LMFDB - Embedded newform 1386.2.g.a.881.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(881,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1386.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.g (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:	$11.0672657201$
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} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 $ x^16 - 4x^14 + 8x^12 + 80x^10 + 1189x^8 - 2028x^6 + 1800x^4 + 1080*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$-2.43348 + 1.00798i$ of defining polynomial
Character	$\chi$	$=$	1386.881
Dual form			1386.2.g.a.881.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +2.01596 q^{5} +(1.78450 - 1.95335i) q^{7} +1.00000i q^{8} -2.01596i q^{10} +1.00000i q^{11} +(-1.95335 - 1.78450i) q^{14} +1.00000 q^{16} +5.92266 q^{17} +0.657927i q^{19} -2.01596 q^{20} +1.00000 q^{22} -7.87575i q^{23} -0.935886 q^{25} +(-1.78450 + 1.95335i) q^{28} -1.32636i q^{29} +10.6942i q^{31} -1.00000i q^{32} -5.92266i q^{34} +(3.59749 - 3.93788i) q^{35} +5.87575 q^{37} +0.657927 q^{38} +2.01596i q^{40} +4.56462 q^{41} -1.56900 q^{43} -1.00000i q^{44} -7.87575 q^{46} +0.532690 q^{47} +(-0.631122 - 6.97149i) q^{49} +0.935886i q^{50} -7.44475i q^{53} +2.01596i q^{55} +(1.95335 + 1.78450i) q^{56} -1.32636 q^{58} +4.03193 q^{59} +0.250475i q^{61} +10.6942 q^{62} -1.00000 q^{64} -7.81164 q^{67} -5.92266 q^{68} +(-3.93788 - 3.59749i) q^{70} +0.609527i q^{71} +0.532690i q^{73} -5.87575i q^{74} -0.657927i q^{76} +(1.95335 + 1.78450i) q^{77} -12.7711 q^{79} +2.01596 q^{80} -4.56462i q^{82} +13.3681 q^{83} +11.9399 q^{85} +1.56900i q^{86} -1.00000 q^{88} +5.22255 q^{89} +7.87575i q^{92} -0.532690i q^{94} +1.32636i q^{95} -2.71607i q^{97} +(-6.97149 + 0.631122i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 - 8 * q^7 + 16 * q^16 + 16 * q^22 + 16 * q^25 + 8 * q^28 - 16 * q^37 + 48 * q^43 - 16 * q^46 + 8 * q^49 - 16 * q^58 - 16 * q^64 + 16 * q^67 - 8 * q^70 - 16 * q^79 + 112 * q^85 - 16 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	2.01596			0.901567			0.450783	−	0.892633i	$-0.351145\pi$
$5$	2.01596			0.901567			0.450783	+	0.892633i	$0.351145\pi$
$6$		0			0
$7$	1.78450	−	1.95335i	0.674477	−	0.738295i
$7$	1.78450	−	1.95335i	0.674477	−	0.738295i
$8$			1.00000i			0.353553i
$9$		0			0
$10$		−	2.01596i		−	0.637504i
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−1.95335	−	1.78450i	−0.522054	−	0.476928i
$15$		0			0
$16$	1.00000			0.250000
$17$	5.92266			1.43646			0.718228	−	0.695808i	$-0.244954\pi$
$17$	5.92266			1.43646			0.718228	+	0.695808i	$0.244954\pi$
$18$		0			0
$19$			0.657927i			0.150939i	0.997148	+	0.0754695i	$0.0240455\pi$
$19$			0.657927i			0.150939i	−0.997148	+	0.0754695i	$0.975954\pi$
$20$	−2.01596			−0.450783
$21$		0			0
$22$	1.00000			0.213201
$23$		−	7.87575i		−	1.64221i	−0.570778	−	0.821104i	$-0.693358\pi$
$23$		−	7.87575i		−	1.64221i	0.570778	−	0.821104i	$-0.306642\pi$
$24$		0			0
$25$	−0.935886			−0.187177
$26$		0			0
$27$		0			0
$28$	−1.78450	+	1.95335i	−0.337239	+	0.369148i
$29$		−	1.32636i		−	0.246299i	−0.992388	−	0.123149i	$-0.960701\pi$
$29$		−	1.32636i		−	0.246299i	0.992388	−	0.123149i	$-0.0392994\pi$
$30$		0			0
$31$			10.6942i			1.92074i	0.278732	+	0.960369i	$0.410086\pi$
$31$			10.6942i			1.92074i	−0.278732	+	0.960369i	$0.589914\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	5.92266i		−	1.01573i
$35$	3.59749	−	3.93788i	0.608087	−	0.665623i
$36$		0			0
$37$	5.87575			0.965968			0.482984	−	0.875629i	$-0.339553\pi$
$37$	5.87575			0.965968			0.482984	+	0.875629i	$0.339553\pi$
$38$	0.657927			0.106730
$39$		0			0
$40$			2.01596i			0.318752i
$41$	4.56462			0.712874			0.356437	−	0.934319i	$-0.383991\pi$
$41$	4.56462			0.712874			0.356437	+	0.934319i	$0.383991\pi$
$42$		0			0
$43$	−1.56900			−0.239270			−0.119635	−	0.992818i	$-0.538172\pi$
$43$	−1.56900			−0.239270			−0.119635	+	0.992818i	$0.538172\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$	−7.87575			−1.16122
$47$	0.532690			0.0777008			0.0388504	−	0.999245i	$-0.487630\pi$
$47$	0.532690			0.0777008			0.0388504	+	0.999245i	$0.487630\pi$
$48$		0			0
$49$	−0.631122	−	6.97149i	−0.0901603	−	0.995927i
$50$			0.935886i			0.132354i
$51$		0			0
$52$		0			0
$53$		−	7.44475i		−	1.02262i	−0.859398	−	0.511308i	$-0.829161\pi$
$53$		−	7.44475i		−	1.02262i	0.859398	−	0.511308i	$-0.170839\pi$
$54$		0			0
$55$			2.01596i			0.271833i
$56$	1.95335	+	1.78450i	0.261027	+	0.238464i
$57$		0			0
$58$	−1.32636			−0.174159
$59$	4.03193			0.524913			0.262456	−	0.964944i	$-0.415467\pi$
$59$	4.03193			0.524913			0.262456	+	0.964944i	$0.415467\pi$
$60$		0			0
$61$			0.250475i			0.0320700i	0.999871	+	0.0160350i	$0.00510432\pi$
$61$			0.250475i			0.0320700i	−0.999871	+	0.0160350i	$0.994896\pi$
$62$	10.6942			1.35817
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−7.81164			−0.954344			−0.477172	−	0.878810i	$-0.658338\pi$
$67$	−7.81164			−0.954344			−0.477172	+	0.878810i	$0.658338\pi$
$68$	−5.92266			−0.718228
$69$		0			0
$70$	−3.93788	−	3.59749i	−0.470666	−	0.429982i
$71$			0.609527i			0.0723376i	0.999346	+	0.0361688i	$0.0115154\pi$
$71$			0.609527i			0.0723376i	−0.999346	+	0.0361688i	$0.988485\pi$
$72$		0			0
$73$			0.532690i			0.0623466i	0.999514	+	0.0311733i	$0.00992438\pi$
$73$			0.532690i			0.0623466i	−0.999514	+	0.0311733i	$0.990076\pi$
$74$		−	5.87575i		−	0.683043i
$75$		0			0
$76$		−	0.657927i		−	0.0754695i
$77$	1.95335	+	1.78450i	0.222604	+	0.203363i
$78$		0			0
$79$	−12.7711			−1.43686			−0.718431	−	0.695598i	$-0.755139\pi$
$79$	−12.7711			−1.43686			−0.718431	+	0.695598i	$0.755139\pi$
