LMFDB - Embedded newform 3969.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.574857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 3x - 3 $ x^5 - 2x^4 - 5x^3 + 9x^2 + 3x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.38687$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38687 q^{2} +3.69714 q^{4} -2.92087 q^{5} -4.05086 q^{8} +O(q^{10})$	$q-2.38687 q^{2} +3.69714 q^{4} -2.92087 q^{5} -4.05086 q^{8} +6.97172 q^{10} -1.35371 q^{11} +1.46600 q^{13} +2.27458 q^{16} -3.31027 q^{17} +2.20659 q^{19} -10.7989 q^{20} +3.23114 q^{22} +2.62830 q^{23} +3.53146 q^{25} -3.49916 q^{26} -1.04344 q^{29} +3.27458 q^{31} +2.67259 q^{32} +7.90119 q^{34} -10.8755 q^{37} -5.26683 q^{38} +11.8320 q^{40} +1.80858 q^{41} +4.34257 q^{43} -5.00488 q^{44} -6.27340 q^{46} +3.97914 q^{47} -8.42913 q^{50} +5.42002 q^{52} +6.45486 q^{53} +3.95402 q^{55} +2.49056 q^{58} -12.2140 q^{59} +0.559734 q^{61} -7.81600 q^{62} -10.9283 q^{64} -4.28200 q^{65} +12.8118 q^{67} -12.2386 q^{68} +12.9177 q^{71} -10.4554 q^{73} +25.9583 q^{74} +8.15807 q^{76} +0.767677 q^{79} -6.64375 q^{80} -4.31684 q^{82} +1.96741 q^{83} +9.66887 q^{85} -10.3652 q^{86} +5.48371 q^{88} -6.40711 q^{89} +9.71719 q^{92} -9.49769 q^{94} -6.44514 q^{95} +8.28285 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^8	$ 5 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{8} + 7 q^{10} - 4 q^{11} + 8 q^{13} - 2 q^{16} - 12 q^{17} - q^{19} - 5 q^{20} + q^{22} - 3 q^{23} + q^{25} - 11 q^{26} - 7 q^{29} + 3 q^{31} + 2 q^{32} - 3 q^{34} - 20 q^{38} + 3 q^{40} - 5 q^{41} + 7 q^{43} + 10 q^{44} - 3 q^{46} - 27 q^{47} - 19 q^{50} + 10 q^{52} + 21 q^{53} + 2 q^{55} + 10 q^{58} - 30 q^{59} + 14 q^{61} - 6 q^{62} - 25 q^{64} + 11 q^{65} + 2 q^{67} - 27 q^{68} - 3 q^{71} - 15 q^{73} + 36 q^{74} - 5 q^{76} + 4 q^{79} - 20 q^{80} + 5 q^{82} - 9 q^{83} + 6 q^{85} + 8 q^{86} + 18 q^{88} - 28 q^{89} - 27 q^{92} + 3 q^{94} + 14 q^{95} + 12 q^{97}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^8 + 7 * q^10 - 4 * q^11 + 8 * q^13 - 2 * q^16 - 12 * q^17 - q^19 - 5 * q^20 + q^22 - 3 * q^23 + q^25 - 11 * q^26 - 7 * q^29 + 3 * q^31 + 2 * q^32 - 3 * q^34 - 20 * q^38 + 3 * q^40 - 5 * q^41 + 7 * q^43 + 10 * q^44 - 3 * q^46 - 27 * q^47 - 19 * q^50 + 10 * q^52 + 21 * q^53 + 2 * q^55 + 10 * q^58 - 30 * q^59 + 14 * q^61 - 6 * q^62 - 25 * q^64 + 11 * q^65 + 2 * q^67 - 27 * q^68 - 3 * q^71 - 15 * q^73 + 36 * q^74 - 5 * q^76 + 4 * q^79 - 20 * q^80 + 5 * q^82 - 9 * q^83 + 6 * q^85 + 8 * q^86 + 18 * q^88 - 28 * q^89 - 27 * q^92 + 3 * q^94 + 14 * q^95 + 12 * q^97

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 $

$2$	−2.38687	−1.68777	−0.843886	−	0.536523i	$-0.819737\pi$
$2$	−2.38687	−1.68777	−0.843886	+	0.536523i	$0.819737\pi$
$3$	0	0
$3$	0	0
$4$	3.69714	1.84857
$5$	−2.92087	−1.30625	−0.653125	−	0.757250i	$-0.726543\pi$
$5$	−2.92087	−1.30625	−0.653125	+	0.757250i	$0.726543\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−4.05086	−1.43219
$9$	0	0
$10$	6.97172	2.20465
$11$	−1.35371	−0.408160	−0.204080	−	0.978954i	$-0.565420\pi$
$11$	−1.35371	−0.408160	−0.204080	+	0.978954i	$0.565420\pi$
$12$	0	0
$13$	1.46600	0.406596	0.203298	−	0.979117i	$-0.434834\pi$
$13$	1.46600	0.406596	0.203298	+	0.979117i	$0.434834\pi$
$14$	0	0
$15$	0	0
$16$	2.27458	0.568645
$17$	−3.31027	−0.802859	−0.401430	−	0.915890i	$-0.631487\pi$
$17$	−3.31027	−0.802859	−0.401430	+	0.915890i	$0.631487\pi$
$18$	0	0
$19$	2.20659	0.506226	0.253113	−	0.967437i	$-0.418546\pi$
$19$	2.20659	0.506226	0.253113	+	0.967437i	$0.418546\pi$
$20$	−10.7989	−2.41470
$21$	0	0
$22$	3.23114	0.688881
$23$	2.62830	0.548038	0.274019	−	0.961724i	$-0.411647\pi$
$23$	2.62830	0.548038	0.274019	+	0.961724i	$0.411647\pi$
$24$	0	0
$25$	3.53146	0.706292
$26$	−3.49916	−0.686241
$27$	0	0
$28$	0	0
$29$	−1.04344	−0.193762	−0.0968810	−	0.995296i	$-0.530887\pi$
$29$	−1.04344	−0.193762	−0.0968810	+	0.995296i	$0.530887\pi$
$30$	0	0
$31$	3.27458	0.588132	0.294066	−	0.955785i	$-0.404991\pi$
$31$	3.27458	0.588132	0.294066	+	0.955785i	$0.404991\pi$
$32$	2.67259	0.472452
$33$	0	0
$34$	7.90119	1.35504
$35$	0	0
$36$	0	0
$37$	−10.8755	−1.78791	−0.893957	−	0.448153i	$-0.852082\pi$
$37$	−10.8755	−1.78791	−0.893957	+	0.448153i	$0.852082\pi$
$38$	−5.26683	−0.854393
$39$	0	0
$40$	11.8320	1.87081
$41$	1.80858	0.282452	0.141226	−	0.989977i	$-0.454896\pi$
$41$	1.80858	0.282452	0.141226	+	0.989977i	$0.454896\pi$
$42$	0	0
$43$	4.34257	0.662236	0.331118	−	0.943589i	$-0.392574\pi$
$43$	4.34257	0.662236	0.331118	+	0.943589i	$0.392574\pi$
$44$	−5.00488	−0.754514
$45$	0	0
$46$	−6.27340	−0.924962
$47$	3.97914	0.580417	0.290209	−	0.956963i	$-0.406275\pi$
$47$	3.97914	0.580417	0.290209	+	0.956963i	$0.406275\pi$
$48$	0	0
$49$	0	0
$50$	−8.42913	−1.19206
$51$	0	0
$52$	5.42002	0.751622
$53$	6.45486	0.886644	0.443322	−	0.896363i	$-0.353800\pi$
$53$	6.45486	0.886644	0.443322	+	0.896363i	$0.353800\pi$
$54$	0	0
$55$	3.95402	0.533160
$56$	0	0
$57$	0	0
$58$	2.49056	0.327026
$59$	−12.2140	−1.59013	−0.795064	−	0.606526i	$-0.792563\pi$
$59$	−12.2140	−1.59013	−0.795064	+	0.606526i	$0.792563\pi$
$60$	0	0
$61$	0.559734	0.0716666	0.0358333	−	0.999358i	$-0.488591\pi$
$61$	0.559734	0.0716666	0.0358333	+	0.999358i	$0.488591\pi$
$62$	−7.81600	−0.992632
