LMFDB - Embedded newform 414.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,2,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("414.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	414.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:	$3.30580664368$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} +2.00000 q^{7} -1.00000 q^{8} +3.64575 q^{10} +3.64575 q^{11} -5.29150 q^{13} -2.00000 q^{14} +1.00000 q^{16} +7.29150 q^{17} +5.64575 q^{19} -3.64575 q^{20} -3.64575 q^{22} +1.00000 q^{23} +8.29150 q^{25} +5.29150 q^{26} +2.00000 q^{28} -1.29150 q^{29} +9.29150 q^{31} -1.00000 q^{32} -7.29150 q^{34} -7.29150 q^{35} -8.93725 q^{37} -5.64575 q^{38} +3.64575 q^{40} +6.00000 q^{41} +5.64575 q^{43} +3.64575 q^{44} -1.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -8.29150 q^{50} -5.29150 q^{52} +3.64575 q^{53} -13.2915 q^{55} -2.00000 q^{56} +1.29150 q^{58} +5.64575 q^{61} -9.29150 q^{62} +1.00000 q^{64} +19.2915 q^{65} +0.937254 q^{67} +7.29150 q^{68} +7.29150 q^{70} -6.00000 q^{71} +3.29150 q^{73} +8.93725 q^{74} +5.64575 q^{76} +7.29150 q^{77} -12.5830 q^{79} -3.64575 q^{80} -6.00000 q^{82} -8.35425 q^{83} -26.5830 q^{85} -5.64575 q^{86} -3.64575 q^{88} -10.5830 q^{91} +1.00000 q^{92} -6.00000 q^{94} -20.5830 q^{95} +9.29150 q^{97} +3.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 4 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 2 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^22 + 2 * q^23 + 6 * q^25 + 4 * q^28 + 8 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 4 * q^35 - 2 * q^37 - 6 * q^38 + 2 * q^40 + 12 * q^41 + 6 * q^43 + 2 * q^44 - 2 * q^46 + 12 * q^47 - 6 * q^49 - 6 * q^50 + 2 * q^53 - 16 * q^55 - 4 * q^56 - 8 * q^58 + 6 * q^61 - 8 * q^62 + 2 * q^64 + 28 * q^65 - 14 * q^67 + 4 * q^68 + 4 * q^70 - 12 * q^71 - 4 * q^73 + 2 * q^74 + 6 * q^76 + 4 * q^77 - 4 * q^79 - 2 * q^80 - 12 * q^82 - 22 * q^83 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 2 * q^92 - 12 * q^94 - 20 * q^95 + 8 * q^97 + 6 * q^98

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
$5$	−3.64575	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$5$	−3.64575	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.64575	1.15289
$11$	3.64575	1.09924	0.549618	−	0.835416i	$-0.314773\pi$
$11$	3.64575	1.09924	0.549618	+	0.835416i	$0.314773\pi$
$12$	0	0
$13$	−5.29150	−1.46760	−0.733799	−	0.679366i	$-0.762255\pi$
$13$	−5.29150	−1.46760	−0.733799	+	0.679366i	$0.762255\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	7.29150	1.76845	0.884225	−	0.467062i	$-0.154688\pi$
$17$	7.29150	1.76845	0.884225	+	0.467062i	$0.154688\pi$
$18$	0	0
$19$	5.64575	1.29522	0.647612	−	0.761970i	$-0.275768\pi$
$19$	5.64575	1.29522	0.647612	+	0.761970i	$0.275768\pi$
$20$	−3.64575	−0.815215
$21$	0	0
$22$	−3.64575	−0.777277
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	8.29150	1.65830
$26$	5.29150	1.03775
$27$	0	0
$28$	2.00000	0.377964
$29$	−1.29150	−0.239826	−0.119913	−	0.992784i	$-0.538262\pi$
$29$	−1.29150	−0.239826	−0.119913	+	0.992784i	$0.538262\pi$
$30$	0	0
$31$	9.29150	1.66880	0.834402	−	0.551157i	$-0.185813\pi$
$31$	9.29150	1.66880	0.834402	+	0.551157i	$0.185813\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.29150	−1.25048
$35$	−7.29150	−1.23249
$36$	0	0
$37$	−8.93725	−1.46928	−0.734638	−	0.678460i	$-0.762648\pi$
$37$	−8.93725	−1.46928	−0.734638	+	0.678460i	$0.762648\pi$
$38$	−5.64575	−0.915862
$39$	0	0
$40$	3.64575	0.576444
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	5.64575	0.860969	0.430485	−	0.902598i	$-0.358343\pi$
$43$	5.64575	0.860969	0.430485	+	0.902598i	$0.358343\pi$
$44$	3.64575	0.549618
$45$	0	0
$46$	−1.00000	−0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−8.29150	−1.17260
$51$	0	0
$52$	−5.29150	−0.733799
$53$	3.64575	0.500782	0.250391	−	0.968145i	$-0.419441\pi$
$53$	3.64575	0.500782	0.250391	+	0.968145i	$0.419441\pi$
$54$	0	0
$55$	−13.2915	−1.79223
$56$	−2.00000	−0.267261
$57$	0	0
$58$	1.29150	0.169583
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	5.64575	0.722864	0.361432	−	0.932398i	$-0.382288\pi$
$61$	5.64575	0.722864	0.361432	+	0.932398i	$0.382288\pi$
$62$	−9.29150	−1.18002
$63$	0	0
$64$	1.00000	0.125000
$65$	19.2915	2.39282
$66$	0	0
$67$	0.937254	0.114504	0.0572519	−	0.998360i	$-0.481766\pi$
$67$	0.937254	0.114504	0.0572519	+	0.998360i	$0.481766\pi$
$68$	7.29150	0.884225
$69$	0	0
$70$	7.29150	0.871501
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	3.29150	0.385241	0.192621	−	0.981273i	$-0.438301\pi$
$73$	3.29150	0.385241	0.192621	+	0.981273i	$0.438301\pi$
$74$	8.93725	1.03893
$75$	0	0
$76$	5.64575	0.647612
$77$	7.29150	0.830944
$78$	0	0
$79$	−12.5830	−1.41570	−0.707849	−	0.706363i	$-0.750334\pi$
