LMFDB - Embedded newform 1386.2.g.a.881.1

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			881.1
Root			$-0.876932 - 2.11710i$ of defining polynomial
Character	$\chi$	$=$	1386.881
Dual form			1386.2.g.a.881.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -4.23420 q^{5} +(-2.59178 + 0.531652i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -4.23420 q^{5} +(-2.59178 + 0.531652i) q^{7} +1.00000i q^{8} +4.23420i q^{10} +1.00000i q^{11} +(0.531652 + 2.59178i) q^{14} +1.00000 q^{16} -5.29751 q^{17} -0.250123i q^{19} +4.23420 q^{20} +1.00000 q^{22} -4.50225i q^{23} +12.9285 q^{25} +(2.59178 - 0.531652i) q^{28} -1.05907i q^{29} +6.32489i q^{31} -1.00000i q^{32} +5.29751i q^{34} +(10.9741 - 2.25112i) q^{35} +2.50225 q^{37} -0.250123 q^{38} -4.23420i q^{40} -1.31343 q^{41} +7.18357 q^{43} -1.00000i q^{44} -4.50225 q^{46} +7.15497 q^{47} +(6.43469 - 2.75586i) q^{49} -12.9285i q^{50} +4.68132i q^{53} -4.23420i q^{55} +(-0.531652 - 2.59178i) q^{56} -1.05907 q^{58} -8.46840 q^{59} -14.8102i q^{61} +6.32489 q^{62} -1.00000 q^{64} +9.42621 q^{67} +5.29751 q^{68} +(-2.25112 - 10.9741i) q^{70} -12.9875i q^{71} +7.15497i q^{73} -2.50225i q^{74} +0.250123i q^{76} +(-0.531652 - 2.59178i) q^{77} -0.377749 q^{79} -4.23420 q^{80} +1.31343i q^{82} +1.84057 q^{83} +22.4307 q^{85} -7.18357i q^{86} -1.00000 q^{88} -1.56355 q^{89} +4.50225i q^{92} -7.15497i q^{94} +1.05907i q^{95} +7.96815i q^{97} +(-2.75586 - 6.43469i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 8 q^{7}+O(q^{10}) $ 16 * q - 16 * q^4 - 8 * q^7	$ 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

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
$5$	−4.23420			−1.89359			−0.946796	−	0.321834i	$-0.895701\pi$
$5$	−4.23420			−1.89359			−0.946796	+	0.321834i	$0.895701\pi$
$6$		0			0
$7$	−2.59178	+	0.531652i	−0.979602	+	0.200946i
$7$	−2.59178	+	0.531652i	−0.979602	+	0.200946i
$8$			1.00000i			0.353553i
$9$		0			0
$10$			4.23420i			1.33897i
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	0.531652	+	2.59178i	0.142090	+	0.692683i
$15$		0			0
$16$	1.00000			0.250000
$17$	−5.29751			−1.28483			−0.642417	−	0.766355i	$-0.722068\pi$
$17$	−5.29751			−1.28483			−0.642417	+	0.766355i	$0.722068\pi$
$18$		0			0
$19$		−	0.250123i		−	0.0573822i	−0.999588	−	0.0286911i	$-0.990866\pi$
$19$		−	0.250123i		−	0.0573822i	0.999588	−	0.0286911i	$-0.00913392\pi$
$20$	4.23420			0.946796
$21$		0			0
$22$	1.00000			0.213201
$23$		−	4.50225i		−	0.938783i	−0.882990	−	0.469392i	$-0.844473\pi$
$23$		−	4.50225i		−	0.938783i	0.882990	−	0.469392i	$-0.155527\pi$
$24$		0			0
$25$	12.9285			2.58569
$26$		0			0
$27$		0			0
$28$	2.59178	−	0.531652i	0.489801	−	0.100473i
$29$		−	1.05907i		−	0.196665i	−0.995154	−	0.0983324i	$-0.968649\pi$
$29$		−	1.05907i		−	0.196665i	0.995154	−	0.0983324i	$-0.0313508\pi$
$30$		0			0
$31$			6.32489i			1.13598i	0.823034	+	0.567992i	$0.192280\pi$
$31$			6.32489i			1.13598i	−0.823034	+	0.567992i	$0.807720\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			5.29751i			0.908515i
$35$	10.9741	−	2.25112i	1.85497	−	0.380509i
$36$		0			0
$37$	2.50225			0.411367			0.205683	−	0.978619i	$-0.434058\pi$
$37$	2.50225			0.411367			0.205683	+	0.978619i	$0.434058\pi$
$38$	−0.250123			−0.0405754
$39$		0			0
$40$		−	4.23420i		−	0.669486i
$41$	−1.31343			−0.205123			−0.102562	−	0.994727i	$-0.532704\pi$
$41$	−1.31343			−0.205123			−0.102562	+	0.994727i	$0.532704\pi$
$42$		0			0
$43$	7.18357			1.09548			0.547742	−	0.836647i	$-0.315488\pi$
$43$	7.18357			1.09548			0.547742	+	0.836647i	$0.315488\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$	−4.50225			−0.663820
$47$	7.15497			1.04366			0.521830	−	0.853049i	$-0.325250\pi$
$47$	7.15497			1.04366			0.521830	+	0.853049i	$0.325250\pi$
$48$		0			0
$49$	6.43469	−	2.75586i	0.919242	−	0.393694i
$50$		−	12.9285i		−	1.82836i
$51$		0			0
$52$		0			0
$53$			4.68132i			0.643029i	0.946905	+	0.321515i	$0.104192\pi$
$53$			4.68132i			0.643029i	−0.946905	+	0.321515i	$0.895808\pi$
$54$		0			0
$55$		−	4.23420i		−	0.570939i
$56$	−0.531652	−	2.59178i	−0.0710450	−	0.346342i
$57$		0			0
$58$	−1.05907			−0.139063
$59$	−8.46840			−1.10249			−0.551246	−	0.834343i	$-0.685847\pi$
$59$	−8.46840			−1.10249			−0.551246	+	0.834343i	$0.685847\pi$
$60$		0			0
$61$		−	14.8102i		−	1.89625i	−0.317896	−	0.948125i	$-0.602976\pi$
$61$		−	14.8102i		−	1.89625i	0.317896	−	0.948125i	$-0.397024\pi$
$62$	6.32489			0.803262
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	9.42621			1.15159			0.575797	−	0.817592i	$-0.304692\pi$
$67$	9.42621			1.15159			0.575797	+	0.817592i	$0.304692\pi$
$68$	5.29751			0.642417
$69$		0			0
$70$	−2.25112	−	10.9741i	−0.269061	−	1.31166i
$71$		−	12.9875i		−	1.54134i	−0.637237	−	0.770668i	$-0.719923\pi$
$71$		−	12.9875i		−	1.54134i	0.637237	−	0.770668i	$-0.280077\pi$
$72$		0			0
$73$			7.15497i			0.837426i	0.908119	+	0.418713i	$0.137519\pi$
$73$			7.15497i			0.837426i	−0.908119	+	0.418713i	$0.862481\pi$
$74$		−	2.50225i		−	0.290880i
$75$		0			0
$76$			0.250123i			0.0286911i
$77$	−0.531652	−	2.59178i	−0.0605874	−	0.295361i
$78$		0			0
$79$	−0.377749			−0.0425001			−0.0212501	−	0.999774i	$-0.506765\pi$
$79$	−0.377749			−0.0425001			−0.0212501	+	0.999774i	$0.506765\pi$
$80$	−4.23420			−0.473398
$81$		0			0
$82$			1.31343i			0.145044i
$83$	1.84057			0.202029			0.101014	−	0.994885i	$-0.467791\pi$
$83$	1.84057			0.202029			0.101014	+	0.994885i	$0.467791\pi$
$84$		0			0
$85$	22.4307			2.43295
$86$		−	7.18357i		−	0.774624i
$87$		0			0
$88$	−1.00000			−0.106600
$89$	−1.56355			−0.165736			−0.0828681	−	0.996561i	$-0.526408\pi$
