LMFDB - Embedded newform 3969.2.a.bb.1.3

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			1.3
Root			$0.495868$ 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)

$f(q)$	$=$	$q+0.495868 q^{2} -1.75411 q^{4} -3.69258 q^{5} -1.86155 q^{8} +O(q^{10})$	$q+0.495868 q^{2} -1.75411 q^{4} -3.69258 q^{5} -1.86155 q^{8} -1.83103 q^{10} +0.892568 q^{11} +1.19671 q^{13} +2.58515 q^{16} +0.249983 q^{17} +2.80827 q^{19} +6.47721 q^{20} +0.442596 q^{22} -2.47772 q^{23} +8.63514 q^{25} +0.593411 q^{26} +4.14255 q^{29} -3.58515 q^{31} +5.00499 q^{32} +0.123959 q^{34} +4.73136 q^{37} +1.39253 q^{38} +6.87391 q^{40} +4.78186 q^{41} +9.97857 q^{43} -1.56567 q^{44} -1.22862 q^{46} -10.1731 q^{47} +4.28189 q^{50} -2.09917 q^{52} -9.88929 q^{53} -3.29588 q^{55} +2.05416 q^{58} +1.81237 q^{59} -10.8041 q^{61} -1.77776 q^{62} -2.68848 q^{64} -4.41895 q^{65} +1.02937 q^{67} -0.438499 q^{68} +4.94533 q^{71} -1.83052 q^{73} +2.34613 q^{74} -4.92604 q^{76} -1.79912 q^{79} -9.54586 q^{80} +2.37117 q^{82} -12.3231 q^{83} -0.923082 q^{85} +4.94806 q^{86} -1.66156 q^{88} +2.40741 q^{89} +4.34620 q^{92} -5.04450 q^{94} -10.3698 q^{95} +11.0442 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.495868	0.350632	0.175316	−	0.984512i	$-0.443905\pi$
$2$	0.495868	0.350632	0.175316	+	0.984512i	$0.443905\pi$
$3$	0	0
$3$	0	0
$4$	−1.75411	−0.877057
$5$	−3.69258	−1.65137	−0.825686	−	0.564130i	$-0.809212\pi$
$5$	−3.69258	−1.65137	−0.825686	+	0.564130i	$0.809212\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.86155	−0.658156
$9$	0	0
$10$	−1.83103	−0.579023
$11$	0.892568	0.269119	0.134560	−	0.990905i	$-0.457038\pi$
$11$	0.892568	0.269119	0.134560	+	0.990905i	$0.457038\pi$
$12$	0	0
$13$	1.19671	0.331908	0.165954	−	0.986134i	$-0.446930\pi$
$13$	1.19671	0.331908	0.165954	+	0.986134i	$0.446930\pi$
$14$	0	0
$15$	0	0
$16$	2.58515	0.646287
$17$	0.249983	0.0606298	0.0303149	−	0.999540i	$-0.490349\pi$
$17$	0.249983	0.0606298	0.0303149	+	0.999540i	$0.490349\pi$
$18$	0	0
$19$	2.80827	0.644262	0.322131	−	0.946695i	$-0.395601\pi$
$19$	2.80827	0.644262	0.322131	+	0.946695i	$0.395601\pi$
$20$	6.47721	1.44835
$21$	0	0
$22$	0.442596	0.0943618
$23$	−2.47772	−0.516639	−0.258320	−	0.966059i	$-0.583169\pi$
$23$	−2.47772	−0.516639	−0.258320	+	0.966059i	$0.583169\pi$
$24$	0	0
$25$	8.63514	1.72703
$26$	0.593411	0.116377
$27$	0	0
$28$	0	0
$29$	4.14255	0.769252	0.384626	−	0.923072i	$-0.374330\pi$
$29$	4.14255	0.769252	0.384626	+	0.923072i	$0.374330\pi$
$30$	0	0
$31$	−3.58515	−0.643912	−0.321956	−	0.946755i	$-0.604340\pi$
$31$	−3.58515	−0.643912	−0.321956	+	0.946755i	$0.604340\pi$
$32$	5.00499	0.884765
$33$	0	0
$34$	0.123959	0.0212587
$35$	0	0
$36$	0	0
$37$	4.73136	0.777830	0.388915	−	0.921274i	$-0.372850\pi$
$37$	4.73136	0.777830	0.388915	+	0.921274i	$0.372850\pi$
$38$	1.39253	0.225899
$39$	0	0
$40$	6.87391	1.08686
$41$	4.78186	0.746801	0.373400	−	0.927670i	$-0.378192\pi$
$41$	4.78186	0.746801	0.373400	+	0.927670i	$0.378192\pi$
$42$	0	0
$43$	9.97857	1.52172	0.760859	−	0.648917i	$-0.224778\pi$
$43$	9.97857	1.52172	0.760859	+	0.648917i	$0.224778\pi$
$44$	−1.56567	−0.236033
$45$	0	0
$46$	−1.22862	−0.181150
$47$	−10.1731	−1.48389	−0.741947	−	0.670459i	$-0.766097\pi$
$47$	−10.1731	−1.48389	−0.741947	+	0.670459i	$0.766097\pi$
$48$	0	0
$49$	0	0
$50$	4.28189	0.605551
$51$	0	0
$52$	−2.09917	−0.291102
$53$	−9.88929	−1.35840	−0.679199	−	0.733954i	$-0.737673\pi$
$53$	−9.88929	−1.35840	−0.679199	+	0.733954i	$0.737673\pi$
$54$	0	0
$55$	−3.29588	−0.444416
$56$	0	0
$57$	0	0
$58$	2.05416	0.269724
$59$	1.81237	0.235951	0.117975	−	0.993017i	$-0.462360\pi$
$59$	1.81237	0.235951	0.117975	+	0.993017i	$0.462360\pi$
$60$	0	0
$61$	−10.8041	−1.38332	−0.691662	−	0.722221i	$-0.743121\pi$
$61$	−10.8041	−1.38332	−0.691662	+	0.722221i	$0.743121\pi$
$62$	−1.77776	−0.225776
$63$	0	0
$64$	−2.68848	−0.336060
$65$	−4.41895	−0.548103
$66$	0	0
$67$	1.02937	0.125757	0.0628787	−	0.998021i	$-0.479972\pi$
$67$	1.02937	0.125757	0.0628787	+	0.998021i	$0.479972\pi$
$68$	−0.438499	−0.0531758
$69$	0	0
$70$	0	0
$71$	4.94533	0.586903	0.293451	−	0.955974i	$-0.405196\pi$
$71$	4.94533	0.586903	0.293451	+	0.955974i	$0.405196\pi$
$72$	0	0
$73$	−1.83052	−0.214247	−0.107123	−	0.994246i	$-0.534164\pi$
$73$	−1.83052	−0.214247	−0.107123	+	0.994246i	$0.534164\pi$
$74$	2.34613	0.272732
$75$	0	0
$76$	−4.92604	−0.565055
$77$	0	0
$78$	0	0
$79$	−1.79912	−0.202417	−0.101209	−	0.994865i	$-0.532271\pi$
$79$	−1.79912	−0.202417	−0.101209	+	0.994865i	$0.532271\pi$
$80$	−9.54586	−1.06726
$81$	0	0
$82$	2.37117	0.261852
$83$	−12.3231	−1.35264	−0.676319	−	0.736609i	$-0.736426\pi$
$83$	−12.3231	−1.35264	−0.676319	+	0.736609i	$0.736426\pi$
$84$	0	0
$85$	−0.923082	−0.100122
$86$	4.94806	0.533563
$87$	0	0
$88$	−1.66156	−0.177123
$89$	2.40741	0.255185	0.127592	−	0.991827i	$-0.459275\pi$
$89$	2.40741	0.255185	0.127592	+	0.991827i	$0.459275\pi$
$90$	0	0
$91$	0	0
$92$	4.34620	0.453122
$93$	0	0
$94$	−5.04450	−0.520300