$79$	−12.5830	−1.41570	−0.707849	+	0.706363i	$0.750334\pi$
$80$	−3.64575	−0.407607
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−8.35425	−0.916998	−0.458499	−	0.888695i	$-0.651613\pi$
$83$	−8.35425	−0.916998	−0.458499	+	0.888695i	$0.651613\pi$
$84$	0	0
$85$	−26.5830	−2.88333
$86$	−5.64575	−0.608797
$87$	0	0
$88$	−3.64575	−0.388638
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−10.5830	−1.10940
$92$	1.00000	0.104257
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−20.5830	−2.11177
$96$	0	0
$97$	9.29150	0.943409	0.471705	−	0.881757i	$-0.343639\pi$
$97$	9.29150	0.943409	0.471705	+	0.881757i	$0.343639\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	8.29150	0.829150
$101$	−1.29150	−0.128509	−0.0642547	−	0.997934i	$-0.520467\pi$
$101$	−1.29150	−0.128509	−0.0642547	+	0.997934i	$0.520467\pi$
$102$	0	0
$103$	−17.2915	−1.70378	−0.851891	−	0.523719i	$-0.824544\pi$
$103$	−17.2915	−1.70378	−0.851891	+	0.523719i	$0.824544\pi$
$104$	5.29150	0.518875
$105$	0	0
$106$	−3.64575	−0.354107
$107$	−8.35425	−0.807636	−0.403818	−	0.914839i	$-0.632317\pi$
$107$	−8.35425	−0.807636	−0.403818	+	0.914839i	$0.632317\pi$
$108$	0	0
$109$	8.22876	0.788172	0.394086	−	0.919074i	$-0.371061\pi$
$109$	8.22876	0.788172	0.394086	+	0.919074i	$0.371061\pi$
$110$	13.2915	1.26730
$111$	0	0
$112$	2.00000	0.188982
$113$	−14.5830	−1.37185	−0.685927	−	0.727670i	$-0.740603\pi$
$113$	−14.5830	−1.37185	−0.685927	+	0.727670i	$0.740603\pi$
$114$	0	0
$115$	−3.64575	−0.339968
$116$	−1.29150	−0.119913
$117$	0	0
$118$	0	0
$119$	14.5830	1.33682
$120$	0	0
$121$	2.29150	0.208318
$122$	−5.64575	−0.511142
$123$	0	0
$124$	9.29150	0.834402
$125$	−12.0000	−1.07331
$126$	0	0
$127$	10.5830	0.939090	0.469545	−	0.882909i	$-0.344418\pi$
$127$	10.5830	0.939090	0.469545	+	0.882909i	$0.344418\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−19.2915	−1.69198
$131$	9.87451	0.862740	0.431370	−	0.902175i	$-0.358030\pi$
$131$	9.87451	0.862740	0.431370	+	0.902175i	$0.358030\pi$
$132$	0	0
$133$	11.2915	0.979097
$134$	−0.937254	−0.0809664
$135$	0	0
$136$	−7.29150	−0.625241
$137$	−9.87451	−0.843636	−0.421818	−	0.906680i	$-0.638608\pi$
$137$	−9.87451	−0.843636	−0.421818	+	0.906680i	$0.638608\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−7.29150	−0.616244
$141$	0	0
$142$	6.00000	0.503509
$143$	−19.2915	−1.61324
$144$	0	0
$145$	4.70850	0.391019
$146$	−3.29150	−0.272407
$147$	0	0
$148$	−8.93725	−0.734638
$149$	15.6458	1.28175	0.640875	−	0.767645i	$-0.278572\pi$
$149$	15.6458	1.28175	0.640875	+	0.767645i	$0.278572\pi$
$150$	0	0
$151$	6.70850	0.545930	0.272965	−	0.962024i	$-0.411996\pi$
$151$	6.70850	0.545930	0.272965	+	0.962024i	$0.411996\pi$
$152$	−5.64575	−0.457931
$153$	0	0
$154$	−7.29150	−0.587566
$155$	−33.8745	−2.72087
$156$	0	0
$157$	−1.64575	−0.131345	−0.0656726	−	0.997841i	$-0.520919\pi$
$157$	−1.64575	−0.131345	−0.0656726	+	0.997841i	$0.520919\pi$
$158$	12.5830	1.00105
$159$	0	0
$160$	3.64575	0.288222
$161$	2.00000	0.157622
$162$	0	0
$163$	−11.2915	−0.884419	−0.442209	−	0.896912i	$-0.645805\pi$
$163$	−11.2915	−0.884419	−0.442209	+	0.896912i	$0.645805\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	8.35425	0.648415
$167$	23.1660	1.79264	0.896320	−	0.443408i	$-0.146231\pi$
$167$	23.1660	1.79264	0.896320	+	0.443408i	$0.146231\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	26.5830	2.03882
$171$	0	0
$172$	5.64575	0.430485
$173$	15.8745	1.20692	0.603458	−	0.797395i	$-0.293789\pi$
$173$	15.8745	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$174$	0	0
$175$	16.5830	1.25356
$176$	3.64575	0.274809
$177$	0	0
$178$	0	0
$179$	4.70850	0.351930	0.175965	−	0.984396i	$-0.443696\pi$
$179$	4.70850	0.351930	0.175965	+	0.984396i	$0.443696\pi$
$180$	0	0
$181$	10.3542	0.769625	0.384813	−	0.922995i	$-0.374266\pi$
$181$	10.3542	0.769625	0.384813	+	0.922995i	$0.374266\pi$
$182$	10.5830	0.784465
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	32.5830	2.39555
$186$	0	0
$187$	26.5830	1.94394
$188$	6.00000	0.437595
$189$	0	0
$190$	20.5830	1.49325
$191$	−7.29150	−0.527595	−0.263797	−	0.964578i	$-0.584975\pi$
$191$	−7.29150	−0.527595	−0.263797	+	0.964578i	$0.584975\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−9.29150	−0.667091
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−8.29150	−0.586298
$201$	0	0
$202$	1.29150	0.0908698
$203$	−2.58301	−0.181291
$204$	0	0
$205$	−21.8745	−1.52778
$206$	17.2915	1.20476
$207$	0	0
$208$	−5.29150	−0.366900
$209$	20.5830	1.42376
$210$	0	0
$211$	0.708497	0.0487750	0.0243875	−	0.999703i	$-0.492236\pi$
$211$	0.708497	0.0487750	0.0243875	+	0.999703i	$0.492236\pi$
$212$	3.64575	0.250391
$213$	0	0
$214$	8.35425	0.571085
