LMFDB - Embedded newform 9800.2.a.cn.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28825 q^{3} +2.23607 q^{9} +O(q^{10})$	$q+2.28825 q^{3} +2.23607 q^{9} -1.00000 q^{11} +0.874032 q^{13} -4.57649 q^{17} -3.16228 q^{19} +1.00000 q^{23} -1.74806 q^{27} -8.70820 q^{29} +2.62210 q^{31} -2.28825 q^{33} +0.236068 q^{37} +2.00000 q^{39} +9.69316 q^{41} -2.23607 q^{43} +1.41421 q^{47} -10.4721 q^{51} -1.23607 q^{53} -7.23607 q^{57} -11.1074 q^{59} +5.11667 q^{61} +7.47214 q^{67} +2.28825 q^{69} +6.70820 q^{71} -2.95595 q^{73} -11.9443 q^{79} -10.7082 q^{81} -8.27895 q^{83} -19.9265 q^{87} +1.95440 q^{89} +6.00000 q^{93} -12.8554 q^{97} -2.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} + 4 q^{23} - 8 q^{29} - 8 q^{37} + 8 q^{39} - 24 q^{51} + 4 q^{53} - 20 q^{57} + 12 q^{67} - 12 q^{79} - 16 q^{81} + 24 q^{93}+O(q^{100}) $ 4 * q - 4 * q^11 + 4 * q^23 - 8 * q^29 - 8 * q^37 + 8 * q^39 - 24 * q^51 + 4 * q^53 - 20 * q^57 + 12 * q^67 - 12 * q^79 - 16 * q^81 + 24 * q^93

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
$3$	2.28825	1.32112	0.660560	−	0.750774i	$-0.270319\pi$
$3$	2.28825	1.32112	0.660560	+	0.750774i	$0.270319\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.23607	0.745356
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	0.874032	0.242413	0.121206	−	0.992627i	$-0.461324\pi$
$13$	0.874032	0.242413	0.121206	+	0.992627i	$0.461324\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.57649	−1.10996	−0.554981	−	0.831863i	$-0.687275\pi$
$17$	−4.57649	−1.10996	−0.554981	+	0.831863i	$0.687275\pi$
$18$	0	0
$19$	−3.16228	−0.725476	−0.362738	−	0.931891i	$-0.618158\pi$
$19$	−3.16228	−0.725476	−0.362738	+	0.931891i	$0.618158\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.74806	−0.336415
$28$	0	0
$29$	−8.70820	−1.61707	−0.808536	−	0.588446i	$-0.799740\pi$
$29$	−8.70820	−1.61707	−0.808536	+	0.588446i	$0.799740\pi$
$30$	0	0
$31$	2.62210	0.470942	0.235471	−	0.971881i	$-0.424337\pi$
$31$	2.62210	0.470942	0.235471	+	0.971881i	$0.424337\pi$
$32$	0	0
$33$	−2.28825	−0.398332
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	9.69316	1.51382	0.756909	−	0.653520i	$-0.226709\pi$
$41$	9.69316	1.51382	0.756909	+	0.653520i	$0.226709\pi$
$42$	0	0
$43$	−2.23607	−0.340997	−0.170499	−	0.985358i	$-0.554538\pi$
$43$	−2.23607	−0.340997	−0.170499	+	0.985358i	$0.554538\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.41421	0.206284	0.103142	−	0.994667i	$-0.467110\pi$
$47$	1.41421	0.206284	0.103142	+	0.994667i	$0.467110\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−10.4721	−1.46639
$52$	0	0
$53$	−1.23607	−0.169787	−0.0848935	−	0.996390i	$-0.527055\pi$
$53$	−1.23607	−0.169787	−0.0848935	+	0.996390i	$0.527055\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.23607	−0.958441
$58$	0	0
$59$	−11.1074	−1.44606	−0.723029	−	0.690818i	$-0.757251\pi$
$59$	−11.1074	−1.44606	−0.723029	+	0.690818i	$0.757251\pi$
$60$	0	0
$61$	5.11667	0.655123	0.327561	−	0.944830i	$-0.393773\pi$
$61$	5.11667	0.655123	0.327561	+	0.944830i	$0.393773\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.47214	0.912867	0.456433	−	0.889758i	$-0.349127\pi$
$67$	7.47214	0.912867	0.456433	+	0.889758i	$0.349127\pi$
$68$	0	0
$69$	2.28825	0.275472
$70$	0	0
$71$	6.70820	0.796117	0.398059	−	0.917360i	$-0.369684\pi$
$71$	6.70820	0.796117	0.398059	+	0.917360i	$0.369684\pi$
$72$	0	0
$73$	−2.95595	−0.345967	−0.172984	−	0.984925i	$-0.555341\pi$
$73$	−2.95595	−0.345967	−0.172984	+	0.984925i	$0.555341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.9443	−1.34384	−0.671918	−	0.740626i	$-0.734529\pi$
$79$	−11.9443	−1.34384	−0.671918	+	0.740626i	$0.734529\pi$
$80$	0	0
$81$	−10.7082	−1.18980
$82$	0	0
$83$	−8.27895	−0.908733	−0.454366	−	0.890815i	$-0.650134\pi$
$83$	−8.27895	−0.908733	−0.454366	+	0.890815i	$0.650134\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−19.9265	−2.13635
$88$	0	0
$89$	1.95440	0.207165	0.103583	−	0.994621i	$-0.466969\pi$
$89$	1.95440	0.207165	0.103583	+	0.994621i	$0.466969\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.8554	−1.30527	−0.652636	−	0.757671i	$-0.726337\pi$
$97$	−12.8554	−1.30527	−0.652636	+	0.757671i	$0.726337\pi$
$98$	0	0
