383$	− 33.2361i	− 1.69828i	0.528165	−	0.849142i	$-0.322880\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 13.1803i	− 0.669994i
$388$	0	0
$389$	−2.88854	−0.146455	−0.0732275	−	0.997315i	$-0.523330\pi$
$389$	−2.88854	−0.146455	−0.0732275	+	0.997315i	$0.523330\pi$
$390$	0	0
$391$	−15.4164	−0.779641
$392$	0	0
$393$	− 54.8328i	− 2.76595i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.05573i	0.454494i	0.973837	+	0.227247i	$0.0729725\pi$
$397$	9.05573i	0.454494i	−0.973837	+	0.227247i	$0.927028\pi$
$398$	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
$403$	4.58359i	0.228325i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.708204i	0.0351044i
$408$	0	0
$409$	−24.4721	−1.21007	−0.605035	−	0.796199i	$-0.706841\pi$
$409$	−24.4721	−1.21007	−0.605035	+	0.796199i	$0.706841\pi$
$410$	0	0
$411$	−35.4164	−1.74696
$412$	0	0
$413$	− 11.7082i	− 0.576123i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 34.4721i	− 1.68811i
$418$	0	0
$419$	−26.1803	−1.27899	−0.639497	−	0.768794i	$-0.720857\pi$
$419$	−26.1803	−1.27899	−0.639497	+	0.768794i	$0.720857\pi$
$420$	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
$423$	− 14.9443i	− 0.726615i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.70820i	0.469813i
$428$	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
$433$	28.3607i	1.36293i	0.731852	+	0.681464i	$0.238656\pi$
$433$	28.3607i	1.36293i	−0.731852	+	0.681464i	$0.761344\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 27.8885i	− 1.33409i
$438$	0	0
$439$	−8.29180	−0.395746	−0.197873	−	0.980228i	$-0.563403\pi$
$439$	−8.29180	−0.395746	−0.197873	+	0.980228i	$0.563403\pi$
$440$	0	0
$441$	7.47214	0.355816
$442$	0	0
$443$	− 19.4164i	− 0.922501i	−0.887270	−	0.461251i	$-0.847401\pi$
$443$	− 19.4164i	− 0.922501i	0.887270	−	0.461251i	$-0.152599\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.7639i	0.603713i
$448$	0	0
$449$	−20.5279	−0.968770	−0.484385	−	0.874855i	$-0.660957\pi$
$449$	−20.5279	−0.968770	−0.484385	+	0.874855i	$0.660957\pi$
$450$	0	0
$451$	1.12461	0.0529559
$452$	0	0
$453$	− 65.4853i	− 3.07677i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 12.5279i	− 0.586029i	−0.956108	−	0.293014i	$-0.905342\pi$
$457$	− 12.5279i	− 0.586029i	0.956108	−	0.293014i	$-0.0946584\pi$
$458$	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
$463$	− 13.8885i	− 0.645455i	−0.946492	−	0.322728i	$-0.895400\pi$
$463$	− 13.8885i	− 0.645455i	0.946492	−	0.322728i	$-0.104600\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.94427i	− 0.321343i	−0.987008	−	0.160671i	$-0.948634\pi$
$467$	− 6.94427i	− 0.321343i	0.987008	−	0.160671i	$-0.0513659\pi$
$468$	0	0
$469$	4.23607	0.195603
$470$	0	0
$471$	2.47214	0.113910
$472$	0	0
$473$	0.416408i	0.0191465i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	63.3050i	2.89853i
$478$	0	0
$479$	−26.1803	−1.19621	−0.598105	−	0.801418i	$-0.704079\pi$
$479$	−26.1803	−1.19621	−0.598105	+	0.801418i	$0.704079\pi$
$480$	0	0
$481$	3.70820	0.169080
$482$	0	0
$483$	− 20.1803i	− 0.918237i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.76393i	− 0.261189i	−0.991436	−	0.130594i	$-0.958311\pi$
$487$	− 5.76393i	− 0.261189i	0.991436	−	0.130594i	$-0.0416885\pi$
