LMFDB - Embedded newform 8281.2.a.bo.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.105456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 13x^{2} + 39 $ x^4 - 13*x^2 + 39
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.88145$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278 q^{2} +2.88145 q^{3} -0.302776 q^{4} -2.88145 q^{5} +3.75389 q^{6} -3.00000 q^{8} +5.30278 q^{9} +O(q^{10})$	\(q+1.30278 q^{2} +2.88145 q^{3} -0.302776 q^{4} -2.88145 q^{5} +3.75389 q^{6} -3.00000 q^{8} +5.30278 q^{9} -3.75389 q^{10} +5.90833 q^{11} -0.872434 q^{12} -8.30278 q^{15} -3.30278 q^{16} -0.872434 q^{17} +6.90833 q^{18} -2.88145 q^{19} +0.872434 q^{20} +7.69722 q^{22} +6.60555 q^{23} -8.64436 q^{24} +3.30278 q^{25} +6.63534 q^{27} +1.30278 q^{29} -10.8167 q^{30} -0.872434 q^{31} +1.69722 q^{32} +17.0246 q^{33} -1.13659 q^{34} -1.60555 q^{36} -1.39445 q^{37} -3.75389 q^{38} +8.64436 q^{40} -7.50778 q^{41} +5.51388 q^{43} -1.78890 q^{44} -15.2797 q^{45} +8.60555 q^{46} +12.3982 q^{47} -9.51680 q^{48} +4.30278 q^{50} -2.51388 q^{51} +9.60555 q^{53} +8.64436 q^{54} -17.0246 q^{55} -8.30278 q^{57} +1.69722 q^{58} -6.63534 q^{59} +2.51388 q^{60} +5.76291 q^{61} -1.13659 q^{62} +8.81665 q^{64} +22.1792 q^{66} -1.00000 q^{67} +0.264152 q^{68} +19.0336 q^{69} -4.00000 q^{71} -15.9083 q^{72} +5.76291 q^{73} -1.81665 q^{74} +9.51680 q^{75} +0.872434 q^{76} +0.605551 q^{79} +9.51680 q^{80} +3.21110 q^{81} -9.78095 q^{82} +6.63534 q^{83} +2.51388 q^{85} +7.18335 q^{86} +3.75389 q^{87} -17.7250 q^{88} -8.64436 q^{89} -19.9060 q^{90} -2.00000 q^{92} -2.51388 q^{93} +16.1521 q^{94} +8.30278 q^{95} +4.89047 q^{96} +7.77193 q^{97} +31.3305 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 + 14 * q^9	$ 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + 2 q^{11} - 26 q^{15} - 6 q^{16} + 6 q^{18} + 38 q^{22} + 12 q^{23} + 6 q^{25} - 2 q^{29} + 14 q^{32} + 8 q^{36} - 20 q^{37} - 14 q^{43} - 36 q^{44} + 20 q^{46} + 10 q^{50} + 26 q^{51} + 24 q^{53} - 26 q^{57} + 14 q^{58} - 26 q^{60} - 8 q^{64} - 4 q^{67} - 16 q^{71} - 42 q^{72} + 36 q^{74} - 12 q^{79} - 16 q^{81} - 26 q^{85} + 72 q^{86} - 6 q^{88} - 8 q^{92} + 26 q^{93} + 26 q^{95} + 46 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 + 14 * q^9 + 2 * q^11 - 26 * q^15 - 6 * q^16 + 6 * q^18 + 38 * q^22 + 12 * q^23 + 6 * q^25 - 2 * q^29 + 14 * q^32 + 8 * q^36 - 20 * q^37 - 14 * q^43 - 36 * q^44 + 20 * q^46 + 10 * q^50 + 26 * q^51 + 24 * q^53 - 26 * q^57 + 14 * q^58 - 26 * q^60 - 8 * q^64 - 4 * q^67 - 16 * q^71 - 42 * q^72 + 36 * q^74 - 12 * q^79 - 16 * q^81 - 26 * q^85 + 72 * q^86 - 6 * q^88 - 8 * q^92 + 26 * q^93 + 26 * q^95 + 46 * 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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	2.88145	1.66361	0.831804	−	0.555069i	$-0.187308\pi$
$3$	2.88145	1.66361	0.831804	+	0.555069i	$0.187308\pi$
$4$	−0.302776	−0.151388
$5$	−2.88145	−1.28863	−0.644313	−	0.764762i	$-0.722856\pi$
$5$	−2.88145	−1.28863	−0.644313	+	0.764762i	$0.722856\pi$
$6$	3.75389	1.53252
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	5.30278	1.76759
$10$	−3.75389	−1.18708
$11$	5.90833	1.78143	0.890714	−	0.454565i	$-0.150205\pi$
$11$	5.90833	1.78143	0.890714	+	0.454565i	$0.150205\pi$
$12$	−0.872434	−0.251850
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−8.30278	−2.14377
$16$	−3.30278	−0.825694
$17$	−0.872434	−0.211596	−0.105798	−	0.994388i	$-0.533740\pi$
$17$	−0.872434	−0.211596	−0.105798	+	0.994388i	$0.533740\pi$
$18$	6.90833	1.62831
$19$	−2.88145	−0.661051	−0.330525	−	0.943797i	$-0.607226\pi$
$19$	−2.88145	−0.661051	−0.330525	+	0.943797i	$0.607226\pi$
$20$	0.872434	0.195082
$21$	0	0
$22$	7.69722	1.64105
$23$	6.60555	1.37735	0.688676	−	0.725069i	$-0.258192\pi$
$23$	6.60555	1.37735	0.688676	+	0.725069i	$0.258192\pi$
$24$	−8.64436	−1.76452
$25$	3.30278	0.660555
$26$	0	0
$27$	6.63534	1.27697
$28$	0	0
$29$	1.30278	0.241919	0.120960	−	0.992657i	$-0.461403\pi$
$29$	1.30278	0.241919	0.120960	+	0.992657i	$0.461403\pi$
$30$	−10.8167	−1.97484
$31$	−0.872434	−0.156694	−0.0783469	−	0.996926i	$-0.524964\pi$
$31$	−0.872434	−0.156694	−0.0783469	+	0.996926i	$0.524964\pi$
$32$	1.69722	0.300030
$33$	17.0246	2.96360
$34$	−1.13659	−0.194923
$35$	0	0
$36$	−1.60555	−0.267592
$37$	−1.39445	−0.229246	−0.114623	−	0.993409i	$-0.536566\pi$
$37$	−1.39445	−0.229246	−0.114623	+	0.993409i	$0.536566\pi$
$38$	−3.75389	−0.608961
$39$	0	0
$40$	8.64436	1.36679
$41$	−7.50778	−1.17252	−0.586259	−	0.810124i	$-0.699400\pi$
$41$	−7.50778	−1.17252	−0.586259	+	0.810124i	$0.699400\pi$
$42$	0	0
$43$	5.51388	0.840859	0.420429	−	0.907325i	$-0.361879\pi$
$43$	5.51388	0.840859	0.420429	+	0.907325i	$0.361879\pi$
$44$	−1.78890	−0.269686
$45$	−15.2797	−2.27776
$46$	8.60555	1.26882
$47$	12.3982	1.80847	0.904235	−	0.427035i	$-0.140442\pi$
$47$	12.3982	1.80847	0.904235	+	0.427035i	$0.140442\pi$
$48$	−9.51680	−1.37363
$49$	0	0
$50$	4.30278	0.608504
$51$	−2.51388	−0.352013
$52$	0	0
$53$	9.60555	1.31942	0.659712	−	0.751519i	$-0.270678\pi$
$53$	9.60555	1.31942	0.659712	+	0.751519i	$0.270678\pi$
$54$	8.64436	1.17635
$55$	−17.0246	−2.29559
$56$	0	0
$57$	−8.30278	−1.09973
$58$	1.69722	0.222856
$59$	−6.63534	−0.863848	−0.431924	−	0.901910i	$-0.642165\pi$
$59$	−6.63534	−0.863848	−0.431924	+	0.901910i	$0.642165\pi$
$60$	2.51388	0.324540
$61$	5.76291	0.737865	0.368932	−	0.929456i	$-0.379723\pi$
$61$	5.76291	0.737865	0.368932	+	0.929456i	$0.379723\pi$
