LMFDB - Embedded newform 1386.2.e.c.307.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(307,1386)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("1386.307");

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.e (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:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			307.4
Root			$0.581861 - 1.28897i$ of defining polynomial
Character	$\chi$	$=$	1386.307
Dual form			1386.2.e.c.307.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{5} +2.64575i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{5} +2.64575i q^{7} +1.00000i q^{8} +2.82843 q^{10} +(-2.64575 - 2.00000i) q^{11} -5.65685 q^{13} +2.64575 q^{14} +1.00000 q^{16} +7.48331 q^{17} +2.82843 q^{19} -2.82843i q^{20} +(-2.00000 + 2.64575i) q^{22} -5.29150 q^{23} -3.00000 q^{25} +5.65685i q^{26} -2.64575i q^{28} -6.00000i q^{29} +7.48331i q^{31} -1.00000i q^{32} -7.48331i q^{34} -7.48331 q^{35} -10.0000 q^{37} -2.82843i q^{38} -2.82843 q^{40} -7.48331 q^{41} +5.29150i q^{43} +(2.64575 + 2.00000i) q^{44} +5.29150i q^{46} -2.82843i q^{47} -7.00000 q^{49} +3.00000i q^{50} +5.65685 q^{52} -10.5830 q^{53} +(5.65685 - 7.48331i) q^{55} -2.64575 q^{56} -6.00000 q^{58} -11.3137i q^{59} +7.48331 q^{62} -1.00000 q^{64} -16.0000i q^{65} -12.0000 q^{67} -7.48331 q^{68} +7.48331i q^{70} +5.29150 q^{71} -8.48528 q^{73} +10.0000i q^{74} -2.82843 q^{76} +(5.29150 - 7.00000i) q^{77} +5.29150i q^{79} +2.82843i q^{80} +7.48331i q^{82} +7.48331 q^{83} +21.1660i q^{85} +5.29150 q^{86} +(2.00000 - 2.64575i) q^{88} -14.9666i q^{91} +5.29150 q^{92} -2.82843 q^{94} +8.00000i q^{95} +14.9666i q^{97} +7.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} - 16 q^{22} - 24 q^{25} - 80 q^{37} - 56 q^{49} - 48 q^{58} - 8 q^{64} - 96 q^{67} + 16 q^{88}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 - 16 * q^22 - 24 * q^25 - 80 * q^37 - 56 * q^49 - 48 * q^58 - 8 * q^64 - 96 * q^67 + 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
$5$			2.82843i			1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$			2.82843i			1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$		0			0
$7$			2.64575i			1.00000i
$7$			2.64575i			1.00000i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	2.82843			0.894427
$11$	−2.64575	−	2.00000i	−0.797724	−	0.603023i
$11$	−2.64575	−	2.00000i	−0.797724	−	0.603023i
$12$		0			0
$13$	−5.65685			−1.56893			−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685			−1.56893			−0.784465	+	0.620174i	$0.787062\pi$
$14$	2.64575			0.707107
$15$		0			0
$16$	1.00000			0.250000
$17$	7.48331			1.81497			0.907485	−	0.420084i	$-0.137999\pi$
$17$	7.48331			1.81497			0.907485	+	0.420084i	$0.137999\pi$
$18$		0			0
$19$	2.82843			0.648886			0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843			0.648886			0.324443	+	0.945905i	$0.394823\pi$
$20$		−	2.82843i		−	0.632456i
$21$		0			0
$22$	−2.00000	+	2.64575i	−0.426401	+	0.564076i
$23$	−5.29150			−1.10335			−0.551677	−	0.834058i	$-0.686012\pi$
$23$	−5.29150			−1.10335			−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$			5.65685i			1.10940i
$27$		0			0
$28$		−	2.64575i		−	0.500000i
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$			7.48331i			1.34404i	0.740532	+	0.672022i	$0.234574\pi$
$31$			7.48331i			1.34404i	−0.740532	+	0.672022i	$0.765426\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	7.48331i		−	1.28338i
$35$	−7.48331			−1.26491
$36$		0			0
$37$	−10.0000			−1.64399			−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000			−1.64399			−0.821995	+	0.569495i	$0.807139\pi$
$38$		−	2.82843i		−	0.458831i
$39$		0			0
$40$	−2.82843			−0.447214
$41$	−7.48331			−1.16870			−0.584349	−	0.811503i	$-0.698650\pi$
$41$	−7.48331			−1.16870			−0.584349	+	0.811503i	$0.698650\pi$
$42$		0			0
$43$			5.29150i			0.806947i	0.914991	+	0.403473i	$0.132197\pi$
$43$			5.29150i			0.806947i	−0.914991	+	0.403473i	$0.867803\pi$
$44$	2.64575	+	2.00000i	0.398862	+	0.301511i
$45$		0			0
$46$			5.29150i			0.780189i
$47$		−	2.82843i		−	0.412568i	−0.978492	−	0.206284i	$-0.933863\pi$
$47$		−	2.82843i		−	0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$			3.00000i			0.424264i
$51$		0			0
$52$	5.65685			0.784465
$53$	−10.5830			−1.45369			−0.726844	−	0.686803i	$-0.759014\pi$
$53$	−10.5830			−1.45369			−0.726844	+	0.686803i	$0.759014\pi$
$54$		0			0
$55$	5.65685	−	7.48331i	0.762770	−	1.00905i
$56$	−2.64575			−0.353553
$57$		0			0
$58$	−6.00000			−0.787839
$59$		−	11.3137i		−	1.47292i	−0.676481	−	0.736460i	$-0.736496\pi$
$59$		−	11.3137i		−	1.47292i	0.676481	−	0.736460i	$-0.263504\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$	7.48331			0.950382
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	16.0000i		−	1.98456i
$66$		0			0
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$	−7.48331			−0.907485
$69$		0			0
$70$			7.48331i			0.894427i
$71$	5.29150			0.627986			0.313993	−	0.949425i	$-0.398333\pi$
$71$	5.29150			0.627986			0.313993	+	0.949425i	$0.398333\pi$
$72$		0			0
$73$	−8.48528			−0.993127			−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528			−0.993127			−0.496564	+	0.868000i	$0.665405\pi$
