LMFDB - Embedded newform 2240.2.k.f.1791.5

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			1791.5
Root			$-1.65334 + 0.722492i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.f.1791.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44498 q^{3} -1.00000i q^{5} +(2.48080 - 0.919594i) q^{7} -0.912023 q^{9} +O(q^{10})$	$q-1.44498 q^{3} -1.00000i q^{5} +(2.48080 - 0.919594i) q^{7} -0.912023 q^{9} -5.66807i q^{11} -5.14588i q^{13} +1.44498i q^{15} -1.65490i q^{17} +1.33951 q^{19} +(-3.58471 + 1.32880i) q^{21} +3.62209i q^{23} -1.00000 q^{25} +5.65281 q^{27} +0.0482240 q^{29} -3.08555 q^{31} +8.19027i q^{33} +(-0.919594 - 2.48080i) q^{35} -11.2462 q^{37} +7.43570i q^{39} -2.37889i q^{41} +6.18334i q^{43} +0.912023i q^{45} +7.75943 q^{47} +(5.30869 - 4.56265i) q^{49} +2.39131i q^{51} -10.5028 q^{53} -5.66807 q^{55} -1.93556 q^{57} +5.32378 q^{59} +2.30924i q^{61} +(-2.26254 + 0.838692i) q^{63} -5.14588 q^{65} -0.207825i q^{67} -5.23385i q^{69} +6.67465i q^{71} -3.42442i q^{73} +1.44498 q^{75} +(-5.21233 - 14.0613i) q^{77} +1.73371i q^{79} -5.43214 q^{81} +12.2384 q^{83} -1.65490 q^{85} -0.0696828 q^{87} +4.28261i q^{89} +(-4.73212 - 12.7659i) q^{91} +4.45856 q^{93} -1.33951i q^{95} -12.1226i q^{97} +5.16942i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.44498			−0.834261			−0.417131	−	0.908846i	$-0.636964\pi$
$3$	−1.44498			−0.834261			−0.417131	+	0.908846i	$0.636964\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	2.48080	−	0.919594i	0.937653	−	0.347574i
$7$	2.48080	−	0.919594i	0.937653	−	0.347574i
$8$		0			0
$9$	−0.912023			−0.304008
$10$		0			0
$11$		−	5.66807i		−	1.70899i	−0.519461	−	0.854494i	$-0.673867\pi$
$11$		−	5.66807i		−	1.70899i	0.519461	−	0.854494i	$-0.326133\pi$
$12$		0			0
$13$		−	5.14588i		−	1.42721i	−0.700549	−	0.713605i	$-0.747061\pi$
$13$		−	5.14588i		−	1.42721i	0.700549	−	0.713605i	$-0.252939\pi$
$14$		0			0
$15$			1.44498i			0.373093i
$16$		0			0
$17$		−	1.65490i		−	0.401373i	−0.979656	−	0.200687i	$-0.935683\pi$
$17$		−	1.65490i		−	0.401373i	0.979656	−	0.200687i	$-0.0643173\pi$
$18$		0			0
$19$	1.33951			0.307304			0.153652	−	0.988125i	$-0.450897\pi$
$19$	1.33951			0.307304			0.153652	+	0.988125i	$0.450897\pi$
$20$		0			0
$21$	−3.58471	+	1.32880i	−0.782247	+	0.289968i
$22$		0			0
$23$			3.62209i			0.755257i	0.925957	+	0.377628i	$0.123260\pi$
$23$			3.62209i			0.755257i	−0.925957	+	0.377628i	$0.876740\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.65281			1.08788
$28$		0			0
$29$	0.0482240			0.00895496			0.00447748	−	0.999990i	$-0.498575\pi$
$29$	0.0482240			0.00895496			0.00447748	+	0.999990i	$0.498575\pi$
$30$		0			0
$31$	−3.08555			−0.554181			−0.277090	−	0.960844i	$-0.589370\pi$
$31$	−3.08555			−0.554181			−0.277090	+	0.960844i	$0.589370\pi$
$32$		0			0
$33$			8.19027i			1.42574i
$34$		0			0
$35$	−0.919594	−	2.48080i	−0.155440	−	0.419331i
$36$		0			0
$37$	−11.2462			−1.84887			−0.924434	−	0.381343i	$-0.875462\pi$
$37$	−11.2462			−1.84887			−0.924434	+	0.381343i	$0.875462\pi$
$38$		0			0
$39$			7.43570i			1.19067i
$40$		0			0
$41$		−	2.37889i		−	0.371520i	−0.982595	−	0.185760i	$-0.940525\pi$
$41$		−	2.37889i		−	0.371520i	0.982595	−	0.185760i	$-0.0594747\pi$
$42$		0			0
$43$			6.18334i			0.942950i	0.881879	+	0.471475i	$0.156278\pi$
$43$			6.18334i			0.942950i	−0.881879	+	0.471475i	$0.843722\pi$
$44$		0			0
$45$			0.912023i			0.135956i
$46$		0			0
$47$	7.75943			1.13183			0.565915	−	0.824464i	$-0.308523\pi$
$47$	7.75943			1.13183			0.565915	+	0.824464i	$0.308523\pi$
$48$		0			0
$49$	5.30869	−	4.56265i	0.758385	−	0.651807i
$50$		0			0
$51$			2.39131i			0.334850i
$52$		0			0
$53$	−10.5028			−1.44267			−0.721333	−	0.692588i	$-0.756470\pi$
$53$	−10.5028			−1.44267			−0.721333	+	0.692588i	$0.756470\pi$
$54$		0			0
$55$	−5.66807			−0.764283
$56$		0			0
$57$	−1.93556			−0.256372
$58$		0			0
$59$	5.32378			0.693097			0.346549	−	0.938032i	$-0.387354\pi$
$59$	5.32378			0.693097			0.346549	+	0.938032i	$0.387354\pi$
$60$		0			0
$61$			2.30924i			0.295668i	0.989012	+	0.147834i	$0.0472301\pi$
$61$			2.30924i			0.295668i	−0.989012	+	0.147834i	$0.952770\pi$
$62$		0			0
$63$	−2.26254	+	0.838692i	−0.285054	+	0.105665i
$64$		0			0
$65$	−5.14588			−0.638267
$66$		0			0
$67$		−	0.207825i		−	0.0253899i	−0.999919	−	0.0126949i	$-0.995959\pi$
$67$		−	0.207825i		−	0.0253899i	0.999919	−	0.0126949i	$-0.00404103\pi$
$68$		0			0
$69$		−	5.23385i		−	0.630082i
$70$		0			0
$71$			6.67465i			0.792135i	0.918221	+	0.396068i	$0.129625\pi$
$71$			6.67465i			0.792135i	−0.918221	+	0.396068i	$0.870375\pi$
$72$		0			0
$73$		−	3.42442i		−	0.400798i	−0.979714	−	0.200399i	$-0.935776\pi$
$73$		−	3.42442i		−	0.400798i	0.979714	−	0.200399i	$-0.0642238\pi$
$74$		0			0
$75$	1.44498			0.166852
$76$		0			0
$77$	−5.21233	−	14.0613i	−0.594000	−	1.60244i
$78$		0			0
$79$			1.73371i			0.195058i	0.995233	+	0.0975289i	$0.0310938\pi$
$79$			1.73371i			0.195058i	−0.995233	+	0.0975289i	$0.968906\pi$
$80$		0			0
$81$	−5.43214			−0.603572