$62$	−1.13659	−0.144347
$63$	0	0
$64$	8.81665	1.10208
$65$	0	0
$66$	22.1792	2.73007
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	0.264152	0.0320331
$69$	19.0336	2.29138
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−15.9083	−1.87481
$73$	5.76291	0.674497	0.337249	−	0.941416i	$-0.390504\pi$
$73$	5.76291	0.674497	0.337249	+	0.941416i	$0.390504\pi$
$74$	−1.81665	−0.211182
$75$	9.51680	1.09890
$76$	0.872434	0.100075
$77$	0	0
$78$	0	0
$79$	0.605551	0.0681298	0.0340649	−	0.999420i	$-0.489155\pi$
$79$	0.605551	0.0681298	0.0340649	+	0.999420i	$0.489155\pi$
$80$	9.51680	1.06401
$81$	3.21110	0.356789
$82$	−9.78095	−1.08012
$83$	6.63534	0.728323	0.364162	−	0.931336i	$-0.381356\pi$
$83$	6.63534	0.728323	0.364162	+	0.931336i	$0.381356\pi$
$84$	0	0
$85$	2.51388	0.272668
$86$	7.18335	0.774600
$87$	3.75389	0.402459
$88$	−17.7250	−1.88949
$89$	−8.64436	−0.916300	−0.458150	−	0.888875i	$-0.651488\pi$
$89$	−8.64436	−0.916300	−0.458150	+	0.888875i	$0.651488\pi$
$90$	−19.9060	−2.09828
$91$	0	0
$92$	−2.00000	−0.208514
$93$	−2.51388	−0.260677
$94$	16.1521	1.66597
$95$	8.30278	0.851847
$96$	4.89047	0.499132
$97$	7.77193	0.789120	0.394560	−	0.918870i	$-0.370897\pi$
$97$	7.77193	0.789120	0.394560	+	0.918870i	$0.370897\pi$
$98$	0	0
$99$	31.3305	3.14884
$100$	−1.00000	−0.100000
$101$	−8.64436	−0.860146	−0.430073	−	0.902794i	$-0.641512\pi$
$101$	−8.64436	−0.860146	−0.430073	+	0.902794i	$0.641512\pi$
$102$	−3.27502	−0.324275
$103$	4.89047	0.481873	0.240936	−	0.970541i	$-0.422545\pi$
$103$	4.89047	0.481873	0.240936	+	0.970541i	$0.422545\pi$
$104$	0	0
$105$	0	0
$106$	12.5139	1.21546
$107$	9.30278	0.899333	0.449667	−	0.893196i	$-0.351543\pi$
$107$	9.30278	0.899333	0.449667	+	0.893196i	$0.351543\pi$
$108$	−2.00902	−0.193318
$109$	6.21110	0.594916	0.297458	−	0.954735i	$-0.403861\pi$
$109$	6.21110	0.594916	0.297458	+	0.954735i	$0.403861\pi$
$110$	−22.1792	−2.11470
$111$	−4.01804	−0.381375
$112$	0	0
$113$	−14.8167	−1.39383	−0.696917	−	0.717152i	$-0.745445\pi$
$113$	−14.8167	−1.39383	−0.696917	+	0.717152i	$0.745445\pi$
$114$	−10.8167	−1.01307
$115$	−19.0336	−1.77489
$116$	−0.394449	−0.0366236
$117$	0	0
$118$	−8.64436	−0.795778
$119$	0	0
$120$	24.9083	2.27381
$121$	23.9083	2.17348
$122$	7.50778	0.679722
$123$	−21.6333	−1.95061
$124$	0.264152	0.0237215
$125$	4.89047	0.437417
$126$	0	0
$127$	−2.90833	−0.258072	−0.129036	−	0.991640i	$-0.541188\pi$
$127$	−2.90833	−0.258072	−0.129036	+	0.991640i	$0.541188\pi$
$128$	8.09167	0.715210
$129$	15.8880	1.39886
$130$	0	0
$131$	17.0246	1.48744	0.743722	−	0.668489i	$-0.233059\pi$
$131$	17.0246	1.48744	0.743722	+	0.668489i	$0.233059\pi$
$132$	−5.15463	−0.448653
$133$	0	0
$134$	−1.30278	−0.112543
$135$	−19.1194	−1.64554
$136$	2.61730	0.224432
$137$	−2.69722	−0.230439	−0.115220	−	0.993340i	$-0.536757\pi$
$137$	−2.69722	−0.230439	−0.115220	+	0.993340i	$0.536757\pi$
$138$	24.7965	2.11082
$139$	17.0246	1.44401	0.722003	−	0.691890i	$-0.243222\pi$
$139$	17.0246	1.44401	0.722003	+	0.691890i	$0.243222\pi$
$140$	0	0
$141$	35.7250	3.00859
$142$	−5.21110	−0.437306
$143$	0	0
$144$	−17.5139	−1.45949
$145$	−3.75389	−0.311743
$146$	7.50778	0.621348
$147$	0	0
$148$	0.422205	0.0347050
$149$	−0.513878	−0.0420985	−0.0210493	−	0.999778i	$-0.506701\pi$
$149$	−0.513878	−0.0420985	−0.0210493	+	0.999778i	$0.506701\pi$
$150$	12.3982	1.01231
$151$	6.21110	0.505452	0.252726	−	0.967538i	$-0.418673\pi$
$151$	6.21110	0.505452	0.252726	+	0.967538i	$0.418673\pi$
$152$	8.64436	0.701150
$153$	−4.62632	−0.374016
$154$	0	0
$155$	2.51388	0.201920
$156$	0	0
$157$	9.51680	0.759523	0.379761	−	0.925084i	$-0.376006\pi$
$157$	9.51680	0.759523	0.379761	+	0.925084i	$0.376006\pi$
$158$	0.788897	0.0627613
$159$	27.6780	2.19500
$160$	−4.89047	−0.386626
$161$	0	0
$162$	4.18335	0.328675
$163$	17.2111	1.34808	0.674039	−	0.738696i	$-0.264558\pi$
$163$	17.2111	1.34808	0.674039	+	0.738696i	$0.264558\pi$
$164$	2.27317	0.177505
$165$	−49.0555	−3.81897
$166$	8.64436	0.670933
$167$	−4.89047	−0.378436	−0.189218	−	0.981935i	$-0.560595\pi$
$167$	−4.89047	−0.378436	−0.189218	+	0.981935i	$0.560595\pi$
$168$	0	0
$169$	0	0
$170$	3.27502	0.251183
$171$	−15.2797	−1.16847
$172$	−1.66947	−0.127296
$173$	23.9241	1.81891	0.909456	−	0.415799i	$-0.136498\pi$
$173$	23.9241	1.81891	0.909456	+	0.415799i	$0.136498\pi$
$174$	4.89047	0.370746
$175$	0	0
$176$	−19.5139	−1.47091
$177$	−19.1194	−1.43710
$178$	−11.2617	−0.844097
$179$	14.3944	1.07589	0.537946	−	0.842979i	$-0.319200\pi$
$179$	14.3944	1.07589	0.537946	+	0.842979i	$0.319200\pi$
$180$	4.62632	0.344826
$181$	9.25264	0.687744	0.343872	−	0.939017i	$-0.388261\pi$
$181$	9.25264	0.687744	0.343872	+	0.939017i	$0.388261\pi$
$182$	0	0
$183$	16.6056	1.22752
$184$	−19.8167	−1.46090
$185$	4.01804	0.295412
$186$	−3.27502	−0.240136
$187$	−5.15463	−0.376944
$188$	−3.75389	−0.273780
$189$	0	0
$190$	10.8167	0.784723
$191$	19.3028	1.39670	0.698350	−	0.715757i	$-0.253918\pi$
$191$	19.3028	1.39670	0.698350	+	0.715757i	$0.253918\pi$
$192$	25.4048	1.83343
$193$	1.81665	0.130766	0.0653828	−	0.997860i	$-0.479173\pi$
$193$	1.81665	0.130766	0.0653828	+	0.997860i	$0.479173\pi$
$194$	10.1251	0.726938
$195$	0	0
$196$	0	0
$197$	11.9083	0.848433	0.424217	−	0.905561i	$-0.360550\pi$
$197$	11.9083	0.848433	0.424217	+	0.905561i	$0.360550\pi$
$198$	40.8167	2.90071
$199$	0.872434	0.0618452	0.0309226	−	0.999522i	$-0.490155\pi$
$199$	0.872434	0.0618452	0.0309226	+	0.999522i	$0.490155\pi$