$215$	−20.5830	−1.40375
$216$	0	0
$217$	18.5830	1.26150
$218$	−8.22876	−0.557322
$219$	0	0
$220$	−13.2915	−0.896113
$221$	−38.5830	−2.59537
$222$	0	0
$223$	11.8745	0.795176	0.397588	−	0.917564i	$-0.369847\pi$
$223$	11.8745	0.795176	0.397588	+	0.917564i	$0.369847\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	14.5830	0.970047
$227$	−22.9373	−1.52240	−0.761200	−	0.648518i	$-0.775389\pi$
$227$	−22.9373	−1.52240	−0.761200	+	0.648518i	$0.775389\pi$
$228$	0	0
$229$	24.9373	1.64790	0.823950	−	0.566662i	$-0.191766\pi$
$229$	24.9373	1.64790	0.823950	+	0.566662i	$0.191766\pi$
$230$	3.64575	0.240394
$231$	0	0
$232$	1.29150	0.0847913
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−21.8745	−1.42694
$236$	0	0
$237$	0	0
$238$	−14.5830	−0.945276
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−5.29150	−0.340856	−0.170428	−	0.985370i	$-0.554515\pi$
$241$	−5.29150	−0.340856	−0.170428	+	0.985370i	$0.554515\pi$
$242$	−2.29150	−0.147303
$243$	0	0
$244$	5.64575	0.361432
$245$	10.9373	0.698756
$246$	0	0
$247$	−29.8745	−1.90087
$248$	−9.29150	−0.590011
$249$	0	0
$250$	12.0000	0.758947
$251$	−20.3542	−1.28475	−0.642374	−	0.766391i	$-0.722051\pi$
$251$	−20.3542	−1.28475	−0.642374	+	0.766391i	$0.722051\pi$
$252$	0	0
$253$	3.64575	0.229206
$254$	−10.5830	−0.664037
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.41699	0.213146	0.106573	−	0.994305i	$-0.466012\pi$
$257$	3.41699	0.213146	0.106573	+	0.994305i	$0.466012\pi$
$258$	0	0
$259$	−17.8745	−1.11067
$260$	19.2915	1.19641
$261$	0	0
$262$	−9.87451	−0.610049
$263$	−9.41699	−0.580677	−0.290338	−	0.956924i	$-0.593768\pi$
$263$	−9.41699	−0.580677	−0.290338	+	0.956924i	$0.593768\pi$
$264$	0	0
$265$	−13.2915	−0.816491
$266$	−11.2915	−0.692326
$267$	0	0
$268$	0.937254	0.0572519
$269$	−32.5830	−1.98662	−0.993310	−	0.115474i	$-0.963161\pi$
$269$	−32.5830	−1.98662	−0.993310	+	0.115474i	$0.963161\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	7.29150	0.442112
$273$	0	0
$274$	9.87451	0.596541
$275$	30.2288	1.82286
$276$	0	0
$277$	−5.29150	−0.317936	−0.158968	−	0.987284i	$-0.550817\pi$
$277$	−5.29150	−0.317936	−0.158968	+	0.987284i	$0.550817\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	7.29150	0.435751
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	15.5203	0.922584	0.461292	−	0.887248i	$-0.347386\pi$
$283$	15.5203	0.922584	0.461292	+	0.887248i	$0.347386\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	19.2915	1.14073
$287$	12.0000	0.708338
$288$	0	0
$289$	36.1660	2.12741
$290$	−4.70850	−0.276492
$291$	0	0
$292$	3.29150	0.192621
$293$	−5.77124	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$293$	−5.77124	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$294$	0	0
$295$	0	0
$296$	8.93725	0.519467
$297$	0	0
$298$	−15.6458	−0.906334
$299$	−5.29150	−0.306015
$300$	0	0
$301$	11.2915	0.650831
$302$	−6.70850	−0.386031
$303$	0	0
$304$	5.64575	0.323806
$305$	−20.5830	−1.17858
$306$	0	0
$307$	−1.87451	−0.106984	−0.0534919	−	0.998568i	$-0.517035\pi$
$307$	−1.87451	−0.106984	−0.0534919	+	0.998568i	$0.517035\pi$
$308$	7.29150	0.415472
$309$	0	0
$310$	33.8745	1.92394
$311$	−29.1660	−1.65385	−0.826926	−	0.562310i	$-0.809913\pi$
$311$	−29.1660	−1.65385	−0.826926	+	0.562310i	$0.809913\pi$
$312$	0	0
$313$	−5.29150	−0.299093	−0.149547	−	0.988755i	$-0.547781\pi$
$313$	−5.29150	−0.299093	−0.149547	+	0.988755i	$0.547781\pi$
$314$	1.64575	0.0928751
$315$	0	0
$316$	−12.5830	−0.707849
$317$	10.7085	0.601449	0.300725	−	0.953711i	$-0.402772\pi$
$317$	10.7085	0.601449	0.300725	+	0.953711i	$0.402772\pi$
$318$	0	0
$319$	−4.70850	−0.263625
$320$	−3.64575	−0.203804
$321$	0	0
$322$	−2.00000	−0.111456
$323$	41.1660	2.29054
$324$	0	0
$325$	−43.8745	−2.43372
$326$	11.2915	0.625378
$327$	0	0
$328$	−6.00000	−0.331295
$329$	12.0000	0.661581
$330$	0	0
$331$	5.87451	0.322892	0.161446	−	0.986882i	$-0.448384\pi$
$331$	5.87451	0.322892	0.161446	+	0.986882i	$0.448384\pi$
$332$	−8.35425	−0.458499
$333$	0	0
$334$	−23.1660	−1.26759
$335$	−3.41699	−0.186690
$336$	0	0
$337$	−27.1660	−1.47983	−0.739913	−	0.672702i	$-0.765134\pi$
$337$	−27.1660	−1.47983	−0.739913	+	0.672702i	$0.765134\pi$
$338$	−15.0000	−0.815892
$339$	0	0
$340$	−26.5830	−1.44167
$341$	33.8745	1.83441
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−5.64575	−0.304399
$345$	0	0
$346$	−15.8745	−0.853419
$347$	4.70850	0.252765	0.126383	−	0.991982i	$-0.459663\pi$
$347$	4.70850	0.252765	0.126383	+	0.991982i	$0.459663\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−16.5830	−0.886399
$351$	0	0
$352$	−3.64575	−0.194319
$353$	14.5830	0.776175	0.388088	−	0.921623i	$-0.373136\pi$
$353$	14.5830	0.776175	0.388088	+	0.921623i	$0.373136\pi$
$354$	0	0
$355$	21.8745	1.16098