$89$	−1.56355			−0.165736			−0.0828681	+	0.996561i	$0.526408\pi$
$90$		0			0
$91$		0			0
$92$			4.50225i			0.469392i
$93$		0			0
$94$		−	7.15497i		−	0.737979i
$95$			1.05907i			0.108659i
$96$		0			0
$97$			7.96815i			0.809044i	0.914528	+	0.404522i	$0.132562\pi$
$97$			7.96815i			0.809044i	−0.914528	+	0.404522i	$0.867438\pi$
$98$	−2.75586	−	6.43469i	−0.278384	−	0.650002i
$99$		0			0
$100$	−12.9285			−1.29285
$101$	15.6065			1.55290			0.776451	−	0.630177i	$-0.217018\pi$
$101$	15.6065			1.55290			0.776451	+	0.630177i	$0.217018\pi$
$102$		0			0
$103$		−	1.84057i		−	0.181357i	−0.995880	−	0.0906784i	$-0.971096\pi$
$103$		−	1.84057i		−	0.181357i	0.995880	−	0.0906784i	$-0.0289035\pi$
$104$		0			0
$105$		0			0
$106$	4.68132			0.454690
$107$			17.2365i			1.66632i	0.553034	+	0.833159i	$0.313470\pi$
$107$			17.2365i			1.66632i	−0.553034	+	0.833159i	$0.686530\pi$
$108$		0			0
$109$	−17.8103			−1.70592			−0.852959	−	0.521978i	$-0.825194\pi$
$109$	−17.8103			−1.70592			−0.852959	+	0.521978i	$0.825194\pi$
$110$	−4.23420			−0.403715
$111$		0			0
$112$	−2.59178	+	0.531652i	−0.244901	+	0.0502364i
$113$		−	9.80396i		−	0.922279i	−0.887328	−	0.461139i	$-0.847441\pi$
$113$		−	9.80396i		−	0.922279i	0.887328	−	0.461139i	$-0.152559\pi$
$114$		0			0
$115$			19.0634i			1.77767i
$116$			1.05907i			0.0983324i
$117$		0			0
$118$			8.46840i			0.779580i
$119$	13.7300	−	2.81643i	1.25863	−	0.258182i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	−14.8102			−1.34085
$123$		0			0
$124$		−	6.32489i		−	0.567992i
$125$	−33.5707			−3.00265
$126$		0			0
$127$	−13.1120			−1.16350			−0.581752	−	0.813366i	$-0.697633\pi$
$127$	−13.1120			−1.16350			−0.581752	+	0.813366i	$0.697633\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$	16.3928			1.43224			0.716121	−	0.697976i	$-0.245916\pi$
$131$	16.3928			1.43224			0.716121	+	0.697976i	$0.245916\pi$
$132$		0			0
$133$	0.132979	+	0.648266i	0.0115307	+	0.0562118i
$134$		−	9.42621i		−	0.814300i
$135$		0			0
$136$		−	5.29751i		−	0.454257i
$137$			17.6688i			1.50955i	0.655983	+	0.754776i	$0.272254\pi$
$137$			17.6688i			1.50955i	−0.655983	+	0.754776i	$0.727746\pi$
$138$		0			0
$139$			16.8840i			1.43208i	0.698059	+	0.716041i	$0.254048\pi$
$139$			16.8840i			1.43208i	−0.698059	+	0.716041i	$0.745952\pi$
$140$	−10.9741	+	2.25112i	−0.927484	+	0.190255i
$141$		0			0
$142$	−12.9875			−1.08989
$143$		0			0
$144$		0			0
$145$			4.48432i			0.372403i
$146$	7.15497			0.592150
$147$		0			0
$148$	−2.50225			−0.205683
$149$		−	10.8694i		−	0.890455i	−0.895418	−	0.445227i	$-0.853123\pi$
$149$		−	10.8694i		−	0.890455i	0.895418	−	0.445227i	$-0.146877\pi$
$150$		0			0
$151$	8.17110			0.664954			0.332477	−	0.943111i	$-0.392116\pi$
$151$	8.17110			0.664954			0.332477	+	0.943111i	$0.392116\pi$
$152$	0.250123			0.0202877
$153$		0			0
$154$	−2.59178	+	0.531652i	−0.208852	+	0.0428418i
$155$		−	26.7809i		−	2.15109i
$156$		0			0
$157$			20.6348i			1.64684i	0.567433	+	0.823420i	$0.307937\pi$
$157$			20.6348i			1.64684i	−0.567433	+	0.823420i	$0.692063\pi$
$158$			0.377749i			0.0300521i
$159$		0			0
$160$			4.23420i			0.334743i
$161$	2.39363	+	11.6688i	0.188644	+	0.919634i
$162$		0			0
$163$	8.62039			0.675201			0.337601	−	0.941289i	$-0.390385\pi$
$163$	8.62039			0.675201			0.337601	+	0.941289i	$0.390385\pi$
$164$	1.31343			0.102562
$165$		0			0
$166$		−	1.84057i		−	0.142856i
$167$	−2.88486			−0.223237			−0.111618	−	0.993751i	$-0.535603\pi$
$167$	−2.88486			−0.223237			−0.111618	+	0.993751i	$0.535603\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		−	22.4307i		−	1.72036i
$171$		0			0
$172$	−7.18357			−0.547742
$173$	11.8535			0.901205			0.450603	−	0.892725i	$-0.351209\pi$
$173$	11.8535			0.901205			0.450603	+	0.892725i	$0.351209\pi$
$174$		0			0
$175$	−33.5078	+	6.87344i	−2.53295	+	0.519584i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$			1.56355i			0.117193i
$179$			13.4262i			1.00352i	0.865006	+	0.501761i	$0.167314\pi$
$179$			13.4262i			1.00352i	−0.865006	+	0.501761i	$0.832686\pi$
$180$		0			0
$181$		−	19.8766i		−	1.47742i	−0.674026	−	0.738708i	$-0.735436\pi$
$181$		−	19.8766i		−	1.47742i	0.674026	−	0.738708i	$-0.264564\pi$
$182$		0			0
$183$		0			0
$184$	4.50225			0.331910
$185$	−10.5950			−0.778961
$186$		0			0
$187$		−	5.29751i		−	0.387392i
$188$	−7.15497			−0.521830
$189$		0			0
$190$	1.05907			0.0768332
$191$			0.253253i			0.0183247i	0.999958	+	0.00916237i	$0.00291651\pi$
$191$			0.253253i			0.0183247i	−0.999958	+	0.00916237i	$0.997083\pi$
$192$		0			0
$193$	12.8524			0.925137			0.462569	−	0.886583i	$-0.346928\pi$
$193$	12.8524			0.925137			0.462569	+	0.886583i	$0.346928\pi$
$194$	7.96815			0.572080
$195$		0			0
$196$	−6.43469	+	2.75586i	−0.459621	+	0.196847i
$197$			25.4352i			1.81218i	0.423082	+	0.906091i	$0.360948\pi$
$197$			25.4352i			1.81218i	−0.423082	+	0.906091i	$0.639052\pi$
$198$		0			0
$199$		−	22.7615i		−	1.61352i	−0.590882	−	0.806758i	$-0.701220\pi$
$199$		−	22.7615i		−	1.61352i	0.590882	−	0.806758i	$-0.298780\pi$
$200$			12.9285i			0.914180i
$201$		0			0
$202$		−	15.6065i		−	1.09807i
$203$	0.563058	+	2.74489i	0.0395189	+	0.192653i
$204$		0			0
$205$	5.56132			0.388419
$206$	−1.84057			−0.128239
$207$		0			0
$208$		0			0
$209$	0.250123			0.0173014
$210$		0			0
$211$	−16.6733			−1.14784			−0.573920	−	0.818911i	$-0.694578\pi$
$211$	−16.6733			−1.14784			−0.573920	+	0.818911i	$0.694578\pi$
$212$		−	4.68132i		−	0.321515i
$213$		0			0
$214$	17.2365			1.17826
$215$	−30.4167			−2.07440
$216$		0			0
$217$	−3.36265	−	16.3928i	−0.228271	−	1.11281i
