LMFDB - Embedded newform 3042.2.b.l.1351.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1351,3042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3042.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,0,0,0,0,12,0,4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.l.1351.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.73205i q^{5} -4.73205i q^{7} -1.00000i q^{8} -1.73205 q^{10} +4.73205i q^{11} +4.73205 q^{14} +1.00000 q^{16} -5.19615 q^{17} +1.26795i q^{19} -1.73205i q^{20} -4.73205 q^{22} +2.19615 q^{23} +2.00000 q^{25} +4.73205i q^{28} +3.00000 q^{29} +2.53590i q^{31} +1.00000i q^{32} -5.19615i q^{34} +8.19615 q^{35} -3.00000i q^{37} -1.26795 q^{38} +1.73205 q^{40} +0.464102i q^{41} -6.19615 q^{43} -4.73205i q^{44} +2.19615i q^{46} -1.26795i q^{47} -15.3923 q^{49} +2.00000i q^{50} -3.00000 q^{53} -8.19615 q^{55} -4.73205 q^{56} +3.00000i q^{58} +13.8564i q^{59} +4.80385 q^{61} -2.53590 q^{62} -1.00000 q^{64} +10.7321i q^{67} +5.19615 q^{68} +8.19615i q^{70} -8.19615i q^{71} +12.1244i q^{73} +3.00000 q^{74} -1.26795i q^{76} +22.3923 q^{77} -12.3923 q^{79} +1.73205i q^{80} -0.464102 q^{82} +11.6603i q^{83} -9.00000i q^{85} -6.19615i q^{86} +4.73205 q^{88} +2.53590i q^{89} -2.19615 q^{92} +1.26795 q^{94} -2.19615 q^{95} +6.00000i q^{97} -15.3923i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64}+ \cdots + 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 12 * q^14 + 4 * q^16 - 12 * q^22 - 12 * q^23 + 8 * q^25 + 12 * q^29 + 12 * q^35 - 12 * q^38 - 4 * q^43 - 20 * q^49 - 12 * q^53 - 12 * q^55 - 12 * q^56 + 40 * q^61 - 24 * q^62 - 4 * q^64 + 12 * q^74 + 48 * q^77 - 8 * q^79 + 12 * q^82 + 12 * q^88 + 12 * q^92 + 12 * q^94 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.73205i	0.774597i	0.921954	+	0.387298i	$0.126592\pi$
$5$	1.73205i	0.774597i	−0.921954	+	0.387298i	$0.873408\pi$
$6$	0	0
$7$	− 4.73205i	− 1.78855i	−0.447521	−	0.894274i	$-0.647693\pi$
$7$	− 4.73205i	− 1.78855i	0.447521	−	0.894274i	$-0.352307\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	−1.73205	−0.547723
$11$	4.73205i	1.42677i	0.700774	+	0.713384i	$0.252838\pi$
$11$	4.73205i	1.42677i	−0.700774	+	0.713384i	$0.747162\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.73205	1.26469
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.19615	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−5.19615	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	0	0
$19$	1.26795i	0.290887i	0.989367	+	0.145444i	$0.0464610\pi$
$19$	1.26795i	0.290887i	−0.989367	+	0.145444i	$0.953539\pi$
$20$	− 1.73205i	− 0.387298i
$21$	0	0
$22$	−4.73205	−1.00888
$23$	2.19615	0.457929	0.228965	−	0.973435i	$-0.426466\pi$
$23$	2.19615	0.457929	0.228965	+	0.973435i	$0.426466\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	4.73205i	0.894274i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	2.53590i	0.455461i	0.973724	+	0.227730i	$0.0731305\pi$
$31$	2.53590i	0.455461i	−0.973724	+	0.227730i	$0.926870\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 5.19615i	− 0.891133i
$35$	8.19615	1.38540
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	−1.26795	−0.205689
$39$	0	0
$40$	1.73205	0.273861
$41$	0.464102i	0.0724805i	0.999343	+	0.0362402i	$0.0115382\pi$
$41$	0.464102i	0.0724805i	−0.999343	+	0.0362402i	$0.988462\pi$
$42$	0	0
$43$	−6.19615	−0.944904	−0.472452	−	0.881356i	$-0.656631\pi$
$43$	−6.19615	−0.944904	−0.472452	+	0.881356i	$0.656631\pi$
$44$	− 4.73205i	− 0.713384i
$45$	0	0
$46$	2.19615i	0.323805i
$47$	− 1.26795i	− 0.184949i	−0.995715	−	0.0924747i	$-0.970522\pi$
$47$	− 1.26795i	− 0.184949i	0.995715	−	0.0924747i	$-0.0294777\pi$
$48$	0	0
$49$	−15.3923	−2.19890
$50$	2.00000i	0.282843i
$51$	0	0
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	−8.19615	−1.10517
$56$	−4.73205	−0.632347
$57$	0	0
$58$	3.00000i	0.393919i
$59$	13.8564i	1.80395i	0.431788	+	0.901975i	$0.357883\pi$
$59$	13.8564i	1.80395i	−0.431788	+	0.901975i	$0.642117\pi$
$60$	0	0
$61$	4.80385	0.615070	0.307535	−	0.951537i	$-0.400496\pi$
$61$	4.80385	0.615070	0.307535	+	0.951537i	$0.400496\pi$
$62$	−2.53590	−0.322059
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	10.7321i	1.31113i	0.755139	+	0.655564i	$0.227569\pi$
$67$	10.7321i	1.31113i	−0.755139	+	0.655564i	$0.772431\pi$
$68$	5.19615	0.630126
$69$	0	0
$70$	8.19615i	0.979628i
$71$	− 8.19615i	− 0.972704i	−0.873763	−	0.486352i	$-0.838327\pi$
$71$	− 8.19615i	− 0.972704i	0.873763	−	0.486352i	$-0.161673\pi$
$72$	0	0
$73$	12.1244i	1.41905i	0.704681	+	0.709524i	$0.251090\pi$
$73$	12.1244i	1.41905i	−0.704681	+	0.709524i	$0.748910\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	− 1.26795i	− 0.145444i
$77$	22.3923	2.55184
$78$	0	0
$79$	−12.3923	−1.39424	−0.697122	−	0.716953i	$-0.745536\pi$
$79$	−12.3923	−1.39424	−0.697122	+	0.716953i	$0.745536\pi$
$80$	1.73205i	0.193649i
$81$	0	0
$82$	−0.464102	−0.0512514
$83$	11.6603i	1.27988i	0.768425	+	0.639940i	$0.221041\pi$
$83$	11.6603i	1.27988i	−0.768425	+	0.639940i	$0.778959\pi$