$95$	−10.3698	−1.06392
$96$	0	0
$97$	11.0442	1.12137	0.560684	−	0.828030i	$-0.310538\pi$
$97$	11.0442	1.12137	0.560684	+	0.828030i	$0.310538\pi$
$98$	0	0
$99$	0	0
$100$	−15.1470	−1.51470
$101$	−2.59964	−0.258674	−0.129337	−	0.991601i	$-0.541285\pi$
$101$	−2.59964	−0.258674	−0.129337	+	0.991601i	$0.541285\pi$
$102$	0	0
$103$	−9.71155	−0.956908	−0.478454	−	0.878113i	$-0.658803\pi$
$103$	−9.71155	−0.956908	−0.478454	+	0.878113i	$0.658803\pi$
$104$	−2.22773	−0.218447
$105$	0	0
$106$	−4.90379	−0.476298
$107$	−10.9005	−1.05379	−0.526896	−	0.849930i	$-0.676644\pi$
$107$	−10.9005	−1.05379	−0.526896	+	0.849930i	$0.676644\pi$
$108$	0	0
$109$	2.12193	0.203244	0.101622	−	0.994823i	$-0.467597\pi$
$109$	2.12193	0.203244	0.101622	+	0.994823i	$0.467597\pi$
$110$	−1.63432	−0.155826
$111$	0	0
$112$	0	0
$113$	15.8264	1.48882	0.744409	−	0.667724i	$-0.232731\pi$
$113$	15.8264	1.48882	0.744409	+	0.667724i	$0.232731\pi$
$114$	0	0
$115$	9.14916	0.853164
$116$	−7.26651	−0.674679
$117$	0	0
$118$	0.898698	0.0827318
$119$	0	0
$120$	0	0
$121$	−10.2033	−0.927575
$122$	−5.35741	−0.485038
$123$	0	0
$124$	6.28876	0.564747
$125$	−13.4230	−1.20059
$126$	0	0
$127$	−1.26946	−0.112647	−0.0563233	−	0.998413i	$-0.517938\pi$
$127$	−1.26946	−0.112647	−0.0563233	+	0.998413i	$0.517938\pi$
$128$	−11.3431	−1.00260
$129$	0	0
$130$	−2.19122	−0.192182
$131$	−15.0289	−1.31308	−0.656540	−	0.754291i	$-0.727981\pi$
$131$	−15.0289	−1.31308	−0.656540	+	0.754291i	$0.727981\pi$
$132$	0	0
$133$	0	0
$134$	0.510432	0.0440946
$135$	0	0
$136$	−0.465355	−0.0399038
$137$	0.488493	0.0417347	0.0208674	−	0.999782i	$-0.493357\pi$
$137$	0.488493	0.0417347	0.0208674	+	0.999782i	$0.493357\pi$
$138$	0	0
$139$	−9.86974	−0.837141	−0.418570	−	0.908184i	$-0.637469\pi$
$139$	−9.86974	−0.837141	−0.418570	+	0.908184i	$0.637469\pi$
$140$	0	0
$141$	0	0
$142$	2.45223	0.205787
$143$	1.06815	0.0893229
$144$	0	0
$145$	−15.2967	−1.27032
$146$	−0.907699	−0.0751217
$147$	0	0
$148$	−8.29934	−0.682202
$149$	−21.0240	−1.72235	−0.861175	−	0.508309i	$-0.830271\pi$
$149$	−21.0240	−1.72235	−0.861175	+	0.508309i	$0.830271\pi$
$150$	0	0
$151$	1.49838	0.121937	0.0609683	−	0.998140i	$-0.480581\pi$
$151$	1.49838	0.121937	0.0609683	+	0.998140i	$0.480581\pi$
$152$	−5.22773	−0.424025
$153$	0	0
$154$	0	0
$155$	13.2384	1.06334
$156$	0	0
$157$	16.6796	1.33118	0.665590	−	0.746317i	$-0.268180\pi$
$157$	16.6796	1.33118	0.665590	+	0.746317i	$0.268180\pi$
$158$	−0.892128	−0.0709739
$159$	0	0
$160$	−18.4813	−1.46108
$161$	0	0
$162$	0	0
$163$	6.68269	0.523429	0.261714	−	0.965145i	$-0.415712\pi$
$163$	6.68269	0.523429	0.261714	+	0.965145i	$0.415712\pi$
$164$	−8.38793	−0.654987
$165$	0	0
$166$	−6.11064	−0.474278
$167$	−17.6310	−1.36433	−0.682163	−	0.731200i	$-0.738961\pi$
$167$	−17.6310	−1.36433	−0.682163	+	0.731200i	$0.738961\pi$
$168$	0	0
$169$	−11.5679	−0.889837
$170$	−0.457727	−0.0351061
$171$	0	0
$172$	−17.5036	−1.33463
$173$	−3.88685	−0.295511	−0.147756	−	0.989024i	$-0.547205\pi$
$173$	−3.88685	−0.295511	−0.147756	+	0.989024i	$0.547205\pi$
$174$	0	0
$175$	0	0
$176$	2.30742	0.173928
$177$	0	0
$178$	1.19376	0.0894759
$179$	7.33516	0.548256	0.274128	−	0.961693i	$-0.411611\pi$
$179$	7.33516	0.548256	0.274128	+	0.961693i	$0.411611\pi$
$180$	0	0
$181$	−11.2566	−0.836693	−0.418346	−	0.908288i	$-0.637390\pi$
$181$	−11.2566	−0.836693	−0.418346	+	0.908288i	$0.637390\pi$
$182$	0	0
$183$	0	0
$184$	4.61238	0.340029
$185$	−17.4709	−1.28449
$186$	0	0
$187$	0.223127	0.0163167
$188$	17.8447	1.30146
$189$	0	0
$190$	−5.14204	−0.373043
$191$	23.8459	1.72543	0.862715	−	0.505690i	$-0.168762\pi$
$191$	23.8459	1.72543	0.862715	+	0.505690i	$0.168762\pi$
$192$	0	0
$193$	5.93456	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$193$	5.93456	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$194$	5.47647	0.393188
$195$	0	0
$196$	0	0
$197$	15.4682	1.10206	0.551032	−	0.834484i	$-0.314234\pi$
$197$	15.4682	1.10206	0.551032	+	0.834484i	$0.314234\pi$
$198$	0	0
$199$	15.4964	1.09851	0.549254	−	0.835655i	$-0.314912\pi$
$199$	15.4964	1.09851	0.549254	+	0.835655i	$0.314912\pi$
$200$	−16.0747	−1.13665
$201$	0	0
$202$	−1.28908	−0.0906994
$203$	0	0
$204$	0	0
$205$	−17.6574	−1.23325
$206$	−4.81565	−0.335522
$207$	0	0
$208$	3.09367	0.214508
$209$	2.50658	0.173384
$210$	0	0
$211$	−1.54380	−0.106279	−0.0531397	−	0.998587i	$-0.516923\pi$
$211$	−1.54380	−0.106279	−0.0531397	+	0.998587i	$0.516923\pi$
$212$	17.3469	1.19139
$213$	0	0
$214$	−5.40522	−0.369493
$215$	−36.8467	−2.51292
$216$	0	0
$217$	0	0
$218$	1.05220	0.0712637
$219$	0	0
$220$	5.78135	0.389779
$221$	0.299157	0.0201235
$222$	0	0
$223$	−5.44342	−0.364518	−0.182259	−	0.983251i	$-0.558341\pi$
$223$	−5.44342	−0.364518	−0.182259	+	0.983251i	$0.558341\pi$
$224$	0	0
$225$	0	0
$226$	7.84779	0.522027
$227$	−16.0764	−1.06703	−0.533513	−	0.845792i	$-0.679128\pi$
$227$	−16.0764	−1.06703	−0.533513	+	0.845792i	$0.679128\pi$
$228$	0	0
$229$	9.96840	0.658730	0.329365	−	0.944203i	$-0.393165\pi$