$99$	−2.23607	−0.224733
$100$	0	0
$101$	−9.02546	−0.898067	−0.449034	−	0.893515i	$-0.648232\pi$
$101$	−9.02546	−0.898067	−0.449034	+	0.893515i	$0.648232\pi$
$102$	0	0
$103$	−10.5672	−1.04122	−0.520608	−	0.853796i	$-0.674295\pi$
$103$	−10.5672	−1.04122	−0.520608	+	0.853796i	$0.674295\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.4164	1.29701	0.648507	−	0.761209i	$-0.275394\pi$
$107$	13.4164	1.29701	0.648507	+	0.761209i	$0.275394\pi$
$108$	0	0
$109$	−2.52786	−0.242125	−0.121063	−	0.992645i	$-0.538630\pi$
$109$	−2.52786	−0.242125	−0.121063	+	0.992645i	$0.538630\pi$
$110$	0	0
$111$	0.540182	0.0512718
$112$	0	0
$113$	−15.1803	−1.42805	−0.714023	−	0.700122i	$-0.753129\pi$
$113$	−15.1803	−1.42805	−0.714023	+	0.700122i	$0.753129\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.95440	0.180684
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	22.1803	1.99993
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−22.1246	−1.96324	−0.981621	−	0.190841i	$-0.938878\pi$
$127$	−22.1246	−1.96324	−0.981621	+	0.190841i	$0.938878\pi$
$128$	0	0
$129$	−5.11667	−0.450498
$130$	0	0
$131$	2.16073	0.188784	0.0943918	−	0.995535i	$-0.469909\pi$
$131$	2.16073	0.188784	0.0943918	+	0.995535i	$0.469909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.05573	−0.0901969	−0.0450985	−	0.998983i	$-0.514360\pi$
$137$	−1.05573	−0.0901969	−0.0450985	+	0.998983i	$0.514360\pi$
$138$	0	0
$139$	−14.9374	−1.26697	−0.633485	−	0.773755i	$-0.718376\pi$
$139$	−14.9374	−1.26697	−0.633485	+	0.773755i	$0.718376\pi$
$140$	0	0
$141$	3.23607	0.272526
$142$	0	0
$143$	−0.874032	−0.0730902
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.6525	1.28230	0.641150	−	0.767415i	$-0.278457\pi$
$149$	15.6525	1.28230	0.641150	+	0.767415i	$0.278457\pi$
$150$	0	0
$151$	−13.7639	−1.12009	−0.560046	−	0.828461i	$-0.689217\pi$
$151$	−13.7639	−1.12009	−0.560046	+	0.828461i	$0.689217\pi$
$152$	0	0
$153$	−10.2333	−0.827317
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−21.6746	−1.72982	−0.864910	−	0.501928i	$-0.832624\pi$
$157$	−21.6746	−1.72982	−0.864910	+	0.501928i	$0.832624\pi$
$158$	0	0
$159$	−2.82843	−0.224309
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.2361	1.03673	0.518364	−	0.855160i	$-0.326541\pi$
$163$	13.2361	1.03673	0.518364	+	0.855160i	$0.326541\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.7642	1.29726	0.648628	−	0.761106i	$-0.275343\pi$
$167$	16.7642	1.29726	0.648628	+	0.761106i	$0.275343\pi$
$168$	0	0
$169$	−12.2361	−0.941236
$170$	0	0
$171$	−7.07107	−0.540738
$172$	0	0
$173$	8.40647	0.639132	0.319566	−	0.947564i	$-0.396463\pi$
$173$	8.40647	0.639132	0.319566	+	0.947564i	$0.396463\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−25.4164	−1.91041
$178$	0	0
$179$	16.1803	1.20938	0.604688	−	0.796463i	$-0.293298\pi$
$179$	16.1803	1.20938	0.604688	+	0.796463i	$0.293298\pi$
$180$	0	0
$181$	13.6020	1.01103	0.505513	−	0.862819i	$-0.331303\pi$
$181$	13.6020	1.01103	0.505513	+	0.862819i	$0.331303\pi$
$182$	0	0
$183$	11.7082	0.865495
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.57649	0.334666
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.7639	−1.21300	−0.606498	−	0.795085i	$-0.707426\pi$
$191$	−16.7639	−1.21300	−0.606498	+	0.795085i	$0.707426\pi$
$192$	0	0
$193$	2.23607	0.160956	0.0804778	−	0.996756i	$-0.474355\pi$
$193$	2.23607	0.160956	0.0804778	+	0.996756i	$0.474355\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.4164	−1.88209	−0.941046	−	0.338280i	$-0.890155\pi$
$197$	−26.4164	−1.88209	−0.941046	+	0.338280i	$0.890155\pi$
$198$	0	0
$199$	−12.5216	−0.887632	−0.443816	−	0.896118i	$-0.646376\pi$
$199$	−12.5216	−0.887632	−0.443816	+	0.896118i	$0.646376\pi$
$200$	0	0
$201$	17.0981	1.20601
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.23607	0.155417
$208$	0	0
$209$	3.16228	0.218739
$210$	0	0
$211$	−1.52786	−0.105182	−0.0525912	−	0.998616i	$-0.516748\pi$
$211$	−1.52786	−0.105182	−0.0525912	+	0.998616i	$0.516748\pi$
$212$	0	0
$213$	15.3500	1.05177
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.76393	−0.457064
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−8.61280	−0.576756	−0.288378	−	0.957517i	$-0.593116\pi$