$74$			10.0000i			1.16248i
$75$		0			0
$76$	−2.82843			−0.324443
$77$	5.29150	−	7.00000i	0.603023	−	0.797724i
$78$		0			0
$79$			5.29150i			0.595341i	0.954669	+	0.297670i	$0.0962096\pi$
$79$			5.29150i			0.595341i	−0.954669	+	0.297670i	$0.903790\pi$
$80$			2.82843i			0.316228i
$81$		0			0
$82$			7.48331i			0.826394i
$83$	7.48331			0.821401			0.410700	−	0.911770i	$-0.365284\pi$
$83$	7.48331			0.821401			0.410700	+	0.911770i	$0.365284\pi$
$84$		0			0
$85$			21.1660i			2.29578i
$86$	5.29150			0.570597
$87$		0			0
$88$	2.00000	−	2.64575i	0.213201	−	0.282038i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	14.9666i		−	1.56893i
$92$	5.29150			0.551677
$93$		0			0
$94$	−2.82843			−0.291730
$95$			8.00000i			0.820783i
$96$		0			0
$97$			14.9666i			1.51963i	0.650139	+	0.759815i	$0.274711\pi$
$97$			14.9666i			1.51963i	−0.650139	+	0.759815i	$0.725289\pi$
$98$			7.00000i			0.707107i
$99$		0			0
$100$	3.00000			0.300000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			7.48331i			0.737353i	0.929558	+	0.368676i	$0.120189\pi$
$103$			7.48331i			0.737353i	−0.929558	+	0.368676i	$0.879811\pi$
$104$		−	5.65685i		−	0.554700i
$105$		0			0
$106$			10.5830i			1.02791i
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$		−	10.5830i		−	1.01367i	−0.862044	−	0.506834i	$-0.830816\pi$
$109$		−	10.5830i		−	1.01367i	0.862044	−	0.506834i	$-0.169184\pi$
$110$	−7.48331	−	5.65685i	−0.713506	−	0.539360i
$111$		0			0
$112$			2.64575i			0.250000i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		−	14.9666i		−	1.39565i
$116$			6.00000i			0.557086i
$117$		0			0
$118$	−11.3137			−1.04151
$119$			19.7990i			1.81497i
$120$		0			0
$121$	3.00000	+	10.5830i	0.272727	+	0.962091i
$122$		0			0
$123$		0			0
$124$		−	7.48331i		−	0.672022i
$125$			5.65685i			0.505964i
$126$		0			0
$127$		−	5.29150i		−	0.469545i	−0.972050	−	0.234772i	$-0.924565\pi$
$127$		−	5.29150i		−	0.469545i	0.972050	−	0.234772i	$-0.0754345\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−16.0000			−1.40329
$131$	−7.48331			−0.653820			−0.326910	−	0.945055i	$-0.606007\pi$
$131$	−7.48331			−0.653820			−0.326910	+	0.945055i	$0.606007\pi$
$132$		0			0
$133$			7.48331i			0.648886i
$134$			12.0000i			1.03664i
$135$		0			0
$136$			7.48331i			0.641689i
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$	19.7990			1.67933			0.839664	−	0.543106i	$-0.182752\pi$
$139$	19.7990			1.67933			0.839664	+	0.543106i	$0.182752\pi$
$140$	7.48331			0.632456
$141$		0			0
$142$		−	5.29150i		−	0.444053i
$143$	14.9666	+	11.3137i	1.25157	+	0.946100i
$144$		0			0
$145$	16.9706			1.40933
$146$			8.48528i			0.702247i
$147$		0			0
$148$	10.0000			0.821995
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		0			0
$151$		−	5.29150i		−	0.430616i	−0.976546	−	0.215308i	$-0.930924\pi$
$151$		−	5.29150i		−	0.430616i	0.976546	−	0.215308i	$-0.0690756\pi$
$152$			2.82843i			0.229416i
$153$		0			0
$154$	−7.00000	−	5.29150i	−0.564076	−	0.426401i
$155$	−21.1660			−1.70009
$156$		0			0
$157$		−	22.4499i		−	1.79170i	−0.444356	−	0.895850i	$-0.646567\pi$
$157$		−	22.4499i		−	1.79170i	0.444356	−	0.895850i	$-0.353433\pi$
$158$	5.29150			0.420969
$159$		0			0
$160$	2.82843			0.223607
$161$		−	14.0000i		−	1.10335i
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	7.48331			0.584349
$165$		0			0
$166$		−	7.48331i		−	0.580818i
$167$	14.9666			1.15815			0.579076	−	0.815273i	$-0.303413\pi$
$167$	14.9666			1.15815			0.579076	+	0.815273i	$0.303413\pi$
$168$		0			0
$169$	19.0000			1.46154
$170$	21.1660			1.62336
$171$		0			0
$172$		−	5.29150i		−	0.403473i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		−	7.93725i		−	0.600000i
$176$	−2.64575	−	2.00000i	−0.199431	−	0.150756i
$177$		0			0
$178$		0			0
$179$	5.29150			0.395505			0.197753	−	0.980252i	$-0.436636\pi$
$179$	5.29150			0.395505			0.197753	+	0.980252i	$0.436636\pi$
$180$		0			0
$181$			22.4499i			1.66869i	0.551242	+	0.834346i	$0.314154\pi$
$181$			22.4499i			1.66869i	−0.551242	+	0.834346i	$0.685846\pi$
$182$	−14.9666			−1.10940
$183$		0			0
$184$		−	5.29150i		−	0.390095i
$185$		−	28.2843i		−	2.07950i
$186$		0			0
$187$	−19.7990	−	14.9666i	−1.44785	−	1.09447i
$188$			2.82843i			0.206284i
$189$		0			0
$190$	8.00000			0.580381
$191$	15.8745			1.14864			0.574320	−	0.818631i	$-0.305267\pi$
$191$	15.8745			1.14864			0.574320	+	0.818631i	$0.305267\pi$
$192$		0			0
$193$		−	21.1660i		−	1.52356i	−0.647834	−	0.761781i	$-0.724325\pi$
$193$		−	21.1660i		−	1.52356i	0.647834	−	0.761781i	$-0.275675\pi$
$194$	14.9666			1.07454
$195$		0			0
$196$	7.00000			0.500000
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$			7.48331i			0.530478i	0.964183	+	0.265239i	$0.0854509\pi$
$199$			7.48331i			0.530478i	−0.964183	+	0.265239i	$0.914549\pi$
$200$		−	3.00000i		−	0.212132i
$201$		0			0
$202$		0			0
$203$	15.8745			1.11417
$204$		0			0
$205$		−	21.1660i		−	1.47830i
$206$	7.48331			0.521387
$207$		0			0
$208$	−5.65685			−0.392232
$209$	−7.48331	−	5.65685i	−0.517632	−	0.391293i
$210$		0			0
$211$			15.8745i			1.09285i	0.837509	+	0.546423i	$0.184011\pi$
$211$			15.8745i			1.09285i	−0.837509	+	0.546423i	$0.815989\pi$
$212$	10.5830			0.726844
