LMFDB - Embedded newform 2790.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.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:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.787711$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.662077 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.662077 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.66208 q^{11} +3.57542 q^{13} -0.662077 q^{14} +1.00000 q^{16} -3.18360 q^{17} -2.23750 q^{19} +1.00000 q^{20} +2.66208 q^{22} +7.42110 q^{23} +1.00000 q^{25} +3.57542 q^{26} -0.662077 q^{28} -3.15084 q^{29} +1.00000 q^{31} +1.00000 q^{32} -3.18360 q^{34} -0.662077 q^{35} +9.94263 q^{37} -2.23750 q^{38} +1.00000 q^{40} -3.15084 q^{41} -3.28055 q^{43} +2.66208 q^{44} +7.42110 q^{46} +9.51805 q^{47} -6.56165 q^{49} +1.00000 q^{50} +3.57542 q^{52} -1.08666 q^{53} +2.66208 q^{55} -0.662077 q^{56} -3.15084 q^{58} -2.61847 q^{59} +9.83191 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.57542 q^{65} -14.4176 q^{67} -3.18360 q^{68} -0.662077 q^{70} +5.81292 q^{71} +11.5617 q^{73} +9.94263 q^{74} -2.23750 q^{76} -1.76250 q^{77} +15.7555 q^{79} +1.00000 q^{80} -3.15084 q^{82} -14.8422 q^{83} -3.18360 q^{85} -3.28055 q^{86} +2.66208 q^{88} -1.61903 q^{89} -2.36721 q^{91} +7.42110 q^{92} +9.51805 q^{94} -2.23750 q^{95} +10.4750 q^{97} -6.56165 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.662077	−0.250242	−0.125121	−	0.992142i	$-0.539932\pi$
$7$	−0.662077	−0.250242	−0.125121	+	0.992142i	$0.539932\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	2.66208	0.802647	0.401323	−	0.915936i	$-0.368550\pi$
$11$	2.66208	0.802647	0.401323	+	0.915936i	$0.368550\pi$
$12$	0	0
$13$	3.57542	0.991643	0.495822	−	0.868424i	$-0.334867\pi$
$13$	3.57542	0.991643	0.495822	+	0.868424i	$0.334867\pi$
$14$	−0.662077	−0.176948
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.18360	−0.772137	−0.386069	−	0.922470i	$-0.626167\pi$
$17$	−3.18360	−0.772137	−0.386069	+	0.922470i	$0.626167\pi$
$18$	0	0
$19$	−2.23750	−0.513317	−0.256659	−	0.966502i	$-0.582622\pi$
$19$	−2.23750	−0.513317	−0.256659	+	0.966502i	$0.582622\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	2.66208	0.567557
$23$	7.42110	1.54741	0.773703	−	0.633548i	$-0.218402\pi$
$23$	7.42110	1.54741	0.773703	+	0.633548i	$0.218402\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.57542	0.701198
$27$	0	0
$28$	−0.662077	−0.125121
$29$	−3.15084	−0.585097	−0.292548	−	0.956251i	$-0.594503\pi$
$29$	−3.15084	−0.585097	−0.292548	+	0.956251i	$0.594503\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.18360	−0.545983
$35$	−0.662077	−0.111912
$36$	0	0
$37$	9.94263	1.63456	0.817279	−	0.576242i	$-0.195482\pi$
$37$	9.94263	1.63456	0.817279	+	0.576242i	$0.195482\pi$
$38$	−2.23750	−0.362970
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.15084	−0.492079	−0.246039	−	0.969260i	$-0.579129\pi$
$41$	−3.15084	−0.492079	−0.246039	+	0.969260i	$0.579129\pi$
$42$	0	0
$43$	−3.28055	−0.500279	−0.250140	−	0.968210i	$-0.580477\pi$
$43$	−3.28055	−0.500279	−0.250140	+	0.968210i	$0.580477\pi$
$44$	2.66208	0.401323
$45$	0	0
$46$	7.42110	1.09418
$47$	9.51805	1.38835	0.694175	−	0.719806i	$-0.255769\pi$
$47$	9.51805	1.38835	0.694175	+	0.719806i	$0.255769\pi$
$48$	0	0
$49$	−6.56165	−0.937379
$50$	1.00000	0.141421
$51$	0	0
$52$	3.57542	0.495822
$53$	−1.08666	−0.149264	−0.0746319	−	0.997211i	$-0.523778\pi$
$53$	−1.08666	−0.149264	−0.0746319	+	0.997211i	$0.523778\pi$
$54$	0	0
$55$	2.66208	0.358954
$56$	−0.662077	−0.0884738
$57$	0	0
$58$	−3.15084	−0.413726
$59$	−2.61847	−0.340896	−0.170448	−	0.985367i	$-0.554521\pi$
$59$	−2.61847	−0.340896	−0.170448	+	0.985367i	$0.554521\pi$
$60$	0	0
$61$	9.83191	1.25885	0.629424	−	0.777062i	$-0.283291\pi$
$61$	9.83191	1.25885	0.629424	+	0.777062i	$0.283291\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	3.57542	0.443476
$66$	0	0
$67$	−14.4176	−1.76139	−0.880697	−	0.473681i	$-0.842925\pi$
$67$	−14.4176	−1.76139	−0.880697	+	0.473681i	$0.842925\pi$
$68$	−3.18360	−0.386069
$69$	0	0
$70$	−0.662077	−0.0791334
$71$	5.81292	0.689867	0.344933	−	0.938627i	$-0.387902\pi$
$71$	5.81292	0.689867	0.344933	+	0.938627i	$0.387902\pi$
$72$	0	0
$73$	11.5617	1.35319	0.676595	−	0.736356i	$-0.263455\pi$
$73$	11.5617	1.35319	0.676595	+	0.736356i	$0.263455\pi$
$74$	9.94263	1.15581
$75$	0	0
$76$	−2.23750	−0.256659
$77$	−1.76250	−0.200856
$78$	0	0
$79$	15.7555	1.77264	0.886319	−	0.463076i	$-0.153254\pi$
$79$	15.7555	1.77264	0.886319	+	0.463076i	$0.153254\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−3.15084	−0.347952
$83$	−14.8422	−1.62914	−0.814572	−	0.580063i	$-0.803028\pi$
$83$	−14.8422	−1.62914	−0.814572	+	0.580063i	$0.803028\pi$
$84$	0	0
$85$	−3.18360	−0.345310
$86$	−3.28055	−0.353751
$87$	0	0
$88$	2.66208	0.283778
$89$	−1.61903	−0.171616	−0.0858082	−	0.996312i	$-0.527347\pi$
$89$	−1.61903	−0.171616	−0.0858082	+	0.996312i	$0.527347\pi$
