LMFDB - Embedded newform 1386.2.g.b.881.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1386.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 72x^{12} + 1296x^{10} + 8245x^{8} + 18780x^{6} + 20808x^{4} - 3672x^{2} + 324 $ x^16 - 12x^14 + 72x^12 + 1296x^10 + 8245x^8 + 18780x^6 + 20808x^4 - 3672*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.10
Root			$3.59322 - 1.48836i$ of defining polynomial
Character	$\chi$	$=$	1386.881
Dual form			1386.2.g.b.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -2.97672 q^{5} +(2.55176 + 0.698947i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -2.97672 q^{5} +(2.55176 + 0.698947i) q^{7} -1.00000i q^{8} -2.97672i q^{10} +1.00000i q^{11} -1.97692i q^{13} +(-0.698947 + 2.55176i) q^{14} +1.00000 q^{16} +1.57883 q^{17} -7.18644i q^{19} +2.97672 q^{20} -1.00000 q^{22} +3.86088 q^{25} +1.97692 q^{26} +(-2.55176 - 0.698947i) q^{28} -1.30027i q^{29} +5.78855i q^{31} +1.00000i q^{32} +1.57883i q^{34} +(-7.59588 - 2.08057i) q^{35} +8.04589 q^{37} +7.18644 q^{38} +2.97672i q^{40} -3.81163 q^{41} +9.14941 q^{43} -1.00000i q^{44} +2.63089 q^{47} +(6.02295 + 3.56709i) q^{49} +3.86088i q^{50} +1.97692i q^{52} +8.94237i q^{53} -2.97672i q^{55} +(0.698947 - 2.55176i) q^{56} +1.30027 q^{58} +9.90729 q^{59} -4.77271i q^{61} -5.78855 q^{62} -1.00000 q^{64} +5.88475i q^{65} +0.699733 q^{67} -1.57883 q^{68} +(2.08057 - 7.59588i) q^{70} -8.48528i q^{71} +12.2311i q^{73} +8.04589i q^{74} +7.18644i q^{76} +(-0.698947 + 2.55176i) q^{77} +11.9644 q^{79} -2.97672 q^{80} -3.81163i q^{82} +17.9286 q^{83} -4.69973 q^{85} +9.14941i q^{86} +1.00000 q^{88} +8.20228 q^{89} +(1.38176 - 5.04463i) q^{91} +2.63089i q^{94} +21.3920i q^{95} +8.41944i q^{97} +(-3.56709 + 6.02295i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 16 q^{22} + 48 q^{25} - 16 q^{37} - 80 q^{43} + 24 q^{49} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 24 q^{70} + 96 q^{79} - 80 q^{85} + 16 q^{88} - 32 q^{91}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^22 + 48 * q^25 - 16 * q^37 - 80 * q^43 + 24 * q^49 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 24 * q^70 + 96 * q^79 - 80 * q^85 + 16 * q^88 - 32 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−2.97672			−1.33123			−0.665615	−	0.746295i	$-0.731831\pi$
$5$	−2.97672			−1.33123			−0.665615	+	0.746295i	$0.731831\pi$
$6$		0			0
$7$	2.55176	+	0.698947i	0.964474	+	0.264177i
$7$	2.55176	+	0.698947i	0.964474	+	0.264177i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	2.97672i		−	0.941322i
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		−	1.97692i		−	0.548299i	−0.961687	−	0.274150i	$-0.911604\pi$
$13$		−	1.97692i		−	0.548299i	0.961687	−	0.274150i	$-0.0883964\pi$
$14$	−0.698947	+	2.55176i	−0.186802	+	0.681986i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.57883			0.382922			0.191461	−	0.981500i	$-0.438677\pi$
$17$	1.57883			0.382922			0.191461	+	0.981500i	$0.438677\pi$
$18$		0			0
$19$		−	7.18644i		−	1.64868i	−0.566093	−	0.824342i	$-0.691546\pi$
$19$		−	7.18644i		−	1.64868i	0.566093	−	0.824342i	$-0.308454\pi$
$20$	2.97672			0.665615
$21$		0			0
$22$	−1.00000			−0.213201
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.86088			0.772175
$26$	1.97692			0.387706
$27$		0			0
$28$	−2.55176	−	0.698947i	−0.482237	−	0.132089i
$29$		−	1.30027i		−	0.241454i	−0.992686	−	0.120727i	$-0.961478\pi$
$29$		−	1.30027i		−	0.241454i	0.992686	−	0.120727i	$-0.0385225\pi$
$30$		0			0
$31$			5.78855i			1.03965i	0.854272	+	0.519827i	$0.174004\pi$
$31$			5.78855i			1.03965i	−0.854272	+	0.519827i	$0.825996\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			1.57883i			0.270767i
$35$	−7.59588	−	2.08057i	−1.28394	−	0.351681i
$36$		0			0
$37$	8.04589			1.32274			0.661368	−	0.750061i	$-0.269976\pi$
$37$	8.04589			1.32274			0.661368	+	0.750061i	$0.269976\pi$
$38$	7.18644			1.16580
$39$		0			0
$40$			2.97672i			0.470661i
$41$	−3.81163			−0.595276			−0.297638	−	0.954679i	$-0.596199\pi$
$41$	−3.81163			−0.595276			−0.297638	+	0.954679i	$0.596199\pi$
$42$		0			0
$43$	9.14941			1.39527			0.697636	−	0.716453i	$-0.254235\pi$
$43$	9.14941			1.39527			0.697636	+	0.716453i	$0.254235\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$		0			0
$47$	2.63089			0.383755			0.191878	−	0.981419i	$-0.438542\pi$
$47$	2.63089			0.383755			0.191878	+	0.981419i	$0.438542\pi$
$48$		0			0
$49$	6.02295	+	3.56709i	0.860421	+	0.509584i
$50$			3.86088i			0.546010i
$51$		0			0
$52$			1.97692i			0.274150i
$53$			8.94237i			1.22833i	0.789178	+	0.614165i	$0.210507\pi$
$53$			8.94237i			1.22833i	−0.789178	+	0.614165i	$0.789493\pi$
$54$		0			0
$55$		−	2.97672i		−	0.401381i
$56$	0.698947	−	2.55176i	0.0934008	−	0.340993i
$57$		0			0
$58$	1.30027			0.170733
$59$	9.90729			1.28982			0.644910	−	0.764259i	$-0.276895\pi$
$59$	9.90729			1.28982			0.644910	+	0.764259i	$0.276895\pi$
$60$		0			0
$61$		−	4.77271i		−	0.611083i	−0.952179	−	0.305541i	$-0.901163\pi$
$61$		−	4.77271i		−	0.611083i	0.952179	−	0.305541i	$-0.0988375\pi$
$62$	−5.78855			−0.735146
$63$		0			0
$64$	−1.00000			−0.125000
$65$			5.88475i			0.729913i
$66$		0			0
$67$	0.699733			0.0854859			0.0427430	−	0.999086i	$-0.486390\pi$
$67$	0.699733			0.0854859			0.0427430	+	0.999086i	$0.486390\pi$
$68$	−1.57883			−0.191461
$69$		0			0
$70$	2.08057	−	7.59588i	0.248676	−	0.907881i
$71$		−	8.48528i		−	1.00702i	−0.863990	−	0.503509i	$-0.832042\pi$
$71$		−	8.48528i		−	1.00702i	0.863990	−	0.503509i	$-0.167958\pi$
$72$		0			0
$73$			12.2311i			1.43154i	0.698337	+	0.715769i	$0.253924\pi$
$73$			12.2311i			1.43154i	−0.698337	+	0.715769i	$0.746076\pi$
$74$			8.04589i			0.935316i
$75$		0			0
$76$			7.18644i			0.824342i
