LMFDB - Embedded newform 1386.2.g.b.881.14

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} - 12x^{14} + 72x^{12} + 1296x^{10} + 8245x^{8} + 18780x^{6} + 20808x^{4} - 3672x^{2} + 324 $ x^16 - 12x^14 + 72x^12 + 1296x^10 + 8245x^8 + 18780x^6 + 20808x^4 - 3672*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.14
Root			$0.579826 + 1.39982i$ of defining polynomial
Character	$\chi$	$=$	1386.881
Dual form			1386.2.g.b.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.79965 q^{5} +(-2.20231 + 1.46623i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +2.79965 q^{5} +(-2.20231 + 1.46623i) q^{7} -1.00000i q^{8} +2.79965i q^{10} +1.00000i q^{11} +4.14713i q^{13} +(-1.46623 - 2.20231i) q^{14} +1.00000 q^{16} -5.73211 q^{17} -1.15965i q^{19} -2.79965 q^{20} -1.00000 q^{22} +2.83802 q^{25} -4.14713 q^{26} +(2.20231 - 1.46623i) q^{28} +10.0479i q^{29} -1.77281i q^{31} +1.00000i q^{32} -5.73211i q^{34} +(-6.16569 + 4.10494i) q^{35} +1.40064 q^{37} +1.15965 q^{38} -2.79965i q^{40} -2.37432 q^{41} -7.00398 q^{43} -1.00000i q^{44} +9.69141 q^{47} +(2.70032 - 6.45819i) q^{49} +2.83802i q^{50} -4.14713i q^{52} +11.8053i q^{53} +2.79965i q^{55} +(1.46623 + 2.20231i) q^{56} -10.0479 q^{58} -13.8936 q^{59} -1.71780i q^{61} +1.77281 q^{62} -1.00000 q^{64} +11.6105i q^{65} +12.0479 q^{67} +5.73211 q^{68} +(-4.10494 - 6.16569i) q^{70} +8.48528i q^{71} +10.2929i q^{73} +1.40064i q^{74} +1.15965i q^{76} +(-1.46623 - 2.20231i) q^{77} +1.43341 q^{79} +2.79965 q^{80} -2.37432i q^{82} -7.55994 q^{83} -16.0479 q^{85} -7.00398i q^{86} +1.00000 q^{88} -2.33096 q^{89} +(-6.08067 - 9.13327i) q^{91} +9.69141i q^{94} -3.24662i q^{95} +7.91860i q^{97} +(6.45819 + 2.70032i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 16 q^{22} + 48 q^{25} - 16 q^{37} - 80 q^{43} + 24 q^{49} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 24 q^{70} + 96 q^{79} - 80 q^{85} + 16 q^{88} - 32 q^{91}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^22 + 48 * q^25 - 16 * q^37 - 80 * q^43 + 24 * q^49 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 24 * q^70 + 96 * q^79 - 80 * q^85 + 16 * q^88 - 32 * q^91

Display 100 coefficients 10 coefficients

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
$5$	2.79965			1.25204			0.626020	−	0.779807i	$-0.284683\pi$
$5$	2.79965			1.25204			0.626020	+	0.779807i	$0.284683\pi$
$6$		0			0
$7$	−2.20231	+	1.46623i	−0.832394	+	0.554184i
$7$	−2.20231	+	1.46623i	−0.832394	+	0.554184i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			2.79965i			0.885326i
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			4.14713i			1.15021i	0.818080	+	0.575104i	$0.195038\pi$
$13$			4.14713i			1.15021i	−0.818080	+	0.575104i	$0.804962\pi$
$14$	−1.46623	−	2.20231i	−0.391867	−	0.588592i
$15$		0			0
$16$	1.00000			0.250000
$17$	−5.73211			−1.39024			−0.695121	−	0.718893i	$-0.744649\pi$
$17$	−5.73211			−1.39024			−0.695121	+	0.718893i	$0.744649\pi$
$18$		0			0
$19$		−	1.15965i		−	0.266042i	−0.991113	−	0.133021i	$-0.957532\pi$
$19$		−	1.15965i		−	0.266042i	0.991113	−	0.133021i	$-0.0424678\pi$
$20$	−2.79965			−0.626020
$21$		0			0
$22$	−1.00000			−0.213201
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	2.83802			0.567605
$26$	−4.14713			−0.813320
$27$		0			0
$28$	2.20231	−	1.46623i	0.416197	−	0.277092i
$29$			10.0479i			1.86585i	0.360074	+	0.932924i	$0.382751\pi$
$29$			10.0479i			1.86585i	−0.360074	+	0.932924i	$0.617249\pi$
$30$		0			0
$31$		−	1.77281i		−	0.318407i	−0.987246	−	0.159203i	$-0.949107\pi$
$31$		−	1.77281i		−	0.318407i	0.987246	−	0.159203i	$-0.0508926\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	5.73211i		−	0.983049i
$35$	−6.16569	+	4.10494i	−1.04219	+	0.693861i
$36$		0			0
$37$	1.40064			0.230264			0.115132	−	0.993350i	$-0.463271\pi$
$37$	1.40064			0.230264			0.115132	+	0.993350i	$0.463271\pi$
$38$	1.15965			0.188120
$39$		0			0
$40$		−	2.79965i		−	0.442663i
$41$	−2.37432			−0.370806			−0.185403	−	0.982663i	$-0.559359\pi$
$41$	−2.37432			−0.370806			−0.185403	+	0.982663i	$0.559359\pi$
$42$		0			0
$43$	−7.00398			−1.06810			−0.534048	−	0.845454i	$-0.679330\pi$
$43$	−7.00398			−1.06810			−0.534048	+	0.845454i	$0.679330\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$		0			0
$47$	9.69141			1.41364			0.706819	−	0.707395i	$-0.250130\pi$
$47$	9.69141			1.41364			0.706819	+	0.707395i	$0.250130\pi$
$48$		0			0
$49$	2.70032	−	6.45819i	0.385760	−	0.922599i
$50$			2.83802i			0.401357i
$51$		0			0
$52$		−	4.14713i		−	0.575104i
$53$			11.8053i			1.62158i	0.585339	+	0.810788i	$0.300961\pi$
$53$			11.8053i			1.62158i	−0.585339	+	0.810788i	$0.699039\pi$
$54$		0			0
$55$			2.79965i			0.377504i
$56$	1.46623	+	2.20231i	0.195934	+	0.294296i
$57$		0			0
$58$	−10.0479			−1.31935
$59$	−13.8936			−1.80879			−0.904394	−	0.426699i	$-0.859677\pi$
$59$	−13.8936			−1.80879			−0.904394	+	0.426699i	$0.859677\pi$
$60$		0			0
$61$		−	1.71780i		−	0.219942i	−0.993935	−	0.109971i	$-0.964924\pi$
$61$		−	1.71780i		−	0.219942i	0.993935	−	0.109971i	$-0.0350757\pi$
$62$	1.77281			0.225148
$63$		0			0
$64$	−1.00000			−0.125000
$65$			11.6105i			1.44011i
$66$		0			0
$67$	12.0479			1.47188			0.735942	−	0.677044i	$-0.236739\pi$
$67$	12.0479			1.47188			0.735942	+	0.677044i	$0.236739\pi$
$68$	5.73211			0.695121
$69$		0			0
$70$	−4.10494	−	6.16569i	−0.490634	−	0.736940i
$71$			8.48528i			1.00702i	0.863990	+	0.503509i	$0.167958\pi$
$71$			8.48528i			1.00702i	−0.863990	+	0.503509i	$0.832042\pi$
$72$		0			0
$73$			10.2929i			1.20469i	0.798234	+	0.602347i	$0.205768\pi$
$73$			10.2929i			1.20469i	−0.798234	+	0.602347i	$0.794232\pi$
$74$			1.40064i			0.162821i
$75$		0			0
$76$			1.15965i			0.133021i
$77$	−1.46623	−	2.20231i	−0.167093	−	0.250976i
