LMFDB - Embedded newform 9660.2.a.w.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9660,2,Mod(1,9660)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9660, 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("9660.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9660.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:	$77.1354883526$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 9x^{4} + 16x^{3} + 21x^{2} - 15x + 1 $ x^6 - 3x^5 - 9x^4 + 16x^3 + 21x^2 - 15*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.70802$ of defining polynomial
Character	$\chi$	$=$	9660.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.35507 q^{11} -3.33335 q^{13} -1.00000 q^{15} +0.650611 q^{17} +1.74371 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.41603 q^{29} -6.02739 q^{31} +4.35507 q^{33} +1.00000 q^{35} -4.08268 q^{37} -3.33335 q^{39} -8.77188 q^{41} +9.48406 q^{43} -1.00000 q^{45} -8.24303 q^{47} +1.00000 q^{49} +0.650611 q^{51} +7.74805 q^{53} -4.35507 q^{55} +1.74371 q^{57} -6.03859 q^{59} -2.56581 q^{61} -1.00000 q^{63} +3.33335 q^{65} +6.05952 q^{67} -1.00000 q^{69} -6.59532 q^{71} -14.4749 q^{73} +1.00000 q^{75} -4.35507 q^{77} +5.27806 q^{79} +1.00000 q^{81} +13.4757 q^{83} -0.650611 q^{85} -3.41603 q^{87} -12.5156 q^{89} +3.33335 q^{91} -6.02739 q^{93} -1.74371 q^{95} +11.4274 q^{97} +4.35507 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9} - 3 q^{13} - 6 q^{15} - 3 q^{17} - 6 q^{21} - 6 q^{23} + 6 q^{25} + 6 q^{27} + 6 q^{29} + 6 q^{31} + 6 q^{35} - 15 q^{37} - 3 q^{39} + 6 q^{41} - 3 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{49} - 3 q^{51} - 3 q^{53} + 6 q^{59} - 18 q^{61} - 6 q^{63} + 3 q^{65} - 9 q^{67} - 6 q^{69} - 9 q^{73} + 6 q^{75} - 18 q^{79} + 6 q^{81} - 3 q^{83} + 3 q^{85} + 6 q^{87} - 6 q^{89} + 3 q^{91} + 6 q^{93} - 24 q^{97}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9 - 3 * q^13 - 6 * q^15 - 3 * q^17 - 6 * q^21 - 6 * q^23 + 6 * q^25 + 6 * q^27 + 6 * q^29 + 6 * q^31 + 6 * q^35 - 15 * q^37 - 3 * q^39 + 6 * q^41 - 3 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^49 - 3 * q^51 - 3 * q^53 + 6 * q^59 - 18 * q^61 - 6 * q^63 + 3 * q^65 - 9 * q^67 - 6 * q^69 - 9 * q^73 + 6 * q^75 - 18 * q^79 + 6 * q^81 - 3 * q^83 + 3 * q^85 + 6 * q^87 - 6 * q^89 + 3 * q^91 + 6 * q^93 - 24 * 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	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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.35507	1.31310	0.656551	−	0.754282i	$-0.272015\pi$
$11$	4.35507	1.31310	0.656551	+	0.754282i	$0.272015\pi$
$12$	0	0
$13$	−3.33335	−0.924506	−0.462253	−	0.886748i	$-0.652959\pi$
$13$	−3.33335	−0.924506	−0.462253	+	0.886748i	$0.652959\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.650611	0.157796	0.0788982	−	0.996883i	$-0.474860\pi$
$17$	0.650611	0.157796	0.0788982	+	0.996883i	$0.474860\pi$
$18$	0	0
$19$	1.74371	0.400035	0.200018	−	0.979792i	$-0.435900\pi$
$19$	1.74371	0.400035	0.200018	+	0.979792i	$0.435900\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.41603	−0.634342	−0.317171	−	0.948368i	$-0.602733\pi$
$29$	−3.41603	−0.634342	−0.317171	+	0.948368i	$0.602733\pi$
$30$	0	0
$31$	−6.02739	−1.08255	−0.541275	−	0.840845i	$-0.682058\pi$
$31$	−6.02739	−1.08255	−0.541275	+	0.840845i	$0.682058\pi$
$32$	0	0
$33$	4.35507	0.758120
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−4.08268	−0.671188	−0.335594	−	0.942007i	$-0.608937\pi$
$37$	−4.08268	−0.671188	−0.335594	+	0.942007i	$0.608937\pi$
$38$	0	0
$39$	−3.33335	−0.533764
$40$	0	0
$41$	−8.77188	−1.36994	−0.684969	−	0.728573i	$-0.740184\pi$
$41$	−8.77188	−1.36994	−0.684969	+	0.728573i	$0.740184\pi$
$42$	0	0
$43$	9.48406	1.44631	0.723153	−	0.690688i	$-0.242692\pi$
$43$	9.48406	1.44631	0.723153	+	0.690688i	$0.242692\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.24303	−1.20237	−0.601185	−	0.799110i	$-0.705304\pi$
$47$	−8.24303	−1.20237	−0.601185	+	0.799110i	$0.705304\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.650611	0.0911038
$52$	0	0
$53$	7.74805	1.06428	0.532138	−	0.846657i	$-0.321389\pi$
$53$	7.74805	1.06428	0.532138	+	0.846657i	$0.321389\pi$
$54$	0	0
$55$	−4.35507	−0.587237
$56$	0	0
$57$	1.74371	0.230960
$58$	0	0
$59$	−6.03859	−0.786157	−0.393079	−	0.919505i	$-0.628590\pi$
$59$	−6.03859	−0.786157	−0.393079	+	0.919505i	$0.628590\pi$
$60$	0	0
$61$	−2.56581	−0.328519	−0.164259	−	0.986417i	$-0.552523\pi$
$61$	−2.56581	−0.328519	−0.164259	+	0.986417i	$0.552523\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.33335	0.413452
$66$	0	0
$67$	6.05952	0.740288	0.370144	−	0.928974i	$-0.379308\pi$
