LMFDB - Embedded newform 3720.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3720,2,Mod(1,3720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("3720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.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:	$29.7043495519$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.688892 q^{7} +1.00000 q^{9} +3.80642 q^{11} +0.688892 q^{13} -1.00000 q^{15} -6.70964 q^{17} -4.00000 q^{19} -0.688892 q^{21} -4.14764 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.969888 q^{29} -1.00000 q^{31} +3.80642 q^{33} +0.688892 q^{35} -5.73975 q^{37} +0.688892 q^{39} -7.05086 q^{41} -6.00000 q^{43} -1.00000 q^{45} +10.3827 q^{47} -6.52543 q^{49} -6.70964 q^{51} +12.3827 q^{53} -3.80642 q^{55} -4.00000 q^{57} -4.77631 q^{59} -6.29529 q^{61} -0.688892 q^{63} -0.688892 q^{65} -0.453829 q^{67} -4.14764 q^{69} -3.45875 q^{71} +13.3526 q^{73} +1.00000 q^{75} -2.62222 q^{77} -8.08742 q^{79} +1.00000 q^{81} +11.7605 q^{83} +6.70964 q^{85} -0.969888 q^{87} -14.0207 q^{89} -0.474572 q^{91} -1.00000 q^{93} +4.00000 q^{95} -0.949145 q^{97} +3.80642 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * q^97 - 2 * q^99

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	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.688892	−0.260377	−0.130188	−	0.991489i	$-0.541558\pi$
$7$	−0.688892	−0.260377	−0.130188	+	0.991489i	$0.541558\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.80642	1.14768	0.573840	−	0.818967i	$-0.305453\pi$
$11$	3.80642	1.14768	0.573840	+	0.818967i	$0.305453\pi$
$12$	0	0
$13$	0.688892	0.191064	0.0955322	−	0.995426i	$-0.469545\pi$
$13$	0.688892	0.191064	0.0955322	+	0.995426i	$0.469545\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.70964	−1.62733	−0.813663	−	0.581337i	$-0.802530\pi$
$17$	−6.70964	−1.62733	−0.813663	+	0.581337i	$0.802530\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−0.688892	−0.150329
$22$	0	0
$23$	−4.14764	−0.864843	−0.432422	−	0.901671i	$-0.642341\pi$
$23$	−4.14764	−0.864843	−0.432422	+	0.901671i	$0.642341\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.969888	−0.180104	−0.0900519	−	0.995937i	$-0.528703\pi$
$29$	−0.969888	−0.180104	−0.0900519	+	0.995937i	$0.528703\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	3.80642	0.662613
$34$	0	0
$35$	0.688892	0.116444
$36$	0	0
$37$	−5.73975	−0.943609	−0.471804	−	0.881703i	$-0.656397\pi$
$37$	−5.73975	−0.943609	−0.471804	+	0.881703i	$0.656397\pi$
$38$	0	0
$39$	0.688892	0.110311
$40$	0	0
$41$	−7.05086	−1.10116	−0.550579	−	0.834783i	$-0.685593\pi$
$41$	−7.05086	−1.10116	−0.550579	+	0.834783i	$0.685593\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	10.3827	1.51447	0.757237	−	0.653141i	$-0.226549\pi$
$47$	10.3827	1.51447	0.757237	+	0.653141i	$0.226549\pi$
$48$	0	0
$49$	−6.52543	−0.932204
$50$	0	0
$51$	−6.70964	−0.939537
$52$	0	0
$53$	12.3827	1.70090	0.850448	−	0.526059i	$-0.176331\pi$
$53$	12.3827	1.70090	0.850448	+	0.526059i	$0.176331\pi$
$54$	0	0
$55$	−3.80642	−0.513258
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−4.77631	−0.621823	−0.310911	−	0.950439i	$-0.600634\pi$
$59$	−4.77631	−0.621823	−0.310911	+	0.950439i	$0.600634\pi$
$60$	0	0
$61$	−6.29529	−0.806029	−0.403014	−	0.915194i	$-0.632038\pi$
$61$	−6.29529	−0.806029	−0.403014	+	0.915194i	$0.632038\pi$
$62$	0	0
$63$	−0.688892	−0.0867923
$64$	0	0
$65$	−0.688892	−0.0854466
$66$	0	0
$67$	−0.453829	−0.0554440	−0.0277220	−	0.999616i	$-0.508825\pi$
$67$	−0.453829	−0.0554440	−0.0277220	+	0.999616i	$0.508825\pi$
$68$	0	0
$69$	−4.14764	−0.499318
$70$	0	0
$71$	−3.45875	−0.410478	−0.205239	−	0.978712i	$-0.565797\pi$
$71$	−3.45875	−0.410478	−0.205239	+	0.978712i	$0.565797\pi$
$72$	0	0
$73$	13.3526	1.56280	0.781402	−	0.624029i	$-0.214505\pi$
$73$	13.3526	1.56280	0.781402	+	0.624029i	$0.214505\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.62222	−0.298829
$78$	0	0
$79$	−8.08742	−0.909906	−0.454953	−	0.890515i	$-0.650344\pi$
$79$	−8.08742	−0.909906	−0.454953	+	0.890515i	$0.650344\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.7605	1.29088	0.645441	−	0.763810i	$-0.276674\pi$
$83$	11.7605	1.29088	0.645441	+	0.763810i	$0.276674\pi$
$84$	0	0
$85$	6.70964	0.727762
$86$	0	0
$87$	−0.969888	−0.103983
$88$	0	0
$89$	−14.0207	−1.48620	−0.743098	−	0.669183i	$-0.766644\pi$
$89$	−14.0207	−1.48620	−0.743098	+	0.669183i	$0.766644\pi$
$90$	0	0
$91$	−0.474572	−0.0497487
