LMFDB - Embedded newform 1960.2.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.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:	$15.6506787962$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.87996$ of defining polynomial
Character	$\chi$	$=$	1960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87996 q^{3} -1.00000 q^{5} +0.534253 q^{9} +O(q^{10})$	$q+1.87996 q^{3} -1.00000 q^{5} +0.534253 q^{9} -3.29417 q^{11} -4.19292 q^{13} -1.87996 q^{15} +1.43737 q^{17} -1.24445 q^{19} -0.272828 q^{23} +1.00000 q^{25} -4.63551 q^{27} -2.36268 q^{29} +3.72717 q^{31} -6.19292 q^{33} +0.169761 q^{37} -7.88252 q^{39} -11.6630 q^{41} -10.1458 q^{43} -0.534253 q^{45} -3.12441 q^{47} +2.70220 q^{51} +9.24701 q^{53} +3.29417 q^{55} -2.33952 q^{57} -9.07107 q^{59} -7.27102 q^{61} +4.19292 q^{65} -13.1439 q^{67} -0.512907 q^{69} +6.87474 q^{71} +15.2496 q^{73} +1.87996 q^{75} +14.9510 q^{79} -10.3173 q^{81} -0.167199 q^{83} -1.43737 q^{85} -4.44175 q^{87} -3.09869 q^{89} +7.00694 q^{93} +1.24445 q^{95} -6.60894 q^{97} -1.75992 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^5 + 6 * q^9	$ 4 q - 2 q^{3} - 4 q^{5} + 6 q^{9} + 2 q^{11} - 10 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{23} + 4 q^{25} - 14 q^{27} - 2 q^{29} + 12 q^{31} - 18 q^{33} + 14 q^{39} - 12 q^{41} - 8 q^{43} - 6 q^{45} + 2 q^{47} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 8 q^{57} - 8 q^{59} - 20 q^{61} + 10 q^{65} - 8 q^{67} - 24 q^{69} + 4 q^{71} - 16 q^{73} - 2 q^{75} + 22 q^{79} - 20 q^{81} - 36 q^{83} + 6 q^{85} + 18 q^{87} - 40 q^{89} - 32 q^{93} - 26 q^{97} + 12 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^5 + 6 * q^9 + 2 * q^11 - 10 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^23 + 4 * q^25 - 14 * q^27 - 2 * q^29 + 12 * q^31 - 18 * q^33 + 14 * q^39 - 12 * q^41 - 8 * q^43 - 6 * q^45 + 2 * q^47 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 - 8 * q^59 - 20 * q^61 + 10 * q^65 - 8 * q^67 - 24 * q^69 + 4 * q^71 - 16 * q^73 - 2 * q^75 + 22 * q^79 - 20 * q^81 - 36 * q^83 + 6 * q^85 + 18 * q^87 - 40 * q^89 - 32 * q^93 - 26 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.87996	1.08540	0.542698	−	0.839928i	$-0.317403\pi$
$3$	1.87996	1.08540	0.542698	+	0.839928i	$0.317403\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.534253	0.178084
$10$	0	0
$11$	−3.29417	−0.993231	−0.496615	−	0.867971i	$-0.665424\pi$
$11$	−3.29417	−0.993231	−0.496615	+	0.867971i	$0.665424\pi$
$12$	0	0
$13$	−4.19292	−1.16291	−0.581453	−	0.813580i	$-0.697516\pi$
$13$	−4.19292	−1.16291	−0.581453	+	0.813580i	$0.697516\pi$
$14$	0	0
$15$	−1.87996	−0.485404
$16$	0	0
$17$	1.43737	0.348614	0.174307	−	0.984691i	$-0.444232\pi$
$17$	1.43737	0.348614	0.174307	+	0.984691i	$0.444232\pi$
$18$	0	0
$19$	−1.24445	−0.285497	−0.142748	−	0.989759i	$-0.545594\pi$
$19$	−1.24445	−0.285497	−0.142748	+	0.989759i	$0.545594\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.272828	−0.0568887	−0.0284443	−	0.999595i	$-0.509055\pi$
$23$	−0.272828	−0.0568887	−0.0284443	+	0.999595i	$0.509055\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.63551	−0.892104
$28$	0	0
$29$	−2.36268	−0.438739	−0.219369	−	0.975642i	$-0.570400\pi$
$29$	−2.36268	−0.438739	−0.219369	+	0.975642i	$0.570400\pi$
$30$	0	0
$31$	3.72717	0.669420	0.334710	−	0.942321i	$-0.391362\pi$
$31$	3.72717	0.669420	0.334710	+	0.942321i	$0.391362\pi$
$32$	0	0
$33$	−6.19292	−1.07805
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.169761	0.0279085	0.0139543	−	0.999903i	$-0.495558\pi$
$37$	0.169761	0.0279085	0.0139543	+	0.999903i	$0.495558\pi$
$38$	0	0
$39$	−7.88252	−1.26221
$40$	0	0
$41$	−11.6630	−1.82146	−0.910730	−	0.413001i	$-0.864480\pi$
$41$	−11.6630	−1.82146	−0.910730	+	0.413001i	$0.864480\pi$
$42$	0	0
$43$	−10.1458	−1.54721	−0.773607	−	0.633666i	$-0.781549\pi$
$43$	−10.1458	−1.54721	−0.773607	+	0.633666i	$0.781549\pi$
$44$	0	0
$45$	−0.534253	−0.0796417
$46$	0	0
$47$	−3.12441	−0.455743	−0.227871	−	0.973691i	$-0.573177\pi$
$47$	−3.12441	−0.455743	−0.227871	+	0.973691i	$0.573177\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.70220	0.378384
$52$	0	0
$53$	9.24701	1.27018	0.635088	−	0.772440i	$-0.280964\pi$
$53$	9.24701	1.27018	0.635088	+	0.772440i	$0.280964\pi$
$54$	0	0
$55$	3.29417	0.444186
$56$	0	0
$57$	−2.33952	−0.309877
$58$	0	0
$59$	−9.07107	−1.18095	−0.590476	−	0.807055i	$-0.701060\pi$
$59$	−9.07107	−1.18095	−0.590476	+	0.807055i	$0.701060\pi$
$60$	0	0
$61$	−7.27102	−0.930958	−0.465479	−	0.885059i	$-0.654118\pi$
$61$	−7.27102	−0.930958	−0.465479	+	0.885059i	$0.654118\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.19292	0.520068
$66$	0	0
$67$	−13.1439	−1.60579	−0.802894	−	0.596121i	$-0.796708\pi$
$67$	−13.1439	−1.60579	−0.802894	+	0.596121i	$0.796708\pi$
