LMFDB - Embedded newform 306.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [306,2,Mod(1,306)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,2,0,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			1.1
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)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.44949 q^{5} +4.44949 q^{7} +1.00000 q^{8} -2.44949 q^{10} +4.89898 q^{11} -2.89898 q^{13} +4.44949 q^{14} +1.00000 q^{16} -1.00000 q^{17} +6.89898 q^{19} -2.44949 q^{20} +4.89898 q^{22} -8.44949 q^{23} +1.00000 q^{25} -2.89898 q^{26} +4.44949 q^{28} +2.44949 q^{29} -5.34847 q^{31} +1.00000 q^{32} -1.00000 q^{34} -10.8990 q^{35} -6.44949 q^{37} +6.89898 q^{38} -2.44949 q^{40} -6.00000 q^{41} -4.00000 q^{43} +4.89898 q^{44} -8.44949 q^{46} -4.89898 q^{47} +12.7980 q^{49} +1.00000 q^{50} -2.89898 q^{52} +10.8990 q^{53} -12.0000 q^{55} +4.44949 q^{56} +2.44949 q^{58} -1.10102 q^{59} -1.55051 q^{61} -5.34847 q^{62} +1.00000 q^{64} +7.10102 q^{65} +6.89898 q^{67} -1.00000 q^{68} -10.8990 q^{70} +1.34847 q^{71} -10.0000 q^{73} -6.44949 q^{74} +6.89898 q^{76} +21.7980 q^{77} -5.34847 q^{79} -2.44949 q^{80} -6.00000 q^{82} +1.10102 q^{83} +2.44949 q^{85} -4.00000 q^{86} +4.89898 q^{88} -3.79796 q^{89} -12.8990 q^{91} -8.44949 q^{92} -4.89898 q^{94} -16.8990 q^{95} -14.8990 q^{97} +12.7980 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	4.44949	1.68175	0.840875	−	0.541230i	$-0.182041\pi$
$7$	4.44949	1.68175	0.840875	+	0.541230i	$0.182041\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.44949	−0.774597
$11$	4.89898	1.47710	0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	0	0
$13$	−2.89898	−0.804032	−0.402016	−	0.915633i	$-0.631690\pi$
$13$	−2.89898	−0.804032	−0.402016	+	0.915633i	$0.631690\pi$
$14$	4.44949	1.18918
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.89898	1.58273	0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898	1.58273	0.791367	+	0.611341i	$0.209370\pi$
$20$	−2.44949	−0.547723
$21$	0	0
$22$	4.89898	1.04447
$23$	−8.44949	−1.76184	−0.880920	−	0.473265i	$-0.843075\pi$
$23$	−8.44949	−1.76184	−0.880920	+	0.473265i	$0.843075\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.89898	−0.568537
$27$	0	0
$28$	4.44949	0.840875
$29$	2.44949	0.454859	0.227429	−	0.973795i	$-0.426968\pi$
$29$	2.44949	0.454859	0.227429	+	0.973795i	$0.426968\pi$
$30$	0	0
$31$	−5.34847	−0.960613	−0.480307	−	0.877101i	$-0.659475\pi$
$31$	−5.34847	−0.960613	−0.480307	+	0.877101i	$0.659475\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−10.8990	−1.84226
$36$	0	0
$37$	−6.44949	−1.06029	−0.530145	−	0.847907i	$-0.677862\pi$
$37$	−6.44949	−1.06029	−0.530145	+	0.847907i	$0.677862\pi$
$38$	6.89898	1.11916
$39$	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
$46$	−8.44949	−1.24581
$47$	−4.89898	−0.714590	−0.357295	−	0.933992i	$-0.616301\pi$
$47$	−4.89898	−0.714590	−0.357295	+	0.933992i	$0.616301\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.89898	−0.402016
$53$	10.8990	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$53$	10.8990	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	4.44949	0.594588
$57$	0	0
$58$	2.44949	0.321634
$59$	−1.10102	−0.143341	−0.0716703	−	0.997428i	$-0.522833\pi$
$59$	−1.10102	−0.143341	−0.0716703	+	0.997428i	$0.522833\pi$
$60$	0	0
$61$	−1.55051	−0.198522	−0.0992612	−	0.995061i	$-0.531648\pi$
$61$	−1.55051	−0.198522	−0.0992612	+	0.995061i	$0.531648\pi$
$62$	−5.34847	−0.679256
$63$	0	0
$64$	1.00000	0.125000
$65$	7.10102	0.880773
$66$	0	0
$67$	6.89898	0.842844	0.421422	−	0.906865i	$-0.361531\pi$
$67$	6.89898	0.842844	0.421422	+	0.906865i	$0.361531\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	−10.8990	−1.30268
$71$	1.34847	0.160034	0.0800169	−	0.996794i	$-0.474503\pi$
$71$	1.34847	0.160034	0.0800169	+	0.996794i	$0.474503\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−6.44949	−0.749738
$75$	0	0
$76$	6.89898	0.791367
$77$	21.7980	2.48411
$78$	0	0
$79$	−5.34847	−0.601750	−0.300875	−	0.953664i	$-0.597279\pi$
$79$	−5.34847	−0.601750	−0.300875	+	0.953664i	$0.597279\pi$
$80$	−2.44949	−0.273861
$81$	0	0
$82$	−6.00000	−0.662589
$83$	1.10102	0.120853	0.0604264	−	0.998173i	$-0.480754\pi$
$83$	1.10102	0.120853	0.0604264	+	0.998173i	$0.480754\pi$
$84$	0	0
$85$	2.44949	0.265684
$86$	−4.00000	−0.431331
$87$	0	0
$88$	4.89898	0.522233
$89$	−3.79796	−0.402583	−0.201291	−	0.979531i	$-0.564514\pi$
$89$	−3.79796	−0.402583	−0.201291	+	0.979531i	$0.564514\pi$