$200$	−9.90833	−0.700625
$201$	−2.88145	−0.203242
$202$	−11.2617	−0.792368
$203$	0	0
$204$	0.761141	0.0532905
$205$	21.6333	1.51094
$206$	6.37119	0.443902
$207$	35.0278	2.43460
$208$	0	0
$209$	−17.0246	−1.17761
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	−2.90833	−0.199745
$213$	−11.5258	−0.789736
$214$	12.1194	0.828467
$215$	−15.8880	−1.08355
$216$	−19.9060	−1.35443
$217$	0	0
$218$	8.09167	0.548037
$219$	16.6056	1.12210
$220$	5.15463	0.347525
$221$	0	0
$222$	−5.23460	−0.351324
$223$	13.5348	0.906360	0.453180	−	0.891419i	$-0.350290\pi$
$223$	13.5348	0.906360	0.453180	+	0.891419i	$0.350290\pi$
$224$	0	0
$225$	17.5139	1.16759
$226$	−19.3028	−1.28400
$227$	−27.4138	−1.81952	−0.909759	−	0.415137i	$-0.863734\pi$
$227$	−27.4138	−1.81952	−0.909759	+	0.415137i	$0.863734\pi$
$228$	2.51388	0.166486
$229$	−20.7785	−1.37308	−0.686540	−	0.727092i	$-0.740871\pi$
$229$	−20.7785	−1.37308	−0.686540	+	0.727092i	$0.740871\pi$
$230$	−24.7965	−1.63503
$231$	0	0
$232$	−3.90833	−0.256594
$233$	−23.9083	−1.56629	−0.783143	−	0.621841i	$-0.786385\pi$
$233$	−23.9083	−1.56629	−0.783143	+	0.621841i	$0.786385\pi$
$234$	0	0
$235$	−35.7250	−2.33044
$236$	2.00902	0.130776
$237$	1.74487	0.113341
$238$	0	0
$239$	−11.6056	−0.750701	−0.375350	−	0.926883i	$-0.622478\pi$
$239$	−11.6056	−0.750701	−0.375350	+	0.926883i	$0.622478\pi$
$240$	27.4222	1.77010
$241$	2.00902	0.129412	0.0647062	−	0.997904i	$-0.479389\pi$
$241$	2.00902	0.129412	0.0647062	+	0.997904i	$0.479389\pi$
$242$	31.1472	2.00222
$243$	−10.6534	−0.683415
$244$	−1.74487	−0.111704
$245$	0	0
$246$	−28.1833	−1.79690
$247$	0	0
$248$	2.61730	0.166199
$249$	19.1194	1.21164
$250$	6.37119	0.402949
$251$	−22.5233	−1.42166	−0.710830	−	0.703364i	$-0.751680\pi$
$251$	−22.5233	−1.42166	−0.710830	+	0.703364i	$0.751680\pi$
$252$	0	0
$253$	39.0278	2.45365
$254$	−3.78890	−0.237737
$255$	7.24362	0.453613
$256$	−7.09167	−0.443230
$257$	−22.7875	−1.42144	−0.710722	−	0.703473i	$-0.751632\pi$
$257$	−22.7875	−1.42144	−0.710722	+	0.703473i	$0.751632\pi$
$258$	20.6985	1.28863
$259$	0	0
$260$	0	0
$261$	6.90833	0.427615
$262$	22.1792	1.37024
$263$	−13.4222	−0.827649	−0.413824	−	0.910357i	$-0.635807\pi$
$263$	−13.4222	−0.827649	−0.413824	+	0.910357i	$0.635807\pi$
$264$	−51.0737	−3.14337
$265$	−27.6780	−1.70024
$266$	0	0
$267$	−24.9083	−1.52436
$268$	0.302776	0.0184950
$269$	9.51680	0.580249	0.290125	−	0.956989i	$-0.406303\pi$
$269$	9.51680	0.580249	0.290125	+	0.956989i	$0.406303\pi$
$270$	−24.9083	−1.51587
$271$	−29.6870	−1.80336	−0.901678	−	0.432409i	$-0.857664\pi$
$271$	−29.6870	−1.80336	−0.901678	+	0.432409i	$0.857664\pi$
$272$	2.88145	0.174714
$273$	0	0
$274$	−3.51388	−0.212281
$275$	19.5139	1.17673
$276$	−5.76291	−0.346886
$277$	14.2111	0.853862	0.426931	−	0.904284i	$-0.359595\pi$
$277$	14.2111	0.853862	0.426931	+	0.904284i	$0.359595\pi$
$278$	22.1792	1.33022
$279$	−4.62632	−0.276971
$280$	0	0
$281$	−2.18335	−0.130248	−0.0651238	−	0.997877i	$-0.520744\pi$
$281$	−2.18335	−0.130248	−0.0651238	+	0.997877i	$0.520744\pi$
$282$	46.5416	2.77151
$283$	4.01804	0.238848	0.119424	−	0.992843i	$-0.461895\pi$
$283$	4.01804	0.238848	0.119424	+	0.992843i	$0.461895\pi$
$284$	1.21110	0.0718657
$285$	23.9241	1.41714
$286$	0	0
$287$	0	0
$288$	9.00000	0.530330
$289$	−16.2389	−0.955227
$290$	−4.89047	−0.287178
$291$	22.3944	1.31279
$292$	−1.74487	−0.102111
$293$	−3.48974	−0.203873	−0.101936	−	0.994791i	$-0.532504\pi$
$293$	−3.48974	−0.203873	−0.101936	+	0.994791i	$0.532504\pi$
$294$	0	0
$295$	19.1194	1.11318
$296$	4.18335	0.243152
$297$	39.2038	2.27483
$298$	−0.669468	−0.0387812
$299$	0	0
$300$	−2.88145	−0.166361
$301$	0	0
$302$	8.09167	0.465623
$303$	−24.9083	−1.43095
$304$	9.51680	0.545826
$305$	−16.6056	−0.950831
$306$	−6.02706	−0.344544
$307$	−15.2797	−0.872059	−0.436029	−	0.899932i	$-0.643616\pi$
$307$	−15.2797	−0.872059	−0.436029	+	0.899932i	$0.643616\pi$
$308$	0	0
$309$	14.0917	0.801647
$310$	3.27502	0.186009
$311$	−17.0246	−0.965375	−0.482687	−	0.875793i	$-0.660339\pi$
$311$	−17.0246	−0.965375	−0.482687	+	0.875793i	$0.660339\pi$
$312$	0	0
$313$	13.2707	0.750103	0.375052	−	0.927004i	$-0.377625\pi$
$313$	13.2707	0.750103	0.375052	+	0.927004i	$0.377625\pi$
$314$	12.3982	0.699674
$315$	0	0
$316$	−0.183346	−0.0103140
$317$	−8.21110	−0.461181	−0.230591	−	0.973051i	$-0.574066\pi$
$317$	−8.21110	−0.461181	−0.230591	+	0.973051i	$0.574066\pi$
$318$	36.0582	2.02204
$319$	7.69722	0.430962
$320$	−25.4048	−1.42017
$321$	26.8055	1.49614
$322$	0	0
$323$	2.51388	0.139876
$324$	−0.972244	−0.0540135
$325$	0	0
$326$	22.4222	1.24185
$327$	17.8970	0.989707
$328$	22.5233	1.24364
$329$	0	0
$330$	−63.9083	−3.51804
$331$	−0.697224	−0.0383229	−0.0191615	−	0.999816i	$-0.506100\pi$
$331$	−0.697224	−0.0383229	−0.0191615	+	0.999816i	$0.506100\pi$
$332$	−2.00902	−0.110259
$333$	−7.39445	−0.405213
$334$	−6.37119	−0.348616
$335$	2.88145	0.157431
$336$	0	0
$337$	−7.11943	−0.387820	−0.193910	−	0.981019i	$-0.562117\pi$
$337$	−7.11943	−0.387820	−0.193910	+	0.981019i	$0.562117\pi$
$338$	0	0
$339$	−42.6935	−2.31879
$340$	−0.761141	−0.0412787
$341$	−5.15463	−0.279139
$342$	−19.9060	−1.07639
$343$	0	0
$344$	−16.5416	−0.891865
$345$	−54.8444	−2.95272
$346$	31.1677	1.67559
$347$	0.788897	0.0423502	0.0211751	−	0.999776i	$-0.493259\pi$
$347$	0.788897	0.0423502	0.0211751	+	0.999776i	$0.493259\pi$
$348$	−1.13659	−0.0609274
$349$	−23.9241	−1.28063	−0.640313	−	0.768114i	$-0.721195\pi$