$213$		0			0
$214$	4.00000			0.273434
$215$	−14.9666			−1.02072
$216$		0			0
$217$	−19.7990			−1.34404
$218$	−10.5830			−0.716772
$219$		0			0
$220$	−5.65685	+	7.48331i	−0.381385	+	0.504525i
$221$	−42.3320			−2.84756
$222$		0			0
$223$			22.4499i			1.50336i	0.659528	+	0.751680i	$0.270756\pi$
$223$			22.4499i			1.50336i	−0.659528	+	0.751680i	$0.729244\pi$
$224$	2.64575			0.176777
$225$		0			0
$226$		0			0
$227$	−22.4499			−1.49006			−0.745028	−	0.667034i	$-0.767564\pi$
$227$	−22.4499			−1.49006			−0.745028	+	0.667034i	$0.767564\pi$
$228$		0			0
$229$			7.48331i			0.494511i	0.968950	+	0.247256i	$0.0795288\pi$
$229$			7.48331i			0.494511i	−0.968950	+	0.247256i	$0.920471\pi$
$230$	−14.9666			−0.986870
$231$		0			0
$232$	6.00000			0.393919
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	8.00000			0.521862
$236$			11.3137i			0.736460i
$237$		0			0
$238$	19.7990			1.28338
$239$			8.00000i			0.517477i	0.965947	+	0.258738i	$0.0833068\pi$
$239$			8.00000i			0.517477i	−0.965947	+	0.258738i	$0.916693\pi$
$240$		0			0
$241$	−8.48528			−0.546585			−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528			−0.546585			−0.273293	+	0.961931i	$0.588113\pi$
$242$	10.5830	−	3.00000i	0.680301	−	0.192847i
$243$		0			0
$244$		0			0
$245$		−	19.7990i		−	1.26491i
$246$		0			0
$247$	−16.0000			−1.01806
$248$	−7.48331			−0.475191
$249$		0			0
$250$	5.65685			0.357771
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		0			0
$253$	14.0000	+	10.5830i	0.880172	+	0.665348i
$254$	−5.29150			−0.332018
$255$		0			0
$256$	1.00000			0.0625000
$257$			16.9706i			1.05859i	0.848436	+	0.529297i	$0.177544\pi$
$257$			16.9706i			1.05859i	−0.848436	+	0.529297i	$0.822456\pi$
$258$		0			0
$259$		−	26.4575i		−	1.64399i
$260$			16.0000i			0.992278i
$261$		0			0
$262$			7.48331i			0.462321i
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		0			0
$265$		−	29.9333i		−	1.83879i
$266$	7.48331			0.458831
$267$		0			0
$268$	12.0000			0.733017
$269$			19.7990i			1.20717i	0.797300	+	0.603583i	$0.206261\pi$
$269$			19.7990i			1.20717i	−0.797300	+	0.603583i	$0.793739\pi$
$270$		0			0
$271$	−16.9706			−1.03089			−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706			−1.03089			−0.515444	+	0.856923i	$0.672373\pi$
$272$	7.48331			0.453743
$273$		0			0
$274$		0			0
$275$	7.93725	+	6.00000i	0.478634	+	0.361814i
$276$		0			0
$277$		−	10.5830i		−	0.635871i	−0.948112	−	0.317936i	$-0.897010\pi$
$277$		−	10.5830i		−	0.635871i	0.948112	−	0.317936i	$-0.102990\pi$
$278$		−	19.7990i		−	1.18746i
$279$		0			0
$280$		−	7.48331i		−	0.447214i
$281$		−	22.0000i		−	1.31241i	−0.754583	−	0.656205i	$-0.772161\pi$
$281$		−	22.0000i		−	1.31241i	0.754583	−	0.656205i	$-0.227839\pi$
$282$		0			0
$283$	8.48528			0.504398			0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528			0.504398			0.252199	+	0.967675i	$0.418846\pi$
$284$	−5.29150			−0.313993
$285$		0			0
$286$	11.3137	−	14.9666i	0.668994	−	0.884995i
$287$		−	19.7990i		−	1.16870i
$288$		0			0
$289$	39.0000			2.29412
$290$		−	16.9706i		−	0.996546i
$291$		0			0
$292$	8.48528			0.496564
$293$	−14.9666			−0.874360			−0.437180	−	0.899374i	$-0.644023\pi$
$293$	−14.9666			−0.874360			−0.437180	+	0.899374i	$0.644023\pi$
$294$		0			0
$295$	32.0000			1.86311
$296$		−	10.0000i		−	0.581238i
$297$		0			0
$298$	10.0000			0.579284
$299$	29.9333			1.73109
$300$		0			0
$301$	−14.0000			−0.806947
$302$	−5.29150			−0.304492
$303$		0			0
$304$	2.82843			0.162221
$305$		0			0
$306$		0			0
$307$	19.7990			1.12999			0.564994	−	0.825095i	$-0.308878\pi$
$307$	19.7990			1.12999			0.564994	+	0.825095i	$0.308878\pi$
$308$	−5.29150	+	7.00000i	−0.301511	+	0.398862i
$309$		0			0
$310$			21.1660i			1.20215i
$311$			19.7990i			1.12270i	0.827579	+	0.561349i	$0.189717\pi$
$311$			19.7990i			1.12270i	−0.827579	+	0.561349i	$0.810283\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−22.4499			−1.26692
$315$		0			0
$316$		−	5.29150i		−	0.297670i
$317$	−31.7490			−1.78320			−0.891601	−	0.452822i	$-0.850417\pi$
$317$	−31.7490			−1.78320			−0.891601	+	0.452822i	$0.850417\pi$
$318$		0			0
$319$	−12.0000	+	15.8745i	−0.671871	+	0.888802i
$320$		−	2.82843i		−	0.158114i
$321$		0			0
$322$	−14.0000			−0.780189
$323$	21.1660			1.17771
$324$		0			0
$325$	16.9706			0.941357
$326$		−	4.00000i		−	0.221540i
$327$		0			0
$328$		−	7.48331i		−	0.413197i
$329$	7.48331			0.412568
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$	−7.48331			−0.410700
$333$		0			0
$334$		−	14.9666i		−	0.818938i
$335$		−	33.9411i		−	1.85440i
$336$		0			0
$337$			21.1660i			1.15299i	0.817102	+	0.576493i	$0.195579\pi$
$337$			21.1660i			1.15299i	−0.817102	+	0.576493i	$0.804421\pi$
$338$		−	19.0000i		−	1.03346i
$339$		0			0
$340$		−	21.1660i		−	1.14789i
$341$	14.9666	−	19.7990i	0.810488	−	1.07218i
$342$		0			0
$343$		−	18.5203i		−	1.00000i
$344$	−5.29150			−0.285299
$345$		0			0
$346$		0			0
$347$		−	4.00000i		−	0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$		0			0
$349$	−5.65685			−0.302804			−0.151402	−	0.988472i	$-0.548379\pi$
$349$	−5.65685			−0.302804			−0.151402	+	0.988472i	$0.548379\pi$
$350$	−7.93725			−0.424264
$351$		0			0