$80$	2.01596			0.225392
$81$		0			0
$82$		−	4.56462i		−	0.504078i
$83$	13.3681			1.46734			0.733670	−	0.679506i	$-0.237806\pi$
$83$	13.3681			1.46734			0.733670	+	0.679506i	$0.237806\pi$
$84$		0			0
$85$	11.9399			1.29506
$86$			1.56900i			0.169190i
$87$		0			0
$88$	−1.00000			−0.106600
$89$	5.22255			0.553589			0.276794	−	0.960929i	$-0.410728\pi$
$89$	5.22255			0.553589			0.276794	+	0.960929i	$0.410728\pi$
$90$		0			0
$91$		0			0
$92$			7.87575i			0.821104i
$93$		0			0
$94$		−	0.532690i		−	0.0549428i
$95$			1.32636i			0.136082i
$96$		0			0
$97$		−	2.71607i		−	0.275776i	−0.990448	−	0.137888i	$-0.955969\pi$
$97$		−	2.71607i		−	0.275776i	0.990448	−	0.137888i	$-0.0440313\pi$
$98$	−6.97149	+	0.631122i	−0.704227	+	0.0637530i
$99$		0			0
$100$	0.935886			0.0935886
$101$	3.41352			0.339658			0.169829	−	0.985474i	$-0.445678\pi$
$101$	3.41352			0.339658			0.169829	+	0.985474i	$0.445678\pi$
$102$		0			0
$103$		−	13.3681i		−	1.31720i	−0.752494	−	0.658599i	$-0.771149\pi$
$103$		−	13.3681i		−	1.31720i	0.752494	−	0.658599i	$-0.228851\pi$
$104$		0			0
$105$		0			0
$106$	−7.44475			−0.723098
$107$		−	14.4002i		−	1.39212i	−0.717982	−	0.696062i	$-0.754934\pi$
$107$		−	14.4002i		−	1.39212i	0.717982	−	0.696062i	$-0.245066\pi$
$108$		0			0
$109$	−3.41140			−0.326753			−0.163376	−	0.986564i	$-0.552238\pi$
$109$	−3.41140			−0.326753			−0.163376	+	0.986564i	$0.552238\pi$
$110$	2.01596			0.192215
$111$		0			0
$112$	1.78450	−	1.95335i	0.168619	−	0.184574i
$113$		−	4.95947i		−	0.466548i	−0.972411	−	0.233274i	$-0.925056\pi$
$113$		−	4.95947i		−	0.466548i	0.972411	−	0.233274i	$-0.0749439\pi$
$114$		0			0
$115$		−	15.8772i		−	1.48056i
$116$			1.32636i			0.123149i
$117$		0			0
$118$		−	4.03193i		−	0.371169i
$119$	10.5690	−	11.5690i	0.968857	−	1.06053i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	0.250475			0.0226769
$123$		0			0
$124$		−	10.6942i		−	0.960369i
$125$	−11.9665			−1.07032
$126$		0			0
$127$	9.50488			0.843422			0.421711	−	0.906730i	$-0.361430\pi$
$127$	9.50488			0.843422			0.421711	+	0.906730i	$0.361430\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$	−19.0838			−1.66736			−0.833681	−	0.552247i	$-0.813771\pi$
$131$	−19.0838			−1.66736			−0.833681	+	0.552247i	$0.813771\pi$
$132$		0			0
$133$	1.28516	+	1.17407i	0.111438	+	0.101805i
$134$			7.81164i			0.674823i
$135$		0			0
$136$			5.92266i			0.507864i
$137$		−	8.05428i		−	0.688124i	−0.938947	−	0.344062i	$-0.888197\pi$
$137$		−	8.05428i		−	0.688124i	0.938947	−	0.344062i	$-0.111803\pi$
$138$		0			0
$139$			19.3725i			1.64315i	0.570099	+	0.821576i	$0.306905\pi$
$139$			19.3725i			1.64315i	−0.570099	+	0.821576i	$0.693095\pi$
$140$	−3.59749	+	3.93788i	−0.304043	+	0.332811i
$141$		0			0
$142$	0.609527			0.0511504
$143$		0			0
$144$		0			0
$145$		−	2.67389i		−	0.222055i
$146$	0.532690			0.0440857
$147$		0			0
$148$	−5.87575			−0.482984
$149$			3.26224i			0.267253i	0.991032	+	0.133627i	$0.0426623\pi$
$149$			3.26224i			0.267253i	−0.991032	+	0.133627i	$0.957338\pi$
$150$		0			0
$151$	−14.1785			−1.15383			−0.576916	−	0.816803i	$-0.695744\pi$
$151$	−14.1785			−1.15383			−0.576916	+	0.816803i	$0.695744\pi$
$152$	−0.657927			−0.0533650
$153$		0			0
$154$	1.78450	−	1.95335i	0.143799	−	0.157405i
$155$			21.5592i			1.73167i
$156$		0			0
$157$			11.7596i			0.938518i	0.883061	+	0.469259i	$0.155479\pi$
$157$			11.7596i			0.938518i	−0.883061	+	0.469259i	$0.844521\pi$
$158$			12.7711i			1.01602i
$159$		0			0
$160$		−	2.01596i		−	0.159376i
$161$	−15.3841	−	14.0543i	−1.21243	−	1.10763i
$162$		0			0
$163$	12.5285			0.981306			0.490653	−	0.871355i	$-0.336758\pi$
$163$	12.5285			0.981306			0.490653	+	0.871355i	$0.336758\pi$
$164$	−4.56462			−0.356437
$165$		0			0
$166$		−	13.3681i		−	1.03757i
$167$	−23.0722			−1.78538			−0.892691	−	0.450669i	$-0.851185\pi$
$167$	−23.0722			−1.78538			−0.892691	+	0.450669i	$0.851185\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		−	11.9399i		−	0.915746i
$171$		0			0
$172$	1.56900			0.119635
$173$	17.7244			1.34756			0.673782	−	0.738930i	$-0.264669\pi$
$173$	17.7244			1.34756			0.673782	+	0.738930i	$0.264669\pi$
$174$		0			0
$175$	−1.67009	+	1.82811i	−0.126247	+	0.138192i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$		−	5.22255i		−	0.391446i
$179$		−	3.81164i		−	0.284895i	−0.989802	−	0.142448i	$-0.954503\pi$
$179$		−	3.81164i		−	0.284895i	0.989802	−	0.142448i	$-0.0454973\pi$
$180$		0			0
$181$			19.1260i			1.42163i	0.703382	+	0.710813i	$0.251673\pi$
$181$			19.1260i			1.42163i	−0.703382	+	0.710813i	$0.748327\pi$
$182$		0			0
$183$		0			0
$184$	7.87575			0.580608
$185$	11.8453			0.870885
$186$		0			0
$187$			5.92266i			0.433108i
$188$	−0.532690			−0.0388504
$189$		0			0
$190$	1.32636			0.0962242
$191$			21.6665i			1.56773i	0.620931	+	0.783865i	$0.286755\pi$
$191$			21.6665i			1.56773i	−0.620931	+	0.783865i	$0.713245\pi$
$192$		0			0
$193$	−21.6233			−1.55648			−0.778239	−	0.627968i	$-0.783887\pi$
$193$	−21.6233			−1.55648			−0.778239	+	0.627968i	$0.783887\pi$
$194$	−2.71607			−0.195003
$195$		0			0
$196$	0.631122	+	6.97149i	0.0450802	+	0.497964i
$197$			21.6914i			1.54545i	0.634743	+	0.772723i	$0.281106\pi$
$197$			21.6914i			1.54545i	−0.634743	+	0.772723i	$0.718894\pi$
$198$		0			0
$199$		−	3.94621i		−	0.279740i	−0.990170	−	0.139870i	$-0.955332\pi$
$199$		−	3.94621i		−	0.279740i	0.990170	−	0.139870i	$-0.0446684\pi$
$200$		−	0.935886i		−	0.0661771i
$201$		0			0
$202$		−	3.41352i		−	0.240175i
$203$	−2.59084	−	2.36689i	−0.181841	−	0.166123i
$204$		0			0
$205$	9.20211			0.642703
$206$	−13.3681			−0.931400
$207$		0			0
$208$		0			0
$209$	−0.657927			−0.0455098
$210$		0			0
