LMFDB - Embedded newform 2240.2.k.f.1791.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1791,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.k (of order $2$, degree $1$, not 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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} + \cdots + 5956 $ x^16 - 4x^15 - 6x^14 + 68x^13 - 126x^12 - 148x^11 + 1006x^10 - 1516x^9 - 179x^8 + 3992x^7 - 5596x^6 - 488x^5 + 16080x^4 - 33776x^3 + 37344x^2 - 22336*x + 5956
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.13
Root			$1.40610 - 1.29043i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.f.1791.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58086 q^{3} -1.00000i q^{5} +(-0.319247 - 2.62642i) q^{7} +3.66083 q^{9} +O(q^{10})$	$q+2.58086 q^{3} -1.00000i q^{5} +(-0.319247 - 2.62642i) q^{7} +3.66083 q^{9} -0.0961900i q^{11} -2.44063i q^{13} -2.58086i q^{15} -2.17371i q^{17} +2.11309 q^{19} +(-0.823932 - 6.77842i) q^{21} -2.75159i q^{23} -1.00000 q^{25} +1.70550 q^{27} -6.89769 q^{29} +2.00611 q^{31} -0.248253i q^{33} +(-2.62642 + 0.319247i) q^{35} +3.95159 q^{37} -6.29893i q^{39} +6.44638i q^{41} -6.88147i q^{43} -3.66083i q^{45} -12.4808 q^{47} +(-6.79616 + 1.67696i) q^{49} -5.61004i q^{51} +13.6314 q^{53} -0.0961900 q^{55} +5.45360 q^{57} +11.2454 q^{59} -8.32907i q^{61} +(-1.16871 - 9.61487i) q^{63} -2.44063 q^{65} -0.286360i q^{67} -7.10146i q^{69} +12.7829i q^{71} -4.38647i q^{73} -2.58086 q^{75} +(-0.252635 + 0.0307084i) q^{77} +9.94679i q^{79} -6.58082 q^{81} +9.06774 q^{83} -2.17371 q^{85} -17.8020 q^{87} -4.14416i q^{89} +(-6.41013 + 0.779166i) q^{91} +5.17749 q^{93} -2.11309i q^{95} -16.3766i q^{97} -0.352135i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^7 + 16 * q^9	$ 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) $ 16 * q - 4 * q^7 + 16 * q^9 - 8 * q^19 - 4 * q^21 - 16 * q^25 + 48 * q^27 - 8 * q^29 - 16 * q^37 - 8 * q^47 - 4 * q^49 - 16 * q^53 + 8 * q^55 + 16 * q^57 - 8 * q^59 - 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 + 64 * q^83 - 16 * q^87 - 64 * q^91 + 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	2.58086			1.49006			0.745030	−	0.667032i	$-0.232435\pi$
$3$	2.58086			1.49006			0.745030	+	0.667032i	$0.232435\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−0.319247	−	2.62642i	−0.120664	−	0.992693i
$7$	−0.319247	−	2.62642i	−0.120664	−	0.992693i
$8$		0			0
$9$	3.66083			1.22028
$10$		0			0
$11$		−	0.0961900i		−	0.0290024i	−0.999895	−	0.0145012i	$-0.995384\pi$
$11$		−	0.0961900i		−	0.0290024i	0.999895	−	0.0145012i	$-0.00461604\pi$
$12$		0			0
$13$		−	2.44063i		−	0.676910i	−0.940983	−	0.338455i	$-0.890096\pi$
$13$		−	2.44063i		−	0.676910i	0.940983	−	0.338455i	$-0.109904\pi$
$14$		0			0
$15$		−	2.58086i		−	0.666375i
$16$		0			0
$17$		−	2.17371i		−	0.527203i	−0.964632	−	0.263601i	$-0.915090\pi$
$17$		−	2.17371i		−	0.527203i	0.964632	−	0.263601i	$-0.0849103\pi$
$18$		0			0
$19$	2.11309			0.484777			0.242389	−	0.970179i	$-0.422069\pi$
$19$	2.11309			0.484777			0.242389	+	0.970179i	$0.422069\pi$
$20$		0			0
$21$	−0.823932	−	6.77842i	−0.179797	−	1.47917i
$22$		0			0
$23$		−	2.75159i		−	0.573746i	−0.957969	−	0.286873i	$-0.907384\pi$
$23$		−	2.75159i		−	0.573746i	0.957969	−	0.286873i	$-0.0926158\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.70550			0.328224
$28$		0			0
$29$	−6.89769			−1.28087			−0.640434	−	0.768013i	$-0.721246\pi$
$29$	−6.89769			−1.28087			−0.640434	+	0.768013i	$0.721246\pi$
$30$		0			0
$31$	2.00611			0.360308			0.180154	−	0.983638i	$-0.442340\pi$
$31$	2.00611			0.360308			0.180154	+	0.983638i	$0.442340\pi$
$32$		0			0
$33$		−	0.248253i		−	0.0432153i
$34$		0			0
$35$	−2.62642	+	0.319247i	−0.443946	+	0.0539627i
$36$		0			0
$37$	3.95159			0.649637			0.324819	−	0.945776i	$-0.394697\pi$
$37$	3.95159			0.649637			0.324819	+	0.945776i	$0.394697\pi$
$38$		0			0
$39$		−	6.29893i		−	1.00864i
$40$		0			0
$41$			6.44638i			1.00676i	0.864066	+	0.503378i	$0.167910\pi$
$41$			6.44638i			1.00676i	−0.864066	+	0.503378i	$0.832090\pi$
$42$		0			0
$43$		−	6.88147i		−	1.04941i	−0.851283	−	0.524707i	$-0.824175\pi$
$43$		−	6.88147i		−	1.04941i	0.851283	−	0.524707i	$-0.175825\pi$
$44$		0			0
$45$		−	3.66083i		−	0.545724i
$46$		0			0
$47$	−12.4808			−1.82052			−0.910259	−	0.414040i	$-0.864117\pi$
$47$	−12.4808			−1.82052			−0.910259	+	0.414040i	$0.864117\pi$
$48$		0			0
$49$	−6.79616	+	1.67696i	−0.970880	+	0.239565i
$50$		0			0
$51$		−	5.61004i		−	0.785563i
$52$		0			0
$53$	13.6314			1.87241			0.936207	−	0.351450i	$-0.114311\pi$
$53$	13.6314			1.87241			0.936207	+	0.351450i	$0.114311\pi$
$54$		0			0
$55$	−0.0961900			−0.0129703
$56$		0			0
$57$	5.45360			0.722346
$58$		0			0
$59$	11.2454			1.46403			0.732015	−	0.681288i	$-0.238580\pi$
$59$	11.2454			1.46403			0.732015	+	0.681288i	$0.238580\pi$
$60$		0			0
$61$		−	8.32907i		−	1.06643i	−0.845980	−	0.533214i	$-0.820984\pi$
$61$		−	8.32907i		−	1.06643i	0.845980	−	0.533214i	$-0.179016\pi$
$62$		0			0
$63$	−1.16871	−	9.61487i	−0.147244	−	1.21136i
$64$		0			0
$65$	−2.44063			−0.302723
$66$		0			0
$67$		−	0.286360i		−	0.0349844i	−0.999847	−	0.0174922i	$-0.994432\pi$
$67$		−	0.286360i		−	0.0349844i	0.999847	−	0.0174922i	$-0.00556823\pi$
$68$		0			0
$69$		−	7.10146i		−	0.854915i
$70$		0			0
$71$			12.7829i			1.51705i	0.651641	+	0.758527i	$0.274081\pi$
$71$			12.7829i			1.51705i	−0.651641	+	0.758527i	$0.725919\pi$
$72$		0			0
$73$		−	4.38647i		−	0.513398i	−0.966491	−	0.256699i	$-0.917365\pi$