$356$	0	0
$357$	0	0
$358$	−4.70850	−0.248852
$359$	9.87451	0.521157	0.260578	−	0.965453i	$-0.416087\pi$
$359$	9.87451	0.521157	0.260578	+	0.965453i	$0.416087\pi$
$360$	0	0
$361$	12.8745	0.677606
$362$	−10.3542	−0.544207
$363$	0	0
$364$	−10.5830	−0.554700
$365$	−12.0000	−0.628109
$366$	0	0
$367$	31.1660	1.62685	0.813426	−	0.581668i	$-0.197600\pi$
$367$	31.1660	1.62685	0.813426	+	0.581668i	$0.197600\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−32.5830	−1.69391
$371$	7.29150	0.378556
$372$	0	0
$373$	−11.0627	−0.572807	−0.286404	−	0.958109i	$-0.592460\pi$
$373$	−11.0627	−0.572807	−0.286404	+	0.958109i	$0.592460\pi$
$374$	−26.5830	−1.37457
$375$	0	0
$376$	−6.00000	−0.309426
$377$	6.83399	0.351968
$378$	0	0
$379$	3.52026	0.180824	0.0904118	−	0.995904i	$-0.471182\pi$
$379$	3.52026	0.180824	0.0904118	+	0.995904i	$0.471182\pi$
$380$	−20.5830	−1.05589
$381$	0	0
$382$	7.29150	0.373066
$383$	−31.7490	−1.62230	−0.811149	−	0.584839i	$-0.801158\pi$
$383$	−31.7490	−1.62230	−0.811149	+	0.584839i	$0.801158\pi$
$384$	0	0
$385$	−26.5830	−1.35480
$386$	22.0000	1.11977
$387$	0	0
$388$	9.29150	0.471705
$389$	−22.9373	−1.16296	−0.581482	−	0.813559i	$-0.697527\pi$
$389$	−22.9373	−1.16296	−0.581482	+	0.813559i	$0.697527\pi$
$390$	0	0
$391$	7.29150	0.368747
$392$	3.00000	0.151523
$393$	0	0
$394$	6.00000	0.302276
$395$	45.8745	2.30820
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	8.29150	0.414575
$401$	−21.8745	−1.09236	−0.546180	−	0.837668i	$-0.683919\pi$
$401$	−21.8745	−1.09236	−0.546180	+	0.837668i	$0.683919\pi$
$402$	0	0
$403$	−49.1660	−2.44913
$404$	−1.29150	−0.0642547
$405$	0	0
$406$	2.58301	0.128192
$407$	−32.5830	−1.61508
$408$	0	0
$409$	−27.1660	−1.34327	−0.671636	−	0.740881i	$-0.734408\pi$
$409$	−27.1660	−1.34327	−0.671636	+	0.740881i	$0.734408\pi$
$410$	21.8745	1.08030
$411$	0	0
$412$	−17.2915	−0.851891
$413$	0	0
$414$	0	0
$415$	30.4575	1.49510
$416$	5.29150	0.259437
$417$	0	0
$418$	−20.5830	−1.00675
$419$	1.06275	0.0519185	0.0259593	−	0.999663i	$-0.491736\pi$
$419$	1.06275	0.0519185	0.0259593	+	0.999663i	$0.491736\pi$
$420$	0	0
$421$	−4.22876	−0.206097	−0.103048	−	0.994676i	$-0.532860\pi$
$421$	−4.22876	−0.206097	−0.103048	+	0.994676i	$0.532860\pi$
$422$	−0.708497	−0.0344891
$423$	0	0
$424$	−3.64575	−0.177053
$425$	60.4575	2.93262
$426$	0	0
$427$	11.2915	0.546434
$428$	−8.35425	−0.403818
$429$	0	0
$430$	20.5830	0.992601
$431$	−2.12549	−0.102381	−0.0511907	−	0.998689i	$-0.516302\pi$
$431$	−2.12549	−0.102381	−0.0511907	+	0.998689i	$0.516302\pi$
$432$	0	0
$433$	−29.2915	−1.40766	−0.703830	−	0.710369i	$-0.748528\pi$
$433$	−29.2915	−1.40766	−0.703830	+	0.710369i	$0.748528\pi$
$434$	−18.5830	−0.892013
$435$	0	0
$436$	8.22876	0.394086
$437$	5.64575	0.270073
$438$	0	0
$439$	−35.7490	−1.70621	−0.853104	−	0.521741i	$-0.825283\pi$
$439$	−35.7490	−1.70621	−0.853104	+	0.521741i	$0.825283\pi$
$440$	13.2915	0.633648
$441$	0	0
$442$	38.5830	1.83521
$443$	−17.1660	−0.815582	−0.407791	−	0.913075i	$-0.633701\pi$
$443$	−17.1660	−0.815582	−0.407791	+	0.913075i	$0.633701\pi$
$444$	0	0
$445$	0	0
$446$	−11.8745	−0.562274
$447$	0	0
$448$	2.00000	0.0944911
$449$	32.5830	1.53769	0.768844	−	0.639437i	$-0.220832\pi$
$449$	32.5830	1.53769	0.768844	+	0.639437i	$0.220832\pi$
$450$	0	0
$451$	21.8745	1.03003
$452$	−14.5830	−0.685927
$453$	0	0
$454$	22.9373	1.07650
$455$	38.5830	1.80880
$456$	0	0
$457$	−31.8745	−1.49103	−0.745513	−	0.666491i	$-0.767796\pi$
$457$	−31.8745	−1.49103	−0.745513	+	0.666491i	$0.767796\pi$
$458$	−24.9373	−1.16524
$459$	0	0
$460$	−3.64575	−0.169984
$461$	−37.7490	−1.75815	−0.879073	−	0.476686i	$-0.841838\pi$
$461$	−37.7490	−1.75815	−0.879073	+	0.476686i	$0.841838\pi$
$462$	0	0
$463$	−21.1660	−0.983668	−0.491834	−	0.870689i	$-0.663673\pi$
$463$	−21.1660	−0.983668	−0.491834	+	0.870689i	$0.663673\pi$
$464$	−1.29150	−0.0599565
$465$	0	0
$466$	0	0
$467$	8.35425	0.386589	0.193294	−	0.981141i	$-0.438083\pi$
$467$	8.35425	0.386589	0.193294	+	0.981141i	$0.438083\pi$
$468$	0	0
$469$	1.87451	0.0865567
$470$	21.8745	1.00900
$471$	0	0
$472$	0	0
$473$	20.5830	0.946408
$474$	0	0
$475$	46.8118	2.14787
$476$	14.5830	0.668411
$477$	0	0
$478$	−24.0000	−1.09773
$479$	21.4170	0.978567	0.489284	−	0.872125i	$-0.337258\pi$
$479$	21.4170	0.978567	0.489284	+	0.872125i	$0.337258\pi$
$480$	0	0
$481$	47.2915	2.15631
$482$	5.29150	0.241021
$483$	0	0
$484$	2.29150	0.104159
$485$	−33.8745	−1.53816
$486$	0	0
$487$	−21.1660	−0.959123	−0.479562	−	0.877508i	$-0.659204\pi$
$487$	−21.1660	−0.959123	−0.479562	+	0.877508i	$0.659204\pi$
$488$	−5.64575	−0.255571
$489$	0	0
$490$	−10.9373	−0.494095
$491$	9.87451	0.445630	0.222815	−	0.974861i	$-0.428475\pi$
$491$	9.87451	0.445630	0.222815	+	0.974861i	$0.428475\pi$