$77$	−0.698947	+	2.55176i	−0.0796524	+	0.290800i
$78$		0			0
$79$	11.9644			1.34610			0.673050	−	0.739597i	$-0.264984\pi$
$79$	11.9644			1.34610			0.673050	+	0.739597i	$0.264984\pi$
$80$	−2.97672			−0.332808
$81$		0			0
$82$		−	3.81163i		−	0.420924i
$83$	17.9286			1.96792			0.983962	−	0.178379i	$-0.0570854\pi$
$83$	17.9286			1.96792			0.983962	+	0.178379i	$0.0570854\pi$
$84$		0			0
$85$	−4.69973			−0.509758
$86$			9.14941i			0.986606i
$87$		0			0
$88$	1.00000			0.106600
$89$	8.20228			0.869440			0.434720	−	0.900566i	$-0.356847\pi$
$89$	8.20228			0.869440			0.434720	+	0.900566i	$0.356847\pi$
$90$		0			0
$91$	1.38176	−	5.04463i	0.144848	−	0.528821i
$92$		0			0
$93$		0			0
$94$			2.63089i			0.271356i
$95$			21.3920i			2.19478i
$96$		0			0
$97$			8.41944i			0.854865i	0.904047	+	0.427432i	$0.140582\pi$
$97$			8.41944i			0.854865i	−0.904047	+	0.427432i	$0.859418\pi$
$98$	−3.56709	+	6.02295i	−0.360330	+	0.608409i
$99$		0			0
$100$	−3.86088			−0.386088
$101$	−10.3964			−1.03448			−0.517238	−	0.855841i	$-0.673040\pi$
$101$	−10.3964			−1.03448			−0.517238	+	0.855841i	$0.673040\pi$
$102$		0			0
$103$		−	13.6129i		−	1.34132i	−0.741764	−	0.670661i	$-0.766011\pi$
$103$		−	13.6129i		−	1.34132i	0.741764	−	0.670661i	$-0.233989\pi$
$104$	−1.97692			−0.193853
$105$		0			0
$106$	−8.94237			−0.868560
$107$			4.16114i			0.402273i	0.979563	+	0.201136i	$0.0644635\pi$
$107$			4.16114i			0.402273i	−0.979563	+	0.201136i	$0.935537\pi$
$108$		0			0
$109$	8.42149			0.806632			0.403316	−	0.915061i	$-0.367858\pi$
$109$	8.42149			0.806632			0.403316	+	0.915061i	$0.367858\pi$
$110$	2.97672			0.283819
$111$		0			0
$112$	2.55176	+	0.698947i	0.241119	+	0.0660443i
$113$		−	1.38176i		−	0.129985i	−0.997886	−	0.0649927i	$-0.979298\pi$
$113$		−	1.38176i		−	0.129985i	0.997886	−	0.0649927i	$-0.0207024\pi$
$114$		0			0
$115$		0			0
$116$			1.30027i			0.120727i
$117$		0			0
$118$			9.90729i			0.912040i
$119$	4.02879	+	1.10352i	0.369318	+	0.101159i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	4.77271			0.432101
$123$		0			0
$124$		−	5.78855i		−	0.519827i
$125$	3.39085			0.303287
$126$		0			0
$127$	−16.4497			−1.45967			−0.729836	−	0.683622i	$-0.760404\pi$
$127$	−16.4497			−1.45967			−0.729836	+	0.683622i	$0.760404\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−5.88475			−0.516126
$131$	−8.84022			−0.772373			−0.386187	−	0.922421i	$-0.626208\pi$
$131$	−8.84022			−0.772373			−0.386187	+	0.922421i	$0.626208\pi$
$132$		0			0
$133$	5.02295	−	18.3381i	0.435545	−	1.59011i
$134$			0.699733i			0.0604477i
$135$		0			0
$136$		−	1.57883i		−	0.135383i
$137$		−	2.66413i		−	0.227612i	−0.993503	−	0.113806i	$-0.963696\pi$
$137$		−	2.66413i		−	0.227612i	0.993503	−	0.113806i	$-0.0363042\pi$
$138$		0			0
$139$		−	18.1685i		−	1.54103i	−0.637422	−	0.770515i	$-0.719999\pi$
$139$		−	18.1685i		−	1.54103i	0.637422	−	0.770515i	$-0.280001\pi$
$140$	7.59588	+	2.08057i	0.641969	+	0.175840i
$141$		0			0
$142$	8.48528			0.712069
$143$	1.97692			0.165318
$144$		0			0
$145$			3.87054i			0.321430i
$146$	−12.2311			−1.01225
$147$		0			0
$148$	−8.04589			−0.661368
$149$		−	10.0459i		−	0.822991i	−0.911412	−	0.411496i	$-0.865007\pi$
$149$		−	10.0459i		−	0.822991i	0.911412	−	0.411496i	$-0.134993\pi$
$150$		0			0
$151$	−10.5489			−0.858455			−0.429228	−	0.903196i	$-0.641214\pi$
$151$	−10.5489			−0.858455			−0.429228	+	0.903196i	$0.641214\pi$
$152$	−7.18644			−0.582898
$153$		0			0
$154$	−2.55176	−	0.698947i	−0.205627	−	0.0563228i
$155$		−	17.2309i		−	1.38402i
$156$		0			0
$157$		−	5.78855i		−	0.461977i	−0.972957	−	0.230988i	$-0.925804\pi$
$157$		−	5.78855i		−	0.461977i	0.972957	−	0.230988i	$-0.0741959\pi$
$158$			11.9644i			0.951836i
$159$		0			0
$160$		−	2.97672i		−	0.235331i
$161$		0			0
$162$		0			0
$163$	−13.7676			−1.07837			−0.539183	−	0.842189i	$-0.681267\pi$
$163$	−13.7676			−1.07837			−0.539183	+	0.842189i	$0.681267\pi$
$164$	3.81163			0.297638
$165$		0			0
$166$			17.9286i			1.39153i
$167$	−6.26056			−0.484456			−0.242228	−	0.970219i	$-0.577878\pi$
$167$	−6.26056			−0.484456			−0.242228	+	0.970219i	$0.577878\pi$
$168$		0			0
$169$	9.09178			0.699368
$170$		−	4.69973i		−	0.360453i
$171$		0			0
$172$	−9.14941			−0.697636
$173$	−1.18073			−0.0897696			−0.0448848	−	0.998992i	$-0.514292\pi$
$173$	−1.18073			−0.0897696			−0.0448848	+	0.998992i	$0.514292\pi$
$174$		0			0
$175$	9.85203	+	2.69855i	0.744743	+	0.203991i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$			8.20228i			0.614787i
$179$			1.57851i			0.117984i	0.998258	+	0.0589918i	$0.0187886\pi$
$179$			1.57851i			0.117984i	−0.998258	+	0.0589918i	$0.981211\pi$
$180$		0			0
$181$		−	1.01584i		−	0.0755067i	−0.999287	−	0.0377533i	$-0.987980\pi$
$181$		−	1.01584i		−	0.0755067i	0.999287	−	0.0377533i	$-0.0120201\pi$
$182$	5.04463	+	1.38176i	0.373933	+	0.102423i
$183$		0			0
$184$		0			0
$185$	−23.9504			−1.76087
$186$		0			0
$187$			1.57883i			0.115455i
$188$	−2.63089			−0.191878
$189$		0			0
$190$	−21.3920			−1.55194
$191$		−	8.48528i		−	0.613973i	−0.951714	−	0.306987i	$-0.900679\pi$
$191$		−	8.48528i		−	0.613973i	0.951714	−	0.306987i	$-0.0993207\pi$
$192$		0			0
$193$	−4.60053			−0.331154			−0.165577	−	0.986197i	$-0.552949\pi$
$193$	−4.60053			−0.331154			−0.165577	+	0.986197i	$0.552949\pi$
$194$	−8.41944			−0.604481
$195$		0			0
$196$	−6.02295	−	3.56709i	−0.430210	−	0.254792i
$197$		−	23.9926i		−	1.70940i	−0.519122	−	0.854700i	$-0.673741\pi$
$197$		−	23.9926i		−	1.70940i	0.519122	−	0.854700i	$-0.326259\pi$
$198$		0			0
$199$			11.3801i			0.806716i	0.915042	+	0.403358i	$0.132157\pi$
$199$			11.3801i			0.806716i	−0.915042	+	0.403358i	$0.867843\pi$
$200$		−	3.86088i		−	0.273005i
$201$		0			0
$202$		−	10.3964i		−	0.731486i