$73$		−	4.38647i		−	0.513398i	0.966491	−	0.256699i	$-0.0826349\pi$
$74$		0			0
$75$	−2.58086			−0.298012
$76$		0			0
$77$	−0.252635	+	0.0307084i	−0.0287905	+	0.00349955i
$78$		0			0
$79$			9.94679i			1.11910i	0.828796	+	0.559551i	$0.189026\pi$
$79$			9.94679i			1.11910i	−0.828796	+	0.559551i	$0.810974\pi$
$80$		0			0
$81$	−6.58082			−0.731203
$82$		0			0
$83$	9.06774			0.995313			0.497657	−	0.867374i	$-0.334194\pi$
$83$	9.06774			0.995313			0.497657	+	0.867374i	$0.334194\pi$
$84$		0			0
$85$	−2.17371			−0.235772
$86$		0			0
$87$	−17.8020			−1.90857
$88$		0			0
$89$		−	4.14416i		−	0.439281i	−0.975581	−	0.219640i	$-0.929512\pi$
$89$		−	4.14416i		−	0.439281i	0.975581	−	0.219640i	$-0.0704884\pi$
$90$		0			0
$91$	−6.41013	+	0.779166i	−0.671964	+	0.0816788i
$92$		0			0
$93$	5.17749			0.536881
$94$		0			0
$95$		−	2.11309i		−	0.216799i
$96$		0			0
$97$		−	16.3766i		−	1.66280i	−0.555678	−	0.831398i	$-0.687541\pi$
$97$		−	16.3766i		−	1.66280i	0.555678	−	0.831398i	$-0.312459\pi$
$98$		0			0
$99$		−	0.352135i		−	0.0353909i
$100$		0			0
$101$		−	5.25054i		−	0.522449i	−0.965278	−	0.261224i	$-0.915874\pi$
$101$		−	5.25054i		−	0.522449i	0.965278	−	0.261224i	$-0.0841262\pi$
$102$		0			0
$103$	7.93477			0.781836			0.390918	−	0.920425i	$-0.372158\pi$
$103$	7.93477			0.781836			0.390918	+	0.920425i	$0.372158\pi$
$104$		0			0
$105$	−6.77842	+	0.823932i	−0.661506	+	0.0804076i
$106$		0			0
$107$		−	7.00937i		−	0.677621i	−0.940855	−	0.338811i	$-0.889975\pi$
$107$		−	7.00937i		−	0.677621i	0.940855	−	0.338811i	$-0.110025\pi$
$108$		0			0
$109$	0.242259			0.0232042			0.0116021	−	0.999933i	$-0.496307\pi$
$109$	0.242259			0.0232042			0.0116021	+	0.999933i	$0.496307\pi$
$110$		0			0
$111$	10.1985			0.967998
$112$		0			0
$113$	−7.40368			−0.696479			−0.348240	−	0.937406i	$-0.613220\pi$
$113$	−7.40368			−0.696479			−0.348240	+	0.937406i	$0.613220\pi$
$114$		0			0
$115$	−2.75159			−0.256587
$116$		0			0
$117$		−	8.93474i		−	0.826017i
$118$		0			0
$119$	−5.70908	+	0.693952i	−0.523350	+	0.0636145i
$120$		0			0
$121$	10.9907			0.999159
$122$		0			0
$123$			16.6372i			1.50013i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			21.4423i			1.90269i	0.308124	+	0.951346i	$0.400299\pi$
$127$			21.4423i			1.90269i	−0.308124	+	0.951346i	$0.599701\pi$
$128$		0			0
$129$		−	17.7601i		−	1.56369i
$130$		0			0
$131$	−12.2773			−1.07267			−0.536337	−	0.844004i	$-0.680192\pi$
$131$	−12.2773			−1.07267			−0.536337	+	0.844004i	$0.680192\pi$
$132$		0			0
$133$	−0.674600	−	5.54987i	−0.0584952	−	0.481235i
$134$		0			0
$135$		−	1.70550i		−	0.146786i
$136$		0			0
$137$	8.71383			0.744473			0.372237	−	0.928138i	$-0.378591\pi$
$137$	8.71383			0.744473			0.372237	+	0.928138i	$0.378591\pi$
$138$		0			0
$139$	15.4497			1.31042			0.655212	−	0.755445i	$-0.272579\pi$
$139$	15.4497			1.31042			0.655212	+	0.755445i	$0.272579\pi$
$140$		0			0
$141$	−32.2113			−2.71268
$142$		0			0
$143$	−0.234765			−0.0196320
$144$		0			0
$145$			6.89769i			0.572822i
$146$		0			0
$147$	−17.5399	+	4.32798i	−1.44667	+	0.356966i
$148$		0			0
$149$	21.8803			1.79250			0.896251	−	0.443548i	$-0.146280\pi$
$149$	21.8803			1.79250			0.896251	+	0.443548i	$0.146280\pi$
$150$		0			0
$151$			11.0963i			0.903003i	0.892270	+	0.451501i	$0.149111\pi$
$151$			11.0963i			0.903003i	−0.892270	+	0.451501i	$0.850889\pi$
$152$		0			0
$153$		−	7.95758i		−	0.643333i
$154$		0			0
$155$		−	2.00611i		−	0.161135i
$156$		0			0
$157$			4.08502i			0.326020i	0.986624	+	0.163010i	$0.0521203\pi$
$157$			4.08502i			0.326020i	−0.986624	+	0.163010i	$0.947880\pi$
$158$		0			0
$159$	35.1806			2.79001
$160$		0			0
$161$	−7.22683	+	0.878438i	−0.569554	+	0.0692306i
$162$		0			0
$163$			14.4556i			1.13225i	0.824318	+	0.566127i	$0.191559\pi$
$163$			14.4556i			1.13225i	−0.824318	+	0.566127i	$0.808441\pi$
$164$		0			0
$165$	−0.248253			−0.0193265
$166$		0			0
$167$	−6.04383			−0.467686			−0.233843	−	0.972274i	$-0.575130\pi$
$167$	−6.04383			−0.467686			−0.233843	+	0.972274i	$0.575130\pi$
$168$		0			0
$169$	7.04331			0.541793
$170$		0			0
$171$	7.73567			0.591562
$172$		0			0
$173$		−	16.6150i		−	1.26321i	−0.775289	−	0.631607i	$-0.782396\pi$
$173$		−	16.6150i		−	1.26321i	0.775289	−	0.631607i	$-0.217604\pi$
$174$		0			0
$175$	0.319247	+	2.62642i	0.0241328	+	0.198539i
$176$		0			0
$177$	29.0229			2.18149
$178$		0			0
$179$			18.9687i			1.41779i	0.705314	+	0.708895i	$0.250806\pi$
$179$			18.9687i			1.41779i	−0.705314	+	0.708895i	$0.749194\pi$
$180$		0			0
$181$		−	3.25354i		−	0.241834i	−0.992663	−	0.120917i	$-0.961417\pi$
$181$		−	3.25354i		−	0.241834i	0.992663	−	0.120917i	$-0.0385834\pi$
$182$		0			0
$183$		−	21.4962i		−	1.58904i
$184$		0			0
$185$		−	3.95159i		−	0.290527i
$186$		0			0
$187$	−0.209089			−0.0152901
$188$		0			0
$189$	−0.544477	−	4.47936i	−0.0396049	−	0.325826i
$190$		0			0
$191$			2.18747i			0.158280i	0.996864	+	0.0791400i	$0.0252174\pi$
$191$			2.18747i			0.158280i	−0.996864	+	0.0791400i	$0.974783\pi$
$192$		0			0
$193$	−15.7569			−1.13421			−0.567103	−	0.823647i	$-0.691936\pi$
$193$	−15.7569			−1.13421			−0.567103	+	0.823647i	$0.691936\pi$
$194$		0			0
$195$	−6.29893			−0.451076
$196$		0			0
$197$	−7.08761			−0.504971			−0.252486	−	0.967601i	$-0.581248\pi$
$197$	−7.08761			−0.504971			−0.252486	+	0.967601i	$0.581248\pi$
$198$		0			0
$199$	−3.86744			−0.274155			−0.137078	−	0.990560i	$-0.543771\pi$
$199$	−3.86744			−0.274155			−0.137078	+	0.990560i	$0.543771\pi$