$67$	6.05952	0.740288	0.370144	+	0.928974i	$0.379308\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.59532	−0.782720	−0.391360	−	0.920238i	$-0.627995\pi$
$71$	−6.59532	−0.782720	−0.391360	+	0.920238i	$0.627995\pi$
$72$	0	0
$73$	−14.4749	−1.69416	−0.847082	−	0.531462i	$-0.821643\pi$
$73$	−14.4749	−1.69416	−0.847082	+	0.531462i	$0.821643\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−4.35507	−0.496306
$78$	0	0
$79$	5.27806	0.593828	0.296914	−	0.954904i	$-0.404042\pi$
$79$	5.27806	0.593828	0.296914	+	0.954904i	$0.404042\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.4757	1.47915	0.739577	−	0.673072i	$-0.235026\pi$
$83$	13.4757	1.47915	0.739577	+	0.673072i	$0.235026\pi$
$84$	0	0
$85$	−0.650611	−0.0705687
$86$	0	0
$87$	−3.41603	−0.366237
$88$	0	0
$89$	−12.5156	−1.32665	−0.663325	−	0.748331i	$-0.730855\pi$
$89$	−12.5156	−1.32665	−0.663325	+	0.748331i	$0.730855\pi$
$90$	0	0
$91$	3.33335	0.349431
$92$	0	0
$93$	−6.02739	−0.625011
$94$	0	0
$95$	−1.74371	−0.178901
$96$	0	0
$97$	11.4274	1.16028	0.580138	−	0.814518i	$-0.302999\pi$
$97$	11.4274	1.16028	0.580138	+	0.814518i	$0.302999\pi$
$98$	0	0
$99$	4.35507	0.437701
$100$	0	0
$101$	−4.72828	−0.470482	−0.235241	−	0.971937i	$-0.575588\pi$
$101$	−4.72828	−0.470482	−0.235241	+	0.971937i	$0.575588\pi$
$102$	0	0
$103$	−15.4953	−1.52679	−0.763397	−	0.645930i	$-0.776470\pi$
$103$	−15.4953	−1.52679	−0.763397	+	0.645930i	$0.776470\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	2.66671	0.257801	0.128900	−	0.991658i	$-0.458855\pi$
$107$	2.66671	0.257801	0.128900	+	0.991658i	$0.458855\pi$
$108$	0	0
$109$	3.53085	0.338194	0.169097	−	0.985599i	$-0.445915\pi$
$109$	3.53085	0.338194	0.169097	+	0.985599i	$0.445915\pi$
$110$	0	0
$111$	−4.08268	−0.387511
$112$	0	0
$113$	8.02739	0.755153	0.377577	−	0.925978i	$-0.376758\pi$
$113$	8.02739	0.755153	0.377577	+	0.925978i	$0.376758\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−3.33335	−0.308169
$118$	0	0
$119$	−0.650611	−0.0596414
$120$	0	0
$121$	7.96659	0.724236
$122$	0	0
$123$	−8.77188	−0.790934
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.46710	−0.751334	−0.375667	−	0.926755i	$-0.622586\pi$
$127$	−8.46710	−0.751334	−0.375667	+	0.926755i	$0.622586\pi$
$128$	0	0
$129$	9.48406	0.835025
$130$	0	0
$131$	8.08062	0.706007	0.353004	−	0.935622i	$-0.385160\pi$
$131$	8.08062	0.706007	0.353004	+	0.935622i	$0.385160\pi$
$132$	0	0
$133$	−1.74371	−0.151199
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−14.6807	−1.25426	−0.627128	−	0.778916i	$-0.715770\pi$
$137$	−14.6807	−1.25426	−0.627128	+	0.778916i	$0.715770\pi$
$138$	0	0
$139$	14.4490	1.22555	0.612775	−	0.790258i	$-0.290053\pi$
$139$	14.4490	1.22555	0.612775	+	0.790258i	$0.290053\pi$
$140$	0	0
$141$	−8.24303	−0.694188
$142$	0	0
$143$	−14.5170	−1.21397
$144$	0	0
$145$	3.41603	0.283686
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−1.63029	−0.133558	−0.0667792	−	0.997768i	$-0.521272\pi$
$149$	−1.63029	−0.133558	−0.0667792	+	0.997768i	$0.521272\pi$
$150$	0	0
$151$	0.851607	0.0693028	0.0346514	−	0.999399i	$-0.488968\pi$
$151$	0.851607	0.0693028	0.0346514	+	0.999399i	$0.488968\pi$
$152$	0	0
$153$	0.650611	0.0525988
$154$	0	0
$155$	6.02739	0.484131
$156$	0	0
$157$	−12.5819	−1.00415	−0.502073	−	0.864825i	$-0.667429\pi$
$157$	−12.5819	−1.00415	−0.502073	+	0.864825i	$0.667429\pi$
$158$	0	0
$159$	7.74805	0.614460
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	9.58923	0.751087	0.375543	−	0.926805i	$-0.377456\pi$
$163$	9.58923	0.751087	0.375543	+	0.926805i	$0.377456\pi$
$164$	0	0
$165$	−4.35507	−0.339041
$166$	0	0
$167$	−8.07767	−0.625069	−0.312535	−	0.949906i	$-0.601178\pi$
$167$	−8.07767	−0.625069	−0.312535	+	0.949906i	$0.601178\pi$
$168$	0	0
$169$	−1.88875	−0.145288
$170$	0	0
$171$	1.74371	0.133345
$172$	0	0
$173$	−13.4000	−1.01878	−0.509392	−	0.860535i	$-0.670129\pi$
$173$	−13.4000	−1.01878	−0.509392	+	0.860535i	$0.670129\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−6.03859	−0.453888
$178$	0	0
$179$	4.90467	0.366592	0.183296	−	0.983058i	$-0.441323\pi$
$179$	4.90467	0.366592	0.183296	+	0.983058i	$0.441323\pi$
$180$	0	0
$181$	16.5023	1.22661	0.613304	−	0.789847i	$-0.289840\pi$
$181$	16.5023	1.22661	0.613304	+	0.789847i	$0.289840\pi$
$182$	0	0
$183$	−2.56581	−0.189670
$184$	0	0
$185$	4.08268	0.300165
$186$	0	0
$187$	2.83345	0.207203
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	7.33130	0.530474	0.265237	−	0.964183i	$-0.414550\pi$
$191$	7.33130	0.530474	0.265237	+	0.964183i	$0.414550\pi$
$192$	0	0
$193$	−25.3539	−1.82502	−0.912508	−	0.409058i	$-0.865857\pi$
$193$	−25.3539	−1.82502	−0.912508	+	0.409058i	$0.865857\pi$
$194$	0	0
$195$	3.33335	0.238706
$196$	0	0
