LMFDB - Embedded newform 306.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [306,2,Mod(1,306)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(306, 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("306.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 306 = 2 \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	306.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:	$2.44342230185$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
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			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	306.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.44949 q^{5} -0.449490 q^{7} +1.00000 q^{8} +2.44949 q^{10} -4.89898 q^{11} +6.89898 q^{13} -0.449490 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.89898 q^{19} +2.44949 q^{20} -4.89898 q^{22} -3.55051 q^{23} +1.00000 q^{25} +6.89898 q^{26} -0.449490 q^{28} -2.44949 q^{29} +9.34847 q^{31} +1.00000 q^{32} -1.00000 q^{34} -1.10102 q^{35} -1.55051 q^{37} -2.89898 q^{38} +2.44949 q^{40} -6.00000 q^{41} -4.00000 q^{43} -4.89898 q^{44} -3.55051 q^{46} +4.89898 q^{47} -6.79796 q^{49} +1.00000 q^{50} +6.89898 q^{52} +1.10102 q^{53} -12.0000 q^{55} -0.449490 q^{56} -2.44949 q^{58} -10.8990 q^{59} -6.44949 q^{61} +9.34847 q^{62} +1.00000 q^{64} +16.8990 q^{65} -2.89898 q^{67} -1.00000 q^{68} -1.10102 q^{70} -13.3485 q^{71} -10.0000 q^{73} -1.55051 q^{74} -2.89898 q^{76} +2.20204 q^{77} +9.34847 q^{79} +2.44949 q^{80} -6.00000 q^{82} +10.8990 q^{83} -2.44949 q^{85} -4.00000 q^{86} -4.89898 q^{88} +15.7980 q^{89} -3.10102 q^{91} -3.55051 q^{92} +4.89898 q^{94} -7.10102 q^{95} -5.10102 q^{97} -6.79796 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 2 q^{17} + 4 q^{19} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} - 2 q^{34} - 12 q^{35} - 8 q^{37} + 4 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 4 * q^13 + 4 * q^14 + 2 * q^16 - 2 * q^17 + 4 * q^19 - 12 * q^23 + 2 * q^25 + 4 * q^26 + 4 * q^28 + 4 * q^31 + 2 * q^32 - 2 * q^34 - 12 * q^35 - 8 * q^37 + 4 * q^38 - 12 * q^41 - 8 * q^43 - 12 * q^46 + 6 * q^49 + 2 * q^50 + 4 * q^52 + 12 * q^53 - 24 * q^55 + 4 * q^56 - 12 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 + 24 * q^65 + 4 * q^67 - 2 * q^68 - 12 * q^70 - 12 * q^71 - 20 * q^73 - 8 * q^74 + 4 * q^76 + 24 * q^77 + 4 * q^79 - 12 * q^82 + 12 * q^83 - 8 * q^86 + 12 * q^89 - 16 * q^91 - 12 * q^92 - 24 * q^95 - 20 * 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
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	−0.449490	−0.169891	−0.0849456	−	0.996386i	$-0.527072\pi$
$7$	−0.449490	−0.169891	−0.0849456	+	0.996386i	$0.527072\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.44949	0.774597
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	6.89898	1.91343	0.956716	−	0.291022i	$-0.0939953\pi$
$13$	6.89898	1.91343	0.956716	+	0.291022i	$0.0939953\pi$
$14$	−0.449490	−0.120131
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.89898	−0.665072	−0.332536	−	0.943091i	$-0.607904\pi$
$19$	−2.89898	−0.665072	−0.332536	+	0.943091i	$0.607904\pi$
$20$	2.44949	0.547723
$21$	0	0
$22$	−4.89898	−1.04447
$23$	−3.55051	−0.740333	−0.370166	−	0.928966i	$-0.620699\pi$
$23$	−3.55051	−0.740333	−0.370166	+	0.928966i	$0.620699\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.89898	1.35300
$27$	0	0
$28$	−0.449490	−0.0849456
$29$	−2.44949	−0.454859	−0.227429	−	0.973795i	$-0.573032\pi$
$29$	−2.44949	−0.454859	−0.227429	+	0.973795i	$0.573032\pi$
$30$	0	0
$31$	9.34847	1.67903	0.839517	−	0.543333i	$-0.182838\pi$
$31$	9.34847	1.67903	0.839517	+	0.543333i	$0.182838\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−1.10102	−0.186106
$36$	0	0
$37$	−1.55051	−0.254902	−0.127451	−	0.991845i	$-0.540680\pi$
$37$	−1.55051	−0.254902	−0.127451	+	0.991845i	$0.540680\pi$
$38$	−2.89898	−0.470277
$39$	0	0
$40$	2.44949	0.387298
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	−3.55051	−0.523494
$47$	4.89898	0.714590	0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898	0.714590	0.357295	+	0.933992i	$0.383699\pi$
$48$	0	0
$49$	−6.79796	−0.971137
$50$	1.00000	0.141421
$51$	0	0
$52$	6.89898	0.956716
$53$	1.10102	0.151237	0.0756184	−	0.997137i	$-0.475907\pi$
$53$	1.10102	0.151237	0.0756184	+	0.997137i	$0.475907\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	−0.449490	−0.0600656
$57$	0	0
$58$	−2.44949	−0.321634
$59$	−10.8990	−1.41893	−0.709463	−	0.704743i	$-0.751063\pi$
$59$	−10.8990	−1.41893	−0.709463	+	0.704743i	$0.751063\pi$
$60$	0	0
$61$	−6.44949	−0.825773	−0.412886	−	0.910783i	$-0.635479\pi$
$61$	−6.44949	−0.825773	−0.412886	+	0.910783i	$0.635479\pi$
$62$	9.34847	1.18726
$63$	0	0
$64$	1.00000	0.125000
$65$	16.8990	2.09606
$66$	0	0
$67$	−2.89898	−0.354167	−0.177083	−	0.984196i	$-0.556666\pi$
