LMFDB - Embedded newform 440.2.b.d.89.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(89,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("440.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.b (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:	$3.51341768894$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.47985531136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19x^6 + 91x^4 + 45*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.2
Root			$-2.67673i$ of defining polynomial
Character	$\chi$	$=$	440.89
Dual form			440.2.b.d.89.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67673i q^{3} +(2.06639 + 0.854430i) q^{5} -4.38559i q^{7} -4.16490 q^{9} +O(q^{10})$	$q-2.67673i q^{3} +(2.06639 + 0.854430i) q^{5} -4.38559i q^{7} -4.16490 q^{9} +1.00000 q^{11} +4.00000i q^{13} +(2.28708 - 5.53116i) q^{15} -5.87995i q^{17} +0.526486 q^{19} -11.7391 q^{21} +6.15025i q^{23} +(3.53990 + 3.53116i) q^{25} +3.11812i q^{27} +0.967873 q^{29} -9.60629 q^{31} -2.67673i q^{33} +(3.74718 - 9.06232i) q^{35} -1.76466i q^{37} +10.7069 q^{39} +2.50564 q^{41} -3.85911i q^{43} +(-8.60629 - 3.55861i) q^{45} +4.91208i q^{47} -12.2334 q^{49} -15.7391 q^{51} -5.29767i q^{53} +(2.06639 + 0.854430i) q^{55} -1.40926i q^{57} +2.63841 q^{59} +8.96787 q^{61} +18.2655i q^{63} +(-3.41772 + 8.26554i) q^{65} +7.97440i q^{67} +16.4626 q^{69} +13.8718 q^{71} +12.7712i q^{73} +(9.45198 - 9.47537i) q^{75} -4.38559i q^{77} +13.2768 q^{79} -4.14832 q^{81} -7.61901i q^{83} +(5.02400 - 12.1502i) q^{85} -2.59074i q^{87} -2.83510 q^{89} +17.5424 q^{91} +25.7135i q^{93} +(1.08792 + 0.449845i) q^{95} +10.4800i q^{97} -4.16490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9}+O(q^{10}) $ 8 * q - 14 * q^9	$ 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65} + 60 q^{69} - 2 q^{71} + 26 q^{75} + 40 q^{79} + 88 q^{81} - 26 q^{85} - 42 q^{89} + 8 q^{91} + 28 q^{95} - 14 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$2$		0			0
$3$		−	2.67673i		−	1.54541i	−0.634764	−	0.772706i	$-0.718903\pi$
$3$		−	2.67673i		−	1.54541i	0.634764	−	0.772706i	$-0.281097\pi$
$4$		0			0
$5$	2.06639	+	0.854430i	0.924116	+	0.382113i
$5$	2.06639	+	0.854430i	0.924116	+	0.382113i
$6$		0			0
$7$		−	4.38559i		−	1.65760i	−0.559546	−	0.828799i	$-0.689025\pi$
$7$		−	4.38559i		−	1.65760i	0.559546	−	0.828799i	$-0.310975\pi$
$8$		0			0
$9$	−4.16490			−1.38830
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	2.28708	−	5.53116i	0.590521	−	1.42814i
$16$		0			0
$17$		−	5.87995i		−	1.42610i	−0.701114	−	0.713049i	$-0.747314\pi$
$17$		−	5.87995i		−	1.42610i	0.701114	−	0.713049i	$-0.252686\pi$
$18$		0			0
$19$	0.526486			0.120784			0.0603920	−	0.998175i	$-0.480765\pi$
$19$	0.526486			0.120784			0.0603920	+	0.998175i	$0.480765\pi$
$20$		0			0
$21$	−11.7391			−2.56167
$22$		0			0
$23$			6.15025i			1.28242i	0.767367	+	0.641208i	$0.221566\pi$
$23$			6.15025i			1.28242i	−0.767367	+	0.641208i	$0.778434\pi$
$24$		0			0
$25$	3.53990	+	3.53116i	0.707980	+	0.706232i
$26$		0			0
$27$			3.11812i			0.600083i
$28$		0			0
$29$	0.967873			0.179730			0.0898648	−	0.995954i	$-0.471357\pi$
$29$	0.967873			0.179730			0.0898648	+	0.995954i	$0.471357\pi$
$30$		0			0
$31$	−9.60629			−1.72534			−0.862670	−	0.505767i	$-0.831209\pi$
$31$	−9.60629			−1.72534			−0.862670	+	0.505767i	$0.831209\pi$
$32$		0			0
$33$		−	2.67673i		−	0.465959i
$34$		0			0
$35$	3.74718	−	9.06232i	0.633389	−	1.53181i
$36$		0			0
$37$		−	1.76466i		−	0.290108i	−0.989424	−	0.145054i	$-0.953664\pi$
$37$		−	1.76466i		−	0.290108i	0.989424	−	0.145054i	$-0.0463355\pi$
$38$		0			0
$39$	10.7069			1.71448
$40$		0			0
$41$	2.50564			0.391315			0.195658	−	0.980672i	$-0.437316\pi$
$41$	2.50564			0.391315			0.195658	+	0.980672i	$0.437316\pi$
$42$		0			0
$43$		−	3.85911i		−	0.588508i	−0.955727	−	0.294254i	$-0.904929\pi$
$43$		−	3.85911i		−	0.588508i	0.955727	−	0.294254i	$-0.0950712\pi$
$44$		0			0
$45$	−8.60629	−	3.55861i	−1.28295	−	0.530487i
$46$		0			0
$47$			4.91208i			0.716500i	0.933626	+	0.358250i	$0.116626\pi$
$47$			4.91208i			0.716500i	−0.933626	+	0.358250i	$0.883374\pi$
$48$		0			0
$49$	−12.2334			−1.74763
$50$		0			0
$51$	−15.7391			−2.20391
$52$		0			0
$53$		−	5.29767i		−	0.727691i	−0.931459	−	0.363845i	$-0.881464\pi$
$53$		−	5.29767i		−	0.727691i	0.931459	−	0.363845i	$-0.118536\pi$
$54$		0			0
$55$	2.06639	+	0.854430i	0.278631	+	0.115211i
$56$		0			0
$57$		−	1.40926i		−	0.186661i
$58$		0			0
$59$	2.63841			0.343492			0.171746	−	0.985141i	$-0.445059\pi$
$59$	2.63841			0.343492			0.171746	+	0.985141i	$0.445059\pi$
$60$		0			0
$61$	8.96787			1.14822			0.574109	−	0.818779i	$-0.305348\pi$
$61$	8.96787			1.14822			0.574109	+	0.818779i	$0.305348\pi$
$62$		0			0
$63$			18.2655i			2.30124i
$64$		0			0
$65$	−3.41772	+	8.26554i	−0.423916	+	1.02521i
$66$		0			0
$67$			7.97440i			0.974228i	0.873338	+	0.487114i	$0.161950\pi$
$67$			7.97440i			0.974228i	−0.873338	+	0.487114i	$0.838050\pi$
$68$		0			0
$69$	16.4626			1.98186
$70$		0			0
$71$	13.8718			1.64628			0.823142	−	0.567836i	$-0.192219\pi$
$71$	13.8718			1.64628			0.823142	+	0.567836i	$0.192219\pi$
$72$		0			0
$73$			12.7712i			1.49475i	0.664400	+	0.747377i	$0.268687\pi$
$73$			12.7712i			1.49475i	−0.664400	+	0.747377i	$0.731313\pi$
$74$		0			0
$75$	9.45198	−	9.47537i	1.09142	−	1.09412i
$76$		0			0
$77$		−	4.38559i		−	0.499785i
$78$		0			0
$79$	13.2768			1.49376			0.746880	−	0.664959i	$-0.231551\pi$
$79$	13.2768			1.49376			0.746880	+	0.664959i	$0.231551\pi$
