LMFDB - Embedded newform 4140.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.14345904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 $ x^5 - 2x^4 - 13x^3 + 34x^2 - 11x - 12
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			1.2
Root			$3.18817$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.334658 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -0.334658 q^{7} -2.12122 q^{11} -2.25511 q^{13} -6.93838 q^{17} +4.12122 q^{19} +1.00000 q^{23} +1.00000 q^{25} +1.92046 q^{29} +0.440822 q^{31} -0.334658 q^{35} +6.04168 q^{37} +10.2493 q^{41} -5.50042 q^{43} +4.65664 q^{47} -6.88800 q^{49} +4.40296 q^{53} -2.12122 q^{55} +10.8322 q^{59} +13.7555 q^{61} -2.25511 q^{65} +6.87008 q^{67} -5.57997 q^{71} +6.25511 q^{73} +0.709884 q^{77} +16.7052 q^{79} -10.2770 q^{83} -6.93838 q^{85} -1.33068 q^{89} +0.754693 q^{91} +4.12122 q^{95} +18.4122 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 4 q^{7}+O(q^{10}) $ 5 * q + 5 * q^5 + 4 * q^7	$ 5 q + 5 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{23} + 5 q^{25} + 2 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 2 q^{41} + 18 q^{43} - 4 q^{47} + 13 q^{49} - 2 q^{53} + 4 q^{55} + 6 q^{59} + 10 q^{61} + 2 q^{65} + 16 q^{67} + 10 q^{71} + 18 q^{73} - 16 q^{77} + 14 q^{79} + 8 q^{83} + 2 q^{85} - 18 q^{89} + 36 q^{91} + 6 q^{95} + 6 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^23 + 5 * q^25 + 2 * q^29 + 8 * q^31 + 4 * q^35 + 8 * q^37 + 2 * q^41 + 18 * q^43 - 4 * q^47 + 13 * q^49 - 2 * q^53 + 4 * q^55 + 6 * q^59 + 10 * q^61 + 2 * q^65 + 16 * q^67 + 10 * q^71 + 18 * q^73 - 16 * q^77 + 14 * q^79 + 8 * q^83 + 2 * q^85 - 18 * q^89 + 36 * q^91 + 6 * q^95 + 6 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.334658	−0.126489	−0.0632445	−	0.997998i	$-0.520145\pi$
$7$	−0.334658	−0.126489	−0.0632445	+	0.997998i	$0.520145\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.12122	−0.639572	−0.319786	−	0.947490i	$-0.603611\pi$
$11$	−2.12122	−0.639572	−0.319786	+	0.947490i	$0.603611\pi$
$12$	0	0
$13$	−2.25511	−0.625456	−0.312728	−	0.949843i	$-0.601243\pi$
$13$	−2.25511	−0.625456	−0.312728	+	0.949843i	$0.601243\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.93838	−1.68280	−0.841402	−	0.540410i	$-0.818269\pi$
$17$	−6.93838	−1.68280	−0.841402	+	0.540410i	$0.818269\pi$
$18$	0	0
$19$	4.12122	0.945473	0.472736	−	0.881204i	$-0.343266\pi$
$19$	4.12122	0.945473	0.472736	+	0.881204i	$0.343266\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.92046	0.356620	0.178310	−	0.983974i	$-0.442937\pi$
$29$	1.92046	0.356620	0.178310	+	0.983974i	$0.442937\pi$
$30$	0	0
$31$	0.440822	0.0791740	0.0395870	−	0.999216i	$-0.487396\pi$
$31$	0.440822	0.0791740	0.0395870	+	0.999216i	$0.487396\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.334658	−0.0565676
$36$	0	0
$37$	6.04168	0.993246	0.496623	−	0.867966i	$-0.334573\pi$
$37$	6.04168	0.993246	0.496623	+	0.867966i	$0.334573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.2493	1.60067	0.800334	−	0.599554i	$-0.204655\pi$
$41$	10.2493	1.60067	0.800334	+	0.599554i	$0.204655\pi$
$42$	0	0
$43$	−5.50042	−0.838806	−0.419403	−	0.907800i	$-0.637761\pi$
$43$	−5.50042	−0.838806	−0.419403	+	0.907800i	$0.637761\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.65664	0.679241	0.339621	−	0.940562i	$-0.389701\pi$
$47$	4.65664	0.679241	0.339621	+	0.940562i	$0.389701\pi$
$48$	0	0
$49$	−6.88800	−0.984001
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.40296	0.604792	0.302396	−	0.953182i	$-0.402213\pi$
$53$	4.40296	0.604792	0.302396	+	0.953182i	$0.402213\pi$
$54$	0	0
$55$	−2.12122	−0.286025
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.8322	1.41023	0.705117	−	0.709091i	$-0.250894\pi$
$59$	10.8322	1.41023	0.705117	+	0.709091i	$0.250894\pi$
$60$	0	0
$61$	13.7555	1.76122	0.880608	−	0.473846i	$-0.157135\pi$
$61$	13.7555	1.76122	0.880608	+	0.473846i	$0.157135\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.25511	−0.279713
$66$	0	0
$67$	6.87008	0.839314	0.419657	−	0.907683i	$-0.362150\pi$
$67$	6.87008	0.839314	0.419657	+	0.907683i	$0.362150\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.57997	−0.662220	−0.331110	−	0.943592i	$-0.607423\pi$
$71$	−5.57997	−0.662220	−0.331110	+	0.943592i	$0.607423\pi$
$72$	0	0
$73$	6.25511	0.732106	0.366053	−	0.930594i	$-0.380709\pi$
$73$	6.25511	0.732106	0.366053	+	0.930594i	$0.380709\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.709884	0.0808988
$78$	0	0
$79$	16.7052	1.87948	0.939739	−	0.341893i	$-0.111068\pi$