$84$	0	0
$85$	− 9.00000i	− 0.976187i
$86$	− 6.19615i	− 0.668148i
$87$	0	0
$88$	4.73205	0.504438
$89$	2.53590i	0.268805i	0.990927	+	0.134402i	$0.0429115\pi$
$89$	2.53590i	0.268805i	−0.990927	+	0.134402i	$0.957089\pi$
$90$	0	0
$91$	0	0
$92$	−2.19615	−0.228965
$93$	0	0
$94$	1.26795	0.130779
$95$	−2.19615	−0.225320
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	− 15.3923i	− 1.55486i
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−1.39230	−0.138540	−0.0692698	−	0.997598i	$-0.522067\pi$
$101$	−1.39230	−0.138540	−0.0692698	+	0.997598i	$0.522067\pi$
$102$	0	0
$103$	4.19615	0.413459	0.206730	−	0.978398i	$-0.433718\pi$
$103$	4.19615	0.413459	0.206730	+	0.978398i	$0.433718\pi$
$104$	0	0
$105$	0	0
$106$	− 3.00000i	− 0.291386i
$107$	−8.19615	−0.792352	−0.396176	−	0.918175i	$-0.629663\pi$
$107$	−8.19615	−0.792352	−0.396176	+	0.918175i	$0.629663\pi$
$108$	0	0
$109$	16.3923i	1.57010i	0.619434	+	0.785049i	$0.287362\pi$
$109$	16.3923i	1.57010i	−0.619434	+	0.785049i	$0.712638\pi$
$110$	− 8.19615i	− 0.781472i
$111$	0	0
$112$	− 4.73205i	− 0.447137i
$113$	−11.1962	−1.05325	−0.526623	−	0.850099i	$-0.676542\pi$
$113$	−11.1962	−1.05325	−0.526623	+	0.850099i	$0.676542\pi$
$114$	0	0
$115$	3.80385i	0.354711i
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−13.8564	−1.27559
$119$	24.5885i	2.25402i
$120$	0	0
$121$	−11.3923	−1.03566
$122$	4.80385i	0.434920i
$123$	0	0
$124$	− 2.53590i	− 0.227730i
$125$	12.1244i	1.08444i
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−16.3923	−1.43220	−0.716101	−	0.697997i	$-0.754075\pi$
$131$	−16.3923	−1.43220	−0.716101	+	0.697997i	$0.754075\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−10.7321	−0.927108
$135$	0	0
$136$	5.19615i	0.445566i
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−8.19615	−0.692701
$141$	0	0
$142$	8.19615	0.687806
$143$	0	0
$144$	0	0
$145$	5.19615i	0.431517i
$146$	−12.1244	−1.00342
$147$	0	0
$148$	3.00000i	0.246598i
$149$	18.1244i	1.48481i	0.669954	+	0.742403i	$0.266314\pi$
$149$	18.1244i	1.48481i	−0.669954	+	0.742403i	$0.733686\pi$
$150$	0	0
$151$	− 7.26795i	− 0.591457i	−0.955272	−	0.295729i	$-0.904438\pi$
$151$	− 7.26795i	− 0.591457i	0.955272	−	0.295729i	$-0.0955624\pi$
$152$	1.26795	0.102844
$153$	0	0
$154$	22.3923i	1.80442i
$155$	−4.39230	−0.352798
$156$	0	0
$157$	−3.19615	−0.255081	−0.127540	−	0.991833i	$-0.540708\pi$
$157$	−3.19615	−0.255081	−0.127540	+	0.991833i	$0.540708\pi$
$158$	− 12.3923i	− 0.985879i
$159$	0	0
$160$	−1.73205	−0.136931
$161$	− 10.3923i	− 0.819028i
$162$	0	0
$163$	− 9.46410i	− 0.741286i	−0.928775	−	0.370643i	$-0.879137\pi$
$163$	− 9.46410i	− 0.741286i	0.928775	−	0.370643i	$-0.120863\pi$
$164$	− 0.464102i	− 0.0362402i
$165$	0	0
$166$	−11.6603	−0.905011
$167$	− 2.53590i	− 0.196234i	−0.995175	−	0.0981169i	$-0.968718\pi$
$167$	− 2.53590i	− 0.196234i	0.995175	−	0.0981169i	$-0.0312819\pi$
$168$	0	0
$169$	0	0
$170$	9.00000	0.690268
$171$	0	0
$172$	6.19615	0.472452
$173$	16.3923	1.24628	0.623142	−	0.782109i	$-0.285856\pi$
$173$	16.3923	1.24628	0.623142	+	0.782109i	$0.285856\pi$
$174$	0	0
$175$	− 9.46410i	− 0.715419i
$176$	4.73205i	0.356692i
$177$	0	0
$178$	−2.53590	−0.190074
$179$	−8.19615	−0.612609	−0.306305	−	0.951934i	$-0.599093\pi$
$179$	−8.19615	−0.612609	−0.306305	+	0.951934i	$0.599093\pi$
$180$	0	0
$181$	11.5885	0.861363	0.430682	−	0.902504i	$-0.358273\pi$
$181$	11.5885	0.861363	0.430682	+	0.902504i	$0.358273\pi$
$182$	0	0
$183$	0	0
$184$	− 2.19615i	− 0.161903i
$185$	5.19615	0.382029
$186$	0	0
$187$	− 24.5885i	− 1.79809i
$188$	1.26795i	0.0924747i
$189$	0	0
$190$	− 2.19615i	− 0.159326i
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	0	0
$193$	12.8038i	0.921641i	0.887493	+	0.460821i	$0.152445\pi$
$193$	12.8038i	0.921641i	−0.887493	+	0.460821i	$0.847555\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	15.3923	1.09945
$197$	− 6.92820i	− 0.493614i	−0.969065	−	0.246807i	$-0.920619\pi$
$197$	− 6.92820i	− 0.493614i	0.969065	−	0.246807i	$-0.0793814\pi$
$198$	0	0
$199$	−8.58846	−0.608820	−0.304410	−	0.952541i	$-0.598459\pi$
$199$	−8.58846	−0.608820	−0.304410	+	0.952541i	$0.598459\pi$
$200$	− 2.00000i	− 0.141421i
$201$	0	0
$202$	− 1.39230i	− 0.0979622i
$203$	− 14.1962i	− 0.996375i
$204$	0	0
$205$	−0.803848	−0.0561432
$206$	4.19615i	0.292360i
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−3.60770	−0.248364	−0.124182	−	0.992259i	$-0.539631\pi$
$211$	−3.60770	−0.248364	−0.124182	+	0.992259i	$0.539631\pi$
$212$	3.00000	0.206041
$213$	0	0
$214$	− 8.19615i	− 0.560277i
$215$	− 10.7321i	− 0.731920i
$216$	0	0
$217$	12.0000	0.814613
$218$	−16.3923	−1.11023
$219$	0	0
$220$	8.19615	0.552584
$221$	0	0
$222$	0	0
$223$	− 18.9282i	− 1.26753i	−0.773527	−	0.633763i	$-0.781509\pi$
$223$	− 18.9282i	− 1.26753i	0.773527	−	0.633763i	$-0.218491\pi$
$224$	4.73205	0.316173
$225$	0	0
$226$	− 11.1962i	− 0.744757i
$227$	9.80385i	0.650704i	0.945593	+	0.325352i	$0.105483\pi$
$227$	9.80385i	0.650704i	−0.945593	+	0.325352i	$0.894517\pi$