$68$	0	0
$69$	−0.512907	−0.0617467
$70$	0	0
$71$	6.87474	0.815882	0.407941	−	0.913008i	$-0.366247\pi$
$71$	6.87474	0.815882	0.407941	+	0.913008i	$0.366247\pi$
$72$	0	0
$73$	15.2496	1.78483	0.892414	−	0.451218i	$-0.149010\pi$
$73$	15.2496	1.78483	0.892414	+	0.451218i	$0.149010\pi$
$74$	0	0
$75$	1.87996	0.217079
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.9510	1.68212	0.841061	−	0.540940i	$-0.181931\pi$
$79$	14.9510	1.68212	0.841061	+	0.540940i	$0.181931\pi$
$80$	0	0
$81$	−10.3173	−1.14637
$82$	0	0
$83$	−0.167199	−0.0183524	−0.00917622	−	0.999958i	$-0.502921\pi$
$83$	−0.167199	−0.0183524	−0.00917622	+	0.999958i	$0.502921\pi$
$84$	0	0
$85$	−1.43737	−0.155905
$86$	0	0
$87$	−4.44175	−0.476205
$88$	0	0
$89$	−3.09869	−0.328461	−0.164230	−	0.986422i	$-0.552514\pi$
$89$	−3.09869	−0.328461	−0.164230	+	0.986422i	$0.552514\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.00694	0.726585
$94$	0	0
$95$	1.24445	0.127678
$96$	0	0
$97$	−6.60894	−0.671037	−0.335518	−	0.942034i	$-0.608911\pi$
$97$	−6.60894	−0.671037	−0.335518	+	0.942034i	$0.608911\pi$
$98$	0	0
$99$	−1.75992	−0.176879
$100$	0	0
$101$	−18.8372	−1.87437	−0.937185	−	0.348834i	$-0.886578\pi$
$101$	−18.8372	−1.87437	−0.937185	+	0.348834i	$0.886578\pi$
$102$	0	0
$103$	1.46756	0.144603	0.0723014	−	0.997383i	$-0.476966\pi$
$103$	1.46756	0.144603	0.0723014	+	0.997383i	$0.476966\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.5625	1.31114	0.655570	−	0.755135i	$-0.272428\pi$
$107$	13.5625	1.31114	0.655570	+	0.755135i	$0.272428\pi$
$108$	0	0
$109$	14.8140	1.41893	0.709463	−	0.704743i	$-0.248938\pi$
$109$	14.8140	1.41893	0.709463	+	0.704743i	$0.248938\pi$
$110$	0	0
$111$	0.319144	0.0302918
$112$	0	0
$113$	−13.1903	−1.24084	−0.620418	−	0.784271i	$-0.713037\pi$
$113$	−13.1903	−1.24084	−0.620418	+	0.784271i	$0.713037\pi$
$114$	0	0
$115$	0.272828	0.0254414
$116$	0	0
$117$	−2.24008	−0.207095
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.148415	−0.0134923
$122$	0	0
$123$	−21.9261	−1.97701
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.86118	0.608831	0.304416	−	0.952539i	$-0.401539\pi$
$127$	6.86118	0.608831	0.304416	+	0.952539i	$0.401539\pi$
$128$	0	0
$129$	−19.0736	−1.67934
$130$	0	0
$131$	0.345708	0.0302047	0.0151023	−	0.999886i	$-0.495193\pi$
$131$	0.345708	0.0302047	0.0151023	+	0.999886i	$0.495193\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.63551	0.398961
$136$	0	0
$137$	11.3137	0.966595	0.483298	−	0.875456i	$-0.339439\pi$
$137$	11.3137	0.966595	0.483298	+	0.875456i	$0.339439\pi$
$138$	0	0
$139$	−2.68885	−0.228066	−0.114033	−	0.993477i	$-0.536377\pi$
$139$	−2.68885	−0.228066	−0.114033	+	0.993477i	$0.536377\pi$
$140$	0	0
$141$	−5.87377	−0.494661
$142$	0	0
$143$	13.8122	1.15503
$144$	0	0
$145$	2.36268	0.196210
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.1340	1.07598	0.537990	−	0.842951i	$-0.319184\pi$
$149$	13.1340	1.07598	0.537990	+	0.842951i	$0.319184\pi$
$150$	0	0
$151$	3.01140	0.245065	0.122532	−	0.992465i	$-0.460898\pi$
$151$	3.01140	0.245065	0.122532	+	0.992465i	$0.460898\pi$
$152$	0	0
$153$	0.767920	0.0620826
$154$	0	0
$155$	−3.72717	−0.299374
$156$	0	0
$157$	−19.4631	−1.55332	−0.776662	−	0.629918i	$-0.783089\pi$
$157$	−19.4631	−1.55332	−0.776662	+	0.629918i	$0.783089\pi$
$158$	0	0
$159$	17.3840	1.37864
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.4922	1.05679	0.528396	−	0.848998i	$-0.322794\pi$
$163$	13.4922	1.05679	0.528396	+	0.848998i	$0.322794\pi$
$164$	0	0
$165$	6.19292	0.482118
$166$	0	0
$167$	−12.3011	−0.951889	−0.475944	−	0.879475i	$-0.657894\pi$
$167$	−12.3011	−0.951889	−0.475944	+	0.879475i	$0.657894\pi$
$168$	0	0
$169$	4.58057	0.352351
$170$	0	0
$171$	−0.664852	−0.0508425
$172$	0	0
$173$	−2.48975	−0.189292	−0.0946461	−	0.995511i	$-0.530172\pi$
$173$	−2.48975	−0.189292	−0.0946461	+	0.995511i	$0.530172\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−17.0533	−1.28180
$178$	0	0
$179$	15.1340	1.13117	0.565584	−	0.824690i	$-0.308651\pi$
$179$	15.1340	1.13117	0.565584	+	0.824690i	$0.308651\pi$
$180$	0	0
$181$	−11.0637	−0.822357	−0.411179	−	0.911555i	$-0.634883\pi$
$181$	−11.0637	−0.822357	−0.411179	+	0.911555i	$0.634883\pi$
$182$	0	0
$183$	−13.6692	−1.01046
$184$	0	0
$185$	−0.169761	−0.0124811
$186$	0	0
$187$	−4.73495	−0.346254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.4517	−1.62455	−0.812274	−	0.583277i	$-0.801770\pi$
$191$	−22.4517	−1.62455	−0.812274	+	0.583277i	$0.801770\pi$
$192$	0	0
$193$	3.11482	0.224210	0.112105	−	0.993696i	$-0.464241\pi$
$193$	3.11482	0.224210	0.112105	+	0.993696i	$0.464241\pi$
$194$	0	0
$195$	7.88252	0.564479
$196$	0	0
$197$	1.38765	0.0988659	0.0494330	−	0.998777i	$-0.484259\pi$
$197$	1.38765	0.0988659	0.0494330	+	0.998777i	$0.484259\pi$