$197$	−0.257593	−0.0183527	−0.00917637	−	0.999958i	$-0.502921\pi$
$197$	−0.257593	−0.0183527	−0.00917637	+	0.999958i	$0.502921\pi$
$198$	0	0
$199$	−4.25016	−0.301286	−0.150643	−	0.988588i	$-0.548134\pi$
$199$	−4.25016	−0.301286	−0.150643	+	0.988588i	$0.548134\pi$
$200$	0	0
$201$	6.05952	0.427406
$202$	0	0
$203$	3.41603	0.239759
$204$	0	0
$205$	8.77188	0.612655
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	7.59398	0.525287
$210$	0	0
$211$	−19.5633	−1.34679	−0.673396	−	0.739282i	$-0.735165\pi$
$211$	−19.5633	−1.34679	−0.673396	+	0.739282i	$0.735165\pi$
$212$	0	0
$213$	−6.59532	−0.451904
$214$	0	0
$215$	−9.48406	−0.646808
$216$	0	0
$217$	6.02739	0.409166
$218$	0	0
$219$	−14.4749	−0.978126
$220$	0	0
$221$	−2.16872	−0.145884
$222$	0	0
$223$	−1.59603	−0.106878	−0.0534392	−	0.998571i	$-0.517018\pi$
$223$	−1.59603	−0.106878	−0.0534392	+	0.998571i	$0.517018\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	9.79359	0.650023	0.325012	−	0.945710i	$-0.394632\pi$
$227$	9.79359	0.650023	0.325012	+	0.945710i	$0.394632\pi$
$228$	0	0
$229$	−3.57679	−0.236361	−0.118181	−	0.992992i	$-0.537706\pi$
$229$	−3.57679	−0.236361	−0.118181	+	0.992992i	$0.537706\pi$
$230$	0	0
$231$	−4.35507	−0.286542
$232$	0	0
$233$	−23.7980	−1.55906	−0.779529	−	0.626366i	$-0.784542\pi$
$233$	−23.7980	−1.55906	−0.779529	+	0.626366i	$0.784542\pi$
$234$	0	0
$235$	8.24303	0.537716
$236$	0	0
$237$	5.27806	0.342847
$238$	0	0
$239$	15.3867	0.995285	0.497643	−	0.867382i	$-0.334199\pi$
$239$	15.3867	0.995285	0.497643	+	0.867382i	$0.334199\pi$
$240$	0	0
$241$	−2.32750	−0.149928	−0.0749639	−	0.997186i	$-0.523884\pi$
$241$	−2.32750	−0.149928	−0.0749639	+	0.997186i	$0.523884\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−5.81241	−0.369835
$248$	0	0
$249$	13.4757	0.853990
$250$	0	0
$251$	−28.2847	−1.78532	−0.892659	−	0.450733i	$-0.851163\pi$
$251$	−28.2847	−1.78532	−0.892659	+	0.450733i	$0.851163\pi$
$252$	0	0
$253$	−4.35507	−0.273801
$254$	0	0
$255$	−0.650611	−0.0407428
$256$	0	0
$257$	−10.2369	−0.638560	−0.319280	−	0.947660i	$-0.603441\pi$
$257$	−10.2369	−0.638560	−0.319280	+	0.947660i	$0.603441\pi$
$258$	0	0
$259$	4.08268	0.253685
$260$	0	0
$261$	−3.41603	−0.211447
$262$	0	0
$263$	−19.4645	−1.20023	−0.600117	−	0.799912i	$-0.704879\pi$
$263$	−19.4645	−1.20023	−0.600117	+	0.799912i	$0.704879\pi$
$264$	0	0
$265$	−7.74805	−0.475959
$266$	0	0
$267$	−12.5156	−0.765942
$268$	0	0
$269$	14.6633	0.894040	0.447020	−	0.894524i	$-0.352485\pi$
$269$	14.6633	0.894040	0.447020	+	0.894524i	$0.352485\pi$
$270$	0	0
$271$	10.9842	0.667240	0.333620	−	0.942708i	$-0.391730\pi$
$271$	10.9842	0.667240	0.333620	+	0.942708i	$0.391730\pi$
$272$	0	0
$273$	3.33335	0.201744
$274$	0	0
$275$	4.35507	0.262620
$276$	0	0
$277$	−19.2783	−1.15832	−0.579161	−	0.815213i	$-0.696620\pi$
$277$	−19.2783	−1.15832	−0.579161	+	0.815213i	$0.696620\pi$
$278$	0	0
$279$	−6.02739	−0.360850
$280$	0	0
$281$	25.9491	1.54799	0.773997	−	0.633189i	$-0.218254\pi$
$281$	25.9491	1.54799	0.773997	+	0.633189i	$0.218254\pi$
$282$	0	0
$283$	−29.6041	−1.75978	−0.879890	−	0.475177i	$-0.842384\pi$
$283$	−29.6041	−1.75978	−0.879890	+	0.475177i	$0.842384\pi$
$284$	0	0
$285$	−1.74371	−0.103289
$286$	0	0
$287$	8.77188	0.517788
$288$	0	0
$289$	−16.5767	−0.975100
$290$	0	0
$291$	11.4274	0.669885
$292$	0	0
$293$	−30.7378	−1.79572	−0.897859	−	0.440282i	$-0.854878\pi$
$293$	−30.7378	−1.79572	−0.897859	+	0.440282i	$0.854878\pi$
$294$	0	0
$295$	6.03859	0.351580
$296$	0	0
$297$	4.35507	0.252707
$298$	0	0
$299$	3.33335	0.192773
$300$	0	0
$301$	−9.48406	−0.546652
$302$	0	0
$303$	−4.72828	−0.271633
$304$	0	0
$305$	2.56581	0.146918
$306$	0	0
$307$	29.2559	1.66972	0.834861	−	0.550461i	$-0.185548\pi$
$307$	29.2559	1.66972	0.834861	+	0.550461i	$0.185548\pi$
$308$	0	0
$309$	−15.4953	−0.881495
$310$	0	0
$311$	17.6896	1.00308	0.501542	−	0.865133i	$-0.332766\pi$
$311$	17.6896	1.00308	0.501542	+	0.865133i	$0.332766\pi$
$312$	0	0
$313$	−12.5819	−0.711172	−0.355586	−	0.934644i	$-0.615719\pi$
$313$	−12.5819	−0.711172	−0.355586	+	0.934644i	$0.615719\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	22.2236	1.24820	0.624099	−	0.781345i	$-0.285466\pi$
$317$	22.2236	1.24820	0.624099	+	0.781345i	$0.285466\pi$
$318$	0	0
$319$	−14.8770	−0.832955
$320$	0	0
$321$	2.66671	0.148841
$322$	0	0
$323$	1.13448	0.0631241
$324$	0	0
$325$	−3.33335	−0.184901
$326$	0	0
$327$	3.53085	0.195256
$328$	0	0
$329$	8.24303	0.454453
$330$	0	0
$331$	−16.0745	−0.883534	−0.441767	−	0.897130i	$-0.645648\pi$
$331$	−16.0745	−0.883534	−0.441767	+	0.897130i	$0.645648\pi$
$332$	0	0
$333$	−4.08268	−0.223729
$334$	0	0
$335$	−6.05952	−0.331067
$336$	0	0