$63$	0	0
$64$	−10.9283	−1.36604
$65$	−4.28200	−0.531117
$66$	0	0
$67$	12.8118	1.56521	0.782603	−	0.622521i	$-0.213891\pi$
$67$	12.8118	1.56521	0.782603	+	0.622521i	$0.213891\pi$
$68$	−12.2386	−1.48414
$69$	0	0
$70$	0	0
$71$	12.9177	1.53305	0.766525	−	0.642214i	$-0.221984\pi$
$71$	12.9177	1.53305	0.766525	+	0.642214i	$0.221984\pi$
$72$	0	0
$73$	−10.4554	−1.22372	−0.611858	−	0.790968i	$-0.709578\pi$
$73$	−10.4554	−1.22372	−0.611858	+	0.790968i	$0.709578\pi$
$74$	25.9583	3.01759
$75$	0	0
$76$	8.15807	0.935794
$77$	0	0
$78$	0	0
$79$	0.767677	0.0863704	0.0431852	−	0.999067i	$-0.486249\pi$
$79$	0.767677	0.0863704	0.0431852	+	0.999067i	$0.486249\pi$
$80$	−6.64375	−0.742793
$81$	0	0
$82$	−4.31684	−0.476715
$83$	1.96741	0.215952	0.107976	−	0.994154i	$-0.465563\pi$
$83$	1.96741	0.215952	0.107976	+	0.994154i	$0.465563\pi$
$84$	0	0
$85$	9.66887	1.04874
$86$	−10.3652	−1.11770
$87$	0	0
$88$	5.48371	0.584565
$89$	−6.40711	−0.679153	−0.339576	−	0.940579i	$-0.610284\pi$
$89$	−6.40711	−0.679153	−0.339576	+	0.940579i	$0.610284\pi$
$90$	0	0
$91$	0	0
$92$	9.71719	1.01309
$93$	0	0
$94$	−9.49769	−0.979612
$95$	−6.44514	−0.661258
$96$	0	0
$97$	8.28285	0.840996	0.420498	−	0.907293i	$-0.361855\pi$
$97$	8.28285	0.840996	0.420498	+	0.907293i	$0.361855\pi$
$98$	0	0
$99$	0	0
$100$	13.0563	1.30563
$101$	−16.2266	−1.61461	−0.807305	−	0.590134i	$-0.799075\pi$
$101$	−16.2266	−1.61461	−0.807305	+	0.590134i	$0.799075\pi$
$102$	0	0
$103$	−2.22683	−0.219416	−0.109708	−	0.993964i	$-0.534992\pi$
$103$	−2.22683	−0.219416	−0.109708	+	0.993964i	$0.534992\pi$
$104$	−5.93857	−0.582325
$105$	0	0
$106$	−15.4069	−1.49645
$107$	17.5081	1.69257	0.846284	−	0.532732i	$-0.178835\pi$
$107$	17.5081	1.69257	0.846284	+	0.532732i	$0.178835\pi$
$108$	0	0
$109$	15.5983	1.49405	0.747025	−	0.664796i	$-0.231482\pi$
$109$	15.5983	1.49405	0.747025	+	0.664796i	$0.231482\pi$
$110$	−9.43773	−0.899852
$111$	0	0
$112$	0	0
$113$	1.68911	0.158898	0.0794491	−	0.996839i	$-0.474684\pi$
$113$	1.68911	0.158898	0.0794491	+	0.996839i	$0.474684\pi$
$114$	0	0
$115$	−7.67690	−0.715875
$116$	−3.85775	−0.358183
$117$	0	0
$118$	29.1532	2.68377
$119$	0	0
$120$	0	0
$121$	−9.16746	−0.833405
$122$	−1.33601	−0.120957
$123$	0	0
$124$	12.1066	1.08720
$125$	4.28942	0.383657
$126$	0	0
$127$	−3.96918	−0.352208	−0.176104	−	0.984372i	$-0.556350\pi$
$127$	−3.96918	−0.352208	−0.176104	+	0.984372i	$0.556350\pi$
$128$	20.7392	1.83310
$129$	0	0
$130$	10.2206	0.896403
$131$	5.32863	0.465565	0.232782	−	0.972529i	$-0.425217\pi$
$131$	5.32863	0.465565	0.232782	+	0.972529i	$0.425217\pi$
$132$	0	0
$133$	0	0
$134$	−30.5800	−2.64171
$135$	0	0
$136$	13.4095	1.14985
$137$	−7.49543	−0.640378	−0.320189	−	0.947354i	$-0.603746\pi$
$137$	−7.49543	−0.640378	−0.320189	+	0.947354i	$0.603746\pi$
$138$	0	0
$139$	−14.0657	−1.19304	−0.596518	−	0.802600i	$-0.703450\pi$
$139$	−14.0657	−1.19304	−0.596518	+	0.802600i	$0.703450\pi$
$140$	0	0
$141$	0	0
$142$	−30.8329	−2.58744
$143$	−1.98455	−0.165956
$144$	0	0
$145$	3.04775	0.253102
$146$	24.9557	2.06535
$147$	0	0
$148$	−40.2081	−3.30509
$149$	2.17971	0.178569	0.0892846	−	0.996006i	$-0.471542\pi$
$149$	2.17971	0.178569	0.0892846	+	0.996006i	$0.471542\pi$
$150$	0	0
$151$	14.0277	1.14156	0.570781	−	0.821102i	$-0.306641\pi$
$151$	14.0277	1.14156	0.570781	+	0.821102i	$0.306641\pi$
$152$	−8.93857	−0.725014
$153$	0	0
$154$	0	0
$155$	−9.56461	−0.768248
$156$	0	0
$157$	2.96623	0.236731	0.118365	−	0.992970i	$-0.462235\pi$
$157$	2.96623	0.236731	0.118365	+	0.992970i	$0.462235\pi$
$158$	−1.83234	−0.145773
$159$	0	0
$160$	−7.80628	−0.617140
$161$	0	0
$162$	0	0
$163$	0.388555	0.0304340	0.0152170	−	0.999884i	$-0.495156\pi$
$163$	0.388555	0.0304340	0.0152170	+	0.999884i	$0.495156\pi$
$164$	6.68657	0.522133
$165$	0	0
$166$	−4.69596	−0.364477
$167$	−7.29778	−0.564719	−0.282360	−	0.959309i	$-0.591117\pi$
$167$	−7.29778	−0.564719	−0.282360	+	0.959309i	$0.591117\pi$
$168$	0	0
$169$	−10.8508	−0.834680
$170$	−23.0783	−1.77003
$171$	0	0
$172$	16.0551	1.22419
$173$	−4.05508	−0.308302	−0.154151	−	0.988047i	$-0.549264\pi$
$173$	−4.05508	−0.308302	−0.154151	+	0.988047i	$0.549264\pi$
$174$	0	0
$175$	0	0
$176$	−3.07913	−0.232098
$177$	0	0
$178$	15.2929	1.14625
$179$	−10.5849	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$179$	−10.5849	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$180$	0	0
$181$	−19.6312	−1.45917	−0.729586	−	0.683889i	$-0.760287\pi$
$181$	−19.6312	−1.45917	−0.729586	+	0.683889i	$0.760287\pi$
$182$	0	0
$183$	0	0
$184$	−10.6469	−0.784896
$185$	31.7657	2.33546
$186$	0	0
$187$	4.48117	0.327695
$188$	14.7115	1.07294
$189$	0	0
$190$	15.3837	1.11605
$191$	8.28714	0.599637	0.299818	−	0.953996i	$-0.403074\pi$
$191$	8.28714	0.599637	0.299818	+	0.953996i	$0.403074\pi$
$192$	0	0
$193$	−18.7848	−1.35216	−0.676082	−	0.736827i	$-0.736323\pi$
$193$	−18.7848	−1.35216	−0.676082	+	0.736827i	$0.736323\pi$
$194$	−19.7701	−1.41941
$195$	0	0
$196$	0	0
$197$	5.99634	0.427222	0.213611	−	0.976919i	$-0.431478\pi$
$197$	5.99634	0.427222	0.213611	+	0.976919i	$0.431478\pi$
$198$	0	0
$199$	−14.4087	−1.02140	−0.510702	−	0.859758i	$-0.670615\pi$
$199$	−14.4087	−1.02140	−0.510702	+	0.859758i	$0.670615\pi$
$200$	−14.3054	−1.01155
$201$	0	0
$202$	38.7308	2.72509
$203$	0	0