$198$	0	0
$199$	0.413463	0.0293096	0.0146548	−	0.999893i	$-0.495335\pi$
$199$	0.413463	0.0293096	0.0146548	+	0.999893i	$0.495335\pi$
$200$	0	0
$201$	−24.7101	−1.74292
$202$	0	0
$203$	0	0
$204$	0	0
$205$	11.6630	0.814582
$206$	0	0
$207$	−0.145759	−0.0101310
$208$	0	0
$209$	4.09944	0.283564
$210$	0	0
$211$	24.6203	1.69493	0.847464	−	0.530853i	$-0.178128\pi$
$211$	24.6203	1.69493	0.847464	+	0.530853i	$0.178128\pi$
$212$	0	0
$213$	12.9242	0.885555
$214$	0	0
$215$	10.1458	0.691935
$216$	0	0
$217$	0	0
$218$	0	0
$219$	28.6686	1.93724
$220$	0	0
$221$	−6.02678	−0.405405
$222$	0	0
$223$	17.9065	1.19911	0.599555	−	0.800334i	$-0.295344\pi$
$223$	17.9065	1.19911	0.599555	+	0.800334i	$0.295344\pi$
$224$	0	0
$225$	0.534253	0.0356168
$226$	0	0
$227$	12.5486	0.832878	0.416439	−	0.909164i	$-0.363278\pi$
$227$	12.5486	0.832878	0.416439	+	0.909164i	$0.363278\pi$
$228$	0	0
$229$	15.8354	1.04643	0.523215	−	0.852201i	$-0.324732\pi$
$229$	15.8354	1.04643	0.523215	+	0.852201i	$0.324732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.3792	−1.53162	−0.765811	−	0.643065i	$-0.777662\pi$
$233$	−23.3792	−1.53162	−0.765811	+	0.643065i	$0.777662\pi$
$234$	0	0
$235$	3.12441	0.203814
$236$	0	0
$237$	28.1073	1.82577
$238$	0	0
$239$	20.9135	1.35278	0.676390	−	0.736544i	$-0.263544\pi$
$239$	20.9135	1.35278	0.676390	+	0.736544i	$0.263544\pi$
$240$	0	0
$241$	3.35746	0.216273	0.108137	−	0.994136i	$-0.465512\pi$
$241$	3.35746	0.216273	0.108137	+	0.994136i	$0.465512\pi$
$242$	0	0
$243$	−5.48966	−0.352162
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.21789	0.332006
$248$	0	0
$249$	−0.314327	−0.0199197
$250$	0	0
$251$	−16.5717	−1.04600	−0.522999	−	0.852333i	$-0.675187\pi$
$251$	−16.5717	−1.04600	−0.522999	+	0.852333i	$0.675187\pi$
$252$	0	0
$253$	0.898744	0.0565036
$254$	0	0
$255$	−2.70220	−0.169218
$256$	0	0
$257$	−13.1414	−0.819737	−0.409869	−	0.912145i	$-0.634425\pi$
$257$	−13.1414	−0.819737	−0.409869	+	0.912145i	$0.634425\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.26227	−0.0781324
$262$	0	0
$263$	−3.75480	−0.231531	−0.115765	−	0.993277i	$-0.536932\pi$
$263$	−3.75480	−0.231531	−0.115765	+	0.993277i	$0.536932\pi$
$264$	0	0
$265$	−9.24701	−0.568040
$266$	0	0
$267$	−5.82542	−0.356510
$268$	0	0
$269$	−16.4952	−1.00573	−0.502866	−	0.864365i	$-0.667721\pi$
$269$	−16.4952	−1.00573	−0.502866	+	0.864365i	$0.667721\pi$
$270$	0	0
$271$	22.5420	1.36933	0.684665	−	0.728857i	$-0.259948\pi$
$271$	22.5420	1.36933	0.684665	+	0.728857i	$0.259948\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.29417	−0.198646
$276$	0	0
$277$	−14.9982	−0.901154	−0.450577	−	0.892738i	$-0.648782\pi$
$277$	−14.9982	−0.901154	−0.450577	+	0.892738i	$0.648782\pi$
$278$	0	0
$279$	1.99125	0.119213
$280$	0	0
$281$	−28.0283	−1.67203	−0.836014	−	0.548709i	$-0.815120\pi$
$281$	−28.0283	−1.67203	−0.836014	+	0.548709i	$0.815120\pi$
$282$	0	0
$283$	26.1715	1.55573	0.777866	−	0.628430i	$-0.216302\pi$
$283$	26.1715	1.55573	0.777866	+	0.628430i	$0.216302\pi$
$284$	0	0
$285$	2.33952	0.138581
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.9340	−0.878468
$290$	0	0
$291$	−12.4246	−0.728340
$292$	0	0
$293$	−15.6061	−0.911716	−0.455858	−	0.890052i	$-0.650668\pi$
$293$	−15.6061	−0.911716	−0.455858	+	0.890052i	$0.650668\pi$
$294$	0	0
$295$	9.07107	0.528138
$296$	0	0
$297$	15.2702	0.886065
$298$	0	0
$299$	1.14395	0.0661562
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−35.4132	−2.03443
$304$	0	0
$305$	7.27102	0.416337
$306$	0	0
$307$	−21.3055	−1.21597	−0.607984	−	0.793949i	$-0.708022\pi$
$307$	−21.3055	−1.21597	−0.607984	+	0.793949i	$0.708022\pi$
$308$	0	0
$309$	2.75895	0.156951
$310$	0	0
$311$	14.8799	0.843760	0.421880	−	0.906652i	$-0.361370\pi$
$311$	14.8799	0.843760	0.421880	+	0.906652i	$0.361370\pi$
$312$	0	0
$313$	−2.16273	−0.122245	−0.0611224	−	0.998130i	$-0.519468\pi$
$313$	−2.16273	−0.122245	−0.0611224	+	0.998130i	$0.519468\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.4595	−1.20528	−0.602642	−	0.798012i	$-0.705885\pi$
$317$	−21.4595	−1.20528	−0.602642	+	0.798012i	$0.705885\pi$
$318$	0	0
$319$	7.78308	0.435769
$320$	0	0
$321$	25.4970	1.42311
$322$	0	0
$323$	−1.78874	−0.0995282
$324$	0	0
$325$	−4.19292	−0.232581
$326$	0	0
$327$	27.8498	1.54010
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.6692	−0.641399	−0.320699	−	0.947181i	$-0.603918\pi$
$331$	−11.6692	−0.641399	−0.320699	+	0.947181i	$0.603918\pi$
$332$	0	0
$333$	0.0906953	0.00497007
$334$	0	0
$335$	13.1439	0.718131
$336$	0	0
$337$	17.1403	0.933693	0.466846	−	0.884338i	$-0.345390\pi$
$337$	17.1403	0.933693	0.466846	+	0.884338i	$0.345390\pi$
$338$	0	0
$339$	−24.7972	−1.34680
$340$	0	0
$341$	−12.2780	−0.664888