$82$		0			0
$83$	12.2384			1.34333			0.671667	−	0.740853i	$-0.265579\pi$
$83$	12.2384			1.34333			0.671667	+	0.740853i	$0.265579\pi$
$84$		0			0
$85$	−1.65490			−0.179500
$86$		0			0
$87$	−0.0696828			−0.00747078
$88$		0			0
$89$			4.28261i			0.453956i	0.973900	+	0.226978i	$0.0728845\pi$
$89$			4.28261i			0.453956i	−0.973900	+	0.226978i	$0.927115\pi$
$90$		0			0
$91$	−4.73212	−	12.7659i	−0.496061	−	1.33823i
$92$		0			0
$93$	4.45856			0.462331
$94$		0			0
$95$		−	1.33951i		−	0.137430i
$96$		0			0
$97$		−	12.1226i		−	1.23086i	−0.788190	−	0.615432i	$-0.788981\pi$
$97$		−	12.1226i		−	1.23086i	0.788190	−	0.615432i	$-0.211019\pi$
$98$		0			0
$99$			5.16942i			0.519546i
$100$		0			0
$101$			12.4565i			1.23947i	0.784812	+	0.619734i	$0.212760\pi$
$101$			12.4565i			1.23947i	−0.784812	+	0.619734i	$0.787240\pi$
$102$		0			0
$103$	−8.57029			−0.844455			−0.422228	−	0.906490i	$-0.638752\pi$
$103$	−8.57029			−0.844455			−0.422228	+	0.906490i	$0.638752\pi$
$104$		0			0
$105$	1.32880	+	3.58471i	0.129677	+	0.349832i
$106$		0			0
$107$		−	18.1310i		−	1.75279i	−0.481592	−	0.876395i	$-0.659941\pi$
$107$		−	18.1310i		−	1.75279i	0.481592	−	0.876395i	$-0.340059\pi$
$108$		0			0
$109$	−13.9272			−1.33398			−0.666991	−	0.745066i	$-0.732418\pi$
$109$	−13.9272			−1.33398			−0.666991	+	0.745066i	$0.732418\pi$
$110$		0			0
$111$	16.2506			1.54244
$112$		0			0
$113$	−5.13757			−0.483302			−0.241651	−	0.970363i	$-0.577689\pi$
$113$	−5.13757			−0.483302			−0.241651	+	0.970363i	$0.577689\pi$
$114$		0			0
$115$	3.62209			0.337761
$116$		0			0
$117$			4.69316i			0.433883i
$118$		0			0
$119$	−1.52184	−	4.10548i	−0.139507	−	0.376349i
$120$		0			0
$121$	−21.1271			−1.92064
$122$		0			0
$123$			3.43745i			0.309945i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	17.1695i		−	1.52355i	−0.647841	−	0.761775i	$-0.724328\pi$
$127$		−	17.1695i		−	1.52355i	0.647841	−	0.761775i	$-0.275672\pi$
$128$		0			0
$129$		−	8.93482i		−	0.786667i
$130$		0			0
$131$	−9.37209			−0.818843			−0.409422	−	0.912345i	$-0.634269\pi$
$131$	−9.37209			−0.818843			−0.409422	+	0.912345i	$0.634269\pi$
$132$		0			0
$133$	3.32304	−	1.23180i	0.288144	−	0.106811i
$134$		0			0
$135$		−	5.65281i		−	0.486516i
$136$		0			0
$137$	20.3631			1.73973			0.869867	−	0.493287i	$-0.164205\pi$
$137$	20.3631			1.73973			0.869867	+	0.493287i	$0.164205\pi$
$138$		0			0
$139$	−20.6541			−1.75186			−0.875930	−	0.482438i	$-0.839752\pi$
$139$	−20.6541			−1.75186			−0.875930	+	0.482438i	$0.839752\pi$
$140$		0			0
$141$	−11.2123			−0.944242
$142$		0			0
$143$	−29.1672			−2.43908
$144$		0			0
$145$		−	0.0482240i		−	0.00400478i
$146$		0			0
$147$	−7.67097	+	6.59295i	−0.632691	+	0.543778i
$148$		0			0
$149$	−10.4375			−0.855076			−0.427538	−	0.903997i	$-0.640619\pi$
$149$	−10.4375			−0.855076			−0.427538	+	0.903997i	$0.640619\pi$
$150$		0			0
$151$			5.01484i			0.408102i	0.978960	+	0.204051i	$0.0654108\pi$
$151$			5.01484i			0.408102i	−0.978960	+	0.204051i	$0.934589\pi$
$152$		0			0
$153$			1.50931i			0.122021i
$154$		0			0
$155$			3.08555i			0.247837i
$156$		0			0
$157$		−	0.287851i		−	0.0229730i	−0.999934	−	0.0114865i	$-0.996344\pi$
$157$		−	0.287851i		−	0.0229730i	0.999934	−	0.0114865i	$-0.00365635\pi$
$158$		0			0
$159$	15.1763			1.20356
$160$		0			0
$161$	3.33085	+	8.98565i	0.262508	+	0.708169i
$162$		0			0
$163$			8.02740i			0.628755i	0.949298	+	0.314377i	$0.101796\pi$
$163$			8.02740i			0.628755i	−0.949298	+	0.314377i	$0.898204\pi$
$164$		0			0
$165$	8.19027			0.637612
$166$		0			0
$167$	−1.74986			−0.135408			−0.0677040	−	0.997705i	$-0.521567\pi$
$167$	−1.74986			−0.135408			−0.0677040	+	0.997705i	$0.521567\pi$
$168$		0			0
$169$	−13.4800			−1.03693
$170$		0			0
$171$	−1.22166			−0.0934227
$172$		0			0
$173$		−	19.0563i		−	1.44882i	−0.689368	−	0.724412i	$-0.742112\pi$
$173$		−	19.0563i		−	1.44882i	0.689368	−	0.724412i	$-0.257888\pi$
$174$		0			0
$175$	−2.48080	+	0.919594i	−0.187531	+	0.0695148i
$176$		0			0
$177$	−7.69277			−0.578224
$178$		0			0
$179$			8.22119i			0.614480i	0.951632	+	0.307240i	$0.0994055\pi$
$179$			8.22119i			0.614480i	−0.951632	+	0.307240i	$0.900594\pi$
$180$		0			0
$181$			25.7060i			1.91071i	0.295461	+	0.955355i	$0.404527\pi$
$181$			25.7060i			1.91071i	−0.295461	+	0.955355i	$0.595473\pi$
$182$		0			0
$183$		−	3.33681i		−	0.246664i
$184$		0			0
$185$			11.2462i			0.826839i
$186$		0			0
$187$	−9.38012			−0.685942
$188$		0			0
$189$	14.0235	−	5.19829i	1.02006	−	0.378120i
$190$		0			0
$191$		−	10.4549i		−	0.756489i	−0.925706	−	0.378244i	$-0.876528\pi$
$191$		−	10.4549i		−	0.756489i	0.925706	−	0.378244i	$-0.123472\pi$
$192$		0			0
$193$	−21.1128			−1.51973			−0.759867	−	0.650079i	$-0.774736\pi$
$193$	−21.1128			−1.51973			−0.759867	+	0.650079i	$0.774736\pi$
$194$		0			0
$195$	7.43570			0.532482
$196$		0			0
$197$	27.9221			1.98937			0.994683	−	0.102988i	$-0.0328403\pi$
$197$	27.9221			1.98937			0.994683	+	0.102988i	$0.0328403\pi$
$198$		0			0
$199$	−21.4173			−1.51823			−0.759117	−	0.650955i	$-0.774369\pi$