$211$	2.30277			0.158529			0.0792647	−	0.996854i	$-0.474743\pi$
$211$	2.30277			0.158529			0.0792647	+	0.996854i	$0.474743\pi$
$212$			7.44475i			0.511308i
$213$		0			0
$214$	−14.4002			−0.984380
$215$	−3.16305			−0.215718
$216$		0			0
$217$	20.8895	+	19.0838i	1.41807	+	1.29549i
$218$			3.41140i			0.231049i
$219$		0			0
$220$		−	2.01596i		−	0.135916i
$221$		0			0
$222$		0			0
$223$		−	18.2571i		−	1.22259i	−0.791404	−	0.611294i	$-0.790649\pi$
$223$		−	18.2571i		−	1.22259i	0.791404	−	0.611294i	$-0.209351\pi$
$224$	−1.95335	−	1.78450i	−0.130513	−	0.119232i
$225$		0			0
$226$	−4.95947			−0.329899
$227$	−20.0739			−1.33235			−0.666177	−	0.745794i	$-0.732070\pi$
$227$	−20.0739			−1.33235			−0.666177	+	0.745794i	$0.732070\pi$
$228$		0			0
$229$			14.6840i			0.970344i	0.874419	+	0.485172i	$0.161243\pi$
$229$			14.6840i			0.970344i	−0.874419	+	0.485172i	$0.838757\pi$
$230$	−15.8772			−1.04691
$231$		0			0
$232$	1.32636			0.0870797
$233$			2.48528i			0.162816i	0.996681	+	0.0814081i	$0.0259417\pi$
$233$			2.48528i			0.162816i	−0.996681	+	0.0814081i	$0.974058\pi$
$234$		0			0
$235$	1.07388			0.0700525
$236$	−4.03193			−0.262456
$237$		0			0
$238$	−11.5690	−	10.5690i	−0.749907	−	0.685085i
$239$			22.3610i			1.44642i	0.690631	+	0.723208i	$0.257333\pi$
$239$			22.3610i			1.44642i	−0.690631	+	0.723208i	$0.742667\pi$
$240$		0			0
$241$		−	3.45706i		−	0.222689i	−0.993782	−	0.111344i	$-0.964484\pi$
$241$		−	3.45706i		−	0.222689i	0.993782	−	0.111344i	$-0.0355156\pi$
$242$			1.00000i			0.0642824i
$243$		0			0
$244$		−	0.250475i		−	0.0160350i
$245$	−1.27232	−	14.0543i	−0.0812855	−	0.897895i
$246$		0			0
$247$		0			0
$248$	−10.6942			−0.679083
$249$		0			0
$250$			11.9665i			0.756830i
$251$	3.78145			0.238683			0.119342	−	0.992853i	$-0.461922\pi$
$251$	3.78145			0.238683			0.119342	+	0.992853i	$0.461922\pi$
$252$		0			0
$253$	7.87575			0.495144
$254$		−	9.50488i		−	0.596390i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−5.22255			−0.325774			−0.162887	−	0.986645i	$-0.552081\pi$
$257$	−5.22255			−0.325774			−0.162887	+	0.986645i	$0.552081\pi$
$258$		0			0
$259$	10.4853	−	11.4774i	0.651524	−	0.713170i
$260$		0			0
$261$		0			0
$262$			19.0838i			1.17900i
$263$			13.7306i			0.846664i	0.905975	+	0.423332i	$0.139140\pi$
$263$			13.7306i			0.846664i	−0.905975	+	0.423332i	$0.860860\pi$
$264$		0			0
$265$		−	15.0084i		−	0.921956i
$266$	1.17407	−	1.28516i	0.0719869	−	0.0787982i
$267$		0			0
$268$	7.81164			0.477172
$269$	29.6093			1.80531			0.902655	−	0.430366i	$-0.141615\pi$
$269$	29.6093			1.80531			0.902655	+	0.430366i	$0.141615\pi$
$270$		0			0
$271$			15.5015i			0.941651i	0.882226	+	0.470825i	$0.156044\pi$
$271$			15.5015i			0.941651i	−0.882226	+	0.470825i	$0.843956\pi$
$272$	5.92266			0.359114
$273$		0			0
$274$	−8.05428			−0.486577
$275$		−	0.935886i		−	0.0564360i
$276$		0			0
$277$	−19.0347			−1.14368			−0.571841	−	0.820364i	$-0.693771\pi$
$277$	−19.0347			−1.14368			−0.571841	+	0.820364i	$0.693771\pi$
$278$	19.3725			1.16188
$279$		0			0
$280$	3.93788	+	3.59749i	0.235333	+	0.214991i
$281$			10.9706i			0.654449i	0.944947	+	0.327224i	$0.106113\pi$
$281$			10.9706i			0.654449i	−0.944947	+	0.327224i	$0.893887\pi$
$282$		0			0
$283$			19.2091i			1.14186i	0.820999	+	0.570930i	$0.193417\pi$
$283$			19.2091i			1.14186i	−0.820999	+	0.570930i	$0.806583\pi$
$284$		−	0.609527i		−	0.0361688i
$285$		0			0
$286$		0			0
$287$	8.14556	−	8.91628i	0.480817	−	0.526311i
$288$		0			0
$289$	18.0779			1.06340
$290$	−2.67389			−0.157016
$291$		0			0
$292$		−	0.532690i		−	0.0311733i
$293$	9.91105			0.579010			0.289505	−	0.957177i	$-0.406509\pi$
$293$	9.91105			0.579010			0.289505	+	0.957177i	$0.406509\pi$
$294$		0			0
$295$	8.12823			0.473244
$296$			5.87575i			0.341521i
$297$		0			0
$298$	3.26224			0.188977
$299$		0			0
$300$		0			0
$301$	−2.79988	+	3.06480i	−0.161382	+	0.176652i
$302$			14.1785i			0.815883i
$303$		0			0
$304$			0.657927i			0.0377347i
$305$			0.504949i			0.0289133i
$306$		0			0
$307$			1.64427i			0.0938433i	0.998899	+	0.0469216i	$0.0149411\pi$
$307$			1.64427i			0.0938433i	−0.998899	+	0.0469216i	$0.985059\pi$
$308$	−1.95335	−	1.78450i	−0.111302	−	0.101681i
$309$		0			0
$310$	21.5592			1.22448
$311$	2.14120			0.121416			0.0607082	−	0.998156i	$-0.480664\pi$
$311$	2.14120			0.121416			0.0607082	+	0.998156i	$0.480664\pi$
$312$		0			0
$313$			12.7932i			0.723116i	0.932350	+	0.361558i	$0.117755\pi$
$313$			12.7932i			0.723116i	−0.932350	+	0.361558i	$0.882245\pi$
$314$	11.7596			0.663632
$315$		0			0
$316$	12.7711			0.718431
$317$			6.51604i			0.365977i	0.983115	+	0.182989i	$0.0585771\pi$
$317$			6.51604i			0.365977i	−0.983115	+	0.182989i	$0.941423\pi$
$318$		0			0
$319$	1.32636			0.0742618
$320$	−2.01596			−0.112696
$321$		0			0
$322$	−14.0543	+	15.3841i	−0.783214	+	0.857321i
$323$			3.89668i			0.216817i
$324$		0			0
$325$		0			0
$326$		−	12.5285i		−	0.693888i
$327$		0			0
$328$			4.56462i			0.252039i
$329$	0.950585	−	1.04053i	0.0524074	−	0.0573662i
$330$		0			0
$331$	−30.2159			−1.66081			−0.830407	−	0.557157i	$-0.811892\pi$
$331$	−30.2159			−1.66081			−0.830407	+	0.557157i	$0.811892\pi$
$332$	−13.3681			−0.733670
$333$		0			0
$334$			23.0722i			1.26246i
$335$	−15.7480			−0.860405
$336$		0			0
$337$	24.8895			1.35582			0.677909	−	0.735146i	$-0.262886\pi$
$337$	24.8895			1.35582			0.677909	+	0.735146i	$0.262886\pi$
$338$		−	13.0000i		−	0.707107i
$339$		0			0
$340$	−11.9399			−0.647530
$341$	−10.6942			−0.579124
$342$		0			0
$343$	−14.7440	−	11.2078i	−0.796100	−	0.605166i
$344$		−	1.56900i		−	0.0845948i
$345$		0			0
$346$		−	17.7244i		−	0.952872i
