LMFDB - Embedded newform 8281.2.a.cz.1.17

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:	$36$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+0.286533 q^{2} -1.75963 q^{3} -1.91790 q^{4} -2.61493 q^{5} -0.504191 q^{6} -1.12261 q^{8} +0.0962853 q^{9} +O(q^{10})$	\(q+0.286533 q^{2} -1.75963 q^{3} -1.91790 q^{4} -2.61493 q^{5} -0.504191 q^{6} -1.12261 q^{8} +0.0962853 q^{9} -0.749265 q^{10} +5.95853 q^{11} +3.37479 q^{12} +4.60130 q^{15} +3.51413 q^{16} +0.218839 q^{17} +0.0275889 q^{18} +6.05260 q^{19} +5.01518 q^{20} +1.70732 q^{22} +1.77006 q^{23} +1.97537 q^{24} +1.83787 q^{25} +5.10945 q^{27} +9.23486 q^{29} +1.31843 q^{30} +6.25342 q^{31} +3.25213 q^{32} -10.4848 q^{33} +0.0627046 q^{34} -0.184665 q^{36} -5.85357 q^{37} +1.73427 q^{38} +2.93554 q^{40} -2.74723 q^{41} +0.366654 q^{43} -11.4279 q^{44} -0.251779 q^{45} +0.507180 q^{46} -10.3582 q^{47} -6.18356 q^{48} +0.526611 q^{50} -0.385075 q^{51} +7.58648 q^{53} +1.46403 q^{54} -15.5812 q^{55} -10.6503 q^{57} +2.64609 q^{58} -2.44206 q^{59} -8.82484 q^{60} -9.37829 q^{61} +1.79181 q^{62} -6.09642 q^{64} -3.00424 q^{66} +12.3721 q^{67} -0.419711 q^{68} -3.11464 q^{69} +1.05223 q^{71} -0.108091 q^{72} -6.01608 q^{73} -1.67724 q^{74} -3.23397 q^{75} -11.6083 q^{76} +12.4367 q^{79} -9.18922 q^{80} -9.27958 q^{81} -0.787172 q^{82} -7.46510 q^{83} -0.572249 q^{85} +0.105058 q^{86} -16.2499 q^{87} -6.68909 q^{88} +0.418197 q^{89} -0.0721431 q^{90} -3.39479 q^{92} -11.0037 q^{93} -2.96796 q^{94} -15.8271 q^{95} -5.72253 q^{96} +13.3273 q^{97} +0.573719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 14 q^{2} + 38 q^{4} + 42 q^{8} + 36 q^{9}+O(q^{10}) $ 36 * q + 14 * q^2 + 38 * q^4 + 42 * q^8 + 36 * q^9	$ 36 q + 14 q^{2} + 38 q^{4} + 42 q^{8} + 36 q^{9} + 52 q^{11} + 44 q^{15} + 42 q^{16} + 46 q^{18} + 68 q^{22} - 8 q^{23} + 32 q^{25} - 8 q^{29} + 118 q^{32} + 66 q^{36} + 28 q^{37} - 16 q^{43} + 130 q^{44} + 48 q^{46} + 90 q^{50} + 4 q^{51} + 52 q^{57} - 34 q^{58} + 100 q^{60} + 62 q^{64} + 40 q^{67} + 188 q^{71} - 42 q^{72} + 96 q^{74} + 8 q^{79} - 4 q^{81} + 64 q^{85} + 58 q^{86} + 156 q^{88} - 76 q^{92} - 16 q^{93} - 64 q^{95} + 156 q^{99}+O(q^{100}) $ 36 * q + 14 * q^2 + 38 * q^4 + 42 * q^8 + 36 * q^9 + 52 * q^11 + 44 * q^15 + 42 * q^16 + 46 * q^18 + 68 * q^22 - 8 * q^23 + 32 * q^25 - 8 * q^29 + 118 * q^32 + 66 * q^36 + 28 * q^37 - 16 * q^43 + 130 * q^44 + 48 * q^46 + 90 * q^50 + 4 * q^51 + 52 * q^57 - 34 * q^58 + 100 * q^60 + 62 * q^64 + 40 * q^67 + 188 * q^71 - 42 * q^72 + 96 * q^74 + 8 * q^79 - 4 * q^81 + 64 * q^85 + 58 * q^86 + 156 * q^88 - 76 * q^92 - 16 * q^93 - 64 * q^95 + 156 * 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 $

$2$	0.286533	0.202609	0.101305	−	0.994855i	$-0.467698\pi$
$2$	0.286533	0.202609	0.101305	+	0.994855i	$0.467698\pi$
$3$	−1.75963	−1.01592	−0.507960	−	0.861380i	$-0.669600\pi$
$3$	−1.75963	−1.01592	−0.507960	+	0.861380i	$0.669600\pi$
$4$	−1.91790	−0.958949
$5$	−2.61493	−1.16943	−0.584717	−	0.811238i	$-0.698794\pi$
$5$	−2.61493	−1.16943	−0.584717	+	0.811238i	$0.698794\pi$
$6$	−0.504191	−0.205835
$7$	0	0
$7$	0	0
$8$	−1.12261	−0.396902
$9$	0.0962853	0.0320951
$10$	−0.749265	−0.236938
$11$	5.95853	1.79656	0.898282	−	0.439419i	$-0.144816\pi$
$11$	5.95853	1.79656	0.898282	+	0.439419i	$0.144816\pi$
$12$	3.37479	0.974217
$13$	0	0
$13$	0	0
$14$	0	0
$15$	4.60130	1.18805
$16$	3.51413	0.878533
$17$	0.218839	0.0530762	0.0265381	−	0.999648i	$-0.491552\pi$
$17$	0.218839	0.0530762	0.0265381	+	0.999648i	$0.491552\pi$
$18$	0.0275889	0.00650277
$19$	6.05260	1.38856	0.694281	−	0.719704i	$-0.255723\pi$
$19$	6.05260	1.38856	0.694281	+	0.719704i	$0.255723\pi$
$20$	5.01518	1.12143
$21$	0	0
$22$	1.70732	0.364001
$23$	1.77006	0.369083	0.184541	−	0.982825i	$-0.440920\pi$
$23$	1.77006	0.369083	0.184541	+	0.982825i	$0.440920\pi$
$24$	1.97537	0.403221
$25$	1.83787	0.367574
$26$	0	0
$27$	5.10945	0.983315
$28$	0	0
$29$	9.23486	1.71487	0.857435	−	0.514591i	$-0.172056\pi$
$29$	9.23486	1.71487	0.857435	+	0.514591i	$0.172056\pi$
$30$	1.31843	0.240711
$31$	6.25342	1.12315	0.561573	−	0.827427i	$-0.310196\pi$
$31$	6.25342	1.12315	0.561573	+	0.827427i	$0.310196\pi$
$32$	3.25213	0.574901
$33$	−10.4848	−1.82517
$34$	0.0627046	0.0107537
$35$	0	0
$36$	−0.184665	−0.0307776
$37$	−5.85357	−0.962321	−0.481160	−	0.876633i	$-0.659785\pi$
$37$	−5.85357	−0.962321	−0.481160	+	0.876633i	$0.659785\pi$
$38$	1.73427	0.281336
$39$	0	0
$40$	2.93554	0.464150
$41$	−2.74723	−0.429045	−0.214522	−	0.976719i	$-0.568820\pi$
$41$	−2.74723	−0.429045	−0.214522	+	0.976719i	$0.568820\pi$
$42$	0	0
$43$	0.366654	0.0559141	0.0279571	−	0.999609i	$-0.491100\pi$
$43$	0.366654	0.0559141	0.0279571	+	0.999609i	$0.491100\pi$
$44$	−11.4279	−1.72281
$45$	−0.251779	−0.0375331
$46$	0.507180	0.0747796
$47$	−10.3582	−1.51089	−0.755447	−	0.655210i	$-0.772580\pi$
$47$	−10.3582	−1.51089	−0.755447	+	0.655210i	$0.772580\pi$
$48$	−6.18356	−0.892520
$49$	0	0
$50$	0.526611	0.0744740
$51$	−0.385075	−0.0539213
$52$	0	0
$53$	7.58648	1.04208	0.521041	−	0.853532i	$-0.325544\pi$
$53$	7.58648	1.04208	0.521041	+	0.853532i	$0.325544\pi$
$54$	1.46403	0.199229
$55$	−15.5812	−2.10096
$56$	0	0
$57$	−10.6503	−1.41067
$58$	2.64609	0.347449
$59$	−2.44206	−0.317929	−0.158965	−	0.987284i	$-0.550816\pi$
$59$	−2.44206	−0.317929	−0.158965	+	0.987284i	$0.550816\pi$
$60$	−8.82484	−1.13928
$61$	−9.37829	−1.20077	−0.600384	−	0.799712i	$-0.704986\pi$