$218$			17.8103i			1.20627i
$219$		0			0
$220$			4.23420i			0.285470i
$221$		0			0
$222$		0			0
$223$		−	19.0085i		−	1.27290i	−0.771317	−	0.636451i	$-0.780402\pi$
$223$		−	19.0085i		−	1.27290i	0.771317	−	0.636451i	$-0.219598\pi$
$224$	0.531652	+	2.59178i	0.0355225	+	0.173171i
$225$		0			0
$226$	−9.80396			−0.652150
$227$	11.1122			0.737540			0.368770	−	0.929521i	$-0.379779\pi$
$227$	11.1122			0.737540			0.368770	+	0.929521i	$0.379779\pi$
$228$		0			0
$229$			1.34032i			0.0885711i	0.999019	+	0.0442856i	$0.0141011\pi$
$229$			1.34032i			0.0885711i	−0.999019	+	0.0442856i	$0.985899\pi$
$230$	19.0634			1.25700
$231$		0			0
$232$	1.05907			0.0695315
$233$		−	14.4853i		−	0.948962i	−0.880266	−	0.474481i	$-0.842636\pi$
$233$		−	14.4853i		−	0.948962i	0.880266	−	0.474481i	$-0.157364\pi$
$234$		0			0
$235$	−30.2956			−1.97627
$236$	8.46840			0.551246
$237$		0			0
$238$	−2.81643	−	13.7300i	−0.182562	−	0.889983i
$239$			2.01696i			0.130467i	0.997870	+	0.0652333i	$0.0207792\pi$
$239$			2.01696i			0.130467i	−0.997870	+	0.0652333i	$0.979221\pi$
$240$		0			0
$241$			12.1395i			0.781977i	0.920396	+	0.390988i	$0.127867\pi$
$241$			12.1395i			0.781977i	−0.920396	+	0.390988i	$0.872133\pi$
$242$			1.00000i			0.0642824i
$243$		0			0
$244$			14.8102i			0.948125i
$245$	−27.2458	+	11.6688i	−1.74067	+	0.745495i
$246$		0			0
$247$		0			0
$248$	−6.32489			−0.401631
$249$		0			0
$250$			33.5707i			2.12320i
$251$	6.34179			0.400290			0.200145	−	0.979766i	$-0.435859\pi$
$251$	6.34179			0.400290			0.200145	+	0.979766i	$0.435859\pi$
$252$		0			0
$253$	4.50225			0.283054
$254$			13.1120i			0.822722i
$255$		0			0
$256$	1.00000			0.0625000
$257$	1.56355			0.0975317			0.0487658	−	0.998810i	$-0.484471\pi$
$257$	1.56355			0.0975317			0.0487658	+	0.998810i	$0.484471\pi$
$258$		0			0
$259$	−6.48528	+	1.33032i	−0.402976	+	0.0826624i
$260$		0			0
$261$		0			0
$262$		−	16.3928i		−	1.01275i
$263$			6.18171i			0.381180i	0.981670	+	0.190590i	$0.0610402\pi$
$263$			6.18171i			0.381180i	−0.981670	+	0.190590i	$0.938960\pi$
$264$		0			0
$265$		−	19.8217i		−	1.21763i
$266$	0.648266	−	0.132979i	0.0397477	−	0.00815344i
$267$		0			0
$268$	−9.42621			−0.575797
$269$	25.0832			1.52935			0.764676	−	0.644415i	$-0.222899\pi$
$269$	25.0832			1.52935			0.764676	+	0.644415i	$0.222899\pi$
$270$		0			0
$271$			3.15188i			0.191463i	0.995407	+	0.0957315i	$0.0305190\pi$
$271$			3.15188i			0.191463i	−0.995407	+	0.0957315i	$0.969481\pi$
$272$	−5.29751			−0.321208
$273$		0			0
$274$	17.6688			1.06741
$275$			12.9285i			0.779615i
$276$		0			0
$277$	1.04211			0.0626142			0.0313071	−	0.999510i	$-0.490033\pi$
$277$	1.04211			0.0626142			0.0313071	+	0.999510i	$0.490033\pi$
$278$	16.8840			1.01263
$279$		0			0
$280$	2.25112	+	10.9741i	0.134530	+	0.655830i
$281$		−	22.9706i		−	1.37031i	−0.728398	−	0.685154i	$-0.759735\pi$
$281$		−	22.9706i		−	1.37031i	0.728398	−	0.685154i	$-0.240265\pi$
$282$		0			0
$283$		−	23.7979i		−	1.41464i	−0.706896	−	0.707318i	$-0.749905\pi$
$283$		−	23.7979i		−	1.41464i	0.706896	−	0.707318i	$-0.250095\pi$
$284$			12.9875i			0.770668i
$285$		0			0
$286$		0			0
$287$	3.40412	−	0.698287i	0.200939	−	0.0412186i
$288$		0			0
$289$	11.0636			0.650798
$290$	4.48432			0.263329
$291$		0			0
$292$		−	7.15497i		−	0.418713i
$293$	13.9801			0.816727			0.408364	−	0.912819i	$-0.366100\pi$
$293$	13.9801			0.816727			0.408364	+	0.912819i	$0.366100\pi$
$294$		0			0
$295$	35.8569			2.08767
$296$			2.50225i			0.145440i
$297$		0			0
$298$	−10.8694			−0.629646
$299$		0			0
$300$		0			0
$301$	−18.6183	+	3.81916i	−1.07314	+	0.220133i
$302$		−	8.17110i		−	0.470194i
$303$		0			0
$304$		−	0.250123i		−	0.0143456i
$305$			62.7093i			3.59073i
$306$		0			0
$307$		−	33.5897i		−	1.91706i	−0.284984	−	0.958532i	$-0.591988\pi$
$307$		−	33.5897i		−	1.91706i	0.284984	−	0.958532i	$-0.408012\pi$
$308$	0.531652	+	2.59178i	0.0302937	+	0.147681i
$309$		0			0
$310$	−26.7809			−1.52105
$311$	−11.6393			−0.660004			−0.330002	−	0.943980i	$-0.607049\pi$
$311$	−11.6393			−0.660004			−0.330002	+	0.943980i	$0.607049\pi$
$312$		0			0
$313$		−	1.83057i		−	0.103470i	−0.998661	−	0.0517350i	$-0.983525\pi$
$313$		−	1.83057i		−	0.103470i	0.998661	−	0.0517350i	$-0.0164751\pi$
$314$	20.6348			1.16449
$315$		0			0
$316$	0.377749			0.0212501
$317$			29.9348i			1.68131i	0.541574	+	0.840653i	$0.317829\pi$
$317$			29.9348i			1.68131i	−0.541574	+	0.840653i	$0.682171\pi$
$318$		0			0
$319$	1.05907			0.0592967
$320$	4.23420			0.236699
$321$		0			0
$322$	11.6688	−	2.39363i	0.650280	−	0.133392i
$323$			1.32503i			0.0737266i
$324$		0			0
$325$		0			0
$326$		−	8.62039i		−	0.477439i
$327$		0			0
$328$		−	1.31343i		−	0.0725219i
$329$	−18.5441	+	3.80396i	−1.02237	+	0.209719i
$330$		0			0
$331$	−5.69643			−0.313104			−0.156552	−	0.987670i	$-0.550038\pi$
$331$	−5.69643			−0.313104			−0.156552	+	0.987670i	$0.550038\pi$
$332$	−1.84057			−0.101014
$333$		0			0
$334$			2.88486i			0.157852i
$335$	−39.9125			−2.18065
$336$		0			0
$337$	0.637355			0.0347189			0.0173595	−	0.999849i	$-0.494474\pi$
$337$	0.637355			0.0347189			0.0173595	+	0.999849i	$0.494474\pi$
$338$		−	13.0000i		−	0.707107i
$339$		0			0
$340$	−22.4307			−1.21648
$341$	−6.32489			−0.342512
$342$		0			0
$343$	−15.2122	+	10.5636i	−0.821380	+	0.570381i
$344$			7.18357i			0.387312i
$345$		0			0
$346$		−	11.8535i		−	0.637248i
$347$			21.4728i			1.15272i	0.817196	+	0.576360i	$0.195528\pi$
$347$			21.4728i			1.15272i	−0.817196	+	0.576360i	$0.804472\pi$
$348$		0			0
$349$			9.52269i			0.509738i	0.966976	+	0.254869i	$0.0820323\pi$
$349$			9.52269i			0.509738i	−0.966976	+	0.254869i	$0.917968\pi$
