LMFDB - Embedded newform 1848.2.v.d.881.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1848,2,Mod(881,1848)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1848, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1848.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.v (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$14.7563542935$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 77x^{12} - 342x^{10} + 1160x^{8} - 3078x^{6} + 6237x^{4} - 8748x^{2} + 6561 $ x^16 - 12x^14 + 77x^12 - 342x^10 + 1160x^8 - 3078x^6 + 6237x^4 - 8748*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.5
Root			$1.63065 + 0.583933i$ of defining polynomial
Character	$\chi$	$=$	1848.881
Dual form			1848.2.v.d.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.583933 - 1.63065i) q^{3} -3.70180 q^{5} +(2.42989 - 1.04672i) q^{7} +(-2.31805 + 1.90438i) q^{9} +O(q^{10})$	\(q+(-0.583933 - 1.63065i) q^{3} -3.70180 q^{5} +(2.42989 - 1.04672i) q^{7} +(-2.31805 + 1.90438i) q^{9} +1.00000i q^{11} -0.623204i q^{13} +(2.16160 + 6.03635i) q^{15} +3.09755 q^{17} +3.80596i q^{19} +(-3.12573 - 3.35109i) q^{21} +7.18299i q^{23} +8.70334 q^{25} +(4.45896 + 2.66789i) q^{27} +0.105417i q^{29} -6.41844i q^{31} +(1.63065 - 0.583933i) q^{33} +(-8.99499 + 3.87475i) q^{35} -5.28841 q^{37} +(-1.01623 + 0.363909i) q^{39} +3.33984 q^{41} +7.71703 q^{43} +(8.58095 - 7.04964i) q^{45} +2.53394 q^{47} +(4.80876 - 5.08683i) q^{49} +(-1.80876 - 5.05103i) q^{51} -10.2639i q^{53} -3.70180i q^{55} +(6.20620 - 2.22243i) q^{57} -5.55295 q^{59} +6.33990i q^{61} +(-3.63925 + 7.05378i) q^{63} +2.30698i q^{65} -0.329112 q^{67} +(11.7130 - 4.19438i) q^{69} -12.7681i q^{71} +6.20128i q^{73} +(-5.08217 - 14.1921i) q^{75} +(1.04672 + 2.42989i) q^{77} +16.6093 q^{79} +(1.74667 - 8.82888i) q^{81} -13.7814 q^{83} -11.4665 q^{85} +(0.171898 - 0.0615564i) q^{87} +7.90715 q^{89} +(-0.652319 - 1.51432i) q^{91} +(-10.4662 + 3.74794i) q^{93} -14.0889i q^{95} +5.93386i q^{97} +(-1.90438 - 2.31805i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7} - 24 q^{9}+O(q^{10}) $ 16 * q + 16 * q^7 - 24 * q^9	$ 16 q + 16 q^{7} - 24 q^{9} + 4 q^{15} - 30 q^{21} + 76 q^{25} - 4 q^{37} + 12 q^{39} + 8 q^{43} + 40 q^{49} + 8 q^{51} + 96 q^{57} + 2 q^{63} - 12 q^{67} + 48 q^{79} - 20 q^{81} + 104 q^{85} + 44 q^{91} - 28 q^{93} - 12 q^{99}+O(q^{100}) $ 16 * q + 16 * q^7 - 24 * q^9 + 4 * q^15 - 30 * q^21 + 76 * q^25 - 4 * q^37 + 12 * q^39 + 8 * q^43 + 40 * q^49 + 8 * q^51 + 96 * q^57 + 2 * q^63 - 12 * q^67 + 48 * q^79 - 20 * q^81 + 104 * q^85 + 44 * q^91 - 28 * q^93 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times$.

$n$	$463$	$617$	$673$	$925$	$1585$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.583933	−	1.63065i	−0.337134	−	0.941457i
$3$	−0.583933	−	1.63065i	−0.337134	−	0.941457i
$4$		0			0
$5$	−3.70180			−1.65550			−0.827748	−	0.561100i	$-0.810378\pi$
$5$	−3.70180			−1.65550			−0.827748	+	0.561100i	$0.810378\pi$
$6$		0			0
$7$	2.42989	−	1.04672i	0.918413	−	0.395622i
$7$	2.42989	−	1.04672i	0.918413	−	0.395622i
$8$		0			0
$9$	−2.31805	+	1.90438i	−0.772682	+	0.634794i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		−	0.623204i		−	0.172846i	−0.996259	−	0.0864229i	$-0.972456\pi$
$13$		−	0.623204i		−	0.172846i	0.996259	−	0.0864229i	$-0.0275436\pi$
$14$		0			0
$15$	2.16160	+	6.03635i	0.558124	+	1.55858i
$16$		0			0
$17$	3.09755			0.751267			0.375633	−	0.926768i	$-0.377425\pi$
$17$	3.09755			0.751267			0.375633	+	0.926768i	$0.377425\pi$
$18$		0			0
$19$			3.80596i			0.873148i	0.899668	+	0.436574i	$0.143808\pi$
$19$			3.80596i			0.873148i	−0.899668	+	0.436574i	$0.856192\pi$
$20$		0			0
$21$	−3.12573	−	3.35109i	−0.682089	−	0.731269i
$22$		0			0
$23$			7.18299i			1.49776i	0.662707	+	0.748879i	$0.269407\pi$
$23$			7.18299i			1.49776i	−0.662707	+	0.748879i	$0.730593\pi$
$24$		0			0
$25$	8.70334			1.74067
$26$		0			0
$27$	4.45896	+	2.66789i	0.858128	+	0.513436i
$28$		0			0
$29$			0.105417i			0.0195754i	0.999952	+	0.00978772i	$0.00311558\pi$
$29$			0.105417i			0.0195754i	−0.999952	+	0.00978772i	$0.996884\pi$
$30$		0			0
$31$		−	6.41844i		−	1.15279i	−0.817172	−	0.576393i	$-0.804460\pi$
$31$		−	6.41844i		−	1.15279i	0.817172	−	0.576393i	$-0.195540\pi$
$32$		0			0
$33$	1.63065	−	0.583933i	0.283860	−	0.101650i
$34$		0			0
$35$	−8.99499	+	3.87475i	−1.52043	+	0.654951i
$36$		0			0
$37$	−5.28841			−0.869409			−0.434705	−	0.900573i	$-0.643147\pi$
$37$	−5.28841			−0.869409			−0.434705	+	0.900573i	$0.643147\pi$
$38$		0			0
$39$	−1.01623	+	0.363909i	−0.162727	+	0.0582721i
$40$		0			0
$41$	3.33984			0.521596			0.260798	−	0.965393i	$-0.416014\pi$
$41$	3.33984			0.521596			0.260798	+	0.965393i	$0.416014\pi$
$42$		0			0
$43$	7.71703			1.17684			0.588418	−	0.808557i	$-0.299751\pi$
$43$	7.71703			1.17684			0.588418	+	0.808557i	$0.299751\pi$
$44$		0			0
$45$	8.58095	−	7.04964i	1.27917	−	1.05090i
$46$		0			0
$47$	2.53394			0.369613			0.184806	−	0.982775i	$-0.440834\pi$
$47$	2.53394			0.369613			0.184806	+	0.982775i	$0.440834\pi$
$48$		0			0
$49$	4.80876	−	5.08683i	0.686966	−	0.726690i
$50$		0			0
$51$	−1.80876	−	5.05103i	−0.253277	−	0.707285i
$52$		0			0
$53$		−	10.2639i		−	1.40986i	−0.709277	−	0.704930i	$-0.750978\pi$
$53$		−	10.2639i		−	1.40986i	0.709277	−	0.704930i	$-0.249022\pi$
$54$		0			0
$55$		−	3.70180i		−	0.499151i
$56$		0			0
$57$	6.20620	−	2.22243i	0.822031	−	0.294368i
$58$		0			0
$59$	−5.55295			−0.722932			−0.361466	−	0.932385i	$-0.617724\pi$
$59$	−5.55295			−0.722932			−0.361466	+	0.932385i	$0.617724\pi$
$60$		0			0
$61$			6.33990i			0.811741i	0.913931	+	0.405871i	$0.133032\pi$
$61$			6.33990i			0.811741i	−0.913931	+	0.405871i	$0.866968\pi$
$62$		0			0
$63$	−3.63925	+	7.05378i	−0.458503	+	0.888693i
$64$		0			0
$65$			2.30698i			0.286145i
$66$		0			0
$67$	−0.329112			−0.0402075			−0.0201037	−	0.999798i	$-0.506400\pi$