$61$	−9.37829	−1.20077	−0.600384	+	0.799712i	$0.704986\pi$
$62$	1.79181	0.227560
$63$	0	0
$64$	−6.09642	−0.762053
$65$	0	0
$66$	−3.00424	−0.369796
$67$	12.3721	1.51150	0.755749	−	0.654862i	$-0.227273\pi$
$67$	12.3721	1.51150	0.755749	+	0.654862i	$0.227273\pi$
$68$	−0.419711	−0.0508974
$69$	−3.11464	−0.374959
$70$	0	0
$71$	1.05223	0.124876	0.0624382	−	0.998049i	$-0.480112\pi$
$71$	1.05223	0.124876	0.0624382	+	0.998049i	$0.480112\pi$
$72$	−0.108091	−0.0127386
$73$	−6.01608	−0.704128	−0.352064	−	0.935976i	$-0.614520\pi$
$73$	−6.01608	−0.704128	−0.352064	+	0.935976i	$0.614520\pi$
$74$	−1.67724	−0.194975
$75$	−3.23397	−0.373426
$76$	−11.6083	−1.33156
$77$	0	0
$78$	0	0
$79$	12.4367	1.39924	0.699621	−	0.714514i	$-0.253352\pi$
$79$	12.4367	1.39924	0.699621	+	0.714514i	$0.253352\pi$
$80$	−9.18922	−1.02739
$81$	−9.27958	−1.03106
$82$	−0.787172	−0.0869286
$83$	−7.46510	−0.819401	−0.409701	−	0.912220i	$-0.634367\pi$
$83$	−7.46510	−0.819401	−0.409701	+	0.912220i	$0.634367\pi$
$84$	0	0
$85$	−0.572249	−0.0620691
$86$	0.105058	0.0113287
$87$	−16.2499	−1.74217
$88$	−6.68909	−0.713059
$89$	0.418197	0.0443288	0.0221644	−	0.999754i	$-0.492944\pi$
$89$	0.418197	0.0443288	0.0221644	+	0.999754i	$0.492944\pi$
$90$	−0.0721431	−0.00760455
$91$	0	0
$92$	−3.39479	−0.353932
$93$	−11.0037	−1.14103
$94$	−2.96796	−0.306121
$95$	−15.8271	−1.62383
$96$	−5.72253	−0.584054
$97$	13.3273	1.35319	0.676593	−	0.736358i	$-0.263456\pi$
$97$	13.3273	1.35319	0.676593	+	0.736358i	$0.263456\pi$
$98$	0	0
$99$	0.573719	0.0576609
$100$	−3.52485	−0.352485
$101$	13.1800	1.31146	0.655732	−	0.754994i	$-0.272360\pi$
$101$	13.1800	1.31146	0.655732	+	0.754994i	$0.272360\pi$
$102$	−0.110337	−0.0109250
$103$	−16.8695	−1.66220	−0.831099	−	0.556124i	$-0.812288\pi$
$103$	−16.8695	−1.66220	−0.831099	+	0.556124i	$0.812288\pi$
$104$	0	0
$105$	0	0
$106$	2.17378	0.211136
$107$	−13.1550	−1.27175	−0.635873	−	0.771794i	$-0.719360\pi$
$107$	−13.1550	−1.27175	−0.635873	+	0.771794i	$0.719360\pi$
$108$	−9.79941	−0.942949
$109$	3.56800	0.341752	0.170876	−	0.985293i	$-0.445340\pi$
$109$	3.56800	0.341752	0.170876	+	0.985293i	$0.445340\pi$
$110$	−4.46452	−0.425675
$111$	10.3001	0.977642
$112$	0	0
$113$	17.6787	1.66307	0.831537	−	0.555469i	$-0.187461\pi$
$113$	17.6787	1.66307	0.831537	+	0.555469i	$0.187461\pi$
$114$	−3.05167	−0.285815
$115$	−4.62858	−0.431618
$116$	−17.7115	−1.64447
$117$	0	0
$118$	−0.699731	−0.0644155
$119$	0	0
$120$	−5.16546	−0.471540
$121$	24.5041	2.22764
$122$	−2.68719	−0.243287
$123$	4.83409	0.435876
$124$	−11.9934	−1.07704
$125$	8.26875	0.739580
$126$	0	0
$127$	−21.1467	−1.87646	−0.938231	−	0.346009i	$-0.887537\pi$
$127$	−21.1467	−1.87646	−0.938231	+	0.346009i	$0.887537\pi$
$128$	−8.25109	−0.729300
$129$	−0.645173	−0.0568043
$130$	0	0
$131$	1.39283	0.121692	0.0608459	−	0.998147i	$-0.480620\pi$
$131$	1.39283	0.121692	0.0608459	+	0.998147i	$0.480620\pi$
$132$	20.1088	1.75024
$133$	0	0
$134$	3.54503	0.306244
$135$	−13.3609	−1.14992
$136$	−0.245670	−0.0210660
$137$	2.11376	0.180591	0.0902954	−	0.995915i	$-0.471219\pi$
$137$	2.11376	0.180591	0.0902954	+	0.995915i	$0.471219\pi$
$138$	−0.892448	−0.0759702
$139$	17.4942	1.48384	0.741919	−	0.670490i	$-0.233916\pi$
$139$	17.4942	1.48384	0.741919	+	0.670490i	$0.233916\pi$
$140$	0	0
$141$	18.2265	1.53495
$142$	0.301498	0.0253011
$143$	0	0
$144$	0.338359	0.0281966
$145$	−24.1485	−2.00543
$146$	−1.72380	−0.142663
$147$	0	0
$148$	11.2266	0.922817
$149$	−1.27446	−0.104407	−0.0522037	−	0.998636i	$-0.516625\pi$
$149$	−1.27446	−0.104407	−0.0522037	+	0.998636i	$0.516625\pi$
$150$	−0.926638	−0.0756597
$151$	4.27852	0.348181	0.174090	−	0.984730i	$-0.444301\pi$
$151$	4.27852	0.348181	0.174090	+	0.984730i	$0.444301\pi$
$152$	−6.79470	−0.551123
$153$	0.0210710	0.00170349
$154$	0	0
$155$	−16.3523	−1.31345
$156$	0	0
$157$	5.02154	0.400762	0.200381	−	0.979718i	$-0.435782\pi$
$157$	5.02154	0.400762	0.200381	+	0.979718i	$0.435782\pi$
$158$	3.56354	0.283500
$159$	−13.3494	−1.05867
$160$	−8.50410	−0.672308
$161$	0	0
$162$	−2.65891	−0.208904
$163$	11.0126	0.862576	0.431288	−	0.902214i	$-0.358059\pi$
$163$	11.0126	0.862576	0.431288	+	0.902214i	$0.358059\pi$
$164$	5.26891	0.411432
$165$	27.4170	2.13441
$166$	−2.13900	−0.166018
$167$	−11.8178	−0.914491	−0.457246	−	0.889340i	$-0.651164\pi$
$167$	−11.8178	−0.914491	−0.457246	+	0.889340i	$0.651164\pi$
$168$	0	0
$169$	0	0
$170$	−0.163968	−0.0125758
$171$	0.582776	0.0445660
$172$	−0.703204	−0.0536188
$173$	11.0000	0.836317	0.418159	−	0.908374i	$-0.362676\pi$
$173$	11.0000	0.836317	0.418159	+	0.908374i	$0.362676\pi$
$174$	−4.65614	−0.352981
$175$	0	0
$176$	20.9391	1.57834
$177$	4.29712	0.322991
$178$	0.119827	0.00898144
$179$	4.80276	0.358975	0.179487	−	0.983760i	$-0.442556\pi$
$179$	4.80276	0.358975	0.179487	+	0.983760i	$0.442556\pi$
$180$	0.482888	0.0359923
$181$	−3.58266	−0.266297	−0.133148	−	0.991096i	$-0.542509\pi$
$181$	−3.58266	−0.266297	−0.133148	+	0.991096i	$0.542509\pi$
$182$	0	0
$183$	16.5023	1.21988
$184$	−1.98708	−0.146490
$185$	15.3067	1.12537
$186$	−3.15292	−0.231183
$187$	1.30396	0.0953549
$188$	19.8659	1.44887
$189$	0	0
$190$	−4.53500	−0.329003
$191$	−11.1821	−0.809111	−0.404555	−	0.914513i	$-0.632574\pi$
$191$	−11.1821	−0.809111	−0.404555	+	0.914513i	$0.632574\pi$
$192$	10.7274	0.774185
$193$	7.41154	0.533495	0.266747	−	0.963767i	$-0.414051\pi$
$193$	7.41154	0.533495	0.266747	+	0.963767i	$0.414051\pi$