$347$		−	9.09481i		−	0.488235i	−0.969746	−	0.244117i	$-0.921502\pi$
$347$		−	9.09481i		−	0.488235i	0.969746	−	0.244117i	$-0.0784982\pi$
$348$		0			0
$349$			31.8335i			1.70401i	0.523534	+	0.852005i	$0.324613\pi$
$349$			31.8335i			1.70401i	−0.523534	+	0.852005i	$0.675387\pi$
$350$	1.82811	+	1.67009i	0.0977165	+	0.0892699i
$351$		0			0
$352$	1.00000			0.0533002
$353$	−10.9052			−0.580424			−0.290212	−	0.956962i	$-0.593726\pi$
$353$	−10.9052			−0.580424			−0.290212	+	0.956962i	$0.593726\pi$
$354$		0			0
$355$			1.22879i			0.0652171i
$356$	−5.22255			−0.276794
$357$		0			0
$358$	−3.81164			−0.201451
$359$			23.6874i			1.25017i	0.780556	+	0.625086i	$0.214936\pi$
$359$			23.6874i			1.25017i	−0.780556	+	0.625086i	$0.785064\pi$
$360$		0			0
$361$	18.5671			0.977217
$362$	19.1260			1.00524
$363$		0			0
$364$		0			0
$365$			1.07388i			0.0562097i
$366$		0			0
$367$		−	13.1176i		−	0.684735i	−0.939566	−	0.342367i	$-0.888771\pi$
$367$		−	13.1176i		−	0.684735i	0.939566	−	0.342367i	$-0.111229\pi$
$368$		−	7.87575i		−	0.410552i
$369$		0			0
$370$		−	11.8453i		−	0.615809i
$371$	−14.5422	−	13.2852i	−0.754992	−	0.689731i
$372$		0			0
$373$	3.38649			0.175346			0.0876729	−	0.996149i	$-0.472057\pi$
$373$	3.38649			0.175346			0.0876729	+	0.996149i	$0.472057\pi$
$374$	5.92266			0.306253
$375$		0			0
$376$			0.532690i			0.0274714i
$377$		0			0
$378$		0			0
$379$	8.52847			0.438078			0.219039	−	0.975716i	$-0.429708\pi$
$379$	8.52847			0.438078			0.219039	+	0.975716i	$0.429708\pi$
$380$		−	1.32636i		−	0.0680408i
$381$		0			0
$382$	21.6665			1.10855
$383$	−10.1180			−0.517005			−0.258503	−	0.966011i	$-0.583229\pi$
$383$	−10.1180			−0.517005			−0.258503	+	0.966011i	$0.583229\pi$
$384$		0			0
$385$	3.93788	+	3.59749i	0.200693	+	0.183345i
$386$			21.6233i			1.10060i
$387$		0			0
$388$			2.71607i			0.137888i
$389$		−	16.7920i		−	0.851390i	−0.904867	−	0.425695i	$-0.860030\pi$
$389$		−	16.7920i		−	0.851390i	0.904867	−	0.425695i	$-0.139970\pi$
$390$		0			0
$391$		−	46.6454i		−	2.35896i
$392$	6.97149	−	0.631122i	0.352113	−	0.0318765i
$393$		0			0
$394$	21.6914			1.09280
$395$	−25.7461			−1.29543
$396$		0			0
$397$			13.4856i			0.676821i	0.940999	+	0.338411i	$0.109889\pi$
$397$			13.4856i			0.676821i	−0.940999	+	0.338411i	$0.890111\pi$
$398$	−3.94621			−0.197806
$399$		0			0
$400$	−0.935886			−0.0467943
$401$			13.5298i			0.675646i	0.941210	+	0.337823i	$0.109690\pi$
$401$			13.5298i			0.675646i	−0.941210	+	0.337823i	$0.890310\pi$
$402$		0			0
$403$		0			0
$404$	−3.41352			−0.169829
$405$		0			0
$406$	−2.36689	+	2.59084i	−0.117467	+	0.128581i
$407$			5.87575i			0.291250i
$408$		0			0
$409$		−	27.0606i		−	1.33806i	−0.743235	−	0.669031i	$-0.766709\pi$
$409$		−	27.0606i		−	1.33806i	0.743235	−	0.669031i	$-0.233291\pi$
$410$		−	9.20211i		−	0.454460i
$411$		0			0
$412$			13.3681i			0.658599i
$413$	7.19498	−	7.87575i	0.354042	−	0.387541i
$414$		0			0
$415$	26.9496			1.32291
$416$		0			0
$417$		0			0
$418$			0.657927i			0.0321803i
$419$	−10.7799			−0.526634			−0.263317	−	0.964709i	$-0.584817\pi$
$419$	−10.7799			−0.526634			−0.263317	+	0.964709i	$0.584817\pi$
$420$		0			0
$421$	23.0948			1.12557			0.562786	−	0.826603i	$-0.309729\pi$
$421$	23.0948			1.12557			0.562786	+	0.826603i	$0.309729\pi$
$422$		−	2.30277i		−	0.112097i
$423$		0			0
$424$	7.44475			0.361549
$425$	−5.54293			−0.268872
$426$		0			0
$427$	0.489264	+	0.446973i	0.0236772	+	0.0216305i
$428$			14.4002i			0.696062i
$429$		0			0
$430$			3.16305i			0.152536i
$431$			13.8508i			0.667172i	0.942720	+	0.333586i	$0.108259\pi$
$431$			13.8508i			0.667172i	−0.942720	+	0.333586i	$0.891741\pi$
$432$		0			0
$433$			36.3968i			1.74912i	0.484919	+	0.874559i	$0.338849\pi$
$433$			36.3968i			1.74912i	−0.484919	+	0.874559i	$0.661151\pi$
$434$	19.0838	−	20.8895i	0.916053	−	1.00273i
$435$		0			0
$436$	3.41140			0.163376
$437$	5.18167			0.247873
$438$		0			0
$439$		−	24.9603i		−	1.19129i	−0.803248	−	0.595645i	$-0.796897\pi$
$439$		−	24.9603i		−	1.19129i	0.803248	−	0.595645i	$-0.203103\pi$
$440$	−2.01596			−0.0961074
$441$		0			0
$442$		0			0
$443$		−	7.20610i		−	0.342372i	−0.985239	−	0.171186i	$-0.945240\pi$
$443$		−	7.20610i		−	0.342372i	0.985239	−	0.171186i	$-0.0547599\pi$
$444$		0			0
$445$	10.5285			0.499097
$446$	−18.2571			−0.864500
$447$		0			0
$448$	−1.78450	+	1.95335i	−0.0843097	+	0.0922869i
$449$		−	31.8490i		−	1.50305i	−0.659707	−	0.751523i	$-0.729320\pi$
$449$		−	31.8490i		−	1.50305i	0.659707	−	0.751523i	$-0.270680\pi$
$450$		0			0
$451$			4.56462i			0.214940i
$452$			4.95947i			0.233274i
$453$		0			0
$454$			20.0739i			0.942116i
$455$		0			0
$456$		0			0
$457$	−10.4853			−0.490481			−0.245240	−	0.969462i	$-0.578867\pi$
$457$	−10.4853			−0.490481			−0.245240	+	0.969462i	$0.578867\pi$
$458$	14.6840			0.686137
$459$		0			0
$460$			15.8772i			0.740280i
$461$	−24.3881			−1.13587			−0.567933	−	0.823075i	$-0.692257\pi$
$461$	−24.3881			−1.13587			−0.567933	+	0.823075i	$0.692257\pi$
$462$		0			0
$463$	−22.6410			−1.05222			−0.526109	−	0.850417i	$-0.676349\pi$
$463$	−22.6410			−1.05222			−0.526109	+	0.850417i	$0.676349\pi$
$464$		−	1.32636i		−	0.0615747i
$465$		0			0
$466$	2.48528			0.115128
$467$	27.6355			1.27882			0.639409	−	0.768867i	$-0.279179\pi$
$467$	27.6355			1.27882			0.639409	+	0.768867i	$0.279179\pi$
$468$		0			0
$469$	−13.9399	+	15.2588i	−0.643683	+	0.704588i
$470$		−	1.07388i		−	0.0495346i
$471$		0			0
$472$			4.03193i			0.185585i
$473$		−	1.56900i		−	0.0721427i
$474$		0			0
$475$		−	0.615745i		−	0.0282523i
$476$	−10.5690	+	11.5690i	−0.484428	+	0.530264i
$477$		0			0
$478$	22.3610			1.02277
$479$	−34.4166			−1.57253			−0.786267	−	0.617887i	$-0.787989\pi$