$349$	−23.9241	−1.28063	−0.640313	+	0.768114i	$0.721195\pi$
$350$	0	0
$351$	0	0
$352$	10.0278	0.534481
$353$	−6.37119	−0.339104	−0.169552	−	0.985521i	$-0.554232\pi$
$353$	−6.37119	−0.339104	−0.169552	+	0.985521i	$0.554232\pi$
$354$	−24.9083	−1.32386
$355$	11.5258	0.611727
$356$	2.61730	0.138717
$357$	0	0
$358$	18.7527	0.991113
$359$	22.9083	1.20906	0.604528	−	0.796584i	$-0.293362\pi$
$359$	22.9083	1.20906	0.604528	+	0.796584i	$0.293362\pi$
$360$	45.8391	2.41593
$361$	−10.6972	−0.563012
$362$	12.0541	0.633550
$363$	68.8907	3.61583
$364$	0	0
$365$	−16.6056	−0.869174
$366$	21.6333	1.13079
$367$	−28.2862	−1.47653	−0.738265	−	0.674511i	$-0.764354\pi$
$367$	−28.2862	−1.47653	−0.738265	+	0.674511i	$0.764354\pi$
$368$	−21.8167	−1.13727
$369$	−39.8120	−2.07253
$370$	5.23460	0.272134
$371$	0	0
$372$	0.761141	0.0394633
$373$	−16.3028	−0.844126	−0.422063	−	0.906567i	$-0.638694\pi$
$373$	−16.3028	−0.844126	−0.422063	+	0.906567i	$0.638694\pi$
$374$	−6.71532	−0.347241
$375$	14.0917	0.727691
$376$	−37.1947	−1.91817
$377$	0	0
$378$	0	0
$379$	−13.1194	−0.673900	−0.336950	−	0.941523i	$-0.609395\pi$
$379$	−13.1194	−0.673900	−0.336950	+	0.941523i	$0.609395\pi$
$380$	−2.51388	−0.128959
$381$	−8.38021	−0.429331
$382$	25.1472	1.28664
$383$	19.2977	0.986069	0.493034	−	0.870010i	$-0.335888\pi$
$383$	19.2977	0.986069	0.493034	+	0.870010i	$0.335888\pi$
$384$	23.3158	1.18983
$385$	0	0
$386$	2.36669	0.120461
$387$	29.2389	1.48629
$388$	−2.35315	−0.119463
$389$	−7.02776	−0.356321	−0.178161	−	0.984001i	$-0.557015\pi$
$389$	−7.02776	−0.356321	−0.178161	+	0.984001i	$0.557015\pi$
$390$	0	0
$391$	−5.76291	−0.291443
$392$	0	0
$393$	49.0555	2.47452
$394$	15.5139	0.781578
$395$	−1.74487	−0.0877938
$396$	−9.48612	−0.476696
$397$	−5.76291	−0.289232	−0.144616	−	0.989488i	$-0.546195\pi$
$397$	−5.76291	−0.289232	−0.144616	+	0.989488i	$0.546195\pi$
$398$	1.13659	0.0569719
$399$	0	0
$400$	−10.9083	−0.545416
$401$	−15.1194	−0.755028	−0.377514	−	0.926004i	$-0.623221\pi$
$401$	−15.1194	−0.755028	−0.377514	+	0.926004i	$0.623221\pi$
$402$	−3.75389	−0.187227
$403$	0	0
$404$	2.61730	0.130216
$405$	−9.25264	−0.459768
$406$	0	0
$407$	−8.23886	−0.408385
$408$	7.54163	0.373367
$409$	16.1521	0.798672	0.399336	−	0.916805i	$-0.369241\pi$
$409$	16.1521	0.798672	0.399336	+	0.916805i	$0.369241\pi$
$410$	28.1833	1.39188
$411$	−7.77193	−0.383361
$412$	−1.48072	−0.0729497
$413$	0	0
$414$	45.6333	2.24275
$415$	−19.1194	−0.938536
$416$	0	0
$417$	49.0555	2.40226
$418$	−22.1792	−1.08482
$419$	8.38021	0.409400	0.204700	−	0.978825i	$-0.434378\pi$
$419$	8.38021	0.409400	0.204700	+	0.978825i	$0.434378\pi$
$420$	0	0
$421$	31.0278	1.51220	0.756100	−	0.654456i	$-0.227102\pi$
$421$	31.0278	1.51220	0.756100	+	0.654456i	$0.227102\pi$
$422$	19.5416	0.951272
$423$	65.7451	3.19664
$424$	−28.8167	−1.39946
$425$	−2.88145	−0.139771
$426$	−15.0156	−0.727506
$427$	0	0
$428$	−2.81665	−0.136148
$429$	0	0
$430$	−20.6985	−0.998169
$431$	−25.9361	−1.24930	−0.624649	−	0.780906i	$-0.714758\pi$
$431$	−25.9361	−1.24930	−0.624649	+	0.780906i	$0.714758\pi$
$432$	−21.9150	−1.05439
$433$	−8.38021	−0.402727	−0.201364	−	0.979517i	$-0.564537\pi$
$433$	−8.38021	−0.402727	−0.201364	+	0.979517i	$0.564537\pi$
$434$	0	0
$435$	−10.8167	−0.518619
$436$	−1.88057	−0.0900630
$437$	−19.0336	−0.910500
$438$	21.6333	1.03368
$439$	−15.2797	−0.729260	−0.364630	−	0.931152i	$-0.618805\pi$
$439$	−15.2797	−0.729260	−0.364630	+	0.931152i	$0.618805\pi$
$440$	51.0737	2.43484
$441$	0	0
$442$	0	0
$443$	30.2389	1.43669	0.718346	−	0.695686i	$-0.244900\pi$
$443$	30.2389	1.43669	0.718346	+	0.695686i	$0.244900\pi$
$444$	1.21656	0.0577356
$445$	24.9083	1.18077
$446$	17.6329	0.834940
$447$	−1.48072	−0.0700355
$448$	0	0
$449$	−8.42221	−0.397468	−0.198734	−	0.980053i	$-0.563683\pi$
$449$	−8.42221	−0.397468	−0.198734	+	0.980053i	$0.563683\pi$
$450$	22.8167	1.07559
$451$	−44.3584	−2.08876
$452$	4.48612	0.211009
$453$	17.8970	0.840875
$454$	−35.7140	−1.67614
$455$	0	0
$456$	24.9083	1.16644
$457$	−13.3944	−0.626566	−0.313283	−	0.949660i	$-0.601429\pi$
$457$	−13.3944	−0.626566	−0.313283	+	0.949660i	$0.601429\pi$
$458$	−27.0697	−1.26488
$459$	−5.78890	−0.270203
$460$	5.76291	0.268697
$461$	12.6624	0.589747	0.294873	−	0.955536i	$-0.404723\pi$
$461$	12.6624	0.589747	0.294873	+	0.955536i	$0.404723\pi$
$462$	0	0
$463$	28.2111	1.31108	0.655541	−	0.755160i	$-0.272441\pi$
$463$	28.2111	1.31108	0.655541	+	0.755160i	$0.272441\pi$
$464$	−4.30278	−0.199751
$465$	7.24362	0.335915
$466$	−31.1472	−1.44287
$467$	−14.1431	−0.654465	−0.327233	−	0.944944i	$-0.606116\pi$
$467$	−14.1431	−0.654465	−0.327233	+	0.944944i	$0.606116\pi$
$468$	0	0
$469$	0	0
$470$	−46.5416	−2.14681
$471$	27.4222	1.26355
$472$	19.9060	0.916249
$473$	32.5778	1.49793
$474$	2.27317	0.104410
$475$	−9.51680	−0.436661
$476$	0	0
$477$	50.9361	2.33220
$478$	−15.1194	−0.691547
$479$	1.13659	0.0519319	0.0259660	−	0.999663i	$-0.491734\pi$
$479$	1.13659	0.0519319	0.0259660	+	0.999663i	$0.491734\pi$
$480$	−14.0917	−0.643194
$481$	0	0
$482$	2.61730	0.119215
$483$	0	0
$484$	−7.23886	−0.329039
$485$	−22.3944	−1.01688
$486$	−13.8790	−0.629563
$487$	27.6972	1.25508	0.627541	−	0.778584i	$-0.284062\pi$
$487$	27.6972	1.25508	0.627541	+	0.778584i	$0.284062\pi$
$488$	−17.2887	−0.782624
$489$	49.5930	2.24267
$490$	0	0
$491$	−12.7250	−0.574270	−0.287135	−	0.957890i	$-0.592703\pi$
$491$	−12.7250	−0.574270	−0.287135	+	0.957890i	$0.592703\pi$