$67$	−2.89898	−0.354167	−0.177083	+	0.984196i	$0.556666\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	−1.10102	−0.131597
$71$	−13.3485	−1.58417	−0.792086	−	0.610410i	$-0.791005\pi$
$71$	−13.3485	−1.58417	−0.792086	+	0.610410i	$0.791005\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−1.55051	−0.180243
$75$	0	0
$76$	−2.89898	−0.332536
$77$	2.20204	0.250946
$78$	0	0
$79$	9.34847	1.05178	0.525892	−	0.850551i	$-0.323731\pi$
$79$	9.34847	1.05178	0.525892	+	0.850551i	$0.323731\pi$
$80$	2.44949	0.273861
$81$	0	0
$82$	−6.00000	−0.662589
$83$	10.8990	1.19632	0.598159	−	0.801377i	$-0.295899\pi$
$83$	10.8990	1.19632	0.598159	+	0.801377i	$0.295899\pi$
$84$	0	0
$85$	−2.44949	−0.265684
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−4.89898	−0.522233
$89$	15.7980	1.67458	0.837290	−	0.546759i	$-0.184139\pi$
$89$	15.7980	1.67458	0.837290	+	0.546759i	$0.184139\pi$
$90$	0	0
$91$	−3.10102	−0.325075
$92$	−3.55051	−0.370166
$93$	0	0
$94$	4.89898	0.505291
$95$	−7.10102	−0.728549
$96$	0	0
$97$	−5.10102	−0.517930	−0.258965	−	0.965887i	$-0.583381\pi$
$97$	−5.10102	−0.517930	−0.258965	+	0.965887i	$0.583381\pi$
$98$	−6.79796	−0.686698
$99$	0	0
$100$	1.00000	0.100000
$101$	10.8990	1.08449	0.542244	−	0.840221i	$-0.317575\pi$
$101$	10.8990	1.08449	0.542244	+	0.840221i	$0.317575\pi$
$102$	0	0
$103$	12.8990	1.27097	0.635487	−	0.772111i	$-0.280799\pi$
$103$	12.8990	1.27097	0.635487	+	0.772111i	$0.280799\pi$
$104$	6.89898	0.676501
$105$	0	0
$106$	1.10102	0.106941
$107$	7.10102	0.686482	0.343241	−	0.939247i	$-0.388475\pi$
$107$	7.10102	0.686482	0.343241	+	0.939247i	$0.388475\pi$
$108$	0	0
$109$	15.3485	1.47012	0.735058	−	0.678004i	$-0.237155\pi$
$109$	15.3485	1.47012	0.735058	+	0.678004i	$0.237155\pi$
$110$	−12.0000	−1.14416
$111$	0	0
$112$	−0.449490	−0.0424728
$113$	10.8990	1.02529	0.512645	−	0.858601i	$-0.328666\pi$
$113$	10.8990	1.02529	0.512645	+	0.858601i	$0.328666\pi$
$114$	0	0
$115$	−8.69694	−0.810994
$116$	−2.44949	−0.227429
$117$	0	0
$118$	−10.8990	−1.00333
$119$	0.449490	0.0412047
$120$	0	0
$121$	13.0000	1.18182
$122$	−6.44949	−0.583909
$123$	0	0
$124$	9.34847	0.839517
$125$	−9.79796	−0.876356
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	16.8990	1.48214
$131$	−7.10102	−0.620419	−0.310210	−	0.950668i	$-0.600399\pi$
$131$	−7.10102	−0.620419	−0.310210	+	0.950668i	$0.600399\pi$
$132$	0	0
$133$	1.30306	0.112990
$134$	−2.89898	−0.250434
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−19.5959	−1.67419	−0.837096	−	0.547056i	$-0.815749\pi$
$137$	−19.5959	−1.67419	−0.837096	+	0.547056i	$0.815749\pi$
$138$	0	0
$139$	12.8990	1.09408	0.547039	−	0.837107i	$-0.315755\pi$
$139$	12.8990	1.09408	0.547039	+	0.837107i	$0.315755\pi$
$140$	−1.10102	−0.0930532
$141$	0	0
$142$	−13.3485	−1.12018
$143$	−33.7980	−2.82633
$144$	0	0
$145$	−6.00000	−0.498273
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−1.55051	−0.127451
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	17.7980	1.44838	0.724189	−	0.689602i	$-0.242214\pi$
$151$	17.7980	1.44838	0.724189	+	0.689602i	$0.242214\pi$
$152$	−2.89898	−0.235138
$153$	0	0
$154$	2.20204	0.177446
$155$	22.8990	1.83929
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	9.34847	0.743724
$159$	0	0
$160$	2.44949	0.193649
$161$	1.59592	0.125776
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	10.8990	0.845925
$167$	3.55051	0.274747	0.137373	−	0.990519i	$-0.456134\pi$
$167$	3.55051	0.274747	0.137373	+	0.990519i	$0.456134\pi$
$168$	0	0
$169$	34.5959	2.66122
$170$	−2.44949	−0.187867
$171$	0	0
$172$	−4.00000	−0.304997
$173$	17.1464	1.30362	0.651809	−	0.758383i	$-0.274010\pi$
$173$	17.1464	1.30362	0.651809	+	0.758383i	$0.274010\pi$
$174$	0	0
$175$	−0.449490	−0.0339782
$176$	−4.89898	−0.369274
$177$	0	0
$178$	15.7980	1.18411
$179$	21.7980	1.62926	0.814628	−	0.579984i	$-0.196941\pi$
$179$	21.7980	1.62926	0.814628	+	0.579984i	$0.196941\pi$
$180$	0	0
$181$	−16.2474	−1.20766	−0.603832	−	0.797112i	$-0.706360\pi$
$181$	−16.2474	−1.20766	−0.603832	+	0.797112i	$0.706360\pi$
$182$	−3.10102	−0.229863
$183$	0	0
$184$	−3.55051	−0.261747
$185$	−3.79796	−0.279231
$186$	0	0
$187$	4.89898	0.358249
$188$	4.89898	0.357295
$189$	0	0
$190$	−7.10102	−0.515162
$191$	−14.6969	−1.06343	−0.531717	−	0.846922i	$-0.678453\pi$
$191$	−14.6969	−1.06343	−0.531717	+	0.846922i	$0.678453\pi$
$192$	0	0
$193$	9.10102	0.655106	0.327553	−	0.944833i	$-0.393776\pi$
$193$	9.10102	0.655106	0.327553	+	0.944833i	$0.393776\pi$
$194$	−5.10102	−0.366232
$195$	0	0
$196$	−6.79796	−0.485568
$197$	0.247449	0.0176300	0.00881500	−	0.999961i	$-0.497194\pi$
$197$	0.247449	0.0176300	0.00881500	+	0.999961i	$0.497194\pi$
$198$	0	0
$199$	−20.0454	−1.42098	−0.710491	−	0.703707i	$-0.751527\pi$
$199$	−20.0454	−1.42098	−0.710491	+	0.703707i	$0.751527\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	10.8990	0.766850