$350$	6.87344	+	33.5078i	0.367401	+	1.79107i
$351$		0			0
$352$	1.00000			0.0533002
$353$	32.3101			1.71969			0.859845	−	0.510555i	$-0.170560\pi$
$353$	32.3101			1.71969			0.859845	+	0.510555i	$0.170560\pi$
$354$		0			0
$355$			54.9918i			2.91866i
$356$	1.56355			0.0828681
$357$		0			0
$358$	13.4262			0.709597
$359$			3.07604i			0.162347i	0.996700	+	0.0811735i	$0.0258668\pi$
$359$			3.07604i			0.162347i	−0.996700	+	0.0811735i	$0.974133\pi$
$360$		0			0
$361$	18.9374			0.996707
$362$	−19.8766			−1.04469
$363$		0			0
$364$		0			0
$365$		−	30.2956i		−	1.58574i
$366$		0			0
$367$		−	16.6508i		−	0.869163i	−0.900633	−	0.434581i	$-0.856896\pi$
$367$		−	16.6508i		−	0.869163i	0.900633	−	0.434581i	$-0.143104\pi$
$368$		−	4.50225i		−	0.234696i
$369$		0			0
$370$			10.5950i			0.550808i
$371$	−2.48884	−	12.1330i	−0.129214	−	0.629913i
$372$		0			0
$373$	−7.37163			−0.381688			−0.190844	−	0.981620i	$-0.561123\pi$
$373$	−7.37163			−0.381688			−0.190844	+	0.981620i	$0.561123\pi$
$374$	−5.29751			−0.273927
$375$		0			0
$376$			7.15497i			0.368990i
$377$		0			0
$378$		0			0
$379$	4.62039			0.237333			0.118667	−	0.992934i	$-0.462138\pi$
$379$	4.62039			0.237333			0.118667	+	0.992934i	$0.462138\pi$
$380$		−	1.05907i		−	0.0543293i
$381$		0			0
$382$	0.253253			0.0129576
$383$	−26.9159			−1.37534			−0.687670	−	0.726023i	$-0.741366\pi$
$383$	−26.9159			−1.37534			−0.687670	+	0.726023i	$0.741366\pi$
$384$		0			0
$385$	2.25112	+	10.9741i	0.114728	+	0.559294i
$386$		−	12.8524i		−	0.654171i
$387$		0			0
$388$		−	7.96815i		−	0.404522i
$389$		−	5.20053i		−	0.263677i	−0.991271	−	0.131839i	$-0.957912\pi$
$389$		−	5.20053i		−	0.263677i	0.991271	−	0.131839i	$-0.0420881\pi$
$390$		0			0
$391$			23.8507i			1.20618i
$392$	2.75586	+	6.43469i	0.139192	+	0.325001i
$393$		0			0
$394$	25.4352			1.28141
$395$	1.59947			0.0804779
$396$		0			0
$397$			7.38608i			0.370697i	0.982673	+	0.185348i	$0.0593413\pi$
$397$			7.38608i			0.370697i	−0.982673	+	0.185348i	$0.940659\pi$
$398$	−22.7615			−1.14093
$399$		0			0
$400$	12.9285			0.646423
$401$			16.0699i			0.802493i	0.915970	+	0.401247i	$0.131423\pi$
$401$			16.0699i			0.802493i	−0.915970	+	0.401247i	$0.868577\pi$
$402$		0			0
$403$		0			0
$404$	−15.6065			−0.776451
$405$		0			0
$406$	2.74489	−	0.563058i	0.136226	−	0.0279441i
$407$			2.50225i			0.124032i
$408$		0			0
$409$		−	22.1625i		−	1.09586i	−0.836523	−	0.547932i	$-0.815415\pi$
$409$		−	22.1625i		−	1.09586i	0.836523	−	0.547932i	$-0.184585\pi$
$410$		−	5.56132i		−	0.274654i
$411$		0			0
$412$			1.84057i			0.0906784i
$413$	21.9483	−	4.50225i	1.08000	−	0.221541i
$414$		0			0
$415$	−7.79335			−0.382560
$416$		0			0
$417$		0			0
$418$		−	0.250123i		−	0.0122339i
$419$	24.9050			1.21669			0.608343	−	0.793674i	$-0.291834\pi$
$419$	24.9050			1.21669			0.608343	+	0.793674i	$0.291834\pi$
$420$		0			0
$421$	−7.47281			−0.364202			−0.182101	−	0.983280i	$-0.558290\pi$
$421$	−7.47281			−0.364202			−0.182101	+	0.983280i	$0.558290\pi$
$422$			16.6733i			0.811646i
$423$		0			0
$424$	−4.68132			−0.227345
$425$	−68.4886			−3.32218
$426$		0			0
$427$	7.87387	+	38.3848i	0.381043	+	1.85757i
$428$		−	17.2365i		−	0.833159i
$429$		0			0
$430$			30.4167i			1.46682i
$431$		−	14.6797i		−	0.707096i	−0.935416	−	0.353548i	$-0.884975\pi$
$431$		−	14.6797i		−	0.707096i	0.935416	−	0.353548i	$-0.115025\pi$
$432$		0			0
$433$			32.4714i			1.56048i	0.625481	+	0.780239i	$0.284903\pi$
$433$			32.4714i			1.56048i	−0.625481	+	0.780239i	$0.715097\pi$
$434$	−16.3928	+	3.36265i	−0.786878	+	0.161412i
$435$		0			0
$436$	17.8103			0.852959
$437$	−1.12612			−0.0538695
$438$		0			0
$439$		−	33.3643i		−	1.59239i	−0.605038	−	0.796197i	$-0.706842\pi$
$439$		−	33.3643i		−	1.59239i	0.605038	−	0.796197i	$-0.293158\pi$
$440$	4.23420			0.201858
$441$		0			0
$442$		0			0
$443$		−	27.9205i		−	1.32654i	−0.748379	−	0.663271i	$-0.769168\pi$
$443$		−	27.9205i		−	1.32654i	0.748379	−	0.663271i	$-0.230832\pi$
$444$		0			0
$445$	6.62039			0.313837
$446$	−19.0085			−0.900078
$447$		0			0
$448$	2.59178	−	0.531652i	0.122450	−	0.0251182i
$449$		−	12.4413i		−	0.587142i	−0.955937	−	0.293571i	$-0.905156\pi$
$449$		−	12.4413i		−	0.587142i	0.955937	−	0.293571i	$-0.0948437\pi$
$450$		0			0
$451$		−	1.31343i		−	0.0618469i
$452$			9.80396i			0.461139i
$453$		0			0
$454$		−	11.1122i		−	0.521519i
$455$		0			0
$456$		0			0
$457$	6.48528			0.303369			0.151684	−	0.988429i	$-0.451530\pi$
$457$	6.48528			0.303369			0.151684	+	0.988429i	$0.451530\pi$
$458$	1.34032			0.0626292
$459$		0			0
$460$		−	19.0634i		−	0.888836i
$461$	−2.38461			−0.111062			−0.0555312	−	0.998457i	$-0.517685\pi$
$461$	−2.38461			−0.111062			−0.0555312	+	0.998457i	$0.517685\pi$
$462$		0			0
$463$	8.35815			0.388436			0.194218	−	0.980958i	$-0.437783\pi$
$463$	8.35815			0.388436			0.194218	+	0.980958i	$0.437783\pi$
$464$		−	1.05907i		−	0.0491662i
$465$		0			0
$466$	−14.4853			−0.671018
$467$	25.8336			1.19544			0.597719	−	0.801706i	$-0.296074\pi$
$467$	25.8336			1.19544			0.597719	+	0.801706i	$0.296074\pi$
$468$		0			0
$469$	−24.4307	+	5.01147i	−1.12810	+	0.231408i
$470$			30.2956i			1.39743i
$471$		0			0
$472$		−	8.46840i		−	0.389790i
$473$			7.18357i			0.330301i
$474$		0			0
$475$		−	3.23371i		−	0.148373i
$476$	−13.7300	+	2.81643i	−0.629313	+	0.129091i
$477$		0			0
$478$	2.01696			0.0922538
$479$	−21.9102			−1.00110			−0.500552	−	0.865706i	$-0.666870\pi$
$479$	−21.9102			−1.00110			−0.500552	+	0.865706i	$0.666870\pi$
$480$		0			0
$481$		0			0
$482$	12.1395			0.552941
$483$		0			0
$484$	1.00000			0.0454545
