LMFDB - Embedded newform 1287.2.e.c.1286.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1286,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1287.1286");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1286.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1287.1286
Dual form			1287.2.e.c.1286.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214i q^{2} +1.82843 q^{4} +3.41421 q^{7} -1.58579i q^{8} +O(q^{10})$	$q-0.414214i q^{2} +1.82843 q^{4} +3.41421 q^{7} -1.58579i q^{8} +(-3.00000 - 1.41421i) q^{11} +(2.00000 + 3.00000i) q^{13} -1.41421i q^{14} +3.00000 q^{16} +4.24264 q^{17} +0.585786 q^{19} +(-0.585786 + 1.24264i) q^{22} +1.17157i q^{23} -5.00000 q^{25} +(1.24264 - 0.828427i) q^{26} +6.24264 q^{28} +4.24264 q^{29} -1.75736i q^{31} -4.41421i q^{32} -1.75736i q^{34} -8.48528i q^{37} -0.242641i q^{38} -1.17157i q^{41} +8.48528i q^{43} +(-5.48528 - 2.58579i) q^{44} +0.485281 q^{46} -6.00000 q^{47} +4.65685 q^{49} +2.07107i q^{50} +(3.65685 + 5.48528i) q^{52} +9.41421i q^{53} -5.41421i q^{56} -1.75736i q^{58} +8.48528 q^{59} -6.00000i q^{61} -0.727922 q^{62} +4.17157 q^{64} +10.2426i q^{67} +7.75736 q^{68} -8.48528 q^{71} -1.17157 q^{73} -3.51472 q^{74} +1.07107 q^{76} +(-10.2426 - 4.82843i) q^{77} -8.48528i q^{79} -0.485281 q^{82} -3.65685i q^{83} +3.51472 q^{86} +(-2.24264 + 4.75736i) q^{88} +(6.82843 + 10.2426i) q^{91} +2.14214i q^{92} +2.48528i q^{94} -12.0000i q^{97} -1.92893i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^7	$ 4 q - 4 q^{4} + 8 q^{7} - 12 q^{11} + 8 q^{13} + 12 q^{16} + 8 q^{19} - 8 q^{22} - 20 q^{25} - 12 q^{26} + 8 q^{28} + 12 q^{44} - 32 q^{46} - 24 q^{47} - 4 q^{49} - 8 q^{52} + 48 q^{62} + 28 q^{64} + 48 q^{68} - 16 q^{73} - 48 q^{74} - 24 q^{76} - 24 q^{77} + 32 q^{82} + 48 q^{86} + 8 q^{88} + 16 q^{91}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^11 + 8 * q^13 + 12 * q^16 + 8 * q^19 - 8 * q^22 - 20 * q^25 - 12 * q^26 + 8 * q^28 + 12 * q^44 - 32 * q^46 - 24 * q^47 - 4 * q^49 - 8 * q^52 + 48 * q^62 + 28 * q^64 + 48 * q^68 - 16 * q^73 - 48 * q^74 - 24 * q^76 - 24 * q^77 + 32 * q^82 + 48 * q^86 + 8 * q^88 + 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times$.

$n$	$496$	$937$	$1145$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		−	0.414214i		−	0.292893i	−0.989219	−	0.146447i	$-0.953216\pi$
$2$		−	0.414214i		−	0.292893i	0.989219	−	0.146447i	$-0.0467837\pi$
$3$		0			0
$3$		0			0
$4$	1.82843			0.914214
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	3.41421			1.29045			0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421			1.29045			0.645226	+	0.763992i	$0.276763\pi$
$8$		−	1.58579i		−	0.560660i
$9$		0			0
$10$		0			0
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		−	1.41421i		−	0.377964i
$15$		0			0
$16$	3.00000			0.750000
$17$	4.24264			1.02899			0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264			1.02899			0.514496	+	0.857493i	$0.327979\pi$
$18$		0			0
$19$	0.585786			0.134389			0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786			0.134389			0.0671943	+	0.997740i	$0.478595\pi$
$20$		0			0
$21$		0			0
$22$	−0.585786	+	1.24264i	−0.124890	+	0.264932i
$23$			1.17157i			0.244290i	0.992512	+	0.122145i	$0.0389773\pi$
$23$			1.17157i			0.244290i	−0.992512	+	0.122145i	$0.961023\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$	1.24264	−	0.828427i	0.243702	−	0.162468i
$27$		0			0
$28$	6.24264			1.17975
$29$	4.24264			0.787839			0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264			0.787839			0.393919	+	0.919145i	$0.371119\pi$
$30$		0			0
$31$		−	1.75736i		−	0.315631i	−0.987469	−	0.157816i	$-0.949555\pi$
$31$		−	1.75736i		−	0.315631i	0.987469	−	0.157816i	$-0.0504451\pi$
$32$		−	4.41421i		−	0.780330i
$33$		0			0
$34$		−	1.75736i		−	0.301385i
$35$		0			0
$36$		0			0
$37$		−	8.48528i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$		−	8.48528i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$		−	0.242641i		−	0.0393615i
$39$		0			0
$40$		0			0
$41$		−	1.17157i		−	0.182969i	−0.995807	−	0.0914845i	$-0.970839\pi$
$41$		−	1.17157i		−	0.182969i	0.995807	−	0.0914845i	$-0.0291612\pi$
$42$		0			0
$43$			8.48528i			1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$			8.48528i			1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−5.48528	−	2.58579i	−0.826937	−	0.389822i
$45$		0			0
$46$	0.485281			0.0715508
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$	4.65685			0.665265
$50$			2.07107i			0.292893i
$51$		0			0
$52$	3.65685	+	5.48528i	0.507114	+	0.760672i
$53$			9.41421i			1.29314i	0.762854	+	0.646571i	$0.223798\pi$
$53$			9.41421i			1.29314i	−0.762854	+	0.646571i	$0.776202\pi$
$54$		0			0
$55$		0			0
$56$		−	5.41421i		−	0.723505i
$57$		0			0
$58$		−	1.75736i		−	0.230753i
$59$	8.48528			1.10469			0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528			1.10469			0.552345	+	0.833616i	$0.313733\pi$
$60$		0			0
$61$		−	6.00000i		−	0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$		−	6.00000i		−	0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$	−0.727922			−0.0924462
$63$		0			0
$64$	4.17157			0.521447
$65$		0			0
$66$		0			0
$67$			10.2426i			1.25134i	0.780089	+	0.625669i	$0.215174\pi$
$67$			10.2426i			1.25134i	−0.780089	+	0.625669i	$0.784826\pi$
$68$	7.75736			0.940718
$69$		0			0
$70$		0			0
$71$	−8.48528			−1.00702			−0.503509	−	0.863990i	$-0.667958\pi$
$71$	−8.48528			−1.00702			−0.503509	+	0.863990i	$0.667958\pi$
$72$		0			0
$73$	−1.17157			−0.137122			−0.0685611	−	0.997647i	$-0.521841\pi$
$73$	−1.17157			−0.137122			−0.0685611	+	0.997647i	$0.521841\pi$
