LMFDB - Embedded newform 2500.2.c.a.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2500,2,Mod(1249,2500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("2500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2500 = 2^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2500.c (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:	$19.9626005053$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	2500.1249
Dual form			2500.2.c.a.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{3} +0.618034i q^{7} +2.61803 q^{9} +O(q^{10})$	$q+0.618034i q^{3} +0.618034i q^{7} +2.61803 q^{9} +1.38197 q^{11} +3.00000i q^{13} +3.47214i q^{17} +0.381966 q^{19} -0.381966 q^{21} +2.61803i q^{23} +3.47214i q^{27} -7.70820 q^{29} +0.472136 q^{31} +0.854102i q^{33} -7.47214i q^{37} -1.85410 q^{39} +1.52786 q^{41} +10.0000i q^{43} -11.3262i q^{47} +6.61803 q^{49} -2.14590 q^{51} +0.472136i q^{53} +0.236068i q^{57} +10.2361 q^{59} -3.32624 q^{61} +1.61803i q^{63} +13.1803i q^{67} -1.61803 q^{69} +6.09017 q^{71} +6.61803i q^{73} +0.854102i q^{77} -13.0000 q^{79} +5.70820 q^{81} +12.9443i q^{83} -4.76393i q^{87} -7.61803 q^{89} -1.85410 q^{91} +0.291796i q^{93} -12.1459i q^{97} +3.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{9}+O(q^{10}) $ 4 * q + 6 * q^9	$ 4 q + 6 q^{9} + 10 q^{11} + 6 q^{19} - 6 q^{21} - 4 q^{29} - 16 q^{31} + 6 q^{39} + 24 q^{41} + 22 q^{49} - 22 q^{51} + 32 q^{59} + 18 q^{61} - 2 q^{69} + 2 q^{71} - 52 q^{79} - 4 q^{81} - 26 q^{89} + 6 q^{91} + 10 q^{99}+O(q^{100}) $ 4 * q + 6 * q^9 + 10 * q^11 + 6 * q^19 - 6 * q^21 - 4 * q^29 - 16 * q^31 + 6 * q^39 + 24 * q^41 + 22 * q^49 - 22 * q^51 + 32 * q^59 + 18 * q^61 - 2 * q^69 + 2 * q^71 - 52 * q^79 - 4 * q^81 - 26 * q^89 + 6 * q^91 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times$.

$n$	$1251$	$1877$
$\chi(n)$	$1$	$-1$

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
$3$	0.618034i	0.356822i	0.983956	+	0.178411i	$0.0570957\pi$
$3$	0.618034i	0.356822i	−0.983956	+	0.178411i	$0.942904\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.618034i	0.233595i	0.993156	+	0.116797i	$0.0372628\pi$
$7$	0.618034i	0.233595i	−0.993156	+	0.116797i	$0.962737\pi$
$8$	0	0
$9$	2.61803	0.872678
$10$	0	0
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0	0
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.47214i	0.842117i	0.907034	+	0.421058i	$0.138341\pi$
$17$	3.47214i	0.842117i	−0.907034	+	0.421058i	$0.861659\pi$
$18$	0	0
$19$	0.381966	0.0876290	0.0438145	−	0.999040i	$-0.486049\pi$
$19$	0.381966	0.0876290	0.0438145	+	0.999040i	$0.486049\pi$
$20$	0	0
$21$	−0.381966	−0.0833518
$22$	0	0
$23$	2.61803i	0.545898i	0.962029	+	0.272949i	$0.0879990\pi$
$23$	2.61803i	0.545898i	−0.962029	+	0.272949i	$0.912001\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.47214i	0.668213i
$28$	0	0
$29$	−7.70820	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$29$	−7.70820	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$30$	0	0
$31$	0.472136	0.0847981	0.0423991	−	0.999101i	$-0.486500\pi$
$31$	0.472136	0.0847981	0.0423991	+	0.999101i	$0.486500\pi$
$32$	0	0
$33$	0.854102i	0.148680i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.47214i	− 1.22841i	−0.789146	−	0.614206i	$-0.789476\pi$
$37$	− 7.47214i	− 1.22841i	0.789146	−	0.614206i	$-0.210524\pi$
$38$	0	0
$39$	−1.85410	−0.296894
$40$	0	0
$41$	1.52786	0.238612	0.119306	−	0.992858i	$-0.461933\pi$
$41$	1.52786	0.238612	0.119306	+	0.992858i	$0.461933\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.3262i	− 1.65210i	−0.563596	−	0.826051i	$-0.690582\pi$
$47$	− 11.3262i	− 1.65210i	0.563596	−	0.826051i	$-0.309418\pi$
$48$	0	0
$49$	6.61803	0.945433
$50$	0	0
$51$	−2.14590	−0.300486
$52$	0	0
$53$	0.472136i	0.0648529i	0.999474	+	0.0324264i	$0.0103235\pi$
$53$	0.472136i	0.0648529i	−0.999474	+	0.0324264i	$0.989677\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.236068i	0.0312680i
$58$	0	0
$59$	10.2361	1.33262	0.666311	−	0.745674i	$-0.267872\pi$
$59$	10.2361	1.33262	0.666311	+	0.745674i	$0.267872\pi$
$60$	0	0
$61$	−3.32624	−0.425881	−0.212941	−	0.977065i	$-0.568304\pi$
$61$	−3.32624	−0.425881	−0.212941	+	0.977065i	$0.568304\pi$
$62$	0	0
$63$	1.61803i	0.203853i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1803i	1.61023i	0.593115	+	0.805117i	$0.297898\pi$
$67$	13.1803i	1.61023i	−0.593115	+	0.805117i	$0.702102\pi$
$68$	0	0
$69$	−1.61803	−0.194788
$70$	0	0
$71$	6.09017	0.722770	0.361385	−	0.932417i	$-0.382304\pi$
$71$	6.09017	0.722770	0.361385	+	0.932417i	$0.382304\pi$
$72$	0	0
$73$	6.61803i	0.774582i	0.921957	+	0.387291i	$0.126589\pi$