$488$	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
$493$	− 12.3607i	− 0.556697i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.70820i	0.390616i
$498$	0	0
$499$	11.0557	0.494922	0.247461	−	0.968898i	$-0.420404\pi$
$499$	11.0557	0.494922	0.247461	+	0.968898i	$0.420404\pi$
$500$	0	0
$501$	−16.9443	−0.757014
$502$	0	0
$503$	− 8.11146i	− 0.361672i	−0.983513	−	0.180836i	$-0.942120\pi$
$503$	− 8.11146i	− 0.361672i	0.983513	−	0.180836i	$-0.0578803\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 37.1246i	− 1.64876i
$508$	0	0
$509$	−40.6525	−1.80189	−0.900945	−	0.433934i	$-0.857125\pi$
$509$	−40.6525	−1.80189	−0.900945	+	0.433934i	$0.857125\pi$
$510$	0	0
$511$	8.76393	0.387694
$512$	0	0
$513$	− 64.7214i	− 2.85752i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.472136i	0.0207645i
$518$	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
$523$	16.3607i	0.715403i	0.933836	+	0.357701i	$0.116439\pi$
$523$	16.3607i	0.715403i	−0.933836	+	0.357701i	$0.883561\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.16718i	− 0.399329i
$528$	0	0
$529$	−15.8885	−0.690806
$530$	0	0
$531$	−87.4853	−3.79654
$532$	0	0
$533$	− 5.88854i	− 0.255061i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	75.7771i	3.27002i
$538$	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
$543$	− 26.4721i	− 1.13603i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.76393i	0.417476i	0.977972	+	0.208738i	$0.0669355\pi$
$547$	9.76393i	0.417476i	−0.977972	+	0.208738i	$0.933064\pi$
$548$	0	0
$549$	72.5410	3.09598
$550$	0	0
$551$	22.3607	0.952597
$552$	0	0
$553$	11.1803i	0.475436i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.11146i	− 0.386065i	−0.981192	−	0.193032i	$-0.938168\pi$
$557$	− 9.11146i	− 0.386065i	0.981192	−	0.193032i	$-0.0618322\pi$
$558$	0	0
$559$	2.18034	0.0922186
$560$	0	0
$561$	1.88854	0.0797344
$562$	0	0
$563$	17.4164i	0.734014i	0.930218	+	0.367007i	$0.119618\pi$
$563$	17.4164i	0.734014i	−0.930218	+	0.367007i	$0.880382\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 24.4164i	− 1.02539i
$568$	0	0
$569$	3.94427	0.165352	0.0826762	−	0.996576i	$-0.473653\pi$
$569$	3.94427	0.165352	0.0826762	+	0.996576i	$0.473653\pi$
$570$	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
$573$	20.9443i	0.874960i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	0	0
$579$	−40.1803	−1.66984
$580$	0	0
$581$	−7.70820	−0.319790
$582$	0	0
$583$	− 2.00000i	− 0.0828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.7639i	1.02212i	0.859546	+	0.511058i	$0.170746\pi$
$587$	24.7639i	1.02212i	−0.859546	+	0.511058i	$0.829254\pi$
$588$	0	0
$589$	16.5836	0.683315
$590$	0	0
$591$	−4.76393	−0.195962
$592$	0	0
$593$	37.3050i	1.53193i	0.642882	+	0.765965i	$0.277739\pi$
$593$	37.3050i	1.53193i	−0.642882	+	0.765965i	$0.722261\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 23.4164i	− 0.958370i
$598$	0	0
$599$	−11.1803	−0.456816	−0.228408	−	0.973565i	$-0.573352\pi$
$599$	−11.1803	−0.456816	−0.228408	+	0.973565i	$0.573352\pi$
$600$	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
$603$	− 31.6525i	− 1.28899i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.12461i	0.289179i	0.989492	+	0.144590i	$0.0461862\pi$
$607$	7.12461i	0.289179i	−0.989492	+	0.144590i	$0.953814\pi$