$229$	9.96840	0.658730	0.329365	+	0.944203i	$0.393165\pi$
$230$	4.53678	0.299146
$231$	0	0
$232$	−7.71155	−0.506288
$233$	16.5409	1.08363	0.541815	−	0.840498i	$-0.317737\pi$
$233$	16.5409	1.08363	0.541815	+	0.840498i	$0.317737\pi$
$234$	0	0
$235$	37.5648	2.45046
$236$	−3.17911	−0.206942
$237$	0	0
$238$	0	0
$239$	−22.0239	−1.42461	−0.712303	−	0.701872i	$-0.752348\pi$
$239$	−22.0239	−1.42461	−0.712303	+	0.701872i	$0.752348\pi$
$240$	0	0
$241$	−16.7201	−1.07703	−0.538517	−	0.842615i	$-0.681015\pi$
$241$	−16.7201	−1.07703	−0.538517	+	0.842615i	$0.681015\pi$
$242$	−5.05950	−0.325237
$243$	0	0
$244$	18.9516	1.21325
$245$	0	0
$246$	0	0
$247$	3.36069	0.213836
$248$	6.67392	0.423794
$249$	0	0
$250$	−6.65606	−0.420966
$251$	−8.53099	−0.538471	−0.269236	−	0.963074i	$-0.586771\pi$
$251$	−8.53099	−0.538471	−0.269236	+	0.963074i	$0.586771\pi$
$252$	0	0
$253$	−2.21153	−0.139038
$254$	−0.629487	−0.0394975
$255$	0	0
$256$	−0.247722	−0.0154826
$257$	−17.1197	−1.06790	−0.533950	−	0.845516i	$-0.679293\pi$
$257$	−17.1197	−1.06790	−0.533950	+	0.845516i	$0.679293\pi$
$258$	0	0
$259$	0	0
$260$	7.75135	0.480718
$261$	0	0
$262$	−7.45235	−0.460408
$263$	−20.5527	−1.26733	−0.633666	−	0.773607i	$-0.718451\pi$
$263$	−20.5527	−1.26733	−0.633666	+	0.773607i	$0.718451\pi$
$264$	0	0
$265$	36.5170	2.24322
$266$	0	0
$267$	0	0
$268$	−1.80563	−0.110297
$269$	−19.8453	−1.20999	−0.604996	−	0.796229i	$-0.706825\pi$
$269$	−19.8453	−1.20999	−0.604996	+	0.796229i	$0.706825\pi$
$270$	0	0
$271$	10.6411	0.646402	0.323201	−	0.946330i	$-0.395241\pi$
$271$	10.6411	0.646402	0.323201	+	0.946330i	$0.395241\pi$
$272$	0.646243	0.0391842
$273$	0	0
$274$	0.242228	0.0146335
$275$	7.70745	0.464777
$276$	0	0
$277$	−24.8813	−1.49497	−0.747487	−	0.664276i	$-0.768740\pi$
$277$	−24.8813	−1.49497	−0.747487	+	0.664276i	$0.768740\pi$
$278$	−4.89409	−0.293528
$279$	0	0
$280$	0	0
$281$	−13.6747	−0.815762	−0.407881	−	0.913035i	$-0.633732\pi$
$281$	−13.6747	−0.815762	−0.407881	+	0.913035i	$0.633732\pi$
$282$	0	0
$283$	−6.32179	−0.375791	−0.187896	−	0.982189i	$-0.560167\pi$
$283$	−6.32179	−0.375791	−0.187896	+	0.982189i	$0.560167\pi$
$284$	−8.67468	−0.514747
$285$	0	0
$286$	0.529660	0.0313194
$287$	0	0
$288$	0	0
$289$	−16.9375	−0.996324
$290$	−7.58515	−0.445415
$291$	0	0
$292$	3.21095	0.187907
$293$	2.63016	0.153655	0.0768277	−	0.997044i	$-0.475521\pi$
$293$	2.63016	0.153655	0.0768277	+	0.997044i	$0.475521\pi$
$294$	0	0
$295$	−6.69233	−0.389642
$296$	−8.80764	−0.511934
$297$	0	0
$298$	−10.4251	−0.603911
$299$	−2.96511	−0.171477
$300$	0	0
$301$	0	0
$302$	0.743000	0.0427548
$303$	0	0
$304$	7.25980	0.416378
$305$	39.8950	2.28438
$306$	0	0
$307$	2.79496	0.159517	0.0797583	−	0.996814i	$-0.474585\pi$
$307$	2.79496	0.159517	0.0797583	+	0.996814i	$0.474585\pi$
$308$	0	0
$309$	0	0
$310$	6.56452	0.372840
$311$	−15.1003	−0.856258	−0.428129	−	0.903718i	$-0.640827\pi$
$311$	−15.1003	−0.856258	−0.428129	+	0.903718i	$0.640827\pi$
$312$	0	0
$313$	25.4785	1.44013	0.720064	−	0.693908i	$-0.244112\pi$
$313$	25.4785	1.44013	0.720064	+	0.693908i	$0.244112\pi$
$314$	8.27090	0.466754
$315$	0	0
$316$	3.15587	0.177531
$317$	−32.5209	−1.82656	−0.913278	−	0.407337i	$-0.866457\pi$
$317$	−32.5209	−1.82656	−0.913278	+	0.407337i	$0.866457\pi$
$318$	0	0
$319$	3.69751	0.207021
$320$	9.92743	0.554960
$321$	0	0
$322$	0	0
$323$	0.702021	0.0390615
$324$	0	0
$325$	10.3338	0.573214
$326$	3.31373	0.183531
$327$	0	0
$328$	−8.90165	−0.491511
$329$	0	0
$330$	0	0
$331$	18.0948	0.994582	0.497291	−	0.867584i	$-0.334328\pi$
$331$	18.0948	0.994582	0.497291	+	0.867584i	$0.334328\pi$
$332$	21.6162	1.18634
$333$	0	0
$334$	−8.74264	−0.478376
$335$	−3.80103	−0.207672
$336$	0	0
$337$	25.0173	1.36278	0.681389	−	0.731921i	$-0.261376\pi$
$337$	25.0173	1.36278	0.681389	+	0.731921i	$0.261376\pi$
$338$	−5.73615	−0.312005
$339$	0	0
$340$	1.61919	0.0878130
$341$	−3.19999	−0.173289
$342$	0	0
$343$	0	0
$344$	−18.5756	−1.00153
$345$	0	0
$346$	−1.92736	−0.103616
$347$	−10.7489	−0.577030	−0.288515	−	0.957475i	$-0.593162\pi$
$347$	−10.7489	−0.577030	−0.288515	+	0.957475i	$0.593162\pi$
$348$	0	0
$349$	−3.28602	−0.175896	−0.0879482	−	0.996125i	$-0.528031\pi$
$349$	−3.28602	−0.175896	−0.0879482	+	0.996125i	$0.528031\pi$
$350$	0	0
$351$	0	0
$352$	4.46729	0.238107
$353$	16.8192	0.895195	0.447598	−	0.894235i	$-0.352280\pi$
$353$	16.8192	0.895195	0.447598	+	0.894235i	$0.352280\pi$
$354$	0	0
$355$	−18.2610	−0.969195
$356$	−4.22287	−0.223812
$357$	0	0
$358$	3.63728	0.192236
$359$	23.7842	1.25528	0.627642	−	0.778502i	$-0.284020\pi$
$359$	23.7842	1.25528	0.627642	+	0.778502i	$0.284020\pi$
$360$	0	0
$361$	−11.1136	−0.584926
$362$	−5.58177	−0.293371
$363$	0	0
$364$	0	0
$365$	6.75936	0.353801
$366$	0	0
$367$	0.689984	0.0360169	0.0180084	−	0.999838i	$-0.494267\pi$
$367$	0.689984	0.0360169	0.0180084	+	0.999838i	$0.494267\pi$
$368$	−6.40526	−0.333897
$369$	0	0
$370$	−8.66327	−0.450382
$371$	0	0
$372$	0	0