$90$	0	0
$91$	−2.36721	−0.248151
$92$	7.42110	0.773703
$93$	0	0
$94$	9.51805	0.981712
$95$	−2.23750	−0.229563
$96$	0	0
$97$	10.4750	1.06357	0.531787	−	0.846878i	$-0.321521\pi$
$97$	10.4750	1.06357	0.531787	+	0.846878i	$0.321521\pi$
$98$	−6.56165	−0.662827
$99$	0	0
$100$	1.00000	0.100000
$101$	10.6047	1.05521	0.527604	−	0.849491i	$-0.323091\pi$
$101$	10.6047	1.05521	0.527604	+	0.849491i	$0.323091\pi$
$102$	0	0
$103$	10.6185	1.04627	0.523135	−	0.852250i	$-0.324763\pi$
$103$	10.6185	1.04627	0.523135	+	0.852250i	$0.324763\pi$
$104$	3.57542	0.350599
$105$	0	0
$106$	−1.08666	−0.105545
$107$	−11.2805	−1.09053	−0.545266	−	0.838263i	$-0.683571\pi$
$107$	−11.2805	−1.09053	−0.545266	+	0.838263i	$0.683571\pi$
$108$	0	0
$109$	−2.19389	−0.210137	−0.105068	−	0.994465i	$-0.533506\pi$
$109$	−2.19389	−0.210137	−0.105068	+	0.994465i	$0.533506\pi$
$110$	2.66208	0.253819
$111$	0	0
$112$	−0.662077	−0.0625604
$113$	−19.0797	−1.79487	−0.897434	−	0.441149i	$-0.854571\pi$
$113$	−19.0797	−1.79487	−0.897434	+	0.441149i	$0.854571\pi$
$114$	0	0
$115$	7.42110	0.692021
$116$	−3.15084	−0.292548
$117$	0	0
$118$	−2.61847	−0.241050
$119$	2.10779	0.193221
$120$	0	0
$121$	−3.91334	−0.355759
$122$	9.83191	0.890140
$123$	0	0
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.72626	0.596859	0.298430	−	0.954432i	$-0.403537\pi$
$127$	6.72626	0.596859	0.298430	+	0.954432i	$0.403537\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.57542	0.313585
$131$	18.5909	1.62430	0.812149	−	0.583450i	$-0.198298\pi$
$131$	18.5909	1.62430	0.812149	+	0.583450i	$0.198298\pi$
$132$	0	0
$133$	1.48140	0.128453
$134$	−14.4176	−1.24549
$135$	0	0
$136$	−3.18360	−0.272992
$137$	−22.3069	−1.90581	−0.952904	−	0.303272i	$-0.901921\pi$
$137$	−22.3069	−1.90581	−0.952904	+	0.303272i	$0.901921\pi$
$138$	0	0
$139$	4.14055	0.351197	0.175599	−	0.984462i	$-0.443814\pi$
$139$	4.14055	0.351197	0.175599	+	0.984462i	$0.443814\pi$
$140$	−0.662077	−0.0559558
$141$	0	0
$142$	5.81292	0.487809
$143$	9.51805	0.795939
$144$	0	0
$145$	−3.15084	−0.261663
$146$	11.5617	0.956849
$147$	0	0
$148$	9.94263	0.817279
$149$	8.43139	0.690727	0.345363	−	0.938469i	$-0.387756\pi$
$149$	8.43139	0.690727	0.345363	+	0.938469i	$0.387756\pi$
$150$	0	0
$151$	2.36721	0.192640	0.0963202	−	0.995350i	$-0.469293\pi$
$151$	2.36721	0.192640	0.0963202	+	0.995350i	$0.469293\pi$
$152$	−2.23750	−0.181485
$153$	0	0
$154$	−1.76250	−0.142026
$155$	1.00000	0.0803219
$156$	0	0
$157$	−17.0797	−1.36311	−0.681554	−	0.731768i	$-0.738696\pi$
$157$	−17.0797	−1.36311	−0.681554	+	0.731768i	$0.738696\pi$
$158$	15.7555	1.25344
$159$	0	0
$160$	1.00000	0.0790569
$161$	−4.91334	−0.387226
$162$	0	0
$163$	25.0935	1.96547	0.982736	−	0.185013i	$-0.0592327\pi$
$163$	25.0935	1.96547	0.982736	+	0.185013i	$0.0592327\pi$
$164$	−3.15084	−0.246039
$165$	0	0
$166$	−14.8422	−1.15198
$167$	−6.59948	−0.510683	−0.255342	−	0.966851i	$-0.582188\pi$
$167$	−6.59948	−0.510683	−0.255342	+	0.966851i	$0.582188\pi$
$168$	0	0
$169$	−0.216363	−0.0166433
$170$	−3.18360	−0.244171
$171$	0	0
$172$	−3.28055	−0.250140
$173$	11.9930	0.911814	0.455907	−	0.890027i	$-0.349315\pi$
$173$	11.9930	0.911814	0.455907	+	0.890027i	$0.349315\pi$
$174$	0	0
$175$	−0.662077	−0.0500483
$176$	2.66208	0.200662
$177$	0	0
$178$	−1.61903	−0.121351
$179$	15.7418	1.17660	0.588298	−	0.808644i	$-0.299798\pi$
$179$	15.7418	1.17660	0.588298	+	0.808644i	$0.299798\pi$
$180$	0	0
$181$	−8.57194	−0.637148	−0.318574	−	0.947898i	$-0.603204\pi$
$181$	−8.57194	−0.637148	−0.318574	+	0.947898i	$0.603204\pi$
$182$	−2.36721	−0.175469
$183$	0	0
$184$	7.42110	0.547091
$185$	9.94263	0.730996
$186$	0	0
$187$	−8.47500	−0.619753
$188$	9.51805	0.694175
$189$	0	0
$190$	−2.23750	−0.162325
$191$	−12.5909	−0.911048	−0.455524	−	0.890223i	$-0.650548\pi$
$191$	−12.5909	−0.911048	−0.455524	+	0.890223i	$0.650548\pi$
$192$	0	0
$193$	12.6483	0.910445	0.455223	−	0.890378i	$-0.349560\pi$
$193$	12.6483	0.910445	0.455223	+	0.890378i	$0.349560\pi$
$194$	10.4750	0.752061
$195$	0	0
$196$	−6.56165	−0.468690
$197$	−21.1508	−1.50694	−0.753468	−	0.657485i	$-0.771620\pi$
$197$	−21.1508	−1.50694	−0.753468	+	0.657485i	$0.771620\pi$
$198$	0	0
$199$	−0.0641862	−0.00455004	−0.00227502	−	0.999997i	$-0.500724\pi$
$199$	−0.0641862	−0.00455004	−0.00227502	+	0.999997i	$0.500724\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	10.6047	0.746144
$203$	2.08610	0.146416
$204$	0	0
$205$	−3.15084	−0.220064
$206$	10.6185	0.739824
$207$	0	0
$208$	3.57542	0.247911
$209$	−5.95639	−0.412012
$210$	0	0
$211$	13.5822	0.935040	0.467520	−	0.883983i	$-0.345148\pi$
$211$	13.5822	0.935040	0.467520	+	0.883983i	$0.345148\pi$
$212$	−1.08666	−0.0746319
$213$	0	0
$214$	−11.2805	−0.771122
$215$	−3.28055	−0.223732
$216$	0	0
$217$	−0.662077	−0.0449447
$218$	−2.19389	−0.148589
$219$	0	0
$220$	2.66208	0.179477
$221$	−11.3827	−0.765685
$222$	0	0