$73$	6.61803i	0.774582i	−0.921957	+	0.387291i	$0.873411\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.854102i	0.0973340i
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	12.9443i	1.42082i	0.703789	+	0.710409i	$0.251490\pi$
$83$	12.9443i	1.42082i	−0.703789	+	0.710409i	$0.748510\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.76393i	− 0.510747i
$88$	0	0
$89$	−7.61803	−0.807510	−0.403755	−	0.914867i	$-0.632295\pi$
$89$	−7.61803	−0.807510	−0.403755	+	0.914867i	$0.632295\pi$
$90$	0	0
$91$	−1.85410	−0.194363
$92$	0	0
$93$	0.291796i	0.0302578i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12.1459i	− 1.23323i	−0.787265	−	0.616615i	$-0.788504\pi$
$97$	− 12.1459i	− 1.23323i	0.787265	−	0.616615i	$-0.211496\pi$
$98$	0	0
$99$	3.61803	0.363626
$100$	0	0
$101$	2.70820	0.269476	0.134738	−	0.990881i	$-0.456981\pi$
$101$	2.70820	0.269476	0.134738	+	0.990881i	$0.456981\pi$
$102$	0	0
$103$	13.0344i	1.28432i	0.766570	+	0.642161i	$0.221962\pi$
$103$	13.0344i	1.28432i	−0.766570	+	0.642161i	$0.778038\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.56231i	0.634402i	0.948358	+	0.317201i	$0.102743\pi$
$107$	6.56231i	0.634402i	−0.948358	+	0.317201i	$0.897257\pi$
$108$	0	0
$109$	15.5623	1.49060	0.745299	−	0.666730i	$-0.232307\pi$
$109$	15.5623	1.49060	0.745299	+	0.666730i	$0.232307\pi$
$110$	0	0
$111$	4.61803	0.438324
$112$	0	0
$113$	13.7639i	1.29480i	0.762150	+	0.647401i	$0.224144\pi$
$113$	13.7639i	1.29480i	−0.762150	+	0.647401i	$0.775856\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.85410i	0.726112i
$118$	0	0
$119$	−2.14590	−0.196714
$120$	0	0
$121$	−9.09017	−0.826379
$122$	0	0
$123$	0.944272i	0.0851421i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.0902i	0.895358i	0.894194	+	0.447679i	$0.147749\pi$
$127$	10.0902i	0.895358i	−0.894194	+	0.447679i	$0.852251\pi$
$128$	0	0
$129$	−6.18034	−0.544149
$130$	0	0
$131$	−16.1803	−1.41368	−0.706841	−	0.707372i	$-0.749881\pi$
$131$	−16.1803	−1.41368	−0.706841	+	0.707372i	$0.749881\pi$
$132$	0	0
$133$	0.236068i	0.0204697i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.2361i	− 1.21627i	−0.793834	−	0.608135i	$-0.791918\pi$
$137$	− 14.2361i	− 1.21627i	0.793834	−	0.608135i	$-0.208082\pi$
$138$	0	0
$139$	−12.7082	−1.07790	−0.538948	−	0.842339i	$-0.681178\pi$
$139$	−12.7082	−1.07790	−0.538948	+	0.842339i	$0.681178\pi$
$140$	0	0
$141$	7.00000	0.589506
$142$	0	0
$143$	4.14590i	0.346697i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.09017i	0.337352i
$148$	0	0
$149$	13.0344	1.06782	0.533912	−	0.845540i	$-0.320722\pi$
$149$	13.0344	1.06782	0.533912	+	0.845540i	$0.320722\pi$
$150$	0	0
$151$	1.18034	0.0960547	0.0480273	−	0.998846i	$-0.484707\pi$
$151$	1.18034	0.0960547	0.0480273	+	0.998846i	$0.484707\pi$
$152$	0	0
$153$	9.09017i	0.734897i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1.47214i	− 0.117489i	−0.998273	−	0.0587446i	$-0.981290\pi$
$157$	− 1.47214i	− 0.117489i	0.998273	−	0.0587446i	$-0.0187098\pi$
$158$	0	0
$159$	−0.291796	−0.0231409
$160$	0	0
$161$	−1.61803	−0.127519
$162$	0	0
$163$	− 1.38197i	− 0.108244i	−0.998534	−	0.0541220i	$-0.982764\pi$
$163$	− 1.38197i	− 0.108244i	0.998534	−	0.0541220i	$-0.0172360\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.7082i	1.13815i	0.822284	+	0.569077i	$0.192700\pi$
$167$	14.7082i	1.13815i	−0.822284	+	0.569077i	$0.807300\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	18.7639i	1.42660i	0.700861	+	0.713298i	$0.252799\pi$
$173$	18.7639i	1.42660i	−0.700861	+	0.713298i	$0.747201\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.32624i	0.475509i
$178$	0	0
$179$	−8.52786	−0.637402	−0.318701	−	0.947855i	$-0.603247\pi$
$179$	−8.52786	−0.637402	−0.318701	+	0.947855i	$0.603247\pi$
$180$	0	0
$181$	9.14590	0.679809	0.339905	−	0.940460i	$-0.389605\pi$
$181$	9.14590	0.679809	0.339905	+	0.940460i	$0.389605\pi$
$182$	0	0
$183$	− 2.05573i	− 0.151964i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.79837i	0.350892i
$188$	0	0
$189$	−2.14590	−0.156091
$190$	0	0
$191$	−22.1246	−1.60088	−0.800440	−	0.599412i	$-0.795401\pi$
$191$	−22.1246	−1.60088	−0.800440	+	0.599412i	$0.795401\pi$
$192$	0	0
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.4721i	− 1.17359i	−0.809735	−	0.586796i	$-0.800389\pi$
$197$	− 16.4721i	− 1.17359i	0.809735	−	0.586796i	$-0.199611\pi$
$198$	0	0
$199$	12.8541	0.911203	0.455602	−	0.890184i	$-0.349424\pi$
$199$	12.8541	0.911203	0.455602	+	0.890184i	$0.349424\pi$
$200$	0	0
$201$	−8.14590	−0.574567
$202$	0	0
$203$	− 4.76393i	− 0.334362i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.85410i	0.476393i
$208$	0	0
$209$	0.527864	0.0365131