$203$	1.10102	0.0772765
$204$	0	0
$205$	−14.6969	−1.02648
$206$	12.8990	0.898714
$207$	0	0
$208$	6.89898	0.478358
$209$	14.2020	0.982376
$210$	0	0
$211$	3.10102	0.213483	0.106742	−	0.994287i	$-0.465958\pi$
$211$	3.10102	0.213483	0.106742	+	0.994287i	$0.465958\pi$
$212$	1.10102	0.0756184
$213$	0	0
$214$	7.10102	0.485416
$215$	−9.79796	−0.668215
$216$	0	0
$217$	−4.20204	−0.285253
$218$	15.3485	1.03953
$219$	0	0
$220$	−12.0000	−0.809040
$221$	−6.89898	−0.464076
$222$	0	0
$223$	10.6969	0.716320	0.358160	−	0.933660i	$-0.383404\pi$
$223$	10.6969	0.716320	0.358160	+	0.933660i	$0.383404\pi$
$224$	−0.449490	−0.0300328
$225$	0	0
$226$	10.8990	0.724989
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−8.69694	−0.573459
$231$	0	0
$232$	−2.44949	−0.160817
$233$	−20.6969	−1.35590	−0.677951	−	0.735107i	$-0.737132\pi$
$233$	−20.6969	−1.35590	−0.677951	+	0.735107i	$0.737132\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	−10.8990	−0.709463
$237$	0	0
$238$	0.449490	0.0291361
$239$	2.69694	0.174450	0.0872252	−	0.996189i	$-0.472200\pi$
$239$	2.69694	0.174450	0.0872252	+	0.996189i	$0.472200\pi$
$240$	0	0
$241$	−12.6969	−0.817882	−0.408941	−	0.912561i	$-0.634102\pi$
$241$	−12.6969	−0.817882	−0.408941	+	0.912561i	$0.634102\pi$
$242$	13.0000	0.835672
$243$	0	0
$244$	−6.44949	−0.412886
$245$	−16.6515	−1.06383
$246$	0	0
$247$	−20.0000	−1.27257
$248$	9.34847	0.593628
$249$	0	0
$250$	−9.79796	−0.619677
$251$	21.7980	1.37587	0.687937	−	0.725770i	$-0.258516\pi$
$251$	21.7980	1.37587	0.687937	+	0.725770i	$0.258516\pi$
$252$	0	0
$253$	17.3939	1.09354
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	0.696938	0.0433056
$260$	16.8990	1.04803
$261$	0	0
$262$	−7.10102	−0.438703
$263$	−28.8990	−1.78199	−0.890994	−	0.454016i	$-0.849991\pi$
$263$	−28.8990	−1.78199	−0.890994	+	0.454016i	$0.849991\pi$
$264$	0	0
$265$	2.69694	0.165672
$266$	1.30306	0.0798958
$267$	0	0
$268$	−2.89898	−0.177083
$269$	2.44949	0.149348	0.0746740	−	0.997208i	$-0.476208\pi$
$269$	2.44949	0.149348	0.0746740	+	0.997208i	$0.476208\pi$
$270$	0	0
$271$	−1.79796	−0.109218	−0.0546091	−	0.998508i	$-0.517391\pi$
$271$	−1.79796	−0.109218	−0.0546091	+	0.998508i	$0.517391\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−19.5959	−1.18383
$275$	−4.89898	−0.295420
$276$	0	0
$277$	−8.65153	−0.519820	−0.259910	−	0.965633i	$-0.583693\pi$
$277$	−8.65153	−0.519820	−0.259910	+	0.965633i	$0.583693\pi$
$278$	12.8990	0.773629
$279$	0	0
$280$	−1.10102	−0.0657986
$281$	3.79796	0.226567	0.113284	−	0.993563i	$-0.463863\pi$
$281$	3.79796	0.226567	0.113284	+	0.993563i	$0.463863\pi$
$282$	0	0
$283$	−18.6969	−1.11142	−0.555709	−	0.831377i	$-0.687553\pi$
$283$	−18.6969	−1.11142	−0.555709	+	0.831377i	$0.687553\pi$
$284$	−13.3485	−0.792086
$285$	0	0
$286$	−33.7980	−1.99852
$287$	2.69694	0.159195
$288$	0	0
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	0	0
$292$	−10.0000	−0.585206
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−26.6969	−1.55436
$296$	−1.55051	−0.0901216
$297$	0	0
$298$	18.0000	1.04271
$299$	−24.4949	−1.41658
$300$	0	0
$301$	1.79796	0.103633
$302$	17.7980	1.02416
$303$	0	0
$304$	−2.89898	−0.166268
$305$	−15.7980	−0.904588
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	2.20204	0.125473
$309$	0	0
$310$	22.8990	1.30057
$311$	6.24745	0.354260	0.177130	−	0.984187i	$-0.443319\pi$
$311$	6.24745	0.354260	0.177130	+	0.984187i	$0.443319\pi$
$312$	0	0
$313$	−2.89898	−0.163860	−0.0819300	−	0.996638i	$-0.526108\pi$
$313$	−2.89898	−0.163860	−0.0819300	+	0.996638i	$0.526108\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	9.34847	0.525892
$317$	29.1464	1.63703	0.818513	−	0.574488i	$-0.194799\pi$
$317$	29.1464	1.63703	0.818513	+	0.574488i	$0.194799\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	2.44949	0.136931
$321$	0	0
$322$	1.59592	0.0889370
$323$	2.89898	0.161304
$324$	0	0
$325$	6.89898	0.382687
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−2.20204	−0.121402
$330$	0	0
$331$	5.79796	0.318685	0.159342	−	0.987223i	$-0.449063\pi$
$331$	5.79796	0.318685	0.159342	+	0.987223i	$0.449063\pi$
$332$	10.8990	0.598159
$333$	0	0
$334$	3.55051	0.194275
$335$	−7.10102	−0.387970
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	34.5959	1.88177
$339$	0	0
$340$	−2.44949	−0.132842
$341$	−45.7980	−2.48010
$342$	0	0
$343$	6.20204	0.334879
$344$	−4.00000	−0.215666
$345$	0	0
$346$	17.1464	0.921798
$347$	−2.20204	−0.118212	−0.0591059	−	0.998252i	$-0.518825\pi$
$347$	−2.20204	−0.118212	−0.0591059	+	0.998252i	$0.518825\pi$
$348$	0	0
$349$	−0.696938	−0.0373063	−0.0186531	−	0.999826i	$-0.505938\pi$
$349$	−0.696938	−0.0373063	−0.0186531	+	0.999826i	$0.505938\pi$
$350$	−0.449490	−0.0240262
$351$	0	0
$352$	−4.89898	−0.261116
$353$	−25.5959	−1.36233	−0.681167	−	0.732128i	$-0.738527\pi$
$353$	−25.5959	−1.36233	−0.681167	+	0.732128i	$0.738527\pi$
