LMFDB - Embedded newform 2450.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.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:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 12x^{2} + 25 $ x^4 - 12*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.63810$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.63810 q^{3} +1.00000 q^{4} -1.63810 q^{6} -1.00000 q^{8} -0.316625 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.63810 q^{3} +1.00000 q^{4} -1.63810 q^{6} -1.00000 q^{8} -0.316625 q^{9} -1.31662 q^{11} +1.63810 q^{12} +6.10463 q^{13} +1.00000 q^{16} +2.60454 q^{17} +0.316625 q^{18} +3.05231 q^{19} +1.31662 q^{22} -4.63325 q^{23} -1.63810 q^{24} -6.10463 q^{26} -5.43297 q^{27} +10.6332 q^{29} +5.65685 q^{31} -1.00000 q^{32} -2.15676 q^{33} -2.60454 q^{34} -0.316625 q^{36} -8.63325 q^{37} -3.05231 q^{38} +10.0000 q^{39} -7.29496 q^{41} +6.63325 q^{43} -1.31662 q^{44} +4.63325 q^{46} -3.27620 q^{47} +1.63810 q^{48} +4.26650 q^{51} +6.10463 q^{52} -8.00000 q^{53} +5.43297 q^{54} +5.00000 q^{57} -10.6332 q^{58} +7.51884 q^{59} +3.27620 q^{61} -5.65685 q^{62} +1.00000 q^{64} +2.15676 q^{66} +11.6332 q^{67} +2.60454 q^{68} -7.58973 q^{69} -2.00000 q^{71} +0.316625 q^{72} +2.60454 q^{73} +8.63325 q^{74} +3.05231 q^{76} -10.0000 q^{78} +14.6332 q^{79} -7.94987 q^{81} +7.29496 q^{82} +12.9518 q^{83} -6.63325 q^{86} +17.4183 q^{87} +1.31662 q^{88} +9.15694 q^{89} -4.63325 q^{92} +9.26650 q^{93} +3.27620 q^{94} -1.63810 q^{96} -4.24264 q^{97} +0.416876 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{16} - 12 q^{18} - 8 q^{22} + 8 q^{23} + 16 q^{29} - 4 q^{32} + 12 q^{36} - 8 q^{37} + 40 q^{39} + 8 q^{44} - 8 q^{46} - 36 q^{51} - 32 q^{53} + 20 q^{57} - 16 q^{58} + 4 q^{64} + 20 q^{67} - 8 q^{71} - 12 q^{72} + 8 q^{74} - 40 q^{78} + 32 q^{79} + 8 q^{81} - 8 q^{88} + 8 q^{92} - 16 q^{93} + 68 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^16 - 12 * q^18 - 8 * q^22 + 8 * q^23 + 16 * q^29 - 4 * q^32 + 12 * q^36 - 8 * q^37 + 40 * q^39 + 8 * q^44 - 8 * q^46 - 36 * q^51 - 32 * q^53 + 20 * q^57 - 16 * q^58 + 4 * q^64 + 20 * q^67 - 8 * q^71 - 12 * q^72 + 8 * q^74 - 40 * q^78 + 32 * q^79 + 8 * q^81 - 8 * q^88 + 8 * q^92 - 16 * q^93 + 68 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.63810	0.945758	0.472879	−	0.881127i	$-0.343215\pi$
$3$	1.63810	0.945758	0.472879	+	0.881127i	$0.343215\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.63810	−0.668752
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−0.316625	−0.105542
$10$	0	0
$11$	−1.31662	−0.396977	−0.198489	−	0.980103i	$-0.563603\pi$
$11$	−1.31662	−0.396977	−0.198489	+	0.980103i	$0.563603\pi$
$12$	1.63810	0.472879
$13$	6.10463	1.69312	0.846560	−	0.532294i	$-0.178670\pi$
$13$	6.10463	1.69312	0.846560	+	0.532294i	$0.178670\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.60454	0.631694	0.315847	−	0.948810i	$-0.397711\pi$
$17$	2.60454	0.631694	0.315847	+	0.948810i	$0.397711\pi$
$18$	0.316625	0.0746292
$19$	3.05231	0.700249	0.350125	−	0.936703i	$-0.386139\pi$
$19$	3.05231	0.700249	0.350125	+	0.936703i	$0.386139\pi$
$20$	0	0
$21$	0	0
$22$	1.31662	0.280705
$23$	−4.63325	−0.966099	−0.483050	−	0.875593i	$-0.660471\pi$
$23$	−4.63325	−0.966099	−0.483050	+	0.875593i	$0.660471\pi$
$24$	−1.63810	−0.334376
$25$	0	0
$26$	−6.10463	−1.19722
$27$	−5.43297	−1.04557
$28$	0	0
$29$	10.6332	1.97454	0.987272	−	0.159038i	$-0.0508393\pi$
$29$	10.6332	1.97454	0.987272	+	0.159038i	$0.0508393\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	−1.00000	−0.176777
$33$	−2.15676	−0.375445
$34$	−2.60454	−0.446675
$35$	0	0
$36$	−0.316625	−0.0527708
$37$	−8.63325	−1.41930	−0.709649	−	0.704556i	$-0.751146\pi$
$37$	−8.63325	−1.41930	−0.709649	+	0.704556i	$0.751146\pi$
$38$	−3.05231	−0.495151
$39$	10.0000	1.60128
$40$	0	0
$41$	−7.29496	−1.13928	−0.569640	−	0.821894i	$-0.692917\pi$
$41$	−7.29496	−1.13928	−0.569640	+	0.821894i	$0.692917\pi$
$42$	0	0
$43$	6.63325	1.01156	0.505781	−	0.862662i	$-0.331205\pi$
$43$	6.63325	1.01156	0.505781	+	0.862662i	$0.331205\pi$
$44$	−1.31662	−0.198489
$45$	0	0
$46$	4.63325	0.683135
$47$	−3.27620	−0.477883	−0.238942	−	0.971034i	$-0.576800\pi$
$47$	−3.27620	−0.477883	−0.238942	+	0.971034i	$0.576800\pi$
$48$	1.63810	0.236440
$49$	0	0
$50$	0	0
$51$	4.26650	0.597429
$52$	6.10463	0.846560
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	5.43297	0.739333
$55$	0	0
$56$	0	0
$57$	5.00000	0.662266
$58$	−10.6332	−1.39621
$59$	7.51884	0.978870	0.489435	−	0.872040i	$-0.337203\pi$
$59$	7.51884	0.978870	0.489435	+	0.872040i	$0.337203\pi$
$60$	0	0
$61$	3.27620	0.419475	0.209737	−	0.977758i	$-0.432739\pi$
$61$	3.27620	0.419475	0.209737	+	0.977758i	$0.432739\pi$
$62$	−5.65685	−0.718421
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.15676	0.265479
$67$	11.6332	1.42123	0.710614	−	0.703582i	$-0.248417\pi$
$67$	11.6332	1.42123	0.710614	+	0.703582i	$0.248417\pi$
$68$	2.60454	0.315847
$69$	−7.58973	−0.913696