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	3.76393i	0.257900i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.291796i	0.0198084i
$218$	0	0
$219$	−4.09017	−0.276388
$220$	0	0
$221$	−10.4164	−0.700683
$222$	0	0
$223$	− 17.5623i	− 1.17606i	−0.808839	−	0.588029i	$-0.799904\pi$
$223$	− 17.5623i	− 1.17606i	0.808839	−	0.588029i	$-0.200096\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.65248i	− 0.441540i	−0.975326	−	0.220770i	$-0.929143\pi$
$227$	− 6.65248i	− 0.441540i	0.975326	−	0.220770i	$-0.0708571\pi$
$228$	0	0
$229$	8.23607	0.544255	0.272127	−	0.962261i	$-0.412273\pi$
$229$	8.23607	0.544255	0.272127	+	0.962261i	$0.412273\pi$
$230$	0	0
$231$	−0.527864	−0.0347309
$232$	0	0
$233$	12.3820i	0.811170i	0.914057	+	0.405585i	$0.132932\pi$
$233$	12.3820i	0.811170i	−0.914057	+	0.405585i	$0.867068\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.03444i	− 0.521893i
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	12.6180	0.812799	0.406400	−	0.913695i	$-0.366784\pi$
$241$	12.6180	0.812799	0.406400	+	0.913695i	$0.366784\pi$
$242$	0	0
$243$	13.9443i	0.894525i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.14590i	0.0729117i
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	5.00000	0.315597	0.157799	−	0.987471i	$-0.449560\pi$
$251$	5.00000	0.315597	0.157799	+	0.987471i	$0.449560\pi$
$252$	0	0
$253$	3.61803i	0.227464i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 22.1803i	− 1.38357i	−0.722103	−	0.691786i	$-0.756824\pi$
$257$	− 22.1803i	− 1.38357i	0.722103	−	0.691786i	$-0.243176\pi$
$258$	0	0
$259$	4.61803	0.286951
$260$	0	0
$261$	−20.1803	−1.24913
$262$	0	0
$263$	− 22.8541i	− 1.40924i	−0.709583	−	0.704622i	$-0.751117\pi$
$263$	− 22.8541i	− 1.40924i	0.709583	−	0.704622i	$-0.248883\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.70820i	− 0.288137i
$268$	0	0
$269$	2.65248	0.161724	0.0808622	−	0.996725i	$-0.474233\pi$
$269$	2.65248	0.161724	0.0808622	+	0.996725i	$0.474233\pi$
$270$	0	0
$271$	21.1803	1.28661	0.643307	−	0.765608i	$-0.277562\pi$
$271$	21.1803	1.28661	0.643307	+	0.765608i	$0.277562\pi$
$272$	0	0
$273$	− 1.14590i	− 0.0693529i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7.29180i	− 0.438122i	−0.975711	−	0.219061i	$-0.929701\pi$
$277$	− 7.29180i	− 0.438122i	0.975711	−	0.219061i	$-0.0702993\pi$
$278$	0	0
$279$	1.23607	0.0740015
$280$	0	0
$281$	5.47214	0.326440	0.163220	−	0.986590i	$-0.447812\pi$
$281$	5.47214	0.326440	0.163220	+	0.986590i	$0.447812\pi$
$282$	0	0
$283$	8.23607i	0.489583i	0.969576	+	0.244792i	$0.0787196\pi$
$283$	8.23607i	0.489583i	−0.969576	+	0.244792i	$0.921280\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.944272i	0.0557386i
$288$	0	0
$289$	4.94427	0.290840
$290$	0	0
$291$	7.50658	0.440043
$292$	0	0
$293$	− 19.2705i	− 1.12580i	−0.826527	−	0.562898i	$-0.809687\pi$
$293$	− 19.2705i	− 1.12580i	0.826527	−	0.562898i	$-0.190313\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.79837i	0.278430i
$298$	0	0
$299$	−7.85410	−0.454214
$300$	0	0
$301$	−6.18034	−0.356229
$302$	0	0
$303$	1.67376i	0.0961551i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.0344i	1.08635i	0.839619	+	0.543176i	$0.182779\pi$
$307$	19.0344i	1.08635i	−0.839619	+	0.543176i	$0.817221\pi$
$308$	0	0
$309$	−8.05573	−0.458274
$310$	0	0
$311$	33.5066	1.89998	0.949992	−	0.312275i	$-0.101091\pi$
$311$	33.5066	1.89998	0.949992	+	0.312275i	$0.101091\pi$
$312$	0	0
$313$	12.8541i	0.726557i	0.931681	+	0.363278i	$0.118343\pi$
$313$	12.8541i	0.726557i	−0.931681	+	0.363278i	$0.881657\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 27.3820i	− 1.53792i	−0.639294	−	0.768962i	$-0.720773\pi$
$317$	− 27.3820i	− 1.53792i	0.639294	−	0.768962i	$-0.279227\pi$
$318$	0	0
$319$	−10.6525	−0.596424
$320$	0	0
$321$	−4.05573	−0.226369
$322$	0	0
$323$	1.32624i	0.0737939i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.61803i	0.531878i
$328$	0	0
$329$	7.00000	0.385922
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	− 19.5623i	− 1.07201i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.0557i	0.711191i	0.934640	+	0.355595i	$0.115722\pi$
$337$	13.0557i	0.711191i	−0.934640	+	0.355595i	$0.884278\pi$
$338$	0	0
$339$	−8.50658	−0.462014
$340$	0	0
$341$	0.652476	0.0353335
$342$	0	0
$343$	8.41641i	0.454443i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.5279i	1.10199i	0.834507	+	0.550997i	$0.185752\pi$
$347$	20.5279i	1.10199i	−0.834507	+	0.550997i	$0.814248\pi$
$348$	0	0
$349$	−24.5967	−1.31663	−0.658317	−	0.752741i	$-0.728731\pi$
$349$	−24.5967	−1.31663	−0.658317	+	0.752741i	$0.728731\pi$
$350$	0	0
$351$	−10.4164	−0.555987
$352$	0	0
$353$	− 27.5967i	− 1.46883i	−0.678702	−	0.734413i	$-0.737457\pi$