$373$	−3.76012	−0.194691	−0.0973457	−	0.995251i	$-0.531035\pi$
$373$	−3.76012	−0.194691	−0.0973457	+	0.995251i	$0.531035\pi$
$374$	0.110642	0.00572114
$375$	0	0
$376$	18.9376	0.976634
$377$	4.95744	0.255321
$378$	0	0
$379$	32.8735	1.68860	0.844300	−	0.535872i	$-0.180017\pi$
$379$	32.8735	1.68860	0.844300	+	0.535872i	$0.180017\pi$
$380$	18.1898	0.933116
$381$	0	0
$382$	11.8244	0.604991
$383$	−1.07267	−0.0548109	−0.0274055	−	0.999624i	$-0.508725\pi$
$383$	−1.07267	−0.0548109	−0.0274055	+	0.999624i	$0.508725\pi$
$384$	0	0
$385$	0	0
$386$	2.94276	0.149782
$387$	0	0
$388$	−19.3728	−0.983505
$389$	23.7436	1.20385	0.601925	−	0.798553i	$-0.294401\pi$
$389$	23.7436	1.20385	0.601925	+	0.798553i	$0.294401\pi$
$390$	0	0
$391$	−0.619387	−0.0313237
$392$	0	0
$393$	0	0
$394$	7.67019	0.386419
$395$	6.64340	0.334266
$396$	0	0
$397$	−0.0320978	−0.00161094	−0.000805471	−	1.00000i	$-0.500256\pi$
$397$	−0.0320978	−0.00161094	−0.000805471		1.00000i	$0.500256\pi$
$398$	7.68415	0.385172
$399$	0	0
$400$	22.3231	1.11616
$401$	−24.5256	−1.22475	−0.612374	−	0.790568i	$-0.709785\pi$
$401$	−24.5256	−1.22475	−0.612374	+	0.790568i	$0.709785\pi$
$402$	0	0
$403$	−4.29039	−0.213719
$404$	4.56007	0.226872
$405$	0	0
$406$	0	0
$407$	4.22306	0.209329
$408$	0	0
$409$	−26.7897	−1.32467	−0.662333	−	0.749210i	$-0.730433\pi$
$409$	−26.7897	−1.32467	−0.662333	+	0.749210i	$0.730433\pi$
$410$	−8.75574	−0.432415
$411$	0	0
$412$	17.0352	0.839263
$413$	0	0
$414$	0	0
$415$	45.5041	2.23371
$416$	5.98952	0.293660
$417$	0	0
$418$	1.24293	0.0607938
$419$	21.0525	1.02848	0.514240	−	0.857646i	$-0.328074\pi$
$419$	21.0525	1.02848	0.514240	+	0.857646i	$0.328074\pi$
$420$	0	0
$421$	14.8907	0.725727	0.362863	−	0.931842i	$-0.381799\pi$
$421$	14.8907	0.725727	0.362863	+	0.931842i	$0.381799\pi$
$422$	−0.765520	−0.0372649
$423$	0	0
$424$	18.4094	0.894038
$425$	2.15864	0.104709
$426$	0	0
$427$	0	0
$428$	19.1207	0.924236
$429$	0	0
$430$	−18.2711	−0.881110
$431$	−15.9038	−0.766061	−0.383031	−	0.923736i	$-0.625120\pi$
$431$	−15.9038	−0.766061	−0.383031	+	0.923736i	$0.625120\pi$
$432$	0	0
$433$	16.3658	0.786490	0.393245	−	0.919434i	$-0.371352\pi$
$433$	16.3658	0.786490	0.393245	+	0.919434i	$0.371352\pi$
$434$	0	0
$435$	0	0
$436$	−3.72210	−0.178256
$437$	−6.95811	−0.332851
$438$	0	0
$439$	15.5447	0.741909	0.370954	−	0.928651i	$-0.379031\pi$
$439$	15.5447	0.741909	0.370954	+	0.928651i	$0.379031\pi$
$440$	6.13543	0.292495
$441$	0	0
$442$	0.148343	0.00705594
$443$	−1.79005	−0.0850480	−0.0425240	−	0.999095i	$-0.513540\pi$
$443$	−1.79005	−0.0850480	−0.0425240	+	0.999095i	$0.513540\pi$
$444$	0	0
$445$	−8.88955	−0.421405
$446$	−2.69922	−0.127812
$447$	0	0
$448$	0	0
$449$	−13.5666	−0.640250	−0.320125	−	0.947375i	$-0.603725\pi$
$449$	−13.5666	−0.640250	−0.320125	+	0.947375i	$0.603725\pi$
$450$	0	0
$451$	4.26814	0.200979
$452$	−27.7612	−1.30578
$453$	0	0
$454$	−7.97176	−0.374133
$455$	0	0
$456$	0	0
$457$	2.56917	0.120181	0.0600905	−	0.998193i	$-0.480861\pi$
$457$	2.56917	0.120181	0.0600905	+	0.998193i	$0.480861\pi$
$458$	4.94301	0.230972
$459$	0	0
$460$	−16.0487	−0.748274
$461$	−36.1869	−1.68539	−0.842695	−	0.538391i	$-0.819033\pi$
$461$	−36.1869	−1.68539	−0.842695	+	0.538391i	$0.819033\pi$
$462$	0	0
$463$	−16.3845	−0.761451	−0.380726	−	0.924688i	$-0.624326\pi$
$463$	−16.3845	−0.761451	−0.380726	+	0.924688i	$0.624326\pi$
$464$	10.7091	0.497158
$465$	0	0
$466$	8.20210	0.379955
$467$	8.70044	0.402608	0.201304	−	0.979529i	$-0.435482\pi$
$467$	8.70044	0.402608	0.201304	+	0.979529i	$0.435482\pi$
$468$	0	0
$469$	0	0
$470$	18.6272	0.859209
$471$	0	0
$472$	−3.37381	−0.155292
$473$	8.90655	0.409524
$474$	0	0
$475$	24.2498	1.11266
$476$	0	0
$477$	0	0
$478$	−10.9209	−0.499513
$479$	−17.7674	−0.811813	−0.405907	−	0.913915i	$-0.633044\pi$
$479$	−17.7674	−0.811813	−0.405907	+	0.913915i	$0.633044\pi$
$480$	0	0
$481$	5.66207	0.258168
$482$	−8.29095	−0.377643
$483$	0	0
$484$	17.8978	0.813536
$485$	−40.7816	−1.85180
$486$	0	0
$487$	−16.6553	−0.754722	−0.377361	−	0.926066i	$-0.623168\pi$
$487$	−16.6553	−0.754722	−0.377361	+	0.926066i	$0.623168\pi$
$488$	20.1123	0.910443
$489$	0	0
$490$	0	0
$491$	−6.42042	−0.289749	−0.144875	−	0.989450i	$-0.546278\pi$
$491$	−6.42042	−0.289749	−0.144875	+	0.989450i	$0.546278\pi$
$492$	0	0
$493$	1.03557	0.0466396
$494$	1.66646	0.0749776
$495$	0	0
$496$	−9.26814	−0.416152
$497$	0	0
$498$	0	0
$499$	11.1459	0.498960	0.249480	−	0.968380i	$-0.419740\pi$
$499$	11.1459	0.498960	0.249480	+	0.968380i	$0.419740\pi$
$500$	23.5456	1.05299
$501$	0	0
$502$	−4.23025	−0.188805
$503$	−17.7223	−0.790200	−0.395100	−	0.918638i	$-0.629290\pi$
$503$	−17.7223	−0.790200	−0.395100	+	0.918638i	$0.629290\pi$
$504$	0	0
$505$	9.59939	0.427167
$506$	−1.09663	−0.0487511
$507$	0	0
$508$	2.22678	0.0987976
$509$	31.0823	1.37770	0.688848	−	0.724906i	$-0.258117\pi$
$509$	31.0823	1.37770	0.688848	+	0.724906i	$0.258117\pi$
$510$	0	0
$511$	0	0
$512$	22.5634	0.997169
$513$	0	0
$514$	−8.48913	−0.374439