$79$	16.7052	1.87948	0.939739	+	0.341893i	$0.111068\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.2770	−1.12805	−0.564024	−	0.825758i	$-0.690748\pi$
$83$	−10.2770	−1.12805	−0.564024	+	0.825758i	$0.690748\pi$
$84$	0	0
$85$	−6.93838	−0.752573
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.33068	−0.141052	−0.0705261	−	0.997510i	$-0.522468\pi$
$89$	−1.33068	−0.141052	−0.0705261	+	0.997510i	$0.522468\pi$
$90$	0	0
$91$	0.754693	0.0791133
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.12122	0.422828
$96$	0	0
$97$	18.4122	1.86947	0.934737	−	0.355341i	$-0.115635\pi$
$97$	18.4122	1.86947	0.934737	+	0.355341i	$0.115635\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.33179	0.331526	0.165763	−	0.986166i	$-0.446991\pi$
$101$	3.33179	0.331526	0.165763	+	0.986166i	$0.446991\pi$
$102$	0	0
$103$	5.25798	0.518084	0.259042	−	0.965866i	$-0.416593\pi$
$103$	5.25798	0.518084	0.259042	+	0.965866i	$0.416593\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.768640	−0.0743073	−0.0371536	−	0.999310i	$-0.511829\pi$
$107$	−0.768640	−0.0743073	−0.0371536	+	0.999310i	$0.511829\pi$
$108$	0	0
$109$	4.12122	0.394741	0.197371	−	0.980329i	$-0.436760\pi$
$109$	4.12122	0.394741	0.197371	+	0.980329i	$0.436760\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.5979	−0.996965	−0.498483	−	0.866900i	$-0.666109\pi$
$113$	−10.5979	−0.996965	−0.498483	+	0.866900i	$0.666109\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.32199	0.212856
$120$	0	0
$121$	−6.50042	−0.590947
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.29096	−0.558232	−0.279116	−	0.960257i	$-0.590041\pi$
$127$	−6.29096	−0.558232	−0.279116	+	0.960257i	$0.590041\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.66932	0.407960	0.203980	−	0.978975i	$-0.434612\pi$
$131$	4.66932	0.407960	0.203980	+	0.978975i	$0.434612\pi$
$132$	0	0
$133$	−1.37920	−0.119592
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.2172	−1.38553	−0.692766	−	0.721162i	$-0.743608\pi$
$137$	−16.2172	−1.38553	−0.692766	+	0.721162i	$0.743608\pi$
$138$	0	0
$139$	11.8880	1.00833	0.504164	−	0.863608i	$-0.331801\pi$
$139$	11.8880	1.00833	0.504164	+	0.863608i	$0.331801\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.78360	0.400024
$144$	0	0
$145$	1.92046	0.159485
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.3637	−1.01287	−0.506435	−	0.862278i	$-0.669037\pi$
$149$	−12.3637	−1.01287	−0.506435	+	0.862278i	$0.669037\pi$
$150$	0	0
$151$	0.499578	0.0406551	0.0203276	−	0.999793i	$-0.493529\pi$
$151$	0.499578	0.0406551	0.0203276	+	0.999793i	$0.493529\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.440822	0.0354077
$156$	0	0
$157$	0.0668715	0.00533692	0.00266846	−	0.999996i	$-0.499151\pi$
$157$	0.0668715	0.00533692	0.00266846	+	0.999996i	$0.499151\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.334658	−0.0263748
$162$	0	0
$163$	21.8768	1.71352	0.856760	−	0.515715i	$-0.172474\pi$
$163$	21.8768	1.71352	0.856760	+	0.515715i	$0.172474\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.4500	−0.808649	−0.404324	−	0.914616i	$-0.632493\pi$
$167$	−10.4500	−0.808649	−0.404324	+	0.914616i	$0.632493\pi$
$168$	0	0
$169$	−7.91446	−0.608805
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.722557	−0.0549350	−0.0274675	−	0.999623i	$-0.508744\pi$
$173$	−0.722557	−0.0549350	−0.0274675	+	0.999623i	$0.508744\pi$
$174$	0	0
$175$	−0.334658	−0.0252978
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.6546	−1.54380	−0.771899	−	0.635745i	$-0.780693\pi$
$179$	−20.6546	−1.54380	−0.771899	+	0.635745i	$0.780693\pi$
$180$	0	0
$181$	−9.62858	−0.715687	−0.357844	−	0.933782i	$-0.616488\pi$
$181$	−9.62858	−0.715687	−0.357844	+	0.933782i	$0.616488\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.04168	0.444193
$186$	0	0
$187$	14.7178	1.07627
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.2658	1.46638	0.733190	−	0.680024i	$-0.238031\pi$
$191$	20.2658	1.46638	0.733190	+	0.680024i	$0.238031\pi$
$192$	0	0
$193$	7.36653	0.530254	0.265127	−	0.964213i	$-0.414586\pi$
$193$	7.36653	0.530254	0.265127	+	0.964213i	$0.414586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.17955	0.0840391	0.0420196	−	0.999117i	$-0.486621\pi$
$197$	1.17955	0.0840391	0.0420196	+	0.999117i	$0.486621\pi$
$198$	0	0
$199$	17.4140	1.23445	0.617224	−	0.786787i	$-0.288257\pi$
$199$	17.4140	1.23445	0.617224	+	0.786787i	$0.288257\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.642697	−0.0451085
$204$	0	0
$205$	10.2493	0.715841