$223$	−17.2598	−1.15580	−0.577902	−	0.816106i	$-0.696128\pi$
$223$	−17.2598	−1.15580	−0.577902	+	0.816106i	$0.696128\pi$
$224$	−0.662077	−0.0442369
$225$	0	0
$226$	−19.0797	−1.26916
$227$	−9.19445	−0.610257	−0.305128	−	0.952311i	$-0.598699\pi$
$227$	−9.19445	−0.610257	−0.305128	+	0.952311i	$0.598699\pi$
$228$	0	0
$229$	16.8314	1.11225	0.556124	−	0.831100i	$-0.312288\pi$
$229$	16.8314	1.11225	0.556124	+	0.831100i	$0.312288\pi$
$230$	7.42110	0.489333
$231$	0	0
$232$	−3.15084	−0.206863
$233$	−24.6253	−1.61326	−0.806628	−	0.591059i	$-0.798710\pi$
$233$	−24.6253	−1.61326	−0.806628	+	0.591059i	$0.798710\pi$
$234$	0	0
$235$	9.51805	0.620889
$236$	−2.61847	−0.170448
$237$	0	0
$238$	2.10779	0.136628
$239$	−25.4905	−1.64884	−0.824422	−	0.565975i	$-0.808500\pi$
$239$	−25.4905	−1.64884	−0.824422	+	0.565975i	$0.808500\pi$
$240$	0	0
$241$	−16.3672	−1.05430	−0.527152	−	0.849771i	$-0.676740\pi$
$241$	−16.3672	−1.05430	−0.527152	+	0.849771i	$0.676740\pi$
$242$	−3.91334	−0.251559
$243$	0	0
$244$	9.83191	0.629424
$245$	−6.56165	−0.419209
$246$	0	0
$247$	−8.00000	−0.509028
$248$	1.00000	0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	−15.7418	−0.993612	−0.496806	−	0.867862i	$-0.665494\pi$
$251$	−15.7418	−0.993612	−0.496806	+	0.867862i	$0.665494\pi$
$252$	0	0
$253$	19.7555	1.24202
$254$	6.72626	0.422043
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.46082	−0.0911232	−0.0455616	−	0.998962i	$-0.514508\pi$
$257$	−1.46082	−0.0911232	−0.0455616	+	0.998962i	$0.514508\pi$
$258$	0	0
$259$	−6.58279	−0.409035
$260$	3.57542	0.221738
$261$	0	0
$262$	18.5909	1.14855
$263$	−26.1336	−1.61147	−0.805733	−	0.592278i	$-0.798229\pi$
$263$	−26.1336	−1.61147	−0.805733	+	0.592278i	$0.798229\pi$
$264$	0	0
$265$	−1.08666	−0.0667528
$266$	1.48140	0.0908303
$267$	0	0
$268$	−14.4176	−0.880697
$269$	−3.52500	−0.214923	−0.107462	−	0.994209i	$-0.534272\pi$
$269$	−3.52500	−0.214923	−0.107462	+	0.994209i	$0.534272\pi$
$270$	0	0
$271$	7.21503	0.438282	0.219141	−	0.975693i	$-0.429675\pi$
$271$	7.21503	0.438282	0.219141	+	0.975693i	$0.429675\pi$
$272$	−3.18360	−0.193034
$273$	0	0
$274$	−22.3069	−1.34761
$275$	2.66208	0.160529
$276$	0	0
$277$	−7.07289	−0.424969	−0.212484	−	0.977164i	$-0.568155\pi$
$277$	−7.07289	−0.424969	−0.212484	+	0.977164i	$0.568155\pi$
$278$	4.14055	0.248334
$279$	0	0
$280$	−0.662077	−0.0395667
$281$	6.30168	0.375927	0.187963	−	0.982176i	$-0.439811\pi$
$281$	6.30168	0.375927	0.187963	+	0.982176i	$0.439811\pi$
$282$	0	0
$283$	6.05737	0.360073	0.180037	−	0.983660i	$-0.442378\pi$
$283$	6.05737	0.360073	0.180037	+	0.983660i	$0.442378\pi$
$284$	5.81292	0.344933
$285$	0	0
$286$	9.51805	0.562814
$287$	2.08610	0.123139
$288$	0	0
$289$	−6.86467	−0.403804
$290$	−3.15084	−0.185024
$291$	0	0
$292$	11.5617	0.676595
$293$	−17.6914	−1.03354	−0.516770	−	0.856124i	$-0.672866\pi$
$293$	−17.6914	−1.03354	−0.516770	+	0.856124i	$0.672866\pi$
$294$	0	0
$295$	−2.61847	−0.152453
$296$	9.94263	0.577903
$297$	0	0
$298$	8.43139	0.488417
$299$	26.5336	1.53448
$300$	0	0
$301$	2.17198	0.125191
$302$	2.36721	0.136217
$303$	0	0
$304$	−2.23750	−0.128329
$305$	9.83191	0.562974
$306$	0	0
$307$	19.2392	1.09804	0.549021	−	0.835809i	$-0.315001\pi$
$307$	19.2392	1.09804	0.549021	+	0.835809i	$0.315001\pi$
$308$	−1.76250	−0.100428
$309$	0	0
$310$	1.00000	0.0567962
$311$	21.4401	1.21576	0.607878	−	0.794030i	$-0.292021\pi$
$311$	21.4401	1.21576	0.607878	+	0.794030i	$0.292021\pi$
$312$	0	0
$313$	−12.5405	−0.708832	−0.354416	−	0.935088i	$-0.615320\pi$
$313$	−12.5405	−0.708832	−0.354416	+	0.935088i	$0.615320\pi$
$314$	−17.0797	−0.963863
$315$	0	0
$316$	15.7555	0.886319
$317$	−16.6483	−0.935062	−0.467531	−	0.883977i	$-0.654856\pi$
$317$	−16.6483	−0.935062	−0.467531	+	0.883977i	$0.654856\pi$
$318$	0	0
$319$	−8.38779	−0.469626
$320$	1.00000	0.0559017
$321$	0	0
$322$	−4.91334	−0.273810
$323$	7.12331	0.396351
$324$	0	0
$325$	3.57542	0.198329
$326$	25.0935	1.38980
$327$	0	0
$328$	−3.15084	−0.173976
$329$	−6.30168	−0.347423
$330$	0	0
$331$	−7.79393	−0.428393	−0.214196	−	0.976791i	$-0.568713\pi$
$331$	−7.79393	−0.428393	−0.214196	+	0.976791i	$0.568713\pi$
$332$	−14.8422	−0.814572
$333$	0	0
$334$	−6.59948	−0.361107
$335$	−14.4176	−0.787719
$336$	0	0
$337$	−20.1939	−1.10003	−0.550016	−	0.835154i	$-0.685378\pi$
$337$	−20.1939	−1.10003	−0.550016	+	0.835154i	$0.685378\pi$
$338$	−0.216363	−0.0117686
$339$	0	0
$340$	−3.18360	−0.172655
$341$	2.66208	0.144160
$342$	0	0
$343$	8.97886	0.484813
$344$	−3.28055	−0.176875
$345$	0	0
$346$	11.9930	0.644750
$347$	−7.71890	−0.414372	−0.207186	−	0.978302i	$-0.566431\pi$
$347$	−7.71890	−0.414372	−0.207186	+	0.978302i	$0.566431\pi$
$348$	0	0
$349$	29.5983	1.58436	0.792180	−	0.610287i	$-0.208946\pi$
$349$	29.5983	1.58436	0.792180	+	0.610287i	$0.208946\pi$
$350$	−0.662077	−0.0353895
$351$	0	0
$352$	2.66208	0.141889
$353$	21.9983	1.17085	0.585425	−	0.810727i	$-0.300928\pi$