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0.316625	0.0373146
$73$	2.60454	0.304838	0.152419	−	0.988316i	$-0.451294\pi$
$73$	2.60454	0.304838	0.152419	+	0.988316i	$0.451294\pi$
$74$	8.63325	1.00359
$75$	0	0
$76$	3.05231	0.350125
$77$	0	0
$78$	−10.0000	−1.13228
$79$	14.6332	1.64637	0.823185	−	0.567774i	$-0.192195\pi$
$79$	14.6332	1.64637	0.823185	+	0.567774i	$0.192195\pi$
$80$	0	0
$81$	−7.94987	−0.883319
$82$	7.29496	0.805593
$83$	12.9518	1.42165	0.710823	−	0.703371i	$-0.248323\pi$
$83$	12.9518	1.42165	0.710823	+	0.703371i	$0.248323\pi$
$84$	0	0
$85$	0	0
$86$	−6.63325	−0.715282
$87$	17.4183	1.86744
$88$	1.31662	0.140353
$89$	9.15694	0.970634	0.485317	−	0.874338i	$-0.338704\pi$
$89$	9.15694	0.970634	0.485317	+	0.874338i	$0.338704\pi$
$90$	0	0
$91$	0	0
$92$	−4.63325	−0.483050
$93$	9.26650	0.960891
$94$	3.27620	0.337914
$95$	0	0
$96$	−1.63810	−0.167188
$97$	−4.24264	−0.430775	−0.215387	−	0.976529i	$-0.569101\pi$
$97$	−4.24264	−0.430775	−0.215387	+	0.976529i	$0.569101\pi$
$98$	0	0
$99$	0.416876	0.0418976
$100$	0	0
$101$	3.72398	0.370550	0.185275	−	0.982687i	$-0.440682\pi$
$101$	3.72398	0.370550	0.185275	+	0.982687i	$0.440682\pi$
$102$	−4.26650	−0.422446
$103$	−11.3137	−1.11477	−0.557386	−	0.830253i	$-0.688196\pi$
$103$	−11.3137	−1.11477	−0.557386	+	0.830253i	$0.688196\pi$
$104$	−6.10463	−0.598608
$105$	0	0
$106$	8.00000	0.777029
$107$	7.63325	0.737934	0.368967	−	0.929442i	$-0.379712\pi$
$107$	7.63325	0.737934	0.368967	+	0.929442i	$0.379712\pi$
$108$	−5.43297	−0.522787
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	−14.1421	−1.34231
$112$	0	0
$113$	−7.31662	−0.688290	−0.344145	−	0.938916i	$-0.611831\pi$
$113$	−7.31662	−0.688290	−0.344145	+	0.938916i	$0.611831\pi$
$114$	−5.00000	−0.468293
$115$	0	0
$116$	10.6332	0.987272
$117$	−1.93288	−0.178695
$118$	−7.51884	−0.692166
$119$	0	0
$120$	0	0
$121$	−9.26650	−0.842409
$122$	−3.27620	−0.296613
$123$	−11.9499	−1.07748
$124$	5.65685	0.508001
$125$	0	0
$126$	0	0
$127$	5.36675	0.476222	0.238111	−	0.971238i	$-0.423472\pi$
$127$	5.36675	0.476222	0.238111	+	0.971238i	$0.423472\pi$
$128$	−1.00000	−0.0883883
$129$	10.8659	0.956692
$130$	0	0
$131$	21.6610	1.89253	0.946264	−	0.323394i	$-0.104824\pi$
$131$	21.6610	1.89253	0.946264	+	0.323394i	$0.104824\pi$
$132$	−2.15676	−0.187722
$133$	0	0
$134$	−11.6332	−1.00496
$135$	0	0
$136$	−2.60454	−0.223337
$137$	5.94987	0.508332	0.254166	−	0.967161i	$-0.418199\pi$
$137$	5.94987	0.508332	0.254166	+	0.967161i	$0.418199\pi$
$138$	7.58973	0.646081
$139$	−17.1236	−1.45240	−0.726201	−	0.687483i	$-0.758716\pi$
$139$	−17.1236	−1.45240	−0.726201	+	0.687483i	$0.758716\pi$
$140$	0	0
$141$	−5.36675	−0.451962
$142$	2.00000	0.167836
$143$	−8.03751	−0.672130
$144$	−0.316625	−0.0263854
$145$	0	0
$146$	−2.60454	−0.215553
$147$	0	0
$148$	−8.63325	−0.709649
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	12.6332	1.02808	0.514040	−	0.857766i	$-0.328148\pi$
$151$	12.6332	1.02808	0.514040	+	0.857766i	$0.328148\pi$
$152$	−3.05231	−0.247575
$153$	−0.824662	−0.0666700
$154$	0	0
$155$	0	0
$156$	10.0000	0.800641
$157$	−9.38083	−0.748672	−0.374336	−	0.927293i	$-0.622129\pi$
$157$	−9.38083	−0.748672	−0.374336	+	0.927293i	$0.622129\pi$
$158$	−14.6332	−1.16416
$159$	−13.1048	−1.03928
$160$	0	0
$161$	0	0
$162$	7.94987	0.624601
$163$	9.31662	0.729734	0.364867	−	0.931060i	$-0.381114\pi$
$163$	9.31662	0.729734	0.364867	+	0.931060i	$0.381114\pi$
$164$	−7.29496	−0.569640
$165$	0	0
$166$	−12.9518	−1.00526
$167$	−15.9332	−1.23295	−0.616475	−	0.787374i	$-0.711440\pi$
$167$	−15.9332	−1.23295	−0.616475	+	0.787374i	$0.711440\pi$
$168$	0	0
$169$	24.2665	1.86665
$170$	0	0
$171$	−0.966438	−0.0739054
$172$	6.63325	0.505781
$173$	−21.1423	−1.60742	−0.803710	−	0.595021i	$-0.797144\pi$
$173$	−21.1423	−1.60742	−0.803710	+	0.595021i	$0.797144\pi$
$174$	−17.4183	−1.32048
$175$	0	0
$176$	−1.31662	−0.0992443
$177$	12.3166	0.925774
$178$	−9.15694	−0.686342
$179$	−18.8997	−1.41263	−0.706317	−	0.707896i	$-0.749645\pi$
$179$	−18.8997	−1.41263	−0.706317	+	0.707896i	$0.749645\pi$
$180$	0	0
$181$	18.3139	1.36126	0.680630	−	0.732627i	$-0.261706\pi$
$181$	18.3139	1.36126	0.680630	+	0.732627i	$0.261706\pi$
$182$	0	0
$183$	5.36675	0.396722
$184$	4.63325	0.341568
$185$	0	0
$186$	−9.26650	−0.679453
$187$	−3.42920	−0.250768
$188$	−3.27620	−0.238942
$189$	0	0
$190$	0	0
$191$	−11.2665	−0.815215	−0.407608	−	0.913157i	$-0.633637\pi$
$191$	−11.2665	−0.815215	−0.407608	+	0.913157i	$0.633637\pi$
$192$	1.63810	0.118220
$193$	−3.31662	−0.238736	−0.119368	−	0.992850i	$-0.538087\pi$
$193$	−3.31662	−0.238736	−0.119368	+	0.992850i	$0.538087\pi$
$194$	4.24264	0.304604
$195$	0	0
$196$	0	0
$197$	6.63325	0.472599	0.236300	−	0.971680i	$-0.424065\pi$
$197$	6.63325	0.472599	0.236300	+	0.971680i	$0.424065\pi$
$198$	−0.416876	−0.0296261
$199$	−0.447775	−0.0317419	−0.0158710	−	0.999874i	$-0.505052\pi$
$199$	−0.447775	−0.0317419	−0.0158710	+	0.999874i	$0.505052\pi$
$200$	0	0
$201$	19.0564	1.34414