$74$	−3.51472			−0.408578
$75$		0			0
$76$	1.07107			0.122860
$77$	−10.2426	−	4.82843i	−1.16726	−	0.550250i
$78$		0			0
$79$		−	8.48528i		−	0.954669i	−0.878722	−	0.477334i	$-0.841603\pi$
$79$		−	8.48528i		−	0.954669i	0.878722	−	0.477334i	$-0.158397\pi$
$80$		0			0
$81$		0			0
$82$	−0.485281			−0.0535904
$83$		−	3.65685i		−	0.401392i	−0.979654	−	0.200696i	$-0.935680\pi$
$83$		−	3.65685i		−	0.401392i	0.979654	−	0.200696i	$-0.0643203\pi$
$84$		0			0
$85$		0			0
$86$	3.51472			0.379002
$87$		0			0
$88$	−2.24264	+	4.75736i	−0.239066	+	0.507136i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	6.82843	+	10.2426i	0.715814	+	1.07372i
$92$			2.14214i			0.223333i
$93$		0			0
$94$			2.48528i			0.256337i
$95$		0			0
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		−	1.92893i		−	0.194852i
$99$		0			0
$100$	−9.14214			−0.914214
$101$	7.75736			0.771886			0.385943	−	0.922523i	$-0.373876\pi$
$101$	7.75736			0.771886			0.385943	+	0.922523i	$0.373876\pi$
$102$		0			0
$103$	3.31371			0.326509			0.163255	−	0.986584i	$-0.447801\pi$
$103$	3.31371			0.326509			0.163255	+	0.986584i	$0.447801\pi$
$104$	4.75736	−	3.17157i	0.466497	−	0.310998i
$105$		0			0
$106$	3.89949			0.378752
$107$	8.48528			0.820303			0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528			0.820303			0.410152	+	0.912017i	$0.365476\pi$
$108$		0			0
$109$	2.34315			0.224433			0.112216	−	0.993684i	$-0.464205\pi$
$109$	2.34315			0.224433			0.112216	+	0.993684i	$0.464205\pi$
$110$		0			0
$111$		0			0
$112$	10.2426			0.967839
$113$		−	11.0711i		−	1.04148i	−0.853716	−	0.520739i	$-0.825656\pi$
$113$		−	11.0711i		−	1.04148i	0.853716	−	0.520739i	$-0.174344\pi$
$114$		0			0
$115$		0			0
$116$	7.75736			0.720253
$117$		0			0
$118$		−	3.51472i		−	0.323556i
$119$	14.4853			1.32786
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$	−2.48528			−0.225007
$123$		0			0
$124$		−	3.21320i		−	0.288554i
$125$		0			0
$126$		0			0
$127$			3.51472i			0.311881i	0.987766	+	0.155940i	$0.0498408\pi$
$127$			3.51472i			0.311881i	−0.987766	+	0.155940i	$0.950159\pi$
$128$		−	10.5563i		−	0.933058i
$129$		0			0
$130$		0			0
$131$	−20.4853			−1.78981			−0.894904	−	0.446259i	$-0.852756\pi$
$131$	−20.4853			−1.78981			−0.894904	+	0.446259i	$0.852756\pi$
$132$		0			0
$133$	2.00000			0.173422
$134$	4.24264			0.366508
$135$		0			0
$136$		−	6.72792i		−	0.576915i
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$		−	8.48528i		−	0.719712i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$		−	8.48528i		−	0.719712i	0.933008	−	0.359856i	$-0.117174\pi$
$140$		0			0
$141$		0			0
$142$			3.51472i			0.294949i
$143$	−1.75736	−	11.8284i	−0.146958	−	0.989143i
$144$		0			0
$145$		0			0
$146$			0.485281i			0.0401622i
$147$		0			0
$148$		−	15.5147i		−	1.27530i
$149$			10.8284i			0.887099i	0.896250	+	0.443550i	$0.146281\pi$
$149$			10.8284i			0.887099i	−0.896250	+	0.443550i	$0.853719\pi$
$150$		0			0
$151$	−2.72792			−0.221995			−0.110998	−	0.993821i	$-0.535405\pi$
$151$	−2.72792			−0.221995			−0.110998	+	0.993821i	$0.535405\pi$
$152$		−	0.928932i		−	0.0753463i
$153$		0			0
$154$	−2.00000	+	4.24264i	−0.161165	+	0.341882i
$155$		0			0
$156$		0			0
$157$	−13.6569			−1.08994			−0.544968	−	0.838457i	$-0.683458\pi$
$157$	−13.6569			−1.08994			−0.544968	+	0.838457i	$0.683458\pi$
$158$	−3.51472			−0.279616
$159$		0			0
$160$		0			0
$161$			4.00000i			0.315244i
$162$		0			0
$163$			1.75736i			0.137647i	0.997629	+	0.0688235i	$0.0219245\pi$
$163$			1.75736i			0.137647i	−0.997629	+	0.0688235i	$0.978075\pi$
$164$		−	2.14214i		−	0.167273i
$165$		0			0
$166$	−1.51472			−0.117565
$167$			18.8284i			1.45699i	0.685052	+	0.728494i	$0.259779\pi$
$167$			18.8284i			1.45699i	−0.685052	+	0.728494i	$0.740221\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$			15.5147i			1.18299i
$173$	0.727922			0.0553429			0.0276714	−	0.999617i	$-0.491191\pi$
$173$	0.727922			0.0553429			0.0276714	+	0.999617i	$0.491191\pi$
$174$		0			0
$175$	−17.0711			−1.29045
$176$	−9.00000	−	4.24264i	−0.678401	−	0.319801i
$177$		0			0
$178$		0			0
$179$		−	5.65685i		−	0.422813i	−0.977398	−	0.211407i	$-0.932196\pi$
$179$		−	5.65685i		−	0.422813i	0.977398	−	0.211407i	$-0.0678044\pi$
$180$		0			0
$181$	−13.3137			−0.989600			−0.494800	−	0.869007i	$-0.664759\pi$
$181$	−13.3137			−0.989600			−0.494800	+	0.869007i	$0.664759\pi$
$182$	4.24264	−	2.82843i	0.314485	−	0.209657i
$183$		0			0
$184$	1.85786			0.136964
$185$		0			0
$186$		0			0
$187$	−12.7279	−	6.00000i	−0.930758	−	0.438763i
$188$	−10.9706			−0.800111
$189$		0			0
$190$		0			0
$191$			22.1421i			1.60215i	0.598565	+	0.801074i	$0.295738\pi$
$191$			22.1421i			1.60215i	−0.598565	+	0.801074i	$0.704262\pi$
$192$		0			0
$193$	5.17157			0.372258			0.186129	−	0.982525i	$-0.440406\pi$
$193$	5.17157			0.372258			0.186129	+	0.982525i	$0.440406\pi$
$194$	−4.97056			−0.356866
$195$		0			0
$196$	8.51472			0.608194
$197$		−	22.1421i		−	1.57756i	−0.614674	−	0.788781i	$-0.710713\pi$
$197$		−	22.1421i		−	1.57756i	0.614674	−	0.788781i	$-0.289287\pi$
$198$		0			0
$199$	−2.34315			−0.166101			−0.0830506	−	0.996545i	$-0.526466\pi$
$199$	−2.34315			−0.166101			−0.0830506	+	0.996545i	$0.526466\pi$
$200$			7.92893i			0.560660i
$201$		0			0
$202$		−	3.21320i		−	0.226080i
$203$	14.4853			1.01667
$204$		0			0
$205$		0			0
$206$		−	1.37258i		−	0.0956324i
$207$		0			0
$208$	6.00000	+	9.00000i	0.416025	+	0.624038i
$209$	−1.75736	−	0.828427i	−0.121559	−	0.0573035i
$210$		0			0
$211$		−	20.4853i		−	1.41026i	−0.709076	−	0.705132i	$-0.750888\pi$
$211$		−	20.4853i		−	1.41026i	0.709076	−	0.705132i	$-0.249112\pi$