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.74202	−0.604698
$210$	0	0
$211$	16.4515	1.13257	0.566283	−	0.824211i	$-0.308381\pi$
$211$	16.4515	1.13257	0.566283	+	0.824211i	$0.308381\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.50042	−0.375126
$216$	0	0
$217$	−0.147525	−0.0100146
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.6468	1.05252
$222$	0	0
$223$	−5.87676	−0.393537	−0.196768	−	0.980450i	$-0.563045\pi$
$223$	−5.87676	−0.393537	−0.196768	+	0.980450i	$0.563045\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.0913	0.669783	0.334892	−	0.942257i	$-0.391300\pi$
$227$	10.0913	0.669783	0.334892	+	0.942257i	$0.391300\pi$
$228$	0	0
$229$	−25.4578	−1.68230	−0.841150	−	0.540801i	$-0.818121\pi$
$229$	−25.4578	−1.68230	−0.841150	+	0.540801i	$0.818121\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.828404	−0.0542706	−0.0271353	−	0.999632i	$-0.508638\pi$
$233$	−0.828404	−0.0542706	−0.0271353	+	0.999632i	$0.508638\pi$
$234$	0	0
$235$	4.65664	0.303766
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.59247	0.555801	0.277900	−	0.960610i	$-0.410361\pi$
$239$	8.59247	0.555801	0.277900	+	0.960610i	$0.410361\pi$
$240$	0	0
$241$	−15.1221	−0.974098	−0.487049	−	0.873375i	$-0.661927\pi$
$241$	−15.1221	−0.974098	−0.487049	+	0.873375i	$0.661927\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.88800	−0.440058
$246$	0	0
$247$	−9.29382	−0.591352
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.8014	1.24985	0.624925	−	0.780685i	$-0.285129\pi$
$251$	19.8014	1.24985	0.624925	+	0.780685i	$0.285129\pi$
$252$	0	0
$253$	−2.12122	−0.133360
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.4705	−1.71356	−0.856781	−	0.515680i	$-0.827539\pi$
$257$	−27.4705	−1.71356	−0.856781	+	0.515680i	$0.827539\pi$
$258$	0	0
$259$	−2.02190	−0.125635
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.9587	1.47736	0.738678	−	0.674059i	$-0.235450\pi$
$263$	23.9587	1.47736	0.738678	+	0.674059i	$0.235450\pi$
$264$	0	0
$265$	4.40296	0.270471
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.1000	−0.920663	−0.460332	−	0.887747i	$-0.652270\pi$
$269$	−15.1000	−0.920663	−0.460332	+	0.887747i	$0.652270\pi$
$270$	0	0
$271$	24.1837	1.46905	0.734527	−	0.678579i	$-0.237404\pi$
$271$	24.1837	1.46905	0.734527	+	0.678579i	$0.237404\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.12122	−0.127914
$276$	0	0
$277$	−18.9001	−1.13560	−0.567798	−	0.823168i	$-0.692205\pi$
$277$	−18.9001	−1.13560	−0.567798	+	0.823168i	$0.692205\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.7914	−0.942035	−0.471017	−	0.882124i	$-0.656113\pi$
$281$	−15.7914	−0.942035	−0.471017	+	0.882124i	$0.656113\pi$
$282$	0	0
$283$	30.2415	1.79767	0.898836	−	0.438285i	$-0.144414\pi$
$283$	30.2415	1.79767	0.898836	+	0.438285i	$0.144414\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.43001	−0.202467
$288$	0	0
$289$	31.1411	1.83183
$290$	0	0
$291$	0	0
$292$	0	0
$293$	32.1063	1.87567	0.937834	−	0.347085i	$-0.112828\pi$
$293$	32.1063	1.87567	0.937834	+	0.347085i	$0.112828\pi$
$294$	0	0
$295$	10.8322	0.630676
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.25511	−0.130417
$300$	0	0
$301$	1.84076	0.106100
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.7555	0.787640
$306$	0	0
$307$	−24.4102	−1.39316	−0.696581	−	0.717478i	$-0.745296\pi$
$307$	−24.4102	−1.39316	−0.696581	+	0.717478i	$0.745296\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.351141	−0.0199114	−0.00995569	−	0.999950i	$-0.503169\pi$
$311$	−0.351141	−0.0199114	−0.00995569	+	0.999950i	$0.503169\pi$
$312$	0	0
$313$	34.6236	1.95704	0.978521	−	0.206149i	$-0.0660931\pi$
$313$	34.6236	1.95704	0.978521	+	0.206149i	$0.0660931\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.3385	0.692997	0.346499	−	0.938050i	$-0.387371\pi$
$317$	12.3385	0.692997	0.346499	+	0.938050i	$0.387371\pi$
$318$	0	0
$319$	−4.07371	−0.228084
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−28.5946	−1.59105
$324$	0	0
$325$	−2.25511	−0.125091
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.55838	−0.0859165
$330$	0	0
$331$	16.3118	0.896580	0.448290	−	0.893888i	$-0.352033\pi$
$331$	16.3118	0.896580	0.448290	+	0.893888i	$0.352033\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.87008	0.375353
$336$	0	0
$337$	20.9476	1.14109	0.570544	−	0.821267i	$-0.306732\pi$
$337$	20.9476	1.14109	0.570544	+	0.821267i	$0.306732\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.935081	−0.0506375
$342$	0	0
$343$	4.64774	0.250954