$202$	−3.72398	−0.262018
$203$	0	0
$204$	4.26650	0.298715
$205$	0	0
$206$	11.3137	0.788263
$207$	1.46700	0.101964
$208$	6.10463	0.423280
$209$	−4.01875	−0.277983
$210$	0	0
$211$	11.6332	0.800866	0.400433	−	0.916326i	$-0.368860\pi$
$211$	11.6332	0.800866	0.400433	+	0.916326i	$0.368860\pi$
$212$	−8.00000	−0.549442
$213$	−3.27620	−0.224482
$214$	−7.63325	−0.521798
$215$	0	0
$216$	5.43297	0.369667
$217$	0	0
$218$	−18.0000	−1.21911
$219$	4.26650	0.288303
$220$	0	0
$221$	15.8997	1.06953
$222$	14.1421	0.949158
$223$	6.10463	0.408796	0.204398	−	0.978888i	$-0.434476\pi$
$223$	6.10463	0.408796	0.204398	+	0.978888i	$0.434476\pi$
$224$	0	0
$225$	0	0
$226$	7.31662	0.486695
$227$	−13.1757	−0.874502	−0.437251	−	0.899340i	$-0.644048\pi$
$227$	−13.1757	−0.874502	−0.437251	+	0.899340i	$0.644048\pi$
$228$	5.00000	0.331133
$229$	−5.20908	−0.344226	−0.172113	−	0.985077i	$-0.555059\pi$
$229$	−5.20908	−0.344226	−0.172113	+	0.985077i	$0.555059\pi$
$230$	0	0
$231$	0	0
$232$	−10.6332	−0.698107
$233$	13.2665	0.869117	0.434559	−	0.900644i	$-0.356904\pi$
$233$	13.2665	0.869117	0.434559	+	0.900644i	$0.356904\pi$
$234$	1.93288	0.126356
$235$	0	0
$236$	7.51884	0.489435
$237$	23.9707	1.55707
$238$	0	0
$239$	−3.36675	−0.217777	−0.108888	−	0.994054i	$-0.534729\pi$
$239$	−3.36675	−0.217777	−0.108888	+	0.994054i	$0.534729\pi$
$240$	0	0
$241$	−20.3998	−1.31406	−0.657032	−	0.753863i	$-0.728188\pi$
$241$	−20.3998	−1.31406	−0.657032	+	0.753863i	$0.728188\pi$
$242$	9.26650	0.595673
$243$	3.27620	0.210168
$244$	3.27620	0.209737
$245$	0	0
$246$	11.9499	0.761896
$247$	18.6332	1.18561
$248$	−5.65685	−0.359211
$249$	21.2164	1.34453
$250$	0	0
$251$	−13.9182	−0.878512	−0.439256	−	0.898362i	$-0.644758\pi$
$251$	−13.9182	−0.878512	−0.439256	+	0.898362i	$0.644758\pi$
$252$	0	0
$253$	6.10025	0.383520
$254$	−5.36675	−0.336740
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.41421	−0.0882162	−0.0441081	−	0.999027i	$-0.514045\pi$
$257$	−1.41421	−0.0882162	−0.0441081	+	0.999027i	$0.514045\pi$
$258$	−10.8659	−0.676483
$259$	0	0
$260$	0	0
$261$	−3.36675	−0.208397
$262$	−21.6610	−1.33822
$263$	23.8997	1.47372	0.736861	−	0.676044i	$-0.236307\pi$
$263$	23.8997	1.47372	0.736861	+	0.676044i	$0.236307\pi$
$264$	2.15676	0.132740
$265$	0	0
$266$	0	0
$267$	15.0000	0.917985
$268$	11.6332	0.710614
$269$	17.4183	1.06201	0.531007	−	0.847367i	$-0.321814\pi$
$269$	17.4183	1.06201	0.531007	+	0.847367i	$0.321814\pi$
$270$	0	0
$271$	−26.7992	−1.62793	−0.813967	−	0.580911i	$-0.802696\pi$
$271$	−26.7992	−1.62793	−0.813967	+	0.580911i	$0.802696\pi$
$272$	2.60454	0.157923
$273$	0	0
$274$	−5.94987	−0.359445
$275$	0	0
$276$	−7.58973	−0.456848
$277$	−6.63325	−0.398553	−0.199277	−	0.979943i	$-0.563859\pi$
$277$	−6.63325	−0.398553	−0.199277	+	0.979943i	$0.563859\pi$
$278$	17.1236	1.02700
$279$	−1.79110	−0.107230
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	5.36675	0.319585
$283$	29.9224	1.77870	0.889350	−	0.457227i	$-0.151157\pi$
$283$	29.9224	1.77870	0.889350	+	0.457227i	$0.151157\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	8.03751	0.475268
$287$	0	0
$288$	0.316625	0.0186573
$289$	−10.2164	−0.600963
$290$	0	0
$291$	−6.94987	−0.407409
$292$	2.60454	0.152419
$293$	14.5899	0.852352	0.426176	−	0.904640i	$-0.359860\pi$
$293$	14.5899	0.852352	0.426176	+	0.904640i	$0.359860\pi$
$294$	0	0
$295$	0	0
$296$	8.63325	0.501797
$297$	7.15318	0.415070
$298$	0	0
$299$	−28.2843	−1.63572
$300$	0	0
$301$	0	0
$302$	−12.6332	−0.726962
$303$	6.10025	0.350450
$304$	3.05231	0.175062
$305$	0	0
$306$	0.824662	0.0471428
$307$	−20.0229	−1.14277	−0.571383	−	0.820684i	$-0.693593\pi$
$307$	−20.0229	−1.14277	−0.571383	+	0.820684i	$0.693593\pi$
$308$	0	0
$309$	−18.5330	−1.05431
$310$	0	0
$311$	−19.2094	−1.08927	−0.544634	−	0.838674i	$-0.683331\pi$
$311$	−19.2094	−1.08927	−0.544634	+	0.838674i	$0.683331\pi$
$312$	−10.0000	−0.566139
$313$	−25.8327	−1.46015	−0.730076	−	0.683366i	$-0.760515\pi$
$313$	−25.8327	−1.46015	−0.730076	+	0.683366i	$0.760515\pi$
$314$	9.38083	0.529391
$315$	0	0
$316$	14.6332	0.823185
$317$	−5.89975	−0.331363	−0.165681	−	0.986179i	$-0.552982\pi$
$317$	−5.89975	−0.331363	−0.165681	+	0.986179i	$0.552982\pi$
$318$	13.1048	0.734881
$319$	−14.0000	−0.783850
$320$	0	0
$321$	12.5040	0.697907
$322$	0	0
$323$	7.94987	0.442343
$324$	−7.94987	−0.441660
$325$	0	0
$326$	−9.31662	−0.516000
$327$	29.4858	1.63057
$328$	7.29496	0.402797
$329$	0	0
$330$	0	0
$331$	29.3166	1.61139	0.805694	−	0.592332i	$-0.201793\pi$
$331$	29.3166	1.61139	0.805694	+	0.592332i	$0.201793\pi$
$332$	12.9518	0.710823
$333$	2.73350	0.149795
$334$	15.9332	0.871828
$335$	0	0
$336$	0	0
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	−24.2665	−1.31992
$339$	−11.9854	−0.650956
$340$	0	0
$341$	−7.44795	−0.403329
$342$	0.966438	0.0522590
$343$	0	0
$344$	−6.63325	−0.357641
$345$	0	0
$346$	21.1423	1.13662
$347$	9.31662	0.500143	0.250071	−	0.968227i	$-0.419546\pi$
$347$	9.31662	0.500143	0.250071	+	0.968227i	$0.419546\pi$