$200$		0			0
$201$		−	0.739054i		−	0.0521288i
$202$		0			0
$203$	2.20207	+	18.1162i	0.154555	+	1.27151i
$204$		0			0
$205$	6.44638			0.450235
$206$		0			0
$207$		−	10.0731i		−	0.700128i
$208$		0			0
$209$		−	0.203259i		−	0.0140597i
$210$		0			0
$211$			5.85197i			0.402866i	0.979502	+	0.201433i	$0.0645599\pi$
$211$			5.85197i			0.402866i	−0.979502	+	0.201433i	$0.935440\pi$
$212$		0			0
$213$			32.9909i			2.26050i
$214$		0			0
$215$	−6.88147			−0.469312
$216$		0			0
$217$	−0.640446	−	5.26889i	−0.0434763	−	0.357676i
$218$		0			0
$219$		−	11.3209i		−	0.764993i
$220$		0			0
$221$	−5.30523			−0.356869
$222$		0			0
$223$	17.2236			1.15338			0.576688	−	0.816964i	$-0.304345\pi$
$223$	17.2236			1.15338			0.576688	+	0.816964i	$0.304345\pi$
$224$		0			0
$225$	−3.66083			−0.244055
$226$		0			0
$227$	−13.8499			−0.919252			−0.459626	−	0.888112i	$-0.652017\pi$
$227$	−13.8499			−0.919252			−0.459626	+	0.888112i	$0.652017\pi$
$228$		0			0
$229$			19.1389i			1.26473i	0.774669	+	0.632367i	$0.217916\pi$
$229$			19.1389i			1.26473i	−0.774669	+	0.632367i	$0.782084\pi$
$230$		0			0
$231$	−0.652016	+	0.0792541i	−0.0428995	+	0.00521454i
$232$		0			0
$233$	8.67086			0.568047			0.284023	−	0.958817i	$-0.408331\pi$
$233$	8.67086			0.568047			0.284023	+	0.958817i	$0.408331\pi$
$234$		0			0
$235$			12.4808i			0.814160i
$236$		0			0
$237$			25.6713i			1.66753i
$238$		0			0
$239$			16.0440i			1.03780i	0.854834	+	0.518901i	$0.173659\pi$
$239$			16.0440i			1.03780i	−0.854834	+	0.518901i	$0.826341\pi$
$240$		0			0
$241$		−	18.7228i		−	1.20604i	−0.797725	−	0.603022i	$-0.793963\pi$
$241$		−	18.7228i		−	1.20604i	0.797725	−	0.603022i	$-0.206037\pi$
$242$		0			0
$243$	−22.1007			−1.41776
$244$		0			0
$245$	1.67696	+	6.79616i	0.107137	+	0.434191i
$246$		0			0
$247$		−	5.15729i		−	0.328150i
$248$		0			0
$249$	23.4025			1.48308
$250$		0			0
$251$	18.9154			1.19393			0.596965	−	0.802267i	$-0.296373\pi$
$251$	18.9154			1.19393			0.596965	+	0.802267i	$0.296373\pi$
$252$		0			0
$253$	−0.264675			−0.0166400
$254$		0			0
$255$	−5.61004			−0.351314
$256$		0			0
$257$			29.2323i			1.82346i	0.410787	+	0.911732i	$0.365254\pi$
$257$			29.2323i			1.82346i	−0.410787	+	0.911732i	$0.634746\pi$
$258$		0			0
$259$	−1.26154	−	10.3785i	−0.0783880	−	0.644891i
$260$		0			0
$261$	−25.2512			−1.56301
$262$		0			0
$263$			13.1498i			0.810851i	0.914128	+	0.405425i	$0.132877\pi$
$263$			13.1498i			0.810851i	−0.914128	+	0.405425i	$0.867123\pi$
$264$		0			0
$265$		−	13.6314i		−	0.837369i
$266$		0			0
$267$		−	10.6955i		−	0.654554i
$268$		0			0
$269$			1.98850i			0.121241i	0.998161	+	0.0606205i	$0.0193079\pi$
$269$			1.98850i			0.121241i	−0.998161	+	0.0606205i	$0.980692\pi$
$270$		0			0
$271$	12.2835			0.746170			0.373085	−	0.927797i	$-0.378300\pi$
$271$	12.2835			0.746170			0.373085	+	0.927797i	$0.378300\pi$
$272$		0			0
$273$	−16.5436	+	2.01092i	−1.00127	+	0.121706i
$274$		0			0
$275$			0.0961900i			0.00580048i
$276$		0			0
$277$	18.6834			1.12258			0.561289	−	0.827620i	$-0.310305\pi$
$277$	18.6834			1.12258			0.561289	+	0.827620i	$0.310305\pi$
$278$		0			0
$279$	7.34403			0.439676
$280$		0			0
$281$	−6.47899			−0.386504			−0.193252	−	0.981149i	$-0.561903\pi$
$281$	−6.47899			−0.386504			−0.193252	+	0.981149i	$0.561903\pi$
$282$		0			0
$283$	−4.57795			−0.272131			−0.136065	−	0.990700i	$-0.543446\pi$
$283$	−4.57795			−0.272131			−0.136065	+	0.990700i	$0.543446\pi$
$284$		0			0
$285$		−	5.45360i		−	0.323043i
$286$		0			0
$287$	16.9309	−	2.05799i	0.999400	−	0.121479i
$288$		0			0
$289$	12.2750			0.722057
$290$		0			0
$291$		−	42.2658i		−	2.47766i
$292$		0			0
$293$			2.98667i			0.174483i	0.996187	+	0.0872415i	$0.0278052\pi$
$293$			2.98667i			0.174483i	−0.996187	+	0.0872415i	$0.972195\pi$
$294$		0			0
$295$		−	11.2454i		−	0.654734i
$296$		0			0
$297$		−	0.164052i		−	0.00951928i
$298$		0			0
$299$	−6.71562			−0.388374
$300$		0			0
$301$	−18.0736	+	2.19689i	−1.04175	+	0.126627i
$302$		0			0
$303$		−	13.5509i		−	0.778479i
$304$		0			0
$305$	−8.32907			−0.476921
$306$		0			0
$307$	−24.3527			−1.38988			−0.694940	−	0.719068i	$-0.744569\pi$
$307$	−24.3527			−1.38988			−0.694940	+	0.719068i	$0.744569\pi$
$308$		0			0
$309$	20.4785			1.16498
$310$		0			0
$311$	−28.1000			−1.59340			−0.796702	−	0.604372i	$-0.793424\pi$
$311$	−28.1000			−1.59340			−0.796702	+	0.604372i	$0.793424\pi$
$312$		0			0
$313$		−	27.9497i		−	1.57981i	−0.613230	−	0.789905i	$-0.710130\pi$
$313$		−	27.9497i		−	1.57981i	0.613230	−	0.789905i	$-0.289870\pi$
$314$		0			0
$315$	−9.61487	+	1.16871i	−0.541737	+	0.0658493i
$316$		0			0
$317$	26.7666			1.50336			0.751681	−	0.659527i	$-0.229243\pi$
$317$	26.7666			1.50336			0.751681	+	0.659527i	$0.229243\pi$
$318$		0			0
$319$			0.663489i			0.0371483i
$320$		0			0
$321$		−	18.0902i		−	1.00970i
$322$		0			0
$323$		−	4.59326i		−	0.255576i
$324$		0			0
$325$			2.44063i			0.135382i
$326$		0			0
$327$	0.625237			0.0345757
$328$		0			0
$329$	3.98448	+	32.7799i	0.219671	+	1.80722i
$330$		0			0
$331$			5.54702i			0.304892i	0.988312	+	0.152446i	$0.0487149\pi$
$331$			5.54702i			0.304892i	−0.988312	+	0.152446i	$0.951285\pi$
$332$		0			0
$333$	14.4661			0.792737
$334$		0			0
$335$	−0.286360			−0.0156455
$336$		0			0
$337$	−17.8804			−0.974007			−0.487004	−	0.873400i	$-0.661910\pi$
$337$	−17.8804			−0.974007			−0.487004	+	0.873400i	$0.661910\pi$
$338$		0			0
$339$	−19.1078			−1.03780
$340$		0			0
$341$		−	0.192968i		−	0.0104498i
$342$		0			0
