LMFDB - Embedded newform 1386.2.g.a.881.13

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			881.13
Root			$-1.12256 - 0.464978i$ of defining polynomial
Character	$\chi$	$=$	1386.881
Dual form			1386.2.g.a.881.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.929956 q^{5} +(-2.07739 - 1.63843i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +0.929956 q^{5} +(-2.07739 - 1.63843i) q^{7} -1.00000i q^{8} +0.929956i q^{10} -1.00000i q^{11} +(1.63843 - 2.07739i) q^{14} +1.00000 q^{16} -2.34690 q^{17} +6.87928i q^{19} -0.929956 q^{20} +1.00000 q^{22} -3.04733i q^{23} -4.13518 q^{25} +(2.07739 + 1.63843i) q^{28} -6.39743i q^{29} -10.0122i q^{31} +1.00000i q^{32} -2.34690i q^{34} +(-1.93188 - 1.52366i) q^{35} -5.04733 q^{37} -6.87928 q^{38} -0.929956i q^{40} -10.1561 q^{41} +6.15479 q^{43} +1.00000i q^{44} +3.04733 q^{46} -12.0160 q^{47} +(1.63112 + 6.80731i) q^{49} -4.13518i q^{50} -11.2021i q^{53} -0.929956i q^{55} +(-1.63843 + 2.07739i) q^{56} +6.39743 q^{58} +1.85991 q^{59} -10.2735i q^{61} +10.0122 q^{62} -1.00000 q^{64} -0.0878550 q^{67} +2.34690 q^{68} +(1.52366 - 1.93188i) q^{70} -11.5326i q^{71} +12.0160i q^{73} -5.04733i q^{74} -6.87928i q^{76} +(-1.63843 + 2.07739i) q^{77} +13.5995 q^{79} +0.929956 q^{80} -10.1561i q^{82} +4.06286 q^{83} -2.18251 q^{85} +6.15479i q^{86} -1.00000 q^{88} -17.0354 q^{89} +3.04733i q^{92} -12.0160i q^{94} +6.39743i q^{95} +15.6185i q^{97} +(-6.80731 + 1.63112i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.929956			0.415889			0.207944	−	0.978141i	$-0.433323\pi$
$5$	0.929956			0.415889			0.207944	+	0.978141i	$0.433323\pi$
$6$		0			0
$7$	−2.07739	−	1.63843i	−0.785181	−	0.619267i
$7$	−2.07739	−	1.63843i	−0.785181	−	0.619267i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			0.929956i			0.294078i
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	1.63843	−	2.07739i	0.437888	−	0.555207i
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.34690			−0.569206			−0.284603	−	0.958645i	$-0.591862\pi$
$17$	−2.34690			−0.569206			−0.284603	+	0.958645i	$0.591862\pi$
$18$		0			0
$19$			6.87928i			1.57822i	0.614255	+	0.789108i	$0.289457\pi$
$19$			6.87928i			1.57822i	−0.614255	+	0.789108i	$0.710543\pi$
$20$	−0.929956			−0.207944
$21$		0			0
$22$	1.00000			0.213201
$23$		−	3.04733i		−	0.635412i	−0.948189	−	0.317706i	$-0.897088\pi$
$23$		−	3.04733i		−	0.635412i	0.948189	−	0.317706i	$-0.102912\pi$
$24$		0			0
$25$	−4.13518			−0.827036
$26$		0			0
$27$		0			0
$28$	2.07739	+	1.63843i	0.392590	+	0.309633i
$29$		−	6.39743i		−	1.18797i	−0.804475	−	0.593986i	$-0.797553\pi$
$29$		−	6.39743i		−	1.18797i	0.804475	−	0.593986i	$-0.202447\pi$
$30$		0			0
$31$		−	10.0122i		−	1.79824i	−0.437700	−	0.899121i	$-0.644207\pi$
$31$		−	10.0122i		−	1.79824i	0.437700	−	0.899121i	$-0.355793\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	2.34690i		−	0.402489i
$35$	−1.93188	−	1.52366i	−0.326548	−	0.257546i
$36$		0			0
$37$	−5.04733			−0.829775			−0.414888	−	0.909873i	$-0.636179\pi$
$37$	−5.04733			−0.829775			−0.414888	+	0.909873i	$0.636179\pi$
$38$	−6.87928			−1.11597
$39$		0			0
$40$		−	0.929956i		−	0.147039i
$41$	−10.1561			−1.58612			−0.793061	−	0.609143i	$-0.791514\pi$
$41$	−10.1561			−1.58612			−0.793061	+	0.609143i	$0.791514\pi$
$42$		0			0
$43$	6.15479			0.938596			0.469298	−	0.883040i	$-0.344507\pi$
$43$	6.15479			0.938596			0.469298	+	0.883040i	$0.344507\pi$
$44$			1.00000i			0.150756i
$45$		0			0
$46$	3.04733			0.449304
$47$	−12.0160			−1.75272			−0.876360	−	0.481657i	$-0.840035\pi$
$47$	−12.0160			−1.75272			−0.876360	+	0.481657i	$0.840035\pi$
$48$		0			0
$49$	1.63112	+	6.80731i	0.233017	+	0.972473i
$50$		−	4.13518i		−	0.584803i
$51$		0			0
$52$		0			0
$53$		−	11.2021i		−	1.53873i	−0.638810	−	0.769364i	$-0.720573\pi$
$53$		−	11.2021i		−	1.53873i	0.638810	−	0.769364i	$-0.279427\pi$
$54$		0			0
$55$		−	0.929956i		−	0.125395i
$56$	−1.63843	+	2.07739i	−0.218944	+	0.277603i
$57$		0			0
$58$	6.39743			0.840023
$59$	1.85991			0.242140			0.121070	−	0.992644i	$-0.461367\pi$
$59$	1.85991			0.242140			0.121070	+	0.992644i	$0.461367\pi$
$60$		0			0
$61$		−	10.2735i		−	1.31539i	−0.753284	−	0.657695i	$-0.771532\pi$
$61$		−	10.2735i		−	1.31539i	0.753284	−	0.657695i	$-0.228468\pi$
$62$	10.0122			1.27155
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−0.0878550			−0.0107332			−0.00536660	−	0.999986i	$-0.501708\pi$
$67$	−0.0878550			−0.0107332			−0.00536660	+	0.999986i	$0.501708\pi$
$68$	2.34690			0.284603
$69$		0			0
$70$	1.52366	−	1.93188i	0.182113	−	0.230904i
$71$		−	11.5326i		−	1.36867i	−0.729168	−	0.684334i	$-0.760093\pi$
$71$		−	11.5326i		−	1.36867i	0.729168	−	0.684334i	$-0.239907\pi$
$72$		0			0
$73$			12.0160i			1.40637i	0.711006	+	0.703186i	$0.248240\pi$
$73$			12.0160i			1.40637i	−0.711006	+	0.703186i	$0.751760\pi$
$74$		−	5.04733i		−	0.586740i
$75$		0			0
$76$		−	6.87928i		−	0.789108i
$77$	−1.63843	+	2.07739i	−0.186716	+	0.236741i
$78$		0			0
$79$	13.5995			1.53007			0.765034	−	0.643990i	$-0.222722\pi$
$79$	13.5995			1.53007			0.765034	+	0.643990i	$0.222722\pi$
$80$	0.929956			0.103972
$81$		0			0
$82$		−	10.1561i		−	1.12156i
$83$	4.06286			0.445957			0.222979	−	0.974823i	$-0.428422\pi$
$83$	4.06286			0.445957			0.222979	+	0.974823i	$0.428422\pi$
$84$		0			0
$85$	−2.18251			−0.236726
$86$			6.15479i			0.663688i
$87$		0			0
$88$	−1.00000			−0.106600
$89$	−17.0354			−1.80575			−0.902875	−	0.429903i	$-0.858548\pi$
$89$	−17.0354			−1.80575			−0.902875	+	0.429903i	$0.858548\pi$
$90$		0			0
$91$		0			0
$92$			3.04733i			0.317706i
$93$		0			0