$348$	17.4183	0.933721
$349$	36.6278	1.96064	0.980320	−	0.197415i	$-0.0632547\pi$
$349$	36.6278	1.96064	0.980320	+	0.197415i	$0.0632547\pi$
$350$	0	0
$351$	−33.1662	−1.77028
$352$	1.31662	0.0701763
$353$	−27.7656	−1.47781	−0.738907	−	0.673807i	$-0.764658\pi$
$353$	−27.7656	−1.47781	−0.738907	+	0.673807i	$0.764658\pi$
$354$	−12.3166	−0.654621
$355$	0	0
$356$	9.15694	0.485317
$357$	0	0
$358$	18.8997	0.998883
$359$	−4.73350	−0.249825	−0.124912	−	0.992168i	$-0.539865\pi$
$359$	−4.73350	−0.249825	−0.124912	+	0.992168i	$0.539865\pi$
$360$	0	0
$361$	−9.68338	−0.509651
$362$	−18.3139	−0.962557
$363$	−15.1795	−0.796715
$364$	0	0
$365$	0	0
$366$	−5.36675	−0.280525
$367$	−10.2764	−0.536423	−0.268211	−	0.963360i	$-0.586433\pi$
$367$	−10.2764	−0.536423	−0.268211	+	0.963360i	$0.586433\pi$
$368$	−4.63325	−0.241525
$369$	2.30976	0.120241
$370$	0	0
$371$	0	0
$372$	9.26650	0.480446
$373$	−5.89975	−0.305477	−0.152739	−	0.988267i	$-0.548809\pi$
$373$	−5.89975	−0.305477	−0.152739	+	0.988267i	$0.548809\pi$
$374$	3.42920	0.177320
$375$	0	0
$376$	3.27620	0.168957
$377$	64.9120	3.34314
$378$	0	0
$379$	11.9499	0.613824	0.306912	−	0.951738i	$-0.400704\pi$
$379$	11.9499	0.613824	0.306912	+	0.951738i	$0.400704\pi$
$380$	0	0
$381$	8.79128	0.450391
$382$	11.2665	0.576444
$383$	9.82861	0.502218	0.251109	−	0.967959i	$-0.419205\pi$
$383$	9.82861	0.502218	0.251109	+	0.967959i	$0.419205\pi$
$384$	−1.63810	−0.0835940
$385$	0	0
$386$	3.31662	0.168812
$387$	−2.10025	−0.106762
$388$	−4.24264	−0.215387
$389$	−15.3668	−0.779125	−0.389563	−	0.921000i	$-0.627374\pi$
$389$	−15.3668	−0.779125	−0.389563	+	0.921000i	$0.627374\pi$
$390$	0	0
$391$	−12.0675	−0.610279
$392$	0	0
$393$	35.4829	1.78987
$394$	−6.63325	−0.334178
$395$	0	0
$396$	0.416876	0.0209488
$397$	2.38065	0.119482	0.0597408	−	0.998214i	$-0.480973\pi$
$397$	2.38065	0.119482	0.0597408	+	0.998214i	$0.480973\pi$
$398$	0.447775	0.0224449
$399$	0	0
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	−19.0564	−0.950449
$403$	34.5330	1.72021
$404$	3.72398	0.185275
$405$	0	0
$406$	0	0
$407$	11.3668	0.563429
$408$	−4.26650	−0.211223
$409$	3.42920	0.169563	0.0847815	−	0.996400i	$-0.472981\pi$
$409$	3.42920	0.169563	0.0847815	+	0.996400i	$0.472981\pi$
$410$	0	0
$411$	9.74650	0.480759
$412$	−11.3137	−0.557386
$413$	0	0
$414$	−1.46700	−0.0720992
$415$	0	0
$416$	−6.10463	−0.299304
$417$	−28.0501	−1.37362
$418$	4.01875	0.196564
$419$	33.1986	1.62186	0.810928	−	0.585146i	$-0.198963\pi$
$419$	33.1986	1.62186	0.810928	+	0.585146i	$0.198963\pi$
$420$	0	0
$421$	−14.5330	−0.708295	−0.354147	−	0.935190i	$-0.615229\pi$
$421$	−14.5330	−0.708295	−0.354147	+	0.935190i	$0.615229\pi$
$422$	−11.6332	−0.566298
$423$	1.03733	0.0504366
$424$	8.00000	0.388514
$425$	0	0
$426$	3.27620	0.158733
$427$	0	0
$428$	7.63325	0.368967
$429$	−13.1662	−0.635672
$430$	0	0
$431$	20.6332	0.993869	0.496934	−	0.867788i	$-0.334459\pi$
$431$	20.6332	0.993869	0.496934	+	0.867788i	$0.334459\pi$
$432$	−5.43297	−0.261394
$433$	−10.5712	−0.508017	−0.254009	−	0.967202i	$-0.581749\pi$
$433$	−10.5712	−0.508017	−0.254009	+	0.967202i	$0.581749\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	−14.1421	−0.676510
$438$	−4.26650	−0.203861
$439$	−9.38083	−0.447723	−0.223861	−	0.974621i	$-0.571866\pi$
$439$	−9.38083	−0.447723	−0.223861	+	0.974621i	$0.571866\pi$
$440$	0	0
$441$	0	0
$442$	−15.8997	−0.756274
$443$	−26.8997	−1.27805	−0.639023	−	0.769188i	$-0.720661\pi$
$443$	−26.8997	−1.27805	−0.639023	+	0.769188i	$0.720661\pi$
$444$	−14.1421	−0.671156
$445$	0	0
$446$	−6.10463	−0.289063
$447$	0	0
$448$	0	0
$449$	−7.73350	−0.364966	−0.182483	−	0.983209i	$-0.558414\pi$
$449$	−7.73350	−0.364966	−0.182483	+	0.983209i	$0.558414\pi$
$450$	0	0
$451$	9.60472	0.452269
$452$	−7.31662	−0.344145
$453$	20.6945	0.972314
$454$	13.1757	0.618366
$455$	0	0
$456$	−5.00000	−0.234146
$457$	−23.3166	−1.09071	−0.545353	−	0.838207i	$-0.683604\pi$
$457$	−23.3166	−1.09071	−0.545353	+	0.838207i	$0.683604\pi$
$458$	5.20908	0.243404
$459$	−14.1504	−0.660483
$460$	0	0
$461$	−3.27620	−0.152588	−0.0762940	−	0.997085i	$-0.524309\pi$
$461$	−3.27620	−0.152588	−0.0762940	+	0.997085i	$0.524309\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	10.6332	0.493636
$465$	0	0
$466$	−13.2665	−0.614559
$467$	−28.2134	−1.30556	−0.652780	−	0.757548i	$-0.726397\pi$
$467$	−28.2134	−1.30556	−0.652780	+	0.757548i	$0.726397\pi$
$468$	−1.93288	−0.0893473
$469$	0	0
$470$	0	0
$471$	−15.3668	−0.708062
$472$	−7.51884	−0.346083
$473$	−8.73350	−0.401567
$474$	−23.9707	−1.10101
$475$	0	0
$476$	0	0
$477$	2.53300	0.115978
$478$	3.36675	0.153992
$479$	−41.8369	−1.91157	−0.955787	−	0.294059i	$-0.904994\pi$
$479$	−41.8369	−1.91157	−0.955787	+	0.294059i	$0.904994\pi$
$480$	0	0
$481$	−52.7028	−2.40304
$482$	20.3998	0.929184
$483$	0	0
$484$	−9.26650	−0.421205
$485$	0	0
$486$	−3.27620	−0.148612
$487$	−4.73350	−0.214495	−0.107248	−	0.994232i	$-0.534204\pi$
$487$	−4.73350	−0.214495	−0.107248	+	0.994232i	$0.534204\pi$