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−0.949145	−0.0963711	−0.0481855	−	0.998838i	$-0.515344\pi$
$97$	−0.949145	−0.0963711	−0.0481855	+	0.998838i	$0.515344\pi$
$98$	0	0
$99$	3.80642	0.382560
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	5.87310	0.578694	0.289347	−	0.957224i	$-0.406562\pi$
$103$	5.87310	0.578694	0.289347	+	0.957224i	$0.406562\pi$
$104$	0	0
$105$	0.688892	0.0672290
$106$	0	0
$107$	−5.39207	−0.521272	−0.260636	−	0.965437i	$-0.583932\pi$
$107$	−5.39207	−0.521272	−0.260636	+	0.965437i	$0.583932\pi$
$108$	0	0
$109$	12.6178	1.20856	0.604282	−	0.796771i	$-0.293460\pi$
$109$	12.6178	1.20856	0.604282	+	0.796771i	$0.293460\pi$
$110$	0	0
$111$	−5.73975	−0.544793
$112$	0	0
$113$	−11.2859	−1.06169	−0.530845	−	0.847469i	$-0.678125\pi$
$113$	−11.2859	−1.06169	−0.530845	+	0.847469i	$0.678125\pi$
$114$	0	0
$115$	4.14764	0.386770
$116$	0	0
$117$	0.688892	0.0636881
$118$	0	0
$119$	4.62222	0.423718
$120$	0	0
$121$	3.48886	0.317169
$122$	0	0
$123$	−7.05086	−0.635754
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.4795	−1.37358	−0.686792	−	0.726854i	$-0.740981\pi$
$127$	−15.4795	−1.37358	−0.686792	+	0.726854i	$0.740981\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−2.96989	−0.259480	−0.129740	−	0.991548i	$-0.541414\pi$
$131$	−2.96989	−0.259480	−0.129740	+	0.991548i	$0.541414\pi$
$132$	0	0
$133$	2.75557	0.238938
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−3.65878	−0.312591	−0.156295	−	0.987710i	$-0.549955\pi$
$137$	−3.65878	−0.312591	−0.156295	+	0.987710i	$0.549955\pi$
$138$	0	0
$139$	−14.6222	−1.24024	−0.620120	−	0.784507i	$-0.712916\pi$
$139$	−14.6222	−1.24024	−0.620120	+	0.784507i	$0.712916\pi$
$140$	0	0
$141$	10.3827	0.874382
$142$	0	0
$143$	2.62222	0.219281
$144$	0	0
$145$	0.969888	0.0805449
$146$	0	0
$147$	−6.52543	−0.538208
$148$	0	0
$149$	−4.56199	−0.373733	−0.186866	−	0.982385i	$-0.559833\pi$
$149$	−4.56199	−0.373733	−0.186866	+	0.982385i	$0.559833\pi$
$150$	0	0
$151$	4.63651	0.377313	0.188657	−	0.982043i	$-0.439587\pi$
$151$	4.63651	0.377313	0.188657	+	0.982043i	$0.439587\pi$
$152$	0	0
$153$	−6.70964	−0.542442
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−3.61285	−0.288337	−0.144168	−	0.989553i	$-0.546051\pi$
$157$	−3.61285	−0.288337	−0.144168	+	0.989553i	$0.546051\pi$
$158$	0	0
$159$	12.3827	0.982013
$160$	0	0
$161$	2.85728	0.225185
$162$	0	0
$163$	−19.9748	−1.56455	−0.782274	−	0.622935i	$-0.785940\pi$
$163$	−19.9748	−1.56455	−0.782274	+	0.622935i	$0.785940\pi$
$164$	0	0
$165$	−3.80642	−0.296330
$166$	0	0
$167$	−20.0415	−1.55086	−0.775428	−	0.631435i	$-0.782466\pi$
$167$	−20.0415	−1.55086	−0.775428	+	0.631435i	$0.782466\pi$
$168$	0	0
$169$	−12.5254	−0.963494
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	15.2859	1.16217	0.581083	−	0.813844i	$-0.302629\pi$
$173$	15.2859	1.16217	0.581083	+	0.813844i	$0.302629\pi$
$174$	0	0
$175$	−0.688892	−0.0520754
$176$	0	0
$177$	−4.77631	−0.359010
$178$	0	0
$179$	−6.19358	−0.462930	−0.231465	−	0.972843i	$-0.574352\pi$
$179$	−6.19358	−0.462930	−0.231465	+	0.972843i	$0.574352\pi$
$180$	0	0
$181$	−9.05086	−0.672745	−0.336372	−	0.941729i	$-0.609200\pi$
$181$	−9.05086	−0.672745	−0.336372	+	0.941729i	$0.609200\pi$
$182$	0	0
$183$	−6.29529	−0.465361
$184$	0	0
$185$	5.73975	0.421995
$186$	0	0
$187$	−25.5397	−1.86765
$188$	0	0
$189$	−0.688892	−0.0501095
$190$	0	0
$191$	−18.0622	−1.30694	−0.653469	−	0.756954i	$-0.726687\pi$
$191$	−18.0622	−1.30694	−0.653469	+	0.756954i	$0.726687\pi$
$192$	0	0
$193$	−15.2257	−1.09597	−0.547985	−	0.836488i	$-0.684605\pi$
$193$	−15.2257	−1.09597	−0.547985	+	0.836488i	$0.684605\pi$
$194$	0	0
$195$	−0.688892	−0.0493326
$196$	0	0
$197$	26.8845	1.91544	0.957720	−	0.287703i	$-0.0928915\pi$
$197$	26.8845	1.91544	0.957720	+	0.287703i	$0.0928915\pi$
$198$	0	0
$199$	−4.34122	−0.307741	−0.153870	−	0.988091i	$-0.549174\pi$
$199$	−4.34122	−0.307741	−0.153870	+	0.988091i	$0.549174\pi$
$200$	0	0
$201$	−0.453829	−0.0320106
$202$	0	0
$203$	0.668149	0.0468948
$204$	0	0
$205$	7.05086	0.492453
$206$	0	0
$207$	−4.14764	−0.288281
$208$	0	0
$209$	−15.2257	−1.05318
$210$	0	0
$211$	−9.67307	−0.665922	−0.332961	−	0.942941i	$-0.608048\pi$
$211$	−9.67307	−0.665922	−0.332961	+	0.942941i	$0.608048\pi$
$212$	0	0
$213$	−3.45875	−0.236990
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	0.688892	0.0467650
$218$	0	0
$219$	13.3526	0.902285
$220$	0	0