$78$		0			0
$79$	1.43341			0.161271			0.0806356	−	0.996744i	$-0.474305\pi$
$79$	1.43341			0.161271			0.0806356	+	0.996744i	$0.474305\pi$
$80$	2.79965			0.313010
$81$		0			0
$82$		−	2.37432i		−	0.262200i
$83$	−7.55994			−0.829812			−0.414906	−	0.909864i	$-0.636185\pi$
$83$	−7.55994			−0.829812			−0.414906	+	0.909864i	$0.636185\pi$
$84$		0			0
$85$	−16.0479			−1.74064
$86$		−	7.00398i		−	0.755258i
$87$		0			0
$88$	1.00000			0.106600
$89$	−2.33096			−0.247081			−0.123541	−	0.992340i	$-0.539425\pi$
$89$	−2.33096			−0.247081			−0.123541	+	0.992340i	$0.539425\pi$
$90$		0			0
$91$	−6.08067	−	9.13327i	−0.637427	−	0.957426i
$92$		0			0
$93$		0			0
$94$			9.69141i			0.999593i
$95$		−	3.24662i		−	0.333096i
$96$		0			0
$97$			7.91860i			0.804012i	0.915637	+	0.402006i	$0.131687\pi$
$97$			7.91860i			0.804012i	−0.915637	+	0.402006i	$0.868313\pi$
$98$	6.45819	+	2.70032i	0.652376	+	0.272774i
$99$		0			0
$100$	−2.83802			−0.283802
$101$	−3.77146			−0.375275			−0.187637	−	0.982238i	$-0.560083\pi$
$101$	−3.77146			−0.375275			−0.187637	+	0.982238i	$0.560083\pi$
$102$		0			0
$103$		−	18.0635i		−	1.77985i	−0.456109	−	0.889924i	$-0.650757\pi$
$103$		−	18.0635i		−	1.77985i	0.456109	−	0.889924i	$-0.349243\pi$
$104$	4.14713			0.406660
$105$		0			0
$106$	−11.8053			−1.14663
$107$		−	8.20987i		−	0.793678i	−0.917888	−	0.396839i	$-0.870107\pi$
$107$		−	8.20987i		−	0.793678i	0.917888	−	0.396839i	$-0.129893\pi$
$108$		0			0
$109$	17.7239			1.69765			0.848823	−	0.528677i	$-0.177312\pi$
$109$	17.7239			1.69765			0.848823	+	0.528677i	$0.177312\pi$
$110$	−2.79965			−0.266936
$111$		0			0
$112$	−2.20231	+	1.46623i	−0.208099	+	0.138546i
$113$			6.08067i			0.572021i	0.958226	+	0.286010i	$0.0923292\pi$
$113$			6.08067i			0.572021i	−0.958226	+	0.286010i	$0.907671\pi$
$114$		0			0
$115$		0			0
$116$		−	10.0479i		−	0.932924i
$117$		0			0
$118$		−	13.8936i		−	1.27901i
$119$	12.6239	−	8.40462i	1.15723	−	0.770450i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	1.71780			0.155522
$123$		0			0
$124$			1.77281i			0.159203i
$125$	−6.05277			−0.541376
$126$		0			0
$127$	11.0519			0.980695			0.490348	−	0.871527i	$-0.336870\pi$
$127$	11.0519			0.980695			0.490348	+	0.871527i	$0.336870\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−11.6105			−1.01831
$131$	−16.3457			−1.42813			−0.714065	−	0.700080i	$-0.753148\pi$
$131$	−16.3457			−1.42813			−0.714065	+	0.700080i	$0.753148\pi$
$132$		0			0
$133$	1.70032	+	2.55391i	0.147436	+	0.221452i
$134$			12.0479i			1.04078i
$135$		0			0
$136$			5.73211i			0.491525i
$137$		−	3.48131i		−	0.297428i	−0.988880	−	0.148714i	$-0.952487\pi$
$137$		−	3.48131i		−	0.297428i	0.988880	−	0.148714i	$-0.0475134\pi$
$138$		0			0
$139$		−	9.53172i		−	0.808470i	−0.914655	−	0.404235i	$-0.867538\pi$
$139$		−	9.53172i		−	0.808470i	0.914655	−	0.404235i	$-0.132462\pi$
$140$	6.16569	−	4.10494i	0.521096	−	0.346930i
$141$		0			0
$142$	−8.48528			−0.712069
$143$	−4.14713			−0.346801
$144$		0			0
$145$			28.1306i			2.33612i
$146$	−10.2929			−0.851848
$147$		0			0
$148$	−1.40064			−0.115132
$149$		−	3.40064i		−	0.278591i	−0.990251	−	0.139296i	$-0.955516\pi$
$149$		−	3.40064i		−	0.278591i	0.990251	−	0.139296i	$-0.0444838\pi$
$150$		0			0
$151$	−17.0918			−1.39091			−0.695456	−	0.718568i	$-0.744798\pi$
$151$	−17.0918			−1.39091			−0.695456	+	0.718568i	$0.744798\pi$
$152$	−1.15965			−0.0940602
$153$		0			0
$154$	2.20231	−	1.46623i	0.177467	−	0.118152i
$155$		−	4.96326i		−	0.398658i
$156$		0			0
$157$			1.77281i			0.141486i	0.997495	+	0.0707430i	$0.0225370\pi$
$157$			1.77281i			0.141486i	−0.997495	+	0.0707430i	$0.977463\pi$
$158$			1.43341i			0.114036i
$159$		0			0
$160$			2.79965i			0.221332i
$161$		0			0
$162$		0			0
$163$	−5.07669			−0.397637			−0.198819	−	0.980036i	$-0.563710\pi$
$163$	−5.07669			−0.397637			−0.198819	+	0.980036i	$0.563710\pi$
$164$	2.37432			0.185403
$165$		0			0
$166$		−	7.55994i		−	0.586765i
$167$	20.0944			1.55495			0.777474	−	0.628915i	$-0.216501\pi$
$167$	20.0944			1.55495			0.777474	+	0.628915i	$0.216501\pi$
$168$		0			0
$169$	−4.19872			−0.322978
$170$		−	16.0479i		−	1.23082i
$171$		0			0
$172$	7.00398			0.534048
$173$	7.31709			0.556308			0.278154	−	0.960536i	$-0.410277\pi$
$173$	7.31709			0.556308			0.278154	+	0.960536i	$0.410277\pi$
$174$		0			0
$175$	−6.25020	+	4.16121i	−0.472471	+	0.314558i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$		−	2.33096i		−	0.174713i
$179$		−	7.72395i		−	0.577315i	−0.957432	−	0.288657i	$-0.906791\pi$
$179$		−	7.72395i		−	0.577315i	0.957432	−	0.288657i	$-0.0932089\pi$
$180$		0			0
$181$			3.49061i			0.259455i	0.991550	+	0.129728i	$0.0414103\pi$
$181$			3.49061i			0.259455i	−0.991550	+	0.129728i	$0.958590\pi$
$182$	9.13327	−	6.08067i	0.677003	−	0.450729i
$183$		0			0
$184$		0			0
$185$	3.92130			0.288300
$186$		0			0
$187$		−	5.73211i		−	0.419174i
$188$	−9.69141			−0.706819
$189$		0			0
$190$	3.24662			0.235534
$191$			8.48528i			0.613973i	0.951714	+	0.306987i	$0.0993207\pi$
$191$			8.48528i			0.613973i	−0.951714	+	0.306987i	$0.900679\pi$
$192$		0			0
$193$	18.0958			1.30256			0.651282	−	0.758836i	$-0.274232\pi$
$193$	18.0958			1.30256			0.651282	+	0.758836i	$0.274232\pi$
$194$	−7.91860			−0.568522
$195$		0			0
$196$	−2.70032	+	6.45819i	−0.192880	+	0.461300i
$197$			23.3424i			1.66308i	0.555466	+	0.831539i	$0.312540\pi$
$197$			23.3424i			1.66308i	−0.555466	+	0.831539i	$0.687460\pi$
$198$		0			0
$199$			9.95705i			0.705837i	0.935654	+	0.352918i	$0.114811\pi$
$199$			9.95705i			0.705837i	−0.935654	+	0.352918i	$0.885189\pi$
$200$		−	2.83802i		−	0.200679i
$201$		0			0
$202$		−	3.77146i		−	0.265359i
$203$	−14.7326	−	22.1286i	−1.03402	−	1.55312i
$204$		0			0
$205$	−6.64726			−0.464265