$223$	−8.61280	−0.576756	−0.288378	+	0.957517i	$0.593116\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.5563	−1.03251	−0.516256	−	0.856435i	$-0.672675\pi$
$227$	−15.5563	−1.03251	−0.516256	+	0.856435i	$0.672675\pi$
$228$	0	0
$229$	22.5486	1.49005	0.745027	−	0.667034i	$-0.232437\pi$
$229$	22.5486	1.49005	0.745027	+	0.667034i	$0.232437\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.4164	1.86162	0.930810	−	0.365502i	$-0.119103\pi$
$233$	28.4164	1.86162	0.930810	+	0.365502i	$0.119103\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−27.3314	−1.77537
$238$	0	0
$239$	9.23607	0.597432	0.298716	−	0.954342i	$-0.403442\pi$
$239$	9.23607	0.597432	0.298716	+	0.954342i	$0.403442\pi$
$240$	0	0
$241$	−2.90724	−0.187272	−0.0936358	−	0.995607i	$-0.529849\pi$
$241$	−2.90724	−0.187272	−0.0936358	+	0.995607i	$0.529849\pi$
$242$	0	0
$243$	−19.2588	−1.23545
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.76393	−0.175865
$248$	0	0
$249$	−18.9443	−1.20054
$250$	0	0
$251$	1.74806	0.110337	0.0551684	−	0.998477i	$-0.482430\pi$
$251$	1.74806	0.110337	0.0551684	+	0.998477i	$0.482430\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.461370	−0.0287795	−0.0143897	−	0.999896i	$-0.504581\pi$
$257$	−0.461370	−0.0287795	−0.0143897	+	0.999896i	$0.504581\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−19.4721	−1.20529
$262$	0	0
$263$	3.29180	0.202981	0.101490	−	0.994837i	$-0.467639\pi$
$263$	3.29180	0.202981	0.101490	+	0.994837i	$0.467639\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.47214	0.273690
$268$	0	0
$269$	−15.6839	−0.956262	−0.478131	−	0.878288i	$-0.658686\pi$
$269$	−15.6839	−0.956262	−0.478131	+	0.878288i	$0.658686\pi$
$270$	0	0
$271$	28.7456	1.74617	0.873087	−	0.487565i	$-0.162115\pi$
$271$	28.7456	1.74617	0.873087	+	0.487565i	$0.162115\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.05573	−0.183601	−0.0918005	−	0.995777i	$-0.529262\pi$
$277$	−3.05573	−0.183601	−0.0918005	+	0.995777i	$0.529262\pi$
$278$	0	0
$279$	5.86319	0.351020
$280$	0	0
$281$	19.6525	1.17237	0.586184	−	0.810178i	$-0.300629\pi$
$281$	19.6525	1.17237	0.586184	+	0.810178i	$0.300629\pi$
$282$	0	0
$283$	5.73567	0.340950	0.170475	−	0.985362i	$-0.445470\pi$
$283$	5.73567	0.340950	0.170475	+	0.985362i	$0.445470\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.94427	0.232016
$290$	0	0
$291$	−29.4164	−1.72442
$292$	0	0
$293$	19.8477	1.15951	0.579757	−	0.814789i	$-0.303147\pi$
$293$	19.8477	1.15951	0.579757	+	0.814789i	$0.303147\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.74806	0.101433
$298$	0	0
$299$	0.874032	0.0505466
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−20.6525	−1.18645
$304$	0	0
$305$	0	0
$306$	0	0
$307$	30.0323	1.71404	0.857018	−	0.515287i	$-0.172314\pi$
$307$	30.0323	1.71404	0.857018	+	0.515287i	$0.172314\pi$
$308$	0	0
$309$	−24.1803	−1.37557
$310$	0	0
$311$	−18.0509	−1.02357	−0.511787	−	0.859112i	$-0.671016\pi$
$311$	−18.0509	−1.02357	−0.511787	+	0.859112i	$0.671016\pi$
$312$	0	0
$313$	10.5672	0.597293	0.298647	−	0.954364i	$-0.403465\pi$
$313$	10.5672	0.597293	0.298647	+	0.954364i	$0.403465\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.5279	−0.591304	−0.295652	−	0.955296i	$-0.595537\pi$
$317$	−10.5279	−0.591304	−0.295652	+	0.955296i	$0.595537\pi$
$318$	0	0
$319$	8.70820	0.487566
$320$	0	0
$321$	30.7000	1.71351
$322$	0	0
$323$	14.4721	0.805251
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.78437	−0.319877
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.0000	−0.824475	−0.412237	−	0.911077i	$-0.635253\pi$
$331$	−15.0000	−0.824475	−0.412237	+	0.911077i	$0.635253\pi$
$332$	0	0
$333$	0.527864	0.0289268
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.6525	1.01607	0.508033	−	0.861338i	$-0.330373\pi$
$337$	18.6525	1.01607	0.508033	+	0.861338i	$0.330373\pi$
$338$	0	0
$339$	−34.7363	−1.88662
$340$	0	0
$341$	−2.62210	−0.141994
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.8885	−0.584528	−0.292264	−	0.956338i	$-0.594409\pi$
$347$	−10.8885	−0.584528	−0.292264	+	0.956338i	$0.594409\pi$
$348$	0	0
$349$	−4.65530	−0.249193	−0.124596	−	0.992208i	$-0.539764\pi$
$349$	−4.65530	−0.249193	−0.124596	+	0.992208i	$0.539764\pi$