$221$	−4.62222	−0.310924
$222$	0	0
$223$	3.96836	0.265741	0.132870	−	0.991133i	$-0.457581\pi$
$223$	3.96836	0.265741	0.132870	+	0.991133i	$0.457581\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	19.3733	1.28585	0.642927	−	0.765928i	$-0.277720\pi$
$227$	19.3733	1.28585	0.642927	+	0.765928i	$0.277720\pi$
$228$	0	0
$229$	−7.67307	−0.507051	−0.253525	−	0.967329i	$-0.581590\pi$
$229$	−7.67307	−0.507051	−0.253525	+	0.967329i	$0.581590\pi$
$230$	0	0
$231$	−2.62222	−0.172529
$232$	0	0
$233$	−16.6780	−1.09261	−0.546306	−	0.837586i	$-0.683966\pi$
$233$	−16.6780	−1.09261	−0.546306	+	0.837586i	$0.683966\pi$
$234$	0	0
$235$	−10.3827	−0.677293
$236$	0	0
$237$	−8.08742	−0.525334
$238$	0	0
$239$	2.13335	0.137995	0.0689976	−	0.997617i	$-0.478020\pi$
$239$	2.13335	0.137995	0.0689976	+	0.997617i	$0.478020\pi$
$240$	0	0
$241$	−14.3368	−0.923513	−0.461756	−	0.887007i	$-0.652781\pi$
$241$	−14.3368	−0.923513	−0.461756	+	0.887007i	$0.652781\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.52543	0.416894
$246$	0	0
$247$	−2.75557	−0.175333
$248$	0	0
$249$	11.7605	0.745291
$250$	0	0
$251$	0.295286	0.0186383	0.00931916	−	0.999957i	$-0.497034\pi$
$251$	0.295286	0.0186383	0.00931916	+	0.999957i	$0.497034\pi$
$252$	0	0
$253$	−15.7877	−0.992563
$254$	0	0
$255$	6.70964	0.420174
$256$	0	0
$257$	27.4336	1.71126	0.855629	−	0.517589i	$-0.173170\pi$
$257$	27.4336	1.71126	0.855629	+	0.517589i	$0.173170\pi$
$258$	0	0
$259$	3.95407	0.245694
$260$	0	0
$261$	−0.969888	−0.0600346
$262$	0	0
$263$	16.7239	1.03124	0.515621	−	0.856817i	$-0.327561\pi$
$263$	16.7239	1.03124	0.515621	+	0.856817i	$0.327561\pi$
$264$	0	0
$265$	−12.3827	−0.760664
$266$	0	0
$267$	−14.0207	−0.858056
$268$	0	0
$269$	2.05239	0.125136	0.0625681	−	0.998041i	$-0.480071\pi$
$269$	2.05239	0.125136	0.0625681	+	0.998041i	$0.480071\pi$
$270$	0	0
$271$	24.3368	1.47835	0.739177	−	0.673511i	$-0.235215\pi$
$271$	24.3368	1.47835	0.739177	+	0.673511i	$0.235215\pi$
$272$	0	0
$273$	−0.474572	−0.0287224
$274$	0	0
$275$	3.80642	0.229536
$276$	0	0
$277$	8.88247	0.533696	0.266848	−	0.963739i	$-0.414018\pi$
$277$	8.88247	0.533696	0.266848	+	0.963739i	$0.414018\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	31.4608	1.87679	0.938396	−	0.345562i	$-0.112312\pi$
$281$	31.4608	1.87679	0.938396	+	0.345562i	$0.112312\pi$
$282$	0	0
$283$	−16.2099	−0.963577	−0.481788	−	0.876288i	$-0.660013\pi$
$283$	−16.2099	−0.963577	−0.481788	+	0.876288i	$0.660013\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	4.85728	0.286716
$288$	0	0
$289$	28.0192	1.64819
$290$	0	0
$291$	−0.949145	−0.0556399
$292$	0	0
$293$	12.4746	0.728772	0.364386	−	0.931248i	$-0.381279\pi$
$293$	12.4746	0.728772	0.364386	+	0.931248i	$0.381279\pi$
$294$	0	0
$295$	4.77631	0.278088
$296$	0	0
$297$	3.80642	0.220871
$298$	0	0
$299$	−2.85728	−0.165241
$300$	0	0
$301$	4.13335	0.238243
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.29529	0.360467
$306$	0	0
$307$	14.4035	0.822048	0.411024	−	0.911624i	$-0.365171\pi$
$307$	14.4035	0.822048	0.411024	+	0.911624i	$0.365171\pi$
$308$	0	0
$309$	5.87310	0.334109
$310$	0	0
$311$	6.77631	0.384249	0.192125	−	0.981371i	$-0.438462\pi$
$311$	6.77631	0.384249	0.192125	+	0.981371i	$0.438462\pi$
$312$	0	0
$313$	25.8829	1.46299	0.731495	−	0.681847i	$-0.238823\pi$
$313$	25.8829	1.46299	0.731495	+	0.681847i	$0.238823\pi$
$314$	0	0
$315$	0.688892	0.0388147
$316$	0	0
$317$	−25.5353	−1.43420	−0.717102	−	0.696968i	$-0.754532\pi$
$317$	−25.5353	−1.43420	−0.717102	+	0.696968i	$0.754532\pi$
$318$	0	0
$319$	−3.69181	−0.206701
$320$	0	0
$321$	−5.39207	−0.300956
$322$	0	0
$323$	26.8385	1.49334
$324$	0	0
$325$	0.688892	0.0382129
$326$	0	0
$327$	12.6178	0.697764
$328$	0	0
$329$	−7.15257	−0.394334
$330$	0	0
$331$	−20.7699	−1.14161	−0.570807	−	0.821084i	$-0.693370\pi$
$331$	−20.7699	−1.14161	−0.570807	+	0.821084i	$0.693370\pi$
$332$	0	0
$333$	−5.73975	−0.314536
$334$	0	0
$335$	0.453829	0.0247953
$336$	0	0
$337$	−2.98418	−0.162559	−0.0812793	−	0.996691i	$-0.525901\pi$
$337$	−2.98418	−0.162559	−0.0812793	+	0.996691i	$0.525901\pi$
$338$	0	0
$339$	−11.2859	−0.612967
$340$	0	0
$341$	−3.80642	−0.206129
$342$	0	0
$343$	9.31756	0.503101
$344$	0	0
$345$	4.14764	0.223302
$346$	0	0
$347$	−13.0825	−0.702305	−0.351153	−	0.936318i	$-0.614210\pi$
$347$	−13.0825	−0.702305	−0.351153	+	0.936318i	$0.614210\pi$
$348$	0	0
$349$	10.8113	0.578718	0.289359	−	0.957221i	$-0.406558\pi$