$80$		0			0
$81$	−4.14832			−0.460924
$82$		0			0
$83$		−	7.61901i		−	0.836295i	−0.908379	−	0.418147i	$-0.862680\pi$
$83$		−	7.61901i		−	0.836295i	0.908379	−	0.418147i	$-0.137320\pi$
$84$		0			0
$85$	5.02400	−	12.1502i	0.544930	−	1.31788i
$86$		0			0
$87$		−	2.59074i		−	0.277756i
$88$		0			0
$89$	−2.83510			−0.300520			−0.150260	−	0.988646i	$-0.548011\pi$
$89$	−2.83510			−0.300520			−0.150260	+	0.988646i	$0.548011\pi$
$90$		0			0
$91$	17.5424			1.83894
$92$		0			0
$93$			25.7135i			2.66636i
$94$		0			0
$95$	1.08792	+	0.449845i	0.111618	+	0.0461531i
$96$		0			0
$97$			10.4800i			1.06409i	0.846717	+	0.532044i	$0.178576\pi$
$97$			10.4800i			1.06409i	−0.846717	+	0.532044i	$0.821424\pi$
$98$		0			0
$99$	−4.16490			−0.418588
$100$		0			0
$101$	8.70693			0.866372			0.433186	−	0.901305i	$-0.357389\pi$
$101$	8.70693			0.866372			0.433186	+	0.901305i	$0.357389\pi$
$102$		0			0
$103$			15.6190i			1.53899i	0.638655	+	0.769493i	$0.279491\pi$
$103$			15.6190i			1.53899i	−0.638655	+	0.769493i	$0.720509\pi$
$104$		0			0
$105$	−24.2574	−	10.0302i	−2.36728	−	0.978847i
$106$		0			0
$107$		−	7.08792i		−	0.685215i	−0.939479	−	0.342608i	$-0.888690\pi$
$107$		−	7.08792i		−	0.685215i	0.939479	−	0.342608i	$-0.111310\pi$
$108$		0			0
$109$	15.5424			1.48869			0.744344	−	0.667796i	$-0.232762\pi$
$109$	15.5424			1.48869			0.744344	+	0.667796i	$0.232762\pi$
$110$		0			0
$111$	−4.72351			−0.448336
$112$		0			0
$113$			8.11530i			0.763423i	0.924282	+	0.381711i	$0.124665\pi$
$113$			8.11530i			0.763423i	−0.924282	+	0.381711i	$0.875335\pi$
$114$		0			0
$115$	−5.25495	+	12.7088i	−0.490027	+	1.18510i
$116$		0			0
$117$		−	16.6596i		−	1.54018i
$118$		0			0
$119$	−25.7871			−2.36390
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	6.70693i		−	0.604744i
$124$		0			0
$125$	4.29767	+	10.3213i	0.384395	+	0.923169i
$126$		0			0
$127$		−	4.32980i		−	0.384207i	−0.981375	−	0.192104i	$-0.938469\pi$
$127$		−	4.32980i		−	0.384207i	0.981375	−	0.192104i	$-0.0615310\pi$
$128$		0			0
$129$	−10.3298			−0.909488
$130$		0			0
$131$	−17.6748			−1.54425			−0.772127	−	0.635468i	$-0.780807\pi$
$131$	−17.6748			−1.54425			−0.772127	+	0.635468i	$0.780807\pi$
$132$		0			0
$133$		−	2.30895i		−	0.200211i
$134$		0			0
$135$	−2.66421	+	6.44324i	−0.229299	+	0.554546i
$136$		0			0
$137$		−	18.4800i		−	1.57886i	−0.613843	−	0.789428i	$-0.710377\pi$
$137$		−	18.4800i		−	1.57886i	0.613843	−	0.789428i	$-0.289623\pi$
$138$		0			0
$139$	5.55861			0.471475			0.235738	−	0.971817i	$-0.424249\pi$
$139$	5.55861			0.471475			0.235738	+	0.971817i	$0.424249\pi$
$140$		0			0
$141$	13.1483			1.10729
$142$		0			0
$143$			4.00000i			0.334497i
$144$		0			0
$145$	2.00000	+	0.826980i	0.166091	+	0.0686769i
$146$		0			0
$147$			32.7456i			2.70081i
$148$		0			0
$149$	−20.9516			−1.71642			−0.858212	−	0.513295i	$-0.828425\pi$
$149$	−20.9516			−1.71642			−0.858212	+	0.513295i	$0.828425\pi$
$150$		0			0
$151$	−1.67020			−0.135919			−0.0679596	−	0.997688i	$-0.521649\pi$
$151$	−1.67020			−0.135919			−0.0679596	+	0.997688i	$0.521649\pi$
$152$		0			0
$153$			24.4894i			1.97985i
$154$		0			0
$155$	−19.8503	−	8.20790i	−1.59441	−	0.659274i
$156$		0			0
$157$			8.94228i			0.713671i	0.934167	+	0.356836i	$0.116144\pi$
$157$			8.94228i			0.713671i	−0.934167	+	0.356836i	$0.883856\pi$
$158$		0			0
$159$	−14.1804			−1.12458
$160$		0			0
$161$	26.9725			2.12573
$162$		0			0
$163$		−	9.43856i		−	0.739285i	−0.929174	−	0.369643i	$-0.879480\pi$
$163$		−	9.43856i		−	0.739285i	0.929174	−	0.369643i	$-0.120520\pi$
$164$		0			0
$165$	2.28708	−	5.53116i	0.178049	−	0.430600i
$166$		0			0
$167$		−	8.96787i		−	0.693955i	−0.937873	−	0.346977i	$-0.887208\pi$
$167$		−	8.96787i		−	0.693955i	0.937873	−	0.346977i	$-0.112792\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−2.19276			−0.167684
$172$		0			0
$173$			2.64653i			0.201212i	0.994926	+	0.100606i	$0.0320782\pi$
$173$			2.64653i			0.201212i	−0.994926	+	0.100606i	$0.967922\pi$
$174$		0			0
$175$	15.4862	−	15.5246i	1.17065	−	1.17355i
$176$		0			0
$177$		−	7.06232i		−	0.530837i
$178$		0			0
$179$	−20.2225			−1.51150			−0.755749	−	0.654861i	$-0.772727\pi$
$179$	−20.2225			−1.51150			−0.755749	+	0.654861i	$0.772727\pi$
$180$		0			0
$181$	15.3453			1.14061			0.570305	−	0.821433i	$-0.306825\pi$
$181$	15.3453			1.14061			0.570305	+	0.821433i	$0.306825\pi$
$182$		0			0
$183$		−	24.0046i		−	1.77447i
$184$		0			0
$185$	1.50777	−	3.64646i	0.110854	−	0.268093i
$186$		0			0
$187$		−	5.87995i		−	0.429985i
$188$		0			0
$189$	13.6748			0.994696
$190$		0			0
$191$	−5.62713			−0.407165			−0.203582	−	0.979058i	$-0.565258\pi$
$191$	−5.62713			−0.407165			−0.203582	+	0.979058i	$0.565258\pi$
$192$		0			0
$193$		−	7.23342i		−	0.520673i	−0.965518	−	0.260336i	$-0.916167\pi$
$193$		−	7.23342i		−	0.520673i	0.965518	−	0.260336i	$-0.0838335\pi$
$194$		0			0
$195$	22.1246	+	9.14832i	1.58438	+	0.655125i
$196$		0			0
$197$		−	5.35347i		−	0.381419i	−0.981647	−	0.190709i	$-0.938921\pi$
$197$		−	5.35347i		−	0.381419i	0.981647	−	0.190709i	$-0.0610788\pi$
$198$		0			0
$199$	−7.23342			−0.512763			−0.256382	−	0.966576i	$-0.582530\pi$
$199$	−7.23342			−0.512763			−0.256382	+	0.966576i	$0.582530\pi$
$200$		0			0
$201$	21.3453			1.50558
$202$		0			0
$203$		−	4.24470i		−	0.297919i
$204$		0			0
$205$	5.17762	+	2.14089i	0.361621	+	0.149527i
$206$		0			0
$207$		−	25.6152i		−	1.78038i
$208$		0			0
$209$	0.526486			0.0364178
$210$		0			0