$199$	−21.4173			−1.51823			−0.759117	+	0.650955i	$0.774369\pi$
$200$		0			0
$201$			0.300304i			0.0211818i
$202$		0			0
$203$	0.119634	−	0.0443465i	0.00839664	−	0.00311251i
$204$		0			0
$205$	−2.37889			−0.166149
$206$		0			0
$207$		−	3.30343i		−	0.229604i
$208$		0			0
$209$		−	7.59242i		−	0.525179i
$210$		0			0
$211$			1.56333i			0.107624i	0.998551	+	0.0538122i	$0.0171372\pi$
$211$			1.56333i			0.107624i	−0.998551	+	0.0538122i	$0.982863\pi$
$212$		0			0
$213$		−	9.64476i		−	0.660848i
$214$		0			0
$215$	6.18334			0.421700
$216$		0			0
$217$	−7.65461	+	2.83745i	−0.519629	+	0.192619i
$218$		0			0
$219$			4.94823i			0.334370i
$220$		0			0
$221$	−8.51593			−0.572843
$222$		0			0
$223$	16.2856			1.09057			0.545283	−	0.838252i	$-0.316422\pi$
$223$	16.2856			1.09057			0.545283	+	0.838252i	$0.316422\pi$
$224$		0			0
$225$	0.912023			0.0608016
$226$		0			0
$227$	−17.2890			−1.14751			−0.573757	−	0.819025i	$-0.694515\pi$
$227$	−17.2890			−1.14751			−0.573757	+	0.819025i	$0.694515\pi$
$228$		0			0
$229$		−	25.9786i		−	1.71671i	−0.513055	−	0.858356i	$-0.671486\pi$
$229$		−	25.9786i		−	1.71671i	0.513055	−	0.858356i	$-0.328514\pi$
$230$		0			0
$231$	7.53173	+	20.3184i	0.495551	+	1.33685i
$232$		0			0
$233$	−8.47013			−0.554896			−0.277448	−	0.960741i	$-0.589489\pi$
$233$	−8.47013			−0.554896			−0.277448	+	0.960741i	$0.589489\pi$
$234$		0			0
$235$		−	7.75943i		−	0.506170i
$236$		0			0
$237$		−	2.50518i		−	0.162729i
$238$		0			0
$239$		−	6.72523i		−	0.435019i	−0.976058	−	0.217510i	$-0.930207\pi$
$239$		−	6.72523i		−	0.435019i	0.976058	−	0.217510i	$-0.0697933\pi$
$240$		0			0
$241$		−	24.8505i		−	1.60076i	−0.599492	−	0.800381i	$-0.704631\pi$
$241$		−	24.8505i		−	1.60076i	0.599492	−	0.800381i	$-0.295369\pi$
$242$		0			0
$243$	−9.10907			−0.584347
$244$		0			0
$245$	−4.56265	−	5.30869i	−0.291497	−	0.339160i
$246$		0			0
$247$		−	6.89293i		−	0.438587i
$248$		0			0
$249$	−17.6842			−1.12069
$250$		0			0
$251$	13.3141			0.840379			0.420189	−	0.907436i	$-0.361964\pi$
$251$	13.3141			0.840379			0.420189	+	0.907436i	$0.361964\pi$
$252$		0			0
$253$	20.5302			1.29073
$254$		0			0
$255$	2.39131			0.149750
$256$		0			0
$257$		−	24.8989i		−	1.55315i	−0.630023	−	0.776577i	$-0.716955\pi$
$257$		−	24.8989i		−	1.55315i	0.630023	−	0.776577i	$-0.283045\pi$
$258$		0			0
$259$	−27.8996	+	10.3420i	−1.73360	+	0.642618i
$260$		0			0
$261$	−0.0439814			−0.00272238
$262$		0			0
$263$			2.70131i			0.166570i	0.996526	+	0.0832850i	$0.0265412\pi$
$263$			2.70131i			0.166570i	−0.996526	+	0.0832850i	$0.973459\pi$
$264$		0			0
$265$			10.5028i			0.645180i
$266$		0			0
$267$		−	6.18830i		−	0.378718i
$268$		0			0
$269$			25.0495i			1.52730i	0.645632	+	0.763648i	$0.276594\pi$
$269$			25.0495i			1.52730i	−0.645632	+	0.763648i	$0.723406\pi$
$270$		0			0
$271$	13.9628			0.848177			0.424088	−	0.905621i	$-0.360595\pi$
$271$	13.9628			0.848177			0.424088	+	0.905621i	$0.360595\pi$
$272$		0			0
$273$	6.83783	+	18.4465i	0.413844	+	1.11643i
$274$		0			0
$275$			5.66807i			0.341798i
$276$		0			0
$277$	13.6112			0.817816			0.408908	−	0.912576i	$-0.365910\pi$
$277$	13.6112			0.817816			0.408908	+	0.912576i	$0.365910\pi$
$278$		0			0
$279$	2.81409			0.168475
$280$		0			0
$281$	24.1166			1.43868			0.719338	−	0.694660i	$-0.244445\pi$
$281$	24.1166			1.43868			0.719338	+	0.694660i	$0.244445\pi$
$282$		0			0
$283$	23.3390			1.38736			0.693680	−	0.720283i	$-0.255988\pi$
$283$	23.3390			1.38736			0.693680	+	0.720283i	$0.255988\pi$
$284$		0			0
$285$			1.93556i			0.114653i
$286$		0			0
$287$	−2.18761	−	5.90153i	−0.129131	−	0.348357i
$288$		0			0
$289$	14.2613			0.838900
$290$		0			0
$291$			17.5170i			1.02686i
$292$		0			0
$293$		−	7.56273i		−	0.441820i	−0.975294	−	0.220910i	$-0.929097\pi$
$293$		−	7.56273i		−	0.441820i	0.975294	−	0.220910i	$-0.0709026\pi$
$294$		0			0
$295$		−	5.32378i		−	0.309962i
$296$		0			0
$297$		−	32.0405i		−	1.85918i
$298$		0			0
$299$	18.6388			1.07791
$300$		0			0
$301$	5.68616	+	15.3396i	0.327745	+	0.884160i
$302$		0			0
$303$		−	17.9994i		−	1.03404i
$304$		0			0
$305$	2.30924			0.132227
$306$		0			0
$307$	25.3775			1.44837			0.724184	−	0.689607i	$-0.242217\pi$
$307$	25.3775			1.44837			0.724184	+	0.689607i	$0.242217\pi$
$308$		0			0
$309$	12.3839			0.704497
$310$		0			0
$311$	0.939191			0.0532566			0.0266283	−	0.999645i	$-0.491523\pi$
$311$	0.939191			0.0532566			0.0266283	+	0.999645i	$0.491523\pi$
$312$		0			0
$313$			20.0637i			1.13407i	0.823694	+	0.567035i	$0.191910\pi$
$313$			20.0637i			1.13407i	−0.823694	+	0.567035i	$0.808090\pi$
$314$		0			0
$315$	0.838692	+	2.26254i	0.0472549	+	0.127480i
$316$		0			0
$317$	−13.9990			−0.786260			−0.393130	−	0.919483i	$-0.628608\pi$
$317$	−13.9990			−0.786260			−0.393130	+	0.919483i	$0.628608\pi$
$318$		0			0
$319$		−	0.273337i		−	0.0153039i
$320$		0			0
$321$			26.1990i			1.46229i
$322$		0			0
$323$		−	2.21675i		−	0.123343i
$324$		0			0
$325$			5.14588i			0.285442i
$326$		0			0
$327$	20.1245			1.11289