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.752671	0.0404055	0.0202027	−	0.999796i	$-0.493569\pi$
$347$	0.752671	0.0404055	0.0202027	+	0.999796i	$0.493569\pi$
$348$	0	0
$349$	−20.4204	−1.09308	−0.546539	−	0.837433i	$-0.684055\pi$
$349$	−20.4204	−1.09308	−0.546539	+	0.837433i	$0.684055\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.24261	−0.0661373	−0.0330686	−	0.999453i	$-0.510528\pi$
$353$	−1.24261	−0.0661373	−0.0330686	+	0.999453i	$0.510528\pi$
$354$	0	0
$355$	−5.57997	−0.296154
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.6394	−0.667082	−0.333541	−	0.942736i	$-0.608244\pi$
$359$	−12.6394	−0.667082	−0.333541	+	0.942736i	$0.608244\pi$
$360$	0	0
$361$	−2.01554	−0.106081
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.25511	0.327408
$366$	0	0
$367$	25.7622	1.34478	0.672389	−	0.740198i	$-0.265268\pi$
$367$	25.7622	1.34478	0.672389	+	0.740198i	$0.265268\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.47349	−0.0764996
$372$	0	0
$373$	−18.7557	−0.971133	−0.485567	−	0.874200i	$-0.661387\pi$
$373$	−18.7557	−0.971133	−0.485567	+	0.874200i	$0.661387\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.33085	−0.223050
$378$	0	0
$379$	6.56347	0.337143	0.168571	−	0.985689i	$-0.446085\pi$
$379$	6.56347	0.337143	0.168571	+	0.985689i	$0.446085\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−33.0101	−1.68674	−0.843368	−	0.537337i	$-0.819430\pi$
$383$	−33.0101	−1.68674	−0.843368	+	0.537337i	$0.819430\pi$
$384$	0	0
$385$	0.709884	0.0361790
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.5819	1.14495	0.572474	−	0.819923i	$-0.305984\pi$
$389$	22.5819	1.14495	0.572474	+	0.819923i	$0.305984\pi$
$390$	0	0
$391$	−6.93838	−0.350889
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.7052	0.840528
$396$	0	0
$397$	−17.2549	−0.866001	−0.433001	−	0.901394i	$-0.642545\pi$
$397$	−17.2549	−0.866001	−0.433001	+	0.901394i	$0.642545\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.96416	0.397711	0.198855	−	0.980029i	$-0.436278\pi$
$401$	7.96416	0.397711	0.198855	+	0.980029i	$0.436278\pi$
$402$	0	0
$403$	−0.994105	−0.0495199
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.8157	−0.635252
$408$	0	0
$409$	−3.42528	−0.169369	−0.0846847	−	0.996408i	$-0.526988\pi$
$409$	−3.42528	−0.169369	−0.0846847	+	0.996408i	$0.526988\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.62509	−0.178379
$414$	0	0
$415$	−10.2770	−0.504479
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.3141	−1.62750	−0.813751	−	0.581214i	$-0.802578\pi$
$419$	−33.3141	−1.62750	−0.813751	+	0.581214i	$0.802578\pi$
$420$	0	0
$421$	6.42485	0.313128	0.156564	−	0.987668i	$-0.449958\pi$
$421$	6.42485	0.313128	0.156564	+	0.987668i	$0.449958\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.93838	−0.336561
$426$	0	0
$427$	−4.60340	−0.222774
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.3733	1.07768	0.538842	−	0.842407i	$-0.318862\pi$
$431$	22.3733	1.07768	0.538842	+	0.842407i	$0.318862\pi$
$432$	0	0
$433$	−28.5150	−1.37035	−0.685173	−	0.728381i	$-0.740273\pi$
$433$	−28.5150	−1.37035	−0.685173	+	0.728381i	$0.740273\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.12122	0.197145
$438$	0	0
$439$	17.4947	0.834976	0.417488	−	0.908682i	$-0.362911\pi$
$439$	17.4947	0.834976	0.417488	+	0.908682i	$0.362911\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.90397	0.185483	0.0927417	−	0.995690i	$-0.470437\pi$
$443$	3.90397	0.185483	0.0927417	+	0.995690i	$0.470437\pi$
$444$	0	0
$445$	−1.33068	−0.0630804
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.6615	0.691917	0.345959	−	0.938250i	$-0.387554\pi$
$449$	14.6615	0.691917	0.345959	+	0.938250i	$0.387554\pi$
$450$	0	0
$451$	−21.7410	−1.02374
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.754693	0.0353805
$456$	0	0
$457$	−0.435257	−0.0203605	−0.0101802	−	0.999948i	$-0.503241\pi$
$457$	−0.435257	−0.0203605	−0.0101802	+	0.999948i	$0.503241\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.0970	1.49491	0.747454	−	0.664314i	$-0.231276\pi$
$461$	32.0970	1.49491	0.747454	+	0.664314i	$0.231276\pi$
$462$	0	0
$463$	−17.0357	−0.791715	−0.395858	−	0.918312i	$-0.629553\pi$
$463$	−17.0357	−0.791715	−0.395858	+	0.918312i	$0.629553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.1352	−0.839196	−0.419598	−	0.907710i	$-0.637829\pi$
$467$	−18.1352	−0.839196	−0.419598	+	0.907710i	$0.637829\pi$
$468$	0	0
$469$	−2.29913	−0.106164
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.6676	0.536477
$474$	0	0
$475$	4.12122	0.189095