$354$	0	0
$355$	−32.6969	−1.73537
$356$	15.7980	0.837290
$357$	0	0
$358$	21.7980	1.15206
$359$	2.20204	0.116219	0.0581096	−	0.998310i	$-0.481493\pi$
$359$	2.20204	0.116219	0.0581096	+	0.998310i	$0.481493\pi$
$360$	0	0
$361$	−10.5959	−0.557680
$362$	−16.2474	−0.853947
$363$	0	0
$364$	−3.10102	−0.162538
$365$	−24.4949	−1.28212
$366$	0	0
$367$	−34.2474	−1.78770	−0.893851	−	0.448364i	$-0.852007\pi$
$367$	−34.2474	−1.78770	−0.893851	+	0.448364i	$0.852007\pi$
$368$	−3.55051	−0.185083
$369$	0	0
$370$	−3.79796	−0.197446
$371$	−0.494897	−0.0256938
$372$	0	0
$373$	−0.696938	−0.0360861	−0.0180431	−	0.999837i	$-0.505744\pi$
$373$	−0.696938	−0.0360861	−0.0180431	+	0.999837i	$0.505744\pi$
$374$	4.89898	0.253320
$375$	0	0
$376$	4.89898	0.252646
$377$	−16.8990	−0.870342
$378$	0	0
$379$	29.7980	1.53062	0.765309	−	0.643663i	$-0.222586\pi$
$379$	29.7980	1.53062	0.765309	+	0.643663i	$0.222586\pi$
$380$	−7.10102	−0.364275
$381$	0	0
$382$	−14.6969	−0.751961
$383$	4.89898	0.250326	0.125163	−	0.992136i	$-0.460055\pi$
$383$	4.89898	0.250326	0.125163	+	0.992136i	$0.460055\pi$
$384$	0	0
$385$	5.39388	0.274897
$386$	9.10102	0.463230
$387$	0	0
$388$	−5.10102	−0.258965
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	3.55051	0.179557
$392$	−6.79796	−0.343349
$393$	0	0
$394$	0.247449	0.0124663
$395$	22.8990	1.15217
$396$	0	0
$397$	27.3485	1.37258	0.686290	−	0.727328i	$-0.259238\pi$
$397$	27.3485	1.37258	0.686290	+	0.727328i	$0.259238\pi$
$398$	−20.0454	−1.00479
$399$	0	0
$400$	1.00000	0.0500000
$401$	−15.7980	−0.788912	−0.394456	−	0.918915i	$-0.629067\pi$
$401$	−15.7980	−0.788912	−0.394456	+	0.918915i	$0.629067\pi$
$402$	0	0
$403$	64.4949	3.21272
$404$	10.8990	0.542244
$405$	0	0
$406$	1.10102	0.0546427
$407$	7.59592	0.376516
$408$	0	0
$409$	−6.20204	−0.306671	−0.153336	−	0.988174i	$-0.549002\pi$
$409$	−6.20204	−0.306671	−0.153336	+	0.988174i	$0.549002\pi$
$410$	−14.6969	−0.725830
$411$	0	0
$412$	12.8990	0.635487
$413$	4.89898	0.241063
$414$	0	0
$415$	26.6969	1.31050
$416$	6.89898	0.338250
$417$	0	0
$418$	14.2020	0.694645
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−26.8990	−1.31098	−0.655488	−	0.755206i	$-0.727537\pi$
$421$	−26.8990	−1.31098	−0.655488	+	0.755206i	$0.727537\pi$
$422$	3.10102	0.150955
$423$	0	0
$424$	1.10102	0.0534703
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	2.89898	0.140291
$428$	7.10102	0.343241
$429$	0	0
$430$	−9.79796	−0.472500
$431$	32.9444	1.58688	0.793438	−	0.608652i	$-0.208289\pi$
$431$	32.9444	1.58688	0.793438	+	0.608652i	$0.208289\pi$
$432$	0	0
$433$	−21.3939	−1.02812	−0.514062	−	0.857753i	$-0.671860\pi$
$433$	−21.3939	−1.02812	−0.514062	+	0.857753i	$0.671860\pi$
$434$	−4.20204	−0.201704
$435$	0	0
$436$	15.3485	0.735058
$437$	10.2929	0.492374
$438$	0	0
$439$	33.3485	1.59164	0.795818	−	0.605536i	$-0.207041\pi$
$439$	33.3485	1.59164	0.795818	+	0.605536i	$0.207041\pi$
$440$	−12.0000	−0.572078
$441$	0	0
$442$	−6.89898	−0.328151
$443$	41.3939	1.96668	0.983341	−	0.181769i	$-0.0581824\pi$
$443$	41.3939	1.96668	0.983341	+	0.181769i	$0.0581824\pi$
$444$	0	0
$445$	38.6969	1.83441
$446$	10.6969	0.506515
$447$	0	0
$448$	−0.449490	−0.0212364
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	29.3939	1.38410
$452$	10.8990	0.512645
$453$	0	0
$454$	0	0
$455$	−7.59592	−0.356102
$456$	0	0
$457$	−21.3939	−1.00076	−0.500382	−	0.865805i	$-0.666807\pi$
$457$	−21.3939	−1.00076	−0.500382	+	0.865805i	$0.666807\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−8.69694	−0.405497
$461$	20.6969	0.963953	0.481976	−	0.876184i	$-0.339919\pi$
$461$	20.6969	0.963953	0.481976	+	0.876184i	$0.339919\pi$
$462$	0	0
$463$	−40.4949	−1.88196	−0.940979	−	0.338466i	$-0.890092\pi$
$463$	−40.4949	−1.88196	−0.940979	+	0.338466i	$0.890092\pi$
$464$	−2.44949	−0.113715
$465$	0	0
$466$	−20.6969	−0.958767
$467$	−7.59592	−0.351497	−0.175749	−	0.984435i	$-0.556235\pi$
$467$	−7.59592	−0.351497	−0.175749	+	0.984435i	$0.556235\pi$
$468$	0	0
$469$	1.30306	0.0601698
$470$	12.0000	0.553519
$471$	0	0
$472$	−10.8990	−0.501666
$473$	19.5959	0.901021
$474$	0	0
$475$	−2.89898	−0.133014
$476$	0.449490	0.0206023
$477$	0	0
$478$	2.69694	0.123355
$479$	−6.24745	−0.285453	−0.142727	−	0.989762i	$-0.545587\pi$
$479$	−6.24745	−0.285453	−0.142727	+	0.989762i	$0.545587\pi$
$480$	0	0
$481$	−10.6969	−0.487738
$482$	−12.6969	−0.578330
$483$	0	0
$484$	13.0000	0.590909
$485$	−12.4949	−0.567364
$486$	0	0
$487$	−0.449490	−0.0203683	−0.0101842	−	0.999948i	$-0.503242\pi$
$487$	−0.449490	−0.0203683	−0.0101842	+	0.999948i	$0.503242\pi$
$488$	−6.44949	−0.291955
$489$	0	0
$490$	−16.6515	−0.752239
$491$	−17.3939	−0.784975	−0.392487	−	0.919757i	$-0.628385\pi$
$491$	−17.3939	−0.784975	−0.392487	+	0.919757i	$0.628385\pi$
$492$	0	0
$493$	2.44949	0.110319
$494$	−20.0000	−0.899843
$495$	0	0