$479$	−34.4166			−1.57253			−0.786267	+	0.617887i	$0.787989\pi$
$480$		0			0
$481$		0			0
$482$	−3.45706			−0.157465
$483$		0			0
$484$	1.00000			0.0454545
$485$		−	5.47551i		−	0.248630i
$486$		0			0
$487$	−15.2662			−0.691779			−0.345889	−	0.938275i	$-0.612423\pi$
$487$	−15.2662			−0.691779			−0.345889	+	0.938275i	$0.612423\pi$
$488$	−0.250475			−0.0113385
$489$		0			0
$490$	−14.0543	+	1.27232i	−0.634908	+	0.0574776i
$491$			23.6273i			1.06628i	0.846026	+	0.533142i	$0.178989\pi$
$491$			23.6273i			1.06628i	−0.846026	+	0.533142i	$0.821011\pi$
$492$		0			0
$493$		−	7.85557i		−	0.353797i
$494$		0			0
$495$		0			0
$496$			10.6942i			0.480185i
$497$	1.19062	+	1.08770i	0.0534065	+	0.0487901i
$498$		0			0
$499$	−36.0484			−1.61375			−0.806875	−	0.590723i	$-0.798843\pi$
$499$	−36.0484			−1.61375			−0.806875	+	0.590723i	$0.798843\pi$
$500$	11.9665			0.535160
$501$		0			0
$502$		−	3.78145i		−	0.168775i
$503$	31.1361			1.38829			0.694145	−	0.719836i	$-0.255783\pi$
$503$	31.1361			1.38829			0.694145	+	0.719836i	$0.255783\pi$
$504$		0			0
$505$	6.88154			0.306225
$506$		−	7.87575i		−	0.350120i
$507$		0			0
$508$	−9.50488			−0.421711
$509$	21.5454			0.954984			0.477492	−	0.878636i	$-0.341546\pi$
$509$	21.5454			0.954984			0.477492	+	0.878636i	$0.341546\pi$
$510$		0			0
$511$	1.04053	+	0.950585i	0.0460302	+	0.0420514i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			5.22255i			0.230357i
$515$		−	26.9496i		−	1.18754i
$516$		0			0
$517$			0.532690i			0.0234277i
$518$	−11.4774	−	10.4853i	−0.504287	−	0.460697i
$519$		0			0
$520$		0			0
$521$	−24.0663			−1.05437			−0.527183	−	0.849752i	$-0.676752\pi$
$521$	−24.0663			−1.05437			−0.527183	+	0.849752i	$0.676752\pi$
$522$		0			0
$523$		−	7.11327i		−	0.311042i	−0.987833	−	0.155521i	$-0.950294\pi$
$523$		−	7.11327i		−	0.311042i	0.987833	−	0.155521i	$-0.0497056\pi$
$524$	19.0838			0.833681
$525$		0			0
$526$	13.7306			0.598682
$527$			63.3382i			2.75905i
$528$		0			0
$529$	−39.0275			−1.69685
$530$	−15.0084			−0.651922
$531$		0			0
$532$	−1.28516	−	1.17407i	−0.0557188	−	0.0509024i
$533$		0			0
$534$		0			0
$535$		−	29.0304i		−	1.25509i
$536$		−	7.81164i		−	0.337411i
$537$		0			0
$538$		−	29.6093i		−	1.27655i
$539$	6.97149	−	0.631122i	0.300283	−	0.0271844i
$540$		0			0
$541$	4.24849			0.182657			0.0913285	−	0.995821i	$-0.470889\pi$
$541$	4.24849			0.182657			0.0913285	+	0.995821i	$0.470889\pi$
$542$	15.5015			0.665848
$543$		0			0
$544$		−	5.92266i		−	0.253932i
$545$	−6.87726			−0.294589
$546$		0			0
$547$	9.69723			0.414624			0.207312	−	0.978275i	$-0.433529\pi$
$547$	9.69723			0.414624			0.207312	+	0.978275i	$0.433529\pi$
$548$			8.05428i			0.344062i
$549$		0			0
$550$	−0.935886			−0.0399063
$551$	0.872648			0.0371760
$552$		0			0
$553$	−22.7900	+	24.9464i	−0.969131	+	1.06083i
$554$			19.0347i			0.808706i
$555$		0			0
$556$		−	19.3725i		−	0.821576i
$557$			21.6914i			0.919093i	0.888154	+	0.459547i	$0.151988\pi$
$557$			21.6914i			0.919093i	−0.888154	+	0.459547i	$0.848012\pi$
$558$		0			0
$559$		0			0
$560$	3.59749	−	3.93788i	0.152022	−	0.166406i
$561$		0			0
$562$	10.9706			0.462765
$563$	−1.68243			−0.0709062			−0.0354531	−	0.999371i	$-0.511287\pi$
$563$	−1.68243			−0.0709062			−0.0354531	+	0.999371i	$0.511287\pi$
$564$		0			0
$565$		−	9.99812i		−	0.420624i
$566$	19.2091			0.807417
$567$		0			0
$568$	−0.609527			−0.0255752
$569$			14.4853i			0.607255i	0.952791	+	0.303627i	$0.0981978\pi$
$569$			14.4853i			0.607255i	−0.952791	+	0.303627i	$0.901802\pi$
$570$		0			0
$571$	−26.7542			−1.11963			−0.559814	−	0.828619i	$-0.689127\pi$
$571$	−26.7542			−1.11963			−0.559814	+	0.828619i	$0.689127\pi$
$572$		0			0
$573$		0			0
$574$	−8.91628	−	8.14556i	−0.372158	−	0.339989i
$575$			7.37081i			0.307384i
$576$		0			0
$577$			42.3630i			1.76359i	0.471629	+	0.881797i	$0.343666\pi$
$577$			42.3630i			1.76359i	−0.471629	+	0.881797i	$0.656334\pi$
$578$		−	18.0779i		−	0.751940i
$579$		0			0
$580$			2.67389i			0.111027i
$581$	23.8554	−	26.1125i	0.989688	−	1.08333i
$582$		0			0
$583$	7.44475			0.308330
$584$	−0.532690			−0.0220429
$585$		0			0
$586$		−	9.91105i		−	0.409422i
$587$	−2.05172			−0.0846835			−0.0423418	−	0.999103i	$-0.513482\pi$
$587$	−2.05172			−0.0846835			−0.0423418	+	0.999103i	$0.513482\pi$
$588$		0			0
$589$	−7.03602			−0.289914
$590$		−	8.12823i		−	0.334634i
$591$		0			0
$592$	5.87575			0.241492
$593$	−40.2640			−1.65344			−0.826721	−	0.562611i	$-0.809797\pi$
$593$	−40.2640			−1.65344			−0.826721	+	0.562611i	$0.809797\pi$
$594$		0			0
$595$	21.3067	−	23.3227i	0.873489	−	0.956137i
$596$		−	3.26224i		−	0.133627i
$597$		0			0
$598$		0			0
$599$		−	35.6273i		−	1.45569i	−0.685741	−	0.727845i	$-0.740522\pi$
$599$		−	35.6273i		−	1.45569i	0.685741	−	0.727845i	$-0.259478\pi$
$600$		0			0
$601$		−	39.6574i		−	1.61766i	−0.588044	−	0.808829i	$-0.700102\pi$
$601$		−	39.6574i		−	1.61766i	0.588044	−	0.808829i	$-0.299898\pi$
$602$	3.06480	+	2.79988i	0.124912	+	0.114115i
$603$		0			0
$604$	14.1785			0.576916
$605$	−2.01596			−0.0819606
$606$		0			0
$607$		−	23.6444i		−	0.959698i	−0.877351	−	0.479849i	$-0.840691\pi$
$607$		−	23.6444i		−	0.959698i	0.877351	−	0.479849i	$-0.159309\pi$
$608$	0.657927			0.0266825
$609$		0			0
$610$	0.504949			0.0204448
$611$		0			0
$612$		0			0
$613$	−36.0654			−1.45667			−0.728333	−	0.685223i	$-0.759705\pi$
$613$	−36.0654			−1.45667			−0.728333	+	0.685223i	$0.759705\pi$
$614$	1.64427			0.0663572
$615$		0			0
$616$	−1.78450	+	1.95335i	−0.0718995	+	0.0787026i
$617$			2.45192i			0.0987108i	0.998781	+	0.0493554i	$0.0157167\pi$
$617$			2.45192i			0.0987108i	−0.998781	+	0.0493554i	$0.984283\pi$