$194$	3.81872	0.274168
$195$	0	0
$196$	0	0
$197$	8.49565	0.605290	0.302645	−	0.953103i	$-0.402130\pi$
$197$	8.49565	0.605290	0.302645	+	0.953103i	$0.402130\pi$
$198$	0.164389	0.0116826
$199$	17.9278	1.27087	0.635433	−	0.772156i	$-0.280822\pi$
$199$	17.9278	1.27087	0.635433	+	0.772156i	$0.280822\pi$
$200$	−2.06321	−0.145891
$201$	−21.7703	−1.53556
$202$	3.77652	0.265715
$203$	0	0
$204$	0.738534	0.0517078
$205$	7.18382	0.501739
$206$	−4.83366	−0.336777
$207$	0.170431	0.0118457
$208$	0	0
$209$	36.0646	2.49464
$210$	0	0
$211$	−10.7187	−0.737907	−0.368954	−	0.929448i	$-0.620284\pi$
$211$	−10.7187	−0.737907	−0.368954	+	0.929448i	$0.620284\pi$
$212$	−14.5501	−0.999304
$213$	−1.85153	−0.126864
$214$	−3.76935	−0.257668
$215$	−0.958774	−0.0653879
$216$	−5.73591	−0.390279
$217$	0	0
$218$	1.02235	0.0692422
$219$	10.5860	0.715339
$220$	29.8831	2.01472
$221$	0	0
$222$	2.95132	0.198080
$223$	2.78910	0.186772	0.0933861	−	0.995630i	$-0.470231\pi$
$223$	2.78910	0.186772	0.0933861	+	0.995630i	$0.470231\pi$
$224$	0	0
$225$	0.176960	0.0117973
$226$	5.06554	0.336955
$227$	−20.9323	−1.38932	−0.694662	−	0.719337i	$-0.744446\pi$
$227$	−20.9323	−1.38932	−0.694662	+	0.719337i	$0.744446\pi$
$228$	20.4262	1.35276
$229$	13.7786	0.910514	0.455257	−	0.890360i	$-0.349547\pi$
$229$	13.7786	0.910514	0.455257	+	0.890360i	$0.349547\pi$
$230$	−1.32624	−0.0874498
$231$	0	0
$232$	−10.3671	−0.680635
$233$	−20.4043	−1.33673	−0.668365	−	0.743833i	$-0.733006\pi$
$233$	−20.4043	−1.33673	−0.668365	+	0.743833i	$0.733006\pi$
$234$	0	0
$235$	27.0859	1.76689
$236$	4.68363	0.304878
$237$	−21.8840	−1.42152
$238$	0	0
$239$	−3.53874	−0.228902	−0.114451	−	0.993429i	$-0.536511\pi$
$239$	−3.53874	−0.228902	−0.114451	+	0.993429i	$0.536511\pi$
$240$	16.1696	1.04374
$241$	−25.9202	−1.66967	−0.834835	−	0.550501i	$-0.814437\pi$
$241$	−25.9202	−1.66967	−0.834835	+	0.550501i	$0.814437\pi$
$242$	7.02123	0.451342
$243$	1.00024	0.0641656
$244$	17.9866	1.15148
$245$	0	0
$246$	1.38513	0.0883125
$247$	0	0
$248$	−7.02013	−0.445779
$249$	13.1358	0.832447
$250$	2.36927	0.149846
$251$	10.1046	0.637794	0.318897	−	0.947789i	$-0.396688\pi$
$251$	10.1046	0.637794	0.318897	+	0.947789i	$0.396688\pi$
$252$	0	0
$253$	10.5469	0.663081
$254$	−6.05922	−0.380189
$255$	1.00694	0.0630573
$256$	9.82864	0.614290
$257$	−10.3741	−0.647119	−0.323560	−	0.946208i	$-0.604880\pi$
$257$	−10.3741	−0.647119	−0.323560	+	0.946208i	$0.604880\pi$
$258$	−0.184864	−0.0115091
$259$	0	0
$260$	0	0
$261$	0.889181	0.0550389
$262$	0.399091	0.0246559
$263$	3.55547	0.219240	0.109620	−	0.993974i	$-0.465037\pi$
$263$	3.55547	0.219240	0.109620	+	0.993974i	$0.465037\pi$
$264$	11.7703	0.724412
$265$	−19.8381	−1.21865
$266$	0	0
$267$	−0.735871	−0.0450346
$268$	−23.7285	−1.44945
$269$	−1.44987	−0.0883999	−0.0441999	−	0.999023i	$-0.514074\pi$
$269$	−1.44987	−0.0883999	−0.0441999	+	0.999023i	$0.514074\pi$
$270$	−3.82833	−0.232985
$271$	−16.4168	−0.997249	−0.498624	−	0.866818i	$-0.666161\pi$
$271$	−16.4168	−0.997249	−0.498624	+	0.866818i	$0.666161\pi$
$272$	0.769029	0.0466292
$273$	0	0
$274$	0.605662	0.0365894
$275$	10.9510	0.660371
$276$	5.97357	0.359566
$277$	15.1568	0.910686	0.455343	−	0.890316i	$-0.349517\pi$
$277$	15.1568	0.910686	0.455343	+	0.890316i	$0.349517\pi$
$278$	5.01266	0.300639
$279$	0.602112	0.0360475
$280$	0	0
$281$	−7.67167	−0.457653	−0.228827	−	0.973467i	$-0.573489\pi$
$281$	−7.67167	−0.457653	−0.228827	+	0.973467i	$0.573489\pi$
$282$	5.22249	0.310995
$283$	−28.5181	−1.69522	−0.847611	−	0.530618i	$-0.821960\pi$
$283$	−28.5181	−1.69522	−0.847611	+	0.530618i	$0.821960\pi$
$284$	−2.01806	−0.119750
$285$	27.8499	1.64968
$286$	0	0
$287$	0	0
$288$	0.313132	0.0184515
$289$	−16.9521	−0.997183
$290$	−6.91936	−0.406319
$291$	−23.4511	−1.37473
$292$	11.5382	0.675223
$293$	32.1769	1.87980	0.939899	−	0.341453i	$-0.110919\pi$
$293$	32.1769	1.87980	0.939899	+	0.341453i	$0.110919\pi$
$294$	0	0
$295$	6.38582	0.371797
$296$	6.57126	0.381947
$297$	30.4448	1.76659
$298$	−0.365174	−0.0211539
$299$	0	0
$300$	6.20242	0.358097
$301$	0	0
$302$	1.22594	0.0705447
$303$	−23.1920	−1.33234
$304$	21.2696	1.21990
$305$	24.5236	1.40422
$306$	0.00603753	0.000345143 0
$307$	−2.29353	−0.130899	−0.0654493	−	0.997856i	$-0.520848\pi$
$307$	−2.29353	−0.130899	−0.0654493	+	0.997856i	$0.520848\pi$
$308$	0	0
$309$	29.6840	1.68866
$310$	−4.68546	−0.266116
$311$	−20.2919	−1.15065	−0.575323	−	0.817926i	$-0.695124\pi$
$311$	−20.2919	−1.15065	−0.575323	+	0.817926i	$0.695124\pi$
$312$	0	0
$313$	−34.2833	−1.93780	−0.968902	−	0.247445i	$-0.920409\pi$
$313$	−34.2833	−1.93780	−0.968902	+	0.247445i	$0.920409\pi$
$314$	1.43884	0.0811983
$315$	0	0
$316$	−23.8524	−1.34180
$317$	17.0814	0.959389	0.479694	−	0.877436i	$-0.340747\pi$
$317$	17.0814	0.959389	0.479694	+	0.877436i	$0.340747\pi$
$318$	−3.82503	−0.214497
$319$	55.0262	3.08088
$320$	15.9417	0.891170
$321$	23.1479	1.29199
$322$	0	0
$323$	1.32454	0.0736996
$324$	17.7973	0.988739
$325$	0	0
$326$	3.15549	0.174766
$327$	−6.27834	−0.347193
$328$	3.08406	0.170289
$329$	0	0
$330$	7.85588	0.432452
$331$	−16.5398	−0.909111	−0.454556	−	0.890718i	$-0.650202\pi$
$331$	−16.5398	−0.909111	−0.454556	+	0.890718i	$0.650202\pi$
$332$	14.3173	0.785764
$333$	−0.563613	−0.0308858
$334$	−3.38620	−0.185285
$335$	−32.3523	−1.76760
$336$	0	0
$337$	12.4007	0.675510	0.337755	−	0.941234i	$-0.390332\pi$
$337$	12.4007	0.675510	0.337755	+	0.941234i	$0.390332\pi$