$211$	−12.2447			−0.842960			−0.421480	−	0.906838i	$-0.638489\pi$
$211$	−12.2447			−0.842960			−0.421480	+	0.906838i	$0.638489\pi$
$212$		0			0
$213$		−	37.1312i		−	2.54419i
$214$		0			0
$215$	3.29733	−	7.97440i	0.224876	−	0.543850i
$216$		0			0
$217$			42.1292i			2.85992i
$218$		0			0
$219$	34.1850			2.31001
$220$		0			0
$221$	23.5198			1.58211
$222$		0			0
$223$			19.4525i			1.30264i	0.758805	+	0.651318i	$0.225784\pi$
$223$			19.4525i			1.30264i	−0.758805	+	0.651318i	$0.774216\pi$
$224$		0			0
$225$	−14.7433	−	14.7069i	−0.982888	−	0.980462i
$226$		0			0
$227$		−	3.90080i		−	0.258905i	−0.991586	−	0.129452i	$-0.958678\pi$
$227$		−	3.90080i		−	0.258905i	0.991586	−	0.129452i	$-0.0413220\pi$
$228$		0			0
$229$	−13.4096			−0.886131			−0.443066	−	0.896489i	$-0.646109\pi$
$229$	−13.4096			−0.886131			−0.443066	+	0.896489i	$0.646109\pi$
$230$		0			0
$231$	−11.7391			−0.772373
$232$		0			0
$233$		−	3.36192i		−	0.220247i	−0.993918	−	0.110123i	$-0.964875\pi$
$233$		−	3.36192i		−	0.220247i	0.993918	−	0.110123i	$-0.0351246\pi$
$234$		0			0
$235$	−4.19702	+	10.1502i	−0.273784	+	0.662129i
$236$		0			0
$237$		−	35.5385i		−	2.30847i
$238$		0			0
$239$	13.2768			0.858806			0.429403	−	0.903113i	$-0.358724\pi$
$239$	13.2768			0.858806			0.429403	+	0.903113i	$0.358724\pi$
$240$		0			0
$241$	−17.7437			−1.14297			−0.571485	−	0.820613i	$-0.693632\pi$
$241$	−17.7437			−1.14297			−0.571485	+	0.820613i	$0.693632\pi$
$242$		0			0
$243$			20.4583i			1.31240i
$244$		0			0
$245$	−25.2790	−	10.4526i	−1.61501	−	0.667792i
$246$		0			0
$247$			2.10594i			0.133998i
$248$		0			0
$249$	−20.3940			−1.29242
$250$		0			0
$251$	8.57416			0.541196			0.270598	−	0.962692i	$-0.412779\pi$
$251$	8.57416			0.541196			0.270598	+	0.962692i	$0.412779\pi$
$252$		0			0
$253$			6.15025i			0.386663i
$254$		0			0
$255$	−32.5230	−	13.4479i	−2.03667	−	0.842141i
$256$		0			0
$257$			2.70693i			0.168854i	0.996430	+	0.0844269i	$0.0269059\pi$
$257$			2.70693i			0.168854i	−0.996430	+	0.0844269i	$0.973094\pi$
$258$		0			0
$259$	−7.73906			−0.480882
$260$		0			0
$261$	−4.03109			−0.249518
$262$		0			0
$263$			14.3213i			0.883092i	0.897239	+	0.441546i	$0.145570\pi$
$263$			14.3213i			0.883092i	−0.897239	+	0.441546i	$0.854430\pi$
$264$		0			0
$265$	4.52649	−	10.9470i	0.278060	−	0.672471i
$266$		0			0
$267$			7.58881i			0.464428i
$268$		0			0
$269$	11.9357			0.727735			0.363868	−	0.931451i	$-0.381456\pi$
$269$	11.9357			0.727735			0.363868	+	0.931451i	$0.381456\pi$
$270$		0			0
$271$	−24.4195			−1.48338			−0.741689	−	0.670744i	$-0.765975\pi$
$271$	−24.4195			−1.48338			−0.741689	+	0.670744i	$0.765975\pi$
$272$		0			0
$273$		−	46.9562i		−	2.84192i
$274$		0			0
$275$	3.53990	+	3.53116i	0.213464	+	0.212937i
$276$		0			0
$277$			23.0074i			1.38238i	0.722672	+	0.691191i	$0.242914\pi$
$277$			23.0074i			1.38238i	−0.722672	+	0.691191i	$0.757086\pi$
$278$		0			0
$279$	40.0092			2.39529
$280$		0			0
$281$	−10.6596			−0.635898			−0.317949	−	0.948108i	$-0.602994\pi$
$281$	−10.6596			−0.635898			−0.317949	+	0.948108i	$0.602994\pi$
$282$		0			0
$283$			10.3771i			0.616857i	0.951248	+	0.308428i	$0.0998030\pi$
$283$			10.3771i			0.616857i	−0.951248	+	0.308428i	$0.900197\pi$
$284$		0			0
$285$	1.20411	−	2.91208i	0.0713256	−	0.172497i
$286$		0			0
$287$		−	10.9887i		−	0.648644i
$288$		0			0
$289$	−17.5738			−1.03375
$290$		0			0
$291$	28.0523			1.64445
$292$		0			0
$293$		−	22.7069i		−	1.32655i	−0.748374	−	0.663277i	$-0.769165\pi$
$293$		−	22.7069i		−	1.32655i	0.748374	−	0.663277i	$-0.230835\pi$
$294$		0			0
$295$	5.45198	+	2.25434i	0.317426	+	0.131253i
$296$		0			0
$297$			3.11812i			0.180932i
$298$		0			0
$299$	−24.6010			−1.42271
$300$		0			0
$301$	−16.9245			−0.975510
$302$		0			0
$303$		−	23.3061i		−	1.33890i
$304$		0			0
$305$	18.5311	+	7.66242i	1.06109	+	0.438749i
$306$		0			0
$307$			5.49436i			0.313580i	0.987632	+	0.156790i	$0.0501145\pi$
$307$			5.49436i			0.313580i	−0.987632	+	0.156790i	$0.949885\pi$
$308$		0			0
$309$	41.8079			2.37837
$310$		0			0
$311$	−24.2447			−1.37479			−0.687395	−	0.726283i	$-0.741246\pi$
$311$	−24.2447			−1.37479			−0.687395	+	0.726283i	$0.741246\pi$
$312$		0			0
$313$		−	11.7562i		−	0.664500i	−0.943191	−	0.332250i	$-0.892192\pi$
$313$		−	11.7562i		−	0.664500i	0.943191	−	0.332250i	$-0.107808\pi$
$314$		0			0
$315$	−15.6066	+	37.7437i	−0.879333	+	2.12661i
$316$		0			0
$317$			9.05387i			0.508516i	0.967136	+	0.254258i	$0.0818312\pi$
$317$			9.05387i			0.508516i	−0.967136	+	0.254258i	$0.918169\pi$
$318$		0			0
$319$	0.967873			0.0541905
$320$		0			0
$321$	−18.9725			−1.05894
$322$		0			0
$323$		−	3.09571i		−	0.172250i
$324$		0			0
$325$	−14.1246	+	14.1596i	−0.783495	+	0.785433i
$326$		0			0
$327$		−	41.6028i		−	2.30064i
$328$		0			0
$329$	21.5424			1.18767
$330$		0			0
$331$	−3.29733			−0.181238			−0.0906190	−	0.995886i	$-0.528885\pi$
$331$	−3.29733			−0.181238			−0.0906190	+	0.995886i	$0.528885\pi$
$332$		0			0
$333$			7.34961i			0.402756i
$334$		0			0
$335$	−6.81357	+	16.4782i	−0.372265	+	0.900300i
$336$		0			0
$337$			3.17302i			0.172845i	0.996259	+	0.0864227i	$0.0275436\pi$
$337$			3.17302i			0.172845i	−0.996259	+	0.0864227i	$0.972456\pi$
$338$		0			0
$339$	21.7225			1.17980
$340$		0			0
$341$	−9.60629			−0.520210
$342$		0			0
$343$			22.9516i			1.23927i
$344$		0			0
$345$	34.0180	+	14.0661i	1.83147	+	0.757294i
$346$		0			0
$347$			4.44139i			0.238426i	0.992869	+	0.119213i	$0.0380372\pi$
$347$			4.44139i			0.238426i	−0.992869	+	0.119213i	$0.961963\pi$