$337$	0.851522	0.0463853	0.0231927	−	0.999731i	$-0.492617\pi$
$337$	0.851522	0.0463853	0.0231927	+	0.999731i	$0.492617\pi$
$338$	0	0
$339$	8.02739	0.435988
$340$	0	0
$341$	−26.2497	−1.42150
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−35.9184	−1.92820	−0.964101	−	0.265535i	$-0.914451\pi$
$347$	−35.9184	−1.92820	−0.964101	+	0.265535i	$0.914451\pi$
$348$	0	0
$349$	−30.5507	−1.63534	−0.817670	−	0.575687i	$-0.804735\pi$
$349$	−30.5507	−1.63534	−0.817670	+	0.575687i	$0.804735\pi$
$350$	0	0
$351$	−3.33335	−0.177921
$352$	0	0
$353$	26.2979	1.39970	0.699848	−	0.714292i	$-0.253251\pi$
$353$	26.2979	1.39970	0.699848	+	0.714292i	$0.253251\pi$
$354$	0	0
$355$	6.59532	0.350043
$356$	0	0
$357$	−0.650611	−0.0344340
$358$	0	0
$359$	21.2650	1.12232	0.561161	−	0.827706i	$-0.310355\pi$
$359$	21.2650	1.12232	0.561161	+	0.827706i	$0.310355\pi$
$360$	0	0
$361$	−15.9595	−0.839972
$362$	0	0
$363$	7.96659	0.418138
$364$	0	0
$365$	14.4749	0.757653
$366$	0	0
$367$	−31.0190	−1.61918	−0.809589	−	0.586997i	$-0.800310\pi$
$367$	−31.0190	−1.61918	−0.809589	+	0.586997i	$0.800310\pi$
$368$	0	0
$369$	−8.77188	−0.456646
$370$	0	0
$371$	−7.74805	−0.402259
$372$	0	0
$373$	3.74177	0.193741	0.0968707	−	0.995297i	$-0.469117\pi$
$373$	3.74177	0.193741	0.0968707	+	0.995297i	$0.469117\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	11.3869	0.586453
$378$	0	0
$379$	15.7559	0.809326	0.404663	−	0.914466i	$-0.367389\pi$
$379$	15.7559	0.809326	0.404663	+	0.914466i	$0.367389\pi$
$380$	0	0
$381$	−8.46710	−0.433783
$382$	0	0
$383$	8.90027	0.454783	0.227391	−	0.973803i	$-0.426980\pi$
$383$	8.90027	0.454783	0.227391	+	0.973803i	$0.426980\pi$
$384$	0	0
$385$	4.35507	0.221955
$386$	0	0
$387$	9.48406	0.482102
$388$	0	0
$389$	32.2672	1.63601	0.818006	−	0.575209i	$-0.195079\pi$
$389$	32.2672	1.63601	0.818006	+	0.575209i	$0.195079\pi$
$390$	0	0
$391$	−0.650611	−0.0329028
$392$	0	0
$393$	8.08062	0.407614
$394$	0	0
$395$	−5.27806	−0.265568
$396$	0	0
$397$	−28.2102	−1.41583	−0.707915	−	0.706297i	$-0.750364\pi$
$397$	−28.2102	−1.41583	−0.707915	+	0.706297i	$0.750364\pi$
$398$	0	0
$399$	−1.74371	−0.0872948
$400$	0	0
$401$	−34.3750	−1.71660	−0.858302	−	0.513144i	$-0.828481\pi$
$401$	−34.3750	−1.71660	−0.858302	+	0.513144i	$0.828481\pi$
$402$	0	0
$403$	20.0914	1.00082
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−17.7803	−0.881338
$408$	0	0
$409$	7.87529	0.389408	0.194704	−	0.980862i	$-0.437625\pi$
$409$	7.87529	0.389408	0.194704	+	0.980862i	$0.437625\pi$
$410$	0	0
$411$	−14.6807	−0.724145
$412$	0	0
$413$	6.03859	0.297140
$414$	0	0
$415$	−13.4757	−0.661497
$416$	0	0
$417$	14.4490	0.707571
$418$	0	0
$419$	15.3576	0.750266	0.375133	−	0.926971i	$-0.377597\pi$
$419$	15.3576	0.750266	0.375133	+	0.926971i	$0.377597\pi$
$420$	0	0
$421$	25.0003	1.21844	0.609221	−	0.793001i	$-0.291482\pi$
$421$	25.0003	1.21844	0.609221	+	0.793001i	$0.291482\pi$
$422$	0	0
$423$	−8.24303	−0.400790
$424$	0	0
$425$	0.650611	0.0315593
$426$	0	0
$427$	2.56581	0.124168
$428$	0	0
$429$	−14.5170	−0.700886
$430$	0	0
$431$	−5.76826	−0.277847	−0.138924	−	0.990303i	$-0.544364\pi$
$431$	−5.76826	−0.277847	−0.138924	+	0.990303i	$0.544364\pi$
$432$	0	0
$433$	−26.0583	−1.25228	−0.626141	−	0.779710i	$-0.715366\pi$
$433$	−26.0583	−1.25228	−0.626141	+	0.779710i	$0.715366\pi$
$434$	0	0
$435$	3.41603	0.163786
$436$	0	0
$437$	−1.74371	−0.0834131
$438$	0	0
$439$	−19.8691	−0.948302	−0.474151	−	0.880443i	$-0.657245\pi$
$439$	−19.8691	−0.948302	−0.474151	+	0.880443i	$0.657245\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−33.4206	−1.58786	−0.793930	−	0.608009i	$-0.791969\pi$
$443$	−33.4206	−1.58786	−0.793930	+	0.608009i	$0.791969\pi$
$444$	0	0
$445$	12.5156	0.593296
$446$	0	0
$447$	−1.63029	−0.0771099
$448$	0	0
$449$	−6.88417	−0.324884	−0.162442	−	0.986718i	$-0.551937\pi$
$449$	−6.88417	−0.324884	−0.162442	+	0.986718i	$0.551937\pi$
$450$	0	0
$451$	−38.2021	−1.79887
$452$	0	0
$453$	0.851607	0.0400120
$454$	0	0
$455$	−3.33335	−0.156270
$456$	0	0
$457$	−4.30044	−0.201166	−0.100583	−	0.994929i	$-0.532071\pi$
$457$	−4.30044	−0.201166	−0.100583	+	0.994929i	$0.532071\pi$
$458$	0	0
$459$	0.650611	0.0303679
$460$	0	0
$461$	28.0554	1.30667	0.653335	−	0.757069i	$-0.273369\pi$
$461$	28.0554	1.30667	0.653335	+	0.757069i	$0.273369\pi$
$462$	0	0
$463$	12.6956	0.590013	0.295007	−	0.955495i	$-0.404678\pi$
$463$	12.6956	0.590013	0.295007	+	0.955495i	$0.404678\pi$
$464$	0	0
$465$	6.02739	0.279513
$466$	0	0
$467$	−15.7584	−0.729214	−0.364607	−	0.931162i	$-0.618797\pi$
$467$	−15.7584	−0.729214	−0.364607	+	0.931162i	$0.618797\pi$
$468$	0	0
$469$	−6.05952	−0.279803
$470$	0	0
$471$	−12.5819	−0.579744
$472$	0	0
$473$	41.3037	1.89915
$474$	0	0