$203$	0.908818	−	3.31797i	0.0637865	−	0.232876i
$204$		0			0
$205$	11.3462			0.792450
$206$	13.6129			0.948458
$207$		0			0
$208$		−	1.97692i		−	0.137075i
$209$	7.18644			0.497097
$210$		0			0
$211$	−13.5429			−0.932332			−0.466166	−	0.884697i	$-0.654365\pi$
$211$	−13.5429			−0.932332			−0.466166	+	0.884697i	$0.654365\pi$
$212$		−	8.94237i		−	0.614165i
$213$		0			0
$214$	−4.16114			−0.284450
$215$	−27.2352			−1.85743
$216$		0			0
$217$	−4.04589	+	14.7710i	−0.274653	+	1.00272i
$218$			8.42149i			0.570375i
$219$		0			0
$220$			2.97672i			0.200691i
$221$		−	3.12122i		−	0.209956i
$222$		0			0
$223$			24.1153i			1.61488i	0.589951	+	0.807439i	$0.299147\pi$
$223$			24.1153i			1.61488i	−0.589951	+	0.807439i	$0.700853\pi$
$224$	−0.698947	+	2.55176i	−0.0467004	+	0.170497i
$225$		0			0
$226$	1.38176			0.0919136
$227$	19.0034			1.26130			0.630649	−	0.776068i	$-0.282789\pi$
$227$	19.0034			1.26130			0.630649	+	0.776068i	$0.282789\pi$
$228$		0			0
$229$			19.0546i			1.25916i	0.776934	+	0.629582i	$0.216774\pi$
$229$			19.0546i			1.25916i	−0.776934	+	0.629582i	$0.783226\pi$
$230$		0			0
$231$		0			0
$232$	−1.30027			−0.0853667
$233$		−	13.9288i		−	0.912505i	−0.889850	−	0.456253i	$-0.849191\pi$
$233$		−	13.9288i		−	0.912505i	0.889850	−	0.456253i	$-0.150809\pi$
$234$		0			0
$235$	−7.83144			−0.510867
$236$	−9.90729			−0.644910
$237$		0			0
$238$	−1.10352	+	4.02879i	−0.0715304	+	0.261148i
$239$			9.76764i			0.631816i	0.948790	+	0.315908i	$0.102309\pi$
$239$			9.76764i			0.631816i	−0.948790	+	0.315908i	$0.897691\pi$
$240$		0			0
$241$		−	10.4326i		−	0.672022i	−0.941858	−	0.336011i	$-0.890922\pi$
$241$		−	10.4326i		−	0.672022i	0.941858	−	0.336011i	$-0.109078\pi$
$242$		−	1.00000i		−	0.0642824i
$243$		0			0
$244$			4.77271i			0.305541i
$245$	−17.9286	−	10.6182i	−1.14542	−	0.678374i
$246$		0			0
$247$	−14.2070			−0.903972
$248$	5.78855			0.367573
$249$		0			0
$250$			3.39085i			0.214456i
$251$	21.6343			1.36554			0.682772	−	0.730632i	$-0.260774\pi$
$251$	21.6343			1.36554			0.682772	+	0.730632i	$0.260774\pi$
$252$		0			0
$253$		0			0
$254$		−	16.4497i		−	1.03214i
$255$		0			0
$256$	1.00000			0.0625000
$257$	23.8831			1.48979			0.744894	−	0.667183i	$-0.232500\pi$
$257$	23.8831			1.48979			0.744894	+	0.667183i	$0.232500\pi$
$258$		0			0
$259$	20.5312	+	5.62365i	1.27574	+	0.349437i
$260$		−	5.88475i		−	0.364956i
$261$		0			0
$262$		−	8.84022i		−	0.546150i
$263$			18.9068i			1.16584i	0.812529	+	0.582921i	$0.198090\pi$
$263$			18.9068i			1.16584i	−0.812529	+	0.582921i	$0.801910\pi$
$264$		0			0
$265$		−	26.6190i		−	1.63519i
$266$	18.3381	+	5.02295i	1.12438	+	0.307977i
$267$		0			0
$268$	−0.699733			−0.0427430
$269$	−8.00531			−0.488092			−0.244046	−	0.969764i	$-0.578475\pi$
$269$	−8.00531			−0.488092			−0.244046	+	0.969764i	$0.578475\pi$
$270$		0			0
$271$			26.0079i			1.57987i	0.613194	+	0.789933i	$0.289884\pi$
$271$			26.0079i			1.57987i	−0.613194	+	0.789933i	$0.710116\pi$
$272$	1.57883			0.0957305
$273$		0			0
$274$	2.66413			0.160946
$275$			3.86088i			0.232820i
$276$		0			0
$277$	26.7915			1.60975			0.804873	−	0.593447i	$-0.202233\pi$
$277$	26.7915			1.60975			0.804873	+	0.593447i	$0.202233\pi$
$278$	18.1685			1.08967
$279$		0			0
$280$	−2.08057	+	7.59588i	−0.124338	+	0.453941i
$281$		−	28.2988i		−	1.68817i	−0.536213	−	0.844083i	$-0.680145\pi$
$281$		−	28.2988i		−	1.68817i	0.536213	−	0.844083i	$-0.319855\pi$
$282$		0			0
$283$		−	14.5765i		−	0.866483i	−0.901278	−	0.433242i	$-0.857370\pi$
$283$		−	14.5765i		−	0.866483i	0.901278	−	0.433242i	$-0.142630\pi$
$284$			8.48528i			0.503509i
$285$		0			0
$286$			1.97692i			0.116898i
$287$	−9.72635	−	2.66413i	−0.574129	−	0.157258i
$288$		0			0
$289$	−14.5073			−0.853371
$290$	−3.87054			−0.227286
$291$		0			0
$292$		−	12.2311i		−	0.715769i
$293$	−15.8834			−0.927919			−0.463959	−	0.885856i	$-0.653572\pi$
$293$	−15.8834			−0.927919			−0.463959	+	0.885856i	$0.653572\pi$
$294$		0			0
$295$	−29.4912			−1.71705
$296$		−	8.04589i		−	0.467658i
$297$		0			0
$298$	10.0459			0.581943
$299$		0			0
$300$		0			0
$301$	23.3471	+	6.39495i	1.34570	+	0.368599i
$302$		−	10.5489i		−	0.607019i
$303$		0			0
$304$		−	7.18644i		−	0.412171i
$305$			14.2070i			0.813492i
$306$		0			0
$307$		−	1.05103i		−	0.0599856i	−0.999550	−	0.0299928i	$-0.990452\pi$
$307$		−	1.05103i		−	0.0599856i	0.999550	−	0.0299928i	$-0.00954844\pi$
$308$	0.698947	−	2.55176i	0.0398262	−	0.145400i
$309$		0			0
$310$	17.2309			0.978650
$311$	18.0785			1.02514			0.512570	−	0.858646i	$-0.328694\pi$
$311$	18.0785			1.02514			0.512570	+	0.858646i	$0.328694\pi$
$312$		0			0
$313$		−	19.8373i		−	1.12127i	−0.828064	−	0.560634i	$-0.810557\pi$
$313$		−	19.8373i		−	1.12127i	0.828064	−	0.560634i	$-0.189443\pi$
$314$	5.78855			0.326667
$315$		0			0
$316$	−11.9644			−0.673050
$317$		−	6.34184i		−	0.356193i	−0.984013	−	0.178097i	$-0.943006\pi$
$317$		−	6.34184i		−	0.356193i	0.984013	−	0.178097i	$-0.0569939\pi$
$318$		0			0
$319$	1.30027			0.0728010
$320$	2.97672			0.166404
$321$		0			0
$322$		0			0
$323$		−	11.3462i		−	0.631317i
$324$		0			0
$325$		−	7.63265i		−	0.423383i
$326$		−	13.7676i		−	0.762519i
$327$		0			0
$328$			3.81163i			0.210462i
$329$	6.71341	+	1.83886i	0.370122	+	0.101379i
$330$		0			0
$331$	−11.7855			−0.647792			−0.323896	−	0.946093i	$-0.604993\pi$
$331$	−11.7855			−0.647792			−0.323896	+	0.946093i	$0.604993\pi$
$332$	−17.9286			−0.983962
$333$		0			0
$334$		−	6.26056i		−	0.342562i
$335$	−2.08291			−0.113802
$336$		0			0
$337$	3.88475			0.211616			0.105808	−	0.994387i	$-0.466257\pi$
$337$	3.88475			0.211616			0.105808	+	0.994387i	$0.466257\pi$
$338$			9.09178i			0.494528i
$339$		0			0
$340$	4.69973			0.254879