$212$			17.2132i			1.18221i
$213$		0			0
$214$		−	3.51472i		−	0.240261i
$215$		0			0
$216$		0			0
$217$		−	6.00000i		−	0.407307i
$218$		−	0.970563i		−	0.0657348i
$219$		0			0
$220$		0			0
$221$	8.48528	+	12.7279i	0.570782	+	0.856173i
$222$		0			0
$223$			22.2426i			1.48948i	0.667356	+	0.744739i	$0.267426\pi$
$223$			22.2426i			1.48948i	−0.667356	+	0.744739i	$0.732574\pi$
$224$		−	15.0711i		−	1.00698i
$225$		0			0
$226$	−4.58579			−0.305042
$227$			18.8284i			1.24969i	0.780750	+	0.624843i	$0.214837\pi$
$227$			18.8284i			1.24969i	−0.780750	+	0.624843i	$0.785163\pi$
$228$		0			0
$229$			8.48528i			0.560723i	0.959894	+	0.280362i	$0.0904544\pi$
$229$			8.48528i			0.560723i	−0.959894	+	0.280362i	$0.909546\pi$
$230$		0			0
$231$		0			0
$232$		−	6.72792i		−	0.441710i
$233$	7.75736			0.508202			0.254101	−	0.967178i	$-0.418221\pi$
$233$	7.75736			0.508202			0.254101	+	0.967178i	$0.418221\pi$
$234$		0			0
$235$		0			0
$236$	15.5147			1.00992
$237$		0			0
$238$		−	6.00000i		−	0.388922i
$239$			21.3137i			1.37867i	0.724443	+	0.689335i	$0.242097\pi$
$239$			21.3137i			1.37867i	−0.724443	+	0.689335i	$0.757903\pi$
$240$		0			0
$241$	−18.8284			−1.21285			−0.606423	−	0.795142i	$-0.707396\pi$
$241$	−18.8284			−1.21285			−0.606423	+	0.795142i	$0.707396\pi$
$242$	3.51472	−	2.89949i	0.225935	−	0.186387i
$243$		0			0
$244$		−	10.9706i		−	0.702318i
$245$		0			0
$246$		0			0
$247$	1.17157	+	1.75736i	0.0745454	+	0.111818i
$248$	−2.78680			−0.176962
$249$		0			0
$250$		0			0
$251$		−	19.3137i		−	1.21907i	−0.792759	−	0.609535i	$-0.791356\pi$
$251$		−	19.3137i		−	1.21907i	0.792759	−	0.609535i	$-0.208644\pi$
$252$		0			0
$253$	1.65685	−	3.51472i	0.104166	−	0.220968i
$254$	1.45584			0.0913478
$255$		0			0
$256$	3.97056			0.248160
$257$			1.89949i			0.118487i	0.998244	+	0.0592436i	$0.0188689\pi$
$257$			1.89949i			0.118487i	−0.998244	+	0.0592436i	$0.981131\pi$
$258$		0			0
$259$		−	28.9706i		−	1.80014i
$260$		0			0
$261$		0			0
$262$			8.48528i			0.524222i
$263$	−28.9706			−1.78640			−0.893201	−	0.449658i	$-0.851546\pi$
$263$	−28.9706			−1.78640			−0.893201	+	0.449658i	$0.851546\pi$
$264$		0			0
$265$		0			0
$266$		−	0.828427i		−	0.0507941i
$267$		0			0
$268$			18.7279i			1.14399i
$269$			5.41421i			0.330110i	0.986284	+	0.165055i	$0.0527802\pi$
$269$			5.41421i			0.330110i	−0.986284	+	0.165055i	$0.947220\pi$
$270$		0			0
$271$	−28.3848			−1.72425			−0.862126	−	0.506694i	$-0.830868\pi$
$271$	−28.3848			−1.72425			−0.862126	+	0.506694i	$0.830868\pi$
$272$	12.7279			0.771744
$273$		0			0
$274$			4.97056i			0.300283i
$275$	15.0000	+	7.07107i	0.904534	+	0.426401i
$276$		0			0
$277$			6.00000i			0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$			6.00000i			0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	−3.51472			−0.210799
$279$		0			0
$280$		0			0
$281$			23.7990i			1.41973i	0.704338	+	0.709864i	$0.251244\pi$
$281$			23.7990i			1.41973i	−0.704338	+	0.709864i	$0.748756\pi$
$282$		0			0
$283$		−	20.4853i		−	1.21772i	−0.793276	−	0.608862i	$-0.791626\pi$
$283$		−	20.4853i		−	1.21772i	0.793276	−	0.608862i	$-0.208374\pi$
$284$	−15.5147			−0.920629
$285$		0			0
$286$	−4.89949	+	0.727922i	−0.289713	+	0.0430429i
$287$		−	4.00000i		−	0.236113i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$	−2.14214			−0.125359
$293$			14.1421i			0.826192i	0.910687	+	0.413096i	$0.135553\pi$
$293$			14.1421i			0.826192i	−0.910687	+	0.413096i	$0.864447\pi$
$294$		0			0
$295$		0			0
$296$	−13.4558			−0.782105
$297$		0			0
$298$	4.48528			0.259825
$299$	−3.51472	+	2.34315i	−0.203261	+	0.135508i
$300$		0			0
$301$			28.9706i			1.66984i
$302$			1.12994i			0.0650209i
$303$		0			0
$304$	1.75736			0.100791
$305$		0			0
$306$		0			0
$307$	−14.7279			−0.840567			−0.420283	−	0.907393i	$-0.638069\pi$
$307$	−14.7279			−0.840567			−0.420283	+	0.907393i	$0.638069\pi$
$308$	−18.7279	−	8.82843i	−1.06712	−	0.503046i
$309$		0			0
$310$		0			0
$311$			17.1716i			0.973711i	0.873483	+	0.486855i	$0.161856\pi$
$311$			17.1716i			0.973711i	−0.873483	+	0.486855i	$0.838144\pi$
$312$		0			0
$313$	9.31371			0.526442			0.263221	−	0.964736i	$-0.415215\pi$
$313$	9.31371			0.526442			0.263221	+	0.964736i	$0.415215\pi$
$314$			5.65685i			0.319235i
$315$		0			0
$316$		−	15.5147i		−	0.872771i
$317$	−4.97056			−0.279175			−0.139587	−	0.990210i	$-0.544578\pi$
$317$	−4.97056			−0.279175			−0.139587	+	0.990210i	$0.544578\pi$
$318$		0			0
$319$	−12.7279	−	6.00000i	−0.712627	−	0.335936i
$320$		0			0
$321$		0			0
$322$	1.65685			0.0923329
$323$	2.48528			0.138285
$324$		0			0
$325$	−10.0000	−	15.0000i	−0.554700	−	0.832050i
$326$	0.727922			0.0403159
$327$		0			0
$328$	−1.85786			−0.102583
$329$	−20.4853			−1.12939
$330$		0			0
$331$		−	6.72792i		−	0.369800i	−0.982757	−	0.184900i	$-0.940804\pi$
$331$		−	6.72792i		−	0.369800i	0.982757	−	0.184900i	$-0.0591961\pi$
$332$		−	6.68629i		−	0.366958i
$333$		0			0
$334$	7.79899			0.426742
$335$		0			0
$336$		0			0
$337$		−	10.9706i		−	0.597605i	−0.954315	−	0.298802i	$-0.903413\pi$
$337$		−	10.9706i		−	0.597605i	0.954315	−	0.298802i	$-0.0965871\pi$
$338$	4.97056	+	2.07107i	0.270363	+	0.112651i
$339$		0			0
$340$		0			0
$341$	−2.48528	+	5.27208i	−0.134586	+	0.285499i
$342$		0			0
$343$	−8.00000			−0.431959
$344$	13.4558			0.725490
$345$		0			0
$346$		−	0.301515i		−	0.0162096i
$347$	20.4853			1.09971			0.549854	−	0.835261i	$-0.314683\pi$
$347$	20.4853			1.09971			0.549854	+	0.835261i	$0.314683\pi$
$348$		0			0
$349$	−10.3431			−0.553656			−0.276828	−	0.960920i	$-0.589283\pi$
$349$	−10.3431			−0.553656			−0.276828	+	0.960920i	$0.589283\pi$