$90$	0	0
$91$	−12.8990	−1.35218
$92$	−8.44949	−0.880920
$93$	0	0
$94$	−4.89898	−0.505291
$95$	−16.8990	−1.73380
$96$	0	0
$97$	−14.8990	−1.51276	−0.756381	−	0.654131i	$-0.773034\pi$
$97$	−14.8990	−1.51276	−0.756381	+	0.654131i	$0.773034\pi$
$98$	12.7980	1.29279
$99$	0	0
$100$	1.00000	0.100000
$101$	1.10102	0.109556	0.0547778	−	0.998499i	$-0.482555\pi$
$101$	1.10102	0.109556	0.0547778	+	0.998499i	$0.482555\pi$
$102$	0	0
$103$	3.10102	0.305553	0.152776	−	0.988261i	$-0.451179\pi$
$103$	3.10102	0.305553	0.152776	+	0.988261i	$0.451179\pi$
$104$	−2.89898	−0.284268
$105$	0	0
$106$	10.8990	1.05860
$107$	16.8990	1.63369	0.816843	−	0.576860i	$-0.195722\pi$
$107$	16.8990	1.63369	0.816843	+	0.576860i	$0.195722\pi$
$108$	0	0
$109$	0.651531	0.0624053	0.0312027	−	0.999513i	$-0.490066\pi$
$109$	0.651531	0.0624053	0.0312027	+	0.999513i	$0.490066\pi$
$110$	−12.0000	−1.14416
$111$	0	0
$112$	4.44949	0.420437
$113$	1.10102	0.103575	0.0517876	−	0.998658i	$-0.483508\pi$
$113$	1.10102	0.103575	0.0517876	+	0.998658i	$0.483508\pi$
$114$	0	0
$115$	20.6969	1.93000
$116$	2.44949	0.227429
$117$	0	0
$118$	−1.10102	−0.101357
$119$	−4.44949	−0.407884
$120$	0	0
$121$	13.0000	1.18182
$122$	−1.55051	−0.140377
$123$	0	0
$124$	−5.34847	−0.480307
$125$	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
$130$	7.10102	0.622801
$131$	−16.8990	−1.47647	−0.738235	−	0.674543i	$-0.764341\pi$
$131$	−16.8990	−1.47647	−0.738235	+	0.674543i	$0.764341\pi$
$132$	0	0
$133$	30.6969	2.66176
$134$	6.89898	0.595981
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	19.5959	1.67419	0.837096	−	0.547056i	$-0.184251\pi$
$137$	19.5959	1.67419	0.837096	+	0.547056i	$0.184251\pi$
$138$	0	0
$139$	3.10102	0.263025	0.131513	−	0.991315i	$-0.458017\pi$
$139$	3.10102	0.263025	0.131513	+	0.991315i	$0.458017\pi$
$140$	−10.8990	−0.921132
$141$	0	0
$142$	1.34847	0.113161
$143$	−14.2020	−1.18763
$144$	0	0
$145$	−6.00000	−0.498273
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−6.44949	−0.530145
$149$	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
$151$	−1.79796	−0.146316	−0.0731579	−	0.997320i	$-0.523308\pi$
$151$	−1.79796	−0.146316	−0.0731579	+	0.997320i	$0.523308\pi$
$152$	6.89898	0.559581
$153$	0	0
$154$	21.7980	1.75653
$155$	13.1010	1.05230
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−5.34847	−0.425501
$159$	0	0
$160$	−2.44949	−0.193649
$161$	−37.5959	−2.96297
$162$	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
$166$	1.10102	0.0854558
$167$	8.44949	0.653841	0.326921	−	0.945052i	$-0.393989\pi$
$167$	8.44949	0.653841	0.326921	+	0.945052i	$0.393989\pi$
$168$	0	0
$169$	−4.59592	−0.353532
$170$	2.44949	0.187867
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−17.1464	−1.30362	−0.651809	−	0.758383i	$-0.725990\pi$
$173$	−17.1464	−1.30362	−0.651809	+	0.758383i	$0.725990\pi$
$174$	0	0
$175$	4.44949	0.336350
$176$	4.89898	0.369274
$177$	0	0
$178$	−3.79796	−0.284669
$179$	2.20204	0.164588	0.0822941	−	0.996608i	$-0.473775\pi$
$179$	2.20204	0.164588	0.0822941	+	0.996608i	$0.473775\pi$
$180$	0	0
$181$	8.24745	0.613028	0.306514	−	0.951866i	$-0.400837\pi$
$181$	8.24745	0.613028	0.306514	+	0.951866i	$0.400837\pi$
$182$	−12.8990	−0.956136
$183$	0	0
$184$	−8.44949	−0.622905
$185$	15.7980	1.16149
$186$	0	0
$187$	−4.89898	−0.358249
$188$	−4.89898	−0.357295
$189$	0	0
$190$	−16.8990	−1.22598
$191$	14.6969	1.06343	0.531717	−	0.846922i	$-0.321547\pi$
$191$	14.6969	1.06343	0.531717	+	0.846922i	$0.321547\pi$
$192$	0	0
$193$	18.8990	1.36038	0.680189	−	0.733037i	$-0.261898\pi$
$193$	18.8990	1.36038	0.680189	+	0.733037i	$0.261898\pi$
$194$	−14.8990	−1.06968
$195$	0	0
$196$	12.7980	0.914140
$197$	−24.2474	−1.72756	−0.863780	−	0.503870i	$-0.831909\pi$
$197$	−24.2474	−1.72756	−0.863780	+	0.503870i	$0.831909\pi$
$198$	0	0
$199$	24.0454	1.70453	0.852267	−	0.523107i	$-0.175227\pi$
$199$	24.0454	1.70453	0.852267	+	0.523107i	$0.175227\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	1.10102	0.0774675
$203$	10.8990	0.764958
$204$	0	0
$205$	14.6969	1.02648
$206$	3.10102	0.216058
$207$	0	0
$208$	−2.89898	−0.201008
$209$	33.7980	2.33785
$210$	0	0
$211$	12.8990	0.888002	0.444001	−	0.896026i	$-0.353559\pi$
$211$	12.8990	0.888002	0.444001	+	0.896026i	$0.353559\pi$
$212$	10.8990	0.748545
$213$	0	0
$214$	16.8990	1.15519
$215$	9.79796	0.668215
$216$	0	0
$217$	−23.7980	−1.61551
$218$	0.651531	0.0441272
$219$	0	0
$220$	−12.0000	−0.809040
$221$	2.89898	0.195006
$222$	0	0