$350$	0	0
$351$	−1.52786	−0.0815513
$352$	0	0
$353$	19.5927	1.04281	0.521406	−	0.853309i	$-0.325408\pi$
$353$	19.5927	1.04281	0.521406	+	0.853309i	$0.325408\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.1803	−0.906744	−0.453372	−	0.891321i	$-0.649779\pi$
$359$	−17.1803	−0.906744	−0.453372	+	0.891321i	$0.649779\pi$
$360$	0	0
$361$	−9.00000	−0.473684
$362$	0	0
$363$	−22.8825	−1.20102
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.2666	−0.640309	−0.320154	−	0.947365i	$-0.603735\pi$
$367$	−12.2666	−0.640309	−0.320154	+	0.947365i	$0.603735\pi$
$368$	0	0
$369$	21.6746	1.12833
$370$	0	0
$371$	0	0
$372$	0	0
$373$	21.1803	1.09668	0.548338	−	0.836257i	$-0.315261\pi$
$373$	21.1803	1.09668	0.548338	+	0.836257i	$0.315261\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.61125	−0.391999
$378$	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
$381$	−50.6265	−2.59368
$382$	0	0
$383$	1.95440	0.0998649	0.0499325	−	0.998753i	$-0.484099\pi$
$383$	1.95440	0.0998649	0.0499325	+	0.998753i	$0.484099\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.00000	−0.254164
$388$	0	0
$389$	−23.1803	−1.17529	−0.587645	−	0.809119i	$-0.699945\pi$
$389$	−23.1803	−1.17529	−0.587645	+	0.809119i	$0.699945\pi$
$390$	0	0
$391$	−4.57649	−0.231443
$392$	0	0
$393$	4.94427	0.249406
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.23179	0.463330	0.231665	−	0.972796i	$-0.425583\pi$
$397$	9.23179	0.463330	0.231665	+	0.972796i	$0.425583\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.29180	−0.364135	−0.182067	−	0.983286i	$-0.558279\pi$
$401$	−7.29180	−0.364135	−0.182067	+	0.983286i	$0.558279\pi$
$402$	0	0
$403$	2.29180	0.114162
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.236068	−0.0117015
$408$	0	0
$409$	19.7202	0.975100	0.487550	−	0.873095i	$-0.337891\pi$
$409$	19.7202	0.975100	0.487550	+	0.873095i	$0.337891\pi$
$410$	0	0
$411$	−2.41577	−0.119161
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−34.1803	−1.67382
$418$	0	0
$419$	6.27585	0.306595	0.153298	−	0.988180i	$-0.451011\pi$
$419$	6.27585	0.306595	0.153298	+	0.988180i	$0.451011\pi$
$420$	0	0
$421$	−35.0689	−1.70915	−0.854576	−	0.519326i	$-0.826183\pi$
$421$	−35.0689	−1.70915	−0.854576	+	0.519326i	$0.826183\pi$
$422$	0	0
$423$	3.16228	0.153755
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−7.81966	−0.376660	−0.188330	−	0.982106i	$-0.560307\pi$
$431$	−7.81966	−0.376660	−0.188330	+	0.982106i	$0.560307\pi$
$432$	0	0
$433$	−18.0509	−0.867472	−0.433736	−	0.901040i	$-0.642805\pi$
$433$	−18.0509	−0.867472	−0.433736	+	0.901040i	$0.642805\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.16228	−0.151272
$438$	0	0
$439$	−36.4056	−1.73754	−0.868772	−	0.495212i	$-0.835090\pi$
$439$	−36.4056	−1.73754	−0.868772	+	0.495212i	$0.835090\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.8197	0.656592	0.328296	−	0.944575i	$-0.393526\pi$
$443$	13.8197	0.656592	0.328296	+	0.944575i	$0.393526\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	35.8167	1.69407
$448$	0	0
$449$	14.7082	0.694123	0.347062	−	0.937842i	$-0.387179\pi$
$449$	14.7082	0.694123	0.347062	+	0.937842i	$0.387179\pi$
$450$	0	0
$451$	−9.69316	−0.456433
$452$	0	0
$453$	−31.4953	−1.47978
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.7639	−0.737406	−0.368703	−	0.929547i	$-0.620198\pi$
$457$	−15.7639	−0.737406	−0.368703	+	0.929547i	$0.620198\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	7.35621	0.342613	0.171306	−	0.985218i	$-0.445201\pi$
$461$	7.35621	0.342613	0.171306	+	0.985218i	$0.445201\pi$
$462$	0	0
$463$	−13.4164	−0.623513	−0.311757	−	0.950162i	$-0.600917\pi$
$463$	−13.4164	−0.623513	−0.311757	+	0.950162i	$0.600917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.8840	1.10522	0.552610	−	0.833440i	$-0.313632\pi$
$467$	23.8840	1.10522	0.552610	+	0.833440i	$0.313632\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−49.5967	−2.28530
$472$	0	0
$473$	2.23607	0.102815
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.76393	−0.126552
$478$	0	0
$479$	26.1723	1.19584	0.597920	−	0.801555i	$-0.295994\pi$
$479$	26.1723	1.19584	0.597920	+	0.801555i	$0.295994\pi$
$480$	0	0
$481$	0.206331	0.00940788