$515$	35.8607	1.58021
$516$	0	0
$517$	−9.08016	−0.399345
$518$	0	0
$519$	0	0
$520$	8.22608	0.360737
$521$	4.75971	0.208527	0.104263	−	0.994550i	$-0.466751\pi$
$521$	4.75971	0.208527	0.104263	+	0.994550i	$0.466751\pi$
$522$	0	0
$523$	40.2515	1.76008	0.880038	−	0.474904i	$-0.157517\pi$
$523$	40.2515	1.76008	0.880038	+	0.474904i	$0.157517\pi$
$524$	26.3624	1.15165
$525$	0	0
$526$	−10.1914	−0.444367
$527$	−0.896226	−0.0390402
$528$	0	0
$529$	−16.8609	−0.733084
$530$	18.1076	0.786545
$531$	0	0
$532$	0	0
$533$	5.72250	0.247869
$534$	0	0
$535$	40.2510	1.74020
$536$	−1.91622	−0.0827680
$537$	0	0
$538$	−9.84067	−0.424261
$539$	0	0
$540$	0	0
$541$	−24.1094	−1.03655	−0.518273	−	0.855215i	$-0.673425\pi$
$541$	−24.1094	−1.03655	−0.518273	+	0.855215i	$0.673425\pi$
$542$	5.27659	0.226649
$543$	0	0
$544$	1.25116	0.0536431
$545$	−7.83538	−0.335631
$546$	0	0
$547$	12.3550	0.528263	0.264131	−	0.964487i	$-0.414915\pi$
$547$	12.3550	0.528263	0.264131	+	0.964487i	$0.414915\pi$
$548$	−0.856872	−0.0366038
$549$	0	0
$550$	3.82188	0.162966
$551$	11.6334	0.495600
$552$	0	0
$553$	0	0
$554$	−12.3379	−0.524186
$555$	0	0
$556$	17.3127	0.734220
$557$	8.07689	0.342229	0.171114	−	0.985251i	$-0.445263\pi$
$557$	8.07689	0.342229	0.171114	+	0.985251i	$0.445263\pi$
$558$	0	0
$559$	11.9415	0.505070
$560$	0	0
$561$	0	0
$562$	−6.78083	−0.286032
$563$	45.2127	1.90549	0.952744	−	0.303774i	$-0.0982467\pi$
$563$	45.2127	1.90549	0.952744	+	0.303774i	$0.0982467\pi$
$564$	0	0
$565$	−58.4401	−2.45859
$566$	−3.13477	−0.131764
$567$	0	0
$568$	−9.20596	−0.386274
$569$	−22.4299	−0.940309	−0.470155	−	0.882584i	$-0.655802\pi$
$569$	−22.4299	−0.940309	−0.470155	+	0.882584i	$0.655802\pi$
$570$	0	0
$571$	−21.8269	−0.913426	−0.456713	−	0.889614i	$-0.650973\pi$
$571$	−21.8269	−0.913426	−0.456713	+	0.889614i	$0.650973\pi$
$572$	−1.87365	−0.0783413
$573$	0	0
$574$	0	0
$575$	−21.3954	−0.892251
$576$	0	0
$577$	−32.2044	−1.34068	−0.670342	−	0.742052i	$-0.733853\pi$
$577$	−32.2044	−1.34068	−0.670342	+	0.742052i	$0.733853\pi$
$578$	−8.39877	−0.349343
$579$	0	0
$580$	26.8322	1.11414
$581$	0	0
$582$	0	0
$583$	−8.82687	−0.365571
$584$	3.40761	0.141008
$585$	0	0
$586$	1.30421	0.0538765
$587$	19.4461	0.802625	0.401313	−	0.915941i	$-0.368554\pi$
$587$	19.4461	0.802625	0.401313	+	0.915941i	$0.368554\pi$
$588$	0	0
$589$	−10.0681	−0.414848
$590$	−3.31851	−0.136621
$591$	0	0
$592$	12.2313	0.502701
$593$	28.8405	1.18434	0.592168	−	0.805815i	$-0.298272\pi$
$593$	28.8405	1.18434	0.592168	+	0.805815i	$0.298272\pi$
$594$	0	0
$595$	0	0
$596$	36.8784	1.51060
$597$	0	0
$598$	−1.47030	−0.0601252
$599$	46.9989	1.92032	0.960161	−	0.279447i	$-0.0901511\pi$
$599$	46.9989	1.92032	0.960161	+	0.279447i	$0.0901511\pi$
$600$	0	0
$601$	−15.6169	−0.637025	−0.318512	−	0.947919i	$-0.603183\pi$
$601$	−15.6169	−0.637025	−0.318512	+	0.947919i	$0.603183\pi$
$602$	0	0
$603$	0	0
$604$	−2.62833	−0.106945
$605$	37.6766	1.53177
$606$	0	0
$607$	28.6532	1.16300	0.581500	−	0.813547i	$-0.302466\pi$
$607$	28.6532	1.16300	0.581500	+	0.813547i	$0.302466\pi$
$608$	14.0554	0.570021
$609$	0	0
$610$	19.7827	0.800977
$611$	−12.1742	−0.492516
$612$	0	0
$613$	−29.3468	−1.18531	−0.592653	−	0.805458i	$-0.701920\pi$
$613$	−29.3468	−1.18531	−0.592653	+	0.805458i	$0.701920\pi$
$614$	1.38593	0.0559316
$615$	0	0
$616$	0	0
$617$	4.12801	0.166188	0.0830938	−	0.996542i	$-0.473520\pi$
$617$	4.12801	0.166188	0.0830938	+	0.996542i	$0.473520\pi$
$618$	0	0
$619$	−22.7130	−0.912912	−0.456456	−	0.889746i	$-0.650881\pi$
$619$	−22.7130	−0.912912	−0.456456	+	0.889746i	$0.650881\pi$
$620$	−23.2217	−0.932608
$621$	0	0
$622$	−7.48774	−0.300231
$623$	0	0
$624$	0	0
$625$	6.38996	0.255598
$626$	12.6340	0.504955
$627$	0	0
$628$	−29.2580	−1.16752
$629$	1.18276	0.0471597
$630$	0	0
$631$	−38.6411	−1.53828	−0.769138	−	0.639082i	$-0.779314\pi$
$631$	−38.6411	−1.53828	−0.769138	+	0.639082i	$0.779314\pi$
$632$	3.34915	0.133222
$633$	0	0
$634$	−16.1261	−0.640449
$635$	4.68759	0.186021
$636$	0	0
$637$	0	0
$638$	1.83348	0.0725881
$639$	0	0
$640$	41.8853	1.65566
$641$	28.4726	1.12460	0.562301	−	0.826933i	$-0.309916\pi$
$641$	28.4726	1.12460	0.562301	+	0.826933i	$0.309916\pi$
$642$	0	0
$643$	−17.0425	−0.672091	−0.336045	−	0.941846i	$-0.609090\pi$
$643$	−17.0425	−0.672091	−0.336045	+	0.941846i	$0.609090\pi$
$644$	0	0
$645$	0	0
$646$	0.348110	0.0136962
$647$	−3.37618	−0.132731	−0.0663657	−	0.997795i	$-0.521140\pi$
$647$	−3.37618	−0.132731	−0.0663657	+	0.997795i	$0.521140\pi$
$648$	0	0
$649$	1.61767	0.0634989
$650$	5.12419	0.200987
$651$	0	0
$652$	−11.7222	−0.459077
$653$	18.3451	0.717899	0.358950	−	0.933357i	$-0.383135\pi$
$653$	18.3451	0.717899	0.358950	+	0.933357i	$0.383135\pi$
$654$	0	0
$655$	55.4954	2.16838
$656$	12.3618	0.482648
$657$	0	0
$658$	0	0
$659$	−27.8495	−1.08486	−0.542432	−	0.840100i	$-0.682496\pi$
$659$	−27.8495	−1.08486	−0.542432	+	0.840100i	$0.682496\pi$
$660$	0	0
$661$	−39.0141	−1.51747	−0.758737	−	0.651397i	$-0.774183\pi$