$353$	− 27.5967i	− 1.46883i	0.678702	−	0.734413i	$-0.262543\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.32624i	− 0.0701920i
$358$	0	0
$359$	−11.3607	−0.599594	−0.299797	−	0.954003i	$-0.596919\pi$
$359$	−11.3607	−0.599594	−0.299797	+	0.954003i	$0.596919\pi$
$360$	0	0
$361$	−18.8541	−0.992321
$362$	0	0
$363$	− 5.61803i	− 0.294870i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.52786i	0.184153i	0.995752	+	0.0920765i	$0.0293504\pi$
$367$	3.52786i	0.184153i	−0.995752	+	0.0920765i	$0.970650\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	−0.291796	−0.0151493
$372$	0	0
$373$	2.09017i	0.108225i	0.998535	+	0.0541124i	$0.0172329\pi$
$373$	2.09017i	0.108225i	−0.998535	+	0.0541124i	$0.982767\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.1246i	− 1.19098i
$378$	0	0
$379$	14.1803	0.728395	0.364198	−	0.931322i	$-0.381343\pi$
$379$	14.1803	0.728395	0.364198	+	0.931322i	$0.381343\pi$
$380$	0	0
$381$	−6.23607	−0.319483
$382$	0	0
$383$	− 8.03444i	− 0.410541i	−0.978705	−	0.205270i	$-0.934193\pi$
$383$	− 8.03444i	− 0.410541i	0.978705	−	0.205270i	$-0.0658074\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.1803i	1.33082i
$388$	0	0
$389$	18.0344	0.914382	0.457191	−	0.889368i	$-0.348855\pi$
$389$	18.0344	0.914382	0.457191	+	0.889368i	$0.348855\pi$
$390$	0	0
$391$	−9.09017	−0.459710
$392$	0	0
$393$	− 10.0000i	− 0.504433i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.5279i	0.879698i	0.898072	+	0.439849i	$0.144968\pi$
$397$	17.5279i	0.879698i	−0.898072	+	0.439849i	$0.855032\pi$
$398$	0	0
$399$	−0.145898	−0.00730404
$400$	0	0
$401$	33.6180	1.67880	0.839402	−	0.543511i	$-0.182905\pi$
$401$	33.6180	1.67880	0.839402	+	0.543511i	$0.182905\pi$
$402$	0	0
$403$	1.41641i	0.0705563i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.3262i	− 0.511853i
$408$	0	0
$409$	−25.5967	−1.26568	−0.632839	−	0.774284i	$-0.718110\pi$
$409$	−25.5967	−1.26568	−0.632839	+	0.774284i	$0.718110\pi$
$410$	0	0
$411$	8.79837	0.433992
$412$	0	0
$413$	6.32624i	0.311294i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 7.85410i	− 0.384617i
$418$	0	0
$419$	27.9443	1.36517	0.682584	−	0.730808i	$-0.260856\pi$
$419$	27.9443	1.36517	0.682584	+	0.730808i	$0.260856\pi$
$420$	0	0
$421$	−9.65248	−0.470433	−0.235216	−	0.971943i	$-0.575580\pi$
$421$	−9.65248	−0.470433	−0.235216	+	0.971943i	$0.575580\pi$
$422$	0	0
$423$	− 29.6525i	− 1.44175i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.05573i	− 0.0994837i
$428$	0	0
$429$	−2.56231	−0.123709
$430$	0	0
$431$	−27.4164	−1.32060	−0.660301	−	0.751001i	$-0.729571\pi$
$431$	−27.4164	−1.32060	−0.660301	+	0.751001i	$0.729571\pi$
$432$	0	0
$433$	− 40.2148i	− 1.93260i	−0.257420	−	0.966300i	$-0.582872\pi$
$433$	− 40.2148i	− 1.93260i	0.257420	−	0.966300i	$-0.417128\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.00000i	0.0478365i
$438$	0	0
$439$	36.7984	1.75629	0.878145	−	0.478394i	$-0.158781\pi$
$439$	36.7984	1.75629	0.878145	+	0.478394i	$0.158781\pi$
$440$	0	0
$441$	17.3262	0.825059
$442$	0	0
$443$	4.65248i	0.221046i	0.993874	+	0.110523i	$0.0352526\pi$
$443$	4.65248i	0.221046i	−0.993874	+	0.110523i	$0.964747\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.05573i	0.381023i
$448$	0	0
$449$	−1.65248	−0.0779852	−0.0389926	−	0.999240i	$-0.512415\pi$
$449$	−1.65248	−0.0779852	−0.0389926	+	0.999240i	$0.512415\pi$
$450$	0	0
$451$	2.11146	0.0994246
$452$	0	0
$453$	0.729490i	0.0342744i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.05573i	0.0961629i	0.998843	+	0.0480815i	$0.0153107\pi$
$457$	2.05573i	0.0961629i	−0.998843	+	0.0480815i	$0.984689\pi$
$458$	0	0
$459$	−12.0557	−0.562713
$460$	0	0
$461$	4.14590	0.193094	0.0965469	−	0.995328i	$-0.469220\pi$
$461$	4.14590	0.193094	0.0965469	+	0.995328i	$0.469220\pi$
$462$	0	0
$463$	− 16.4377i	− 0.763924i	−0.924178	−	0.381962i	$-0.875248\pi$
$463$	− 16.4377i	− 0.763924i	0.924178	−	0.381962i	$-0.124752\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 32.5623i	− 1.50680i	−0.657560	−	0.753402i	$-0.728412\pi$
$467$	− 32.5623i	− 1.50680i	0.657560	−	0.753402i	$-0.271588\pi$
$468$	0	0
$469$	−8.14590	−0.376143
$470$	0	0
$471$	0.909830	0.0419228
$472$	0	0
$473$	13.8197i	0.635429i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.23607i	0.0565957i
$478$	0	0
$479$	39.1246	1.78765	0.893825	−	0.448417i	$-0.148012\pi$
$479$	39.1246	1.78765	0.893825	+	0.448417i	$0.148012\pi$
$480$	0	0
$481$	22.4164	1.02210
$482$	0	0
$483$	− 1.00000i	− 0.0455016i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	29.2918i	1.32734i	0.748026	+	0.663669i	$0.231002\pi$
$487$	29.2918i	1.32734i	−0.748026	+	0.663669i	$0.768998\pi$
$488$	0	0
$489$	0.854102	0.0386238
$490$	0	0