$343$	6.57405	+	17.3142i	0.354965	+	0.934880i
$344$		0			0
$345$	−7.10146			−0.382330
$346$		0			0
$347$		−	7.98633i		−	0.428729i	−0.976754	−	0.214364i	$-0.931232\pi$
$347$		−	7.98633i		−	0.428729i	0.976754	−	0.214364i	$-0.0687680\pi$
$348$		0			0
$349$		−	9.19042i		−	0.491952i	−0.969276	−	0.245976i	$-0.920892\pi$
$349$		−	9.19042i		−	0.491952i	0.969276	−	0.245976i	$-0.0791084\pi$
$350$		0			0
$351$		−	4.16250i		−	0.222178i
$352$		0			0
$353$			19.6743i			1.04716i	0.851977	+	0.523579i	$0.175403\pi$
$353$			19.6743i			1.04716i	−0.851977	+	0.523579i	$0.824597\pi$
$354$		0			0
$355$	12.7829			0.678447
$356$		0			0
$357$	−14.7343	+	1.79099i	−0.779823	+	0.0947893i
$358$		0			0
$359$		−	5.29652i		−	0.279540i	−0.990184	−	0.139770i	$-0.955364\pi$
$359$		−	5.29652i		−	0.279540i	0.990184	−	0.139770i	$-0.0446363\pi$
$360$		0			0
$361$	−14.5348			−0.764991
$362$		0			0
$363$	28.3656			1.48881
$364$		0			0
$365$	−4.38647			−0.229598
$366$		0			0
$367$	−22.0652			−1.15179			−0.575896	−	0.817523i	$-0.695347\pi$
$367$	−22.0652			−1.15179			−0.575896	+	0.817523i	$0.695347\pi$
$368$		0			0
$369$			23.5991i			1.22852i
$370$		0			0
$371$	−4.35178	−	35.8017i	−0.225933	−	1.85873i
$372$		0			0
$373$	23.6025			1.22209			0.611045	−	0.791596i	$-0.290750\pi$
$373$	23.6025			1.22209			0.611045	+	0.791596i	$0.290750\pi$
$374$		0			0
$375$			2.58086i			0.133275i
$376$		0			0
$377$			16.8347i			0.867032i
$378$		0			0
$379$		−	24.1233i		−	1.23913i	−0.784945	−	0.619565i	$-0.787309\pi$
$379$		−	24.1233i		−	1.23913i	0.784945	−	0.619565i	$-0.212691\pi$
$380$		0			0
$381$			55.3394i			2.83512i
$382$		0			0
$383$	22.3930			1.14423			0.572114	−	0.820174i	$-0.306123\pi$
$383$	22.3930			1.14423			0.572114	+	0.820174i	$0.306123\pi$
$384$		0			0
$385$	0.0307084	+	0.252635i	0.00156505	+	0.0128755i
$386$		0			0
$387$		−	25.1919i		−	1.28057i
$388$		0			0
$389$	−22.8594			−1.15902			−0.579509	−	0.814966i	$-0.696756\pi$
$389$	−22.8594			−1.15902			−0.579509	+	0.814966i	$0.696756\pi$
$390$		0			0
$391$	−5.98116			−0.302480
$392$		0			0
$393$	−31.6860			−1.59835
$394$		0			0
$395$	9.94679			0.500477
$396$		0			0
$397$		−	37.9051i		−	1.90240i	−0.308572	−	0.951201i	$-0.599851\pi$
$397$		−	37.9051i		−	1.90240i	0.308572	−	0.951201i	$-0.400149\pi$
$398$		0			0
$399$	−1.74105	−	14.3234i	−0.0871614	−	0.717068i
$400$		0			0
$401$	−0.968706			−0.0483749			−0.0241874	−	0.999707i	$-0.507700\pi$
$401$	−0.968706			−0.0483749			−0.0241874	+	0.999707i	$0.507700\pi$
$402$		0			0
$403$		−	4.89618i		−	0.243896i
$404$		0			0
$405$			6.58082i			0.327004i
$406$		0			0
$407$		−	0.380104i		−	0.0188410i
$408$		0			0
$409$		−	34.4020i		−	1.70107i	−0.525917	−	0.850536i	$-0.676278\pi$
$409$		−	34.4020i		−	1.70107i	0.525917	−	0.850536i	$-0.323722\pi$
$410$		0			0
$411$	22.4892			1.10931
$412$		0			0
$413$	−3.59008	−	29.5352i	−0.176656	−	1.45333i
$414$		0			0
$415$		−	9.06774i		−	0.445118i
$416$		0			0
$417$	39.8734			1.95261
$418$		0			0
$419$	23.4628			1.14623			0.573115	−	0.819475i	$-0.305735\pi$
$419$	23.4628			1.14623			0.573115	+	0.819475i	$0.305735\pi$
$420$		0			0
$421$	−29.6064			−1.44293			−0.721465	−	0.692451i	$-0.756531\pi$
$421$	−29.6064			−1.44293			−0.721465	+	0.692451i	$0.756531\pi$
$422$		0			0
$423$	−45.6902			−2.22153
$424$		0			0
$425$			2.17371i			0.105441i
$426$		0			0
$427$	−21.8756	+	2.65904i	−1.05864	+	0.128680i
$428$		0			0
$429$	−0.605894			−0.0292528
$430$		0			0
$431$		−	10.2335i		−	0.492931i	−0.969152	−	0.246465i	$-0.920731\pi$
$431$		−	10.2335i		−	0.492931i	0.969152	−	0.246465i	$-0.0792691\pi$
$432$		0			0
$433$			36.3963i			1.74909i	0.484942	+	0.874547i	$0.338841\pi$
$433$			36.3963i			1.74909i	−0.484942	+	0.874547i	$0.661159\pi$
$434$		0			0
$435$			17.8020i			0.853538i
$436$		0			0
$437$		−	5.81437i		−	0.278139i
$438$		0			0
$439$	24.0302			1.14690			0.573450	−	0.819241i	$-0.305605\pi$
$439$	24.0302			1.14690			0.573450	+	0.819241i	$0.305605\pi$
$440$		0			0
$441$	−24.8796	+	6.13905i	−1.18474	+	0.292336i
$442$		0			0
$443$			16.7426i			0.795466i	0.917501	+	0.397733i	$0.130203\pi$
$443$			16.7426i			0.795466i	−0.917501	+	0.397733i	$0.869797\pi$
$444$		0			0
$445$	−4.14416			−0.196452
$446$		0			0
$447$	56.4699			2.67093
$448$		0			0
$449$	−11.7762			−0.555753			−0.277876	−	0.960617i	$-0.589631\pi$
$449$	−11.7762			−0.555753			−0.277876	+	0.960617i	$0.589631\pi$
$450$		0			0
$451$	0.620078			0.0291983
$452$		0			0
$453$			28.6379i			1.34553i
$454$		0			0
$455$	0.779166	+	6.41013i	0.0365279	+	0.300511i
$456$		0			0
$457$	−5.52642			−0.258515			−0.129258	−	0.991611i	$-0.541259\pi$
$457$	−5.52642			−0.258515			−0.129258	+	0.991611i	$0.541259\pi$
$458$		0			0
$459$		−	3.70727i		−	0.173041i
$460$		0			0
$461$			19.2213i			0.895227i	0.894227	+	0.447613i	$0.147726\pi$
$461$			19.2213i			0.895227i	−0.894227	+	0.447613i	$0.852274\pi$
$462$		0			0
$463$		−	16.2231i		−	0.753949i	−0.926224	−	0.376975i	$-0.876964\pi$
$463$		−	16.2231i		−	0.753949i	0.926224	−	0.376975i	$-0.123036\pi$
$464$		0			0
$465$		−	5.17749i		−	0.240100i
$466$		0			0
$467$	36.1653			1.67353			0.836766	−	0.547561i	$-0.184443\pi$
$467$	36.1653			1.67353			0.836766	+	0.547561i	$0.184443\pi$
$468$		0			0
$469$	−0.752101	+	0.0914197i	−0.0347288	+	0.00422137i
$470$		0			0
$471$			10.5429i			0.485789i
$472$		0			0
$473$	−0.661929			−0.0304355
$474$		0			0
$475$	−2.11309			−0.0969554
$476$		0			0
$477$	49.9021			2.28486
$478$		0			0
$479$	−21.2579			−0.971299			−0.485649	−	0.874154i	$-0.661417\pi$
