LMFDB - Embedded newform 8281.2.a.ci.1.7

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:	$1$
Dimension:	$8$
Coefficient field:	8.8.8446345216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 $ x^8 - 8x^6 + 19x^4 - 14*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.7
Root			$-1.51373$ 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)

$f(q)$	$=$	$q+2.43210 q^{2} -0.753592 q^{3} +3.91511 q^{4} -0.341537 q^{5} -1.83281 q^{6} +4.65773 q^{8} -2.43210 q^{9} +O(q^{10})$	\(q+2.43210 q^{2} -0.753592 q^{3} +3.91511 q^{4} -0.341537 q^{5} -1.83281 q^{6} +4.65773 q^{8} -2.43210 q^{9} -0.830652 q^{10} -2.43210 q^{11} -2.95039 q^{12} +0.257380 q^{15} +3.49784 q^{16} +1.94823 q^{17} -5.91511 q^{18} +6.29039 q^{19} -1.33715 q^{20} -5.91511 q^{22} -3.68948 q^{23} -3.51003 q^{24} -4.88335 q^{25} +4.09359 q^{27} +4.44135 q^{29} +0.625973 q^{30} +1.97532 q^{31} -0.808361 q^{32} +1.83281 q^{33} +4.73830 q^{34} -9.52192 q^{36} -9.62867 q^{37} +15.2988 q^{38} -1.59079 q^{40} -12.5359 q^{41} -8.40736 q^{43} -9.52192 q^{44} +0.830652 q^{45} -8.97318 q^{46} +9.00530 q^{47} -2.63594 q^{48} -11.8768 q^{50} -1.46817 q^{51} +1.49226 q^{53} +9.95601 q^{54} +0.830652 q^{55} -4.74039 q^{57} +10.8018 q^{58} -0.626991 q^{59} +1.00767 q^{60} +1.14319 q^{61} +4.80418 q^{62} -8.96169 q^{64} +4.45758 q^{66} -5.59199 q^{67} +7.62754 q^{68} +2.78036 q^{69} +9.49719 q^{71} -11.3280 q^{72} -11.9187 q^{73} -23.4179 q^{74} +3.68006 q^{75} +24.6275 q^{76} +4.47167 q^{79} -1.19464 q^{80} +4.21140 q^{81} -30.4885 q^{82} -1.41231 q^{83} -0.665394 q^{85} -20.4475 q^{86} -3.34697 q^{87} -11.3280 q^{88} -12.4444 q^{89} +2.02023 q^{90} -14.4447 q^{92} -1.48859 q^{93} +21.9018 q^{94} -2.14840 q^{95} +0.609174 q^{96} -10.2770 q^{97} +5.91511 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9	$ 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{15} + 4 q^{16} - 28 q^{18} - 28 q^{22} - 12 q^{23} - 12 q^{25} - 8 q^{29} - 28 q^{30} - 4 q^{36} + 8 q^{37} - 32 q^{43} - 4 q^{44} + 4 q^{46} - 36 q^{50} - 44 q^{51} - 4 q^{53} - 48 q^{57} + 48 q^{58} + 64 q^{60} - 32 q^{64} - 20 q^{67} - 8 q^{71} - 28 q^{72} - 76 q^{74} - 4 q^{79} - 56 q^{81} - 36 q^{85} + 4 q^{86} - 28 q^{88} - 80 q^{92} - 8 q^{93} - 52 q^{95} + 28 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^15 + 4 * q^16 - 28 * q^18 - 28 * q^22 - 12 * q^23 - 12 * q^25 - 8 * q^29 - 28 * q^30 - 4 * q^36 + 8 * q^37 - 32 * q^43 - 4 * q^44 + 4 * q^46 - 36 * q^50 - 44 * q^51 - 4 * q^53 - 48 * q^57 + 48 * q^58 + 64 * q^60 - 32 * q^64 - 20 * q^67 - 8 * q^71 - 28 * q^72 - 76 * q^74 - 4 * q^79 - 56 * q^81 - 36 * q^85 + 4 * q^86 - 28 * q^88 - 80 * q^92 - 8 * q^93 - 52 * q^95 + 28 * 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 $

$2$	2.43210	1.71975	0.859877	−	0.510502i	$-0.170540\pi$
$2$	2.43210	1.71975	0.859877	+	0.510502i	$0.170540\pi$
$3$	−0.753592	−0.435087	−0.217543	−	0.976051i	$-0.569804\pi$
$3$	−0.753592	−0.435087	−0.217543	+	0.976051i	$0.569804\pi$
$4$	3.91511	1.95755
$5$	−0.341537	−0.152740	−0.0763700	−	0.997080i	$-0.524333\pi$
$5$	−0.341537	−0.152740	−0.0763700	+	0.997080i	$0.524333\pi$
$6$	−1.83281	−0.748242
$7$	0	0
$7$	0	0
$8$	4.65773	1.64675
$9$	−2.43210	−0.810700
$10$	−0.830652	−0.262675
$11$	−2.43210	−0.733305	−0.366653	−	0.930358i	$-0.619496\pi$
$11$	−2.43210	−0.733305	−0.366653	+	0.930358i	$0.619496\pi$
$12$	−2.95039	−0.851705
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.257380	0.0664551
$16$	3.49784	0.874460
$17$	1.94823	0.472516	0.236258	−	0.971690i	$-0.424079\pi$
$17$	1.94823	0.472516	0.236258	+	0.971690i	$0.424079\pi$
$18$	−5.91511	−1.39420
$19$	6.29039	1.44311	0.721557	−	0.692355i	$-0.243427\pi$
$19$	6.29039	1.44311	0.721557	+	0.692355i	$0.243427\pi$
$20$	−1.33715	−0.298997
$21$	0	0
$22$	−5.91511	−1.26110
$23$	−3.68948	−0.769309	−0.384655	−	0.923061i	$-0.625679\pi$
$23$	−3.68948	−0.769309	−0.384655	+	0.923061i	$0.625679\pi$
$24$	−3.51003	−0.716481
$25$	−4.88335	−0.976670
$26$	0	0
$27$	4.09359	0.787811
$28$	0	0
$29$	4.44135	0.824738	0.412369	−	0.911017i	$-0.364701\pi$
$29$	4.44135	0.824738	0.412369	+	0.911017i	$0.364701\pi$
$30$	0.625973	0.114286
$31$	1.97532	0.354778	0.177389	−	0.984141i	$-0.443235\pi$
$31$	1.97532	0.354778	0.177389	+	0.984141i	$0.443235\pi$
$32$	−0.808361	−0.142899
$33$	1.83281	0.319051
$34$	4.73830	0.812611
$35$	0	0
$36$	−9.52192	−1.58699
$37$	−9.62867	−1.58294	−0.791472	−	0.611206i	$-0.790685\pi$
$37$	−9.62867	−1.58294	−0.791472	+	0.611206i	$0.790685\pi$
$38$	15.2988	2.48180
$39$	0	0
$40$	−1.59079	−0.251525
$41$	−12.5359	−1.95777	−0.978887	−	0.204403i	$-0.934475\pi$
$41$	−12.5359	−1.95777	−0.978887	+	0.204403i	$0.934475\pi$
$42$	0	0
$43$	−8.40736	−1.28211	−0.641055	−	0.767495i	$-0.721503\pi$
$43$	−8.40736	−1.28211	−0.641055	+	0.767495i	$0.721503\pi$
$44$	−9.52192	−1.43548
$45$	0.830652	0.123826
$46$	−8.97318	−1.32302
$47$	9.00530	1.31356	0.656779	−	0.754083i	$-0.271919\pi$
$47$	9.00530	1.31356	0.656779	+	0.754083i	$0.271919\pi$
$48$	−2.63594	−0.380466
$49$	0	0
$50$	−11.8768	−1.67963
$51$	−1.46817	−0.205585
$52$	0	0
$53$	1.49226	0.204977	0.102489	−	0.994734i	$-0.467319\pi$
$53$	1.49226	0.204977	0.102489	+	0.994734i	$0.467319\pi$
$54$	9.95601	1.35484
$55$	0.830652	0.112005
$56$	0	0
$57$	−4.74039	−0.627879
$58$	10.8018	1.41835
$59$	−0.626991	−0.0816273	−0.0408136	−	0.999167i	$-0.512995\pi$
$59$	−0.626991	−0.0816273	−0.0408136	+	0.999167i	$0.512995\pi$
$60$	1.00767	0.130089