$67$	−0.329112			−0.0402075			−0.0201037	+	0.999798i	$0.506400\pi$
$68$		0			0
$69$	11.7130	−	4.19438i	1.41007	−	0.504944i
$70$		0			0
$71$		−	12.7681i		−	1.51529i	−0.652667	−	0.757645i	$-0.726350\pi$
$71$		−	12.7681i		−	1.51529i	0.652667	−	0.757645i	$-0.273650\pi$
$72$		0			0
$73$			6.20128i			0.725805i	0.931827	+	0.362903i	$0.118214\pi$
$73$			6.20128i			0.725805i	−0.931827	+	0.362903i	$0.881786\pi$
$74$		0			0
$75$	−5.08217	−	14.1921i	−0.586838	−	1.63876i
$76$		0			0
$77$	1.04672	+	2.42989i	0.119285	+	0.276912i
$78$		0			0
$79$	16.6093			1.86869			0.934345	−	0.356370i	$-0.115986\pi$
$79$	16.6093			1.86869			0.934345	+	0.356370i	$0.115986\pi$
$80$		0			0
$81$	1.74667	−	8.82888i	0.194074	−	0.980987i
$82$		0			0
$83$	−13.7814			−1.51271			−0.756353	−	0.654163i	$-0.773021\pi$
$83$	−13.7814			−1.51271			−0.756353	+	0.654163i	$0.773021\pi$
$84$		0			0
$85$	−11.4665			−1.24372
$86$		0			0
$87$	0.171898	−	0.0615564i	0.0184294	−	0.00659954i
$88$		0			0
$89$	7.90715			0.838156			0.419078	−	0.907950i	$-0.362353\pi$
$89$	7.90715			0.838156			0.419078	+	0.907950i	$0.362353\pi$
$90$		0			0
$91$	−0.652319	−	1.51432i	−0.0683816	−	0.158744i
$92$		0			0
$93$	−10.4662	+	3.74794i	−1.08530	+	0.388643i
$94$		0			0
$95$		−	14.0889i		−	1.44549i
$96$		0			0
$97$			5.93386i			0.602492i	0.953546	+	0.301246i	$0.0974025\pi$
$97$			5.93386i			0.602492i	−0.953546	+	0.301246i	$0.902597\pi$
$98$		0			0
$99$	−1.90438	−	2.31805i	−0.191397	−	0.232972i
$100$		0			0
$101$	13.6746			1.36067			0.680336	−	0.732900i	$-0.261834\pi$
$101$	13.6746			1.36067			0.680336	+	0.732900i	$0.261834\pi$
$102$		0			0
$103$		−	14.0833i		−	1.38767i	−0.720135	−	0.693834i	$-0.755920\pi$
$103$		−	14.0833i		−	1.38767i	0.720135	−	0.693834i	$-0.244080\pi$
$104$		0			0
$105$	11.5708	+	12.4051i	1.12920	+	1.21061i
$106$		0			0
$107$		−	4.93275i		−	0.476867i	−0.971159	−	0.238433i	$-0.923366\pi$
$107$		−	4.93275i		−	0.476867i	0.971159	−	0.238433i	$-0.0766338\pi$
$108$		0			0
$109$	17.1237			1.64016			0.820078	−	0.572252i	$-0.193930\pi$
$109$	17.1237			1.64016			0.820078	+	0.572252i	$0.193930\pi$
$110$		0			0
$111$	3.08808	+	8.62355i	0.293107	+	0.818511i
$112$		0			0
$113$			2.50412i			0.235568i	0.993039	+	0.117784i	$0.0375791\pi$
$113$			2.50412i			0.235568i	−0.993039	+	0.117784i	$0.962421\pi$
$114$		0			0
$115$		−	26.5900i		−	2.47953i
$116$		0			0
$117$	1.18682	+	1.44462i	0.109721	+	0.133555i
$118$		0			0
$119$	7.52672	−	3.24226i	0.689973	−	0.297218i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	−1.95024	−	5.44612i	−0.175848	−	0.491060i
$124$		0			0
$125$	−13.7091			−1.22617
$126$		0			0
$127$	9.40415			0.834483			0.417242	−	0.908796i	$-0.362997\pi$
$127$	9.40415			0.834483			0.417242	+	0.908796i	$0.362997\pi$
$128$		0			0
$129$	−4.50623	−	12.5838i	−0.396751	−	1.10794i
$130$		0			0
$131$	11.3226			0.989257			0.494629	−	0.869104i	$-0.335304\pi$
$131$	11.3226			0.989257			0.494629	+	0.869104i	$0.335304\pi$
$132$		0			0
$133$	3.98377	+	9.24808i	0.345437	+	0.801911i
$134$		0			0
$135$	−16.5062	−	9.87601i	−1.42063	−	0.849992i
$136$		0			0
$137$		−	4.53658i		−	0.387586i	−0.981042	−	0.193793i	$-0.937921\pi$
$137$		−	4.53658i		−	0.387586i	0.981042	−	0.193793i	$-0.0620790\pi$
$138$		0			0
$139$			0.595413i			0.0505022i	0.999681	+	0.0252511i	$0.00803854\pi$
$139$			0.595413i			0.0505022i	−0.999681	+	0.0252511i	$0.991961\pi$
$140$		0			0
$141$	−1.47965	−	4.13197i	−0.124609	−	0.347974i
$142$		0			0
$143$	0.623204			0.0521149
$144$		0			0
$145$		−	0.390233i		−	0.0324071i
$146$		0			0
$147$	−11.1028	−	4.87105i	−0.915746	−	0.401757i
$148$		0			0
$149$			3.22914i			0.264541i	0.991214	+	0.132271i	$0.0422268\pi$
$149$			3.22914i			0.264541i	−0.991214	+	0.132271i	$0.957773\pi$
$150$		0			0
$151$	−17.1212			−1.39330			−0.696651	−	0.717410i	$-0.745327\pi$
$151$	−17.1212			−1.39330			−0.696651	+	0.717410i	$0.745327\pi$
$152$		0			0
$153$	−7.18026	+	5.89892i	−0.580490	+	0.476899i
$154$		0			0
$155$			23.7598i			1.90843i
$156$		0			0
$157$		−	4.18021i		−	0.333617i	−0.985989	−	0.166808i	$-0.946654\pi$
$157$		−	4.18021i		−	0.333617i	0.985989	−	0.166808i	$-0.0533462\pi$
$158$		0			0
$159$	−16.7369	+	5.99345i	−1.32732	+	0.475311i
$160$		0			0
$161$	7.51857	+	17.4539i	0.592546	+	1.37556i
$162$		0			0
$163$	−21.4606			−1.68093			−0.840463	−	0.541869i	$-0.817717\pi$
$163$	−21.4606			−1.68093			−0.840463	+	0.541869i	$0.817717\pi$
$164$		0			0
$165$	−6.03635	+	2.16160i	−0.469929	+	0.168281i
$166$		0			0
$167$	−0.795794			−0.0615804			−0.0307902	−	0.999526i	$-0.509802\pi$
$167$	−0.795794			−0.0615804			−0.0307902	+	0.999526i	$0.509802\pi$
$168$		0			0
$169$	12.6116			0.970124
$170$		0			0
$171$	−7.24800	−	8.82240i	−0.554269	−	0.674665i
$172$		0			0
$173$	15.5946			1.18564			0.592820	−	0.805335i	$-0.298015\pi$
$173$	15.5946			1.18564			0.592820	+	0.805335i	$0.298015\pi$
$174$		0			0
$175$	21.1482	−	9.10995i	1.59865	−	0.688648i
$176$		0			0
$177$	3.24255	+	9.05492i	0.243725	+	0.680609i
$178$		0			0
$179$			0.536579i			0.0401058i	0.999799	+	0.0200529i	$0.00638347\pi$
$179$			0.536579i			0.0401058i	−0.999799	+	0.0200529i	$0.993617\pi$
$180$		0			0
$181$		−	12.3077i		−	0.914826i	−0.889254	−	0.457413i	$-0.848776\pi$
$181$		−	12.3077i		−	0.914826i	0.889254	−	0.457413i	$-0.151224\pi$
$182$		0			0
$183$	10.3382	−	3.70207i	0.764219	−	0.273665i
$184$		0			0
$185$	19.5767			1.43930
$186$		0			0
$187$			3.09755i			0.226515i
$188$		0			0
$189$	13.6273	+	1.81542i	0.991243	+	0.132052i
$190$		0			0
$191$			4.35312i			0.314981i	0.987520	+	0.157491i	$0.0503404\pi$
$191$			4.35312i			0.314981i	−0.987520	+	0.157491i	$0.949660\pi$
$192$		0			0
$193$	1.40415			0.101073			0.0505365	−	0.998722i	$-0.483907\pi$
$193$	1.40415			0.101073			0.0505365	+	0.998722i	$0.483907\pi$
$194$		0			0
$195$	3.76188	−	1.34712i	0.269394	−	0.0964693i