$479$	−21.2579			−0.971299			−0.485649	+	0.874154i	$0.661417\pi$
$480$		0			0
$481$		−	9.64438i		−	0.439746i
$482$		0			0
$483$	−18.6514	+	2.26712i	−0.848669	+	0.103158i
$484$		0			0
$485$	−16.3766			−0.743625
$486$		0			0
$487$		−	0.766893i		−	0.0347513i	−0.999849	−	0.0173756i	$-0.994469\pi$
$487$		−	0.766893i		−	0.0347513i	0.999849	−	0.0173756i	$-0.00553111\pi$
$488$		0			0
$489$			37.3080i			1.68712i
$490$		0			0
$491$			24.4594i			1.10384i	0.833898	+	0.551919i	$0.186104\pi$
$491$			24.4594i			1.10384i	−0.833898	+	0.551919i	$0.813896\pi$
$492$		0			0
$493$			14.9936i			0.675277i
$494$		0			0
$495$	−0.352135			−0.0158273
$496$		0			0
$497$	33.5733	−	4.08092i	1.50597	−	0.183054i
$498$		0			0
$499$			32.2272i			1.44269i	0.692578	+	0.721343i	$0.256475\pi$
$499$			32.2272i			1.44269i	−0.692578	+	0.721343i	$0.743525\pi$
$500$		0			0
$501$	−15.5983			−0.696879
$502$		0			0
$503$	−3.28701			−0.146561			−0.0732804	−	0.997311i	$-0.523347\pi$
$503$	−3.28701			−0.146561			−0.0732804	+	0.997311i	$0.523347\pi$
$504$		0			0
$505$	−5.25054			−0.233646
$506$		0			0
$507$	18.1778			0.807304
$508$		0			0
$509$			36.0143i			1.59631i	0.602455	+	0.798153i	$0.294189\pi$
$509$			36.0143i			1.59631i	−0.602455	+	0.798153i	$0.705811\pi$
$510$		0			0
$511$	−11.5207	+	1.40037i	−0.509647	+	0.0619487i
$512$		0			0
$513$	3.60389			0.159115
$514$		0			0
$515$		−	7.93477i		−	0.349648i
$516$		0			0
$517$			1.20053i			0.0527993i
$518$		0			0
$519$		−	42.8809i		−	1.88226i
$520$		0			0
$521$			37.6167i			1.64802i	0.566578	+	0.824008i	$0.308267\pi$
$521$			37.6167i			1.64802i	−0.566578	+	0.824008i	$0.691733\pi$
$522$		0			0
$523$	1.03753			0.0453682			0.0226841	−	0.999743i	$-0.492779\pi$
$523$	1.03753			0.0453682			0.0226841	+	0.999743i	$0.492779\pi$
$524$		0			0
$525$	0.823932	+	6.77842i	0.0359594	+	0.295834i
$526$		0			0
$527$		−	4.36071i		−	0.189955i
$528$		0			0
$529$	15.4288			0.670816
$530$		0			0
$531$	41.1676			1.78652
$532$		0			0
$533$	15.7333			0.681483
$534$		0			0
$535$	−7.00937			−0.303041
$536$		0			0
$537$			48.9556i			2.11259i
$538$		0			0
$539$	0.161306	+	0.653723i	0.00694796	+	0.0281578i
$540$		0			0
$541$	−6.93689			−0.298240			−0.149120	−	0.988819i	$-0.547644\pi$
$541$	−6.93689			−0.298240			−0.149120	+	0.988819i	$0.547644\pi$
$542$		0			0
$543$		−	8.39692i		−	0.360346i
$544$		0			0
$545$		−	0.242259i		−	0.0103772i
$546$		0			0
$547$			23.8129i			1.01817i	0.860717	+	0.509084i	$0.170016\pi$
$547$			23.8129i			1.01817i	−0.860717	+	0.509084i	$0.829984\pi$
$548$		0			0
$549$		−	30.4913i		−	1.30134i
$550$		0			0
$551$	−14.5755			−0.620936
$552$		0			0
$553$	26.1245	−	3.17549i	1.11092	−	0.135035i
$554$		0			0
$555$		−	10.1985i		−	0.432902i
$556$		0			0
$557$	−27.3648			−1.15949			−0.579743	−	0.814799i	$-0.696847\pi$
$557$	−27.3648			−1.15949			−0.579743	+	0.814799i	$0.696847\pi$
$558$		0			0
$559$	−16.7951			−0.710359
$560$		0			0
$561$	−0.539630			−0.0227832
$562$		0			0
$563$	−7.58823			−0.319806			−0.159903	−	0.987133i	$-0.551118\pi$
$563$	−7.58823			−0.319806			−0.159903	+	0.987133i	$0.551118\pi$
$564$		0			0
$565$			7.40368i			0.311475i
$566$		0			0
$567$	2.10091	+	17.2840i	0.0882300	+	0.725860i
$568$		0			0
$569$	−6.93366			−0.290674			−0.145337	−	0.989382i	$-0.546427\pi$
$569$	−6.93366			−0.290674			−0.145337	+	0.989382i	$0.546427\pi$
$570$		0			0
$571$		−	4.07049i		−	0.170345i	−0.996366	−	0.0851724i	$-0.972856\pi$
$571$		−	4.07049i		−	0.170345i	0.996366	−	0.0851724i	$-0.0271441\pi$
$572$		0			0
$573$			5.64556i			0.235846i
$574$		0			0
$575$			2.75159i			0.114749i
$576$		0			0
$577$			9.41228i			0.391838i	0.980620	+	0.195919i	$0.0627690\pi$
$577$			9.41228i			0.391838i	−0.980620	+	0.195919i	$0.937231\pi$
$578$		0			0
$579$	−40.6663			−1.69003
$580$		0			0
$581$	−2.89485	−	23.8157i	−0.120099	−	0.988041i
$582$		0			0
$583$		−	1.31120i		−	0.0543045i
$584$		0			0
$585$	−8.93474			−0.369406
$586$		0			0
$587$	24.5572			1.01358			0.506792	−	0.862069i	$-0.330831\pi$
$587$	24.5572			1.01358			0.506792	+	0.862069i	$0.330831\pi$
$588$		0			0
$589$	4.23910			0.174669
$590$		0			0
$591$	−18.2921			−0.752437
$592$		0			0
$593$			19.0241i			0.781225i	0.920555	+	0.390612i	$0.127737\pi$
$593$			19.0241i			0.781225i	−0.920555	+	0.390612i	$0.872263\pi$
$594$		0			0
$595$	0.693952	+	5.70908i	0.0284493	+	0.234049i
$596$		0			0
$597$	−9.98131			−0.408508
$598$		0			0
$599$			4.44908i			0.181785i	0.995861	+	0.0908923i	$0.0289719\pi$
$599$			4.44908i			0.181785i	−0.995861	+	0.0908923i	$0.971028\pi$
$600$		0			0
$601$			13.5897i			0.554335i	0.960822	+	0.277167i	$0.0893956\pi$
$601$			13.5897i			0.554335i	−0.960822	+	0.277167i	$0.910604\pi$
$602$		0			0
$603$		−	1.04831i		−	0.0426906i
$604$		0			0
$605$		−	10.9907i		−	0.446837i
$606$		0			0
$607$	−1.10701			−0.0449320			−0.0224660	−	0.999748i	$-0.507152\pi$
$607$	−1.10701			−0.0449320			−0.0224660	+	0.999748i	$0.507152\pi$
$608$		0			0
$609$	5.68323	+	46.7554i	0.230296	+	1.89462i
$610$		0			0
$611$			30.4611i			1.23233i
$612$		0			0
$613$	−32.5780			−1.31581			−0.657906	−	0.753100i	$-0.728558\pi$
$613$	−32.5780			−1.31581			−0.657906	+	0.753100i	$0.728558\pi$
$614$		0			0
$615$	16.6372			0.670877
$616$		0			0
$617$	11.6569			0.469289			0.234644	−	0.972081i	$-0.424607\pi$
$617$	11.6569			0.469289			0.234644	+	0.972081i	$0.424607\pi$
$618$		0			0
$619$	−11.5426			−0.463936			−0.231968	−	0.972723i	$-0.574516\pi$