$61$	1.14319	0.146371	0.0731855	−	0.997318i	$-0.476683\pi$
$61$	1.14319	0.146371	0.0731855	+	0.997318i	$0.476683\pi$
$62$	4.80418	0.610131
$63$	0	0
$64$	−8.96169	−1.12021
$65$	0	0
$66$	4.45758	0.548690
$67$	−5.59199	−0.683170	−0.341585	−	0.939851i	$-0.610964\pi$
$67$	−5.59199	−0.683170	−0.341585	+	0.939851i	$0.610964\pi$
$68$	7.62754	0.924975
$69$	2.78036	0.334716
$70$	0	0
$71$	9.49719	1.12711	0.563554	−	0.826079i	$-0.309434\pi$
$71$	9.49719	1.12711	0.563554	+	0.826079i	$0.309434\pi$
$72$	−11.3280	−1.33502
$73$	−11.9187	−1.39498	−0.697488	−	0.716597i	$-0.745699\pi$
$73$	−11.9187	−1.39498	−0.697488	+	0.716597i	$0.745699\pi$
$74$	−23.4179	−2.72227
$75$	3.68006	0.424936
$76$	24.6275	2.82497
$77$	0	0
$78$	0	0
$79$	4.47167	0.503102	0.251551	−	0.967844i	$-0.419059\pi$
$79$	4.47167	0.503102	0.251551	+	0.967844i	$0.419059\pi$
$80$	−1.19464	−0.133565
$81$	4.21140	0.467934
$82$	−30.4885	−3.36689
$83$	−1.41231	−0.155021	−0.0775104	−	0.996992i	$-0.524697\pi$
$83$	−1.41231	−0.155021	−0.0775104	+	0.996992i	$0.524697\pi$
$84$	0	0
$85$	−0.665394	−0.0721721
$86$	−20.4475	−2.20491
$87$	−3.34697	−0.358833
$88$	−11.3280	−1.20757
$89$	−12.4444	−1.31910	−0.659551	−	0.751660i	$-0.729253\pi$
$89$	−12.4444	−1.31910	−0.659551	+	0.751660i	$0.729253\pi$
$90$	2.02023	0.212951
$91$	0	0
$92$	−14.4447	−1.50596
$93$	−1.48859	−0.154359
$94$	21.9018	2.25900
$95$	−2.14840	−0.220421
$96$	0.609174	0.0621736
$97$	−10.2770	−1.04347	−0.521736	−	0.853107i	$-0.674715\pi$
$97$	−10.2770	−1.04347	−0.521736	+	0.853107i	$0.674715\pi$
$98$	0	0
$99$	5.91511	0.594490
$100$	−19.1188	−1.91188
$101$	−15.0537	−1.49790	−0.748948	−	0.662629i	$-0.769441\pi$
$101$	−15.0537	−1.49790	−0.748948	+	0.662629i	$0.769441\pi$
$102$	−3.57074	−0.353556
$103$	−17.6176	−1.73591	−0.867957	−	0.496639i	$-0.834567\pi$
$103$	−17.6176	−1.73591	−0.867957	+	0.496639i	$0.834567\pi$
$104$	0	0
$105$	0	0
$106$	3.62932	0.352511
$107$	6.38454	0.617217	0.308608	−	0.951189i	$-0.400137\pi$
$107$	6.38454	0.617217	0.308608	+	0.951189i	$0.400137\pi$
$108$	16.0268	1.54218
$109$	−8.17472	−0.782996	−0.391498	−	0.920179i	$-0.628043\pi$
$109$	−8.17472	−0.782996	−0.391498	+	0.920179i	$0.628043\pi$
$110$	2.02023	0.192621
$111$	7.25609	0.688717
$112$	0	0
$113$	−9.62165	−0.905129	−0.452564	−	0.891732i	$-0.649491\pi$
$113$	−9.62165	−0.905129	−0.452564	+	0.891732i	$0.649491\pi$
$114$	−11.5291	−1.07980
$115$	1.26009	0.117504
$116$	17.3884	1.61447
$117$	0	0
$118$	−1.52490	−0.140379
$119$	0	0
$120$	1.19880	0.109435
$121$	−5.08489	−0.462263
$122$	2.78036	0.251722
$123$	9.44693	0.851801
$124$	7.73360	0.694497
$125$	3.37553	0.301917
$126$	0	0
$127$	9.01976	0.800375	0.400187	−	0.916433i	$-0.368945\pi$
$127$	9.01976	0.800375	0.400187	+	0.916433i	$0.368945\pi$
$128$	−20.1790	−1.78359
$129$	6.33572	0.557829
$130$	0	0
$131$	0.192483	0.0168173	0.00840867	−	0.999965i	$-0.497323\pi$
$131$	0.192483	0.0168173	0.00840867	+	0.999965i	$0.497323\pi$
$132$	7.17565	0.624560
$133$	0	0
$134$	−13.6003	−1.17488
$135$	−1.39811	−0.120330
$136$	9.07434	0.778118
$137$	4.87680	0.416653	0.208326	−	0.978059i	$-0.433198\pi$
$137$	4.87680	0.416653	0.208326	+	0.978059i	$0.433198\pi$
$138$	6.76212	0.575629
$139$	11.0740	0.939286	0.469643	−	0.882856i	$-0.344383\pi$
$139$	11.0740	0.939286	0.469643	+	0.882856i	$0.344383\pi$
$140$	0	0
$141$	−6.78632	−0.571511
$142$	23.0981	1.93835
$143$	0	0
$144$	−8.50709	−0.708924
$145$	−1.51689	−0.125971
$146$	−28.9874	−2.39901
$147$	0	0
$148$	−37.6973	−3.09869
$149$	15.9087	1.30329	0.651646	−	0.758524i	$-0.274079\pi$
$149$	15.9087	1.30329	0.651646	+	0.758524i	$0.274079\pi$
$150$	8.95026	0.730786
$151$	−10.5904	−0.861831	−0.430916	−	0.902392i	$-0.641809\pi$
$151$	−10.5904	−0.861831	−0.430916	+	0.902392i	$0.641809\pi$
$152$	29.2989	2.37645
$153$	−4.73830	−0.383069
$154$	0	0
$155$	−0.674646	−0.0541889
$156$	0	0
$157$	−9.12388	−0.728165	−0.364082	−	0.931367i	$-0.618617\pi$
$157$	−9.12388	−0.728165	−0.364082	+	0.931367i	$0.618617\pi$
$158$	10.8755	0.865211
$159$	−1.12455	−0.0891829
$160$	0.276085	0.0218264
$161$	0	0
$162$	10.2425	0.804730
$163$	−10.9639	−0.858761	−0.429380	−	0.903124i	$-0.641268\pi$
$163$	−10.9639	−0.858761	−0.429380	+	0.903124i	$0.641268\pi$
$164$	−49.0792	−3.83245
$165$	−0.625973	−0.0487319
$166$	−3.43487	−0.266598
$167$	18.2777	1.41437	0.707185	−	0.707029i	$-0.249965\pi$
$167$	18.2777	1.41437	0.707185	+	0.707029i	$0.249965\pi$
$168$	0	0
$169$	0	0
$170$	−1.61830	−0.124118
$171$	−15.2988	−1.16993
$172$	−32.9157	−2.50980
$173$	8.19835	0.623309	0.311655	−	0.950195i	$-0.399117\pi$
$173$	8.19835	0.623309	0.311655	+	0.950195i	$0.399117\pi$
$174$	−8.14015	−0.617104
$175$	0	0
$176$	−8.50709	−0.641246
$177$	0.472495	0.0355149
$178$	−30.2659	−2.26853
$179$	−15.5537	−1.16254	−0.581268	−	0.813712i	$-0.697443\pi$
$179$	−15.5537	−1.16254	−0.581268	+	0.813712i	$0.697443\pi$
$180$	3.25209	0.242396
$181$	6.67302	0.496001	0.248001	−	0.968760i	$-0.420226\pi$
$181$	6.67302	0.496001	0.248001	+	0.968760i	$0.420226\pi$
$182$	0	0
$183$	−0.861502	−0.0636841
$184$	−17.1846	−1.26686
$185$	3.28855	0.241779
$186$	−3.62039	−0.265460
$187$	−4.73830	−0.346499
$188$	35.2567	2.57136
$189$	0	0
$190$	−5.22512	−0.379070
$191$	−18.7459	−1.35641	−0.678204	−	0.734874i	$-0.737241\pi$
$191$	−18.7459	−1.35641	−0.678204	+	0.734874i	$0.737241\pi$
$192$	6.75346	0.487389
$193$	−8.17309	−0.588312	−0.294156	−	0.955757i	$-0.595039\pi$
$193$	−8.17309	−0.588312	−0.294156	+	0.955757i	$0.595039\pi$
$194$	−24.9947	−1.79451
$195$	0	0
$196$	0	0