$618$		0			0
$619$		−	12.9107i		−	0.518925i	−0.965753	−	0.259462i	$-0.916455\pi$
$619$		−	12.9107i		−	0.518925i	0.965753	−	0.259462i	$-0.0835453\pi$
$620$		−	21.5592i		−	0.865837i
$621$		0			0
$622$		−	2.14120i		−	0.0858544i
$623$	9.31963	−	10.2014i	0.373383	−	0.408712i
$624$		0			0
$625$	−19.4447			−0.777788
$626$	12.7932			0.511321
$627$		0			0
$628$		−	11.7596i		−	0.469259i
$629$	34.8001			1.38757
$630$		0			0
$631$	−6.53245			−0.260053			−0.130026	−	0.991511i	$-0.541506\pi$
$631$	−6.53245			−0.260053			−0.130026	+	0.991511i	$0.541506\pi$
$632$		−	12.7711i		−	0.508008i
$633$		0			0
$634$	6.51604			0.258785
$635$	19.1615			0.760401
$636$		0			0
$637$		0			0
$638$		−	1.32636i		−	0.0525110i
$639$		0			0
$640$			2.01596i			0.0796880i
$641$			6.03336i			0.238303i	0.992876	+	0.119152i	$0.0380175\pi$
$641$			6.03336i			0.238303i	−0.992876	+	0.119152i	$0.961983\pi$
$642$		0			0
$643$		−	9.21360i		−	0.363349i	−0.983359	−	0.181675i	$-0.941848\pi$
$643$		−	9.21360i		−	0.363349i	0.983359	−	0.181675i	$-0.0581517\pi$
$644$	15.3841	+	14.0543i	0.606217	+	0.553816i
$645$		0			0
$646$	3.89668			0.153313
$647$	1.72732			0.0679080			0.0339540	−	0.999423i	$-0.489190\pi$
$647$	1.72732			0.0679080			0.0339540	+	0.999423i	$0.489190\pi$
$648$		0			0
$649$			4.03193i			0.158267i
$650$		0			0
$651$		0			0
$652$	−12.5285			−0.490653
$653$		−	41.9575i		−	1.64193i	−0.570982	−	0.820963i	$-0.693437\pi$
$653$		−	41.9575i		−	1.64193i	0.570982	−	0.820963i	$-0.306563\pi$
$654$		0			0
$655$	−38.4723			−1.50324
$656$	4.56462			0.178218
$657$		0			0
$658$	−1.04053	−	0.950585i	−0.0405640	−	0.0370577i
$659$			10.3963i			0.404981i	0.979284	+	0.202490i	$0.0649035\pi$
$659$			10.3963i			0.404981i	−0.979284	+	0.202490i	$0.935097\pi$
$660$		0			0
$661$		−	34.0922i		−	1.32603i	−0.748605	−	0.663016i	$-0.769276\pi$
$661$		−	34.0922i		−	1.32603i	0.748605	−	0.663016i	$-0.230724\pi$
$662$			30.2159i			1.17437i
$663$		0			0
$664$			13.3681i			0.518783i
$665$	2.59084	+	2.36689i	0.100468	+	0.0917839i
$666$		0			0
$667$	−10.4461			−0.404474
$668$	23.0722			0.892691
$669$		0			0
$670$			15.7480i			0.608398i
$671$	−0.250475			−0.00966948
$672$		0			0
$673$	9.66513			0.372563			0.186282	−	0.982496i	$-0.440356\pi$
$673$	9.66513			0.372563			0.186282	+	0.982496i	$0.440356\pi$
$674$		−	24.8895i		−	0.958708i
$675$		0			0
$676$	−13.0000			−0.500000
$677$	−45.8609			−1.76258			−0.881288	−	0.472579i	$-0.843323\pi$
$677$	−45.8609			−1.76258			−0.881288	+	0.472579i	$0.843323\pi$
$678$		0			0
$679$	−5.30543	−	4.84683i	−0.203604	−	0.186004i
$680$			11.9399i			0.457873i
$681$		0			0
$682$			10.6942i			0.409503i
$683$			3.93190i			0.150450i	0.997167	+	0.0752250i	$0.0239675\pi$
$683$			3.93190i			0.150450i	−0.997167	+	0.0752250i	$0.976032\pi$
$684$		0			0
$685$		−	16.2371i		−	0.620389i
$686$	−11.2078	+	14.7440i	−0.427917	+	0.562927i
$687$		0			0
$688$	−1.56900			−0.0598175
$689$		0			0
$690$		0			0
$691$			17.6940i			0.673113i	0.941663	+	0.336557i	$0.109262\pi$
$691$			17.6940i			0.673113i	−0.941663	+	0.336557i	$0.890738\pi$
$692$	−17.7244			−0.673782
$693$		0			0
$694$	−9.09481			−0.345234
$695$			39.0542i			1.48141i
$696$		0			0
$697$	27.0347			1.02401
$698$	31.8335			1.20492
$699$		0			0
$700$	1.67009	−	1.82811i	0.0631234	−	0.0690960i
$701$		−	27.9189i		−	1.05448i	−0.849715	−	0.527242i	$-0.823226\pi$
$701$		−	27.9189i		−	1.05448i	0.849715	−	0.527242i	$-0.176774\pi$
$702$		0			0
$703$			3.86582i			0.145802i
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$			10.9052i			0.410422i
$707$	6.09143	−	6.66779i	0.229092	−	0.250768i
$708$		0			0
$709$	−25.4951			−0.957487			−0.478743	−	0.877955i	$-0.658908\pi$
$709$	−25.4951			−0.957487			−0.478743	+	0.877955i	$0.658908\pi$
$710$	1.22879			0.0461155
$711$		0			0
$712$			5.22255i			0.195723i
$713$	84.2250			3.15425
$714$		0			0
$715$		0			0
$716$			3.81164i			0.142448i
$717$		0			0
$718$	23.6874			0.884006
$719$	43.0169			1.60426			0.802130	−	0.597150i	$-0.203700\pi$
$719$	43.0169			1.60426			0.802130	+	0.597150i	$0.203700\pi$
$720$		0			0
$721$	−26.1125	−	23.8554i	−0.972482	−	0.888421i
$722$		−	18.5671i		−	0.690997i
$723$		0			0
$724$		−	19.1260i		−	0.710813i
$725$			1.24132i			0.0461015i
$726$		0			0
$727$		−	30.2686i		−	1.12260i	−0.827613	−	0.561299i	$-0.810302\pi$
$727$		−	30.2686i		−	1.12260i	0.827613	−	0.561299i	$-0.189698\pi$
$728$		0			0
$729$		0			0
$730$	1.07388			0.0397462
$731$	−9.29264			−0.343701
$732$		0			0
$733$		−	7.56291i		−	0.279342i	−0.990198	−	0.139671i	$-0.955395\pi$
$733$		−	7.56291i		−	0.279342i	0.990198	−	0.139671i	$-0.0446046\pi$
$734$	−13.1176			−0.484181
$735$		0			0
$736$	−7.87575			−0.290304
$737$		−	7.81164i		−	0.287745i
$738$		0			0
$739$	−17.1923			−0.632428			−0.316214	−	0.948688i	$-0.602412\pi$
$739$	−17.1923			−0.632428			−0.316214	+	0.948688i	$0.602412\pi$
$740$	−11.8453			−0.435442
$741$		0			0
$742$	−13.2852	+	14.5422i	−0.487714	+	0.533860i
$743$			23.9843i			0.879899i	0.898022	+	0.439950i	$0.145004\pi$
$743$			23.9843i			0.879899i	−0.898022	+	0.439950i	$0.854996\pi$
$744$		0			0
$745$			6.57657i			0.240947i
$746$		−	3.38649i		−	0.123988i
$747$		0			0
$748$		−	5.92266i		−	0.216554i
$749$	−28.1287	−	25.6972i	−1.02780	−	0.938956i
$750$		0			0
$751$	−32.4736			−1.18498			−0.592489	−	0.805579i	$-0.701855\pi$
$751$	−32.4736			−1.18498			−0.592489	+	0.805579i	$0.701855\pi$
$752$	0.532690			0.0194252
$753$		0			0
$754$		0			0
$755$	−28.5834			−1.04026
$756$		0			0
$757$	40.0315			1.45497			0.727485	−	0.686124i	$-0.240689\pi$
$757$	40.0315			1.45497			0.727485	+	0.686124i	$0.240689\pi$
$758$		−	8.52847i		−	0.309768i