$353$	21.9983	1.17085	0.585425	+	0.810727i	$0.300928\pi$
$354$	0	0
$355$	5.81292	0.308518
$356$	−1.61903	−0.0858082
$357$	0	0
$358$	15.7418	0.831979
$359$	−18.6345	−0.983494	−0.491747	−	0.870738i	$-0.663641\pi$
$359$	−18.6345	−0.983494	−0.491747	+	0.870738i	$0.663641\pi$
$360$	0	0
$361$	−13.9936	−0.736505
$362$	−8.57194	−0.450531
$363$	0	0
$364$	−2.36721	−0.124075
$365$	11.5617	0.605165
$366$	0	0
$367$	24.4107	1.27423	0.637113	−	0.770770i	$-0.280128\pi$
$367$	24.4107	1.27423	0.637113	+	0.770770i	$0.280128\pi$
$368$	7.42110	0.386852
$369$	0	0
$370$	9.94263	0.516893
$371$	0.719451	0.0373520
$372$	0	0
$373$	−9.58223	−0.496149	−0.248075	−	0.968741i	$-0.579798\pi$
$373$	−9.58223	−0.496149	−0.248075	+	0.968741i	$0.579798\pi$
$374$	−8.47500	−0.438232
$375$	0	0
$376$	9.51805	0.490856
$377$	−11.2656	−0.580207
$378$	0	0
$379$	20.0367	1.02921	0.514607	−	0.857426i	$-0.327938\pi$
$379$	20.0367	1.02921	0.514607	+	0.857426i	$0.327938\pi$
$380$	−2.23750	−0.114781
$381$	0	0
$382$	−12.5909	−0.644208
$383$	7.76639	0.396844	0.198422	−	0.980117i	$-0.436418\pi$
$383$	7.76639	0.396844	0.198422	+	0.980117i	$0.436418\pi$
$384$	0	0
$385$	−1.76250	−0.0898254
$386$	12.6483	0.643782
$387$	0	0
$388$	10.4750	0.531787
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−23.6258	−1.19481
$392$	−6.56165	−0.331414
$393$	0	0
$394$	−21.1508	−1.06556
$395$	15.7555	0.792748
$396$	0	0
$397$	9.45386	0.474476	0.237238	−	0.971452i	$-0.423758\pi$
$397$	9.45386	0.474476	0.237238	+	0.971452i	$0.423758\pi$
$398$	−0.0641862	−0.00321736
$399$	0	0
$400$	1.00000	0.0500000
$401$	32.5679	1.62636	0.813182	−	0.582010i	$-0.197733\pi$
$401$	32.5679	1.62636	0.813182	+	0.582010i	$0.197733\pi$
$402$	0	0
$403$	3.57542	0.178104
$404$	10.6047	0.527604
$405$	0	0
$406$	2.08610	0.103531
$407$	26.4680	1.31197
$408$	0	0
$409$	−0.994935	−0.0491964	−0.0245982	−	0.999697i	$-0.507831\pi$
$409$	−0.994935	−0.0491964	−0.0245982	+	0.999697i	$0.507831\pi$
$410$	−3.15084	−0.155609
$411$	0	0
$412$	10.6185	0.523135
$413$	1.73363	0.0853064
$414$	0	0
$415$	−14.8422	−0.728575
$416$	3.57542	0.175299
$417$	0	0
$418$	−5.95639	−0.291337
$419$	−3.68321	−0.179937	−0.0899684	−	0.995945i	$-0.528677\pi$
$419$	−3.68321	−0.179937	−0.0899684	+	0.995945i	$0.528677\pi$
$420$	0	0
$421$	−38.8628	−1.89406	−0.947028	−	0.321151i	$-0.895930\pi$
$421$	−38.8628	−1.89406	−0.947028	+	0.321151i	$0.895930\pi$
$422$	13.5822	0.661173
$423$	0	0
$424$	−1.08666	−0.0527727
$425$	−3.18360	−0.154427
$426$	0	0
$427$	−6.50949	−0.315016
$428$	−11.2805	−0.545266
$429$	0	0
$430$	−3.28055	−0.158202
$431$	10.0574	0.484447	0.242223	−	0.970221i	$-0.422123\pi$
$431$	10.0574	0.484447	0.242223	+	0.970221i	$0.422123\pi$
$432$	0	0
$433$	−17.3011	−0.831439	−0.415720	−	0.909493i	$-0.636470\pi$
$433$	−17.3011	−0.831439	−0.415720	+	0.909493i	$0.636470\pi$
$434$	−0.662077	−0.0317807
$435$	0	0
$436$	−2.19389	−0.105068
$437$	−16.6047	−0.794311
$438$	0	0
$439$	−15.6258	−0.745781	−0.372890	−	0.927875i	$-0.621633\pi$
$439$	−15.6258	−0.745781	−0.372890	+	0.927875i	$0.621633\pi$
$440$	2.66208	0.126910
$441$	0	0
$442$	−11.3827	−0.541421
$443$	−10.4589	−0.496919	−0.248459	−	0.968642i	$-0.579924\pi$
$443$	−10.4589	−0.496919	−0.248459	+	0.968642i	$0.579924\pi$
$444$	0	0
$445$	−1.61903	−0.0767492
$446$	−17.2598	−0.817276
$447$	0	0
$448$	−0.662077	−0.0312802
$449$	−26.2513	−1.23887	−0.619437	−	0.785046i	$-0.712639\pi$
$449$	−26.2513	−1.23887	−0.619437	+	0.785046i	$0.712639\pi$
$450$	0	0
$451$	−8.38779	−0.394965
$452$	−19.0797	−0.897434
$453$	0	0
$454$	−9.19445	−0.431517
$455$	−2.36721	−0.110976
$456$	0	0
$457$	−41.4905	−1.94084	−0.970422	−	0.241414i	$-0.922389\pi$
$457$	−41.4905	−1.94084	−0.970422	+	0.241414i	$0.922389\pi$
$458$	16.8314	0.786478
$459$	0	0
$460$	7.42110	0.346011
$461$	−29.6844	−1.38254	−0.691270	−	0.722596i	$-0.742949\pi$
$461$	−29.6844	−1.38254	−0.691270	+	0.722596i	$0.742949\pi$
$462$	0	0
$463$	23.9632	1.11366	0.556832	−	0.830625i	$-0.312017\pi$
$463$	23.9632	1.11366	0.556832	+	0.830625i	$0.312017\pi$
$464$	−3.15084	−0.146274
$465$	0	0
$466$	−24.6253	−1.14074
$467$	−23.5386	−1.08924	−0.544619	−	0.838684i	$-0.683326\pi$
$467$	−23.5386	−1.08924	−0.544619	+	0.838684i	$0.683326\pi$
$468$	0	0
$469$	9.54558	0.440774
$470$	9.51805	0.439035
$471$	0	0
$472$	−2.61847	−0.120525
$473$	−8.73308	−0.401547
$474$	0	0
$475$	−2.23750	−0.102663
$476$	2.10779	0.0966105
$477$	0	0
$478$	−25.4905	−1.16591
$479$	−11.1371	−0.508866	−0.254433	−	0.967090i	$-0.581889\pi$
$479$	−11.1371	−0.508866	−0.254433	+	0.967090i	$0.581889\pi$
$480$	0	0
$481$	35.5491	1.62090
$482$	−16.3672	−0.745506
$483$	0	0
$484$	−3.91334	−0.177879
$485$	10.4750	0.475645
$486$	0	0
$487$	−21.0279	−0.952867	−0.476434	−	0.879210i	$-0.658071\pi$
$487$	−21.0279	−0.952867	−0.476434	+	0.879210i	$0.658071\pi$
$488$	9.83191	0.445070
$489$	0	0
$490$	−6.56165	−0.296425
$491$	12.8629	0.580496	0.290248	−	0.956952i	$-0.406262\pi$