$475$	1.74371	0.0800070
$476$	0	0
$477$	7.74805	0.354759
$478$	0	0
$479$	22.9974	1.05078	0.525390	−	0.850862i	$-0.323920\pi$
$479$	22.9974	1.05078	0.525390	+	0.850862i	$0.323920\pi$
$480$	0	0
$481$	13.6090	0.620518
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−11.4274	−0.518891
$486$	0	0
$487$	−34.3600	−1.55700	−0.778501	−	0.627644i	$-0.784019\pi$
$487$	−34.3600	−1.55700	−0.778501	+	0.627644i	$0.784019\pi$
$488$	0	0
$489$	9.58923	0.433640
$490$	0	0
$491$	0.149907	0.00676520	0.00338260	−	0.999994i	$-0.498923\pi$
$491$	0.149907	0.00676520	0.00338260	+	0.999994i	$0.498923\pi$
$492$	0	0
$493$	−2.22251	−0.100097
$494$	0	0
$495$	−4.35507	−0.195746
$496$	0	0
$497$	6.59532	0.295840
$498$	0	0
$499$	31.7307	1.42046	0.710230	−	0.703970i	$-0.248591\pi$
$499$	31.7307	1.42046	0.710230	+	0.703970i	$0.248591\pi$
$500$	0	0
$501$	−8.07767	−0.360884
$502$	0	0
$503$	−37.7919	−1.68506	−0.842528	−	0.538653i	$-0.818933\pi$
$503$	−37.7919	−1.68506	−0.842528	+	0.538653i	$0.818933\pi$
$504$	0	0
$505$	4.72828	0.210406
$506$	0	0
$507$	−1.88875	−0.0838821
$508$	0	0
$509$	15.9948	0.708955	0.354478	−	0.935064i	$-0.384659\pi$
$509$	15.9948	0.708955	0.354478	+	0.935064i	$0.384659\pi$
$510$	0	0
$511$	14.4749	0.640334
$512$	0	0
$513$	1.74371	0.0769868
$514$	0	0
$515$	15.4953	0.682803
$516$	0	0
$517$	−35.8989	−1.57883
$518$	0	0
$519$	−13.4000	−0.588195
$520$	0	0
$521$	−14.8742	−0.651649	−0.325824	−	0.945430i	$-0.605642\pi$
$521$	−14.8742	−0.651649	−0.325824	+	0.945430i	$0.605642\pi$
$522$	0	0
$523$	−32.1200	−1.40451	−0.702255	−	0.711925i	$-0.747823\pi$
$523$	−32.1200	−1.40451	−0.702255	+	0.711925i	$0.747823\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−3.92148	−0.170823
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.03859	−0.262052
$532$	0	0
$533$	29.2398	1.26652
$534$	0	0
$535$	−2.66671	−0.115292
$536$	0	0
$537$	4.90467	0.211652
$538$	0	0
$539$	4.35507	0.187586
$540$	0	0
$541$	−44.3870	−1.90835	−0.954174	−	0.299253i	$-0.903262\pi$
$541$	−44.3870	−1.90835	−0.954174	+	0.299253i	$0.903262\pi$
$542$	0	0
$543$	16.5023	0.708183
$544$	0	0
$545$	−3.53085	−0.151245
$546$	0	0
$547$	−12.0213	−0.513992	−0.256996	−	0.966413i	$-0.582733\pi$
$547$	−12.0213	−0.513992	−0.256996	+	0.966413i	$0.582733\pi$
$548$	0	0
$549$	−2.56581	−0.109506
$550$	0	0
$551$	−5.95658	−0.253759
$552$	0	0
$553$	−5.27806	−0.224446
$554$	0	0
$555$	4.08268	0.173300
$556$	0	0
$557$	−22.0809	−0.935600	−0.467800	−	0.883834i	$-0.654953\pi$
$557$	−22.0809	−0.935600	−0.467800	+	0.883834i	$0.654953\pi$
$558$	0	0
$559$	−31.6138	−1.33712
$560$	0	0
$561$	2.83345	0.119628
$562$	0	0
$563$	12.4677	0.525452	0.262726	−	0.964871i	$-0.415378\pi$
$563$	12.4677	0.525452	0.262726	+	0.964871i	$0.415378\pi$
$564$	0	0
$565$	−8.02739	−0.337715
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	27.4414	1.15040	0.575201	−	0.818012i	$-0.304924\pi$
$569$	27.4414	1.15040	0.575201	+	0.818012i	$0.304924\pi$
$570$	0	0
$571$	−15.6142	−0.653435	−0.326718	−	0.945122i	$-0.605943\pi$
$571$	−15.6142	−0.653435	−0.326718	+	0.945122i	$0.605943\pi$
$572$	0	0
$573$	7.33130	0.306269
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−4.54453	−0.189191	−0.0945957	−	0.995516i	$-0.530156\pi$
$577$	−4.54453	−0.189191	−0.0945957	+	0.995516i	$0.530156\pi$
$578$	0	0
$579$	−25.3539	−1.05367
$580$	0	0
$581$	−13.4757	−0.559067
$582$	0	0
$583$	33.7433	1.39750
$584$	0	0
$585$	3.33335	0.137817
$586$	0	0
$587$	−35.9087	−1.48211	−0.741056	−	0.671443i	$-0.765675\pi$
$587$	−35.9087	−1.48211	−0.741056	+	0.671443i	$0.765675\pi$
$588$	0	0
$589$	−10.5100	−0.433058
$590$	0	0
$591$	−0.257593	−0.0105960
$592$	0	0
$593$	29.1788	1.19823	0.599115	−	0.800663i	$-0.295519\pi$
$593$	29.1788	1.19823	0.599115	+	0.800663i	$0.295519\pi$
$594$	0	0
$595$	0.650611	0.0266725
$596$	0	0
$597$	−4.25016	−0.173947
$598$	0	0
$599$	−33.6138	−1.37342	−0.686712	−	0.726930i	$-0.740946\pi$
$599$	−33.6138	−1.37342	−0.686712	+	0.726930i	$0.740946\pi$
$600$	0	0
$601$	17.9763	0.733267	0.366633	−	0.930365i	$-0.380510\pi$
$601$	17.9763	0.733267	0.366633	+	0.930365i	$0.380510\pi$
$602$	0	0
$603$	6.05952	0.246763
$604$	0	0
$605$	−7.96659	−0.323888
$606$	0	0
$607$	−12.5607	−0.509825	−0.254912	−	0.966964i	$-0.582047\pi$
$607$	−12.5607	−0.509825	−0.254912	+	0.966964i	$0.582047\pi$
$608$	0	0
$609$	3.41603	0.138425
$610$	0	0
$611$	27.4769	1.11160
$612$	0	0
$613$	−19.4416	−0.785240	−0.392620	−	0.919701i	$-0.628431\pi$
$613$	−19.4416	−0.785240	−0.392620	+	0.919701i	$0.628431\pi$
$614$	0	0
$615$	8.77188	0.353716
$616$	0	0
$617$	16.8031	0.676468	0.338234	−	0.941062i	$-0.390170\pi$
$617$	16.8031	0.676468	0.338234	+	0.941062i	$0.390170\pi$
$618$	0	0