$223$	−18.6969	−1.25204	−0.626020	−	0.779807i	$-0.715317\pi$
$223$	−18.6969	−1.25204	−0.626020	+	0.779807i	$0.715317\pi$
$224$	4.44949	0.297294
$225$	0	0
$226$	1.10102	0.0732388
$227$	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	$0.478950\pi$
$230$	20.6969	1.36472
$231$	0	0
$232$	2.44949	0.160817
$233$	8.69694	0.569755	0.284878	−	0.958564i	$-0.408047\pi$
$233$	8.69694	0.569755	0.284878	+	0.958564i	$0.408047\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	−1.10102	−0.0716703
$237$	0	0
$238$	−4.44949	−0.288418
$239$	−26.6969	−1.72688	−0.863441	−	0.504450i	$-0.831695\pi$
$239$	−26.6969	−1.72688	−0.863441	+	0.504450i	$0.831695\pi$
$240$	0	0
$241$	16.6969	1.07554	0.537772	−	0.843090i	$-0.319266\pi$
$241$	16.6969	1.07554	0.537772	+	0.843090i	$0.319266\pi$
$242$	13.0000	0.835672
$243$	0	0
$244$	−1.55051	−0.0992612
$245$	−31.3485	−2.00278
$246$	0	0
$247$	−20.0000	−1.27257
$248$	−5.34847	−0.339628
$249$	0	0
$250$	9.79796	0.619677
$251$	2.20204	0.138992	0.0694958	−	0.997582i	$-0.477861\pi$
$251$	2.20204	0.138992	0.0694958	+	0.997582i	$0.477861\pi$
$252$	0	0
$253$	−41.3939	−2.60241
$254$	−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
$259$	−28.6969	−1.78314
$260$	7.10102	0.440387
$261$	0	0
$262$	−16.8990	−1.04402
$263$	−19.1010	−1.17782	−0.588910	−	0.808199i	$-0.700443\pi$
$263$	−19.1010	−1.17782	−0.588910	+	0.808199i	$0.700443\pi$
$264$	0	0
$265$	−26.6969	−1.63998
$266$	30.6969	1.88215
$267$	0	0
$268$	6.89898	0.421422
$269$	−2.44949	−0.149348	−0.0746740	−	0.997208i	$-0.523792\pi$
$269$	−2.44949	−0.149348	−0.0746740	+	0.997208i	$0.523792\pi$
$270$	0	0
$271$	17.7980	1.08115	0.540575	−	0.841296i	$-0.318207\pi$
$271$	17.7980	1.08115	0.540575	+	0.841296i	$0.318207\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	19.5959	1.18383
$275$	4.89898	0.295420
$276$	0	0
$277$	−23.3485	−1.40287	−0.701437	−	0.712732i	$-0.747458\pi$
$277$	−23.3485	−1.40287	−0.701437	+	0.712732i	$0.747458\pi$
$278$	3.10102	0.185987
$279$	0	0
$280$	−10.8990	−0.651339
$281$	−15.7980	−0.942427	−0.471214	−	0.882019i	$-0.656184\pi$
$281$	−15.7980	−0.942427	−0.471214	+	0.882019i	$0.656184\pi$
$282$	0	0
$283$	10.6969	0.635867	0.317933	−	0.948113i	$-0.397011\pi$
$283$	10.6969	0.635867	0.317933	+	0.948113i	$0.397011\pi$
$284$	1.34847	0.0800169
$285$	0	0
$286$	−14.2020	−0.839784
$287$	−26.6969	−1.57587
$288$	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
$295$	2.69694	0.157022
$296$	−6.44949	−0.374869
$297$	0	0
$298$	18.0000	1.04271
$299$	24.4949	1.41658
$300$	0	0
$301$	−17.7980	−1.02586
$302$	−1.79796	−0.103461
$303$	0	0
$304$	6.89898	0.395684
$305$	3.79796	0.217470
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	21.7980	1.24205
$309$	0	0
$310$	13.1010	0.744088
$311$	−18.2474	−1.03472	−0.517359	−	0.855768i	$-0.673085\pi$
$311$	−18.2474	−1.03472	−0.517359	+	0.855768i	$0.673085\pi$
$312$	0	0
$313$	6.89898	0.389953	0.194977	−	0.980808i	$-0.437537\pi$
$313$	6.89898	0.389953	0.194977	+	0.980808i	$0.437537\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−5.34847	−0.300875
$317$	−5.14643	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$317$	−5.14643	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	−2.44949	−0.136931
$321$	0	0
$322$	−37.5959	−2.09514
$323$	−6.89898	−0.383869
$324$	0	0
$325$	−2.89898	−0.160806
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−21.7980	−1.20176
$330$	0	0
$331$	−13.7980	−0.758404	−0.379202	−	0.925314i	$-0.623802\pi$
$331$	−13.7980	−0.758404	−0.379202	+	0.925314i	$0.623802\pi$
$332$	1.10102	0.0604264
$333$	0	0
$334$	8.44949	0.462336
$335$	−16.8990	−0.923290
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−4.59592	−0.249985
$339$	0	0
$340$	2.44949	0.132842
$341$	−26.2020	−1.41892
$342$	0	0
$343$	25.7980	1.39296
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−17.1464	−0.921798
$347$	−21.7980	−1.17018	−0.585088	−	0.810970i	$-0.698940\pi$
$347$	−21.7980	−1.17018	−0.585088	+	0.810970i	$0.698940\pi$
$348$	0	0
$349$	28.6969	1.53611	0.768056	−	0.640383i	$-0.221224\pi$
$349$	28.6969	1.53611	0.768056	+	0.640383i	$0.221224\pi$
$350$	4.44949	0.237835
$351$	0	0
$352$	4.89898	0.261116
$353$	13.5959	0.723638	0.361819	−	0.932248i	$-0.382156\pi$
$353$	13.5959	0.723638	0.361819	+	0.932248i	$0.382156\pi$
$354$	0	0
$355$	−3.30306	−0.175308
$356$	−3.79796	−0.201291
$357$	0	0
$358$	2.20204	0.116381
$359$	21.7980	1.15045	0.575226	−	0.817994i	$-0.304914\pi$
$359$	21.7980	1.15045	0.575226	+	0.817994i	$0.304914\pi$