$228$	0	0
$229$	− 19.8564i	− 1.31215i	−0.754696	−	0.656074i	$-0.772216\pi$
$229$	− 19.8564i	− 1.31215i	0.754696	−	0.656074i	$-0.227784\pi$
$230$	−3.80385	−0.250818
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	2.19615	0.143261
$236$	− 13.8564i	− 0.901975i
$237$	0	0
$238$	−24.5885	−1.59383
$239$	24.5885i	1.59050i	0.606285	+	0.795248i	$0.292659\pi$
$239$	24.5885i	1.59050i	−0.606285	+	0.795248i	$0.707341\pi$
$240$	0	0
$241$	− 0.803848i	− 0.0517804i	−0.999665	−	0.0258902i	$-0.991758\pi$
$241$	− 0.803848i	− 0.0517804i	0.999665	−	0.0258902i	$-0.00824202\pi$
$242$	− 11.3923i	− 0.732325i
$243$	0	0
$244$	−4.80385	−0.307535
$245$	− 26.6603i	− 1.70326i
$246$	0	0
$247$	0	0
$248$	2.53590	0.161030
$249$	0	0
$250$	−12.1244	−0.766812
$251$	−4.39230	−0.277240	−0.138620	−	0.990346i	$-0.544267\pi$
$251$	−4.39230	−0.277240	−0.138620	+	0.990346i	$0.544267\pi$
$252$	0	0
$253$	10.3923i	0.653359i
$254$	4.00000i	0.250982i
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.8038	0.798682	0.399341	−	0.916802i	$-0.369239\pi$
$257$	12.8038	0.798682	0.399341	+	0.916802i	$0.369239\pi$
$258$	0	0
$259$	−14.1962	−0.882106
$260$	0	0
$261$	0	0
$262$	− 16.3923i	− 1.01272i
$263$	2.19615	0.135421	0.0677103	−	0.997705i	$-0.478431\pi$
$263$	2.19615	0.135421	0.0677103	+	0.997705i	$0.478431\pi$
$264$	0	0
$265$	− 5.19615i	− 0.319197i
$266$	6.00000i	0.367884i
$267$	0	0
$268$	− 10.7321i	− 0.655564i
$269$	28.3923	1.73111	0.865555	−	0.500814i	$-0.166966\pi$
$269$	28.3923	1.73111	0.865555	+	0.500814i	$0.166966\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−5.19615	−0.315063
$273$	0	0
$274$	9.00000	0.543710
$275$	9.46410i	0.570707i
$276$	0	0
$277$	−15.1962	−0.913048	−0.456524	−	0.889711i	$-0.650906\pi$
$277$	−15.1962	−0.913048	−0.456524	+	0.889711i	$0.650906\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	− 8.19615i	− 0.489814i
$281$	24.4641i	1.45941i	0.683764	+	0.729703i	$0.260342\pi$
$281$	24.4641i	1.45941i	−0.683764	+	0.729703i	$0.739658\pi$
$282$	0	0
$283$	30.1962	1.79497	0.897487	−	0.441040i	$-0.145390\pi$
$283$	30.1962	1.79497	0.897487	+	0.441040i	$0.145390\pi$
$284$	8.19615i	0.486352i
$285$	0	0
$286$	0	0
$287$	2.19615	0.129635
$288$	0	0
$289$	10.0000	0.588235
$290$	−5.19615	−0.305129
$291$	0	0
$292$	− 12.1244i	− 0.709524i
$293$	− 14.6603i	− 0.856461i	−0.903669	−	0.428231i	$-0.859137\pi$
$293$	− 14.6603i	− 0.856461i	0.903669	−	0.428231i	$-0.140863\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−18.1244	−1.04992
$299$	0	0
$300$	0	0
$301$	29.3205i	1.69001i
$302$	7.26795	0.418223
$303$	0	0
$304$	1.26795i	0.0727219i
$305$	8.32051i	0.476431i
$306$	0	0
$307$	− 10.7321i	− 0.612510i	−0.951949	−	0.306255i	$-0.900924\pi$
$307$	− 10.7321i	− 0.612510i	0.951949	−	0.306255i	$-0.0990761\pi$
$308$	−22.3923	−1.27592
$309$	0	0
$310$	− 4.39230i	− 0.249466i
$311$	2.19615	0.124532	0.0622662	−	0.998060i	$-0.480167\pi$
$311$	2.19615	0.124532	0.0622662	+	0.998060i	$0.480167\pi$
$312$	0	0
$313$	−24.3923	−1.37873	−0.689367	−	0.724412i	$-0.742111\pi$
$313$	−24.3923	−1.37873	−0.689367	+	0.724412i	$0.742111\pi$
$314$	− 3.19615i	− 0.180369i
$315$	0	0
$316$	12.3923	0.697122
$317$	− 6.12436i	− 0.343978i	−0.985099	−	0.171989i	$-0.944981\pi$
$317$	− 6.12436i	− 0.343978i	0.985099	−	0.171989i	$-0.0550194\pi$
$318$	0	0
$319$	14.1962i	0.794832i
$320$	− 1.73205i	− 0.0968246i
$321$	0	0
$322$	10.3923	0.579141
$323$	− 6.58846i	− 0.366592i
$324$	0	0
$325$	0	0
$326$	9.46410	0.524168
$327$	0	0
$328$	0.464102	0.0256257
$329$	−6.00000	−0.330791
$330$	0	0
$331$	12.0000i	0.659580i	0.944054	+	0.329790i	$0.106978\pi$
$331$	12.0000i	0.659580i	−0.944054	+	0.329790i	$0.893022\pi$
$332$	− 11.6603i	− 0.639940i
$333$	0	0
$334$	2.53590	0.138758
$335$	−18.5885	−1.01560
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	0	0
$339$	0	0
$340$	9.00000i	0.488094i
$341$	−12.0000	−0.649836
$342$	0	0
$343$	39.7128i	2.14429i
$344$	6.19615i	0.334074i
$345$	0	0
$346$	16.3923i	0.881256i
$347$	12.5885	0.675784	0.337892	−	0.941185i	$-0.390286\pi$
$347$	12.5885	0.675784	0.337892	+	0.941185i	$0.390286\pi$
$348$	0	0
$349$	− 2.53590i	− 0.135744i	−0.997694	−	0.0678718i	$-0.978379\pi$
$349$	− 2.53590i	− 0.135744i	0.997694	−	0.0678718i	$-0.0216209\pi$
$350$	9.46410	0.505878
$351$	0	0
$352$	−4.73205	−0.252219
$353$	− 5.78461i	− 0.307884i	−0.988080	−	0.153942i	$-0.950803\pi$
$353$	− 5.78461i	− 0.307884i	0.988080	−	0.153942i	$-0.0491968\pi$
$354$	0	0
$355$	14.1962	0.753454
$356$	− 2.53590i	− 0.134402i
$357$	0	0
$358$	− 8.19615i	− 0.433180i
$359$	22.0526i	1.16389i	0.813228	+	0.581945i	$0.197708\pi$
$359$	22.0526i	1.16389i	−0.813228	+	0.581945i	$0.802292\pi$
$360$	0	0
$361$	17.3923	0.915384
$362$	11.5885i	0.609076i
$363$	0	0
$364$	0	0
$365$	−21.0000	−1.09919
$366$	0	0
$367$	−24.1962	−1.26303	−0.631514	−	0.775364i	$-0.717566\pi$
$367$	−24.1962	−1.26303	−0.631514	+	0.775364i	$0.717566\pi$
$368$	2.19615	0.114482
$369$	0	0
$370$	5.19615i	0.270135i
$371$	14.1962i	0.737028i
$372$	0	0