$204$	0	0
$205$	−5.28261	−0.368954
$206$	5.31515	0.370324
$207$	0	0
$208$	3.33454	0.231209
$209$	−2.98709	−0.206621
$210$	0	0
$211$	13.8484	0.953360	0.476680	−	0.879077i	$-0.341840\pi$
$211$	13.8484	0.953360	0.476680	+	0.879077i	$0.341840\pi$
$212$	23.8646	1.63902
$213$	0	0
$214$	−41.7894	−2.85667
$215$	−12.6841	−0.865047
$216$	0	0
$217$	0	0
$218$	−37.2312	−2.52161
$219$	0	0
$220$	14.6186	0.985584
$221$	−4.85287	−0.326439
$222$	0	0
$223$	−4.67513	−0.313070	−0.156535	−	0.987672i	$-0.550032\pi$
$223$	−4.67513	−0.313070	−0.156535	+	0.987672i	$0.550032\pi$
$224$	0	0
$225$	0	0
$226$	−4.03169	−0.268184
$227$	19.7126	1.30837	0.654187	−	0.756333i	$-0.273011\pi$
$227$	19.7126	1.30837	0.654187	+	0.756333i	$0.273011\pi$
$228$	0	0
$229$	28.0728	1.85510	0.927552	−	0.373694i	$-0.121909\pi$
$229$	28.0728	1.85510	0.927552	+	0.373694i	$0.121909\pi$
$230$	18.3238	1.20823
$231$	0	0
$232$	4.22683	0.277505
$233$	13.8023	0.904216	0.452108	−	0.891963i	$-0.350672\pi$
$233$	13.8023	0.904216	0.452108	+	0.891963i	$0.350672\pi$
$234$	0	0
$235$	−11.6225	−0.758171
$236$	−45.1569	−2.93947
$237$	0	0
$238$	0	0
$239$	−11.0614	−0.715501	−0.357751	−	0.933817i	$-0.616456\pi$
$239$	−11.0614	−0.715501	−0.357751	+	0.933817i	$0.616456\pi$
$240$	0	0
$241$	−23.1697	−1.49249	−0.746247	−	0.665669i	$-0.768146\pi$
$241$	−23.1697	−1.49249	−0.746247	+	0.665669i	$0.768146\pi$
$242$	21.8815	1.40660
$243$	0	0
$244$	2.06942	0.132481
$245$	0	0
$246$	0	0
$247$	3.23486	0.205829
$248$	−13.2649	−0.842320
$249$	0	0
$250$	−10.2383	−0.647525
$251$	−7.78402	−0.491323	−0.245662	−	0.969356i	$-0.579005\pi$
$251$	−7.78402	−0.491323	−0.245662	+	0.969356i	$0.579005\pi$
$252$	0	0
$253$	−3.55796	−0.223687
$254$	9.47392	0.594447
$255$	0	0
$256$	−27.6452	−1.72782
$257$	10.3760	0.647235	0.323618	−	0.946188i	$-0.395101\pi$
$257$	10.3760	0.647235	0.323618	+	0.946188i	$0.395101\pi$
$258$	0	0
$259$	0	0
$260$	−15.8312	−0.981807
$261$	0	0
$262$	−12.7187	−0.785767
$263$	−19.1331	−1.17980	−0.589898	−	0.807478i	$-0.700832\pi$
$263$	−19.1331	−1.17980	−0.589898	+	0.807478i	$0.700832\pi$
$264$	0	0
$265$	−18.8538	−1.15818
$266$	0	0
$267$	0	0
$268$	47.3669	2.89340
$269$	8.83681	0.538790	0.269395	−	0.963030i	$-0.413176\pi$
$269$	8.83681	0.538790	0.269395	+	0.963030i	$0.413176\pi$
$270$	0	0
$271$	18.3391	1.11402	0.557010	−	0.830506i	$-0.311948\pi$
$271$	18.3391	1.11402	0.557010	+	0.830506i	$0.311948\pi$
$272$	−7.52949	−0.456542
$273$	0	0
$274$	17.8906	1.08081
$275$	−4.78059	−0.288280
$276$	0	0
$277$	5.10482	0.306719	0.153360	−	0.988170i	$-0.450991\pi$
$277$	5.10482	0.306719	0.153360	+	0.988170i	$0.450991\pi$
$278$	33.5730	2.01357
$279$	0	0
$280$	0	0
$281$	1.70636	0.101793	0.0508964	−	0.998704i	$-0.483792\pi$
$281$	1.70636	0.101793	0.0508964	+	0.998704i	$0.483792\pi$
$282$	0	0
$283$	−12.4883	−0.742352	−0.371176	−	0.928562i	$-0.621045\pi$
$283$	−12.4883	−0.742352	−0.371176	+	0.928562i	$0.621045\pi$
$284$	47.7586	2.83395
$285$	0	0
$286$	4.73686	0.280096
$287$	0	0
$288$	0	0
$289$	−6.04208	−0.355417
$290$	−7.27458	−0.427178
$291$	0	0
$292$	−38.6552	−2.26213
$293$	5.20405	0.304024	0.152012	−	0.988379i	$-0.451425\pi$
$293$	5.20405	0.304024	0.152012	+	0.988379i	$0.451425\pi$
$294$	0	0
$295$	35.6755	2.07711
$296$	44.0549	2.56064
$297$	0	0
$298$	−5.20269	−0.301384
$299$	3.85309	0.222830
$300$	0	0
$301$	0	0
$302$	−33.4824	−1.92669
$303$	0	0
$304$	5.01906	0.287863
$305$	−1.63491	−0.0936145
$306$	0	0
$307$	5.00136	0.285442	0.142721	−	0.989763i	$-0.454415\pi$
$307$	5.00136	0.285442	0.142721	+	0.989763i	$0.454415\pi$
$308$	0	0
$309$	0	0
$310$	22.8295	1.29663
$311$	−32.3968	−1.83706	−0.918528	−	0.395355i	$-0.870621\pi$
$311$	−32.3968	−1.83706	−0.918528	+	0.395355i	$0.870621\pi$
$312$	0	0
$313$	1.51907	0.0858629	0.0429315	−	0.999078i	$-0.486330\pi$
$313$	1.51907	0.0858629	0.0429315	+	0.999078i	$0.486330\pi$
$314$	−7.08000	−0.399548
$315$	0	0
$316$	2.83821	0.159662
$317$	−21.5089	−1.20806	−0.604029	−	0.796962i	$-0.706439\pi$
$317$	−21.5089	−1.20806	−0.604029	+	0.796962i	$0.706439\pi$
$318$	0	0
$319$	1.41252	0.0790860
$320$	31.9200	1.78439
$321$	0	0
$322$	0	0
$323$	−7.30441	−0.406428
$324$	0	0
$325$	5.17713	0.287175
$326$	−0.927430	−0.0513656
$327$	0	0
$328$	−7.32629	−0.404527
$329$	0	0
$330$	0	0
$331$	19.4780	1.07061	0.535305	−	0.844659i	$-0.320197\pi$
$331$	19.4780	1.07061	0.535305	+	0.844659i	$0.320197\pi$
$332$	7.27381	0.399202
$333$	0	0
$334$	17.4188	0.953116
$335$	−37.4215	−2.04455
$336$	0	0
$337$	−9.69484	−0.528112	−0.264056	−	0.964507i	$-0.585060\pi$
$337$	−9.69484	−0.528112	−0.264056	+	0.964507i	$0.585060\pi$
$338$	25.8995	1.40875
$339$	0	0
$340$	35.7472	1.93866
$341$	−4.43285	−0.240052
$342$	0	0
$343$	0	0
$344$	−17.5912	−0.948451
$345$	0	0
$346$	9.67895	0.520344
$347$	2.02604	0.108763	0.0543817	−	0.998520i	$-0.482681\pi$
$347$	2.02604	0.108763	0.0543817	+	0.998520i	$0.482681\pi$
$348$	0	0
$349$	−16.2915	−0.872066	−0.436033	−	0.899931i	$-0.643617\pi$
$349$	−16.2915	−0.872066	−0.436033	+	0.899931i	$0.643617\pi$
$350$	0	0
$351$	0	0
$352$	−3.61792	−0.192836
$353$	17.0614	0.908089	0.454045	−	0.890979i	$-0.349981\pi$
$353$	17.0614	0.908089	0.454045	+	0.890979i	$0.349981\pi$
$354$	0	0
$355$	−37.7309	−2.00255
$356$	−23.6880	−1.25546
$357$	0	0
$358$	25.2647	1.33528