$492$	6.55004	0.295299
$493$	−1.13659	−0.0511892
$494$	0	0
$495$	−90.2775	−4.05767
$496$	2.88145	0.129381
$497$	0	0
$498$	24.9083	1.11617
$499$	−31.3305	−1.40255	−0.701274	−	0.712892i	$-0.747385\pi$
$499$	−31.3305	−1.40255	−0.701274	+	0.712892i	$0.747385\pi$
$500$	−1.48072	−0.0662196
$501$	−14.0917	−0.629570
$502$	−29.3428	−1.30964
$503$	−25.9331	−1.15630	−0.578150	−	0.815931i	$-0.696225\pi$
$503$	−25.9331	−1.15630	−0.578150	+	0.815931i	$0.696225\pi$
$504$	0	0
$505$	24.9083	1.10841
$506$	50.8444	2.26031
$507$	0	0
$508$	0.880571	0.0390690
$509$	2.61730	0.116010	0.0580049	−	0.998316i	$-0.481526\pi$
$509$	2.61730	0.116010	0.0580049	+	0.998316i	$0.481526\pi$
$510$	9.43682	0.417869
$511$	0	0
$512$	−25.4222	−1.12351
$513$	−19.1194	−0.844143
$514$	−29.6870	−1.30944
$515$	−14.0917	−0.620953
$516$	−4.81049	−0.211770
$517$	73.2529	3.22166
$518$	0	0
$519$	68.9361	3.02596
$520$	0	0
$521$	−28.8145	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$521$	−28.8145	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$522$	9.00000	0.393919
$523$	−9.25264	−0.404590	−0.202295	−	0.979325i	$-0.564840\pi$
$523$	−9.25264	−0.404590	−0.202295	+	0.979325i	$0.564840\pi$
$524$	−5.15463	−0.225181
$525$	0	0
$526$	−17.4861	−0.762431
$527$	0.761141	0.0331558
$528$	−56.2283	−2.44702
$529$	20.6333	0.897100
$530$	−36.0582	−1.56627
$531$	−35.1857	−1.52693
$532$	0	0
$533$	0	0
$534$	−32.4500	−1.40425
$535$	−26.8055	−1.15890
$536$	3.00000	0.129580
$537$	41.4769	1.78986
$538$	12.3982	0.534526
$539$	0	0
$540$	5.78890	0.249114
$541$	18.9361	0.814126	0.407063	−	0.913400i	$-0.366553\pi$
$541$	18.9361	0.814126	0.407063	+	0.913400i	$0.366553\pi$
$542$	−38.6755	−1.66125
$543$	26.6611	1.14414
$544$	−1.48072	−0.0634852
$545$	−17.8970	−0.766623
$546$	0	0
$547$	29.0000	1.23995	0.619975	−	0.784621i	$-0.287143\pi$
$547$	29.0000	1.23995	0.619975	+	0.784621i	$0.287143\pi$
$548$	0.816654	0.0348857
$549$	30.5594	1.30424
$550$	25.4222	1.08401
$551$	−3.75389	−0.159921
$552$	−57.1008	−2.43037
$553$	0	0
$554$	18.5139	0.786579
$555$	11.5778	0.491450
$556$	−5.15463	−0.218605
$557$	−16.9083	−0.716429	−0.358214	−	0.933639i	$-0.616614\pi$
$557$	−16.9083	−0.716429	−0.358214	+	0.933639i	$0.616614\pi$
$558$	−6.02706	−0.255146
$559$	0	0
$560$	0	0
$561$	−14.8528	−0.627086
$562$	−2.84441	−0.119984
$563$	−19.0336	−0.802170	−0.401085	−	0.916041i	$-0.631367\pi$
$563$	−19.0336	−0.802170	−0.401085	+	0.916041i	$0.631367\pi$
$564$	−10.8167	−0.455463
$565$	42.6935	1.79613
$566$	5.23460	0.220027
$567$	0	0
$568$	12.0000	0.503509
$569$	−27.3944	−1.14844	−0.574218	−	0.818703i	$-0.694694\pi$
$569$	−27.3944	−1.14844	−0.574218	+	0.818703i	$0.694694\pi$
$570$	31.1677	1.30547
$571$	21.7250	0.909162	0.454581	−	0.890705i	$-0.349789\pi$
$571$	21.7250	0.909162	0.454581	+	0.890705i	$0.349789\pi$
$572$	0	0
$573$	55.6201	2.32356
$574$	0	0
$575$	21.8167	0.909817
$576$	46.7527	1.94803
$577$	−33.7050	−1.40316	−0.701579	−	0.712592i	$-0.747521\pi$
$577$	−33.7050	−1.40316	−0.701579	+	0.712592i	$0.747521\pi$
$578$	−21.1556	−0.879957
$579$	5.23460	0.217543
$580$	1.13659	0.0471942
$581$	0	0
$582$	29.1749	1.20934
$583$	56.7527	2.35046
$584$	−17.2887	−0.715412
$585$	0	0
$586$	−4.54634	−0.187808
$587$	28.5504	1.17840	0.589200	−	0.807987i	$-0.299443\pi$
$587$	28.5504	1.17840	0.589200	+	0.807987i	$0.299443\pi$
$588$	0	0
$589$	2.51388	0.103583
$590$	24.9083	1.02546
$591$	34.3133	1.41146
$592$	4.60555	0.189287
$593$	22.7875	0.935770	0.467885	−	0.883789i	$-0.345016\pi$
$593$	22.7875	0.935770	0.467885	+	0.883789i	$0.345016\pi$
$594$	51.0737	2.09558
$595$	0	0
$596$	0.155590	0.00637321
$597$	2.51388	0.102886
$598$	0	0
$599$	7.51388	0.307009	0.153504	−	0.988148i	$-0.450944\pi$
$599$	7.51388	0.307009	0.153504	+	0.988148i	$0.450944\pi$
$600$	−28.5504	−1.16556
$601$	−29.4228	−1.20018	−0.600091	−	0.799932i	$-0.704869\pi$
$601$	−29.4228	−1.20018	−0.600091	+	0.799932i	$0.704869\pi$
$602$	0	0
$603$	−5.30278	−0.215946
$604$	−1.88057	−0.0765193
$605$	−68.8907	−2.80081
$606$	−32.4500	−1.31819
$607$	20.4343	0.829404	0.414702	−	0.909957i	$-0.363886\pi$
$607$	20.4343	0.829404	0.414702	+	0.909957i	$0.363886\pi$
$608$	−4.89047	−0.198335
$609$	0	0
$610$	−21.6333	−0.875907
$611$	0	0
$612$	1.40074	0.0566215
$613$	−21.0917	−0.851885	−0.425942	−	0.904750i	$-0.640057\pi$
$613$	−21.0917	−0.851885	−0.425942	+	0.904750i	$0.640057\pi$
$614$	−19.9060	−0.803342
$615$	62.3354	2.51360
$616$	0	0
$617$	41.8444	1.68459	0.842296	−	0.539015i	$-0.181203\pi$
$617$	41.8444	1.68459	0.842296	+	0.539015i	$0.181203\pi$
$618$	18.3583	0.738479
$619$	−22.5233	−0.905289	−0.452644	−	0.891691i	$-0.649519\pi$
$619$	−22.5233	−0.905289	−0.452644	+	0.891691i	$0.649519\pi$
$620$	−0.761141	−0.0305682
$621$	43.8301	1.75884
$622$	−22.1792	−0.889305
$623$	0	0
$624$	0	0
$625$	−30.6056	−1.22422
$626$	17.2887	0.690996
$627$	−49.0555	−1.95909
$628$	−2.88145	−0.114983
$629$	1.21656	0.0485076
$630$	0	0
$631$	−12.0917	−0.481362	−0.240681	−	0.970604i	$-0.577371\pi$
$631$	−12.0917	−0.481362	−0.240681	+	0.970604i	$0.577371\pi$
$632$	−1.81665	−0.0722626
$633$	43.2218	1.71791
$634$	−10.6972	−0.424841
$635$	8.38021	0.332558
$636$	−8.38021	−0.332297
$637$	0	0
$638$	10.0278	0.397003
$639$	−21.2111	−0.839098
$640$	−23.3158	−0.921637
$641$	3.51388	0.138790	0.0693949	−	0.997589i	$-0.477893\pi$
$641$	3.51388	0.138790	0.0693949	+	0.997589i	$0.477893\pi$
$642$	34.9216	1.37824
$643$	8.38021	0.330483	0.165242	−	0.986253i	$-0.447160\pi$