$94$		−	12.0160i		−	1.23936i
$95$			6.39743i			0.656362i
$96$		0			0
$97$			15.6185i			1.58582i	0.609342	+	0.792908i	$0.291434\pi$
$97$			15.6185i			1.58582i	−0.609342	+	0.792908i	$0.708566\pi$
$98$	−6.80731	+	1.63112i	−0.687642	+	0.164768i
$99$		0			0
$100$	4.13518			0.413518
$101$	4.54985			0.452727			0.226363	−	0.974043i	$-0.427316\pi$
$101$	4.54985			0.452727			0.226363	+	0.974043i	$0.427316\pi$
$102$		0			0
$103$			4.06286i			0.400326i	0.979763	+	0.200163i	$0.0641471\pi$
$103$			4.06286i			0.400326i	−0.979763	+	0.200163i	$0.935853\pi$
$104$		0			0
$105$		0			0
$106$	11.2021			1.08805
$107$		−	5.57182i		−	0.538648i	−0.963050	−	0.269324i	$-0.913200\pi$
$107$		−	5.57182i		−	0.538648i	0.963050	−	0.269324i	$-0.0868002\pi$
$108$		0			0
$109$	−15.6597			−1.49992			−0.749962	−	0.661481i	$-0.769928\pi$
$109$	−15.6597			−1.49992			−0.749962	+	0.661481i	$0.769928\pi$
$110$	0.929956			0.0886678
$111$		0			0
$112$	−2.07739	−	1.63843i	−0.196295	−	0.154817i
$113$		−	13.6874i		−	1.28760i	−0.765193	−	0.643801i	$-0.777357\pi$
$113$		−	13.6874i		−	1.28760i	0.765193	−	0.643801i	$-0.222643\pi$
$114$		0			0
$115$		−	2.83388i		−	0.264261i
$116$			6.39743i			0.593986i
$117$		0			0
$118$			1.85991i			0.171219i
$119$	4.87542	+	3.84521i	0.446929	+	0.352490i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	10.2735			0.930121
$123$		0			0
$124$			10.0122i			0.899121i
$125$	−8.49532			−0.759844
$126$		0			0
$127$	4.98040			0.441939			0.220969	−	0.975281i	$-0.429078\pi$
$127$	4.98040			0.441939			0.220969	+	0.975281i	$0.429078\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		0			0
$131$	20.7992			1.81724			0.908619	−	0.417625i	$-0.137138\pi$
$131$	20.7992			1.81724			0.908619	+	0.417625i	$0.137138\pi$
$132$		0			0
$133$	11.2712	−	14.2910i	0.977336	−	1.23918i
$134$		−	0.0878550i		−	0.00758951i
$135$		0			0
$136$			2.34690i			0.201245i
$137$			0.330496i			0.0282362i	0.999900	+	0.0141181i	$0.00449407\pi$
$137$			0.330496i			0.0282362i	−0.999900	+	0.0141181i	$0.995506\pi$
$138$		0			0
$139$		−	19.0944i		−	1.61957i	−0.586728	−	0.809784i	$-0.699584\pi$
$139$		−	19.0944i		−	1.61957i	0.586728	−	0.809784i	$-0.300416\pi$
$140$	1.93188	+	1.52366i	0.163274	+	0.128773i
$141$		0			0
$142$	11.5326			0.967795
$143$		0			0
$144$		0			0
$145$		−	5.94932i		−	0.494064i
$146$	−12.0160			−0.994455
$147$		0			0
$148$	5.04733			0.414888
$149$			1.26224i			0.103407i	0.998662	+	0.0517035i	$0.0164651\pi$
$149$			1.26224i			0.103407i	−0.998662	+	0.0517035i	$0.983535\pi$
$150$		0			0
$151$	−17.3778			−1.41419			−0.707094	−	0.707120i	$-0.749994\pi$
$151$	−17.3778			−1.41419			−0.707094	+	0.707120i	$0.749994\pi$
$152$	6.87928			0.557983
$153$		0			0
$154$	−2.07739	−	1.63843i	−0.167401	−	0.132028i
$155$		−	9.31089i		−	0.747869i
$156$		0			0
$157$			14.0199i			1.11891i	0.828861	+	0.559455i	$0.188989\pi$
$157$			14.0199i			1.11891i	−0.828861	+	0.559455i	$0.811011\pi$
$158$			13.5995i			1.08192i
$159$		0			0
$160$			0.929956i			0.0735195i
$161$	−4.99282	+	6.33050i	−0.393489	+	0.498913i
$162$		0			0
$163$	−13.8422			−1.08420			−0.542102	−	0.840313i	$-0.682371\pi$
$163$	−13.8422			−1.08420			−0.542102	+	0.840313i	$0.682371\pi$
$164$	10.1561			0.793061
$165$		0			0
$166$			4.06286i			0.315339i
$167$	6.69765			0.518279			0.259140	−	0.965840i	$-0.416561\pi$
$167$	6.69765			0.518279			0.259140	+	0.965840i	$0.416561\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		−	2.18251i		−	0.167391i
$171$		0			0
$172$	−6.15479			−0.469298
$173$	5.20100			0.395425			0.197712	−	0.980260i	$-0.436649\pi$
$173$	5.20100			0.395425			0.197712	+	0.980260i	$0.436649\pi$
$174$		0			0
$175$	8.59040	+	6.77519i	0.649373	+	0.512156i
$176$		−	1.00000i		−	0.0753778i
$177$		0			0
$178$		−	17.0354i		−	1.27686i
$179$		−	3.91215i		−	0.292407i	−0.989255	−	0.146204i	$-0.953294\pi$
$179$		−	3.91215i		−	0.292407i	0.989255	−	0.146204i	$-0.0467055\pi$
$180$		0			0
$181$		−	0.768549i		−	0.0571258i	−0.999592	−	0.0285629i	$-0.990907\pi$
$181$		−	0.768549i		−	0.0571258i	0.999592	−	0.0285629i	$-0.00909309\pi$
$182$		0			0
$183$		0			0
$184$	−3.04733			−0.224652
$185$	−4.69379			−0.345094
$186$		0			0
$187$			2.34690i			0.171622i
$188$	12.0160			0.876360
$189$		0			0
$190$	−6.39743			−0.464118
$191$			20.1518i			1.45813i	0.684445	+	0.729065i	$0.260045\pi$
$191$			20.1518i			1.45813i	−0.684445	+	0.729065i	$0.739955\pi$
$192$		0			0
$193$	−6.17571			−0.444537			−0.222269	−	0.974985i	$-0.571346\pi$
$193$	−6.17571			−0.444537			−0.222269	+	0.974985i	$0.571346\pi$
$194$	−15.6185			−1.12134
$195$		0			0
$196$	−1.63112	−	6.80731i	−0.116509	−	0.486236i
$197$			14.2772i			1.01721i	0.861001	+	0.508603i	$0.169838\pi$
$197$			14.2772i			1.01721i	−0.861001	+	0.508603i	$0.830162\pi$
$198$		0			0
$199$		−	7.46620i		−	0.529265i	−0.964349	−	0.264632i	$-0.914749\pi$
$199$		−	7.46620i		−	0.529265i	0.964349	−	0.264632i	$-0.0852506\pi$
$200$			4.13518i			0.292402i
$201$		0			0
$202$			4.54985i			0.320126i
$203$	−10.4817	+	13.2900i	−0.735672	+	0.932773i
$204$		0			0
$205$	−9.44475			−0.659650
$206$	−4.06286			−0.283073
$207$		0			0
$208$		0			0
$209$	6.87928			0.475850
$210$		0			0
$211$	16.4252			1.13075			0.565377	−	0.824833i	$-0.308731\pi$
$211$	16.4252			1.13075			0.565377	+	0.824833i	$0.308731\pi$
$212$			11.2021i			0.769364i
$213$		0			0
$214$	5.57182			0.380881
$215$	5.72368			0.390352
$216$		0			0
$217$	−16.4042	+	20.7992i	−1.11359	+	1.41194i
$218$		−	15.6597i		−	1.06061i
$219$		0			0
$220$			0.929956i			0.0626976i
$221$		0			0
$222$		0			0