$360$	0	0
$361$	28.5959	1.50505
$362$	8.24745	0.433476
$363$	0	0
$364$	−12.8990	−0.676090
$365$	24.4949	1.28212
$366$	0	0
$367$	−9.75255	−0.509079	−0.254540	−	0.967062i	$-0.581924\pi$
$367$	−9.75255	−0.509079	−0.254540	+	0.967062i	$0.581924\pi$
$368$	−8.44949	−0.440460
$369$	0	0
$370$	15.7980	0.821297
$371$	48.4949	2.51773
$372$	0	0
$373$	28.6969	1.48587	0.742936	−	0.669363i	$-0.233433\pi$
$373$	28.6969	1.48587	0.742936	+	0.669363i	$0.233433\pi$
$374$	−4.89898	−0.253320
$375$	0	0
$376$	−4.89898	−0.252646
$377$	−7.10102	−0.365721
$378$	0	0
$379$	10.2020	0.524044	0.262022	−	0.965062i	$-0.415611\pi$
$379$	10.2020	0.524044	0.262022	+	0.965062i	$0.415611\pi$
$380$	−16.8990	−0.866899
$381$	0	0
$382$	14.6969	0.751961
$383$	−4.89898	−0.250326	−0.125163	−	0.992136i	$-0.539945\pi$
$383$	−4.89898	−0.250326	−0.125163	+	0.992136i	$0.539945\pi$
$384$	0	0
$385$	−53.3939	−2.72120
$386$	18.8990	0.961933
$387$	0	0
$388$	−14.8990	−0.756381
$389$	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
$391$	8.44949	0.427309
$392$	12.7980	0.646395
$393$	0	0
$394$	−24.2474	−1.22157
$395$	13.1010	0.659184
$396$	0	0
$397$	12.6515	0.634962	0.317481	−	0.948265i	$-0.397163\pi$
$397$	12.6515	0.634962	0.317481	+	0.948265i	$0.397163\pi$
$398$	24.0454	1.20529
$399$	0	0
$400$	1.00000	0.0500000
$401$	3.79796	0.189661	0.0948305	−	0.995493i	$-0.469769\pi$
$401$	3.79796	0.189661	0.0948305	+	0.995493i	$0.469769\pi$
$402$	0	0
$403$	15.5051	0.772364
$404$	1.10102	0.0547778
$405$	0	0
$406$	10.8990	0.540907
$407$	−31.5959	−1.56615
$408$	0	0
$409$	−25.7980	−1.27563	−0.637813	−	0.770191i	$-0.720161\pi$
$409$	−25.7980	−1.27563	−0.637813	+	0.770191i	$0.720161\pi$
$410$	14.6969	0.725830
$411$	0	0
$412$	3.10102	0.152776
$413$	−4.89898	−0.241063
$414$	0	0
$415$	−2.69694	−0.132388
$416$	−2.89898	−0.142134
$417$	0	0
$418$	33.7980	1.65311
$419$	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
$421$	−17.1010	−0.833453	−0.416726	−	0.909032i	$-0.636823\pi$
$421$	−17.1010	−0.833453	−0.416726	+	0.909032i	$0.636823\pi$
$422$	12.8990	0.627912
$423$	0	0
$424$	10.8990	0.529301
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−6.89898	−0.333865
$428$	16.8990	0.816843
$429$	0	0
$430$	9.79796	0.472500
$431$	−20.9444	−1.00886	−0.504428	−	0.863454i	$-0.668297\pi$
$431$	−20.9444	−1.00886	−0.504428	+	0.863454i	$0.668297\pi$
$432$	0	0
$433$	37.3939	1.79704	0.898518	−	0.438938i	$-0.144645\pi$
$433$	37.3939	1.79704	0.898518	+	0.438938i	$0.144645\pi$
$434$	−23.7980	−1.14234
$435$	0	0
$436$	0.651531	0.0312027
$437$	−58.2929	−2.78853
$438$	0	0
$439$	18.6515	0.890189	0.445094	−	0.895484i	$-0.353170\pi$
$439$	18.6515	0.890189	0.445094	+	0.895484i	$0.353170\pi$
$440$	−12.0000	−0.572078
$441$	0	0
$442$	2.89898	0.137890
$443$	−17.3939	−0.826408	−0.413204	−	0.910638i	$-0.635590\pi$
$443$	−17.3939	−0.826408	−0.413204	+	0.910638i	$0.635590\pi$
$444$	0	0
$445$	9.30306	0.441007
$446$	−18.6969	−0.885326
$447$	0	0
$448$	4.44949	0.210219
$449$	−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
$452$	1.10102	0.0517876
$453$	0	0
$454$	0	0
$455$	31.5959	1.48124
$456$	0	0
$457$	37.3939	1.74921	0.874606	−	0.484835i	$-0.161120\pi$
$457$	37.3939	1.74921	0.874606	+	0.484835i	$0.161120\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	20.6969	0.965000
$461$	−8.69694	−0.405057	−0.202528	−	0.979276i	$-0.564916\pi$
$461$	−8.69694	−0.405057	−0.202528	+	0.979276i	$0.564916\pi$
$462$	0	0
$463$	8.49490	0.394791	0.197396	−	0.980324i	$-0.436752\pi$
$463$	8.49490	0.394791	0.197396	+	0.980324i	$0.436752\pi$
$464$	2.44949	0.113715
$465$	0	0
$466$	8.69694	0.402878
$467$	31.5959	1.46208	0.731042	−	0.682332i	$-0.239034\pi$
$467$	31.5959	1.46208	0.731042	+	0.682332i	$0.239034\pi$
$468$	0	0
$469$	30.6969	1.41745
$470$	12.0000	0.553519
$471$	0	0
$472$	−1.10102	−0.0506786
$473$	−19.5959	−0.901021
$474$	0	0
$475$	6.89898	0.316547
$476$	−4.44949	−0.203942
$477$	0	0
$478$	−26.6969	−1.22109
$479$	18.2474	0.833747	0.416874	−	0.908964i	$-0.363126\pi$
$479$	18.2474	0.833747	0.416874	+	0.908964i	$0.363126\pi$
$480$	0	0
$481$	18.6969	0.852507
$482$	16.6969	0.760525
$483$	0	0
$484$	13.0000	0.590909
$485$	36.4949	1.65715
$486$	0	0
$487$	4.44949	0.201626	0.100813	−	0.994905i	$-0.467856\pi$
$487$	4.44949	0.201626	0.100813	+	0.994905i	$0.467856\pi$
$488$	−1.55051	−0.0701883
$489$	0	0
$490$	−31.3485	−1.41618
$491$	41.3939	1.86808	0.934040	−	0.357169i	$-0.116258\pi$
$491$	41.3939	1.86808	0.934040	+	0.357169i	$0.116258\pi$