$491$	12.8629	0.580496	0.290248	+	0.956952i	$0.406262\pi$
$492$	0	0
$493$	10.0310	0.451775
$494$	−8.00000	−0.359937
$495$	0	0
$496$	1.00000	0.0449013
$497$	−3.84860	−0.172633
$498$	0	0
$499$	2.70165	0.120943	0.0604713	−	0.998170i	$-0.480740\pi$
$499$	2.70165	0.120943	0.0604713	+	0.998170i	$0.480740\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−15.7418	−0.702590
$503$	21.4525	0.956521	0.478260	−	0.878218i	$-0.341267\pi$
$503$	21.4525	0.956521	0.478260	+	0.878218i	$0.341267\pi$
$504$	0	0
$505$	10.6047	0.471903
$506$	19.7555	0.878241
$507$	0	0
$508$	6.72626	0.298430
$509$	−19.1233	−0.847626	−0.423813	−	0.905750i	$-0.639309\pi$
$509$	−19.1233	−0.847626	−0.423813	+	0.905750i	$0.639309\pi$
$510$	0	0
$511$	−7.65471	−0.338624
$512$	1.00000	0.0441942
$513$	0	0
$514$	−1.46082	−0.0644339
$515$	10.6185	0.467906
$516$	0	0
$517$	25.3378	1.11435
$518$	−6.58279	−0.289231
$519$	0	0
$520$	3.57542	0.156793
$521$	4.50253	0.197260	0.0986298	−	0.995124i	$-0.468554\pi$
$521$	4.50253	0.197260	0.0986298	+	0.995124i	$0.468554\pi$
$522$	0	0
$523$	−23.4469	−1.02526	−0.512631	−	0.858609i	$-0.671329\pi$
$523$	−23.4469	−1.02526	−0.512631	+	0.858609i	$0.671329\pi$
$524$	18.5909	0.812149
$525$	0	0
$526$	−26.1336	−1.13948
$527$	−3.18360	−0.138680
$528$	0	0
$529$	32.0727	1.39447
$530$	−1.08666	−0.0472013
$531$	0	0
$532$	1.48140	0.0642267
$533$	−11.2656	−0.487967
$534$	0	0
$535$	−11.2805	−0.487701
$536$	−14.4176	−0.622747
$537$	0	0
$538$	−3.52500	−0.151974
$539$	−17.4676	−0.752384
$540$	0	0
$541$	10.5405	0.453172	0.226586	−	0.973991i	$-0.427244\pi$
$541$	10.5405	0.453172	0.226586	+	0.973991i	$0.427244\pi$
$542$	7.21503	0.309912
$543$	0	0
$544$	−3.18360	−0.136496
$545$	−2.19389	−0.0939761
$546$	0	0
$547$	−26.4176	−1.12954	−0.564768	−	0.825250i	$-0.691034\pi$
$547$	−26.4176	−1.12954	−0.564768	+	0.825250i	$0.691034\pi$
$548$	−22.3069	−0.952904
$549$	0	0
$550$	2.66208	0.113511
$551$	7.05001	0.300340
$552$	0	0
$553$	−10.4314	−0.443588
$554$	−7.07289	−0.300498
$555$	0	0
$556$	4.14055	0.175599
$557$	−6.71945	−0.284712	−0.142356	−	0.989816i	$-0.545468\pi$
$557$	−6.71945	−0.284712	−0.142356	+	0.989816i	$0.545468\pi$
$558$	0	0
$559$	−11.7293	−0.496098
$560$	−0.662077	−0.0279779
$561$	0	0
$562$	6.30168	0.265821
$563$	−9.29662	−0.391806	−0.195903	−	0.980623i	$-0.562764\pi$
$563$	−9.29662	−0.391806	−0.195903	+	0.980623i	$0.562764\pi$
$564$	0	0
$565$	−19.0797	−0.802689
$566$	6.05737	0.254610
$567$	0	0
$568$	5.81292	0.243905
$569$	−18.6345	−0.781201	−0.390600	−	0.920560i	$-0.627733\pi$
$569$	−18.6345	−0.781201	−0.390600	+	0.920560i	$0.627733\pi$
$570$	0	0
$571$	−15.0700	−0.630658	−0.315329	−	0.948982i	$-0.602115\pi$
$571$	−15.0700	−0.630658	−0.315329	+	0.948982i	$0.602115\pi$
$572$	9.51805	0.397970
$573$	0	0
$574$	2.08610	0.0870722
$575$	7.42110	0.309481
$576$	0	0
$577$	−41.8577	−1.74256	−0.871280	−	0.490787i	$-0.836709\pi$
$577$	−41.8577	−1.74256	−0.871280	+	0.490787i	$0.836709\pi$
$578$	−6.86467	−0.285533
$579$	0	0
$580$	−3.15084	−0.130832
$581$	9.82669	0.407680
$582$	0	0
$583$	−2.89276	−0.119806
$584$	11.5617	0.478425
$585$	0	0
$586$	−17.6914	−0.730823
$587$	21.4525	0.885441	0.442720	−	0.896660i	$-0.354013\pi$
$587$	21.4525	0.885441	0.442720	+	0.896660i	$0.354013\pi$
$588$	0	0
$589$	−2.23750	−0.0921945
$590$	−2.61847	−0.107801
$591$	0	0
$592$	9.94263	0.408639
$593$	25.1016	1.03080	0.515400	−	0.856950i	$-0.327643\pi$
$593$	25.1016	1.03080	0.515400	+	0.856950i	$0.327643\pi$
$594$	0	0
$595$	2.10779	0.0864110
$596$	8.43139	0.345363
$597$	0	0
$598$	26.5336	1.08504
$599$	29.4388	1.20284	0.601418	−	0.798935i	$-0.294603\pi$
$599$	29.4388	1.20284	0.601418	+	0.798935i	$0.294603\pi$
$600$	0	0
$601$	3.40330	0.138824	0.0694118	−	0.997588i	$-0.477888\pi$
$601$	3.40330	0.138824	0.0694118	+	0.997588i	$0.477888\pi$
$602$	2.17198	0.0885232
$603$	0	0
$604$	2.36721	0.0963202
$605$	−3.91334	−0.159100
$606$	0	0
$607$	−48.0448	−1.95008	−0.975039	−	0.222033i	$-0.928731\pi$
$607$	−48.0448	−1.95008	−0.975039	+	0.222033i	$0.928731\pi$
$608$	−2.23750	−0.0907426
$609$	0	0
$610$	9.83191	0.398083
$611$	34.0310	1.37675
$612$	0	0
$613$	−23.3323	−0.942383	−0.471191	−	0.882031i	$-0.656176\pi$
$613$	−23.3323	−0.942383	−0.471191	+	0.882031i	$0.656176\pi$
$614$	19.2392	0.776433
$615$	0	0
$616$	−1.76250	−0.0710132
$617$	23.2736	0.936960	0.468480	−	0.883474i	$-0.344802\pi$
$617$	23.2736	0.936960	0.468480	+	0.883474i	$0.344802\pi$
$618$	0	0
$619$	9.82496	0.394898	0.197449	−	0.980313i	$-0.436734\pi$
$619$	9.82496	0.394898	0.197449	+	0.980313i	$0.436734\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	21.4401	0.859669
$623$	1.07192	0.0429456
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.5405	−0.501220
$627$	0	0
$628$	−17.0797	−0.681554
$629$	−31.6534	−1.26210
$630$	0	0
$631$	38.0297	1.51394	0.756969	−	0.653451i	$-0.226679\pi$
$631$	38.0297	1.51394	0.756969	+	0.653451i	$0.226679\pi$