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$		−	3.65105i		−	0.258816i	−0.991591	−	0.129408i	$-0.958692\pi$
$199$		−	3.65105i		−	0.258816i	0.991591	−	0.129408i	$-0.0413077\pi$
$200$		0			0
$201$	0.192179	+	0.536667i	0.0135553	+	0.0378536i
$202$		0			0
$203$	0.110342	+	0.256152i	0.00774448	+	0.0179783i
$204$		0			0
$205$	−12.3634			−0.863501
$206$		0			0
$207$	−13.6792	−	16.6505i	−0.950767	−	1.15729i
$208$		0			0
$209$	−3.80596			−0.263264
$210$		0			0
$211$	17.8840			1.23119			0.615593	−	0.788065i	$-0.288917\pi$
$211$	17.8840			1.23119			0.615593	+	0.788065i	$0.288917\pi$
$212$		0			0
$213$	−20.8202	+	7.45569i	−1.42658	+	0.510855i
$214$		0			0
$215$	−28.5669			−1.94825
$216$		0			0
$217$	−6.71830	−	15.5961i	−0.456068	−	1.05873i
$218$		0			0
$219$	10.1121	−	3.62113i	0.683314	−	0.244693i
$220$		0			0
$221$		−	1.93041i		−	0.129853i
$222$		0			0
$223$			17.6125i			1.17942i	0.807615	+	0.589710i	$0.200758\pi$
$223$			17.6125i			1.17942i	−0.807615	+	0.589710i	$0.799242\pi$
$224$		0			0
$225$	−20.1747	+	16.5745i	−1.34498	+	1.10497i
$226$		0			0
$227$	20.9767			1.39227			0.696136	−	0.717910i	$-0.254901\pi$
$227$	20.9767			1.39227			0.696136	+	0.717910i	$0.254901\pi$
$228$		0			0
$229$		−	7.35904i		−	0.486299i	−0.969989	−	0.243149i	$-0.921819\pi$
$229$		−	7.35904i		−	0.486299i	0.969989	−	0.243149i	$-0.0781805\pi$
$230$		0			0
$231$	3.35109	−	3.12573i	0.220486	−	0.205658i
$232$		0			0
$233$		−	4.19688i		−	0.274947i	−0.990505	−	0.137473i	$-0.956102\pi$
$233$		−	4.19688i		−	0.274947i	0.990505	−	0.137473i	$-0.0438982\pi$
$234$		0			0
$235$	−9.38014			−0.611893
$236$		0			0
$237$	−9.69870	−	27.0839i	−0.629998	−	1.75929i
$238$		0			0
$239$			21.0276i			1.36016i	0.733137	+	0.680080i	$0.238055\pi$
$239$			21.0276i			1.36016i	−0.733137	+	0.680080i	$0.761945\pi$
$240$		0			0
$241$			26.4255i			1.70221i	0.524992	+	0.851107i	$0.324068\pi$
$241$			26.4255i			1.70221i	−0.524992	+	0.851107i	$0.675932\pi$
$242$		0			0
$243$	−15.4168	+	2.30726i	−0.988986	+	0.148011i
$244$		0			0
$245$	−17.8011	+	18.8304i	−1.13727	+	1.20303i
$246$		0			0
$247$	2.37189			0.150920
$248$		0			0
$249$	8.04742	+	22.4727i	0.509984	+	1.42415i
$250$		0			0
$251$	−24.0994			−1.52114			−0.760570	−	0.649256i	$-0.775080\pi$
$251$	−24.0994			−1.52114			−0.760570	+	0.649256i	$0.775080\pi$
$252$		0			0
$253$	−7.18299			−0.451591
$254$		0			0
$255$	6.69568	+	18.6979i	0.419300	+	1.17091i
$256$		0			0
$257$	28.6041			1.78428			0.892138	−	0.451763i	$-0.149205\pi$
$257$	28.6041			1.78428			0.892138	+	0.451763i	$0.149205\pi$
$258$		0			0
$259$	−12.8503	+	5.53548i	−0.798477	+	0.343958i
$260$		0			0
$261$	−0.200754	−	0.244361i	−0.0124264	−	0.0151256i
$262$		0			0
$263$		−	8.30698i		−	0.512230i	−0.966646	−	0.256115i	$-0.917557\pi$
$263$		−	8.30698i		−	0.512230i	0.966646	−	0.256115i	$-0.0824426\pi$
$264$		0			0
$265$			37.9951i			2.33402i
$266$		0			0
$267$	−4.61724	−	12.8938i	−0.282570	−	0.789087i
$268$		0			0
$269$	−14.8358			−0.904554			−0.452277	−	0.891877i	$-0.649388\pi$
$269$	−14.8358			−0.904554			−0.452277	+	0.891877i	$0.649388\pi$
$270$		0			0
$271$			28.3145i			1.71999i	0.510306	+	0.859993i	$0.329532\pi$
$271$			28.3145i			1.71999i	−0.510306	+	0.859993i	$0.670468\pi$
$272$		0			0
$273$	−2.08842	+	1.94797i	−0.126397	+	0.117896i
$274$		0			0
$275$			8.70334i			0.524831i
$276$		0			0
$277$	5.33456			0.320522			0.160261	−	0.987075i	$-0.448766\pi$
$277$	5.33456			0.320522			0.160261	+	0.987075i	$0.448766\pi$
$278$		0			0
$279$	12.2232	+	14.8782i	0.731781	+	0.890737i
$280$		0			0
$281$		−	3.34514i		−	0.199554i	−0.995010	−	0.0997772i	$-0.968187\pi$
$281$		−	3.34514i		−	0.199554i	0.995010	−	0.0997772i	$-0.0318130\pi$
$282$		0			0
$283$			13.6994i			0.814346i	0.913351	+	0.407173i	$0.133485\pi$
$283$			13.6994i			0.814346i	−0.913351	+	0.407173i	$0.866515\pi$
$284$		0			0
$285$	−22.9741	+	8.22698i	−1.36087	+	0.487324i
$286$		0			0
$287$	8.11547	−	3.49588i	0.479041	−	0.206355i
$288$		0			0
$289$	−7.40518			−0.435599
$290$		0			0
$291$	9.67605	−	3.46497i	0.567220	−	0.203120i
$292$		0			0
$293$	−0.667290			−0.0389835			−0.0194918	−	0.999810i	$-0.506205\pi$
$293$	−0.667290			−0.0389835			−0.0194918	+	0.999810i	$0.506205\pi$
$294$		0			0
$295$	20.5559			1.19681
$296$		0			0
$297$	−2.66789	+	4.45896i	−0.154807	+	0.258735i
$298$		0			0
$299$	4.47647			0.258881
$300$		0			0
$301$	18.7516	−	8.07756i	1.08082	−	0.465583i
$302$		0			0
$303$	−7.98504	−	22.2985i	−0.458729	−	1.28101i
$304$		0			0
$305$		−	23.4691i		−	1.34383i
$306$		0			0
$307$			23.8143i			1.35915i	0.733605	+	0.679576i	$0.237836\pi$
$307$			23.8143i			1.35915i	−0.733605	+	0.679576i	$0.762164\pi$
$308$		0			0
$309$	−22.9649	+	8.22370i	−1.30643	+	0.467830i
$310$		0			0
$311$	−7.44151			−0.421969			−0.210985	−	0.977489i	$-0.567667\pi$
$311$	−7.44151			−0.421969			−0.210985	+	0.977489i	$0.567667\pi$
$312$		0			0
$313$			17.7193i			1.00156i	0.865576	+	0.500778i	$0.166953\pi$
$313$			17.7193i			1.00156i	−0.865576	+	0.500778i	$0.833047\pi$
$314$		0			0
$315$	13.4718	−	26.1117i	0.759050	−	1.47123i
$316$		0			0
$317$		−	34.4062i		−	1.93245i	−0.257707	−	0.966223i	$-0.582967\pi$
$317$		−	34.4062i		−	1.93245i	0.257707	−	0.966223i	$-0.417033\pi$
$318$		0			0
$319$	−0.105417			−0.00590222
$320$		0			0
$321$	−8.04359	+	2.88039i	−0.448949	+	0.160768i
$322$		0			0
$323$			11.7892i			0.655967i
$324$		0			0
$325$		−	5.42396i		−	0.300867i
$326$		0			0
$327$	−9.99910	−	27.9228i	−0.552951	−	1.54414i
$328$		0			0
$329$	6.15720	−	2.65232i	0.339457	−	0.146227i
$330$		0			0
$331$	27.9242			1.53485			0.767427	−	0.641136i	$-0.221536\pi$
$331$	27.9242			1.53485			0.767427	+	0.641136i	$0.221536\pi$
$332$		0			0
$333$	12.2588	−	10.0711i	0.671777	−	0.551895i
$334$		0			0
$335$	1.21831			0.0665633
$336$		0			0
$337$	25.5412			1.39132			0.695659	−	0.718372i	$-0.255113\pi$
$337$	25.5412			1.39132			0.695659	+	0.718372i	$0.255113\pi$