$492$	0	0
$493$	−2.44949	−0.110319
$494$	−20.0000	−0.899843
$495$	0	0
$496$	−5.34847	−0.240153
$497$	6.00000	0.269137
$498$	0	0
$499$	5.30306	0.237398	0.118699	−	0.992930i	$-0.462128\pi$
$499$	5.30306	0.237398	0.118699	+	0.992930i	$0.462128\pi$
$500$	9.79796	0.438178
$501$	0	0
$502$	2.20204	0.0982819
$503$	18.2474	0.813614	0.406807	−	0.913514i	$-0.366642\pi$
$503$	18.2474	0.813614	0.406807	+	0.913514i	$0.366642\pi$
$504$	0	0
$505$	−2.69694	−0.120012
$506$	−41.3939	−1.84018
$507$	0	0
$508$	−4.00000	−0.177471
$509$	15.7980	0.700232	0.350116	−	0.936706i	$-0.386142\pi$
$509$	15.7980	0.700232	0.350116	+	0.936706i	$0.386142\pi$
$510$	0	0
$511$	−44.4949	−1.96834
$512$	1.00000	0.0441942
$513$	0	0
$514$	24.0000	1.05859
$515$	−7.59592	−0.334716
$516$	0	0
$517$	−24.0000	−1.05552
$518$	−28.6969	−1.26087
$519$	0	0
$520$	7.10102	0.311400
$521$	−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
$523$	−12.6969	−0.555198	−0.277599	−	0.960697i	$-0.589539\pi$
$523$	−12.6969	−0.555198	−0.277599	+	0.960697i	$0.589539\pi$
$524$	−16.8990	−0.738235
$525$	0	0
$526$	−19.1010	−0.832844
$527$	5.34847	0.232983
$528$	0	0
$529$	48.3939	2.10408
$530$	−26.6969	−1.15964
$531$	0	0
$532$	30.6969	1.33088
$533$	17.3939	0.753412
$534$	0	0
$535$	−41.3939	−1.78961
$536$	6.89898	0.297991
$537$	0	0
$538$	−2.44949	−0.105605
$539$	62.6969	2.70055
$540$	0	0
$541$	12.6515	0.543932	0.271966	−	0.962307i	$-0.412326\pi$
$541$	12.6515	0.543932	0.271966	+	0.962307i	$0.412326\pi$
$542$	17.7980	0.764488
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	−1.59592	−0.0683616
$546$	0	0
$547$	−11.1010	−0.474645	−0.237323	−	0.971431i	$-0.576270\pi$
$547$	−11.1010	−0.474645	−0.237323	+	0.971431i	$0.576270\pi$
$548$	19.5959	0.837096
$549$	0	0
$550$	4.89898	0.208893
$551$	16.8990	0.719921
$552$	0	0
$553$	−23.7980	−1.01199
$554$	−23.3485	−0.991981
$555$	0	0
$556$	3.10102	0.131513
$557$	−22.8990	−0.970261	−0.485130	−	0.874442i	$-0.661228\pi$
$557$	−22.8990	−0.970261	−0.485130	+	0.874442i	$0.661228\pi$
$558$	0	0
$559$	11.5959	0.490455
$560$	−10.8990	−0.460566
$561$	0	0
$562$	−15.7980	−0.666397
$563$	−10.8990	−0.459337	−0.229669	−	0.973269i	$-0.573764\pi$
$563$	−10.8990	−0.459337	−0.229669	+	0.973269i	$0.573764\pi$
$564$	0	0
$565$	−2.69694	−0.113461
$566$	10.6969	0.449626
$567$	0	0
$568$	1.34847	0.0565805
$569$	−39.1918	−1.64301	−0.821504	−	0.570203i	$-0.806864\pi$
$569$	−39.1918	−1.64301	−0.821504	+	0.570203i	$0.806864\pi$
$570$	0	0
$571$	−25.7980	−1.07961	−0.539805	−	0.841790i	$-0.681502\pi$
$571$	−25.7980	−1.07961	−0.539805	+	0.841790i	$0.681502\pi$
$572$	−14.2020	−0.593817
$573$	0	0
$574$	−26.6969	−1.11431
$575$	−8.44949	−0.352368
$576$	0	0
$577$	33.5959	1.39862	0.699308	−	0.714820i	$-0.253492\pi$
$577$	33.5959	1.39862	0.699308	+	0.714820i	$0.253492\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−6.00000	−0.249136
$581$	4.89898	0.203244
$582$	0	0
$583$	53.3939	2.21135
$584$	−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
$589$	−36.8990	−1.52040
$590$	2.69694	0.111031
$591$	0	0
$592$	−6.44949	−0.265072
$593$	−2.20204	−0.0904270	−0.0452135	−	0.998977i	$-0.514397\pi$
$593$	−2.20204	−0.0904270	−0.0452135	+	0.998977i	$0.514397\pi$
$594$	0	0
$595$	10.8990	0.446815
$596$	18.0000	0.737309
$597$	0	0
$598$	24.4949	1.00167
$599$	−14.2020	−0.580280	−0.290140	−	0.956984i	$-0.593702\pi$
$599$	−14.2020	−0.580280	−0.290140	+	0.956984i	$0.593702\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−17.7980	−0.725391
$603$	0	0
$604$	−1.79796	−0.0731579
$605$	−31.8434	−1.29462
$606$	0	0
$607$	31.1464	1.26419	0.632097	−	0.774889i	$-0.282194\pi$
$607$	31.1464	1.26419	0.632097	+	0.774889i	$0.282194\pi$
$608$	6.89898	0.279791
$609$	0	0
$610$	3.79796	0.153775
$611$	14.2020	0.574553
$612$	0	0
$613$	−7.79796	−0.314957	−0.157478	−	0.987522i	$-0.550336\pi$
$613$	−7.79796	−0.314957	−0.157478	+	0.987522i	$0.550336\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	21.7980	0.878265
$617$	−3.30306	−0.132976	−0.0664881	−	0.997787i	$-0.521179\pi$
$617$	−3.30306	−0.132976	−0.0664881	+	0.997787i	$0.521179\pi$
$618$	0	0
$619$	25.3939	1.02067	0.510333	−	0.859977i	$-0.329522\pi$
$619$	25.3939	1.02067	0.510333	+	0.859977i	$0.329522\pi$
$620$	13.1010	0.526150
$621$	0	0
$622$	−18.2474	−0.731656
$623$	−16.8990	−0.677043
$624$	0	0
$625$	−29.0000	−1.16000
$626$	6.89898	0.275739
$627$	0	0
$628$	−10.0000	−0.399043
$629$	6.44949	0.257158
$630$	0	0
$631$	20.4949	0.815889	0.407944	−	0.913007i	$-0.366246\pi$