$206$	18.0635			1.25854
$207$		0			0
$208$			4.14713i			0.287552i
$209$	1.15965			0.0802148
$210$		0			0
$211$	6.29054			0.433058			0.216529	−	0.976276i	$-0.430526\pi$
$211$	6.29054			0.433058			0.216529	+	0.976276i	$0.430526\pi$
$212$		−	11.8053i		−	0.810788i
$213$		0			0
$214$	8.20987			0.561215
$215$	−19.6087			−1.33730
$216$		0			0
$217$	2.59936	+	3.90428i	0.176456	+	0.265040i
$218$			17.7239i			1.20042i
$219$		0			0
$220$		−	2.79965i		−	0.188752i
$221$		−	23.7718i		−	1.59907i
$222$		0			0
$223$		−	7.74778i		−	0.518830i	−0.965766	−	0.259415i	$-0.916470\pi$
$223$		−	7.74778i		−	0.518830i	0.965766	−	0.259415i	$-0.0835297\pi$
$224$	−1.46623	−	2.20231i	−0.0979668	−	0.147148i
$225$		0			0
$226$	−6.08067			−0.404480
$227$	14.7057			0.976051			0.488025	−	0.872829i	$-0.337717\pi$
$227$	14.7057			0.976051			0.488025	+	0.872829i	$0.337717\pi$
$228$		0			0
$229$		−	12.0429i		−	0.795820i	−0.917424	−	0.397910i	$-0.869736\pi$
$229$		−	12.0429i		−	0.795820i	0.917424	−	0.397910i	$-0.130264\pi$
$230$		0			0
$231$		0			0
$232$	10.0479			0.659677
$233$			7.13318i			0.467310i	0.972320	+	0.233655i	$0.0750687\pi$
$233$			7.13318i			0.467310i	−0.972320	+	0.233655i	$0.924931\pi$
$234$		0			0
$235$	27.1325			1.76993
$236$	13.8936			0.904394
$237$		0			0
$238$	8.40462	+	12.6239i	0.544790	+	0.818285i
$239$			1.07669i			0.0696453i	0.999394	+	0.0348226i	$0.0110866\pi$
$239$			1.07669i			0.0696453i	−0.999394	+	0.0348226i	$0.988913\pi$
$240$		0			0
$241$		−	22.6856i		−	1.46131i	−0.682748	−	0.730654i	$-0.739215\pi$
$241$		−	22.6856i		−	1.46131i	0.682748	−	0.730654i	$-0.260785\pi$
$242$		−	1.00000i		−	0.0642824i
$243$		0			0
$244$			1.71780i			0.109971i
$245$	7.55994	−	18.0807i	0.482987	−	1.15513i
$246$		0			0
$247$	4.80923			0.306004
$248$	−1.77281			−0.112574
$249$		0			0
$250$		−	6.05277i		−	0.382811i
$251$	24.3971			1.53993			0.769966	−	0.638085i	$-0.220273\pi$
$251$	24.3971			1.53993			0.769966	+	0.638085i	$0.220273\pi$
$252$		0			0
$253$		0			0
$254$			11.0519i			0.693456i
$255$		0			0
$256$	1.00000			0.0625000
$257$	27.6654			1.72572			0.862861	−	0.505441i	$-0.168670\pi$
$257$	27.6654			1.72572			0.862861	+	0.505441i	$0.168670\pi$
$258$		0			0
$259$	−3.08464	+	2.05367i	−0.191670	+	0.127609i
$260$		−	11.6105i		−	0.720053i
$261$		0			0
$262$		−	16.3457i		−	1.00984i
$263$			11.2387i			0.693006i	0.938049	+	0.346503i	$0.112631\pi$
$263$			11.2387i			0.693006i	−0.938049	+	0.346503i	$0.887369\pi$
$264$		0			0
$265$			33.0506i			2.03028i
$266$	−2.55391	+	1.70032i	−0.156590	+	0.104253i
$267$		0			0
$268$	−12.0479			−0.735942
$269$	−11.1717			−0.681152			−0.340576	−	0.940217i	$-0.610622\pi$
$269$	−11.1717			−0.681152			−0.340576	+	0.940217i	$0.610622\pi$
$270$		0			0
$271$		−	16.2948i		−	0.989836i	−0.868940	−	0.494918i	$-0.835198\pi$
$271$		−	16.2948i		−	0.989836i	0.868940	−	0.494918i	$-0.164802\pi$
$272$	−5.73211			−0.347560
$273$		0			0
$274$	3.48131			0.210313
$275$			2.83802i			0.171139i
$276$		0			0
$277$	24.8492			1.49304			0.746521	−	0.665362i	$-0.231723\pi$
$277$	24.8492			1.49304			0.746521	+	0.665362i	$0.231723\pi$
$278$	9.53172			0.571675
$279$		0			0
$280$	4.10494	+	6.16569i	0.245317	+	0.368470i
$281$			4.00795i			0.239094i	0.992829	+	0.119547i	$0.0381443\pi$
$281$			4.00795i			0.239094i	−0.992829	+	0.119547i	$0.961856\pi$
$282$		0			0
$283$		−	0.496831i		−	0.0295335i	−0.999891	−	0.0147668i	$-0.995299\pi$
$283$		−	0.496831i		−	0.0295335i	0.999891	−	0.0147668i	$-0.00470058\pi$
$284$		−	8.48528i		−	0.503509i
$285$		0			0
$286$		−	4.14713i		−	0.245225i
$287$	5.22898	−	3.48131i	0.308657	−	0.205495i
$288$		0			0
$289$	15.8571			0.932772
$290$	−28.1306			−1.65188
$291$		0			0
$292$		−	10.2929i		−	0.602347i
$293$	12.6507			0.739065			0.369532	−	0.929218i	$-0.379518\pi$
$293$	12.6507			0.739065			0.369532	+	0.929218i	$0.379518\pi$
$294$		0			0
$295$	−38.8971			−2.26468
$296$		−	1.40064i		−	0.0814106i
$297$		0			0
$298$	3.40064			0.196994
$299$		0			0
$300$		0			0
$301$	15.4249	−	10.2695i	0.889077	−	0.591922i
$302$		−	17.0918i		−	0.983524i
$303$		0			0
$304$		−	1.15965i		−	0.0665106i
$305$		−	4.80923i		−	0.275376i
$306$		0			0
$307$			25.4011i			1.44972i	0.688896	+	0.724860i	$0.258096\pi$
$307$			25.4011i			1.44972i	−0.688896	+	0.724860i	$0.741904\pi$
$308$	1.46623	+	2.20231i	0.0835464	+	0.125488i
$309$		0			0
$310$	4.96326			0.281894
$311$	34.2764			1.94363			0.971817	−	0.235738i	$-0.0757506\pi$
$311$	34.2764			1.94363			0.971817	+	0.235738i	$0.0757506\pi$
$312$		0			0
$313$			20.9450i			1.18388i	0.805981	+	0.591941i	$0.201638\pi$
$313$			20.9450i			1.18388i	−0.805981	+	0.591941i	$0.798362\pi$
$314$	−1.77281			−0.100046
$315$		0			0
$316$	−1.43341			−0.0806356
$317$		−	31.9010i		−	1.79174i	−0.444315	−	0.895871i	$-0.646553\pi$
$317$		−	31.9010i		−	1.79174i	0.444315	−	0.895871i	$-0.353447\pi$
$318$		0			0
$319$	−10.0479			−0.562574
$320$	−2.79965			−0.156505
$321$		0			0
$322$		0			0
$323$			6.64726i			0.369863i
$324$		0			0
$325$			11.7697i			0.652864i
$326$		−	5.07669i		−	0.281172i
$327$		0			0
$328$			2.37432i			0.131100i
$329$	−21.3435	+	14.2099i	−1.17670	+	0.783416i
$330$		0			0
$331$	16.5332			0.908746			0.454373	−	0.890812i	$-0.349863\pi$
$331$	16.5332			0.908746			0.454373	+	0.890812i	$0.349863\pi$
$332$	7.55994			0.414906
$333$		0			0
$334$			20.0944i			1.09951i
$335$	33.7299			1.84286
$336$		0			0
$337$	9.61051			0.523518			0.261759	−	0.965133i	$-0.415697\pi$
$337$	9.61051			0.523518			0.261759	+	0.965133i	$0.415697\pi$
$338$		−	4.19872i		−	0.228380i
$339$		0			0
$340$	16.0479			0.870319
$341$	1.77281			0.0960033
$342$		0			0
$343$	3.52228	+	18.1822i	0.190185	+	0.981748i
$344$			7.00398i			0.377629i
$345$		0			0
$346$			7.31709i			0.393369i