$485$		−	33.7388i		−	1.53200i
$486$		0			0
$487$	−25.4898			−1.15505			−0.577526	−	0.816372i	$-0.695982\pi$
$487$	−25.4898			−1.15505			−0.577526	+	0.816372i	$0.695982\pi$
$488$	14.8102			0.670426
$489$		0			0
$490$	11.6688	+	27.2458i	0.527145	+	1.23084i
$491$			13.5067i			0.609551i	0.952424	+	0.304775i	$0.0985814\pi$
$491$			13.5067i			0.609551i	−0.952424	+	0.304775i	$0.901419\pi$
$492$		0			0
$493$			5.61044i			0.252682i
$494$		0			0
$495$		0			0
$496$			6.32489i			0.283996i
$497$	6.90485	+	33.6609i	0.309725	+	1.50990i
$498$		0			0
$499$	4.90700			0.219667			0.109834	−	0.993950i	$-0.464968\pi$
$499$	4.90700			0.219667			0.109834	+	0.993950i	$0.464968\pi$
$500$	33.5707			1.50133
$501$		0			0
$502$		−	6.34179i		−	0.283048i
$503$	−14.0519			−0.626545			−0.313273	−	0.949663i	$-0.601425\pi$
$503$	−14.0519			−0.626545			−0.313273	+	0.949663i	$0.601425\pi$
$504$		0			0
$505$	−66.0810			−2.94056
$506$		−	4.50225i		−	0.200149i
$507$		0			0
$508$	13.1120			0.581752
$509$	42.0201			1.86251			0.931253	−	0.364373i	$-0.118717\pi$
$509$	42.0201			1.86251			0.931253	+	0.364373i	$0.118717\pi$
$510$		0			0
$511$	−3.80396	−	18.5441i	−0.168277	−	0.820345i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	1.56355i		−	0.0689653i
$515$			7.79335i			0.343416i
$516$		0			0
$517$			7.15497i			0.314675i
$518$	1.33032	+	6.48528i	0.0584511	+	0.284947i
$519$		0			0
$520$		0			0
$521$	43.4053			1.90162			0.950811	−	0.309773i	$-0.100253\pi$
$521$	43.4053			1.90162			0.950811	+	0.309773i	$0.100253\pi$
$522$		0			0
$523$		−	1.60734i		−	0.0702843i	−0.999382	−	0.0351421i	$-0.988812\pi$
$523$		−	1.60734i		−	0.0702843i	0.999382	−	0.0351421i	$-0.0111884\pi$
$524$	−16.3928			−0.716121
$525$		0			0
$526$	6.18171			0.269535
$527$		−	33.5062i		−	1.45955i
$528$		0			0
$529$	2.72978			0.118686
$530$	−19.8217			−0.860998
$531$		0			0
$532$	−0.132979	−	0.648266i	−0.00576536	−	0.0281059i
$533$		0			0
$534$		0			0
$535$		−	72.9829i		−	3.15533i
$536$			9.42621i			0.407150i
$537$		0			0
$538$		−	25.0832i		−	1.08142i
$539$	2.75586	+	6.43469i	0.118703	+	0.277162i
$540$		0			0
$541$	10.9955			0.472734			0.236367	−	0.971664i	$-0.424043\pi$
$541$	10.9955			0.472734			0.236367	+	0.971664i	$0.424043\pi$
$542$	3.15188			0.135385
$543$		0			0
$544$			5.29751i			0.227129i
$545$	75.4124			3.23031
$546$		0			0
$547$	28.6733			1.22598			0.612992	−	0.790089i	$-0.289966\pi$
$547$	28.6733			1.22598			0.612992	+	0.790089i	$0.289966\pi$
$548$		−	17.6688i		−	0.754776i
$549$		0			0
$550$	12.9285			0.551271
$551$	−0.264899			−0.0112851
$552$		0			0
$553$	0.979045	−	0.200831i	0.0416332	−	0.00854022i
$554$		−	1.04211i		−	0.0442749i
$555$		0			0
$556$		−	16.8840i		−	0.716041i
$557$			25.4352i			1.07772i	0.842394	+	0.538862i	$0.181146\pi$
$557$			25.4352i			1.07772i	−0.842394	+	0.538862i	$0.818854\pi$
$558$		0			0
$559$		0			0
$560$	10.9741	−	2.25112i	0.463742	−	0.0951273i
$561$		0			0
$562$	−22.9706			−0.968955
$563$	−14.4973			−0.610987			−0.305493	−	0.952194i	$-0.598821\pi$
$563$	−14.4973			−0.610987			−0.305493	+	0.952194i	$0.598821\pi$
$564$		0			0
$565$			41.5119i			1.74642i
$566$	−23.7979			−1.00030
$567$		0			0
$568$	12.9875			0.544945
$569$		−	2.48528i		−	0.104188i	−0.998642	−	0.0520942i	$-0.983410\pi$
$569$		−	2.48528i		−	0.104188i	0.998642	−	0.0520942i	$-0.0165896\pi$
$570$		0			0
$571$	−37.9141			−1.58666			−0.793328	−	0.608794i	$-0.791654\pi$
$571$	−37.9141			−1.58666			−0.793328	+	0.608794i	$0.791654\pi$
$572$		0			0
$573$		0			0
$574$	−0.698287	−	3.40412i	−0.0291459	−	0.142085i
$575$		−	58.2071i		−	2.42740i
$576$		0			0
$577$		−	0.572077i		−	0.0238159i	−0.999929	−	0.0119079i	$-0.996209\pi$
$577$		−	0.572077i		−	0.0238159i	0.999929	−	0.0119079i	$-0.00379051\pi$
$578$		−	11.0636i		−	0.460184i
$579$		0			0
$580$		−	4.48432i		−	0.186201i
$581$	−4.77036	+	0.978544i	−0.197908	+	0.0405968i
$582$		0			0
$583$	−4.68132			−0.193881
$584$	−7.15497			−0.296075
$585$		0			0
$586$		−	13.9801i		−	0.577513i
$587$	19.0296			0.785436			0.392718	−	0.919659i	$-0.371535\pi$
$587$	19.0296			0.785436			0.392718	+	0.919659i	$0.371535\pi$
$588$		0			0
$589$	1.58200			0.0651853
$590$		−	35.8569i		−	1.47621i
$591$		0			0
$592$	2.50225			0.102842
$593$	−7.58339			−0.311412			−0.155706	−	0.987803i	$-0.549765\pi$
$593$	−7.58339			−0.311412			−0.155706	+	0.987803i	$0.549765\pi$
$594$		0			0
$595$	−58.1355	+	11.9253i	−2.38332	+	0.488891i
$596$			10.8694i			0.445227i
$597$		0			0
$598$		0			0
$599$		−	25.5067i		−	1.04218i	−0.853503	−	0.521089i	$-0.825526\pi$
$599$		−	25.5067i		−	1.04218i	0.853503	−	0.521089i	$-0.174474\pi$
$600$		0			0
$601$			32.8631i			1.34051i	0.742129	+	0.670257i	$0.233816\pi$
$601$			32.8631i			1.34051i	−0.742129	+	0.670257i	$0.766184\pi$
$602$	3.81916	+	18.6183i	0.155657	+	0.758824i
$603$		0			0
$604$	−8.17110			−0.332477
$605$	4.23420			0.172145
$606$		0			0
$607$		−	33.8646i		−	1.37452i	−0.726411	−	0.687261i	$-0.758813\pi$
$607$		−	33.8646i		−	1.37452i	0.726411	−	0.687261i	$-0.241187\pi$
$608$	−0.250123			−0.0101438
$609$		0			0
$610$	62.7093			2.53903
$611$		0			0
$612$		0			0
$613$	28.4434			1.14882			0.574408	−	0.818569i	$-0.305232\pi$
$613$	28.4434			1.14882			0.574408	+	0.818569i	$0.305232\pi$
$614$	−33.5897			−1.35557
$615$		0			0
$616$	2.59178	−	0.531652i	0.104426	−	0.0214209i
$617$			12.0064i			0.483358i	0.970356	+	0.241679i	$0.0776980\pi$
$617$			12.0064i			0.483358i	−0.970356	+	0.241679i	$0.922302\pi$
$618$		0			0
$619$		−	3.71494i		−	0.149316i	−0.997209	−	0.0746579i	$-0.976214\pi$
$619$		−	3.71494i		−	0.149316i	0.997209	−	0.0746579i	$-0.0237865\pi$