$223$		−	6.81504i		−	0.456369i	−0.973618	−	0.228184i	$-0.926721\pi$
$223$		−	6.81504i		−	0.456369i	0.973618	−	0.228184i	$-0.0732789\pi$
$224$	1.63843	−	2.07739i	0.109472	−	0.138802i
$225$		0			0
$226$	13.6874			0.910472
$227$	0.0265489			0.00176211			0.000881055	−	1.00000i	$-0.499720\pi$
$227$	0.0265489			0.00176211			0.000881055		1.00000i	$0.499720\pi$
$228$		0			0
$229$			9.69570i			0.640710i	0.947298	+	0.320355i	$0.103802\pi$
$229$			9.69570i			0.640710i	−0.947298	+	0.320355i	$0.896198\pi$
$230$	2.83388			0.186860
$231$		0			0
$232$	−6.39743			−0.420012
$233$		−	2.48528i		−	0.162816i	−0.996681	−	0.0814081i	$-0.974058\pi$
$233$		−	2.48528i		−	0.162816i	0.996681	−	0.0814081i	$-0.0259417\pi$
$234$		0			0
$235$	−11.1744			−0.728937
$236$	−1.85991			−0.121070
$237$		0			0
$238$	−3.84521	+	4.87542i	−0.249248	+	0.316027i
$239$		−	11.4380i		−	0.739860i	−0.929060	−	0.369930i	$-0.879382\pi$
$239$		−	11.4380i		−	0.739860i	0.929060	−	0.369930i	$-0.120618\pi$
$240$		0			0
$241$		−	7.69184i		−	0.495475i	−0.968827	−	0.247738i	$-0.920313\pi$
$241$		−	7.69184i		−	0.495475i	0.968827	−	0.247738i	$-0.0796871\pi$
$242$		−	1.00000i		−	0.0642824i
$243$		0			0
$244$			10.2735i			0.657695i
$245$	1.51687	+	6.33050i	0.0969094	+	0.404441i
$246$		0			0
$247$		0			0
$248$	−10.0122			−0.635775
$249$		0			0
$250$		−	8.49532i		−	0.537291i
$251$	−8.41361			−0.531063			−0.265531	−	0.964102i	$-0.585547\pi$
$251$	−8.41361			−0.531063			−0.265531	+	0.964102i	$0.585547\pi$
$252$		0			0
$253$	−3.04733			−0.191584
$254$			4.98040i			0.312498i
$255$		0			0
$256$	1.00000			0.0625000
$257$	17.0354			1.06264			0.531320	−	0.847171i	$-0.321696\pi$
$257$	17.0354			1.06264			0.531320	+	0.847171i	$0.321696\pi$
$258$		0			0
$259$	10.4853	+	8.26967i	0.651524	+	0.513852i
$260$		0			0
$261$		0			0
$262$			20.7992i			1.28498i
$263$			31.2869i			1.92923i	0.263653	+	0.964617i	$0.415072\pi$
$263$			31.2869i			1.92923i	−0.263653	+	0.964617i	$0.584928\pi$
$264$		0			0
$265$		−	10.4175i		−	0.639940i
$266$	14.2910	+	11.2712i	0.876235	+	0.691081i
$267$		0			0
$268$	0.0878550			0.00536660
$269$	−3.68213			−0.224504			−0.112252	−	0.993680i	$-0.535806\pi$
$269$	−3.68213			−0.224504			−0.112252	+	0.993680i	$0.535806\pi$
$270$		0			0
$271$			18.2442i			1.10825i	0.832432	+	0.554127i	$0.186948\pi$
$271$			18.2442i			1.10825i	−0.832432	+	0.554127i	$0.813052\pi$
$272$	−2.34690			−0.142301
$273$		0			0
$274$	−0.330496			−0.0199660
$275$			4.13518i			0.249361i
$276$		0			0
$277$	−15.8354			−0.951456			−0.475728	−	0.879592i	$-0.657815\pi$
$277$	−15.8354			−0.951456			−0.475728	+	0.879592i	$0.657815\pi$
$278$	19.0944			1.14521
$279$		0			0
$280$	−1.52366	+	1.93188i	−0.0910563	+	0.115452i
$281$		−	10.9706i		−	0.654449i	−0.944947	−	0.327224i	$-0.893887\pi$
$281$		−	10.9706i		−	0.654449i	0.944947	−	0.327224i	$-0.106113\pi$
$282$		0			0
$283$			15.6625i			0.931038i	0.885038	+	0.465519i	$0.154132\pi$
$283$			15.6625i			0.931038i	−0.885038	+	0.465519i	$0.845868\pi$
$284$			11.5326i			0.684334i
$285$		0			0
$286$		0			0
$287$	21.0983	+	16.6401i	1.24539	+	0.982232i
$288$		0			0
$289$	−11.4921			−0.676005
$290$	5.94932			0.349356
$291$		0			0
$292$		−	12.0160i		−	0.703186i
$293$	11.7547			0.686717			0.343359	−	0.939204i	$-0.388435\pi$
$293$	11.7547			0.686717			0.343359	+	0.939204i	$0.388435\pi$
$294$		0			0
$295$	1.72964			0.100703
$296$			5.04733i			0.293370i
$297$		0			0
$298$	−1.26224			−0.0731198
$299$		0			0
$300$		0			0
$301$	−12.7859	−	10.0842i	−0.736967	−	0.581241i
$302$		−	17.3778i		−	0.999981i
$303$		0			0
$304$			6.87928i			0.394554i
$305$		−	9.55393i		−	0.547056i
$306$		0			0
$307$			22.5327i			1.28601i	0.765863	+	0.643004i	$0.222312\pi$
$307$			22.5327i			1.28601i	−0.765863	+	0.643004i	$0.777688\pi$
$308$	1.63843	−	2.07739i	0.0933580	−	0.118370i
$309$		0			0
$310$	9.31089			0.528823
$311$	6.06672			0.344012			0.172006	−	0.985096i	$-0.444975\pi$
$311$	6.06672			0.344012			0.172006	+	0.985096i	$0.444975\pi$
$312$		0			0
$313$			5.48889i			0.310250i	0.987895	+	0.155125i	$0.0495781\pi$
$313$			5.48889i			0.310250i	−0.987895	+	0.155125i	$0.950422\pi$
$314$	−14.0199			−0.791189
$315$		0			0
$316$	−13.5995			−0.765034
$317$		−	34.2119i		−	1.92153i	−0.277362	−	0.960765i	$-0.589460\pi$
$317$		−	34.2119i		−	1.92153i	0.277362	−	0.960765i	$-0.410540\pi$
$318$		0			0
$319$	−6.39743			−0.358187
$320$	−0.929956			−0.0519861
$321$		0			0
$322$	−6.33050	−	4.99282i	−0.352785	−	0.278239i
$323$		−	16.1450i		−	0.898329i
$324$		0			0
$325$		0			0
$326$		−	13.8422i		−	0.766647i
$327$		0			0
$328$			10.1561i			0.560779i
$329$	24.9620	+	19.6874i	1.37620	+	1.08540i
$330$		0			0
$331$	14.8017			0.813572			0.406786	−	0.913523i	$-0.366649\pi$
$331$	14.8017			0.813572			0.406786	+	0.913523i	$0.366649\pi$
$332$	−4.06286			−0.222979
$333$		0			0
$334$			6.69765i			0.366479i
$335$	−0.0817013			−0.00446382
$336$		0			0
$337$	−12.4042			−0.675701			−0.337851	−	0.941200i	$-0.609700\pi$
$337$	−12.4042			−0.675701			−0.337851	+	0.941200i	$0.609700\pi$
$338$			13.0000i			0.707107i
$339$		0			0
$340$	2.18251			0.118363
$341$	−10.0122			−0.542190
$342$		0			0
$343$	7.76479	−	16.8139i	0.419259	−	0.907867i
$344$		−	6.15479i		−	0.331844i
$345$		0			0
$346$			5.20100i			0.279608i
$347$			20.0179i			1.07462i	0.843386	+	0.537308i	$0.180559\pi$
$347$			20.0179i			1.07462i	−0.843386	+	0.537308i	$0.819441\pi$
$348$		0			0
$349$			14.0464i			0.751889i	0.926642	+	0.375945i	$0.122682\pi$
$349$			14.0464i			0.751889i	−0.926642	+	0.375945i	$0.877318\pi$
$350$	−6.77519	+	8.59040i	−0.362149	+	0.459176i
$351$		0			0