$496$	9.34847	0.419759
$497$	6.00000	0.269137
$498$	0	0
$499$	34.6969	1.55325	0.776624	−	0.629964i	$-0.216930\pi$
$499$	34.6969	1.55325	0.776624	+	0.629964i	$0.216930\pi$
$500$	−9.79796	−0.438178
$501$	0	0
$502$	21.7980	0.972891
$503$	−6.24745	−0.278560	−0.139280	−	0.990253i	$-0.544479\pi$
$503$	−6.24745	−0.278560	−0.139280	+	0.990253i	$0.544479\pi$
$504$	0	0
$505$	26.6969	1.18800
$506$	17.3939	0.773252
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−3.79796	−0.168342	−0.0841708	−	0.996451i	$-0.526824\pi$
$509$	−3.79796	−0.168342	−0.0841708	+	0.996451i	$0.526824\pi$
$510$	0	0
$511$	4.49490	0.198843
$512$	1.00000	0.0441942
$513$	0	0
$514$	24.0000	1.05859
$515$	31.5959	1.39228
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0.696938	0.0306217
$519$	0	0
$520$	16.8990	0.741069
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	16.6969	0.730106	0.365053	−	0.930987i	$-0.381051\pi$
$523$	16.6969	0.730106	0.365053	+	0.930987i	$0.381051\pi$
$524$	−7.10102	−0.310210
$525$	0	0
$526$	−28.8990	−1.26006
$527$	−9.34847	−0.407226
$528$	0	0
$529$	−10.3939	−0.451908
$530$	2.69694	0.117148
$531$	0	0
$532$	1.30306	0.0564949
$533$	−41.3939	−1.79297
$534$	0	0
$535$	17.3939	0.752003
$536$	−2.89898	−0.125217
$537$	0	0
$538$	2.44949	0.105605
$539$	33.3031	1.43446
$540$	0	0
$541$	27.3485	1.17580	0.587901	−	0.808933i	$-0.299954\pi$
$541$	27.3485	1.17580	0.587901	+	0.808933i	$0.299954\pi$
$542$	−1.79796	−0.0772290
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	37.5959	1.61043
$546$	0	0
$547$	−20.8990	−0.893576	−0.446788	−	0.894640i	$-0.647432\pi$
$547$	−20.8990	−0.893576	−0.446788	+	0.894640i	$0.647432\pi$
$548$	−19.5959	−0.837096
$549$	0	0
$550$	−4.89898	−0.208893
$551$	7.10102	0.302514
$552$	0	0
$553$	−4.20204	−0.178689
$554$	−8.65153	−0.367568
$555$	0	0
$556$	12.8990	0.547039
$557$	−13.1010	−0.555108	−0.277554	−	0.960710i	$-0.589524\pi$
$557$	−13.1010	−0.555108	−0.277554	+	0.960710i	$0.589524\pi$
$558$	0	0
$559$	−27.5959	−1.16718
$560$	−1.10102	−0.0465266
$561$	0	0
$562$	3.79796	0.160207
$563$	−1.10102	−0.0464025	−0.0232012	−	0.999731i	$-0.507386\pi$
$563$	−1.10102	−0.0464025	−0.0232012	+	0.999731i	$0.507386\pi$
$564$	0	0
$565$	26.6969	1.12315
$566$	−18.6969	−0.785891
$567$	0	0
$568$	−13.3485	−0.560089
$569$	39.1918	1.64301	0.821504	−	0.570203i	$-0.193136\pi$
$569$	39.1918	1.64301	0.821504	+	0.570203i	$0.193136\pi$
$570$	0	0
$571$	−6.20204	−0.259547	−0.129774	−	0.991544i	$-0.541425\pi$
$571$	−6.20204	−0.259547	−0.129774	+	0.991544i	$0.541425\pi$
$572$	−33.7980	−1.41316
$573$	0	0
$574$	2.69694	0.112568
$575$	−3.55051	−0.148067
$576$	0	0
$577$	−5.59592	−0.232961	−0.116481	−	0.993193i	$-0.537161\pi$
$577$	−5.59592	−0.232961	−0.116481	+	0.993193i	$0.537161\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−4.89898	−0.203244
$582$	0	0
$583$	−5.39388	−0.223392
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−27.1010	−1.11668
$590$	−26.6969	−1.09910
$591$	0	0
$592$	−1.55051	−0.0637256
$593$	−21.7980	−0.895135	−0.447567	−	0.894250i	$-0.647710\pi$
$593$	−21.7980	−0.895135	−0.447567	+	0.894250i	$0.647710\pi$
$594$	0	0
$595$	1.10102	0.0451374
$596$	18.0000	0.737309
$597$	0	0
$598$	−24.4949	−1.00167
$599$	−33.7980	−1.38095	−0.690474	−	0.723358i	$-0.742598\pi$
$599$	−33.7980	−1.38095	−0.690474	+	0.723358i	$0.742598\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	1.79796	0.0732793
$603$	0	0
$604$	17.7980	0.724189
$605$	31.8434	1.29462
$606$	0	0
$607$	−3.14643	−0.127710	−0.0638548	−	0.997959i	$-0.520339\pi$
$607$	−3.14643	−0.127710	−0.0638548	+	0.997959i	$0.520339\pi$
$608$	−2.89898	−0.117569
$609$	0	0
$610$	−15.7980	−0.639641
$611$	33.7980	1.36732
$612$	0	0
$613$	11.7980	0.476515	0.238258	−	0.971202i	$-0.423424\pi$
$613$	11.7980	0.476515	0.238258	+	0.971202i	$0.423424\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	2.20204	0.0887228
$617$	−32.6969	−1.31633	−0.658165	−	0.752874i	$-0.728667\pi$
$617$	−32.6969	−1.31633	−0.658165	+	0.752874i	$0.728667\pi$
$618$	0	0
$619$	−33.3939	−1.34221	−0.671107	−	0.741361i	$-0.734181\pi$
$619$	−33.3939	−1.34221	−0.671107	+	0.741361i	$0.734181\pi$
$620$	22.8990	0.919645
$621$	0	0
$622$	6.24745	0.250500
$623$	−7.10102	−0.284496
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−2.89898	−0.115867
$627$	0	0
$628$	−10.0000	−0.399043
$629$	1.55051	0.0618229
$630$	0	0
$631$	−28.4949	−1.13436	−0.567182	−	0.823592i	$-0.691966\pi$
$631$	−28.4949	−1.13436	−0.567182	+	0.823592i	$0.691966\pi$
$632$	9.34847	0.371862
$633$	0	0
$634$	29.1464	1.15755
$635$	−9.79796	−0.388820
$636$	0	0
$637$	−46.8990	−1.85821
$638$	12.0000	0.475085
$639$	0	0
$640$	2.44949	0.0968246
$641$	−3.79796	−0.150010	−0.0750052	−	0.997183i	$-0.523897\pi$
$641$	−3.79796	−0.150010	−0.0750052	+	0.997183i	$0.523897\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	1.59592	0.0628880