$620$			26.7809i			1.07555i
$621$		0			0
$622$			11.6393i			0.466693i
$623$	4.05239	−	0.831266i	0.162355	−	0.0333040i
$624$		0			0
$625$	77.5027			3.10011
$626$	−1.83057			−0.0731644
$627$		0			0
$628$		−	20.6348i		−	0.823420i
$629$	−13.2557			−0.528538
$630$		0			0
$631$	−26.9795			−1.07404			−0.537019	−	0.843570i	$-0.680450\pi$
$631$	−26.9795			−1.07404			−0.537019	+	0.843570i	$0.680450\pi$
$632$		−	0.377749i		−	0.0150261i
$633$		0			0
$634$	29.9348			1.18886
$635$	55.5189			2.20320
$636$		0			0
$637$		0			0
$638$		−	1.05907i		−	0.0419291i
$639$		0			0
$640$		−	4.23420i		−	0.167371i
$641$		−	20.4916i		−	0.809371i	−0.914456	−	0.404685i	$-0.867381\pi$
$641$		−	20.4916i		−	0.809371i	0.914456	−	0.404685i	$-0.132619\pi$
$642$		0			0
$643$			9.59452i			0.378371i	0.981941	+	0.189185i	$0.0605847\pi$
$643$			9.59452i			0.378371i	−0.981941	+	0.189185i	$0.939415\pi$
$644$	−2.39363	−	11.6688i	−0.0943222	−	0.459817i
$645$		0			0
$646$	1.32503			0.0521326
$647$	−37.5110			−1.47471			−0.737354	−	0.675507i	$-0.763925\pi$
$647$	−37.5110			−1.47471			−0.737354	+	0.675507i	$0.763925\pi$
$648$		0			0
$649$		−	8.46840i		−	0.332414i
$650$		0			0
$651$		0			0
$652$	−8.62039			−0.337601
$653$			28.8964i			1.13080i	0.824816	+	0.565401i	$0.191279\pi$
$653$			28.8964i			1.13080i	−0.824816	+	0.565401i	$0.808721\pi$
$654$		0			0
$655$	−69.4102			−2.71208
$656$	−1.31343			−0.0512808
$657$		0			0
$658$	3.80396	+	18.5441i	0.148294	+	0.722926i
$659$		−	45.5957i		−	1.77616i	−0.459693	−	0.888078i	$-0.652041\pi$
$659$		−	45.5957i		−	1.77616i	0.459693	−	0.888078i	$-0.347959\pi$
$660$		0			0
$661$		−	3.42890i		−	0.133369i	−0.997774	−	0.0666843i	$-0.978758\pi$
$661$		−	3.42890i		−	0.133369i	0.997774	−	0.0666843i	$-0.0212420\pi$
$662$			5.69643i			0.221398i
$663$		0			0
$664$			1.84057i			0.0714280i
$665$	−0.563058	−	2.74489i	−0.0218345	−	0.106442i
$666$		0			0
$667$	−4.76820			−0.184626
$668$	2.88486			0.111618
$669$		0			0
$670$			39.9125i			1.54195i
$671$	14.8102			0.571741
$672$		0			0
$673$	−23.2069			−0.894558			−0.447279	−	0.894394i	$-0.647607\pi$
$673$	−23.2069			−0.894558			−0.447279	+	0.894394i	$0.647607\pi$
$674$		−	0.637355i		−	0.0245500i
$675$		0			0
$676$	−13.0000			−0.500000
$677$	−8.06674			−0.310030			−0.155015	−	0.987912i	$-0.549543\pi$
$677$	−8.06674			−0.310030			−0.155015	+	0.987912i	$0.549543\pi$
$678$		0			0
$679$	−4.23629	−	20.6517i	−0.162574	−	0.792541i
$680$			22.4307i			0.860178i
$681$		0			0
$682$			6.32489i			0.242193i
$683$		−	34.2876i		−	1.31198i	−0.754770	−	0.655989i	$-0.772252\pi$
$683$		−	34.2876i		−	1.31198i	0.754770	−	0.655989i	$-0.227748\pi$
$684$		0			0
$685$		−	74.8135i		−	2.85848i
$686$	10.5636	+	15.2122i	0.403320	+	0.580804i
$687$		0			0
$688$	7.18357			0.273871
$689$		0			0
$690$		0			0
$691$		−	49.1840i		−	1.87105i	−0.353261	−	0.935525i	$-0.614927\pi$
$691$		−	49.1840i		−	1.87105i	0.353261	−	0.935525i	$-0.385073\pi$
$692$	−11.8535			−0.450603
$693$		0			0
$694$	21.4728			0.815097
$695$		−	71.4902i		−	2.71178i
$696$		0			0
$697$	6.95789			0.263549
$698$	9.52269			0.360439
$699$		0			0
$700$	33.5078	−	6.87344i	1.26647	−	0.259792i
$701$		−	37.6079i		−	1.42043i	−0.703984	−	0.710216i	$-0.748597\pi$
$701$		−	37.6079i		−	1.42043i	0.703984	−	0.710216i	$-0.251403\pi$
$702$		0			0
$703$		−	0.625870i		−	0.0236051i
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$		−	32.3101i		−	1.21600i
$707$	−40.4486	+	8.29722i	−1.52123	+	0.312049i
$708$		0			0
$709$	36.7093			1.37865			0.689324	−	0.724453i	$-0.257908\pi$
$709$	36.7093			1.37865			0.689324	+	0.724453i	$0.257908\pi$
$710$	54.9918			2.06381
$711$		0			0
$712$		−	1.56355i		−	0.0585966i
$713$	28.4762			1.06644
$714$		0			0
$715$		0			0
$716$		−	13.4262i		−	0.501761i
$717$		0			0
$718$	3.07604			0.114797
$719$	50.7486			1.89260			0.946301	−	0.323286i	$-0.104788\pi$
$719$	50.7486			1.89260			0.946301	+	0.323286i	$0.104788\pi$
$720$		0			0
$721$	0.978544	+	4.77036i	0.0364429	+	0.177658i
$722$		−	18.9374i		−	0.704778i
$723$		0			0
$724$			19.8766i			0.738708i
$725$		−	13.6922i		−	0.508514i
$726$		0			0
$727$		−	0.570936i		−	0.0211748i	−0.999944	−	0.0105874i	$-0.996630\pi$
$727$		−	0.570936i		−	0.0211748i	0.999944	−	0.0105874i	$-0.00337014\pi$
$728$		0			0
$729$		0			0
$730$	−30.2956			−1.12129
$731$	−38.0550			−1.40751
$732$		0			0
$733$		−	12.6836i		−	0.468479i	−0.972179	−	0.234239i	$-0.924740\pi$
$733$		−	12.6836i		−	0.468479i	0.972179	−	0.234239i	$-0.0752600\pi$
$734$	−16.6508			−0.614591
$735$		0			0
$736$	−4.50225			−0.165955
$737$			9.42621i			0.347219i
$738$		0			0
$739$	26.0360			0.957749			0.478875	−	0.877883i	$-0.341045\pi$
$739$	26.0360			0.957749			0.478875	+	0.877883i	$0.341045\pi$
$740$	10.5950			0.389480
$741$		0			0
$742$	−12.1330	+	2.48884i	−0.445416	+	0.0913681i
$743$		−	30.8355i		−	1.13124i	−0.824665	−	0.565622i	$-0.808636\pi$
$743$		−	30.8355i		−	1.13124i	0.824665	−	0.565622i	$-0.191364\pi$
$744$		0			0
$745$			46.0231i			1.68616i
$746$			7.37163i			0.269895i
$747$		0			0
$748$			5.29751i			0.193696i
$749$	−9.16384	−	44.6733i	−0.334839	−	1.63233i
$750$		0			0
$751$	14.9616			0.545956			0.272978	−	0.962020i	$-0.411991\pi$
$751$	14.9616			0.545956			0.272978	+	0.962020i	$0.411991\pi$
$752$	7.15497			0.260915
$753$		0			0
$754$		0			0
$755$	−34.5981			−1.25915
$756$		0			0
$757$	22.6294			0.822479			0.411239	−	0.911527i	$-0.365096\pi$
$757$	22.6294			0.822479			0.411239	+	0.911527i	$0.365096\pi$
$758$		−	4.62039i		−	0.167820i
$759$		0			0
$760$	−1.05907			−0.0384166
$761$	−35.1870			−1.27553			−0.637764	−	0.770231i	$-0.720141\pi$