$197$	8.72012	0.621283	0.310641	−	0.950527i	$-0.399456\pi$
$197$	8.72012	0.621283	0.310641	+	0.950527i	$0.399456\pi$
$198$	14.3861	1.02238
$199$	17.4666	1.23818	0.619089	−	0.785321i	$-0.287502\pi$
$199$	17.4666	1.23818	0.619089	+	0.785321i	$0.287502\pi$
$200$	−22.7453	−1.60834
$201$	4.21408	0.297238
$202$	−36.6120	−2.57601
$203$	0	0
$204$	−5.74805	−0.402444
$205$	4.28146	0.299030
$206$	−42.8478	−2.98535
$207$	8.97318	0.623679
$208$	0	0
$209$	−15.2988	−1.05824
$210$	0	0
$211$	−23.2569	−1.60107	−0.800535	−	0.599286i	$-0.795451\pi$
$211$	−23.2569	−1.60107	−0.800535	+	0.599286i	$0.795451\pi$
$212$	5.84235	0.401254
$213$	−7.15701	−0.490390
$214$	15.5278	1.06146
$215$	2.87143	0.195830
$216$	19.0668	1.29733
$217$	0	0
$218$	−19.8817	−1.34656
$219$	8.98182	0.606935
$220$	3.25209	0.219256
$221$	0	0
$222$	17.6475	1.18442
$223$	29.2729	1.96026	0.980128	−	0.198368i	$-0.0635640\pi$
$223$	29.2729	1.96026	0.980128	+	0.198368i	$0.0635640\pi$
$224$	0	0
$225$	11.8768	0.791786
$226$	−23.4008	−1.55660
$227$	−19.8110	−1.31490	−0.657452	−	0.753496i	$-0.728366\pi$
$227$	−19.8110	−1.31490	−0.657452	+	0.753496i	$0.728366\pi$
$228$	−18.5591	−1.22911
$229$	1.32821	0.0877709	0.0438855	−	0.999037i	$-0.486026\pi$
$229$	1.32821	0.0877709	0.0438855	+	0.999037i	$0.486026\pi$
$230$	3.06467	0.202079
$231$	0	0
$232$	20.6866	1.35814
$233$	1.51634	0.0993389	0.0496695	−	0.998766i	$-0.484183\pi$
$233$	1.51634	0.0993389	0.0496695	+	0.998766i	$0.484183\pi$
$234$	0	0
$235$	−3.07564	−0.200633
$236$	−2.45474	−0.159790
$237$	−3.36981	−0.218893
$238$	0	0
$239$	22.4793	1.45406	0.727032	−	0.686603i	$-0.240899\pi$
$239$	22.4793	1.45406	0.727032	+	0.686603i	$0.240899\pi$
$240$	0.900273	0.0581123
$241$	−13.3106	−0.857409	−0.428704	−	0.903445i	$-0.641030\pi$
$241$	−13.3106	−0.857409	−0.428704	+	0.903445i	$0.641030\pi$
$242$	−12.3670	−0.794979
$243$	−15.4544	−0.991403
$244$	4.47573	0.286529
$245$	0	0
$246$	22.9759	1.46489
$247$	0	0
$248$	9.20051	0.584233
$249$	1.06430	0.0674475
$250$	8.20963	0.519222
$251$	−15.9034	−1.00381	−0.501906	−	0.864922i	$-0.667368\pi$
$251$	−15.9034	−1.00381	−0.501906	+	0.864922i	$0.667368\pi$
$252$	0	0
$253$	8.97318	0.564139
$254$	21.9370	1.37645
$255$	0.501436	0.0314011
$256$	−31.1539	−1.94712
$257$	29.2397	1.82392	0.911960	−	0.410280i	$-0.134569\pi$
$257$	29.2397	1.82392	0.911960	+	0.410280i	$0.134569\pi$
$258$	15.4091	0.959329
$259$	0	0
$260$	0	0
$261$	−10.8018	−0.668615
$262$	0.468138	0.0289217
$263$	−1.70435	−0.105095	−0.0525475	−	0.998618i	$-0.516734\pi$
$263$	−1.70435	−0.105095	−0.0525475	+	0.998618i	$0.516734\pi$
$264$	8.53673	0.525399
$265$	−0.509661	−0.0313083
$266$	0	0
$267$	9.37798	0.573923
$268$	−21.8932	−1.33734
$269$	−8.37874	−0.510861	−0.255430	−	0.966827i	$-0.582217\pi$
$269$	−8.37874	−0.510861	−0.255430	+	0.966827i	$0.582217\pi$
$270$	−3.40035	−0.206938
$271$	12.1575	0.738518	0.369259	−	0.929327i	$-0.379612\pi$
$271$	12.1575	0.738518	0.369259	+	0.929327i	$0.379612\pi$
$272$	6.81461	0.413196
$273$	0	0
$274$	11.8609	0.716540
$275$	11.8768	0.716198
$276$	10.8854	0.655225
$277$	10.3181	0.619957	0.309979	−	0.950744i	$-0.399678\pi$
$277$	10.3181	0.619957	0.309979	+	0.950744i	$0.399678\pi$
$278$	26.9331	1.61534
$279$	−4.80418	−0.287619
$280$	0	0
$281$	−2.59677	−0.154910	−0.0774551	−	0.996996i	$-0.524679\pi$
$281$	−2.59677	−0.154910	−0.0774551	+	0.996996i	$0.524679\pi$
$282$	−16.5050	−0.982859
$283$	−4.60368	−0.273660	−0.136830	−	0.990595i	$-0.543691\pi$
$283$	−4.60368	−0.273660	−0.136830	+	0.990595i	$0.543691\pi$
$284$	37.1825	2.20637
$285$	1.61902	0.0959023
$286$	0	0
$287$	0	0
$288$	1.96601	0.115848
$289$	−13.2044	−0.776729
$290$	−3.68922	−0.216638
$291$	7.74466	0.454000
$292$	−46.6629	−2.73074
$293$	1.96119	0.114574	0.0572870	−	0.998358i	$-0.481755\pi$
$293$	1.96119	0.114574	0.0572870	+	0.998358i	$0.481755\pi$
$294$	0	0
$295$	0.214141	0.0124678
$296$	−44.8477	−2.60672
$297$	−9.95601	−0.577706
$298$	38.6915	2.24134
$299$	0	0
$300$	14.4078	0.831835
$301$	0	0
$302$	−25.7568	−1.48214
$303$	11.3443	0.651714
$304$	22.0028	1.26194
$305$	−0.390443	−0.0223567
$306$	−11.5240	−0.658784
$307$	−7.37658	−0.421004	−0.210502	−	0.977593i	$-0.567510\pi$
$307$	−7.37658	−0.421004	−0.210502	+	0.977593i	$0.567510\pi$
$308$	0	0
$309$	13.2765	0.755273
$310$	−1.64081	−0.0931915
$311$	−14.1618	−0.803040	−0.401520	−	0.915850i	$-0.631518\pi$
$311$	−14.1618	−0.803040	−0.401520	+	0.915850i	$0.631518\pi$
$312$	0	0
$313$	26.7152	1.51003	0.755017	−	0.655705i	$-0.227629\pi$
$313$	26.7152	1.51003	0.755017	+	0.655705i	$0.227629\pi$
$314$	−22.1902	−1.25226
$315$	0	0
$316$	17.5070	0.984848
$317$	21.4362	1.20398	0.601989	−	0.798504i	$-0.294375\pi$
$317$	21.4362	1.20398	0.601989	+	0.798504i	$0.294375\pi$
$318$	−2.73503	−0.153373
$319$	−10.8018	−0.604785
$320$	3.06075	0.171101
$321$	−4.81134	−0.268543
$322$	0	0
$323$	12.2551	0.681895
$324$	16.4881	0.916005
$325$	0	0
$326$	−26.6654	−1.47686
$327$	6.16040	0.340671
$328$	−58.3886	−3.22397
$329$	0	0
$330$	−1.52243	−0.0838069
$331$	10.6138	0.583389	0.291695	−	0.956512i	$-0.405781\pi$
$331$	10.6138	0.583389	0.291695	+	0.956512i	$0.405781\pi$
$332$	−5.52933	−0.303462
$333$	23.4179	1.28329
$334$	44.4531	2.43237
$335$	1.90987	0.104347
$336$	0	0
$337$	6.75587	0.368016	0.184008	−	0.982925i	$-0.441093\pi$
$337$	6.75587	0.368016	0.184008	+	0.982925i	$0.441093\pi$
$338$	0	0
$339$	7.25080	0.393809
$340$	−2.60509	−0.141281
$341$	−4.80418	−0.260161
$342$	−37.2083	−2.01199
$343$	0	0