$488$	−3.27620	−0.148307
$489$	15.2616	0.690152
$490$	0	0
$491$	5.26650	0.237674	0.118837	−	0.992914i	$-0.462083\pi$
$491$	5.26650	0.237674	0.118837	+	0.992914i	$0.462083\pi$
$492$	−11.9499	−0.538742
$493$	27.6947	1.24731
$494$	−18.6332	−0.838350
$495$	0	0
$496$	5.65685	0.254000
$497$	0	0
$498$	−21.2164	−0.950728
$499$	13.2665	0.593890	0.296945	−	0.954895i	$-0.404032\pi$
$499$	13.2665	0.593890	0.296945	+	0.954895i	$0.404032\pi$
$500$	0	0
$501$	−26.1003	−1.16607
$502$	13.9182	0.621202
$503$	15.4855	0.690463	0.345231	−	0.938518i	$-0.387800\pi$
$503$	15.4855	0.690463	0.345231	+	0.938518i	$0.387800\pi$
$504$	0	0
$505$	0	0
$506$	−6.10025	−0.271189
$507$	39.7510	1.76540
$508$	5.36675	0.238111
$509$	11.7615	0.521319	0.260659	−	0.965431i	$-0.416060\pi$
$509$	11.7615	0.521319	0.260659	+	0.965431i	$0.416060\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−16.5831	−0.732163
$514$	1.41421	0.0623783
$515$	0	0
$516$	10.8659	0.478346
$517$	4.31353	0.189709
$518$	0	0
$519$	−34.6332	−1.52023
$520$	0	0
$521$	−30.2284	−1.32433	−0.662164	−	0.749359i	$-0.730362\pi$
$521$	−30.2284	−1.32433	−0.662164	+	0.749359i	$0.730362\pi$
$522$	3.36675	0.147359
$523$	−1.26121	−0.0551491	−0.0275745	−	0.999620i	$-0.508778\pi$
$523$	−1.26121	−0.0551491	−0.0275745	+	0.999620i	$0.508778\pi$
$524$	21.6610	0.946264
$525$	0	0
$526$	−23.8997	−1.04208
$527$	14.7335	0.641801
$528$	−2.15676	−0.0938611
$529$	−1.53300	−0.0666521
$530$	0	0
$531$	−2.38065	−0.103311
$532$	0	0
$533$	−44.5330	−1.92894
$534$	−15.0000	−0.649113
$535$	0	0
$536$	−11.6332	−0.502480
$537$	−30.9597	−1.33601
$538$	−17.4183	−0.750958
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	26.7992	1.15112
$543$	30.0000	1.28742
$544$	−2.60454	−0.111669
$545$	0	0
$546$	0	0
$547$	−16.0501	−0.686254	−0.343127	−	0.939289i	$-0.611486\pi$
$547$	−16.0501	−0.686254	−0.343127	+	0.939289i	$0.611486\pi$
$548$	5.94987	0.254166
$549$	−1.03733	−0.0442720
$550$	0	0
$551$	32.4560	1.38267
$552$	7.58973	0.323040
$553$	0	0
$554$	6.63325	0.281820
$555$	0	0
$556$	−17.1236	−0.726201
$557$	−37.2665	−1.57903	−0.789516	−	0.613730i	$-0.789668\pi$
$557$	−37.2665	−1.57903	−0.789516	+	0.613730i	$0.789668\pi$
$558$	1.79110	0.0758233
$559$	40.4935	1.71269
$560$	0	0
$561$	−5.61738	−0.237166
$562$	4.00000	0.168730
$563$	18.8326	0.793697	0.396849	−	0.917884i	$-0.370104\pi$
$563$	18.8326	0.793697	0.396849	+	0.917884i	$0.370104\pi$
$564$	−5.36675	−0.225981
$565$	0	0
$566$	−29.9224	−1.25773
$567$	0	0
$568$	2.00000	0.0839181
$569$	−22.2665	−0.933460	−0.466730	−	0.884400i	$-0.654568\pi$
$569$	−22.2665	−0.933460	−0.466730	+	0.884400i	$0.654568\pi$
$570$	0	0
$571$	−10.6332	−0.444988	−0.222494	−	0.974934i	$-0.571420\pi$
$571$	−10.6332	−0.444988	−0.222494	+	0.974934i	$0.571420\pi$
$572$	−8.03751	−0.336065
$573$	−18.4557	−0.770996
$574$	0	0
$575$	0	0
$576$	−0.316625	−0.0131927
$577$	−13.0227	−0.542142	−0.271071	−	0.962559i	$-0.587378\pi$
$577$	−13.0227	−0.542142	−0.271071	+	0.962559i	$0.587378\pi$
$578$	10.2164	0.424945
$579$	−5.43297	−0.225786
$580$	0	0
$581$	0	0
$582$	6.94987	0.288082
$583$	10.5330	0.436232
$584$	−2.60454	−0.107777
$585$	0	0
$586$	−14.5899	−0.602704
$587$	−15.7093	−0.648394	−0.324197	−	0.945990i	$-0.605094\pi$
$587$	−15.7093	−0.648394	−0.324197	+	0.945990i	$0.605094\pi$
$588$	0	0
$589$	17.2665	0.711454
$590$	0	0
$591$	10.8659	0.446965
$592$	−8.63325	−0.354824
$593$	25.2320	1.03615	0.518076	−	0.855335i	$-0.326648\pi$
$593$	25.2320	1.03615	0.518076	+	0.855335i	$0.326648\pi$
$594$	−7.15318	−0.293498
$595$	0	0
$596$	0	0
$597$	−0.733501	−0.0300202
$598$	28.2843	1.15663
$599$	−33.2665	−1.35923	−0.679616	−	0.733568i	$-0.737854\pi$
$599$	−33.2665	−1.35923	−0.679616	+	0.733568i	$0.737854\pi$
$600$	0	0
$601$	34.0941	1.39073	0.695364	−	0.718658i	$-0.255243\pi$
$601$	34.0941	1.39073	0.695364	+	0.718658i	$0.255243\pi$
$602$	0	0
$603$	−3.68338	−0.149999
$604$	12.6332	0.514040
$605$	0	0
$606$	−6.10025	−0.247806
$607$	11.3137	0.459209	0.229605	−	0.973284i	$-0.426257\pi$
$607$	11.3137	0.459209	0.229605	+	0.973284i	$0.426257\pi$
$608$	−3.05231	−0.123788
$609$	0	0
$610$	0	0
$611$	−20.0000	−0.809113
$612$	−0.824662	−0.0333350
$613$	18.6332	0.752590	0.376295	−	0.926500i	$-0.377198\pi$
$613$	18.6332	0.752590	0.376295	+	0.926500i	$0.377198\pi$
$614$	20.0229	0.808058
$615$	0	0
$616$	0	0
$617$	−11.2665	−0.453572	−0.226786	−	0.973945i	$-0.572822\pi$
$617$	−11.2665	−0.453572	−0.226786	+	0.973945i	$0.572822\pi$
$618$	18.5330	0.745507
$619$	−20.6237	−0.828935	−0.414467	−	0.910064i	$-0.636032\pi$
$619$	−20.6237	−0.828935	−0.414467	+	0.910064i	$0.636032\pi$
$620$	0	0
$621$	25.1723	1.01013
$622$	19.2094	0.770228
$623$	0	0
$624$	10.0000	0.400320
$625$	0	0
$626$	25.8327	1.03248
$627$	−6.58312	−0.262905
$628$	−9.38083	−0.374336
$629$	−22.4856	−0.896561
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	−14.6332	−0.582079
$633$	19.0564	0.757425
$634$	5.89975	0.234309
$635$	0	0
$636$	−13.1048	−0.519639