$619$	36.6970	1.47498	0.737489	−	0.675359i	$-0.236011\pi$
$619$	36.6970	1.47498	0.737489	+	0.675359i	$0.236011\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	12.5156	0.501427
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.59398	0.303274
$628$	0	0
$629$	−2.65624	−0.105911
$630$	0	0
$631$	−21.2653	−0.846557	−0.423279	−	0.906000i	$-0.639121\pi$
$631$	−21.2653	−0.846557	−0.423279	+	0.906000i	$0.639121\pi$
$632$	0	0
$633$	−19.5633	−0.777571
$634$	0	0
$635$	8.46710	0.336007
$636$	0	0
$637$	−3.33335	−0.132072
$638$	0	0
$639$	−6.59532	−0.260907
$640$	0	0
$641$	−10.9986	−0.434417	−0.217208	−	0.976125i	$-0.569695\pi$
$641$	−10.9986	−0.434417	−0.217208	+	0.976125i	$0.569695\pi$
$642$	0	0
$643$	32.8935	1.29719	0.648596	−	0.761133i	$-0.275357\pi$
$643$	32.8935	1.29719	0.648596	+	0.761133i	$0.275357\pi$
$644$	0	0
$645$	−9.48406	−0.373435
$646$	0	0
$647$	18.0306	0.708857	0.354429	−	0.935083i	$-0.384675\pi$
$647$	18.0306	0.708857	0.354429	+	0.935083i	$0.384675\pi$
$648$	0	0
$649$	−26.2985	−1.03230
$650$	0	0
$651$	6.02739	0.236232
$652$	0	0
$653$	36.1214	1.41354	0.706770	−	0.707443i	$-0.250151\pi$
$653$	36.1214	1.41354	0.706770	+	0.707443i	$0.250151\pi$
$654$	0	0
$655$	−8.08062	−0.315736
$656$	0	0
$657$	−14.4749	−0.564721
$658$	0	0
$659$	−4.77538	−0.186023	−0.0930113	−	0.995665i	$-0.529649\pi$
$659$	−4.77538	−0.186023	−0.0930113	+	0.995665i	$0.529649\pi$
$660$	0	0
$661$	−5.81528	−0.226188	−0.113094	−	0.993584i	$-0.536076\pi$
$661$	−5.81528	−0.226188	−0.113094	+	0.993584i	$0.536076\pi$
$662$	0	0
$663$	−2.16872	−0.0842260
$664$	0	0
$665$	1.74371	0.0676183
$666$	0	0
$667$	3.41603	0.132269
$668$	0	0
$669$	−1.59603	−0.0617063
$670$	0	0
$671$	−11.1743	−0.431378
$672$	0	0
$673$	22.4476	0.865291	0.432646	−	0.901564i	$-0.357580\pi$
$673$	22.4476	0.865291	0.432646	+	0.901564i	$0.357580\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−13.3314	−0.512369	−0.256185	−	0.966628i	$-0.582465\pi$
$677$	−13.3314	−0.512369	−0.256185	+	0.966628i	$0.582465\pi$
$678$	0	0
$679$	−11.4274	−0.438543
$680$	0	0
$681$	9.79359	0.375291
$682$	0	0
$683$	50.1862	1.92032	0.960161	−	0.279448i	$-0.0901516\pi$
$683$	50.1862	1.92032	0.960161	+	0.279448i	$0.0901516\pi$
$684$	0	0
$685$	14.6807	0.560920
$686$	0	0
$687$	−3.57679	−0.136463
$688$	0	0
$689$	−25.8270	−0.983930
$690$	0	0
$691$	−35.8569	−1.36406	−0.682030	−	0.731324i	$-0.738903\pi$
$691$	−35.8569	−1.36406	−0.682030	+	0.731324i	$0.738903\pi$
$692$	0	0
$693$	−4.35507	−0.165435
$694$	0	0
$695$	−14.4490	−0.548082
$696$	0	0
$697$	−5.70708	−0.216171
$698$	0	0
$699$	−23.7980	−0.900123
$700$	0	0
$701$	−38.1876	−1.44232	−0.721162	−	0.692766i	$-0.756392\pi$
$701$	−38.1876	−1.44232	−0.721162	+	0.692766i	$0.756392\pi$
$702$	0	0
$703$	−7.11902	−0.268499
$704$	0	0
$705$	8.24303	0.310450
$706$	0	0
$707$	4.72828	0.177825
$708$	0	0
$709$	22.6320	0.849961	0.424981	−	0.905202i	$-0.360281\pi$
$709$	22.6320	0.849961	0.424981	+	0.905202i	$0.360281\pi$
$710$	0	0
$711$	5.27806	0.197943
$712$	0	0
$713$	6.02739	0.225727
$714$	0	0
$715$	14.5170	0.542904
$716$	0	0
$717$	15.3867	0.574628
$718$	0	0
$719$	10.6581	0.397481	0.198741	−	0.980052i	$-0.436315\pi$
$719$	10.6581	0.397481	0.198741	+	0.980052i	$0.436315\pi$
$720$	0	0
$721$	15.4953	0.577074
$722$	0	0
$723$	−2.32750	−0.0865608
$724$	0	0
$725$	−3.41603	−0.126868
$726$	0	0
$727$	43.0461	1.59649	0.798246	−	0.602331i	$-0.205761\pi$
$727$	43.0461	1.59649	0.798246	+	0.602331i	$0.205761\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.17044	0.228222
$732$	0	0
$733$	−43.0634	−1.59058	−0.795292	−	0.606226i	$-0.792683\pi$
$733$	−43.0634	−1.59058	−0.795292	+	0.606226i	$0.792683\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	26.3896	0.972073
$738$	0	0
$739$	50.5602	1.85989	0.929944	−	0.367701i	$-0.119855\pi$
$739$	50.5602	1.85989	0.929944	+	0.367701i	$0.119855\pi$
$740$	0	0
$741$	−5.81241	−0.213524
$742$	0	0
$743$	20.4856	0.751543	0.375771	−	0.926712i	$-0.377378\pi$
$743$	20.4856	0.751543	0.375771	+	0.926712i	$0.377378\pi$
$744$	0	0
$745$	1.63029	0.0597291
$746$	0	0
$747$	13.4757	0.493051
$748$	0	0
$749$	−2.66671	−0.0974395
$750$	0	0
$751$	−5.24292	−0.191317	−0.0956585	−	0.995414i	$-0.530496\pi$
$751$	−5.24292	−0.191317	−0.0956585	+	0.995414i	$0.530496\pi$
$752$	0	0
$753$	−28.2847	−1.03075
$754$	0	0
$755$	−0.851607	−0.0309931
$756$	0	0
$757$	−24.3951	−0.886655	−0.443328	−	0.896360i	$-0.646202\pi$
$757$	−24.3951	−0.886655	−0.443328	+	0.896360i	$0.646202\pi$
$758$	0	0
$759$	−4.35507	−0.158079
$760$	0	0
$761$	6.37909	0.231242	0.115621	−	0.993293i	$-0.463114\pi$
$761$	6.37909	0.231242	0.115621	+	0.993293i	$0.463114\pi$
$762$	0	0
$763$	−3.53085	−0.127825
$764$	0	0