$661$	−39.0141	−1.51747	−0.758737	+	0.651397i	$0.774183\pi$
$662$	8.97265	0.348732
$663$	0	0
$664$	22.9400	0.890247
$665$	0	0
$666$	0	0
$667$	−10.2641	−0.397426
$668$	30.9268	1.19659
$669$	0	0
$670$	−1.88481	−0.0728165
$671$	−9.64340	−0.372280
$672$	0	0
$673$	−49.2309	−1.89771	−0.948856	−	0.315711i	$-0.897757\pi$
$673$	−49.2309	−1.89771	−0.948856	+	0.315711i	$0.897757\pi$
$674$	12.4053	0.477833
$675$	0	0
$676$	20.2914	0.780438
$677$	−23.3915	−0.899010	−0.449505	−	0.893278i	$-0.648400\pi$
$677$	−23.3915	−0.899010	−0.449505	+	0.893278i	$0.648400\pi$
$678$	0	0
$679$	0	0
$680$	1.71836	0.0658961
$681$	0	0
$682$	−1.58677	−0.0607607
$683$	−30.3264	−1.16041	−0.580204	−	0.814471i	$-0.697027\pi$
$683$	−30.3264	−1.16041	−0.580204	+	0.814471i	$0.697027\pi$
$684$	0	0
$685$	−1.80380	−0.0689196
$686$	0	0
$687$	0	0
$688$	25.7961	0.983466
$689$	−11.8346	−0.450863
$690$	0	0
$691$	4.11330	0.156477	0.0782387	−	0.996935i	$-0.475070\pi$
$691$	4.11330	0.156477	0.0782387	+	0.996935i	$0.475070\pi$
$692$	6.81797	0.259180
$693$	0	0
$694$	−5.33003	−0.202325
$695$	36.4448	1.38243
$696$	0	0
$697$	1.19538	0.0452784
$698$	−1.62943	−0.0616749
$699$	0	0
$700$	0	0
$701$	−29.1835	−1.10225	−0.551123	−	0.834424i	$-0.685800\pi$
$701$	−29.1835	−1.10225	−0.551123	+	0.834424i	$0.685800\pi$
$702$	0	0
$703$	13.2869	0.501127
$704$	−2.39965	−0.0904403
$705$	0	0
$706$	8.34010	0.313884
$707$	0	0
$708$	0	0
$709$	−42.4617	−1.59468	−0.797342	−	0.603528i	$-0.793761\pi$
$709$	−42.4617	−1.59468	−0.797342	+	0.603528i	$0.793761\pi$
$710$	−9.05507	−0.339831
$711$	0	0
$712$	−4.48150	−0.167951
$713$	8.88298	0.332670
$714$	0	0
$715$	−3.94421	−0.147505
$716$	−12.8667	−0.480852
$717$	0	0
$718$	11.7938	0.440142
$719$	11.1425	0.415546	0.207773	−	0.978177i	$-0.433379\pi$
$719$	11.1425	0.415546	0.207773	+	0.978177i	$0.433379\pi$
$720$	0	0
$721$	0	0
$722$	−5.51088	−0.205094
$723$	0	0
$724$	19.7453	0.733828
$725$	35.7715	1.32852
$726$	0	0
$727$	−28.6820	−1.06376	−0.531878	−	0.846821i	$-0.678513\pi$
$727$	−28.6820	−1.06376	−0.531878	+	0.846821i	$0.678513\pi$
$728$	0	0
$729$	0	0
$730$	3.35175	0.124054
$731$	2.49447	0.0922614
$732$	0	0
$733$	25.0528	0.925348	0.462674	−	0.886529i	$-0.346890\pi$
$733$	25.0528	0.925348	0.462674	+	0.886529i	$0.346890\pi$
$734$	0.342141	0.0126287
$735$	0	0
$736$	−12.4009	−0.457104
$737$	0.918782	0.0338438
$738$	0	0
$739$	−27.5216	−1.01240	−0.506198	−	0.862417i	$-0.668950\pi$
$739$	−27.5216	−1.01240	−0.506198	+	0.862417i	$0.668950\pi$
$740$	30.6460	1.12657
$741$	0	0
$742$	0	0
$743$	−14.0122	−0.514057	−0.257028	−	0.966404i	$-0.582743\pi$
$743$	−14.0122	−0.514057	−0.257028	+	0.966404i	$0.582743\pi$
$744$	0	0
$745$	77.6326	2.84424
$746$	−1.86452	−0.0682650
$747$	0	0
$748$	−0.391390	−0.0143106
$749$	0	0
$750$	0	0
$751$	−52.2594	−1.90697	−0.953486	−	0.301436i	$-0.902534\pi$
$751$	−52.2594	−1.90697	−0.953486	+	0.301436i	$0.902534\pi$
$752$	−26.2989	−0.959021
$753$	0	0
$754$	2.45824	0.0895237
$755$	−5.53289	−0.201363
$756$	0	0
$757$	−43.3447	−1.57539	−0.787694	−	0.616066i	$-0.788725\pi$
$757$	−43.3447	−1.57539	−0.787694	+	0.616066i	$0.788725\pi$
$758$	16.3009	0.592077
$759$	0	0
$760$	19.3038	0.700223
$761$	−17.2510	−0.625348	−0.312674	−	0.949860i	$-0.601225\pi$
$761$	−17.2510	−0.625348	−0.312674	+	0.949860i	$0.601225\pi$
$762$	0	0
$763$	0	0
$764$	−41.8285	−1.51330
$765$	0	0
$766$	−0.531903	−0.0192184
$767$	2.16889	0.0783139
$768$	0	0
$769$	−21.3454	−0.769734	−0.384867	−	0.922972i	$-0.625753\pi$
$769$	−21.3454	−0.769734	−0.384867	+	0.922972i	$0.625753\pi$
$770$	0	0
$771$	0	0
$772$	−10.4099	−0.374660
$773$	13.1471	0.472870	0.236435	−	0.971647i	$-0.424021\pi$
$773$	13.1471	0.472870	0.236435	+	0.971647i	$0.424021\pi$
$774$	0	0
$775$	−30.9583	−1.11205
$776$	−20.5593	−0.738036
$777$	0	0
$778$	11.7737	0.422108
$779$	13.4288	0.481136
$780$	0	0
$781$	4.41405	0.157947
$782$	−0.307134	−0.0109831
$783$	0	0
$784$	0	0
$785$	−61.5909	−2.19827
$786$	0	0
$787$	28.1301	1.00273	0.501364	−	0.865236i	$-0.332832\pi$
$787$	28.1301	1.00273	0.501364	+	0.865236i	$0.332832\pi$
$788$	−27.1330	−0.966573
$789$	0	0
$790$	3.29425	0.117204
$791$	0	0
$792$	0	0
$793$	−12.9294	−0.459136
$794$	−0.0159163	−0.000564847 0
$795$	0	0
$796$	−27.1824	−0.963455
$797$	−25.7365	−0.911635	−0.455817	−	0.890073i	$-0.650653\pi$
$797$	−25.7365	−0.911635	−0.455817	+	0.890073i	$0.650653\pi$
$798$	0	0
$799$	−2.54309	−0.0899681
$800$	43.2188	1.52801
$801$	0	0
$802$	−12.1615	−0.429436
$803$	−1.63387	−0.0576580
$804$	0	0
$805$	0	0
$806$	−2.12747	−0.0749368
$807$	0	0
$808$	4.83936	0.170248
$809$	31.8705	1.12051	0.560254	−	0.828321i	$-0.310704\pi$
$809$	31.8705	1.12051	0.560254	+	0.828321i	$0.310704\pi$
$810$	0	0
$811$	−43.3860	−1.52349	−0.761744	−	0.647878i	$-0.775657\pi$
$811$	−43.3860	−1.52349	−0.761744	+	0.647878i	$0.775657\pi$
$812$	0	0
$813$	0	0
$814$	2.09408	0.0733975
$815$	−24.6764	−0.864375
$816$	0	0
$817$	28.0226	0.980385
$818$	−13.2842	−0.464470
$819$	0	0
$820$	30.9731	1.08163