$338$	0	0
$339$	−31.1079	−1.68955
$340$	1.09752	0.0595211
$341$	37.2612	2.01781
$342$	0.166985	0.00902950
$343$	0	0
$344$	−0.411608	−0.0221924
$345$	8.14458	0.438489
$346$	3.15187	0.169446
$347$	17.2792	0.927597	0.463799	−	0.885941i	$-0.346486\pi$
$347$	17.2792	0.927597	0.463799	+	0.885941i	$0.346486\pi$
$348$	31.1657	1.67066
$349$	20.6042	1.10292	0.551458	−	0.834202i	$-0.314072\pi$
$349$	20.6042	1.10292	0.551458	+	0.834202i	$0.314072\pi$
$350$	0	0
$351$	0	0
$352$	19.3779	1.03285
$353$	−35.5777	−1.89361	−0.946805	−	0.321807i	$-0.895710\pi$
$353$	−35.5777	−1.89361	−0.946805	+	0.321807i	$0.895710\pi$
$354$	1.23127	0.0654410
$355$	−2.75150	−0.146035
$356$	−0.802060	−0.0425091
$357$	0	0
$358$	1.37615	0.0727317
$359$	17.4589	0.921443	0.460722	−	0.887545i	$-0.347591\pi$
$359$	17.4589	0.921443	0.460722	+	0.887545i	$0.347591\pi$
$360$	0.282650	0.0148969
$361$	17.6340	0.928104
$362$	−1.02655	−0.0539543
$363$	−43.1180	−2.26311
$364$	0	0
$365$	15.7316	0.823431
$366$	4.72845	0.247160
$367$	11.5335	0.602043	0.301021	−	0.953617i	$-0.402672\pi$
$367$	11.5335	0.602043	0.301021	+	0.953617i	$0.402672\pi$
$368$	6.22022	0.324251
$369$	−0.264518	−0.0137702
$370$	4.38587	0.228011
$371$	0	0
$372$	21.1039	1.09419
$373$	3.33440	0.172649	0.0863243	−	0.996267i	$-0.472488\pi$
$373$	3.33440	0.172649	0.0863243	+	0.996267i	$0.472488\pi$
$374$	0.373627	0.0193198
$375$	−14.5499	−0.751354
$376$	11.6282	0.599676
$377$	0	0
$378$	0	0
$379$	26.3440	1.35320	0.676600	−	0.736350i	$-0.263452\pi$
$379$	26.3440	1.35320	0.676600	+	0.736350i	$0.263452\pi$
$380$	30.3549	1.55717
$381$	37.2102	1.90634
$382$	−3.20405	−0.163934
$383$	12.8458	0.656390	0.328195	−	0.944610i	$-0.393560\pi$
$383$	12.8458	0.656390	0.328195	+	0.944610i	$0.393560\pi$
$384$	14.5188	0.740911
$385$	0	0
$386$	2.12365	0.108091
$387$	0.0353033	0.00179457
$388$	−25.5605	−1.29764
$389$	−16.4820	−0.835671	−0.417836	−	0.908523i	$-0.637211\pi$
$389$	−16.4820	−0.835671	−0.417836	+	0.908523i	$0.637211\pi$
$390$	0	0
$391$	0.387358	0.0195895
$392$	0	0
$393$	−2.45085	−0.123629
$394$	2.43429	0.122638
$395$	−32.5212	−1.63632
$396$	−1.10033	−0.0552939
$397$	18.5970	0.933354	0.466677	−	0.884428i	$-0.345451\pi$
$397$	18.5970	0.933354	0.466677	+	0.884428i	$0.345451\pi$
$398$	5.13690	0.257489
$399$	0	0
$400$	6.45852	0.322926
$401$	8.67718	0.433318	0.216659	−	0.976247i	$-0.430484\pi$
$401$	8.67718	0.433318	0.216659	+	0.976247i	$0.430484\pi$
$402$	−6.23792	−0.311119
$403$	0	0
$404$	−25.2780	−1.25763
$405$	24.2655	1.20576
$406$	0	0
$407$	−34.8787	−1.72887
$408$	0.432288	0.0214014
$409$	7.18020	0.355038	0.177519	−	0.984117i	$-0.443193\pi$
$409$	7.18020	0.355038	0.177519	+	0.984117i	$0.443193\pi$
$410$	2.05840	0.101657
$411$	−3.71943	−0.183466
$412$	32.3539	1.59396
$413$	0	0
$414$	0.0488340	0.00240006
$415$	19.5207	0.958235
$416$	0	0
$417$	−30.7832	−1.50746
$418$	10.3337	0.505438
$419$	−0.422635	−0.0206471	−0.0103235	−	0.999947i	$-0.503286\pi$
$419$	−0.422635	−0.0206471	−0.0103235	+	0.999947i	$0.503286\pi$
$420$	0	0
$421$	−30.1100	−1.46747	−0.733736	−	0.679434i	$-0.762225\pi$
$421$	−30.1100	−1.46747	−0.733736	+	0.679434i	$0.762225\pi$
$422$	−3.07127	−0.149507
$423$	−0.997339	−0.0484923
$424$	−8.51663	−0.413604
$425$	0.402198	0.0195095
$426$	−0.530523	−0.0257039
$427$	0	0
$428$	25.2300	1.21954
$429$	0	0
$430$	−0.274721	−0.0132482
$431$	32.8798	1.58376	0.791882	−	0.610674i	$-0.209101\pi$
$431$	32.8798	1.58376	0.791882	+	0.610674i	$0.209101\pi$
$432$	17.9553	0.863875
$433$	−25.4416	−1.22265	−0.611323	−	0.791381i	$-0.709362\pi$
$433$	−25.4416	−1.22265	−0.611323	+	0.791381i	$0.709362\pi$
$434$	0	0
$435$	42.4924	2.03736
$436$	−6.84306	−0.327723
$437$	10.7135	0.512494
$438$	3.03325	0.144934
$439$	8.74852	0.417544	0.208772	−	0.977964i	$-0.433053\pi$
$439$	8.74852	0.417544	0.208772	+	0.977964i	$0.433053\pi$
$440$	17.4915	0.833876
$441$	0	0
$442$	0	0
$443$	−20.6735	−0.982226	−0.491113	−	0.871096i	$-0.663410\pi$
$443$	−20.6735	−0.982226	−0.491113	+	0.871096i	$0.663410\pi$
$444$	−19.7545	−0.937509
$445$	−1.09356	−0.0518396
$446$	0.799171	0.0378418
$447$	2.24257	0.106070
$448$	0	0
$449$	14.2756	0.673709	0.336854	−	0.941557i	$-0.390637\pi$
$449$	14.2756	0.673709	0.336854	+	0.941557i	$0.390637\pi$
$450$	0.0507049	0.00239025
$451$	−16.3694	−0.770807
$452$	−33.9060	−1.59480
$453$	−7.52859	−0.353724
$454$	−5.99779	−0.281490
$455$	0	0
$456$	11.9561	0.559897
$457$	25.9786	1.21523	0.607613	−	0.794233i	$-0.292127\pi$
$457$	25.9786	1.21523	0.607613	+	0.794233i	$0.292127\pi$
$458$	3.94802	0.184479
$459$	1.11815	0.0521906
$460$	8.87715	0.413899
$461$	−5.01118	−0.233394	−0.116697	−	0.993168i	$-0.537231\pi$
$461$	−5.01118	−0.233394	−0.116697	+	0.993168i	$0.537231\pi$
$462$	0	0
$463$	0.202461	0.00940917	0.00470459	−	0.999989i	$-0.498502\pi$
$463$	0.202461	0.00940917	0.00470459	+	0.999989i	$0.498502\pi$
$464$	32.4525	1.50657
$465$	28.7739	1.33436
$466$	−5.84651	−0.270834
$467$	18.8880	0.874031	0.437015	−	0.899454i	$-0.356036\pi$
$467$	18.8880	0.874031	0.437015	+	0.899454i	$0.356036\pi$
$468$	0	0
$469$	0	0
$470$	7.76100	0.357988
$471$	−8.83603	−0.407143
$472$	2.74148	0.126187
$473$	2.18472	0.100453
$474$	−6.27049	−0.288013
$475$	11.1239	0.510400
$476$	0	0
$477$	0.730466	0.0334457
$478$	−1.01397	−0.0463777
$479$	2.03861	0.0931463	0.0465731	−	0.998915i	$-0.485170\pi$
$479$	2.03861	0.0931463	0.0465731	+	0.998915i	$0.485170\pi$