$765$	−0.650611	−0.0235229
$766$	0	0
$767$	20.1288	0.726807
$768$	0	0
$769$	44.8712	1.61810	0.809049	−	0.587741i	$-0.199983\pi$
$769$	44.8712	1.61810	0.809049	+	0.587741i	$0.199983\pi$
$770$	0	0
$771$	−10.2369	−0.368673
$772$	0	0
$773$	22.7723	0.819064	0.409532	−	0.912296i	$-0.365692\pi$
$773$	22.7723	0.819064	0.409532	+	0.912296i	$0.365692\pi$
$774$	0	0
$775$	−6.02739	−0.216510
$776$	0	0
$777$	4.08268	0.146465
$778$	0	0
$779$	−15.2956	−0.548023
$780$	0	0
$781$	−28.7230	−1.02779
$782$	0	0
$783$	−3.41603	−0.122079
$784$	0	0
$785$	12.5819	0.449068
$786$	0	0
$787$	23.8802	0.851238	0.425619	−	0.904902i	$-0.360056\pi$
$787$	23.8802	0.851238	0.425619	+	0.904902i	$0.360056\pi$
$788$	0	0
$789$	−19.4645	−0.692955
$790$	0	0
$791$	−8.02739	−0.285421
$792$	0	0
$793$	8.55277	0.303718
$794$	0	0
$795$	−7.74805	−0.274795
$796$	0	0
$797$	8.26292	0.292688	0.146344	−	0.989234i	$-0.453249\pi$
$797$	8.26292	0.292688	0.146344	+	0.989234i	$0.453249\pi$
$798$	0	0
$799$	−5.36301	−0.189730
$800$	0	0
$801$	−12.5156	−0.442217
$802$	0	0
$803$	−63.0393	−2.22461
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	14.6633	0.516174
$808$	0	0
$809$	29.2891	1.02975	0.514874	−	0.857266i	$-0.327839\pi$
$809$	29.2891	1.02975	0.514874	+	0.857266i	$0.327839\pi$
$810$	0	0
$811$	−9.99809	−0.351080	−0.175540	−	0.984472i	$-0.556167\pi$
$811$	−9.99809	−0.351080	−0.175540	+	0.984472i	$0.556167\pi$
$812$	0	0
$813$	10.9842	0.385231
$814$	0	0
$815$	−9.58923	−0.335896
$816$	0	0
$817$	16.5375	0.578573
$818$	0	0
$819$	3.33335	0.116477
$820$	0	0
$821$	20.6893	0.722061	0.361031	−	0.932554i	$-0.382425\pi$
$821$	20.6893	0.722061	0.361031	+	0.932554i	$0.382425\pi$
$822$	0	0
$823$	12.6311	0.440294	0.220147	−	0.975467i	$-0.429346\pi$
$823$	12.6311	0.440294	0.220147	+	0.975467i	$0.429346\pi$
$824$	0	0
$825$	4.35507	0.151624
$826$	0	0
$827$	3.55548	0.123636	0.0618180	−	0.998087i	$-0.480310\pi$
$827$	3.55548	0.123636	0.0618180	+	0.998087i	$0.480310\pi$
$828$	0	0
$829$	25.4064	0.882402	0.441201	−	0.897408i	$-0.354553\pi$
$829$	25.4064	0.882402	0.441201	+	0.897408i	$0.354553\pi$
$830$	0	0
$831$	−19.2783	−0.668758
$832$	0	0
$833$	0.650611	0.0225423
$834$	0	0
$835$	8.07767	0.279539
$836$	0	0
$837$	−6.02739	−0.208337
$838$	0	0
$839$	31.8855	1.10081	0.550404	−	0.834898i	$-0.314474\pi$
$839$	31.8855	1.10081	0.550404	+	0.834898i	$0.314474\pi$
$840$	0	0
$841$	−17.3307	−0.597611
$842$	0	0
$843$	25.9491	0.893735
$844$	0	0
$845$	1.88875	0.0649748
$846$	0	0
$847$	−7.96659	−0.273735
$848$	0	0
$849$	−29.6041	−1.01601
$850$	0	0
$851$	4.08268	0.139952
$852$	0	0
$853$	−52.6793	−1.80370	−0.901852	−	0.432045i	$-0.857792\pi$
$853$	−52.6793	−1.80370	−0.901852	+	0.432045i	$0.857792\pi$
$854$	0	0
$855$	−1.74371	−0.0596337
$856$	0	0
$857$	37.9539	1.29648	0.648241	−	0.761435i	$-0.275505\pi$
$857$	37.9539	1.29648	0.648241	+	0.761435i	$0.275505\pi$
$858$	0	0
$859$	−37.0238	−1.26324	−0.631618	−	0.775280i	$-0.717609\pi$
$859$	−37.0238	−1.26324	−0.631618	+	0.775280i	$0.717609\pi$
$860$	0	0
$861$	8.77188	0.298945
$862$	0	0
$863$	24.3556	0.829075	0.414537	−	0.910032i	$-0.363943\pi$
$863$	24.3556	0.829075	0.414537	+	0.910032i	$0.363943\pi$
$864$	0	0
$865$	13.4000	0.455614
$866$	0	0
$867$	−16.5767	−0.562974
$868$	0	0
$869$	22.9863	0.779757
$870$	0	0
$871$	−20.1985	−0.684401
$872$	0	0
$873$	11.4274	0.386758
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−0.978593	−0.0330448	−0.0165224	−	0.999863i	$-0.505259\pi$
$877$	−0.978593	−0.0330448	−0.0165224	+	0.999863i	$0.505259\pi$
$878$	0	0
$879$	−30.7378	−1.03676
$880$	0	0
$881$	40.2370	1.35562	0.677810	−	0.735237i	$-0.262929\pi$
$881$	40.2370	1.35562	0.677810	+	0.735237i	$0.262929\pi$
$882$	0	0
$883$	−31.5762	−1.06262	−0.531312	−	0.847176i	$-0.678301\pi$
$883$	−31.5762	−1.06262	−0.531312	+	0.847176i	$0.678301\pi$
$884$	0	0
$885$	6.03859	0.202985
$886$	0	0
$887$	−15.0910	−0.506705	−0.253353	−	0.967374i	$-0.581533\pi$
$887$	−15.0910	−0.506705	−0.253353	+	0.967374i	$0.581533\pi$
$888$	0	0
$889$	8.46710	0.283977
$890$	0	0
$891$	4.35507	0.145900
$892$	0	0
$893$	−14.3735	−0.480990
$894$	0	0
$895$	−4.90467	−0.163945
$896$	0	0
$897$	3.33335	0.111297
$898$	0	0
$899$	20.5898	0.686707
$900$	0	0
$901$	5.04097	0.167939
$902$	0	0
$903$	−9.48406	−0.315610
$904$	0	0
$905$	−16.5023	−0.548556
$906$	0	0
$907$	−37.7884	−1.25474	−0.627372	−	0.778719i	$-0.715870\pi$
$907$	−37.7884	−1.25474	−0.627372	+	0.778719i	$0.715870\pi$
$908$	0	0
$909$	−4.72828	−0.156827
$910$	0	0
$911$	4.48329	0.148538	0.0742691	−	0.997238i	$-0.476338\pi$
$911$	4.48329	0.148538	0.0742691	+	0.997238i	$0.476338\pi$
$912$	0	0
$913$	58.6877	1.94228
$914$	0	0
$915$	2.56581	0.0848232
$916$	0	0