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	0	0
$489$	30.2874	1.36964
$490$	0	0
$491$	−26.1246	−1.17899	−0.589494	−	0.807773i	$-0.700673\pi$
$491$	−26.1246	−1.17899	−0.589494	+	0.807773i	$0.700673\pi$
$492$	0	0
$493$	39.8530	1.79489
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−30.9443	−1.38526	−0.692628	−	0.721295i	$-0.743547\pi$
$499$	−30.9443	−1.38526	−0.692628	+	0.721295i	$0.743547\pi$
$500$	0	0
$501$	38.3607	1.71383
$502$	0	0
$503$	27.1251	1.20945	0.604724	−	0.796435i	$-0.293283\pi$
$503$	27.1251	1.20945	0.604724	+	0.796435i	$0.293283\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−27.9991	−1.24348
$508$	0	0
$509$	0.618993	0.0274364	0.0137182	−	0.999906i	$-0.495633\pi$
$509$	0.618993	0.0274364	0.0137182	+	0.999906i	$0.495633\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.52786	0.244061
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.41421	−0.0621970
$518$	0	0
$519$	19.2361	0.844370
$520$	0	0
$521$	−2.23954	−0.0981159	−0.0490580	−	0.998796i	$-0.515622\pi$
$521$	−2.23954	−0.0981159	−0.0490580	+	0.998796i	$0.515622\pi$
$522$	0	0
$523$	−11.7264	−0.512758	−0.256379	−	0.966576i	$-0.582530\pi$
$523$	−11.7264	−0.512758	−0.256379	+	0.966576i	$0.582530\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−24.8369	−1.07783
$532$	0	0
$533$	8.47214	0.366969
$534$	0	0
$535$	0	0
$536$	0	0
$537$	37.0246	1.59773
$538$	0	0
$539$	0	0
$540$	0	0
$541$	34.0132	1.46234	0.731170	−	0.682195i	$-0.238975\pi$
$541$	34.0132	1.46234	0.731170	+	0.682195i	$0.238975\pi$
$542$	0	0
$543$	31.1246	1.33568
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.6525	1.52439	0.762195	−	0.647348i	$-0.224122\pi$
$547$	35.6525	1.52439	0.762195	+	0.647348i	$0.224122\pi$
$548$	0	0
$549$	11.4412	0.488300
$550$	0	0
$551$	27.5378	1.17315
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.7639	−1.17640	−0.588198	−	0.808717i	$-0.700162\pi$
$557$	−27.7639	−1.17640	−0.588198	+	0.808717i	$0.700162\pi$
$558$	0	0
$559$	−1.95440	−0.0826621
$560$	0	0
$561$	10.4721	0.442134
$562$	0	0
$563$	12.7279	0.536418	0.268209	−	0.963361i	$-0.413568\pi$
$563$	12.7279	0.536418	0.268209	+	0.963361i	$0.413568\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.8885	−0.959538	−0.479769	−	0.877395i	$-0.659279\pi$
$569$	−22.8885	−0.959538	−0.479769	+	0.877395i	$0.659279\pi$
$570$	0	0
$571$	10.1246	0.423702	0.211851	−	0.977302i	$-0.432051\pi$
$571$	10.1246	0.423702	0.211851	+	0.977302i	$0.432051\pi$
$572$	0	0
$573$	−38.3600	−1.60251
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.7124	1.11205	0.556027	−	0.831164i	$-0.312325\pi$
$577$	26.7124	1.11205	0.556027	+	0.831164i	$0.312325\pi$
$578$	0	0
$579$	5.11667	0.212642
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.23607	0.0511927
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.5834	1.05594	0.527969	−	0.849264i	$-0.322954\pi$
$587$	25.5834	1.05594	0.527969	+	0.849264i	$0.322954\pi$
$588$	0	0
$589$	−8.29180	−0.341658
$590$	0	0
$591$	−60.4472	−2.48647
$592$	0	0
$593$	−13.0618	−0.536383	−0.268191	−	0.963366i	$-0.586426\pi$
$593$	−13.0618	−0.536383	−0.268191	+	0.963366i	$0.586426\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−28.6525	−1.17267
$598$	0	0
$599$	8.05573	0.329148	0.164574	−	0.986365i	$-0.447375\pi$
$599$	8.05573	0.329148	0.164574	+	0.986365i	$0.447375\pi$
$600$	0	0
$601$	26.7912	1.09284	0.546419	−	0.837512i	$-0.315991\pi$
$601$	26.7912	1.09284	0.546419	+	0.837512i	$0.315991\pi$
$602$	0	0
$603$	16.7082	0.680411
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.61125	−0.308931	−0.154466	−	0.987998i	$-0.549366\pi$
$607$	−7.61125	−0.308931	−0.154466	+	0.987998i	$0.549366\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.23607	0.0500060
$612$	0	0
$613$	−23.9443	−0.967100	−0.483550	−	0.875317i	$-0.660653\pi$
$613$	−23.9443	−0.967100	−0.483550	+	0.875317i	$0.660653\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.3050	0.575896	0.287948	−	0.957646i	$-0.407027\pi$
$617$	14.3050	0.575896	0.287948	+	0.957646i	$0.407027\pi$
$618$	0	0
$619$	3.95750	0.159065	0.0795326	−	0.996832i	$-0.474657\pi$
$619$	3.95750	0.159065	0.0795326	+	0.996832i	$0.474657\pi$
$620$	0	0
$621$	−1.74806	−0.0701474