$328$		0			0
$329$	19.2496	−	7.13553i	1.06126	−	0.393395i
$330$		0			0
$331$			17.0893i			0.939312i	0.882850	+	0.469656i	$0.155622\pi$
$331$			17.0893i			0.939312i	−0.882850	+	0.469656i	$0.844378\pi$
$332$		0			0
$333$	10.2568			0.562070
$334$		0			0
$335$	−0.207825			−0.0113547
$336$		0			0
$337$	0.384299			0.0209341			0.0104670	−	0.999945i	$-0.496668\pi$
$337$	0.384299			0.0209341			0.0104670	+	0.999945i	$0.496668\pi$
$338$		0			0
$339$	7.42371			0.403201
$340$		0			0
$341$			17.4891i			0.947088i
$342$		0			0
$343$	8.97399	−	16.2008i	0.484550	−	0.874764i
$344$		0			0
$345$	−5.23385			−0.281781
$346$		0			0
$347$		−	4.51811i		−	0.242545i	−0.992619	−	0.121273i	$-0.961303\pi$
$347$		−	4.51811i		−	0.242545i	0.992619	−	0.121273i	$-0.0386975\pi$
$348$		0			0
$349$			19.7108i			1.05510i	0.849525	+	0.527549i	$0.176889\pi$
$349$			19.7108i			1.05510i	−0.849525	+	0.527549i	$0.823111\pi$
$350$		0			0
$351$		−	29.0887i		−	1.55264i
$352$		0			0
$353$		−	8.79118i		−	0.467907i	−0.972248	−	0.233954i	$-0.924834\pi$
$353$		−	8.79118i		−	0.467907i	0.972248	−	0.233954i	$-0.0751664\pi$
$354$		0			0
$355$	6.67465			0.354254
$356$		0			0
$357$	2.19903	+	5.93235i	0.116385	+	0.313973i
$358$		0			0
$359$			26.3042i			1.38828i	0.719839	+	0.694141i	$0.244216\pi$
$359$			26.3042i			1.38828i	−0.719839	+	0.694141i	$0.755784\pi$
$360$		0			0
$361$	−17.2057			−0.905564
$362$		0			0
$363$	30.5283			1.60232
$364$		0			0
$365$	−3.42442			−0.179242
$366$		0			0
$367$	−16.1194			−0.841428			−0.420714	−	0.907193i	$-0.638220\pi$
$367$	−16.1194			−0.841428			−0.420714	+	0.907193i	$0.638220\pi$
$368$		0			0
$369$			2.16960i			0.112945i
$370$		0			0
$371$	−26.0552	+	9.65829i	−1.35272	+	0.501434i
$372$		0			0
$373$	−8.18477			−0.423791			−0.211896	−	0.977292i	$-0.567964\pi$
$373$	−8.18477			−0.423791			−0.211896	+	0.977292i	$0.567964\pi$
$374$		0			0
$375$		−	1.44498i		−	0.0746186i
$376$		0			0
$377$		−	0.248154i		−	0.0127806i
$378$		0			0
$379$		−	21.5932i		−	1.10917i	−0.832128	−	0.554583i	$-0.812878\pi$
$379$		−	21.5932i		−	1.10917i	0.832128	−	0.554583i	$-0.187122\pi$
$380$		0			0
$381$			24.8097i			1.27104i
$382$		0			0
$383$	−14.5100			−0.741424			−0.370712	−	0.928748i	$-0.620886\pi$
$383$	−14.5100			−0.741424			−0.370712	+	0.928748i	$0.620886\pi$
$384$		0			0
$385$	−14.0613	+	5.21233i	−0.716632	+	0.265645i
$386$		0			0
$387$		−	5.63935i		−	0.286664i
$388$		0			0
$389$	6.86183			0.347909			0.173954	−	0.984754i	$-0.444345\pi$
$389$	6.86183			0.347909			0.173954	+	0.984754i	$0.444345\pi$
$390$		0			0
$391$	5.99420			0.303140
$392$		0			0
$393$	13.5425			0.683129
$394$		0			0
$395$	1.73371			0.0872325
$396$		0			0
$397$		−	16.5352i		−	0.829877i	−0.909849	−	0.414939i	$-0.863803\pi$
$397$		−	16.5352i		−	0.829877i	0.909849	−	0.414939i	$-0.136197\pi$
$398$		0			0
$399$	−4.80174	+	1.77993i	−0.240387	+	0.0891081i
$400$		0			0
$401$	27.9269			1.39460			0.697300	−	0.716779i	$-0.254384\pi$
$401$	27.9269			1.39460			0.697300	+	0.716779i	$0.254384\pi$
$402$		0			0
$403$			15.8778i			0.790932i
$404$		0			0
$405$			5.43214i			0.269925i
$406$		0			0
$407$			63.7444i			3.15969i
$408$		0			0
$409$		−	0.782697i		−	0.0387019i	−0.999813	−	0.0193509i	$-0.993840\pi$
$409$		−	0.782697i		−	0.0387019i	0.999813	−	0.0193509i	$-0.00615998\pi$
$410$		0			0
$411$	−29.4243			−1.45139
$412$		0			0
$413$	13.2072	−	4.89572i	0.649884	−	0.240903i
$414$		0			0
$415$		−	12.2384i		−	0.600757i
$416$		0			0
$417$	29.8449			1.46151
$418$		0			0
$419$	35.5686			1.73764			0.868820	−	0.495129i	$-0.164879\pi$
$419$	35.5686			1.73764			0.868820	+	0.495129i	$0.164879\pi$
$420$		0			0
$421$	−18.1499			−0.884571			−0.442286	−	0.896874i	$-0.645832\pi$
$421$	−18.1499			−0.884571			−0.442286	+	0.896874i	$0.645832\pi$
$422$		0			0
$423$	−7.07679			−0.344085
$424$		0			0
$425$			1.65490i			0.0802746i
$426$		0			0
$427$	2.12357	+	5.72875i	0.102767	+	0.277234i
$428$		0			0
$429$	42.1461			2.03483
$430$		0			0
$431$			25.2831i			1.21784i	0.793230	+	0.608922i	$0.208398\pi$
$431$			25.2831i			1.21784i	−0.793230	+	0.608922i	$0.791602\pi$
$432$		0			0
$433$		−	5.25385i		−	0.252484i	−0.991999	−	0.126242i	$-0.959708\pi$
$433$		−	5.25385i		−	0.252484i	0.991999	−	0.126242i	$-0.0402916\pi$
$434$		0			0
$435$			0.0696828i			0.00334103i
$436$		0			0
$437$			4.85180i			0.232093i
$438$		0			0
$439$	2.80378			0.133817			0.0669086	−	0.997759i	$-0.478686\pi$
$439$	2.80378			0.133817			0.0669086	+	0.997759i	$0.478686\pi$
$440$		0			0
$441$	−4.84165	+	4.16124i	−0.230555	+	0.198154i
$442$		0			0
$443$		−	12.0548i		−	0.572739i	−0.958119	−	0.286369i	$-0.907552\pi$
$443$		−	12.0548i		−	0.572739i	0.958119	−	0.286369i	$-0.0924484\pi$
$444$		0			0
$445$	4.28261			0.203015
$446$		0			0
$447$	15.0821			0.713357
$448$		0			0
$449$	7.59847			0.358594			0.179297	−	0.983795i	$-0.442618\pi$
$449$	7.59847			0.358594			0.179297	+	0.983795i	$0.442618\pi$
$450$		0			0
$451$	−13.4837			−0.634923
$452$		0			0
$453$		−	7.24636i		−	0.340464i
$454$		0			0