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.2046	1.01455	0.507276	−	0.861783i	$-0.330652\pi$
$479$	22.2046	1.01455	0.507276	+	0.861783i	$0.330652\pi$
$480$	0	0
$481$	−13.6247	−0.621232
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.4122	0.836054
$486$	0	0
$487$	7.98733	0.361940	0.180970	−	0.983489i	$-0.442076\pi$
$487$	7.98733	0.361940	0.180970	+	0.983489i	$0.442076\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.3122	0.600770	0.300385	−	0.953818i	$-0.402885\pi$
$491$	13.3122	0.600770	0.300385	+	0.953818i	$0.402885\pi$
$492$	0	0
$493$	−13.3249	−0.600121
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.86738	0.0837635
$498$	0	0
$499$	−14.9732	−0.670293	−0.335147	−	0.942166i	$-0.608786\pi$
$499$	−14.9732	−0.670293	−0.335147	+	0.942166i	$0.608786\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.2038	−1.88178	−0.940888	−	0.338718i	$-0.890007\pi$
$503$	−42.2038	−1.88178	−0.940888	+	0.338718i	$0.890007\pi$
$504$	0	0
$505$	3.33179	0.148263
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.5966	−0.558335	−0.279168	−	0.960242i	$-0.590058\pi$
$509$	−12.5966	−0.558335	−0.279168	+	0.960242i	$0.590058\pi$
$510$	0	0
$511$	−2.09333	−0.0926033
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.25798	0.231694
$516$	0	0
$517$	−9.87777	−0.434424
$518$	0	0
$519$	0	0
$520$	0	0
$521$	39.6602	1.73754	0.868772	−	0.495212i	$-0.164910\pi$
$521$	39.6602	1.73754	0.868772	+	0.495212i	$0.164910\pi$
$522$	0	0
$523$	−7.62063	−0.333227	−0.166614	−	0.986022i	$-0.553283\pi$
$523$	−7.62063	−0.333227	−0.166614	+	0.986022i	$0.553283\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.05859	−0.133234
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23.1133	−1.00115
$534$	0	0
$535$	−0.768640	−0.0332312
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.6110	0.629339
$540$	0	0
$541$	16.1020	0.692277	0.346139	−	0.938183i	$-0.387493\pi$
$541$	16.1020	0.692277	0.346139	+	0.938183i	$0.387493\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.12122	0.176534
$546$	0	0
$547$	30.8139	1.31751	0.658753	−	0.752359i	$-0.271084\pi$
$547$	30.8139	1.31751	0.658753	+	0.752359i	$0.271084\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.91462	0.337174
$552$	0	0
$553$	−5.59052	−0.237733
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.54954	−0.404627	−0.202313	−	0.979321i	$-0.564846\pi$
$557$	−9.54954	−0.404627	−0.202313	+	0.979321i	$0.564846\pi$
$558$	0	0
$559$	12.4041	0.524637
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.6843	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$563$	31.6843	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$564$	0	0
$565$	−10.5979	−0.445856
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−40.3500	−1.69156	−0.845780	−	0.533533i	$-0.820864\pi$
$569$	−40.3500	−1.69156	−0.845780	+	0.533533i	$0.820864\pi$
$570$	0	0
$571$	19.4732	0.814928	0.407464	−	0.913221i	$-0.366413\pi$
$571$	19.4732	0.814928	0.407464	+	0.913221i	$0.366413\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	5.17381	0.215389	0.107694	−	0.994184i	$-0.465653\pi$
$577$	5.17381	0.215389	0.107694	+	0.994184i	$0.465653\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.43929	0.142686
$582$	0	0
$583$	−9.33964	−0.386808
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.1892	−0.833298	−0.416649	−	0.909068i	$-0.636796\pi$
$587$	−20.1892	−0.833298	−0.416649	+	0.909068i	$0.636796\pi$
$588$	0	0
$589$	1.81673	0.0748569
$590$	0	0
$591$	0	0
$592$	0	0
$593$	40.1978	1.65073	0.825364	−	0.564602i	$-0.190970\pi$
$593$	40.1978	1.65073	0.825364	+	0.564602i	$0.190970\pi$
$594$	0	0
$595$	2.32199	0.0951921
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11.1599	0.455982	0.227991	−	0.973663i	$-0.426784\pi$
$599$	11.1599	0.455982	0.227991	+	0.973663i	$0.426784\pi$
$600$	0	0
$601$	−3.60669	−0.147120	−0.0735599	−	0.997291i	$-0.523436\pi$
$601$	−3.60669	−0.147120	−0.0735599	+	0.997291i	$0.523436\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.50042	−0.264280
$606$	0	0
$607$	−30.3968	−1.23377	−0.616884	−	0.787054i	$-0.711605\pi$
$607$	−30.3968	−1.23377	−0.616884	+	0.787054i	$0.711605\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.5013	−0.424836
$612$	0	0
$613$	−27.2908	−1.10226	−0.551132	−	0.834418i	$-0.685804\pi$
$613$	−27.2908	−1.10226	−0.551132	+	0.834418i	$0.685804\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.7815	−0.595081	−0.297541	−	0.954709i	$-0.596166\pi$
$617$	−14.7815	−0.595081	−0.297541	+	0.954709i	$0.596166\pi$
$618$	0	0