$631$	20.4949	0.815889	0.407944	+	0.913007i	$0.366246\pi$
$632$	−5.34847	−0.212751
$633$	0	0
$634$	−5.14643	−0.204391
$635$	9.79796	0.388820
$636$	0	0
$637$	−37.1010	−1.47000
$638$	12.0000	0.475085
$639$	0	0
$640$	−2.44949	−0.0968246
$641$	15.7980	0.623982	0.311991	−	0.950085i	$-0.399004\pi$
$641$	15.7980	0.623982	0.311991	+	0.950085i	$0.399004\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−37.5959	−1.48149
$645$	0	0
$646$	−6.89898	−0.271437
$647$	24.4949	0.962994	0.481497	−	0.876448i	$-0.340093\pi$
$647$	24.4949	0.962994	0.481497	+	0.876448i	$0.340093\pi$
$648$	0	0
$649$	−5.39388	−0.211728
$650$	−2.89898	−0.113707
$651$	0	0
$652$	−4.00000	−0.156652
$653$	16.6515	0.651625	0.325812	−	0.945434i	$-0.394362\pi$
$653$	16.6515	0.651625	0.325812	+	0.945434i	$0.394362\pi$
$654$	0	0
$655$	41.3939	1.61739
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−21.7980	−0.849773
$659$	10.8990	0.424564	0.212282	−	0.977208i	$-0.431910\pi$
$659$	10.8990	0.424564	0.212282	+	0.977208i	$0.431910\pi$
$660$	0	0
$661$	−46.4949	−1.80844	−0.904221	−	0.427065i	$-0.859548\pi$
$661$	−46.4949	−1.80844	−0.904221	+	0.427065i	$0.859548\pi$
$662$	−13.7980	−0.536273
$663$	0	0
$664$	1.10102	0.0427279
$665$	−75.1918	−2.91581
$666$	0	0
$667$	−20.6969	−0.801389
$668$	8.44949	0.326921
$669$	0	0
$670$	−16.8990	−0.652865
$671$	−7.59592	−0.293237
$672$	0	0
$673$	−22.4949	−0.867115	−0.433557	−	0.901126i	$-0.642742\pi$
$673$	−22.4949	−0.867115	−0.433557	+	0.901126i	$0.642742\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−4.59592	−0.176766
$677$	14.9444	0.574359	0.287180	−	0.957877i	$-0.407282\pi$
$677$	14.9444	0.574359	0.287180	+	0.957877i	$0.407282\pi$
$678$	0	0
$679$	−66.2929	−2.54409
$680$	2.44949	0.0939336
$681$	0	0
$682$	−26.2020	−1.00333
$683$	−27.1918	−1.04047	−0.520233	−	0.854024i	$-0.674155\pi$
$683$	−27.1918	−1.04047	−0.520233	+	0.854024i	$0.674155\pi$
$684$	0	0
$685$	−48.0000	−1.83399
$686$	25.7980	0.984971
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−31.5959	−1.20371
$690$	0	0
$691$	−1.30306	−0.0495708	−0.0247854	−	0.999693i	$-0.507890\pi$
$691$	−1.30306	−0.0495708	−0.0247854	+	0.999693i	$0.507890\pi$
$692$	−17.1464	−0.651809
$693$	0	0
$694$	−21.7980	−0.827439
$695$	−7.59592	−0.288130
$696$	0	0
$697$	6.00000	0.227266
$698$	28.6969	1.08620
$699$	0	0
$700$	4.44949	0.168175
$701$	20.6969	0.781713	0.390856	−	0.920452i	$-0.372179\pi$
$701$	20.6969	0.781713	0.390856	+	0.920452i	$0.372179\pi$
$702$	0	0
$703$	−44.4949	−1.67816
$704$	4.89898	0.184637
$705$	0	0
$706$	13.5959	0.511689
$707$	4.89898	0.184245
$708$	0	0
$709$	34.9444	1.31236	0.656182	−	0.754603i	$-0.272170\pi$
$709$	34.9444	1.31236	0.656182	+	0.754603i	$0.272170\pi$
$710$	−3.30306	−0.123962
$711$	0	0
$712$	−3.79796	−0.142335
$713$	45.1918	1.69245
$714$	0	0
$715$	34.7878	1.30099
$716$	2.20204	0.0822941
$717$	0	0
$718$	21.7980	0.813493
$719$	15.5505	0.579936	0.289968	−	0.957036i	$-0.406355\pi$
$719$	15.5505	0.579936	0.289968	+	0.957036i	$0.406355\pi$
$720$	0	0
$721$	13.7980	0.513863
$722$	28.5959	1.06423
$723$	0	0
$724$	8.24745	0.306514
$725$	2.44949	0.0909718
$726$	0	0
$727$	−33.3939	−1.23851	−0.619255	−	0.785190i	$-0.712565\pi$
$727$	−33.3939	−1.23851	−0.619255	+	0.785190i	$0.712565\pi$
$728$	−12.8990	−0.478068
$729$	0	0
$730$	24.4949	0.906597
$731$	4.00000	0.147945
$732$	0	0
$733$	11.3031	0.417488	0.208744	−	0.977970i	$-0.433062\pi$
$733$	11.3031	0.417488	0.208744	+	0.977970i	$0.433062\pi$
$734$	−9.75255	−0.359973
$735$	0	0
$736$	−8.44949	−0.311452
$737$	33.7980	1.24496
$738$	0	0
$739$	52.6969	1.93849	0.969244	−	0.246101i	$-0.0791496\pi$
$739$	52.6969	1.93849	0.969244	+	0.246101i	$0.0791496\pi$
$740$	15.7980	0.580745
$741$	0	0
$742$	48.4949	1.78030
$743$	−5.75255	−0.211041	−0.105520	−	0.994417i	$-0.533651\pi$
$743$	−5.75255	−0.211041	−0.105520	+	0.994417i	$0.533651\pi$
$744$	0	0
$745$	−44.0908	−1.61536
$746$	28.6969	1.05067
$747$	0	0
$748$	−4.89898	−0.179124
$749$	75.1918	2.74745
$750$	0	0
$751$	14.2474	0.519897	0.259948	−	0.965623i	$-0.416294\pi$
$751$	14.2474	0.519897	0.259948	+	0.965623i	$0.416294\pi$
$752$	−4.89898	−0.178647
$753$	0	0
$754$	−7.10102	−0.258604
$755$	4.40408	0.160281
$756$	0	0
$757$	11.7980	0.428804	0.214402	−	0.976745i	$-0.431220\pi$
$757$	11.7980	0.428804	0.214402	+	0.976745i	$0.431220\pi$
$758$	10.2020	0.370555
$759$	0	0
$760$	−16.8990	−0.612990
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	2.89898	0.104950