$373$	23.9808	1.24168	0.620838	−	0.783939i	$-0.286792\pi$
$373$	23.9808	1.24168	0.620838	+	0.783939i	$0.286792\pi$
$374$	24.5885	1.27144
$375$	0	0
$376$	−1.26795	−0.0653895
$377$	0	0
$378$	0	0
$379$	− 18.2487i	− 0.937373i	−0.883364	−	0.468687i	$-0.844727\pi$
$379$	− 18.2487i	− 0.937373i	0.883364	−	0.468687i	$-0.155273\pi$
$380$	2.19615	0.112660
$381$	0	0
$382$	− 20.7846i	− 1.06343i
$383$	− 11.3205i	− 0.578451i	−0.957261	−	0.289225i	$-0.906602\pi$
$383$	− 11.3205i	− 0.578451i	0.957261	−	0.289225i	$-0.0933977\pi$
$384$	0	0
$385$	38.7846i	1.97665i
$386$	−12.8038	−0.651699
$387$	0	0
$388$	− 6.00000i	− 0.304604i
$389$	−13.3923	−0.679017	−0.339508	−	0.940603i	$-0.610261\pi$
$389$	−13.3923	−0.679017	−0.339508	+	0.940603i	$0.610261\pi$
$390$	0	0
$391$	−11.4115	−0.577107
$392$	15.3923i	0.777429i
$393$	0	0
$394$	6.92820	0.349038
$395$	− 21.4641i	− 1.07998i
$396$	0	0
$397$	16.3923i	0.822706i	0.911476	+	0.411353i	$0.134944\pi$
$397$	16.3923i	0.822706i	−0.911476	+	0.411353i	$0.865056\pi$
$398$	− 8.58846i	− 0.430500i
$399$	0	0
$400$	2.00000	0.100000
$401$	21.0000i	1.04869i	0.851506	+	0.524345i	$0.175690\pi$
$401$	21.0000i	1.04869i	−0.851506	+	0.524345i	$0.824310\pi$
$402$	0	0
$403$	0	0
$404$	1.39230	0.0692698
$405$	0	0
$406$	14.1962	0.704543
$407$	14.1962	0.703677
$408$	0	0
$409$	− 3.33975i	− 0.165140i	−0.996585	−	0.0825699i	$-0.973687\pi$
$409$	− 3.33975i	− 0.165140i	0.996585	−	0.0825699i	$-0.0263128\pi$
$410$	− 0.803848i	− 0.0396992i
$411$	0	0
$412$	−4.19615	−0.206730
$413$	65.5692	3.22645
$414$	0	0
$415$	−20.1962	−0.991390
$416$	0	0
$417$	0	0
$418$	− 6.00000i	− 0.293470i
$419$	−16.3923	−0.800816	−0.400408	−	0.916337i	$-0.631132\pi$
$419$	−16.3923	−0.800816	−0.400408	+	0.916337i	$0.631132\pi$
$420$	0	0
$421$	0.464102i	0.0226189i	0.999936	+	0.0113095i	$0.00359999\pi$
$421$	0.464102i	0.0226189i	−0.999936	+	0.0113095i	$0.996400\pi$
$422$	− 3.60770i	− 0.175620i
$423$	0	0
$424$	3.00000i	0.145693i
$425$	−10.3923	−0.504101
$426$	0	0
$427$	− 22.7321i	− 1.10008i
$428$	8.19615	0.396176
$429$	0	0
$430$	10.7321	0.517545
$431$	− 27.8038i	− 1.33926i	−0.742693	−	0.669632i	$-0.766452\pi$
$431$	− 27.8038i	− 1.33926i	0.742693	−	0.669632i	$-0.233548\pi$
$432$	0	0
$433$	33.7846	1.62358	0.811792	−	0.583946i	$-0.198492\pi$
$433$	33.7846	1.62358	0.811792	+	0.583946i	$0.198492\pi$
$434$	12.0000i	0.576018i
$435$	0	0
$436$	− 16.3923i	− 0.785049i
$437$	2.78461i	0.133206i
$438$	0	0
$439$	−16.5885	−0.791724	−0.395862	−	0.918310i	$-0.629554\pi$
$439$	−16.5885	−0.791724	−0.395862	+	0.918310i	$0.629554\pi$
$440$	8.19615i	0.390736i
$441$	0	0
$442$	0	0
$443$	4.39230	0.208685	0.104342	−	0.994541i	$-0.466726\pi$
$443$	4.39230	0.208685	0.104342	+	0.994541i	$0.466726\pi$
$444$	0	0
$445$	−4.39230	−0.208215
$446$	18.9282	0.896276
$447$	0	0
$448$	4.73205i	0.223568i
$449$	33.4641i	1.57927i	0.613578	+	0.789634i	$0.289730\pi$
$449$	33.4641i	1.57927i	−0.613578	+	0.789634i	$0.710270\pi$
$450$	0	0
$451$	−2.19615	−0.103413
$452$	11.1962	0.526623
$453$	0	0
$454$	−9.80385	−0.460117
$455$	0	0
$456$	0	0
$457$	19.9808i	0.934661i	0.884083	+	0.467330i	$0.154784\pi$
$457$	19.9808i	0.934661i	−0.884083	+	0.467330i	$0.845216\pi$
$458$	19.8564	0.927829
$459$	0	0
$460$	− 3.80385i	− 0.177355i
$461$	19.9808i	0.930597i	0.885154	+	0.465298i	$0.154053\pi$
$461$	19.9808i	0.930597i	−0.885154	+	0.465298i	$0.845947\pi$
$462$	0	0
$463$	26.1962i	1.21744i	0.793386	+	0.608719i	$0.208316\pi$
$463$	26.1962i	1.21744i	−0.793386	+	0.608719i	$0.791684\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	− 18.0000i	− 0.833834i
$467$	36.5885	1.69311	0.846556	−	0.532300i	$-0.178672\pi$
$467$	36.5885	1.69311	0.846556	+	0.532300i	$0.178672\pi$
$468$	0	0
$469$	50.7846	2.34502
$470$	2.19615i	0.101301i
$471$	0	0
$472$	13.8564	0.637793
$473$	− 29.3205i	− 1.34816i
$474$	0	0
$475$	2.53590i	0.116355i
$476$	− 24.5885i	− 1.12701i
$477$	0	0
$478$	−24.5885	−1.12465
$479$	− 35.3205i	− 1.61384i	−0.590664	−	0.806918i	$-0.701134\pi$
$479$	− 35.3205i	− 1.61384i	0.590664	−	0.806918i	$-0.298866\pi$
$480$	0	0
$481$	0	0
$482$	0.803848	0.0366143
$483$	0	0
$484$	11.3923	0.517832
$485$	−10.3923	−0.471890
$486$	0	0
$487$	9.12436i	0.413464i	0.978398	+	0.206732i	$0.0662828\pi$
$487$	9.12436i	0.413464i	−0.978398	+	0.206732i	$0.933717\pi$
$488$	− 4.80385i	− 0.217460i
$489$	0	0
$490$	26.6603	1.20439
$491$	0.588457	0.0265567	0.0132784	−	0.999912i	$-0.495773\pi$
$491$	0.588457	0.0265567	0.0132784	+	0.999912i	$0.495773\pi$
$492$	0	0
$493$	−15.5885	−0.702069
$494$	0	0
$495$	0	0
$496$	2.53590i	0.113865i
$497$	−38.7846	−1.73973
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 12.1244i	− 0.542218i
$501$	0	0
$502$	− 4.39230i	− 0.196038i
$503$	18.5885	0.828818	0.414409	−	0.910091i	$-0.363988\pi$
$503$	18.5885	0.828818	0.414409	+	0.910091i	$0.363988\pi$
$504$	0	0
$505$	− 2.41154i	− 0.107312i
$506$	−10.3923	−0.461994
$507$	0	0
$508$	−4.00000	−0.177471
$509$	− 9.33975i	− 0.413977i	−0.978343	−	0.206988i	$-0.933634\pi$