$359$	−2.96726	−0.156606	−0.0783030	−	0.996930i	$-0.524950\pi$
$359$	−2.96726	−0.156606	−0.0783030	+	0.996930i	$0.524950\pi$
$360$	0	0
$361$	−14.1310	−0.743736
$362$	46.8570	2.46275
$363$	0	0
$364$	0	0
$365$	30.5389	1.59848
$366$	0	0
$367$	−10.1575	−0.530216	−0.265108	−	0.964219i	$-0.585408\pi$
$367$	−10.1575	−0.530216	−0.265108	+	0.964219i	$0.585408\pi$
$368$	5.97827	0.311639
$369$	0	0
$370$	−75.8207	−3.94173
$371$	0	0
$372$	0	0
$373$	−25.4846	−1.31954	−0.659771	−	0.751467i	$-0.729347\pi$
$373$	−25.4846	−1.31954	−0.659771	+	0.751467i	$0.729347\pi$
$374$	−10.6960	−0.553075
$375$	0	0
$376$	−16.1189	−0.831271
$377$	−1.52969	−0.0787829
$378$	0	0
$379$	9.85497	0.506216	0.253108	−	0.967438i	$-0.418547\pi$
$379$	9.85497	0.506216	0.253108	+	0.967438i	$0.418547\pi$
$380$	−23.8286	−1.22238
$381$	0	0
$382$	−19.7803	−1.01205
$383$	−27.3127	−1.39561	−0.697806	−	0.716286i	$-0.745840\pi$
$383$	−27.3127	−1.39561	−0.697806	+	0.716286i	$0.745840\pi$
$384$	0	0
$385$	0	0
$386$	44.8370	2.28214
$387$	0	0
$388$	30.6229	1.55464
$389$	4.18446	0.212161	0.106080	−	0.994358i	$-0.466170\pi$
$389$	4.18446	0.212161	0.106080	+	0.994358i	$0.466170\pi$
$390$	0	0
$391$	−8.70038	−0.439997
$392$	0	0
$393$	0	0
$394$	−14.3125	−0.721053
$395$	−2.24228	−0.112821
$396$	0	0
$397$	−30.6709	−1.53933	−0.769664	−	0.638450i	$-0.779576\pi$
$397$	−30.6709	−1.53933	−0.769664	+	0.638450i	$0.779576\pi$
$398$	34.3916	1.72390
$399$	0	0
$400$	8.03259	0.401629
$401$	−6.84803	−0.341974	−0.170987	−	0.985273i	$-0.554696\pi$
$401$	−6.84803	−0.341974	−0.170987	+	0.985273i	$0.554696\pi$
$402$	0	0
$403$	4.80055	0.239132
$404$	−59.9922	−2.98472
$405$	0	0
$406$	0	0
$407$	14.7223	0.729756
$408$	0	0
$409$	−18.2698	−0.903384	−0.451692	−	0.892174i	$-0.649179\pi$
$409$	−18.2698	−0.903384	−0.451692	+	0.892174i	$0.649179\pi$
$410$	12.6089	0.622709
$411$	0	0
$412$	−8.23291	−0.405606
$413$	0	0
$414$	0	0
$415$	−5.74655	−0.282087
$416$	3.91802	0.192097
$417$	0	0
$418$	7.12979	0.348729
$419$	−22.4619	−1.09734	−0.548669	−	0.836040i	$-0.684865\pi$
$419$	−22.4619	−1.09734	−0.548669	+	0.836040i	$0.684865\pi$
$420$	0	0
$421$	−20.8354	−1.01546	−0.507728	−	0.861517i	$-0.669515\pi$
$421$	−20.8354	−1.01546	−0.507728	+	0.861517i	$0.669515\pi$
$422$	−33.0542	−1.60905
$423$	0	0
$424$	−26.1477	−1.26985
$425$	−11.6901	−0.567053
$426$	0	0
$427$	0	0
$428$	64.7298	3.12883
$429$	0	0
$430$	30.2752	1.46000
$431$	20.2427	0.975055	0.487527	−	0.873108i	$-0.337899\pi$
$431$	20.2427	0.975055	0.487527	+	0.873108i	$0.337899\pi$
$432$	0	0
$433$	−21.6764	−1.04170	−0.520851	−	0.853648i	$-0.674385\pi$
$433$	−21.6764	−1.04170	−0.520851	+	0.853648i	$0.674385\pi$
$434$	0	0
$435$	0	0
$436$	57.6693	2.76186
$437$	5.79956	0.277431
$438$	0	0
$439$	−35.4781	−1.69328	−0.846639	−	0.532168i	$-0.821377\pi$
$439$	−35.4781	−1.69328	−0.846639	+	0.532168i	$0.821377\pi$
$440$	−16.0172	−0.763589
$441$	0	0
$442$	11.5832	0.550955
$443$	−19.2063	−0.912517	−0.456258	−	0.889847i	$-0.650811\pi$
$443$	−19.2063	−0.912517	−0.456258	+	0.889847i	$0.650811\pi$
$444$	0	0
$445$	18.7143	0.887144
$446$	11.1589	0.528390
$447$	0	0
$448$	0	0
$449$	−29.6082	−1.39730	−0.698648	−	0.715465i	$-0.746215\pi$
$449$	−29.6082	−1.39730	−0.698648	+	0.715465i	$0.746215\pi$
$450$	0	0
$451$	−2.44830	−0.115286
$452$	6.24488	0.293735
$453$	0	0
$454$	−47.0515	−2.20823
$455$	0	0
$456$	0	0
$457$	−9.56196	−0.447290	−0.223645	−	0.974671i	$-0.571796\pi$
$457$	−9.56196	−0.447290	−0.223645	+	0.974671i	$0.571796\pi$
$458$	−67.0062	−3.13099
$459$	0	0
$460$	−28.3826	−1.32335
$461$	−21.8374	−1.01707	−0.508536	−	0.861041i	$-0.669813\pi$
$461$	−21.8374	−1.01707	−0.508536	+	0.861041i	$0.669813\pi$
$462$	0	0
$463$	−26.1489	−1.21524	−0.607621	−	0.794227i	$-0.707876\pi$
$463$	−26.1489	−1.21524	−0.607621	+	0.794227i	$0.707876\pi$
$464$	−2.37339	−0.110182
$465$	0	0
$466$	−32.9442	−1.52611
$467$	34.9527	1.61742	0.808709	−	0.588209i	$-0.200167\pi$
$467$	34.9527	1.61742	0.808709	+	0.588209i	$0.200167\pi$
$468$	0	0
$469$	0	0
$470$	27.7415	1.27962
$471$	0	0
$472$	49.4772	2.27737
$473$	−5.87861	−0.270299
$474$	0	0
$475$	7.79247	0.357543
$476$	0	0
$477$	0	0
$478$	26.4021	1.20760
$479$	−29.8109	−1.36209	−0.681047	−	0.732240i	$-0.738475\pi$
$479$	−29.8109	−1.36209	−0.681047	+	0.732240i	$0.738475\pi$
$480$	0	0
$481$	−15.9434	−0.726959
$482$	55.3031	2.51899
$483$	0	0
$484$	−33.8934	−1.54061
$485$	−24.1931	−1.09855
$486$	0	0
$487$	22.4506	1.01733	0.508667	−	0.860964i	$-0.330139\pi$
$487$	22.4506	1.01733	0.508667	+	0.860964i	$0.330139\pi$
$488$	−2.26740	−0.102640
$489$	0	0
$490$	0	0
$491$	−35.0444	−1.58153	−0.790767	−	0.612118i	$-0.790318\pi$
$491$	−35.0444	−1.58153	−0.790767	+	0.612118i	$0.790318\pi$
$492$	0	0
$493$	3.45407	0.155564
$494$	−7.72119	−0.347393
$495$	0	0
$496$	7.44830	0.334438
$497$	0	0
$498$	0	0
$499$	−8.93520	−0.399994	−0.199997	−	0.979796i	$-0.564093\pi$
$499$	−8.93520	−0.399994	−0.199997	+	0.979796i	$0.564093\pi$
$500$	15.8586	0.709217
$501$	0	0
$502$	18.5794	0.829241
$503$	−12.6403	−0.563603	−0.281802	−	0.959473i	$-0.590932\pi$
$503$	−12.6403	−0.563603	−0.281802	+	0.959473i	$0.590932\pi$
$504$	0	0
$505$	47.3958	2.10909
$506$	8.49239	0.377533
$507$	0	0
$508$	−14.6746	−0.651082
$509$	−28.1110	−1.24600	−0.623000	−	0.782222i	$-0.714086\pi$