$608$	0	0
$609$	16.1803	0.655660
$610$	0	0
$611$	2.47214	0.100012
$612$	0	0
$613$	44.4164i	1.79396i	0.442069	+	0.896981i	$0.354245\pi$
$613$	44.4164i	1.79396i	−0.442069	+	0.896981i	$0.645755\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.94427i	− 0.239307i	−0.992816	−	0.119654i	$-0.961822\pi$
$617$	− 5.94427i	− 0.239307i	0.992816	−	0.119654i	$-0.0381784\pi$
$618$	0	0
$619$	−11.7082	−0.470592	−0.235296	−	0.971924i	$-0.575606\pi$
$619$	−11.7082	−0.470592	−0.235296	+	0.971924i	$0.575606\pi$
$620$	0	0
$621$	−90.2492	−3.62158
$622$	0	0
$623$	17.2361i	0.690548i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.41641i	0.136438i
$628$	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
$633$	38.8328i	1.54347i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.23607i	0.0489748i
$638$	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
$643$	18.4721i	0.728470i	0.931307	+	0.364235i	$0.118669\pi$
$643$	18.4721i	0.728470i	−0.931307	+	0.364235i	$0.881331\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.8885i	0.781899i	0.920412	+	0.390950i	$0.127853\pi$
$647$	19.8885i	0.781899i	−0.920412	+	0.390950i	$0.872147\pi$
$648$	0	0
$649$	2.76393	0.108494
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	− 25.0557i	− 0.980506i	−0.871580	−	0.490253i	$-0.836904\pi$
$653$	− 25.0557i	− 0.980506i	0.871580	−	0.490253i	$-0.163096\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 65.4853i	− 2.55482i
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	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
$663$	− 9.88854i	− 0.384039i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 31.1803i	− 1.20731i
$668$	0	0
$669$	65.3050	2.52484
$670$	0	0
$671$	−2.29180	−0.0884738
$672$	0	0
$673$	− 19.5279i	− 0.752744i	−0.926469	−	0.376372i	$-0.877171\pi$
$673$	− 19.5279i	− 0.752744i	0.926469	−	0.376372i	$-0.122829\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.3607i	− 0.551926i	−0.961168	−	0.275963i	$-0.911003\pi$
$677$	− 14.3607i	− 0.551926i	0.961168	−	0.275963i	$-0.0889967\pi$
$678$	0	0
$679$	5.23607	0.200942
$680$	0	0
$681$	−69.3050	−2.65577
$682$	0	0
$683$	14.1246i	0.540463i	0.962795	+	0.270232i	$0.0871003\pi$
$683$	14.1246i	0.540463i	−0.962795	+	0.270232i	$0.912900\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 14.4721i	− 0.552146i
$688$	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
$693$	1.76393i	0.0670062i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.7771i	0.446089i
$698$	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
$703$	− 13.4164i	− 0.506009i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.76393i	− 0.179166i
$708$	0	0
$709$	12.1115	0.454855	0.227428	−	0.973795i	$-0.426968\pi$
$709$	12.1115	0.454855	0.227428	+	0.973795i	$0.426968\pi$
$710$	0	0
$711$	83.5410	3.13303
$712$	0	0
$713$	− 23.1246i	− 0.866024i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.8885i	0.668060i
$718$	0	0
$719$	16.1803	0.603425	0.301712	−	0.953399i	$-0.402442\pi$
$719$	16.1803	0.603425	0.301712	+	0.953399i	$0.402442\pi$
$720$	0	0
$721$	8.47214	0.315519
$722$	0	0
$723$	11.4164i	0.424581i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.05573i	0.113331i	0.998393	+	0.0566653i	$0.0180468\pi$