$509$	− 9.33975i	− 0.413977i	0.978343	−	0.206988i	$-0.0663663\pi$
$510$	0	0
$511$	57.3731	2.53804
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	12.8038i	0.564754i
$515$	7.26795i	0.320264i
$516$	0	0
$517$	6.00000	0.263880
$518$	− 14.1962i	− 0.623743i
$519$	0	0
$520$	0	0
$521$	18.8038	0.823812	0.411906	−	0.911226i	$-0.364863\pi$
$521$	18.8038	0.823812	0.411906	+	0.911226i	$0.364863\pi$
$522$	0	0
$523$	1.41154	0.0617225	0.0308612	−	0.999524i	$-0.490175\pi$
$523$	1.41154	0.0617225	0.0308612	+	0.999524i	$0.490175\pi$
$524$	16.3923	0.716101
$525$	0	0
$526$	2.19615i	0.0957568i
$527$	− 13.1769i	− 0.573995i
$528$	0	0
$529$	−18.1769	−0.790301
$530$	5.19615	0.225706
$531$	0	0
$532$	−6.00000	−0.260133
$533$	0	0
$534$	0	0
$535$	− 14.1962i	− 0.613753i
$536$	10.7321	0.463554
$537$	0	0
$538$	28.3923i	1.22408i
$539$	− 72.8372i	− 3.13732i
$540$	0	0
$541$	− 16.8564i	− 0.724714i	−0.932039	−	0.362357i	$-0.881972\pi$
$541$	− 16.8564i	− 0.724714i	0.932039	−	0.362357i	$-0.118028\pi$
$542$	0	0
$543$	0	0
$544$	− 5.19615i	− 0.222783i
$545$	−28.3923	−1.21619
$546$	0	0
$547$	−6.19615	−0.264928	−0.132464	−	0.991188i	$-0.542289\pi$
$547$	−6.19615	−0.264928	−0.132464	+	0.991188i	$0.542289\pi$
$548$	9.00000i	0.384461i
$549$	0	0
$550$	−9.46410	−0.403551
$551$	3.80385i	0.162049i
$552$	0	0
$553$	58.6410i	2.49367i
$554$	− 15.1962i	− 0.645623i
$555$	0	0
$556$	4.00000	0.169638
$557$	22.2679i	0.943523i	0.881726	+	0.471762i	$0.156382\pi$
$557$	22.2679i	0.943523i	−0.881726	+	0.471762i	$0.843618\pi$
$558$	0	0
$559$	0	0
$560$	8.19615	0.346351
$561$	0	0
$562$	−24.4641	−1.03196
$563$	8.78461	0.370227	0.185114	−	0.982717i	$-0.440735\pi$
$563$	8.78461	0.370227	0.185114	+	0.982717i	$0.440735\pi$
$564$	0	0
$565$	− 19.3923i	− 0.815840i
$566$	30.1962i	1.26924i
$567$	0	0
$568$	−8.19615	−0.343903
$569$	−32.7846	−1.37440	−0.687201	−	0.726467i	$-0.741161\pi$
$569$	−32.7846	−1.37440	−0.687201	+	0.726467i	$0.741161\pi$
$570$	0	0
$571$	13.8038	0.577673	0.288837	−	0.957378i	$-0.406732\pi$
$571$	13.8038	0.577673	0.288837	+	0.957378i	$0.406732\pi$
$572$	0	0
$573$	0	0
$574$	2.19615i	0.0916656i
$575$	4.39230	0.183172
$576$	0	0
$577$	− 16.2679i	− 0.677244i	−0.940923	−	0.338622i	$-0.890039\pi$
$577$	− 16.2679i	− 0.677244i	0.940923	−	0.338622i	$-0.109961\pi$
$578$	10.0000i	0.415945i
$579$	0	0
$580$	− 5.19615i	− 0.215758i
$581$	55.1769	2.28912
$582$	0	0
$583$	− 14.1962i	− 0.587945i
$584$	12.1244	0.501709
$585$	0	0
$586$	14.6603	0.605610
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−3.21539	−0.132488
$590$	− 24.0000i	− 0.988064i
$591$	0	0
$592$	− 3.00000i	− 0.123299i
$593$	− 46.8564i	− 1.92416i	−0.272764	−	0.962081i	$-0.587938\pi$
$593$	− 46.8564i	− 1.92416i	0.272764	−	0.962081i	$-0.412062\pi$
$594$	0	0
$595$	−42.5885	−1.74596
$596$	− 18.1244i	− 0.742403i
$597$	0	0
$598$	0	0
$599$	4.39230	0.179465	0.0897324	−	0.995966i	$-0.471399\pi$
$599$	4.39230	0.179465	0.0897324	+	0.995966i	$0.471399\pi$
$600$	0	0
$601$	21.7846	0.888613	0.444306	−	0.895875i	$-0.353450\pi$
$601$	21.7846	0.888613	0.444306	+	0.895875i	$0.353450\pi$
$602$	−29.3205	−1.19501
$603$	0	0
$604$	7.26795i	0.295729i
$605$	− 19.7321i	− 0.802222i
$606$	0	0
$607$	−48.7846	−1.98011	−0.990053	−	0.140694i	$-0.955067\pi$
$607$	−48.7846	−1.98011	−0.990053	+	0.140694i	$0.955067\pi$
$608$	−1.26795	−0.0514221
$609$	0	0
$610$	−8.32051	−0.336888
$611$	0	0
$612$	0	0
$613$	− 40.8564i	− 1.65017i	−0.565005	−	0.825087i	$-0.691126\pi$
$613$	− 40.8564i	− 1.65017i	0.565005	−	0.825087i	$-0.308874\pi$
$614$	10.7321	0.433110
$615$	0	0
$616$	− 22.3923i	− 0.902212i
$617$	− 10.6077i	− 0.427050i	−0.976938	−	0.213525i	$-0.931506\pi$
$617$	− 10.6077i	− 0.427050i	0.976938	−	0.213525i	$-0.0684944\pi$
$618$	0	0
$619$	7.60770i	0.305779i	0.988243	+	0.152890i	$0.0488579\pi$
$619$	7.60770i	0.305779i	−0.988243	+	0.152890i	$0.951142\pi$
$620$	4.39230	0.176399
$621$	0	0
$622$	2.19615i	0.0880577i
$623$	12.0000	0.480770
$624$	0	0
$625$	−11.0000	−0.440000
$626$	− 24.3923i	− 0.974913i
$627$	0	0
$628$	3.19615	0.127540
$629$	15.5885i	0.621552i
$630$	0	0
$631$	− 25.8564i	− 1.02933i	−0.857392	−	0.514664i	$-0.827917\pi$
$631$	− 25.8564i	− 1.02933i	0.857392	−	0.514664i	$-0.172083\pi$
$632$	12.3923i	0.492939i
$633$	0	0
$634$	6.12436	0.243229
$635$	6.92820i	0.274937i
$636$	0	0
$637$	0	0
$638$	−14.1962	−0.562031
$639$	0	0
$640$	1.73205	0.0684653
$641$	−30.8038	−1.21668	−0.608339	−	0.793677i	$-0.708164\pi$
$641$	−30.8038	−1.21668	−0.608339	+	0.793677i	$0.708164\pi$
$642$	0	0
$643$	− 27.7128i	− 1.09289i	−0.837496	−	0.546443i	$-0.815981\pi$
$643$	− 27.7128i	− 1.09289i	0.837496	−	0.546443i	$-0.184019\pi$
$644$	10.3923i	0.409514i
$645$	0	0
$646$	6.58846	0.259219
$647$	13.1769	0.518038	0.259019	−	0.965872i	$-0.416601\pi$
$647$	13.1769	0.518038	0.259019	+	0.965872i	$0.416601\pi$
$648$	0	0
$649$	−65.5692	−2.57382
$650$	0	0
$651$	0	0
$652$	9.46410i	0.370643i
$653$	49.1769	1.92444	0.962221	−	0.272271i	$-0.0877746\pi$
$653$	49.1769	1.92444	0.962221	+	0.272271i	$0.0877746\pi$