$509$	−28.1110	−1.24600	−0.623000	+	0.782222i	$0.714086\pi$
$510$	0	0
$511$	0	0
$512$	24.5070	1.08307
$513$	0	0
$514$	−24.7661	−1.09238
$515$	6.50427	0.286613
$516$	0	0
$517$	−5.38662	−0.236903
$518$	0	0
$519$	0	0
$520$	17.3458	0.760662
$521$	−8.47536	−0.371312	−0.185656	−	0.982615i	$-0.559441\pi$
$521$	−8.47536	−0.371312	−0.185656	+	0.982615i	$0.559441\pi$
$522$	0	0
$523$	−33.4473	−1.46255	−0.731273	−	0.682085i	$-0.761074\pi$
$523$	−33.4473	−1.46255	−0.731273	+	0.682085i	$0.761074\pi$
$524$	19.7007	0.860630
$525$	0	0
$526$	45.6681	1.99123
$527$	−10.8398	−0.472187
$528$	0	0
$529$	−16.0921	−0.699655
$530$	45.0015	1.95474
$531$	0	0
$532$	0	0
$533$	2.65138	0.114844
$534$	0	0
$535$	−51.1387	−2.21092
$536$	−51.8987	−2.24168
$537$	0	0
$538$	−21.0923	−0.909354
$539$	0	0
$540$	0	0
$541$	18.2586	0.784998	0.392499	−	0.919752i	$-0.371611\pi$
$541$	18.2586	0.784998	0.392499	+	0.919752i	$0.371611\pi$
$542$	−43.7730	−1.88021
$543$	0	0
$544$	−8.84701	−0.379312
$545$	−45.5606	−1.95160
$546$	0	0
$547$	5.77199	0.246792	0.123396	−	0.992357i	$-0.460621\pi$
$547$	5.77199	0.246792	0.123396	+	0.992357i	$0.460621\pi$
$548$	−27.7117	−1.18378
$549$	0	0
$550$	11.4106	0.486551
$551$	−2.30244	−0.0980874
$552$	0	0
$553$	0	0
$554$	−12.1845	−0.517672
$555$	0	0
$556$	−52.0029	−2.20541
$557$	−33.3821	−1.41445	−0.707223	−	0.706991i	$-0.750052\pi$
$557$	−33.3821	−1.41445	−0.707223	+	0.706991i	$0.750052\pi$
$558$	0	0
$559$	6.36623	0.269263
$560$	0	0
$561$	0	0
$562$	−4.07286	−0.171803
$563$	2.19131	0.0923528	0.0461764	−	0.998933i	$-0.485296\pi$
$563$	2.19131	0.0923528	0.0461764	+	0.998933i	$0.485296\pi$
$564$	0	0
$565$	−4.93367	−0.207561
$566$	29.8079	1.25292
$567$	0	0
$568$	−52.3278	−2.19563
$569$	18.9860	0.795936	0.397968	−	0.917399i	$-0.369716\pi$
$569$	18.9860	0.795936	0.397968	+	0.917399i	$0.369716\pi$
$570$	0	0
$571$	−21.7380	−0.909709	−0.454854	−	0.890566i	$-0.650309\pi$
$571$	−21.7380	−0.909709	−0.454854	+	0.890566i	$0.650309\pi$
$572$	−7.33717	−0.306782
$573$	0	0
$574$	0	0
$575$	9.28172	0.387074
$576$	0	0
$577$	30.9032	1.28652	0.643258	−	0.765649i	$-0.277582\pi$
$577$	30.9032	1.28652	0.643258	+	0.765649i	$0.277582\pi$
$578$	14.4217	0.599862
$579$	0	0
$580$	11.2680	0.467877
$581$	0	0
$582$	0	0
$583$	−8.73804	−0.361893
$584$	42.3535	1.75260
$585$	0	0
$586$	−12.4214	−0.513122
$587$	18.3666	0.758072	0.379036	−	0.925382i	$-0.376256\pi$
$587$	18.3666	0.758072	0.379036	+	0.925382i	$0.376256\pi$
$588$	0	0
$589$	7.22565	0.297728
$590$	−85.1527	−3.50568
$591$	0	0
$592$	−24.7371	−1.01669
$593$	−27.7550	−1.13976	−0.569880	−	0.821728i	$-0.693010\pi$
$593$	−27.7550	−1.13976	−0.569880	+	0.821728i	$0.693010\pi$
$594$	0	0
$595$	0	0
$596$	8.05871	0.330098
$597$	0	0
$598$	−9.19682	−0.376086
$599$	0.402823	0.0164589	0.00822945	−	0.999966i	$-0.497380\pi$
$599$	0.402823	0.0164589	0.00822945	+	0.999966i	$0.497380\pi$
$600$	0	0
$601$	−24.7466	−1.00943	−0.504717	−	0.863285i	$-0.668403\pi$
$601$	−24.7466	−1.00943	−0.504717	+	0.863285i	$0.668403\pi$
$602$	0	0
$603$	0	0
$604$	51.8626	2.11026
$605$	26.7769	1.08864
$606$	0	0
$607$	24.0697	0.976957	0.488479	−	0.872576i	$-0.337552\pi$
$607$	24.0697	0.976957	0.488479	+	0.872576i	$0.337552\pi$
$608$	5.89730	0.239167
$609$	0	0
$610$	3.90231	0.158000
$611$	5.83343	0.235995
$612$	0	0
$613$	−20.3815	−0.823200	−0.411600	−	0.911365i	$-0.635030\pi$
$613$	−20.3815	−0.823200	−0.411600	+	0.911365i	$0.635030\pi$
$614$	−11.9376	−0.481762
$615$	0	0
$616$	0	0
$617$	41.8629	1.68534	0.842669	−	0.538431i	$-0.180983\pi$
$617$	41.8629	1.68534	0.842669	+	0.538431i	$0.180983\pi$
$618$	0	0
$619$	14.8219	0.595743	0.297871	−	0.954606i	$-0.403723\pi$
$619$	14.8219	0.595743	0.297871	+	0.954606i	$0.403723\pi$
$620$	−35.3617	−1.42016
$621$	0	0
$622$	77.3270	3.10053
$623$	0	0
$624$	0	0
$625$	−30.1861	−1.20744
$626$	−3.62582	−0.144917
$627$	0	0
$628$	10.9666	0.437614
$629$	36.0007	1.43544
$630$	0	0
$631$	−21.0294	−0.837169	−0.418585	−	0.908178i	$-0.637474\pi$
$631$	−21.0294	−0.837169	−0.418585	+	0.908178i	$0.637474\pi$
$632$	−3.10975	−0.123699
$633$	0	0
$634$	51.3388	2.03893
$635$	11.5935	0.460072
$636$	0	0
$637$	0	0
$638$	−3.37150	−0.133479
$639$	0	0
$640$	−60.5764	−2.39449
$641$	11.9318	0.471279	0.235640	−	0.971840i	$-0.424281\pi$
$641$	11.9318	0.471279	0.235640	+	0.971840i	$0.424281\pi$
$642$	0	0
$643$	39.9355	1.57490	0.787452	−	0.616377i	$-0.211400\pi$
$643$	39.9355	1.57490	0.787452	+	0.616377i	$0.211400\pi$
$644$	0	0
$645$	0	0
$646$	17.4347	0.685958
$647$	−0.988954	−0.0388798	−0.0194399	−	0.999811i	$-0.506188\pi$
$647$	−0.988954	−0.0388798	−0.0194399	+	0.999811i	$0.506188\pi$
$648$	0	0
$649$	16.5343	0.649027
$650$	−12.3571	−0.484686
$651$	0	0
$652$	1.43654	0.0562594
$653$	22.7147	0.888894	0.444447	−	0.895805i	$-0.353400\pi$
$653$	22.7147	0.888894	0.444447	+	0.895805i	$0.353400\pi$
$654$	0	0
$655$	−15.5642	−0.608144
$656$	4.11376	0.160615
$657$	0	0
$658$	0	0
$659$	38.3885	1.49540	0.747702	−	0.664035i	$-0.231157\pi$
$659$	38.3885	1.49540	0.747702	+	0.664035i	$0.231157\pi$
$660$	0	0
$661$	33.9258	1.31956	0.659780	−	0.751459i	$-0.270649\pi$
$661$	33.9258	1.31956	0.659780	+	0.751459i	$0.270649\pi$
$662$	−46.4915	−1.80694
$663$	0	0
$664$	−7.96972	−0.309285
$665$	0	0
$666$	0	0
$667$	−2.74247	−0.106189