$344$	−39.1592	−2.11132
$345$	−0.949597	−0.0511246
$346$	19.9392	1.07194
$347$	−16.0204	−0.860021	−0.430010	−	0.902824i	$-0.641490\pi$
$347$	−16.0204	−0.860021	−0.430010	+	0.902824i	$0.641490\pi$
$348$	−13.1037	−0.702434
$349$	−16.0384	−0.858518	−0.429259	−	0.903182i	$-0.641225\pi$
$349$	−16.0384	−0.858518	−0.429259	+	0.903182i	$0.641225\pi$
$350$	0	0
$351$	0	0
$352$	1.96601	0.104789
$353$	−3.84311	−0.204548	−0.102274	−	0.994756i	$-0.532612\pi$
$353$	−3.84311	−0.204548	−0.102274	+	0.994756i	$0.532612\pi$
$354$	1.14916	0.0610769
$355$	−3.24364	−0.172155
$356$	−48.7210	−2.58221
$357$	0	0
$358$	−37.8281	−1.99928
$359$	−21.1335	−1.11538	−0.557692	−	0.830048i	$-0.688313\pi$
$359$	−21.1335	−1.11538	−0.557692	+	0.830048i	$0.688313\pi$
$360$	3.86895	0.203912
$361$	20.5690	1.08258
$362$	16.2294	0.853000
$363$	3.83194	0.201125
$364$	0	0
$365$	4.07067	0.213069
$366$	−2.09526	−0.109521
$367$	14.7392	0.769381	0.384690	−	0.923046i	$-0.374308\pi$
$367$	14.7392	0.769381	0.384690	+	0.923046i	$0.374308\pi$
$368$	−12.9052	−0.672730
$369$	30.4885	1.58717
$370$	7.99807	0.415800
$371$	0	0
$372$	−5.82798	−0.302166
$373$	12.9266	0.669314	0.334657	−	0.942340i	$-0.391380\pi$
$373$	12.9266	0.669314	0.334657	+	0.942340i	$0.391380\pi$
$374$	−11.5240	−0.595892
$375$	−2.54377	−0.131360
$376$	41.9442	2.16311
$377$	0	0
$378$	0	0
$379$	−26.8358	−1.37846	−0.689231	−	0.724542i	$-0.742051\pi$
$379$	−26.8358	−1.37846	−0.689231	+	0.724542i	$0.742051\pi$
$380$	−8.41121	−0.431486
$381$	−6.79722	−0.348232
$382$	−45.5919	−2.33269
$383$	2.90782	0.148583	0.0742914	−	0.997237i	$-0.476331\pi$
$383$	2.90782	0.148583	0.0742914	+	0.997237i	$0.476331\pi$
$384$	15.2067	0.776015
$385$	0	0
$386$	−19.8778	−1.01175
$387$	20.4475	1.03941
$388$	−40.2355	−2.04265
$389$	−16.7010	−0.846773	−0.423386	−	0.905949i	$-0.639159\pi$
$389$	−16.7010	−0.846773	−0.423386	+	0.905949i	$0.639159\pi$
$390$	0	0
$391$	−7.18797	−0.363511
$392$	0	0
$393$	−0.145054	−0.00731700
$394$	21.2082	1.06845
$395$	−1.52724	−0.0768438
$396$	23.1583	1.16375
$397$	24.0984	1.20947	0.604733	−	0.796428i	$-0.293280\pi$
$397$	24.0984	1.20947	0.604733	+	0.796428i	$0.293280\pi$
$398$	42.4806	2.12936
$399$	0	0
$400$	−17.0812	−0.854059
$401$	1.84490	0.0921297	0.0460649	−	0.998938i	$-0.485332\pi$
$401$	1.84490	0.0921297	0.0460649	+	0.998938i	$0.485332\pi$
$402$	10.2491	0.511176
$403$	0	0
$404$	−58.9367	−2.93221
$405$	−1.43835	−0.0714722
$406$	0	0
$407$	23.4179	1.16078
$408$	−6.83835	−0.338549
$409$	25.6703	1.26931	0.634657	−	0.772794i	$-0.281141\pi$
$409$	25.6703	1.26931	0.634657	+	0.772794i	$0.281141\pi$
$410$	10.4129	0.514259
$411$	−3.67512	−0.181280
$412$	−68.9748	−3.39814
$413$	0	0
$414$	21.8237	1.07257
$415$	0.482355	0.0236779
$416$	0	0
$417$	−8.34529	−0.408671
$418$	−37.2083	−1.81992
$419$	26.2398	1.28190	0.640950	−	0.767582i	$-0.278541\pi$
$419$	26.2398	1.28190	0.640950	+	0.767582i	$0.278541\pi$
$420$	0	0
$421$	23.6637	1.15330	0.576650	−	0.816992i	$-0.304360\pi$
$421$	23.6637	1.15330	0.576650	+	0.816992i	$0.304360\pi$
$422$	−56.5630	−2.75344
$423$	−21.9018	−1.06490
$424$	6.95053	0.337548
$425$	−9.51391	−0.461493
$426$	−17.4065	−0.843350
$427$	0	0
$428$	24.9961	1.20823
$429$	0	0
$430$	6.98359	0.336779
$431$	−23.0177	−1.10872	−0.554361	−	0.832276i	$-0.687037\pi$
$431$	−23.0177	−1.10872	−0.554361	+	0.832276i	$0.687037\pi$
$432$	14.3187	0.688909
$433$	25.8607	1.24279	0.621394	−	0.783498i	$-0.286567\pi$
$433$	25.8607	1.24279	0.621394	+	0.783498i	$0.286567\pi$
$434$	0	0
$435$	1.14311	0.0548081
$436$	−32.0049	−1.53276
$437$	−23.2082	−1.11020
$438$	21.8447	1.04378
$439$	35.6771	1.70277	0.851387	−	0.524537i	$-0.175762\pi$
$439$	35.6771	1.70277	0.851387	+	0.524537i	$0.175762\pi$
$440$	3.86895	0.184445
$441$	0	0
$442$	0	0
$443$	−6.85881	−0.325872	−0.162936	−	0.986637i	$-0.552096\pi$
$443$	−6.85881	−0.325872	−0.162936	+	0.986637i	$0.552096\pi$
$444$	28.4084	1.34820
$445$	4.25022	0.201480
$446$	71.1945	3.37116
$447$	−11.9887	−0.567045
$448$	0	0
$449$	−9.98150	−0.471056	−0.235528	−	0.971868i	$-0.575682\pi$
$449$	−9.98150	−0.471056	−0.235528	+	0.971868i	$0.575682\pi$
$450$	28.8855	1.36168
$451$	30.4885	1.43565
$452$	−37.6698	−1.77184
$453$	7.98081	0.374971
$454$	−48.1824	−2.26131
$455$	0	0
$456$	−22.0794	−1.03396
$457$	−8.77311	−0.410389	−0.205194	−	0.978721i	$-0.565783\pi$
$457$	−8.77311	−0.410389	−0.205194	+	0.978721i	$0.565783\pi$
$458$	3.23035	0.150944
$459$	7.97526	0.372253
$460$	4.93340	0.230021
$461$	−6.88543	−0.320687	−0.160343	−	0.987061i	$-0.551260\pi$
$461$	−6.88543	−0.320687	−0.160343	+	0.987061i	$0.551260\pi$
$462$	0	0
$463$	−13.9526	−0.648432	−0.324216	−	0.945983i	$-0.605100\pi$
$463$	−13.9526	−0.648432	−0.324216	+	0.945983i	$0.605100\pi$
$464$	15.5351	0.721200
$465$	0.508408	0.0235768
$466$	3.68790	0.170838
$467$	28.8113	1.33323	0.666613	−	0.745404i	$-0.267743\pi$
$467$	28.8113	1.33323	0.666613	+	0.745404i	$0.267743\pi$
$468$	0	0
$469$	0	0
$470$	−7.48027	−0.345039
$471$	6.87568	0.316815
$472$	−2.92035	−0.134420
$473$	20.4475	0.940179
$474$	−8.19572	−0.376442
$475$	−30.7182	−1.40945
$476$	0	0
$477$	−3.62932	−0.166175
$478$	54.6719	2.50063
$479$	24.5871	1.12341	0.561707	−	0.827336i	$-0.310145\pi$
$479$	24.5871	1.12341	0.561707	+	0.827336i	$0.310145\pi$
$480$	−0.208056	−0.00949639
$481$	0	0
$482$	−32.3726	−1.47453
$483$	0	0
$484$	−19.9079	−0.904904
$485$	3.50998	0.159380
$486$	−37.5867	−1.70497
$487$	−2.57316	−0.116601	−0.0583004	−	0.998299i	$-0.518568\pi$