$654$	0	0
$655$	− 28.3923i	− 1.10938i
$656$	0.464102i	0.0181201i
$657$	0	0
$658$	− 6.00000i	− 0.233904i
$659$	−25.1769	−0.980753	−0.490377	−	0.871511i	$-0.663141\pi$
$659$	−25.1769	−0.980753	−0.490377	+	0.871511i	$0.663141\pi$
$660$	0	0
$661$	− 9.00000i	− 0.350059i	−0.984563	−	0.175030i	$-0.943998\pi$
$661$	− 9.00000i	− 0.350059i	0.984563	−	0.175030i	$-0.0560022\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	11.6603	0.452506
$665$	10.3923i	0.402996i
$666$	0	0
$667$	6.58846	0.255106
$668$	2.53590i	0.0981169i
$669$	0	0
$670$	− 18.5885i	− 0.718135i
$671$	22.7321i	0.877561i
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	31.0000i	1.19408i
$675$	0	0
$676$	0	0
$677$	−4.39230	−0.168810	−0.0844050	−	0.996432i	$-0.526899\pi$
$677$	−4.39230	−0.168810	−0.0844050	+	0.996432i	$0.526899\pi$
$678$	0	0
$679$	28.3923	1.08960
$680$	−9.00000	−0.345134
$681$	0	0
$682$	− 12.0000i	− 0.459504i
$683$	27.7128i	1.06040i	0.847872	+	0.530201i	$0.177883\pi$
$683$	27.7128i	1.06040i	−0.847872	+	0.530201i	$0.822117\pi$
$684$	0	0
$685$	15.5885	0.595604
$686$	−39.7128	−1.51624
$687$	0	0
$688$	−6.19615	−0.236226
$689$	0	0
$690$	0	0
$691$	− 19.5167i	− 0.742449i	−0.928543	−	0.371224i	$-0.878938\pi$
$691$	− 19.5167i	− 0.742449i	0.928543	−	0.371224i	$-0.121062\pi$
$692$	−16.3923	−0.623142
$693$	0	0
$694$	12.5885i	0.477851i
$695$	− 6.92820i	− 0.262802i
$696$	0	0
$697$	− 2.41154i	− 0.0913437i
$698$	2.53590	0.0959852
$699$	0	0
$700$	9.46410i	0.357709i
$701$	4.39230	0.165895	0.0829475	−	0.996554i	$-0.473567\pi$
$701$	4.39230	0.165895	0.0829475	+	0.996554i	$0.473567\pi$
$702$	0	0
$703$	3.80385	0.143465
$704$	− 4.73205i	− 0.178346i
$705$	0	0
$706$	5.78461	0.217707
$707$	6.58846i	0.247784i
$708$	0	0
$709$	− 3.24871i	− 0.122008i	−0.998138	−	0.0610040i	$-0.980570\pi$
$709$	− 3.24871i	− 0.122008i	0.998138	−	0.0610040i	$-0.0194302\pi$
$710$	14.1962i	0.532772i
$711$	0	0
$712$	2.53590	0.0950368
$713$	5.56922i	0.208569i
$714$	0	0
$715$	0	0
$716$	8.19615	0.306305
$717$	0	0
$718$	−22.0526	−0.822994
$719$	52.3923	1.95390	0.976952	−	0.213461i	$-0.0684736\pi$
$719$	52.3923	1.95390	0.976952	+	0.213461i	$0.0684736\pi$
$720$	0	0
$721$	− 19.8564i	− 0.739491i
$722$	17.3923i	0.647275i
$723$	0	0
$724$	−11.5885	−0.430682
$725$	6.00000	0.222834
$726$	0	0
$727$	24.1962	0.897386	0.448693	−	0.893686i	$-0.351890\pi$
$727$	24.1962	0.897386	0.448693	+	0.893686i	$0.351890\pi$
$728$	0	0
$729$	0	0
$730$	− 21.0000i	− 0.777245i
$731$	32.1962	1.19082
$732$	0	0
$733$	− 14.3205i	− 0.528940i	−0.964394	−	0.264470i	$-0.914803\pi$
$733$	− 14.3205i	− 0.528940i	0.964394	−	0.264470i	$-0.0851970\pi$
$734$	− 24.1962i	− 0.893096i
$735$	0	0
$736$	2.19615i	0.0809513i
$737$	−50.7846	−1.87068
$738$	0	0
$739$	18.9282i	0.696285i	0.937442	+	0.348143i	$0.113188\pi$
$739$	18.9282i	0.696285i	−0.937442	+	0.348143i	$0.886812\pi$
$740$	−5.19615	−0.191014
$741$	0	0
$742$	−14.1962	−0.521157
$743$	− 4.39230i	− 0.161138i	−0.996749	−	0.0805690i	$-0.974326\pi$
$743$	− 4.39230i	− 0.161138i	0.996749	−	0.0805690i	$-0.0256737\pi$
$744$	0	0
$745$	−31.3923	−1.15013
$746$	23.9808i	0.877998i
$747$	0	0
$748$	24.5885i	0.899043i
$749$	38.7846i	1.41716i
$750$	0	0
$751$	24.9808	0.911561	0.455780	−	0.890092i	$-0.349360\pi$
$751$	24.9808	0.911561	0.455780	+	0.890092i	$0.349360\pi$
$752$	− 1.26795i	− 0.0462373i
$753$	0	0
$754$	0	0
$755$	12.5885	0.458141
$756$	0	0
$757$	18.7846	0.682738	0.341369	−	0.939929i	$-0.389109\pi$
$757$	18.7846	0.682738	0.341369	+	0.939929i	$0.389109\pi$
$758$	18.2487	0.662823
$759$	0	0
$760$	2.19615i	0.0796628i
$761$	− 4.39230i	− 0.159221i	−0.996826	−	0.0796105i	$-0.974632\pi$
$761$	− 4.39230i	− 0.159221i	0.996826	−	0.0796105i	$-0.0253676\pi$
$762$	0	0
$763$	77.5692	2.80819
$764$	20.7846	0.751961
$765$	0	0
$766$	11.3205	0.409027
$767$	0	0
$768$	0	0
$769$	33.7128i	1.21572i	0.794046	+	0.607858i	$0.207971\pi$
$769$	33.7128i	1.21572i	−0.794046	+	0.607858i	$0.792029\pi$
$770$	−38.7846	−1.39770
$771$	0	0
$772$	− 12.8038i	− 0.460821i
$773$	50.7846i	1.82660i	0.407293	+	0.913298i	$0.366473\pi$
$773$	50.7846i	1.82660i	−0.407293	+	0.913298i	$0.633527\pi$
$774$	0	0
$775$	5.07180i	0.182184i
$776$	6.00000	0.215387
$777$	0	0
$778$	− 13.3923i	− 0.480137i
$779$	−0.588457	−0.0210837
$780$	0	0
$781$	38.7846	1.38782
$782$	− 11.4115i	− 0.408076i
$783$	0	0
$784$	−15.3923	−0.549725
$785$	− 5.53590i	− 0.197585i
$786$	0	0
$787$	14.5359i	0.518149i	0.965857	+	0.259074i	$0.0834175\pi$
$787$	14.5359i	0.518149i	−0.965857	+	0.259074i	$0.916583\pi$
$788$	6.92820i	0.246807i
$789$	0	0
$790$	21.4641	0.763658
$791$	52.9808i	1.88378i
$792$	0	0
$793$	0	0
$794$	−16.3923	−0.581741
$795$	0	0
$796$	8.58846	0.304410
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	6.58846i	0.233083i
$800$	2.00000i	0.0707107i
$801$	0	0
$802$	−21.0000	−0.741536
$803$	−57.3731	−2.02465
$804$	0	0
$805$	18.0000	0.634417
$806$	0	0
$807$	0	0
$808$	1.39230i	0.0489811i
$809$	47.1962	1.65933	0.829664	−	0.558263i	$-0.188532\pi$