$727$	3.05573i	0.113331i	−0.998393	+	0.0566653i	$0.981953\pi$
$728$	0	0
$729$	−41.9443	−1.55349
$730$	0	0
$731$	−4.36068	−0.161286
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.00000i	0.0368355i
$738$	0	0
$739$	25.6525	0.943642	0.471821	−	0.881694i	$-0.343597\pi$
$739$	25.6525	0.943642	0.471821	+	0.881694i	$0.343597\pi$
$740$	0	0
$741$	17.8885	0.657152
$742$	0	0
$743$	− 10.4721i	− 0.384185i	−0.981377	−	0.192093i	$-0.938473\pi$
$743$	− 10.4721i	− 0.384185i	0.981377	−	0.192093i	$-0.0615274\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	57.5967i	2.10735i
$748$	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
$753$	20.9443i	0.763252i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19.5836i	0.711778i	0.934528	+	0.355889i	$0.115822\pi$
$757$	19.5836i	0.711778i	−0.934528	+	0.355889i	$0.884178\pi$
$758$	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
$763$	8.41641i	0.304694i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 14.4721i	− 0.522559i
$768$	0	0
$769$	43.0132	1.55109	0.775547	−	0.631290i	$-0.217474\pi$
$769$	43.0132	1.55109	0.775547	+	0.631290i	$0.217474\pi$
$770$	0	0
$771$	−40.9443	−1.47457
$772$	0	0
$773$	− 50.1803i	− 1.80486i	−0.430835	−	0.902431i	$-0.641781\pi$
$773$	− 50.1803i	− 1.80486i	0.430835	−	0.902431i	$-0.358219\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 9.70820i	− 0.348280i
$778$	0	0
$779$	−21.3050	−0.763329
$780$	0	0
$781$	−2.05573	−0.0735597
$782$	0	0
$783$	− 72.3607i	− 2.58596i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 40.7639i	− 1.45308i	−0.687126	−	0.726539i	$-0.741128\pi$
$787$	− 40.7639i	− 1.45308i	0.687126	−	0.726539i	$-0.258872\pi$
$788$	0	0
$789$	52.5410	1.87051
$790$	0	0
$791$	14.4164	0.512588
$792$	0	0
$793$	12.0000i	0.426132i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 35.4164i	− 1.25451i	−0.778813	−	0.627257i	$-0.784178\pi$
$797$	− 35.4164i	− 1.25451i	0.778813	−	0.627257i	$-0.215822\pi$
$798$	0	0
$799$	−4.94427	−0.174916
$800$	0	0
$801$	128.790	4.55058
$802$	0	0
$803$	2.06888i	0.0730093i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 37.8885i	− 1.33374i
$808$	0	0
$809$	−29.4721	−1.03619	−0.518093	−	0.855325i	$-0.673358\pi$
$809$	−29.4721	−1.03619	−0.518093	+	0.855325i	$0.673358\pi$
$810$	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
$813$	76.7214i	2.69074i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.88854i	− 0.275985i
$818$	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
$823$	− 31.6525i	− 1.10334i	−0.834064	−	0.551668i	$-0.813992\pi$
$823$	− 31.6525i	− 1.10334i	0.834064	−	0.551668i	$-0.186008\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 41.5410i	− 1.44452i	−0.691620	−	0.722261i	$-0.743103\pi$
$827$	− 41.5410i	− 1.44452i	0.691620	−	0.722261i	$-0.256897\pi$
$828$	0	0
$829$	7.63932	0.265325	0.132662	−	0.991161i	$-0.457647\pi$
$829$	7.63932	0.265325	0.132662	+	0.991161i	$0.457647\pi$
$830$	0	0
$831$	−64.3607	−2.23265
$832$	0	0
$833$	− 2.47214i	− 0.0856544i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 53.6656i	− 1.85496i
$838$	0	0
$839$	−30.6525	−1.05824	−0.529120	−	0.848547i	$-0.677478\pi$