$349$	10.8113	0.578718	0.289359	+	0.957221i	$0.406558\pi$
$350$	0	0
$351$	0.688892	0.0367703
$352$	0	0
$353$	27.7418	1.47654	0.738272	−	0.674503i	$-0.235642\pi$
$353$	27.7418	1.47654	0.738272	+	0.674503i	$0.235642\pi$
$354$	0	0
$355$	3.45875	0.183571
$356$	0	0
$357$	4.62222	0.244634
$358$	0	0
$359$	2.11261	0.111499	0.0557496	−	0.998445i	$-0.482245\pi$
$359$	2.11261	0.111499	0.0557496	+	0.998445i	$0.482245\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	3.48886	0.183118
$364$	0	0
$365$	−13.3526	−0.698907
$366$	0	0
$367$	−34.1847	−1.78443	−0.892213	−	0.451615i	$-0.850848\pi$
$367$	−34.1847	−1.78443	−0.892213	+	0.451615i	$0.850848\pi$
$368$	0	0
$369$	−7.05086	−0.367053
$370$	0	0
$371$	−8.53035	−0.442874
$372$	0	0
$373$	−0.797056	−0.0412700	−0.0206350	−	0.999787i	$-0.506569\pi$
$373$	−0.797056	−0.0412700	−0.0206350	+	0.999787i	$0.506569\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−0.668149	−0.0344114
$378$	0	0
$379$	−14.6351	−0.751756	−0.375878	−	0.926669i	$-0.622659\pi$
$379$	−14.6351	−0.751756	−0.375878	+	0.926669i	$0.622659\pi$
$380$	0	0
$381$	−15.4795	−0.793039
$382$	0	0
$383$	−18.0370	−0.921650	−0.460825	−	0.887491i	$-0.652446\pi$
$383$	−18.0370	−0.921650	−0.460825	+	0.887491i	$0.652446\pi$
$384$	0	0
$385$	2.62222	0.133640
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	36.2558	1.83824	0.919121	−	0.393975i	$-0.128900\pi$
$389$	36.2558	1.83824	0.919121	+	0.393975i	$0.128900\pi$
$390$	0	0
$391$	27.8292	1.40738
$392$	0	0
$393$	−2.96989	−0.149811
$394$	0	0
$395$	8.08742	0.406922
$396$	0	0
$397$	9.70471	0.487066	0.243533	−	0.969893i	$-0.421694\pi$
$397$	9.70471	0.487066	0.243533	+	0.969893i	$0.421694\pi$
$398$	0	0
$399$	2.75557	0.137951
$400$	0	0
$401$	21.7540	1.08634	0.543172	−	0.839621i	$-0.317223\pi$
$401$	21.7540	1.08634	0.543172	+	0.839621i	$0.317223\pi$
$402$	0	0
$403$	−0.688892	−0.0343162
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−21.8479	−1.08296
$408$	0	0
$409$	−9.90813	−0.489926	−0.244963	−	0.969532i	$-0.578776\pi$
$409$	−9.90813	−0.489926	−0.244963	+	0.969532i	$0.578776\pi$
$410$	0	0
$411$	−3.65878	−0.180474
$412$	0	0
$413$	3.29036	0.161908
$414$	0	0
$415$	−11.7605	−0.577300
$416$	0	0
$417$	−14.6222	−0.716053
$418$	0	0
$419$	−17.2237	−0.841432	−0.420716	−	0.907192i	$-0.638221\pi$
$419$	−17.2237	−0.841432	−0.420716	+	0.907192i	$0.638221\pi$
$420$	0	0
$421$	5.43356	0.264816	0.132408	−	0.991195i	$-0.457729\pi$
$421$	5.43356	0.264816	0.132408	+	0.991195i	$0.457729\pi$
$422$	0	0
$423$	10.3827	0.504824
$424$	0	0
$425$	−6.70964	−0.325465
$426$	0	0
$427$	4.33677	0.209871
$428$	0	0
$429$	2.62222	0.126602
$430$	0	0
$431$	7.16346	0.345052	0.172526	−	0.985005i	$-0.444807\pi$
$431$	7.16346	0.345052	0.172526	+	0.985005i	$0.444807\pi$
$432$	0	0
$433$	26.8637	1.29099	0.645494	−	0.763765i	$-0.276651\pi$
$433$	26.8637	1.29099	0.645494	+	0.763765i	$0.276651\pi$
$434$	0	0
$435$	0.969888	0.0465026
$436$	0	0
$437$	16.5906	0.793635
$438$	0	0
$439$	3.34614	0.159703	0.0798513	−	0.996807i	$-0.474555\pi$
$439$	3.34614	0.159703	0.0798513	+	0.996807i	$0.474555\pi$
$440$	0	0
$441$	−6.52543	−0.310735
$442$	0	0
$443$	−11.3733	−0.540364	−0.270182	−	0.962809i	$-0.587084\pi$
$443$	−11.3733	−0.540364	−0.270182	+	0.962809i	$0.587084\pi$
$444$	0	0
$445$	14.0207	0.664647
$446$	0	0
$447$	−4.56199	−0.215775
$448$	0	0
$449$	20.1225	0.949637	0.474819	−	0.880084i	$-0.342514\pi$
$449$	20.1225	0.949637	0.474819	+	0.880084i	$0.342514\pi$
$450$	0	0
$451$	−26.8385	−1.26378
$452$	0	0
$453$	4.63651	0.217842
$454$	0	0
$455$	0.474572	0.0222483
$456$	0	0
$457$	2.42219	0.113305	0.0566525	−	0.998394i	$-0.481957\pi$
$457$	2.42219	0.113305	0.0566525	+	0.998394i	$0.481957\pi$
$458$	0	0
$459$	−6.70964	−0.313179
$460$	0	0
$461$	−3.41726	−0.159158	−0.0795789	−	0.996829i	$-0.525358\pi$
$461$	−3.41726	−0.159158	−0.0795789	+	0.996829i	$0.525358\pi$
$462$	0	0
$463$	−6.56199	−0.304962	−0.152481	−	0.988306i	$-0.548726\pi$
$463$	−6.56199	−0.304962	−0.152481	+	0.988306i	$0.548726\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−1.52543	−0.0705884	−0.0352942	−	0.999377i	$-0.511237\pi$
$467$	−1.52543	−0.0705884	−0.0352942	+	0.999377i	$0.511237\pi$
$468$	0	0
$469$	0.312639	0.0144363
$470$	0	0
$471$	−3.61285	−0.166471
$472$	0	0
$473$	−22.8385	−1.05012
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	12.3827	0.566965
$478$	0	0
$479$	28.4494	1.29989	0.649943	−	0.759983i	$-0.274793\pi$