$455$	−12.7659	+	4.73212i	−0.598473	+	0.221845i
$456$		0			0
$457$	−13.4671			−0.629966			−0.314983	−	0.949097i	$-0.601999\pi$
$457$	−13.4671			−0.629966			−0.314983	+	0.949097i	$0.601999\pi$
$458$		0			0
$459$		−	9.35485i		−	0.436647i
$460$		0			0
$461$			11.3606i			0.529117i	0.964370	+	0.264558i	$0.0852262\pi$
$461$			11.3606i			0.529117i	−0.964370	+	0.264558i	$0.914774\pi$
$462$		0			0
$463$			2.25805i			0.104940i	0.998622	+	0.0524702i	$0.0167095\pi$
$463$			2.25805i			0.104940i	−0.998622	+	0.0524702i	$0.983291\pi$
$464$		0			0
$465$		−	4.45856i		−	0.206761i
$466$		0			0
$467$	−28.9266			−1.33856			−0.669282	−	0.743009i	$-0.733398\pi$
$467$	−28.9266			−1.33856			−0.669282	+	0.743009i	$0.733398\pi$
$468$		0			0
$469$	−0.191115	−	0.515571i	−0.00882486	−	0.0238069i
$470$		0			0
$471$			0.415940i			0.0191655i
$472$		0			0
$473$	35.0476			1.61149
$474$		0			0
$475$	−1.33951			−0.0614607
$476$		0			0
$477$	9.57877			0.438582
$478$		0			0
$479$	0.476369			0.0217659			0.0108829	−	0.999941i	$-0.496536\pi$
$479$	0.476369			0.0217659			0.0108829	+	0.999941i	$0.496536\pi$
$480$		0			0
$481$			57.8717i			2.63872i
$482$		0			0
$483$	−4.81302	−	12.9841i	−0.219000	−	0.590798i
$484$		0			0
$485$	−12.1226			−0.550459
$486$		0			0
$487$		−	12.2827i		−	0.556582i	−0.960497	−	0.278291i	$-0.910232\pi$
$487$		−	12.2827i		−	0.556582i	0.960497	−	0.278291i	$-0.0897679\pi$
$488$		0			0
$489$		−	11.5995i		−	0.524546i
$490$		0			0
$491$		−	1.06404i		−	0.0480195i	−0.999712	−	0.0240098i	$-0.992357\pi$
$491$		−	1.06404i		−	0.0480195i	0.999712	−	0.0240098i	$-0.00764328\pi$
$492$		0			0
$493$		−	0.0798060i		−	0.00359428i
$494$		0			0
$495$	5.16942			0.232348
$496$		0			0
$497$	6.13797	+	16.5584i	0.275326	+	0.742748i
$498$		0			0
$499$			16.3062i			0.729968i	0.931014	+	0.364984i	$0.118926\pi$
$499$			16.3062i			0.729968i	−0.931014	+	0.364984i	$0.881074\pi$
$500$		0			0
$501$	2.52851			0.112966
$502$		0			0
$503$	−0.483813			−0.0215721			−0.0107861	−	0.999942i	$-0.503433\pi$
$503$	−0.483813			−0.0215721			−0.0107861	+	0.999942i	$0.503433\pi$
$504$		0			0
$505$	12.4565			0.554307
$506$		0			0
$507$	19.4784			0.865068
$508$		0			0
$509$		−	38.0294i		−	1.68562i	−0.538209	−	0.842811i	$-0.680899\pi$
$509$		−	38.0294i		−	1.68562i	0.538209	−	0.842811i	$-0.319101\pi$
$510$		0			0
$511$	−3.14908	−	8.49528i	−0.139307	−	0.375809i
$512$		0			0
$513$	7.57197			0.334311
$514$		0			0
$515$			8.57029i			0.377652i
$516$		0			0
$517$		−	43.9811i		−	1.93428i
$518$		0			0
$519$			27.5360i			1.20870i
$520$		0			0
$521$			39.2374i			1.71902i	0.511119	+	0.859510i	$0.329231\pi$
$521$			39.2374i			1.71902i	−0.511119	+	0.859510i	$0.670769\pi$
$522$		0			0
$523$	25.4346			1.11218			0.556088	−	0.831123i	$-0.312302\pi$
$523$	25.4346			1.11218			0.556088	+	0.831123i	$0.312302\pi$
$524$		0			0
$525$	3.58471	−	1.32880i	0.156449	−	0.0579935i
$526$		0			0
$527$			5.10628i			0.222433i
$528$		0			0
$529$	9.88050			0.429587
$530$		0			0
$531$	−4.85541			−0.210707
$532$		0			0
$533$	−12.2415			−0.530237
$534$		0			0
$535$	−18.1310			−0.783872
$536$		0			0
$537$		−	11.8795i		−	0.512637i
$538$		0			0
$539$	−25.8614	−	30.0901i	−1.11393	−	1.29607i
$540$		0			0
$541$	−43.2141			−1.85792			−0.928960	−	0.370181i	$-0.879296\pi$
$541$	−43.2141			−1.85792			−0.928960	+	0.370181i	$0.879296\pi$
$542$		0			0
$543$		−	37.1447i		−	1.59403i
$544$		0			0
$545$			13.9272i			0.596575i
$546$		0			0
$547$		−	8.50599i		−	0.363690i	−0.983327	−	0.181845i	$-0.941793\pi$
$547$		−	8.50599i		−	0.363690i	0.983327	−	0.181845i	$-0.0582069\pi$
$548$		0			0
$549$		−	2.10608i		−	0.0898854i
$550$		0			0
$551$	0.0645963			0.00275189
$552$		0			0
$553$	1.59431	+	4.30098i	0.0677970	+	0.182896i
$554$		0			0
$555$		−	16.2506i		−	0.689800i
$556$		0			0
$557$	16.9999			0.720309			0.360155	−	0.932893i	$-0.382724\pi$
$557$	16.9999			0.720309			0.360155	+	0.932893i	$0.382724\pi$
$558$		0			0
$559$	31.8187			1.34579
$560$		0			0
$561$	13.5541			0.572255
$562$		0			0
$563$	−8.94755			−0.377094			−0.188547	−	0.982064i	$-0.560378\pi$
$563$	−8.94755			−0.377094			−0.188547	+	0.982064i	$0.560378\pi$
$564$		0			0
$565$			5.13757i			0.216139i
$566$		0			0
$567$	−13.4760	+	4.99537i	−0.565940	+	0.209786i
$568$		0			0
$569$	19.8378			0.831644			0.415822	−	0.909446i	$-0.363494\pi$
$569$	19.8378			0.831644			0.415822	+	0.909446i	$0.363494\pi$
$570$		0			0
$571$		−	31.9383i		−	1.33657i	−0.743903	−	0.668287i	$-0.767028\pi$
$571$		−	31.9383i		−	1.33657i	0.743903	−	0.668287i	$-0.232972\pi$
$572$		0			0
$573$			15.1071i			0.631109i
$574$		0			0
$575$		−	3.62209i		−	0.151051i
$576$		0			0
$577$		−	31.1339i		−	1.29612i	−0.761589	−	0.648061i	$-0.775580\pi$
$577$		−	31.1339i		−	1.29612i	0.761589	−	0.648061i	$-0.224420\pi$
$578$		0			0
$579$	30.5077			1.26786
$580$		0			0
$581$	30.3609	−	11.2543i	1.25958	−	0.466908i
$582$		0			0
$583$			59.5305i			2.46550i
$584$		0			0
$585$	4.69316			0.194038
$586$		0			0
$587$	15.4665			0.638371			0.319186	−	0.947692i	$-0.396591\pi$