$491$	−33.4164	−1.50806	−0.754031	−	0.656839i	$-0.771893\pi$
$491$	−33.4164	−1.50806	−0.754031	+	0.656839i	$0.771893\pi$
$492$	0	0
$493$	− 26.7639i	− 1.20539i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.76393i	0.168835i
$498$	0	0
$499$	−28.2361	−1.26402	−0.632010	−	0.774960i	$-0.717770\pi$
$499$	−28.2361	−1.26402	−0.632010	+	0.774960i	$0.717770\pi$
$500$	0	0
$501$	−9.09017	−0.406119
$502$	0	0
$503$	− 44.4853i	− 1.98350i	−0.128183	−	0.991751i	$-0.540914\pi$
$503$	− 44.4853i	− 1.98350i	0.128183	−	0.991751i	$-0.459086\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.47214i	0.109791i
$508$	0	0
$509$	−5.27051	−0.233611	−0.116806	−	0.993155i	$-0.537265\pi$
$509$	−5.27051	−0.233611	−0.116806	+	0.993155i	$0.537265\pi$
$510$	0	0
$511$	−4.09017	−0.180938
$512$	0	0
$513$	1.32624i	0.0585548i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 15.6525i	− 0.688395i
$518$	0	0
$519$	−11.5967	−0.509041
$520$	0	0
$521$	7.94427	0.348045	0.174022	−	0.984742i	$-0.444323\pi$
$521$	7.94427	0.348045	0.174022	+	0.984742i	$0.444323\pi$
$522$	0	0
$523$	− 20.1803i	− 0.882425i	−0.897403	−	0.441212i	$-0.854549\pi$
$523$	− 20.1803i	− 0.882425i	0.897403	−	0.441212i	$-0.145451\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.63932i	0.0714099i
$528$	0	0
$529$	16.1459	0.701996
$530$	0	0
$531$	26.7984	1.16295
$532$	0	0
$533$	4.58359i	0.198537i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 5.27051i	− 0.227439i
$538$	0	0
$539$	9.14590	0.393942
$540$	0	0
$541$	28.0689	1.20678	0.603388	−	0.797448i	$-0.293817\pi$
$541$	28.0689	1.20678	0.603388	+	0.797448i	$0.293817\pi$
$542$	0	0
$543$	5.65248i	0.242571i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.3607i	1.85397i	0.375100	+	0.926984i	$0.377608\pi$
$547$	43.3607i	1.85397i	−0.375100	+	0.926984i	$0.622392\pi$
$548$	0	0
$549$	−8.70820	−0.371657
$550$	0	0
$551$	−2.94427	−0.125430
$552$	0	0
$553$	− 8.03444i	− 0.341659i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 20.4508i	− 0.866530i	−0.901267	−	0.433265i	$-0.857361\pi$
$557$	− 20.4508i	− 0.866530i	0.901267	−	0.433265i	$-0.142639\pi$
$558$	0	0
$559$	−30.0000	−1.26886
$560$	0	0
$561$	−2.96556	−0.125206
$562$	0	0
$563$	− 30.0132i	− 1.26490i	−0.774600	−	0.632452i	$-0.782049\pi$
$563$	− 30.0132i	− 1.26490i	0.774600	−	0.632452i	$-0.217951\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.52786i	0.148156i
$568$	0	0
$569$	−22.2361	−0.932184	−0.466092	−	0.884736i	$-0.654339\pi$
$569$	−22.2361	−0.932184	−0.466092	+	0.884736i	$0.654339\pi$
$570$	0	0
$571$	11.9443	0.499852	0.249926	−	0.968265i	$-0.419594\pi$
$571$	11.9443	0.499852	0.249926	+	0.968265i	$0.419594\pi$
$572$	0	0
$573$	− 13.6738i	− 0.571230i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22.1246i	− 0.921060i	−0.887644	−	0.460530i	$-0.847659\pi$
$577$	− 22.1246i	− 0.921060i	0.887644	−	0.460530i	$-0.152341\pi$
$578$	0	0
$579$	3.70820	0.154108
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	0.652476i	0.0270228i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.20163i	− 0.255969i	−0.991776	−	0.127984i	$-0.959149\pi$
$587$	− 6.20163i	− 0.255969i	0.991776	−	0.127984i	$-0.0408507\pi$
$588$	0	0
$589$	0.180340	0.00743078
$590$	0	0
$591$	10.1803	0.418763
$592$	0	0
$593$	− 29.8328i	− 1.22509i	−0.790437	−	0.612543i	$-0.790146\pi$
$593$	− 29.8328i	− 1.22509i	0.790437	−	0.612543i	$-0.209854\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.94427i	0.325137i
$598$	0	0
$599$	−23.8197	−0.973245	−0.486622	−	0.873612i	$-0.661771\pi$
$599$	−23.8197	−0.973245	−0.486622	+	0.873612i	$0.661771\pi$
$600$	0	0
$601$	−19.5967	−0.799368	−0.399684	−	0.916653i	$-0.630880\pi$
$601$	−19.5967	−0.799368	−0.399684	+	0.916653i	$0.630880\pi$
$602$	0	0
$603$	34.5066i	1.40522i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 17.5279i	− 0.711434i	−0.934594	−	0.355717i	$-0.884237\pi$
$607$	− 17.5279i	− 0.711434i	0.934594	−	0.355717i	$-0.115763\pi$
$608$	0	0
$609$	2.94427	0.119308
$610$	0	0
$611$	33.9787	1.37463
$612$	0	0
$613$	− 14.7639i	− 0.596310i	−0.954518	−	0.298155i	$-0.903629\pi$
$613$	− 14.7639i	− 0.596310i	0.954518	−	0.298155i	$-0.0963712\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 47.6312i	− 1.91756i	−0.284150	−	0.958780i	$-0.591711\pi$
$617$	− 47.6312i	− 1.91756i	0.284150	−	0.958780i	$-0.408289\pi$
$618$	0	0
$619$	30.1246	1.21081	0.605405	−	0.795917i	$-0.293011\pi$
$619$	30.1246	1.21081	0.605405	+	0.795917i	$0.293011\pi$
$620$	0	0
$621$	−9.09017	−0.364776
$622$	0	0
$623$	− 4.70820i	− 0.188630i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.326238i	0.0130287i
$628$	0	0
$629$	25.9443	1.03447
$630$	0	0
$631$	0.819660	0.0326302	0.0163151	−	0.999867i	$-0.494807\pi$