$839$	−30.6525	−1.05824	−0.529120	+	0.848547i	$0.677478\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	49.7082i	1.71204i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.9443i	0.376050i
$848$	0	0
$849$	56.3607	1.93429
$850$	0	0
$851$	−18.7082	−0.641309
$852$	0	0
$853$	− 27.4164i	− 0.938720i	−0.883007	−	0.469360i	$-0.844485\pi$
$853$	− 27.4164i	− 0.938720i	0.883007	−	0.469360i	$-0.155515\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15.8197i	− 0.540389i	−0.962806	−	0.270195i	$-0.912912\pi$
$857$	− 15.8197i	− 0.540389i	0.962806	−	0.270195i	$-0.0870881\pi$
$858$	0	0
$859$	22.3607	0.762937	0.381468	−	0.924382i	$-0.375419\pi$
$859$	22.3607	0.762937	0.381468	+	0.924382i	$0.375419\pi$
$860$	0	0
$861$	−15.4164	−0.525390
$862$	0	0
$863$	18.3475i	0.624557i	0.949991	+	0.312278i	$0.101092\pi$
$863$	18.3475i	0.624557i	−0.949991	+	0.312278i	$0.898908\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 35.2361i	− 1.19668i
$868$	0	0
$869$	−2.63932	−0.0895328
$870$	0	0
$871$	5.23607	0.177417
$872$	0	0
$873$	− 39.1246i	− 1.32417i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.3607i	1.02521i	0.858625	+	0.512604i	$0.171319\pi$
$877$	30.3607i	1.02521i	−0.858625	+	0.512604i	$0.828681\pi$
$878$	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
$883$	− 1.40325i	− 0.0472232i	−0.999721	−	0.0236116i	$-0.992483\pi$
$883$	− 1.40325i	− 0.0472232i	0.999721	−	0.0236116i	$-0.00751650\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.3475i	0.716780i	0.933572	+	0.358390i	$0.116674\pi$
$887$	21.3475i	0.716780i	−0.933572	+	0.358390i	$0.883326\pi$
$888$	0	0
$889$	−13.6525	−0.457889
$890$	0	0
$891$	5.76393	0.193099
$892$	0	0
$893$	− 8.94427i	− 0.299309i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 24.9443i	− 0.832865i
$898$	0	0
$899$	18.5410	0.618378
$900$	0	0
$901$	20.9443	0.697755
$902$	0	0
$903$	− 5.70820i	− 0.189957i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 34.8328i	− 1.15660i	−0.815823	−	0.578302i	$-0.803715\pi$
$907$	− 34.8328i	− 1.15660i	0.815823	−	0.578302i	$-0.196285\pi$
$908$	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
$913$	− 1.81966i	− 0.0602220i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.9443i	− 0.559549i
$918$	0	0
$919$	−27.7639	−0.915848	−0.457924	−	0.888991i	$-0.651407\pi$
$919$	−27.7639	−0.915848	−0.457924	+	0.888991i	$0.651407\pi$
$920$	0	0
$921$	−14.8328	−0.488758
$922$	0	0
$923$	10.7639i	0.354299i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 63.3050i	− 2.07921i
$928$	0	0
$929$	−38.2918	−1.25631	−0.628157	−	0.778087i	$-0.716190\pi$
$929$	−38.2918	−1.25631	−0.628157	+	0.778087i	$0.716190\pi$
$930$	0	0
$931$	4.47214	0.146568
$932$	0	0
$933$	78.8328i	2.58087i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.2361i	1.15111i	0.817762	+	0.575556i	$0.195214\pi$
$937$	35.2361i	1.15111i	−0.817762	+	0.575556i	$0.804786\pi$
$938$	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
$943$	29.7082i	0.967432i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34.8328i	− 1.13191i	−0.824435	−	0.565957i	$-0.808507\pi$
$947$	− 34.8328i	− 1.13191i	0.824435	−	0.565957i	$-0.191493\pi$
$948$	0	0
$949$	10.8328	0.351648
$950$	0	0
$951$	82.0689	2.66127
$952$	0	0
$953$	− 3.47214i	− 0.112474i	−0.998417	−	0.0562368i	$-0.982090\pi$