$637$	0	0
$638$	14.0000	0.554265
$639$	0.633250	0.0250510
$640$	0	0
$641$	−28.5330	−1.12699	−0.563493	−	0.826121i	$-0.690543\pi$
$641$	−28.5330	−1.12699	−0.563493	+	0.826121i	$0.690543\pi$
$642$	−12.5040	−0.493495
$643$	−38.6315	−1.52348	−0.761740	−	0.647883i	$-0.775654\pi$
$643$	−38.6315	−1.52348	−0.761740	+	0.647883i	$0.775654\pi$
$644$	0	0
$645$	0	0
$646$	−7.94987	−0.312784
$647$	−15.9332	−0.626400	−0.313200	−	0.949687i	$-0.601401\pi$
$647$	−15.9332	−0.626400	−0.313200	+	0.949687i	$0.601401\pi$
$648$	7.94987	0.312301
$649$	−9.89949	−0.388589
$650$	0	0
$651$	0	0
$652$	9.31662	0.364867
$653$	31.1662	1.21963	0.609815	−	0.792544i	$-0.291244\pi$
$653$	31.1662	1.21963	0.609815	+	0.792544i	$0.291244\pi$
$654$	−29.4858	−1.15299
$655$	0	0
$656$	−7.29496	−0.284820
$657$	−0.824662	−0.0321731
$658$	0	0
$659$	−46.5831	−1.81462	−0.907310	−	0.420461i	$-0.861868\pi$
$659$	−46.5831	−1.81462	−0.907310	+	0.420461i	$0.861868\pi$
$660$	0	0
$661$	13.6944	0.532649	0.266324	−	0.963883i	$-0.414191\pi$
$661$	13.6944	0.532649	0.266324	+	0.963883i	$0.414191\pi$
$662$	−29.3166	−1.13942
$663$	26.0454	1.01152
$664$	−12.9518	−0.502628
$665$	0	0
$666$	−2.73350	−0.105921
$667$	−49.2665	−1.90761
$668$	−15.9332	−0.616475
$669$	10.0000	0.386622
$670$	0	0
$671$	−4.31353	−0.166522
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	27.0000	1.04000
$675$	0	0
$676$	24.2665	0.933327
$677$	−27.2469	−1.04719	−0.523593	−	0.851969i	$-0.675409\pi$
$677$	−27.2469	−1.04719	−0.523593	+	0.851969i	$0.675409\pi$
$678$	11.9854	0.460295
$679$	0	0
$680$	0	0
$681$	−21.5831	−0.827067
$682$	7.44795	0.285197
$683$	−50.1662	−1.91956	−0.959779	−	0.280756i	$-0.909415\pi$
$683$	−50.1662	−1.91956	−0.959779	+	0.280756i	$0.909415\pi$
$684$	−0.966438	−0.0369527
$685$	0	0
$686$	0	0
$687$	−8.53300	−0.325554
$688$	6.63325	0.252890
$689$	−48.8370	−1.86054
$690$	0	0
$691$	−24.7842	−0.942835	−0.471417	−	0.881910i	$-0.656257\pi$
$691$	−24.7842	−0.942835	−0.471417	+	0.881910i	$0.656257\pi$
$692$	−21.1423	−0.803710
$693$	0	0
$694$	−9.31662	−0.353654
$695$	0	0
$696$	−17.4183	−0.660240
$697$	−19.0000	−0.719676
$698$	−36.6278	−1.38638
$699$	21.7319	0.821975
$700$	0	0
$701$	−51.7995	−1.95644	−0.978220	−	0.207571i	$-0.933444\pi$
$701$	−51.7995	−1.95644	−0.978220	+	0.207571i	$0.933444\pi$
$702$	33.1662	1.25178
$703$	−26.3514	−0.993862
$704$	−1.31662	−0.0496222
$705$	0	0
$706$	27.7656	1.04497
$707$	0	0
$708$	12.3166	0.462887
$709$	34.5330	1.29691	0.648457	−	0.761251i	$-0.275415\pi$
$709$	34.5330	1.29691	0.648457	+	0.761251i	$0.275415\pi$
$710$	0	0
$711$	−4.63325	−0.173760
$712$	−9.15694	−0.343171
$713$	−26.2096	−0.981558
$714$	0	0
$715$	0	0
$716$	−18.8997	−0.706317
$717$	−5.51508	−0.205964
$718$	4.73350	0.176653
$719$	−0.895550	−0.0333984	−0.0166992	−	0.999861i	$-0.505316\pi$
$719$	−0.895550	−0.0333984	−0.0166992	+	0.999861i	$0.505316\pi$
$720$	0	0
$721$	0	0
$722$	9.68338	0.360378
$723$	−33.4169	−1.24279
$724$	18.3139	0.680630
$725$	0	0
$726$	15.1795	0.563363
$727$	−15.6272	−0.579582	−0.289791	−	0.957090i	$-0.593586\pi$
$727$	−15.6272	−0.579582	−0.289791	+	0.957090i	$0.593586\pi$
$728$	0	0
$729$	29.2164	1.08209
$730$	0	0
$731$	17.2766	0.638997
$732$	5.36675	0.198361
$733$	−26.3514	−0.973311	−0.486655	−	0.873594i	$-0.661783\pi$
$733$	−26.3514	−0.973311	−0.486655	+	0.873594i	$0.661783\pi$
$734$	10.2764	0.379308
$735$	0	0
$736$	4.63325	0.170784
$737$	−15.3166	−0.564195
$738$	−2.30976	−0.0850236
$739$	−33.1662	−1.22004	−0.610020	−	0.792386i	$-0.708839\pi$
$739$	−33.1662	−1.22004	−0.610020	+	0.792386i	$0.708839\pi$
$740$	0	0
$741$	30.5231	1.12130
$742$	0	0
$743$	−4.63325	−0.169977	−0.0849887	−	0.996382i	$-0.527085\pi$
$743$	−4.63325	−0.169977	−0.0849887	+	0.996382i	$0.527085\pi$
$744$	−9.26650	−0.339726
$745$	0	0
$746$	5.89975	0.216005
$747$	−4.10086	−0.150043
$748$	−3.42920	−0.125384
$749$	0	0
$750$	0	0
$751$	14.6332	0.533975	0.266987	−	0.963700i	$-0.413972\pi$
$751$	14.6332	0.533975	0.266987	+	0.963700i	$0.413972\pi$
$752$	−3.27620	−0.119471
$753$	−22.7995	−0.830860
$754$	−64.9120	−2.36396
$755$	0	0
$756$	0	0
$757$	−13.3668	−0.485823	−0.242911	−	0.970048i	$-0.578102\pi$
$757$	−13.3668	−0.485823	−0.242911	+	0.970048i	$0.578102\pi$
$758$	−11.9499	−0.434039
$759$	9.99283	0.362717
$760$	0	0
$761$	−0.152999	−0.00554622	−0.00277311	−	0.999996i	$-0.500883\pi$
$761$	−0.152999	−0.00554622	−0.00277311	+	0.999996i	$0.500883\pi$
$762$	−8.79128	−0.318474
$763$	0	0
$764$	−11.2665	−0.407608
$765$	0	0
$766$	−9.82861	−0.355122
$767$	45.8997	1.65734
$768$	1.63810	0.0591099
$769$	−52.3371	−1.88732	−0.943662	−	0.330910i	$-0.892645\pi$
$769$	−52.3371	−1.88732	−0.943662	+	0.330910i	$0.892645\pi$
$770$	0	0
$771$	−2.31662	−0.0834312
$772$	−3.31662	−0.119368
$773$	1.79110	0.0644214	0.0322107	−	0.999481i	$-0.489745\pi$
$773$	1.79110	0.0644214	0.0322107	+	0.999481i	$0.489745\pi$
$774$	2.10025	0.0754920
$775$	0	0
$776$	4.24264	0.152302
$777$	0	0
$778$	15.3668	0.550925
$779$	−22.2665	−0.797780
$780$	0	0