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.512907	0.0276140
$346$	0	0
$347$	5.20070	0.279188	0.139594	−	0.990209i	$-0.455420\pi$
$347$	5.20070	0.279188	0.139594	+	0.990209i	$0.455420\pi$
$348$	0	0
$349$	33.8645	1.81272	0.906362	−	0.422501i	$-0.138848\pi$
$349$	33.8645	1.81272	0.906362	+	0.422501i	$0.138848\pi$
$350$	0	0
$351$	19.4363	1.03743
$352$	0	0
$353$	23.2451	1.23721	0.618606	−	0.785701i	$-0.287698\pi$
$353$	23.2451	1.23721	0.618606	+	0.785701i	$0.287698\pi$
$354$	0	0
$355$	−6.87474	−0.364873
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.2843	−1.07056	−0.535281	−	0.844674i	$-0.679794\pi$
$359$	−20.2843	−1.07056	−0.535281	+	0.844674i	$0.679794\pi$
$360$	0	0
$361$	−17.4513	−0.918491
$362$	0	0
$363$	−0.279014	−0.0146445
$364$	0	0
$365$	−15.2496	−0.798199
$366$	0	0
$367$	13.2738	0.692887	0.346443	−	0.938071i	$-0.387389\pi$
$367$	13.2738	0.692887	0.346443	+	0.938071i	$0.387389\pi$
$368$	0	0
$369$	−6.23101	−0.324373
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−35.7083	−1.84891	−0.924453	−	0.381297i	$-0.875478\pi$
$373$	−35.7083	−1.84891	−0.924453	+	0.381297i	$0.875478\pi$
$374$	0	0
$375$	−1.87996	−0.0970808
$376$	0	0
$377$	9.90652	0.510212
$378$	0	0
$379$	−12.6878	−0.651728	−0.325864	−	0.945417i	$-0.605655\pi$
$379$	−12.6878	−0.651728	−0.325864	+	0.945417i	$0.605655\pi$
$380$	0	0
$381$	12.8987	0.660823
$382$	0	0
$383$	−36.5707	−1.86867	−0.934337	−	0.356391i	$-0.884007\pi$
$383$	−36.5707	−1.86867	−0.934337	+	0.356391i	$0.884007\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.42040	−0.275534
$388$	0	0
$389$	30.1085	1.52656	0.763282	−	0.646066i	$-0.223587\pi$
$389$	30.1085	1.52656	0.763282	+	0.646066i	$0.223587\pi$
$390$	0	0
$391$	−0.392156	−0.0198322
$392$	0	0
$393$	0.649918	0.0327840
$394$	0	0
$395$	−14.9510	−0.752268
$396$	0	0
$397$	7.46756	0.374786	0.187393	−	0.982285i	$-0.439996\pi$
$397$	7.46756	0.374786	0.187393	+	0.982285i	$0.439996\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.7331	−0.585925	−0.292963	−	0.956124i	$-0.594641\pi$
$401$	−11.7331	−0.585925	−0.292963	+	0.956124i	$0.594641\pi$
$402$	0	0
$403$	−15.6277	−0.778473
$404$	0	0
$405$	10.3173	0.512672
$406$	0	0
$407$	−0.559222	−0.0277196
$408$	0	0
$409$	−16.8601	−0.833679	−0.416840	−	0.908980i	$-0.636862\pi$
$409$	−16.8601	−0.833679	−0.416840	+	0.908980i	$0.636862\pi$
$410$	0	0
$411$	21.2693	1.04914
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.167199	0.00820746
$416$	0	0
$417$	−5.05494	−0.247541
$418$	0	0
$419$	10.3884	0.507507	0.253753	−	0.967269i	$-0.418335\pi$
$419$	10.3884	0.507507	0.253753	+	0.967269i	$0.418335\pi$
$420$	0	0
$421$	13.7758	0.671393	0.335696	−	0.941970i	$-0.391028\pi$
$421$	13.7758	0.671393	0.335696	+	0.941970i	$0.391028\pi$
$422$	0	0
$423$	−1.66923	−0.0811606
$424$	0	0
$425$	1.43737	0.0697228
$426$	0	0
$427$	0	0
$428$	0	0
$429$	25.9664	1.25367
$430$	0	0
$431$	30.4568	1.46705	0.733527	−	0.679661i	$-0.237873\pi$
$431$	30.4568	1.46705	0.733527	+	0.679661i	$0.237873\pi$
$432$	0	0
$433$	−25.9182	−1.24555	−0.622774	−	0.782402i	$-0.713994\pi$
$433$	−25.9182	−1.24555	−0.622774	+	0.782402i	$0.713994\pi$
$434$	0	0
$435$	4.44175	0.212965
$436$	0	0
$437$	0.339522	0.0162415
$438$	0	0
$439$	26.9414	1.28584	0.642922	−	0.765931i	$-0.277722\pi$
$439$	26.9414	1.28584	0.642922	+	0.765931i	$0.277722\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.3858	−0.778515	−0.389257	−	0.921129i	$-0.627268\pi$
$443$	−16.3858	−0.778515	−0.389257	+	0.921129i	$0.627268\pi$
$444$	0	0
$445$	3.09869	0.146892
$446$	0	0
$447$	24.6914	1.16786
$448$	0	0
$449$	14.2152	0.670858	0.335429	−	0.942065i	$-0.391119\pi$
$449$	14.2152	0.670858	0.335429	+	0.942065i	$0.391119\pi$
$450$	0	0
$451$	38.4201	1.80913
$452$	0	0
$453$	5.66132	0.265992
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.5093	−0.678716	−0.339358	−	0.940657i	$-0.610210\pi$
$457$	−14.5093	−0.678716	−0.339358	+	0.940657i	$0.610210\pi$
$458$	0	0
$459$	−6.66295	−0.311000
$460$	0	0
$461$	17.8933	0.833374	0.416687	−	0.909050i	$-0.363191\pi$
$461$	17.8933	0.833374	0.416687	+	0.909050i	$0.363191\pi$
$462$	0	0
$463$	15.2657	0.709457	0.354729	−	0.934969i	$-0.384573\pi$
$463$	15.2657	0.709457	0.354729	+	0.934969i	$0.384573\pi$
$464$	0	0
$465$	−7.00694	−0.324939
$466$	0	0
$467$	−23.5492	−1.08973	−0.544863	−	0.838525i	$-0.683418\pi$
$467$	−23.5492	−1.08973	−0.544863	+	0.838525i	$0.683418\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−36.5898	−1.68597
$472$	0	0
$473$	33.4219	1.53674
$474$	0	0
$475$	−1.24445	−0.0570994
$476$	0	0
$477$	4.94024	0.226198
$478$	0	0
$479$	−4.65867	−0.212860	−0.106430	−	0.994320i	$-0.533942\pi$
$479$	−4.65867	−0.212860	−0.106430	+	0.994320i	$0.533942\pi$
$480$	0	0
$481$	−0.711794	−0.0324550
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.60894	0.300097