$587$	15.4665			0.638371			0.319186	+	0.947692i	$0.396591\pi$
$588$		0			0
$589$	−4.13311			−0.170302
$590$		0			0
$591$	−40.3469			−1.65965
$592$		0			0
$593$			13.3740i			0.549204i	0.961558	+	0.274602i	$0.0885461\pi$
$593$			13.3740i			0.549204i	−0.961558	+	0.274602i	$0.911454\pi$
$594$		0			0
$595$	−4.10548	+	1.52184i	−0.168308	+	0.0623894i
$596$		0			0
$597$	30.9477			1.26660
$598$		0			0
$599$		−	32.1853i		−	1.31506i	−0.753430	−	0.657528i	$-0.771602\pi$
$599$		−	32.1853i		−	1.31506i	0.753430	−	0.657528i	$-0.228398\pi$
$600$		0			0
$601$		−	27.3375i		−	1.11512i	−0.830136	−	0.557561i	$-0.811737\pi$
$601$		−	27.3375i		−	1.11512i	0.830136	−	0.557561i	$-0.188263\pi$
$602$		0			0
$603$			0.189541i			0.00771872i
$604$		0			0
$605$			21.1271i			0.858937i
$606$		0			0
$607$	30.3275			1.23095			0.615477	−	0.788155i	$-0.288963\pi$
$607$	30.3275			1.23095			0.615477	+	0.788155i	$0.288963\pi$
$608$		0			0
$609$	−0.172869	+	0.0640799i	−0.00700500	+	0.00259665i
$610$		0			0
$611$		−	39.9291i		−	1.61536i
$612$		0			0
$613$	−2.89751			−0.117029			−0.0585146	−	0.998287i	$-0.518636\pi$
$613$	−2.89751			−0.117029			−0.0585146	+	0.998287i	$0.518636\pi$
$614$		0			0
$615$	3.43745			0.138612
$616$		0			0
$617$	21.1763			0.852526			0.426263	−	0.904599i	$-0.359830\pi$
$617$	21.1763			0.852526			0.426263	+	0.904599i	$0.359830\pi$
$618$		0			0
$619$	−31.4589			−1.26444			−0.632220	−	0.774789i	$-0.717856\pi$
$619$	−31.4589			−1.26444			−0.632220	+	0.774789i	$0.717856\pi$
$620$		0			0
$621$			20.4750i			0.821632i
$622$		0			0
$623$	3.93826	+	10.6243i	0.157783	+	0.425653i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			10.9709i			0.438136i
$628$		0			0
$629$			18.6114i			0.742086i
$630$		0			0
$631$		−	24.3798i		−	0.970545i	−0.874363	−	0.485273i	$-0.838720\pi$
$631$		−	24.3798i		−	0.970545i	0.874363	−	0.485273i	$-0.161280\pi$
$632$		0			0
$633$		−	2.25899i		−	0.0897868i
$634$		0			0
$635$	−17.1695			−0.681353
$636$		0			0
$637$	−23.4788	−	27.3179i	−0.930265	−	1.08237i
$638$		0			0
$639$		−	6.08744i		−	0.240815i
$640$		0			0
$641$	−6.42059			−0.253598			−0.126799	−	0.991928i	$-0.540470\pi$
$641$	−6.42059			−0.253598			−0.126799	+	0.991928i	$0.540470\pi$
$642$		0			0
$643$	−21.3031			−0.840111			−0.420056	−	0.907498i	$-0.637989\pi$
$643$	−21.3031			−0.840111			−0.420056	+	0.907498i	$0.637989\pi$
$644$		0			0
$645$	−8.93482			−0.351808
$646$		0			0
$647$	−32.0503			−1.26003			−0.630014	−	0.776584i	$-0.716951\pi$
$647$	−32.0503			−1.26003			−0.630014	+	0.776584i	$0.716951\pi$
$648$		0			0
$649$		−	30.1756i		−	1.18450i
$650$		0			0
$651$	11.0608	−	4.10007i	0.433506	−	0.160694i
$652$		0			0
$653$	20.8031			0.814088			0.407044	−	0.913408i	$-0.366560\pi$
$653$	20.8031			0.814088			0.407044	+	0.913408i	$0.366560\pi$
$654$		0			0
$655$			9.37209i			0.366198i
$656$		0			0
$657$			3.12315i			0.121846i
$658$		0			0
$659$		−	24.0443i		−	0.936634i	−0.883561	−	0.468317i	$-0.844861\pi$
$659$		−	24.0443i		−	0.936634i	0.883561	−	0.468317i	$-0.155139\pi$
$660$		0			0
$661$			3.26973i			0.127178i	0.997976	+	0.0635888i	$0.0202546\pi$
$661$			3.26973i			0.127178i	−0.997976	+	0.0635888i	$0.979745\pi$
$662$		0			0
$663$	12.3054			0.477901
$664$		0			0
$665$	−1.23180	−	3.32304i	−0.0477672	−	0.128862i
$666$		0			0
$667$			0.174671i			0.00676330i
$668$		0			0
$669$	−23.5324			−0.909817
$670$		0			0
$671$	13.0889			0.505293
$672$		0			0
$673$	−3.99925			−0.154160			−0.0770798	−	0.997025i	$-0.524560\pi$
$673$	−3.99925			−0.154160			−0.0770798	+	0.997025i	$0.524560\pi$
$674$		0			0
$675$	−5.65281			−0.217577
$676$		0			0
$677$			9.79714i			0.376535i	0.982118	+	0.188267i	$0.0602871\pi$
$677$			9.79714i			0.376535i	−0.982118	+	0.188267i	$0.939713\pi$
$678$		0			0
$679$	−11.1479	−	30.0737i	−0.427816	−	1.15412i
$680$		0			0
$681$	24.9824			0.957327
$682$		0			0
$683$		−	41.4747i		−	1.58698i	−0.608580	−	0.793492i	$-0.708261\pi$
$683$		−	41.4747i		−	1.58698i	0.608580	−	0.793492i	$-0.291739\pi$
$684$		0			0
$685$		−	20.3631i		−	0.778032i
$686$		0			0
$687$			37.5386i			1.43219i
$688$		0			0
$689$			54.0460i			2.05899i
$690$		0			0
$691$	28.2780			1.07574			0.537872	−	0.843026i	$-0.319228\pi$
$691$	28.2780			1.07574			0.537872	+	0.843026i	$0.319228\pi$
$692$		0			0
$693$	4.75377	+	12.8243i	0.180581	+	0.487153i
$694$		0			0
$695$			20.6541i			0.783456i
$696$		0			0
$697$	−3.93683			−0.149118
$698$		0			0
$699$	12.2392			0.462929
$700$		0			0
$701$	−32.3111			−1.22037			−0.610186	−	0.792258i	$-0.708905\pi$
$701$	−32.3111			−1.22037			−0.610186	+	0.792258i	$0.708905\pi$
$702$		0			0
$703$	−15.0644			−0.568164
$704$		0			0
$705$			11.2123i			0.422278i
$706$		0			0
$707$	11.4549	+	30.9020i	0.430807	+	1.16219i
$708$		0			0
$709$	0.463813			0.0174189			0.00870943	−	0.999962i	$-0.497228\pi$
$709$	0.463813			0.0174189			0.00870943	+	0.999962i	$0.497228\pi$
$710$		0			0
$711$		−	1.58118i		−	0.0592991i
$712$		0			0
$713$		−	11.1761i		−	0.418549i
$714$		0			0
$715$			29.1672i			1.09079i