$781$	2.63325	0.0942251
$782$	12.0675	0.431532
$783$	−57.7701	−2.06453
$784$	0	0
$785$	0	0
$786$	−35.4829	−1.26563
$787$	47.1168	1.67953	0.839767	−	0.542947i	$-0.182692\pi$
$787$	47.1168	1.67953	0.839767	+	0.542947i	$0.182692\pi$
$788$	6.63325	0.236300
$789$	39.1502	1.39378
$790$	0	0
$791$	0	0
$792$	−0.416876	−0.0148130
$793$	20.0000	0.710221
$794$	−2.38065	−0.0844862
$795$	0	0
$796$	−0.447775	−0.0158710
$797$	22.1796	0.785643	0.392822	−	0.919615i	$-0.371499\pi$
$797$	22.1796	0.785643	0.392822	+	0.919615i	$0.371499\pi$
$798$	0	0
$799$	−8.53300	−0.301876
$800$	0	0
$801$	−2.89932	−0.102442
$802$	3.00000	0.105934
$803$	−3.42920	−0.121014
$804$	19.0564	0.672069
$805$	0	0
$806$	−34.5330	−1.21637
$807$	28.5330	1.00441
$808$	−3.72398	−0.131009
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	7.51884	0.264022	0.132011	−	0.991248i	$-0.457857\pi$
$811$	7.51884	0.264022	0.132011	+	0.991248i	$0.457857\pi$
$812$	0	0
$813$	−43.8997	−1.53963
$814$	−11.3668	−0.398404
$815$	0	0
$816$	4.26650	0.149357
$817$	20.2468	0.708345
$818$	−3.42920	−0.119899
$819$	0	0
$820$	0	0
$821$	49.2665	1.71941	0.859706	−	0.510789i	$-0.170647\pi$
$821$	49.2665	1.71941	0.859706	+	0.510789i	$0.170647\pi$
$822$	−9.74650	−0.339948
$823$	−4.53300	−0.158010	−0.0790052	−	0.996874i	$-0.525174\pi$
$823$	−4.53300	−0.158010	−0.0790052	+	0.996874i	$0.525174\pi$
$824$	11.3137	0.394132
$825$	0	0
$826$	0	0
$827$	5.63325	0.195887	0.0979436	−	0.995192i	$-0.468774\pi$
$827$	5.63325	0.195887	0.0979436	+	0.995192i	$0.468774\pi$
$828$	1.46700	0.0509818
$829$	6.69418	0.232499	0.116249	−	0.993220i	$-0.462913\pi$
$829$	6.69418	0.232499	0.116249	+	0.993220i	$0.462913\pi$
$830$	0	0
$831$	−10.8659	−0.376935
$832$	6.10463	0.211640
$833$	0	0
$834$	28.0501	0.971296
$835$	0	0
$836$	−4.01875	−0.138991
$837$	−30.7335	−1.06231
$838$	−33.1986	−1.14683
$839$	2.38065	0.0821892	0.0410946	−	0.999155i	$-0.486915\pi$
$839$	2.38065	0.0821892	0.0410946	+	0.999155i	$0.486915\pi$
$840$	0	0
$841$	84.0660	2.89883
$842$	14.5330	0.500840
$843$	−6.55240	−0.225677
$844$	11.6332	0.400433
$845$	0	0
$846$	−1.03733	−0.0356640
$847$	0	0
$848$	−8.00000	−0.274721
$849$	49.0159	1.68222
$850$	0	0
$851$	40.0000	1.37118
$852$	−3.27620	−0.112241
$853$	36.3218	1.24363	0.621817	−	0.783163i	$-0.286395\pi$
$853$	36.3218	1.24363	0.621817	+	0.783163i	$0.286395\pi$
$854$	0	0
$855$	0	0
$856$	−7.63325	−0.260899
$857$	−37.3703	−1.27655	−0.638273	−	0.769810i	$-0.720351\pi$
$857$	−37.3703	−1.27655	−0.638273	+	0.769810i	$0.720351\pi$
$858$	13.1662	0.449488
$859$	5.80985	0.198230	0.0991148	−	0.995076i	$-0.468399\pi$
$859$	5.80985	0.198230	0.0991148	+	0.995076i	$0.468399\pi$
$860$	0	0
$861$	0	0
$862$	−20.6332	−0.702771
$863$	49.8997	1.69861	0.849304	−	0.527905i	$-0.177022\pi$
$863$	49.8997	1.69861	0.849304	+	0.527905i	$0.177022\pi$
$864$	5.43297	0.184833
$865$	0	0
$866$	10.5712	0.359223
$867$	−16.7355	−0.568366
$868$	0	0
$869$	−19.2665	−0.653571
$870$	0	0
$871$	71.0167	2.40631
$872$	−18.0000	−0.609557
$873$	1.34333	0.0454647
$874$	14.1421	0.478365
$875$	0	0
$876$	4.26650	0.144152
$877$	−21.2665	−0.718119	−0.359059	−	0.933315i	$-0.616902\pi$
$877$	−21.2665	−0.718119	−0.359059	+	0.933315i	$0.616902\pi$
$878$	9.38083	0.316588
$879$	23.8997	0.806119
$880$	0	0
$881$	−21.9670	−0.740086	−0.370043	−	0.929015i	$-0.620657\pi$
$881$	−21.9670	−0.740086	−0.370043	+	0.929015i	$0.620657\pi$
$882$	0	0
$883$	2.36675	0.0796475	0.0398237	−	0.999207i	$-0.487320\pi$
$883$	2.36675	0.0796475	0.0398237	+	0.999207i	$0.487320\pi$
$884$	15.8997	0.534766
$885$	0	0
$886$	26.8997	0.903715
$887$	1.48510	0.0498648	0.0249324	−	0.999689i	$-0.492063\pi$
$887$	1.48510	0.0498648	0.0249324	+	0.999689i	$0.492063\pi$
$888$	14.1421	0.474579
$889$	0	0
$890$	0	0
$891$	10.4670	0.350658
$892$	6.10463	0.204398
$893$	−10.0000	−0.334637
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−46.3325	−1.54700
$898$	7.73350	0.258070
$899$	60.1507	2.00614
$900$	0	0
$901$	−20.8363	−0.694158
$902$	−9.60472	−0.319802
$903$	0	0
$904$	7.31662	0.243347
$905$	0	0
$906$	−20.6945	−0.687530
$907$	21.2665	0.706143	0.353071	−	0.935596i	$-0.385137\pi$
$907$	21.2665	0.706143	0.353071	+	0.935596i	$0.385137\pi$
$908$	−13.1757	−0.437251
$909$	−1.17910	−0.0391084
$910$	0	0
$911$	23.1662	0.767532	0.383766	−	0.923430i	$-0.374627\pi$
$911$	23.1662	0.767532	0.383766	+	0.923430i	$0.374627\pi$
$912$	5.00000	0.165567
$913$	−17.0527	−0.564361
$914$	23.3166	0.771245
$915$	0	0
$916$	−5.20908	−0.172113
$917$	0	0
$918$	14.1504	0.467032
$919$	18.0000	0.593765	0.296883	−	0.954914i	$-0.404053\pi$
$919$	18.0000	0.593765	0.296883	+	0.954914i	$0.404053\pi$
$920$	0	0
$921$	−32.7995	−1.08078
$922$	3.27620	0.107896
$923$	−12.2093	−0.401873
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	3.58220	0.117655
$928$	−10.6332	−0.349054
$929$	−24.7954	−0.813511	−0.406755	−	0.913537i	$-0.633340\pi$
$929$	−24.7954	−0.813511	−0.406755	+	0.913537i	$0.633340\pi$
$930$	0	0
$931$	0	0
$932$	13.2665	0.434559