$821$	16.3935	0.572139	0.286069	−	0.958209i	$-0.407651\pi$
$821$	16.3935	0.572139	0.286069	+	0.958209i	$0.407651\pi$
$822$	0	0
$823$	−26.3780	−0.919478	−0.459739	−	0.888054i	$-0.652057\pi$
$823$	−26.3780	−0.919478	−0.459739	+	0.888054i	$0.652057\pi$
$824$	18.0785	0.629794
$825$	0	0
$826$	0	0
$827$	−36.7225	−1.27697	−0.638484	−	0.769635i	$-0.720438\pi$
$827$	−36.7225	−1.27697	−0.638484	+	0.769635i	$0.720438\pi$
$828$	0	0
$829$	24.3158	0.844522	0.422261	−	0.906474i	$-0.361237\pi$
$829$	24.3158	0.844522	0.422261	+	0.906474i	$0.361237\pi$
$830$	22.5640	0.783209
$831$	0	0
$832$	−3.21734	−0.111541
$833$	0	0
$834$	0	0
$835$	65.1038	2.25301
$836$	−4.39682	−0.152067
$837$	0	0
$838$	10.4392	0.360618
$839$	25.6810	0.886606	0.443303	−	0.896372i	$-0.353807\pi$
$839$	25.6810	0.886606	0.443303	+	0.896372i	$0.353807\pi$
$840$	0	0
$841$	−11.8393	−0.408251
$842$	7.38381	0.254463
$843$	0	0
$844$	2.70800	0.0932131
$845$	42.7153	1.46945
$846$	0	0
$847$	0	0
$848$	−25.5653	−0.877915
$849$	0	0
$850$	1.07040	0.0367144
$851$	−11.7230	−0.401858
$852$	0	0
$853$	28.9743	0.992061	0.496031	−	0.868305i	$-0.334790\pi$
$853$	28.9743	0.992061	0.496031	+	0.868305i	$0.334790\pi$
$854$	0	0
$855$	0	0
$856$	20.2918	0.693559
$857$	−25.3868	−0.867197	−0.433598	−	0.901106i	$-0.642756\pi$
$857$	−25.3868	−0.867197	−0.433598	+	0.901106i	$0.642756\pi$
$858$	0	0
$859$	5.95783	0.203279	0.101639	−	0.994821i	$-0.467591\pi$
$859$	5.95783	0.203279	0.101639	+	0.994821i	$0.467591\pi$
$860$	64.6333	2.20398
$861$	0	0
$862$	−7.88621	−0.268605
$863$	16.3909	0.557953	0.278977	−	0.960298i	$-0.410005\pi$
$863$	16.3909	0.557953	0.278977	+	0.960298i	$0.410005\pi$
$864$	0	0
$865$	14.3525	0.487999
$866$	8.11528	0.275768
$867$	0	0
$868$	0	0
$869$	−1.60584	−0.0544744
$870$	0	0
$871$	1.23186	0.0417399
$872$	−3.95006	−0.133766
$873$	0	0
$874$	−3.45030	−0.116708
$875$	0	0
$876$	0	0
$877$	35.2539	1.19044	0.595220	−	0.803563i	$-0.297065\pi$
$877$	35.2539	1.19044	0.595220	+	0.803563i	$0.297065\pi$
$878$	7.70813	0.260137
$879$	0	0
$880$	−8.52033	−0.287220
$881$	26.2582	0.884661	0.442331	−	0.896852i	$-0.354152\pi$
$881$	26.2582	0.884661	0.442331	+	0.896852i	$0.354152\pi$
$882$	0	0
$883$	10.0087	0.336821	0.168410	−	0.985717i	$-0.446137\pi$
$883$	10.0087	0.336821	0.168410	+	0.985717i	$0.446137\pi$
$884$	−0.524756	−0.0176495
$885$	0	0
$886$	−0.887631	−0.0298205
$887$	−15.9056	−0.534059	−0.267030	−	0.963688i	$-0.586042\pi$
$887$	−15.9056	−0.534059	−0.267030	+	0.963688i	$0.586042\pi$
$888$	0	0
$889$	0	0
$890$	−4.40804	−0.147758
$891$	0	0
$892$	9.54838	0.319703
$893$	−28.5688	−0.956017
$894$	0	0
$895$	−27.0857	−0.905374
$896$	0	0
$897$	0	0
$898$	−6.72727	−0.224492
$899$	−14.8517	−0.495331
$900$	0	0
$901$	−2.47215	−0.0823594
$902$	2.11643	0.0704695
$903$	0	0
$904$	−29.4615	−0.979875
$905$	41.5657	1.38169
$906$	0	0
$907$	−17.0925	−0.567547	−0.283773	−	0.958891i	$-0.591586\pi$
$907$	−17.0925	−0.567547	−0.283773	+	0.958891i	$0.591586\pi$
$908$	28.1998	0.935843
$909$	0	0
$910$	0	0
$911$	29.8869	0.990199	0.495099	−	0.868836i	$-0.335132\pi$
$911$	29.8869	0.990199	0.495099	+	0.868836i	$0.335132\pi$
$912$	0	0
$913$	−10.9992	−0.364021
$914$	1.27397	0.0421393
$915$	0	0
$916$	−17.4857	−0.577744
$917$	0	0
$918$	0	0
$919$	−23.6567	−0.780362	−0.390181	−	0.920738i	$-0.627588\pi$
$919$	−23.6567	−0.780362	−0.390181	+	0.920738i	$0.627588\pi$
$920$	−17.0316	−0.561515
$921$	0	0
$922$	−17.9439	−0.590952
$923$	5.91813	0.194798
$924$	0	0
$925$	40.8559	1.34333
$926$	−8.12454	−0.266989
$927$	0	0
$928$	20.7334	0.680607
$929$	12.6176	0.413970	0.206985	−	0.978344i	$-0.433635\pi$
$929$	12.6176	0.413970	0.206985	+	0.978344i	$0.433635\pi$
$930$	0	0
$931$	0	0
$932$	−29.0146	−0.950405
$933$	0	0
$934$	4.31427	0.141167
$935$	−0.823914	−0.0269449
$936$	0	0
$937$	26.3440	0.860622	0.430311	−	0.902681i	$-0.358404\pi$
$937$	26.3440	0.860622	0.430311	+	0.902681i	$0.358404\pi$
$938$	0	0
$939$	0	0
$940$	−65.8930	−2.14919
$941$	50.9397	1.66059	0.830294	−	0.557326i	$-0.188173\pi$
$941$	50.9397	1.66059	0.830294	+	0.557326i	$0.188173\pi$
$942$	0	0
$943$	−11.8481	−0.385827
$944$	4.68525	0.152492
$945$	0	0
$946$	4.41648	0.143592
$947$	−27.6798	−0.899474	−0.449737	−	0.893161i	$-0.648482\pi$
$947$	−27.6798	−0.899474	−0.449737	+	0.893161i	$0.648482\pi$
$948$	0	0
$949$	−2.19061	−0.0711102
$950$	12.0247	0.390134
$951$	0	0
$952$	0	0
$953$	27.4017	0.887628	0.443814	−	0.896119i	$-0.353625\pi$
$953$	27.4017	0.887628	0.443814	+	0.896119i	$0.353625\pi$
$954$	0	0
$955$	−88.0530	−2.84933
$956$	38.6324	1.24946
$957$	0	0
$958$	−8.81029	−0.284648
$959$	0	0
$960$	0	0
$961$	−18.1467	−0.585378
$962$	2.80764	0.0905219
$963$	0	0
$964$	29.3289	0.944621
$965$	−21.9138	−0.705431
$966$	0	0
$967$	−18.1814	−0.584674	−0.292337	−	0.956315i	$-0.594433\pi$
$967$	−18.1814	−0.584674	−0.292337	+	0.956315i	$0.594433\pi$
$968$	18.9940	0.610489
$969$	0	0
$970$	−20.2223	−0.649299
$971$	−39.4832	−1.26708	−0.633538	−	0.773712i	$-0.718398\pi$