$486$	0	0
$487$	11.5210	0.522068	0.261034	−	0.965330i	$-0.415937\pi$
$487$	11.5210	0.522068	0.261034	+	0.965330i	$0.415937\pi$
$488$	0	0
$489$	25.3648	1.14704
$490$	0	0
$491$	−14.2771	−0.644317	−0.322158	−	0.946686i	$-0.604408\pi$
$491$	−14.2771	−0.644317	−0.322158	+	0.946686i	$0.604408\pi$
$492$	0	0
$493$	−3.39605	−0.152950
$494$	0	0
$495$	1.75992	0.0791026
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.9790	1.65541	0.827703	−	0.561167i	$-0.189647\pi$
$499$	36.9790	1.65541	0.827703	+	0.561167i	$0.189647\pi$
$500$	0	0
$501$	−23.1256	−1.03318
$502$	0	0
$503$	8.08985	0.360709	0.180354	−	0.983602i	$-0.442276\pi$
$503$	8.08985	0.360709	0.180354	+	0.983602i	$0.442276\pi$
$504$	0	0
$505$	18.8372	0.838243
$506$	0	0
$507$	8.61129	0.382441
$508$	0	0
$509$	28.0806	1.24465	0.622325	−	0.782759i	$-0.286188\pi$
$509$	28.0806	1.24465	0.622325	+	0.782759i	$0.286188\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.76867	0.254693
$514$	0	0
$515$	−1.46756	−0.0646684
$516$	0	0
$517$	10.2924	0.452658
$518$	0	0
$519$	−4.68063	−0.205457
$520$	0	0
$521$	12.5810	0.551182	0.275591	−	0.961275i	$-0.411126\pi$
$521$	12.5810	0.551182	0.275591	+	0.961275i	$0.411126\pi$
$522$	0	0
$523$	1.48453	0.0649140	0.0324570	−	0.999473i	$-0.489667\pi$
$523$	1.48453	0.0649140	0.0324570	+	0.999473i	$0.489667\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.35733	0.233369
$528$	0	0
$529$	−22.9256	−0.996764
$530$	0	0
$531$	−4.84624	−0.210309
$532$	0	0
$533$	48.9022	2.11819
$534$	0	0
$535$	−13.5625	−0.586360
$536$	0	0
$537$	28.4513	1.22777
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.9135	0.813153	0.406577	−	0.913617i	$-0.366722\pi$
$541$	18.9135	0.813153	0.406577	+	0.913617i	$0.366722\pi$
$542$	0	0
$543$	−20.7993	−0.892583
$544$	0	0
$545$	−14.8140	−0.634563
$546$	0	0
$547$	−17.4403	−0.745693	−0.372846	−	0.927893i	$-0.621618\pi$
$547$	−17.4403	−0.745693	−0.372846	+	0.927893i	$0.621618\pi$
$548$	0	0
$549$	−3.88456	−0.165789
$550$	0	0
$551$	2.94024	0.125259
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.319144	−0.0135469
$556$	0	0
$557$	19.2106	0.813981	0.406990	−	0.913432i	$-0.366578\pi$
$557$	19.2106	0.813981	0.406990	+	0.913432i	$0.366578\pi$
$558$	0	0
$559$	42.5403	1.79926
$560$	0	0
$561$	−8.90152	−0.375823
$562$	0	0
$563$	−33.7952	−1.42430	−0.712150	−	0.702028i	$-0.752278\pi$
$563$	−33.7952	−1.42430	−0.712150	+	0.702028i	$0.752278\pi$
$564$	0	0
$565$	13.1903	0.554919
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.63635	0.194366	0.0971830	−	0.995267i	$-0.469017\pi$
$569$	4.63635	0.194366	0.0971830	+	0.995267i	$0.469017\pi$
$570$	0	0
$571$	43.8681	1.83582	0.917912	−	0.396785i	$-0.129874\pi$
$571$	43.8681	1.83582	0.917912	+	0.396785i	$0.129874\pi$
$572$	0	0
$573$	−42.2083	−1.76328
$574$	0	0
$575$	−0.272828	−0.0113777
$576$	0	0
$577$	−9.27755	−0.386230	−0.193115	−	0.981176i	$-0.561859\pi$
$577$	−9.27755	−0.386230	−0.193115	+	0.981176i	$0.561859\pi$
$578$	0	0
$579$	5.85574	0.243356
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−30.4613	−1.26158
$584$	0	0
$585$	2.24008	0.0926158
$586$	0	0
$587$	−28.9971	−1.19684	−0.598420	−	0.801183i	$-0.704204\pi$
$587$	−28.9971	−1.19684	−0.598420	+	0.801183i	$0.704204\pi$
$588$	0	0
$589$	−4.63829	−0.191117
$590$	0	0
$591$	2.60873	0.107309
$592$	0	0
$593$	25.5228	1.04809	0.524047	−	0.851689i	$-0.324422\pi$
$593$	25.5228	1.04809	0.524047	+	0.851689i	$0.324422\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.777294	0.0318125
$598$	0	0
$599$	31.0364	1.26811	0.634057	−	0.773287i	$-0.281389\pi$
$599$	31.0364	1.26811	0.634057	+	0.773287i	$0.281389\pi$
$600$	0	0
$601$	−3.56479	−0.145411	−0.0727054	−	0.997353i	$-0.523163\pi$
$601$	−3.56479	−0.145411	−0.0727054	+	0.997353i	$0.523163\pi$
$602$	0	0
$603$	−7.02219	−0.285966
$604$	0	0
$605$	0.148415	0.00603393
$606$	0	0
$607$	6.72620	0.273008	0.136504	−	0.990640i	$-0.456413\pi$
$607$	6.72620	0.273008	0.136504	+	0.990640i	$0.456413\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.1004	0.529986
$612$	0	0
$613$	−19.7422	−0.797381	−0.398691	−	0.917085i	$-0.630535\pi$
$613$	−19.7422	−0.797381	−0.398691	+	0.917085i	$0.630535\pi$
$614$	0	0
$615$	21.9261	0.884144
$616$	0	0
$617$	−16.3958	−0.660069	−0.330035	−	0.943969i	$-0.607060\pi$
$617$	−16.3958	−0.660069	−0.330035	+	0.943969i	$0.607060\pi$
$618$	0	0
$619$	−35.5510	−1.42892	−0.714458	−	0.699678i	$-0.753327\pi$
$619$	−35.5510	−1.42892	−0.714458	+	0.699678i	$0.753327\pi$
$620$	0	0
$621$	1.26470	0.0507506
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.70679	0.307780
$628$	0	0
$629$	0.244010	0.00972930
$630$	0	0
$631$	−9.58569	−0.381600	−0.190800	−	0.981629i	$-0.561108\pi$
$631$	−9.58569	−0.381600	−0.190800	+	0.981629i	$0.561108\pi$