$338$		0			0
$339$	4.08335	−	1.46224i	0.221777	−	0.0794179i
$340$		0			0
$341$	6.41844			0.347578
$342$		0			0
$343$	6.36030	−	17.3939i	0.343424	−	0.939181i
$344$		0			0
$345$	−43.3590	+	15.5268i	−2.33437	+	0.835934i
$346$		0			0
$347$			23.4201i			1.25726i	0.777705	+	0.628629i	$0.216384\pi$
$347$			23.4201i			1.25726i	−0.777705	+	0.628629i	$0.783616\pi$
$348$		0			0
$349$			22.4007i			1.19908i	0.800345	+	0.599540i	$0.204650\pi$
$349$			22.4007i			1.19908i	−0.800345	+	0.599540i	$0.795350\pi$
$350$		0			0
$351$	1.66264	−	2.77884i	0.0887453	−	0.148324i
$352$		0			0
$353$	30.3038			1.61291			0.806454	−	0.591296i	$-0.201384\pi$
$353$	30.3038			1.61291			0.806454	+	0.591296i	$0.201384\pi$
$354$		0			0
$355$			47.2648i			2.50856i
$356$		0			0
$357$	−9.68210	−	10.3802i	−0.512431	−	0.549378i
$358$		0			0
$359$		−	25.7305i		−	1.35800i	−0.734137	−	0.679001i	$-0.762413\pi$
$359$		−	25.7305i		−	1.35800i	0.734137	−	0.679001i	$-0.237587\pi$
$360$		0			0
$361$	4.51464			0.237613
$362$		0			0
$363$	0.583933	+	1.63065i	0.0306485	+	0.0855870i
$364$		0			0
$365$		−	22.9559i		−	1.20157i
$366$		0			0
$367$		−	17.6948i		−	0.923659i	−0.886969	−	0.461829i	$-0.847193\pi$
$367$		−	17.6948i		−	0.923659i	0.886969	−	0.461829i	$-0.152807\pi$
$368$		0			0
$369$	−7.74191	+	6.36034i	−0.403028	+	0.331106i
$370$		0			0
$371$	−10.7435	−	24.9403i	−0.557772	−	1.29483i
$372$		0			0
$373$	11.8840			0.615330			0.307665	−	0.951495i	$-0.400452\pi$
$373$	11.8840			0.615330			0.307665	+	0.951495i	$0.400452\pi$
$374$		0			0
$375$	8.00516	+	22.3547i	0.413385	+	1.15439i
$376$		0			0
$377$	0.0656963			0.00338353
$378$		0			0
$379$	19.3091			0.991840			0.495920	−	0.868368i	$-0.334831\pi$
$379$	19.3091			0.991840			0.495920	+	0.868368i	$0.334831\pi$
$380$		0			0
$381$	−5.49139	−	15.3349i	−0.281332	−	0.785630i
$382$		0			0
$383$	−0.0701875			−0.00358641			−0.00179321	−	0.999998i	$-0.500571\pi$
$383$	−0.0701875			−0.00358641			−0.00179321	+	0.999998i	$0.500571\pi$
$384$		0			0
$385$	−3.87475	−	8.99499i	−0.197475	−	0.458427i
$386$		0			0
$387$	−17.8884	+	14.6962i	−0.909320	+	0.747048i
$388$		0			0
$389$			26.2397i			1.33040i	0.746664	+	0.665202i	$0.231654\pi$
$389$			26.2397i			1.33040i	−0.746664	+	0.665202i	$0.768346\pi$
$390$		0			0
$391$			22.2497i			1.12522i
$392$		0			0
$393$	−6.61162	−	18.4632i	−0.333512	−	0.931343i
$394$		0			0
$395$	−61.4843			−3.09361
$396$		0			0
$397$		−	23.4489i		−	1.17687i	−0.808546	−	0.588433i	$-0.799745\pi$
$397$		−	23.4489i		−	1.17687i	0.808546	−	0.588433i	$-0.200255\pi$
$398$		0			0
$399$	12.7541	−	11.8964i	0.638506	−	0.595565i
$400$		0			0
$401$			21.5851i			1.07791i	0.842336	+	0.538953i	$0.181180\pi$
$401$			21.5851i			1.07791i	−0.842336	+	0.538953i	$0.818820\pi$
$402$		0			0
$403$	−4.00000			−0.199254
$404$		0			0
$405$	−6.46582	+	32.6828i	−0.321289	+	1.62402i
$406$		0			0
$407$		−	5.28841i		−	0.262137i
$408$		0			0
$409$		−	25.1513i		−	1.24365i	−0.783156	−	0.621825i	$-0.786391\pi$
$409$		−	25.1513i		−	1.24365i	0.783156	−	0.621825i	$-0.213609\pi$
$410$		0			0
$411$	−7.39758	+	2.64906i	−0.364896	+	0.130668i
$412$		0			0
$413$	−13.4931	+	5.81237i	−0.663950	+	0.286008i
$414$		0			0
$415$	51.0161			2.50428
$416$		0			0
$417$	0.970910	−	0.347681i	0.0475457	−	0.0170260i
$418$		0			0
$419$	36.9050			1.80292			0.901462	−	0.432857i	$-0.142495\pi$
$419$	36.9050			1.80292			0.901462	+	0.432857i	$0.142495\pi$
$420$		0			0
$421$	−7.76262			−0.378327			−0.189163	−	0.981946i	$-0.560578\pi$
$421$	−7.76262			−0.378327			−0.189163	+	0.981946i	$0.560578\pi$
$422$		0			0
$423$	−5.87378	+	4.82558i	−0.285593	+	0.234628i
$424$		0			0
$425$	26.9591			1.30771
$426$		0			0
$427$	6.63609	+	15.4053i	0.321143	+	0.745514i
$428$		0			0
$429$	−0.363909	−	1.01623i	−0.0175697	−	0.0490640i
$430$		0			0
$431$			23.0972i			1.11255i	0.830998	+	0.556276i	$0.187770\pi$
$431$			23.0972i			1.11255i	−0.830998	+	0.556276i	$0.812230\pi$
$432$		0			0
$433$			30.7645i			1.47845i	0.673459	+	0.739225i	$0.264808\pi$
$433$			30.7645i			1.47845i	−0.673459	+	0.739225i	$0.735192\pi$
$434$		0			0
$435$	−0.636334	+	0.227870i	−0.0305099	+	0.0109255i
$436$		0			0
$437$	−27.3382			−1.30776
$438$		0			0
$439$		−	30.8236i		−	1.47113i	−0.677453	−	0.735566i	$-0.736916\pi$
$439$		−	30.8236i		−	1.47113i	0.677453	−	0.735566i	$-0.263084\pi$
$440$		0			0
$441$	−1.45967	+	20.9492i	−0.0695081	+	0.997581i
$442$		0			0
$443$		−	7.27997i		−	0.345882i	−0.984932	−	0.172941i	$-0.944673\pi$
$443$		−	7.27997i		−	0.345882i	0.984932	−	0.172941i	$-0.0553269\pi$
$444$		0			0
$445$	−29.2707			−1.38756
$446$		0			0
$447$	5.26560	−	1.88560i	0.249054	−	0.0891858i
$448$		0			0
$449$		−	12.9789i		−	0.612512i	−0.951949	−	0.306256i	$-0.900924\pi$
$449$		−	12.9789i		−	0.612512i	0.951949	−	0.306256i	$-0.0990763\pi$
$450$		0			0
$451$			3.33984i			0.157267i
$452$		0			0
$453$	9.99762	+	27.9187i	0.469729	+	1.31173i
$454$		0			0
$455$	2.41476	+	5.60571i	0.113206	+	0.262800i
$456$		0			0
$457$	−5.98351			−0.279897			−0.139948	−	0.990159i	$-0.544694\pi$
$457$	−5.98351			−0.279897			−0.139948	+	0.990159i	$0.544694\pi$
$458$		0			0
$459$	13.8119	+	8.26394i	0.644683	+	0.385727i
$460$		0			0
$461$	−16.4313			−0.765283			−0.382641	−	0.923897i	$-0.624985\pi$
$461$	−16.4313			−0.765283			−0.382641	+	0.923897i	$0.624985\pi$
$462$		0			0
$463$	−15.0678			−0.700261			−0.350130	−	0.936701i	$-0.613863\pi$
$463$	−15.0678			−0.700261			−0.350130	+	0.936701i	$0.613863\pi$
$464$		0			0
$465$	38.7440	−	13.8741i	1.79671	−	0.643397i
$466$		0			0
$467$	0.156602			0.00724669			0.00362334	−	0.999993i	$-0.498847\pi$
$467$	0.156602			0.00724669			0.00362334	+	0.999993i	$0.498847\pi$
$468$		0			0
$469$	−0.799708	+	0.344488i	−0.0369271	+	0.0159070i
$470$		0			0
$471$	−6.81646	+	2.44096i	−0.314086	+	0.112474i
$472$		0			0
$473$			7.71703i			0.354830i
$474$		0			0
$475$			33.1246i			1.51986i