$479$	28.4494	1.29989	0.649943	+	0.759983i	$0.274793\pi$
$480$	0	0
$481$	−3.95407	−0.180290
$482$	0	0
$483$	2.85728	0.130011
$484$	0	0
$485$	0.949145	0.0430984
$486$	0	0
$487$	−31.5714	−1.43063	−0.715317	−	0.698800i	$-0.753718\pi$
$487$	−31.5714	−1.43063	−0.715317	+	0.698800i	$0.753718\pi$
$488$	0	0
$489$	−19.9748	−0.903292
$490$	0	0
$491$	1.40636	0.0634683	0.0317342	−	0.999496i	$-0.489897\pi$
$491$	1.40636	0.0634683	0.0317342	+	0.999496i	$0.489897\pi$
$492$	0	0
$493$	6.50760	0.293087
$494$	0	0
$495$	−3.80642	−0.171086
$496$	0	0
$497$	2.38271	0.106879
$498$	0	0
$499$	−31.6128	−1.41519	−0.707593	−	0.706621i	$-0.750219\pi$
$499$	−31.6128	−1.41519	−0.707593	+	0.706621i	$0.750219\pi$
$500$	0	0
$501$	−20.0415	−0.895388
$502$	0	0
$503$	29.9956	1.33744	0.668718	−	0.743516i	$-0.266843\pi$
$503$	29.9956	1.33744	0.668718	+	0.743516i	$0.266843\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.5254	−0.556274
$508$	0	0
$509$	−5.76694	−0.255615	−0.127808	−	0.991799i	$-0.540794\pi$
$509$	−5.76694	−0.255615	−0.127808	+	0.991799i	$0.540794\pi$
$510$	0	0
$511$	−9.19850	−0.406918
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−5.87310	−0.258800
$516$	0	0
$517$	39.5210	1.73813
$518$	0	0
$519$	15.2859	0.670977
$520$	0	0
$521$	29.1526	1.27720	0.638599	−	0.769540i	$-0.279514\pi$
$521$	29.1526	1.27720	0.638599	+	0.769540i	$0.279514\pi$
$522$	0	0
$523$	19.1842	0.838867	0.419433	−	0.907786i	$-0.362229\pi$
$523$	19.1842	0.838867	0.419433	+	0.907786i	$0.362229\pi$
$524$	0	0
$525$	−0.688892	−0.0300657
$526$	0	0
$527$	6.70964	0.292276
$528$	0	0
$529$	−5.79706	−0.252046
$530$	0	0
$531$	−4.77631	−0.207274
$532$	0	0
$533$	−4.85728	−0.210392
$534$	0	0
$535$	5.39207	0.233120
$536$	0	0
$537$	−6.19358	−0.267273
$538$	0	0
$539$	−24.8385	−1.06987
$540$	0	0
$541$	−36.7511	−1.58005	−0.790027	−	0.613072i	$-0.789934\pi$
$541$	−36.7511	−1.58005	−0.790027	+	0.613072i	$0.789934\pi$
$542$	0	0
$543$	−9.05086	−0.388409
$544$	0	0
$545$	−12.6178	−0.540486
$546$	0	0
$547$	−31.7081	−1.35574	−0.677870	−	0.735181i	$-0.737097\pi$
$547$	−31.7081	−1.35574	−0.677870	+	0.735181i	$0.737097\pi$
$548$	0	0
$549$	−6.29529	−0.268676
$550$	0	0
$551$	3.87955	0.165275
$552$	0	0
$553$	5.57136	0.236918
$554$	0	0
$555$	5.73975	0.243639
$556$	0	0
$557$	33.0781	1.40156	0.700781	−	0.713376i	$-0.252835\pi$
$557$	33.0781	1.40156	0.700781	+	0.713376i	$0.252835\pi$
$558$	0	0
$559$	−4.13335	−0.174822
$560$	0	0
$561$	−25.5397	−1.07829
$562$	0	0
$563$	−27.4019	−1.15485	−0.577427	−	0.816443i	$-0.695943\pi$
$563$	−27.4019	−1.15485	−0.577427	+	0.816443i	$0.695943\pi$
$564$	0	0
$565$	11.2859	0.474802
$566$	0	0
$567$	−0.688892	−0.0289308
$568$	0	0
$569$	37.2652	1.56224	0.781119	−	0.624383i	$-0.214649\pi$
$569$	37.2652	1.56224	0.781119	+	0.624383i	$0.214649\pi$
$570$	0	0
$571$	8.99063	0.376246	0.188123	−	0.982145i	$-0.439760\pi$
$571$	8.99063	0.376246	0.188123	+	0.982145i	$0.439760\pi$
$572$	0	0
$573$	−18.0622	−0.754561
$574$	0	0
$575$	−4.14764	−0.172969
$576$	0	0
$577$	−5.01921	−0.208953	−0.104476	−	0.994527i	$-0.533317\pi$
$577$	−5.01921	−0.208953	−0.104476	+	0.994527i	$0.533317\pi$
$578$	0	0
$579$	−15.2257	−0.632758
$580$	0	0
$581$	−8.10171	−0.336116
$582$	0	0
$583$	47.1338	1.95208
$584$	0	0
$585$	−0.688892	−0.0284822
$586$	0	0
$587$	−39.8292	−1.64393	−0.821963	−	0.569541i	$-0.807121\pi$
$587$	−39.8292	−1.64393	−0.821963	+	0.569541i	$0.807121\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	26.8845	1.10588
$592$	0	0
$593$	9.74620	0.400228	0.200114	−	0.979773i	$-0.435869\pi$
$593$	9.74620	0.400228	0.200114	+	0.979773i	$0.435869\pi$
$594$	0	0
$595$	−4.62222	−0.189492
$596$	0	0
$597$	−4.34122	−0.177674
$598$	0	0
$599$	6.98862	0.285547	0.142774	−	0.989755i	$-0.454398\pi$
$599$	6.98862	0.285547	0.142774	+	0.989755i	$0.454398\pi$
$600$	0	0
$601$	20.8256	0.849495	0.424748	−	0.905312i	$-0.360363\pi$
$601$	20.8256	0.849495	0.424748	+	0.905312i	$0.360363\pi$
$602$	0	0
$603$	−0.453829	−0.0184813
$604$	0	0
$605$	−3.48886	−0.141842
$606$	0	0
$607$	37.3022	1.51405	0.757025	−	0.653386i	$-0.226652\pi$
$607$	37.3022	1.51405	0.757025	+	0.653386i	$0.226652\pi$
$608$	0	0
$609$	0.668149	0.0270747
$610$	0	0
$611$	7.15257	0.289362
$612$	0	0
$613$	19.9017	0.803821	0.401911	−	0.915679i	$-0.368346\pi$
$613$	19.9017	0.803821	0.401911	+	0.915679i	$0.368346\pi$
$614$	0	0
$615$	7.05086	0.284318
$616$	0	0
$617$	−34.7940	−1.40075	−0.700377	−	0.713773i	$-0.746985\pi$