$352$	−2.00000	+	2.64575i	−0.106600	+	0.141019i
$353$			28.2843i			1.50542i	0.658352	+	0.752710i	$0.271254\pi$
$353$			28.2843i			1.50542i	−0.658352	+	0.752710i	$0.728746\pi$
$354$		0			0
$355$			14.9666i			0.794346i
$356$		0			0
$357$		0			0
$358$		−	5.29150i		−	0.279665i
$359$		−	8.00000i		−	0.422224i	−0.977462	−	0.211112i	$-0.932292\pi$
$359$		−	8.00000i		−	0.422224i	0.977462	−	0.211112i	$-0.0677085\pi$
$360$		0			0
$361$	−11.0000			−0.578947
$362$	22.4499			1.17994
$363$		0			0
$364$			14.9666i			0.784465i
$365$		−	24.0000i		−	1.25622i
$366$		0			0
$367$			22.4499i			1.17188i	0.810355	+	0.585939i	$0.199274\pi$
$367$			22.4499i			1.17188i	−0.810355	+	0.585939i	$0.800726\pi$
$368$	−5.29150			−0.275839
$369$		0			0
$370$	−28.2843			−1.47043
$371$		−	28.0000i		−	1.45369i
$372$		0			0
$373$		−	10.5830i		−	0.547967i	−0.961734	−	0.273984i	$-0.911659\pi$
$373$		−	10.5830i		−	0.547967i	0.961734	−	0.273984i	$-0.0883414\pi$
$374$	−14.9666	+	19.7990i	−0.773906	+	1.02378i
$375$		0			0
$376$	2.82843			0.145865
$377$			33.9411i			1.74806i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		−	8.00000i		−	0.410391i
$381$		0			0
$382$		−	15.8745i		−	0.812210i
$383$			19.7990i			1.01168i	0.862627	+	0.505841i	$0.168818\pi$
$383$			19.7990i			1.01168i	−0.862627	+	0.505841i	$0.831182\pi$
$384$		0			0
$385$	19.7990	+	14.9666i	1.00905	+	0.762770i
$386$	−21.1660			−1.07732
$387$		0			0
$388$		−	14.9666i		−	0.759815i
$389$	10.5830			0.536580			0.268290	−	0.963338i	$-0.413542\pi$
$389$	10.5830			0.536580			0.268290	+	0.963338i	$0.413542\pi$
$390$		0			0
$391$	−39.5980			−2.00256
$392$		−	7.00000i		−	0.353553i
$393$		0			0
$394$	−2.00000			−0.100759
$395$	−14.9666			−0.753053
$396$		0			0
$397$		−	7.48331i		−	0.375577i	−0.982210	−	0.187788i	$-0.939868\pi$
$397$		−	7.48331i		−	0.375577i	0.982210	−	0.187788i	$-0.0601319\pi$
$398$	7.48331			0.375105
$399$		0			0
$400$	−3.00000			−0.150000
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		−	42.3320i		−	2.10871i
$404$		0			0
$405$		0			0
$406$		−	15.8745i		−	0.787839i
$407$	26.4575	+	20.0000i	1.31145	+	0.991363i
$408$		0			0
$409$	−19.7990			−0.978997			−0.489499	−	0.872004i	$-0.662820\pi$
$409$	−19.7990			−0.978997			−0.489499	+	0.872004i	$0.662820\pi$
$410$	−21.1660			−1.04531
$411$		0			0
$412$		−	7.48331i		−	0.368676i
$413$	29.9333			1.47292
$414$		0			0
$415$			21.1660i			1.03900i
$416$			5.65685i			0.277350i
$417$		0			0
$418$	−5.65685	+	7.48331i	−0.276686	+	0.366021i
$419$			5.65685i			0.276355i	0.990407	+	0.138178i	$0.0441245\pi$
$419$			5.65685i			0.276355i	−0.990407	+	0.138178i	$0.955875\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$	15.8745			0.772759
$423$		0			0
$424$		−	10.5830i		−	0.513956i
$425$	−22.4499			−1.08898
$426$		0			0
$427$		0			0
$428$		−	4.00000i		−	0.193347i
$429$		0			0
$430$			14.9666i			0.721755i
$431$			16.0000i			0.770693i	0.922772	+	0.385346i	$0.125918\pi$
$431$			16.0000i			0.770693i	−0.922772	+	0.385346i	$0.874082\pi$
$432$		0			0
$433$		−	29.9333i		−	1.43850i	−0.694751	−	0.719250i	$-0.744485\pi$
$433$		−	29.9333i		−	1.43850i	0.694751	−	0.719250i	$-0.255515\pi$
$434$			19.7990i			0.950382i
$435$		0			0
$436$			10.5830i			0.506834i
$437$	−14.9666			−0.715951
$438$		0			0
$439$	−22.6274			−1.07995			−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274			−1.07995			−0.539974	+	0.841682i	$0.681566\pi$
$440$	7.48331	+	5.65685i	0.356753	+	0.269680i
$441$		0			0
$442$			42.3320i			2.01353i
$443$	−15.8745			−0.754221			−0.377110	−	0.926168i	$-0.623082\pi$
$443$	−15.8745			−0.754221			−0.377110	+	0.926168i	$0.623082\pi$
$444$		0			0
$445$		0			0
$446$	22.4499			1.06304
$447$		0			0
$448$		−	2.64575i		−	0.125000i
$449$	−21.1660			−0.998886			−0.499443	−	0.866347i	$-0.666462\pi$
$449$	−21.1660			−0.998886			−0.499443	+	0.866347i	$0.666462\pi$
$450$		0			0
$451$	19.7990	+	14.9666i	0.932298	+	0.704751i
$452$		0			0
$453$		0			0
$454$			22.4499i			1.05363i
$455$	42.3320			1.98456
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	7.48331			0.349672
$459$		0			0
$460$			14.9666i			0.697823i
$461$	29.9333			1.39413			0.697065	−	0.717008i	$-0.254489\pi$
$461$	29.9333			1.39413			0.697065	+	0.717008i	$0.254489\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$		−	6.00000i		−	0.278543i
$465$		0			0
$466$	−6.00000			−0.277945
$467$		−	16.9706i		−	0.785304i	−0.919687	−	0.392652i	$-0.871558\pi$
$467$		−	16.9706i		−	0.785304i	0.919687	−	0.392652i	$-0.128442\pi$
$468$		0			0
$469$		−	31.7490i		−	1.46603i
$470$		−	8.00000i		−	0.369012i
$471$		0			0
$472$	11.3137			0.520756
$473$	10.5830	−	14.0000i	0.486607	−	0.643721i
$474$		0			0
$475$	−8.48528			−0.389331
$476$		−	19.7990i		−	0.907485i
$477$		0			0
$478$	8.00000			0.365911
$479$	−14.9666			−0.683843			−0.341921	−	0.939729i	$-0.611078\pi$
$479$	−14.9666			−0.683843			−0.341921	+	0.939729i	$0.611078\pi$
$480$		0			0
$481$	56.5685			2.57930
$482$			8.48528i			0.386494i
$483$		0			0
$484$	−3.00000	−	10.5830i	−0.136364	−	0.481046i
$485$	−42.3320			−1.92220
$486$		0			0
$487$	−24.0000			−1.08754			−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000			−1.08754			−0.543772	+	0.839233i	$0.683004\pi$