$476$		0			0
$477$	19.5464	+	23.7923i	0.894970	+	1.08937i
$478$		0			0
$479$	18.4027			0.840840			0.420420	−	0.907330i	$-0.361883\pi$
$479$	18.4027			0.840840			0.420420	+	0.907330i	$0.361883\pi$
$480$		0			0
$481$			3.29576i			0.150274i
$482$		0			0
$483$	24.0709	−	22.4521i	1.09526	−	1.02160i
$484$		0			0
$485$		−	21.9660i		−	0.997423i
$486$		0			0
$487$	−37.8428			−1.71482			−0.857411	−	0.514632i	$-0.827929\pi$
$487$	−37.8428			−1.71482			−0.857411	+	0.514632i	$0.827929\pi$
$488$		0			0
$489$	12.5316	+	34.9948i	0.566696	+	1.58252i
$490$		0			0
$491$			3.99156i			0.180136i	0.995936	+	0.0900682i	$0.0287085\pi$
$491$			3.99156i			0.180136i	−0.995936	+	0.0900682i	$0.971291\pi$
$492$		0			0
$493$			0.326534i			0.0147064i
$494$		0			0
$495$	7.04964	+	8.58095i	0.316858	+	0.385685i
$496$		0			0
$497$	−13.3646	−	31.0250i	−0.599483	−	1.39166i
$498$		0			0
$499$	−36.7812			−1.64655			−0.823277	−	0.567640i	$-0.807857\pi$
$499$	−36.7812			−1.64655			−0.823277	+	0.567640i	$0.807857\pi$
$500$		0			0
$501$	0.464690	+	1.29766i	0.0207608	+	0.0579753i
$502$		0			0
$503$	−40.7233			−1.81576			−0.907881	−	0.419229i	$-0.862301\pi$
$503$	−40.7233			−1.81576			−0.907881	+	0.419229i	$0.862301\pi$
$504$		0			0
$505$	−50.6206			−2.25259
$506$		0			0
$507$	−7.36433	−	20.5651i	−0.327062	−	0.913330i
$508$		0			0
$509$	22.2717			0.987175			0.493587	−	0.869696i	$-0.335685\pi$
$509$	22.2717			0.987175			0.493587	+	0.869696i	$0.335685\pi$
$510$		0			0
$511$	6.49100	+	15.0685i	0.287145	+	0.666589i
$512$		0			0
$513$	−10.1539	+	16.9706i	−0.448306	+	0.749272i
$514$		0			0
$515$			52.1336i			2.29728i
$516$		0			0
$517$			2.53394i			0.111442i
$518$		0			0
$519$	−9.10622	−	25.4294i	−0.399719	−	1.11623i
$520$		0			0
$521$	9.66158			0.423282			0.211641	−	0.977348i	$-0.432119\pi$
$521$	9.66158			0.423282			0.211641	+	0.977348i	$0.432119\pi$
$522$		0			0
$523$			4.73427i			0.207015i	0.994629	+	0.103508i	$0.0330066\pi$
$523$			4.73427i			0.207015i	−0.994629	+	0.103508i	$0.966993\pi$
$524$		0			0
$525$	−27.2043	−	29.1657i	−1.18729	−	1.27290i
$526$		0			0
$527$		−	19.8815i		−	0.866050i
$528$		0			0
$529$	−28.5954			−1.24328
$530$		0			0
$531$	12.8720	−	10.5749i	0.558596	−	0.458913i
$532$		0			0
$533$		−	2.08140i		−	0.0901556i
$534$		0			0
$535$			18.2601i			0.789451i
$536$		0			0
$537$	0.874974	−	0.313326i	0.0377579	−	0.0135210i
$538$		0			0
$539$	5.08683	+	4.80876i	0.219105	+	0.207128i
$540$		0			0
$541$	36.8408			1.58391			0.791954	−	0.610581i	$-0.209064\pi$
$541$	36.8408			1.58391			0.791954	+	0.610581i	$0.209064\pi$
$542$		0			0
$543$	−20.0696	+	7.18688i	−0.861270	+	0.308419i
$544$		0			0
$545$	−63.3886			−2.71527
$546$		0			0
$547$	−45.0676			−1.92695			−0.963475	−	0.267799i	$-0.913704\pi$
$547$	−45.0676			−1.92695			−0.963475	+	0.267799i	$0.913704\pi$
$548$		0			0
$549$	−12.0736	−	14.6962i	−0.515288	−	0.627218i
$550$		0			0
$551$	−0.401213			−0.0170923
$552$		0			0
$553$	40.3588	−	17.3852i	1.71623	−	0.739295i
$554$		0			0
$555$	−11.4314	−	31.9227i	−0.485238	−	1.35504i
$556$		0			0
$557$		−	23.6492i		−	1.00205i	−0.865434	−	0.501023i	$-0.832957\pi$
$557$		−	23.6492i		−	1.00205i	0.865434	−	0.501023i	$-0.167043\pi$
$558$		0			0
$559$		−	4.80929i		−	0.203411i
$560$		0			0
$561$	5.05103	−	1.80876i	0.213254	−	0.0763660i
$562$		0			0
$563$	−16.5801			−0.698769			−0.349384	−	0.936979i	$-0.613609\pi$
$563$	−16.5801			−0.698769			−0.349384	+	0.936979i	$0.613609\pi$
$564$		0			0
$565$		−	9.26977i		−	0.389982i
$566$		0			0
$567$	−4.99713	−	23.2815i	−0.209860	−	0.977731i
$568$		0			0
$569$			11.2532i			0.471757i	0.971783	+	0.235878i	$0.0757967\pi$
$569$			11.2532i			0.471757i	−0.971783	+	0.235878i	$0.924203\pi$
$570$		0			0
$571$	−1.60156			−0.0670231			−0.0335115	−	0.999438i	$-0.510669\pi$
$571$	−1.60156			−0.0670231			−0.0335115	+	0.999438i	$0.510669\pi$
$572$		0			0
$573$	7.09843	−	2.54193i	0.296541	−	0.106191i
$574$		0			0
$575$			62.5161i			2.60710i
$576$		0			0
$577$		−	25.9863i		−	1.08182i	−0.841080	−	0.540911i	$-0.818080\pi$
$577$		−	25.9863i		−	1.08182i	0.841080	−	0.540911i	$-0.181920\pi$
$578$		0			0
$579$	−0.819929	−	2.28968i	−0.0340751	−	0.0951558i
$580$		0			0
$581$	−33.4874	+	14.4253i	−1.38929	+	0.598460i
$582$		0			0
$583$	10.2639			0.425089
$584$		0			0
$585$	−4.39337	−	5.34768i	−0.181643	−	0.221099i
$586$		0			0
$587$	1.10779			0.0457234			0.0228617	−	0.999739i	$-0.492722\pi$
$587$	1.10779			0.0457234			0.0228617	+	0.999739i	$0.492722\pi$
$588$		0			0
$589$	24.4284			1.00655
$590$		0			0
$591$	−3.26130	+	1.16787i	−0.134152	+	0.0480396i
$592$		0			0
$593$	1.77927			0.0730657			0.0365328	−	0.999332i	$-0.488369\pi$
$593$	1.77927			0.0730657			0.0365328	+	0.999332i	$0.488369\pi$
$594$		0			0
$595$	−27.8624	+	12.0022i	−1.14225	+	0.492043i
$596$		0			0
$597$	−5.95359	+	2.13197i	−0.243664	+	0.0872556i
$598$		0			0
$599$		−	35.0449i		−	1.43189i	−0.698155	−	0.715947i	$-0.745995\pi$
$599$		−	35.0449i		−	1.43189i	0.698155	−	0.715947i	$-0.254005\pi$
$600$		0			0
$601$			23.9559i			0.977180i	0.872514	+	0.488590i	$0.162489\pi$
$601$			23.9559i			0.977180i	−0.872514	+	0.488590i	$0.837511\pi$
$602$		0			0
$603$	0.762897	−	0.626755i	0.0310676	−	0.0255234i
$604$		0			0
$605$	3.70180			0.150500
$606$		0			0
$607$			27.4838i			1.11553i	0.829998	+	0.557766i	$0.188342\pi$
$607$			27.4838i			1.11553i	−0.829998	+	0.557766i	$0.811658\pi$
$608$		0			0
$609$	0.353262	−	0.329505i	0.0143149	−	0.0133522i
$610$		0			0
$611$		−	1.57916i		−	0.0638860i
$612$		0			0
$613$	20.6367			0.833507			0.416753	−	0.909020i	$-0.363168\pi$
$613$	20.6367			0.833507			0.416753	+	0.909020i	$0.363168\pi$
$614$		0			0
$615$	7.21942	+	20.1605i	0.291115	+	0.812949i
$616$		0			0
$617$		−	25.7031i		−	1.03477i	−0.855754	−	0.517384i	$-0.826906\pi$
$617$		−	25.7031i		−	1.03477i	0.855754	−	0.517384i	$-0.173094\pi$
$618$		0			0