$761$	−35.1870			−1.27553			−0.637764	+	0.770231i	$0.720141\pi$
$762$		0			0
$763$	46.1605	−	9.46889i	1.67112	−	0.342797i
$764$		−	0.253253i		−	0.00916237i
$765$		0			0
$766$			26.9159i			0.972513i
$767$		0			0
$768$		0			0
$769$			31.4721i			1.13491i	0.823403	+	0.567456i	$0.192072\pi$
$769$			31.4721i			1.13491i	−0.823403	+	0.567456i	$0.807928\pi$
$770$	10.9741	−	2.25112i	0.395480	−	0.0811248i
$771$		0			0
$772$	−12.8524			−0.462569
$773$	29.3365			1.05516			0.527580	−	0.849506i	$-0.323100\pi$
$773$	29.3365			1.05516			0.527580	+	0.849506i	$0.323100\pi$
$774$		0			0
$775$			81.7711i			2.93731i
$776$	−7.96815			−0.286040
$777$		0			0
$778$	−5.20053			−0.186448
$779$			0.328519i			0.0117704i
$780$		0			0
$781$	12.9875			0.464730
$782$	23.8507			0.852898
$783$		0			0
$784$	6.43469	−	2.75586i	0.229810	−	0.0984235i
$785$		−	87.3721i		−	3.11844i
$786$		0			0
$787$			27.7101i			0.987759i	0.869530	+	0.493879i	$0.164422\pi$
$787$			27.7101i			0.987759i	−0.869530	+	0.493879i	$0.835578\pi$
$788$		−	25.4352i		−	0.906091i
$789$		0			0
$790$		−	1.59947i		−	0.0569065i
$791$	5.21230	+	25.4097i	0.185328	+	0.903467i
$792$		0			0
$793$		0			0
$794$	7.38608			0.262122
$795$		0			0
$796$			22.7615i			0.806758i
$797$	21.4739			0.760646			0.380323	−	0.924854i	$-0.375813\pi$
$797$	21.4739			0.760646			0.380323	+	0.924854i	$0.375813\pi$
$798$		0			0
$799$	−37.9035			−1.34093
$800$		−	12.9285i		−	0.457090i
$801$		0			0
$802$	16.0699			0.567448
$803$	−7.15497			−0.252494
$804$		0			0
$805$	−10.1351	−	49.4083i	−0.357216	−	1.74141i
$806$		0			0
$807$		0			0
$808$			15.6065i			0.549034i
$809$		−	9.89084i		−	0.347743i	−0.984768	−	0.173872i	$-0.944372\pi$
$809$		−	9.89084i		−	0.347743i	0.984768	−	0.173872i	$-0.0556278\pi$
$810$		0			0
$811$			14.7574i			0.518202i	0.965850	+	0.259101i	$0.0834262\pi$
$811$			14.7574i			0.518202i	−0.965850	+	0.259101i	$0.916574\pi$
$812$	−0.563058	−	2.74489i	−0.0197595	−	0.0963266i
$813$		0			0
$814$	2.50225			0.0877037
$815$	−36.5005			−1.27856
$816$		0			0
$817$		−	1.79678i		−	0.0628613i
$818$	−22.1625			−0.774893
$819$		0			0
$820$	−5.56132			−0.194210
$821$			33.3716i			1.16468i	0.812946	+	0.582339i	$0.197862\pi$
$821$			33.3716i			1.16468i	−0.812946	+	0.582339i	$0.802138\pi$
$822$		0			0
$823$	3.02944			0.105600			0.0527998	−	0.998605i	$-0.483185\pi$
$823$	3.02944			0.105600			0.0527998	+	0.998605i	$0.483185\pi$
$824$	1.84057			0.0641193
$825$		0			0
$826$	−4.50225	−	21.9483i	−0.156653	−	0.763678i
$827$			0.468317i			0.0162850i	0.999967	+	0.00814249i	$0.00259186\pi$
$827$			0.468317i			0.0162850i	−0.999967	+	0.00814249i	$0.997408\pi$
$828$		0			0
$829$			44.4855i			1.54505i	0.634987	+	0.772523i	$0.281005\pi$
$829$			44.4855i			1.54505i	−0.634987	+	0.772523i	$0.718995\pi$
$830$			7.79335i			0.270511i
$831$		0			0
$832$		0			0
$833$	−34.0878	+	14.5992i	−1.18107	+	0.505831i
$834$		0			0
$835$	12.2151			0.422720
$836$	−0.250123			−0.00865069
$837$		0			0
$838$		−	24.9050i		−	0.860327i
$839$	−18.4095			−0.635567			−0.317783	−	0.948163i	$-0.602938\pi$
$839$	−18.4095			−0.635567			−0.317783	+	0.948163i	$0.602938\pi$
$840$		0			0
$841$	27.8784			0.961323
$842$			7.47281i			0.257530i
$843$		0			0
$844$	16.6733			0.573920
$845$	−55.0446			−1.89359
$846$		0			0
$847$	2.59178	−	0.531652i	0.0890548	−	0.0182678i
$848$			4.68132i			0.160757i
$849$		0			0
$850$			68.4886i			2.34914i
$851$		−	11.2657i		−	0.386184i
$852$		0			0
$853$			39.0555i			1.33723i	0.743607	+	0.668617i	$0.233113\pi$
$853$			39.0555i			1.33723i	−0.743607	+	0.668617i	$0.766887\pi$
$854$	38.3848	−	7.87387i	1.31350	−	0.269438i
$855$		0			0
$856$	−17.2365			−0.589132
$857$	−52.9808			−1.80979			−0.904895	−	0.425635i	$-0.860051\pi$
$857$	−52.9808			−1.80979			−0.904895	+	0.425635i	$0.860051\pi$
$858$		0			0
$859$			57.1564i			1.95015i	0.221870	+	0.975076i	$0.428784\pi$
$859$			57.1564i			1.95015i	−0.221870	+	0.975076i	$0.571216\pi$
$860$	30.4167			1.03720
$861$		0			0
$862$	−14.6797			−0.499992
$863$			36.4683i			1.24140i	0.784050	+	0.620698i	$0.213151\pi$
$863$			36.4683i			1.24140i	−0.784050	+	0.620698i	$0.786849\pi$
$864$		0			0
$865$	−50.1901			−1.70652
$866$	32.4714			1.10342
$867$		0			0
$868$	3.36265	+	16.3928i	0.114136	+	0.556407i
$869$		−	0.377749i		−	0.0128143i
$870$		0			0
$871$		0			0
$872$		−	17.8103i		−	0.603133i
$873$		0			0
$874$			1.12612i			0.0380915i
$875$	87.0079	−	17.8479i	2.94141	−	0.603370i
$876$		0			0
$877$	26.4566			0.893378			0.446689	−	0.894689i	$-0.352603\pi$
$877$	26.4566			0.893378			0.446689	+	0.894689i	$0.352603\pi$
$878$	−33.3643			−1.12599
$879$		0			0
$880$		−	4.23420i		−	0.142735i
$881$	11.5527			0.389220			0.194610	−	0.980881i	$-0.437656\pi$
$881$	11.5527			0.389220			0.194610	+	0.980881i	$0.437656\pi$
$882$		0			0
$883$	43.1274			1.45135			0.725676	−	0.688037i	$-0.241527\pi$
$883$	43.1274			1.45135			0.725676	+	0.688037i	$0.241527\pi$
$884$		0			0
$885$		0			0
$886$	−27.9205			−0.938007
$887$	11.6293			0.390474			0.195237	−	0.980756i	$-0.437452\pi$
$887$	11.6293			0.390474			0.195237	+	0.980756i	$0.437452\pi$
$888$		0			0
$889$	33.9835	−	6.97104i	1.13977	−	0.233801i
$890$		−	6.62039i		−	0.221916i
$891$		0			0
$892$			19.0085i			0.636451i
$893$		−	1.78963i		−	0.0598875i
$894$		0			0
$895$		−	56.8493i		−	1.90026i
$896$	−0.531652	−	2.59178i	−0.0177613	−	0.0865854i
$897$		0			0
$898$	−12.4413			−0.415172
$899$	6.69852			0.223408
$900$		0			0
$901$		−	24.7993i		−	0.826186i
$902$	−1.31343			−0.0437324
$903$		0			0
$904$	9.80396			0.326075
$905$			84.1615i			2.79762i
$906$		0			0