$632$	15.7555	0.626722
$633$	0	0
$634$	−16.6483	−0.661189
$635$	6.72626	0.266924
$636$	0	0
$637$	−23.4607	−0.929546
$638$	−8.38779	−0.332076
$639$	0	0
$640$	1.00000	0.0395285
$641$	47.1520	1.86239	0.931197	−	0.364517i	$-0.118766\pi$
$641$	47.1520	1.86239	0.931197	+	0.364517i	$0.118766\pi$
$642$	0	0
$643$	19.7935	0.780581	0.390290	−	0.920692i	$-0.372375\pi$
$643$	19.7935	0.780581	0.390290	+	0.920692i	$0.372375\pi$
$644$	−4.91334	−0.193613
$645$	0	0
$646$	7.12331	0.280263
$647$	−10.3505	−0.406921	−0.203460	−	0.979083i	$-0.565219\pi$
$647$	−10.3505	−0.406921	−0.203460	+	0.979083i	$0.565219\pi$
$648$	0	0
$649$	−6.97058	−0.273619
$650$	3.57542	0.140240
$651$	0	0
$652$	25.0935	0.982736
$653$	15.5180	0.607268	0.303634	−	0.952789i	$-0.401800\pi$
$653$	15.5180	0.607268	0.303634	+	0.952789i	$0.401800\pi$
$654$	0	0
$655$	18.5909	0.726408
$656$	−3.15084	−0.123020
$657$	0	0
$658$	−6.30168	−0.245665
$659$	−12.2020	−0.475324	−0.237662	−	0.971348i	$-0.576381\pi$
$659$	−12.2020	−0.475324	−0.237662	+	0.971348i	$0.576381\pi$
$660$	0	0
$661$	−14.2594	−0.554627	−0.277313	−	0.960779i	$-0.589444\pi$
$661$	−14.2594	−0.554627	−0.277313	+	0.960779i	$0.589444\pi$
$662$	−7.79393	−0.302920
$663$	0	0
$664$	−14.8422	−0.575989
$665$	1.48140	0.0574461
$666$	0	0
$667$	−23.3827	−0.905383
$668$	−6.59948	−0.255342
$669$	0	0
$670$	−14.4176	−0.557001
$671$	26.1733	1.01041
$672$	0	0
$673$	1.99305	0.0768263	0.0384131	−	0.999262i	$-0.487770\pi$
$673$	1.99305	0.0768263	0.0384131	+	0.999262i	$0.487770\pi$
$674$	−20.1939	−0.777840
$675$	0	0
$676$	−0.216363	−0.00832166
$677$	−19.9917	−0.768344	−0.384172	−	0.923262i	$-0.625513\pi$
$677$	−19.9917	−0.768344	−0.384172	+	0.923262i	$0.625513\pi$
$678$	0	0
$679$	−6.93526	−0.266151
$680$	−3.18360	−0.122086
$681$	0	0
$682$	2.66208	0.101936
$683$	−19.2255	−0.735642	−0.367821	−	0.929897i	$-0.619896\pi$
$683$	−19.2255	−0.735642	−0.367821	+	0.929897i	$0.619896\pi$
$684$	0	0
$685$	−22.3069	−0.852303
$686$	8.97886	0.342815
$687$	0	0
$688$	−3.28055	−0.125070
$689$	−3.88525	−0.148016
$690$	0	0
$691$	−9.80049	−0.372828	−0.186414	−	0.982471i	$-0.559687\pi$
$691$	−9.80049	−0.372828	−0.186414	+	0.982471i	$0.559687\pi$
$692$	11.9930	0.455907
$693$	0	0
$694$	−7.71890	−0.293005
$695$	4.14055	0.157060
$696$	0	0
$697$	10.0310	0.379952
$698$	29.5983	1.12031
$699$	0	0
$700$	−0.662077	−0.0250242
$701$	0.865228	0.0326792	0.0163396	−	0.999866i	$-0.494799\pi$
$701$	0.865228	0.0326792	0.0163396	+	0.999866i	$0.494799\pi$
$702$	0	0
$703$	−22.2466	−0.839047
$704$	2.66208	0.100331
$705$	0	0
$706$	21.9983	0.827916
$707$	−7.02114	−0.264057
$708$	0	0
$709$	−50.5650	−1.89901	−0.949504	−	0.313755i	$-0.898413\pi$
$709$	−50.5650	−1.89901	−0.949504	+	0.313755i	$0.898413\pi$
$710$	5.81292	0.218155
$711$	0	0
$712$	−1.61903	−0.0606756
$713$	7.42110	0.277922
$714$	0	0
$715$	9.51805	0.355955
$716$	15.7418	0.588298
$717$	0	0
$718$	−18.6345	−0.695435
$719$	−30.9706	−1.15501	−0.577504	−	0.816388i	$-0.695973\pi$
$719$	−30.9706	−1.15501	−0.577504	+	0.816388i	$0.695973\pi$
$720$	0	0
$721$	−7.03025	−0.261820
$722$	−13.9936	−0.520788
$723$	0	0
$724$	−8.57194	−0.318574
$725$	−3.15084	−0.117019
$726$	0	0
$727$	−43.4112	−1.61003	−0.805017	−	0.593252i	$-0.797844\pi$
$727$	−43.4112	−1.61003	−0.805017	+	0.593252i	$0.797844\pi$
$728$	−2.36721	−0.0877345
$729$	0	0
$730$	11.5617	0.427916
$731$	10.4440	0.386284
$732$	0	0
$733$	19.6258	0.724897	0.362448	−	0.932004i	$-0.381941\pi$
$733$	19.6258	0.724897	0.362448	+	0.932004i	$0.381941\pi$
$734$	24.4107	0.901014
$735$	0	0
$736$	7.42110	0.273545
$737$	−38.3808	−1.41378
$738$	0	0
$739$	−17.0033	−0.625478	−0.312739	−	0.949839i	$-0.601246\pi$
$739$	−17.0033	−0.625478	−0.312739	+	0.949839i	$0.601246\pi$
$740$	9.94263	0.365498
$741$	0	0
$742$	0.719451	0.0264119
$743$	12.2323	0.448759	0.224379	−	0.974502i	$-0.427965\pi$
$743$	12.2323	0.448759	0.224379	+	0.974502i	$0.427965\pi$
$744$	0	0
$745$	8.43139	0.308902
$746$	−9.58223	−0.350831
$747$	0	0
$748$	−8.47500	−0.309877
$749$	7.46860	0.272897
$750$	0	0
$751$	−21.2369	−0.774947	−0.387474	−	0.921881i	$-0.626652\pi$
$751$	−21.2369	−0.774947	−0.387474	+	0.921881i	$0.626652\pi$
$752$	9.51805	0.347087
$753$	0	0
$754$	−11.2656	−0.410269
$755$	2.36721	0.0861514
$756$	0	0
$757$	17.2874	0.628320	0.314160	−	0.949370i	$-0.398277\pi$
$757$	17.2874	0.628320	0.314160	+	0.949370i	$0.398277\pi$
$758$	20.0367	0.727764
$759$	0	0
$760$	−2.23750	−0.0811626
$761$	−13.8921	−0.503587	−0.251794	−	0.967781i	$-0.581020\pi$
$761$	−13.8921	−0.503587	−0.251794	+	0.967781i	$0.581020\pi$
$762$	0	0
$763$	1.45253	0.0525850
$764$	−12.5909	−0.455524
$765$	0	0
$766$	7.76639	0.280611
$767$	−9.36214	−0.338047
$768$	0	0
$769$	−13.8564	−0.499674	−0.249837	−	0.968288i	$-0.580377\pi$
$769$	−13.8564	−0.499674	−0.249837	+	0.968288i	$0.580377\pi$
$770$	−1.76250	−0.0635161
$771$	0	0
$772$	12.6483	0.455223