$645$	0	0
$646$	2.89898	0.114059
$647$	−24.4949	−0.962994	−0.481497	−	0.876448i	$-0.659907\pi$
$647$	−24.4949	−0.962994	−0.481497	+	0.876448i	$0.659907\pi$
$648$	0	0
$649$	53.3939	2.09589
$650$	6.89898	0.270600
$651$	0	0
$652$	−4.00000	−0.156652
$653$	31.3485	1.22676	0.613380	−	0.789788i	$-0.289809\pi$
$653$	31.3485	1.22676	0.613380	+	0.789788i	$0.289809\pi$
$654$	0	0
$655$	−17.3939	−0.679635
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−2.20204	−0.0858445
$659$	1.10102	0.0428897	0.0214448	−	0.999770i	$-0.493173\pi$
$659$	1.10102	0.0428897	0.0214448	+	0.999770i	$0.493173\pi$
$660$	0	0
$661$	2.49490	0.0970403	0.0485201	−	0.998822i	$-0.484549\pi$
$661$	2.49490	0.0970403	0.0485201	+	0.998822i	$0.484549\pi$
$662$	5.79796	0.225344
$663$	0	0
$664$	10.8990	0.422962
$665$	3.19184	0.123774
$666$	0	0
$667$	8.69694	0.336747
$668$	3.55051	0.137373
$669$	0	0
$670$	−7.10102	−0.274336
$671$	31.5959	1.21975
$672$	0	0
$673$	26.4949	1.02130	0.510652	−	0.859788i	$-0.329404\pi$
$673$	26.4949	1.02130	0.510652	+	0.859788i	$0.329404\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	34.5959	1.33061
$677$	−38.9444	−1.49675	−0.748377	−	0.663273i	$-0.769167\pi$
$677$	−38.9444	−1.49675	−0.748377	+	0.663273i	$0.769167\pi$
$678$	0	0
$679$	2.29286	0.0879918
$680$	−2.44949	−0.0939336
$681$	0	0
$682$	−45.7980	−1.75369
$683$	51.1918	1.95880	0.979401	−	0.201927i	$-0.0647204\pi$
$683$	51.1918	1.95880	0.979401	+	0.201927i	$0.0647204\pi$
$684$	0	0
$685$	−48.0000	−1.83399
$686$	6.20204	0.236795
$687$	0	0
$688$	−4.00000	−0.152499
$689$	7.59592	0.289381
$690$	0	0
$691$	−30.6969	−1.16777	−0.583883	−	0.811838i	$-0.698467\pi$
$691$	−30.6969	−1.16777	−0.583883	+	0.811838i	$0.698467\pi$
$692$	17.1464	0.651809
$693$	0	0
$694$	−2.20204	−0.0835883
$695$	31.5959	1.19850
$696$	0	0
$697$	6.00000	0.227266
$698$	−0.696938	−0.0263795
$699$	0	0
$700$	−0.449490	−0.0169891
$701$	−8.69694	−0.328479	−0.164239	−	0.986421i	$-0.552517\pi$
$701$	−8.69694	−0.328479	−0.164239	+	0.986421i	$0.552517\pi$
$702$	0	0
$703$	4.49490	0.169528
$704$	−4.89898	−0.184637
$705$	0	0
$706$	−25.5959	−0.963315
$707$	−4.89898	−0.184245
$708$	0	0
$709$	−18.9444	−0.711471	−0.355736	−	0.934587i	$-0.615770\pi$
$709$	−18.9444	−0.711471	−0.355736	+	0.934587i	$0.615770\pi$
$710$	−32.6969	−1.22709
$711$	0	0
$712$	15.7980	0.592054
$713$	−33.1918	−1.24304
$714$	0	0
$715$	−82.7878	−3.09609
$716$	21.7980	0.814628
$717$	0	0
$718$	2.20204	0.0821794
$719$	20.4495	0.762637	0.381319	−	0.924444i	$-0.375470\pi$
$719$	20.4495	0.762637	0.381319	+	0.924444i	$0.375470\pi$
$720$	0	0
$721$	−5.79796	−0.215927
$722$	−10.5959	−0.394339
$723$	0	0
$724$	−16.2474	−0.603832
$725$	−2.44949	−0.0909718
$726$	0	0
$727$	25.3939	0.941807	0.470903	−	0.882185i	$-0.343928\pi$
$727$	25.3939	0.941807	0.470903	+	0.882185i	$0.343928\pi$
$728$	−3.10102	−0.114931
$729$	0	0
$730$	−24.4949	−0.906597
$731$	4.00000	0.147945
$732$	0	0
$733$	40.6969	1.50318	0.751588	−	0.659633i	$-0.229288\pi$
$733$	40.6969	1.50318	0.751588	+	0.659633i	$0.229288\pi$
$734$	−34.2474	−1.26410
$735$	0	0
$736$	−3.55051	−0.130874
$737$	14.2020	0.523139
$738$	0	0
$739$	23.3031	0.857217	0.428608	−	0.903490i	$-0.359004\pi$
$739$	23.3031	0.857217	0.428608	+	0.903490i	$0.359004\pi$
$740$	−3.79796	−0.139616
$741$	0	0
$742$	−0.494897	−0.0181683
$743$	−30.2474	−1.10967	−0.554836	−	0.831960i	$-0.687219\pi$
$743$	−30.2474	−1.10967	−0.554836	+	0.831960i	$0.687219\pi$
$744$	0	0
$745$	44.0908	1.61536
$746$	−0.696938	−0.0255167
$747$	0	0
$748$	4.89898	0.179124
$749$	−3.19184	−0.116627
$750$	0	0
$751$	−10.2474	−0.373935	−0.186967	−	0.982366i	$-0.559866\pi$
$751$	−10.2474	−0.373935	−0.186967	+	0.982366i	$0.559866\pi$
$752$	4.89898	0.178647
$753$	0	0
$754$	−16.8990	−0.615425
$755$	43.5959	1.58662
$756$	0	0
$757$	−7.79796	−0.283422	−0.141711	−	0.989908i	$-0.545260\pi$
$757$	−7.79796	−0.283422	−0.141711	+	0.989908i	$0.545260\pi$
$758$	29.7980	1.08231
$759$	0	0
$760$	−7.10102	−0.257581
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−6.89898	−0.249760
$764$	−14.6969	−0.531717
$765$	0	0
$766$	4.89898	0.177007
$767$	−75.1918	−2.71502
$768$	0	0
$769$	19.3939	0.699361	0.349681	−	0.936869i	$-0.386290\pi$
$769$	19.3939	0.699361	0.349681	+	0.936869i	$0.386290\pi$
$770$	5.39388	0.194382
$771$	0	0
$772$	9.10102	0.327553
$773$	−25.5959	−0.920621	−0.460311	−	0.887758i	$-0.652262\pi$
$773$	−25.5959	−0.920621	−0.460311	+	0.887758i	$0.652262\pi$
$774$	0	0
$775$	9.34847	0.335807
$776$	−5.10102	−0.183116
$777$	0	0
$778$	6.00000	0.215110
$779$	17.3939	0.623200
$780$	0	0
$781$	65.3939	2.33998
$782$	3.55051	0.126966
$783$	0	0
$784$	−6.79796	−0.242784
$785$	−24.4949	−0.874260
$786$	0	0
$787$	32.4949	1.15832	0.579159	−	0.815215i	$-0.303381\pi$
$787$	32.4949	1.15832	0.579159	+	0.815215i	$0.303381\pi$
$788$	0.247449	0.00881500