$350$			7.07107i			0.377964i
$351$		0			0
$352$	−6.24264	+	13.2426i	−0.332734	+	0.705835i
$353$	16.9706			0.903252			0.451626	−	0.892207i	$-0.350844\pi$
$353$	16.9706			0.903252			0.451626	+	0.892207i	$0.350844\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−2.34315			−0.123839
$359$		−	30.1421i		−	1.59084i	−0.606058	−	0.795421i	$-0.707250\pi$
$359$		−	30.1421i		−	1.59084i	0.606058	−	0.795421i	$-0.292750\pi$
$360$		0			0
$361$	−18.6569			−0.981940
$362$			5.51472i			0.289847i
$363$		0			0
$364$	12.4853	+	18.7279i	0.654407	+	0.981610i
$365$		0			0
$366$		0			0
$367$	−31.3137			−1.63456			−0.817281	−	0.576239i	$-0.804520\pi$
$367$	−31.3137			−1.63456			−0.817281	+	0.576239i	$0.804520\pi$
$368$			3.51472i			0.183217i
$369$		0			0
$370$		0			0
$371$			32.1421i			1.66874i
$372$		0			0
$373$			24.0000i			1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$			24.0000i			1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$	−2.48528	+	5.27208i	−0.128511	+	0.272613i
$375$		0			0
$376$			9.51472i			0.490684i
$377$	8.48528	+	12.7279i	0.437014	+	0.655521i
$378$		0			0
$379$			18.7279i			0.961989i	0.876724	+	0.480994i	$0.159724\pi$
$379$			18.7279i			0.961989i	−0.876724	+	0.480994i	$0.840276\pi$
$380$		0			0
$381$		0			0
$382$	9.17157			0.469258
$383$	−18.0000			−0.919757			−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000			−0.919757			−0.459879	+	0.887982i	$0.652107\pi$
$384$		0			0
$385$		0			0
$386$		−	2.14214i		−	0.109032i
$387$		0			0
$388$		−	21.9411i		−	1.11389i
$389$			4.44365i			0.225302i	0.993635	+	0.112651i	$0.0359342\pi$
$389$			4.44365i			0.225302i	−0.993635	+	0.112651i	$0.964066\pi$
$390$		0			0
$391$			4.97056i			0.251372i
$392$		−	7.38478i		−	0.372988i
$393$		0			0
$394$	−9.17157			−0.462057
$395$		0			0
$396$		0			0
$397$		−	25.4558i		−	1.27759i	−0.769376	−	0.638796i	$-0.779433\pi$
$397$		−	25.4558i		−	1.27759i	0.769376	−	0.638796i	$-0.220567\pi$
$398$			0.970563i			0.0486499i
$399$		0			0
$400$	−15.0000			−0.750000
$401$	−28.9706			−1.44672			−0.723360	−	0.690471i	$-0.757403\pi$
$401$	−28.9706			−1.44672			−0.723360	+	0.690471i	$0.757403\pi$
$402$		0			0
$403$	5.27208	−	3.51472i	0.262621	−	0.175081i
$404$	14.1838			0.705669
$405$		0			0
$406$		−	6.00000i		−	0.297775i
$407$	−12.0000	+	25.4558i	−0.594818	+	1.26180i
$408$		0			0
$409$	−21.4558			−1.06092			−0.530462	−	0.847709i	$-0.677981\pi$
$409$	−21.4558			−1.06092			−0.530462	+	0.847709i	$0.677981\pi$
$410$		0			0
$411$		0			0
$412$	6.05887			0.298499
$413$	28.9706			1.42555
$414$		0			0
$415$		0			0
$416$	13.2426	−	8.82843i	0.649274	−	0.432849i
$417$		0			0
$418$	−0.343146	+	0.727922i	−0.0167838	+	0.0356038i
$419$			14.6274i			0.714596i	0.933990	+	0.357298i	$0.116302\pi$
$419$			14.6274i			0.714596i	−0.933990	+	0.357298i	$0.883698\pi$
$420$		0			0
$421$		−	3.51472i		−	0.171297i	−0.996325	−	0.0856485i	$-0.972704\pi$
$421$		−	3.51472i		−	0.171297i	0.996325	−	0.0856485i	$-0.0272962\pi$
$422$	−8.48528			−0.413057
$423$		0			0
$424$	14.9289			0.725013
$425$	−21.2132			−1.02899
$426$		0			0
$427$		−	20.4853i		−	0.991352i
$428$	15.5147			0.749932
$429$		0			0
$430$		0			0
$431$		−	6.14214i		−	0.295856i	−0.988998	−	0.147928i	$-0.952740\pi$
$431$		−	6.14214i		−	0.295856i	0.988998	−	0.147928i	$-0.0472604\pi$
$432$		0			0
$433$	4.68629			0.225209			0.112604	−	0.993640i	$-0.464081\pi$
$433$	4.68629			0.225209			0.112604	+	0.993640i	$0.464081\pi$
$434$	−2.48528			−0.119297
$435$		0			0
$436$	4.28427			0.205179
$437$			0.686292i			0.0328298i
$438$		0			0
$439$			3.51472i			0.167748i	0.996476	+	0.0838742i	$0.0267294\pi$
$439$			3.51472i			0.167748i	−0.996476	+	0.0838742i	$0.973271\pi$
$440$		0			0
$441$		0			0
$442$	5.27208	−	3.51472i	0.250767	−	0.167178i
$443$		−	14.3431i		−	0.681463i	−0.940161	−	0.340732i	$-0.889325\pi$
$443$		−	14.3431i		−	0.681463i	0.940161	−	0.340732i	$-0.110675\pi$
$444$		0			0
$445$		0			0
$446$	9.21320			0.436258
$447$		0			0
$448$	14.2426			0.672902
$449$	−28.9706			−1.36721			−0.683603	−	0.729854i	$-0.739588\pi$
$449$	−28.9706			−1.36721			−0.683603	+	0.729854i	$0.739588\pi$
$450$		0			0
$451$	−1.65685	+	3.51472i	−0.0780182	+	0.165502i
$452$		−	20.2426i		−	0.952134i
$453$		0			0
$454$	7.79899			0.366025
$455$		0			0
$456$		0			0
$457$	34.8284			1.62921			0.814603	−	0.580020i	$-0.196955\pi$
$457$	34.8284			1.62921			0.814603	+	0.580020i	$0.196955\pi$
$458$	3.51472			0.164232
$459$		0			0
$460$		0			0
$461$		−	17.1716i		−	0.799760i	−0.916568	−	0.399880i	$-0.869052\pi$
$461$		−	17.1716i		−	0.799760i	0.916568	−	0.399880i	$-0.130948\pi$
$462$		0			0
$463$			13.7574i			0.639359i	0.947526	+	0.319679i	$0.103575\pi$
$463$			13.7574i			0.639359i	−0.947526	+	0.319679i	$0.896425\pi$
$464$	12.7279			0.590879
$465$		0			0
$466$		−	3.21320i		−	0.148849i
$467$		−	41.6569i		−	1.92765i	−0.266537	−	0.963825i	$-0.585879\pi$
$467$		−	41.6569i		−	1.92765i	0.266537	−	0.963825i	$-0.414121\pi$
$468$		0			0
$469$			34.9706i			1.61479i
$470$		0			0
$471$		0			0
$472$		−	13.4558i		−	0.619355i
$473$	12.0000	−	25.4558i	0.551761	−	1.17046i
$474$		0			0
$475$	−2.92893			−0.134389
$476$	26.4853			1.21395
$477$		0			0
$478$	8.82843			0.403803
$479$			40.6274i			1.85631i	0.372189	+	0.928157i	$0.378607\pi$
$479$			40.6274i			1.85631i	−0.372189	+	0.928157i	$0.621393\pi$
$480$		0			0
$481$	25.4558	−	16.9706i	1.16069	−	0.773791i
$482$			7.79899i			0.355234i
$483$		0			0
$484$	12.7990	+	15.5147i	0.581772	+	0.705214i
$485$		0			0
$486$		0			0
$487$		−	15.2132i		−	0.689376i	−0.938717	−	0.344688i	$-0.887985\pi$