$617$	−34.7940	−1.40075	−0.700377	+	0.713773i	$0.746985\pi$
$618$	0	0
$619$	11.2587	0.452526	0.226263	−	0.974066i	$-0.427349\pi$
$619$	11.2587	0.452526	0.226263	+	0.974066i	$0.427349\pi$
$620$	0	0
$621$	−4.14764	−0.166439
$622$	0	0
$623$	9.65878	0.386971
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−15.2257	−0.608056
$628$	0	0
$629$	38.5116	1.53556
$630$	0	0
$631$	2.75557	0.109697	0.0548487	−	0.998495i	$-0.482532\pi$
$631$	2.75557	0.109697	0.0548487	+	0.998495i	$0.482532\pi$
$632$	0	0
$633$	−9.67307	−0.384470
$634$	0	0
$635$	15.4795	0.614285
$636$	0	0
$637$	−4.49532	−0.178111
$638$	0	0
$639$	−3.45875	−0.136826
$640$	0	0
$641$	−40.6528	−1.60569	−0.802845	−	0.596188i	$-0.796681\pi$
$641$	−40.6528	−1.60569	−0.802845	+	0.596188i	$0.796681\pi$
$642$	0	0
$643$	19.1842	0.756551	0.378276	−	0.925693i	$-0.376517\pi$
$643$	19.1842	0.756551	0.378276	+	0.925693i	$0.376517\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	−23.5210	−0.924705	−0.462353	−	0.886696i	$-0.652995\pi$
$647$	−23.5210	−0.924705	−0.462353	+	0.886696i	$0.652995\pi$
$648$	0	0
$649$	−18.1807	−0.713654
$650$	0	0
$651$	0.688892	0.0269998
$652$	0	0
$653$	42.0973	1.64739	0.823697	−	0.567031i	$-0.191908\pi$
$653$	42.0973	1.64739	0.823697	+	0.567031i	$0.191908\pi$
$654$	0	0
$655$	2.96989	0.116043
$656$	0	0
$657$	13.3526	0.520934
$658$	0	0
$659$	21.7067	0.845574	0.422787	−	0.906229i	$-0.361052\pi$
$659$	21.7067	0.845574	0.422787	+	0.906229i	$0.361052\pi$
$660$	0	0
$661$	22.1245	0.860542	0.430271	−	0.902700i	$-0.358418\pi$
$661$	22.1245	0.860542	0.430271	+	0.902700i	$0.358418\pi$
$662$	0	0
$663$	−4.62222	−0.179512
$664$	0	0
$665$	−2.75557	−0.106856
$666$	0	0
$667$	4.02275	0.155762
$668$	0	0
$669$	3.96836	0.153426
$670$	0	0
$671$	−23.9625	−0.925063
$672$	0	0
$673$	34.8450	1.34318	0.671588	−	0.740925i	$-0.265613\pi$
$673$	34.8450	1.34318	0.671588	+	0.740925i	$0.265613\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−8.94470	−0.343773	−0.171886	−	0.985117i	$-0.554986\pi$
$677$	−8.94470	−0.343773	−0.171886	+	0.985117i	$0.554986\pi$
$678$	0	0
$679$	0.653858	0.0250928
$680$	0	0
$681$	19.3733	0.742388
$682$	0	0
$683$	25.7703	0.986074	0.493037	−	0.870008i	$-0.335887\pi$
$683$	25.7703	0.986074	0.493037	+	0.870008i	$0.335887\pi$
$684$	0	0
$685$	3.65878	0.139795
$686$	0	0
$687$	−7.67307	−0.292746
$688$	0	0
$689$	8.53035	0.324980
$690$	0	0
$691$	−30.9175	−1.17616	−0.588079	−	0.808804i	$-0.700115\pi$
$691$	−30.9175	−1.17616	−0.588079	+	0.808804i	$0.700115\pi$
$692$	0	0
$693$	−2.62222	−0.0996097
$694$	0	0
$695$	14.6222	0.554652
$696$	0	0
$697$	47.3087	1.79194
$698$	0	0
$699$	−16.6780	−0.630820
$700$	0	0
$701$	−3.45091	−0.130339	−0.0651696	−	0.997874i	$-0.520759\pi$
$701$	−3.45091	−0.130339	−0.0651696	+	0.997874i	$0.520759\pi$
$702$	0	0
$703$	22.9590	0.865915
$704$	0	0
$705$	−10.3827	−0.391035
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−27.7146	−1.04084	−0.520421	−	0.853910i	$-0.674225\pi$
$709$	−27.7146	−1.04084	−0.520421	+	0.853910i	$0.674225\pi$
$710$	0	0
$711$	−8.08742	−0.303302
$712$	0	0
$713$	4.14764	0.155330
$714$	0	0
$715$	−2.62222	−0.0980653
$716$	0	0
$717$	2.13335	0.0796715
$718$	0	0
$719$	36.0098	1.34294	0.671470	−	0.741031i	$-0.265663\pi$
$719$	36.0098	1.34294	0.671470	+	0.741031i	$0.265663\pi$
$720$	0	0
$721$	−4.04593	−0.150678
$722$	0	0
$723$	−14.3368	−0.533190
$724$	0	0
$725$	−0.969888	−0.0360208
$726$	0	0
$727$	−43.7625	−1.62306	−0.811531	−	0.584310i	$-0.801365\pi$
$727$	−43.7625	−1.62306	−0.811531	+	0.584310i	$0.801365\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.2578	1.48899
$732$	0	0
$733$	19.5111	0.720660	0.360330	−	0.932825i	$-0.382664\pi$
$733$	19.5111	0.720660	0.360330	+	0.932825i	$0.382664\pi$
$734$	0	0
$735$	6.52543	0.240694
$736$	0	0
$737$	−1.72746	−0.0636320
$738$	0	0
$739$	−42.0558	−1.54705	−0.773523	−	0.633768i	$-0.781507\pi$
$739$	−42.0558	−1.54705	−0.773523	+	0.633768i	$0.781507\pi$
$740$	0	0
$741$	−2.75557	−0.101228
$742$	0	0
$743$	34.4385	1.26343	0.631713	−	0.775203i	$-0.282352\pi$
$743$	34.4385	1.26343	0.631713	+	0.775203i	$0.282352\pi$
$744$	0	0
$745$	4.56199	0.167138
$746$	0	0
$747$	11.7605	0.430294
$748$	0	0
$749$	3.71456	0.135727
$750$	0	0
$751$	1.29481	0.0472483	0.0236241	−	0.999721i	$-0.492480\pi$
$751$	1.29481	0.0472483	0.0236241	+	0.999721i	$0.492480\pi$
$752$	0	0
$753$	0.295286	0.0107608
$754$	0	0
$755$	−4.63651	−0.168740