$716$		0			0
$717$			9.71785i			0.362920i
$718$		0			0
$719$	−0.133597			−0.00498231			−0.00249116	−	0.999997i	$-0.500793\pi$
$719$	−0.133597			−0.00498231			−0.00249116	+	0.999997i	$0.500793\pi$
$720$		0			0
$721$	−21.2611	+	7.88119i	−0.791806	+	0.293511i
$722$		0			0
$723$			35.9085i			1.33545i
$724$		0			0
$725$	−0.0482240			−0.00179099
$726$		0			0
$727$	11.1815			0.414700			0.207350	−	0.978267i	$-0.433516\pi$
$727$	11.1815			0.414700			0.207350	+	0.978267i	$0.433516\pi$
$728$		0			0
$729$	29.4589			1.09107
$730$		0			0
$731$	10.2328			0.378475
$732$		0			0
$733$			39.9801i			1.47670i	0.674418	+	0.738350i	$0.264395\pi$
$733$			39.9801i			1.47670i	−0.674418	+	0.738350i	$0.735605\pi$
$734$		0			0
$735$	6.59295	+	7.67097i	0.243185	+	0.282948i
$736$		0			0
$737$	−1.17797			−0.0433910
$738$		0			0
$739$			16.7898i			0.617622i	0.951123	+	0.308811i	$0.0999311\pi$
$739$			16.7898i			0.617622i	−0.951123	+	0.308811i	$0.900069\pi$
$740$		0			0
$741$			9.96017i			0.365896i
$742$		0			0
$743$			24.1106i			0.884534i	0.896883	+	0.442267i	$0.145826\pi$
$743$			24.1106i			0.884534i	−0.896883	+	0.442267i	$0.854174\pi$
$744$		0			0
$745$			10.4375i			0.382402i
$746$		0			0
$747$	−11.1617			−0.408384
$748$		0			0
$749$	−16.6732	−	44.9793i	−0.609224	−	1.64351i
$750$		0			0
$751$			38.0487i			1.38842i	0.719774	+	0.694208i	$0.244245\pi$
$751$			38.0487i			1.38842i	−0.719774	+	0.694208i	$0.755755\pi$
$752$		0			0
$753$	−19.2387			−0.701096
$754$		0			0
$755$	5.01484			0.182509
$756$		0			0
$757$	1.19151			0.0433060			0.0216530	−	0.999766i	$-0.493107\pi$
$757$	1.19151			0.0433060			0.0216530	+	0.999766i	$0.493107\pi$
$758$		0			0
$759$	−29.6659			−1.07680
$760$		0			0
$761$			10.0562i			0.364538i	0.983249	+	0.182269i	$0.0583441\pi$
$761$			10.0562i			0.364538i	−0.983249	+	0.182269i	$0.941656\pi$
$762$		0			0
$763$	−34.5505	+	12.8074i	−1.25081	+	0.463658i
$764$		0			0
$765$	1.50931			0.0545692
$766$		0			0
$767$		−	27.3955i		−	0.989195i
$768$		0			0
$769$		−	31.1778i		−	1.12430i	−0.827035	−	0.562150i	$-0.809974\pi$
$769$		−	31.1778i		−	1.12430i	0.827035	−	0.562150i	$-0.190026\pi$
$770$		0			0
$771$			35.9785i			1.29574i
$772$		0			0
$773$		−	40.4974i		−	1.45659i	−0.685264	−	0.728294i	$-0.740313\pi$
$773$		−	40.4974i		−	1.45659i	0.685264	−	0.728294i	$-0.259687\pi$
$774$		0			0
$775$	3.08555			0.110836
$776$		0			0
$777$	40.3144	−	14.9440i	1.44627	−	0.536112i
$778$		0			0
$779$		−	3.18653i		−	0.114169i
$780$		0			0
$781$	37.8324			1.35375
$782$		0			0
$783$	0.272601			0.00974196
$784$		0			0
$785$	−0.287851			−0.0102738
$786$		0			0
$787$	9.24513			0.329553			0.164777	−	0.986331i	$-0.447310\pi$
$787$	9.24513			0.329553			0.164777	+	0.986331i	$0.447310\pi$
$788$		0			0
$789$		−	3.90335i		−	0.138963i
$790$		0			0
$791$	−12.7453	+	4.72448i	−0.453170	+	0.167983i
$792$		0			0
$793$	11.8831			0.421980
$794$		0			0
$795$		−	15.1763i		−	0.538249i
$796$		0			0
$797$			3.51772i			0.124604i	0.998057	+	0.0623020i	$0.0198442\pi$
$797$			3.51772i			0.124604i	−0.998057	+	0.0623020i	$0.980156\pi$
$798$		0			0
$799$		−	12.8411i		−	0.454286i
$800$		0			0
$801$		−	3.90584i		−	0.138006i
$802$		0			0
$803$	−19.4099			−0.684959
$804$		0			0
$805$	8.98565	−	3.33085i	0.316703	−	0.117397i
$806$		0			0
$807$		−	36.1961i		−	1.27416i
$808$		0			0
$809$	−2.36429			−0.0831240			−0.0415620	−	0.999136i	$-0.513233\pi$
$809$	−2.36429			−0.0831240			−0.0415620	+	0.999136i	$0.513233\pi$
$810$		0			0
$811$	38.3645			1.34716			0.673580	−	0.739114i	$-0.264756\pi$
$811$	38.3645			1.34716			0.673580	+	0.739114i	$0.264756\pi$
$812$		0			0
$813$	−20.1759			−0.707601
$814$		0			0
$815$	8.02740			0.281188
$816$		0			0
$817$			8.28262i			0.289772i
$818$		0			0
$819$	4.31580	+	11.6428i	0.150806	+	0.406831i
$820$		0			0
$821$	6.57030			0.229305			0.114652	−	0.993406i	$-0.463425\pi$
$821$	6.57030			0.229305			0.114652	+	0.993406i	$0.463425\pi$
$822$		0			0
$823$			21.8232i			0.760708i	0.924841	+	0.380354i	$0.124198\pi$
$823$			21.8232i			0.760708i	−0.924841	+	0.380354i	$0.875802\pi$
$824$		0			0
$825$		−	8.19027i		−	0.285149i
$826$		0			0
$827$			5.11783i			0.177965i	0.996033	+	0.0889823i	$0.0283614\pi$
$827$			5.11783i			0.177965i	−0.996033	+	0.0889823i	$0.971639\pi$
$828$		0			0
$829$			3.88700i			0.135001i	0.997719	+	0.0675006i	$0.0215025\pi$
$829$			3.88700i			0.135001i	−0.997719	+	0.0675006i	$0.978498\pi$
$830$		0			0
$831$	−19.6679			−0.682272
$832$		0			0
$833$	−7.55075	−	8.78537i	−0.261618	−	0.304395i
$834$		0			0
$835$			1.74986i			0.0605563i
$836$		0			0
$837$	−17.4420			−0.602884
$838$		0			0
$839$	14.4205			0.497851			0.248925	−	0.968523i	$-0.419923\pi$
$839$	14.4205			0.497851			0.248925	+	0.968523i	$0.419923\pi$
$840$		0			0
$841$	−28.9977			−0.999920
$842$		0			0
$843$	−34.8481			−1.20023
$844$		0			0
$845$			13.4800i			0.463727i
$846$		0			0
$847$	−52.4119	+	19.4283i	−1.80090	+	0.667565i
$848$		0			0
$849$	−33.7245			−1.15742
$850$		0			0
$851$		−	40.7348i		−	1.39637i