$347$			1.40859i			0.0756171i	0.999285	+	0.0378086i	$0.0120377\pi$
$347$			1.40859i			0.0756171i	−0.999285	+	0.0378086i	$0.987962\pi$
$348$		0			0
$349$		−	32.2992i		−	1.72894i	−0.502687	−	0.864469i	$-0.667655\pi$
$349$		−	32.2992i		−	1.72894i	0.502687	−	0.864469i	$-0.332345\pi$
$350$	−4.16121	−	6.25020i	−0.222426	−	0.334087i
$351$		0			0
$352$	−1.00000			−0.0533002
$353$	−18.6994			−0.995271			−0.497635	−	0.867386i	$-0.665798\pi$
$353$	−18.6994			−0.995271			−0.497635	+	0.867386i	$0.665798\pi$
$354$		0			0
$355$			23.7558i			1.26083i
$356$	2.33096			0.123541
$357$		0			0
$358$	7.72395			0.408223
$359$			7.44854i			0.393119i	0.980492	+	0.196559i	$0.0629768\pi$
$359$			7.44854i			0.393119i	−0.980492	+	0.196559i	$0.937023\pi$
$360$		0			0
$361$	17.6552			0.929221
$362$	−3.49061			−0.183463
$363$		0			0
$364$	6.08067	+	9.13327i	0.318713	+	0.478713i
$365$			28.8165i			1.50833i
$366$		0			0
$367$			18.3291i			0.956772i	0.878150	+	0.478386i	$0.158778\pi$
$367$			18.3291i			0.956772i	−0.878150	+	0.478386i	$0.841222\pi$
$368$		0			0
$369$		0			0
$370$			3.92130i			0.203859i
$371$	−17.3093	−	25.9988i	−0.898652	−	1.34979i
$372$		0			0
$373$	13.5540			0.701801			0.350900	−	0.936413i	$-0.385876\pi$
$373$	13.5540			0.701801			0.350900	+	0.936413i	$0.385876\pi$
$374$	5.73211			0.296401
$375$		0			0
$376$		−	9.69141i		−	0.499796i
$377$	−41.6700			−2.14611
$378$		0			0
$379$	−9.07669			−0.466238			−0.233119	−	0.972448i	$-0.574893\pi$
$379$	−9.07669			−0.466238			−0.233119	+	0.972448i	$0.574893\pi$
$380$			3.24662i			0.166548i
$381$		0			0
$382$	−8.48528			−0.434145
$383$	−13.9866			−0.714681			−0.357340	−	0.933974i	$-0.616316\pi$
$383$	−13.9866			−0.714681			−0.357340	+	0.933974i	$0.616316\pi$
$384$		0			0
$385$	−4.10494	−	6.16569i	−0.209207	−	0.314232i
$386$			18.0958i			0.921052i
$387$		0			0
$388$		−	7.91860i		−	0.402006i
$389$			20.1292i			1.02059i	0.859999	+	0.510296i	$0.170464\pi$
$389$			20.1292i			1.02059i	−0.859999	+	0.510296i	$0.829536\pi$
$390$		0			0
$391$		0			0
$392$	−6.45819	−	2.70032i	−0.326188	−	0.136387i
$393$		0			0
$394$	−23.3424			−1.17597
$395$	4.01304			0.201918
$396$		0			0
$397$			22.3206i			1.12024i	0.828411	+	0.560121i	$0.189245\pi$
$397$			22.3206i			1.12024i	−0.828411	+	0.560121i	$0.810755\pi$
$398$	−9.95705			−0.499102
$399$		0			0
$400$	2.83802			0.141901
$401$			8.68003i			0.433460i	0.976232	+	0.216730i	$0.0695391\pi$
$401$			8.68003i			0.433460i	−0.976232	+	0.216730i	$0.930461\pi$
$402$		0			0
$403$	7.35210			0.366234
$404$	3.77146			0.187637
$405$		0			0
$406$	22.1286	−	14.7326i	1.09822	−	0.731165i
$407$			1.40064i			0.0694272i
$408$		0			0
$409$		−	22.9512i		−	1.13487i	−0.823420	−	0.567433i	$-0.807937\pi$
$409$		−	22.9512i		−	1.13487i	0.823420	−	0.567433i	$-0.192063\pi$
$410$		−	6.64726i		−	0.328285i
$411$		0			0
$412$			18.0635i			0.889924i
$413$	30.5979	−	20.3712i	1.50562	−	1.00240i
$414$		0			0
$415$	−21.1652			−1.03896
$416$	−4.14713			−0.203330
$417$		0			0
$418$			1.15965i			0.0567204i
$419$	15.0099			0.733279			0.366640	−	0.930363i	$-0.380508\pi$
$419$	15.0099			0.733279			0.366640	+	0.930363i	$0.380508\pi$
$420$		0			0
$421$	−13.4006			−0.653107			−0.326554	−	0.945179i	$-0.605887\pi$
$421$	−13.4006			−0.653107			−0.326554	+	0.945179i	$0.605887\pi$
$422$			6.29054i			0.306219i
$423$		0			0
$424$	11.8053			0.573314
$425$	−16.2679			−0.789108
$426$		0			0
$427$	2.51869	+	3.78312i	0.121888	+	0.183078i
$428$			8.20987i			0.396839i
$429$		0			0
$430$		−	19.6087i		−	0.945614i
$431$		−	28.2092i		−	1.35879i	−0.733772	−	0.679395i	$-0.762242\pi$
$431$		−	28.2092i		−	1.35879i	0.733772	−	0.679395i	$-0.237758\pi$
$432$		0			0
$433$		−	9.48079i		−	0.455618i	−0.973706	−	0.227809i	$-0.926844\pi$
$433$		−	9.48079i		−	0.455618i	0.973706	−	0.227809i	$-0.0731561\pi$
$434$	−3.90428	+	2.59936i	−0.187412	+	0.124773i
$435$		0			0
$436$	−17.7239			−0.848823
$437$		0			0
$438$		0			0
$439$			28.0246i			1.33754i	0.743468	+	0.668771i	$0.233179\pi$
$439$			28.0246i			1.33754i	−0.743468	+	0.668771i	$0.766821\pi$
$440$	2.79965			0.133468
$441$		0			0
$442$	23.7718			1.13071
$443$			4.43738i			0.210827i	0.994429	+	0.105413i	$0.0336165\pi$
$443$			4.43738i			0.210827i	−0.994429	+	0.105413i	$0.966383\pi$
$444$		0			0
$445$	−6.52587			−0.309356
$446$	7.74778			0.366868
$447$		0			0
$448$	2.20231	−	1.46623i	0.104049	−	0.0692730i
$449$		−	29.5298i		−	1.39360i	−0.717266	−	0.696800i	$-0.754607\pi$
$449$		−	29.5298i		−	1.39360i	0.717266	−	0.696800i	$-0.245393\pi$
$450$		0			0
$451$		−	2.37432i		−	0.111802i
$452$		−	6.08067i		−	0.286010i
$453$		0			0
$454$			14.7057i			0.690172i
$455$	−17.0237	−	25.5699i	−0.798084	−	1.19874i
$456$		0			0
$457$	13.2866			0.621519			0.310760	−	0.950489i	$-0.399417\pi$
$457$	13.2866			0.621519			0.310760	+	0.950489i	$0.399417\pi$
$458$	12.0429			0.562730
$459$		0			0
$460$		0			0
$461$	−40.7035			−1.89575			−0.947876	−	0.318640i	$-0.896774\pi$
$461$	−40.7035			−1.89575			−0.947876	+	0.318640i	$0.896774\pi$
$462$		0			0
$463$	−20.9706			−0.974585			−0.487292	−	0.873239i	$-0.662015\pi$
$463$	−20.9706			−0.974585			−0.487292	+	0.873239i	$0.662015\pi$
$464$			10.0479i			0.466462i
$465$		0			0
$466$	−7.13318			−0.330438
$467$	−4.13065			−0.191144			−0.0955718	−	0.995423i	$-0.530468\pi$
$467$	−4.13065			−0.191144			−0.0955718	+	0.995423i	$0.530468\pi$
$468$		0			0
$469$	−26.5332	+	17.6650i	−1.22519	+	0.815695i
$470$			27.1325i			1.25153i
$471$		0			0
$472$			13.8936i			0.639503i
$473$		−	7.00398i		−	0.322043i
$474$		0			0
$475$		−	3.29112i		−	0.151007i
$476$	−12.6239	+	8.40462i	−0.578615	+	0.385225i
$477$		0			0
$478$	−1.07669			−0.0492467
$479$	13.7670			0.629032			0.314516	−	0.949252i	$-0.398158\pi$