$764$	14.6969	0.531717
$765$	0	0
$766$	−4.89898	−0.177007
$767$	3.19184	0.115251
$768$	0	0
$769$	−39.3939	−1.42058	−0.710290	−	0.703909i	$-0.751436\pi$
$769$	−39.3939	−1.42058	−0.710290	+	0.703909i	$0.751436\pi$
$770$	−53.3939	−1.92418
$771$	0	0
$772$	18.8990	0.680189
$773$	13.5959	0.489011	0.244506	−	0.969648i	$-0.421374\pi$
$773$	13.5959	0.489011	0.244506	+	0.969648i	$0.421374\pi$
$774$	0	0
$775$	−5.34847	−0.192123
$776$	−14.8990	−0.534842
$777$	0	0
$778$	6.00000	0.215110
$779$	−41.3939	−1.48309
$780$	0	0
$781$	6.60612	0.236386
$782$	8.44949	0.302153
$783$	0	0
$784$	12.7980	0.457070
$785$	24.4949	0.874260
$786$	0	0
$787$	−16.4949	−0.587980	−0.293990	−	0.955809i	$-0.594983\pi$
$787$	−16.4949	−0.587980	−0.293990	+	0.955809i	$0.594983\pi$
$788$	−24.2474	−0.863780
$789$	0	0
$790$	13.1010	0.466113
$791$	4.89898	0.174188
$792$	0	0
$793$	4.49490	0.159618
$794$	12.6515	0.448986
$795$	0	0
$796$	24.0454	0.852267
$797$	−8.69694	−0.308061	−0.154031	−	0.988066i	$-0.549225\pi$
$797$	−8.69694	−0.308061	−0.154031	+	0.988066i	$0.549225\pi$
$798$	0	0
$799$	4.89898	0.173313
$800$	1.00000	0.0353553
$801$	0	0
$802$	3.79796	0.134111
$803$	−48.9898	−1.72881
$804$	0	0
$805$	92.0908	3.24577
$806$	15.5051	0.546144
$807$	0	0
$808$	1.10102	0.0387338
$809$	−3.79796	−0.133529	−0.0667646	−	0.997769i	$-0.521268\pi$
$809$	−3.79796	−0.133529	−0.0667646	+	0.997769i	$0.521268\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	10.8990	0.382479
$813$	0	0
$814$	−31.5959	−1.10744
$815$	9.79796	0.343208
$816$	0	0
$817$	−27.5959	−0.965459
$818$	−25.7980	−0.902004
$819$	0	0
$820$	14.6969	0.513239
$821$	−18.8536	−0.657994	−0.328997	−	0.944331i	$-0.606711\pi$
$821$	−18.8536	−0.657994	−0.328997	+	0.944331i	$0.606711\pi$
$822$	0	0
$823$	−24.9444	−0.869507	−0.434753	−	0.900550i	$-0.643164\pi$
$823$	−24.9444	−0.869507	−0.434753	+	0.900550i	$0.643164\pi$
$824$	3.10102	0.108029
$825$	0	0
$826$	−4.89898	−0.170457
$827$	−28.8990	−1.00492	−0.502458	−	0.864602i	$-0.667571\pi$
$827$	−28.8990	−1.00492	−0.502458	+	0.864602i	$0.667571\pi$
$828$	0	0
$829$	29.1918	1.01387	0.506937	−	0.861983i	$-0.330778\pi$
$829$	29.1918	1.01387	0.506937	+	0.861983i	$0.330778\pi$
$830$	−2.69694	−0.0936121
$831$	0	0
$832$	−2.89898	−0.100504
$833$	−12.7980	−0.443423
$834$	0	0
$835$	−20.6969	−0.716247
$836$	33.7980	1.16893
$837$	0	0
$838$	24.0000	0.829066
$839$	−44.9444	−1.55165	−0.775826	−	0.630947i	$-0.782667\pi$
$839$	−44.9444	−1.55165	−0.775826	+	0.630947i	$0.782667\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	−17.1010	−0.589340
$843$	0	0
$844$	12.8990	0.444001
$845$	11.2577	0.387275
$846$	0	0
$847$	57.8434	1.98752
$848$	10.8990	0.374272
$849$	0	0
$850$	−1.00000	−0.0342997
$851$	54.4949	1.86806
$852$	0	0
$853$	10.4495	0.357784	0.178892	−	0.983869i	$-0.442749\pi$
$853$	10.4495	0.357784	0.178892	+	0.983869i	$0.442749\pi$
$854$	−6.89898	−0.236078
$855$	0	0
$856$	16.8990	0.577595
$857$	27.7980	0.949560	0.474780	−	0.880104i	$-0.342528\pi$
$857$	27.7980	0.949560	0.474780	+	0.880104i	$0.342528\pi$
$858$	0	0
$859$	5.79796	0.197824	0.0989119	−	0.995096i	$-0.468464\pi$
$859$	5.79796	0.197824	0.0989119	+	0.995096i	$0.468464\pi$
$860$	9.79796	0.334108
$861$	0	0
$862$	−20.9444	−0.713369
$863$	−57.7980	−1.96747	−0.983733	−	0.179638i	$-0.942507\pi$
$863$	−57.7980	−1.96747	−0.983733	+	0.179638i	$0.942507\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	37.3939	1.27070
$867$	0	0
$868$	−23.7980	−0.807755
$869$	−26.2020	−0.888843
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0.651531	0.0220636
$873$	0	0
$874$	−58.2929	−1.97179
$875$	43.5959	1.47381
$876$	0	0
$877$	54.0454	1.82498	0.912492	−	0.409095i	$-0.134155\pi$
$877$	54.0454	1.82498	0.912492	+	0.409095i	$0.134155\pi$
$878$	18.6515	0.629459
$879$	0	0
$880$	−12.0000	−0.404520
$881$	−32.6969	−1.10159	−0.550794	−	0.834641i	$-0.685675\pi$
$881$	−32.6969	−1.10159	−0.550794	+	0.834641i	$0.685675\pi$
$882$	0	0
$883$	−24.6969	−0.831118	−0.415559	−	0.909566i	$-0.636414\pi$
$883$	−24.6969	−0.831118	−0.415559	+	0.909566i	$0.636414\pi$
$884$	2.89898	0.0975032
$885$	0	0
$886$	−17.3939	−0.584359
$887$	−52.0454	−1.74751	−0.873757	−	0.486363i	$-0.838323\pi$
$887$	−52.0454	−1.74751	−0.873757	+	0.486363i	$0.838323\pi$
$888$	0	0
$889$	−17.7980	−0.596924
$890$	9.30306	0.311839
$891$	0	0
$892$	−18.6969	−0.626020
$893$	−33.7980	−1.13101
$894$	0	0
$895$	−5.39388	−0.180297
$896$	4.44949	0.148647
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−13.1010	−0.436943