$632$	0	0
$633$	46.2851	1.83967
$634$	0	0
$635$	−6.86118	−0.272278
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.67285	0.145296
$640$	0	0
$641$	19.5145	0.770778	0.385389	−	0.922754i	$-0.374067\pi$
$641$	19.5145	0.770778	0.385389	+	0.922754i	$0.374067\pi$
$642$	0	0
$643$	18.9895	0.748872	0.374436	−	0.927253i	$-0.377836\pi$
$643$	18.9895	0.748872	0.374436	+	0.927253i	$0.377836\pi$
$644$	0	0
$645$	19.0736	0.751023
$646$	0	0
$647$	−41.2570	−1.62198	−0.810989	−	0.585061i	$-0.801070\pi$
$647$	−41.2570	−1.62198	−0.810989	+	0.585061i	$0.801070\pi$
$648$	0	0
$649$	29.8817	1.17296
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.7495	−0.772857	−0.386429	−	0.922319i	$-0.626291\pi$
$653$	−19.7495	−0.772857	−0.386429	+	0.922319i	$0.626291\pi$
$654$	0	0
$655$	−0.345708	−0.0135079
$656$	0	0
$657$	8.14713	0.317850
$658$	0	0
$659$	10.1830	0.396672	0.198336	−	0.980134i	$-0.436446\pi$
$659$	10.1830	0.396672	0.198336	+	0.980134i	$0.436446\pi$
$660$	0	0
$661$	2.79086	0.108552	0.0542759	−	0.998526i	$-0.482715\pi$
$661$	2.79086	0.108552	0.0542759	+	0.998526i	$0.482715\pi$
$662$	0	0
$663$	−11.3301	−0.440025
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.644606	0.0249593
$668$	0	0
$669$	33.6636	1.30151
$670$	0	0
$671$	23.9520	0.924657
$672$	0	0
$673$	18.2879	0.704947	0.352473	−	0.935822i	$-0.385341\pi$
$673$	18.2879	0.704947	0.352473	+	0.935822i	$0.385341\pi$
$674$	0	0
$675$	−4.63551	−0.178421
$676$	0	0
$677$	−24.0523	−0.924404	−0.462202	−	0.886775i	$-0.652941\pi$
$677$	−24.0523	−0.924404	−0.462202	+	0.886775i	$0.652941\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	23.5908	0.904002
$682$	0	0
$683$	−38.2200	−1.46245	−0.731224	−	0.682137i	$-0.761051\pi$
$683$	−38.2200	−1.46245	−0.731224	+	0.682137i	$0.761051\pi$
$684$	0	0
$685$	−11.3137	−0.432275
$686$	0	0
$687$	29.7699	1.13579
$688$	0	0
$689$	−38.7720	−1.47709
$690$	0	0
$691$	−26.4623	−1.00667	−0.503337	−	0.864090i	$-0.667895\pi$
$691$	−26.4623	−1.00667	−0.503337	+	0.864090i	$0.667895\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.68885	0.101994
$696$	0	0
$697$	−16.7641	−0.634986
$698$	0	0
$699$	−43.9520	−1.66242
$700$	0	0
$701$	−34.5136	−1.30356	−0.651780	−	0.758408i	$-0.725977\pi$
$701$	−34.5136	−1.30356	−0.651780	+	0.758408i	$0.725977\pi$
$702$	0	0
$703$	−0.211260	−0.00796780
$704$	0	0
$705$	5.87377	0.221219
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.40900	0.240695	0.120347	−	0.992732i	$-0.461599\pi$
$709$	6.40900	0.240695	0.120347	+	0.992732i	$0.461599\pi$
$710$	0	0
$711$	7.98763	0.299559
$712$	0	0
$713$	−1.01688	−0.0380824
$714$	0	0
$715$	−13.8122	−0.516547
$716$	0	0
$717$	39.3165	1.46830
$718$	0	0
$719$	−38.6070	−1.43980	−0.719900	−	0.694078i	$-0.755812\pi$
$719$	−38.6070	−1.43980	−0.719900	+	0.694078i	$0.755812\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6.31190	0.234742
$724$	0	0
$725$	−2.36268	−0.0877477
$726$	0	0
$727$	22.5280	0.835516	0.417758	−	0.908558i	$-0.362816\pi$
$727$	22.5280	0.835516	0.417758	+	0.908558i	$0.362816\pi$
$728$	0	0
$729$	20.6317	0.764136
$730$	0	0
$731$	−14.5832	−0.539380
$732$	0	0
$733$	7.50150	0.277074	0.138537	−	0.990357i	$-0.455760\pi$
$733$	7.50150	0.277074	0.138537	+	0.990357i	$0.455760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	43.2985	1.59492
$738$	0	0
$739$	35.0864	1.29067	0.645336	−	0.763899i	$-0.276717\pi$
$739$	35.0864	1.29067	0.645336	+	0.763899i	$0.276717\pi$
$740$	0	0
$741$	9.80943	0.360358
$742$	0	0
$743$	−2.42902	−0.0891122	−0.0445561	−	0.999007i	$-0.514187\pi$
$743$	−2.42902	−0.0891122	−0.0445561	+	0.999007i	$0.514187\pi$
$744$	0	0
$745$	−13.1340	−0.481193
$746$	0	0
$747$	−0.0893263	−0.00326828
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.9598	0.764833	0.382417	−	0.923990i	$-0.375092\pi$
$751$	20.9598	0.764833	0.382417	+	0.923990i	$0.375092\pi$
$752$	0	0
$753$	−31.1542	−1.13532
$754$	0	0
$755$	−3.01140	−0.109596
$756$	0	0
$757$	−35.9748	−1.30753	−0.653763	−	0.756699i	$-0.726811\pi$
$757$	−35.9748	−1.30753	−0.653763	+	0.756699i	$0.726811\pi$
$758$	0	0
$759$	1.68960	0.0613288
$760$	0	0
$761$	2.45752	0.0890852	0.0445426	−	0.999007i	$-0.485817\pi$
$761$	2.45752	0.0890852	0.0445426	+	0.999007i	$0.485817\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.767920	−0.0277642
$766$	0	0
$767$	38.0343	1.37334
$768$	0	0
$769$	−21.6994	−0.782501	−0.391250	−	0.920284i	$-0.627957\pi$
$769$	−21.6994	−0.782501	−0.391250	+	0.920284i	$0.627957\pi$
$770$	0	0
$771$	−24.7053	−0.889739
$772$	0	0
$773$	−15.6980	−0.564618	−0.282309	−	0.959324i	$-0.591100\pi$
$773$	−15.6980	−0.564618	−0.282309	+	0.959324i	$0.591100\pi$
$774$	0	0
$775$	3.72717	0.133884
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.5141	0.520022
$780$	0	0
$781$	−22.6466	−0.810359
$782$	0	0