$631$	0.819660	0.0326302	0.0163151	+	0.999867i	$0.494807\pi$
$632$	0	0
$633$	− 0.618034i	− 0.0245646i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	19.8541i	0.786648i
$638$	0	0
$639$	15.9443	0.630746
$640$	0	0
$641$	2.65248	0.104766	0.0523832	−	0.998627i	$-0.483318\pi$
$641$	2.65248	0.104766	0.0523832	+	0.998627i	$0.483318\pi$
$642$	0	0
$643$	22.3050i	0.879621i	0.898090	+	0.439811i	$0.144954\pi$
$643$	22.3050i	0.879621i	−0.898090	+	0.439811i	$0.855046\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.83282i	− 0.386568i	−0.981143	−	0.193284i	$-0.938086\pi$
$647$	− 9.83282i	− 0.386568i	0.981143	−	0.193284i	$-0.0619139\pi$
$648$	0	0
$649$	14.1459	0.555275
$650$	0	0
$651$	−0.180340	−0.00706808
$652$	0	0
$653$	− 29.0344i	− 1.13621i	−0.822958	−	0.568103i	$-0.807678\pi$
$653$	− 29.0344i	− 1.13621i	0.822958	−	0.568103i	$-0.192322\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	17.3262i	0.675961i
$658$	0	0
$659$	−10.2705	−0.400082	−0.200041	−	0.979788i	$-0.564108\pi$
$659$	−10.2705	−0.400082	−0.200041	+	0.979788i	$0.564108\pi$
$660$	0	0
$661$	−38.1803	−1.48504	−0.742522	−	0.669822i	$-0.766370\pi$
$661$	−38.1803	−1.48504	−0.742522	+	0.669822i	$0.766370\pi$
$662$	0	0
$663$	− 6.43769i	− 0.250019i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 20.1803i	− 0.781386i
$668$	0	0
$669$	10.8541	0.419644
$670$	0	0
$671$	−4.59675	−0.177455
$672$	0	0
$673$	− 23.4721i	− 0.904784i	−0.891819	−	0.452392i	$-0.850571\pi$
$673$	− 23.4721i	− 0.904784i	0.891819	−	0.452392i	$-0.149429\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 41.2705i	− 1.58615i	−0.609121	−	0.793077i	$-0.708478\pi$
$677$	− 41.2705i	− 1.58615i	0.609121	−	0.793077i	$-0.291522\pi$
$678$	0	0
$679$	7.50658	0.288076
$680$	0	0
$681$	4.11146	0.157551
$682$	0	0
$683$	22.2016i	0.849522i	0.905306	+	0.424761i	$0.139642\pi$
$683$	22.2016i	0.849522i	−0.905306	+	0.424761i	$0.860358\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.09017i	0.194202i
$688$	0	0
$689$	−1.41641	−0.0539608
$690$	0	0
$691$	−35.9443	−1.36738	−0.683692	−	0.729770i	$-0.739627\pi$
$691$	−35.9443	−1.36738	−0.683692	+	0.729770i	$0.739627\pi$
$692$	0	0
$693$	2.23607i	0.0849412i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.30495i	0.200939i
$698$	0	0
$699$	−7.65248	−0.289443
$700$	0	0
$701$	17.3607	0.655704	0.327852	−	0.944729i	$-0.393675\pi$
$701$	17.3607	0.655704	0.327852	+	0.944729i	$0.393675\pi$
$702$	0	0
$703$	− 2.85410i	− 0.107644i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.67376i	0.0629483i
$708$	0	0
$709$	−18.2361	−0.684870	−0.342435	−	0.939542i	$-0.611252\pi$
$709$	−18.2361	−0.684870	−0.342435	+	0.939542i	$0.611252\pi$
$710$	0	0
$711$	−34.0344	−1.27639
$712$	0	0
$713$	1.23607i	0.0462911i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.47214i	0.0923236i
$718$	0	0
$719$	−22.3820	−0.834706	−0.417353	−	0.908744i	$-0.637042\pi$
$719$	−22.3820	−0.834706	−0.417353	+	0.908744i	$0.637042\pi$
$720$	0	0
$721$	−8.05573	−0.300011
$722$	0	0
$723$	7.79837i	0.290025i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.3262i	1.27309i	0.771240	+	0.636545i	$0.219637\pi$
$727$	34.3262i	1.27309i	−0.771240	+	0.636545i	$0.780363\pi$
$728$	0	0
$729$	8.50658	0.315058
$730$	0	0
$731$	−34.7214	−1.28422
$732$	0	0
$733$	− 33.2492i	− 1.22809i	−0.789272	−	0.614044i	$-0.789542\pi$
$733$	− 33.2492i	− 1.22809i	0.789272	−	0.614044i	$-0.210458\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.2148i	0.670950i
$738$	0	0
$739$	−32.2361	−1.18582	−0.592911	−	0.805268i	$-0.702022\pi$
$739$	−32.2361	−1.18582	−0.592911	+	0.805268i	$0.702022\pi$
$740$	0	0
$741$	−0.708204	−0.0260165
$742$	0	0
$743$	− 31.3262i	− 1.14925i	−0.818417	−	0.574624i	$-0.805148\pi$
$743$	− 31.3262i	− 1.14925i	0.818417	−	0.574624i	$-0.194852\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	33.8885i	1.23992i
$748$	0	0
$749$	−4.05573	−0.148193
$750$	0	0
$751$	−1.96556	−0.0717242	−0.0358621	−	0.999357i	$-0.511418\pi$
$751$	−1.96556	−0.0717242	−0.0358621	+	0.999357i	$0.511418\pi$
$752$	0	0
$753$	3.09017i	0.112612i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.47214i	0.271579i	0.990738	+	0.135790i	$0.0433571\pi$
$757$	7.47214i	0.271579i	−0.990738	+	0.135790i	$0.956643\pi$
$758$	0	0
$759$	−2.23607	−0.0811641
$760$	0	0
$761$	22.9787	0.832978	0.416489	−	0.909141i	$-0.363260\pi$
$761$	22.9787	0.832978	0.416489	+	0.909141i	$0.363260\pi$
$762$	0	0
$763$	9.61803i	0.348196i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.7082i	1.10881i
$768$	0	0
$769$	29.1803	1.05227	0.526135	−	0.850401i	$-0.323641\pi$
$769$	29.1803	1.05227	0.526135	+	0.850401i	$0.323641\pi$