$348$		0			0
$349$	−29.2380			−1.56508			−0.782538	−	0.622603i	$-0.786075\pi$
$349$	−29.2380			−1.56508			−0.782538	+	0.622603i	$0.786075\pi$
$350$		0			0
$351$	−12.4725			−0.665732
$352$		0			0
$353$			1.04927i			0.0558469i	0.999610	+	0.0279234i	$0.00888946\pi$
$353$			1.04927i			0.0558469i	−0.999610	+	0.0279234i	$0.991111\pi$
$354$		0			0
$355$	28.6645	+	11.8525i	1.52136	+	0.629065i
$356$		0			0
$357$			69.0251i			3.65319i
$358$		0			0
$359$	−9.69565			−0.511717			−0.255858	−	0.966714i	$-0.582358\pi$
$359$	−9.69565			−0.511717			−0.255858	+	0.966714i	$0.582358\pi$
$360$		0			0
$361$	−18.7228			−0.985411
$362$		0			0
$363$		−	2.67673i		−	0.140492i
$364$		0			0
$365$	−10.9121	+	26.3902i	−0.571164	+	1.38133i
$366$		0			0
$367$		−	11.3921i		−	0.594664i	−0.954774	−	0.297332i	$-0.903903\pi$
$367$		−	11.3921i		−	0.594664i	0.954774	−	0.297332i	$-0.0960968\pi$
$368$		0			0
$369$	−10.4357			−0.543263
$370$		0			0
$371$	−23.2334			−1.20622
$372$		0			0
$373$		−	6.47634i		−	0.335332i	−0.985844	−	0.167666i	$-0.946377\pi$
$373$		−	6.47634i		−	0.335332i	0.985844	−	0.167666i	$-0.0536230\pi$
$374$		0			0
$375$	27.6275	−	11.5037i	1.42668	−	0.594049i
$376$		0			0
$377$			3.87149i			0.199392i
$378$		0			0
$379$	33.4993			1.72074			0.860372	−	0.509667i	$-0.170232\pi$
$379$	33.4993			1.72074			0.860372	+	0.509667i	$0.170232\pi$
$380$		0			0
$381$	−11.5897			−0.593759
$382$		0			0
$383$			30.8702i			1.57740i	0.614781	+	0.788698i	$0.289244\pi$
$383$			30.8702i			1.57740i	−0.614781	+	0.788698i	$0.710756\pi$
$384$		0			0
$385$	3.74718	−	9.06232i	0.190974	−	0.461859i
$386$		0			0
$387$			16.0728i			0.817026i
$388$		0			0
$389$	23.3453			1.18366			0.591828	−	0.806064i	$-0.298406\pi$
$389$	23.3453			1.18366			0.591828	+	0.806064i	$0.298406\pi$
$390$		0			0
$391$	36.1632			1.82885
$392$		0			0
$393$			47.3107i			2.38651i
$394$		0			0
$395$	27.4350	+	11.3441i	1.38041	+	0.570784i
$396$		0			0
$397$		−	19.4781i		−	0.977579i	−0.872402	−	0.488789i	$-0.837439\pi$
$397$		−	19.4781i		−	0.977579i	0.872402	−	0.488789i	$-0.162561\pi$
$398$		0			0
$399$	−6.18045			−0.309409
$400$		0			0
$401$	2.24470			0.112095			0.0560474	−	0.998428i	$-0.482150\pi$
$401$	2.24470			0.112095			0.0560474	+	0.998428i	$0.482150\pi$
$402$		0			0
$403$		−	38.4251i		−	1.91409i
$404$		0			0
$405$	−8.57203	−	3.54445i	−0.425947	−	0.176125i
$406$		0			0
$407$		−	1.76466i		−	0.0874707i
$408$		0			0
$409$	−29.4781			−1.45760			−0.728799	−	0.684727i	$-0.759921\pi$
$409$	−29.4781			−1.45760			−0.728799	+	0.684727i	$0.759921\pi$
$410$		0			0
$411$	−49.4661			−2.43998
$412$		0			0
$413$		−	11.5710i		−	0.569372i
$414$		0			0
$415$	6.50991	−	15.7438i	0.319559	−	0.772833i
$416$		0			0
$417$		−	14.8789i		−	0.728624i
$418$		0			0
$419$	−7.64831			−0.373644			−0.186822	−	0.982394i	$-0.559819\pi$
$419$	−7.64831			−0.373644			−0.186822	+	0.982394i	$0.559819\pi$
$420$		0			0
$421$	29.4781			1.43668			0.718338	−	0.695695i	$-0.244903\pi$
$421$	29.4781			1.43668			0.718338	+	0.695695i	$0.244903\pi$
$422$		0			0
$423$		−	20.4583i		−	0.994717i
$424$		0			0
$425$	20.7631	−	20.8144i	1.00716	−	1.00965i
$426$		0			0
$427$		−	39.3294i		−	1.90328i
$428$		0			0
$429$	10.7069			0.516935
$430$		0			0
$431$	−2.08970			−0.100657			−0.0503286	−	0.998733i	$-0.516027\pi$
$431$	−2.08970			−0.100657			−0.0503286	+	0.998733i	$0.516027\pi$
$432$		0			0
$433$		−	5.17777i		−	0.248828i	−0.992230	−	0.124414i	$-0.960295\pi$
$433$		−	5.17777i		−	0.248828i	0.992230	−	0.124414i	$-0.0397051\pi$
$434$		0			0
$435$	2.21360	−	5.35347i	0.106134	−	0.256679i
$436$		0			0
$437$			3.23802i			0.154895i
$438$		0			0
$439$	−27.6066			−1.31759			−0.658796	−	0.752322i	$-0.728934\pi$
$439$	−27.6066			−1.31759			−0.658796	+	0.752322i	$0.728934\pi$
$440$		0			0
$441$	50.9509			2.42623
$442$		0			0
$443$		−	14.1502i		−	0.672299i	−0.941809	−	0.336149i	$-0.890875\pi$
$443$		−	14.1502i		−	0.672299i	0.941809	−	0.336149i	$-0.109125\pi$
$444$		0			0
$445$	−5.85841	−	2.42240i	−0.277715	−	0.114833i
$446$		0			0
$447$			56.0819i			2.65258i
$448$		0			0
$449$	6.46257			0.304987			0.152494	−	0.988304i	$-0.451270\pi$
$449$	6.46257			0.304987			0.152494	+	0.988304i	$0.451270\pi$
$450$		0			0
$451$	2.50564			0.117986
$452$		0			0
$453$			4.47069i			0.210051i
$454$		0			0
$455$	36.2493	+	14.9887i	1.69939	+	0.702682i
$456$		0			0
$457$			38.5582i			1.80368i	0.432071	+	0.901839i	$0.357783\pi$
$457$			38.5582i			1.80368i	−0.432071	+	0.901839i	$0.642217\pi$
$458$		0			0
$459$	18.3344			0.855776
$460$		0			0
$461$	−18.4234			−0.858064			−0.429032	−	0.903289i	$-0.641145\pi$
$461$	−18.4234			−0.858064			−0.429032	+	0.903289i	$0.641145\pi$
$462$		0			0
$463$			7.09163i			0.329576i	0.986329	+	0.164788i	$0.0526940\pi$
$463$			7.09163i			0.329576i	−0.986329	+	0.164788i	$0.947306\pi$
$464$		0			0
$465$	−21.9703	+	53.1339i	−1.01885	+	2.46403i
$466$		0			0
$467$		−	40.4328i		−	1.87101i	−0.353319	−	0.935503i	$-0.614947\pi$
$467$		−	40.4328i		−	1.87101i	0.353319	−	0.935503i	$-0.385053\pi$
$468$		0			0
$469$	34.9725			1.61488
$470$		0			0
$471$	23.9361			1.10292
$472$		0			0
$473$		−	3.85911i		−	0.177442i
$474$		0			0
$475$	1.86371	+	1.85911i	0.0855127	+	0.0853016i
$476$		0			0
$477$			22.0643i			1.01025i
$478$		0			0
$479$	10.7486			0.491117			0.245559	−	0.969382i	$-0.421029\pi$
$479$	10.7486			0.491117			0.245559	+	0.969382i	$0.421029\pi$
$480$		0			0
$481$	7.05862			0.321845
$482$		0			0
$483$		−	72.1981i		−	3.28513i