$480$	14.9640	0.683012
$481$	0	0
$482$	−7.42701	−0.338291
$483$	0	0
$484$	−46.9963	−2.13620
$485$	−34.8501	−1.58246
$486$	0.286603	0.0130006
$487$	20.0121	0.906835	0.453417	−	0.891298i	$-0.350205\pi$
$487$	20.0121	0.906835	0.453417	+	0.891298i	$0.350205\pi$
$488$	10.5281	0.476587
$489$	−19.3781	−0.876309
$490$	0	0
$491$	6.17892	0.278851	0.139425	−	0.990233i	$-0.455474\pi$
$491$	6.17892	0.278851	0.139425	+	0.990233i	$0.455474\pi$
$492$	−9.27130	−0.417983
$493$	2.02095	0.0910189
$494$	0	0
$495$	−1.50024	−0.0674306
$496$	21.9753	0.986722
$497$	0	0
$498$	3.76384	0.168662
$499$	20.7564	0.929184	0.464592	−	0.885525i	$-0.346201\pi$
$499$	20.7564	0.929184	0.464592	+	0.885525i	$0.346201\pi$
$500$	−15.8586	−0.709220
$501$	20.7950	0.929051
$502$	2.89529	0.129223
$503$	20.9063	0.932167	0.466083	−	0.884741i	$-0.345665\pi$
$503$	20.9063	0.932167	0.466083	+	0.884741i	$0.345665\pi$
$504$	0	0
$505$	−34.4649	−1.53367
$506$	3.02205	0.134346
$507$	0	0
$508$	40.5571	1.79943
$509$	0.607139	0.0269110	0.0134555	−	0.999909i	$-0.495717\pi$
$509$	0.607139	0.0269110	0.0134555	+	0.999909i	$0.495717\pi$
$510$	0.288523	0.0127760
$511$	0	0
$512$	19.3184	0.853761
$513$	30.9255	1.36539
$514$	−2.97253	−0.131113
$515$	44.1125	1.94383
$516$	1.23738	0.0544725
$517$	−61.7194	−2.71442
$518$	0	0
$519$	−19.3560	−0.849632
$520$	0	0
$521$	21.3481	0.935275	0.467638	−	0.883920i	$-0.345105\pi$
$521$	21.3481	0.935275	0.467638	+	0.883920i	$0.345105\pi$
$522$	0.254780	0.0111514
$523$	5.87130	0.256734	0.128367	−	0.991727i	$-0.459026\pi$
$523$	5.87130	0.256734	0.128367	+	0.991727i	$0.459026\pi$
$524$	−2.67130	−0.116696
$525$	0	0
$526$	1.01876	0.0444201
$527$	1.36849	0.0596124
$528$	−36.8449	−1.60347
$529$	−19.8669	−0.863778
$530$	−5.68428	−0.246909
$531$	−0.235135	−0.0102040
$532$	0	0
$533$	0	0
$534$	−0.210851	−0.00912443
$535$	34.3995	1.48722
$536$	−13.8891	−0.599916
$537$	−8.45106	−0.364690
$538$	−0.415434	−0.0179106
$539$	0	0
$540$	25.6248	1.10272
$541$	−1.34846	−0.0579748	−0.0289874	−	0.999580i	$-0.509228\pi$
$541$	−1.34846	−0.0579748	−0.0289874	+	0.999580i	$0.509228\pi$
$542$	−4.70395	−0.202052
$543$	6.30414	0.270536
$544$	0.711693	0.0305136
$545$	−9.33007	−0.399656
$546$	0	0
$547$	−22.0396	−0.942346	−0.471173	−	0.882041i	$-0.656169\pi$
$547$	−22.0396	−0.942346	−0.471173	+	0.882041i	$0.656169\pi$
$548$	−4.05398	−0.173177
$549$	−0.902992	−0.0385387
$550$	3.13783	0.133797
$551$	55.8949	2.38120
$552$	3.49652	0.148822
$553$	0	0
$554$	4.34294	0.184514
$555$	−26.9341	−1.14329
$556$	−33.5521	−1.42292
$557$	−44.9693	−1.90541	−0.952704	−	0.303899i	$-0.901712\pi$
$557$	−44.9693	−1.90541	−0.952704	+	0.303899i	$0.901712\pi$
$558$	0.172525	0.00730356
$559$	0	0
$560$	0	0
$561$	−2.29448	−0.0968730
$562$	−2.19819	−0.0927249
$563$	−8.51559	−0.358889	−0.179445	−	0.983768i	$-0.557430\pi$
$563$	−8.51559	−0.358889	−0.179445	+	0.983768i	$0.557430\pi$
$564$	−34.9566	−1.47194
$565$	−46.2287	−1.94485
$566$	−8.17136	−0.343468
$567$	0	0
$568$	−1.18124	−0.0495636
$569$	13.2852	0.556946	0.278473	−	0.960444i	$-0.410172\pi$
$569$	13.2852	0.556946	0.278473	+	0.960444i	$0.410172\pi$
$570$	7.97991	0.334241
$571$	−10.1106	−0.423114	−0.211557	−	0.977366i	$-0.567853\pi$
$571$	−10.1106	−0.423114	−0.211557	+	0.977366i	$0.567853\pi$
$572$	0	0
$573$	19.6764	0.821992
$574$	0	0
$575$	3.25314	0.135665
$576$	−0.586996	−0.0244582
$577$	21.6179	0.899963	0.449982	−	0.893038i	$-0.351431\pi$
$577$	21.6179	0.899963	0.449982	+	0.893038i	$0.351431\pi$
$578$	−4.85734	−0.202039
$579$	−13.0415	−0.541988
$580$	46.3145	1.92310
$581$	0	0
$582$	−6.71952	−0.278533
$583$	45.2042	1.87217
$584$	6.75369	0.279470
$585$	0	0
$586$	9.21976	0.380865
$587$	39.0725	1.61270	0.806348	−	0.591441i	$-0.201441\pi$
$587$	39.0725	1.61270	0.806348	+	0.591441i	$0.201441\pi$
$588$	0	0
$589$	37.8494	1.55956
$590$	1.82975	0.0753296
$591$	−14.9492	−0.614927
$592$	−20.5702	−0.845431
$593$	1.27127	0.0522048	0.0261024	−	0.999659i	$-0.491690\pi$
$593$	1.27127	0.0522048	0.0261024	+	0.999659i	$0.491690\pi$
$594$	8.72345	0.357928
$595$	0	0
$596$	2.44428	0.100121
$597$	−31.5462	−1.29110
$598$	0	0
$599$	25.6737	1.04900	0.524499	−	0.851411i	$-0.324252\pi$
$599$	25.6737	1.04900	0.524499	+	0.851411i	$0.324252\pi$
$600$	3.63048	0.148214
$601$	23.2625	0.948899	0.474450	−	0.880283i	$-0.342647\pi$
$601$	23.2625	0.948899	0.474450	+	0.880283i	$0.342647\pi$
$602$	0	0
$603$	1.19125	0.0485116
$604$	−8.20576	−0.333888
$605$	−64.0765	−2.60508
$606$	−6.64526	−0.269945
$607$	11.5447	0.468586	0.234293	−	0.972166i	$-0.424722\pi$
$607$	11.5447	0.468586	0.234293	+	0.972166i	$0.424722\pi$
$608$	19.6838	0.798286
$609$	0	0
$610$	7.02682	0.284508
$611$	0	0
$612$	−0.0404120	−0.00163356
$613$	−1.64984	−0.0666366	−0.0333183	−	0.999445i	$-0.510608\pi$
$613$	−1.64984	−0.0666366	−0.0333183	+	0.999445i	$0.510608\pi$
$614$	−0.657172	−0.0265213
$615$	−12.6408	−0.509728
$616$	0	0
$617$	−20.7349	−0.834757	−0.417379	−	0.908733i	$-0.637051\pi$
$617$	−20.7349	−0.834757	−0.417379	+	0.908733i	$0.637051\pi$
$618$	8.50544	0.342139
$619$	19.2997	0.775719	0.387860	−	0.921718i	$-0.373214\pi$
$619$	19.2997	0.775719	0.387860	+	0.921718i	$0.373214\pi$
$620$	31.3620	1.25953
$621$	9.04403	0.362924
$622$	−5.81429	−0.233132
$623$	0	0
$624$	0	0
$625$	−30.8116	−1.23246
$626$	−9.82329	−0.392617
$627$	−63.4602	−2.53436
$628$	−9.63080	−0.384311
$629$	−1.28099	−0.0510764
$630$	0	0
$631$	5.83037	0.232103	0.116052	−	0.993243i	$-0.462976\pi$