$479$	13.7670			0.629032			0.314516	+	0.949252i	$0.398158\pi$
$480$		0			0
$481$			5.80864i			0.264851i
$482$	22.6856			1.03330
$483$		0			0
$484$	1.00000			0.0454545
$485$			22.1693i			1.00666i
$486$		0			0
$487$	26.7516			1.21223			0.606116	−	0.795376i	$-0.292727\pi$
$487$	26.7516			1.21223			0.606116	+	0.795376i	$0.292727\pi$
$488$	−1.71780			−0.0777611
$489$		0			0
$490$	18.0807	+	7.55994i	0.816801	+	0.341523i
$491$			15.5620i			0.702302i	0.936319	+	0.351151i	$0.114210\pi$
$491$			15.5620i			0.702302i	−0.936319	+	0.351151i	$0.885790\pi$
$492$		0			0
$493$		−	57.5957i		−	2.59398i
$494$			4.80923i			0.216378i
$495$		0			0
$496$		−	1.77281i		−	0.0796017i
$497$	−12.4414	−	18.6872i	−0.558073	−	0.838236i
$498$		0			0
$499$	−36.0479			−1.61373			−0.806863	−	0.590739i	$-0.798836\pi$
$499$	−36.0479			−1.61373			−0.806863	+	0.590739i	$0.798836\pi$
$500$	6.05277			0.270688
$501$		0			0
$502$			24.3971i			1.08890i
$503$	−13.0031			−0.579780			−0.289890	−	0.957060i	$-0.593619\pi$
$503$	−13.0031			−0.579780			−0.289890	+	0.957060i	$0.593619\pi$
$504$		0			0
$505$	−10.5588			−0.469859
$506$		0			0
$507$		0			0
$508$	−11.0519			−0.490348
$509$	9.30409			0.412396			0.206198	−	0.978510i	$-0.433891\pi$
$509$	9.30409			0.412396			0.206198	+	0.978510i	$0.433891\pi$
$510$		0			0
$511$	−15.0918	−	22.6682i	−0.667623	−	1.00278i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			27.6654i			1.22027i
$515$		−	50.5714i		−	2.22844i
$516$		0			0
$517$			9.69141i			0.426228i
$518$	−2.05367	−	3.08464i	−0.0902329	−	0.135531i
$519$		0			0
$520$	11.6105			0.509155
$521$	21.9794			0.962936			0.481468	−	0.876464i	$-0.340104\pi$
$521$	21.9794			0.962936			0.481468	+	0.876464i	$0.340104\pi$
$522$		0			0
$523$		−	24.5953i		−	1.07548i	−0.843111	−	0.537739i	$-0.819279\pi$
$523$		−	24.5953i		−	1.07548i	0.843111	−	0.537739i	$-0.180721\pi$
$524$	16.3457			0.714065
$525$		0			0
$526$	−11.2387			−0.490029
$527$			10.1620i			0.442663i
$528$		0			0
$529$	23.0000			1.00000
$530$	−33.0506			−1.43562
$531$		0			0
$532$	−1.70032	−	2.55391i	−0.0737182	−	0.110726i
$533$		−	9.84662i		−	0.426504i
$534$		0			0
$535$		−	22.9847i		−	0.993717i
$536$		−	12.0479i		−	0.520390i
$537$		0			0
$538$		−	11.1717i		−	0.481647i
$539$	6.45819	+	2.70032i	0.278174	+	0.116311i
$540$		0			0
$541$	42.7989			1.84007			0.920034	−	0.391838i	$-0.128161\pi$
$541$	42.7989			1.84007			0.920034	+	0.391838i	$0.128161\pi$
$542$	16.2948			0.699920
$543$		0			0
$544$		−	5.73211i		−	0.245762i
$545$	49.6208			2.12552
$546$		0			0
$547$	−25.0918			−1.07285			−0.536424	−	0.843948i	$-0.680225\pi$
$547$	−25.0918			−1.07285			−0.536424	+	0.843948i	$0.680225\pi$
$548$			3.48131i			0.148714i
$549$		0			0
$550$	−2.83802			−0.121014
$551$	11.6521			0.496395
$552$		0			0
$553$	−3.15681	+	2.10171i	−0.134241	+	0.0893739i
$554$			24.8492i			1.05574i
$555$		0			0
$556$			9.53172i			0.404235i
$557$			18.7613i			0.794943i	0.917614	+	0.397472i	$0.130112\pi$
$557$			18.7613i			0.794943i	−0.917614	+	0.397472i	$0.869888\pi$
$558$		0			0
$559$		−	29.0464i		−	1.22853i
$560$	−6.16569	+	4.10494i	−0.260548	+	0.173465i
$561$		0			0
$562$	−4.00795			−0.169065
$563$	−15.9155			−0.670760			−0.335380	−	0.942083i	$-0.608865\pi$
$563$	−15.9155			−0.670760			−0.335380	+	0.942083i	$0.608865\pi$
$564$		0			0
$565$			17.0237i			0.716193i
$566$	0.496831			0.0208833
$567$		0			0
$568$	8.48528			0.356034
$569$			28.4773i			1.19383i	0.802304	+	0.596916i	$0.203607\pi$
$569$			28.4773i			1.19383i	−0.802304	+	0.596916i	$0.796393\pi$
$570$		0			0
$571$	−28.2985			−1.18425			−0.592127	−	0.805844i	$-0.701712\pi$
$571$	−28.2985			−1.18425			−0.592127	+	0.805844i	$0.701712\pi$
$572$	4.14713			0.173400
$573$		0			0
$574$	3.48131	+	5.22898i	0.145307	+	0.218253i
$575$		0			0
$576$		0			0
$577$		−	11.4642i		−	0.477262i	−0.971110	−	0.238631i	$-0.923301\pi$
$577$		−	11.4642i		−	0.477262i	0.971110	−	0.238631i	$-0.0766986\pi$
$578$			15.8571i			0.659570i
$579$		0			0
$580$		−	28.1306i		−	1.16806i
$581$	16.6493	−	11.0846i	0.690730	−	0.459868i
$582$		0			0
$583$	−11.8053			−0.488924
$584$	10.2929			0.425924
$585$		0			0
$586$			12.6507i			0.522598i
$587$	−21.5921			−0.891201			−0.445601	−	0.895232i	$-0.647010\pi$
$587$	−21.5921			−0.891201			−0.445601	+	0.895232i	$0.647010\pi$
$588$		0			0
$589$	−2.05585			−0.0847097
$590$		−	38.8971i		−	1.60137i
$591$		0			0
$592$	1.40064			0.0575660
$593$	1.62298			0.0666478			0.0333239	−	0.999445i	$-0.489391\pi$
$593$	1.62298			0.0666478			0.0333239	+	0.999445i	$0.489391\pi$
$594$		0			0
$595$	35.3424	−	23.5300i	1.44890	−	0.964634i
$596$			3.40064i			0.139296i
$597$		0			0
$598$		0			0
$599$		−	14.8172i		−	0.605414i	−0.953084	−	0.302707i	$-0.902110\pi$
$599$		−	14.8172i		−	0.605414i	0.953084	−	0.302707i	$-0.0978903\pi$
$600$		0			0
$601$		−	33.4092i		−	1.36279i	−0.731916	−	0.681395i	$-0.761374\pi$
$601$		−	33.4092i		−	1.36279i	0.731916	−	0.681395i	$-0.238626\pi$
$602$	10.2695	+	15.4249i	0.418552	+	0.628672i
$603$		0			0
$604$	17.0918			0.695456
$605$	−2.79965			−0.113822
$606$		0			0
$607$		−	13.2804i		−	0.539035i	−0.962996	−	0.269517i	$-0.913136\pi$
$607$		−	13.2804i		−	0.539035i	0.962996	−	0.269517i	$-0.0868642\pi$
$608$	1.15965			0.0470301
$609$		0			0
$610$	4.80923			0.194720
$611$			40.1916i			1.62598i
$612$		0			0
$613$	−26.9706			−1.08933			−0.544665	−	0.838653i	$-0.683343\pi$
$613$	−26.9706			−1.08933			−0.544665	+	0.838653i	$0.683343\pi$
$614$	−25.4011			−1.02511
$615$		0			0
$616$	−2.20231	+	1.46623i	−0.0887335	+	0.0590762i
$617$		−	15.1095i		−	0.608284i	−0.952627	−	0.304142i	$-0.901630\pi$
$617$		−	15.1095i		−	0.608284i	0.952627	−	0.304142i	$-0.0983697\pi$
$618$		0			0
$619$		−	2.58494i		−	0.103898i	−0.998650	−	0.0519488i	$-0.983457\pi$