$487$	−2.57316	−0.116601	−0.0583004	+	0.998299i	$0.518568\pi$
$488$	5.32469	0.241037
$489$	8.26233	0.373635
$490$	0	0
$491$	14.0379	0.633523	0.316762	−	0.948505i	$-0.397404\pi$
$491$	14.0379	0.633523	0.316762	+	0.948505i	$0.397404\pi$
$492$	36.9857	1.66745
$493$	8.65279	0.389702
$494$	0	0
$495$	−2.02023	−0.0908025
$496$	6.90936	0.310239
$497$	0	0
$498$	2.58849	0.115993
$499$	13.5345	0.605888	0.302944	−	0.953008i	$-0.402030\pi$
$499$	13.5345	0.605888	0.302944	+	0.953008i	$0.402030\pi$
$500$	13.2156	0.591018
$501$	−13.7739	−0.615373
$502$	−38.6786	−1.72631
$503$	8.27754	0.369077	0.184539	−	0.982825i	$-0.440921\pi$
$503$	8.27754	0.369077	0.184539	+	0.982825i	$0.440921\pi$
$504$	0	0
$505$	5.14138	0.228789
$506$	21.8237	0.970180
$507$	0	0
$508$	35.3133	1.56678
$509$	0.166218	0.00736750	0.00368375	−	0.999993i	$-0.498827\pi$
$509$	0.166218	0.00736750	0.00368375	+	0.999993i	$0.498827\pi$
$510$	1.21954	0.0540022
$511$	0	0
$512$	−35.4115	−1.56498
$513$	25.7502	1.13690
$514$	71.1137	3.13669
$515$	6.01707	0.265144
$516$	24.8050	1.09198
$517$	−21.9018	−0.963239
$518$	0	0
$519$	−6.17821	−0.271193
$520$	0	0
$521$	7.06180	0.309383	0.154691	−	0.987963i	$-0.450562\pi$
$521$	7.06180	0.309383	0.154691	+	0.987963i	$0.450562\pi$
$522$	−26.2711	−1.14985
$523$	−23.3912	−1.02283	−0.511414	−	0.859335i	$-0.670878\pi$
$523$	−23.3912	−1.02283	−0.511414	+	0.859335i	$0.670878\pi$
$524$	0.753592	0.0329208
$525$	0	0
$526$	−4.14516	−0.180737
$527$	3.84839	0.167639
$528$	6.41088	0.278998
$529$	−9.38775	−0.408163
$530$	−1.23955	−0.0538425
$531$	1.52490	0.0661752
$532$	0	0
$533$	0	0
$534$	22.8082	0.987006
$535$	−2.18056	−0.0942737
$536$	−26.0459	−1.12501
$537$	11.7211	0.505804
$538$	−20.3779	−0.878554
$539$	0	0
$540$	−5.47375	−0.235553
$541$	26.3079	1.13107	0.565533	−	0.824726i	$-0.308671\pi$
$541$	26.3079	1.13107	0.565533	+	0.824726i	$0.308671\pi$
$542$	29.5683	1.27007
$543$	−5.02873	−0.215804
$544$	−1.57488	−0.0675222
$545$	2.79197	0.119595
$546$	0	0
$547$	41.7636	1.78568	0.892841	−	0.450371i	$-0.148708\pi$
$547$	41.7636	1.78568	0.892841	+	0.450371i	$0.148708\pi$
$548$	19.0932	0.815620
$549$	−2.78036	−0.118663
$550$	28.8855	1.23168
$551$	27.9378	1.19019
$552$	12.9502	0.551196
$553$	0	0
$554$	25.0948	1.06617
$555$	−2.47822	−0.105195
$556$	43.3560	1.83870
$557$	7.30987	0.309729	0.154865	−	0.987936i	$-0.450506\pi$
$557$	7.30987	0.309729	0.154865	+	0.987936i	$0.450506\pi$
$558$	−11.6842	−0.494633
$559$	0	0
$560$	0	0
$561$	3.57074	0.150757
$562$	−6.31560	−0.266407
$563$	−44.7737	−1.88699	−0.943493	−	0.331393i	$-0.892481\pi$
$563$	−44.7737	−1.88699	−0.943493	+	0.331393i	$0.892481\pi$
$564$	−26.5692	−1.11876
$565$	3.28615	0.138249
$566$	−11.1966	−0.470628
$567$	0	0
$568$	44.2353	1.85607
$569$	−42.5127	−1.78222	−0.891112	−	0.453784i	$-0.850074\pi$
$569$	−42.5127	−1.78222	−0.891112	+	0.453784i	$0.850074\pi$
$570$	3.93761	0.164928
$571$	−40.8648	−1.71014	−0.855069	−	0.518515i	$-0.826485\pi$
$571$	−40.8648	−1.71014	−0.855069	+	0.518515i	$0.826485\pi$
$572$	0	0
$573$	14.1268	0.590155
$574$	0	0
$575$	18.0170	0.751362
$576$	21.7957	0.908155
$577$	21.7280	0.904550	0.452275	−	0.891879i	$-0.350613\pi$
$577$	21.7280	0.904550	0.452275	+	0.891879i	$0.350613\pi$
$578$	−32.1144	−1.33578
$579$	6.15918	0.255967
$580$	−5.93877	−0.246594
$581$	0	0
$582$	18.8358	0.780769
$583$	−3.62932	−0.150311
$584$	−55.5139	−2.29718
$585$	0	0
$586$	4.76981	0.197039
$587$	−20.4816	−0.845365	−0.422683	−	0.906278i	$-0.638911\pi$
$587$	−20.4816	−0.845365	−0.422683	+	0.906278i	$0.638911\pi$
$588$	0	0
$589$	12.4255	0.511986
$590$	0.520811	0.0214415
$591$	−6.57141	−0.270312
$592$	−33.6795	−1.38422
$593$	5.63861	0.231550	0.115775	−	0.993275i	$-0.463065\pi$
$593$	5.63861	0.231550	0.115775	+	0.993275i	$0.463065\pi$
$594$	−24.2140	−0.993512
$595$	0	0
$596$	62.2842	2.55126
$597$	−13.1627	−0.538715
$598$	0	0
$599$	39.6719	1.62095	0.810474	−	0.585774i	$-0.199209\pi$
$599$	39.6719	1.62095	0.810474	+	0.585774i	$0.199209\pi$
$600$	17.1407	0.699766
$601$	−16.8267	−0.686374	−0.343187	−	0.939267i	$-0.611507\pi$
$601$	−16.8267	−0.686374	−0.343187	+	0.939267i	$0.611507\pi$
$602$	0	0
$603$	13.6003	0.553845
$604$	−41.4624	−1.68708
$605$	1.73668	0.0706061
$606$	27.5905	1.12079
$607$	−22.4980	−0.913164	−0.456582	−	0.889681i	$-0.650927\pi$
$607$	−22.4980	−0.913164	−0.456582	+	0.889681i	$0.650927\pi$
$608$	−5.08490	−0.206220
$609$	0	0
$610$	−0.949597	−0.0384480
$611$	0	0
$612$	−18.5509	−0.749877
$613$	27.4269	1.10776	0.553882	−	0.832595i	$-0.313146\pi$
$613$	27.4269	1.10776	0.553882	+	0.832595i	$0.313146\pi$
$614$	−17.9406	−0.724022
$615$	−3.22648	−0.130104
$616$	0	0
$617$	10.6379	0.428267	0.214133	−	0.976804i	$-0.431307\pi$
$617$	10.6379	0.428267	0.214133	+	0.976804i	$0.431307\pi$
$618$	32.2897	1.29888
$619$	45.4677	1.82750	0.913751	−	0.406274i	$-0.133172\pi$
$619$	45.4677	1.82750	0.913751	+	0.406274i	$0.133172\pi$
$620$	−2.64131	−0.106078
$621$	−15.1032	−0.606071
$622$	−34.4428	−1.38103
$623$	0	0
$624$	0	0
$625$	23.2639	0.930556
$626$	64.9741	2.59689
$627$	11.5291	0.460427
$628$	−35.7210	−1.42542
$629$	−18.7589	−0.747966
$630$	0	0
$631$	29.5984	1.17829	0.589146	−	0.808027i	$-0.299464\pi$
$631$	29.5984	1.17829	0.589146	+	0.808027i	$0.299464\pi$
$632$	20.8278	0.828485
$633$	17.5262	0.696604
$634$	52.1350	2.07055
$635$	−3.08058	−0.122249
$636$	−4.40275	−0.174580
$637$	0	0
$638$	−26.2711	−1.04008
$639$	−23.0981	−0.913747
$640$	6.89188	0.272425
$641$	−42.5646	−1.68120	−0.840601	−	0.541654i	$-0.817798\pi$