$352$	1.00000			0.0533002
$353$	−24.4751			−1.30268			−0.651338	−	0.758788i	$-0.725792\pi$
$353$	−24.4751			−1.30268			−0.651338	+	0.758788i	$0.725792\pi$
$354$		0			0
$355$		−	10.7248i		−	0.569214i
$356$	17.0354			0.902875
$357$		0			0
$358$	3.91215			0.206763
$359$		−	5.04053i		−	0.266029i	−0.991114	−	0.133014i	$-0.957534\pi$
$359$		−	5.04053i		−	0.266029i	0.991114	−	0.133014i	$-0.0424657\pi$
$360$		0			0
$361$	−28.3245			−1.49076
$362$	0.768549			0.0403940
$363$		0			0
$364$		0			0
$365$			11.1744i			0.584894i
$366$		0			0
$367$		−	6.21066i		−	0.324194i	−0.986775	−	0.162097i	$-0.948174\pi$
$367$		−	6.21066i		−	0.324194i	0.986775	−	0.162097i	$-0.0518257\pi$
$368$		−	3.04733i		−	0.158853i
$369$		0			0
$370$		−	4.69379i		−	0.244019i
$371$	−18.3538	+	23.2712i	−0.952883	+	1.20818i
$372$		0			0
$373$	9.78508			0.506652			0.253326	−	0.967381i	$-0.418475\pi$
$373$	9.78508			0.506652			0.253326	+	0.967381i	$0.418475\pi$
$374$	−2.34690			−0.121355
$375$		0			0
$376$			12.0160i			0.619680i
$377$		0			0
$378$		0			0
$379$	−17.8422			−0.916491			−0.458246	−	0.888826i	$-0.651522\pi$
$379$	−17.8422			−0.916491			−0.458246	+	0.888826i	$0.651522\pi$
$380$		−	6.39743i		−	0.328181i
$381$		0			0
$382$	−20.1518			−1.03105
$383$	−34.2699			−1.75111			−0.875556	−	0.483118i	$-0.839504\pi$
$383$	−34.2699			−1.75111			−0.875556	+	0.483118i	$0.839504\pi$
$384$		0			0
$385$	−1.52366	+	1.93188i	−0.0776531	+	0.0984579i
$386$		−	6.17571i		−	0.314335i
$387$		0			0
$388$		−	15.6185i		−	0.792908i
$389$			13.5927i			0.689179i	0.938753	+	0.344590i	$0.111982\pi$
$389$			13.5927i			0.689179i	−0.938753	+	0.344590i	$0.888018\pi$
$390$		0			0
$391$			7.15176i			0.361680i
$392$	6.80731	−	1.63112i	0.343821	−	0.0823841i
$393$		0			0
$394$	−14.2772			−0.719273
$395$	12.6470			0.636338
$396$		0			0
$397$			19.1741i			0.962322i	0.876632	+	0.481161i	$0.159785\pi$
$397$			19.1741i			0.962322i	−0.876632	+	0.481161i	$0.840215\pi$
$398$	7.46620			0.374247
$399$		0			0
$400$	−4.13518			−0.206759
$401$		−	14.8550i		−	0.741823i	−0.928668	−	0.370911i	$-0.879045\pi$
$401$		−	14.8550i		−	0.741823i	0.928668	−	0.370911i	$-0.120955\pi$
$402$		0			0
$403$		0			0
$404$	−4.54985			−0.226363
$405$		0			0
$406$	−13.2900	−	10.4817i	−0.659570	−	0.520198i
$407$			5.04733i			0.250187i
$408$		0			0
$409$			7.40395i			0.366102i	0.983103	+	0.183051i	$0.0585973\pi$
$409$			7.40395i			0.366102i	−0.983103	+	0.183051i	$0.941403\pi$
$410$		−	9.44475i		−	0.466443i
$411$		0			0
$412$		−	4.06286i		−	0.200163i
$413$	−3.86377	−	3.04733i	−0.190124	−	0.149949i
$414$		0			0
$415$	3.77828			0.185469
$416$		0			0
$417$		0			0
$418$			6.87928i			0.336477i
$419$	−19.3383			−0.944738			−0.472369	−	0.881401i	$-0.656601\pi$
$419$	−19.3383			−0.944738			−0.472369	+	0.881401i	$0.656601\pi$
$420$		0			0
$421$	34.0179			1.65793			0.828965	−	0.559300i	$-0.188930\pi$
$421$	34.0179			1.65793			0.828965	+	0.559300i	$0.188930\pi$
$422$			16.4252i			0.799564i
$423$		0			0
$424$	−11.2021			−0.544023
$425$	9.70484			0.470754
$426$		0			0
$427$	−16.8324	+	21.3422i	−0.814577	+	1.03282i
$428$			5.57182i			0.269324i
$429$		0			0
$430$			5.72368i			0.276020i
$431$			2.92191i			0.140744i	0.997521	+	0.0703718i	$0.0224186\pi$
$431$			2.92191i			0.140744i	−0.997521	+	0.0703718i	$0.977581\pi$
$432$		0			0
$433$		−	9.60690i		−	0.461678i	−0.972992	−	0.230839i	$-0.925853\pi$
$433$		−	9.60690i		−	0.461678i	0.972992	−	0.230839i	$-0.0741471\pi$
$434$	−20.7992	−	16.4042i	−0.998396	−	0.787428i
$435$		0			0
$436$	15.6597			0.749962
$437$	20.9634			1.00282
$438$		0			0
$439$		−	36.6616i		−	1.74976i	−0.484338	−	0.874881i	$-0.660939\pi$
$439$		−	36.6616i		−	1.74976i	0.484338	−	0.874881i	$-0.339061\pi$
$440$	−0.929956			−0.0443339
$441$		0			0
$442$		0			0
$443$		−	28.7624i		−	1.36655i	−0.730163	−	0.683273i	$-0.760556\pi$
$443$		−	28.7624i		−	1.36655i	0.730163	−	0.683273i	$-0.239444\pi$
$444$		0			0
$445$	−15.8422			−0.750991
$446$	6.81504			0.322702
$447$		0			0
$448$	2.07739	+	1.63843i	0.0981476	+	0.0774083i
$449$		−	24.0916i		−	1.13695i	−0.822699	−	0.568477i	$-0.807533\pi$
$449$		−	24.0916i		−	1.13695i	0.822699	−	0.568477i	$-0.192467\pi$
$450$		0			0
$451$			10.1561i			0.478234i
$452$			13.6874i			0.643801i
$453$		0			0
$454$			0.0265489i			0.00124600i
$455$		0			0
$456$		0			0
$457$	−10.4853			−0.490481			−0.245240	−	0.969462i	$-0.578867\pi$
$457$	−10.4853			−0.490481			−0.245240	+	0.969462i	$0.578867\pi$
$458$	−9.69570			−0.453050
$459$		0			0
$460$			2.83388i			0.132130i
$461$	20.4562			0.952741			0.476370	−	0.879245i	$-0.341952\pi$
$461$	20.4562			0.952741			0.476370	+	0.879245i	$0.341952\pi$
$462$		0			0
$463$	36.4989			1.69625			0.848123	−	0.529799i	$-0.177733\pi$
$463$	36.4989			1.69625			0.848123	+	0.529799i	$0.177733\pi$
$464$		−	6.39743i		−	0.296993i
$465$		0			0
$466$	2.48528			0.115128
$467$	16.9557			0.784616			0.392308	−	0.919834i	$-0.371677\pi$
$467$	16.9557			0.784616			0.392308	+	0.919834i	$0.371677\pi$
$468$		0			0
$469$	0.182509	+	0.143944i	0.00842750	+	0.00664671i
$470$		−	11.1744i		−	0.515436i
$471$		0			0
$472$		−	1.85991i		−	0.0856094i
$473$		−	6.15479i		−	0.282997i
$474$		0			0
$475$		−	28.4471i		−	1.30524i
$476$	−4.87542	−	3.84521i	−0.223465	−	0.176245i
$477$		0			0
$478$	11.4380			0.523160
$479$	31.9385			1.45931			0.729653	−	0.683817i	$-0.239681\pi$
$479$	31.9385			1.45931			0.729653	+	0.683817i	$0.239681\pi$
$480$		0			0
$481$		0			0
$482$	7.69184			0.350354
$483$		0			0
$484$	1.00000			0.0454545
$485$			14.5245i			0.659523i
$486$		0			0