$756$	0	0
$757$	16.5970	0.603229	0.301615	−	0.953430i	$-0.402474\pi$
$757$	16.5970	0.603229	0.301615	+	0.953430i	$0.402474\pi$
$758$	0	0
$759$	−15.7877	−0.573057
$760$	0	0
$761$	1.44002	0.0522005	0.0261003	−	0.999659i	$-0.491691\pi$
$761$	1.44002	0.0522005	0.0261003	+	0.999659i	$0.491691\pi$
$762$	0	0
$763$	−8.69228	−0.314682
$764$	0	0
$765$	6.70964	0.242587
$766$	0	0
$767$	−3.29036	−0.118808
$768$	0	0
$769$	13.1887	0.475595	0.237798	−	0.971315i	$-0.423575\pi$
$769$	13.1887	0.475595	0.237798	+	0.971315i	$0.423575\pi$
$770$	0	0
$771$	27.4336	0.987996
$772$	0	0
$773$	−46.4055	−1.66909	−0.834544	−	0.550941i	$-0.814269\pi$
$773$	−46.4055	−1.66909	−0.834544	+	0.550941i	$0.814269\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	3.95407	0.141851
$778$	0	0
$779$	28.2034	1.01049
$780$	0	0
$781$	−13.1655	−0.471098
$782$	0	0
$783$	−0.969888	−0.0346610
$784$	0	0
$785$	3.61285	0.128948
$786$	0	0
$787$	45.2672	1.61360	0.806800	−	0.590824i	$-0.201197\pi$
$787$	45.2672	1.61360	0.806800	+	0.590824i	$0.201197\pi$
$788$	0	0
$789$	16.7239	0.595388
$790$	0	0
$791$	7.77478	0.276439
$792$	0	0
$793$	−4.33677	−0.154003
$794$	0	0
$795$	−12.3827	−0.439169
$796$	0	0
$797$	−20.3956	−0.722450	−0.361225	−	0.932479i	$-0.617641\pi$
$797$	−20.3956	−0.722450	−0.361225	+	0.932479i	$0.617641\pi$
$798$	0	0
$799$	−69.6642	−2.46454
$800$	0	0
$801$	−14.0207	−0.495399
$802$	0	0
$803$	50.8256	1.79360
$804$	0	0
$805$	−2.85728	−0.100706
$806$	0	0
$807$	2.05239	0.0722474
$808$	0	0
$809$	−38.4780	−1.35281	−0.676407	−	0.736528i	$-0.736464\pi$
$809$	−38.4780	−1.35281	−0.676407	+	0.736528i	$0.736464\pi$
$810$	0	0
$811$	−11.0094	−0.386591	−0.193296	−	0.981141i	$-0.561918\pi$
$811$	−11.0094	−0.386591	−0.193296	+	0.981141i	$0.561918\pi$
$812$	0	0
$813$	24.3368	0.853528
$814$	0	0
$815$	19.9748	0.699687
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	−0.474572	−0.0165829
$820$	0	0
$821$	−52.2973	−1.82519	−0.912594	−	0.408867i	$-0.865924\pi$
$821$	−52.2973	−1.82519	−0.912594	+	0.408867i	$0.865924\pi$
$822$	0	0
$823$	−52.3595	−1.82514	−0.912569	−	0.408922i	$-0.865905\pi$
$823$	−52.3595	−1.82514	−0.912569	+	0.408922i	$0.865905\pi$
$824$	0	0
$825$	3.80642	0.132523
$826$	0	0
$827$	−30.9576	−1.07650	−0.538251	−	0.842785i	$-0.680915\pi$
$827$	−30.9576	−1.07650	−0.538251	+	0.842785i	$0.680915\pi$
$828$	0	0
$829$	−36.6766	−1.27383	−0.636916	−	0.770933i	$-0.719790\pi$
$829$	−36.6766	−1.27383	−0.636916	+	0.770933i	$0.719790\pi$
$830$	0	0
$831$	8.88247	0.308129
$832$	0	0
$833$	43.7832	1.51700
$834$	0	0
$835$	20.0415	0.693564
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	35.7067	1.23273	0.616366	−	0.787459i	$-0.288604\pi$
$839$	35.7067	1.23273	0.616366	+	0.787459i	$0.288604\pi$
$840$	0	0
$841$	−28.0593	−0.967563
$842$	0	0
$843$	31.4608	1.08357
$844$	0	0
$845$	12.5254	0.430888
$846$	0	0
$847$	−2.40345	−0.0825835
$848$	0	0
$849$	−16.2099	−0.556321
$850$	0	0
$851$	23.8064	0.816074
$852$	0	0
$853$	43.2386	1.48046	0.740231	−	0.672353i	$-0.234716\pi$
$853$	43.2386	1.48046	0.740231	+	0.672353i	$0.234716\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	45.6543	1.55952	0.779761	−	0.626077i	$-0.215340\pi$
$857$	45.6543	1.55952	0.779761	+	0.626077i	$0.215340\pi$
$858$	0	0
$859$	5.12399	0.174828	0.0874141	−	0.996172i	$-0.472140\pi$
$859$	5.12399	0.174828	0.0874141	+	0.996172i	$0.472140\pi$
$860$	0	0
$861$	4.85728	0.165536
$862$	0	0
$863$	−26.6923	−0.908616	−0.454308	−	0.890845i	$-0.650113\pi$
$863$	−26.6923	−0.908616	−0.454308	+	0.890845i	$0.650113\pi$
$864$	0	0
$865$	−15.2859	−0.519737
$866$	0	0
$867$	28.0192	0.951582
$868$	0	0
$869$	−30.7841	−1.04428
$870$	0	0
$871$	−0.312639	−0.0105934
$872$	0	0
$873$	−0.949145	−0.0321237
$874$	0	0
$875$	0.688892	0.0232888
$876$	0	0
$877$	−15.2672	−0.515536	−0.257768	−	0.966207i	$-0.582987\pi$
$877$	−15.2672	−0.515536	−0.257768	+	0.966207i	$0.582987\pi$
$878$	0	0
$879$	12.4746	0.420757
$880$	0	0
$881$	−23.6434	−0.796568	−0.398284	−	0.917262i	$-0.630394\pi$
$881$	−23.6434	−0.796568	−0.398284	+	0.917262i	$0.630394\pi$
$882$	0	0
$883$	4.26364	0.143483	0.0717415	−	0.997423i	$-0.477144\pi$
$883$	4.26364	0.143483	0.0717415	+	0.997423i	$0.477144\pi$
$884$	0	0
$885$	4.77631	0.160554
$886$	0	0
$887$	21.0464	0.706669	0.353335	−	0.935497i	$-0.385048\pi$
$887$	21.0464	0.706669	0.353335	+	0.935497i	$0.385048\pi$
$888$	0	0
$889$	10.6637	0.357649
$890$	0	0
$891$	3.80642	0.127520
$892$	0	0