$487$		−	15.2132i		−	0.689376i	0.938717	−	0.344688i	$-0.112015\pi$
$488$	−9.51472			−0.430711
$489$		0			0
$490$		0			0
$491$	−3.51472			−0.158617			−0.0793085	−	0.996850i	$-0.525271\pi$
$491$	−3.51472			−0.158617			−0.0793085	+	0.996850i	$0.525271\pi$
$492$		0			0
$493$	18.0000			0.810679
$494$	0.727922	−	0.485281i	0.0327508	−	0.0218338i
$495$		0			0
$496$		−	5.27208i		−	0.236723i
$497$	−28.9706			−1.29951
$498$		0			0
$499$			13.7574i			0.615864i	0.951408	+	0.307932i	$0.0996369\pi$
$499$			13.7574i			0.615864i	−0.951408	+	0.307932i	$0.900363\pi$
$500$		0			0
$501$		0			0
$502$	−8.00000			−0.357057
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$		0			0
$506$	−1.45584	−	0.686292i	−0.0647202	−	0.0305094i
$507$		0			0
$508$			6.42641i			0.285126i
$509$	12.0000			0.531891			0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000			0.531891			0.265945	+	0.963988i	$0.414316\pi$
$510$		0			0
$511$	−4.00000			−0.176950
$512$		−	22.7574i		−	1.00574i
$513$		0			0
$514$	0.786797			0.0347041
$515$		0			0
$516$		0			0
$517$	18.0000	+	8.48528i	0.791639	+	0.373182i
$518$	−12.0000			−0.527250
$519$		0			0
$520$		0			0
$521$		−	18.5858i		−	0.814258i	−0.913371	−	0.407129i	$-0.866530\pi$
$521$		−	18.5858i		−	0.814258i	0.913371	−	0.407129i	$-0.133470\pi$
$522$		0			0
$523$		−	13.4558i		−	0.588383i	−0.955746	−	0.294191i	$-0.904950\pi$
$523$		−	13.4558i		−	0.588383i	0.955746	−	0.294191i	$-0.0950503\pi$
$524$	−37.4558			−1.63627
$525$		0			0
$526$			12.0000i			0.523225i
$527$		−	7.45584i		−	0.324782i
$528$		0			0
$529$	21.6274			0.940322
$530$		0			0
$531$		0			0
$532$	3.65685			0.158545
$533$	3.51472	−	2.34315i	0.152239	−	0.101493i
$534$		0			0
$535$		0			0
$536$	16.2426			0.701575
$537$		0			0
$538$	2.24264			0.0966871
$539$	−13.9706	−	6.58579i	−0.601755	−	0.283670i
$540$		0			0
$541$	35.5980			1.53048			0.765238	−	0.643747i	$-0.222621\pi$
$541$	35.5980			1.53048			0.765238	+	0.643747i	$0.222621\pi$
$542$			11.7574i			0.505022i
$543$		0			0
$544$		−	18.7279i		−	0.802953i
$545$		0			0
$546$		0			0
$547$		−	8.48528i		−	0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$		−	8.48528i		−	0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$	−21.9411			−0.937278
$549$		0			0
$550$	2.92893	−	6.21320i	0.124890	−	0.264932i
$551$	2.48528			0.105877
$552$		0			0
$553$		−	28.9706i		−	1.23195i
$554$	2.48528			0.105589
$555$		0			0
$556$		−	15.5147i		−	0.657971i
$557$		−	42.1421i		−	1.78562i	−0.450434	−	0.892810i	$-0.648731\pi$
$557$		−	42.1421i		−	1.78562i	0.450434	−	0.892810i	$-0.351269\pi$
$558$		0			0
$559$	−25.4558	+	16.9706i	−1.07667	+	0.717778i
$560$		0			0
$561$		0			0
$562$	9.85786			0.415829
$563$	−3.51472			−0.148128			−0.0740639	−	0.997254i	$-0.523597\pi$
$563$	−3.51472			−0.148128			−0.0740639	+	0.997254i	$0.523597\pi$
$564$		0			0
$565$		0			0
$566$	−8.48528			−0.356663
$567$		0			0
$568$			13.4558i			0.564595i
$569$	40.2426			1.68706			0.843530	−	0.537083i	$-0.180474\pi$
$569$	40.2426			1.68706			0.843530	+	0.537083i	$0.180474\pi$
$570$		0			0
$571$			44.4853i			1.86165i	0.365464	+	0.930826i	$0.380910\pi$
$571$			44.4853i			1.86165i	−0.365464	+	0.930826i	$0.619090\pi$
$572$	−3.21320	−	21.6274i	−0.134351	−	0.904288i
$573$		0			0
$574$	−1.65685			−0.0691558
$575$		−	5.85786i		−	0.244290i
$576$		0			0
$577$			40.9706i			1.70563i	0.522216	+	0.852813i	$0.325106\pi$
$577$			40.9706i			1.70563i	−0.522216	+	0.852813i	$0.674894\pi$
$578$		−	0.414214i		−	0.0172290i
$579$		0			0
$580$		0			0
$581$		−	12.4853i		−	0.517977i
$582$		0			0
$583$	13.3137	−	28.2426i	0.551397	−	1.16969i
$584$			1.85786i			0.0768790i
$585$		0			0
$586$	5.85786			0.241986
$587$	32.4853			1.34081			0.670406	−	0.741995i	$-0.266120\pi$
$587$	32.4853			1.34081			0.670406	+	0.741995i	$0.266120\pi$
$588$		0			0
$589$		−	1.02944i		−	0.0424172i
$590$		0			0
$591$		0			0
$592$		−	25.4558i		−	1.04623i
$593$		−	18.1421i		−	0.745008i	−0.928031	−	0.372504i	$-0.878499\pi$
$593$		−	18.1421i		−	0.745008i	0.928031	−	0.372504i	$-0.121501\pi$
$594$		0			0
$595$		0			0
$596$			19.7990i			0.810998i
$597$		0			0
$598$	0.970563	+	1.45584i	0.0396893	+	0.0595339i
$599$			17.1716i			0.701611i	0.936448	+	0.350806i	$0.114092\pi$
$599$			17.1716i			0.701611i	−0.936448	+	0.350806i	$0.885908\pi$
$600$		0			0
$601$			16.9706i			0.692244i	0.938190	+	0.346122i	$0.112502\pi$
$601$			16.9706i			0.692244i	−0.938190	+	0.346122i	$0.887498\pi$
$602$	12.0000			0.489083
$603$		0			0
$604$	−4.98781			−0.202951
$605$		0			0
$606$		0			0
$607$			32.4853i			1.31854i	0.751908	+	0.659268i	$0.229134\pi$
$607$			32.4853i			1.31854i	−0.751908	+	0.659268i	$0.770866\pi$
$608$		−	2.58579i		−	0.104867i
$609$		0			0
$610$		0			0
$611$	−12.0000	−	18.0000i	−0.485468	−	0.728202i
$612$		0			0
$613$	31.3137			1.26475			0.632374	−	0.774663i	$-0.282080\pi$
$613$	31.3137			1.26475			0.632374	+	0.774663i	$0.282080\pi$
$614$			6.10051i			0.246196i
$615$		0			0
$616$	−7.65685	+	16.2426i	−0.308503	+	0.654435i
$617$	16.9706			0.683209			0.341605	−	0.939844i	$-0.389030\pi$
$617$	16.9706			0.683209			0.341605	+	0.939844i	$0.389030\pi$
$618$		0			0
$619$			27.2132i			1.09379i	0.837201	+	0.546895i	$0.184191\pi$
$619$			27.2132i			1.09379i	−0.837201	+	0.546895i	$0.815809\pi$
$620$		0			0
$621$		0			0
$622$	7.11270			0.285193
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		−	3.85786i		−	0.154191i
$627$		0			0
$628$	−24.9706			−0.996434
$629$		−	36.0000i		−	1.43541i
$630$		0			0
$631$		−	42.7279i		−	1.70097i	−0.525998	−	0.850486i	$-0.676308\pi$
$631$		−	42.7279i		−	1.70097i	0.525998	−	0.850486i	$-0.323692\pi$