$643$	8.38021	0.330483	0.165242	+	0.986253i	$0.447160\pi$
$644$	0	0
$645$	−45.7805	−1.80261
$646$	3.27502	0.128854
$647$	−2.27317	−0.0893676	−0.0446838	−	0.999001i	$-0.514228\pi$
$647$	−2.27317	−0.0893676	−0.0446838	+	0.999001i	$0.514228\pi$
$648$	−9.63331	−0.378432
$649$	−39.2038	−1.53888
$650$	0	0
$651$	0	0
$652$	−5.21110	−0.204083
$653$	21.7527	0.851250	0.425625	−	0.904900i	$-0.360054\pi$
$653$	21.7527	0.851250	0.425625	+	0.904900i	$0.360054\pi$
$654$	23.3158	0.911719
$655$	−49.0555	−1.91676
$656$	24.7965	0.968141
$657$	30.5594	1.19224
$658$	0	0
$659$	23.6333	0.920623	0.460311	−	0.887757i	$-0.347738\pi$
$659$	23.6333	0.920623	0.460311	+	0.887757i	$0.347738\pi$
$660$	14.8528	0.578145
$661$	9.78095	0.380435	0.190217	−	0.981742i	$-0.439081\pi$
$661$	9.78095	0.380435	0.190217	+	0.981742i	$0.439081\pi$
$662$	−0.908327	−0.0353031
$663$	0	0
$664$	−19.9060	−0.772504
$665$	0	0
$666$	−9.63331	−0.373283
$667$	8.60555	0.333208
$668$	1.48072	0.0572906
$669$	39.0000	1.50783
$670$	3.75389	0.145025
$671$	34.0491	1.31445
$672$	0	0
$673$	12.2111	0.470703	0.235352	−	0.971910i	$-0.424376\pi$
$673$	12.2111	0.470703	0.235352	+	0.971910i	$0.424376\pi$
$674$	−9.27502	−0.357260
$675$	21.9150	0.843510
$676$	0	0
$677$	6.37119	0.244865	0.122432	−	0.992477i	$-0.460931\pi$
$677$	6.37119	0.244865	0.122432	+	0.992477i	$0.460931\pi$
$678$	−55.6201	−2.13608
$679$	0	0
$680$	−7.54163	−0.289208
$681$	−78.9916	−3.02696
$682$	−6.71532	−0.257143
$683$	−3.60555	−0.137963	−0.0689813	−	0.997618i	$-0.521975\pi$
$683$	−3.60555	−0.137963	−0.0689813	+	0.997618i	$0.521975\pi$
$684$	4.62632	0.176892
$685$	7.77193	0.296950
$686$	0	0
$687$	−59.8722	−2.28427
$688$	−18.2111	−0.694292
$689$	0	0
$690$	−71.4500	−2.72005
$691$	−32.9126	−1.25205	−0.626026	−	0.779802i	$-0.715320\pi$
$691$	−32.9126	−1.25205	−0.626026	+	0.779802i	$0.715320\pi$
$692$	−7.24362	−0.275361
$693$	0	0
$694$	1.02776	0.0390131
$695$	−49.0555	−1.86078
$696$	−11.2617	−0.426872
$697$	6.55004	0.248100
$698$	−31.1677	−1.17971
$699$	−68.8907	−2.60569
$700$	0	0
$701$	−27.0278	−1.02082	−0.510412	−	0.859930i	$-0.670507\pi$
$701$	−27.0278	−1.02082	−0.510412	+	0.859930i	$0.670507\pi$
$702$	0	0
$703$	4.01804	0.151543
$704$	52.0917	1.96328
$705$	−102.940	−3.87694
$706$	−8.30023	−0.312383
$707$	0	0
$708$	5.78890	0.217560
$709$	−0.275019	−0.0103286	−0.00516428	−	0.999987i	$-0.501644\pi$
$709$	−0.275019	−0.0103286	−0.00516428	+	0.999987i	$0.501644\pi$
$710$	15.0156	0.563524
$711$	3.21110	0.120426
$712$	25.9331	0.971883
$713$	−5.76291	−0.215823
$714$	0	0
$715$	0	0
$716$	−4.35829	−0.162877
$717$	−33.4409	−1.24887
$718$	29.8444	1.11378
$719$	21.6509	0.807442	0.403721	−	0.914882i	$-0.367717\pi$
$719$	21.6509	0.807442	0.403721	+	0.914882i	$0.367717\pi$
$720$	50.4654	1.88074
$721$	0	0
$722$	−13.9361	−0.518647
$723$	5.78890	0.215291
$724$	−2.80148	−0.104116
$725$	4.30278	0.159801
$726$	89.7492	3.33090
$727$	−23.3958	−0.867701	−0.433850	−	0.900985i	$-0.642845\pi$
$727$	−23.3958	−0.867701	−0.433850	+	0.900985i	$0.642845\pi$
$728$	0	0
$729$	−40.3305	−1.49372
$730$	−21.6333	−0.800685
$731$	−4.81049	−0.177923
$732$	−5.02776	−0.185831
$733$	40.0762	1.48025	0.740124	−	0.672470i	$-0.234767\pi$
$733$	40.0762	1.48025	0.740124	+	0.672470i	$0.234767\pi$
$734$	−36.8506	−1.36018
$735$	0	0
$736$	11.2111	0.413247
$737$	−5.90833	−0.217636
$738$	−51.8662	−1.90922
$739$	18.7889	0.691161	0.345580	−	0.938389i	$-0.387682\pi$
$739$	18.7889	0.691161	0.345580	+	0.938389i	$0.387682\pi$
$740$	−1.21656	−0.0447218
$741$	0	0
$742$	0	0
$743$	37.6972	1.38298	0.691489	−	0.722387i	$-0.256955\pi$
$743$	37.6972	1.38298	0.691489	+	0.722387i	$0.256955\pi$
$744$	7.54163	0.276490
$745$	1.48072	0.0542492
$746$	−21.2389	−0.777610
$747$	35.1857	1.28738
$748$	1.56069	0.0570647
$749$	0	0
$750$	18.3583	0.670350
$751$	−4.39445	−0.160356	−0.0801779	−	0.996781i	$-0.525549\pi$
$751$	−4.39445	−0.160356	−0.0801779	+	0.996781i	$0.525549\pi$
$752$	−40.9486	−1.49324
$753$	−64.8999	−2.36508
$754$	0	0
$755$	−17.8970	−0.651339
$756$	0	0
$757$	−44.2389	−1.60789	−0.803944	−	0.594705i	$-0.797269\pi$
$757$	−44.2389	−1.60789	−0.803944	+	0.594705i	$0.797269\pi$
$758$	−17.0917	−0.620798
$759$	112.457	4.08192
$760$	−24.9083	−0.903520
$761$	4.28219	0.155229	0.0776147	−	0.996983i	$-0.475270\pi$
$761$	4.28219	0.155229	0.0776147	+	0.996983i	$0.475270\pi$
$762$	−10.9175	−0.395500
$763$	0	0
$764$	−5.84441	−0.211443
$765$	13.3305	0.481966
$766$	25.1406	0.908368
$767$	0	0
$768$	−20.4343	−0.737360
$769$	30.9035	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$769$	30.9035	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$770$	0	0
$771$	−65.6611	−2.36473
$772$	−0.550039	−0.0197963
$773$	34.9216	1.25604	0.628021	−	0.778196i	$-0.283865\pi$
$773$	34.9216	1.25604	0.628021	+	0.778196i	$0.283865\pi$
$774$	38.0917	1.36918
$775$	−2.88145	−0.103505
$776$	−23.3158	−0.836988
$777$	0	0
$778$	−9.15559	−0.328244
$779$	21.6333	0.775094
$780$	0	0
$781$	−23.6333	−0.845666
$782$	−7.50778	−0.268478
$783$	8.64436	0.308924
$784$	0	0
$785$	−27.4222	−0.978740
$786$	63.9083	2.27953
$787$	−41.5569	−1.48134	−0.740672	−	0.671867i	$-0.765493\pi$
$787$	−41.5569	−1.48134	−0.740672	+	0.671867i	$0.765493\pi$
$788$	−3.60555	−0.128442
$789$	−38.6755	−1.37688
$790$	−2.27317	−0.0808758
$791$	0	0
$792$	−93.9916	−3.33985
$793$	0	0
$794$	−7.50778	−0.266441
$795$	−79.7527	−2.82854
$796$	−0.264152	−0.00936261
$797$	34.5774	1.22480	0.612398	−	0.790550i	$-0.290205\pi$