$619$		−	30.5912i		−	1.22957i	−0.788697	−	0.614783i	$-0.789244\pi$
$619$		−	30.5912i		−	1.22957i	0.788697	−	0.614783i	$-0.210756\pi$
$620$		0			0
$621$	−19.1635	+	32.0287i	−0.769003	+	1.28527i
$622$		0			0
$623$	19.2135	−	8.27655i	0.769773	−	0.331593i
$624$		0			0
$625$	7.23148			0.289259
$626$		0			0
$627$	2.22243	+	6.20620i	0.0887551	+	0.247852i
$628$		0			0
$629$	−16.3811			−0.653158
$630$		0			0
$631$	−36.3557			−1.44730			−0.723648	−	0.690169i	$-0.757536\pi$
$631$	−36.3557			−1.44730			−0.723648	+	0.690169i	$0.757536\pi$
$632$		0			0
$633$	−10.4430	−	29.1626i	−0.415074	−	1.15911i
$634$		0			0
$635$	−34.8123			−1.38148
$636$		0			0
$637$	−3.17013	−	2.99684i	−0.125605	−	0.118739i
$638$		0			0
$639$	24.3152	+	29.5969i	0.961896	+	1.17084i
$640$		0			0
$641$			10.3582i			0.409124i	0.978854	+	0.204562i	$0.0655771\pi$
$641$			10.3582i			0.409124i	−0.978854	+	0.204562i	$0.934423\pi$
$642$		0			0
$643$		−	19.0184i		−	0.750014i	−0.927022	−	0.375007i	$-0.877640\pi$
$643$		−	19.0184i		−	0.750014i	0.927022	−	0.375007i	$-0.122360\pi$
$644$		0			0
$645$	16.6812	+	46.5827i	0.656820	+	1.83419i
$646$		0			0
$647$	25.0602			0.985218			0.492609	−	0.870251i	$-0.336043\pi$
$647$	25.0602			0.985218			0.492609	+	0.870251i	$0.336043\pi$
$648$		0			0
$649$		−	5.55295i		−	0.217972i
$650$		0			0
$651$	−21.5088	+	20.0623i	−0.842997	+	0.786303i
$652$		0			0
$653$		−	32.0933i		−	1.25591i	−0.778250	−	0.627955i	$-0.783892\pi$
$653$		−	32.0933i		−	1.25591i	0.778250	−	0.627955i	$-0.216108\pi$
$654$		0			0
$655$	−41.9139			−1.63771
$656$		0			0
$657$	−11.8096	−	14.3749i	−0.460737	−	0.560817i
$658$		0			0
$659$			4.93782i			0.192350i	0.995364	+	0.0961752i	$0.0306609\pi$
$659$			4.93782i			0.192350i	−0.995364	+	0.0961752i	$0.969339\pi$
$660$		0			0
$661$			19.9752i			0.776947i	0.921460	+	0.388473i	$0.126997\pi$
$661$			19.9752i			0.776947i	−0.921460	+	0.388473i	$0.873003\pi$
$662$		0			0
$663$	−3.14782	+	1.12723i	−0.122251	+	0.0437779i
$664$		0			0
$665$	−14.7471	−	34.2346i	−0.571869	−	1.32756i
$666$		0			0
$667$	−0.757209			−0.0293193
$668$		0			0
$669$	28.7199	−	10.2845i	1.11037	−	0.397622i
$670$		0			0
$671$	−6.33990			−0.244749
$672$		0			0
$673$	−12.7737			−0.492390			−0.246195	−	0.969220i	$-0.579180\pi$
$673$	−12.7737			−0.492390			−0.246195	+	0.969220i	$0.579180\pi$
$674$		0			0
$675$	38.8079	+	23.2196i	1.49372	+	0.893723i
$676$		0			0
$677$	−9.73533			−0.374159			−0.187080	−	0.982345i	$-0.559902\pi$
$677$	−9.73533			−0.374159			−0.187080	+	0.982345i	$0.559902\pi$
$678$		0			0
$679$	6.21108	+	14.4186i	0.238359	+	0.553336i
$680$		0			0
$681$	−12.2490	−	34.2057i	−0.469382	−	1.31076i
$682$		0			0
$683$			43.7031i			1.67225i	0.548537	+	0.836126i	$0.315185\pi$
$683$			43.7031i			1.67225i	−0.548537	+	0.836126i	$0.684815\pi$
$684$		0			0
$685$			16.7935i			0.641648i
$686$		0			0
$687$	−12.0000	+	4.29718i	−0.457829	+	0.163948i
$688$		0			0
$689$	−6.39653			−0.243688
$690$		0			0
$691$		−	16.0293i		−	0.609782i	−0.952387	−	0.304891i	$-0.901380\pi$
$691$		−	16.0293i		−	0.609782i	0.952387	−	0.304891i	$-0.0986201\pi$
$692$		0			0
$693$	−7.05378	−	3.63925i	−0.267951	−	0.138244i
$694$		0			0
$695$		−	2.20410i		−	0.0836063i
$696$		0			0
$697$	10.3453			0.391858
$698$		0			0
$699$	−6.84365	+	2.45070i	−0.258851	+	0.0926938i
$700$		0			0
$701$		−	37.9860i		−	1.43471i	−0.696706	−	0.717357i	$-0.745352\pi$
$701$		−	37.9860i		−	1.43471i	0.696706	−	0.717357i	$-0.254648\pi$
$702$		0			0
$703$		−	20.1275i		−	0.759123i
$704$		0			0
$705$	5.47737	+	15.2957i	0.206290	+	0.576071i
$706$		0			0
$707$	33.2278	−	14.3134i	1.24966	−	0.538313i
$708$		0			0
$709$	33.3364			1.25198			0.625988	−	0.779833i	$-0.284696\pi$
$709$	33.3364			1.25198			0.625988	+	0.779833i	$0.284696\pi$
$710$		0			0
$711$	−38.5011	+	31.6304i	−1.44390	+	1.18623i
$712$		0			0
$713$	46.1036			1.72659
$714$		0			0
$715$	−2.30698			−0.0862761
$716$		0			0
$717$	34.2886	−	12.2787i	1.28053	−	0.458556i
$718$		0			0
$719$	−39.7616			−1.48286			−0.741428	−	0.671032i	$-0.765851\pi$
$719$	−39.7616			−1.48286			−0.741428	+	0.671032i	$0.765851\pi$
$720$		0			0
$721$	−14.7412	−	34.2209i	−0.548993	−	1.27445i
$722$		0			0
$723$	43.0907	−	15.4307i	1.60256	−	0.573874i
$724$		0			0
$725$			0.917480i			0.0340744i
$726$		0			0
$727$		−	28.5278i		−	1.05804i	−0.848611	−	0.529018i	$-0.822560\pi$
$727$		−	28.5278i		−	1.05804i	0.848611	−	0.529018i	$-0.177440\pi$
$728$		0			0
$729$	12.7647	+	23.7921i	0.472766	+	0.881188i
$730$		0			0
$731$	23.9039			0.884118
$732$		0			0
$733$			33.0504i			1.22074i	0.792115	+	0.610372i	$0.208980\pi$
$733$			33.0504i			1.22074i	−0.792115	+	0.610372i	$0.791020\pi$
$734$		0			0
$735$	41.1005	+	18.0317i	1.51601	+	0.665108i
$736$		0			0
$737$		−	0.329112i		−	0.0121230i
$738$		0			0
$739$	−7.28867			−0.268118			−0.134059	−	0.990973i	$-0.542801\pi$
$739$	−7.28867			−0.268118			−0.134059	+	0.990973i	$0.542801\pi$
$740$		0			0
$741$	−1.38502	−	3.86773i	−0.0508802	−	0.142085i
$742$		0			0
$743$		−	34.5060i		−	1.26590i	−0.774192	−	0.632951i	$-0.781843\pi$
$743$		−	34.5060i		−	1.26590i	0.774192	−	0.632951i	$-0.218157\pi$
$744$		0			0
$745$		−	11.9536i		−	0.437947i
$746$		0			0
$747$	31.9459	−	26.2451i	1.16884	−	0.960256i
$748$		0			0
$749$	−5.16320	−	11.9860i	−0.188659	−	0.437961i
$750$		0			0
$751$	−19.1699			−0.699518			−0.349759	−	0.936840i	$-0.613737\pi$
$751$	−19.1699			−0.699518			−0.349759	+	0.936840i	$0.613737\pi$
$752$		0			0
$753$	14.0724	+	39.2977i	0.512827	+	1.43209i
$754$		0			0
$755$	63.3792			2.30661
$756$		0			0
$757$	10.2158			0.371300			0.185650	−	0.982616i	$-0.440561\pi$
$757$	10.2158			0.371300			0.185650	+	0.982616i	$0.440561\pi$
$758$		0			0
$759$	4.19438	+	11.7130i	0.152246	+	0.425153i
$760$		0			0
$761$	31.6429			1.14705			0.573527	−	0.819187i	$-0.305575\pi$
$761$	31.6429			1.14705			0.573527	+	0.819187i	$0.305575\pi$
$762$		0			0