$783$	10.9522	0.391400
$784$	0	0
$785$	19.4631	0.694668
$786$	0	0
$787$	24.5074	0.873594	0.436797	−	0.899560i	$-0.356113\pi$
$787$	24.5074	0.873594	0.436797	+	0.899560i	$0.356113\pi$
$788$	0	0
$789$	−7.05887	−0.251302
$790$	0	0
$791$	0	0
$792$	0	0
$793$	30.4868	1.08262
$794$	0	0
$795$	−17.3840	−0.616548
$796$	0	0
$797$	−5.73557	−0.203164	−0.101582	−	0.994827i	$-0.532390\pi$
$797$	−5.73557	−0.203164	−0.101582	+	0.994827i	$0.532390\pi$
$798$	0	0
$799$	−4.49094	−0.158878
$800$	0	0
$801$	−1.65549	−0.0584937
$802$	0	0
$803$	−50.2348	−1.77275
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.0104	−1.09162
$808$	0	0
$809$	3.48431	0.122502	0.0612510	−	0.998122i	$-0.480491\pi$
$809$	3.48431	0.122502	0.0612510	+	0.998122i	$0.480491\pi$
$810$	0	0
$811$	−25.9953	−0.912819	−0.456409	−	0.889770i	$-0.650865\pi$
$811$	−25.9953	−0.912819	−0.456409	+	0.889770i	$0.650865\pi$
$812$	0	0
$813$	42.3781	1.48627
$814$	0	0
$815$	−13.4922	−0.472612
$816$	0	0
$817$	12.6259	0.441725
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.4988	−1.76242	−0.881210	−	0.472725i	$-0.843271\pi$
$821$	−50.4988	−1.76242	−0.881210	+	0.472725i	$0.843271\pi$
$822$	0	0
$823$	−44.8699	−1.56407	−0.782034	−	0.623236i	$-0.785818\pi$
$823$	−44.8699	−1.56407	−0.782034	+	0.623236i	$0.785818\pi$
$824$	0	0
$825$	−6.19292	−0.215610
$826$	0	0
$827$	−15.7323	−0.547066	−0.273533	−	0.961863i	$-0.588192\pi$
$827$	−15.7323	−0.547066	−0.273533	+	0.961863i	$0.588192\pi$
$828$	0	0
$829$	51.7726	1.79814	0.899068	−	0.437808i	$-0.144245\pi$
$829$	51.7726	1.79814	0.899068	+	0.437808i	$0.144245\pi$
$830$	0	0
$831$	−28.1960	−0.978109
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.3011	0.425697
$836$	0	0
$837$	−17.2773	−0.597192
$838$	0	0
$839$	2.86118	0.0987788	0.0493894	−	0.998780i	$-0.484272\pi$
$839$	2.86118	0.0987788	0.0493894	+	0.998780i	$0.484272\pi$
$840$	0	0
$841$	−23.4177	−0.807508
$842$	0	0
$843$	−52.6921	−1.81481
$844$	0	0
$845$	−4.58057	−0.157576
$846$	0	0
$847$	0	0
$848$	0	0
$849$	49.2014	1.68859
$850$	0	0
$851$	−0.0463156	−0.00158768
$852$	0	0
$853$	−45.8645	−1.57037	−0.785185	−	0.619261i	$-0.787432\pi$
$853$	−45.8645	−1.57037	−0.785185	+	0.619261i	$0.787432\pi$
$854$	0	0
$855$	0.664852	0.0227375
$856$	0	0
$857$	23.1501	0.790793	0.395397	−	0.918510i	$-0.370607\pi$
$857$	23.1501	0.790793	0.395397	+	0.918510i	$0.370607\pi$
$858$	0	0
$859$	8.46384	0.288783	0.144391	−	0.989521i	$-0.453878\pi$
$859$	8.46384	0.288783	0.144391	+	0.989521i	$0.453878\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.0923	−1.19456	−0.597278	−	0.802034i	$-0.703751\pi$
$863$	−35.0923	−1.19456	−0.597278	+	0.802034i	$0.703751\pi$
$864$	0	0
$865$	2.48975	0.0846540
$866$	0	0
$867$	−28.0753	−0.953486
$868$	0	0
$869$	−49.2513	−1.67074
$870$	0	0
$871$	55.1115	1.86738
$872$	0	0
$873$	−3.53085	−0.119501
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.3372	0.652971	0.326486	−	0.945202i	$-0.394136\pi$
$877$	19.3372	0.652971	0.326486	+	0.945202i	$0.394136\pi$
$878$	0	0
$879$	−29.3388	−0.989573
$880$	0	0
$881$	33.0573	1.11373	0.556865	−	0.830603i	$-0.312004\pi$
$881$	33.0573	1.11373	0.556865	+	0.830603i	$0.312004\pi$
$882$	0	0
$883$	−28.6238	−0.963267	−0.481634	−	0.876373i	$-0.659956\pi$
$883$	−28.6238	−0.963267	−0.481634	+	0.876373i	$0.659956\pi$
$884$	0	0
$885$	17.0533	0.573239
$886$	0	0
$887$	−19.1289	−0.642285	−0.321142	−	0.947031i	$-0.604067\pi$
$887$	−19.1289	−0.642285	−0.321142	+	0.947031i	$0.604067\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	33.9871	1.13861
$892$	0	0
$893$	3.88818	0.130113
$894$	0	0
$895$	−15.1340	−0.505874
$896$	0	0
$897$	2.15058	0.0718057
$898$	0	0
$899$	−8.80611	−0.293700
$900$	0	0
$901$	13.2914	0.442801
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.0637	0.367769
$906$	0	0
$907$	−50.9781	−1.69270	−0.846350	−	0.532627i	$-0.821205\pi$
$907$	−50.9781	−1.69270	−0.846350	+	0.532627i	$0.821205\pi$
$908$	0	0
$909$	−10.0638	−0.333796
$910$	0	0
$911$	−5.52585	−0.183080	−0.0915399	−	0.995801i	$-0.529179\pi$
$911$	−5.52585	−0.183080	−0.0915399	+	0.995801i	$0.529179\pi$
$912$	0	0
$913$	0.550782	0.0182282
$914$	0	0
$915$	13.6692	0.451891
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−40.6326	−1.34035	−0.670173	−	0.742205i	$-0.733780\pi$
$919$	−40.6326	−1.34035	−0.670173	+	0.742205i	$0.733780\pi$
$920$	0	0
$921$	−40.0535	−1.31981
$922$	0	0
$923$	−28.8252	−0.948794
$924$	0	0
$925$	0.169761	0.00558171
$926$	0	0
$927$	0.784047	0.0257515
$928$	0	0
$929$	15.5879	0.511423	0.255711	−	0.966753i	$-0.417690\pi$
$929$	15.5879	0.511423	0.255711	+	0.966753i	$0.417690\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	27.9736	0.915814
$934$	0	0
$935$	4.73495	0.154849
$936$	0	0
$937$	20.4794	0.669034	0.334517	−	0.942390i	$-0.391427\pi$