$907$	−33.6413			−1.11704			−0.558520	−	0.829491i	$-0.688631\pi$
$907$	−33.6413			−1.11704			−0.558520	+	0.829491i	$0.688631\pi$
$908$	−11.1122			−0.368770
$909$		0			0
$910$		0			0
$911$			1.13062i			0.0374590i	0.999825	+	0.0187295i	$0.00596214\pi$
$911$			1.13062i			0.0374590i	−0.999825	+	0.0187295i	$0.994038\pi$
$912$		0			0
$913$			1.84057i			0.0609140i
$914$		−	6.48528i		−	0.214514i
$915$		0			0
$916$		−	1.34032i		−	0.0442856i
$917$	−42.4865	+	8.71525i	−1.40303	+	0.287803i
$918$		0			0
$919$	−25.2095			−0.831585			−0.415793	−	0.909459i	$-0.636496\pi$
$919$	−25.2095			−0.831585			−0.415793	+	0.909459i	$0.636496\pi$
$920$	−19.0634			−0.628502
$921$		0			0
$922$			2.38461i			0.0785330i
$923$		0			0
$924$		0			0
$925$	32.3502			1.06367
$926$		−	8.35815i		−	0.274666i
$927$		0			0
$928$	−1.05907			−0.0347657
$929$	39.1521			1.28454			0.642269	−	0.766479i	$-0.277993\pi$
$929$	39.1521			1.28454			0.642269	+	0.766479i	$0.277993\pi$
$930$		0			0
$931$	−0.689304	−	1.60947i	−0.0225910	−	0.0527481i
$932$			14.4853i			0.474481i
$933$		0			0
$934$		−	25.8336i		−	0.845302i
$935$			22.4307i			0.733562i
$936$		0			0
$937$			10.9079i			0.356347i	0.983999	+	0.178174i	$0.0570188\pi$
$937$			10.9079i			0.356347i	−0.983999	+	0.178174i	$0.942981\pi$
$938$	5.01147	+	24.4307i	0.163630	+	0.797691i
$939$		0			0
$940$	30.2956			0.988133
$941$	−20.8940			−0.681124			−0.340562	−	0.940222i	$-0.610617\pi$
$941$	−20.8940			−0.681124			−0.340562	+	0.940222i	$0.610617\pi$
$942$		0			0
$943$			5.91338i			0.192566i
$944$	−8.46840			−0.275623
$945$		0			0
$946$	7.18357			0.233558
$947$			35.5999i			1.15684i	0.815738	+	0.578421i	$0.196331\pi$
$947$			35.5999i			1.15684i	−0.815738	+	0.578421i	$0.803669\pi$
$948$		0			0
$949$		0			0
$950$	−3.23371			−0.104915
$951$		0			0
$952$	2.81643	+	13.7300i	0.0912811	+	0.444992i
$953$		−	42.7343i		−	1.38430i	−0.721754	−	0.692149i	$-0.756664\pi$
$953$		−	42.7343i		−	1.38430i	0.721754	−	0.692149i	$-0.243336\pi$
$954$		0			0
$955$		−	1.07232i		−	0.0346996i
$956$		−	2.01696i		−	0.0652333i
$957$		0			0
$958$			21.9102i			0.707887i
$959$	−9.39369	−	45.7938i	−0.303338	−	1.47876i
$960$		0			0
$961$	−9.00429			−0.290461
$962$		0			0
$963$		0			0
$964$		−	12.1395i		−	0.390988i
$965$	−54.4197			−1.75183
$966$		0			0
$967$	−20.5891			−0.662102			−0.331051	−	0.943613i	$-0.607403\pi$
$967$	−20.5891			−0.662102			−0.331051	+	0.943613i	$0.607403\pi$
$968$		−	1.00000i		−	0.0321412i
$969$		0			0
$970$	−33.7388			−1.08329
$971$	44.8252			1.43851			0.719255	−	0.694747i	$-0.244483\pi$
$971$	44.8252			1.43851			0.719255	+	0.694747i	$0.244483\pi$
$972$		0			0
$973$	−8.97641	−	43.7597i	−0.287771	−	1.40287i
$974$			25.4898i			0.816745i
$975$		0			0
$976$		−	14.8102i		−	0.474063i
$977$		−	23.9936i		−	0.767625i	−0.923411	−	0.383812i	$-0.874611\pi$
$977$		−	23.9936i		−	0.767625i	0.923411	−	0.383812i	$-0.125389\pi$
$978$		0			0
$979$		−	1.56355i		−	0.0499713i
$980$	27.2458	−	11.6688i	0.870334	−	0.372748i
$981$		0			0
$982$	13.5067			0.431017
$983$	21.9314			0.699502			0.349751	−	0.936843i	$-0.386266\pi$
$983$	21.9314			0.699502			0.349751	+	0.936843i	$0.386266\pi$
$984$		0			0
$985$		−	107.698i		−	3.43153i
$986$	5.61044			0.178673
$987$		0			0
$988$		0			0
$989$		−	32.3422i		−	1.02842i
$990$		0			0
$991$	−53.5957			−1.70252			−0.851261	−	0.524742i	$-0.824162\pi$
$991$	−53.5957			−1.70252			−0.851261	+	0.524742i	$0.824162\pi$
$992$	6.32489			0.200816
$993$		0			0
$994$	33.6609	−	6.90485i	1.06766	−	0.219009i
$995$			96.3765i			3.05534i
$996$		0			0
$997$		−	32.3191i		−	1.02356i	−0.859118	−	0.511778i	$-0.828987\pi$
$997$		−	32.3191i		−	1.02356i	0.859118	−	0.511778i	$-0.171013\pi$
$998$		−	4.90700i		−	0.155328i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.g.a.881.1	✓	16
3.2	odd	2	inner	1386.2.g.a.881.16	yes	16
7.6	odd	2	inner	1386.2.g.a.881.8	yes	16
21.20	even	2	inner	1386.2.g.a.881.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.g.a.881.1	✓	16	1.1	even	1	trivial
1386.2.g.a.881.8	yes	16	7.6	odd	2	inner
1386.2.g.a.881.9	yes	16	21.20	even	2	inner
1386.2.g.a.881.16	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.23420 q^{5} +(-2.59178 + 0.531652i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.23420 q^{5} +(-2.59178 + 0.531652i) q^{7} +1.00000i q^{8} +4.23420i q^{10} +1.00000i q^{11} +(0.531652 + 2.59178i) q^{14} +1.00000 q^{16} -5.29751 q^{17} -0.250123i q^{19} +4.23420 q^{20} +1.00000 q^{22} -4.50225i q^{23} +12.9285 q^{25} +(2.59178 - 0.531652i) q^{28} -1.05907i q^{29} +6.32489i q^{31} -1.00000i q^{32} +5.29751i q^{34} +(10.9741 - 2.25112i) q^{35} +2.50225 q^{37} -0.250123 q^{38} -4.23420i q^{40} -1.31343 q^{41} +7.18357 q^{43} -1.00000i q^{44} -4.50225 q^{46} +7.15497 q^{47} +(6.43469 - 2.75586i) q^{49} -12.9285i q^{50} +4.68132i q^{53} -4.23420i q^{55} +(-0.531652 - 2.59178i) q^{56} -1.05907 q^{58} -8.46840 q^{59} -14.8102i q^{61} +6.32489 q^{62} -1.00000 q^{64} +9.42621 q^{67} +5.29751 q^{68} +(-2.25112 - 10.9741i) q^{70} -12.9875i q^{71} +7.15497i q^{73} -2.50225i q^{74} +0.250123i q^{76} +(-0.531652 - 2.59178i) q^{77} -0.377749 q^{79} -4.23420 q^{80} +1.31343i q^{82} +1.84057 q^{83} +22.4307 q^{85} -7.18357i q^{86} -1.00000 q^{88} -1.56355 q^{89} +4.50225i q^{92} -7.15497i q^{94} +1.05907i q^{95} +7.96815i q^{97} +(-2.75586 - 6.43469i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) 16 * q - 16 * q^4 - 8 * q^7	\( 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) \) 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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