$492$	0	0
$493$	−9.41699	−0.424120
$494$	29.8745	1.34412
$495$	0	0
$496$	9.29150	0.417201
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−43.0405	−1.92676	−0.963379	−	0.268143i	$-0.913590\pi$
$499$	−43.0405	−1.92676	−0.963379	+	0.268143i	$0.913590\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	20.3542	0.908455
$503$	−33.8745	−1.51039	−0.755195	−	0.655500i	$-0.772458\pi$
$503$	−33.8745	−1.51039	−0.755195	+	0.655500i	$0.772458\pi$
$504$	0	0
$505$	4.70850	0.209525
$506$	−3.64575	−0.162073
$507$	0	0
$508$	10.5830	0.469545
$509$	3.87451	0.171735	0.0858673	−	0.996307i	$-0.472634\pi$
$509$	3.87451	0.171735	0.0858673	+	0.996307i	$0.472634\pi$
$510$	0	0
$511$	6.58301	0.291215
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.41699	−0.150717
$515$	63.0405	2.77790
$516$	0	0
$517$	21.8745	0.962040
$518$	17.8745	0.785361
$519$	0	0
$520$	−19.2915	−0.845988
$521$	24.4575	1.07150	0.535752	−	0.844376i	$-0.320028\pi$
$521$	24.4575	1.07150	0.535752	+	0.844376i	$0.320028\pi$
$522$	0	0
$523$	−20.9373	−0.915522	−0.457761	−	0.889075i	$-0.651348\pi$
$523$	−20.9373	−0.915522	−0.457761	+	0.889075i	$0.651348\pi$
$524$	9.87451	0.431370
$525$	0	0
$526$	9.41699	0.410600
$527$	67.7490	2.95119
$528$	0	0
$529$	1.00000	0.0434783
$530$	13.2915	0.577346
$531$	0	0
$532$	11.2915	0.489549
$533$	−31.7490	−1.37520
$534$	0	0
$535$	30.4575	1.31679
$536$	−0.937254	−0.0404832
$537$	0	0
$538$	32.5830	1.40475
$539$	−10.9373	−0.471101
$540$	0	0
$541$	−0.583005	−0.0250654	−0.0125327	−	0.999921i	$-0.503989\pi$
$541$	−0.583005	−0.0250654	−0.0125327	+	0.999921i	$0.503989\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	−7.29150	−0.312621
$545$	−30.0000	−1.28506
$546$	0	0
$547$	27.2915	1.16690	0.583450	−	0.812149i	$-0.301702\pi$
$547$	27.2915	1.16690	0.583450	+	0.812149i	$0.301702\pi$
$548$	−9.87451	−0.421818
$549$	0	0
$550$	−30.2288	−1.28896
$551$	−7.29150	−0.310628
$552$	0	0
$553$	−25.1660	−1.07017
$554$	5.29150	0.224814
$555$	0	0
$556$	−4.00000	−0.169638
$557$	16.1033	0.682317	0.341159	−	0.940006i	$-0.389181\pi$
$557$	16.1033	0.682317	0.341159	+	0.940006i	$0.389181\pi$
$558$	0	0
$559$	−29.8745	−1.26356
$560$	−7.29150	−0.308122
$561$	0	0
$562$	0	0
$563$	−23.3948	−0.985972	−0.492986	−	0.870037i	$-0.664095\pi$
$563$	−23.3948	−0.985972	−0.492986	+	0.870037i	$0.664095\pi$
$564$	0	0
$565$	53.1660	2.23671
$566$	−15.5203	−0.652365
$567$	0	0
$568$	6.00000	0.251754
$569$	−19.2915	−0.808742	−0.404371	−	0.914595i	$-0.632510\pi$
$569$	−19.2915	−0.808742	−0.404371	+	0.914595i	$0.632510\pi$
$570$	0	0
$571$	3.06275	0.128172	0.0640860	−	0.997944i	$-0.479587\pi$
$571$	3.06275	0.128172	0.0640860	+	0.997944i	$0.479587\pi$
$572$	−19.2915	−0.806618
$573$	0	0
$574$	−12.0000	−0.500870
$575$	8.29150	0.345780
$576$	0	0
$577$	39.2915	1.63573	0.817863	−	0.575413i	$-0.195158\pi$
$577$	39.2915	1.63573	0.817863	+	0.575413i	$0.195158\pi$
$578$	−36.1660	−1.50431
$579$	0	0
$580$	4.70850	0.195510
$581$	−16.7085	−0.693185
$582$	0	0
$583$	13.2915	0.550478
$584$	−3.29150	−0.136203
$585$	0	0
$586$	5.77124	0.238408
$587$	−31.2915	−1.29154	−0.645769	−	0.763533i	$-0.723463\pi$
$587$	−31.2915	−1.29154	−0.645769	+	0.763533i	$0.723463\pi$
$588$	0	0
$589$	52.4575	2.16147
$590$	0	0
$591$	0	0
$592$	−8.93725	−0.367319
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−53.1660	−2.17959
$596$	15.6458	0.640875
$597$	0	0
$598$	5.29150	0.216386
$599$	9.41699	0.384768	0.192384	−	0.981320i	$-0.438378\pi$
$599$	9.41699	0.384768	0.192384	+	0.981320i	$0.438378\pi$
$600$	0	0
$601$	20.4575	0.834479	0.417240	−	0.908796i	$-0.362998\pi$
$601$	20.4575	0.834479	0.417240	+	0.908796i	$0.362998\pi$
$602$	−11.2915	−0.460207
$603$	0	0
$604$	6.70850	0.272965
$605$	−8.35425	−0.339649
$606$	0	0
$607$	4.12549	0.167449	0.0837243	−	0.996489i	$-0.473318\pi$
$607$	4.12549	0.167449	0.0837243	+	0.996489i	$0.473318\pi$
$608$	−5.64575	−0.228965
$609$	0	0
$610$	20.5830	0.833382
$611$	−31.7490	−1.28443
$612$	0	0
$613$	12.9373	0.522531	0.261265	−	0.965267i	$-0.415860\pi$
$613$	12.9373	0.522531	0.261265	+	0.965267i	$0.415860\pi$
$614$	1.87451	0.0756490
$615$	0	0
$616$	−7.29150	−0.293783
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	17.6458	0.709243	0.354621	−	0.935010i	$-0.384610\pi$
$619$	17.6458	0.709243	0.354621	+	0.935010i	$0.384610\pi$
$620$	−33.8745	−1.36043
$621$	0	0
$622$	29.1660	1.16945
$623$	0	0
$624$	0	0
$625$	2.29150	0.0916601
$626$	5.29150	0.211491
$627$	0	0
$628$	−1.64575	−0.0656726
$629$	−65.1660	−2.59834
$630$	0	0
$631$	−7.87451	−0.313479	−0.156740	−	0.987640i	$-0.550098\pi$
$631$	−7.87451	−0.313479	−0.156740	+	0.987640i	$0.550098\pi$
$632$	12.5830	0.500525
$633$	0	0
$634$	−10.7085	−0.425289
$635$	−38.5830	−1.53112