$619$	−11.5426			−0.463936			−0.231968	+	0.972723i	$0.574516\pi$
$620$		0			0
$621$		−	4.69284i		−	0.188317i
$622$		0			0
$623$	−10.8843	+	1.32301i	−0.436071	+	0.0530054i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	0.524582i		−	0.0209498i
$628$		0			0
$629$		−	8.58962i		−	0.342490i
$630$		0			0
$631$		−	33.7609i		−	1.34400i	−0.740552	−	0.672000i	$-0.765436\pi$
$631$		−	33.7609i		−	1.34400i	0.740552	−	0.672000i	$-0.234564\pi$
$632$		0			0
$633$			15.1031i			0.600295i
$634$		0			0
$635$	21.4423			0.850910
$636$		0			0
$637$	4.09283	+	16.5869i	0.162164	+	0.657198i
$638$		0			0
$639$			46.7961i			1.85122i
$640$		0			0
$641$	−6.15841			−0.243242			−0.121621	−	0.992577i	$-0.538809\pi$
$641$	−6.15841			−0.243242			−0.121621	+	0.992577i	$0.538809\pi$
$642$		0			0
$643$	−0.578014			−0.0227946			−0.0113973	−	0.999935i	$-0.503628\pi$
$643$	−0.578014			−0.0227946			−0.0113973	+	0.999935i	$0.503628\pi$
$644$		0			0
$645$	−17.7601			−0.699303
$646$		0			0
$647$	−15.5224			−0.610249			−0.305124	−	0.952313i	$-0.598698\pi$
$647$	−15.5224			−0.610249			−0.305124	+	0.952313i	$0.598698\pi$
$648$		0			0
$649$		−	1.08170i		−	0.0424604i
$650$		0			0
$651$	−1.65290	−	13.5983i	−0.0647823	−	0.532958i
$652$		0			0
$653$	19.6566			0.769224			0.384612	−	0.923078i	$-0.374335\pi$
$653$	19.6566			0.769224			0.384612	+	0.923078i	$0.374335\pi$
$654$		0			0
$655$			12.2773i			0.479714i
$656$		0			0
$657$		−	16.0581i		−	0.626487i
$658$		0			0
$659$			32.1914i			1.25400i	0.779019	+	0.627000i	$0.215717\pi$
$659$			32.1914i			1.25400i	−0.779019	+	0.627000i	$0.784283\pi$
$660$		0			0
$661$		−	21.9375i		−	0.853271i	−0.904424	−	0.426636i	$-0.859699\pi$
$661$		−	21.9375i		−	0.853271i	0.904424	−	0.426636i	$-0.140301\pi$
$662$		0			0
$663$	−13.6921			−0.531755
$664$		0			0
$665$	−5.54987	+	0.674600i	−0.215215	+	0.0261599i
$666$		0			0
$667$			18.9796i			0.734893i
$668$		0			0
$669$	44.4516			1.71860
$670$		0			0
$671$	−0.801174			−0.0309290
$672$		0			0
$673$	18.9652			0.731056			0.365528	−	0.930800i	$-0.380889\pi$
$673$	18.9652			0.731056			0.365528	+	0.930800i	$0.380889\pi$
$674$		0			0
$675$	−1.70550			−0.0656448
$676$		0			0
$677$		−	30.5065i		−	1.17246i	−0.810144	−	0.586231i	$-0.800611\pi$
$677$		−	30.5065i		−	1.17246i	0.810144	−	0.586231i	$-0.199389\pi$
$678$		0			0
$679$	−43.0119	+	5.22820i	−1.65065	+	0.200640i
$680$		0			0
$681$	−35.7447			−1.36974
$682$		0			0
$683$			29.6121i			1.13307i	0.824036	+	0.566537i	$0.191717\pi$
$683$			29.6121i			1.13307i	−0.824036	+	0.566537i	$0.808283\pi$
$684$		0			0
$685$		−	8.71383i		−	0.332938i
$686$		0			0
$687$			49.3947i			1.88453i
$688$		0			0
$689$		−	33.2692i		−	1.26745i
$690$		0			0
$691$	−42.2229			−1.60623			−0.803117	−	0.595821i	$-0.796827\pi$
$691$	−42.2229			−1.60623			−0.803117	+	0.595821i	$0.796827\pi$
$692$		0			0
$693$	−0.924855	+	0.112418i	−0.0351323	+	0.00427042i
$694$		0			0
$695$		−	15.4497i		−	0.586039i
$696$		0			0
$697$	14.0126			0.530764
$698$		0			0
$699$	22.3782			0.846423
$700$		0			0
$701$	−7.60356			−0.287183			−0.143591	−	0.989637i	$-0.545865\pi$
$701$	−7.60356			−0.287183			−0.143591	+	0.989637i	$0.545865\pi$
$702$		0			0
$703$	8.35008			0.314929
$704$		0			0
$705$			32.2113i			1.21315i
$706$		0			0
$707$	−13.7901	+	1.67622i	−0.518631	+	0.0630408i
$708$		0			0
$709$	−3.76601			−0.141435			−0.0707177	−	0.997496i	$-0.522529\pi$
$709$	−3.76601			−0.141435			−0.0707177	+	0.997496i	$0.522529\pi$
$710$		0			0
$711$			36.4135i			1.36561i
$712$		0			0
$713$		−	5.52000i		−	0.206725i
$714$		0			0
$715$			0.234765i			0.00877970i
$716$		0			0
$717$			41.4074i			1.54639i
$718$		0			0
$719$	−16.6389			−0.620526			−0.310263	−	0.950651i	$-0.600417\pi$
$719$	−16.6389			−0.620526			−0.310263	+	0.950651i	$0.600417\pi$
$720$		0			0
$721$	−2.53316	−	20.8400i	−0.0943396	−	0.776124i
$722$		0			0
$723$		−	48.3209i		−	1.79708i
$724$		0			0
$725$	6.89769			0.256174
$726$		0			0
$727$	39.9344			1.48109			0.740543	−	0.672009i	$-0.234569\pi$
$727$	39.9344			1.48109			0.740543	+	0.672009i	$0.234569\pi$
$728$		0			0
$729$	−37.2962			−1.38134
$730$		0			0
$731$	−14.9583			−0.553254
$732$		0			0
$733$		−	50.4219i		−	1.86238i	−0.364539	−	0.931188i	$-0.618774\pi$
$733$		−	50.4219i		−	1.86238i	0.364539	−	0.931188i	$-0.381226\pi$
$734$		0			0
$735$	4.32798	+	17.5399i	0.159640	+	0.646970i
$736$		0			0
$737$	−0.0275450			−0.00101463
$738$		0			0
$739$		−	41.8406i		−	1.53913i	−0.638568	−	0.769566i	$-0.720473\pi$
$739$		−	41.8406i		−	1.53913i	0.638568	−	0.769566i	$-0.279527\pi$
$740$		0			0
$741$		−	13.3102i		−	0.488963i
$742$		0			0
$743$		−	1.00362i		−	0.0368192i	−0.999831	−	0.0184096i	$-0.994140\pi$
$743$		−	1.00362i		−	0.0368192i	0.999831	−	0.0184096i	$-0.00586029\pi$
$744$		0			0
$745$		−	21.8803i		−	0.801631i
$746$		0			0
$747$	33.1954			1.21456
$748$		0			0
$749$	−18.4095	+	2.23772i	−0.672670	+	0.0817646i
$750$		0			0
$751$			9.62558i			0.351242i	0.984458	+	0.175621i	$0.0561934\pi$
$751$			9.62558i			0.351242i	−0.984458	+	0.175621i	$0.943807\pi$
$752$		0			0
$753$	48.8180			1.77903
$754$		0			0
$755$	11.0963			0.403835
$756$		0			0
$757$	50.2015			1.82460			0.912302	−	0.409519i	$-0.134303\pi$
$757$	50.2015			1.82460			0.912302	+	0.409519i	$0.134303\pi$
$758$		0			0
$759$	−0.683090			−0.0247946
$760$		0			0
$761$			7.43989i			0.269696i	0.990866	+	0.134848i	$0.0430546\pi$
$761$			7.43989i			0.269696i	−0.990866	+	0.134848i	$0.956945\pi$
$762$		0			0
$763$	−0.0773406	−	0.636274i	−0.00279992	−	0.0230347i