$900$	0	0
$901$	−10.8990	−0.363098
$902$	−29.3939	−0.978709
$903$	0	0
$904$	1.10102	0.0366194
$905$	−20.2020	−0.671539
$906$	0	0
$907$	−8.89898	−0.295486	−0.147743	−	0.989026i	$-0.547201\pi$
$907$	−8.89898	−0.295486	−0.147743	+	0.989026i	$0.547201\pi$
$908$	0	0
$909$	0	0
$910$	31.5959	1.04739
$911$	−1.34847	−0.0446768	−0.0223384	−	0.999750i	$-0.507111\pi$
$911$	−1.34847	−0.0446768	−0.0223384	+	0.999750i	$0.507111\pi$
$912$	0	0
$913$	5.39388	0.178511
$914$	37.3939	1.23688
$915$	0	0
$916$	2.00000	0.0660819
$917$	−75.1918	−2.48305
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	20.6969	0.682358
$921$	0	0
$922$	−8.69694	−0.286418
$923$	−3.90918	−0.128672
$924$	0	0
$925$	−6.44949	−0.212058
$926$	8.49490	0.279160
$927$	0	0
$928$	2.44949	0.0804084
$929$	27.7980	0.912021	0.456011	−	0.889974i	$-0.349278\pi$
$929$	27.7980	0.912021	0.456011	+	0.889974i	$0.349278\pi$
$930$	0	0
$931$	88.2929	2.89368
$932$	8.69694	0.284878
$933$	0	0
$934$	31.5959	1.03385
$935$	12.0000	0.392442
$936$	0	0
$937$	−1.79796	−0.0587368	−0.0293684	−	0.999569i	$-0.509350\pi$
$937$	−1.79796	−0.0587368	−0.0293684	+	0.999569i	$0.509350\pi$
$938$	30.6969	1.00229
$939$	0	0
$940$	12.0000	0.391397
$941$	−5.14643	−0.167769	−0.0838844	−	0.996475i	$-0.526733\pi$
$941$	−5.14643	−0.167769	−0.0838844	+	0.996475i	$0.526733\pi$
$942$	0	0
$943$	50.6969	1.65092
$944$	−1.10102	−0.0358352
$945$	0	0
$946$	−19.5959	−0.637118
$947$	0.494897	0.0160820	0.00804100	−	0.999968i	$-0.497440\pi$
$947$	0.494897	0.0160820	0.00804100	+	0.999968i	$0.497440\pi$
$948$	0	0
$949$	28.9898	0.941049
$950$	6.89898	0.223832
$951$	0	0
$952$	−4.44949	−0.144209
$953$	−49.5959	−1.60657	−0.803285	−	0.595595i	$-0.796916\pi$
$953$	−49.5959	−1.60657	−0.803285	+	0.595595i	$0.796916\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	−26.6969	−0.863441
$957$	0	0
$958$	18.2474	0.589548
$959$	87.1918	2.81557
$960$	0	0
$961$	−2.39388	−0.0772218
$962$	18.6969	0.602813
$963$	0	0
$964$	16.6969	0.537772
$965$	−46.2929	−1.49022
$966$	0	0
$967$	−48.0908	−1.54650	−0.773248	−	0.634103i	$-0.781369\pi$
$967$	−48.0908	−1.54650	−0.773248	+	0.634103i	$0.781369\pi$
$968$	13.0000	0.417836
$969$	0	0
$970$	36.4949	1.17178
$971$	28.2929	0.907961	0.453980	−	0.891012i	$-0.350004\pi$
$971$	28.2929	0.907961	0.453980	+	0.891012i	$0.350004\pi$
$972$	0	0
$973$	13.7980	0.442342
$974$	4.44949	0.142571
$975$	0	0
$976$	−1.55051	−0.0496306
$977$	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
$979$	−18.6061	−0.594654
$980$	−31.3485	−1.00139
$981$	0	0
$982$	41.3939	1.32093
$983$	−39.5505	−1.26147	−0.630733	−	0.776000i	$-0.717246\pi$
$983$	−39.5505	−1.26147	−0.630733	+	0.776000i	$0.717246\pi$
$984$	0	0
$985$	59.3939	1.89245
$986$	−2.44949	−0.0780076
$987$	0	0
$988$	−20.0000	−0.636285
$989$	33.7980	1.07471
$990$	0	0
$991$	−8.04541	−0.255571	−0.127785	−	0.991802i	$-0.540787\pi$
$991$	−8.04541	−0.255571	−0.127785	+	0.991802i	$0.540787\pi$
$992$	−5.34847	−0.169814
$993$	0	0
$994$	6.00000	0.190308
$995$	−58.8990	−1.86722
$996$	0	0
$997$	12.6515	0.400678	0.200339	−	0.979727i	$-0.435796\pi$
$997$	12.6515	0.400678	0.200339	+	0.979727i	$0.435796\pi$
$998$	5.30306	0.167865
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.44949 q^{5} +4.44949 q^{7} +1.00000 q^{8} -2.44949 q^{10} +4.89898 q^{11} -2.89898 q^{13} +4.44949 q^{14} +1.00000 q^{16} -1.00000 q^{17} +6.89898 q^{19} -2.44949 q^{20} +4.89898 q^{22} -8.44949 q^{23} +1.00000 q^{25} -2.89898 q^{26} +4.44949 q^{28} +2.44949 q^{29} -5.34847 q^{31} +1.00000 q^{32} -1.00000 q^{34} -10.8990 q^{35} -6.44949 q^{37} +6.89898 q^{38} -2.44949 q^{40} -6.00000 q^{41} -4.00000 q^{43} +4.89898 q^{44} -8.44949 q^{46} -4.89898 q^{47} +12.7980 q^{49} +1.00000 q^{50} -2.89898 q^{52} +10.8990 q^{53} -12.0000 q^{55} +4.44949 q^{56} +2.44949 q^{58} -1.10102 q^{59} -1.55051 q^{61} -5.34847 q^{62} +1.00000 q^{64} +7.10102 q^{65} +6.89898 q^{67} -1.00000 q^{68} -10.8990 q^{70} +1.34847 q^{71} -10.0000 q^{73} -6.44949 q^{74} +6.89898 q^{76} +21.7980 q^{77} -5.34847 q^{79} -2.44949 q^{80} -6.00000 q^{82} +1.10102 q^{83} +2.44949 q^{85} -4.00000 q^{86} +4.89898 q^{88} -3.79796 q^{89} -12.8990 q^{91} -8.44949 q^{92} -4.89898 q^{94} -16.8990 q^{95} -14.8990 q^{97} +12.7980 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 2 q^{17} + 4 q^{19} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} - 2 q^{34} - 12 q^{35} - 8 q^{37} + 4 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more