$789$	0	0
$790$	22.8990	0.814709
$791$	−4.89898	−0.174188
$792$	0	0
$793$	−44.4949	−1.58006
$794$	27.3485	0.970561
$795$	0	0
$796$	−20.0454	−0.710491
$797$	20.6969	0.733123	0.366562	−	0.930394i	$-0.380535\pi$
$797$	20.6969	0.733123	0.366562	+	0.930394i	$0.380535\pi$
$798$	0	0
$799$	−4.89898	−0.173313
$800$	1.00000	0.0353553
$801$	0	0
$802$	−15.7980	−0.557845
$803$	48.9898	1.72881
$804$	0	0
$805$	3.90918	0.137781
$806$	64.4949	2.27174
$807$	0	0
$808$	10.8990	0.383425
$809$	15.7980	0.555427	0.277713	−	0.960664i	$-0.410423\pi$
$809$	15.7980	0.555427	0.277713	+	0.960664i	$0.410423\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	1.10102	0.0386382
$813$	0	0
$814$	7.59592	0.266237
$815$	−9.79796	−0.343208
$816$	0	0
$817$	11.5959	0.405690
$818$	−6.20204	−0.216849
$819$	0	0
$820$	−14.6969	−0.513239
$821$	−53.1464	−1.85482	−0.927412	−	0.374042i	$-0.877971\pi$
$821$	−53.1464	−1.85482	−0.927412	+	0.374042i	$0.877971\pi$
$822$	0	0
$823$	28.9444	1.00894	0.504469	−	0.863430i	$-0.331688\pi$
$823$	28.9444	1.00894	0.504469	+	0.863430i	$0.331688\pi$
$824$	12.8990	0.449357
$825$	0	0
$826$	4.89898	0.170457
$827$	−19.1010	−0.664208	−0.332104	−	0.943243i	$-0.607758\pi$
$827$	−19.1010	−0.664208	−0.332104	+	0.943243i	$0.607758\pi$
$828$	0	0
$829$	−49.1918	−1.70850	−0.854252	−	0.519860i	$-0.825984\pi$
$829$	−49.1918	−1.70850	−0.854252	+	0.519860i	$0.825984\pi$
$830$	26.6969	0.926664
$831$	0	0
$832$	6.89898	0.239179
$833$	6.79796	0.235535
$834$	0	0
$835$	8.69694	0.300970
$836$	14.2020	0.491188
$837$	0	0
$838$	24.0000	0.829066
$839$	8.94439	0.308795	0.154397	−	0.988009i	$-0.450656\pi$
$839$	8.94439	0.308795	0.154397	+	0.988009i	$0.450656\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	−26.8990	−0.927000
$843$	0	0
$844$	3.10102	0.106742
$845$	84.7423	2.91523
$846$	0	0
$847$	−5.84337	−0.200780
$848$	1.10102	0.0378092
$849$	0	0
$850$	−1.00000	−0.0342997
$851$	5.50510	0.188712
$852$	0	0
$853$	5.55051	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$853$	5.55051	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$854$	2.89898	0.0992010
$855$	0	0
$856$	7.10102	0.242708
$857$	8.20204	0.280176	0.140088	−	0.990139i	$-0.455261\pi$
$857$	8.20204	0.280176	0.140088	+	0.990139i	$0.455261\pi$
$858$	0	0
$859$	−13.7980	−0.470780	−0.235390	−	0.971901i	$-0.575637\pi$
$859$	−13.7980	−0.470780	−0.235390	+	0.971901i	$0.575637\pi$
$860$	−9.79796	−0.334108
$861$	0	0
$862$	32.9444	1.12209
$863$	−38.2020	−1.30041	−0.650206	−	0.759758i	$-0.725318\pi$
$863$	−38.2020	−1.30041	−0.650206	+	0.759758i	$0.725318\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	−21.3939	−0.726994
$867$	0	0
$868$	−4.20204	−0.142627
$869$	−45.7980	−1.55359
$870$	0	0
$871$	−20.0000	−0.677674
$872$	15.3485	0.519765
$873$	0	0
$874$	10.2929	0.348161
$875$	4.40408	0.148885
$876$	0	0
$877$	9.95459	0.336143	0.168071	−	0.985775i	$-0.446246\pi$
$877$	9.95459	0.336143	0.168071	+	0.985775i	$0.446246\pi$
$878$	33.3485	1.12546
$879$	0	0
$880$	−12.0000	−0.404520
$881$	−3.30306	−0.111283	−0.0556415	−	0.998451i	$-0.517720\pi$
$881$	−3.30306	−0.111283	−0.0556415	+	0.998451i	$0.517720\pi$
$882$	0	0
$883$	4.69694	0.158065	0.0790323	−	0.996872i	$-0.474817\pi$
$883$	4.69694	0.158065	0.0790323	+	0.996872i	$0.474817\pi$
$884$	−6.89898	−0.232038
$885$	0	0
$886$	41.3939	1.39065
$887$	−7.95459	−0.267089	−0.133545	−	0.991043i	$-0.542636\pi$
$887$	−7.95459	−0.267089	−0.133545	+	0.991043i	$0.542636\pi$
$888$	0	0
$889$	1.79796	0.0603016
$890$	38.6969	1.29712
$891$	0	0
$892$	10.6969	0.358160
$893$	−14.2020	−0.475253
$894$	0	0
$895$	53.3939	1.78476
$896$	−0.449490	−0.0150164
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−22.8990	−0.763724
$900$	0	0
$901$	−1.10102	−0.0366803
$902$	29.3939	0.978709
$903$	0	0
$904$	10.8990	0.362495
$905$	−39.7980	−1.32293
$906$	0	0
$907$	0.898979	0.0298501	0.0149251	−	0.999889i	$-0.495249\pi$
$907$	0.898979	0.0298501	0.0149251	+	0.999889i	$0.495249\pi$
$908$	0	0
$909$	0	0
$910$	−7.59592	−0.251802
$911$	13.3485	0.442255	0.221127	−	0.975245i	$-0.429026\pi$
$911$	13.3485	0.442255	0.221127	+	0.975245i	$0.429026\pi$
$912$	0	0
$913$	−53.3939	−1.76708
$914$	−21.3939	−0.707647
$915$	0	0
$916$	2.00000	0.0660819
$917$	3.19184	0.105404
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−8.69694	−0.286730
$921$	0	0
$922$	20.6969	0.681617
$923$	−92.0908	−3.03121
$924$	0	0
$925$	−1.55051	−0.0509805
$926$	−40.4949	−1.33074
$927$	0	0
$928$	−2.44949	−0.0804084
$929$	8.20204	0.269100	0.134550	−	0.990907i	$-0.457041\pi$
$929$	8.20204	0.269100	0.134550	+	0.990907i	$0.457041\pi$
$930$	0	0
$931$	19.7071	0.645876
$932$	−20.6969	−0.677951
$933$	0	0
$934$	−7.59592	−0.248546
$935$	12.0000	0.392442
$936$	0	0
$937$	17.7980	0.581434	0.290717	−	0.956809i	$-0.406106\pi$
$937$	17.7980	0.581434	0.290717	+	0.956809i	$0.406106\pi$