$636$	0	0
$637$	15.8745	0.628971
$638$	4.70850	0.186411
$639$	0	0
$640$	3.64575	0.144111
$641$	24.4575	0.966014	0.483007	−	0.875617i	$-0.339545\pi$
$641$	24.4575	0.966014	0.483007	+	0.875617i	$0.339545\pi$
$642$	0	0
$643$	29.6458	1.16911	0.584557	−	0.811353i	$-0.301268\pi$
$643$	29.6458	1.16911	0.584557	+	0.811353i	$0.301268\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	−41.1660	−1.61966
$647$	−29.1660	−1.14663	−0.573317	−	0.819334i	$-0.694344\pi$
$647$	−29.1660	−1.14663	−0.573317	+	0.819334i	$0.694344\pi$
$648$	0	0
$649$	0	0
$650$	43.8745	1.72090
$651$	0	0
$652$	−11.2915	−0.442209
$653$	15.8745	0.621217	0.310609	−	0.950538i	$-0.399467\pi$
$653$	15.8745	0.621217	0.310609	+	0.950538i	$0.399467\pi$
$654$	0	0
$655$	−36.0000	−1.40664
$656$	6.00000	0.234261
$657$	0	0
$658$	−12.0000	−0.467809
$659$	22.9373	0.893509	0.446754	−	0.894657i	$-0.352580\pi$
$659$	22.9373	0.893509	0.446754	+	0.894657i	$0.352580\pi$
$660$	0	0
$661$	22.3542	0.869480	0.434740	−	0.900556i	$-0.356840\pi$
$661$	22.3542	0.869480	0.434740	+	0.900556i	$0.356840\pi$
$662$	−5.87451	−0.228319
$663$	0	0
$664$	8.35425	0.324208
$665$	−41.1660	−1.59635
$666$	0	0
$667$	−1.29150	−0.0500072
$668$	23.1660	0.896320
$669$	0	0
$670$	3.41699	0.132010
$671$	20.5830	0.794598
$672$	0	0
$673$	−23.2915	−0.897821	−0.448911	−	0.893577i	$-0.648188\pi$
$673$	−23.2915	−0.897821	−0.448911	+	0.893577i	$0.648188\pi$
$674$	27.1660	1.04640
$675$	0	0
$676$	15.0000	0.576923
$677$	−22.9373	−0.881550	−0.440775	−	0.897618i	$-0.645296\pi$
$677$	−22.9373	−0.881550	−0.440775	+	0.897618i	$0.645296\pi$
$678$	0	0
$679$	18.5830	0.713150
$680$	26.5830	1.01941
$681$	0	0
$682$	−33.8745	−1.29712
$683$	−2.12549	−0.0813297	−0.0406648	−	0.999173i	$-0.512948\pi$
$683$	−2.12549	−0.0813297	−0.0406648	+	0.999173i	$0.512948\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	20.0000	0.763604
$687$	0	0
$688$	5.64575	0.215242
$689$	−19.2915	−0.734948
$690$	0	0
$691$	−13.8745	−0.527811	−0.263906	−	0.964549i	$-0.585011\pi$
$691$	−13.8745	−0.527811	−0.263906	+	0.964549i	$0.585011\pi$
$692$	15.8745	0.603458
$693$	0	0
$694$	−4.70850	−0.178732
$695$	14.5830	0.553165
$696$	0	0
$697$	43.7490	1.65711
$698$	−2.00000	−0.0757011
$699$	0	0
$700$	16.5830	0.626779
$701$	−34.9373	−1.31956	−0.659781	−	0.751458i	$-0.729351\pi$
$701$	−34.9373	−1.31956	−0.659781	+	0.751458i	$0.729351\pi$
$702$	0	0
$703$	−50.4575	−1.90304
$704$	3.64575	0.137404
$705$	0	0
$706$	−14.5830	−0.548839
$707$	−2.58301	−0.0971439
$708$	0	0
$709$	−23.0627	−0.866140	−0.433070	−	0.901360i	$-0.642570\pi$
$709$	−23.0627	−0.866140	−0.433070	+	0.901360i	$0.642570\pi$
$710$	−21.8745	−0.820936
$711$	0	0
$712$	0	0
$713$	9.29150	0.347970
$714$	0	0
$715$	70.3320	2.63027
$716$	4.70850	0.175965
$717$	0	0
$718$	−9.87451	−0.368513
$719$	43.7490	1.63156	0.815781	−	0.578360i	$-0.196307\pi$
$719$	43.7490	1.63156	0.815781	+	0.578360i	$0.196307\pi$
$720$	0	0
$721$	−34.5830	−1.28794
$722$	−12.8745	−0.479140
$723$	0	0
$724$	10.3542	0.384813
$725$	−10.7085	−0.397704
$726$	0	0
$727$	−44.3320	−1.64418	−0.822092	−	0.569355i	$-0.807193\pi$
$727$	−44.3320	−1.64418	−0.822092	+	0.569355i	$0.807193\pi$
$728$	10.5830	0.392232
$729$	0	0
$730$	12.0000	0.444140
$731$	41.1660	1.52258
$732$	0	0
$733$	37.3948	1.38121	0.690604	−	0.723233i	$-0.257345\pi$
$733$	37.3948	1.38121	0.690604	+	0.723233i	$0.257345\pi$
$734$	−31.1660	−1.15036
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	3.41699	0.125867
$738$	0	0
$739$	10.5830	0.389302	0.194651	−	0.980873i	$-0.437643\pi$
$739$	10.5830	0.389302	0.194651	+	0.980873i	$0.437643\pi$
$740$	32.5830	1.19778
$741$	0	0
$742$	−7.29150	−0.267679
$743$	43.7490	1.60500	0.802498	−	0.596655i	$-0.203504\pi$
$743$	43.7490	1.60500	0.802498	+	0.596655i	$0.203504\pi$
$744$	0	0
$745$	−57.0405	−2.08980
$746$	11.0627	0.405036
$747$	0	0
$748$	26.5830	0.971971
$749$	−16.7085	−0.610515
$750$	0	0
$751$	−10.4575	−0.381600	−0.190800	−	0.981629i	$-0.561108\pi$
$751$	−10.4575	−0.381600	−0.190800	+	0.981629i	$0.561108\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	−6.83399	−0.248879
$755$	−24.4575	−0.890100
$756$	0	0
$757$	32.2288	1.17137	0.585687	−	0.810537i	$-0.300825\pi$
$757$	32.2288	1.17137	0.585687	+	0.810537i	$0.300825\pi$
$758$	−3.52026	−0.127862
$759$	0	0
$760$	20.5830	0.746624
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	16.4575	0.595802
$764$	−7.29150	−0.263797
$765$	0	0
$766$	31.7490	1.14714
$767$	0	0
$768$	0	0
$769$	−49.0405	−1.76845	−0.884223	−	0.467065i	$-0.845312\pi$
$769$	−49.0405	−1.76845	−0.884223	+	0.467065i	$0.845312\pi$
$770$	26.5830	0.957985
$771$	0	0
$772$	−22.0000	−0.791797
$773$	3.64575	0.131129	0.0655643	−	0.997848i	$-0.479115\pi$