$484$		0			0
$485$	−8.95446	+	21.6558i	−0.406601	+	0.983340i
$486$		0			0
$487$			38.6396i			1.75093i	0.483282	+	0.875465i	$0.339445\pi$
$487$			38.6396i			1.75093i	−0.483282	+	0.875465i	$0.660555\pi$
$488$		0			0
$489$	−25.2645			−1.14250
$490$		0			0
$491$	32.9460			1.48683			0.743416	−	0.668830i	$-0.233204\pi$
$491$	32.9460			1.48683			0.743416	+	0.668830i	$0.233204\pi$
$492$		0			0
$493$		−	5.69105i		−	0.256312i
$494$		0			0
$495$	−8.60629	−	3.55861i	−0.386824	−	0.159948i
$496$		0			0
$497$		−	60.8362i		−	2.72888i
$498$		0			0
$499$	−26.9562			−1.20673			−0.603363	−	0.797466i	$-0.706173\pi$
$499$	−26.9562			−1.20673			−0.603363	+	0.797466i	$0.706173\pi$
$500$		0			0
$501$	−24.0046			−1.07245
$502$		0			0
$503$			14.0766i			0.627646i	0.949481	+	0.313823i	$0.101610\pi$
$503$			14.0766i			0.627646i	−0.949481	+	0.313823i	$0.898390\pi$
$504$		0			0
$505$	17.9919	+	7.43946i	0.800628	+	0.331052i
$506$		0			0
$507$			8.03020i			0.356634i
$508$		0			0
$509$	−1.28109			−0.0567833			−0.0283917	−	0.999597i	$-0.509039\pi$
$509$	−1.28109			−0.0567833			−0.0283917	+	0.999597i	$0.509039\pi$
$510$		0			0
$511$	56.0092			2.47770
$512$		0			0
$513$			1.64165i			0.0724804i
$514$		0			0
$515$	−13.3453	+	32.2749i	−0.588066	+	1.42220i
$516$		0			0
$517$			4.91208i			0.216033i
$518$		0			0
$519$	7.08407			0.310956
$520$		0			0
$521$	−29.2394			−1.28100			−0.640501	−	0.767958i	$-0.721273\pi$
$521$	−29.2394			−1.28100			−0.640501	+	0.767958i	$0.721273\pi$
$522$		0			0
$523$		−	2.50564i		−	0.109564i	−0.998498	−	0.0547820i	$-0.982554\pi$
$523$		−	2.50564i		−	0.109564i	0.998498	−	0.0547820i	$-0.0174464\pi$
$524$		0			0
$525$	−41.5551	−	41.4525i	−1.81361	−	1.80914i
$526$		0			0
$527$			56.4845i			2.46050i
$528$		0			0
$529$	−14.8255			−0.644589
$530$		0			0
$531$	−10.9887			−0.476870
$532$		0			0
$533$			10.0226i			0.434125i
$534$		0			0
$535$	6.05613	−	14.6464i	0.261829	−	0.633218i
$536$		0			0
$537$			54.1301i			2.33589i
$538$		0			0
$539$	−12.2334			−0.526931
$540$		0			0
$541$	−3.27222			−0.140684			−0.0703420	−	0.997523i	$-0.522409\pi$
$541$	−3.27222			−0.140684			−0.0703420	+	0.997523i	$0.522409\pi$
$542$		0			0
$543$		−	41.0754i		−	1.76271i
$544$		0			0
$545$	32.1165	+	13.2799i	1.37572	+	0.568847i
$546$		0			0
$547$		−	3.03673i		−	0.129841i	−0.997890	−	0.0649205i	$-0.979321\pi$
$547$		−	3.03673i		−	0.129841i	0.997890	−	0.0649205i	$-0.0206794\pi$
$548$		0			0
$549$	−37.3503			−1.59407
$550$		0			0
$551$	0.509572			0.0217085
$552$		0			0
$553$		−	58.2267i		−	2.47605i
$554$		0			0
$555$	−9.76059	−	4.03591i	−0.414314	−	0.171315i
$556$		0			0
$557$			1.52366i			0.0645596i	0.999479	+	0.0322798i	$0.0102768\pi$
$557$			1.52366i			0.0645596i	−0.999479	+	0.0322798i	$0.989723\pi$
$558$		0			0
$559$	15.4364			0.652891
$560$		0			0
$561$	−15.7391			−0.664504
$562$		0			0
$563$			27.9195i			1.17667i	0.808618	+	0.588333i	$0.200216\pi$
$563$			27.9195i			1.17667i	−0.808618	+	0.588333i	$0.799784\pi$
$564$		0			0
$565$	−6.93395	+	16.7693i	−0.291713	+	0.705491i
$566$		0			0
$567$			18.1928i			0.764027i
$568$		0			0
$569$	13.7437			0.576164			0.288082	−	0.957606i	$-0.406982\pi$
$569$	13.7437			0.576164			0.288082	+	0.957606i	$0.406982\pi$
$570$		0			0
$571$	−1.42618			−0.0596836			−0.0298418	−	0.999555i	$-0.509500\pi$
$571$	−1.42618			−0.0596836			−0.0298418	+	0.999555i	$0.509500\pi$
$572$		0			0
$573$			15.0623i			0.629238i
$574$		0			0
$575$	−21.7175	+	21.7713i	−0.905683	+	0.907924i
$576$		0			0
$577$		−	6.35154i		−	0.264418i	−0.991222	−	0.132209i	$-0.957793\pi$
$577$		−	6.35154i		−	0.264418i	0.991222	−	0.132209i	$-0.0422070\pi$
$578$		0			0
$579$	−19.3619			−0.804654
$580$		0			0
$581$	−33.4139			−1.38624
$582$		0			0
$583$		−	5.29767i		−	0.219407i
$584$		0			0
$585$	14.2344	−	34.4251i	0.588522	−	1.42330i
$586$		0			0
$587$		−	22.1681i		−	0.914974i	−0.889216	−	0.457487i	$-0.848750\pi$
$587$		−	22.1681i		−	0.914974i	0.889216	−	0.457487i	$-0.151250\pi$
$588$		0			0
$589$	−5.05757			−0.208394
$590$		0			0
$591$	−14.3298			−0.589449
$592$		0			0
$593$			1.05297i			0.0432403i	0.999766	+	0.0216202i	$0.00688245\pi$
$593$			1.05297i			0.0432403i	−0.999766	+	0.0216202i	$0.993118\pi$
$594$		0			0
$595$	−53.2860	−	22.0332i	−2.18451	−	0.903274i
$596$		0			0
$597$			19.3619i			0.792431i
$598$		0			0
$599$	5.12747			0.209503			0.104751	−	0.994498i	$-0.466595\pi$
$599$	5.12747			0.209503			0.104751	+	0.994498i	$0.466595\pi$
$600$		0			0
$601$	−1.71821			−0.0700874			−0.0350437	−	0.999386i	$-0.511157\pi$
$601$	−1.71821			−0.0700874			−0.0350437	+	0.999386i	$0.511157\pi$
$602$		0			0
$603$		−	33.2126i		−	1.35252i
$604$		0			0
$605$	2.06639	+	0.854430i	0.0840105	+	0.0347375i
$606$		0			0
$607$		−	4.42728i		−	0.179698i	−0.995955	−	0.0898489i	$-0.971362\pi$
$607$		−	4.42728i		−	0.179698i	0.995955	−	0.0898489i	$-0.0286384\pi$
$608$		0			0
$609$	−11.3619			−0.460408
$610$		0			0
$611$	−19.6483			−0.794886
$612$		0			0
$613$		−	33.8334i		−	1.36652i	−0.730177	−	0.683258i	$-0.760562\pi$
$613$		−	33.8334i		−	1.36652i	0.730177	−	0.683258i	$-0.239438\pi$
$614$		0			0
$615$	5.73060	−	13.8591i	0.231080	−	0.558853i
$616$		0			0
$617$		−	40.3778i		−	1.62555i	−0.582578	−	0.812775i	$-0.697956\pi$
$617$		−	40.3778i		−	1.62555i	0.582578	−	0.812775i	$-0.302044\pi$
$618$		0			0
$619$	1.62713			0.0653999			0.0327000	−	0.999465i	$-0.489589\pi$
$619$	1.62713			0.0653999			0.0327000	+	0.999465i	$0.489589\pi$
$620$		0			0