$632$	−13.4558			−0.535245
$633$		0			0
$634$			2.05887i			0.0817684i
$635$		0			0
$636$		0			0
$637$	9.31371	+	13.9706i	0.369023	+	0.553534i
$638$	−2.48528	+	5.27208i	−0.0983932	+	0.208724i
$639$		0			0
$640$		0			0
$641$		−	3.07107i		−	0.121300i	−0.998159	−	0.0606499i	$-0.980683\pi$
$641$		−	3.07107i		−	0.121300i	0.998159	−	0.0606499i	$-0.0193173\pi$
$642$		0			0
$643$		−	23.6985i		−	0.934577i	−0.884105	−	0.467289i	$-0.845231\pi$
$643$		−	23.6985i		−	0.934577i	0.884105	−	0.467289i	$-0.154769\pi$
$644$			7.31371i			0.288200i
$645$		0			0
$646$		−	1.02944i		−	0.0405027i
$647$		−	23.7990i		−	0.935635i	−0.883825	−	0.467817i	$-0.845040\pi$
$647$		−	23.7990i		−	0.935635i	0.883825	−	0.467817i	$-0.154960\pi$
$648$		0			0
$649$	−25.4558	−	12.0000i	−0.999229	−	0.471041i
$650$	−6.21320	+	4.14214i	−0.243702	+	0.162468i
$651$		0			0
$652$			3.21320i			0.125839i
$653$		−	37.4142i		−	1.46413i	−0.681234	−	0.732066i	$-0.738556\pi$
$653$		−	37.4142i		−	1.46413i	0.681234	−	0.732066i	$-0.261444\pi$
$654$		0			0
$655$		0			0
$656$		−	3.51472i		−	0.137227i
$657$		0			0
$658$			8.48528i			0.330791i
$659$	−25.4558			−0.991619			−0.495809	−	0.868431i	$-0.665129\pi$
$659$	−25.4558			−0.991619			−0.495809	+	0.868431i	$0.665129\pi$
$660$		0			0
$661$		−	20.4853i		−	0.796785i	−0.917215	−	0.398393i	$-0.869568\pi$
$661$		−	20.4853i		−	0.796785i	0.917215	−	0.398393i	$-0.130432\pi$
$662$	−2.78680			−0.108312
$663$		0			0
$664$	−5.79899			−0.225044
$665$		0			0
$666$		0			0
$667$			4.97056i			0.192461i
$668$			34.4264i			1.33200i
$669$		0			0
$670$		0			0
$671$	−8.48528	+	18.0000i	−0.327571	+	0.694882i
$672$		0			0
$673$			10.9706i			0.422884i	0.977391	+	0.211442i	$0.0678160\pi$
$673$			10.9706i			0.422884i	−0.977391	+	0.211442i	$0.932184\pi$
$674$	−4.54416			−0.175034
$675$		0			0
$676$	−9.14214	+	21.9411i	−0.351621	+	0.843889i
$677$	40.2426			1.54665			0.773325	−	0.634010i	$-0.218592\pi$
$677$	40.2426			1.54665			0.773325	+	0.634010i	$0.218592\pi$
$678$		0			0
$679$		−	40.9706i		−	1.57231i
$680$		0			0
$681$		0			0
$682$	2.18377	+	1.02944i	0.0836207	+	0.0394192i
$683$	−27.9411			−1.06914			−0.534569	−	0.845125i	$-0.679526\pi$
$683$	−27.9411			−1.06914			−0.534569	+	0.845125i	$0.679526\pi$
$684$		0			0
$685$		0			0
$686$			3.31371i			0.126518i
$687$		0			0
$688$			25.4558i			0.970495i
$689$	−28.2426	+	18.8284i	−1.07596	+	0.717306i
$690$		0			0
$691$			23.6985i			0.901533i	0.892642	+	0.450766i	$0.148849\pi$
$691$			23.6985i			0.901533i	−0.892642	+	0.450766i	$0.851151\pi$
$692$	1.33095			0.0505952
$693$		0			0
$694$		−	8.48528i		−	0.322097i
$695$		0			0
$696$		0			0
$697$		−	4.97056i		−	0.188273i
$698$			4.28427i			0.162162i
$699$		0			0
$700$	−31.2132			−1.17975
$701$	28.2426			1.06671			0.533355	−	0.845892i	$-0.320931\pi$
$701$	28.2426			1.06671			0.533355	+	0.845892i	$0.320931\pi$
$702$		0			0
$703$		−	4.97056i		−	0.187468i
$704$	−12.5147	−	5.89949i	−0.471666	−	0.222346i
$705$		0			0
$706$		−	7.02944i		−	0.264556i
$707$	26.4853			0.996082
$708$		0			0
$709$		−	49.4558i		−	1.85735i	−0.370890	−	0.928677i	$-0.620947\pi$
$709$		−	49.4558i		−	1.85735i	0.370890	−	0.928677i	$-0.379053\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.05887			0.0771055
$714$		0			0
$715$		0			0
$716$		−	10.3431i		−	0.386542i
$717$		0			0
$718$	−12.4853			−0.465947
$719$			19.7990i			0.738378i	0.929354	+	0.369189i	$0.120364\pi$
$719$			19.7990i			0.738378i	−0.929354	+	0.369189i	$0.879636\pi$
$720$		0			0
$721$	11.3137			0.421345
$722$			7.72792i			0.287603i
$723$		0			0
$724$	−24.3431			−0.904706
$725$	−21.2132			−0.787839
$726$		0			0
$727$	−25.6569			−0.951560			−0.475780	−	0.879564i	$-0.657834\pi$
$727$	−25.6569			−0.951560			−0.475780	+	0.879564i	$0.657834\pi$
$728$	16.2426	−	10.8284i	0.601992	−	0.401328i
$729$		0			0
$730$		0			0
$731$			36.0000i			1.33151i
$732$		0			0
$733$	47.5980			1.75807			0.879036	−	0.476756i	$-0.158187\pi$
$733$	47.5980			1.75807			0.879036	+	0.476756i	$0.158187\pi$
$734$			12.9706i			0.478752i
$735$		0			0
$736$	5.17157			0.190627
$737$	14.4853	−	30.7279i	0.533572	−	1.13188i
$738$		0			0
$739$	−10.0416			−0.369387			−0.184694	−	0.982796i	$-0.559129\pi$
$739$	−10.0416			−0.369387			−0.184694	+	0.982796i	$0.559129\pi$
$740$		0			0
$741$		0			0
$742$	13.3137			0.488762
$743$			6.68629i			0.245296i	0.992450	+	0.122648i	$0.0391387\pi$
$743$			6.68629i			0.245296i	−0.992450	+	0.122648i	$0.960861\pi$
$744$		0			0
$745$		0			0
$746$	9.94113			0.363970
$747$		0			0
$748$	−23.2721	−	10.9706i	−0.850911	−	0.401124i
$749$	28.9706			1.05856
$750$		0			0
$751$	−24.2843			−0.886146			−0.443073	−	0.896486i	$-0.646112\pi$
$751$	−24.2843			−0.886146			−0.443073	+	0.896486i	$0.646112\pi$
$752$	−18.0000			−0.656392
$753$		0			0
$754$	5.27208	−	3.51472i	0.191998	−	0.127999i
$755$		0			0
$756$		0			0
$757$	33.3137			1.21081			0.605404	−	0.795919i	$-0.293012\pi$
$757$	33.3137			1.21081			0.605404	+	0.795919i	$0.293012\pi$
$758$	7.75736			0.281760
$759$		0			0
$760$		0			0
$761$		−	13.1716i		−	0.477469i	−0.971085	−	0.238735i	$-0.923267\pi$
$761$		−	13.1716i		−	0.477469i	0.971085	−	0.238735i	$-0.0767326\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$			40.4853i			1.46471i
$765$		0			0
$766$			7.45584i			0.269391i
$767$	16.9706	+	25.4558i	0.612772	+	0.919157i
$768$		0			0
$769$	−13.8579			−0.499727			−0.249864	−	0.968281i	$-0.580386\pi$
$769$	−13.8579			−0.499727			−0.249864	+	0.968281i	$0.580386\pi$
$770$		0			0
$771$		0			0
$772$	9.45584			0.340323
$773$	−45.9411			−1.65239			−0.826194	−	0.563386i	$-0.809498\pi$