$341$	−5.78855			−0.313468
$342$		0			0
$343$	12.8759	+	13.3121i	0.695233	+	0.718784i
$344$		−	9.14941i		−	0.493303i
$345$		0			0
$346$		−	1.18073i		−	0.0634767i
$347$		−	24.2529i		−	1.30197i	−0.759093	−	0.650983i	$-0.774357\pi$
$347$		−	24.2529i		−	1.30197i	0.759093	−	0.650983i	$-0.225643\pi$
$348$		0			0
$349$			1.87239i			0.100227i	0.998744	+	0.0501134i	$0.0159583\pi$
$349$			1.87239i			0.100227i	−0.998744	+	0.0501134i	$0.984042\pi$
$350$	−2.69855	+	9.85203i	−0.144244	+	0.526613i
$351$		0			0
$352$	−1.00000			−0.0533002
$353$	−26.1351			−1.39103			−0.695515	−	0.718512i	$-0.744824\pi$
$353$	−26.1351			−1.39103			−0.695515	+	0.718512i	$0.744824\pi$
$354$		0			0
$355$			25.2583i			1.34057i
$356$	−8.20228			−0.434720
$357$		0			0
$358$	−1.57851			−0.0834270
$359$			2.74562i			0.144908i	0.997372	+	0.0724542i	$0.0230831\pi$
$359$			2.74562i			0.144908i	−0.997372	+	0.0724542i	$0.976917\pi$
$360$		0			0
$361$	−32.6450			−1.71816
$362$	1.01584			0.0533913
$363$		0			0
$364$	−1.38176	+	5.04463i	−0.0724241	+	0.264410i
$365$		−	36.4085i		−	1.90571i
$366$		0			0
$367$			22.3622i			1.16730i	0.812007	+	0.583648i	$0.198375\pi$
$367$			22.3622i			1.16730i	−0.812007	+	0.583648i	$0.801625\pi$
$368$		0			0
$369$		0			0
$370$		−	23.9504i		−	1.24512i
$371$	−6.25025	+	22.8188i	−0.324497	+	1.18469i
$372$		0			0
$373$	37.5812			1.94588			0.972940	−	0.231059i	$-0.0742189\pi$
$373$	37.5812			1.94588			0.972940	+	0.231059i	$0.0742189\pi$
$374$	−1.57883			−0.0816392
$375$		0			0
$376$		−	2.63089i		−	0.135678i
$377$	−2.57053			−0.132389
$378$		0			0
$379$	−17.7676			−0.912663			−0.456331	−	0.889810i	$-0.650837\pi$
$379$	−17.7676			−0.912663			−0.456331	+	0.889810i	$0.650837\pi$
$380$		−	21.3920i		−	1.09739i
$381$		0			0
$382$	8.48528			0.434145
$383$	−7.69156			−0.393020			−0.196510	−	0.980502i	$-0.562961\pi$
$383$	−7.69156			−0.393020			−0.196510	+	0.980502i	$0.562961\pi$
$384$		0			0
$385$	2.08057	−	7.59588i	0.106036	−	0.387122i
$386$		−	4.60053i		−	0.234161i
$387$		0			0
$388$		−	8.41944i		−	0.427432i
$389$			15.2206i			0.771716i	0.922558	+	0.385858i	$0.126095\pi$
$389$			15.2206i			0.771716i	−0.922558	+	0.385858i	$0.873905\pi$
$390$		0			0
$391$		0			0
$392$	3.56709	−	6.02295i	0.180165	−	0.304205i
$393$		0			0
$394$	23.9926			1.20873
$395$	−35.6147			−1.79197
$396$		0			0
$397$		−	9.48651i		−	0.476114i	−0.971251	−	0.238057i	$-0.923489\pi$
$397$		−	9.48651i		−	0.476114i	0.971251	−	0.238057i	$-0.0765106\pi$
$398$	−11.3801			−0.570434
$399$		0			0
$400$	3.86088			0.193044
$401$		−	5.42765i		−	0.271044i	−0.990774	−	0.135522i	$-0.956729\pi$
$401$		−	5.42765i		−	0.271044i	0.990774	−	0.135522i	$-0.0432712\pi$
$402$		0			0
$403$	11.4435			0.570042
$404$	10.3964			0.517238
$405$		0			0
$406$	3.31797	+	0.908818i	0.164668	+	0.0451039i
$407$			8.04589i			0.398820i
$408$		0			0
$409$		−	19.1818i		−	0.948480i	−0.880396	−	0.474240i	$-0.842723\pi$
$409$		−	19.1818i		−	0.948480i	0.880396	−	0.474240i	$-0.157277\pi$
$410$			11.3462i			0.560347i
$411$		0			0
$412$			13.6129i			0.670661i
$413$	25.2810	+	6.92467i	1.24400	+	0.340741i
$414$		0			0
$415$	−53.3686			−2.61976
$416$	1.97692			0.0969265
$417$		0			0
$418$			7.18644i			0.351500i
$419$	−14.7348			−0.719840			−0.359920	−	0.932983i	$-0.617196\pi$
$419$	−14.7348			−0.719840			−0.359920	+	0.932983i	$0.617196\pi$
$420$		0			0
$421$	−20.0459			−0.976977			−0.488488	−	0.872570i	$-0.662452\pi$
$421$	−20.0459			−0.976977			−0.488488	+	0.872570i	$0.662452\pi$
$422$		−	13.5429i		−	0.659258i
$423$		0			0
$424$	8.94237			0.434280
$425$	6.09566			0.295683
$426$		0			0
$427$	3.33587	−	12.1788i	0.161434	−	0.589374i
$428$		−	4.16114i		−	0.201136i
$429$		0			0
$430$		−	27.2352i		−	1.31340i
$431$		−	1.93620i		−	0.0932637i	−0.998912	−	0.0466319i	$-0.985151\pi$
$431$		−	1.93620i		−	0.0932637i	0.998912	−	0.0466319i	$-0.0148488\pi$
$432$		0			0
$433$			16.6796i			0.801570i	0.916172	+	0.400785i	$0.131263\pi$
$433$			16.6796i			0.801570i	−0.916172	+	0.400785i	$0.868737\pi$
$434$	−14.7710	−	4.04589i	−0.709030	−	0.194209i
$435$		0			0
$436$	−8.42149			−0.403316
$437$		0			0
$438$		0			0
$439$		−	20.4163i		−	0.974416i	−0.873286	−	0.487208i	$-0.838015\pi$
$439$		−	20.4163i		−	0.974416i	0.873286	−	0.487208i	$-0.161985\pi$
$440$	−2.97672			−0.141910
$441$		0			0
$442$	3.12122			0.148461
$443$		−	1.18501i		−	0.0563017i	−0.999604	−	0.0281509i	$-0.991038\pi$
$443$		−	1.18501i		−	0.0563017i	0.999604	−	0.0281509i	$-0.00896188\pi$
$444$		0			0
$445$	−24.4159			−1.15743
$446$	−24.1153			−1.14189
$447$		0			0
$448$	−2.55176	−	0.698947i	−0.120559	−	0.0330222i
$449$		−	31.2665i		−	1.47556i	−0.675042	−	0.737779i	$-0.735875\pi$
$449$		−	31.2665i		−	1.47556i	0.675042	−	0.737779i	$-0.264125\pi$
$450$		0			0
$451$		−	3.81163i		−	0.179483i
$452$			1.38176i			0.0649927i
$453$		0			0
$454$			19.0034i			0.891873i
$455$	−4.11313	+	15.0165i	−0.192826	+	0.703982i
$456$		0			0
$457$	9.60650			0.449373			0.224687	−	0.974431i	$-0.427864\pi$
$457$	9.60650			0.449373			0.224687	+	0.974431i	$0.427864\pi$
$458$	−19.0546			−0.890364
$459$		0			0
$460$		0			0
$461$	26.9488			1.25513			0.627564	−	0.778565i	$-0.284052\pi$
$461$	26.9488			1.25513			0.627564	+	0.778565i	$0.284052\pi$
$462$		0			0
$463$	12.9706			0.602793			0.301397	−	0.953499i	$-0.402547\pi$
$463$	12.9706			0.602793			0.301397	+	0.953499i	$0.402547\pi$
$464$		−	1.30027i		−	0.0603634i
$465$		0			0
$466$	13.9288			0.645239
$467$	33.9030			1.56885			0.784423	−	0.620227i	$-0.212959\pi$
$467$	33.9030			1.56885			0.784423	+	0.620227i	$0.212959\pi$
$468$		0			0
$469$	1.78555	+	0.489076i	0.0824490	+	0.0225834i
$470$		−	7.83144i		−	0.361237i
$471$		0			0
$472$		−	9.90729i		−	0.456020i