$759$		0			0
$760$	−1.32636			−0.0481121
$761$	20.6923			0.750097			0.375048	−	0.927005i	$-0.377626\pi$
$761$	20.6923			0.750097			0.375048	+	0.927005i	$0.377626\pi$
$762$		0			0
$763$	−6.08764	+	6.66364i	−0.220387	+	0.241240i
$764$		−	21.6665i		−	0.783865i
$765$		0			0
$766$			10.1180i			0.365578i
$767$		0			0
$768$		0			0
$769$		−	33.6030i		−	1.21176i	−0.795557	−	0.605878i	$-0.792822\pi$
$769$		−	33.6030i		−	1.21176i	0.795557	−	0.605878i	$-0.207178\pi$
$770$	3.59749	−	3.93788i	0.129644	−	0.141911i
$771$		0			0
$772$	21.6233			0.778239
$773$	13.9825			0.502916			0.251458	−	0.967868i	$-0.419090\pi$
$773$	13.9825			0.502916			0.251458	+	0.967868i	$0.419090\pi$
$774$		0			0
$775$		−	10.0086i		−	0.359518i
$776$	2.71607			0.0975014
$777$		0			0
$778$	−16.7920			−0.602024
$779$			3.00319i			0.107600i
$780$		0			0
$781$	−0.609527			−0.0218106
$782$	−46.6454			−1.66804
$783$		0			0
$784$	−0.631122	−	6.97149i	−0.0225401	−	0.248982i
$785$			23.7069i			0.846137i
$786$		0			0
$787$			20.4800i			0.730034i	0.931001	+	0.365017i	$0.118937\pi$
$787$			20.4800i			0.730034i	−0.931001	+	0.365017i	$0.881063\pi$
$788$		−	21.6914i		−	0.772723i
$789$		0			0
$790$			25.7461i			0.916006i
$791$	−9.68757	−	8.85018i	−0.344450	−	0.314676i
$792$		0			0
$793$		0			0
$794$	13.4856			0.478585
$795$		0			0
$796$			3.94621i			0.139870i
$797$	−38.1741			−1.35220			−0.676098	−	0.736812i	$-0.736330\pi$
$797$	−38.1741			−1.35220			−0.676098	+	0.736812i	$0.736330\pi$
$798$		0			0
$799$	3.15494			0.111614
$800$			0.935886i			0.0330886i
$801$		0			0
$802$	13.5298			0.477754
$803$	−0.532690			−0.0187982
$804$		0			0
$805$	−31.0138	−	28.3329i	−1.09309	−	0.998605i
$806$		0			0
$807$		0			0
$808$			3.41352i			0.120087i
$809$		−	22.8503i		−	0.803374i	−0.915777	−	0.401687i	$-0.868424\pi$
$809$		−	22.8503i		−	0.803374i	0.915777	−	0.401687i	$-0.131576\pi$
$810$		0			0
$811$			27.1859i			0.954624i	0.878734	+	0.477312i	$0.158389\pi$
$811$			27.1859i			0.954624i	−0.878734	+	0.477312i	$0.841611\pi$
$812$	2.59084	+	2.36689i	0.0909206	+	0.0830614i
$813$		0			0
$814$	5.87575			0.205945
$815$	25.2570			0.884713
$816$		0			0
$817$		−	1.03229i		−	0.0361152i
$818$	−27.0606			−0.946152
$819$		0			0
$820$	−9.20211			−0.321352
$821$			22.6135i			0.789217i	0.918849	+	0.394608i	$0.129120\pi$
$821$			22.6135i			0.789217i	−0.918849	+	0.394608i	$0.870880\pi$
$822$		0			0
$823$	36.9706			1.28871			0.644356	−	0.764725i	$-0.277125\pi$
$823$	36.9706			1.28871			0.644356	+	0.764725i	$0.277125\pi$
$824$	13.3681			0.465700
$825$		0			0
$826$	−7.87575	−	7.19498i	−0.274033	−	0.250345i
$827$		−	36.8463i		−	1.28127i	−0.767845	−	0.640636i	$-0.778671\pi$
$827$		−	36.8463i		−	1.28127i	0.767845	−	0.640636i	$-0.221329\pi$
$828$		0			0
$829$		−	34.8858i		−	1.21163i	−0.795604	−	0.605817i	$-0.792846\pi$
$829$		−	34.8858i		−	1.21163i	0.795604	−	0.605817i	$-0.207154\pi$
$830$		−	26.9496i		−	0.935436i
$831$		0			0
$832$		0			0
$833$	−3.73792	−	41.2897i	−0.129511	−	1.43061i
$834$		0			0
$835$	−46.5128			−1.60964
$836$	0.657927			0.0227549
$837$		0			0
$838$			10.7799i			0.372387i
$839$	−41.3715			−1.42830			−0.714152	−	0.699991i	$-0.753187\pi$
$839$	−41.3715			−1.42830			−0.714152	+	0.699991i	$0.753187\pi$
$840$		0			0
$841$	27.2408			0.939337
$842$		−	23.0948i		−	0.795900i
$843$		0			0
$844$	−2.30277			−0.0792647
$845$	26.2075			0.901567
$846$		0			0
$847$	−1.78450	+	1.95335i	−0.0613161	+	0.0671178i
$848$		−	7.44475i		−	0.255654i
$849$		0			0
$850$			5.54293i			0.190121i
$851$		−	46.2760i		−	1.58632i
$852$		0			0
$853$			6.66093i			0.228066i	0.993477	+	0.114033i	$0.0363770\pi$
$853$			6.66093i			0.228066i	−0.993477	+	0.114033i	$0.963623\pi$
$854$	0.446973	−	0.489264i	0.0152951	−	0.0167423i
$855$		0			0
$856$	14.4002			0.492190
$857$	19.6691			0.671885			0.335943	−	0.941882i	$-0.390945\pi$
$857$	19.6691			0.671885			0.335943	+	0.941882i	$0.390945\pi$
$858$		0			0
$859$		−	33.8062i		−	1.15345i	−0.816937	−	0.576726i	$-0.804330\pi$
$859$		−	33.8062i		−	1.15345i	0.816937	−	0.576726i	$-0.195670\pi$
$860$	3.16305			0.107859
$861$		0			0
$862$	13.8508			0.471762
$863$		−	0.846317i		−	0.0288090i	−0.999896	−	0.0144045i	$-0.995415\pi$
$863$		−	0.846317i		−	0.0288090i	0.999896	−	0.0144045i	$-0.00458525\pi$
$864$		0			0
$865$	35.7318			1.21492
$866$	36.3968			1.23681
$867$		0			0
$868$	−20.8895	−	19.0838i	−0.709036	−	0.647747i
$869$		−	12.7711i		−	0.433230i
$870$		0			0
$871$		0			0
$872$		−	3.41140i		−	0.115524i
$873$		0			0
$874$		−	5.18167i		−	0.175273i
$875$	−21.3543	+	23.3748i	−0.721906	+	0.790212i
$876$		0			0
$877$	49.8039			1.68176			0.840879	−	0.541222i	$-0.182038\pi$
$877$	49.8039			1.68176			0.840879	+	0.541222i	$0.182038\pi$
$878$	−24.9603			−0.842369
$879$		0			0
$880$			2.01596i			0.0679582i
$881$	39.1207			1.31801			0.659004	−	0.752139i	$-0.270978\pi$
$881$	39.1207			1.31801			0.659004	+	0.752139i	$0.270978\pi$
$882$		0			0
$883$	24.4501			0.822810			0.411405	−	0.911453i	$-0.365038\pi$
$883$	24.4501			0.822810			0.411405	+	0.911453i	$0.365038\pi$
$884$		0			0
$885$		0			0
$886$	−7.20610			−0.242094
$887$	−28.3025			−0.950307			−0.475153	−	0.879903i	$-0.657607\pi$
$887$	−28.3025			−0.950307			−0.475153	+	0.879903i	$0.657607\pi$
$888$		0			0
$889$	16.9615	−	18.5663i	0.568869	−	0.622695i
$890$		−	10.5285i		−	0.352915i
$891$		0			0
$892$			18.2571i			0.611294i
$893$			0.350471i			0.0117281i
$894$		0			0
$895$		−	7.68413i		−	0.256852i
$896$	1.95335	+	1.78450i	0.0652567	+	0.0596159i
$897$		0			0
$898$	−31.8490			−1.06281
$899$	14.1844			0.473075
$900$		0			0
$901$		−	44.0927i		−	1.46894i
$902$	4.56462			0.151985
$903$		0			0