$773$	−45.9411			−1.65239			−0.826194	+	0.563386i	$0.809498\pi$
$774$		0			0
$775$			8.78680i			0.315631i
$776$	−19.0294			−0.683117
$777$		0			0
$778$	1.84062			0.0659894
$779$		−	0.686292i		−	0.0245889i
$780$		0			0
$781$	25.4558	+	12.0000i	0.910882	+	0.429394i
$782$	2.05887			0.0736252
$783$		0			0
$784$	13.9706			0.498949
$785$		0			0
$786$		0			0
$787$	−34.0416			−1.21345			−0.606727	−	0.794911i	$-0.707518\pi$
$787$	−34.0416			−1.21345			−0.606727	+	0.794911i	$0.707518\pi$
$788$		−	40.4853i		−	1.44223i
$789$		0			0
$790$		0			0
$791$		−	37.7990i		−	1.34398i
$792$		0			0
$793$	18.0000	−	12.0000i	0.639199	−	0.426132i
$794$	−10.5442			−0.374198
$795$		0			0
$796$	−4.28427			−0.151852
$797$		−	37.4142i		−	1.32528i	−0.748938	−	0.662640i	$-0.769436\pi$
$797$		−	37.4142i		−	1.32528i	0.748938	−	0.662640i	$-0.230564\pi$
$798$		0			0
$799$	−25.4558			−0.900563
$800$			22.0711i			0.780330i
$801$		0			0
$802$			12.0000i			0.423735i
$803$	3.51472	+	1.65685i	0.124032	+	0.0584691i
$804$		0			0
$805$		0			0
$806$	−1.45584	−	2.18377i	−0.0512799	−	0.0769199i
$807$		0			0
$808$		−	12.3015i		−	0.432766i
$809$	46.6690			1.64080			0.820398	−	0.571793i	$-0.193752\pi$
$809$	46.6690			1.64080			0.820398	+	0.571793i	$0.193752\pi$
$810$		0			0
$811$	32.3848			1.13718			0.568592	−	0.822620i	$-0.307488\pi$
$811$	32.3848			1.13718			0.568592	+	0.822620i	$0.307488\pi$
$812$	26.4853			0.929451
$813$		0			0
$814$	10.5442	+	4.97056i	0.369572	+	0.174218i
$815$		0			0
$816$		0			0
$817$			4.97056i			0.173898i
$818$			8.88730i			0.310737i
$819$		0			0
$820$		0			0
$821$		−	38.8284i		−	1.35512i	−0.735467	−	0.677561i	$-0.763037\pi$
$821$		−	38.8284i		−	1.35512i	0.735467	−	0.677561i	$-0.236963\pi$
$822$		0			0
$823$	22.3431			0.778833			0.389417	−	0.921062i	$-0.372677\pi$
$823$	22.3431			0.778833			0.389417	+	0.921062i	$0.372677\pi$
$824$		−	5.25483i		−	0.183061i
$825$		0			0
$826$		−	12.0000i		−	0.417533i
$827$		−	12.2010i		−	0.424271i	−0.977240	−	0.212135i	$-0.931958\pi$
$827$		−	12.2010i		−	0.424271i	0.977240	−	0.212135i	$-0.0680418\pi$
$828$		0			0
$829$	33.3137			1.15703			0.578516	−	0.815671i	$-0.303632\pi$
$829$	33.3137			1.15703			0.578516	+	0.815671i	$0.303632\pi$
$830$		0			0
$831$		0			0
$832$	8.34315	+	12.5147i	0.289247	+	0.433870i
$833$	19.7574			0.684552
$834$		0			0
$835$		0			0
$836$	−3.21320	−	1.51472i	−0.111131	−	0.0523876i
$837$		0			0
$838$	6.05887			0.209300
$839$	−49.4558			−1.70741			−0.853703	−	0.520761i	$-0.825648\pi$
$839$	−49.4558			−1.70741			−0.853703	+	0.520761i	$0.825648\pi$
$840$		0			0
$841$	−11.0000			−0.379310
$842$	−1.45584			−0.0501717
$843$		0			0
$844$		−	37.4558i		−	1.28928i
$845$		0			0
$846$		0			0
$847$	23.8995	+	28.9706i	0.821196	+	0.995440i
$848$			28.2426i			0.969856i
$849$		0			0
$850$			8.78680i			0.301385i
$851$	9.94113			0.340777
$852$		0			0
$853$	53.2548			1.82341			0.911705	−	0.410845i	$-0.134766\pi$
$853$	53.2548			1.82341			0.911705	+	0.410845i	$0.134766\pi$
$854$	−8.48528			−0.290360
$855$		0			0
$856$		−	13.4558i		−	0.459911i
$857$	−41.6985			−1.42439			−0.712197	−	0.701980i	$-0.752300\pi$
$857$	−41.6985			−1.42439			−0.712197	+	0.701980i	$0.752300\pi$
$858$		0			0
$859$	−31.3137			−1.06841			−0.534205	−	0.845355i	$-0.679389\pi$
$859$	−31.3137			−1.06841			−0.534205	+	0.845355i	$0.679389\pi$
$860$		0			0
$861$		0			0
$862$	−2.54416			−0.0866543
$863$	15.9411			0.542642			0.271321	−	0.962489i	$-0.412539\pi$
$863$	15.9411			0.542642			0.271321	+	0.962489i	$0.412539\pi$
$864$		0			0
$865$		0			0
$866$		−	1.94113i		−	0.0659621i
$867$		0			0
$868$		−	10.9706i		−	0.372365i
$869$	−12.0000	+	25.4558i	−0.407072	+	0.863530i
$870$		0			0
$871$	−30.7279	+	20.4853i	−1.04118	+	0.694117i
$872$		−	3.71573i		−	0.125830i
$873$		0			0
$874$	0.284271			0.00961562
$875$		0			0
$876$		0			0
$877$	53.2548			1.79829			0.899144	−	0.437653i	$-0.144190\pi$
$877$	53.2548			1.79829			0.899144	+	0.437653i	$0.144190\pi$
$878$	1.45584			0.0491324
$879$		0			0
$880$		0			0
$881$			9.41421i			0.317173i	0.987345	+	0.158586i	$0.0506937\pi$
$881$			9.41421i			0.317173i	−0.987345	+	0.158586i	$0.949306\pi$
$882$		0			0
$883$	26.6274			0.896084			0.448042	−	0.894013i	$-0.352122\pi$
$883$	26.6274			0.896084			0.448042	+	0.894013i	$0.352122\pi$
$884$	15.5147	+	23.2721i	0.521816	+	0.782725i
$885$		0			0
$886$	−5.94113			−0.199596
$887$	4.97056			0.166895			0.0834476	−	0.996512i	$-0.473407\pi$
$887$	4.97056			0.166895			0.0834476	+	0.996512i	$0.473407\pi$
$888$		0			0
$889$			12.0000i			0.402467i
$890$		0			0
$891$		0			0
$892$			40.6690i			1.36170i
$893$	−3.51472			−0.117616
$894$		0			0
$895$		0			0
$896$		−	36.0416i		−	1.20407i
$897$		0			0
$898$			12.0000i			0.400445i
$899$		−	7.45584i		−	0.248666i
$900$		0			0
$901$			39.9411i			1.33063i
$902$	1.45584	+	0.686292i	0.0484743	+	0.0228510i
$903$		0			0
$904$	−17.5563			−0.583915
$905$		0			0
$906$		0			0
$907$	8.28427			0.275075			0.137537	−	0.990497i	$-0.456081\pi$
$907$	8.28427			0.275075			0.137537	+	0.990497i	$0.456081\pi$
$908$			34.4264i			1.14248i
$909$		0			0
$910$		0			0
$911$			20.2010i			0.669289i	0.942344	+	0.334645i	$0.108616\pi$
$911$			20.2010i			0.669289i	−0.942344	+	0.334645i	$0.891384\pi$
$912$		0			0
$913$	−5.17157	+	10.9706i	−0.171154	+	0.363073i
$914$		−	14.4264i		−	0.477183i
$915$		0			0
$916$			15.5147i			0.512621i
$917$	−69.9411			−2.30966
$918$		0			0
$919$		−	8.48528i		−	0.279904i	−0.990158	−	0.139952i	$-0.955305\pi$
$919$		−	8.48528i		−	0.279904i	0.990158	−	0.139952i	$-0.0446948\pi$
$920$		0			0
$921$		0			0
$922$	−7.11270			−0.234244