$488$		0			0
$489$		0			0
$490$	−19.7990			−0.894427
$491$			36.0000i			1.62466i	0.583200	+	0.812329i	$0.301800\pi$
$491$			36.0000i			1.62466i	−0.583200	+	0.812329i	$0.698200\pi$
$492$		0			0
$493$		−	44.8999i		−	2.02219i
$494$			16.0000i			0.719874i
$495$		0			0
$496$			7.48331i			0.336011i
$497$			14.0000i			0.627986i
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$		−	5.65685i		−	0.252982i
$501$		0			0
$502$	−5.65685			−0.252478
$503$	−14.9666			−0.667329			−0.333665	−	0.942692i	$-0.608285\pi$
$503$	−14.9666			−0.667329			−0.333665	+	0.942692i	$0.608285\pi$
$504$		0			0
$505$		0			0
$506$	10.5830	−	14.0000i	0.470472	−	0.622376i
$507$		0			0
$508$			5.29150i			0.234772i
$509$		−	19.7990i		−	0.877575i	−0.898591	−	0.438787i	$-0.855408\pi$
$509$		−	19.7990i		−	0.877575i	0.898591	−	0.438787i	$-0.144592\pi$
$510$		0			0
$511$		−	22.4499i		−	0.993127i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	16.9706			0.748539
$515$	−21.1660			−0.932686
$516$		0			0
$517$	−5.65685	+	7.48331i	−0.248788	+	0.329116i
$518$	−26.4575			−1.16248
$519$		0			0
$520$	16.0000			0.701646
$521$		−	28.2843i		−	1.23916i	−0.784935	−	0.619578i	$-0.787304\pi$
$521$		−	28.2843i		−	1.23916i	0.784935	−	0.619578i	$-0.212696\pi$
$522$		0			0
$523$	36.7696			1.60782			0.803910	−	0.594751i	$-0.202749\pi$
$523$	36.7696			1.60782			0.803910	+	0.594751i	$0.202749\pi$
$524$	7.48331			0.326910
$525$		0			0
$526$	16.0000			0.697633
$527$			56.0000i			2.43940i
$528$		0			0
$529$	5.00000			0.217391
$530$	−29.9333			−1.30022
$531$		0			0
$532$		−	7.48331i		−	0.324443i
$533$	42.3320			1.83360
$534$		0			0
$535$	−11.3137			−0.489134
$536$		−	12.0000i		−	0.518321i
$537$		0			0
$538$	19.7990			0.853595
$539$	18.5203	+	14.0000i	0.797724	+	0.603023i
$540$		0			0
$541$			10.5830i			0.454999i	0.973778	+	0.227499i	$0.0730550\pi$
$541$			10.5830i			0.454999i	−0.973778	+	0.227499i	$0.926945\pi$
$542$			16.9706i			0.728948i
$543$		0			0
$544$		−	7.48331i		−	0.320844i
$545$	29.9333			1.28220
$546$		0			0
$547$			26.4575i			1.13124i	0.824665	+	0.565621i	$0.191363\pi$
$547$			26.4575i			1.13124i	−0.824665	+	0.565621i	$0.808637\pi$
$548$		0			0
$549$		0			0
$550$	6.00000	−	7.93725i	0.255841	−	0.338446i
$551$		−	16.9706i		−	0.722970i
$552$		0			0
$553$	−14.0000			−0.595341
$554$	−10.5830			−0.449629
$555$		0			0
$556$	−19.7990			−0.839664
$557$			26.0000i			1.10166i	0.834619	+	0.550828i	$0.185688\pi$
$557$			26.0000i			1.10166i	−0.834619	+	0.550828i	$0.814312\pi$
$558$		0			0
$559$		−	29.9333i		−	1.26604i
$560$	−7.48331			−0.316228
$561$		0			0
$562$	−22.0000			−0.928014
$563$	7.48331			0.315384			0.157692	−	0.987488i	$-0.449595\pi$
$563$	7.48331			0.315384			0.157692	+	0.987488i	$0.449595\pi$
$564$		0			0
$565$		0			0
$566$		−	8.48528i		−	0.356663i
$567$		0			0
$568$			5.29150i			0.222027i
$569$		−	6.00000i		−	0.251533i	−0.992060	−	0.125767i	$-0.959861\pi$
$569$		−	6.00000i		−	0.251533i	0.992060	−	0.125767i	$-0.0401390\pi$
$570$		0			0
$571$			5.29150i			0.221442i	0.993852	+	0.110721i	$0.0353161\pi$
$571$			5.29150i			0.221442i	−0.993852	+	0.110721i	$0.964684\pi$
$572$	−14.9666	−	11.3137i	−0.625786	−	0.473050i
$573$		0			0
$574$	−19.7990			−0.826394
$575$	15.8745			0.662013
$576$		0			0
$577$			14.9666i			0.623069i	0.950235	+	0.311534i	$0.100843\pi$
$577$			14.9666i			0.623069i	−0.950235	+	0.311534i	$0.899157\pi$
$578$		−	39.0000i		−	1.62219i
$579$		0			0
$580$	−16.9706			−0.704664
$581$			19.7990i			0.821401i
$582$		0			0
$583$	28.0000	+	21.1660i	1.15964	+	0.876607i
$584$		−	8.48528i		−	0.351123i
$585$		0			0
$586$			14.9666i			0.618266i
$587$			33.9411i			1.40090i	0.713701	+	0.700450i	$0.247017\pi$
$587$			33.9411i			1.40090i	−0.713701	+	0.700450i	$0.752983\pi$
$588$		0			0
$589$			21.1660i			0.872130i
$590$		−	32.0000i		−	1.31742i
$591$		0			0
$592$	−10.0000			−0.410997
$593$	−7.48331			−0.307303			−0.153651	−	0.988125i	$-0.549103\pi$
$593$	−7.48331			−0.307303			−0.153651	+	0.988125i	$0.549103\pi$
$594$		0			0
$595$	−56.0000			−2.29578
$596$		−	10.0000i		−	0.409616i
$597$		0			0
$598$		−	29.9333i		−	1.22406i
$599$	47.6235			1.94584			0.972922	−	0.231133i	$-0.0742432\pi$
$599$	47.6235			1.94584			0.972922	+	0.231133i	$0.0742432\pi$
$600$		0			0
$601$	25.4558			1.03837			0.519183	−	0.854663i	$-0.326236\pi$
$601$	25.4558			1.03837			0.519183	+	0.854663i	$0.326236\pi$
$602$			14.0000i			0.570597i
$603$		0			0
$604$			5.29150i			0.215308i
$605$	−29.9333	+	8.48528i	−1.21696	+	0.344976i
$606$		0			0
$607$	−16.9706			−0.688814			−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706			−0.688814			−0.344407	+	0.938820i	$0.611920\pi$
$608$		−	2.82843i		−	0.114708i
$609$		0			0
$610$		0			0
$611$			16.0000i			0.647291i
$612$		0			0
$613$			31.7490i			1.28233i	0.767403	+	0.641165i	$0.221549\pi$
$613$			31.7490i			1.28233i	−0.767403	+	0.641165i	$0.778451\pi$
$614$		−	19.7990i		−	0.799022i
$615$		0			0
$616$	7.00000	+	5.29150i	0.282038	+	0.213201i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$			29.9333i			1.20312i	0.798828	+	0.601560i	$0.205454\pi$
$619$			29.9333i			1.20312i	−0.798828	+	0.601560i	$0.794546\pi$