$763$	41.6088	−	17.9237i	1.50634	−	0.648882i
$764$		0			0
$765$	26.5799	−	21.8366i	0.960999	−	0.789505i
$766$		0			0
$767$			3.46062i			0.124956i
$768$		0			0
$769$		−	44.5386i		−	1.60610i	−0.595910	−	0.803051i	$-0.703209\pi$
$769$		−	44.5386i		−	1.60610i	0.595910	−	0.803051i	$-0.296791\pi$
$770$		0			0
$771$	−16.7029	−	46.6433i	−0.601540	−	1.67982i
$772$		0			0
$773$	14.8133			0.532798			0.266399	−	0.963863i	$-0.414166\pi$
$773$	14.8133			0.532798			0.266399	+	0.963863i	$0.414166\pi$
$774$		0			0
$775$		−	55.8619i		−	2.00662i
$776$		0			0
$777$	16.5301	+	17.7220i	0.593015	+	0.635772i
$778$		0			0
$779$			12.7113i			0.455431i
$780$		0			0
$781$	12.7681			0.456877
$782$		0			0
$783$	−0.281241	+	0.470050i	−0.0100507	+	0.0167982i
$784$		0			0
$785$			15.4743i			0.552302i
$786$		0			0
$787$		−	50.9666i		−	1.81676i	−0.418141	−	0.908382i	$-0.637318\pi$
$787$		−	50.9666i		−	1.81676i	0.418141	−	0.908382i	$-0.362682\pi$
$788$		0			0
$789$	−13.5458	+	4.85072i	−0.482243	+	0.172690i
$790$		0			0
$791$	2.62111	+	6.08475i	0.0931960	+	0.216349i
$792$		0			0
$793$	3.95105			0.140306
$794$		0			0
$795$	61.9567	−	22.1866i	2.19738	−	0.786876i
$796$		0			0
$797$	−51.2155			−1.81415			−0.907073	−	0.420972i	$-0.861689\pi$
$797$	−51.2155			−1.81415			−0.907073	+	0.420972i	$0.861689\pi$
$798$		0			0
$799$	7.84900			0.277678
$800$		0			0
$801$	−18.3291	+	15.0582i	−0.647628	+	0.532056i
$802$		0			0
$803$	−6.20128			−0.218839
$804$		0			0
$805$	−27.8323	−	64.6109i	−0.980959	−	2.27724i
$806$		0			0
$807$	8.66310	+	24.1920i	0.304956	+	0.851599i
$808$		0			0
$809$		−	16.6266i		−	0.584559i	−0.956333	−	0.292279i	$-0.905586\pi$
$809$		−	16.6266i		−	0.584559i	0.956333	−	0.292279i	$-0.0944137\pi$
$810$		0			0
$811$			27.1091i			0.951928i	0.879465	+	0.475964i	$0.157901\pi$
$811$			27.1091i			0.951928i	−0.879465	+	0.475964i	$0.842099\pi$
$812$		0			0
$813$	46.1711	−	16.5338i	1.61929	−	0.579865i
$814$		0			0
$815$	79.4430			2.78277
$816$		0			0
$817$			29.3707i			1.02755i
$818$		0			0
$819$	4.39595	+	2.26800i	0.153607	+	0.0792502i
$820$		0			0
$821$			8.68478i			0.303101i	0.988450	+	0.151550i	$0.0484266\pi$
$821$			8.68478i			0.303101i	−0.988450	+	0.151550i	$0.951573\pi$
$822$		0			0
$823$	−0.291975			−0.0101776			−0.00508881	−	0.999987i	$-0.501620\pi$
$823$	−0.291975			−0.0101776			−0.00508881	+	0.999987i	$0.501620\pi$
$824$		0			0
$825$	14.1921	−	5.08217i	0.494106	−	0.176938i
$826$		0			0
$827$			10.8358i			0.376797i	0.982093	+	0.188398i	$0.0603296\pi$
$827$			10.8358i			0.376797i	−0.982093	+	0.188398i	$0.939670\pi$
$828$		0			0
$829$			41.4578i			1.43989i	0.694032	+	0.719944i	$0.255833\pi$
$829$			41.4578i			1.43989i	−0.694032	+	0.719944i	$0.744167\pi$
$830$		0			0
$831$	−3.11502	−	8.69880i	−0.108059	−	0.301758i
$832$		0			0
$833$	14.8954	−	15.7567i	0.516094	−	0.545938i
$834$		0			0
$835$	2.94587			0.101946
$836$		0			0
$837$	17.1237	−	28.6196i	0.591882	−	0.989238i
$838$		0			0
$839$	−18.3302			−0.632830			−0.316415	−	0.948621i	$-0.602479\pi$
$839$	−18.3302			−0.632830			−0.316415	+	0.948621i	$0.602479\pi$
$840$		0			0
$841$	28.9889			0.999617
$842$		0			0
$843$	−5.45476	+	1.95334i	−0.187872	+	0.0672765i
$844$		0			0
$845$	−46.6857			−1.60604
$846$		0			0
$847$	−2.42989	+	1.04672i	−0.0834921	+	0.0359657i
$848$		0			0
$849$	22.3390	−	7.99954i	0.766672	−	0.274543i
$850$		0			0
$851$		−	37.9866i		−	1.30216i
$852$		0			0
$853$		−	22.2649i		−	0.762337i	−0.924506	−	0.381169i	$-0.875522\pi$
$853$		−	22.2649i		−	0.762337i	0.924506	−	0.381169i	$-0.124478\pi$
$854$		0			0
$855$	26.8307	+	32.6588i	0.917590	+	1.11691i
$856$		0			0
$857$	17.0538			0.582547			0.291273	−	0.956640i	$-0.405921\pi$
$857$	17.0538			0.582547			0.291273	+	0.956640i	$0.405921\pi$
$858$		0			0
$859$			12.0827i			0.412256i	0.978525	+	0.206128i	$0.0660864\pi$
$859$			12.0827i			0.412256i	−0.978525	+	0.206128i	$0.933914\pi$
$860$		0			0
$861$	−10.4394	−	11.1921i	−0.355775	−	0.381427i
$862$		0			0
$863$			52.0800i			1.77282i	0.462899	+	0.886411i	$0.346809\pi$
$863$			52.0800i			1.77282i	−0.462899	+	0.886411i	$0.653191\pi$
$864$		0			0
$865$	−57.7283			−1.96282
$866$		0			0
$867$	4.32412	+	12.0753i	0.146855	+	0.410097i
$868$		0			0
$869$			16.6093i			0.563431i
$870$		0			0
$871$			0.205104i			0.00694969i
$872$		0			0
$873$	−11.3003	−	13.7549i	−0.382458	−	0.465534i
$874$		0			0
$875$	−33.3115	+	14.3495i	−1.12614	+	0.485102i
$876$		0			0
$877$	54.7197			1.84775			0.923876	−	0.382692i	$-0.125003\pi$
$877$	54.7197			1.84775			0.923876	+	0.382692i	$0.125003\pi$
$878$		0			0
$879$	0.389653	+	1.08812i	0.0131427	+	0.0367013i
$880$		0			0
$881$	−10.6997			−0.360481			−0.180241	−	0.983623i	$-0.557688\pi$
$881$	−10.6997			−0.360481			−0.180241	+	0.983623i	$0.557688\pi$
$882$		0			0
$883$	−21.6559			−0.728779			−0.364389	−	0.931247i	$-0.618722\pi$
$883$	−21.6559			−0.728779			−0.364389	+	0.931247i	$0.618722\pi$
$884$		0			0
$885$	−12.0033	−	33.5195i	−0.403485	−	1.12675i
$886$		0			0
$887$	47.0036			1.57823			0.789113	−	0.614248i	$-0.210541\pi$
$887$	47.0036			1.57823			0.789113	+	0.614248i	$0.210541\pi$
$888$		0			0
$889$	22.8511	−	9.84350i	0.766401	−	0.330140i
$890$		0			0
$891$	8.82888	+	1.74667i	0.295779	+	0.0585156i
$892$		0			0
$893$			9.64407i			0.322727i
$894$		0			0
$895$		−	1.98631i		−	0.0663951i
$896$		0			0
$897$	−2.61396	−	7.29956i	−0.0872775	−	0.243725i
$898$		0			0
$899$	0.676613			0.0225663
$900$		0			0
$901$		−	31.7931i		−	1.05918i
$902$		0			0
$903$	−24.1213	−	25.8605i	−0.802708	−	0.860584i
$904$		0			0
$905$			45.5608i			1.51449i
$906$		0			0
$907$	45.0300			1.49520			0.747599	−	0.664151i	$-0.231207\pi$
$907$	45.0300			1.49520			0.747599	+	0.664151i	$0.231207\pi$
$908$		0			0
$909$	−31.6983	+	26.0416i	−1.05137	+	0.863746i
$910$		0			0
$911$		−	44.6898i		−	1.48064i	−0.672256	−	0.740319i	$-0.734675\pi$