$631$	5.83037	0.232103	0.116052	+	0.993243i	$0.462976\pi$
$632$	−13.9616	−0.555362
$633$	18.8610	0.749656
$634$	4.89440	0.194381
$635$	55.2971	2.19440
$636$	25.6027	1.01521
$637$	0	0
$638$	15.7668	0.624215
$639$	0.101314	0.00400792
$640$	21.5760	0.852868
$641$	−26.8412	−1.06016	−0.530081	−	0.847947i	$-0.677839\pi$
$641$	−26.8412	−1.06016	−0.530081	+	0.847947i	$0.677839\pi$
$642$	6.63265	0.261770
$643$	16.6560	0.656849	0.328425	−	0.944530i	$-0.393482\pi$
$643$	16.6560	0.656849	0.328425	+	0.944530i	$0.393482\pi$
$644$	0	0
$645$	1.68708	0.0664289
$646$	0.379526	0.0149322
$647$	20.4062	0.802252	0.401126	−	0.916023i	$-0.368619\pi$
$647$	20.4062	0.802252	0.401126	+	0.916023i	$0.368619\pi$
$648$	10.4173	0.409231
$649$	−14.5511	−0.571180
$650$	0	0
$651$	0	0
$652$	−21.1211	−0.827167
$653$	0.387885	0.0151791	0.00758955	−	0.999971i	$-0.497584\pi$
$653$	0.387885	0.0151791	0.00758955	+	0.999971i	$0.497584\pi$
$654$	−1.79895	−0.0703446
$655$	−3.64215	−0.142310
$656$	−9.65413	−0.376930
$657$	−0.579259	−0.0225991
$658$	0	0
$659$	−37.7788	−1.47165	−0.735826	−	0.677171i	$-0.763206\pi$
$659$	−37.7788	−1.47165	−0.735826	+	0.677171i	$0.763206\pi$
$660$	−52.5830	−2.04679
$661$	17.4652	0.679318	0.339659	−	0.940549i	$-0.389688\pi$
$661$	17.4652	0.679318	0.339659	+	0.940549i	$0.389688\pi$
$662$	−4.73921	−0.184195
$663$	0	0
$664$	8.38038	0.325222
$665$	0	0
$666$	−0.161494	−0.00625775
$667$	16.3462	0.632929
$668$	22.6654	0.876951
$669$	−4.90778	−0.189746
$670$	−9.27001	−0.358132
$671$	−55.8809	−2.15726
$672$	0	0
$673$	−36.9671	−1.42498	−0.712489	−	0.701683i	$-0.752432\pi$
$673$	−36.9671	−1.42498	−0.712489	+	0.701683i	$0.752432\pi$
$674$	3.55322	0.136865
$675$	9.39052	0.361441
$676$	0	0
$677$	6.96519	0.267694	0.133847	−	0.991002i	$-0.457267\pi$
$677$	6.96519	0.267694	0.133847	+	0.991002i	$0.457267\pi$
$678$	−8.91345	−0.342319
$679$	0	0
$680$	0.642411	0.0246353
$681$	36.8330	1.41144
$682$	10.6766	0.408827
$683$	−6.03407	−0.230887	−0.115444	−	0.993314i	$-0.536829\pi$
$683$	−6.03407	−0.230887	−0.115444	+	0.993314i	$0.536829\pi$
$684$	−1.11771	−0.0427366
$685$	−5.52734	−0.211189
$686$	0	0
$687$	−24.2451	−0.925010
$688$	1.28847	0.0491224
$689$	0	0
$690$	2.33369	0.0888421
$691$	−40.7958	−1.55195	−0.775973	−	0.630767i	$-0.782741\pi$
$691$	−40.7958	−1.55195	−0.775973	+	0.630767i	$0.782741\pi$
$692$	−21.0970	−0.801986
$693$	0	0
$694$	4.95107	0.187940
$695$	−45.7461	−1.73525
$696$	18.2423	0.691471
$697$	−0.601200	−0.0227721
$698$	5.90378	0.223461
$699$	35.9039	1.35801
$700$	0	0
$701$	−23.2342	−0.877543	−0.438772	−	0.898599i	$-0.644586\pi$
$701$	−23.2342	−0.877543	−0.438772	+	0.898599i	$0.644586\pi$
$702$	0	0
$703$	−35.4293	−1.33624
$704$	−36.3257	−1.36908
$705$	−47.6611	−1.79502
$706$	−10.1942	−0.383664
$707$	0	0
$708$	−8.24143	−0.309732
$709$	41.1902	1.54693	0.773465	−	0.633839i	$-0.218522\pi$
$709$	41.1902	1.54693	0.773465	+	0.633839i	$0.218522\pi$
$710$	−0.788396	−0.0295880
$711$	1.19747	0.0449088
$712$	−0.469472	−0.0175942
$713$	11.0689	0.414534
$714$	0	0
$715$	0	0
$716$	−9.21120	−0.344239
$717$	6.22686	0.232546
$718$	5.00254	0.186693
$719$	−22.9454	−0.855718	−0.427859	−	0.903845i	$-0.640732\pi$
$719$	−22.9454	−0.855718	−0.427859	+	0.903845i	$0.640732\pi$
$720$	−0.884787	−0.0329741
$721$	0	0
$722$	5.05272	0.188043
$723$	45.6099	1.69625
$724$	6.87117	0.255365
$725$	16.9725	0.630342
$726$	−12.3547	−0.458527
$727$	−16.3256	−0.605484	−0.302742	−	0.953073i	$-0.597902\pi$
$727$	−16.3256	−0.605484	−0.302742	+	0.953073i	$0.597902\pi$
$728$	0	0
$729$	26.0787	0.965878
$730$	4.50763	0.166835
$731$	0.0802381	0.00296771
$732$	−31.6497	−1.16981
$733$	−13.2798	−0.490499	−0.245250	−	0.969460i	$-0.578870\pi$
$733$	−13.2798	−0.490499	−0.245250	+	0.969460i	$0.578870\pi$
$734$	3.30472	0.121980
$735$	0	0
$736$	5.75646	0.212186
$737$	73.7198	2.71550
$738$	−0.0757930	−0.00278998
$739$	44.8777	1.65085	0.825427	−	0.564509i	$-0.190935\pi$
$739$	44.8777	1.65085	0.825427	+	0.564509i	$0.190935\pi$
$740$	−29.3567	−1.07917
$741$	0	0
$742$	0	0
$743$	−7.32828	−0.268849	−0.134424	−	0.990924i	$-0.542919\pi$
$743$	−7.32828	−0.268849	−0.134424	+	0.990924i	$0.542919\pi$
$744$	12.3528	0.452876
$745$	3.33261	0.122098
$746$	0.955416	0.0349803
$747$	−0.718779	−0.0262988
$748$	−2.50086	−0.0914405
$749$	0	0
$750$	−4.16903	−0.152232
$751$	−8.33954	−0.304314	−0.152157	−	0.988356i	$-0.548622\pi$
$751$	−8.33954	−0.304314	−0.152157	+	0.988356i	$0.548622\pi$
$752$	−36.4000	−1.32737
$753$	−17.7803	−0.647948
$754$	0	0
$755$	−11.1880	−0.407174
$756$	0	0
$757$	−40.7984	−1.48284	−0.741422	−	0.671039i	$-0.765848\pi$
$757$	−40.7984	−1.48284	−0.741422	+	0.671039i	$0.765848\pi$
$758$	7.54843	0.274171
$759$	−18.5587	−0.673638
$760$	17.7677	0.644501
$761$	−28.5995	−1.03673	−0.518365	−	0.855160i	$-0.673459\pi$
$761$	−28.5995	−1.03673	−0.518365	+	0.855160i	$0.673459\pi$
$762$	10.6620	0.386242
$763$	0	0
$764$	21.4462	0.775896
$765$	−0.0550991	−0.00199211
$766$	3.68075	0.132991
$767$	0	0
$768$	−17.2947	−0.624070
$769$	15.1940	0.547911	0.273955	−	0.961742i	$-0.411668\pi$
$769$	15.1940	0.547911	0.273955	+	0.961742i	$0.411668\pi$
$770$	0	0
$771$	18.2546	0.657422
$772$	−14.2146	−0.511594
$773$	6.01128	0.216211	0.108105	−	0.994139i	$-0.465522\pi$
$773$	6.01128	0.216211	0.108105	+	0.994139i	$0.465522\pi$
$774$	0.0101156	0.000363597 0
$775$	11.4930	0.412840
$776$	−14.9614	−0.537081
$777$	0	0
$778$	−4.72264	−0.169315