$773$	−26.4027	−0.949641	−0.474820	−	0.880083i	$-0.657487\pi$
$773$	−26.4027	−0.949641	−0.474820	+	0.880083i	$0.657487\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	10.4750	0.376030
$777$	0	0
$778$	−12.0000	−0.430221
$779$	7.05001	0.252593
$780$	0	0
$781$	15.4744	0.553719
$782$	−23.6258	−0.844858
$783$	0	0
$784$	−6.56165	−0.234345
$785$	−17.0797	−0.609601
$786$	0	0
$787$	−37.9289	−1.35202	−0.676009	−	0.736893i	$-0.736292\pi$
$787$	−37.9289	−1.35202	−0.676009	+	0.736893i	$0.736292\pi$
$788$	−21.1508	−0.753468
$789$	0	0
$790$	15.7555	0.560557
$791$	12.6322	0.449151
$792$	0	0
$793$	35.1532	1.24833
$794$	9.45386	0.335505
$795$	0	0
$796$	−0.0641862	−0.00227502
$797$	25.4836	0.902674	0.451337	−	0.892354i	$-0.350947\pi$
$797$	25.4836	0.902674	0.451337	+	0.892354i	$0.350947\pi$
$798$	0	0
$799$	−30.3017	−1.07200
$800$	1.00000	0.0353553
$801$	0	0
$802$	32.5679	1.15001
$803$	30.7780	1.08613
$804$	0	0
$805$	−4.91334	−0.173173
$806$	3.57542	0.125939
$807$	0	0
$808$	10.6047	0.373072
$809$	21.6327	0.760564	0.380282	−	0.924871i	$-0.375827\pi$
$809$	21.6327	0.760564	0.380282	+	0.924871i	$0.375827\pi$
$810$	0	0
$811$	4.52556	0.158914	0.0794569	−	0.996838i	$-0.474681\pi$
$811$	4.52556	0.158914	0.0794569	+	0.996838i	$0.474681\pi$
$812$	2.08610	0.0732078
$813$	0	0
$814$	26.4680	0.927704
$815$	25.0935	0.878986
$816$	0	0
$817$	7.34022	0.256802
$818$	−0.994935	−0.0347871
$819$	0	0
$820$	−3.15084	−0.110032
$821$	−9.82669	−0.342954	−0.171477	−	0.985188i	$-0.554854\pi$
$821$	−9.82669	−0.342954	−0.171477	+	0.985188i	$0.554854\pi$
$822$	0	0
$823$	12.1815	0.424619	0.212310	−	0.977202i	$-0.431902\pi$
$823$	12.1815	0.424619	0.212310	+	0.977202i	$0.431902\pi$
$824$	10.6185	0.369912
$825$	0	0
$826$	1.73363	0.0603207
$827$	11.6258	0.404270	0.202135	−	0.979358i	$-0.435212\pi$
$827$	11.6258	0.404270	0.202135	+	0.979358i	$0.435212\pi$
$828$	0	0
$829$	3.03221	0.105313	0.0526564	−	0.998613i	$-0.483231\pi$
$829$	3.03221	0.105313	0.0526564	+	0.998613i	$0.483231\pi$
$830$	−14.8422	−0.515180
$831$	0	0
$832$	3.57542	0.123955
$833$	20.8897	0.723785
$834$	0	0
$835$	−6.59948	−0.228384
$836$	−5.95639	−0.206006
$837$	0	0
$838$	−3.68321	−0.127234
$839$	30.0860	1.03868	0.519341	−	0.854567i	$-0.326177\pi$
$839$	30.0860	1.03868	0.519341	+	0.854567i	$0.326177\pi$
$840$	0	0
$841$	−19.0722	−0.657662
$842$	−38.8628	−1.33930
$843$	0	0
$844$	13.5822	0.467520
$845$	−0.216363	−0.00744312
$846$	0	0
$847$	2.59094	0.0890256
$848$	−1.08666	−0.0373159
$849$	0	0
$850$	−3.18360	−0.109197
$851$	73.7852	2.52933
$852$	0	0
$853$	−12.8641	−0.440459	−0.220230	−	0.975448i	$-0.570681\pi$
$853$	−12.8641	−0.440459	−0.220230	+	0.975448i	$0.570681\pi$
$854$	−6.50949	−0.222750
$855$	0	0
$856$	−11.2805	−0.385561
$857$	39.9000	1.36296	0.681479	−	0.731838i	$-0.261337\pi$
$857$	39.9000	1.36296	0.681479	+	0.731838i	$0.261337\pi$
$858$	0	0
$859$	20.6260	0.703750	0.351875	−	0.936047i	$-0.385544\pi$
$859$	20.6260	0.703750	0.351875	+	0.936047i	$0.385544\pi$
$860$	−3.28055	−0.111866
$861$	0	0
$862$	10.0574	0.342555
$863$	27.2272	0.926825	0.463412	−	0.886143i	$-0.346625\pi$
$863$	27.2272	0.926825	0.463412	+	0.886143i	$0.346625\pi$
$864$	0	0
$865$	11.9930	0.407776
$866$	−17.3011	−0.587916
$867$	0	0
$868$	−0.662077	−0.0224724
$869$	41.9425	1.42280
$870$	0	0
$871$	−51.5491	−1.74667
$872$	−2.19389	−0.0742946
$873$	0	0
$874$	−16.6047	−0.561663
$875$	−0.662077	−0.0223823
$876$	0	0
$877$	9.88636	0.333839	0.166919	−	0.985971i	$-0.446618\pi$
$877$	9.88636	0.333839	0.166919	+	0.985971i	$0.446618\pi$
$878$	−15.6258	−0.527347
$879$	0	0
$880$	2.66208	0.0897386
$881$	−3.82899	−0.129002	−0.0645010	−	0.997918i	$-0.520546\pi$
$881$	−3.82899	−0.129002	−0.0645010	+	0.997918i	$0.520546\pi$
$882$	0	0
$883$	39.1600	1.31784	0.658919	−	0.752214i	$-0.271014\pi$
$883$	39.1600	1.31784	0.658919	+	0.752214i	$0.271014\pi$
$884$	−11.3827	−0.382842
$885$	0	0
$886$	−10.4589	−0.351375
$887$	2.56916	0.0862640	0.0431320	−	0.999069i	$-0.486266\pi$
$887$	2.56916	0.0862640	0.0431320	+	0.999069i	$0.486266\pi$
$888$	0	0
$889$	−4.45331	−0.149359
$890$	−1.61903	−0.0542699
$891$	0	0
$892$	−17.2598	−0.577902
$893$	−21.2966	−0.712664
$894$	0	0
$895$	15.7418	0.526190
$896$	−0.662077	−0.0221185
$897$	0	0
$898$	−26.2513	−0.876016
$899$	−3.15084	−0.105086
$900$	0	0
$901$	3.45948	0.115252
$902$	−8.38779	−0.279283
$903$	0	0
$904$	−19.0797	−0.634581
$905$	−8.57194	−0.284941
$906$	0	0
$907$	22.7642	0.755874	0.377937	−	0.925831i	$-0.376634\pi$
$907$	22.7642	0.755874	0.377937	+	0.925831i	$0.376634\pi$
$908$	−9.19445	−0.305128
$909$	0	0
$910$	−2.36721	−0.0784721
$911$	45.1955	1.49739	0.748697	−	0.662913i	$-0.230680\pi$
$911$	45.1955	1.49739	0.748697	+	0.662913i	$0.230680\pi$
$912$	0	0
$913$	−39.5111	−1.30763
$914$	−41.4905	−1.37238
$915$	0	0
$916$	16.8314	0.556124
$917$	−12.3086	−0.406467
$918$	0	0
$919$	54.6344	1.80222	0.901111	−	0.433588i	$-0.142753\pi$