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.23607	0.288981
$628$	0	0
$629$	−1.08036	−0.0430769
$630$	0	0
$631$	−25.4721	−1.01403	−0.507015	−	0.861937i	$-0.669251\pi$
$631$	−25.4721	−1.01403	−0.507015	+	0.861937i	$0.669251\pi$
$632$	0	0
$633$	−3.49613	−0.138959
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15.0000	0.593391
$640$	0	0
$641$	−36.3050	−1.43396	−0.716980	−	0.697094i	$-0.754476\pi$
$641$	−36.3050	−1.43396	−0.716980	+	0.697094i	$0.754476\pi$
$642$	0	0
$643$	24.2179	0.955059	0.477530	−	0.878616i	$-0.341532\pi$
$643$	24.2179	0.955059	0.477530	+	0.878616i	$0.341532\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.6429	0.811557	0.405778	−	0.913971i	$-0.367000\pi$
$647$	20.6429	0.811557	0.405778	+	0.913971i	$0.367000\pi$
$648$	0	0
$649$	11.1074	0.436003
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.47214	−0.175008	−0.0875041	−	0.996164i	$-0.527889\pi$
$653$	−4.47214	−0.175008	−0.0875041	+	0.996164i	$0.527889\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.60970	−0.257869
$658$	0	0
$659$	25.4164	0.990083	0.495041	−	0.868869i	$-0.335153\pi$
$659$	25.4164	0.990083	0.495041	+	0.868869i	$0.335153\pi$
$660$	0	0
$661$	−50.4202	−1.96112	−0.980560	−	0.196222i	$-0.937133\pi$
$661$	−50.4202	−1.96112	−0.980560	+	0.196222i	$0.937133\pi$
$662$	0	0
$663$	−9.15298	−0.355472
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.70820	−0.337183
$668$	0	0
$669$	−19.7082	−0.761963
$670$	0	0
$671$	−5.11667	−0.197527
$672$	0	0
$673$	−44.0689	−1.69873	−0.849365	−	0.527805i	$-0.823015\pi$
$673$	−44.0689	−1.69873	−0.849365	+	0.527805i	$0.823015\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.81913	0.338947	0.169473	−	0.985535i	$-0.445793\pi$
$677$	8.81913	0.338947	0.169473	+	0.985535i	$0.445793\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−35.5967	−1.36407
$682$	0	0
$683$	16.0557	0.614355	0.307178	−	0.951652i	$-0.400615\pi$
$683$	16.0557	0.614355	0.307178	+	0.951652i	$0.400615\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	51.5967	1.96854
$688$	0	0
$689$	−1.08036	−0.0411586
$690$	0	0
$691$	−13.8570	−0.527145	−0.263572	−	0.964640i	$-0.584901\pi$
$691$	−13.8570	−0.527145	−0.263572	+	0.964640i	$0.584901\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−44.3607	−1.68028
$698$	0	0
$699$	65.0237	2.45942
$700$	0	0
$701$	−45.3050	−1.71114	−0.855572	−	0.517683i	$-0.826794\pi$
$701$	−45.3050	−1.71114	−0.855572	+	0.517683i	$0.826794\pi$
$702$	0	0
$703$	−0.746512	−0.0281553
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−26.9443	−1.01191	−0.505957	−	0.862559i	$-0.668860\pi$
$709$	−26.9443	−1.01191	−0.505957	+	0.862559i	$0.668860\pi$
$710$	0	0
$711$	−26.7082	−1.00164
$712$	0	0
$713$	2.62210	0.0981983
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.1344	0.789278
$718$	0	0
$719$	−3.78127	−0.141018	−0.0705088	−	0.997511i	$-0.522462\pi$
$719$	−3.78127	−0.141018	−0.0705088	+	0.997511i	$0.522462\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.65248	−0.247408
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.5980	1.46861	0.734304	−	0.678821i	$-0.237509\pi$
$727$	39.5980	1.46861	0.734304	+	0.678821i	$0.237509\pi$
$728$	0	0
$729$	−11.9443	−0.442380
$730$	0	0
$731$	10.2333	0.378494
$732$	0	0
$733$	44.4295	1.64104	0.820521	−	0.571617i	$-0.193684\pi$
$733$	44.4295	1.64104	0.820521	+	0.571617i	$0.193684\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.47214	−0.275240
$738$	0	0
$739$	−49.3607	−1.81576	−0.907881	−	0.419228i	$-0.862301\pi$
$739$	−49.3607	−1.81576	−0.907881	+	0.419228i	$0.862301\pi$
$740$	0	0
$741$	−6.32456	−0.232338
$742$	0	0
$743$	22.1803	0.813718	0.406859	−	0.913491i	$-0.366624\pi$
$743$	22.1803	0.813718	0.406859	+	0.913491i	$0.366624\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−18.5123	−0.677329
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−6.11146	−0.223010	−0.111505	−	0.993764i	$-0.535567\pi$
$751$	−6.11146	−0.223010	−0.111505	+	0.993764i	$0.535567\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.05573	−0.220099	−0.110050	−	0.993926i	$-0.535101\pi$
$757$	−6.05573	−0.220099	−0.110050	+	0.993926i	$0.535101\pi$
$758$	0	0
$759$	−2.28825	−0.0830581
$760$	0	0