$619$		−	2.58494i		−	0.103898i	0.998650	−	0.0519488i	$-0.0165433\pi$
$620$			4.96326i			0.199329i
$621$		0			0
$622$			34.2764i			1.37436i
$623$	5.13350	−	3.41773i	0.205669	−	0.136929i
$624$		0			0
$625$	−31.1357			−1.24543
$626$	−20.9450			−0.837131
$627$		0			0
$628$		−	1.77281i		−	0.0707430i
$629$	−8.02863			−0.320122
$630$		0			0
$631$	−10.4118			−0.414487			−0.207243	−	0.978289i	$-0.566449\pi$
$631$	−10.4118			−0.414487			−0.207243	+	0.978289i	$0.566449\pi$
$632$		−	1.43341i		−	0.0570180i
$633$		0			0
$634$	31.9010			1.26695
$635$	30.9413			1.22787
$636$		0			0
$637$	26.7830	+	11.1986i	1.06118	+	0.443704i
$638$		−	10.0479i		−	0.397800i
$639$		0			0
$640$		−	2.79965i		−	0.110666i
$641$		−	10.5745i		−	0.417669i	−0.977951	−	0.208835i	$-0.933033\pi$
$641$		−	10.5745i		−	0.417669i	0.977951	−	0.208835i	$-0.0669670\pi$
$642$		0			0
$643$		−	36.8220i		−	1.45212i	−0.687632	−	0.726059i	$-0.741350\pi$
$643$		−	36.8220i		−	1.45212i	0.687632	−	0.726059i	$-0.258650\pi$
$644$		0			0
$645$		0			0
$646$	−6.64726			−0.261533
$647$	37.4463			1.47217			0.736083	−	0.676891i	$-0.236673\pi$
$647$	37.4463			1.47217			0.736083	+	0.676891i	$0.236673\pi$
$648$		0			0
$649$		−	13.8936i		−	0.545370i
$650$	−11.7697			−0.461644
$651$		0			0
$652$	5.07669			0.198819
$653$			32.0624i			1.25470i	0.778738	+	0.627349i	$0.215860\pi$
$653$			32.0624i			1.25470i	−0.778738	+	0.627349i	$0.784140\pi$
$654$		0			0
$655$	−45.7622			−1.78808
$656$	−2.37432			−0.0927016
$657$		0			0
$658$	−14.2099	−	21.3435i	−0.553958	−	0.832055i
$659$			27.4479i			1.06922i	0.845099	+	0.534609i	$0.179541\pi$
$659$			27.4479i			1.06922i	−0.845099	+	0.534609i	$0.820459\pi$
$660$		0			0
$661$			44.5143i			1.73140i	0.500560	+	0.865702i	$0.333127\pi$
$661$			44.5143i			1.73140i	−0.500560	+	0.865702i	$0.666873\pi$
$662$			16.5332i			0.642580i
$663$		0			0
$664$			7.55994i			0.293383i
$665$	4.76030	+	7.15005i	0.184596	+	0.277267i
$666$		0			0
$667$		0			0
$668$	−20.0944			−0.777474
$669$		0			0
$670$			33.7299i			1.30310i
$671$	1.71780			0.0663149
$672$		0			0
$673$	−35.6185			−1.37299			−0.686495	−	0.727134i	$-0.740852\pi$
$673$	−35.6185			−1.37299			−0.686495	+	0.727134i	$0.740852\pi$
$674$			9.61051i			0.370183i
$675$		0			0
$676$	4.19872			0.161489
$677$	−42.4271			−1.63061			−0.815303	−	0.579035i	$-0.803429\pi$
$677$	−42.4271			−1.63061			−0.815303	+	0.579035i	$0.803429\pi$
$678$		0			0
$679$	−11.6105	−	17.4392i	−0.445571	−	0.669255i
$680$			16.0479i			0.615409i
$681$		0			0
$682$			1.77281i			0.0678846i
$683$			32.4676i			1.24234i	0.783676	+	0.621170i	$0.213342\pi$
$683$			32.4676i			1.24234i	−0.783676	+	0.621170i	$0.786658\pi$
$684$		0			0
$685$		−	9.74643i		−	0.372392i
$686$	−18.1822	+	3.52228i	−0.694201	+	0.134481i
$687$		0			0
$688$	−7.00398			−0.267024
$689$	−48.9580			−1.86515
$690$		0			0
$691$		−	25.4678i		−	0.968842i	−0.874835	−	0.484421i	$-0.839030\pi$
$691$		−	25.4678i		−	0.968842i	0.874835	−	0.484421i	$-0.160970\pi$
$692$	−7.31709			−0.278154
$693$		0			0
$694$	−1.40859			−0.0534694
$695$		−	26.6855i		−	1.01224i
$696$		0			0
$697$	13.6099			0.515511
$698$	32.2992			1.22254
$699$		0			0
$700$	6.25020	−	4.16121i	0.236236	−	0.157279i
$701$		−	39.6185i		−	1.49637i	−0.663491	−	0.748184i	$-0.730926\pi$
$701$		−	39.6185i		−	1.49637i	0.663491	−	0.748184i	$-0.269074\pi$
$702$		0			0
$703$		−	1.62426i		−	0.0612599i
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$		−	18.6994i		−	0.703763i
$707$	8.30593	−	5.52985i	0.312376	−	0.207971i
$708$		0			0
$709$	−25.2210			−0.947195			−0.473598	−	0.880741i	$-0.657045\pi$
$709$	−25.2210			−0.947195			−0.473598	+	0.880741i	$0.657045\pi$
$710$	−23.7558			−0.891539
$711$		0			0
$712$			2.33096i			0.0873565i
$713$		0			0
$714$		0			0
$715$	−11.6105			−0.434209
$716$			7.72395i			0.288657i
$717$		0			0
$718$	−7.44854			−0.277977
$719$	36.9383			1.37757			0.688784	−	0.724967i	$-0.258145\pi$
$719$	36.9383			1.37757			0.688784	+	0.724967i	$0.258145\pi$
$720$		0			0
$721$	26.4853	+	39.7814i	0.986363	+	1.48154i
$722$			17.6552i			0.657059i
$723$		0			0
$724$		−	3.49061i		−	0.129728i
$725$			28.5162i			1.05906i
$726$		0			0
$727$		−	8.33280i		−	0.309046i	−0.987989	−	0.154523i	$-0.950616\pi$
$727$		−	8.33280i		−	0.309046i	0.987989	−	0.154523i	$-0.0493841\pi$
$728$	−9.13327	+	6.08067i	−0.338501	+	0.225364i
$729$		0			0
$730$	−28.8165			−1.06655
$731$	40.1476			1.48491
$732$		0			0
$733$		−	42.6927i		−	1.57689i	−0.615104	−	0.788446i	$-0.710886\pi$
$733$		−	42.6927i		−	1.57689i	0.615104	−	0.788446i	$-0.289114\pi$
$734$	−18.3291			−0.676540
$735$		0			0
$736$		0			0
$737$			12.0479i			0.443790i
$738$		0			0
$739$	−6.45187			−0.237336			−0.118668	−	0.992934i	$-0.537862\pi$
$739$	−6.45187			−0.237336			−0.118668	+	0.992934i	$0.537862\pi$
$740$	−3.92130			−0.144150
$741$		0			0
$742$	25.9988	−	17.3093i	0.954446	−	0.635443i
$743$			27.0112i			0.990943i	0.868624	+	0.495472i	$0.165005\pi$
$743$			27.0112i			0.990943i	−0.868624	+	0.495472i	$0.834995\pi$
$744$		0			0
$745$		−	9.52059i		−	0.348808i
$746$			13.5540i			0.496248i
$747$		0			0
$748$			5.73211i			0.209587i
$749$	12.0376	+	18.0807i	0.439844	+	0.660653i
$750$		0			0
$751$	−36.0303			−1.31476			−0.657381	−	0.753558i	$-0.728336\pi$
$751$	−36.0303			−1.31476			−0.657381	+	0.753558i	$0.728336\pi$
$752$	9.69141			0.353409
$753$		0			0
$754$		−	41.6700i		−	1.51753i
$755$	−47.8511			−1.74148
$756$		0			0
$757$	−16.4288			−0.597115			−0.298557	−	0.954392i	$-0.596505\pi$
$757$	−16.4288			−0.597115			−0.298557	+	0.954392i	$0.596505\pi$
$758$		−	9.07669i		−	0.329680i
$759$		0			0
$760$	−3.24662			−0.117767
$761$	−13.4091			−0.486081			−0.243040	−	0.970016i	$-0.578145\pi$