$487$	6.57994			0.298165			0.149083	−	0.988825i	$-0.452368\pi$
$487$	6.57994			0.298165			0.149083	+	0.988825i	$0.452368\pi$
$488$	−10.2735			−0.465061
$489$		0			0
$490$	−6.33050	+	1.51687i	−0.285983	+	0.0685253i
$491$			9.14198i			0.412572i	0.978492	+	0.206286i	$0.0661377\pi$
$491$			9.14198i			0.412572i	−0.978492	+	0.206286i	$0.933862\pi$
$492$		0			0
$493$			15.0141i			0.676201i
$494$		0			0
$495$		0			0
$496$		−	10.0122i		−	0.449560i
$497$	−18.8953	+	23.9578i	−0.847571	+	1.07465i
$498$		0			0
$499$	−6.47848			−0.290017			−0.145008	−	0.989430i	$-0.546321\pi$
$499$	−6.47848			−0.290017			−0.145008	+	0.989430i	$0.546321\pi$
$500$	8.49532			0.379922
$501$		0			0
$502$		−	8.41361i		−	0.375518i
$503$	−2.97782			−0.132775			−0.0663873	−	0.997794i	$-0.521147\pi$
$503$	−2.97782			−0.132775			−0.0663873	+	0.997794i	$0.521147\pi$
$504$		0			0
$505$	4.23116			0.188284
$506$		−	3.04733i		−	0.135470i
$507$		0			0
$508$	−4.98040			−0.220969
$509$	−7.40196			−0.328086			−0.164043	−	0.986453i	$-0.552454\pi$
$509$	−7.40196			−0.328086			−0.164043	+	0.986453i	$0.552454\pi$
$510$		0			0
$511$	19.6874	−	24.9620i	0.870919	−	1.10426i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			17.0354i			0.751400i
$515$			3.77828i			0.166491i
$516$		0			0
$517$			12.0160i			0.528465i
$518$	−8.26967	+	10.4853i	−0.363348	+	0.460697i
$519$		0			0
$520$		0			0
$521$	−6.02271			−0.263860			−0.131930	−	0.991259i	$-0.542117\pi$
$521$	−6.02271			−0.263860			−0.131930	+	0.991259i	$0.542117\pi$
$522$		0			0
$523$		−	21.2422i		−	0.928857i	−0.885611	−	0.464429i	$-0.846260\pi$
$523$		−	21.2422i		−	0.928857i	0.885611	−	0.464429i	$-0.153740\pi$
$524$	−20.7992			−0.908619
$525$		0			0
$526$	−31.2869			−1.36418
$527$			23.4976i			1.02357i
$528$		0			0
$529$	13.7138			0.596252
$530$	10.4175			0.452506
$531$		0			0
$532$	−11.2712	+	14.2910i	−0.488668	+	0.619592i
$533$		0			0
$534$		0			0
$535$		−	5.18154i		−	0.224018i
$536$			0.0878550i			0.00379476i
$537$		0			0
$538$		−	3.68213i		−	0.158748i
$539$	6.80731	−	1.63112i	0.293212	−	0.0702574i
$540$		0			0
$541$	26.0947			1.12190			0.560948	−	0.827851i	$-0.310437\pi$
$541$	26.0947			1.12190			0.560948	+	0.827851i	$0.310437\pi$
$542$	−18.2442			−0.783654
$543$		0			0
$544$		−	2.34690i		−	0.100622i
$545$	−14.5628			−0.623802
$546$		0			0
$547$	−4.42515			−0.189206			−0.0946029	−	0.995515i	$-0.530158\pi$
$547$	−4.42515			−0.189206			−0.0946029	+	0.995515i	$0.530158\pi$
$548$		−	0.330496i		−	0.0141181i
$549$		0			0
$550$	−4.13518			−0.176325
$551$	44.0097			1.87488
$552$		0			0
$553$	−28.2516	−	22.2818i	−1.20138	−	0.947520i
$554$		−	15.8354i		−	0.672781i
$555$		0			0
$556$			19.0944i			0.809784i
$557$			14.2772i			0.604943i	0.953158	+	0.302471i	$0.0978117\pi$
$557$			14.2772i			0.604943i	−0.953158	+	0.302471i	$0.902188\pi$
$558$		0			0
$559$		0			0
$560$	−1.93188	−	1.52366i	−0.0816370	−	0.0643865i
$561$		0			0
$562$	10.9706			0.462765
$563$	−7.08746			−0.298701			−0.149350	−	0.988784i	$-0.547718\pi$
$563$	−7.08746			−0.298701			−0.149350	+	0.988784i	$0.547718\pi$
$564$		0			0
$565$		−	12.7287i		−	0.535499i
$566$	−15.6625			−0.658343
$567$		0			0
$568$	−11.5326			−0.483898
$569$		−	14.4853i		−	0.607255i	−0.952791	−	0.303627i	$-0.901802\pi$
$569$		−	14.4853i		−	0.607255i	0.952791	−	0.303627i	$-0.0981978\pi$
$570$		0			0
$571$	40.1095			1.67853			0.839265	−	0.543722i	$-0.182985\pi$
$571$	40.1095			1.67853			0.839265	+	0.543722i	$0.182985\pi$
$572$		0			0
$573$		0			0
$574$	−16.6401	+	21.0983i	−0.694543	+	0.880625i
$575$			12.6013i			0.525509i
$576$		0			0
$577$			4.98168i			0.207390i	0.994609	+	0.103695i	$0.0330666\pi$
$577$			4.98168i			0.207390i	−0.994609	+	0.103695i	$0.966933\pi$
$578$		−	11.4921i		−	0.478008i
$579$		0			0
$580$			5.94932i			0.247032i
$581$	−8.44016	−	6.65670i	−0.350157	−	0.276166i
$582$		0			0
$583$	−11.2021			−0.463944
$584$	12.0160			0.497227
$585$		0			0
$586$			11.7547i			0.485582i
$587$	39.6855			1.63800			0.818998	−	0.573797i	$-0.194530\pi$
$587$	39.6855			1.63800			0.818998	+	0.573797i	$0.194530\pi$
$588$		0			0
$589$	68.8766			2.83801
$590$			1.72964i			0.0712080i
$591$		0			0
$592$	−5.04733			−0.207444
$593$	−10.5194			−0.431980			−0.215990	−	0.976396i	$-0.569298\pi$
$593$	−10.5194			−0.431980			−0.215990	+	0.976396i	$0.569298\pi$
$594$		0			0
$595$	4.53393	+	3.57588i	0.185873	+	0.146597i
$596$		−	1.26224i		−	0.0517035i
$597$		0			0
$598$		0			0
$599$			2.85802i			0.116775i	0.998294	+	0.0583877i	$0.0185960\pi$
$599$			2.85802i			0.116775i	−0.998294	+	0.0583877i	$0.981404\pi$
$600$		0			0
$601$			33.5307i			1.36775i	0.729600	+	0.683874i	$0.239706\pi$
$601$			33.5307i			1.36775i	−0.729600	+	0.683874i	$0.760294\pi$
$602$	10.0842	−	12.7859i	0.411000	−	0.521115i
$603$		0			0
$604$	17.3778			0.707094
$605$	−0.929956			−0.0378081
$606$		0			0
$607$		−	22.9030i		−	0.929605i	−0.885414	−	0.464803i	$-0.846125\pi$
$607$		−	22.9030i		−	0.929605i	0.885414	−	0.464803i	$-0.153875\pi$
$608$	−6.87928			−0.278992
$609$		0			0
$610$	9.55393			0.386827
$611$		0			0
$612$		0			0
$613$	−46.9885			−1.89785			−0.948923	−	0.315508i	$-0.897825\pi$
$613$	−46.9885			−1.89785			−0.948923	+	0.315508i	$0.897825\pi$
$614$	−22.5327			−0.909345
$615$		0			0
$616$	2.07739	+	1.63843i	0.0837005	+	0.0660141i
$617$		−	33.3471i		−	1.34250i	−0.741230	−	0.671251i	$-0.765757\pi$
$617$		−	33.3471i		−	1.34250i	0.741230	−	0.671251i	$-0.234243\pi$
$618$		0			0
$619$		−	28.7259i		−	1.15459i	−0.816535	−	0.577295i	$-0.804108\pi$
$619$		−	28.7259i		−	1.15459i	0.816535	−	0.577295i	$-0.195892\pi$
$620$			9.31089i			0.373934i