$770$	0	0
$771$	13.7082	0.493689
$772$	0	0
$773$	48.4164i	1.74142i	0.491799	+	0.870709i	$0.336339\pi$
$773$	48.4164i	1.74142i	−0.491799	+	0.870709i	$0.663661\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.85410i	0.102390i
$778$	0	0
$779$	0.583592	0.0209094
$780$	0	0
$781$	8.41641	0.301163
$782$	0	0
$783$	− 26.7639i	− 0.956465i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16.1246i	0.574780i	0.957814	+	0.287390i	$0.0927876\pi$
$787$	16.1246i	0.574780i	−0.957814	+	0.287390i	$0.907212\pi$
$788$	0	0
$789$	14.1246	0.502849
$790$	0	0
$791$	−8.50658	−0.302459
$792$	0	0
$793$	− 9.97871i	− 0.354355i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 25.5623i	− 0.905463i	−0.891647	−	0.452732i	$-0.850450\pi$
$797$	− 25.5623i	− 0.905463i	0.891647	−	0.452732i	$-0.149550\pi$
$798$	0	0
$799$	39.3262	1.39126
$800$	0	0
$801$	−19.9443	−0.704696
$802$	0	0
$803$	9.14590i	0.322752i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.63932i	0.0577068i
$808$	0	0
$809$	−31.1591	−1.09549	−0.547747	−	0.836644i	$-0.684514\pi$
$809$	−31.1591	−1.09549	−0.547747	+	0.836644i	$0.684514\pi$
$810$	0	0
$811$	16.8885	0.593037	0.296518	−	0.955027i	$-0.404174\pi$
$811$	16.8885	0.593037	0.296518	+	0.955027i	$0.404174\pi$
$812$	0	0
$813$	13.0902i	0.459092i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.81966i	0.133633i
$818$	0	0
$819$	−4.85410	−0.169616
$820$	0	0
$821$	1.52786	0.0533228	0.0266614	−	0.999645i	$-0.491512\pi$
$821$	1.52786	0.0533228	0.0266614	+	0.999645i	$0.491512\pi$
$822$	0	0
$823$	− 16.4164i	− 0.572240i	−0.958194	−	0.286120i	$-0.907634\pi$
$823$	− 16.4164i	− 0.572240i	0.958194	−	0.286120i	$-0.0923656\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 34.0689i	− 1.18469i	−0.805684	−	0.592346i	$-0.798202\pi$
$827$	− 34.0689i	− 1.18469i	0.805684	−	0.592346i	$-0.201798\pi$
$828$	0	0
$829$	−6.87539	−0.238792	−0.119396	−	0.992847i	$-0.538096\pi$
$829$	−6.87539	−0.238792	−0.119396	+	0.992847i	$0.538096\pi$
$830$	0	0
$831$	4.50658	0.156331
$832$	0	0
$833$	22.9787i	0.796165i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.63932i	0.0566632i
$838$	0	0
$839$	−5.09017	−0.175732	−0.0878661	−	0.996132i	$-0.528005\pi$
$839$	−5.09017	−0.175732	−0.0878661	+	0.996132i	$0.528005\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	0	0
$843$	3.38197i	0.116481i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.61803i	− 0.193038i
$848$	0	0
$849$	−5.09017	−0.174694
$850$	0	0
$851$	19.5623	0.670587
$852$	0	0
$853$	28.0902i	0.961789i	0.876778	+	0.480895i	$0.159688\pi$
$853$	28.0902i	0.961789i	−0.876778	+	0.480895i	$0.840312\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 10.2918i	− 0.351561i	−0.984429	−	0.175781i	$-0.943755\pi$
$857$	− 10.2918i	− 0.351561i	0.984429	−	0.175781i	$-0.0562449\pi$
$858$	0	0
$859$	33.7214	1.15056	0.575279	−	0.817957i	$-0.304893\pi$
$859$	33.7214	1.15056	0.575279	+	0.817957i	$0.304893\pi$
$860$	0	0
$861$	−0.583592	−0.0198888
$862$	0	0
$863$	− 22.7082i	− 0.772996i	−0.922290	−	0.386498i	$-0.873685\pi$
$863$	− 22.7082i	− 0.772996i	0.922290	−	0.386498i	$-0.126315\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.05573i	0.103778i
$868$	0	0
$869$	−17.9656	−0.609440
$870$	0	0
$871$	−39.5410	−1.33980
$872$	0	0
$873$	− 31.7984i	− 1.07621i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.7426i	1.17318i	0.809886	+	0.586588i	$0.199529\pi$
$877$	34.7426i	1.17318i	−0.809886	+	0.586588i	$0.800471\pi$
$878$	0	0
$879$	11.9098	0.401709
$880$	0	0
$881$	−46.7426	−1.57480	−0.787400	−	0.616443i	$-0.788573\pi$
$881$	−46.7426	−1.57480	−0.787400	+	0.616443i	$0.788573\pi$
$882$	0	0
$883$	− 8.34752i	− 0.280917i	−0.990087	−	0.140458i	$-0.955142\pi$
$883$	− 8.34752i	− 0.280917i	0.990087	−	0.140458i	$-0.0448576\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 29.2361i	− 0.981651i	−0.871258	−	0.490826i	$-0.836695\pi$
$887$	− 29.2361i	− 0.981651i	0.871258	−	0.490826i	$-0.163305\pi$
$888$	0	0
$889$	−6.23607	−0.209151
$890$	0	0
$891$	7.88854	0.264276
$892$	0	0
$893$	− 4.32624i	− 0.144772i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.85410i	− 0.162074i
$898$	0	0
$899$	−3.63932	−0.121378
$900$	0	0
$901$	−1.63932	−0.0546137
$902$	0	0
$903$	− 3.81966i	− 0.127110i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.5623i	0.549942i	0.961452	+	0.274971i	$0.0886683\pi$
$907$	16.5623i	0.549942i	−0.961452	+	0.274971i	$0.911332\pi$
$908$	0	0
$909$	7.09017	0.235166
$910$	0	0
$911$	20.8197	0.689786	0.344893	−	0.938642i	$-0.387915\pi$
$911$	20.8197	0.689786	0.344893	+	0.938642i	$0.387915\pi$
$912$	0	0
$913$	17.8885i	0.592024i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.0000i	− 0.330229i