$933$	−31.4670	−1.03018
$934$	28.2134	0.923170
$935$	0	0
$936$	1.93288	0.0631781
$937$	−3.12320	−0.102031	−0.0510153	−	0.998698i	$-0.516246\pi$
$937$	−3.12320	−0.102031	−0.0510153	+	0.998698i	$0.516246\pi$
$938$	0	0
$939$	−42.3166	−1.38095
$940$	0	0
$941$	−15.1795	−0.494836	−0.247418	−	0.968909i	$-0.579582\pi$
$941$	−15.1795	−0.494836	−0.247418	+	0.968909i	$0.579582\pi$
$942$	15.3668	0.500676
$943$	33.7993	1.10066
$944$	7.51884	0.244717
$945$	0	0
$946$	8.73350	0.283951
$947$	46.6332	1.51538	0.757688	−	0.652616i	$-0.226329\pi$
$947$	46.6332	1.51538	0.757688	+	0.652616i	$0.226329\pi$
$948$	23.9707	0.778534
$949$	15.8997	0.516128
$950$	0	0
$951$	−9.66438	−0.313389
$952$	0	0
$953$	−31.7335	−1.02795	−0.513974	−	0.857805i	$-0.671827\pi$
$953$	−31.7335	−1.02795	−0.513974	+	0.857805i	$0.671827\pi$
$954$	−2.53300	−0.0820088
$955$	0	0
$956$	−3.36675	−0.108888
$957$	−22.9334	−0.741332
$958$	41.8369	1.35169
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	52.7028	1.69921
$963$	−2.41688	−0.0778827
$964$	−20.3998	−0.657032
$965$	0	0
$966$	0	0
$967$	7.89975	0.254039	0.127019	−	0.991900i	$-0.459459\pi$
$967$	7.89975	0.254039	0.127019	+	0.991900i	$0.459459\pi$
$968$	9.26650	0.297837
$969$	13.0227	0.418349
$970$	0	0
$971$	20.0229	0.642565	0.321282	−	0.946983i	$-0.395886\pi$
$971$	20.0229	0.642565	0.321282	+	0.946983i	$0.395886\pi$
$972$	3.27620	0.105084
$973$	0	0
$974$	4.73350	0.151671
$975$	0	0
$976$	3.27620	0.104869
$977$	27.3166	0.873936	0.436968	−	0.899477i	$-0.356052\pi$
$977$	27.3166	0.873936	0.436968	+	0.899477i	$0.356052\pi$
$978$	−15.2616	−0.488011
$979$	−12.0563	−0.385320
$980$	0	0
$981$	−5.69925	−0.181963
$982$	−5.26650	−0.168061
$983$	−12.3510	−0.393937	−0.196968	−	0.980410i	$-0.563110\pi$
$983$	−12.3510	−0.393937	−0.196968	+	0.980410i	$0.563110\pi$
$984$	11.9499	0.380948
$985$	0	0
$986$	−27.6947	−0.881980
$987$	0	0
$988$	18.6332	0.592803
$989$	−30.7335	−0.977268
$990$	0	0
$991$	−30.5330	−0.969913	−0.484956	−	0.874538i	$-0.661165\pi$
$991$	−30.5330	−0.969913	−0.484956	+	0.874538i	$0.661165\pi$
$992$	−5.65685	−0.179605
$993$	48.0236	1.52398
$994$	0	0
$995$	0	0
$996$	21.2164	0.672267
$997$	−47.4937	−1.50414	−0.752070	−	0.659083i	$-0.770945\pi$
$997$	−47.4937	−1.50414	−0.752070	+	0.659083i	$0.770945\pi$
$998$	−13.2665	−0.419944
$999$	46.9042	1.48398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bt.1.3	yes	4
5.2	odd	4		2450.2.c.x.99.2		8
5.3	odd	4		2450.2.c.x.99.7		8
5.4	even	2		2450.2.a.bu.1.2	yes	4
7.6	odd	2	inner	2450.2.a.bt.1.2	✓	4
35.13	even	4		2450.2.c.x.99.6		8
35.27	even	4		2450.2.c.x.99.3		8
35.34	odd	2		2450.2.a.bu.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.2.a.bt.1.2	✓	4	7.6	odd	2	inner
2450.2.a.bt.1.3	yes	4	1.1	even	1	trivial
2450.2.a.bu.1.2	yes	4	5.4	even	2
2450.2.a.bu.1.3	yes	4	35.34	odd	2
2450.2.c.x.99.2		8	5.2	odd	4
2450.2.c.x.99.3		8	35.27	even	4
2450.2.c.x.99.6		8	35.13	even	4
2450.2.c.x.99.7		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.63810 q^{3} +1.00000 q^{4} -1.63810 q^{6} -1.00000 q^{8} -0.316625 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.63810 q^{3} +1.00000 q^{4} -1.63810 q^{6} -1.00000 q^{8} -0.316625 q^{9} -1.31662 q^{11} +1.63810 q^{12} +6.10463 q^{13} +1.00000 q^{16} +2.60454 q^{17} +0.316625 q^{18} +3.05231 q^{19} +1.31662 q^{22} -4.63325 q^{23} -1.63810 q^{24} -6.10463 q^{26} -5.43297 q^{27} +10.6332 q^{29} +5.65685 q^{31} -1.00000 q^{32} -2.15676 q^{33} -2.60454 q^{34} -0.316625 q^{36} -8.63325 q^{37} -3.05231 q^{38} +10.0000 q^{39} -7.29496 q^{41} +6.63325 q^{43} -1.31662 q^{44} +4.63325 q^{46} -3.27620 q^{47} +1.63810 q^{48} +4.26650 q^{51} +6.10463 q^{52} -8.00000 q^{53} +5.43297 q^{54} +5.00000 q^{57} -10.6332 q^{58} +7.51884 q^{59} +3.27620 q^{61} -5.65685 q^{62} +1.00000 q^{64} +2.15676 q^{66} +11.6332 q^{67} +2.60454 q^{68} -7.58973 q^{69} -2.00000 q^{71} +0.316625 q^{72} +2.60454 q^{73} +8.63325 q^{74} +3.05231 q^{76} -10.0000 q^{78} +14.6332 q^{79} -7.94987 q^{81} +7.29496 q^{82} +12.9518 q^{83} -6.63325 q^{86} +17.4183 q^{87} +1.31662 q^{88} +9.15694 q^{89} -4.63325 q^{92} +9.26650 q^{93} +3.27620 q^{94} -1.63810 q^{96} -4.24264 q^{97} +0.416876 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{16} - 12 q^{18} - 8 q^{22} + 8 q^{23} + 16 q^{29} - 4 q^{32} + 12 q^{36} - 8 q^{37} + 40 q^{39} + 8 q^{44} - 8 q^{46} - 36 q^{51} - 32 q^{53} + 20 q^{57} - 16 q^{58} + 4 q^{64} + 20 q^{67} - 8 q^{71} - 12 q^{72} + 8 q^{74} - 40 q^{78} + 32 q^{79} + 8 q^{81} - 8 q^{88} + 8 q^{92} - 16 q^{93} + 68 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^16 - 12 * q^18 - 8 * q^22 + 8 * q^23 + 16 * q^29 - 4 * q^32 + 12 * q^36 - 8 * q^37 + 40 * q^39 + 8 * q^44 - 8 * q^46 - 36 * q^51 - 32 * q^53 + 20 * q^57 - 16 * q^58 + 4 * q^64 + 20 * q^67 - 8 * q^71 - 12 * q^72 + 8 * q^74 - 40 * q^78 + 32 * q^79 + 8 * q^81 - 8 * q^88 + 8 * q^92 - 16 * q^93 + 68 * q^99

Introduction

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