$641$	−42.5646	−1.68120	−0.840601	+	0.541654i	$0.817798\pi$
$642$	−11.7016	−0.461827
$643$	−21.9961	−0.867441	−0.433721	−	0.901047i	$-0.642800\pi$
$643$	−21.9961	−0.867441	−0.433721	+	0.901047i	$0.642800\pi$
$644$	0	0
$645$	−2.16388	−0.0852029
$646$	29.8057	1.17269
$647$	−34.8051	−1.36833	−0.684166	−	0.729327i	$-0.739833\pi$
$647$	−34.8051	−1.36833	−0.684166	+	0.729327i	$0.739833\pi$
$648$	19.6156	0.770572
$649$	1.52490	0.0598577
$650$	0	0
$651$	0	0
$652$	−42.9249	−1.68107
$653$	−50.8167	−1.98861	−0.994306	−	0.106559i	$-0.966017\pi$
$653$	−50.8167	−1.98861	−0.994306	+	0.106559i	$0.966017\pi$
$654$	14.9827	0.585870
$655$	−0.0657402	−0.00256868
$656$	−43.8485	−1.71199
$657$	28.9874	1.13091
$658$	0	0
$659$	−14.7441	−0.574347	−0.287173	−	0.957879i	$-0.592716\pi$
$659$	−14.7441	−0.574347	−0.287173	+	0.957879i	$0.592716\pi$
$660$	−2.45075	−0.0953953
$661$	18.1245	0.704963	0.352481	−	0.935819i	$-0.385338\pi$
$661$	18.1245	0.704963	0.352481	+	0.935819i	$0.385338\pi$
$662$	25.8139	1.00329
$663$	0	0
$664$	−6.57814	−0.255281
$665$	0	0
$666$	56.9546	2.20695
$667$	−16.3863	−0.634479
$668$	71.5590	2.76870
$669$	−22.0598	−0.852881
$670$	4.64499	0.179452
$671$	−2.78036	−0.107335
$672$	0	0
$673$	20.9147	0.806204	0.403102	−	0.915155i	$-0.367932\pi$
$673$	20.9147	0.806204	0.403102	+	0.915155i	$0.367932\pi$
$674$	16.4309	0.632896
$675$	−19.9904	−0.769432
$676$	0	0
$677$	38.2179	1.46883	0.734416	−	0.678700i	$-0.237456\pi$
$677$	38.2179	1.46883	0.734416	+	0.678700i	$0.237456\pi$
$678$	17.6347	0.677255
$679$	0	0
$680$	−3.09922	−0.118850
$681$	14.9294	0.572097
$682$	−11.6842	−0.447413
$683$	−23.8253	−0.911649	−0.455825	−	0.890070i	$-0.650656\pi$
$683$	−23.8253	−0.911649	−0.455825	+	0.890070i	$0.650656\pi$
$684$	−59.8966	−2.29020
$685$	−1.66561	−0.0636396
$686$	0	0
$687$	−1.00093	−0.0381879
$688$	−29.4076	−1.12115
$689$	0	0
$690$	−2.30951	−0.0879217
$691$	19.1413	0.728168	0.364084	−	0.931366i	$-0.381382\pi$
$691$	19.1413	0.728168	0.364084	+	0.931366i	$0.381382\pi$
$692$	32.0974	1.22016
$693$	0	0
$694$	−38.9632	−1.47902
$695$	−3.78219	−0.143467
$696$	−15.5893	−0.590909
$697$	−24.4228	−0.925080
$698$	−39.0071	−1.47644
$699$	−1.14270	−0.0432210
$700$	0	0
$701$	−27.2956	−1.03094	−0.515471	−	0.856907i	$-0.672383\pi$
$701$	−27.2956	−1.03094	−0.515471	+	0.856907i	$0.672383\pi$
$702$	0	0
$703$	−60.5681	−2.28437
$704$	21.7957	0.821457
$705$	2.31778	0.0872927
$706$	−9.34683	−0.351773
$707$	0	0
$708$	1.84987	0.0695223
$709$	6.17166	0.231781	0.115891	−	0.993262i	$-0.463028\pi$
$709$	6.17166	0.231781	0.115891	+	0.993262i	$0.463028\pi$
$710$	−7.88886	−0.296064
$711$	−10.8755	−0.407865
$712$	−57.9625	−2.17224
$713$	−7.28791	−0.272934
$714$	0	0
$715$	0	0
$716$	−60.8943	−2.27573
$717$	−16.9402	−0.632644
$718$	−51.3988	−1.91819
$719$	−2.72133	−0.101488	−0.0507442	−	0.998712i	$-0.516159\pi$
$719$	−2.72133	−0.101488	−0.0507442	+	0.998712i	$0.516159\pi$
$720$	2.90549	0.108281
$721$	0	0
$722$	50.0258	1.86177
$723$	10.0307	0.373047
$724$	26.1256	0.970949
$725$	−21.6887	−0.805497
$726$	9.31965	0.345885
$727$	9.47153	0.351280	0.175640	−	0.984455i	$-0.443801\pi$
$727$	9.47153	0.351280	0.175640	+	0.984455i	$0.443801\pi$
$728$	0	0
$729$	−0.987863	−0.0365875
$730$	9.90027	0.366426
$731$	−16.3795	−0.605818
$732$	−3.37287	−0.124665
$733$	−3.49707	−0.129167	−0.0645836	−	0.997912i	$-0.520572\pi$
$733$	−3.49707	−0.129167	−0.0645836	+	0.997912i	$0.520572\pi$
$734$	35.8472	1.32315
$735$	0	0
$736$	2.98243	0.109934
$737$	13.6003	0.500972
$738$	74.1510	2.72954
$739$	32.0303	1.17825	0.589126	−	0.808041i	$-0.299472\pi$
$739$	32.0303	1.17825	0.589126	+	0.808041i	$0.299472\pi$
$740$	12.8750	0.473295
$741$	0	0
$742$	0	0
$743$	34.9186	1.28104	0.640519	−	0.767942i	$-0.278719\pi$
$743$	34.9186	1.28104	0.640519	+	0.767942i	$0.278719\pi$
$744$	−6.93343	−0.254192
$745$	−5.43341	−0.199065
$746$	31.4387	1.15105
$747$	3.43487	0.125675
$748$	−18.5509	−0.678289
$749$	0	0
$750$	−6.18671	−0.225907
$751$	26.9972	0.985143	0.492571	−	0.870272i	$-0.336057\pi$
$751$	26.9972	0.985143	0.492571	+	0.870272i	$0.336057\pi$
$752$	31.4991	1.14865
$753$	11.9847	0.436745
$754$	0	0
$755$	3.61700	0.131636
$756$	0	0
$757$	−52.5899	−1.91141	−0.955707	−	0.294319i	$-0.904907\pi$
$757$	−52.5899	−1.91141	−0.955707	+	0.294319i	$0.904907\pi$
$758$	−65.2673	−2.37062
$759$	−6.76212	−0.245449
$760$	−10.0067	−0.362980
$761$	−13.9286	−0.504912	−0.252456	−	0.967608i	$-0.581238\pi$
$761$	−13.9286	−0.504912	−0.252456	+	0.967608i	$0.581238\pi$
$762$	−16.5315	−0.598874
$763$	0	0
$764$	−73.3923	−2.65524
$765$	1.61830	0.0585099
$766$	7.07211	0.255526
$767$	0	0
$768$	23.4774	0.847166
$769$	−13.7909	−0.497312	−0.248656	−	0.968592i	$-0.579989\pi$
$769$	−13.7909	−0.497312	−0.248656	+	0.968592i	$0.579989\pi$
$770$	0	0
$771$	−22.0348	−0.793563
$772$	−31.9985	−1.15165
$773$	−50.4870	−1.81589	−0.907946	−	0.419087i	$-0.862350\pi$
$773$	−50.4870	−1.81589	−0.907946	+	0.419087i	$0.862350\pi$
$774$	49.7304	1.78752
$775$	−9.64620	−0.346502
$776$	−47.8674	−1.71834
$777$	0	0
$778$	−40.6184	−1.45624
$779$	−78.8555	−2.82529
$780$	0	0
$781$	−23.0981	−0.826515
$782$	−17.4818	−0.625150
$783$	18.1811	0.649738
$784$	0	0
$785$	3.11614	0.111220
$786$	−0.352785	−0.0125834
$787$	−10.8638	−0.387252	−0.193626	−	0.981075i	$-0.562025\pi$
$787$	−10.8638	−0.387252	−0.193626	+	0.981075i	$0.562025\pi$
$788$	34.1402	1.21619
$789$	1.28439	0.0457254
$790$	−3.71440	−0.132152
$791$	0	0
$792$	27.5509	0.978980
$793$	0	0