$764$		0			0
$765$	−7.95758			−0.287707
$766$		0			0
$767$		−	27.4460i		−	0.991016i
$768$		0			0
$769$		−	41.4969i		−	1.49642i	−0.663463	−	0.748209i	$-0.730914\pi$
$769$		−	41.4969i		−	1.49642i	0.663463	−	0.748209i	$-0.269086\pi$
$770$		0			0
$771$			75.4445i			2.71707i
$772$		0			0
$773$			32.2305i			1.15925i	0.814884	+	0.579624i	$0.196801\pi$
$773$			32.2305i			1.15925i	−0.814884	+	0.579624i	$0.803199\pi$
$774$		0			0
$775$	−2.00611			−0.0720617
$776$		0			0
$777$	−3.25584	−	26.7855i	−0.116803	−	0.960925i
$778$		0			0
$779$			13.6218i			0.488052i
$780$		0			0
$781$	1.22959			0.0439982
$782$		0			0
$783$	−11.7640			−0.420412
$784$		0			0
$785$	4.08502			0.145801
$786$		0			0
$787$	37.5361			1.33802			0.669008	−	0.743255i	$-0.266719\pi$
$787$	37.5361			1.33802			0.669008	+	0.743255i	$0.266719\pi$
$788$		0			0
$789$			33.9377i			1.20822i
$790$		0			0
$791$	2.36361	+	19.4452i	0.0840401	+	0.691391i
$792$		0			0
$793$	−20.3282			−0.721876
$794$		0			0
$795$		−	35.1806i		−	1.24773i
$796$		0			0
$797$			33.5055i			1.18682i	0.804899	+	0.593412i	$0.202220\pi$
$797$			33.5055i			1.18682i	−0.804899	+	0.593412i	$0.797780\pi$
$798$		0			0
$799$			27.1297i			0.959781i
$800$		0			0
$801$		−	15.1711i		−	0.536043i
$802$		0			0
$803$	−0.421935			−0.0148898
$804$		0			0
$805$	0.878438	+	7.22683i	0.0309609	+	0.254712i
$806$		0			0
$807$			5.13204i			0.180656i
$808$		0			0
$809$	−44.4512			−1.56282			−0.781411	−	0.624017i	$-0.785500\pi$
$809$	−44.4512			−1.56282			−0.781411	+	0.624017i	$0.785500\pi$
$810$		0			0
$811$	−26.5900			−0.933701			−0.466850	−	0.884336i	$-0.654611\pi$
$811$	−26.5900			−0.933701			−0.466850	+	0.884336i	$0.654611\pi$
$812$		0			0
$813$	31.7020			1.11184
$814$		0			0
$815$	14.4556			0.506359
$816$		0			0
$817$		−	14.5412i		−	0.508732i
$818$		0			0
$819$	−23.4664	+	2.85239i	−0.819981	+	0.0996706i
$820$		0			0
$821$	−3.00415			−0.104845			−0.0524227	−	0.998625i	$-0.516694\pi$
$821$	−3.00415			−0.104845			−0.0524227	+	0.998625i	$0.516694\pi$
$822$		0			0
$823$			20.4150i			0.711624i	0.934558	+	0.355812i	$0.115796\pi$
$823$			20.4150i			0.711624i	−0.934558	+	0.355812i	$0.884204\pi$
$824$		0			0
$825$			0.248253i			0.00864305i
$826$		0			0
$827$		−	15.8666i		−	0.551735i	−0.961196	−	0.275868i	$-0.911035\pi$
$827$		−	15.8666i		−	0.551735i	0.961196	−	0.275868i	$-0.0889652\pi$
$828$		0			0
$829$		−	17.7081i		−	0.615028i	−0.951543	−	0.307514i	$-0.900503\pi$
$829$		−	17.7081i		−	0.615028i	0.951543	−	0.307514i	$-0.0994972\pi$
$830$		0			0
$831$	48.2193			1.67271
$832$		0			0
$833$	3.64522	+	14.7729i	0.126299	+	0.511851i
$834$		0			0
$835$			6.04383i			0.209155i
$836$		0			0
$837$	3.42143			0.118262
$838$		0			0
$839$	−46.0811			−1.59090			−0.795449	−	0.606021i	$-0.792765\pi$
$839$	−46.0811			−1.59090			−0.795449	+	0.606021i	$0.792765\pi$
$840$		0			0
$841$	18.5781			0.640624
$842$		0			0
$843$	−16.7213			−0.575914
$844$		0			0
$845$		−	7.04331i		−	0.242297i
$846$		0			0
$847$	−3.50877	−	28.8663i	−0.120563	−	0.991858i
$848$		0			0
$849$	−11.8150			−0.405491
$850$		0			0
$851$		−	10.8732i		−	0.372727i
$852$		0			0
$853$			5.53228i			0.189422i	0.995505	+	0.0947109i	$0.0301927\pi$
$853$			5.53228i			0.189422i	−0.995505	+	0.0947109i	$0.969807\pi$
$854$		0			0
$855$		−	7.73567i		−	0.264554i
$856$		0			0
$857$			42.8657i			1.46426i	0.681163	+	0.732132i	$0.261475\pi$
$857$			42.8657i			1.46426i	−0.681163	+	0.732132i	$0.738525\pi$
$858$		0			0
$859$	−21.4601			−0.732209			−0.366105	−	0.930574i	$-0.619309\pi$
$859$	−21.4601			−0.732209			−0.366105	+	0.930574i	$0.619309\pi$
$860$		0			0
$861$	43.6963	−	5.31138i	1.48916	−	0.181011i
$862$		0			0
$863$			13.8881i			0.472755i	0.971661	+	0.236377i	$0.0759602\pi$
$863$			13.8881i			0.472755i	−0.971661	+	0.236377i	$0.924040\pi$
$864$		0			0
$865$	−16.6150			−0.564926
$866$		0			0
$867$	31.6800			1.07591
$868$		0			0
$869$	0.956782			0.0324566
$870$		0			0
$871$	−0.698899			−0.0236813
$872$		0			0
$873$		−	59.9520i		−	2.02907i
$874$		0			0
$875$	2.62642	−	0.319247i	0.0887892	−	0.0107925i
$876$		0			0
$877$	37.3202			1.26021			0.630106	−	0.776509i	$-0.283011\pi$
$877$	37.3202			1.26021			0.630106	+	0.776509i	$0.283011\pi$
$878$		0			0
$879$			7.70816i			0.259990i
$880$		0			0
$881$			18.1185i			0.610427i	0.952284	+	0.305214i	$0.0987279\pi$
$881$			18.1185i			0.610427i	−0.952284	+	0.305214i	$0.901272\pi$
$882$		0			0
$883$		−	23.0468i		−	0.775586i	−0.921746	−	0.387793i	$-0.873237\pi$
$883$		−	23.0468i		−	0.775586i	0.921746	−	0.387793i	$-0.126763\pi$
$884$		0			0
$885$		−	29.0229i		−	0.975593i
$886$		0			0
$887$	−10.9214			−0.366703			−0.183352	−	0.983047i	$-0.558695\pi$
$887$	−10.9214			−0.366703			−0.183352	+	0.983047i	$0.558695\pi$
$888$		0			0
$889$	56.3164	−	6.84539i	1.88879	−	0.229587i
$890$		0			0
$891$			0.633010i			0.0212066i
$892$		0			0
$893$	−26.3732			−0.882545
$894$		0			0
$895$	18.9687			0.634055
$896$		0			0
$897$	−17.3321			−0.578701
$898$		0			0
$899$	−13.8375			−0.461508
$900$		0			0
$901$		−	29.6307i		−	0.987141i
$902$		0			0
$903$	−46.6455	+	5.66986i	−1.55226	+	0.188681i
$904$		0			0
$905$	−3.25354			−0.108151
$906$		0			0
$907$			43.8749i			1.45684i	0.685129	+	0.728421i	$0.259746\pi$
$907$			43.8749i			1.45684i	−0.685129	+	0.728421i	$0.740254\pi$
$908$		0			0
$909$		−	19.2213i		−	0.637531i
$910$		0			0
$911$		−	10.5105i		−	0.348228i	−0.984725	−	0.174114i	$-0.944294\pi$
$911$		−	10.5105i		−	0.348228i	0.984725	−	0.174114i	$-0.0557062\pi$