$620$	21.1660			0.850047
$621$		0			0
$622$	19.7990			0.793867
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$		0			0
$628$			22.4499i			0.895850i
$629$	−74.8331			−2.98379
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	−5.29150			−0.210485
$633$		0			0
$634$			31.7490i			1.26091i
$635$	14.9666			0.593933
$636$		0			0
$637$	39.5980			1.56893
$638$	15.8745	+	12.0000i	0.628478	+	0.475085i
$639$		0			0
$640$	−2.82843			−0.111803
$641$	−21.1660			−0.836007			−0.418004	−	0.908445i	$-0.637270\pi$
$641$	−21.1660			−0.836007			−0.418004	+	0.908445i	$0.637270\pi$
$642$		0			0
$643$			14.9666i			0.590226i	0.955462	+	0.295113i	$0.0953573\pi$
$643$			14.9666i			0.590226i	−0.955462	+	0.295113i	$0.904643\pi$
$644$			14.0000i			0.551677i
$645$		0			0
$646$		−	21.1660i		−	0.832766i
$647$		−	31.1127i		−	1.22317i	−0.791180	−	0.611583i	$-0.790533\pi$
$647$		−	31.1127i		−	1.22317i	0.791180	−	0.611583i	$-0.209467\pi$
$648$		0			0
$649$	−22.6274	+	29.9333i	−0.888204	+	1.17498i
$650$		−	16.9706i		−	0.665640i
$651$		0			0
$652$	−4.00000			−0.156652
$653$	31.7490			1.24243			0.621217	−	0.783638i	$-0.286638\pi$
$653$	31.7490			1.24243			0.621217	+	0.783638i	$0.286638\pi$
$654$		0			0
$655$		−	21.1660i		−	0.827024i
$656$	−7.48331			−0.292174
$657$		0			0
$658$		−	7.48331i		−	0.291730i
$659$		−	36.0000i		−	1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$		−	36.0000i		−	1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$		0			0
$661$			22.4499i			0.873202i	0.899655	+	0.436601i	$0.143818\pi$
$661$			22.4499i			0.873202i	−0.899655	+	0.436601i	$0.856182\pi$
$662$		−	20.0000i		−	0.777322i
$663$		0			0
$664$			7.48331i			0.290409i
$665$	−21.1660			−0.820783
$666$		0			0
$667$			31.7490i			1.22933i
$668$	−14.9666			−0.579076
$669$		0			0
$670$	−33.9411			−1.31126
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	21.1660			0.815284
$675$		0			0
$676$	−19.0000			−0.730769
$677$	14.9666			0.575214			0.287607	−	0.957748i	$-0.407140\pi$
$677$	14.9666			0.575214			0.287607	+	0.957748i	$0.407140\pi$
$678$		0			0
$679$	−39.5980			−1.51963
$680$	−21.1660			−0.811679
$681$		0			0
$682$	−19.7990	−	14.9666i	−0.758143	−	0.573102i
$683$	−15.8745			−0.607421			−0.303711	−	0.952764i	$-0.598226\pi$
$683$	−15.8745			−0.607421			−0.303711	+	0.952764i	$0.598226\pi$
$684$		0			0
$685$		0			0
$686$	−18.5203			−0.707107
$687$		0			0
$688$			5.29150i			0.201737i
$689$	59.8665			2.28073
$690$		0			0
$691$		−	44.8999i		−	1.70807i	−0.520214	−	0.854036i	$-0.674148\pi$
$691$		−	44.8999i		−	1.70807i	0.520214	−	0.854036i	$-0.325852\pi$
$692$		0			0
$693$		0			0
$694$	−4.00000			−0.151838
$695$			56.0000i			2.12420i
$696$		0			0
$697$	−56.0000			−2.12115
$698$			5.65685i			0.214115i
$699$		0			0
$700$			7.93725i			0.300000i
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$		0			0
$703$	−28.2843			−1.06676
$704$	2.64575	+	2.00000i	0.0997155	+	0.0753778i
$705$		0			0
$706$	28.2843			1.06449
$707$		0			0
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$	14.9666			0.561688
$711$		0			0
$712$		0			0
$713$		−	39.5980i		−	1.48296i
$714$		0			0
$715$	−32.0000	+	42.3320i	−1.19673	+	1.58313i
$716$	−5.29150			−0.197753
$717$		0			0
$718$	−8.00000			−0.298557
$719$			2.82843i			0.105483i	0.998608	+	0.0527413i	$0.0167959\pi$
$719$			2.82843i			0.105483i	−0.998608	+	0.0527413i	$0.983204\pi$
$720$		0			0
$721$	−19.7990			−0.737353
$722$			11.0000i			0.409378i
$723$		0			0
$724$		−	22.4499i		−	0.834346i
$725$			18.0000i			0.668503i
$726$		0			0
$727$			22.4499i			0.832622i	0.909222	+	0.416311i	$0.136677\pi$
$727$			22.4499i			0.832622i	−0.909222	+	0.416311i	$0.863323\pi$
$728$	14.9666			0.554700
$729$		0			0
$730$	−24.0000			−0.888280
$731$			39.5980i			1.46458i
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	22.4499			0.828643
$735$		0			0
$736$			5.29150i			0.195047i
$737$	31.7490	+	24.0000i	1.16949	+	0.884051i
$738$		0			0
$739$		−	15.8745i		−	0.583953i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$		−	15.8745i		−	0.583953i	0.956425	−	0.291977i	$-0.0943129\pi$
$740$			28.2843i			1.03975i
$741$		0			0
$742$	−28.0000			−1.02791
$743$			48.0000i			1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$			48.0000i			1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$		0			0
$745$	−28.2843			−1.03626
$746$	−10.5830			−0.387471
$747$		0			0
$748$	19.7990	+	14.9666i	0.723923	+	0.547234i
$749$	−10.5830			−0.386695
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		−	2.82843i		−	0.103142i
$753$		0			0
$754$	33.9411			1.23606
$755$	14.9666			0.544691
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		−	20.0000i		−	0.726433i
$759$		0			0
$760$	−8.00000			−0.290191
$761$	−7.48331			−0.271270			−0.135635	−	0.990759i	$-0.543307\pi$
$761$	−7.48331			−0.271270			−0.135635	+	0.990759i	$0.543307\pi$
$762$		0			0
$763$	28.0000			1.01367
$764$	−15.8745			−0.574320
$765$		0			0
$766$	19.7990			0.715367
$767$			64.0000i			2.31091i
$768$		0			0
$769$	25.4558			0.917961			0.458981	−	0.888446i	$-0.348215\pi$
$769$	25.4558			0.917961			0.458981	+	0.888446i	$0.348215\pi$
$770$	14.9666	−	19.7990i	0.539360	−	0.713506i