$797$	34.5774	1.22480	0.612398	+	0.790550i	$0.290205\pi$
$798$	0	0
$799$	−10.8167	−0.382666
$800$	5.60555	0.198186
$801$	−45.8391	−1.61965
$802$	−19.6972	−0.695533
$803$	34.0491	1.20157
$804$	0.872434	0.0307684
$805$	0	0
$806$	0	0
$807$	27.4222	0.965307
$808$	25.9331	0.912323
$809$	−16.0278	−0.563506	−0.281753	−	0.959487i	$-0.590916\pi$
$809$	−16.0278	−0.563506	−0.281753	+	0.959487i	$0.590916\pi$
$810$	−12.0541	−0.423539
$811$	−21.9150	−0.769541	−0.384771	−	0.923012i	$-0.625719\pi$
$811$	−21.9150	−0.769541	−0.384771	+	0.923012i	$0.625719\pi$
$812$	0	0
$813$	−85.5416	−3.00008
$814$	−10.7334	−0.376205
$815$	−49.5930	−1.73717
$816$	8.30278	0.290655
$817$	−15.8880	−0.555850
$818$	21.0426	0.735738
$819$	0	0
$820$	−6.55004	−0.228737
$821$	−1.84441	−0.0643704	−0.0321852	−	0.999482i	$-0.510247\pi$
$821$	−1.84441	−0.0643704	−0.0321852	+	0.999482i	$0.510247\pi$
$822$	−10.1251	−0.353153
$823$	−44.2666	−1.54304	−0.771519	−	0.636207i	$-0.780503\pi$
$823$	−44.2666	−1.54304	−0.771519	+	0.636207i	$0.780503\pi$
$824$	−14.6714	−0.511103
$825$	56.2283	1.95762
$826$	0	0
$827$	−51.7527	−1.79962	−0.899809	−	0.436283i	$-0.856295\pi$
$827$	−51.7527	−1.79962	−0.899809	+	0.436283i	$0.856295\pi$
$828$	−10.6056	−0.368568
$829$	−15.6238	−0.542638	−0.271319	−	0.962489i	$-0.587460\pi$
$829$	−15.6238	−0.542638	−0.271319	+	0.962489i	$0.587460\pi$
$830$	−24.9083	−0.864581
$831$	40.9486	1.42049
$832$	0	0
$833$	0	0
$834$	63.9083	2.21296
$835$	14.0917	0.487662
$836$	5.15463	0.178276
$837$	−5.78890	−0.200094
$838$	10.9175	0.377140
$839$	23.6599	0.816831	0.408415	−	0.912796i	$-0.366081\pi$
$839$	23.6599	0.816831	0.408415	+	0.912796i	$0.366081\pi$
$840$	0	0
$841$	−27.3028	−0.941475
$842$	40.4222	1.39304
$843$	−6.29121	−0.216681
$844$	−4.54163	−0.156330
$845$	0	0
$846$	85.6512	2.94475
$847$	0	0
$848$	−31.7250	−1.08944
$849$	11.5778	0.397349
$850$	−3.75389	−0.128757
$851$	−9.21110	−0.315753
$852$	3.48974	0.119556
$853$	14.1431	0.484251	0.242126	−	0.970245i	$-0.422155\pi$
$853$	14.1431	0.484251	0.242126	+	0.970245i	$0.422155\pi$
$854$	0	0
$855$	44.0278	1.50572
$856$	−27.9083	−0.953887
$857$	−47.5840	−1.62544	−0.812719	−	0.582656i	$-0.802013\pi$
$857$	−47.5840	−1.62544	−0.812719	+	0.582656i	$0.802013\pi$
$858$	0	0
$859$	24.7965	0.846046	0.423023	−	0.906119i	$-0.360969\pi$
$859$	24.7965	0.846046	0.423023	+	0.906119i	$0.360969\pi$
$860$	4.81049	0.164037
$861$	0	0
$862$	−33.7889	−1.15085
$863$	11.8167	0.402244	0.201122	−	0.979566i	$-0.435541\pi$
$863$	11.8167	0.402244	0.201122	+	0.979566i	$0.435541\pi$
$864$	11.2617	0.383130
$865$	−68.9361	−2.34390
$866$	−10.9175	−0.370993
$867$	−46.7915	−1.58912
$868$	0	0
$869$	3.57779	0.121368
$870$	−14.0917	−0.477752
$871$	0	0
$872$	−18.6333	−0.631003
$873$	41.2128	1.39484
$874$	−24.7965	−0.838754
$875$	0	0
$876$	−5.02776	−0.169872
$877$	38.3944	1.29649	0.648244	−	0.761432i	$-0.275504\pi$
$877$	38.3944	1.29649	0.648244	+	0.761432i	$0.275504\pi$
$878$	−19.9060	−0.671796
$879$	−10.0555	−0.339164
$880$	56.2283	1.89546
$881$	35.4499	1.19434	0.597168	−	0.802116i	$-0.296292\pi$
$881$	35.4499	1.19434	0.597168	+	0.802116i	$0.296292\pi$
$882$	0	0
$883$	−31.6056	−1.06361	−0.531806	−	0.846866i	$-0.678486\pi$
$883$	−31.6056	−1.06361	−0.531806	+	0.846866i	$0.678486\pi$
$884$	0	0
$885$	55.0918	1.85189
$886$	39.3944	1.32348
$887$	46.1033	1.54800	0.773998	−	0.633188i	$-0.218254\pi$
$887$	46.1033	1.54800	0.773998	+	0.633188i	$0.218254\pi$
$888$	12.0541	0.404510
$889$	0	0
$890$	32.4500	1.08773
$891$	18.9722	0.635594
$892$	−4.09802	−0.137212
$893$	−35.7250	−1.19549
$894$	−1.92904	−0.0645168
$895$	−41.4769	−1.38642
$896$	0	0
$897$	0	0
$898$	−10.9722	−0.366149
$899$	−1.13659	−0.0379073
$900$	−5.30278	−0.176759
$901$	−8.38021	−0.279185
$902$	−57.7890	−1.92416
$903$	0	0
$904$	44.4500	1.47838
$905$	−26.6611	−0.886244
$906$	23.3158	0.774615
$907$	18.8444	0.625718	0.312859	−	0.949800i	$-0.398713\pi$
$907$	18.8444	0.625718	0.312859	+	0.949800i	$0.398713\pi$
$908$	8.30023	0.275453
$909$	−45.8391	−1.52039
$910$	0	0
$911$	−10.9361	−0.362329	−0.181164	−	0.983453i	$-0.557987\pi$
$911$	−10.9361	−0.362329	−0.181164	+	0.983453i	$0.557987\pi$
$912$	27.4222	0.908040
$913$	39.2038	1.29746
$914$	−17.4500	−0.577193
$915$	−47.8481	−1.58181
$916$	6.29121	0.207867
$917$	0	0
$918$	−7.54163	−0.248911
$919$	−11.4500	−0.377699	−0.188850	−	0.982006i	$-0.560476\pi$
$919$	−11.4500	−0.377699	−0.188850	+	0.982006i	$0.560476\pi$
$920$	57.1008	1.88256
$921$	−44.0278	−1.45076
$922$	16.4963	0.543276
$923$	0	0
$924$	0	0
$925$	−4.60555	−0.151430
$926$	36.7527	1.20777
$927$	25.9331	0.851754
$928$	2.21110	0.0725830
$929$	−8.38021	−0.274946	−0.137473	−	0.990506i	$-0.543898\pi$
$929$	−8.38021	−0.274946	−0.137473	+	0.990506i	$0.543898\pi$
$930$	9.43682	0.309445
$931$	0	0
$932$	7.23886	0.237117
$933$	−49.0555	−1.60601
$934$	−18.4253	−0.602895
$935$	14.8528	0.485739
$936$	0	0
$937$	46.4474	1.51737	0.758685	−	0.651458i	$-0.225842\pi$
$937$	46.4474	1.51737	0.758685	+	0.651458i	$0.225842\pi$
$938$	0	0
$939$	38.2389	1.24788
$940$	10.8167	0.352800
$941$	−44.7025	−1.45726	−0.728630	−	0.684907i	$-0.759843\pi$
$941$	−44.7025	−1.45726	−0.728630	+	0.684907i	$0.759843\pi$
$942$	35.7250	1.16398
$943$	−49.5930	−1.61497
$944$	21.9150	0.713274
$945$	0	0
$946$	42.4416	1.37989
$947$	−53.1194	−1.72615	−0.863075	−	0.505076i	$-0.831464\pi$
$947$	−53.1194	−1.72615	−0.863075	+	0.505076i	$0.831464\pi$
$948$	−0.528304	−0.0171585