$938$	1.30306	0.0425465
$939$	0	0
$940$	12.0000	0.391397
$941$	29.1464	0.950146	0.475073	−	0.879946i	$-0.342422\pi$
$941$	29.1464	0.950146	0.475073	+	0.879946i	$0.342422\pi$
$942$	0	0
$943$	21.3031	0.693723
$944$	−10.8990	−0.354732
$945$	0	0
$946$	19.5959	0.637118
$947$	−48.4949	−1.57587	−0.787936	−	0.615757i	$-0.788850\pi$
$947$	−48.4949	−1.57587	−0.787936	+	0.615757i	$0.788850\pi$
$948$	0	0
$949$	−68.9898	−2.23950
$950$	−2.89898	−0.0940553
$951$	0	0
$952$	0.449490	0.0145680
$953$	−10.4041	−0.337021	−0.168511	−	0.985700i	$-0.553896\pi$
$953$	−10.4041	−0.337021	−0.168511	+	0.985700i	$0.553896\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	2.69694	0.0872252
$957$	0	0
$958$	−6.24745	−0.201846
$959$	8.80816	0.284430
$960$	0	0
$961$	56.3939	1.81916
$962$	−10.6969	−0.344883
$963$	0	0
$964$	−12.6969	−0.408941
$965$	22.2929	0.717632
$966$	0	0
$967$	40.0908	1.28923	0.644617	−	0.764506i	$-0.277017\pi$
$967$	40.0908	1.28923	0.644617	+	0.764506i	$0.277017\pi$
$968$	13.0000	0.417836
$969$	0	0
$970$	−12.4949	−0.401187
$971$	−40.2929	−1.29306	−0.646530	−	0.762889i	$-0.723780\pi$
$971$	−40.2929	−1.29306	−0.646530	+	0.762889i	$0.723780\pi$
$972$	0	0
$973$	−5.79796	−0.185874
$974$	−0.449490	−0.0144026
$975$	0	0
$976$	−6.44949	−0.206443
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	0	0
$979$	−77.3939	−2.47352
$980$	−16.6515	−0.531914
$981$	0	0
$982$	−17.3939	−0.555061
$983$	−44.4495	−1.41772	−0.708859	−	0.705350i	$-0.750790\pi$
$983$	−44.4495	−1.41772	−0.708859	+	0.705350i	$0.750790\pi$
$984$	0	0
$985$	0.606123	0.0193127
$986$	2.44949	0.0780076
$987$	0	0
$988$	−20.0000	−0.636285
$989$	14.2020	0.451599
$990$	0	0
$991$	36.0454	1.14502	0.572510	−	0.819898i	$-0.305970\pi$
$991$	36.0454	1.14502	0.572510	+	0.819898i	$0.305970\pi$
$992$	9.34847	0.296814
$993$	0	0
$994$	6.00000	0.190308
$995$	−49.1010	−1.55661
$996$	0	0
$997$	27.3485	0.866135	0.433067	−	0.901362i	$-0.357431\pi$
$997$	27.3485	0.866135	0.433067	+	0.901362i	$0.357431\pi$
$998$	34.6969	1.09831
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	306.2.a.f.1.2	yes	2
3.2	odd	2		306.2.a.e.1.1	✓	2
4.3	odd	2		2448.2.a.w.1.2		2
5.4	even	2		7650.2.a.cq.1.2		2
8.3	odd	2		9792.2.a.cp.1.1		2
8.5	even	2		9792.2.a.ct.1.1		2
12.11	even	2		2448.2.a.x.1.1		2
15.14	odd	2		7650.2.a.cz.1.2		2
17.16	even	2		5202.2.a.z.1.1		2
24.5	odd	2		9792.2.a.cu.1.2		2
24.11	even	2		9792.2.a.cq.1.2		2
51.50	odd	2		5202.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
306.2.a.e.1.1	✓	2	3.2	odd	2
306.2.a.f.1.2	yes	2	1.1	even	1	trivial
2448.2.a.w.1.2		2	4.3	odd	2
2448.2.a.x.1.1		2	12.11	even	2
5202.2.a.q.1.2		2	51.50	odd	2
5202.2.a.z.1.1		2	17.16	even	2
7650.2.a.cq.1.2		2	5.4	even	2
7650.2.a.cz.1.2		2	15.14	odd	2
9792.2.a.cp.1.1		2	8.3	odd	2
9792.2.a.cq.1.2		2	24.11	even	2
9792.2.a.ct.1.1		2	8.5	even	2
9792.2.a.cu.1.2		2	24.5	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	306.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.44949 q^{5} -0.449490 q^{7} +1.00000 q^{8} +2.44949 q^{10} -4.89898 q^{11} +6.89898 q^{13} -0.449490 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.89898 q^{19} +2.44949 q^{20} -4.89898 q^{22} -3.55051 q^{23} +1.00000 q^{25} +6.89898 q^{26} -0.449490 q^{28} -2.44949 q^{29} +9.34847 q^{31} +1.00000 q^{32} -1.00000 q^{34} -1.10102 q^{35} -1.55051 q^{37} -2.89898 q^{38} +2.44949 q^{40} -6.00000 q^{41} -4.00000 q^{43} -4.89898 q^{44} -3.55051 q^{46} +4.89898 q^{47} -6.79796 q^{49} +1.00000 q^{50} +6.89898 q^{52} +1.10102 q^{53} -12.0000 q^{55} -0.449490 q^{56} -2.44949 q^{58} -10.8990 q^{59} -6.44949 q^{61} +9.34847 q^{62} +1.00000 q^{64} +16.8990 q^{65} -2.89898 q^{67} -1.00000 q^{68} -1.10102 q^{70} -13.3485 q^{71} -10.0000 q^{73} -1.55051 q^{74} -2.89898 q^{76} +2.20204 q^{77} +9.34847 q^{79} +2.44949 q^{80} -6.00000 q^{82} +10.8990 q^{83} -2.44949 q^{85} -4.00000 q^{86} -4.89898 q^{88} +15.7980 q^{89} -3.10102 q^{91} -3.55051 q^{92} +4.89898 q^{94} -7.10102 q^{95} -5.10102 q^{97} -6.79796 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 2 q^{17} + 4 q^{19} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} - 2 q^{34} - 12 q^{35} - 8 q^{37} + 4 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 4 * q^13 + 4 * q^14 + 2 * q^16 - 2 * q^17 + 4 * q^19 - 12 * q^23 + 2 * q^25 + 4 * q^26 + 4 * q^28 + 4 * q^31 + 2 * q^32 - 2 * q^34 - 12 * q^35 - 8 * q^37 + 4 * q^38 - 12 * q^41 - 8 * q^43 - 12 * q^46 + 6 * q^49 + 2 * q^50 + 4 * q^52 + 12 * q^53 - 24 * q^55 + 4 * q^56 - 12 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 + 24 * q^65 + 4 * q^67 - 2 * q^68 - 12 * q^70 - 12 * q^71 - 20 * q^73 - 8 * q^74 + 4 * q^76 + 24 * q^77 + 4 * q^79 - 12 * q^82 + 12 * q^83 - 8 * q^86 + 12 * q^89 - 16 * q^91 - 12 * q^92 - 24 * q^95 - 20 * q^97 + 6 * q^98

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