$917$	−8.08062	−0.266846
$918$	0	0
$919$	14.1856	0.467940	0.233970	−	0.972244i	$-0.424828\pi$
$919$	14.1856	0.467940	0.233970	+	0.972244i	$0.424828\pi$
$920$	0	0
$921$	29.2559	0.964014
$922$	0	0
$923$	21.9845	0.723630
$924$	0	0
$925$	−4.08268	−0.134238
$926$	0	0
$927$	−15.4953	−0.508931
$928$	0	0
$929$	−10.9369	−0.358829	−0.179414	−	0.983774i	$-0.557420\pi$
$929$	−10.9369	−0.358829	−0.179414	+	0.983774i	$0.557420\pi$
$930$	0	0
$931$	1.74371	0.0571479
$932$	0	0
$933$	17.6896	0.579131
$934$	0	0
$935$	−2.83345	−0.0926638
$936$	0	0
$937$	−40.7279	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$937$	−40.7279	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$938$	0	0
$939$	−12.5819	−0.410595
$940$	0	0
$941$	24.1754	0.788095	0.394048	−	0.919090i	$-0.371075\pi$
$941$	24.1754	0.788095	0.394048	+	0.919090i	$0.371075\pi$
$942$	0	0
$943$	8.77188	0.285652
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	7.70790	0.250473	0.125237	−	0.992127i	$-0.460031\pi$
$947$	7.70790	0.250473	0.125237	+	0.992127i	$0.460031\pi$
$948$	0	0
$949$	48.2501	1.56627
$950$	0	0
$951$	22.2236	0.720648
$952$	0	0
$953$	25.1809	0.815690	0.407845	−	0.913051i	$-0.366280\pi$
$953$	25.1809	0.815690	0.407845	+	0.913051i	$0.366280\pi$
$954$	0	0
$955$	−7.33130	−0.237235
$956$	0	0
$957$	−14.8770	−0.480907
$958$	0	0
$959$	14.6807	0.474064
$960$	0	0
$961$	5.32939	0.171916
$962$	0	0
$963$	2.66671	0.0859335
$964$	0	0
$965$	25.3539	0.816172
$966$	0	0
$967$	6.44299	0.207193	0.103596	−	0.994619i	$-0.466965\pi$
$967$	6.44299	0.207193	0.103596	+	0.994619i	$0.466965\pi$
$968$	0	0
$969$	1.13448	0.0364447
$970$	0	0
$971$	−26.2943	−0.843824	−0.421912	−	0.906637i	$-0.638641\pi$
$971$	−26.2943	−0.843824	−0.421912	+	0.906637i	$0.638641\pi$
$972$	0	0
$973$	−14.4490	−0.463214
$974$	0	0
$975$	−3.33335	−0.106753
$976$	0	0
$977$	3.70718	0.118603	0.0593016	−	0.998240i	$-0.481113\pi$
$977$	3.70718	0.118603	0.0593016	+	0.998240i	$0.481113\pi$
$978$	0	0
$979$	−54.5062	−1.74203
$980$	0	0
$981$	3.53085	0.112731
$982$	0	0
$983$	48.1362	1.53531	0.767653	−	0.640866i	$-0.221425\pi$
$983$	48.1362	1.53531	0.767653	+	0.640866i	$0.221425\pi$
$984$	0	0
$985$	0.257593	0.00820760
$986$	0	0
$987$	8.24303	0.262379
$988$	0	0
$989$	−9.48406	−0.301576
$990$	0	0
$991$	−4.88229	−0.155091	−0.0775455	−	0.996989i	$-0.524708\pi$
$991$	−4.88229	−0.155091	−0.0775455	+	0.996989i	$0.524708\pi$
$992$	0	0
$993$	−16.0745	−0.510108
$994$	0	0
$995$	4.25016	0.134739
$996$	0	0
$997$	47.7725	1.51297	0.756485	−	0.654010i	$-0.226915\pi$
$997$	47.7725	1.51297	0.756485	+	0.654010i	$0.226915\pi$
$998$	0	0
$999$	−4.08268	−0.129170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9660.2.a.w.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9660.2.a.w.1.6	✓	6	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9660.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.35507 q^{11} -3.33335 q^{13} -1.00000 q^{15} +0.650611 q^{17} +1.74371 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.41603 q^{29} -6.02739 q^{31} +4.35507 q^{33} +1.00000 q^{35} -4.08268 q^{37} -3.33335 q^{39} -8.77188 q^{41} +9.48406 q^{43} -1.00000 q^{45} -8.24303 q^{47} +1.00000 q^{49} +0.650611 q^{51} +7.74805 q^{53} -4.35507 q^{55} +1.74371 q^{57} -6.03859 q^{59} -2.56581 q^{61} -1.00000 q^{63} +3.33335 q^{65} +6.05952 q^{67} -1.00000 q^{69} -6.59532 q^{71} -14.4749 q^{73} +1.00000 q^{75} -4.35507 q^{77} +5.27806 q^{79} +1.00000 q^{81} +13.4757 q^{83} -0.650611 q^{85} -3.41603 q^{87} -12.5156 q^{89} +3.33335 q^{91} -6.02739 q^{93} -1.74371 q^{95} +11.4274 q^{97} +4.35507 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{5} - 6 q^{7} + 6 q^{9} - 3 q^{13} - 6 q^{15} - 3 q^{17} - 6 q^{21} - 6 q^{23} + 6 q^{25} + 6 q^{27} + 6 q^{29} + 6 q^{31} + 6 q^{35} - 15 q^{37} - 3 q^{39} + 6 q^{41} - 3 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{49} - 3 q^{51} - 3 q^{53} + 6 q^{59} - 18 q^{61} - 6 q^{63} + 3 q^{65} - 9 q^{67} - 6 q^{69} - 9 q^{73} + 6 q^{75} - 18 q^{79} + 6 q^{81} - 3 q^{83} + 3 q^{85} + 6 q^{87} - 6 q^{89} + 3 q^{91} + 6 q^{93} - 24 q^{97}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^5 - 6 * q^7 + 6 * q^9 - 3 * q^13 - 6 * q^15 - 3 * q^17 - 6 * q^21 - 6 * q^23 + 6 * q^25 + 6 * q^27 + 6 * q^29 + 6 * q^31 + 6 * q^35 - 15 * q^37 - 3 * q^39 + 6 * q^41 - 3 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^49 - 3 * q^51 - 3 * q^53 + 6 * q^59 - 18 * q^61 - 6 * q^63 + 3 * q^65 - 9 * q^67 - 6 * q^69 - 9 * q^73 + 6 * q^75 - 18 * q^79 + 6 * q^81 - 3 * q^83 + 3 * q^85 + 6 * q^87 - 6 * q^89 + 3 * q^91 + 6 * q^93 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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