$779$	−16.6279	−0.595755
$780$	0	0
$781$	6.26973	0.224348
$782$	0.110991	0.00396902
$783$	47.1851	1.68626
$784$	0	0
$785$	−13.1310	−0.468665
$786$	−0.702251	−0.0250485
$787$	−9.61614	−0.342779	−0.171389	−	0.985203i	$-0.554826\pi$
$787$	−9.61614	−0.342779	−0.171389	+	0.985203i	$0.554826\pi$
$788$	−16.2938	−0.580443
$789$	−6.25630	−0.222730
$790$	−9.31841	−0.331534
$791$	0	0
$792$	−0.644061	−0.0228857
$793$	0	0
$794$	5.32864	0.189106
$795$	34.9077	1.23805
$796$	−34.3836	−1.21870
$797$	4.53499	0.160638	0.0803188	−	0.996769i	$-0.474406\pi$
$797$	4.53499	0.160638	0.0803188	+	0.996769i	$0.474406\pi$
$798$	0	0
$799$	−2.26677	−0.0801925
$800$	5.97700	0.211319
$801$	0.0402663	0.00142274
$802$	2.48630	0.0877943
$803$	−35.8470	−1.26501
$804$	41.7533	1.47253
$805$	0	0
$806$	0	0
$807$	2.55122	0.0898072
$808$	−14.7960	−0.520522
$809$	17.6434	0.620308	0.310154	−	0.950686i	$-0.399619\pi$
$809$	17.6434	0.620308	0.310154	+	0.950686i	$0.399619\pi$
$810$	6.95286	0.244299
$811$	19.0412	0.668627	0.334314	−	0.942462i	$-0.391495\pi$
$811$	19.0412	0.668627	0.334314	+	0.942462i	$0.391495\pi$
$812$	0	0
$813$	28.8874	1.01313
$814$	−9.99389	−0.350286
$815$	−28.7973	−1.00873
$816$	−1.35320	−0.0473716
$817$	2.21921	0.0776403
$818$	2.05736	0.0719340
$819$	0	0
$820$	−13.7778	−0.481143
$821$	30.1089	1.05081	0.525404	−	0.850853i	$-0.323914\pi$
$821$	30.1089	1.05081	0.525404	+	0.850853i	$0.323914\pi$
$822$	−1.06574	−0.0371719
$823$	−19.2986	−0.672708	−0.336354	−	0.941736i	$-0.609194\pi$
$823$	−19.2986	−0.672708	−0.336354	+	0.941736i	$0.609194\pi$
$824$	18.9378	0.659729
$825$	−19.2697	−0.670884
$826$	0	0
$827$	−31.0627	−1.08016	−0.540078	−	0.841615i	$-0.681605\pi$
$827$	−31.0627	−1.08016	−0.540078	+	0.841615i	$0.681605\pi$
$828$	−0.326869	−0.0113595
$829$	−18.0469	−0.626794	−0.313397	−	0.949622i	$-0.601467\pi$
$829$	−18.0469	−0.626794	−0.313397	+	0.949622i	$0.601467\pi$
$830$	5.59334	0.194148
$831$	−26.6704	−0.925185
$832$	0	0
$833$	0	0
$834$	−8.82041	−0.305426
$835$	30.9028	1.06944
$836$	−69.1683	−2.39223
$837$	31.9515	1.10441
$838$	−0.121099	−0.00418329
$839$	41.9247	1.44740	0.723701	−	0.690114i	$-0.242440\pi$
$839$	41.9247	1.44740	0.723701	+	0.690114i	$0.242440\pi$
$840$	0	0
$841$	56.2827	1.94078
$842$	−8.62752	−0.297324
$843$	13.4993	0.464940
$844$	20.5574	0.707616
$845$	0	0
$846$	−0.285770	−0.00982499
$847$	0	0
$848$	26.6599	0.915504
$849$	50.1811	1.72221
$850$	0.115243	0.00395280
$851$	−10.3612	−0.355176
$852$	3.55104	0.121657
$853$	9.58374	0.328141	0.164071	−	0.986449i	$-0.447538\pi$
$853$	9.58374	0.328141	0.164071	+	0.986449i	$0.447538\pi$
$854$	0	0
$855$	−1.52392	−0.0521170
$856$	14.7679	0.504758
$857$	−25.1348	−0.858590	−0.429295	−	0.903164i	$-0.641238\pi$
$857$	−25.1348	−0.858590	−0.429295	+	0.903164i	$0.641238\pi$
$858$	0	0
$859$	37.7081	1.28658	0.643291	−	0.765622i	$-0.277568\pi$
$859$	37.7081	1.28658	0.643291	+	0.765622i	$0.277568\pi$
$860$	1.83883	0.0627037
$861$	0	0
$862$	9.42115	0.320886
$863$	3.76048	0.128008	0.0640040	−	0.997950i	$-0.479613\pi$
$863$	3.76048	0.128008	0.0640040	+	0.997950i	$0.479613\pi$
$864$	16.6166	0.565309
$865$	−28.7643	−0.978017
$866$	−7.28986	−0.247720
$867$	29.8294	1.01306
$868$	0	0
$869$	74.1047	2.51383
$870$	12.1755	0.412787
$871$	0	0
$872$	−4.00546	−0.135642
$873$	1.28323	0.0434306
$874$	3.06976	0.103836
$875$	0	0
$876$	−20.3030	−0.685974
$877$	9.32654	0.314935	0.157468	−	0.987524i	$-0.449667\pi$
$877$	9.32654	0.314935	0.157468	+	0.987524i	$0.449667\pi$
$878$	2.50674	0.0845984
$879$	−56.6194	−1.90973
$880$	−54.7543	−1.84577
$881$	40.8381	1.37587	0.687936	−	0.725771i	$-0.258517\pi$
$881$	40.8381	1.37587	0.687936	+	0.725771i	$0.258517\pi$
$882$	0	0
$883$	−32.5428	−1.09515	−0.547577	−	0.836755i	$-0.684450\pi$
$883$	−32.5428	−1.09515	−0.547577	+	0.836755i	$0.684450\pi$
$884$	0	0
$885$	−11.2367	−0.377716
$886$	−5.92363	−0.199008
$887$	40.6605	1.36525	0.682623	−	0.730770i	$-0.260839\pi$
$887$	40.6605	1.36525	0.682623	+	0.730770i	$0.260839\pi$
$888$	−11.5630	−0.388028
$889$	0	0
$890$	−0.313341	−0.0105032
$891$	−55.2927	−1.85237
$892$	−5.34922	−0.179105
$893$	−62.6938	−2.09797
$894$	0.642569	0.0214907
$895$	−12.5589	−0.419797
$896$	0	0
$897$	0	0
$898$	4.09044	0.136500
$899$	57.7494	1.92605
$900$	−0.339391	−0.0113130
$901$	1.66022	0.0553098
$902$	−4.69039	−0.156173
$903$	0	0
$904$	−19.8463	−0.660077
$905$	9.36841	0.311416
$906$	−2.15719	−0.0716679
$907$	−20.3000	−0.674050	−0.337025	−	0.941496i	$-0.609421\pi$
$907$	−20.3000	−0.674050	−0.337025	+	0.941496i	$0.609421\pi$
$908$	40.1460	1.33229
$909$	1.26904	0.0420916
$910$	0	0
$911$	−25.0324	−0.829361	−0.414681	−	0.909967i	$-0.636107\pi$
$911$	−25.0324	−0.829361	−0.414681	+	0.909967i	$0.636107\pi$
$912$	−37.4266	−1.23932
$913$	−44.4810	−1.47211
$914$	7.44372	0.246216
$915$	−43.1524	−1.42657
$916$	−26.4259	−0.873136
$917$	0	0
$918$	0.320386	0.0105743
$919$	−9.37526	−0.309261	−0.154631	−	0.987972i	$-0.549419\pi$
$919$	−9.37526	−0.309261	−0.154631	+	0.987972i	$0.549419\pi$
$920$	5.19608	0.171310
$921$	4.03576	0.132983
$922$	−1.43587	−0.0472878
$923$	0	0
$924$	0	0
$925$	−10.7581	−0.353724
$926$	0.0580118	0.00190639
$927$	−1.62428	−0.0533484
$928$	30.0330	0.985881
$929$	37.3013	1.22382	0.611908	−	0.790929i	$-0.290402\pi$
$929$	37.3013	1.22382	0.611908	+	0.790929i	$0.290402\pi$
$930$	8.24467	0.270353
$931$	0	0
$932$	39.1334	1.28186
$933$	35.7061	1.16897
$934$	5.41202	0.177087