$771$		0			0
$772$			21.1660i			0.761781i
$773$		−	31.1127i		−	1.11905i	−0.828815	−	0.559523i	$-0.810984\pi$
$773$		−	31.1127i		−	1.11905i	0.828815	−	0.559523i	$-0.189016\pi$
$774$		0			0
$775$		−	22.4499i		−	0.806426i
$776$	−14.9666			−0.537271
$777$		0			0
$778$		−	10.5830i		−	0.379419i
$779$	−21.1660			−0.758351
$780$		0			0
$781$	−14.0000	−	10.5830i	−0.500959	−	0.378690i
$782$			39.5980i			1.41602i
$783$		0			0
$784$	−7.00000			−0.250000
$785$	63.4980			2.26634
$786$		0			0
$787$	31.1127			1.10905			0.554524	−	0.832168i	$-0.312900\pi$
$787$	31.1127			1.10905			0.554524	+	0.832168i	$0.312900\pi$
$788$			2.00000i			0.0712470i
$789$		0			0
$790$			14.9666i			0.532489i
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−7.48331			−0.265573
$795$		0			0
$796$		−	7.48331i		−	0.265239i
$797$			25.4558i			0.901692i	0.892602	+	0.450846i	$0.148878\pi$
$797$			25.4558i			0.901692i	−0.892602	+	0.450846i	$0.851122\pi$
$798$		0			0
$799$		−	21.1660i		−	0.748800i
$800$			3.00000i			0.106066i
$801$		0			0
$802$		0			0
$803$	22.4499	+	16.9706i	0.792241	+	0.598878i
$804$		0			0
$805$	39.5980			1.39565
$806$	−42.3320			−1.49108
$807$		0			0
$808$		0			0
$809$		−	54.0000i		−	1.89854i	−0.314464	−	0.949269i	$-0.601825\pi$
$809$		−	54.0000i		−	1.89854i	0.314464	−	0.949269i	$-0.398175\pi$
$810$		0			0
$811$	−25.4558			−0.893876			−0.446938	−	0.894565i	$-0.647485\pi$
$811$	−25.4558			−0.893876			−0.446938	+	0.894565i	$0.647485\pi$
$812$	−15.8745			−0.557086
$813$		0			0
$814$	20.0000	−	26.4575i	0.701000	−	0.927335i
$815$			11.3137i			0.396302i
$816$		0			0
$817$			14.9666i			0.523616i
$818$			19.7990i			0.692255i
$819$		0			0
$820$			21.1660i			0.739149i
$821$		−	10.0000i		−	0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$		−	10.0000i		−	0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$		0			0
$823$	−16.0000			−0.557725			−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000			−0.557725			−0.278862	+	0.960331i	$0.589957\pi$
$824$	−7.48331			−0.260694
$825$		0			0
$826$		−	29.9333i		−	1.04151i
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$		−	37.4166i		−	1.29953i	−0.760135	−	0.649766i	$-0.774867\pi$
$829$		−	37.4166i		−	1.29953i	0.760135	−	0.649766i	$-0.225133\pi$
$830$	21.1660			0.734683
$831$		0			0
$832$	5.65685			0.196116
$833$	−52.3832			−1.81497
$834$		0			0
$835$			42.3320i			1.46496i
$836$	7.48331	+	5.65685i	0.258816	+	0.195646i
$837$		0			0
$838$	5.65685			0.195413
$839$		−	14.1421i		−	0.488241i	−0.969745	−	0.244120i	$-0.921501\pi$
$839$		−	14.1421i		−	0.488241i	0.969745	−	0.244120i	$-0.0784992\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$			6.00000i			0.206774i
$843$		0			0
$844$		−	15.8745i		−	0.546423i
$845$			53.7401i			1.84872i
$846$		0			0
$847$	−28.0000	+	7.93725i	−0.962091	+	0.272727i
$848$	−10.5830			−0.363422
$849$		0			0
$850$			22.4499i			0.770027i
$851$	52.9150			1.81390
$852$		0			0
$853$	−39.5980			−1.35581			−0.677905	−	0.735150i	$-0.737112\pi$
$853$	−39.5980			−1.35581			−0.677905	+	0.735150i	$0.737112\pi$
$854$		0			0
$855$		0			0
$856$	−4.00000			−0.136717
$857$	7.48331			0.255625			0.127813	−	0.991798i	$-0.459204\pi$
$857$	7.48331			0.255625			0.127813	+	0.991798i	$0.459204\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$	14.9666			0.510358
$861$		0			0
$862$	16.0000			0.544962
$863$	−15.8745			−0.540375			−0.270187	−	0.962808i	$-0.587086\pi$
$863$	−15.8745			−0.540375			−0.270187	+	0.962808i	$0.587086\pi$
$864$		0			0
$865$		0			0
$866$	−29.9333			−1.01717
$867$		0			0
$868$	19.7990			0.672022
$869$	10.5830	−	14.0000i	0.359004	−	0.474917i
$870$		0			0
$871$	67.8823			2.30010
$872$	10.5830			0.358386
$873$		0			0
$874$			14.9666i			0.506254i
$875$	−14.9666			−0.505964
$876$		0			0
$877$			52.9150i			1.78681i	0.449249	+	0.893407i	$0.351692\pi$
$877$			52.9150i			1.78681i	−0.449249	+	0.893407i	$0.648308\pi$
$878$			22.6274i			0.763638i
$879$		0			0
$880$	5.65685	−	7.48331i	0.190693	−	0.252262i
$881$			5.65685i			0.190584i	0.995449	+	0.0952921i	$0.0303785\pi$
$881$			5.65685i			0.190584i	−0.995449	+	0.0952921i	$0.969621\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$	42.3320			1.42378
$885$		0			0
$886$			15.8745i			0.533315i
$887$	14.9666			0.502530			0.251265	−	0.967918i	$-0.419153\pi$
$887$	14.9666			0.502530			0.251265	+	0.967918i	$0.419153\pi$
$888$		0			0
$889$	14.0000			0.469545
$890$		0			0
$891$		0			0
$892$		−	22.4499i		−	0.751680i
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$			14.9666i			0.500279i
$896$	−2.64575			−0.0883883
$897$		0			0
$898$			21.1660i			0.706319i
$899$	44.8999			1.49750
$900$		0			0
$901$	−79.1960			−2.63840
$902$	14.9666	−	19.7990i	0.498334	−	0.659234i
$903$		0			0
$904$		0			0
$905$	−63.4980			−2.11075
$906$		0			0
$907$	4.00000			0.132818			0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000			0.132818			0.0664089	+	0.997792i	$0.478846\pi$
$908$	22.4499			0.745028
$909$		0			0
$910$		−	42.3320i		−	1.40329i
$911$	−15.8745			−0.525946			−0.262973	−	0.964803i	$-0.584703\pi$
$911$	−15.8745			−0.525946			−0.262973	+	0.964803i	$0.584703\pi$
$912$		0			0
$913$	−19.7990	−	14.9666i	−0.655251	−	0.495323i