$761$	11.9814	0.434326	0.217163	−	0.976135i	$-0.430320\pi$
$761$	11.9814	0.434326	0.217163	+	0.976135i	$0.430320\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.70820	−0.350543
$768$	0	0
$769$	−20.7518	−0.748330	−0.374165	−	0.927362i	$-0.622071\pi$
$769$	−20.7518	−0.748330	−0.374165	+	0.927362i	$0.622071\pi$
$770$	0	0
$771$	−1.05573	−0.0380211
$772$	0	0
$773$	25.2495	0.908162	0.454081	−	0.890960i	$-0.349968\pi$
$773$	25.2495	0.908162	0.454081	+	0.890960i	$0.349968\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30.6525	−1.09824
$780$	0	0
$781$	−6.70820	−0.240038
$782$	0	0
$783$	15.2225	0.544008
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.87558	0.0668573	0.0334287	−	0.999441i	$-0.489357\pi$
$787$	1.87558	0.0668573	0.0334287	+	0.999441i	$0.489357\pi$
$788$	0	0
$789$	7.53244	0.268162
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.47214	0.158810
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.0323	−1.06380	−0.531900	−	0.846807i	$-0.678522\pi$
$797$	−30.0323	−1.06380	−0.531900	+	0.846807i	$0.678522\pi$
$798$	0	0
$799$	−6.47214	−0.228968
$800$	0	0
$801$	4.37016	0.154412
$802$	0	0
$803$	2.95595	0.104313
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−35.8885	−1.26334
$808$	0	0
$809$	−5.29180	−0.186050	−0.0930248	−	0.995664i	$-0.529654\pi$
$809$	−5.29180	−0.186050	−0.0930248	+	0.995664i	$0.529654\pi$
$810$	0	0
$811$	−10.3122	−0.362109	−0.181054	−	0.983473i	$-0.557951\pi$
$811$	−10.3122	−0.362109	−0.181054	+	0.983473i	$0.557951\pi$
$812$	0	0
$813$	65.7771	2.30690
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.07107	0.247385
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.4164	0.607837	0.303918	−	0.952698i	$-0.401705\pi$
$821$	17.4164	0.607837	0.303918	+	0.952698i	$0.401705\pi$
$822$	0	0
$823$	−52.3050	−1.82324	−0.911618	−	0.411038i	$-0.865166\pi$
$823$	−52.3050	−1.82324	−0.911618	+	0.411038i	$0.865166\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.2361	0.703677	0.351839	−	0.936061i	$-0.385557\pi$
$827$	20.2361	0.703677	0.351839	+	0.936061i	$0.385557\pi$
$828$	0	0
$829$	8.89794	0.309038	0.154519	−	0.987990i	$-0.450617\pi$
$829$	8.89794	0.309038	0.154519	+	0.987990i	$0.450617\pi$
$830$	0	0
$831$	−6.99226	−0.242559
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.58359	−0.158432
$838$	0	0
$839$	10.1058	0.348892	0.174446	−	0.984667i	$-0.444187\pi$
$839$	10.1058	0.348892	0.174446	+	0.984667i	$0.444187\pi$
$840$	0	0
$841$	46.8328	1.61492
$842$	0	0
$843$	44.9697	1.54884
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	13.1246	0.450436
$850$	0	0
$851$	0.236068	0.00809231
$852$	0	0
$853$	27.7140	0.948909	0.474454	−	0.880280i	$-0.342645\pi$
$853$	27.7140	0.948909	0.474454	+	0.880280i	$0.342645\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	52.3746	1.78908	0.894541	−	0.446985i	$-0.147502\pi$
$857$	52.3746	1.78908	0.894541	+	0.446985i	$0.147502\pi$
$858$	0	0
$859$	46.5114	1.58695	0.793475	−	0.608603i	$-0.208270\pi$
$859$	46.5114	1.58695	0.793475	+	0.608603i	$0.208270\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.1246	0.821211	0.410606	−	0.911813i	$-0.365317\pi$
$863$	24.1246	0.821211	0.410606	+	0.911813i	$0.365317\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.02546	0.306521
$868$	0	0
$869$	11.9443	0.405182
$870$	0	0
$871$	6.53089	0.221291
$872$	0	0
$873$	−28.7456	−0.972893
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.7082	0.395358	0.197679	−	0.980267i	$-0.436660\pi$
$877$	11.7082	0.395358	0.197679	+	0.980267i	$0.436660\pi$
$878$	0	0
$879$	45.4164	1.53186
$880$	0	0
$881$	20.0053	0.673996	0.336998	−	0.941505i	$-0.390588\pi$
$881$	20.0053	0.673996	0.336998	+	0.941505i	$0.390588\pi$
$882$	0	0
$883$	27.3607	0.920760	0.460380	−	0.887722i	$-0.347713\pi$
$883$	27.3607	0.920760	0.460380	+	0.887722i	$0.347713\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.0146	0.470565	0.235282	−	0.971927i	$-0.424399\pi$
$887$	14.0146	0.470565	0.235282	+	0.971927i	$0.424399\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.7082	0.358738
$892$	0	0
$893$	−4.47214	−0.149654
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	−22.8337	−0.761548
$900$	0	0
$901$	5.65685	0.188457