$473$			9.14941i			0.420690i
$474$		0			0
$475$		−	27.7460i		−	1.27307i
$476$	−4.02879	−	1.10352i	−0.184659	−	0.0505796i
$477$		0			0
$478$	−9.76764			−0.446762
$479$	−20.7109			−0.946304			−0.473152	−	0.880981i	$-0.656884\pi$
$479$	−20.7109			−0.946304			−0.473152	+	0.880981i	$0.656884\pi$
$480$		0			0
$481$		−	15.9061i		−	0.725255i
$482$	10.4326			0.475191
$483$		0			0
$484$	1.00000			0.0454545
$485$		−	25.0623i		−	1.13802i
$486$		0			0
$487$	−32.3429			−1.46559			−0.732797	−	0.680447i	$-0.761786\pi$
$487$	−32.3429			−1.46559			−0.732797	+	0.680447i	$0.761786\pi$
$488$	−4.77271			−0.216050
$489$		0			0
$490$	10.6182	−	17.9286i	0.479683	−	0.809933i
$491$			7.28236i			0.328648i	0.986406	+	0.164324i	$0.0525443\pi$
$491$			7.28236i			0.328648i	−0.986406	+	0.164324i	$0.947456\pi$
$492$		0			0
$493$		−	2.05290i		−	0.0924579i
$494$		−	14.2070i		−	0.639205i
$495$		0			0
$496$			5.78855i			0.259914i
$497$	5.93076	−	21.6524i	0.266031	−	0.971242i
$498$		0			0
$499$	−24.6997			−1.10571			−0.552856	−	0.833277i	$-0.686462\pi$
$499$	−24.6997			−1.10571			−0.552856	+	0.833277i	$0.686462\pi$
$500$	−3.39085			−0.151644
$501$		0			0
$502$			21.6343i			0.965585i
$503$	−16.8936			−0.753250			−0.376625	−	0.926366i	$-0.622916\pi$
$503$	−16.8936			−0.753250			−0.376625	+	0.926366i	$0.622916\pi$
$504$		0			0
$505$	30.9471			1.37713
$506$		0			0
$507$		0			0
$508$	16.4497			0.729836
$509$	37.5793			1.66568			0.832838	−	0.553517i	$-0.186715\pi$
$509$	37.5793			1.66568			0.832838	+	0.553517i	$0.186715\pi$
$510$		0			0
$511$	−8.54887	+	31.2107i	−0.378180	+	1.38068i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			23.8831i			1.05344i
$515$			40.5219i			1.78561i
$516$		0			0
$517$			2.63089i			0.115707i
$518$	−5.62365	+	20.5312i	−0.247089	+	0.902088i
$519$		0			0
$520$	5.88475			0.258063
$521$	5.80874			0.254485			0.127243	−	0.991872i	$-0.459387\pi$
$521$	5.80874			0.254485			0.127243	+	0.991872i	$0.459387\pi$
$522$		0			0
$523$			36.3907i			1.59125i	0.605787	+	0.795627i	$0.292858\pi$
$523$			36.3907i			1.59125i	−0.605787	+	0.795627i	$0.707142\pi$
$524$	8.84022			0.386187
$525$		0			0
$526$	−18.9068			−0.824374
$527$			9.13912i			0.398106i
$528$		0			0
$529$	23.0000			1.00000
$530$	26.6190			1.15625
$531$		0			0
$532$	−5.02295	+	18.3381i	−0.217772	+	0.795056i
$533$			7.53529i			0.326390i
$534$		0			0
$535$		−	12.3866i		−	0.535518i
$536$		−	0.699733i		−	0.0302238i
$537$		0			0
$538$		−	8.00531i		−	0.345133i
$539$	−3.56709	+	6.02295i	−0.153645	+	0.259427i
$540$		0			0
$541$	−41.5458			−1.78619			−0.893096	−	0.449866i	$-0.851472\pi$
$541$	−41.5458			−1.78619			−0.893096	+	0.449866i	$0.851472\pi$
$542$	−26.0079			−1.11713
$543$		0			0
$544$			1.57883i			0.0676917i
$545$	−25.0684			−1.07381
$546$		0			0
$547$	−18.5489			−0.793093			−0.396546	−	0.918015i	$-0.629791\pi$
$547$	−18.5489			−0.793093			−0.396546	+	0.918015i	$0.629791\pi$
$548$			2.66413i			0.113806i
$549$		0			0
$550$	−3.86088			−0.164628
$551$	−9.34430			−0.398081
$552$		0			0
$553$	30.5302	+	8.36248i	1.29828	+	0.355609i
$554$			26.7915i			1.13826i
$555$		0			0
$556$			18.1685i			0.770515i
$557$			11.0932i			0.470035i	0.971991	+	0.235018i	$0.0755148\pi$
$557$			11.0932i			0.470035i	−0.971991	+	0.235018i	$0.924485\pi$
$558$		0			0
$559$		−	18.0877i		−	0.765027i
$560$	−7.59588	−	2.08057i	−0.320984	−	0.0879202i
$561$		0			0
$562$	28.2988			1.19371
$563$	38.8727			1.63829			0.819145	−	0.573587i	$-0.194448\pi$
$563$	38.8727			1.63829			0.819145	+	0.573587i	$0.194448\pi$
$564$		0			0
$565$			4.11313i			0.173041i
$566$	14.5765			0.612696
$567$		0			0
$568$	−8.48528			−0.356034
$569$			43.8135i			1.83676i	0.395701	+	0.918379i	$0.370502\pi$
$569$			43.8135i			1.83676i	−0.395701	+	0.918379i	$0.629498\pi$
$570$		0			0
$571$	23.8417			0.997745			0.498873	−	0.866675i	$-0.333748\pi$
$571$	23.8417			0.997745			0.498873	+	0.866675i	$0.333748\pi$
$572$	−1.97692			−0.0826592
$573$		0			0
$574$	2.66413	−	9.72635i	0.111199	−	0.405970i
$575$		0			0
$576$		0			0
$577$			3.15766i			0.131455i	0.997838	+	0.0657275i	$0.0209368\pi$
$577$			3.15766i			0.131455i	−0.997838	+	0.0657275i	$0.979063\pi$
$578$		−	14.5073i		−	0.603424i
$579$		0			0
$580$		−	3.87054i		−	0.160715i
$581$	45.7496	+	12.5312i	1.89801	+	0.519881i
$582$		0			0
$583$	−8.94237			−0.370355
$584$	12.2311			0.506125
$585$		0			0
$586$		−	15.8834i		−	0.656138i
$587$	−40.7572			−1.68223			−0.841115	−	0.540857i	$-0.818100\pi$
$587$	−40.7572			−1.68223			−0.841115	+	0.540857i	$0.818100\pi$
$588$		0			0
$589$	41.5991			1.71406
$590$		−	29.4912i		−	1.21414i
$591$		0			0
$592$	8.04589			0.330684
$593$	28.5582			1.17274			0.586372	−	0.810042i	$-0.300556\pi$
$593$	28.5582			1.17274			0.586372	+	0.810042i	$0.300556\pi$
$594$		0			0
$595$	−11.9926	−	3.28487i	−0.491648	−	0.134666i
$596$			10.0459i			0.411496i
$597$		0			0
$598$		0			0
$599$			36.5059i			1.49159i	0.666176	+	0.745794i	$0.267930\pi$
$599$			36.5059i			1.49159i	−0.666176	+	0.745794i	$0.732070\pi$
$600$		0			0
$601$			0.270884i			0.0110496i	0.999985	+	0.00552480i	$0.00175861\pi$
$601$			0.270884i			0.0110496i	−0.999985	+	0.00552480i	$0.998241\pi$
$602$	−6.39495	+	23.3471i	−0.260639	+	0.951556i
$603$		0			0
$604$	10.5489			0.429228
$605$	2.97672			0.121021
$606$		0			0
$607$		−	3.06771i		−	0.124514i	−0.998060	−	0.0622572i	$-0.980170\pi$
$607$		−	3.06771i		−	0.124514i	0.998060	−	0.0622572i	$-0.0198299\pi$
$608$	7.18644			0.291449
$609$		0			0
$610$	−14.2070			−0.575226
$611$		−	5.20107i		−	0.210413i
$612$		0			0
$613$	6.97056			0.281538			0.140769	−	0.990042i	$-0.455042\pi$
$613$	6.97056			0.281538			0.140769	+	0.990042i	$0.455042\pi$
$614$	1.05103			0.0424163
$615$		0			0