$919$	54.6344	1.80222	0.901111	+	0.433588i	$0.142753\pi$
$920$	7.42110	0.244666
$921$	0	0
$922$	−29.6844	−0.977604
$923$	20.7836	0.684102
$924$	0	0
$925$	9.94263	0.326912
$926$	23.9632	0.787480
$927$	0	0
$928$	−3.15084	−0.103431
$929$	−55.5982	−1.82412	−0.912058	−	0.410061i	$-0.865508\pi$
$929$	−55.5982	−1.82412	−0.912058	+	0.410061i	$0.865508\pi$
$930$	0	0
$931$	14.6817	0.481173
$932$	−24.6253	−0.806628
$933$	0	0
$934$	−23.5386	−0.770207
$935$	−8.47500	−0.277162
$936$	0	0
$937$	30.9500	1.01109	0.505546	−	0.862800i	$-0.331291\pi$
$937$	30.9500	1.01109	0.505546	+	0.862800i	$0.331291\pi$
$938$	9.54558	0.311674
$939$	0	0
$940$	9.51805	0.310444
$941$	−19.1233	−0.623402	−0.311701	−	0.950180i	$-0.600899\pi$
$941$	−19.1233	−0.623402	−0.311701	+	0.950180i	$0.600899\pi$
$942$	0	0
$943$	−23.3827	−0.761446
$944$	−2.61847	−0.0852240
$945$	0	0
$946$	−8.73308	−0.283937
$947$	−10.3397	−0.335994	−0.167997	−	0.985787i	$-0.553730\pi$
$947$	−10.3397	−0.335994	−0.167997	+	0.985787i	$0.553730\pi$
$948$	0	0
$949$	41.3378	1.34188
$950$	−2.23750	−0.0725940
$951$	0	0
$952$	2.10779	0.0683139
$953$	−25.3922	−0.822535	−0.411268	−	0.911515i	$-0.634914\pi$
$953$	−25.3922	−0.822535	−0.411268	+	0.911515i	$0.634914\pi$
$954$	0	0
$955$	−12.5909	−0.407433
$956$	−25.4905	−0.824422
$957$	0	0
$958$	−11.1371	−0.359823
$959$	14.7689	0.476913
$960$	0	0
$961$	1.00000	0.0322581
$962$	35.5491	1.14615
$963$	0	0
$964$	−16.3672	−0.527152
$965$	12.6483	0.407163
$966$	0	0
$967$	13.0718	0.420360	0.210180	−	0.977663i	$-0.432595\pi$
$967$	13.0718	0.420360	0.210180	+	0.977663i	$0.432595\pi$
$968$	−3.91334	−0.125780
$969$	0	0
$970$	10.4750	0.336332
$971$	47.5409	1.52566	0.762831	−	0.646598i	$-0.223809\pi$
$971$	47.5409	1.52566	0.762831	+	0.646598i	$0.223809\pi$
$972$	0	0
$973$	−2.74137	−0.0878842
$974$	−21.0279	−0.673779
$975$	0	0
$976$	9.83191	0.314712
$977$	−23.0155	−0.736332	−0.368166	−	0.929760i	$-0.620014\pi$
$977$	−23.0155	−0.736332	−0.368166	+	0.929760i	$0.620014\pi$
$978$	0	0
$979$	−4.30997	−0.137747
$980$	−6.56165	−0.209604
$981$	0	0
$982$	12.8629	0.410472
$983$	−24.8474	−0.792510	−0.396255	−	0.918141i	$-0.629690\pi$
$983$	−24.8474	−0.792510	−0.396255	+	0.918141i	$0.629690\pi$
$984$	0	0
$985$	−21.1508	−0.673922
$986$	10.0310	0.319453
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−24.3453	−0.774135
$990$	0	0
$991$	39.1383	1.24327	0.621634	−	0.783308i	$-0.286469\pi$
$991$	39.1383	1.24327	0.621634	+	0.783308i	$0.286469\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	−3.84860	−0.122070
$995$	−0.0641862	−0.00203484
$996$	0	0
$997$	32.3603	1.02486	0.512430	−	0.858729i	$-0.328746\pi$
$997$	32.3603	1.02486	0.512430	+	0.858729i	$0.328746\pi$
$998$	2.70165	0.0855193
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bk.1.2	yes	4
3.2	odd	2		2790.2.a.bj.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2790.2.a.bj.1.2	✓	4	3.2	odd	2
2790.2.a.bk.1.2	yes	4	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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.662077 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.662077 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.66208 q^{11} +3.57542 q^{13} -0.662077 q^{14} +1.00000 q^{16} -3.18360 q^{17} -2.23750 q^{19} +1.00000 q^{20} +2.66208 q^{22} +7.42110 q^{23} +1.00000 q^{25} +3.57542 q^{26} -0.662077 q^{28} -3.15084 q^{29} +1.00000 q^{31} +1.00000 q^{32} -3.18360 q^{34} -0.662077 q^{35} +9.94263 q^{37} -2.23750 q^{38} +1.00000 q^{40} -3.15084 q^{41} -3.28055 q^{43} +2.66208 q^{44} +7.42110 q^{46} +9.51805 q^{47} -6.56165 q^{49} +1.00000 q^{50} +3.57542 q^{52} -1.08666 q^{53} +2.66208 q^{55} -0.662077 q^{56} -3.15084 q^{58} -2.61847 q^{59} +9.83191 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.57542 q^{65} -14.4176 q^{67} -3.18360 q^{68} -0.662077 q^{70} +5.81292 q^{71} +11.5617 q^{73} +9.94263 q^{74} -2.23750 q^{76} -1.76250 q^{77} +15.7555 q^{79} +1.00000 q^{80} -3.15084 q^{82} -14.8422 q^{83} -3.18360 q^{85} -3.28055 q^{86} +2.66208 q^{88} -1.61903 q^{89} -2.36721 q^{91} +7.42110 q^{92} +9.51805 q^{94} -2.23750 q^{95} +10.4750 q^{97} -6.56165 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 5 * q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 6 * q^13 + 5 * q^14 + 4 * q^16 + 7 * q^19 + 4 * q^20 + 3 * q^22 + q^23 + 4 * q^25 + 6 * q^26 + 5 * q^28 + 4 * q^29 + 4 * q^31 + 4 * q^32 + 5 * q^35 + 6 * q^37 + 7 * q^38 + 4 * q^40 + 4 * q^41 + 13 * q^43 + 3 * q^44 + q^46 - 4 * q^47 + 5 * q^49 + 4 * q^50 + 6 * q^52 - 5 * q^53 + 3 * q^55 + 5 * q^56 + 4 * q^58 + 8 * q^59 - 4 * q^61 + 4 * q^62 + 4 * q^64 + 6 * q^65 + 8 * q^67 + 5 * q^70 - q^71 + 15 * q^73 + 6 * q^74 + 7 * q^76 - 23 * q^77 + 5 * q^79 + 4 * q^80 + 4 * q^82 - 2 * q^83 + 13 * q^86 + 3 * q^88 - 9 * q^89 + 16 * q^91 + q^92 - 4 * q^94 + 7 * q^95 + 10 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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