$761$	−13.4091			−0.486081			−0.243040	+	0.970016i	$0.578145\pi$
$762$		0			0
$763$	−39.0336	+	25.9874i	−1.41311	+	0.940808i
$764$		−	8.48528i		−	0.306987i
$765$		0			0
$766$		−	13.9866i		−	0.505356i
$767$		−	57.6185i		−	2.08048i
$768$		0			0
$769$			6.16227i			0.222217i	0.993808	+	0.111109i	$0.0354401\pi$
$769$			6.16227i			0.222217i	−0.993808	+	0.111109i	$0.964560\pi$
$770$	6.16569	−	4.10494i	0.222196	−	0.147932i
$771$		0			0
$772$	−18.0958			−0.651282
$773$	−27.7603			−0.998467			−0.499233	−	0.866468i	$-0.666385\pi$
$773$	−27.7603			−0.998467			−0.499233	+	0.866468i	$0.666385\pi$
$774$		0			0
$775$		−	5.03129i		−	0.180729i
$776$	7.91860			0.284261
$777$		0			0
$778$	−20.1292			−0.721667
$779$			2.75338i			0.0986502i
$780$		0			0
$781$	−8.48528			−0.303627
$782$		0			0
$783$		0			0
$784$	2.70032	−	6.45819i	0.0964400	−	0.230650i
$785$			4.96326i			0.177146i
$786$		0			0
$787$		−	7.49957i		−	0.267331i	−0.991027	−	0.133665i	$-0.957325\pi$
$787$		−	7.49957i		−	0.267331i	0.991027	−	0.133665i	$-0.0426747\pi$
$788$		−	23.3424i		−	0.831539i
$789$		0			0
$790$			4.01304i			0.142778i
$791$	−8.91567	−	13.3915i	−0.317005	−	0.476147i
$792$		0			0
$793$	7.12394			0.252979
$794$	−22.3206			−0.792130
$795$		0			0
$796$		−	9.95705i		−	0.352918i
$797$	−45.9168			−1.62645			−0.813227	−	0.581946i	$-0.802292\pi$
$797$	−45.9168			−1.62645			−0.813227	+	0.581946i	$0.802292\pi$
$798$		0			0
$799$	−55.5523			−1.96530
$800$			2.83802i			0.100339i
$801$		0			0
$802$	−8.68003			−0.306502
$803$	−10.2929			−0.363229
$804$		0			0
$805$		0			0
$806$			7.35210i			0.258967i
$807$		0			0
$808$			3.77146i			0.132680i
$809$			38.0958i			1.33938i	0.742642	+	0.669688i	$0.233572\pi$
$809$			38.0958i			1.33938i	−0.742642	+	0.669688i	$0.766428\pi$
$810$		0			0
$811$			19.7929i			0.695024i	0.937676	+	0.347512i	$0.112973\pi$
$811$			19.7929i			0.695024i	−0.937676	+	0.347512i	$0.887027\pi$
$812$	14.7326	+	22.1286i	0.517012	+	0.776560i
$813$		0			0
$814$	−1.40064			−0.0490924
$815$	−14.2129			−0.497858
$816$		0			0
$817$			8.12217i			0.284159i
$818$	22.9512			0.802471
$819$		0			0
$820$	6.64726			0.232132
$821$			7.99205i			0.278924i	0.990227	+	0.139462i	$0.0445374\pi$
$821$			7.99205i			0.278924i	−0.990227	+	0.139462i	$0.955463\pi$
$822$		0			0
$823$	−17.6026			−0.613587			−0.306793	−	0.951776i	$-0.599256\pi$
$823$	−17.6026			−0.613587			−0.306793	+	0.951776i	$0.599256\pi$
$824$	−18.0635			−0.629271
$825$		0			0
$826$	20.3712	+	30.5979i	0.708805	+	1.06464i
$827$			23.5779i			0.819883i	0.912112	+	0.409941i	$0.134451\pi$
$827$			23.5779i			0.819883i	−0.912112	+	0.409941i	$0.865549\pi$
$828$		0			0
$829$			40.5385i			1.40796i	0.710220	+	0.703980i	$0.248595\pi$
$829$			40.5385i			1.40796i	−0.710220	+	0.703980i	$0.751405\pi$
$830$		−	21.1652i		−	0.734654i
$831$		0			0
$832$		−	4.14713i		−	0.143776i
$833$	−15.4785	+	37.0191i	−0.536300	+	1.28264i
$834$		0			0
$835$	56.2571			1.94686
$836$	−1.15965			−0.0401074
$837$		0			0
$838$			15.0099i			0.518507i
$839$	−10.5510			−0.364260			−0.182130	−	0.983274i	$-0.558299\pi$
$839$	−10.5510			−0.364260			−0.182130	+	0.983274i	$0.558299\pi$
$840$		0			0
$841$	−71.9602			−2.48139
$842$		−	13.4006i		−	0.461817i
$843$		0			0
$844$	−6.29054			−0.216529
$845$	−11.7549			−0.404382
$846$		0			0
$847$	2.20231	−	1.46623i	0.0756722	−	0.0503804i
$848$			11.8053i			0.405394i
$849$		0			0
$850$		−	16.2679i		−	0.557984i
$851$		0			0
$852$		0			0
$853$			43.9835i			1.50597i	0.658040	+	0.752983i	$0.271386\pi$
$853$			43.9835i			1.50597i	−0.658040	+	0.752983i	$0.728614\pi$
$854$	−3.78312	+	2.51869i	−0.129456	+	0.0861879i
$855$		0			0
$856$	−8.20987			−0.280608
$857$	14.2920			0.488206			0.244103	−	0.969749i	$-0.421506\pi$
$857$	14.2920			0.488206			0.244103	+	0.969749i	$0.421506\pi$
$858$		0			0
$859$		−	43.9893i		−	1.50089i	−0.660930	−	0.750447i	$-0.729838\pi$
$859$		−	43.9893i		−	1.50089i	0.660930	−	0.750447i	$-0.270162\pi$
$860$	19.6087			0.668650
$861$		0			0
$862$	28.2092			0.960810
$863$		−	2.81718i		−	0.0958980i	−0.998850	−	0.0479490i	$-0.984732\pi$
$863$		−	2.81718i		−	0.0958980i	0.998850	−	0.0479490i	$-0.0152685\pi$
$864$		0			0
$865$	20.4853			0.696520
$866$	9.48079			0.322170
$867$		0			0
$868$	−2.59936	−	3.90428i	−0.0882280	−	0.132520i
$869$			1.43341i			0.0486251i
$870$		0			0
$871$			49.9642i			1.69297i
$872$		−	17.7239i		−	0.600209i
$873$		0			0
$874$		0			0
$875$	13.3301	−	8.87477i	0.450638	−	0.300022i
$876$		0			0
$877$	39.8277			1.34489			0.672443	−	0.740149i	$-0.265245\pi$
$877$	39.8277			1.34489			0.672443	+	0.740149i	$0.265245\pi$
$878$	−28.0246			−0.945785
$879$		0			0
$880$			2.79965i			0.0943761i
$881$	−10.4589			−0.352370			−0.176185	−	0.984357i	$-0.556376\pi$
$881$	−10.4589			−0.352370			−0.176185	+	0.984357i	$0.556376\pi$
$882$		0			0
$883$	55.2762			1.86019			0.930097	−	0.367315i	$-0.119723\pi$
$883$	55.2762			1.86019			0.930097	+	0.367315i	$0.119723\pi$
$884$			23.7718i			0.799534i
$885$		0			0
$886$	−4.43738			−0.149077
$887$	−32.5813			−1.09397			−0.546987	−	0.837141i	$-0.684225\pi$
$887$	−32.5813			−1.09397			−0.546987	+	0.837141i	$0.684225\pi$
$888$		0			0
$889$	−24.3396	+	16.2046i	−0.816325	+	0.543486i
$890$		−	6.52587i		−	0.218748i
$891$		0			0
$892$			7.74778i			0.259415i
$893$		−	11.2387i		−	0.376088i
$894$		0			0
$895$		−	21.6243i		−	0.722821i
$896$	1.46623	+	2.20231i	0.0489834	+	0.0735739i
$897$		0			0
$898$	29.5298			0.985424
$899$	17.8131			0.594099
$900$		0			0
$901$		−	67.6691i		−	2.25438i
$902$	2.37432			0.0790562
$903$		0			0
$904$	6.08067			0.202240
$905$			9.77249i			0.324848i
$906$		0			0
$907$	7.88657			0.261869			0.130935	−	0.991391i	$-0.458202\pi$
$907$	7.88657			0.261869			0.130935	+	0.991391i	$0.458202\pi$
$908$	−14.7057			−0.488025