$668$	−26.9809	−1.04392
$669$	0	0
$670$	89.3201	3.45074
$671$	−0.757720	−0.0292514
$672$	0	0
$673$	32.2060	1.24145	0.620725	−	0.784028i	$-0.286838\pi$
$673$	32.2060	1.24145	0.620725	+	0.784028i	$0.286838\pi$
$674$	23.1403	0.891332
$675$	0	0
$676$	−40.1171	−1.54297
$677$	−37.9684	−1.45924	−0.729622	−	0.683850i	$-0.760304\pi$
$677$	−37.9684	−1.45924	−0.729622	+	0.683850i	$0.760304\pi$
$678$	0	0
$679$	0	0
$680$	−39.1672	−1.50199
$681$	0	0
$682$	10.5806	0.405153
$683$	−15.1871	−0.581120	−0.290560	−	0.956857i	$-0.593842\pi$
$683$	−15.1871	−0.581120	−0.290560	+	0.956857i	$0.593842\pi$
$684$	0	0
$685$	21.8932	0.836495
$686$	0	0
$687$	0	0
$688$	9.87754	0.376578
$689$	9.46285	0.360506
$690$	0	0
$691$	2.69148	0.102389	0.0511943	−	0.998689i	$-0.483697\pi$
$691$	2.69148	0.102389	0.0511943	+	0.998689i	$0.483697\pi$
$692$	−14.9922	−0.569919
$693$	0	0
$694$	−4.83589	−0.183568
$695$	41.0840	1.55841
$696$	0	0
$697$	−5.98689	−0.226770
$698$	38.8858	1.47185
$699$	0	0
$700$	0	0
$701$	−11.8515	−0.447625	−0.223813	−	0.974632i	$-0.571850\pi$
$701$	−11.8515	−0.447625	−0.223813	+	0.974632i	$0.571850\pi$
$702$	0	0
$703$	−23.9976	−0.905088
$704$	14.7938	0.557562
$705$	0	0
$706$	−40.7234	−1.53265
$707$	0	0
$708$	0	0
$709$	−41.0333	−1.54104	−0.770520	−	0.637416i	$-0.780003\pi$
$709$	−41.0333	−1.54104	−0.770520	+	0.637416i	$0.780003\pi$
$710$	90.0587	3.37984
$711$	0	0
$712$	25.9543	0.972679
$713$	8.60657	0.322318
$714$	0	0
$715$	5.79660	0.216781
$716$	−39.1337	−1.46250
$717$	0	0
$718$	7.08246	0.264315
$719$	−20.9109	−0.779845	−0.389923	−	0.920848i	$-0.627498\pi$
$719$	−20.9109	−0.779845	−0.389923	+	0.920848i	$0.627498\pi$
$720$	0	0
$721$	0	0
$722$	33.7288	1.25526
$723$	0	0
$724$	−72.5792	−2.69738
$725$	−3.68487	−0.136853
$726$	0	0
$727$	−2.64330	−0.0980347	−0.0490173	−	0.998798i	$-0.515609\pi$
$727$	−2.64330	−0.0980347	−0.0490173	+	0.998798i	$0.515609\pi$
$728$	0	0
$729$	0	0
$730$	−72.8924	−2.69787
$731$	−14.3751	−0.531683
$732$	0	0
$733$	14.1489	0.522602	0.261301	−	0.965257i	$-0.415848\pi$
$733$	14.1489	0.522602	0.261301	+	0.965257i	$0.415848\pi$
$734$	24.2446	0.894884
$735$	0	0
$736$	7.02436	0.258921
$737$	−17.3435	−0.638855
$738$	0	0
$739$	15.7181	0.578200	0.289100	−	0.957299i	$-0.406644\pi$
$739$	15.7181	0.578200	0.289100	+	0.957299i	$0.406644\pi$
$740$	117.442	4.31727
$741$	0	0
$742$	0	0
$743$	−21.0991	−0.774051	−0.387026	−	0.922069i	$-0.626497\pi$
$743$	−21.0991	−0.774051	−0.387026	+	0.922069i	$0.626497\pi$
$744$	0	0
$745$	−6.36665	−0.233256
$746$	60.8283	2.22708
$747$	0	0
$748$	16.5675	0.605768
$749$	0	0
$750$	0	0
$751$	13.0370	0.475725	0.237863	−	0.971299i	$-0.423553\pi$
$751$	13.0370	0.475725	0.237863	+	0.971299i	$0.423553\pi$
$752$	9.05088	0.330052
$753$	0	0
$754$	3.65116	0.132968
$755$	−40.9732	−1.49117
$756$	0	0
$757$	−12.6856	−0.461065	−0.230532	−	0.973065i	$-0.574047\pi$
$757$	−12.6856	−0.461065	−0.230532	+	0.973065i	$0.574047\pi$
$758$	−23.5225	−0.854377
$759$	0	0
$760$	26.1084	0.947050
$761$	−6.04077	−0.218978	−0.109489	−	0.993988i	$-0.534921\pi$
$761$	−6.04077	−0.218978	−0.109489	+	0.993988i	$0.534921\pi$
$762$	0	0
$763$	0	0
$764$	30.6388	1.10847
$765$	0	0
$766$	65.1918	2.35547
$767$	−17.9058	−0.646540
$768$	0	0
$769$	−0.216258	−0.00779848	−0.00389924	−	0.999992i	$-0.501241\pi$
$769$	−0.216258	−0.00779848	−0.00389924	+	0.999992i	$0.501241\pi$
$770$	0	0
$771$	0	0
$772$	−69.4503	−2.49957
$773$	−37.6264	−1.35333	−0.676663	−	0.736293i	$-0.736575\pi$
$773$	−37.6264	−1.35333	−0.676663	+	0.736293i	$0.736575\pi$
$774$	0	0
$775$	11.5640	0.415393
$776$	−33.5527	−1.20447
$777$	0	0
$778$	−9.98776	−0.358078
$779$	3.99078	0.142985
$780$	0	0
$781$	−17.4869	−0.625730
$782$	20.7667	0.742614
$783$	0	0
$784$	0	0
$785$	−8.66396	−0.309230
$786$	0	0
$787$	30.8135	1.09838	0.549191	−	0.835697i	$-0.314936\pi$
$787$	30.8135	1.09838	0.549191	+	0.835697i	$0.314936\pi$
$788$	22.1693	0.789750
$789$	0	0
$790$	5.35203	0.190417
$791$	0	0
$792$	0	0
$793$	0.820571	0.0291393
$794$	73.2074	2.59803
$795$	0	0
$796$	−53.2710	−1.88814
$797$	35.9583	1.27371	0.636855	−	0.770984i	$-0.280235\pi$
$797$	35.9583	1.27371	0.636855	+	0.770984i	$0.280235\pi$
$798$	0	0
$799$	−13.1720	−0.465994
$800$	9.43814	0.333689
$801$	0	0
$802$	16.3454	0.577174
$803$	14.1537	0.499472
$804$	0	0
$805$	0	0
$806$	−11.4583	−0.403600
$807$	0	0
$808$	65.7318	2.31244
$809$	38.9636	1.36989	0.684943	−	0.728596i	$-0.259827\pi$
$809$	38.9636	1.36989	0.684943	+	0.728596i	$0.259827\pi$
$810$	0	0
$811$	−28.2811	−0.993082	−0.496541	−	0.868013i	$-0.665397\pi$
$811$	−28.2811	−0.993082	−0.496541	+	0.868013i	$0.665397\pi$
$812$	0	0
$813$	0	0
$814$	−35.1401	−1.23166
$815$	−1.13492	−0.0397544
$816$	0	0
$817$	9.58227	0.335241
$818$	43.6076	1.52471
$819$	0	0
$820$	−19.5306	−0.682037
$821$	41.5834	1.45127	0.725635	−	0.688080i	$-0.241546\pi$
$821$	41.5834	1.45127	0.725635	+	0.688080i	$0.241546\pi$
$822$	0	0
$823$	8.45998	0.294896	0.147448	−	0.989070i	$-0.452894\pi$
$823$	8.45998	0.294896	0.147448	+	0.989070i	$0.452894\pi$
$824$	9.02057	0.314247
$825$	0	0
$826$	0	0
$827$	44.2823	1.53985	0.769923	−	0.638137i	$-0.220294\pi$
$827$	44.2823	1.53985	0.769923	+	0.638137i	$0.220294\pi$
$828$	0	0
$829$	16.6327	0.577679	0.288839	−	0.957378i	$-0.406731\pi$