$912$		0			0
$913$		−	0.872226i		−	0.0288665i
$914$		0			0
$915$	−21.4962			−0.710641
$916$		0			0
$917$	3.91950	+	32.2454i	0.129433	+	1.06484i
$918$		0			0
$919$		−	16.6088i		−	0.547873i	−0.961748	−	0.273936i	$-0.911674\pi$
$919$		−	16.6088i		−	0.547873i	0.961748	−	0.273936i	$-0.0883258\pi$
$920$		0			0
$921$	−62.8507			−2.07100
$922$		0			0
$923$	31.1984			1.02691
$924$		0			0
$925$	−3.95159			−0.129927
$926$		0			0
$927$	29.0478			0.954056
$928$		0			0
$929$		−	4.56827i		−	0.149880i	−0.997188	−	0.0749401i	$-0.976123\pi$
$929$		−	4.56827i		−	0.149880i	0.997188	−	0.0749401i	$-0.0238766\pi$
$930$		0			0
$931$	−14.3609	+	3.54357i	−0.470660	+	0.116136i
$932$		0			0
$933$	−72.5221			−2.37427
$934$		0			0
$935$			0.209089i			0.00683796i
$936$		0			0
$937$		−	45.8979i		−	1.49942i	−0.661767	−	0.749710i	$-0.730193\pi$
$937$		−	45.8979i		−	1.49942i	0.661767	−	0.749710i	$-0.269807\pi$
$938$		0			0
$939$		−	72.1341i		−	2.35401i
$940$		0			0
$941$			3.53734i			0.115314i	0.998336	+	0.0576569i	$0.0183629\pi$
$941$			3.53734i			0.115314i	−0.998336	+	0.0576569i	$0.981637\pi$
$942$		0			0
$943$	17.7378			0.577622
$944$		0			0
$945$	−4.47936	+	0.544477i	−0.145714	+	0.0177118i
$946$		0			0
$947$		−	25.1805i		−	0.818255i	−0.912477	−	0.409127i	$-0.865833\pi$
$947$		−	25.1805i		−	0.818255i	0.912477	−	0.409127i	$-0.134167\pi$
$948$		0			0
$949$	−10.7058			−0.347524
$950$		0			0
$951$	69.0808			2.24010
$952$		0			0
$953$	55.0349			1.78276			0.891378	−	0.453260i	$-0.149739\pi$
$953$	55.0349			1.78276			0.891378	+	0.453260i	$0.149739\pi$
$954$		0			0
$955$	2.18747			0.0707849
$956$		0			0
$957$			1.71237i			0.0553531i
$958$		0			0
$959$	−2.78187	−	22.8862i	−0.0898313	−	0.739033i
$960$		0			0
$961$	−26.9755			−0.870178
$962$		0			0
$963$		−	25.6601i		−	0.826885i
$964$		0			0
$965$			15.7569i			0.507232i
$966$		0			0
$967$			12.8454i			0.413081i	0.978438	+	0.206540i	$0.0662205\pi$
$967$			12.8454i			0.413081i	−0.978438	+	0.206540i	$0.933780\pi$
$968$		0			0
$969$		−	11.8545i		−	0.380823i
$970$		0			0
$971$	2.14824			0.0689404			0.0344702	−	0.999406i	$-0.489026\pi$
$971$	2.14824			0.0689404			0.0344702	+	0.999406i	$0.489026\pi$
$972$		0			0
$973$	−4.93227	−	40.5773i	−0.158121	−	1.30085i
$974$		0			0
$975$			6.29893i			0.201727i
$976$		0			0
$977$	−20.0936			−0.642850			−0.321425	−	0.946935i	$-0.604162\pi$
$977$	−20.0936			−0.642850			−0.321425	+	0.946935i	$0.604162\pi$
$978$		0			0
$979$	−0.398627			−0.0127402
$980$		0			0
$981$	0.886869			0.0283156
$982$		0			0
$983$	47.7437			1.52279			0.761393	−	0.648291i	$-0.224516\pi$
$983$	47.7437			1.52279			0.761393	+	0.648291i	$0.224516\pi$
$984$		0			0
$985$			7.08761i			0.225830i
$986$		0			0
$987$	10.2834	+	84.6003i	0.327323	+	2.69286i
$988$		0			0
$989$	−18.9350			−0.602097
$990$		0			0
$991$		−	15.4637i		−	0.491219i	−0.969369	−	0.245610i	$-0.921012\pi$
$991$		−	15.4637i		−	0.491219i	0.969369	−	0.245610i	$-0.0789881\pi$
$992$		0			0
$993$			14.3161i			0.454306i
$994$		0			0
$995$			3.86744i			0.122606i
$996$		0			0
$997$		−	3.20881i		−	0.101624i	−0.998708	−	0.0508121i	$-0.983819\pi$
$997$		−	3.20881i		−	0.101624i	0.998708	−	0.0508121i	$-0.0161809\pi$
$998$		0			0
$999$	6.73944			0.213227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.f.1791.13		16
4.3	odd	2		2240.2.k.g.1791.3		16
7.6	odd	2		2240.2.k.g.1791.4		16
8.3	odd	2		1120.2.k.b.671.14	yes	16
8.5	even	2		1120.2.k.a.671.4	yes	16
28.27	even	2	inner	2240.2.k.f.1791.14		16
56.13	odd	2		1120.2.k.b.671.13	yes	16
56.27	even	2		1120.2.k.a.671.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.k.a.671.3	✓	16	56.27	even	2
1120.2.k.a.671.4	yes	16	8.5	even	2
1120.2.k.b.671.13	yes	16	56.13	odd	2
1120.2.k.b.671.14	yes	16	8.3	odd	2
2240.2.k.f.1791.13		16	1.1	even	1	trivial
2240.2.k.f.1791.14		16	28.27	even	2	inner
2240.2.k.g.1791.3		16	4.3	odd	2
2240.2.k.g.1791.4		16	7.6	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.58086 q^{3} -1.00000i q^{5} +(-0.319247 - 2.62642i) q^{7} +3.66083 q^{9} +O(q^{10})\)	\(q+2.58086 q^{3} -1.00000i q^{5} +(-0.319247 - 2.62642i) q^{7} +3.66083 q^{9} -0.0961900i q^{11} -2.44063i q^{13} -2.58086i q^{15} -2.17371i q^{17} +2.11309 q^{19} +(-0.823932 - 6.77842i) q^{21} -2.75159i q^{23} -1.00000 q^{25} +1.70550 q^{27} -6.89769 q^{29} +2.00611 q^{31} -0.248253i q^{33} +(-2.62642 + 0.319247i) q^{35} +3.95159 q^{37} -6.29893i q^{39} +6.44638i q^{41} -6.88147i q^{43} -3.66083i q^{45} -12.4808 q^{47} +(-6.79616 + 1.67696i) q^{49} -5.61004i q^{51} +13.6314 q^{53} -0.0961900 q^{55} +5.45360 q^{57} +11.2454 q^{59} -8.32907i q^{61} +(-1.16871 - 9.61487i) q^{63} -2.44063 q^{65} -0.286360i q^{67} -7.10146i q^{69} +12.7829i q^{71} -4.38647i q^{73} -2.58086 q^{75} +(-0.252635 + 0.0307084i) q^{77} +9.94679i q^{79} -6.58082 q^{81} +9.06774 q^{83} -2.17371 q^{85} -17.8020 q^{87} -4.14416i q^{89} +(-6.41013 + 0.779166i) q^{91} +5.17749 q^{93} -2.11309i q^{95} -16.3766i q^{97} -0.352135i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^7 + 16 * q^9	\( 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) \) 16 * q - 4 * q^7 + 16 * q^9 - 8 * q^19 - 4 * q^21 - 16 * q^25 + 48 * q^27 - 8 * q^29 - 16 * q^37 - 8 * q^47 - 4 * q^49 - 16 * q^53 + 8 * q^55 + 16 * q^57 - 8 * q^59 - 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 + 64 * q^83 - 16 * q^87 - 64 * q^91 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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