$621$	−19.1772			−0.769555
$622$		0			0
$623$			12.4336i			0.498142i
$624$		0			0
$625$	0.0617854	+	24.9999i	0.00247142	+	0.999997i
$626$		0			0
$627$		−	1.40926i		−	0.0562805i
$628$		0			0
$629$	−10.3761			−0.413722
$630$		0			0
$631$	3.71223			0.147781			0.0738907	−	0.997266i	$-0.476458\pi$
$631$	3.71223			0.147781			0.0738907	+	0.997266i	$0.476458\pi$
$632$		0			0
$633$			32.7758i			1.30272i
$634$		0			0
$635$	3.69951	−	8.94703i	0.146810	−	0.355052i
$636$		0			0
$637$		−	48.9337i		−	1.93882i
$638$		0			0
$639$	−57.7748			−2.28553
$640$		0			0
$641$	32.5562			1.28589			0.642946	−	0.765911i	$-0.277712\pi$
$641$	32.5562			1.28589			0.642946	+	0.765911i	$0.277712\pi$
$642$		0			0
$643$			27.9017i			1.10034i	0.835054	+	0.550168i	$0.185436\pi$
$643$			27.9017i			1.10034i	−0.835054	+	0.550168i	$0.814564\pi$
$644$		0			0
$645$	−21.3453	−	8.82608i	−0.840472	−	0.347527i
$646$		0			0
$647$		−	28.7873i		−	1.13174i	−0.824493	−	0.565872i	$-0.808540\pi$
$647$		−	28.7873i		−	1.13174i	0.824493	−	0.565872i	$-0.191460\pi$
$648$		0			0
$649$	2.63841			0.103567
$650$		0			0
$651$	112.769			4.41976
$652$		0			0
$653$			34.7152i			1.35851i	0.733901	+	0.679256i	$0.237697\pi$
$653$			34.7152i			1.35851i	−0.733901	+	0.679256i	$0.762303\pi$
$654$		0			0
$655$	−36.5230	−	15.1019i	−1.42707	−	0.590079i
$656$		0			0
$657$		−	53.1907i		−	2.07517i
$658$		0			0
$659$	25.5632			0.995801			0.497901	−	0.867234i	$-0.334104\pi$
$659$	25.5632			0.995801			0.497901	+	0.867234i	$0.334104\pi$
$660$		0			0
$661$	1.40960			0.0548269			0.0274135	−	0.999624i	$-0.491273\pi$
$661$	1.40960			0.0548269			0.0274135	+	0.999624i	$0.491273\pi$
$662$		0			0
$663$		−	62.9562i		−	2.44502i
$664$		0			0
$665$	1.97284	−	4.77118i	0.0765033	−	0.185019i
$666$		0			0
$667$			5.95266i			0.230488i
$668$		0			0
$669$	52.0692			2.01311
$670$		0			0
$671$	8.96787			0.346201
$672$		0			0
$673$			0.0557959i			0.00215078i	0.999999	+	0.00107539i	$0.000342307\pi$
$673$			0.0557959i			0.00215078i	−0.999999	+	0.00107539i	$0.999658\pi$
$674$		0			0
$675$	−11.0106	+	11.0378i	−0.423798	+	0.424846i
$676$		0			0
$677$		−	20.8316i		−	0.800623i	−0.916379	−	0.400311i	$-0.868902\pi$
$677$		−	20.8316i		−	0.800623i	0.916379	−	0.400311i	$-0.131098\pi$
$678$		0			0
$679$	45.9612			1.76383
$680$		0			0
$681$	−10.4414			−0.400115
$682$		0			0
$683$			21.5085i			0.822999i	0.911410	+	0.411499i	$0.134995\pi$
$683$			21.5085i			0.822999i	−0.911410	+	0.411499i	$0.865005\pi$
$684$		0			0
$685$	15.7899	−	38.1869i	0.603301	−	1.45905i
$686$		0			0
$687$			35.8939i			1.36944i
$688$		0			0
$689$	21.1907			0.807301
$690$		0			0
$691$	5.49862			0.209178			0.104589	−	0.994516i	$-0.466647\pi$
$691$	5.49862			0.209178			0.104589	+	0.994516i	$0.466647\pi$
$692$		0			0
$693$			18.2655i			0.693851i
$694$		0			0
$695$	11.4862	+	4.74944i	0.435698	+	0.180157i
$696$		0			0
$697$		−	14.7330i		−	0.558054i
$698$		0			0
$699$	−8.99897			−0.340372
$700$		0			0
$701$	−6.00460			−0.226791			−0.113395	−	0.993550i	$-0.536173\pi$
$701$	−6.00460			−0.226791			−0.113395	+	0.993550i	$0.536173\pi$
$702$		0			0
$703$		−	0.929066i		−	0.0350404i
$704$		0			0
$705$	27.1695	+	11.2343i	1.02326	+	0.423109i
$706$		0			0
$707$		−	38.1850i		−	1.43610i
$708$		0			0
$709$	−17.7168			−0.665369			−0.332685	−	0.943038i	$-0.607954\pi$
$709$	−17.7168			−0.665369			−0.332685	+	0.943038i	$0.607954\pi$
$710$		0			0
$711$	−55.2966			−2.07379
$712$		0			0
$713$		−	59.0810i		−	2.21260i
$714$		0			0
$715$	−3.41772	+	8.26554i	−0.127815	+	0.309114i
$716$		0			0
$717$		−	35.5385i		−	1.32721i
$718$		0			0
$719$	−30.5308			−1.13860			−0.569302	−	0.822128i	$-0.692787\pi$
$719$	−30.5308			−1.13860			−0.569302	+	0.822128i	$0.692787\pi$
$720$		0			0
$721$	68.4986			2.55102
$722$		0			0
$723$			47.4950i			1.76636i
$724$		0			0
$725$	3.42618	+	3.41772i	0.127245	+	0.126931i
$726$		0			0
$727$			43.9063i			1.62839i	0.580589	+	0.814197i	$0.302823\pi$
$727$			43.9063i			1.62839i	−0.580589	+	0.814197i	$0.697177\pi$
$728$		0			0
$729$	42.3165			1.56728
$730$		0			0
$731$	−22.6914			−0.839270
$732$		0			0
$733$		−	20.6427i		−	0.762455i	−0.924481	−	0.381227i	$-0.875502\pi$
$733$		−	20.6427i		−	0.762455i	0.924481	−	0.381227i	$-0.124498\pi$
$734$		0			0
$735$	−27.9788	+	67.6650i	−1.03201	+	2.49586i
$736$		0			0
$737$			7.97440i			0.293741i
$738$		0			0
$739$	7.73446			0.284517			0.142258	−	0.989830i	$-0.454564\pi$
$739$	7.73446			0.284517			0.142258	+	0.989830i	$0.454564\pi$
$740$		0			0
$741$	5.63705			0.207082
$742$		0			0
$743$		−	29.1755i		−	1.07034i	−0.844743	−	0.535172i	$-0.820247\pi$
$743$		−	29.1755i		−	1.07034i	0.844743	−	0.535172i	$-0.179753\pi$
$744$		0			0
$745$	−43.2941	−	17.9017i	−1.58617	−	0.655867i
$746$		0			0
$747$			31.7324i			1.16103i
$748$		0			0
$749$	−31.0847			−1.13581
$750$		0			0
$751$	−9.38310			−0.342394			−0.171197	−	0.985237i	$-0.554763\pi$
$751$	−9.38310			−0.342394			−0.171197	+	0.985237i	$0.554763\pi$
$752$		0			0
$753$		−	22.9507i		−	0.836371i
$754$		0			0
$755$	−3.45129	−	1.42707i	−0.125605	−	0.0519365i
$756$		0			0
$757$		−	12.8997i		−	0.468847i	−0.972135	−	0.234424i	$-0.924680\pi$
$757$		−	12.8997i		−	0.468847i	0.972135	−	0.234424i	$-0.0753203\pi$
$758$		0			0
$759$	16.4626			0.597553
$760$		0			0
$761$	15.6963			0.568991			0.284496	−	0.958677i	$-0.408174\pi$
$761$	15.6963			0.568991			0.284496	+	0.958677i	$0.408174\pi$
$762$		0			0
$763$		−	68.1625i		−	2.46765i
$764$		0			0
$765$	−20.9245	+	50.6045i	−0.756526	+	1.82961i