$773$	3.64575	0.131129	0.0655643	+	0.997848i	$0.479115\pi$
$774$	0	0
$775$	77.0405	2.76738
$776$	−9.29150	−0.333546
$777$	0	0
$778$	22.9373	0.822340
$779$	33.8745	1.21368
$780$	0	0
$781$	−21.8745	−0.782731
$782$	−7.29150	−0.260744
$783$	0	0
$784$	−3.00000	−0.107143
$785$	6.00000	0.214149
$786$	0	0
$787$	−49.6458	−1.76968	−0.884840	−	0.465895i	$-0.845732\pi$
$787$	−49.6458	−1.76968	−0.884840	+	0.465895i	$0.845732\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−45.8745	−1.63214
$791$	−29.1660	−1.03702
$792$	0	0
$793$	−29.8745	−1.06087
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	14.0000	0.496217
$797$	1.52026	0.0538503	0.0269252	−	0.999637i	$-0.491428\pi$
$797$	1.52026	0.0538503	0.0269252	+	0.999637i	$0.491428\pi$
$798$	0	0
$799$	43.7490	1.54773
$800$	−8.29150	−0.293149
$801$	0	0
$802$	21.8745	0.772416
$803$	12.0000	0.423471
$804$	0	0
$805$	−7.29150	−0.256992
$806$	49.1660	1.73180
$807$	0	0
$808$	1.29150	0.0454349
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−1.87451	−0.0658229	−0.0329114	−	0.999458i	$-0.510478\pi$
$811$	−1.87451	−0.0658229	−0.0329114	+	0.999458i	$0.510478\pi$
$812$	−2.58301	−0.0906457
$813$	0	0
$814$	32.5830	1.14203
$815$	41.1660	1.44198
$816$	0	0
$817$	31.8745	1.11515
$818$	27.1660	0.949837
$819$	0	0
$820$	−21.8745	−0.763891
$821$	−39.8745	−1.39163	−0.695815	−	0.718221i	$-0.744957\pi$
$821$	−39.8745	−1.39163	−0.695815	+	0.718221i	$0.744957\pi$
$822$	0	0
$823$	−2.70850	−0.0944123	−0.0472061	−	0.998885i	$-0.515032\pi$
$823$	−2.70850	−0.0944123	−0.0472061	+	0.998885i	$0.515032\pi$
$824$	17.2915	0.602378
$825$	0	0
$826$	0	0
$827$	32.8118	1.14098	0.570488	−	0.821306i	$-0.306754\pi$
$827$	32.8118	1.14098	0.570488	+	0.821306i	$0.306754\pi$
$828$	0	0
$829$	−36.5830	−1.27058	−0.635290	−	0.772274i	$-0.719119\pi$
$829$	−36.5830	−1.27058	−0.635290	+	0.772274i	$0.719119\pi$
$830$	−30.4575	−1.05720
$831$	0	0
$832$	−5.29150	−0.183450
$833$	−21.8745	−0.757907
$834$	0	0
$835$	−84.4575	−2.92277
$836$	20.5830	0.711878
$837$	0	0
$838$	−1.06275	−0.0367120
$839$	−19.7490	−0.681812	−0.340906	−	0.940097i	$-0.610734\pi$
$839$	−19.7490	−0.681812	−0.340906	+	0.940097i	$0.610734\pi$
$840$	0	0
$841$	−27.3320	−0.942483
$842$	4.22876	0.145733
$843$	0	0
$844$	0.708497	0.0243875
$845$	−54.6863	−1.88126
$846$	0	0
$847$	4.58301	0.157474
$848$	3.64575	0.125196
$849$	0	0
$850$	−60.4575	−2.07368
$851$	−8.93725	−0.306365
$852$	0	0
$853$	−15.1660	−0.519274	−0.259637	−	0.965706i	$-0.583603\pi$
$853$	−15.1660	−0.519274	−0.259637	+	0.965706i	$0.583603\pi$
$854$	−11.2915	−0.386387
$855$	0	0
$856$	8.35425	0.285542
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	34.5830	1.17996	0.589978	−	0.807419i	$-0.299136\pi$
$859$	34.5830	1.17996	0.589978	+	0.807419i	$0.299136\pi$
$860$	−20.5830	−0.701875
$861$	0	0
$862$	2.12549	0.0723945
$863$	53.1660	1.80979	0.904896	−	0.425633i	$-0.139948\pi$
$863$	53.1660	1.80979	0.904896	+	0.425633i	$0.139948\pi$
$864$	0	0
$865$	−57.8745	−1.96779
$866$	29.2915	0.995366
$867$	0	0
$868$	18.5830	0.630748
$869$	−45.8745	−1.55619
$870$	0	0
$871$	−4.95948	−0.168046
$872$	−8.22876	−0.278661
$873$	0	0
$874$	−5.64575	−0.190970
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−2.70850	−0.0914595	−0.0457297	−	0.998954i	$-0.514561\pi$
$877$	−2.70850	−0.0914595	−0.0457297	+	0.998954i	$0.514561\pi$
$878$	35.7490	1.20647
$879$	0	0
$880$	−13.2915	−0.448056
$881$	−31.7490	−1.06965	−0.534826	−	0.844962i	$-0.679623\pi$
$881$	−31.7490	−1.06965	−0.534826	+	0.844962i	$0.679623\pi$
$882$	0	0
$883$	3.29150	0.110768	0.0553839	−	0.998465i	$-0.482362\pi$
$883$	3.29150	0.110768	0.0553839	+	0.998465i	$0.482362\pi$
$884$	−38.5830	−1.29769
$885$	0	0
$886$	17.1660	0.576703
$887$	−37.7490	−1.26749	−0.633744	−	0.773543i	$-0.718483\pi$
$887$	−37.7490	−1.26749	−0.633744	+	0.773543i	$0.718483\pi$
$888$	0	0
$889$	21.1660	0.709885
$890$	0	0
$891$	0	0
$892$	11.8745	0.397588
$893$	33.8745	1.13357
$894$	0	0
$895$	−17.1660	−0.573796
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−32.5830	−1.08731
$899$	−12.0000	−0.400222
$900$	0	0
$901$	26.5830	0.885608
$902$	−21.8745	−0.728341
$903$	0	0
$904$	14.5830	0.485024
$905$	−37.7490	−1.25482
$906$	0	0
$907$	−8.47974	−0.281565	−0.140783	−	0.990041i	$-0.544962\pi$
$907$	−8.47974	−0.281565	−0.140783	+	0.990041i	$0.544962\pi$
$908$	−22.9373	−0.761200
$909$	0	0
$910$	−38.5830	−1.27901
$911$	−7.29150	−0.241578	−0.120789	−	0.992678i	$-0.538542\pi$
$911$	−7.29150	−0.241578	−0.120789	+	0.992678i	$0.538542\pi$
$912$	0	0
$913$	−30.4575	−1.00800
$914$	31.8745	1.05432
$915$	0	0
$916$	24.9373	0.823950
$917$	19.7490	0.652170
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	3.64575	0.120197
$921$	0	0
$922$	37.7490	1.24320
$923$	31.7490	1.04503