$971$	−39.4832	−1.26708	−0.633538	+	0.773712i	$0.718398\pi$
$972$	0	0
$973$	0	0
$974$	−8.25881	−0.264629
$975$	0	0
$976$	−27.9302	−0.894024
$977$	−11.9156	−0.381215	−0.190608	−	0.981666i	$-0.561046\pi$
$977$	−11.9156	−0.381215	−0.190608	+	0.981666i	$0.561046\pi$
$978$	0	0
$979$	2.14878	0.0686752
$980$	0	0
$981$	0	0
$982$	−3.18368	−0.101595
$983$	−18.4779	−0.589354	−0.294677	−	0.955597i	$-0.595212\pi$
$983$	−18.4779	−0.589354	−0.294677	+	0.955597i	$0.595212\pi$
$984$	0	0
$985$	−57.1176	−1.81992
$986$	0.513505	0.0163533
$987$	0	0
$988$	−5.89504	−0.187546
$989$	−24.7241	−0.786179
$990$	0	0
$991$	12.6970	0.403334	0.201667	−	0.979454i	$-0.435364\pi$
$991$	12.6970	0.403334	0.201667	+	0.979454i	$0.435364\pi$
$992$	−17.9436	−0.569710
$993$	0	0
$994$	0	0
$995$	−57.2215	−1.81405
$996$	0	0
$997$	−41.9533	−1.32868	−0.664338	−	0.747432i	$-0.731286\pi$
$997$	−41.9533	−1.32868	−0.664338	+	0.747432i	$0.731286\pi$
$998$	5.52690	0.174951
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bb.1.3		5
3.2	odd	2		3969.2.a.ba.1.3		5
7.3	odd	6		567.2.e.e.163.3		10
7.5	odd	6		567.2.e.e.487.3		10
7.6	odd	2		3969.2.a.bc.1.3		5
9.2	odd	6		441.2.f.f.148.3		10
9.4	even	3		1323.2.f.f.883.3		10
9.5	odd	6		441.2.f.f.295.3		10
9.7	even	3		1323.2.f.f.442.3		10
21.5	even	6		567.2.e.f.487.3		10
21.17	even	6		567.2.e.f.163.3		10
21.20	even	2		3969.2.a.z.1.3		5
63.2	odd	6		441.2.g.f.67.3		10
63.4	even	3		1323.2.g.f.667.3		10
63.5	even	6		63.2.h.b.25.3	yes	10
63.11	odd	6		441.2.h.f.373.3		10
63.13	odd	6		1323.2.f.e.883.3		10
63.16	even	3		1323.2.g.f.361.3		10
63.20	even	6		441.2.f.e.148.3		10
63.23	odd	6		441.2.h.f.214.3		10
63.25	even	3		1323.2.h.f.226.3		10
63.31	odd	6		189.2.g.b.100.3		10
63.32	odd	6		441.2.g.f.79.3		10
63.34	odd	6		1323.2.f.e.442.3		10
63.38	even	6		63.2.h.b.58.3	yes	10
63.40	odd	6		189.2.h.b.46.3		10
63.41	even	6		441.2.f.e.295.3		10
63.47	even	6		63.2.g.b.4.3	✓	10
63.52	odd	6		189.2.h.b.37.3		10
63.58	even	3		1323.2.h.f.802.3		10
63.59	even	6		63.2.g.b.16.3	yes	10
63.61	odd	6		189.2.g.b.172.3		10
252.31	even	6		3024.2.t.i.289.5		10
252.47	odd	6		1008.2.t.i.193.1		10
252.59	odd	6		1008.2.t.i.961.1		10
252.103	even	6		3024.2.q.i.2881.1		10
252.115	even	6		3024.2.q.i.2305.1		10
252.131	odd	6		1008.2.q.i.529.3		10
252.187	even	6		3024.2.t.i.1873.5		10
252.227	odd	6		1008.2.q.i.625.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.b.4.3	✓	10	63.47	even	6
63.2.g.b.16.3	yes	10	63.59	even	6
63.2.h.b.25.3	yes	10	63.5	even	6
63.2.h.b.58.3	yes	10	63.38	even	6
189.2.g.b.100.3		10	63.31	odd	6
189.2.g.b.172.3		10	63.61	odd	6
189.2.h.b.37.3		10	63.52	odd	6
189.2.h.b.46.3		10	63.40	odd	6
441.2.f.e.148.3		10	63.20	even	6
441.2.f.e.295.3		10	63.41	even	6
441.2.f.f.148.3		10	9.2	odd	6
441.2.f.f.295.3		10	9.5	odd	6
441.2.g.f.67.3		10	63.2	odd	6
441.2.g.f.79.3		10	63.32	odd	6
441.2.h.f.214.3		10	63.23	odd	6
441.2.h.f.373.3		10	63.11	odd	6
567.2.e.e.163.3		10	7.3	odd	6
567.2.e.e.487.3		10	7.5	odd	6
567.2.e.f.163.3		10	21.17	even	6
567.2.e.f.487.3		10	21.5	even	6
1008.2.q.i.529.3		10	252.131	odd	6
1008.2.q.i.625.3		10	252.227	odd	6
1008.2.t.i.193.1		10	252.47	odd	6
1008.2.t.i.961.1		10	252.59	odd	6
1323.2.f.e.442.3		10	63.34	odd	6
1323.2.f.e.883.3		10	63.13	odd	6
1323.2.f.f.442.3		10	9.7	even	3
1323.2.f.f.883.3		10	9.4	even	3
1323.2.g.f.361.3		10	63.16	even	3
1323.2.g.f.667.3		10	63.4	even	3
1323.2.h.f.226.3		10	63.25	even	3
1323.2.h.f.802.3		10	63.58	even	3
3024.2.q.i.2305.1		10	252.115	even	6
3024.2.q.i.2881.1		10	252.103	even	6
3024.2.t.i.289.5		10	252.31	even	6
3024.2.t.i.1873.5		10	252.187	even	6
3969.2.a.z.1.3		5	21.20	even	2
3969.2.a.ba.1.3		5	3.2	odd	2
3969.2.a.bb.1.3		5	1.1	even	1	trivial
3969.2.a.bc.1.3		5	7.6	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)

\(f(q)\)	\(=\)	\(q+0.495868 q^{2} -1.75411 q^{4} -3.69258 q^{5} -1.86155 q^{8} +O(q^{10})\)	\(q+0.495868 q^{2} -1.75411 q^{4} -3.69258 q^{5} -1.86155 q^{8} -1.83103 q^{10} +0.892568 q^{11} +1.19671 q^{13} +2.58515 q^{16} +0.249983 q^{17} +2.80827 q^{19} +6.47721 q^{20} +0.442596 q^{22} -2.47772 q^{23} +8.63514 q^{25} +0.593411 q^{26} +4.14255 q^{29} -3.58515 q^{31} +5.00499 q^{32} +0.123959 q^{34} +4.73136 q^{37} +1.39253 q^{38} +6.87391 q^{40} +4.78186 q^{41} +9.97857 q^{43} -1.56567 q^{44} -1.22862 q^{46} -10.1731 q^{47} +4.28189 q^{50} -2.09917 q^{52} -9.88929 q^{53} -3.29588 q^{55} +2.05416 q^{58} +1.81237 q^{59} -10.8041 q^{61} -1.77776 q^{62} -2.68848 q^{64} -4.41895 q^{65} +1.02937 q^{67} -0.438499 q^{68} +4.94533 q^{71} -1.83052 q^{73} +2.34613 q^{74} -4.92604 q^{76} -1.79912 q^{79} -9.54586 q^{80} +2.37117 q^{82} -12.3231 q^{83} -0.923082 q^{85} +4.94806 q^{86} -1.66156 q^{88} +2.40741 q^{89} +4.34620 q^{92} -5.04450 q^{94} -10.3698 q^{95} +11.0442 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 4 q^{4} - 4 q^{5} + 3 q^{8}+O(q^{10}) \) 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

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