$935$	−3.40976	−0.111511
$936$	0	0
$937$	−23.6563	−0.772818	−0.386409	−	0.922327i	$-0.626285\pi$
$937$	−23.6563	−0.772818	−0.386409	+	0.922327i	$0.626285\pi$
$938$	0	0
$939$	60.3257	1.96866
$940$	−51.9480	−1.69436
$941$	30.1491	0.982832	0.491416	−	0.870925i	$-0.336479\pi$
$941$	30.1491	0.982832	0.491416	+	0.870925i	$0.336479\pi$
$942$	−2.53182	−0.0824910
$943$	−4.86275	−0.158353
$944$	−8.58173	−0.279311
$945$	0	0
$946$	0.625994	0.0203528
$947$	−23.4974	−0.763562	−0.381781	−	0.924253i	$-0.624689\pi$
$947$	−23.4974	−0.763562	−0.381781	+	0.924253i	$0.624689\pi$
$948$	41.9713	1.36317
$949$	0	0
$950$	3.18737	0.103412
$951$	−30.0569	−0.974663
$952$	0	0
$953$	−23.2344	−0.752637	−0.376318	−	0.926490i	$-0.622810\pi$
$953$	−23.2344	−0.752637	−0.376318	+	0.926490i	$0.622810\pi$
$954$	0.209303	0.00677642
$955$	29.2405	0.946201
$956$	6.78694	0.219505
$957$	−96.8256	−3.12993
$958$	0.584128	0.0188723
$959$	0	0
$960$	−28.0515	−0.905358
$961$	8.10523	0.261459
$962$	0	0
$963$	−1.26664	−0.0408168
$964$	49.7124	1.60113
$965$	−19.3807	−0.623886
$966$	0	0
$967$	−28.4927	−0.916263	−0.458132	−	0.888884i	$-0.651481\pi$
$967$	−28.4927	−0.916263	−0.458132	+	0.888884i	$0.651481\pi$
$968$	−27.5085	−0.884156
$969$	−2.33070	−0.0748730
$970$	−9.98569	−0.320621
$971$	7.41150	0.237846	0.118923	−	0.992903i	$-0.462056\pi$
$971$	7.41150	0.237846	0.118923	+	0.992903i	$0.462056\pi$
$972$	−1.91837	−0.0615316
$973$	0	0
$974$	5.73413	0.183733
$975$	0	0
$976$	−32.9566	−1.05491
$977$	14.8730	0.475829	0.237915	−	0.971286i	$-0.423536\pi$
$977$	14.8730	0.475829	0.237915	+	0.971286i	$0.423536\pi$
$978$	−5.55248	−0.177549
$979$	2.49184	0.0796396
$980$	0	0
$981$	0.343545	0.0109686
$982$	1.77047	0.0564978
$983$	−10.9972	−0.350757	−0.175379	−	0.984501i	$-0.556115\pi$
$983$	−10.9972	−0.350757	−0.175379	+	0.984501i	$0.556115\pi$
$984$	−5.42679	−0.173000
$985$	−22.2156	−0.707846
$986$	0.579068	0.0184413
$987$	0	0
$988$	0	0
$989$	0.648998	0.0206369
$990$	−0.429867	−0.0136621
$991$	−16.8726	−0.535977	−0.267988	−	0.963422i	$-0.586359\pi$
$991$	−16.8726	−0.535977	−0.267988	+	0.963422i	$0.586359\pi$
$992$	20.3369	0.645698
$993$	29.1039	0.923585
$994$	0	0
$995$	−46.8799	−1.48619
$996$	−25.1931	−0.798274
$997$	−0.833613	−0.0264008	−0.0132004	−	0.999913i	$-0.504202\pi$
$997$	−0.833613	−0.0264008	−0.0132004	+	0.999913i	$0.504202\pi$
$998$	5.94739	0.188261
$999$	−29.9085	−0.946264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cz.1.17	yes	36
7.6	odd	2	inner	8281.2.a.cz.1.18	yes	36
13.12	even	2		8281.2.a.cy.1.19	✓	36
91.90	odd	2		8281.2.a.cy.1.20	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8281.2.a.cy.1.19	✓	36	13.12	even	2
8281.2.a.cy.1.20	yes	36	91.90	odd	2
8281.2.a.cz.1.17	yes	36	1.1	even	1	trivial
8281.2.a.cz.1.18	yes	36	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+0.286533 q^{2} -1.75963 q^{3} -1.91790 q^{4} -2.61493 q^{5} -0.504191 q^{6} -1.12261 q^{8} +0.0962853 q^{9} +O(q^{10})\)	\(q+0.286533 q^{2} -1.75963 q^{3} -1.91790 q^{4} -2.61493 q^{5} -0.504191 q^{6} -1.12261 q^{8} +0.0962853 q^{9} -0.749265 q^{10} +5.95853 q^{11} +3.37479 q^{12} +4.60130 q^{15} +3.51413 q^{16} +0.218839 q^{17} +0.0275889 q^{18} +6.05260 q^{19} +5.01518 q^{20} +1.70732 q^{22} +1.77006 q^{23} +1.97537 q^{24} +1.83787 q^{25} +5.10945 q^{27} +9.23486 q^{29} +1.31843 q^{30} +6.25342 q^{31} +3.25213 q^{32} -10.4848 q^{33} +0.0627046 q^{34} -0.184665 q^{36} -5.85357 q^{37} +1.73427 q^{38} +2.93554 q^{40} -2.74723 q^{41} +0.366654 q^{43} -11.4279 q^{44} -0.251779 q^{45} +0.507180 q^{46} -10.3582 q^{47} -6.18356 q^{48} +0.526611 q^{50} -0.385075 q^{51} +7.58648 q^{53} +1.46403 q^{54} -15.5812 q^{55} -10.6503 q^{57} +2.64609 q^{58} -2.44206 q^{59} -8.82484 q^{60} -9.37829 q^{61} +1.79181 q^{62} -6.09642 q^{64} -3.00424 q^{66} +12.3721 q^{67} -0.419711 q^{68} -3.11464 q^{69} +1.05223 q^{71} -0.108091 q^{72} -6.01608 q^{73} -1.67724 q^{74} -3.23397 q^{75} -11.6083 q^{76} +12.4367 q^{79} -9.18922 q^{80} -9.27958 q^{81} -0.787172 q^{82} -7.46510 q^{83} -0.572249 q^{85} +0.105058 q^{86} -16.2499 q^{87} -6.68909 q^{88} +0.418197 q^{89} -0.0721431 q^{90} -3.39479 q^{92} -11.0037 q^{93} -2.96796 q^{94} -15.8271 q^{95} -5.72253 q^{96} +13.3273 q^{97} +0.573719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 14 q^{2} + 38 q^{4} + 42 q^{8} + 36 q^{9}+O(q^{10}) \) 36 * q + 14 * q^2 + 38 * q^4 + 42 * q^8 + 36 * q^9	\( 36 q + 14 q^{2} + 38 q^{4} + 42 q^{8} + 36 q^{9} + 52 q^{11} + 44 q^{15} + 42 q^{16} + 46 q^{18} + 68 q^{22} - 8 q^{23} + 32 q^{25} - 8 q^{29} + 118 q^{32} + 66 q^{36} + 28 q^{37} - 16 q^{43} + 130 q^{44} + 48 q^{46} + 90 q^{50} + 4 q^{51} + 52 q^{57} - 34 q^{58} + 100 q^{60} + 62 q^{64} + 40 q^{67} + 188 q^{71} - 42 q^{72} + 96 q^{74} + 8 q^{79} - 4 q^{81} + 64 q^{85} + 58 q^{86} + 156 q^{88} - 76 q^{92} - 16 q^{93} - 64 q^{95} + 156 q^{99}+O(q^{100}) \) 36 * q + 14 * q^2 + 38 * q^4 + 42 * q^8 + 36 * q^9 + 52 * q^11 + 44 * q^15 + 42 * q^16 + 46 * q^18 + 68 * q^22 - 8 * q^23 + 32 * q^25 - 8 * q^29 + 118 * q^32 + 66 * q^36 + 28 * q^37 - 16 * q^43 + 130 * q^44 + 48 * q^46 + 90 * q^50 + 4 * q^51 + 52 * q^57 - 34 * q^58 + 100 * q^60 + 62 * q^64 + 40 * q^67 + 188 * q^71 - 42 * q^72 + 96 * q^74 + 8 * q^79 - 4 * q^81 + 64 * q^85 + 58 * q^86 + 156 * q^88 - 76 * q^92 - 16 * q^93 - 64 * q^95 + 156 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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