$621$		0			0
$622$			6.06672i			0.243253i
$623$	35.3892	+	27.9113i	1.41784	+	1.11824i
$624$		0			0
$625$	12.7756			0.511026
$626$	−5.48889			−0.219380
$627$		0			0
$628$		−	14.0199i		−	0.559455i
$629$	11.8456			0.472313
$630$		0			0
$631$	37.1599			1.47931			0.739656	−	0.672986i	$-0.234988\pi$
$631$	37.1599			1.47931			0.739656	+	0.672986i	$0.234988\pi$
$632$		−	13.5995i		−	0.540961i
$633$		0			0
$634$	34.2119			1.35873
$635$	4.63155			0.183797
$636$		0			0
$637$		0			0
$638$		−	6.39743i		−	0.253277i
$639$		0			0
$640$		−	0.929956i		−	0.0367597i
$641$			24.8618i			0.981981i	0.871165	+	0.490991i	$0.163365\pi$
$641$			24.8618i			0.981981i	−0.871165	+	0.490991i	$0.836635\pi$
$642$		0			0
$643$			22.8233i			0.900064i	0.893012	+	0.450032i	$0.148587\pi$
$643$			22.8233i			0.900064i	−0.893012	+	0.450032i	$0.851413\pi$
$644$	4.99282	−	6.33050i	0.196745	−	0.249456i
$645$		0			0
$646$	16.1450			0.635215
$647$	−38.9637			−1.53182			−0.765911	−	0.642947i	$-0.777712\pi$
$647$	−38.9637			−1.53182			−0.765911	+	0.642947i	$0.777712\pi$
$648$		0			0
$649$		−	1.85991i		−	0.0730079i
$650$		0			0
$651$		0			0
$652$	13.8422			0.542102
$653$		−	29.4306i		−	1.15171i	−0.817552	−	0.575855i	$-0.804669\pi$
$653$		−	29.4306i		−	1.15171i	0.817552	−	0.575855i	$-0.195331\pi$
$654$		0			0
$655$	19.3424			0.755769
$656$	−10.1561			−0.396530
$657$		0			0
$658$	−19.6874	+	24.9620i	−0.767495	+	0.973122i
$659$		−	7.74588i		−	0.301736i	−0.988554	−	0.150868i	$-0.951793\pi$
$659$		−	7.74588i		−	0.301736i	0.988554	−	0.150868i	$-0.0482069\pi$
$660$		0			0
$661$		−	31.2167i		−	1.21419i	−0.794629	−	0.607095i	$-0.792335\pi$
$661$		−	31.2167i		−	1.21419i	0.794629	−	0.607095i	$-0.207665\pi$
$662$			14.8017i			0.575283i
$663$		0			0
$664$		−	4.06286i		−	0.157670i
$665$	10.4817	−	13.2900i	0.406463	−	0.515363i
$666$		0			0
$667$	−19.4951			−0.754851
$668$	−6.69765			−0.259140
$669$		0			0
$670$		−	0.0817013i		−	0.00315639i
$671$	−10.2735			−0.396605
$672$		0			0
$673$	40.5603			1.56348			0.781742	−	0.623602i	$-0.214332\pi$
$673$	40.5603			1.56348			0.781742	+	0.623602i	$0.214332\pi$
$674$		−	12.4042i		−	0.477793i
$675$		0			0
$676$	−13.0000			−0.500000
$677$	−42.7038			−1.64124			−0.820619	−	0.571475i	$-0.806371\pi$
$677$	−42.7038			−1.64124			−0.820619	+	0.571475i	$0.806371\pi$
$678$		0			0
$679$	25.5897	−	32.4457i	0.982043	−	1.24515i
$680$			2.18251i			0.0836954i
$681$		0			0
$682$		−	10.0122i		−	0.383386i
$683$		−	24.4529i		−	0.935663i	−0.883818	−	0.467832i	$-0.845035\pi$
$683$		−	24.4529i		−	0.935663i	0.883818	−	0.467832i	$-0.154965\pi$
$684$		0			0
$685$			0.307346i			0.0117431i
$686$	16.8139	+	7.76479i	0.641959	+	0.296461i
$687$		0			0
$688$	6.15479			0.234649
$689$		0			0
$690$		0			0
$691$		−	3.95461i		−	0.150441i	−0.997167	−	0.0752203i	$-0.976034\pi$
$691$		−	3.95461i		−	0.150441i	0.997167	−	0.0752203i	$-0.0239660\pi$
$692$	−5.20100			−0.197712
$693$		0			0
$694$	−20.0179			−0.759869
$695$		−	17.7570i		−	0.673560i
$696$		0			0
$697$	23.8354			0.902829
$698$	−14.0464			−0.531666
$699$		0			0
$700$	−8.59040	−	6.77519i	−0.324687	−	0.256078i
$701$		−	9.37479i		−	0.354081i	−0.984204	−	0.177040i	$-0.943348\pi$
$701$		−	9.37479i		−	0.354081i	0.984204	−	0.177040i	$-0.0566523\pi$
$702$		0			0
$703$		−	34.7220i		−	1.30956i
$704$			1.00000i			0.0376889i
$705$		0			0
$706$		−	24.4751i		−	0.921131i
$707$	−9.45182	−	7.45459i	−0.355472	−	0.280359i
$708$		0			0
$709$	−16.4461			−0.617645			−0.308823	−	0.951120i	$-0.599935\pi$
$709$	−16.4461			−0.617645			−0.308823	+	0.951120i	$0.599935\pi$
$710$	10.7248			0.402495
$711$		0			0
$712$			17.0354i			0.638429i
$713$	−30.5104			−1.14262
$714$		0			0
$715$		0			0
$716$			3.91215i			0.146204i
$717$		0			0
$718$	5.04053			0.188111
$719$	−3.80861			−0.142037			−0.0710187	−	0.997475i	$-0.522625\pi$
$719$	−3.80861			−0.142037			−0.0710187	+	0.997475i	$0.522625\pi$
$720$		0			0
$721$	6.65670	−	8.44016i	0.247908	−	0.314328i
$722$		−	28.3245i		−	1.05413i
$723$		0			0
$724$			0.768549i			0.0285629i
$725$			26.4545i			0.982496i
$726$		0			0
$727$		−	44.3709i		−	1.64563i	−0.568313	−	0.822813i	$-0.692404\pi$
$727$		−	44.3709i		−	1.64563i	0.568313	−	0.822813i	$-0.307596\pi$
$728$		0			0
$729$		0			0
$730$	−11.1744			−0.413583
$731$	−14.4446			−0.534254
$732$		0			0
$733$		−	16.8272i		−	0.621528i	−0.950487	−	0.310764i	$-0.899415\pi$
$733$		−	16.8272i		−	0.621528i	0.950487	−	0.310764i	$-0.100585\pi$
$734$	6.21066			0.229240
$735$		0			0
$736$	3.04733			0.112326
$737$			0.0878550i			0.00323618i
$738$		0			0
$739$	5.97908			0.219944			0.109972	−	0.993935i	$-0.464924\pi$
$739$	5.97908			0.219944			0.109972	+	0.993935i	$0.464924\pi$
$740$	4.69379			0.172547
$741$		0			0
$742$	−23.2712	−	18.3538i	−0.854312	−	0.673790i
$743$			2.38634i			0.0875462i	0.999041	+	0.0437731i	$0.0139379\pi$
$743$			2.38634i			0.0875462i	−0.999041	+	0.0437731i	$0.986062\pi$
$744$		0			0
$745$			1.17383i			0.0430059i
$746$			9.78508i			0.358257i
$747$		0			0
$748$		−	2.34690i		−	0.0858110i
$749$	−9.12901	+	11.5748i	−0.333567	+	0.422936i
$750$		0			0
$751$	11.2187			0.409378			0.204689	−	0.978827i	$-0.434382\pi$
$751$	11.2187			0.409378			0.204689	+	0.978827i	$0.434382\pi$
$752$	−12.0160			−0.438180
$753$		0			0
$754$		0			0
$755$	−16.1606			−0.588145
$756$		0			0
$757$	−30.0315			−1.09151			−0.545757	−	0.837944i	$-0.683758\pi$
$757$	−30.0315			−1.09151			−0.545757	+	0.837944i	$0.683758\pi$
$758$		−	17.8422i		−	0.648057i
$759$		0			0
$760$	6.39743			0.232059
$761$	−2.71648			−0.0984725			−0.0492362	−	0.998787i	$-0.515679\pi$