$924$	0	0
$925$	−74.1033	−2.43650
$926$	21.1660	0.695558
$927$	0	0
$928$	1.29150	0.0423957
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−16.9373	−0.555096
$932$	0	0
$933$	0	0
$934$	−8.35425	−0.273359
$935$	−96.9150	−3.16946
$936$	0	0
$937$	−7.41699	−0.242303	−0.121151	−	0.992634i	$-0.538659\pi$
$937$	−7.41699	−0.242303	−0.121151	+	0.992634i	$0.538659\pi$
$938$	−1.87451	−0.0612049
$939$	0	0
$940$	−21.8745	−0.713468
$941$	20.8118	0.678444	0.339222	−	0.940706i	$-0.389836\pi$
$941$	20.8118	0.678444	0.339222	+	0.940706i	$0.389836\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	0	0
$945$	0	0
$946$	−20.5830	−0.669211
$947$	33.8745	1.10077	0.550387	−	0.834910i	$-0.314480\pi$
$947$	33.8745	1.10077	0.550387	+	0.834910i	$0.314480\pi$
$948$	0	0
$949$	−17.4170	−0.565380
$950$	−46.8118	−1.51877
$951$	0	0
$952$	−14.5830	−0.472638
$953$	51.0405	1.65336	0.826682	−	0.562669i	$-0.190225\pi$
$953$	51.0405	1.65336	0.826682	+	0.562669i	$0.190225\pi$
$954$	0	0
$955$	26.5830	0.860206
$956$	24.0000	0.776215
$957$	0	0
$958$	−21.4170	−0.691952
$959$	−19.7490	−0.637729
$960$	0	0
$961$	55.3320	1.78490
$962$	−47.2915	−1.52474
$963$	0	0
$964$	−5.29150	−0.170428
$965$	80.2065	2.58194
$966$	0	0
$967$	−7.87451	−0.253227	−0.126614	−	0.991952i	$-0.540411\pi$
$967$	−7.87451	−0.253227	−0.126614	+	0.991952i	$0.540411\pi$
$968$	−2.29150	−0.0736517
$969$	0	0
$970$	33.8745	1.08764
$971$	6.22876	0.199890	0.0999452	−	0.994993i	$-0.468133\pi$
$971$	6.22876	0.199890	0.0999452	+	0.994993i	$0.468133\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	21.1660	0.678203
$975$	0	0
$976$	5.64575	0.180716
$977$	9.87451	0.315913	0.157957	−	0.987446i	$-0.449509\pi$
$977$	9.87451	0.315913	0.157957	+	0.987446i	$0.449509\pi$
$978$	0	0
$979$	0	0
$980$	10.9373	0.349378
$981$	0	0
$982$	−9.87451	−0.315108
$983$	9.87451	0.314948	0.157474	−	0.987523i	$-0.449665\pi$
$983$	9.87451	0.314948	0.157474	+	0.987523i	$0.449665\pi$
$984$	0	0
$985$	21.8745	0.696980
$986$	9.41699	0.299898
$987$	0	0
$988$	−29.8745	−0.950435
$989$	5.64575	0.179524
$990$	0	0
$991$	38.4575	1.22164	0.610822	−	0.791768i	$-0.290839\pi$
$991$	38.4575	1.22164	0.610822	+	0.791768i	$0.290839\pi$
$992$	−9.29150	−0.295006
$993$	0	0
$994$	12.0000	0.380617
$995$	−51.0405	−1.61809
$996$	0	0
$997$	−19.4170	−0.614942	−0.307471	−	0.951557i	$-0.599483\pi$
$997$	−19.4170	−0.614942	−0.307471	+	0.951557i	$0.599483\pi$
$998$	43.0405	1.36242
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.2.a.e.1.1	✓	2
3.2	odd	2		414.2.a.g.1.2	yes	2
4.3	odd	2		3312.2.a.v.1.1		2
12.11	even	2		3312.2.a.z.1.2		2
23.22	odd	2		9522.2.a.bb.1.2		2
69.68	even	2		9522.2.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.a.e.1.1	✓	2	1.1	even	1	trivial
414.2.a.g.1.2	yes	2	3.2	odd	2
3312.2.a.v.1.1		2	4.3	odd	2
3312.2.a.z.1.2		2	12.11	even	2
9522.2.a.bb.1.2		2	23.22	odd	2
9522.2.a.bc.1.1		2	69.68	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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} +2.00000 q^{7} -1.00000 q^{8} +3.64575 q^{10} +3.64575 q^{11} -5.29150 q^{13} -2.00000 q^{14} +1.00000 q^{16} +7.29150 q^{17} +5.64575 q^{19} -3.64575 q^{20} -3.64575 q^{22} +1.00000 q^{23} +8.29150 q^{25} +5.29150 q^{26} +2.00000 q^{28} -1.29150 q^{29} +9.29150 q^{31} -1.00000 q^{32} -7.29150 q^{34} -7.29150 q^{35} -8.93725 q^{37} -5.64575 q^{38} +3.64575 q^{40} +6.00000 q^{41} +5.64575 q^{43} +3.64575 q^{44} -1.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -8.29150 q^{50} -5.29150 q^{52} +3.64575 q^{53} -13.2915 q^{55} -2.00000 q^{56} +1.29150 q^{58} +5.64575 q^{61} -9.29150 q^{62} +1.00000 q^{64} +19.2915 q^{65} +0.937254 q^{67} +7.29150 q^{68} +7.29150 q^{70} -6.00000 q^{71} +3.29150 q^{73} +8.93725 q^{74} +5.64575 q^{76} +7.29150 q^{77} -12.5830 q^{79} -3.64575 q^{80} -6.00000 q^{82} -8.35425 q^{83} -26.5830 q^{85} -5.64575 q^{86} -3.64575 q^{88} -10.5830 q^{91} +1.00000 q^{92} -6.00000 q^{94} -20.5830 q^{95} +9.29150 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 4 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 2 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^22 + 2 * q^23 + 6 * q^25 + 4 * q^28 + 8 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 4 * q^35 - 2 * q^37 - 6 * q^38 + 2 * q^40 + 12 * q^41 + 6 * q^43 + 2 * q^44 - 2 * q^46 + 12 * q^47 - 6 * q^49 - 6 * q^50 + 2 * q^53 - 16 * q^55 - 4 * q^56 - 8 * q^58 + 6 * q^61 - 8 * q^62 + 2 * q^64 + 28 * q^65 - 14 * q^67 + 4 * q^68 + 4 * q^70 - 12 * q^71 - 4 * q^73 + 2 * q^74 + 6 * q^76 + 4 * q^77 - 4 * q^79 - 2 * q^80 - 12 * q^82 - 22 * q^83 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 2 * q^92 - 12 * q^94 - 20 * q^95 + 8 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more