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	−20.1815	−0.669380
$910$	0	0
$911$	−35.3607	−1.17155	−0.585776	−	0.810473i	$-0.699210\pi$
$911$	−35.3607	−1.17155	−0.585776	+	0.810473i	$0.699210\pi$
$912$	0	0
$913$	8.27895	0.273993
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	46.5967	1.53708	0.768542	−	0.639799i	$-0.220982\pi$
$919$	46.5967	1.53708	0.768542	+	0.639799i	$0.220982\pi$
$920$	0	0
$921$	68.7214	2.26445
$922$	0	0
$923$	5.86319	0.192989
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−23.6290	−0.776077
$928$	0	0
$929$	6.19704	0.203318	0.101659	−	0.994819i	$-0.467585\pi$
$929$	6.19704	0.203318	0.101659	+	0.994819i	$0.467585\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−41.3050	−1.35226
$934$	0	0
$935$	0	0
$936$	0	0
$937$	57.4612	1.87717	0.938587	−	0.345041i	$-0.112135\pi$
$937$	57.4612	1.87717	0.938587	+	0.345041i	$0.112135\pi$
$938$	0	0
$939$	24.1803	0.789096
$940$	0	0
$941$	−43.8105	−1.42818	−0.714091	−	0.700053i	$-0.753160\pi$
$941$	−43.8105	−1.42818	−0.714091	+	0.700053i	$0.753160\pi$
$942$	0	0
$943$	9.69316	0.315653
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.88854	−0.191352	−0.0956760	−	0.995413i	$-0.530501\pi$
$947$	−5.88854	−0.191352	−0.0956760	+	0.995413i	$0.530501\pi$
$948$	0	0
$949$	−2.58359	−0.0838669
$950$	0	0
$951$	−24.0903	−0.781183
$952$	0	0
$953$	−13.7639	−0.445857	−0.222929	−	0.974835i	$-0.571562\pi$
$953$	−13.7639	−0.445857	−0.222929	+	0.974835i	$0.571562\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.9265	0.644133
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.1246	−0.778213
$962$	0	0
$963$	30.0000	0.966736
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.4721	0.465393	0.232696	−	0.972549i	$-0.425245\pi$
$967$	14.4721	0.465393	0.232696	+	0.972549i	$0.425245\pi$
$968$	0	0
$969$	33.1158	1.06383
$970$	0	0
$971$	49.5949	1.59158	0.795788	−	0.605575i	$-0.207057\pi$
$971$	49.5949	1.59158	0.795788	+	0.605575i	$0.207057\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−54.5967	−1.74670	−0.873352	−	0.487089i	$-0.838059\pi$
$977$	−54.5967	−1.74670	−0.873352	+	0.487089i	$0.838059\pi$
$978$	0	0
$979$	−1.95440	−0.0624627
$980$	0	0
$981$	−5.65248	−0.180470
$982$	0	0
$983$	−45.4612	−1.44999	−0.724993	−	0.688756i	$-0.758157\pi$
$983$	−45.4612	−1.44999	−0.724993	+	0.688756i	$0.758157\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.23607	−0.0711028
$990$	0	0
$991$	−45.6525	−1.45020	−0.725099	−	0.688644i	$-0.758206\pi$
$991$	−45.6525	−1.45020	−0.725099	+	0.688644i	$0.758206\pi$
$992$	0	0
$993$	−34.3237	−1.08923
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.5595	−0.556113	−0.278057	−	0.960565i	$-0.589690\pi$
$997$	−17.5595	−0.556113	−0.278057	+	0.960565i	$0.589690\pi$
$998$	0	0
$999$	−0.412662	−0.0130560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cn.1.4	yes	4
5.4	even	2		9800.2.a.cm.1.1	✓	4
7.6	odd	2	inner	9800.2.a.cn.1.1	yes	4
35.34	odd	2		9800.2.a.cm.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9800.2.a.cm.1.1	✓	4	5.4	even	2
9800.2.a.cm.1.4	yes	4	35.34	odd	2
9800.2.a.cn.1.1	yes	4	7.6	odd	2	inner
9800.2.a.cn.1.4	yes	4	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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28825 q^{3} +2.23607 q^{9} +O(q^{10})\)	\(q+2.28825 q^{3} +2.23607 q^{9} -1.00000 q^{11} +0.874032 q^{13} -4.57649 q^{17} -3.16228 q^{19} +1.00000 q^{23} -1.74806 q^{27} -8.70820 q^{29} +2.62210 q^{31} -2.28825 q^{33} +0.236068 q^{37} +2.00000 q^{39} +9.69316 q^{41} -2.23607 q^{43} +1.41421 q^{47} -10.4721 q^{51} -1.23607 q^{53} -7.23607 q^{57} -11.1074 q^{59} +5.11667 q^{61} +7.47214 q^{67} +2.28825 q^{69} +6.70820 q^{71} -2.95595 q^{73} -11.9443 q^{79} -10.7082 q^{81} -8.27895 q^{83} -19.9265 q^{87} +1.95440 q^{89} +6.00000 q^{93} -12.8554 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} + 4 q^{23} - 8 q^{29} - 8 q^{37} + 8 q^{39} - 24 q^{51} + 4 q^{53} - 20 q^{57} + 12 q^{67} - 12 q^{79} - 16 q^{81} + 24 q^{93}+O(q^{100}) \) 4 * q - 4 * q^11 + 4 * q^23 - 8 * q^29 - 8 * q^37 + 8 * q^39 - 24 * q^51 + 4 * q^53 - 20 * q^57 + 12 * q^67 - 12 * q^79 - 16 * q^81 + 24 * q^93

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