$829$	16.6327	0.577679	0.288839	+	0.957378i	$0.406731\pi$
$830$	13.7163	0.476099
$831$	0	0
$832$	−16.0209	−0.555425
$833$	0	0
$834$	0	0
$835$	21.3158	0.737665
$836$	−11.0437	−0.381954
$837$	0	0
$838$	53.6137	1.85205
$839$	−29.6012	−1.02195	−0.510974	−	0.859596i	$-0.670715\pi$
$839$	−29.6012	−1.02195	−0.510974	+	0.859596i	$0.670715\pi$
$840$	0	0
$841$	−27.9112	−0.962456
$842$	49.7314	1.71386
$843$	0	0
$844$	51.1994	1.76236
$845$	31.6938	1.09030
$846$	0	0
$847$	0	0
$848$	14.6821	0.504186
$849$	0	0
$850$	27.9027	0.957055
$851$	−28.5839	−0.979844
$852$	0	0
$853$	30.1238	1.03142	0.515710	−	0.856763i	$-0.327528\pi$
$853$	30.1238	1.03142	0.515710	+	0.856763i	$0.327528\pi$
$854$	0	0
$855$	0	0
$856$	−70.9227	−2.42409
$857$	37.0894	1.26695	0.633475	−	0.773763i	$-0.281628\pi$
$857$	37.0894	1.26695	0.633475	+	0.773763i	$0.281628\pi$
$858$	0	0
$859$	−3.78333	−0.129085	−0.0645427	−	0.997915i	$-0.520559\pi$
$859$	−3.78333	−0.129085	−0.0645427	+	0.997915i	$0.520559\pi$
$860$	−46.8949	−1.59910
$861$	0	0
$862$	−48.3166	−1.64567
$863$	−0.427118	−0.0145393	−0.00726963	−	0.999974i	$-0.502314\pi$
$863$	−0.427118	−0.0145393	−0.00726963	+	0.999974i	$0.502314\pi$
$864$	0	0
$865$	11.8444	0.402720
$866$	51.7388	1.75815
$867$	0	0
$868$	0	0
$869$	−1.03922	−0.0352530
$870$	0	0
$871$	18.7821	0.636407
$872$	−63.1866	−2.13977
$873$	0	0
$874$	−13.8428	−0.468240
$875$	0	0
$876$	0	0
$877$	11.2608	0.380249	0.190124	−	0.981760i	$-0.439111\pi$
$877$	11.2608	0.380249	0.190124	+	0.981760i	$0.439111\pi$
$878$	84.6816	2.85786
$879$	0	0
$880$	8.99374	0.303179
$881$	−35.4810	−1.19538	−0.597692	−	0.801726i	$-0.703916\pi$
$881$	−35.4810	−1.19538	−0.597692	+	0.801726i	$0.703916\pi$
$882$	0	0
$883$	−5.30092	−0.178390	−0.0891952	−	0.996014i	$-0.528429\pi$
$883$	−5.30092	−0.178390	−0.0891952	+	0.996014i	$0.528429\pi$
$884$	−17.9418	−0.603447
$885$	0	0
$886$	45.8428	1.54012
$887$	57.5664	1.93289	0.966446	−	0.256870i	$-0.0826913\pi$
$887$	57.5664	1.93289	0.966446	+	0.256870i	$0.0826913\pi$
$888$	0	0
$889$	0	0
$890$	−44.6686	−1.49730
$891$	0	0
$892$	−17.2846	−0.578732
$893$	8.78032	0.293822
$894$	0	0
$895$	30.9169	1.03344
$896$	0	0
$897$	0	0
$898$	70.6708	2.35832
$899$	−3.41683	−0.113958
$900$	0	0
$901$	−21.3674	−0.711850
$902$	5.84377	0.194576
$903$	0	0
$904$	−6.84235	−0.227573
$905$	57.3400	1.90605
$906$	0	0
$907$	20.8972	0.693879	0.346939	−	0.937888i	$-0.387221\pi$
$907$	20.8972	0.693879	0.346939	+	0.937888i	$0.387221\pi$
$908$	72.8804	2.41862
$909$	0	0
$910$	0	0
$911$	−22.7639	−0.754201	−0.377101	−	0.926172i	$-0.623079\pi$
$911$	−22.7639	−0.754201	−0.377101	+	0.926172i	$0.623079\pi$
$912$	0	0
$913$	−2.66332	−0.0881430
$914$	22.8231	0.754923
$915$	0	0
$916$	103.789	3.42929
$917$	0	0
$918$	0	0
$919$	−37.3030	−1.23051	−0.615257	−	0.788327i	$-0.710948\pi$
$919$	−37.3030	−1.23051	−0.615257	+	0.788327i	$0.710948\pi$
$920$	31.0980	1.02527
$921$	0	0
$922$	52.1231	1.71658
$923$	18.9374	0.623332
$924$	0	0
$925$	−38.4062	−1.26279
$926$	62.4140	2.05105
$927$	0	0
$928$	−2.78869	−0.0915432
$929$	5.66725	0.185937	0.0929683	−	0.995669i	$-0.470364\pi$
$929$	5.66725	0.185937	0.0929683	+	0.995669i	$0.470364\pi$
$930$	0	0
$931$	0	0
$932$	51.0289	1.67151
$933$	0	0
$934$	−83.4275	−2.72983
$935$	−13.0889	−0.428052
$936$	0	0
$937$	−7.64754	−0.249834	−0.124917	−	0.992167i	$-0.539866\pi$
$937$	−7.64754	−0.249834	−0.124917	+	0.992167i	$0.539866\pi$
$938$	0	0
$939$	0	0
$940$	−42.9702	−1.40153
$941$	20.4552	0.666819	0.333410	−	0.942782i	$-0.391801\pi$
$941$	20.4552	0.666819	0.333410	+	0.942782i	$0.391801\pi$
$942$	0	0
$943$	4.75348	0.154795
$944$	−27.7817	−0.904219
$945$	0	0
$946$	14.0315	0.456202
$947$	−4.76687	−0.154902	−0.0774512	−	0.996996i	$-0.524678\pi$
$947$	−4.76687	−0.154902	−0.0774512	+	0.996996i	$0.524678\pi$
$948$	0	0
$949$	−15.3277	−0.497558
$950$	−18.5996	−0.603451
$951$	0	0
$952$	0	0
$953$	−48.9412	−1.58536	−0.792680	−	0.609638i	$-0.791315\pi$
$953$	−48.9412	−1.58536	−0.792680	+	0.609638i	$0.791315\pi$
$954$	0	0
$955$	−24.2056	−0.783276
$956$	−40.8955	−1.32266
$957$	0	0
$958$	71.1546	2.29890
$959$	0	0
$960$	0	0
$961$	−20.2771	−0.654101
$962$	38.0549	1.22694
$963$	0	0
$964$	−85.6619	−2.75898
$965$	54.8680	1.76626
$966$	0	0
$967$	5.91712	0.190282	0.0951409	−	0.995464i	$-0.469670\pi$
$967$	5.91712	0.190282	0.0951409	+	0.995464i	$0.469670\pi$
$968$	37.1361	1.19360
$969$	0	0
$970$	57.7458	1.85410
$971$	−28.9775	−0.929933	−0.464966	−	0.885328i	$-0.653934\pi$
$971$	−28.9775	−0.929933	−0.464966	+	0.885328i	$0.653934\pi$
$972$	0	0
$973$	0	0
$974$	−53.5866	−1.71703
$975$	0	0
$976$	1.27316	0.0407528
$977$	22.8455	0.730893	0.365447	−	0.930832i	$-0.380916\pi$
$977$	22.8455	0.730893	0.365447	+	0.930832i	$0.380916\pi$
$978$	0	0
$979$	8.67340	0.277203
$980$	0	0
$981$	0	0
$982$	83.6465	2.66927
$983$	−31.2703	−0.997367	−0.498684	−	0.866784i	$-0.666183\pi$
$983$	−31.2703	−0.997367	−0.498684	+	0.866784i	$0.666183\pi$
$984$	0	0
$985$	−17.5145	−0.558059
$986$	−8.24442	−0.262556
$987$	0	0
$988$	11.9598	0.380490
$989$	11.4136	0.362930
$990$	0	0
$991$	−7.01463	−0.222827	−0.111414	−	0.993774i	$-0.535538\pi$
$991$	−7.01463	−0.222827	−0.111414	+	0.993774i	$0.535538\pi$
$992$	8.75161	0.277864
$993$	0	0
$994$	0	0
$995$	42.0858	1.33421
$996$	0	0
$997$	−21.2878	−0.674191	−0.337095	−	0.941470i	$-0.609445\pi$