$904$	4.95947			0.164950
$905$			38.5574i			1.28169i
$906$		0			0
$907$	42.3244			1.40536			0.702680	−	0.711506i	$-0.251987\pi$
$907$	42.3244			1.40536			0.702680	+	0.711506i	$0.251987\pi$
$908$	20.0739			0.666177
$909$		0			0
$910$		0			0
$911$			15.2622i			0.505661i	0.967511	+	0.252830i	$0.0813615\pi$
$911$			15.2622i			0.505661i	−0.967511	+	0.252830i	$0.918639\pi$
$912$		0			0
$913$			13.3681i			0.442420i
$914$			10.4853i			0.346822i
$915$		0			0
$916$		−	14.6840i		−	0.485172i
$917$	−34.0551	+	37.2773i	−1.12460	+	1.23101i
$918$		0			0
$919$	−50.2951			−1.65908			−0.829540	−	0.558447i	$-0.811398\pi$
$919$	−50.2951			−1.65908			−0.829540	+	0.558447i	$0.811398\pi$
$920$	15.8772			0.523457
$921$		0			0
$922$			24.3881i			0.803179i
$923$		0			0
$924$		0			0
$925$	−5.49903			−0.180807
$926$			22.6410i			0.744030i
$927$		0			0
$928$	−1.32636			−0.0435399
$929$	−8.43957			−0.276893			−0.138447	−	0.990370i	$-0.544211\pi$
$929$	−8.43957			−0.276893			−0.138447	+	0.990370i	$0.544211\pi$
$930$		0			0
$931$	4.58674	−	0.415233i	0.150324	−	0.0136087i
$932$		−	2.48528i		−	0.0814081i
$933$		0			0
$934$		−	27.6355i		−	0.904261i
$935$			11.9399i			0.390475i
$936$		0			0
$937$		−	13.7782i		−	0.450115i	−0.974345	−	0.225057i	$-0.927743\pi$
$937$		−	13.7782i		−	0.450115i	0.974345	−	0.225057i	$-0.0722570\pi$
$938$	15.2588	+	13.9399i	0.498219	+	0.455153i
$939$		0			0
$940$	−1.07388			−0.0350262
$941$	28.6705			0.934631			0.467315	−	0.884091i	$-0.345221\pi$
$941$	28.6705			0.934631			0.467315	+	0.884091i	$0.345221\pi$
$942$		0			0
$943$		−	35.9498i		−	1.17069i
$944$	4.03193			0.131228
$945$		0			0
$946$	−1.56900			−0.0510126
$947$			19.0609i			0.619397i	0.950835	+	0.309698i	$0.100228\pi$
$947$			19.0609i			0.619397i	−0.950835	+	0.309698i	$0.899772\pi$
$948$		0			0
$949$		0			0
$950$	−0.615745			−0.0199774
$951$		0			0
$952$	11.5690	+	10.5690i	0.374953	+	0.342543i
$953$		−	7.72400i		−	0.250205i	−0.992144	−	0.125102i	$-0.960074\pi$
$953$		−	7.72400i		−	0.250205i	0.992144	−	0.125102i	$-0.0399260\pi$
$954$		0			0
$955$			43.6788i			1.41341i
$956$		−	22.3610i		−	0.723208i
$957$		0			0
$958$			34.4166i			1.11195i
$959$	−15.7328	−	14.3729i	−0.508039	−	0.464124i
$960$		0			0
$961$	−83.3663			−2.68923
$962$		0			0
$963$		0			0
$964$			3.45706i			0.111344i
$965$	−43.5918			−1.40327
$966$		0			0
$967$	−41.7666			−1.34312			−0.671561	−	0.740949i	$-0.734376\pi$
$967$	−41.7666			−1.34312			−0.671561	+	0.740949i	$0.734376\pi$
$968$		−	1.00000i		−	0.0321412i
$969$		0			0
$970$	−5.47551			−0.175808
$971$	52.8054			1.69461			0.847303	−	0.531110i	$-0.178225\pi$
$971$	52.8054			1.69461			0.847303	+	0.531110i	$0.178225\pi$
$972$		0			0
$973$	37.8411	+	34.5702i	1.21313	+	1.10827i
$974$			15.2662i			0.489161i
$975$		0			0
$976$			0.250475i			0.00801751i
$977$		−	33.5481i		−	1.07330i	−0.843806	−	0.536649i	$-0.819690\pi$
$977$		−	33.5481i		−	1.07330i	0.843806	−	0.536649i	$-0.180310\pi$
$978$		0			0
$979$			5.22255i			0.166913i
$980$	1.27232	+	14.0543i	0.0406428	+	0.448948i
$981$		0			0
$982$	23.6273			0.753976
$983$	14.1077			0.449967			0.224984	−	0.974363i	$-0.427767\pi$
$983$	14.1077			0.449967			0.224984	+	0.974363i	$0.427767\pi$
$984$		0			0
$985$			43.7291i			1.39332i
$986$	−7.85557			−0.250172
$987$		0			0
$988$		0			0
$989$			12.3571i			0.392931i
$990$		0			0
$991$	2.39626			0.0761197			0.0380599	−	0.999275i	$-0.487882\pi$
$991$	2.39626			0.0761197			0.0380599	+	0.999275i	$0.487882\pi$
$992$	10.6942			0.339542
$993$		0			0
$994$	1.08770	−	1.19062i	0.0344998	−	0.0377641i
$995$		−	7.95543i		−	0.252204i
$996$		0			0
$997$			50.6773i			1.60497i	0.596674	+	0.802483i	$0.296488\pi$
$997$			50.6773i			1.60497i	−0.596674	+	0.802483i	$0.703512\pi$
$998$			36.0484i			1.14109i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.g.a.881.7	yes	16
3.2	odd	2	inner	1386.2.g.a.881.10	yes	16
7.6	odd	2	inner	1386.2.g.a.881.2	✓	16
21.20	even	2	inner	1386.2.g.a.881.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.g.a.881.2	✓	16	7.6	odd	2	inner
1386.2.g.a.881.7	yes	16	1.1	even	1	trivial
1386.2.g.a.881.10	yes	16	3.2	odd	2	inner
1386.2.g.a.881.15	yes	16	21.20	even	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.01596 q^{5} +(1.78450 - 1.95335i) q^{7} +1.00000i q^{8} -2.01596i q^{10} +1.00000i q^{11} +(-1.95335 - 1.78450i) q^{14} +1.00000 q^{16} +5.92266 q^{17} +0.657927i q^{19} -2.01596 q^{20} +1.00000 q^{22} -7.87575i q^{23} -0.935886 q^{25} +(-1.78450 + 1.95335i) q^{28} -1.32636i q^{29} +10.6942i q^{31} -1.00000i q^{32} -5.92266i q^{34} +(3.59749 - 3.93788i) q^{35} +5.87575 q^{37} +0.657927 q^{38} +2.01596i q^{40} +4.56462 q^{41} -1.56900 q^{43} -1.00000i q^{44} -7.87575 q^{46} +0.532690 q^{47} +(-0.631122 - 6.97149i) q^{49} +0.935886i q^{50} -7.44475i q^{53} +2.01596i q^{55} +(1.95335 + 1.78450i) q^{56} -1.32636 q^{58} +4.03193 q^{59} +0.250475i q^{61} +10.6942 q^{62} -1.00000 q^{64} -7.81164 q^{67} -5.92266 q^{68} +(-3.93788 - 3.59749i) q^{70} +0.609527i q^{71} +0.532690i q^{73} -5.87575i q^{74} -0.657927i q^{76} +(1.95335 + 1.78450i) q^{77} -12.7711 q^{79} +2.01596 q^{80} -4.56462i q^{82} +13.3681 q^{83} +11.9399 q^{85} +1.56900i q^{86} -1.00000 q^{88} +5.22255 q^{89} +7.87575i q^{92} -0.532690i q^{94} +1.32636i q^{95} -2.71607i q^{97} +(-6.97149 + 0.631122i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 - 8 * q^7 + 16 * q^16 + 16 * q^22 + 16 * q^25 + 8 * q^28 - 16 * q^37 + 48 * q^43 - 16 * q^46 + 8 * q^49 - 16 * q^58 - 16 * q^64 + 16 * q^67 - 8 * q^70 - 16 * q^79 + 112 * q^85 - 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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