$761$	−2.71648			−0.0984725			−0.0492362	+	0.998787i	$0.515679\pi$
$762$		0			0
$763$	32.5313	+	25.6572i	1.17771	+	0.928853i
$764$		−	20.1518i		−	0.729065i
$765$		0			0
$766$		−	34.2699i		−	1.23822i
$767$		0			0
$768$		0			0
$769$		−	31.4424i		−	1.13384i	−0.823773	−	0.566920i	$-0.808135\pi$
$769$		−	31.4424i		−	1.13384i	0.823773	−	0.566920i	$-0.191865\pi$
$770$	−1.93188	−	1.52366i	−0.0696203	−	0.0549090i
$771$		0			0
$772$	6.17571			0.222269
$773$	9.42527			0.339003			0.169502	−	0.985530i	$-0.445784\pi$
$773$	9.42527			0.339003			0.169502	+	0.985530i	$0.445784\pi$
$774$		0			0
$775$			41.4022i			1.48721i
$776$	15.6185			0.560670
$777$		0			0
$778$	−13.5927			−0.487323
$779$		−	69.8669i		−	2.50324i
$780$		0			0
$781$	−11.5326			−0.412669
$782$	−7.15176			−0.255746
$783$		0			0
$784$	1.63112	+	6.80731i	0.0582544	+	0.243118i
$785$			13.0379i			0.465342i
$786$		0			0
$787$		−	16.6301i		−	0.592800i	−0.955064	−	0.296400i	$-0.904214\pi$
$787$		−	16.6301i		−	0.592800i	0.955064	−	0.296400i	$-0.0957862\pi$
$788$		−	14.2772i		−	0.508603i
$789$		0			0
$790$			12.6470i			0.449959i
$791$	−22.4258	+	28.4341i	−0.797369	+	1.01100i
$792$		0			0
$793$		0			0
$794$	−19.1741			−0.680464
$795$		0			0
$796$			7.46620i			0.264632i
$797$	−20.5847			−0.729149			−0.364574	−	0.931174i	$-0.618786\pi$
$797$	−20.5847			−0.729149			−0.364574	+	0.931174i	$0.618786\pi$
$798$		0			0
$799$	28.2004			0.997658
$800$		−	4.13518i		−	0.146201i
$801$		0			0
$802$	14.8550			0.524548
$803$	12.0160			0.424037
$804$		0			0
$805$	−4.64310	+	5.88708i	−0.163648	+	0.207492i
$806$		0			0
$807$		0			0
$808$		−	4.54985i		−	0.160063i
$809$		−	5.39446i		−	0.189659i	−0.995494	−	0.0948295i	$-0.969769\pi$
$809$		−	5.39446i		−	0.189659i	0.995494	−	0.0948295i	$-0.0302306\pi$
$810$		0			0
$811$		−	12.5407i		−	0.440364i	−0.975459	−	0.220182i	$-0.929335\pi$
$811$		−	12.5407i		−	0.440364i	0.975459	−	0.220182i	$-0.0706651\pi$
$812$	10.4817	−	13.2900i	0.367836	−	0.466386i
$813$		0			0
$814$	−5.04733			−0.176909
$815$	−12.8726			−0.450908
$816$		0			0
$817$			42.3405i			1.48131i
$818$	−7.40395			−0.258873
$819$		0			0
$820$	9.44475			0.329825
$821$		−	16.2149i		−	0.565905i	−0.959134	−	0.282952i	$-0.908686\pi$
$821$		−	16.2149i		−	0.565905i	0.959134	−	0.282952i	$-0.0913138\pi$
$822$		0			0
$823$	36.9706			1.28871			0.644356	−	0.764725i	$-0.277125\pi$
$823$	36.9706			1.28871			0.644356	+	0.764725i	$0.277125\pi$
$824$	4.06286			0.141537
$825$		0			0
$826$	3.04733	−	3.86377i	0.106030	−	0.134438i
$827$			25.9232i			0.901439i	0.892666	+	0.450720i	$0.148833\pi$
$827$			25.9232i			0.901439i	−0.892666	+	0.450720i	$0.851167\pi$
$828$		0			0
$829$			21.1717i			0.735322i	0.929960	+	0.367661i	$0.119841\pi$
$829$			21.1717i			0.735322i	−0.929960	+	0.367661i	$0.880159\pi$
$830$			3.77828i			0.131146i
$831$		0			0
$832$		0			0
$833$	−3.82807	−	15.9760i	−0.132635	−	0.553537i
$834$		0			0
$835$	6.22852			0.215547
$836$	−6.87928			−0.237925
$837$		0			0
$838$		−	19.3383i		−	0.668030i
$839$	−8.05511			−0.278093			−0.139047	−	0.990286i	$-0.544404\pi$
$839$	−8.05511			−0.278093			−0.139047	+	0.990286i	$0.544404\pi$
$840$		0			0
$841$	−11.9271			−0.411278
$842$			34.0179i			1.17233i
$843$		0			0
$844$	−16.4252			−0.565377
$845$	12.0894			0.415889
$846$		0			0
$847$	2.07739	+	1.63843i	0.0713801	+	0.0562970i
$848$		−	11.2021i		−	0.384682i
$849$		0			0
$850$			9.70484i			0.332873i
$851$			15.3809i			0.527249i
$852$		0			0
$853$		−	41.9618i		−	1.43674i	−0.695660	−	0.718372i	$-0.744888\pi$
$853$		−	41.9618i		−	1.43674i	0.695660	−	0.718372i	$-0.255112\pi$
$854$	−21.3422	−	16.8324i	−0.730313	−	0.575993i
$855$		0			0
$856$	−5.57182			−0.190441
$857$	42.8834			1.46487			0.732434	−	0.680838i	$-0.238384\pi$
$857$	42.8834			1.46487			0.732434	+	0.680838i	$0.238384\pi$
$858$		0			0
$859$		−	45.3532i		−	1.54743i	−0.633532	−	0.773716i	$-0.718396\pi$
$859$		−	45.3532i		−	1.54743i	0.633532	−	0.773716i	$-0.281604\pi$
$860$	−5.72368			−0.195176
$861$		0			0
$862$	−2.92191			−0.0995208
$863$		−	10.0768i		−	0.343017i	−0.985183	−	0.171509i	$-0.945136\pi$
$863$		−	10.0768i		−	0.343017i	0.985183	−	0.171509i	$-0.0548641\pi$
$864$		0			0
$865$	4.83670			0.164453
$866$	9.60690			0.326456
$867$		0			0
$868$	16.4042	−	20.7992i	0.556796	−	0.705972i
$869$		−	13.5995i		−	0.461333i
$870$		0			0
$871$		0			0
$872$			15.6597i			0.530303i
$873$		0			0
$874$			20.9634i			0.709098i
$875$	17.6481	+	13.9189i	0.596615	+	0.470546i
$876$		0			0
$877$	−18.9339			−0.639351			−0.319676	−	0.947527i	$-0.603574\pi$
$877$	−18.9339			−0.639351			−0.319676	+	0.947527i	$0.603574\pi$
$878$	36.6616			1.23727
$879$		0			0
$880$		−	0.929956i		−	0.0313488i
$881$	53.5991			1.80580			0.902900	−	0.429850i	$-0.141434\pi$
$881$	53.5991			1.80580			0.902900	+	0.429850i	$0.141434\pi$
$882$		0			0
$883$	16.1774			0.544412			0.272206	−	0.962239i	$-0.412247\pi$
$883$	16.1774			0.544412			0.272206	+	0.962239i	$0.412247\pi$
$884$		0			0
$885$		0			0
$886$	28.7624			0.966293
$887$	−4.64069			−0.155819			−0.0779096	−	0.996960i	$-0.524825\pi$
$887$	−4.64069			−0.155819			−0.0779096	+	0.996960i	$0.524825\pi$
$888$		0			0
$889$	−10.3462	−	8.16001i	−0.347002	−	0.273678i
$890$		−	15.8422i		−	0.531031i
$891$		0			0
$892$			6.81504i			0.228184i
$893$		−	82.6617i		−	2.76617i
$894$		0			0
$895$		−	3.63812i		−	0.121609i
$896$	−1.63843	+	2.07739i	−0.0547360	+	0.0694008i
$897$		0			0
$898$	24.0916			0.803948
$899$	−64.0522			−2.13626
$900$		0			0
$901$			26.2902i			0.875853i
$902$	−10.1561			−0.338162
$903$		0			0
$904$	−13.6874			−0.455236