$794$	58.6098	2.07998
$795$	0.384077	0.0136218
$796$	68.3838	2.42380
$797$	7.91681	0.280428	0.140214	−	0.990121i	$-0.455221\pi$
$797$	7.91681	0.280428	0.140214	+	0.990121i	$0.455221\pi$
$798$	0	0
$799$	17.5444	0.620677
$800$	3.94751	0.139566
$801$	30.2659	1.06939
$802$	4.48697	0.158440
$803$	28.9874	1.02294
$804$	16.4986	0.581859
$805$	0	0
$806$	0	0
$807$	6.31415	0.222269
$808$	−70.1158	−2.46667
$809$	−29.6389	−1.04205	−0.521023	−	0.853542i	$-0.674450\pi$
$809$	−29.6389	−1.04205	−0.521023	+	0.853542i	$0.674450\pi$
$810$	−3.49821	−0.122915
$811$	−15.8344	−0.556022	−0.278011	−	0.960578i	$-0.589675\pi$
$811$	−15.8344	−0.556022	−0.278011	+	0.960578i	$0.589675\pi$
$812$	0	0
$813$	−9.16183	−0.321319
$814$	56.9546	1.99626
$815$	3.74459	0.131167
$816$	−5.13544	−0.179776
$817$	−52.8856	−1.85023
$818$	62.4327	2.18291
$819$	0	0
$820$	16.7624	0.585368
$821$	17.7394	0.619110	0.309555	−	0.950882i	$-0.399820\pi$
$821$	17.7394	0.619110	0.309555	+	0.950882i	$0.399820\pi$
$822$	−8.93824	−0.311757
$823$	−8.68200	−0.302635	−0.151318	−	0.988485i	$-0.548352\pi$
$823$	−8.68200	−0.302635	−0.151318	+	0.988485i	$0.548352\pi$
$824$	−82.0580	−2.85863
$825$	−8.95026	−0.311608
$826$	0	0
$827$	14.3121	0.497681	0.248840	−	0.968545i	$-0.419951\pi$
$827$	14.3121	0.497681	0.248840	+	0.968545i	$0.419951\pi$
$828$	35.1309	1.22088
$829$	−25.3066	−0.878933	−0.439467	−	0.898259i	$-0.644833\pi$
$829$	−25.3066	−0.878933	−0.439467	+	0.898259i	$0.644833\pi$
$830$	1.17314	0.0407201
$831$	−7.77567	−0.269735
$832$	0	0
$833$	0	0
$834$	−20.2966	−0.702813
$835$	−6.24250	−0.216031
$836$	−59.8966	−2.07157
$837$	8.08615	0.279498
$838$	63.8179	2.20455
$839$	−13.0426	−0.450280	−0.225140	−	0.974326i	$-0.572284\pi$
$839$	−13.0426	−0.450280	−0.225140	+	0.974326i	$0.572284\pi$
$840$	0	0
$841$	−9.27440	−0.319807
$842$	57.5525	1.98339
$843$	1.95690	0.0673994
$844$	−91.0531	−3.13418
$845$	0	0
$846$	−53.2673	−1.83137
$847$	0	0
$848$	5.21968	0.179245
$849$	3.46930	0.119066
$850$	−23.1388	−0.793653
$851$	35.5248	1.21777
$852$	−28.0204	−0.959964
$853$	−18.8926	−0.646869	−0.323435	−	0.946251i	$-0.604838\pi$
$853$	−18.8926	−0.646869	−0.323435	+	0.946251i	$0.604838\pi$
$854$	0	0
$855$	5.22512	0.178695
$856$	29.7374	1.01640
$857$	−24.6439	−0.841819	−0.420909	−	0.907103i	$-0.638289\pi$
$857$	−24.6439	−0.841819	−0.420909	+	0.907103i	$0.638289\pi$
$858$	0	0
$859$	−2.57141	−0.0877355	−0.0438677	−	0.999037i	$-0.513968\pi$
$859$	−2.57141	−0.0877355	−0.0438677	+	0.999037i	$0.513968\pi$
$860$	11.2419	0.383347
$861$	0	0
$862$	−55.9813	−1.90673
$863$	39.5407	1.34598	0.672991	−	0.739651i	$-0.265009\pi$
$863$	39.5407	1.34598	0.672991	+	0.739651i	$0.265009\pi$
$864$	−3.30909	−0.112578
$865$	−2.80004	−0.0952043
$866$	62.8959	2.13729
$867$	9.95072	0.337944
$868$	0	0
$869$	−10.8755	−0.368927
$870$	2.78016	0.0942564
$871$	0	0
$872$	−38.0756	−1.28940
$873$	24.9947	0.845942
$874$	−56.4448	−1.90927
$875$	0	0
$876$	35.1648	1.18811
$877$	21.0455	0.710655	0.355328	−	0.934742i	$-0.384369\pi$
$877$	21.0455	0.710655	0.355328	+	0.934742i	$0.384369\pi$
$878$	86.7702	2.92835
$879$	−1.47794	−0.0498497
$880$	2.90549	0.0979439
$881$	5.01184	0.168853	0.0844266	−	0.996430i	$-0.473094\pi$
$881$	5.01184	0.168853	0.0844266	+	0.996430i	$0.473094\pi$
$882$	0	0
$883$	−7.13079	−0.239970	−0.119985	−	0.992776i	$-0.538285\pi$
$883$	−7.13079	−0.239970	−0.119985	+	0.992776i	$0.538285\pi$
$884$	0	0
$885$	−0.161375	−0.00542455
$886$	−16.6813	−0.560419
$887$	−6.72602	−0.225838	−0.112919	−	0.993604i	$-0.536020\pi$
$887$	−6.72602	−0.225838	−0.112919	+	0.993604i	$0.536020\pi$
$888$	33.7969	1.13415
$889$	0	0
$890$	10.3369	0.346495
$891$	−10.2425	−0.343138
$892$	114.606	3.83730
$893$	56.6468	1.89561
$894$	−29.1576	−0.975177
$895$	5.31216	0.177566
$896$	0	0
$897$	0	0
$898$	−24.2760	−0.810100
$899$	8.77310	0.292599
$900$	46.4989	1.54996
$901$	2.90727	0.0968551
$902$	74.1510	2.46896
$903$	0	0
$904$	−44.8150	−1.49053
$905$	−2.27908	−0.0757593
$906$	19.4101	0.644858
$907$	29.5725	0.981938	0.490969	−	0.871177i	$-0.336643\pi$
$907$	29.5725	0.981938	0.490969	+	0.871177i	$0.336643\pi$
$908$	−77.5623	−2.57399
$909$	36.6120	1.21434
$910$	0	0
$911$	−20.6132	−0.682947	−0.341473	−	0.939891i	$-0.610926\pi$
$911$	−20.6132	−0.682947	−0.341473	+	0.939891i	$0.610926\pi$
$912$	−16.5811	−0.549055
$913$	3.43487	0.113678
$914$	−21.3371	−0.705768
$915$	0.294235	0.00972711
$916$	5.20010	0.171816
$917$	0	0
$918$	19.3966	0.640184
$919$	−4.17904	−0.137854	−0.0689269	−	0.997622i	$-0.521958\pi$
$919$	−4.17904	−0.137854	−0.0689269	+	0.997622i	$0.521958\pi$
$920$	5.86917	0.193501
$921$	5.55893	0.183173
$922$	−16.7461	−0.551502
$923$	0	0
$924$	0	0
$925$	47.0202	1.54601
$926$	−33.9341	−1.11514
$927$	42.8478	1.40731
$928$	−3.59021	−0.117855
$929$	48.0912	1.57782	0.788910	−	0.614509i	$-0.210646\pi$
$929$	48.0912	1.57782	0.788910	+	0.614509i	$0.210646\pi$
$930$	1.23650	0.0405464
$931$	0	0
$932$	5.93664	0.194461
$933$	10.6722	0.349392
$934$	70.0718	2.29282
$935$	1.61830	0.0529242
$936$	0	0
$937$	−12.5441	−0.409798	−0.204899	−	0.978783i	$-0.565687\pi$
$937$	−12.5441	−0.409798	−0.204899	+	0.978783i	$0.565687\pi$
$938$	0	0
$939$	−20.1324	−0.656995
$940$	−12.0415	−0.392749
$941$	30.5888	0.997167	0.498583	−	0.866842i	$-0.333854\pi$
$941$	30.5888	0.997167	0.498583	+	0.866842i	$0.333854\pi$
$942$	16.7223	0.544843
$943$	46.2508	1.50613
$944$	−2.19311	−0.0713798
$945$	0	0
$946$	49.7304	1.61688
$947$	−7.78348	−0.252929	−0.126465	−	0.991971i	$-0.540363\pi$