$909$		0			0
$910$	25.5699	−	17.0237i	0.847635	−	0.564331i
$911$		−	19.0756i		−	0.632003i	−0.948759	−	0.316001i	$-0.897660\pi$
$911$		−	19.0756i		−	0.632003i	0.948759	−	0.316001i	$-0.102340\pi$
$912$		0			0
$913$		−	7.55994i		−	0.250198i
$914$			13.2866i			0.439480i
$915$		0			0
$916$			12.0429i			0.397910i
$917$	35.9982	−	23.9666i	1.18877	−	0.791446i
$918$		0			0
$919$	18.0624			0.595823			0.297911	−	0.954593i	$-0.403710\pi$
$919$	18.0624			0.595823			0.297911	+	0.954593i	$0.403710\pi$
$920$		0			0
$921$		0			0
$922$		−	40.7035i		−	1.34050i
$923$	−35.1896			−1.15828
$924$		0			0
$925$	3.97505			0.130699
$926$		−	20.9706i		−	0.689135i
$927$		0			0
$928$	−10.0479			−0.329838
$929$	25.8667			0.848659			0.424329	−	0.905508i	$-0.360510\pi$
$929$	25.8667			0.848659			0.424329	+	0.905508i	$0.360510\pi$
$930$		0			0
$931$	−7.48926	−	3.13143i	−0.245450	−	0.102629i
$932$		−	7.13318i		−	0.233655i
$933$		0			0
$934$		−	4.13065i		−	0.135159i
$935$		−	16.0479i		−	0.524822i
$936$		0			0
$937$			36.2920i			1.18561i	0.805347	+	0.592804i	$0.201979\pi$
$937$			36.2920i			1.18561i	−0.805347	+	0.592804i	$0.798021\pi$
$938$	−17.6650	−	26.5332i	−0.576784	−	0.866339i
$939$		0			0
$940$	−27.1325			−0.884966
$941$	−33.6811			−1.09797			−0.548987	−	0.835831i	$-0.684986\pi$
$941$	−33.6811			−1.09797			−0.548987	+	0.835831i	$0.684986\pi$
$942$		0			0
$943$		0			0
$944$	−13.8936			−0.452197
$945$		0			0
$946$	7.00398			0.227719
$947$			25.3510i			0.823797i	0.911230	+	0.411898i	$0.135134\pi$
$947$			25.3510i			0.823797i	−0.911230	+	0.411898i	$0.864866\pi$
$948$		0			0
$949$	−42.6861			−1.38565
$950$	3.29112			0.106778
$951$		0			0
$952$	−8.40462	−	12.6239i	−0.272395	−	0.409142i
$953$			22.1916i			0.718856i	0.933173	+	0.359428i	$0.117028\pi$
$953$			22.1916i			0.718856i	−0.933173	+	0.359428i	$0.882972\pi$
$954$		0			0
$955$			23.7558i			0.768719i
$956$		−	1.07669i		−	0.0348226i
$957$		0			0
$958$			13.7670i			0.444793i
$959$	5.10441	+	7.66691i	0.164830	+	0.247577i
$960$		0			0
$961$	27.8571			0.898617
$962$	−5.80864			−0.187278
$963$		0			0
$964$			22.6856i			0.730654i
$965$	50.6618			1.63086
$966$		0			0
$967$	16.9554			0.545250			0.272625	−	0.962120i	$-0.412108\pi$
$967$	16.9554			0.545250			0.272625	+	0.962120i	$0.412108\pi$
$968$			1.00000i			0.0321412i
$969$		0			0
$970$	−22.1693			−0.711813
$971$	21.0152			0.674411			0.337206	−	0.941431i	$-0.390518\pi$
$971$	21.0152			0.674411			0.337206	+	0.941431i	$0.390518\pi$
$972$		0			0
$973$	13.9757	+	20.9918i	0.448041	+	0.672966i
$974$			26.7516i			0.857178i
$975$		0			0
$976$		−	1.71780i		−	0.0549854i
$977$		−	33.1318i		−	1.05998i	−0.848004	−	0.529990i	$-0.822196\pi$
$977$		−	33.1318i		−	1.05998i	0.848004	−	0.529990i	$-0.177804\pi$
$978$		0			0
$979$		−	2.33096i		−	0.0744979i
$980$	−7.55994	+	18.0807i	−0.241494	+	0.577566i
$981$		0			0
$982$	−15.5620			−0.496602
$983$	5.13952			0.163925			0.0819626	−	0.996635i	$-0.473881\pi$
$983$	5.13952			0.163925			0.0819626	+	0.996635i	$0.473881\pi$
$984$		0			0
$985$			65.3505i			2.08224i
$986$	57.5957			1.83422
$987$		0			0
$988$	−4.80923			−0.153002
$989$		0			0
$990$		0			0
$991$	−26.6546			−0.846710			−0.423355	−	0.905964i	$-0.639148\pi$
$991$	−26.6546			−0.846710			−0.423355	+	0.905964i	$0.639148\pi$
$992$	1.77281			0.0562869
$993$		0			0
$994$	18.6872	−	12.4414i	0.592722	−	0.394617i
$995$			27.8762i			0.883736i
$996$		0			0
$997$			35.6122i			1.12785i	0.825826	+	0.563925i	$0.190709\pi$
$997$			35.6122i			1.12785i	−0.825826	+	0.563925i	$0.809291\pi$
$998$		−	36.0479i		−	1.14108i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.g.b.881.14	yes	16
3.2	odd	2	inner	1386.2.g.b.881.3	✓	16
7.6	odd	2	inner	1386.2.g.b.881.11	yes	16
21.20	even	2	inner	1386.2.g.b.881.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.g.b.881.3	✓	16	3.2	odd	2	inner
1386.2.g.b.881.6	yes	16	21.20	even	2	inner
1386.2.g.b.881.11	yes	16	7.6	odd	2	inner
1386.2.g.b.881.14	yes	16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.79965 q^{5} +(-2.20231 + 1.46623i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.79965 q^{5} +(-2.20231 + 1.46623i) q^{7} -1.00000i q^{8} +2.79965i q^{10} +1.00000i q^{11} +4.14713i q^{13} +(-1.46623 - 2.20231i) q^{14} +1.00000 q^{16} -5.73211 q^{17} -1.15965i q^{19} -2.79965 q^{20} -1.00000 q^{22} +2.83802 q^{25} -4.14713 q^{26} +(2.20231 - 1.46623i) q^{28} +10.0479i q^{29} -1.77281i q^{31} +1.00000i q^{32} -5.73211i q^{34} +(-6.16569 + 4.10494i) q^{35} +1.40064 q^{37} +1.15965 q^{38} -2.79965i q^{40} -2.37432 q^{41} -7.00398 q^{43} -1.00000i q^{44} +9.69141 q^{47} +(2.70032 - 6.45819i) q^{49} +2.83802i q^{50} -4.14713i q^{52} +11.8053i q^{53} +2.79965i q^{55} +(1.46623 + 2.20231i) q^{56} -10.0479 q^{58} -13.8936 q^{59} -1.71780i q^{61} +1.77281 q^{62} -1.00000 q^{64} +11.6105i q^{65} +12.0479 q^{67} +5.73211 q^{68} +(-4.10494 - 6.16569i) q^{70} +8.48528i q^{71} +10.2929i q^{73} +1.40064i q^{74} +1.15965i q^{76} +(-1.46623 - 2.20231i) q^{77} +1.43341 q^{79} +2.79965 q^{80} -2.37432i q^{82} -7.55994 q^{83} -16.0479 q^{85} -7.00398i q^{86} +1.00000 q^{88} -2.33096 q^{89} +(-6.08067 - 9.13327i) q^{91} +9.69141i q^{94} -3.24662i q^{95} +7.91860i q^{97} +(6.45819 + 2.70032i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{22} + 48 q^{25} - 16 q^{37} - 80 q^{43} + 24 q^{49} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 24 q^{70} + 96 q^{79} - 80 q^{85} + 16 q^{88} - 32 q^{91}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^22 + 48 * q^25 - 16 * q^37 - 80 * q^43 + 24 * q^49 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 24 * q^70 + 96 * q^79 - 80 * q^85 + 16 * q^88 - 32 * q^91

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