$809$	47.1962	1.65933	0.829664	+	0.558263i	$0.188532\pi$
$810$	0	0
$811$	4.39230i	0.154235i	0.997022	+	0.0771173i	$0.0245716\pi$
$811$	4.39230i	0.154235i	−0.997022	+	0.0771173i	$0.975428\pi$
$812$	14.1962i	0.498187i
$813$	0	0
$814$	14.1962i	0.497575i
$815$	16.3923	0.574197
$816$	0	0
$817$	− 7.85641i	− 0.274861i
$818$	3.33975	0.116771
$819$	0	0
$820$	0.803848	0.0280716
$821$	40.6410i	1.41838i	0.705017	+	0.709191i	$0.250939\pi$
$821$	40.6410i	1.41838i	−0.705017	+	0.709191i	$0.749061\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	− 4.19615i	− 0.146180i
$825$	0	0
$826$	65.5692i	2.28144i
$827$	− 32.1051i	− 1.11640i	−0.829705	−	0.558202i	$-0.811491\pi$
$827$	− 32.1051i	− 1.11640i	0.829705	−	0.558202i	$-0.188509\pi$
$828$	0	0
$829$	11.9808	0.416109	0.208055	−	0.978117i	$-0.433287\pi$
$829$	11.9808	0.416109	0.208055	+	0.978117i	$0.433287\pi$
$830$	− 20.1962i	− 0.701019i
$831$	0	0
$832$	0	0
$833$	79.9808	2.77117
$834$	0	0
$835$	4.39230	0.152002
$836$	6.00000	0.207514
$837$	0	0
$838$	− 16.3923i	− 0.566263i
$839$	− 12.0000i	− 0.414286i	−0.978311	−	0.207143i	$-0.933583\pi$
$839$	− 12.0000i	− 0.414286i	0.978311	−	0.207143i	$-0.0664165\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−0.464102	−0.0159940
$843$	0	0
$844$	3.60770	0.124182
$845$	0	0
$846$	0	0
$847$	53.9090i	1.85233i
$848$	−3.00000	−0.103020
$849$	0	0
$850$	− 10.3923i	− 0.356453i
$851$	− 6.58846i	− 0.225849i
$852$	0	0
$853$	− 9.00000i	− 0.308154i	−0.988059	−	0.154077i	$-0.950760\pi$
$853$	− 9.00000i	− 0.308154i	0.988059	−	0.154077i	$-0.0492404\pi$
$854$	22.7321	0.777875
$855$	0	0
$856$	8.19615i	0.280139i
$857$	−54.3731	−1.85735	−0.928674	−	0.370896i	$-0.879051\pi$
$857$	−54.3731	−1.85735	−0.928674	+	0.370896i	$0.879051\pi$
$858$	0	0
$859$	10.5885	0.361273	0.180637	−	0.983550i	$-0.442184\pi$
$859$	10.5885	0.361273	0.180637	+	0.983550i	$0.442184\pi$
$860$	10.7321i	0.365960i
$861$	0	0
$862$	27.8038	0.947003
$863$	− 4.48334i	− 0.152615i	−0.997084	−	0.0763073i	$-0.975687\pi$
$863$	− 4.48334i	− 0.152615i	0.997084	−	0.0763073i	$-0.0243130\pi$
$864$	0	0
$865$	28.3923i	0.965367i
$866$	33.7846i	1.14805i
$867$	0	0
$868$	−12.0000	−0.407307
$869$	− 58.6410i	− 1.98926i
$870$	0	0
$871$	0	0
$872$	16.3923	0.555113
$873$	0	0
$874$	−2.78461	−0.0941908
$875$	57.3731	1.93956
$876$	0	0
$877$	43.3923i	1.46525i	0.680630	+	0.732627i	$0.261706\pi$
$877$	43.3923i	1.46525i	−0.680630	+	0.732627i	$0.738294\pi$
$878$	− 16.5885i	− 0.559833i
$879$	0	0
$880$	−8.19615	−0.276292
$881$	37.9808	1.27960	0.639802	−	0.768540i	$-0.279016\pi$
$881$	37.9808	1.27960	0.639802	+	0.768540i	$0.279016\pi$
$882$	0	0
$883$	24.7846	0.834069	0.417034	−	0.908891i	$-0.363070\pi$
$883$	24.7846	0.834069	0.417034	+	0.908891i	$0.363070\pi$
$884$	0	0
$885$	0	0
$886$	4.39230i	0.147562i
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	− 18.9282i	− 0.634832i
$890$	− 4.39230i	− 0.147230i
$891$	0	0
$892$	18.9282i	0.633763i
$893$	1.60770	0.0537995
$894$	0	0
$895$	− 14.1962i	− 0.474525i
$896$	−4.73205	−0.158087
$897$	0	0
$898$	−33.4641	−1.11671
$899$	7.60770i	0.253731i
$900$	0	0
$901$	15.5885	0.519327
$902$	− 2.19615i	− 0.0731239i
$903$	0	0
$904$	11.1962i	0.372378i
$905$	20.0718i	0.667209i
$906$	0	0
$907$	41.1769	1.36726	0.683629	−	0.729830i	$-0.260401\pi$
$907$	41.1769	1.36726	0.683629	+	0.729830i	$0.260401\pi$
$908$	− 9.80385i	− 0.325352i
$909$	0	0
$910$	0	0
$911$	−37.1769	−1.23173	−0.615863	−	0.787853i	$-0.711193\pi$
$911$	−37.1769	−1.23173	−0.615863	+	0.787853i	$0.711193\pi$
$912$	0	0
$913$	−55.1769	−1.82609
$914$	−19.9808	−0.660905
$915$	0	0
$916$	19.8564i	0.656074i
$917$	77.5692i	2.56156i
$918$	0	0
$919$	32.3923	1.06852	0.534262	−	0.845319i	$-0.320590\pi$
$919$	32.3923	1.06852	0.534262	+	0.845319i	$0.320590\pi$
$920$	3.80385	0.125409
$921$	0	0
$922$	−19.9808	−0.658031
$923$	0	0
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	−26.1962	−0.860859
$927$	0	0
$928$	3.00000i	0.0984798i
$929$	34.6077i	1.13544i	0.823221	+	0.567721i	$0.192175\pi$
$929$	34.6077i	1.13544i	−0.823221	+	0.567721i	$0.807825\pi$
$930$	0	0
$931$	− 19.5167i	− 0.639633i
$932$	18.0000	0.589610
$933$	0	0
$934$	36.5885i	1.19721i
$935$	42.5885	1.39279
$936$	0	0
$937$	5.39230	0.176159	0.0880795	−	0.996113i	$-0.471927\pi$
$937$	5.39230	0.176159	0.0880795	+	0.996113i	$0.471927\pi$
$938$	50.7846i	1.65818i
$939$	0	0
$940$	−2.19615	−0.0716306
$941$	− 2.78461i	− 0.0907757i	−0.998969	−	0.0453878i	$-0.985548\pi$
$941$	− 2.78461i	− 0.0907757i	0.998969	−	0.0453878i	$-0.0144524\pi$
$942$	0	0
$943$	1.01924i	0.0331910i
$944$	13.8564i	0.450988i
$945$	0	0
$946$	29.3205	0.953292
$947$	− 42.9282i	− 1.39498i	−0.716595	−	0.697490i	$-0.754300\pi$
$947$	− 42.9282i	− 1.39498i	0.716595	−	0.697490i	$-0.245700\pi$
$948$	0	0
$949$	0	0
$950$	−2.53590	−0.0822754
$951$	0	0
$952$	24.5885	0.796916
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	− 36.0000i	− 1.16493i
$956$	− 24.5885i	− 0.795248i
$957$	0	0
$958$	35.3205	1.14115
$959$	−42.5885	−1.37525
$960$	0	0
$961$	24.5692	0.792555