$947$	−7.78348	−0.252929	−0.126465	+	0.991971i	$0.540363\pi$
$948$	−13.1932	−0.428494
$949$	0	0
$950$	−74.7096	−2.42390
$951$	−16.1542	−0.523835
$952$	0	0
$953$	−38.4304	−1.24488	−0.622442	−	0.782666i	$-0.713859\pi$
$953$	−38.4304	−1.24488	−0.622442	+	0.782666i	$0.713859\pi$
$954$	−8.82686	−0.285780
$955$	6.40243	0.207178
$956$	88.0088	2.84641
$957$	8.14015	0.263134
$958$	59.7983	1.93200
$959$	0	0
$960$	−2.30656	−0.0744438
$961$	−27.0981	−0.874132
$962$	0	0
$963$	−15.5278	−0.500377
$964$	−52.1123	−1.67842
$965$	2.79141	0.0898588
$966$	0	0
$967$	7.80008	0.250834	0.125417	−	0.992104i	$-0.459973\pi$
$967$	7.80008	0.250834	0.125417	+	0.992104i	$0.459973\pi$
$968$	−23.6840	−0.761234
$969$	−9.23538	−0.296683
$970$	8.53661	0.274094
$971$	−57.7914	−1.85462	−0.927308	−	0.374300i	$-0.877883\pi$
$971$	−57.7914	−1.85462	−0.927308	+	0.374300i	$0.877883\pi$
$972$	−60.5058	−1.94072
$973$	0	0
$974$	−6.25817	−0.200525
$975$	0	0
$976$	3.99871	0.127996
$977$	8.67414	0.277510	0.138755	−	0.990327i	$-0.455690\pi$
$977$	8.67414	0.277510	0.138755	+	0.990327i	$0.455690\pi$
$978$	20.0948	0.642561
$979$	30.2659	0.967304
$980$	0	0
$981$	19.8817	0.634775
$982$	34.1417	1.08950
$983$	−37.1121	−1.18369	−0.591846	−	0.806051i	$-0.701601\pi$
$983$	−37.1121	−1.18369	−0.591846	+	0.806051i	$0.701601\pi$
$984$	44.0012	1.40271
$985$	−2.97824	−0.0948947
$986$	21.0444	0.670192
$987$	0	0
$988$	0	0
$989$	31.0188	0.986340
$990$	−4.91339	−0.156158
$991$	−45.2637	−1.43785	−0.718924	−	0.695089i	$-0.755365\pi$
$991$	−45.2637	−1.43785	−0.718924	+	0.695089i	$0.755365\pi$
$992$	−1.59677	−0.0506976
$993$	−7.99850	−0.253825
$994$	0	0
$995$	−5.96551	−0.189119
$996$	4.16686	0.132032
$997$	44.2554	1.40158	0.700791	−	0.713367i	$-0.252831\pi$
$997$	44.2554	1.40158	0.700791	+	0.713367i	$0.252831\pi$
$998$	32.9173	1.04198
$999$	−39.4158	−1.24706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ci.1.7		8
7.6	odd	2	inner	8281.2.a.ci.1.8		8
13.3	even	3		637.2.f.l.295.2	yes	16
13.9	even	3		637.2.f.l.393.2	yes	16
13.12	even	2		8281.2.a.cl.1.1		8
91.3	odd	6		637.2.g.m.373.2		16
91.9	even	3		637.2.g.m.263.1		16
91.16	even	3		637.2.h.m.165.8		16
91.48	odd	6		637.2.f.l.393.1	yes	16
91.55	odd	6		637.2.f.l.295.1	✓	16
91.61	odd	6		637.2.g.m.263.2		16
91.68	odd	6		637.2.h.m.165.7		16
91.74	even	3		637.2.h.m.471.8		16
91.81	even	3		637.2.g.m.373.1		16
91.87	odd	6		637.2.h.m.471.7		16
91.90	odd	2		8281.2.a.cl.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.f.l.295.1	✓	16	91.55	odd	6
637.2.f.l.295.2	yes	16	13.3	even	3
637.2.f.l.393.1	yes	16	91.48	odd	6
637.2.f.l.393.2	yes	16	13.9	even	3
637.2.g.m.263.1		16	91.9	even	3
637.2.g.m.263.2		16	91.61	odd	6
637.2.g.m.373.1		16	91.81	even	3
637.2.g.m.373.2		16	91.3	odd	6
637.2.h.m.165.7		16	91.68	odd	6
637.2.h.m.165.8		16	91.16	even	3
637.2.h.m.471.7		16	91.87	odd	6
637.2.h.m.471.8		16	91.74	even	3
8281.2.a.ci.1.7		8	1.1	even	1	trivial
8281.2.a.ci.1.8		8	7.6	odd	2	inner
8281.2.a.cl.1.1		8	13.12	even	2
8281.2.a.cl.1.2		8	91.90	odd	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)

\(f(q)\)	\(=\)	\(q+2.43210 q^{2} -0.753592 q^{3} +3.91511 q^{4} -0.341537 q^{5} -1.83281 q^{6} +4.65773 q^{8} -2.43210 q^{9} +O(q^{10})\)	\(q+2.43210 q^{2} -0.753592 q^{3} +3.91511 q^{4} -0.341537 q^{5} -1.83281 q^{6} +4.65773 q^{8} -2.43210 q^{9} -0.830652 q^{10} -2.43210 q^{11} -2.95039 q^{12} +0.257380 q^{15} +3.49784 q^{16} +1.94823 q^{17} -5.91511 q^{18} +6.29039 q^{19} -1.33715 q^{20} -5.91511 q^{22} -3.68948 q^{23} -3.51003 q^{24} -4.88335 q^{25} +4.09359 q^{27} +4.44135 q^{29} +0.625973 q^{30} +1.97532 q^{31} -0.808361 q^{32} +1.83281 q^{33} +4.73830 q^{34} -9.52192 q^{36} -9.62867 q^{37} +15.2988 q^{38} -1.59079 q^{40} -12.5359 q^{41} -8.40736 q^{43} -9.52192 q^{44} +0.830652 q^{45} -8.97318 q^{46} +9.00530 q^{47} -2.63594 q^{48} -11.8768 q^{50} -1.46817 q^{51} +1.49226 q^{53} +9.95601 q^{54} +0.830652 q^{55} -4.74039 q^{57} +10.8018 q^{58} -0.626991 q^{59} +1.00767 q^{60} +1.14319 q^{61} +4.80418 q^{62} -8.96169 q^{64} +4.45758 q^{66} -5.59199 q^{67} +7.62754 q^{68} +2.78036 q^{69} +9.49719 q^{71} -11.3280 q^{72} -11.9187 q^{73} -23.4179 q^{74} +3.68006 q^{75} +24.6275 q^{76} +4.47167 q^{79} -1.19464 q^{80} +4.21140 q^{81} -30.4885 q^{82} -1.41231 q^{83} -0.665394 q^{85} -20.4475 q^{86} -3.34697 q^{87} -11.3280 q^{88} -12.4444 q^{89} +2.02023 q^{90} -14.4447 q^{92} -1.48859 q^{93} +21.9018 q^{94} -2.14840 q^{95} +0.609174 q^{96} -10.2770 q^{97} +5.91511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9	\( 8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{15} + 4 q^{16} - 28 q^{18} - 28 q^{22} - 12 q^{23} - 12 q^{25} - 8 q^{29} - 28 q^{30} - 4 q^{36} + 8 q^{37} - 32 q^{43} - 4 q^{44} + 4 q^{46} - 36 q^{50} - 44 q^{51} - 4 q^{53} - 48 q^{57} + 48 q^{58} + 64 q^{60} - 32 q^{64} - 20 q^{67} - 8 q^{71} - 28 q^{72} - 76 q^{74} - 4 q^{79} - 56 q^{81} - 36 q^{85} + 4 q^{86} - 28 q^{88} - 80 q^{92} - 8 q^{93} - 52 q^{95} + 28 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^15 + 4 * q^16 - 28 * q^18 - 28 * q^22 - 12 * q^23 - 12 * q^25 - 8 * q^29 - 28 * q^30 - 4 * q^36 + 8 * q^37 - 32 * q^43 - 4 * q^44 + 4 * q^46 - 36 * q^50 - 44 * q^51 - 4 * q^53 - 48 * q^57 + 48 * q^58 + 64 * q^60 - 32 * q^64 - 20 * q^67 - 8 * q^71 - 28 * q^72 - 76 * q^74 - 4 * q^79 - 56 * q^81 - 36 * q^85 + 4 * q^86 - 28 * q^88 - 80 * q^92 - 8 * q^93 - 52 * q^95 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more