$949$	0	0
$950$	−12.3982	−0.402252
$951$	−23.6599	−0.767225
$952$	0	0
$953$	41.6056	1.34774	0.673868	−	0.738852i	$-0.264632\pi$
$953$	41.6056	1.34774	0.673868	+	0.738852i	$0.264632\pi$
$954$	66.3583	2.14843
$955$	−55.6201	−1.79982
$956$	3.51388	0.113647
$957$	22.1792	0.716952
$958$	1.48072	0.0478398
$959$	0	0
$960$	−73.2027	−2.36261
$961$	−30.2389	−0.975447
$962$	0	0
$963$	49.3305	1.58965
$964$	−0.608282	−0.0195915
$965$	−5.23460	−0.168508
$966$	0	0
$967$	−22.4500	−0.721942	−0.360971	−	0.932577i	$-0.617555\pi$
$967$	−22.4500	−0.721942	−0.360971	+	0.932577i	$0.617555\pi$
$968$	−71.7250	−2.30533
$969$	7.24362	0.232699
$970$	−29.1749	−0.936751
$971$	10.1251	0.324929	0.162465	−	0.986714i	$-0.448056\pi$
$971$	10.1251	0.324929	0.162465	+	0.986714i	$0.448056\pi$
$972$	3.22558	0.103461
$973$	0	0
$974$	36.0833	1.15618
$975$	0	0
$976$	−19.0336	−0.609250
$977$	40.0278	1.28060	0.640301	−	0.768124i	$-0.278810\pi$
$977$	40.0278	1.28060	0.640301	+	0.768124i	$0.278810\pi$
$978$	64.6085	2.06595
$979$	−51.0737	−1.63232
$980$	0	0
$981$	32.9361	1.05157
$982$	−16.5778	−0.529019
$983$	13.0065	0.414844	0.207422	−	0.978252i	$-0.433493\pi$
$983$	13.0065	0.414844	0.207422	+	0.978252i	$0.433493\pi$
$984$	64.8999	2.06893
$985$	−34.3133	−1.09331
$986$	−1.48072	−0.0471556
$987$	0	0
$988$	0	0
$989$	36.4222	1.15816
$990$	−117.611	−3.73793
$991$	−46.3305	−1.47174	−0.735869	−	0.677124i	$-0.763226\pi$
$991$	−46.3305	−1.47174	−0.735869	+	0.677124i	$0.763226\pi$
$992$	−1.48072	−0.0470128
$993$	−2.00902	−0.0637543
$994$	0	0
$995$	−2.51388	−0.0796953
$996$	−5.78890	−0.183428
$997$	−44.1742	−1.39901	−0.699506	−	0.714627i	$-0.746596\pi$
$997$	−44.1742	−1.39901	−0.699506	+	0.714627i	$0.746596\pi$
$998$	−40.8167	−1.29203
$999$	−9.25264	−0.292741

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bo.1.4		4
7.6	odd	2	inner	8281.2.a.bo.1.3		4
13.4	even	6		637.2.f.h.393.3	yes	8
13.10	even	6		637.2.f.h.295.3	✓	8
13.12	even	2		8281.2.a.bu.1.2		4
91.4	even	6		637.2.h.j.471.1		8
91.10	odd	6		637.2.g.i.373.3		8
91.17	odd	6		637.2.h.j.471.2		8
91.23	even	6		637.2.h.j.165.1		8
91.30	even	6		637.2.g.i.263.4		8
91.62	odd	6		637.2.f.h.295.4	yes	8
91.69	odd	6		637.2.f.h.393.4	yes	8
91.75	odd	6		637.2.h.j.165.2		8
91.82	odd	6		637.2.g.i.263.3		8
91.88	even	6		637.2.g.i.373.4		8
91.90	odd	2		8281.2.a.bu.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.f.h.295.3	✓	8	13.10	even	6
637.2.f.h.295.4	yes	8	91.62	odd	6
637.2.f.h.393.3	yes	8	13.4	even	6
637.2.f.h.393.4	yes	8	91.69	odd	6
637.2.g.i.263.3		8	91.82	odd	6
637.2.g.i.263.4		8	91.30	even	6
637.2.g.i.373.3		8	91.10	odd	6
637.2.g.i.373.4		8	91.88	even	6
637.2.h.j.165.1		8	91.23	even	6
637.2.h.j.165.2		8	91.75	odd	6
637.2.h.j.471.1		8	91.4	even	6
637.2.h.j.471.2		8	91.17	odd	6
8281.2.a.bo.1.3		4	7.6	odd	2	inner
8281.2.a.bo.1.4		4	1.1	even	1	trivial
8281.2.a.bu.1.1		4	91.90	odd	2
8281.2.a.bu.1.2		4	13.12	even	2

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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} +2.88145 q^{3} -0.302776 q^{4} -2.88145 q^{5} +3.75389 q^{6} -3.00000 q^{8} +5.30278 q^{9} +O(q^{10})\)	\(q+1.30278 q^{2} +2.88145 q^{3} -0.302776 q^{4} -2.88145 q^{5} +3.75389 q^{6} -3.00000 q^{8} +5.30278 q^{9} -3.75389 q^{10} +5.90833 q^{11} -0.872434 q^{12} -8.30278 q^{15} -3.30278 q^{16} -0.872434 q^{17} +6.90833 q^{18} -2.88145 q^{19} +0.872434 q^{20} +7.69722 q^{22} +6.60555 q^{23} -8.64436 q^{24} +3.30278 q^{25} +6.63534 q^{27} +1.30278 q^{29} -10.8167 q^{30} -0.872434 q^{31} +1.69722 q^{32} +17.0246 q^{33} -1.13659 q^{34} -1.60555 q^{36} -1.39445 q^{37} -3.75389 q^{38} +8.64436 q^{40} -7.50778 q^{41} +5.51388 q^{43} -1.78890 q^{44} -15.2797 q^{45} +8.60555 q^{46} +12.3982 q^{47} -9.51680 q^{48} +4.30278 q^{50} -2.51388 q^{51} +9.60555 q^{53} +8.64436 q^{54} -17.0246 q^{55} -8.30278 q^{57} +1.69722 q^{58} -6.63534 q^{59} +2.51388 q^{60} +5.76291 q^{61} -1.13659 q^{62} +8.81665 q^{64} +22.1792 q^{66} -1.00000 q^{67} +0.264152 q^{68} +19.0336 q^{69} -4.00000 q^{71} -15.9083 q^{72} +5.76291 q^{73} -1.81665 q^{74} +9.51680 q^{75} +0.872434 q^{76} +0.605551 q^{79} +9.51680 q^{80} +3.21110 q^{81} -9.78095 q^{82} +6.63534 q^{83} +2.51388 q^{85} +7.18335 q^{86} +3.75389 q^{87} -17.7250 q^{88} -8.64436 q^{89} -19.9060 q^{90} -2.00000 q^{92} -2.51388 q^{93} +16.1521 q^{94} +8.30278 q^{95} +4.89047 q^{96} +7.77193 q^{97} +31.3305 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 + 14 * q^9	\( 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + 2 q^{11} - 26 q^{15} - 6 q^{16} + 6 q^{18} + 38 q^{22} + 12 q^{23} + 6 q^{25} - 2 q^{29} + 14 q^{32} + 8 q^{36} - 20 q^{37} - 14 q^{43} - 36 q^{44} + 20 q^{46} + 10 q^{50} + 26 q^{51} + 24 q^{53} - 26 q^{57} + 14 q^{58} - 26 q^{60} - 8 q^{64} - 4 q^{67} - 16 q^{71} - 42 q^{72} + 36 q^{74} - 12 q^{79} - 16 q^{81} - 26 q^{85} + 72 q^{86} - 6 q^{88} - 8 q^{92} + 26 q^{93} + 26 q^{95} + 46 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 + 14 * q^9 + 2 * q^11 - 26 * q^15 - 6 * q^16 + 6 * q^18 + 38 * q^22 + 12 * q^23 + 6 * q^25 - 2 * q^29 + 14 * q^32 + 8 * q^36 - 20 * q^37 - 14 * q^43 - 36 * q^44 + 20 * q^46 + 10 * q^50 + 26 * q^51 + 24 * q^53 - 26 * q^57 + 14 * q^58 - 26 * q^60 - 8 * q^64 - 4 * q^67 - 16 * q^71 - 42 * q^72 + 36 * q^74 - 12 * q^79 - 16 * q^81 - 26 * q^85 + 72 * q^86 - 6 * q^88 - 8 * q^92 + 26 * q^93 + 26 * q^95 + 46 * q^99

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