$962$	0	0
$963$	0	0
$964$	0.803848i	0.0258902i
$965$	−22.1769	−0.713900
$966$	0	0
$967$	− 14.8756i	− 0.478368i	−0.970974	−	0.239184i	$-0.923120\pi$
$967$	− 14.8756i	− 0.478368i	0.970974	−	0.239184i	$-0.0768800\pi$
$968$	11.3923i	0.366163i
$969$	0	0
$970$	− 10.3923i	− 0.333677i
$971$	13.1769	0.422867	0.211434	−	0.977392i	$-0.432187\pi$
$971$	13.1769	0.422867	0.211434	+	0.977392i	$0.432187\pi$
$972$	0	0
$973$	18.9282i	0.606810i
$974$	−9.12436	−0.292363
$975$	0	0
$976$	4.80385	0.153767
$977$	16.8564i	0.539284i	0.962961	+	0.269642i	$0.0869054\pi$
$977$	16.8564i	0.539284i	−0.962961	+	0.269642i	$0.913095\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	26.6603i	0.851631i
$981$	0	0
$982$	0.588457i	0.0187784i
$983$	− 20.7846i	− 0.662926i	−0.943468	−	0.331463i	$-0.892458\pi$
$983$	− 20.7846i	− 0.662926i	0.943468	−	0.331463i	$-0.107542\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	− 15.5885i	− 0.496438i
$987$	0	0
$988$	0	0
$989$	−13.6077	−0.432700
$990$	0	0
$991$	29.3731	0.933066	0.466533	−	0.884504i	$-0.345503\pi$
$991$	29.3731	0.933066	0.466533	+	0.884504i	$0.345503\pi$
$992$	−2.53590	−0.0805149
$993$	0	0
$994$	− 38.7846i	− 1.23017i
$995$	− 14.8756i	− 0.471590i
$996$	0	0
$997$	13.1962	0.417926	0.208963	−	0.977924i	$-0.432991\pi$
$997$	13.1962	0.417926	0.208963	+	0.977924i	$0.432991\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.l.1351.4		4
3.2	odd	2		1014.2.b.d.337.1		4
13.3	even	3		234.2.l.a.199.1		4
13.4	even	6		234.2.l.a.127.1		4
13.5	odd	4		3042.2.a.v.1.2		2
13.8	odd	4		3042.2.a.s.1.1		2
13.12	even	2	inner	3042.2.b.l.1351.1		4
39.2	even	12		1014.2.e.j.529.1		4
39.5	even	4		1014.2.a.h.1.1		2
39.8	even	4		1014.2.a.j.1.2		2
39.11	even	12		1014.2.e.h.529.2		4
39.17	odd	6		78.2.i.b.49.2	yes	4
39.20	even	12		1014.2.e.h.991.2		4
39.23	odd	6		1014.2.i.f.823.1		4
39.29	odd	6		78.2.i.b.43.2	✓	4
39.32	even	12		1014.2.e.j.991.1		4
39.35	odd	6		1014.2.i.f.361.1		4
39.38	odd	2		1014.2.b.d.337.4		4
52.3	odd	6		1872.2.by.k.433.2		4
52.43	odd	6		1872.2.by.k.1297.2		4
156.47	odd	4		8112.2.a.bx.1.2		2
156.83	odd	4		8112.2.a.bq.1.1		2
156.95	even	6		624.2.bv.d.49.2		4
156.107	even	6		624.2.bv.d.433.2		4
195.17	even	12		1950.2.y.h.49.2		4
195.29	odd	6		1950.2.bc.c.901.1		4
195.68	even	12		1950.2.y.h.199.2		4
195.107	even	12		1950.2.y.a.199.1		4
195.134	odd	6		1950.2.bc.c.751.1		4
195.173	even	12		1950.2.y.a.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.b.43.2	✓	4	39.29	odd	6
78.2.i.b.49.2	yes	4	39.17	odd	6
234.2.l.a.127.1		4	13.4	even	6
234.2.l.a.199.1		4	13.3	even	3
624.2.bv.d.49.2		4	156.95	even	6
624.2.bv.d.433.2		4	156.107	even	6
1014.2.a.h.1.1		2	39.5	even	4
1014.2.a.j.1.2		2	39.8	even	4
1014.2.b.d.337.1		4	3.2	odd	2
1014.2.b.d.337.4		4	39.38	odd	2
1014.2.e.h.529.2		4	39.11	even	12
1014.2.e.h.991.2		4	39.20	even	12
1014.2.e.j.529.1		4	39.2	even	12
1014.2.e.j.991.1		4	39.32	even	12
1014.2.i.f.361.1		4	39.35	odd	6
1014.2.i.f.823.1		4	39.23	odd	6
1872.2.by.k.433.2		4	52.3	odd	6
1872.2.by.k.1297.2		4	52.43	odd	6
1950.2.y.a.49.1		4	195.173	even	12
1950.2.y.a.199.1		4	195.107	even	12
1950.2.y.h.49.2		4	195.17	even	12
1950.2.y.h.199.2		4	195.68	even	12
1950.2.bc.c.751.1		4	195.134	odd	6
1950.2.bc.c.901.1		4	195.29	odd	6
3042.2.a.s.1.1		2	13.8	odd	4
3042.2.a.v.1.2		2	13.5	odd	4
3042.2.b.l.1351.1		4	13.12	even	2	inner
3042.2.b.l.1351.4		4	1.1	even	1	trivial
8112.2.a.bq.1.1		2	156.83	odd	4
8112.2.a.bx.1.2		2	156.47	odd	4

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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.73205i q^{5} -4.73205i q^{7} -1.00000i q^{8} -1.73205 q^{10} +4.73205i q^{11} +4.73205 q^{14} +1.00000 q^{16} -5.19615 q^{17} +1.26795i q^{19} -1.73205i q^{20} -4.73205 q^{22} +2.19615 q^{23} +2.00000 q^{25} +4.73205i q^{28} +3.00000 q^{29} +2.53590i q^{31} +1.00000i q^{32} -5.19615i q^{34} +8.19615 q^{35} -3.00000i q^{37} -1.26795 q^{38} +1.73205 q^{40} +0.464102i q^{41} -6.19615 q^{43} -4.73205i q^{44} +2.19615i q^{46} -1.26795i q^{47} -15.3923 q^{49} +2.00000i q^{50} -3.00000 q^{53} -8.19615 q^{55} -4.73205 q^{56} +3.00000i q^{58} +13.8564i q^{59} +4.80385 q^{61} -2.53590 q^{62} -1.00000 q^{64} +10.7321i q^{67} +5.19615 q^{68} +8.19615i q^{70} -8.19615i q^{71} +12.1244i q^{73} +3.00000 q^{74} -1.26795i q^{76} +22.3923 q^{77} -12.3923 q^{79} +1.73205i q^{80} -0.464102 q^{82} +11.6603i q^{83} -9.00000i q^{85} -6.19615i q^{86} +4.73205 q^{88} +2.53590i q^{89} -2.19615 q^{92} +1.26795 q^{94} -2.19615 q^{95} +6.00000i q^{97} -15.3923i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64}+ \cdots + 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 12 * q^14 + 4 * q^16 - 12 * q^22 - 12 * q^23 + 8 * q^25 + 12 * q^29 + 12 * q^35 - 12 * q^38 - 4 * q^43 - 20 * q^49 - 12 * q^53 - 12 * q^55 - 12 * q^56 + 40 * q^61 - 24 * q^62 - 4 * q^64 + 12 * q^74 + 48 * q^77 - 8 * q^79 + 12 * q^82 + 12 * q^88 + 12 * q^92 + 12 * q^94 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more