$914$		0			0
$915$		0			0
$916$		−	7.48331i		−	0.247256i
$917$		−	19.7990i		−	0.653820i
$918$		0			0
$919$		−	37.0405i		−	1.22185i	−0.791687	−	0.610927i	$-0.790797\pi$
$919$		−	37.0405i		−	1.22185i	0.791687	−	0.610927i	$-0.209203\pi$
$920$	14.9666			0.493435
$921$		0			0
$922$		−	29.9333i		−	0.985799i
$923$	−29.9333			−0.985265
$924$		0			0
$925$	30.0000			0.986394
$926$		−	8.00000i		−	0.262896i
$927$		0			0
$928$	−6.00000			−0.196960
$929$		−	22.6274i		−	0.742381i	−0.928557	−	0.371191i	$-0.878950\pi$
$929$		−	22.6274i		−	0.742381i	0.928557	−	0.371191i	$-0.121050\pi$
$930$		0			0
$931$	−19.7990			−0.648886
$932$			6.00000i			0.196537i
$933$		0			0
$934$	−16.9706			−0.555294
$935$	42.3320	−	56.0000i	1.38441	−	1.83140i
$936$		0			0
$937$	−19.7990			−0.646805			−0.323402	−	0.946262i	$-0.604827\pi$
$937$	−19.7990			−0.646805			−0.323402	+	0.946262i	$0.604827\pi$
$938$	−31.7490			−1.03664
$939$		0			0
$940$	−8.00000			−0.260931
$941$	−14.9666			−0.487898			−0.243949	−	0.969788i	$-0.578443\pi$
$941$	−14.9666			−0.487898			−0.243949	+	0.969788i	$0.578443\pi$
$942$		0			0
$943$	39.5980			1.28949
$944$		−	11.3137i		−	0.368230i
$945$		0			0
$946$	−14.0000	−	10.5830i	−0.455179	−	0.344083i
$947$	−26.4575			−0.859754			−0.429877	−	0.902888i	$-0.641443\pi$
$947$	−26.4575			−0.859754			−0.429877	+	0.902888i	$0.641443\pi$
$948$		0			0
$949$	48.0000			1.55815
$950$			8.48528i			0.275299i
$951$		0			0
$952$	−19.7990			−0.641689
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$			44.8999i			1.45293i
$956$		−	8.00000i		−	0.258738i
$957$		0			0
$958$			14.9666i			0.483550i
$959$		0			0
$960$		0			0
$961$	−25.0000			−0.806452
$962$		−	56.5685i		−	1.82384i
$963$		0			0
$964$	8.48528			0.273293
$965$	59.8665			1.92717
$966$		0			0
$967$			47.6235i			1.53147i	0.643157	+	0.765735i	$0.277624\pi$
$967$			47.6235i			1.53147i	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−10.5830	+	3.00000i	−0.340151	+	0.0964237i
$969$		0			0
$970$			42.3320i			1.35920i
$971$			22.6274i			0.726148i	0.931760	+	0.363074i	$0.118273\pi$
$971$			22.6274i			0.726148i	−0.931760	+	0.363074i	$0.881727\pi$
$972$		0			0
$973$			52.3832i			1.67933i
$974$			24.0000i			0.769010i
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$			19.7990i			0.632456i
$981$		0			0
$982$	36.0000			1.14881
$983$		−	19.7990i		−	0.631490i	−0.948844	−	0.315745i	$-0.897746\pi$
$983$		−	19.7990i		−	0.631490i	0.948844	−	0.315745i	$-0.102254\pi$
$984$		0			0
$985$	5.65685			0.180242
$986$	−44.8999			−1.42990
$987$		0			0
$988$	16.0000			0.509028
$989$		−	28.0000i		−	0.890348i
$990$		0			0
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$	7.48331			0.237595
$993$		0			0
$994$	14.0000			0.444053
$995$	−21.1660			−0.671008
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$			4.00000i			0.126618i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.e.c.307.4	yes	8
3.2	odd	2	inner	1386.2.e.c.307.6	yes	8
7.6	odd	2	inner	1386.2.e.c.307.2	yes	8
11.10	odd	2	inner	1386.2.e.c.307.7	yes	8
21.20	even	2	inner	1386.2.e.c.307.8	yes	8
33.32	even	2	inner	1386.2.e.c.307.1	✓	8
77.76	even	2	inner	1386.2.e.c.307.5	yes	8
231.230	odd	2	inner	1386.2.e.c.307.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.e.c.307.1	✓	8	33.32	even	2	inner
1386.2.e.c.307.2	yes	8	7.6	odd	2	inner
1386.2.e.c.307.3	yes	8	231.230	odd	2	inner
1386.2.e.c.307.4	yes	8	1.1	even	1	trivial
1386.2.e.c.307.5	yes	8	77.76	even	2	inner
1386.2.e.c.307.6	yes	8	3.2	odd	2	inner
1386.2.e.c.307.7	yes	8	11.10	odd	2	inner
1386.2.e.c.307.8	yes	8	21.20	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{5} +2.64575i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{5} +2.64575i q^{7} +1.00000i q^{8} +2.82843 q^{10} +(-2.64575 - 2.00000i) q^{11} -5.65685 q^{13} +2.64575 q^{14} +1.00000 q^{16} +7.48331 q^{17} +2.82843 q^{19} -2.82843i q^{20} +(-2.00000 + 2.64575i) q^{22} -5.29150 q^{23} -3.00000 q^{25} +5.65685i q^{26} -2.64575i q^{28} -6.00000i q^{29} +7.48331i q^{31} -1.00000i q^{32} -7.48331i q^{34} -7.48331 q^{35} -10.0000 q^{37} -2.82843i q^{38} -2.82843 q^{40} -7.48331 q^{41} +5.29150i q^{43} +(2.64575 + 2.00000i) q^{44} +5.29150i q^{46} -2.82843i q^{47} -7.00000 q^{49} +3.00000i q^{50} +5.65685 q^{52} -10.5830 q^{53} +(5.65685 - 7.48331i) q^{55} -2.64575 q^{56} -6.00000 q^{58} -11.3137i q^{59} +7.48331 q^{62} -1.00000 q^{64} -16.0000i q^{65} -12.0000 q^{67} -7.48331 q^{68} +7.48331i q^{70} +5.29150 q^{71} -8.48528 q^{73} +10.0000i q^{74} -2.82843 q^{76} +(5.29150 - 7.00000i) q^{77} +5.29150i q^{79} +2.82843i q^{80} +7.48331i q^{82} +7.48331 q^{83} +21.1660i q^{85} +5.29150 q^{86} +(2.00000 - 2.64575i) q^{88} -14.9666i q^{91} +5.29150 q^{92} -2.82843 q^{94} +8.00000i q^{95} +14.9666i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} - 16 q^{22} - 24 q^{25} - 80 q^{37} - 56 q^{49} - 48 q^{58} - 8 q^{64} - 96 q^{67} + 16 q^{88}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 - 16 * q^22 - 24 * q^25 - 80 * q^37 - 56 * q^49 - 48 * q^58 - 8 * q^64 - 96 * q^67 + 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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