$616$	2.55176	+	0.698947i	0.102813	+	0.0281614i
$617$		−	27.6861i		−	1.11460i	−0.830310	−	0.557301i	$-0.811837\pi$
$617$		−	27.6861i		−	1.11460i	0.830310	−	0.557301i	$-0.188163\pi$
$618$		0			0
$619$		−	23.1221i		−	0.929357i	−0.885480	−	0.464678i	$-0.846170\pi$
$619$		−	23.1221i		−	0.929357i	0.885480	−	0.464678i	$-0.153830\pi$
$620$			17.2309i			0.692010i
$621$		0			0
$622$			18.0785i			0.724883i
$623$	20.9302	+	5.73296i	0.838553	+	0.229686i
$624$		0			0
$625$	−29.3980			−1.17592
$626$	19.8373			0.792856
$627$		0			0
$628$			5.78855i			0.230988i
$629$	12.7031			0.506505
$630$		0			0
$631$	−17.9765			−0.715634			−0.357817	−	0.933792i	$-0.616479\pi$
$631$	−17.9765			−0.715634			−0.357817	+	0.933792i	$0.616479\pi$
$632$		−	11.9644i		−	0.475918i
$633$		0			0
$634$	6.34184			0.251867
$635$	48.9661			1.94316
$636$		0			0
$637$	7.05186	−	11.9069i	0.279405	−	0.471768i
$638$			1.30027i			0.0514781i
$639$		0			0
$640$			2.97672i			0.117665i
$641$			32.2632i			1.27432i	0.770731	+	0.637160i	$0.219891\pi$
$641$			32.2632i			1.27432i	−0.770731	+	0.637160i	$0.780109\pi$
$642$		0			0
$643$			16.2226i			0.639757i	0.947459	+	0.319878i	$0.103642\pi$
$643$			16.2226i			0.639757i	−0.947459	+	0.319878i	$0.896358\pi$
$644$		0			0
$645$		0			0
$646$	11.3462			0.446409
$647$	18.8747			0.742042			0.371021	−	0.928625i	$-0.379008\pi$
$647$	18.8747			0.742042			0.371021	+	0.928625i	$0.379008\pi$
$648$		0			0
$649$			9.90729i			0.388895i
$650$	7.63265			0.299377
$651$		0			0
$652$	13.7676			0.539183
$653$		−	8.42169i		−	0.329566i	−0.986330	−	0.164783i	$-0.947308\pi$
$653$		−	8.42169i		−	0.329566i	0.986330	−	0.164783i	$-0.0526924\pi$
$654$		0			0
$655$	26.3149			1.02821
$656$	−3.81163			−0.148819
$657$		0			0
$658$	−1.83886	+	6.71341i	−0.0716861	+	0.261716i
$659$			8.84297i			0.344473i	0.985056	+	0.172237i	$0.0550994\pi$
$659$			8.84297i			0.344473i	−0.985056	+	0.172237i	$0.944901\pi$
$660$		0			0
$661$			40.8709i			1.58969i	0.606811	+	0.794846i	$0.292448\pi$
$661$			40.8709i			1.58969i	−0.606811	+	0.794846i	$0.707552\pi$
$662$		−	11.7855i		−	0.458058i
$663$		0			0
$664$		−	17.9286i		−	0.695766i
$665$	−14.9519	+	54.5873i	−0.579810	+	2.11681i
$666$		0			0
$667$		0			0
$668$	6.26056			0.242228
$669$		0			0
$670$		−	2.08291i		−	0.0804698i
$671$	4.77271			0.184248
$672$		0			0
$673$	2.41407			0.0930556			0.0465278	−	0.998917i	$-0.485184\pi$
$673$	2.41407			0.0930556			0.0465278	+	0.998917i	$0.485184\pi$
$674$			3.88475i			0.149635i
$675$		0			0
$676$	−9.09178			−0.349684
$677$	−42.0425			−1.61582			−0.807911	−	0.589304i	$-0.799402\pi$
$677$	−42.0425			−1.61582			−0.807911	+	0.589304i	$0.799402\pi$
$678$		0			0
$679$	−5.88475	+	21.4844i	−0.225836	+	0.824495i
$680$			4.69973i			0.180227i
$681$		0			0
$682$		−	5.78855i		−	0.221655i
$683$		−	3.62256i		−	0.138613i	−0.997595	−	0.0693066i	$-0.977921\pi$
$683$		−	3.62256i		−	0.138613i	0.997595	−	0.0693066i	$-0.0220787\pi$
$684$		0			0
$685$			7.93037i			0.303004i
$686$	−13.3121	+	12.8759i	−0.508257	+	0.491604i
$687$		0			0
$688$	9.14941			0.348818
$689$	17.6784			0.673492
$690$		0			0
$691$			34.1875i			1.30055i	0.759698	+	0.650276i	$0.225347\pi$
$691$			34.1875i			1.30055i	−0.759698	+	0.650276i	$0.774653\pi$
$692$	1.18073			0.0448848
$693$		0			0
$694$	24.2529			0.920628
$695$			54.0825i			2.05147i
$696$		0			0
$697$	−6.01790			−0.227944
$698$	−1.87239			−0.0708711
$699$		0			0
$700$	−9.85203	−	2.69855i	−0.372372	−	0.101996i
$701$		−	1.58593i		−	0.0598998i	−0.999551	−	0.0299499i	$-0.990465\pi$
$701$		−	1.58593i		−	0.0598998i	0.999551	−	0.0299499i	$-0.00953477\pi$
$702$		0			0
$703$		−	57.8213i		−	2.18077i
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$		−	26.1351i		−	0.983606i
$707$	−26.5290	−	7.26651i	−0.997726	−	0.273285i
$708$		0			0
$709$	−13.7695			−0.517124			−0.258562	−	0.965995i	$-0.583249\pi$
$709$	−13.7695			−0.517124			−0.258562	+	0.965995i	$0.583249\pi$
$710$	−25.2583			−0.947928
$711$		0			0
$712$		−	8.20228i		−	0.307394i
$713$		0			0
$714$		0			0
$715$	−5.88475			−0.220077
$716$		−	1.57851i		−	0.0589918i
$717$		0			0
$718$	−2.74562			−0.102466
$719$	−43.7741			−1.63250			−0.816249	−	0.577700i	$-0.803950\pi$
$719$	−43.7741			−1.63250			−0.816249	+	0.577700i	$0.803950\pi$
$720$		0			0
$721$	9.51472	−	34.7369i	0.354347	−	1.29367i
$722$		−	32.6450i		−	1.21492i
$723$		0			0
$724$			1.01584i			0.0377533i
$725$		−	5.02017i		−	0.186445i
$726$		0			0
$727$			46.4412i			1.72241i	0.508258	+	0.861205i	$0.330290\pi$
$727$			46.4412i			1.72241i	−0.508258	+	0.861205i	$0.669710\pi$
$728$	−5.04463	−	1.38176i	−0.186966	−	0.0512116i
$729$		0			0
$730$	36.4085			1.34754
$731$	14.4453			0.534280
$732$		0			0
$733$		−	50.7917i		−	1.87603i	−0.346589	−	0.938017i	$-0.612660\pi$
$733$		−	50.7917i		−	1.87603i	0.346589	−	0.938017i	$-0.387340\pi$
$734$	−22.3622			−0.825402
$735$		0			0
$736$		0			0
$737$			0.699733i			0.0257750i
$738$		0			0
$739$	28.3064			1.04127			0.520635	−	0.853780i	$-0.325695\pi$
$739$	28.3064			1.04127			0.520635	+	0.853780i	$0.325695\pi$
$740$	23.9504			0.880434
$741$		0			0
$742$	−22.8188	−	6.25025i	−0.837704	−	0.229454i
$743$			27.9306i			1.02468i	0.858784	+	0.512338i	$0.171220\pi$
$743$			27.9306i			1.02468i	−0.858784	+	0.512338i	$0.828780\pi$
$744$		0			0
$745$			29.9038i			1.09559i
$746$			37.5812i			1.37594i
$747$		0			0
$748$		−	1.57883i		−	0.0577277i
$749$	−2.90842	+	10.6182i	−0.106271	+	0.387982i
$750$		0			0
$751$	−5.56246			−0.202977			−0.101488	−	0.994837i	$-0.532361\pi$
$751$	−5.56246			−0.202977			−0.101488	+	0.994837i	$0.532361\pi$
$752$	2.63089			0.0959388
$753$		0			0
$754$		−	2.57053i		−	0.0936131i
$755$	31.4011			1.14280
$756$		0			0