$893$	−41.5308	−1.38978
$894$	0	0
$895$	6.19358	0.207028
$896$	0	0
$897$	−2.85728	−0.0954018
$898$	0	0
$899$	0.969888	0.0323476
$900$	0	0
$901$	−83.0835	−2.76791
$902$	0	0
$903$	4.13335	0.137549
$904$	0	0
$905$	9.05086	0.300861
$906$	0	0
$907$	9.75848	0.324025	0.162013	−	0.986789i	$-0.448201\pi$
$907$	9.75848	0.324025	0.162013	+	0.986789i	$0.448201\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11.9269	0.395155	0.197577	−	0.980287i	$-0.436693\pi$
$911$	11.9269	0.395155	0.197577	+	0.980287i	$0.436693\pi$
$912$	0	0
$913$	44.7654	1.48152
$914$	0	0
$915$	6.29529	0.208116
$916$	0	0
$917$	2.04593	0.0675627
$918$	0	0
$919$	14.0731	0.464230	0.232115	−	0.972688i	$-0.425435\pi$
$919$	14.0731	0.464230	0.232115	+	0.972688i	$0.425435\pi$
$920$	0	0
$921$	14.4035	0.474610
$922$	0	0
$923$	−2.38271	−0.0784277
$924$	0	0
$925$	−5.73975	−0.188722
$926$	0	0
$927$	5.87310	0.192898
$928$	0	0
$929$	25.0114	0.820597	0.410298	−	0.911951i	$-0.365425\pi$
$929$	25.0114	0.820597	0.410298	+	0.911951i	$0.365425\pi$
$930$	0	0
$931$	26.1017	0.855449
$932$	0	0
$933$	6.77631	0.221847
$934$	0	0
$935$	25.5397	0.835238
$936$	0	0
$937$	20.4987	0.669664	0.334832	−	0.942278i	$-0.391321\pi$
$937$	20.4987	0.669664	0.334832	+	0.942278i	$0.391321\pi$
$938$	0	0
$939$	25.8829	0.844658
$940$	0	0
$941$	9.59210	0.312694	0.156347	−	0.987702i	$-0.450028\pi$
$941$	9.59210	0.312694	0.156347	+	0.987702i	$0.450028\pi$
$942$	0	0
$943$	29.2444	0.952330
$944$	0	0
$945$	0.688892	0.0224097
$946$	0	0
$947$	10.8113	0.351322	0.175661	−	0.984451i	$-0.443794\pi$
$947$	10.8113	0.351322	0.175661	+	0.984451i	$0.443794\pi$
$948$	0	0
$949$	9.19850	0.298596
$950$	0	0
$951$	−25.5353	−0.828038
$952$	0	0
$953$	−1.26178	−0.0408732	−0.0204366	−	0.999791i	$-0.506506\pi$
$953$	−1.26178	−0.0408732	−0.0204366	+	0.999791i	$0.506506\pi$
$954$	0	0
$955$	18.0622	0.584480
$956$	0	0
$957$	−3.69181	−0.119339
$958$	0	0
$959$	2.52051	0.0813914
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−5.39207	−0.173757
$964$	0	0
$965$	15.2257	0.490132
$966$	0	0
$967$	−58.8899	−1.89377	−0.946885	−	0.321571i	$-0.895789\pi$
$967$	−58.8899	−1.89377	−0.946885	+	0.321571i	$0.895789\pi$
$968$	0	0
$969$	26.8385	0.862178
$970$	0	0
$971$	−30.5797	−0.981348	−0.490674	−	0.871343i	$-0.663249\pi$
$971$	−30.5797	−0.981348	−0.490674	+	0.871343i	$0.663249\pi$
$972$	0	0
$973$	10.0731	0.322930
$974$	0	0
$975$	0.688892	0.0220622
$976$	0	0
$977$	−12.8385	−0.410741	−0.205371	−	0.978684i	$-0.565840\pi$
$977$	−12.8385	−0.410741	−0.205371	+	0.978684i	$0.565840\pi$
$978$	0	0
$979$	−53.3689	−1.70568
$980$	0	0
$981$	12.6178	0.402854
$982$	0	0
$983$	0.0918659	0.00293007	0.00146503	−	0.999999i	$-0.499534\pi$
$983$	0.0918659	0.00293007	0.00146503	+	0.999999i	$0.499534\pi$
$984$	0	0
$985$	−26.8845	−0.856611
$986$	0	0
$987$	−7.15257	−0.227669
$988$	0	0
$989$	24.8859	0.791324
$990$	0	0
$991$	−30.4385	−0.966910	−0.483455	−	0.875369i	$-0.660618\pi$
$991$	−30.4385	−0.966910	−0.483455	+	0.875369i	$0.660618\pi$
$992$	0	0
$993$	−20.7699	−0.659112
$994$	0	0
$995$	4.34122	0.137626
$996$	0	0
$997$	−37.2672	−1.18026	−0.590132	−	0.807307i	$-0.700924\pi$
$997$	−37.2672	−1.18026	−0.590132	+	0.807307i	$0.700924\pi$
$998$	0	0
$999$	−5.73975	−0.181598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.o.1.2	✓	3
4.3	odd	2		7440.2.a.bo.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.o.1.2	✓	3	1.1	even	1	trivial
7440.2.a.bo.1.2		3	4.3	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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.688892 q^{7} +1.00000 q^{9} +3.80642 q^{11} +0.688892 q^{13} -1.00000 q^{15} -6.70964 q^{17} -4.00000 q^{19} -0.688892 q^{21} -4.14764 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.969888 q^{29} -1.00000 q^{31} +3.80642 q^{33} +0.688892 q^{35} -5.73975 q^{37} +0.688892 q^{39} -7.05086 q^{41} -6.00000 q^{43} -1.00000 q^{45} +10.3827 q^{47} -6.52543 q^{49} -6.70964 q^{51} +12.3827 q^{53} -3.80642 q^{55} -4.00000 q^{57} -4.77631 q^{59} -6.29529 q^{61} -0.688892 q^{63} -0.688892 q^{65} -0.453829 q^{67} -4.14764 q^{69} -3.45875 q^{71} +13.3526 q^{73} +1.00000 q^{75} -2.62222 q^{77} -8.08742 q^{79} +1.00000 q^{81} +11.7605 q^{83} +6.70964 q^{85} -0.969888 q^{87} -14.0207 q^{89} -0.474572 q^{91} -1.00000 q^{93} +4.00000 q^{95} -0.949145 q^{97} +3.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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