$766$		0			0
$767$			10.5536i			0.381070i
$768$		0			0
$769$	−34.1123			−1.23012			−0.615060	−	0.788480i	$-0.710868\pi$
$769$	−34.1123			−1.23012			−0.615060	+	0.788480i	$0.710868\pi$
$770$		0			0
$771$	7.24573			0.260949
$772$		0			0
$773$		−	52.2539i		−	1.87944i	−0.341942	−	0.939721i	$-0.611085\pi$
$773$		−	52.2539i		−	1.87944i	0.341942	−	0.939721i	$-0.388915\pi$
$774$		0			0
$775$	−34.0053	−	33.9214i	−1.22151	−	1.21849i
$776$		0			0
$777$			20.7154i			0.743160i
$778$		0			0
$779$	1.31918			0.0472647
$780$		0			0
$781$	13.8718			0.496373
$782$		0			0
$783$			3.01795i			0.107853i
$784$		0			0
$785$	−7.64055	+	18.4782i	−0.272703	+	0.659515i
$786$		0			0
$787$		−	6.89902i		−	0.245924i	−0.992411	−	0.122962i	$-0.960761\pi$
$787$		−	6.89902i		−	0.245924i	0.992411	−	0.122962i	$-0.0392392\pi$
$788$		0			0
$789$	38.3344			1.36474
$790$		0			0
$791$	35.5904			1.26545
$792$		0			0
$793$			35.8715i			1.27383i
$794$		0			0
$795$	−29.3023	−	12.1162i	−1.03924	−	0.429717i
$796$		0			0
$797$			0.544296i			0.0192800i	0.999954	+	0.00963998i	$0.00306855\pi$
$797$			0.544296i			0.0192800i	−0.999954	+	0.00963998i	$0.996931\pi$
$798$		0			0
$799$	28.8828			1.02180
$800$		0			0
$801$	11.8079			0.417212
$802$		0			0
$803$			12.7712i			0.450685i
$804$		0			0
$805$	55.7355	+	23.0461i	1.96442	+	0.812268i
$806$		0			0
$807$		−	31.9488i		−	1.12465i
$808$		0			0
$809$	−26.3354			−0.925905			−0.462952	−	0.886383i	$-0.653210\pi$
$809$	−26.3354			−0.925905			−0.462952	+	0.886383i	$0.653210\pi$
$810$		0			0
$811$	−24.4686			−0.859207			−0.429604	−	0.903018i	$-0.641347\pi$
$811$	−24.4686			−0.859207			−0.429604	+	0.903018i	$0.641347\pi$
$812$		0			0
$813$			65.3645i			2.29243i
$814$		0			0
$815$	8.06459	−	19.5037i	0.282490	−	0.683185i
$816$		0			0
$817$		−	2.03176i		−	0.0710824i
$818$		0			0
$819$	−73.0622			−2.55300
$820$		0			0
$821$	42.8503			1.49549			0.747743	−	0.663989i	$-0.231138\pi$
$821$	42.8503			1.49549			0.747743	+	0.663989i	$0.231138\pi$
$822$		0			0
$823$			18.7417i			0.653296i	0.945146	+	0.326648i	$0.105919\pi$
$823$			18.7417i			0.653296i	−0.945146	+	0.326648i	$0.894081\pi$
$824$		0			0
$825$	9.45198	−	9.47537i	0.329076	−	0.329890i
$826$		0			0
$827$			30.6681i			1.06644i	0.845978	+	0.533218i	$0.179017\pi$
$827$			30.6681i			1.06644i	−0.845978	+	0.533218i	$0.820983\pi$
$828$		0			0
$829$	19.8758			0.690314			0.345157	−	0.938545i	$-0.387826\pi$
$829$	19.8758			0.690314			0.345157	+	0.938545i	$0.387826\pi$
$830$		0			0
$831$	61.5847			2.13635
$832$		0			0
$833$			71.9319i			2.49229i
$834$		0			0
$835$	7.66242	−	18.5311i	0.265169	−	0.641295i
$836$		0			0
$837$		−	29.9536i		−	1.03535i
$838$		0			0
$839$	48.3178			1.66812			0.834058	−	0.551677i	$-0.186012\pi$
$839$	48.3178			1.66812			0.834058	+	0.551677i	$0.186012\pi$
$840$		0			0
$841$	−28.0632			−0.967697
$842$		0			0
$843$			28.5329i			0.982725i
$844$		0			0
$845$	−6.19916	−	2.56329i	−0.213257	−	0.0881798i
$846$		0			0
$847$		−	4.38559i		−	0.150691i
$848$		0			0
$849$	27.7768			0.953298
$850$		0			0
$851$	10.8531			0.372038
$852$		0			0
$853$			39.4686i			1.35138i	0.737186	+	0.675690i	$0.236154\pi$
$853$			39.4686i			1.35138i	−0.737186	+	0.675690i	$0.763846\pi$
$854$		0			0
$855$	−4.53109	−	1.87356i	−0.154960	−	0.0640743i
$856$		0			0
$857$		−	16.7666i		−	0.572736i	−0.958120	−	0.286368i	$-0.907552\pi$
$857$		−	16.7666i		−	0.572736i	0.958120	−	0.286368i	$-0.0924479\pi$
$858$		0			0
$859$	−0.350976			−0.0119751			−0.00598757	−	0.999982i	$-0.501906\pi$
$859$	−0.350976			−0.0119751			−0.00598757	+	0.999982i	$0.501906\pi$
$860$		0			0
$861$	−29.4139			−1.00242
$862$		0			0
$863$		−	35.2087i		−	1.19852i	−0.800555	−	0.599259i	$-0.795462\pi$
$863$		−	35.2087i		−	1.19852i	0.800555	−	0.599259i	$-0.204538\pi$
$864$		0			0
$865$	−2.26128	+	5.46876i	−0.0768857	+	0.185943i
$866$		0			0
$867$			47.0404i			1.59758i
$868$		0			0
$869$	13.2768			0.450385
$870$		0			0
$871$	−31.8976			−1.08081
$872$		0			0
$873$		−	43.6483i		−	1.47727i
$874$		0			0
$875$	45.2652	−	18.8478i	1.53024	−	0.637173i
$876$		0			0
$877$			8.61790i			0.291006i	0.989358	+	0.145503i	$0.0464800\pi$
$877$			8.61790i			0.291006i	−0.989358	+	0.145503i	$0.953520\pi$
$878$		0			0
$879$	−60.7804			−2.05007
$880$		0			0
$881$	−42.5162			−1.43241			−0.716204	−	0.697891i	$-0.754122\pi$
$881$	−42.5162			−1.43241			−0.716204	+	0.697891i	$0.754122\pi$
$882$		0			0
$883$			25.1568i			0.846593i	0.905991	+	0.423296i	$0.139127\pi$
$883$			25.1568i			0.846593i	−0.905991	+	0.423296i	$0.860873\pi$
$884$		0			0
$885$	6.03426	−	14.5935i	0.202839	−	0.490555i
$886$		0			0
$887$		−	6.27505i		−	0.210696i	−0.994435	−	0.105348i	$-0.966404\pi$
$887$		−	6.27505i		−	0.210696i	0.994435	−	0.105348i	$-0.0335956\pi$
$888$		0			0
$889$	−18.9887			−0.636861
$890$		0			0
$891$	−4.14832			−0.138974
$892$		0			0
$893$			2.58614i			0.0865418i
$894$		0			0
$895$	−41.7874	−	17.2787i	−1.39680	−	0.577562i
$896$		0			0
$897$			65.8503i			2.19868i
$898$		0			0
$899$	−9.29767			−0.310095
$900$		0			0
$901$	−31.1500			−1.03776
$902$		0			0
$903$			45.3023i			1.50757i
$904$		0			0
$905$	31.7094	+	13.1115i	1.05406	+	0.435842i
$906$		0			0
$907$		−	11.0925i		−	0.368321i	−0.982896	−	0.184161i	$-0.941043\pi$
$907$		−	11.0925i		−	0.368321i	0.982896	−	0.184161i	$-0.0589566\pi$
$908$		0			0
$909$	−36.2635			−1.20278
$910$		0			0
$911$	−32.8175			−1.08729			−0.543646	−	0.839315i	$-0.682956\pi$
$911$	−32.8175			−1.08729			−0.543646	+	0.839315i	$0.682956\pi$