$757$	−29.2112			−1.06170			−0.530849	−	0.847467i	$-0.678127\pi$
$757$	−29.2112			−1.06170			−0.530849	+	0.847467i	$0.678127\pi$
$758$		−	17.7676i		−	0.645350i
$759$		0			0
$760$	21.3920			0.775971
$761$	−52.8517			−1.91587			−0.957936	−	0.286983i	$-0.907348\pi$
$761$	−52.8517			−1.91587			−0.957936	+	0.286983i	$0.907348\pi$
$762$		0			0
$763$	21.4896	+	5.88617i	0.777976	+	0.213094i
$764$			8.48528i			0.306987i
$765$		0			0
$766$		−	7.69156i		−	0.277907i
$767$		−	19.5859i		−	0.707207i
$768$		0			0
$769$			46.1341i			1.66364i	0.555047	+	0.831819i	$0.312700\pi$
$769$			46.1341i			1.66364i	−0.555047	+	0.831819i	$0.687300\pi$
$770$	7.59588	+	2.08057i	0.273736	+	0.0749786i
$771$		0			0
$772$	4.60053			0.165577
$773$	−0.0976257			−0.00351135			−0.00175568	−	0.999998i	$-0.500559\pi$
$773$	−0.0976257			−0.00351135			−0.00175568	+	0.999998i	$0.500559\pi$
$774$		0			0
$775$			22.3489i			0.802795i
$776$	8.41944			0.302240
$777$		0			0
$778$	−15.2206			−0.545686
$779$			27.3920i			0.981422i
$780$		0			0
$781$	8.48528			0.303627
$782$		0			0
$783$		0			0
$784$	6.02295	+	3.56709i	0.215105	+	0.127396i
$785$			17.2309i			0.614997i
$786$		0			0
$787$		−	8.77882i		−	0.312931i	−0.987683	−	0.156466i	$-0.949990\pi$
$787$		−	8.77882i		−	0.312931i	0.987683	−	0.156466i	$-0.0500100\pi$
$788$			23.9926i			0.854700i
$789$		0			0
$790$		−	35.6147i		−	1.26711i
$791$	0.965780	−	3.52593i	0.0343392	−	0.125368i
$792$		0			0
$793$	−9.43527			−0.335056
$794$	9.48651			0.336664
$795$		0			0
$796$		−	11.3801i		−	0.403358i
$797$	−31.3094			−1.10904			−0.554518	−	0.832172i	$-0.687097\pi$
$797$	−31.3094			−1.10904			−0.554518	+	0.832172i	$0.687097\pi$
$798$		0			0
$799$	4.15373			0.146948
$800$			3.86088i			0.136503i
$801$		0			0
$802$	5.42765			0.191657
$803$	−12.2311			−0.431625
$804$		0			0
$805$		0			0
$806$			11.4435i			0.403080i
$807$		0			0
$808$			10.3964i			0.365743i
$809$			15.3995i			0.541416i	0.962661	+	0.270708i	$0.0872578\pi$
$809$			15.3995i			0.541416i	−0.962661	+	0.270708i	$0.912742\pi$
$810$		0			0
$811$		−	4.18953i		−	0.147114i	−0.997291	−	0.0735572i	$-0.976565\pi$
$811$		−	4.18953i		−	0.147114i	0.997291	−	0.0735572i	$-0.0234351\pi$
$812$	−0.908818	+	3.31797i	−0.0318933	+	0.116438i
$813$		0			0
$814$	−8.04589			−0.282008
$815$	40.9825			1.43555
$816$		0			0
$817$		−	65.7517i		−	2.30036i
$818$	19.1818			0.670677
$819$		0			0
$820$	−11.3462			−0.396225
$821$			40.2988i			1.40644i	0.710973	+	0.703219i	$0.248255\pi$
$821$			40.2988i			1.40644i	−0.710973	+	0.703219i	$0.751745\pi$
$822$		0			0
$823$	−44.1836			−1.54014			−0.770071	−	0.637958i	$-0.779779\pi$
$823$	−44.1836			−1.54014			−0.770071	+	0.637958i	$0.779779\pi$
$824$	−13.6129			−0.474229
$825$		0			0
$826$	−6.92467	+	25.2810i	−0.240940	+	0.879639i
$827$		−	49.3153i		−	1.71486i	−0.514601	−	0.857430i	$-0.672060\pi$
$827$		−	49.3153i		−	1.71486i	0.514601	−	0.857430i	$-0.327940\pi$
$828$		0			0
$829$			4.73502i			0.164454i	0.996614	+	0.0822271i	$0.0262033\pi$
$829$			4.73502i			0.164454i	−0.996614	+	0.0822271i	$0.973797\pi$
$830$		−	53.3686i		−	1.85245i
$831$		0			0
$832$			1.97692i			0.0685374i
$833$	9.50919	+	5.63182i	0.329474	+	0.195131i
$834$		0			0
$835$	18.6359			0.644923
$836$	−7.18644			−0.248548
$837$		0			0
$838$		−	14.7348i		−	0.509004i
$839$	1.85386			0.0640025			0.0320012	−	0.999488i	$-0.489812\pi$
$839$	1.85386			0.0640025			0.0320012	+	0.999488i	$0.489812\pi$
$840$		0			0
$841$	27.3093			0.941700
$842$		−	20.0459i		−	0.690827i
$843$		0			0
$844$	13.5429			0.466166
$845$	−27.0637			−0.931020
$846$		0			0
$847$	−2.55176	−	0.698947i	−0.0876795	−	0.0240161i
$848$			8.94237i			0.307082i
$849$		0			0
$850$			6.09566i			0.209079i
$851$		0			0
$852$		0			0
$853$		−	47.2751i		−	1.61867i	−0.587349	−	0.809334i	$-0.699828\pi$
$853$		−	47.2751i		−	1.61867i	0.587349	−	0.809334i	$-0.300172\pi$
$854$	12.1788	+	3.33587i	0.416750	+	0.114151i
$855$		0			0
$856$	4.16114			0.142225
$857$	3.21656			0.109876			0.0549379	−	0.998490i	$-0.482504\pi$
$857$	3.21656			0.109876			0.0549379	+	0.998490i	$0.482504\pi$
$858$		0			0
$859$		−	16.9434i		−	0.578102i	−0.957314	−	0.289051i	$-0.906660\pi$
$859$		−	16.9434i		−	0.578102i	0.957314	−	0.289051i	$-0.0933397\pi$
$860$	27.2352			0.928714
$861$		0			0
$862$	1.93620			0.0659474
$863$			48.5059i			1.65116i	0.564286	+	0.825579i	$0.309151\pi$
$863$			48.5059i			1.65116i	−0.564286	+	0.825579i	$0.690849\pi$
$864$		0			0
$865$	3.51472			0.119504
$866$	−16.6796			−0.566796
$867$		0			0
$868$	4.04589	−	14.7710i	0.137326	−	0.501360i
$869$			11.9644i			0.405864i
$870$		0			0
$871$		−	1.38332i		−	0.0468719i
$872$		−	8.42149i		−	0.285188i
$873$		0			0
$874$		0			0
$875$	8.65264	+	2.37003i	0.292513	+	0.0801216i
$876$		0			0
$877$	−24.4779			−0.826559			−0.413279	−	0.910604i	$-0.635617\pi$
$877$	−24.4779			−0.826559			−0.413279	+	0.910604i	$0.635617\pi$
$878$	20.4163			0.689016
$879$		0			0
$880$		−	2.97672i		−	0.100345i
$881$	9.73549			0.327997			0.163998	−	0.986461i	$-0.447561\pi$
$881$	9.73549			0.327997			0.163998	+	0.986461i	$0.447561\pi$
$882$		0			0
$883$	−13.7322			−0.462127			−0.231063	−	0.972939i	$-0.574220\pi$
$883$	−13.7322			−0.462127			−0.231063	+	0.972939i	$0.574220\pi$
$884$			3.12122i			0.104978i
$885$		0			0
$886$	1.18501			0.0398113
$887$	−38.8030			−1.30288			−0.651438	−	0.758702i	$-0.725834\pi$
$887$	−38.8030			−1.30288			−0.651438	+	0.758702i	$0.725834\pi$
$888$		0			0
$889$	−41.9756	−	11.4975i	−1.40782	−	0.385612i
$890$		−	24.4159i		−	0.818423i
$891$		0			0
$892$		−	24.1153i		−	0.807439i
$893$		−	18.9068i		−	0.632691i
$894$		0			0
$895$		−	4.69880i		−	0.157063i
$896$	0.698947	−	2.55176i	0.0233502	−	0.0852483i
$897$		0			0
$898$	31.2665			1.04338