$918$	0	0
$919$	−42.0132	−1.38589	−0.692943	−	0.720992i	$-0.743686\pi$
$919$	−42.0132	−1.38589	−0.692943	+	0.720992i	$0.743686\pi$
$920$	0	0
$921$	−11.7639	−0.387635
$922$	0	0
$923$	18.2705i	0.601381i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	34.1246i	1.12080i
$928$	0	0
$929$	−36.0132	−1.18155	−0.590777	−	0.806835i	$-0.701179\pi$
$929$	−36.0132	−1.18155	−0.590777	+	0.806835i	$0.701179\pi$
$930$	0	0
$931$	2.52786	0.0828474
$932$	0	0
$933$	20.7082i	0.677956i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 36.5967i	− 1.19556i	−0.801659	−	0.597782i	$-0.796049\pi$
$937$	− 36.5967i	− 1.19556i	0.801659	−	0.597782i	$-0.203951\pi$
$938$	0	0
$939$	−7.94427	−0.259252
$940$	0	0
$941$	−23.6525	−0.771049	−0.385524	−	0.922698i	$-0.625979\pi$
$941$	−23.6525	−0.771049	−0.385524	+	0.922698i	$0.625979\pi$
$942$	0	0
$943$	4.00000i	0.130258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5.29180i	− 0.171960i	−0.996297	−	0.0859801i	$-0.972598\pi$
$947$	− 5.29180i	− 0.171960i	0.996297	−	0.0859801i	$-0.0274022\pi$
$948$	0	0
$949$	−19.8541	−0.644491
$950$	0	0
$951$	16.9230	0.548765
$952$	0	0
$953$	− 11.5623i	− 0.374540i	−0.982308	−	0.187270i	$-0.940036\pi$
$953$	− 11.5623i	− 0.374540i	0.982308	−	0.187270i	$-0.0599639\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.58359i	− 0.212817i
$958$	0	0
$959$	8.79837	0.284114
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0	0
$963$	17.1803i	0.553629i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.3607i	0.815544i	0.913084	+	0.407772i	$0.133694\pi$
$967$	25.3607i	0.815544i	−0.913084	+	0.407772i	$0.866306\pi$
$968$	0	0
$969$	−0.819660	−0.0263313
$970$	0	0
$971$	41.4508	1.33022	0.665111	−	0.746745i	$-0.268384\pi$
$971$	41.4508	1.33022	0.665111	+	0.746745i	$0.268384\pi$
$972$	0	0
$973$	− 7.85410i	− 0.251791i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.8197i	0.538109i	0.963125	+	0.269054i	$0.0867111\pi$
$977$	16.8197i	0.538109i	−0.963125	+	0.269054i	$0.913289\pi$
$978$	0	0
$979$	−10.5279	−0.336472
$980$	0	0
$981$	40.7426	1.30081
$982$	0	0
$983$	25.8885i	0.825716i	0.910795	+	0.412858i	$0.135469\pi$
$983$	25.8885i	0.825716i	−0.910795	+	0.412858i	$0.864531\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.32624i	0.137706i
$988$	0	0
$989$	−26.1803	−0.832486
$990$	0	0
$991$	10.8754	0.345468	0.172734	−	0.984969i	$-0.444740\pi$
$991$	10.8754	0.345468	0.172734	+	0.984969i	$0.444740\pi$
$992$	0	0
$993$	1.85410i	0.0588381i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 21.0344i	− 0.666167i	−0.942897	−	0.333084i	$-0.891911\pi$
$997$	− 21.0344i	− 0.666167i	0.942897	−	0.333084i	$-0.108089\pi$
$998$	0	0
$999$	25.9443	0.820840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2500.2.c.a.1249.3		4
5.2	odd	4		2500.2.a.a.1.2	✓	2
5.3	odd	4		2500.2.a.b.1.1	yes	2
5.4	even	2	inner	2500.2.c.a.1249.2		4
20.3	even	4		10000.2.a.e.1.2		2
20.7	even	4		10000.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2500.2.a.a.1.2	✓	2	5.2	odd	4
2500.2.a.b.1.1	yes	2	5.3	odd	4
2500.2.c.a.1249.2		4	5.4	even	2	inner
2500.2.c.a.1249.3		4	1.1	even	1	trivial
10000.2.a.e.1.2		2	20.3	even	4
10000.2.a.j.1.1		2	20.7	even	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:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2500.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{3} +0.618034i q^{7} +2.61803 q^{9} +O(q^{10})\)	\(q+0.618034i q^{3} +0.618034i q^{7} +2.61803 q^{9} +1.38197 q^{11} +3.00000i q^{13} +3.47214i q^{17} +0.381966 q^{19} -0.381966 q^{21} +2.61803i q^{23} +3.47214i q^{27} -7.70820 q^{29} +0.472136 q^{31} +0.854102i q^{33} -7.47214i q^{37} -1.85410 q^{39} +1.52786 q^{41} +10.0000i q^{43} -11.3262i q^{47} +6.61803 q^{49} -2.14590 q^{51} +0.472136i q^{53} +0.236068i q^{57} +10.2361 q^{59} -3.32624 q^{61} +1.61803i q^{63} +13.1803i q^{67} -1.61803 q^{69} +6.09017 q^{71} +6.61803i q^{73} +0.854102i q^{77} -13.0000 q^{79} +5.70820 q^{81} +12.9443i q^{83} -4.76393i q^{87} -7.61803 q^{89} -1.85410 q^{91} +0.291796i q^{93} -12.1459i q^{97} +3.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{9}+O(q^{10}) \) 4 * q + 6 * q^9	\( 4 q + 6 q^{9} + 10 q^{11} + 6 q^{19} - 6 q^{21} - 4 q^{29} - 16 q^{31} + 6 q^{39} + 24 q^{41} + 22 q^{49} - 22 q^{51} + 32 q^{59} + 18 q^{61} - 2 q^{69} + 2 q^{71} - 52 q^{79} - 4 q^{81} - 26 q^{89} + 6 q^{91} + 10 q^{99}+O(q^{100}) \) 4 * q + 6 * q^9 + 10 * q^11 + 6 * q^19 - 6 * q^21 - 4 * q^29 - 16 * q^31 + 6 * q^39 + 24 * q^41 + 22 * q^49 - 22 * q^51 + 32 * q^59 + 18 * q^61 - 2 * q^69 + 2 * q^71 - 52 * q^79 - 4 * q^81 - 26 * q^89 + 6 * q^91 + 10 * q^99

Introduction

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