$619$	−25.3266	−1.01796	−0.508982	−	0.860777i	$-0.669978\pi$
$619$	−25.3266	−1.01796	−0.508982	+	0.860777i	$0.669978\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.445324	0.0178415
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−41.9194	−1.67144
$630$	0	0
$631$	−4.76519	−0.189699	−0.0948497	−	0.995492i	$-0.530237\pi$
$631$	−4.76519	−0.189699	−0.0948497	+	0.995492i	$0.530237\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.29096	−0.249649
$636$	0	0
$637$	15.5332	0.615449
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.8964	0.509377	0.254688	−	0.967023i	$-0.418027\pi$
$641$	12.8964	0.509377	0.254688	+	0.967023i	$0.418027\pi$
$642$	0	0
$643$	−12.0474	−0.475103	−0.237552	−	0.971375i	$-0.576345\pi$
$643$	−12.0474	−0.475103	−0.237552	+	0.971375i	$0.576345\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.9728	−1.33561	−0.667804	−	0.744337i	$-0.732766\pi$
$647$	−33.9728	−1.33561	−0.667804	+	0.744337i	$0.732766\pi$
$648$	0	0
$649$	−22.9775	−0.901947
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.4722	−0.801140	−0.400570	−	0.916266i	$-0.631188\pi$
$653$	−20.4722	−0.801140	−0.400570	+	0.916266i	$0.631188\pi$
$654$	0	0
$655$	4.66932	0.182445
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.08538	−0.159144	−0.0795718	−	0.996829i	$-0.525355\pi$
$659$	−4.08538	−0.159144	−0.0795718	+	0.996829i	$0.525355\pi$
$660$	0	0
$661$	23.5770	0.917040	0.458520	−	0.888684i	$-0.348380\pi$
$661$	23.5770	0.917040	0.458520	+	0.888684i	$0.348380\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.37920	−0.0534831
$666$	0	0
$667$	1.92046	0.0743604
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−29.1785	−1.12642
$672$	0	0
$673$	14.3385	0.552707	0.276354	−	0.961056i	$-0.410874\pi$
$673$	14.3385	0.552707	0.276354	+	0.961056i	$0.410874\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.2265	−0.854233	−0.427116	−	0.904197i	$-0.640471\pi$
$677$	−22.2265	−0.854233	−0.427116	+	0.904197i	$0.640471\pi$
$678$	0	0
$679$	−6.16179	−0.236468
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.17176	−0.312684	−0.156342	−	0.987703i	$-0.549970\pi$
$683$	−8.17176	−0.312684	−0.156342	+	0.987703i	$0.549970\pi$
$684$	0	0
$685$	−16.2172	−0.619629
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.92917	−0.378271
$690$	0	0
$691$	3.55081	0.135079	0.0675396	−	0.997717i	$-0.478485\pi$
$691$	3.55081	0.135079	0.0675396	+	0.997717i	$0.478485\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.8880	0.450938
$696$	0	0
$697$	−71.1134	−2.69361
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.2484	1.52016	0.760080	−	0.649830i	$-0.225160\pi$
$701$	40.2484	1.52016	0.760080	+	0.649830i	$0.225160\pi$
$702$	0	0
$703$	24.8991	0.939087
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.11501	−0.0419343
$708$	0	0
$709$	1.81099	0.0680133	0.0340067	−	0.999422i	$-0.489173\pi$
$709$	1.81099	0.0680133	0.0340067	+	0.999422i	$0.489173\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.440822	0.0165089
$714$	0	0
$715$	4.78360	0.178896
$716$	0	0
$717$	0	0
$718$	0	0
$719$	51.6544	1.92638	0.963191	−	0.268817i	$-0.0866328\pi$
$719$	51.6544	1.92638	0.963191	+	0.268817i	$0.0866328\pi$
$720$	0	0
$721$	−1.75963	−0.0655319
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.92046	0.0713239
$726$	0	0
$727$	40.1715	1.48988	0.744940	−	0.667132i	$-0.232478\pi$
$727$	40.1715	1.48988	0.744940	+	0.667132i	$0.232478\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	38.1640	1.41155
$732$	0	0
$733$	−25.2890	−0.934071	−0.467035	−	0.884239i	$-0.654678\pi$
$733$	−25.2890	−0.934071	−0.467035	+	0.884239i	$0.654678\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.5730	−0.536802
$738$	0	0
$739$	8.09730	0.297864	0.148932	−	0.988847i	$-0.452416\pi$
$739$	8.09730	0.297864	0.148932	+	0.988847i	$0.452416\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.97954	−0.256055	−0.128027	−	0.991771i	$-0.540865\pi$
$743$	−6.97954	−0.256055	−0.128027	+	0.991771i	$0.540865\pi$
$744$	0	0
$745$	−12.3637	−0.452970
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.257232	0.00939905
$750$	0	0
$751$	−5.03076	−0.183575	−0.0917875	−	0.995779i	$-0.529258\pi$
$751$	−5.03076	−0.183575	−0.0917875	+	0.995779i	$0.529258\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.499578	0.0181815
$756$	0	0
$757$	−3.84977	−0.139922	−0.0699612	−	0.997550i	$-0.522288\pi$
$757$	−3.84977	−0.139922	−0.0699612	+	0.997550i	$0.522288\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−41.4084	−1.50105	−0.750527	−	0.660840i	$-0.770200\pi$