$937$	20.4794	0.669034	0.334517	+	0.942390i	$0.391427\pi$
$938$	0	0
$939$	−4.06585	−0.132684
$940$	0	0
$941$	−16.7872	−0.547248	−0.273624	−	0.961837i	$-0.588222\pi$
$941$	−16.7872	−0.547248	−0.273624	+	0.961837i	$0.588222\pi$
$942$	0	0
$943$	3.18201	0.103620
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.43928	−0.111761	−0.0558807	−	0.998437i	$-0.517797\pi$
$947$	−3.43928	−0.111761	−0.0558807	+	0.998437i	$0.517797\pi$
$948$	0	0
$949$	−63.9402	−2.07559
$950$	0	0
$951$	−40.3430	−1.30821
$952$	0	0
$953$	−30.8610	−0.999686	−0.499843	−	0.866116i	$-0.666609\pi$
$953$	−30.8610	−0.999686	−0.499843	+	0.866116i	$0.666609\pi$
$954$	0	0
$955$	22.4517	0.726520
$956$	0	0
$957$	14.6319	0.472982
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17.1082	−0.551877
$962$	0	0
$963$	7.24582	0.233493
$964$	0	0
$965$	−3.11482	−0.100270
$966$	0	0
$967$	−34.3195	−1.10364	−0.551820	−	0.833964i	$-0.686066\pi$
$967$	−34.3195	−1.10364	−0.551820	+	0.833964i	$0.686066\pi$
$968$	0	0
$969$	−3.36276	−0.108027
$970$	0	0
$971$	−31.6203	−1.01475	−0.507373	−	0.861727i	$-0.669383\pi$
$971$	−31.6203	−1.01475	−0.507373	+	0.861727i	$0.669383\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.88252	−0.252443
$976$	0	0
$977$	−18.9454	−0.606116	−0.303058	−	0.952972i	$-0.598008\pi$
$977$	−18.9454	−0.606116	−0.303058	+	0.952972i	$0.598008\pi$
$978$	0	0
$979$	10.2076	0.326237
$980$	0	0
$981$	7.91443	0.252688
$982$	0	0
$983$	−15.2187	−0.485402	−0.242701	−	0.970101i	$-0.578033\pi$
$983$	−15.2187	−0.485402	−0.242701	+	0.970101i	$0.578033\pi$
$984$	0	0
$985$	−1.38765	−0.0442142
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.76805	0.0880189
$990$	0	0
$991$	31.3069	0.994496	0.497248	−	0.867608i	$-0.334344\pi$
$991$	31.3069	0.994496	0.497248	+	0.867608i	$0.334344\pi$
$992$	0	0
$993$	−21.9377	−0.696172
$994$	0	0
$995$	−0.413463	−0.0131077
$996$	0	0
$997$	−40.1054	−1.27015	−0.635076	−	0.772450i	$-0.719031\pi$
$997$	−40.1054	−1.27015	−0.635076	+	0.772450i	$0.719031\pi$
$998$	0	0
$999$	−0.786929	−0.0248973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.a.x.1.4	✓	4
4.3	odd	2		3920.2.a.ce.1.1		4
5.4	even	2		9800.2.a.cs.1.1		4
7.2	even	3		1960.2.q.y.361.1		8
7.3	odd	6		1960.2.q.x.961.4		8
7.4	even	3		1960.2.q.y.961.1		8
7.5	odd	6		1960.2.q.x.361.4		8
7.6	odd	2		1960.2.a.y.1.1	yes	4
28.27	even	2		3920.2.a.cd.1.4		4
35.34	odd	2		9800.2.a.cl.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.a.x.1.4	✓	4	1.1	even	1	trivial
1960.2.a.y.1.1	yes	4	7.6	odd	2
1960.2.q.x.361.4		8	7.5	odd	6
1960.2.q.x.961.4		8	7.3	odd	6
1960.2.q.y.361.1		8	7.2	even	3
1960.2.q.y.961.1		8	7.4	even	3
3920.2.a.cd.1.4		4	28.27	even	2
3920.2.a.ce.1.1		4	4.3	odd	2
9800.2.a.cl.1.4		4	35.34	odd	2
9800.2.a.cs.1.1		4	5.4	even	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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.a (trivial)

\(f(q)\)	\(=\)	\(q+1.87996 q^{3} -1.00000 q^{5} +0.534253 q^{9} +O(q^{10})\)	\(q+1.87996 q^{3} -1.00000 q^{5} +0.534253 q^{9} -3.29417 q^{11} -4.19292 q^{13} -1.87996 q^{15} +1.43737 q^{17} -1.24445 q^{19} -0.272828 q^{23} +1.00000 q^{25} -4.63551 q^{27} -2.36268 q^{29} +3.72717 q^{31} -6.19292 q^{33} +0.169761 q^{37} -7.88252 q^{39} -11.6630 q^{41} -10.1458 q^{43} -0.534253 q^{45} -3.12441 q^{47} +2.70220 q^{51} +9.24701 q^{53} +3.29417 q^{55} -2.33952 q^{57} -9.07107 q^{59} -7.27102 q^{61} +4.19292 q^{65} -13.1439 q^{67} -0.512907 q^{69} +6.87474 q^{71} +15.2496 q^{73} +1.87996 q^{75} +14.9510 q^{79} -10.3173 q^{81} -0.167199 q^{83} -1.43737 q^{85} -4.44175 q^{87} -3.09869 q^{89} +7.00694 q^{93} +1.24445 q^{95} -6.60894 q^{97} -1.75992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^5 + 6 * q^9	\( 4 q - 2 q^{3} - 4 q^{5} + 6 q^{9} + 2 q^{11} - 10 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{23} + 4 q^{25} - 14 q^{27} - 2 q^{29} + 12 q^{31} - 18 q^{33} + 14 q^{39} - 12 q^{41} - 8 q^{43} - 6 q^{45} + 2 q^{47} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 8 q^{57} - 8 q^{59} - 20 q^{61} + 10 q^{65} - 8 q^{67} - 24 q^{69} + 4 q^{71} - 16 q^{73} - 2 q^{75} + 22 q^{79} - 20 q^{81} - 36 q^{83} + 6 q^{85} + 18 q^{87} - 40 q^{89} - 32 q^{93} - 26 q^{97} + 12 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^5 + 6 * q^9 + 2 * q^11 - 10 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^23 + 4 * q^25 - 14 * q^27 - 2 * q^29 + 12 * q^31 - 18 * q^33 + 14 * q^39 - 12 * q^41 - 8 * q^43 - 6 * q^45 + 2 * q^47 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 - 8 * q^59 - 20 * q^61 + 10 * q^65 - 8 * q^67 - 24 * q^69 + 4 * q^71 - 16 * q^73 - 2 * q^75 + 22 * q^79 - 20 * q^81 - 36 * q^83 + 6 * q^85 + 18 * q^87 - 40 * q^89 - 32 * q^93 - 26 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more