$997$	−21.2878	−0.674191	−0.337095	+	0.941470i	$0.609445\pi$
$998$	21.3271	0.675099
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.z.1.1		5
3.2	odd	2		3969.2.a.bc.1.5		5
7.2	even	3		567.2.e.f.487.5		10
7.4	even	3		567.2.e.f.163.5		10
7.6	odd	2		3969.2.a.ba.1.1		5
9.2	odd	6		1323.2.f.e.442.1		10
9.4	even	3		441.2.f.e.295.5		10
9.5	odd	6		1323.2.f.e.883.1		10
9.7	even	3		441.2.f.e.148.5		10
21.2	odd	6		567.2.e.e.487.1		10
21.11	odd	6		567.2.e.e.163.1		10
21.20	even	2		3969.2.a.bb.1.5		5
63.2	odd	6		189.2.g.b.172.1		10
63.4	even	3		63.2.g.b.16.5	yes	10
63.5	even	6		1323.2.h.f.802.5		10
63.11	odd	6		189.2.h.b.37.5		10
63.13	odd	6		441.2.f.f.295.5		10
63.16	even	3		63.2.g.b.4.5	✓	10
63.20	even	6		1323.2.f.f.442.1		10
63.23	odd	6		189.2.h.b.46.5		10
63.25	even	3		63.2.h.b.58.1	yes	10
63.31	odd	6		441.2.g.f.79.5		10
63.32	odd	6		189.2.g.b.100.1		10
63.34	odd	6		441.2.f.f.148.5		10
63.38	even	6		1323.2.h.f.226.5		10
63.40	odd	6		441.2.h.f.214.1		10
63.41	even	6		1323.2.f.f.883.1		10
63.47	even	6		1323.2.g.f.361.1		10
63.52	odd	6		441.2.h.f.373.1		10
63.58	even	3		63.2.h.b.25.1	yes	10
63.59	even	6		1323.2.g.f.667.1		10
63.61	odd	6		441.2.g.f.67.5		10
252.11	even	6		3024.2.q.i.2305.2		10
252.23	even	6		3024.2.q.i.2881.2		10
252.67	odd	6		1008.2.t.i.961.3		10
252.79	odd	6		1008.2.t.i.193.3		10
252.95	even	6		3024.2.t.i.289.4		10
252.151	odd	6		1008.2.q.i.625.4		10
252.191	even	6		3024.2.t.i.1873.4		10
252.247	odd	6		1008.2.q.i.529.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.b.4.5	✓	10	63.16	even	3
63.2.g.b.16.5	yes	10	63.4	even	3
63.2.h.b.25.1	yes	10	63.58	even	3
63.2.h.b.58.1	yes	10	63.25	even	3
189.2.g.b.100.1		10	63.32	odd	6
189.2.g.b.172.1		10	63.2	odd	6
189.2.h.b.37.5		10	63.11	odd	6
189.2.h.b.46.5		10	63.23	odd	6
441.2.f.e.148.5		10	9.7	even	3
441.2.f.e.295.5		10	9.4	even	3
441.2.f.f.148.5		10	63.34	odd	6
441.2.f.f.295.5		10	63.13	odd	6
441.2.g.f.67.5		10	63.61	odd	6
441.2.g.f.79.5		10	63.31	odd	6
441.2.h.f.214.1		10	63.40	odd	6
441.2.h.f.373.1		10	63.52	odd	6
567.2.e.e.163.1		10	21.11	odd	6
567.2.e.e.487.1		10	21.2	odd	6
567.2.e.f.163.5		10	7.4	even	3
567.2.e.f.487.5		10	7.2	even	3
1008.2.q.i.529.4		10	252.247	odd	6
1008.2.q.i.625.4		10	252.151	odd	6
1008.2.t.i.193.3		10	252.79	odd	6
1008.2.t.i.961.3		10	252.67	odd	6
1323.2.f.e.442.1		10	9.2	odd	6
1323.2.f.e.883.1		10	9.5	odd	6
1323.2.f.f.442.1		10	63.20	even	6
1323.2.f.f.883.1		10	63.41	even	6
1323.2.g.f.361.1		10	63.47	even	6
1323.2.g.f.667.1		10	63.59	even	6
1323.2.h.f.226.5		10	63.38	even	6
1323.2.h.f.802.5		10	63.5	even	6
3024.2.q.i.2305.2		10	252.11	even	6
3024.2.q.i.2881.2		10	252.23	even	6
3024.2.t.i.289.4		10	252.95	even	6
3024.2.t.i.1873.4		10	252.191	even	6
3969.2.a.z.1.1		5	1.1	even	1	trivial
3969.2.a.ba.1.1		5	7.6	odd	2
3969.2.a.bb.1.5		5	21.20	even	2
3969.2.a.bc.1.5		5	3.2	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38687 q^{2} +3.69714 q^{4} -2.92087 q^{5} -4.05086 q^{8} +O(q^{10})\)	\(q-2.38687 q^{2} +3.69714 q^{4} -2.92087 q^{5} -4.05086 q^{8} +6.97172 q^{10} -1.35371 q^{11} +1.46600 q^{13} +2.27458 q^{16} -3.31027 q^{17} +2.20659 q^{19} -10.7989 q^{20} +3.23114 q^{22} +2.62830 q^{23} +3.53146 q^{25} -3.49916 q^{26} -1.04344 q^{29} +3.27458 q^{31} +2.67259 q^{32} +7.90119 q^{34} -10.8755 q^{37} -5.26683 q^{38} +11.8320 q^{40} +1.80858 q^{41} +4.34257 q^{43} -5.00488 q^{44} -6.27340 q^{46} +3.97914 q^{47} -8.42913 q^{50} +5.42002 q^{52} +6.45486 q^{53} +3.95402 q^{55} +2.49056 q^{58} -12.2140 q^{59} +0.559734 q^{61} -7.81600 q^{62} -10.9283 q^{64} -4.28200 q^{65} +12.8118 q^{67} -12.2386 q^{68} +12.9177 q^{71} -10.4554 q^{73} +25.9583 q^{74} +8.15807 q^{76} +0.767677 q^{79} -6.64375 q^{80} -4.31684 q^{82} +1.96741 q^{83} +9.66887 q^{85} -10.3652 q^{86} +5.48371 q^{88} -6.40711 q^{89} +9.71719 q^{92} -9.49769 q^{94} -6.44514 q^{95} +8.28285 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^8	\( 5 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{8} + 7 q^{10} - 4 q^{11} + 8 q^{13} - 2 q^{16} - 12 q^{17} - q^{19} - 5 q^{20} + q^{22} - 3 q^{23} + q^{25} - 11 q^{26} - 7 q^{29} + 3 q^{31} + 2 q^{32} - 3 q^{34} - 20 q^{38} + 3 q^{40} - 5 q^{41} + 7 q^{43} + 10 q^{44} - 3 q^{46} - 27 q^{47} - 19 q^{50} + 10 q^{52} + 21 q^{53} + 2 q^{55} + 10 q^{58} - 30 q^{59} + 14 q^{61} - 6 q^{62} - 25 q^{64} + 11 q^{65} + 2 q^{67} - 27 q^{68} - 3 q^{71} - 15 q^{73} + 36 q^{74} - 5 q^{76} + 4 q^{79} - 20 q^{80} + 5 q^{82} - 9 q^{83} + 6 q^{85} + 8 q^{86} + 18 q^{88} - 28 q^{89} - 27 q^{92} + 3 q^{94} + 14 q^{95} + 12 q^{97}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^8 + 7 * q^10 - 4 * q^11 + 8 * q^13 - 2 * q^16 - 12 * q^17 - q^19 - 5 * q^20 + q^22 - 3 * q^23 + q^25 - 11 * q^26 - 7 * q^29 + 3 * q^31 + 2 * q^32 - 3 * q^34 - 20 * q^38 + 3 * q^40 - 5 * q^41 + 7 * q^43 + 10 * q^44 - 3 * q^46 - 27 * q^47 - 19 * q^50 + 10 * q^52 + 21 * q^53 + 2 * q^55 + 10 * q^58 - 30 * q^59 + 14 * q^61 - 6 * q^62 - 25 * q^64 + 11 * q^65 + 2 * q^67 - 27 * q^68 - 3 * q^71 - 15 * q^73 + 36 * q^74 - 5 * q^76 + 4 * q^79 - 20 * q^80 + 5 * q^82 - 9 * q^83 + 6 * q^85 + 8 * q^86 + 18 * q^88 - 28 * q^89 - 27 * q^92 + 3 * q^94 + 14 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more