$761$	−41.4084	−1.50105	−0.750527	+	0.660840i	$0.770200\pi$
$762$	0	0
$763$	−1.37920	−0.0499304
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.4279	−0.882040
$768$	0	0
$769$	−20.2299	−0.729510	−0.364755	−	0.931104i	$-0.618847\pi$
$769$	−20.2299	−0.729510	−0.364755	+	0.931104i	$0.618847\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.6528	−0.994601	−0.497300	−	0.867578i	$-0.665675\pi$
$773$	−27.6528	−0.994601	−0.497300	+	0.867578i	$0.665675\pi$
$774$	0	0
$775$	0.440822	0.0158348
$776$	0	0
$777$	0	0
$778$	0	0
$779$	42.2396	1.51339
$780$	0	0
$781$	11.8363	0.423538
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.0668715	0.00238674
$786$	0	0
$787$	−3.56252	−0.126990	−0.0634951	−	0.997982i	$-0.520225\pi$
$787$	−3.56252	−0.126990	−0.0634951	+	0.997982i	$0.520225\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.54667	0.126105
$792$	0	0
$793$	−31.0203	−1.10156
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.20920	−0.255363	−0.127681	−	0.991815i	$-0.540753\pi$
$797$	−7.20920	−0.255363	−0.127681	+	0.991815i	$0.540753\pi$
$798$	0	0
$799$	−32.3096	−1.14303
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−13.2685	−0.468234
$804$	0	0
$805$	−0.334658	−0.0117952
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.2452	1.06337	0.531683	−	0.846943i	$-0.321560\pi$
$809$	30.2452	1.06337	0.531683	+	0.846943i	$0.321560\pi$
$810$	0	0
$811$	3.32739	0.116840	0.0584202	−	0.998292i	$-0.481394\pi$
$811$	3.32739	0.116840	0.0584202	+	0.998292i	$0.481394\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.8768	0.766309
$816$	0	0
$817$	−22.6685	−0.793069
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.50531	0.0874359	0.0437179	−	0.999044i	$-0.486080\pi$
$821$	2.50531	0.0874359	0.0437179	+	0.999044i	$0.486080\pi$
$822$	0	0
$823$	12.8639	0.448408	0.224204	−	0.974542i	$-0.428022\pi$
$823$	12.8639	0.448408	0.224204	+	0.974542i	$0.428022\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−52.5222	−1.82637	−0.913187	−	0.407541i	$-0.866387\pi$
$827$	−52.5222	−1.82637	−0.913187	+	0.407541i	$0.866387\pi$
$828$	0	0
$829$	−29.6030	−1.02815	−0.514077	−	0.857744i	$-0.671865\pi$
$829$	−29.6030	−1.02815	−0.514077	+	0.857744i	$0.671865\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	47.7916	1.65588
$834$	0	0
$835$	−10.4500	−0.361639
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−51.6360	−1.78267	−0.891336	−	0.453343i	$-0.850231\pi$
$839$	−51.6360	−1.78267	−0.891336	+	0.453343i	$0.850231\pi$
$840$	0	0
$841$	−25.3118	−0.872822
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.91446	−0.272266
$846$	0	0
$847$	2.17542	0.0747483
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.04168	0.207106
$852$	0	0
$853$	−19.3687	−0.663173	−0.331587	−	0.943425i	$-0.607584\pi$
$853$	−19.3687	−0.663173	−0.331587	+	0.943425i	$0.607584\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.9929	0.853741	0.426870	−	0.904313i	$-0.359616\pi$
$857$	24.9929	0.853741	0.426870	+	0.904313i	$0.359616\pi$
$858$	0	0
$859$	33.4904	1.14268	0.571339	−	0.820715i	$-0.306424\pi$
$859$	33.4904	1.14268	0.571339	+	0.820715i	$0.306424\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.4269	−0.423016	−0.211508	−	0.977376i	$-0.567837\pi$
$863$	−12.4269	−0.423016	−0.211508	+	0.977376i	$0.567837\pi$
$864$	0	0
$865$	−0.722557	−0.0245677
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−35.4353	−1.20206
$870$	0	0
$871$	−15.4928	−0.524954
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.334658	−0.0113135
$876$	0	0
$877$	−49.8039	−1.68176	−0.840879	−	0.541223i	$-0.817962\pi$
$877$	−49.8039	−1.68176	−0.840879	+	0.541223i	$0.817962\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.6372	0.493139	0.246570	−	0.969125i	$-0.420697\pi$
$881$	14.6372	0.493139	0.246570	+	0.969125i	$0.420697\pi$
$882$	0	0
$883$	−19.1495	−0.644431	−0.322216	−	0.946666i	$-0.604428\pi$
$883$	−19.1495	−0.644431	−0.322216	+	0.946666i	$0.604428\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.38614	−0.248002	−0.124001	−	0.992282i	$-0.539573\pi$
$887$	−7.38614	−0.248002	−0.124001	+	0.992282i	$0.539573\pi$
$888$	0	0
$889$	2.10532	0.0706102
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19.1911	0.642204
$894$	0	0
$895$	−20.6546	−0.690408
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.846580	0.0282350
$900$	0	0
$901$	−30.5494	−1.01775
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.62858	−0.320065
$906$	0	0
$907$	−34.0630	−1.13104	−0.565521	−	0.824734i	$-0.691325\pi$