$899$	7.52666			0.251028
$900$		0			0
$901$			14.1185i			0.470354i
$902$	3.81163			0.126913
$903$		0			0
$904$	−1.38176			−0.0459568
$905$			3.02387i			0.100517i
$906$		0			0
$907$	11.4633			0.380631			0.190316	−	0.981723i	$-0.439049\pi$
$907$	11.4633			0.380631			0.190316	+	0.981723i	$0.439049\pi$
$908$	−19.0034			−0.630649
$909$		0			0
$910$	−15.0165	−	4.11313i	−0.497791	−	0.136349i
$911$			42.0646i			1.39366i	0.717235	+	0.696831i	$0.245407\pi$
$911$			42.0646i			1.39366i	−0.717235	+	0.696831i	$0.754593\pi$
$912$		0			0
$913$			17.9286i			0.593351i
$914$			9.60650i			0.317755i
$915$		0			0
$916$		−	19.0546i		−	0.629582i
$917$	−22.5581	−	6.17885i	−0.744934	−	0.204043i
$918$		0			0
$919$	−22.4217			−0.739623			−0.369812	−	0.929107i	$-0.620578\pi$
$919$	−22.4217			−0.739623			−0.369812	+	0.929107i	$0.620578\pi$
$920$		0			0
$921$		0			0
$922$			26.9488i			0.887510i
$923$	−16.7747			−0.552147
$924$		0			0
$925$	31.0642			1.02138
$926$			12.9706i			0.426239i
$927$		0			0
$928$	1.30027			0.0426834
$929$	59.3011			1.94561			0.972803	−	0.231635i	$-0.0744075\pi$
$929$	59.3011			1.94561			0.972803	+	0.231635i	$0.0744075\pi$
$930$		0			0
$931$	25.6347	−	43.2836i	0.840143	−	1.41856i
$932$			13.9288i			0.456253i
$933$		0			0
$934$			33.9030i			1.10934i
$935$		−	4.69973i		−	0.153698i
$936$		0			0
$937$		−	4.45792i		−	0.145634i	−0.997345	−	0.0728170i	$-0.976801\pi$
$937$		−	4.45792i		−	0.145634i	0.997345	−	0.0728170i	$-0.0231989\pi$
$938$	−0.489076	+	1.78555i	−0.0159689	+	0.0583002i
$939$		0			0
$940$	7.83144			0.255433
$941$	−2.04938			−0.0668077			−0.0334039	−	0.999442i	$-0.510635\pi$
$941$	−2.04938			−0.0668077			−0.0334039	+	0.999442i	$0.510635\pi$
$942$		0			0
$943$		0			0
$944$	9.90729			0.322455
$945$		0			0
$946$	−9.14941			−0.297473
$947$		−	40.3887i		−	1.31246i	−0.754562	−	0.656229i	$-0.772151\pi$
$947$		−	40.3887i		−	1.31246i	0.754562	−	0.656229i	$-0.227849\pi$
$948$		0			0
$949$	24.1799			0.784912
$950$	27.7460			0.900198
$951$		0			0
$952$	1.10352	−	4.02879i	0.0357652	−	0.130574i
$953$		−	23.2011i		−	0.751556i	−0.926710	−	0.375778i	$-0.877375\pi$
$953$		−	23.2011i		−	0.751556i	0.926710	−	0.375778i	$-0.122625\pi$
$954$		0			0
$955$			25.2583i			0.817340i
$956$		−	9.76764i		−	0.315908i
$957$		0			0
$958$		−	20.7109i		−	0.669138i
$959$	1.86208	−	6.79821i	0.0601298	−	0.219526i
$960$		0			0
$961$	−2.50730			−0.0808807
$962$	15.9061			0.512833
$963$		0			0
$964$			10.4326i			0.336011i
$965$	13.6945			0.440842
$966$		0			0
$967$	−1.75179			−0.0563338			−0.0281669	−	0.999603i	$-0.508967\pi$
$967$	−1.75179			−0.0563338			−0.0281669	+	0.999603i	$0.508967\pi$
$968$			1.00000i			0.0321412i
$969$		0			0
$970$	25.0623			0.804703
$971$	−27.7355			−0.890076			−0.445038	−	0.895512i	$-0.646810\pi$
$971$	−27.7355			−0.890076			−0.445038	+	0.895512i	$0.646810\pi$
$972$		0			0
$973$	12.6988	−	46.3616i	0.407105	−	1.48628i
$974$		−	32.3429i		−	1.03633i
$975$		0			0
$976$		−	4.77271i		−	0.152771i
$977$		−	47.5474i		−	1.52118i	−0.649234	−	0.760588i	$-0.724911\pi$
$977$		−	47.5474i		−	1.52118i	0.649234	−	0.760588i	$-0.275089\pi$
$978$		0			0
$979$			8.20228i			0.262146i
$980$	17.9286	+	10.6182i	0.572709	+	0.339187i
$981$		0			0
$982$	−7.28236			−0.232390
$983$	−2.08706			−0.0665669			−0.0332835	−	0.999446i	$-0.510596\pi$
$983$	−2.08706			−0.0665669			−0.0332835	+	0.999446i	$0.510596\pi$
$984$		0			0
$985$			71.4193i			2.27561i
$986$	2.05290			0.0653776
$987$		0			0
$988$	14.2070			0.451986
$989$		0			0
$990$		0			0
$991$	37.5476			1.19274			0.596370	−	0.802710i	$-0.296609\pi$
$991$	37.5476			1.19274			0.596370	+	0.802710i	$0.296609\pi$
$992$	−5.78855			−0.183787
$993$		0			0
$994$	21.6524	+	5.93076i	0.686772	+	0.188112i
$995$		−	33.8755i		−	1.07392i
$996$		0			0
$997$			41.6535i			1.31918i	0.751626	+	0.659590i	$0.229270\pi$
$997$			41.6535i			1.31918i	−0.751626	+	0.659590i	$0.770730\pi$
$998$		−	24.6997i		−	0.781856i
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.97672 q^{5} +(2.55176 + 0.698947i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.97672 q^{5} +(2.55176 + 0.698947i) q^{7} -1.00000i q^{8} -2.97672i q^{10} +1.00000i q^{11} -1.97692i q^{13} +(-0.698947 + 2.55176i) q^{14} +1.00000 q^{16} +1.57883 q^{17} -7.18644i q^{19} +2.97672 q^{20} -1.00000 q^{22} +3.86088 q^{25} +1.97692 q^{26} +(-2.55176 - 0.698947i) q^{28} -1.30027i q^{29} +5.78855i q^{31} +1.00000i q^{32} +1.57883i q^{34} +(-7.59588 - 2.08057i) q^{35} +8.04589 q^{37} +7.18644 q^{38} +2.97672i q^{40} -3.81163 q^{41} +9.14941 q^{43} -1.00000i q^{44} +2.63089 q^{47} +(6.02295 + 3.56709i) q^{49} +3.86088i q^{50} +1.97692i q^{52} +8.94237i q^{53} -2.97672i q^{55} +(0.698947 - 2.55176i) q^{56} +1.30027 q^{58} +9.90729 q^{59} -4.77271i q^{61} -5.78855 q^{62} -1.00000 q^{64} +5.88475i q^{65} +0.699733 q^{67} -1.57883 q^{68} +(2.08057 - 7.59588i) q^{70} -8.48528i q^{71} +12.2311i q^{73} +8.04589i q^{74} +7.18644i q^{76} +(-0.698947 + 2.55176i) q^{77} +11.9644 q^{79} -2.97672 q^{80} -3.81163i q^{82} +17.9286 q^{83} -4.69973 q^{85} +9.14941i q^{86} +1.00000 q^{88} +8.20228 q^{89} +(1.38176 - 5.04463i) q^{91} +2.63089i q^{94} +21.3920i q^{95} +8.41944i q^{97} +(-3.56709 + 6.02295i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{22} + 48 q^{25} - 16 q^{37} - 80 q^{43} + 24 q^{49} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 24 q^{70} + 96 q^{79} - 80 q^{85} + 16 q^{88} - 32 q^{91}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^22 + 48 * q^25 - 16 * q^37 - 80 * q^43 + 24 * q^49 + 16 * q^58 - 16 * q^64 + 16 * q^67 + 24 * q^70 + 96 * q^79 - 80 * q^85 + 16 * q^88 - 32 * q^91

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