$923$	−16.9706	−	25.4558i	−0.558593	−	0.837889i
$924$		0			0
$925$			42.4264i			1.39497i
$926$	5.69848			0.187264
$927$		0			0
$928$		−	18.7279i		−	0.614774i
$929$	21.9411			0.719865			0.359932	−	0.932978i	$-0.382800\pi$
$929$	21.9411			0.719865			0.359932	+	0.932978i	$0.382800\pi$
$930$		0			0
$931$	2.72792			0.0894040
$932$	14.1838			0.464605
$933$		0			0
$934$	−17.2548			−0.564595
$935$		0			0
$936$		0			0
$937$		−	10.9706i		−	0.358393i	−0.983813	−	0.179196i	$-0.942650\pi$
$937$		−	10.9706i		−	0.358393i	0.983813	−	0.179196i	$-0.0573497\pi$
$938$	14.4853			0.472961
$939$		0			0
$940$		0			0
$941$		−	17.1716i		−	0.559777i	−0.960032	−	0.279889i	$-0.909702\pi$
$941$		−	17.1716i		−	0.559777i	0.960032	−	0.279889i	$-0.0902975\pi$
$942$		0			0
$943$	1.37258			0.0446975
$944$	25.4558			0.828517
$945$		0			0
$946$	−10.5442	−	4.97056i	−0.342820	−	0.161607i
$947$	−32.4853			−1.05563			−0.527815	−	0.849359i	$-0.676989\pi$
$947$	−32.4853			−1.05563			−0.527815	+	0.849359i	$0.676989\pi$
$948$		0			0
$949$	−2.34315	−	3.51472i	−0.0760617	−	0.114093i
$950$			1.21320i			0.0393615i
$951$		0			0
$952$		−	22.9706i		−	0.744480i
$953$	26.1838			0.848175			0.424088	−	0.905621i	$-0.360595\pi$
$953$	26.1838			0.848175			0.424088	+	0.905621i	$0.360595\pi$
$954$		0			0
$955$		0			0
$956$			38.9706i			1.26040i
$957$		0			0
$958$	16.8284			0.543702
$959$	−40.9706			−1.32301
$960$		0			0
$961$	27.9117			0.900377
$962$	−7.02944	−	10.5442i	−0.226638	−	0.339957i
$963$		0			0
$964$	−34.4264			−1.10880
$965$		0			0
$966$		0			0
$967$	0.585786			0.0188376			0.00941881	−	0.999956i	$-0.497002\pi$
$967$	0.585786			0.0188376			0.00941881	+	0.999956i	$0.497002\pi$
$968$	13.4558	−	11.1005i	0.432487	−	0.356784i
$969$		0			0
$970$		0			0
$971$		−	8.28427i		−	0.265855i	−0.991126	−	0.132927i	$-0.957562\pi$
$971$		−	8.28427i		−	0.265855i	0.991126	−	0.132927i	$-0.0424377\pi$
$972$		0			0
$973$		−	28.9706i		−	0.928754i
$974$	−6.30152			−0.201914
$975$		0			0
$976$		−	18.0000i		−	0.576166i
$977$	40.9706			1.31076			0.655382	−	0.755297i	$-0.272508\pi$
$977$	40.9706			1.31076			0.655382	+	0.755297i	$0.272508\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$			1.45584i			0.0464579i
$983$	32.4853			1.03612			0.518060	−	0.855344i	$-0.326654\pi$
$983$	32.4853			1.03612			0.518060	+	0.855344i	$0.326654\pi$
$984$		0			0
$985$		0			0
$986$		−	7.45584i		−	0.237442i
$987$		0			0
$988$	2.14214	+	3.21320i	0.0681504	+	0.102226i
$989$	−9.94113			−0.316109
$990$		0			0
$991$	−2.34315			−0.0744325			−0.0372162	−	0.999307i	$-0.511849\pi$
$991$	−2.34315			−0.0744325			−0.0372162	+	0.999307i	$0.511849\pi$
$992$	−7.75736			−0.246296
$993$		0			0
$994$			12.0000i			0.380617i
$995$		0			0
$996$		0			0
$997$		−	7.02944i		−	0.222625i	−0.993785	−	0.111312i	$-0.964495\pi$
$997$		−	7.02944i		−	0.222625i	0.993785	−	0.111312i	$-0.0355054\pi$
$998$	5.69848			0.180382
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.e.c.1286.2	yes	4
3.2	odd	2		1287.2.e.d.1286.3	yes	4
11.10	odd	2		1287.2.e.a.1286.3	yes	4
13.12	even	2		1287.2.e.b.1286.3	yes	4
33.32	even	2		1287.2.e.b.1286.2	yes	4
39.38	odd	2		1287.2.e.a.1286.2	✓	4
143.142	odd	2		1287.2.e.d.1286.2	yes	4
429.428	even	2	inner	1287.2.e.c.1286.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1287.2.e.a.1286.2	✓	4	39.38	odd	2
1287.2.e.a.1286.3	yes	4	11.10	odd	2
1287.2.e.b.1286.2	yes	4	33.32	even	2
1287.2.e.b.1286.3	yes	4	13.12	even	2
1287.2.e.c.1286.2	yes	4	1.1	even	1	trivial
1287.2.e.c.1286.3	yes	4	429.428	even	2	inner
1287.2.e.d.1286.2	yes	4	143.142	odd	2
1287.2.e.d.1286.3	yes	4	3.2	odd	2

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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.414214i q^{2} +1.82843 q^{4} +3.41421 q^{7} -1.58579i q^{8} +O(q^{10})\)	\(q-0.414214i q^{2} +1.82843 q^{4} +3.41421 q^{7} -1.58579i q^{8} +(-3.00000 - 1.41421i) q^{11} +(2.00000 + 3.00000i) q^{13} -1.41421i q^{14} +3.00000 q^{16} +4.24264 q^{17} +0.585786 q^{19} +(-0.585786 + 1.24264i) q^{22} +1.17157i q^{23} -5.00000 q^{25} +(1.24264 - 0.828427i) q^{26} +6.24264 q^{28} +4.24264 q^{29} -1.75736i q^{31} -4.41421i q^{32} -1.75736i q^{34} -8.48528i q^{37} -0.242641i q^{38} -1.17157i q^{41} +8.48528i q^{43} +(-5.48528 - 2.58579i) q^{44} +0.485281 q^{46} -6.00000 q^{47} +4.65685 q^{49} +2.07107i q^{50} +(3.65685 + 5.48528i) q^{52} +9.41421i q^{53} -5.41421i q^{56} -1.75736i q^{58} +8.48528 q^{59} -6.00000i q^{61} -0.727922 q^{62} +4.17157 q^{64} +10.2426i q^{67} +7.75736 q^{68} -8.48528 q^{71} -1.17157 q^{73} -3.51472 q^{74} +1.07107 q^{76} +(-10.2426 - 4.82843i) q^{77} -8.48528i q^{79} -0.485281 q^{82} -3.65685i q^{83} +3.51472 q^{86} +(-2.24264 + 4.75736i) q^{88} +(6.82843 + 10.2426i) q^{91} +2.14214i q^{92} +2.48528i q^{94} -12.0000i q^{97} -1.92893i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^7	\( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{11} + 8 q^{13} + 12 q^{16} + 8 q^{19} - 8 q^{22} - 20 q^{25} - 12 q^{26} + 8 q^{28} + 12 q^{44} - 32 q^{46} - 24 q^{47} - 4 q^{49} - 8 q^{52} + 48 q^{62} + 28 q^{64} + 48 q^{68} - 16 q^{73} - 48 q^{74} - 24 q^{76} - 24 q^{77} + 32 q^{82} + 48 q^{86} + 8 q^{88} + 16 q^{91}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^11 + 8 * q^13 + 12 * q^16 + 8 * q^19 - 8 * q^22 - 20 * q^25 - 12 * q^26 + 8 * q^28 + 12 * q^44 - 32 * q^46 - 24 * q^47 - 4 * q^49 - 8 * q^52 + 48 * q^62 + 28 * q^64 + 48 * q^68 - 16 * q^73 - 48 * q^74 - 24 * q^76 - 24 * q^77 + 32 * q^82 + 48 * q^86 + 8 * q^88 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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