$905$		−	0.714716i		−	0.0237580i
$906$		0			0
$907$	−18.1407			−0.602351			−0.301175	−	0.953569i	$-0.597379\pi$
$907$	−18.1407			−0.602351			−0.301175	+	0.953569i	$0.597379\pi$
$908$	−0.0265489			−0.000881055
$909$		0			0
$910$		0			0
$911$		−	10.7378i		−	0.355758i	−0.984052	−	0.177879i	$-0.943076\pi$
$911$		−	10.7378i		−	0.355758i	0.984052	−	0.177879i	$-0.0569235\pi$
$912$		0			0
$913$		−	4.06286i		−	0.134461i
$914$		−	10.4853i		−	0.346822i
$915$		0			0
$916$		−	9.69570i		−	0.320355i
$917$	−43.2082	−	34.0780i	−1.42686	−	1.12536i
$918$		0			0
$919$	−3.40343			−0.112269			−0.0561344	−	0.998423i	$-0.517878\pi$
$919$	−3.40343			−0.112269			−0.0561344	+	0.998423i	$0.517878\pi$
$920$	−2.83388			−0.0934302
$921$		0			0
$922$			20.4562i			0.673689i
$923$		0			0
$924$		0			0
$925$	20.8716			0.686255
$926$			36.4989i			1.19943i
$927$		0			0
$928$	6.39743			0.210006
$929$	−19.1301			−0.627639			−0.313819	−	0.949483i	$-0.601609\pi$
$929$	−19.1301			−0.627639			−0.313819	+	0.949483i	$0.601609\pi$
$930$		0			0
$931$	−46.8294	+	11.2209i	−1.53477	+	0.367752i
$932$			2.48528i			0.0814081i
$933$		0			0
$934$			16.9557i			0.554808i
$935$			2.18251i			0.0713757i
$936$		0			0
$937$			12.6672i			0.413819i	0.978360	+	0.206910i	$0.0663406\pi$
$937$			12.6672i			0.413819i	−0.978360	+	0.206910i	$0.933659\pi$
$938$	−0.143944	+	0.182509i	−0.00469993	+	0.00595914i
$939$		0			0
$940$	11.1744			0.364468
$941$	−8.32277			−0.271314			−0.135657	−	0.990756i	$-0.543315\pi$
$941$	−8.32277			−0.271314			−0.135657	+	0.990756i	$0.543315\pi$
$942$		0			0
$943$			30.9491i			1.00784i
$944$	1.85991			0.0605350
$945$		0			0
$946$	6.15479			0.200109
$947$			51.0021i			1.65734i	0.559735	+	0.828672i	$0.310903\pi$
$947$			51.0021i			1.65734i	−0.559735	+	0.828672i	$0.689097\pi$
$948$		0			0
$949$		0			0
$950$	28.4471			0.922945
$951$		0			0
$952$	3.84521	−	4.87542i	0.124624	−	0.158013i
$953$			38.6191i			1.25100i	0.780225	+	0.625498i	$0.215104\pi$
$953$			38.6191i			1.25100i	−0.780225	+	0.625498i	$0.784896\pi$
$954$		0			0
$955$			18.7402i			0.606420i
$956$			11.4380i			0.369930i
$957$		0			0
$958$			31.9385i			1.03189i
$959$	0.541493	−	0.686569i	0.0174857	−	0.0221705i
$960$		0			0
$961$	−69.2439			−2.23367
$962$		0			0
$963$		0			0
$964$			7.69184i			0.247738i
$965$	−5.74314			−0.184878
$966$		0			0
$967$	−21.2456			−0.683213			−0.341606	−	0.939843i	$-0.610971\pi$
$967$	−21.2456			−0.683213			−0.341606	+	0.939843i	$0.610971\pi$
$968$			1.00000i			0.0321412i
$969$		0			0
$970$	−14.5245			−0.466353
$971$	28.5665			0.916742			0.458371	−	0.888761i	$-0.348433\pi$
$971$	28.5665			0.916742			0.458371	+	0.888761i	$0.348433\pi$
$972$		0			0
$973$	−31.2848	+	39.6666i	−1.00294	+	1.27165i
$974$			6.57994i			0.210835i
$975$		0			0
$976$		−	10.2735i		−	0.328848i
$977$			2.65294i			0.0848749i	0.999099	+	0.0424375i	$0.0135123\pi$
$977$			2.65294i			0.0848749i	−0.999099	+	0.0424375i	$0.986488\pi$
$978$		0			0
$979$			17.0354i			0.544454i
$980$	−1.51687	−	6.33050i	−0.0484547	−	0.202220i
$981$		0			0
$982$	−9.14198			−0.291732
$983$	14.5620			0.464457			0.232228	−	0.972661i	$-0.425398\pi$
$983$	14.5620			0.464457			0.232228	+	0.972661i	$0.425398\pi$
$984$		0			0
$985$			13.2771i			0.423045i
$986$	−15.0141			−0.478146
$987$		0			0
$988$		0			0
$989$		−	18.7556i		−	0.596395i
$990$		0			0
$991$	−0.254124			−0.00807252			−0.00403626	−	0.999992i	$-0.501285\pi$
$991$	−0.254124			−0.00807252			−0.00403626	+	0.999992i	$0.501285\pi$
$992$	10.0122			0.317887
$993$		0			0
$994$	−23.9578	−	18.8953i	−0.759894	−	0.599323i
$995$		−	6.94323i		−	0.220115i
$996$		0			0
$997$		−	9.01167i		−	0.285402i	−0.989766	−	0.142701i	$-0.954421\pi$
$997$		−	9.01167i		−	0.285402i	0.989766	−	0.142701i	$-0.0455788\pi$
$998$		−	6.47848i		−	0.205073i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.g.a.881.13	yes	16
3.2	odd	2	inner	1386.2.g.a.881.4	✓	16
7.6	odd	2	inner	1386.2.g.a.881.12	yes	16
21.20	even	2	inner	1386.2.g.a.881.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.g.a.881.4	✓	16	3.2	odd	2	inner
1386.2.g.a.881.5	yes	16	21.20	even	2	inner
1386.2.g.a.881.12	yes	16	7.6	odd	2	inner
1386.2.g.a.881.13	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.929956 q^{5} +(-2.07739 - 1.63843i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.929956 q^{5} +(-2.07739 - 1.63843i) q^{7} -1.00000i q^{8} +0.929956i q^{10} -1.00000i q^{11} +(1.63843 - 2.07739i) q^{14} +1.00000 q^{16} -2.34690 q^{17} +6.87928i q^{19} -0.929956 q^{20} +1.00000 q^{22} -3.04733i q^{23} -4.13518 q^{25} +(2.07739 + 1.63843i) q^{28} -6.39743i q^{29} -10.0122i q^{31} +1.00000i q^{32} -2.34690i q^{34} +(-1.93188 - 1.52366i) q^{35} -5.04733 q^{37} -6.87928 q^{38} -0.929956i q^{40} -10.1561 q^{41} +6.15479 q^{43} +1.00000i q^{44} +3.04733 q^{46} -12.0160 q^{47} +(1.63112 + 6.80731i) q^{49} -4.13518i q^{50} -11.2021i q^{53} -0.929956i q^{55} +(-1.63843 + 2.07739i) q^{56} +6.39743 q^{58} +1.85991 q^{59} -10.2735i q^{61} +10.0122 q^{62} -1.00000 q^{64} -0.0878550 q^{67} +2.34690 q^{68} +(1.52366 - 1.93188i) q^{70} -11.5326i q^{71} +12.0160i q^{73} -5.04733i q^{74} -6.87928i q^{76} +(-1.63843 + 2.07739i) q^{77} +13.5995 q^{79} +0.929956 q^{80} -10.1561i q^{82} +4.06286 q^{83} -2.18251 q^{85} +6.15479i q^{86} -1.00000 q^{88} -17.0354 q^{89} +3.04733i q^{92} -12.0160i q^{94} +6.39743i q^{95} +15.6185i q^{97} +(-6.80731 + 1.63112i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) 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

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