$907$	−34.0630	−1.13104	−0.565521	+	0.824734i	$0.691325\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13.8330	0.458306	0.229153	−	0.973390i	$-0.426404\pi$
$911$	13.8330	0.458306	0.229153	+	0.973390i	$0.426404\pi$
$912$	0	0
$913$	21.7998	0.721468
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.56263	−0.0516024
$918$	0	0
$919$	26.5797	0.876783	0.438392	−	0.898784i	$-0.355548\pi$
$919$	26.5797	0.876783	0.438392	+	0.898784i	$0.355548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.5835	0.414190
$924$	0	0
$925$	6.04168	0.198649
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.2511	−1.09093	−0.545467	−	0.838132i	$-0.683648\pi$
$929$	−33.2511	−1.09093	−0.545467	+	0.838132i	$0.683648\pi$
$930$	0	0
$931$	−28.3870	−0.930346
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.7178	0.481325
$936$	0	0
$937$	−1.26964	−0.0414775	−0.0207387	−	0.999785i	$-0.506602\pi$
$937$	−1.26964	−0.0414775	−0.0207387	+	0.999785i	$0.506602\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39.8534	1.29918	0.649592	−	0.760283i	$-0.274940\pi$
$941$	39.8534	1.29918	0.649592	+	0.760283i	$0.274940\pi$
$942$	0	0
$943$	10.2493	0.333763
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.8798	0.678503	0.339251	−	0.940696i	$-0.389826\pi$
$947$	20.8798	0.678503	0.339251	+	0.940696i	$0.389826\pi$
$948$	0	0
$949$	−14.1060	−0.457900
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.38902	0.109781	0.0548906	−	0.998492i	$-0.482519\pi$
$953$	3.38902	0.109781	0.0548906	+	0.998492i	$0.482519\pi$
$954$	0	0
$955$	20.2658	0.655785
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.42724	0.175255
$960$	0	0
$961$	−30.8057	−0.993731
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.36653	0.237137
$966$	0	0
$967$	61.7169	1.98468	0.992340	−	0.123536i	$-0.0394235\pi$
$967$	61.7169	1.98468	0.992340	+	0.123536i	$0.0394235\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.2441	1.48404	0.742022	−	0.670375i	$-0.233867\pi$
$971$	46.2441	1.48404	0.742022	+	0.670375i	$0.233867\pi$
$972$	0	0
$973$	−3.97842	−0.127542
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.9871	−0.447486	−0.223743	−	0.974648i	$-0.571828\pi$
$977$	−13.9871	−0.447486	−0.223743	+	0.974648i	$0.571828\pi$
$978$	0	0
$979$	2.82267	0.0902130
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.2324	1.02805	0.514026	−	0.857774i	$-0.328153\pi$
$983$	32.2324	1.02805	0.514026	+	0.857774i	$0.328153\pi$
$984$	0	0
$985$	1.17955	0.0375834
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.50042	−0.174903
$990$	0	0
$991$	−34.1460	−1.08468	−0.542342	−	0.840158i	$-0.682462\pi$
$991$	−34.1460	−1.08468	−0.542342	+	0.840158i	$0.682462\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.4140	0.552062
$996$	0	0
$997$	32.4286	1.02702	0.513511	−	0.858083i	$-0.328344\pi$
$997$	32.4286	1.02702	0.513511	+	0.858083i	$0.328344\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.u.1.2	yes	5
3.2	odd	2		4140.2.a.t.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.a.t.1.2	✓	5	3.2	odd	2
4140.2.a.u.1.2	yes	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.334658 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -0.334658 q^{7} -2.12122 q^{11} -2.25511 q^{13} -6.93838 q^{17} +4.12122 q^{19} +1.00000 q^{23} +1.00000 q^{25} +1.92046 q^{29} +0.440822 q^{31} -0.334658 q^{35} +6.04168 q^{37} +10.2493 q^{41} -5.50042 q^{43} +4.65664 q^{47} -6.88800 q^{49} +4.40296 q^{53} -2.12122 q^{55} +10.8322 q^{59} +13.7555 q^{61} -2.25511 q^{65} +6.87008 q^{67} -5.57997 q^{71} +6.25511 q^{73} +0.709884 q^{77} +16.7052 q^{79} -10.2770 q^{83} -6.93838 q^{85} -1.33068 q^{89} +0.754693 q^{91} +4.12122 q^{95} +18.4122 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 4 q^{7}+O(q^{10}) \) 5 * q + 5 * q^5 + 4 * q^7	\( 5 q + 5 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{23} + 5 q^{25} + 2 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 2 q^{41} + 18 q^{43} - 4 q^{47} + 13 q^{49} - 2 q^{53} + 4 q^{55} + 6 q^{59} + 10 q^{61} + 2 q^{65} + 16 q^{67} + 10 q^{71} + 18 q^{73} - 16 q^{77} + 14 q^{79} + 8 q^{83} + 2 q^{85} - 18 q^{89} + 36 q^{91} + 6 q^{95} + 6 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^23 + 5 * q^25 + 2 * q^29 + 8 * q^31 + 4 * q^35 + 8 * q^37 + 2 * q^41 + 18 * q^43 - 4 * q^47 + 13 * q^49 - 2 * q^53 + 4 * q^55 + 6 * q^59 + 10 * q^61 + 2 * q^65 + 16 * q^67 + 10 * q^71 + 18 * q^73 - 16 * q^77 + 14 * q^79 + 8 * q^83 + 2 * q^85 - 18 * q^89 + 36 * q^91 + 6 * q^95 + 6 * q^97

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