$911$		−	44.6898i		−	1.48064i	0.672256	−	0.740319i	$-0.265325\pi$
$912$		0			0
$913$		−	13.7814i		−	0.456098i
$914$		0			0
$915$	−38.2699	+	13.7044i	−1.26516	+	0.453052i
$916$		0			0
$917$	27.5126	−	11.8515i	0.908547	−	0.391372i
$918$		0			0
$919$	18.3685			0.605922			0.302961	−	0.953003i	$-0.402025\pi$
$919$	18.3685			0.605922			0.302961	+	0.953003i	$0.402025\pi$
$920$		0			0
$921$	38.8328	−	13.9059i	1.27958	−	0.458216i
$922$		0			0
$923$	−7.95711			−0.261911
$924$		0			0
$925$	−46.0269			−1.51335
$926$		0			0
$927$	26.8200	+	32.6457i	0.880883	+	1.07223i
$928$		0			0
$929$	−6.54350			−0.214685			−0.107343	−	0.994222i	$-0.534234\pi$
$929$	−6.54350			−0.214685			−0.107343	+	0.994222i	$0.534234\pi$
$930$		0			0
$931$	19.3603	+	18.3020i	0.634508	+	0.599823i
$932$		0			0
$933$	4.34534	+	12.1345i	0.142260	+	0.397266i
$934$		0			0
$935$		−	11.4665i		−	0.374995i
$936$		0			0
$937$		−	48.3947i		−	1.58099i	−0.612471	−	0.790493i	$-0.709824\pi$
$937$		−	48.3947i		−	1.58099i	0.612471	−	0.790493i	$-0.290176\pi$
$938$		0			0
$939$	28.8940	−	10.3469i	0.942921	−	0.337658i
$940$		0			0
$941$	−19.0783			−0.621933			−0.310967	−	0.950421i	$-0.600653\pi$
$941$	−19.0783			−0.621933			−0.310967	+	0.950421i	$0.600653\pi$
$942$		0			0
$943$			23.9901i			0.781225i
$944$		0			0
$945$	−50.4457	−	6.72032i	−1.64100	−	0.218612i
$946$		0			0
$947$			42.2511i			1.37298i	0.727141	+	0.686488i	$0.240848\pi$
$947$			42.2511i			1.37298i	−0.727141	+	0.686488i	$0.759152\pi$
$948$		0			0
$949$	3.86467			0.125452
$950$		0			0
$951$	−56.1045	+	20.0909i	−1.81931	+	0.651493i
$952$		0			0
$953$		−	17.6044i		−	0.570262i	−0.958489	−	0.285131i	$-0.907963\pi$
$953$		−	17.6044i		−	0.570262i	0.958489	−	0.285131i	$-0.0920372\pi$
$954$		0			0
$955$		−	16.1144i		−	0.521450i
$956$		0			0
$957$	0.0615564	+	0.171898i	0.00198984	+	0.00555668i
$958$		0			0
$959$	−4.74852	−	11.0234i	−0.153338	−	0.355964i
$960$		0			0
$961$	−10.1964			−0.328917
$962$		0			0
$963$	9.39383	+	11.4343i	0.302712	+	0.368466i
$964$		0			0
$965$	−5.19789			−0.167326
$966$		0			0
$967$	−29.3712			−0.944513			−0.472257	−	0.881461i	$-0.656560\pi$
$967$	−29.3712			−0.944513			−0.472257	+	0.881461i	$0.656560\pi$
$968$		0			0
$969$	19.2240	−	6.88408i	0.617564	−	0.221148i
$970$		0			0
$971$	26.0318			0.835400			0.417700	−	0.908585i	$-0.362836\pi$
$971$	26.0318			0.835400			0.417700	+	0.908585i	$0.362836\pi$
$972$		0			0
$973$	0.623229	+	1.44679i	0.0199798	+	0.0463819i
$974$		0			0
$975$	−8.84458	+	3.16723i	−0.283253	+	0.101432i
$976$		0			0
$977$			60.6495i			1.94035i	0.242406	+	0.970175i	$0.422064\pi$
$977$			60.6495i			1.94035i	−0.242406	+	0.970175i	$0.577936\pi$
$978$		0			0
$979$			7.90715i			0.252713i
$980$		0			0
$981$	−39.6936	+	32.6101i	−1.26732	+	1.04116i
$982$		0			0
$983$	36.9893			1.17977			0.589887	−	0.807486i	$-0.299172\pi$
$983$	36.9893			1.17977			0.589887	+	0.807486i	$0.299172\pi$
$984$		0			0
$985$			7.40361i			0.235898i
$986$		0			0
$987$	−7.92040	−	8.49146i	−0.252109	−	0.270286i
$988$		0			0
$989$			55.4314i			1.76262i
$990$		0			0
$991$	17.4447			0.554148			0.277074	−	0.960849i	$-0.410635\pi$
$991$	17.4447			0.554148			0.277074	+	0.960849i	$0.410635\pi$
$992$		0			0
$993$	−16.3059	−	45.5347i	−0.517451	−	1.44500i
$994$		0			0
$995$			13.5155i			0.428469i
$996$		0			0
$997$		−	50.5937i		−	1.60232i	−0.598452	−	0.801159i	$-0.704217\pi$
$997$		−	50.5937i		−	1.60232i	0.598452	−	0.801159i	$-0.295783\pi$
$998$		0			0
$999$	−23.5808	−	14.1089i	−0.746064	−	0.446386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.v.d.881.5	✓	16
3.2	odd	2	inner	1848.2.v.d.881.11	yes	16
7.6	odd	2	inner	1848.2.v.d.881.12	yes	16
21.20	even	2	inner	1848.2.v.d.881.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.v.d.881.5	✓	16	1.1	even	1	trivial
1848.2.v.d.881.6	yes	16	21.20	even	2	inner
1848.2.v.d.881.11	yes	16	3.2	odd	2	inner
1848.2.v.d.881.12	yes	16	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.v (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.583933 - 1.63065i) q^{3} -3.70180 q^{5} +(2.42989 - 1.04672i) q^{7} +(-2.31805 + 1.90438i) q^{9} +O(q^{10})\)	\(q+(-0.583933 - 1.63065i) q^{3} -3.70180 q^{5} +(2.42989 - 1.04672i) q^{7} +(-2.31805 + 1.90438i) q^{9} +1.00000i q^{11} -0.623204i q^{13} +(2.16160 + 6.03635i) q^{15} +3.09755 q^{17} +3.80596i q^{19} +(-3.12573 - 3.35109i) q^{21} +7.18299i q^{23} +8.70334 q^{25} +(4.45896 + 2.66789i) q^{27} +0.105417i q^{29} -6.41844i q^{31} +(1.63065 - 0.583933i) q^{33} +(-8.99499 + 3.87475i) q^{35} -5.28841 q^{37} +(-1.01623 + 0.363909i) q^{39} +3.33984 q^{41} +7.71703 q^{43} +(8.58095 - 7.04964i) q^{45} +2.53394 q^{47} +(4.80876 - 5.08683i) q^{49} +(-1.80876 - 5.05103i) q^{51} -10.2639i q^{53} -3.70180i q^{55} +(6.20620 - 2.22243i) q^{57} -5.55295 q^{59} +6.33990i q^{61} +(-3.63925 + 7.05378i) q^{63} +2.30698i q^{65} -0.329112 q^{67} +(11.7130 - 4.19438i) q^{69} -12.7681i q^{71} +6.20128i q^{73} +(-5.08217 - 14.1921i) q^{75} +(1.04672 + 2.42989i) q^{77} +16.6093 q^{79} +(1.74667 - 8.82888i) q^{81} -13.7814 q^{83} -11.4665 q^{85} +(0.171898 - 0.0615564i) q^{87} +7.90715 q^{89} +(-0.652319 - 1.51432i) q^{91} +(-10.4662 + 3.74794i) q^{93} -14.0889i q^{95} +5.93386i q^{97} +(-1.90438 - 2.31805i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7} - 24 q^{9}+O(q^{10}) \) 16 * q + 16 * q^7 - 24 * q^9	\( 16 q + 16 q^{7} - 24 q^{9} + 4 q^{15} - 30 q^{21} + 76 q^{25} - 4 q^{37} + 12 q^{39} + 8 q^{43} + 40 q^{49} + 8 q^{51} + 96 q^{57} + 2 q^{63} - 12 q^{67} + 48 q^{79} - 20 q^{81} + 104 q^{85} + 44 q^{91} - 28 q^{93} - 12 q^{99}+O(q^{100}) \) 16 * q + 16 * q^7 - 24 * q^9 + 4 * q^15 - 30 * q^21 + 76 * q^25 - 4 * q^37 + 12 * q^39 + 8 * q^43 + 40 * q^49 + 8 * q^51 + 96 * q^57 + 2 * q^63 - 12 * q^67 + 48 * q^79 - 20 * q^81 + 104 * q^85 + 44 * q^91 - 28 * q^93 - 12 * q^99

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