$953$	− 3.47214i	− 0.112474i	0.998417	−	0.0562368i	$-0.0179102\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.81966i	0.123472i
$958$	0	0
$959$	−10.9443	−0.353409
$960$	0	0
$961$	−17.2492	−0.556427
$962$	0	0
$963$	59.7771i	1.92629i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.1115i	0.453794i	0.973919	+	0.226897i	$0.0728580\pi$
$967$	14.1115i	0.453794i	−0.973919	+	0.226897i	$0.927142\pi$
$968$	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
$973$	− 10.6525i	− 0.341503i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11.4721i	− 0.367026i	−0.983017	−	0.183513i	$-0.941253\pi$
$977$	− 11.4721i	− 0.367026i	0.983017	−	0.183513i	$-0.0587470\pi$
$978$	0	0
$979$	−4.06888	−0.130042
$980$	0	0
$981$	62.8885	2.00788
$982$	0	0
$983$	− 34.5410i	− 1.10169i	−0.834608	−	0.550844i	$-0.814306\pi$
$983$	− 34.5410i	− 1.10169i	0.834608	−	0.550844i	$-0.185694\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.47214i	− 0.206010i
$988$	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
$993$	− 79.9574i	− 2.53737i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 45.4164i	− 1.43835i	−0.694828	−	0.719176i	$-0.744519\pi$
$997$	− 45.4164i	− 1.43835i	0.694828	−	0.719176i	$-0.255481\pi$
$998$	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.g.s.449.1		4
4.3	odd	2		175.2.b.c.99.2		4
5.2	odd	4		2800.2.a.bh.1.1		2
5.3	odd	4		2800.2.a.bp.1.2		2
5.4	even	2	inner	2800.2.g.s.449.4		4
12.11	even	2		1575.2.d.k.1324.3		4
20.3	even	4		175.2.a.e.1.1	yes	2
20.7	even	4		175.2.a.d.1.2	✓	2
20.19	odd	2		175.2.b.c.99.3		4
28.27	even	2		1225.2.b.k.99.2		4
60.23	odd	4		1575.2.a.n.1.2		2
60.47	odd	4		1575.2.a.s.1.1		2
60.59	even	2		1575.2.d.k.1324.2		4
140.27	odd	4		1225.2.a.n.1.2		2
140.83	odd	4		1225.2.a.u.1.1		2
140.139	even	2		1225.2.b.k.99.3		4

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

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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +O(q^{10})\)	\(q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +0.236068 q^{11} -1.23607i q^{13} +2.47214i q^{17} -4.47214 q^{19} -3.23607 q^{21} +6.23607i q^{23} +14.4721i q^{27} -5.00000 q^{29} -3.70820 q^{31} -0.763932i q^{33} +3.00000i q^{37} -4.00000 q^{39} +4.76393 q^{41} +1.76393i q^{43} +2.00000i q^{47} -1.00000 q^{49} +8.00000 q^{51} -8.47214i q^{53} +14.4721i q^{57} +11.7082 q^{59} -9.70820 q^{61} +7.47214i q^{63} +4.23607i q^{67} +20.1803 q^{69} -8.70820 q^{71} +8.76393i q^{73} -0.236068i q^{77} -11.1803 q^{79} +24.4164 q^{81} -7.70820i q^{83} +16.1803i q^{87} -17.2361 q^{89} -1.23607 q^{91} +12.0000i q^{93} +5.23607i q^{97} -1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 8 q^{11} - 4 q^{21} - 20 q^{29} + 12 q^{31} - 16 q^{39} + 28 q^{41} - 4 q^{49} + 32 q^{51} + 20 q^{59} - 12 q^{61} + 36 q^{69} - 8 q^{71} + 44 q^{81} - 60 q^{89} + 4 q^{91} - 16 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 8 * q^11 - 4 * q^21 - 20 * q^29 + 12 * q^31 - 16 * q^39 + 28 * q^41 - 4 * q^49 + 32 * q^51 + 20 * q^59 - 12 * q^61 + 36 * q^69 - 8 * q^71 + 44 * q^81 - 60 * q^89 + 4 * q^91 - 16 * q^99

Introduction

L-functions

Modular forms

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Database

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