$912$		0			0
$913$		−	7.61901i		−	0.252152i
$914$		0			0
$915$	20.5102	−	49.6028i	0.678048	−	1.63982i
$916$		0			0
$917$			77.5145i			2.55975i
$918$		0			0
$919$	−0.658922			−0.0217358			−0.0108679	−	0.999941i	$-0.503459\pi$
$919$	−0.658922			−0.0217358			−0.0108679	+	0.999941i	$0.503459\pi$
$920$		0			0
$921$	14.7069			0.484610
$922$		0			0
$923$			55.4873i			1.82639i
$924$		0			0
$925$	6.23128	−	6.24670i	0.204883	−	0.205390i
$926$		0			0
$927$		−	65.0516i		−	2.13657i
$928$		0			0
$929$	28.0519			0.920354			0.460177	−	0.887827i	$-0.347786\pi$
$929$	28.0519			0.920354			0.460177	+	0.887827i	$0.347786\pi$
$930$		0			0
$931$	−6.44072			−0.211086
$932$		0			0
$933$			64.8966i			2.12462i
$934$		0			0
$935$	5.02400	−	12.1502i	0.164303	−	0.397356i
$936$		0			0
$937$		−	26.6652i		−	0.871115i	−0.900161	−	0.435558i	$-0.856551\pi$
$937$		−	26.6652i		−	0.871115i	0.900161	−	0.435558i	$-0.143449\pi$
$938$		0			0
$939$	−31.4682			−1.02693
$940$		0			0
$941$	−0.222135			−0.00724139			−0.00362069	−	0.999993i	$-0.501153\pi$
$941$	−0.222135			−0.00724139			−0.00362069	+	0.999993i	$0.501153\pi$
$942$		0			0
$943$			15.4103i			0.501829i
$944$		0			0
$945$	28.2574	+	11.6842i	0.919214	+	0.380086i
$946$		0			0
$947$			4.99028i			0.162162i	0.996707	+	0.0810812i	$0.0258373\pi$
$947$			4.99028i			0.162162i	−0.996707	+	0.0810812i	$0.974163\pi$
$948$		0			0
$949$	−51.0847			−1.65828
$950$		0			0
$951$	24.2348			0.785867
$952$		0			0
$953$		−	9.66807i		−	0.313179i	−0.987664	−	0.156590i	$-0.949950\pi$
$953$		−	9.66807i		−	0.313179i	0.987664	−	0.156590i	$-0.0500500\pi$
$954$		0			0
$955$	−11.6278	−	4.80799i	−0.376268	−	0.155583i
$956$		0			0
$957$		−	2.59074i		−	0.0837467i
$958$		0			0
$959$	−81.0459			−2.61711
$960$		0			0
$961$	61.2807			1.97680
$962$		0			0
$963$			29.5205i			0.951284i
$964$		0			0
$965$	6.18045	−	14.9470i	0.198956	−	0.481162i
$966$		0			0
$967$			19.8033i			0.636832i	0.947951	+	0.318416i	$0.103151\pi$
$967$			19.8033i			0.636832i	−0.947951	+	0.318416i	$0.896849\pi$
$968$		0			0
$969$	−8.28639			−0.266197
$970$		0			0
$971$	39.0410			1.25289			0.626443	−	0.779468i	$-0.284510\pi$
$971$	39.0410			1.25289			0.626443	+	0.779468i	$0.284510\pi$
$972$		0			0
$973$		−	24.3778i		−	0.781517i
$974$		0			0
$975$	37.9015	+	37.8079i	1.21382	+	1.21082i
$976$		0			0
$977$			50.8314i			1.62624i	0.582095	+	0.813121i	$0.302233\pi$
$977$			50.8314i			1.62624i	−0.582095	+	0.813121i	$0.697767\pi$
$978$		0			0
$979$	−2.83510			−0.0906103
$980$		0			0
$981$	−64.7324			−2.06675
$982$		0			0
$983$		−	21.7569i		−	0.693936i	−0.937877	−	0.346968i	$-0.887211\pi$
$983$		−	21.7569i		−	0.693936i	0.937877	−	0.346968i	$-0.112789\pi$
$984$		0			0
$985$	4.57416	−	11.0623i	0.145745	−	0.352475i
$986$		0			0
$987$		−	57.6632i		−	1.83544i
$988$		0			0
$989$	23.7345			0.754712
$990$		0			0
$991$	−33.0622			−1.05025			−0.525127	−	0.851024i	$-0.675982\pi$
$991$	−33.0622			−1.05025			−0.525127	+	0.851024i	$0.675982\pi$
$992$		0			0
$993$			8.82608i			0.280087i
$994$		0			0
$995$	−14.9470	−	6.18045i	−0.473853	−	0.195933i
$996$		0			0
$997$			19.4008i			0.614430i	0.951640	+	0.307215i	$0.0993970\pi$
$997$			19.4008i			0.614430i	−0.951640	+	0.307215i	$0.900603\pi$
$998$		0			0
$999$	5.50241			0.174088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.b.d.89.2	✓	8
3.2	odd	2		3960.2.d.f.3169.1		8
4.3	odd	2		880.2.b.j.529.7		8
5.2	odd	4		2200.2.a.y.1.1		4
5.3	odd	4		2200.2.a.x.1.4		4
5.4	even	2	inner	440.2.b.d.89.7	yes	8
15.14	odd	2		3960.2.d.f.3169.2		8
20.3	even	4		4400.2.a.ce.1.1		4
20.7	even	4		4400.2.a.cb.1.4		4
20.19	odd	2		880.2.b.j.529.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.2	✓	8	1.1	even	1	trivial
440.2.b.d.89.7	yes	8	5.4	even	2	inner
880.2.b.j.529.2		8	20.19	odd	2
880.2.b.j.529.7		8	4.3	odd	2
2200.2.a.x.1.4		4	5.3	odd	4
2200.2.a.y.1.1		4	5.2	odd	4
3960.2.d.f.3169.1		8	3.2	odd	2
3960.2.d.f.3169.2		8	15.14	odd	2
4400.2.a.cb.1.4		4	20.7	even	4
4400.2.a.ce.1.1		4	20.3	even	4

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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.67673i q^{3} +(2.06639 + 0.854430i) q^{5} -4.38559i q^{7} -4.16490 q^{9} +O(q^{10})\)	\(q-2.67673i q^{3} +(2.06639 + 0.854430i) q^{5} -4.38559i q^{7} -4.16490 q^{9} +1.00000 q^{11} +4.00000i q^{13} +(2.28708 - 5.53116i) q^{15} -5.87995i q^{17} +0.526486 q^{19} -11.7391 q^{21} +6.15025i q^{23} +(3.53990 + 3.53116i) q^{25} +3.11812i q^{27} +0.967873 q^{29} -9.60629 q^{31} -2.67673i q^{33} +(3.74718 - 9.06232i) q^{35} -1.76466i q^{37} +10.7069 q^{39} +2.50564 q^{41} -3.85911i q^{43} +(-8.60629 - 3.55861i) q^{45} +4.91208i q^{47} -12.2334 q^{49} -15.7391 q^{51} -5.29767i q^{53} +(2.06639 + 0.854430i) q^{55} -1.40926i q^{57} +2.63841 q^{59} +8.96787 q^{61} +18.2655i q^{63} +(-3.41772 + 8.26554i) q^{65} +7.97440i q^{67} +16.4626 q^{69} +13.8718 q^{71} +12.7712i q^{73} +(9.45198 - 9.47537i) q^{75} -4.38559i q^{77} +13.2768 q^{79} -4.14832 q^{81} -7.61901i q^{83} +(5.02400 - 12.1502i) q^{85} -2.59074i q^{87} -2.83510 q^{89} +17.5424 q^{91} +25.7135i q^{93} +(1.08792 + 0.449845i) q^{95} +10.4800i q^{97} -4.16490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9}+O(q^{10}) \) 8 * q - 14 * q^9	\( 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65} + 60 q^{69} - 2 q^{71} + 26 q^{75} + 40 q^{79} + 88 q^{81} - 26 q^{85} - 42 q^{89} + 8 q^{91} + 28 q^{95} - 14 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more