$852$		0			0
$853$		−	15.0531i		−	0.515408i	−0.966224	−	0.257704i	$-0.917034\pi$
$853$		−	15.0531i		−	0.515408i	0.966224	−	0.257704i	$-0.0829659\pi$
$854$		0			0
$855$			1.22166i			0.0417799i
$856$		0			0
$857$		−	12.0124i		−	0.410335i	−0.978727	−	0.205167i	$-0.934226\pi$
$857$		−	12.0124i		−	0.410335i	0.978727	−	0.205167i	$-0.0657739\pi$
$858$		0			0
$859$	−24.1209			−0.822995			−0.411498	−	0.911411i	$-0.634994\pi$
$859$	−24.1209			−0.822995			−0.411498	+	0.911411i	$0.634994\pi$
$860$		0			0
$861$	3.16106	+	8.52762i	0.107729	+	0.290620i
$862$		0			0
$863$		−	44.3179i		−	1.50860i	−0.656530	−	0.754300i	$-0.727976\pi$
$863$		−	44.3179i		−	1.50860i	0.656530	−	0.754300i	$-0.272024\pi$
$864$		0			0
$865$	−19.0563			−0.647933
$866$		0			0
$867$	−20.6073			−0.699862
$868$		0			0
$869$	9.82680			0.333351
$870$		0			0
$871$	−1.06944			−0.0362367
$872$		0			0
$873$			11.0561i			0.374192i
$874$		0			0
$875$	0.919594	+	2.48080i	0.0310880	+	0.0838662i
$876$		0			0
$877$	29.6049			0.999685			0.499843	−	0.866116i	$-0.333391\pi$
$877$	29.6049			0.999685			0.499843	+	0.866116i	$0.333391\pi$
$878$		0			0
$879$			10.9280i			0.368593i
$880$		0			0
$881$		−	2.62314i		−	0.0883758i	−0.999023	−	0.0441879i	$-0.985930\pi$
$881$		−	2.62314i		−	0.0883758i	0.999023	−	0.0441879i	$-0.0140700\pi$
$882$		0			0
$883$		−	54.4982i		−	1.83401i	−0.398875	−	0.917005i	$-0.630599\pi$
$883$		−	54.4982i		−	1.83401i	0.398875	−	0.917005i	$-0.369401\pi$
$884$		0			0
$885$			7.69277i			0.258590i
$886$		0			0
$887$	−9.43247			−0.316712			−0.158356	−	0.987382i	$-0.550619\pi$
$887$	−9.43247			−0.316712			−0.158356	+	0.987382i	$0.550619\pi$
$888$		0			0
$889$	−15.7890	−	42.5941i	−0.529547	−	1.42856i
$890$		0			0
$891$			30.7898i			1.03150i
$892$		0			0
$893$	10.3938			0.347815
$894$		0			0
$895$	8.22119			0.274804
$896$		0			0
$897$	−26.9328			−0.899259
$898$		0			0
$899$	−0.148797			−0.00496267
$900$		0			0
$901$			17.3811i			0.579048i
$902$		0			0
$903$	−8.21641	−	22.1655i	−0.273425	−	0.737620i
$904$		0			0
$905$	25.7060			0.854495
$906$		0			0
$907$			38.3317i			1.27278i	0.771367	+	0.636391i	$0.219574\pi$
$907$			38.3317i			1.27278i	−0.771367	+	0.636391i	$0.780426\pi$
$908$		0			0
$909$		−	11.3606i		−	0.376808i
$910$		0			0
$911$		−	52.7714i		−	1.74840i	−0.485570	−	0.874198i	$-0.661388\pi$
$911$		−	52.7714i		−	1.74840i	0.485570	−	0.874198i	$-0.338612\pi$
$912$		0			0
$913$		−	69.3679i		−	2.29574i
$914$		0			0
$915$	−3.33681			−0.110312
$916$		0			0
$917$	−23.2502	+	8.61852i	−0.767790	+	0.284609i
$918$		0			0
$919$		−	18.1019i		−	0.597126i	−0.954390	−	0.298563i	$-0.903493\pi$
$919$		−	18.1019i		−	0.597126i	0.954390	−	0.298563i	$-0.0965073\pi$
$920$		0			0
$921$	−36.6700			−1.20832
$922$		0			0
$923$	34.3469			1.13054
$924$		0			0
$925$	11.2462			0.369773
$926$		0			0
$927$	7.81630			0.256721
$928$		0			0
$929$			29.0129i			0.951884i	0.879477	+	0.475942i	$0.157893\pi$
$929$			29.0129i			0.951884i	−0.879477	+	0.475942i	$0.842107\pi$
$930$		0			0
$931$	7.11102	−	6.11170i	0.233054	−	0.200303i
$932$		0			0
$933$	−1.35711			−0.0444299
$934$		0			0
$935$			9.38012i			0.306763i
$936$		0			0
$937$			47.2033i			1.54207i	0.636795	+	0.771033i	$0.280260\pi$
$937$			47.2033i			1.54207i	−0.636795	+	0.771033i	$0.719740\pi$
$938$		0			0
$939$		−	28.9918i		−	0.946111i
$940$		0			0
$941$			9.79631i			0.319350i	0.987170	+	0.159675i	$0.0510447\pi$
$941$			9.79631i			0.319350i	−0.987170	+	0.159675i	$0.948955\pi$
$942$		0			0
$943$	8.61654			0.280593
$944$		0			0
$945$	−5.19829	−	14.0235i	−0.169100	−	0.456183i
$946$		0			0
$947$		−	16.3789i		−	0.532243i	−0.963939	−	0.266122i	$-0.914258\pi$
$947$		−	16.3789i		−	0.532243i	0.963939	−	0.266122i	$-0.0857423\pi$
$948$		0			0
$949$	−17.6216			−0.572022
$950$		0			0
$951$	20.2283			0.655946
$952$		0			0
$953$	−26.2951			−0.851780			−0.425890	−	0.904775i	$-0.640039\pi$
$953$	−26.2951			−0.851780			−0.425890	+	0.904775i	$0.640039\pi$
$954$		0			0
$955$	−10.4549			−0.338312
$956$		0			0
$957$			0.394967i			0.0127675i
$958$		0			0
$959$	50.5166	−	18.7258i	1.63127	−	0.604686i
$960$		0			0
$961$	−21.4794			−0.692884
$962$		0			0
$963$			16.5359i			0.532862i
$964$		0			0
$965$			21.1128i			0.679645i
$966$		0			0
$967$			12.9593i			0.416742i	0.978050	+	0.208371i	$0.0668161\pi$
$967$			12.9593i			0.416742i	−0.978050	+	0.208371i	$0.933184\pi$
$968$		0			0
$969$			3.20317i			0.102901i
$970$		0			0
$971$	58.7681			1.88596			0.942979	−	0.332853i	$-0.108011\pi$
$971$	58.7681			1.88596			0.942979	+	0.332853i	$0.108011\pi$
$972$		0			0
$973$	−51.2387	+	18.9934i	−1.64264	+	0.608901i
$974$		0			0
$975$		−	7.43570i		−	0.238133i
$976$		0			0
$977$	25.6628			0.821026			0.410513	−	0.911855i	$-0.365350\pi$
$977$	25.6628			0.821026			0.410513	+	0.911855i	$0.365350\pi$
$978$		0			0
$979$	24.2742			0.775805
$980$		0			0
$981$	12.7019			0.405541
$982$		0			0
$983$	27.8771			0.889142			0.444571	−	0.895744i	$-0.353356\pi$
$983$	27.8771			0.889142			0.444571	+	0.895744i	$0.353356\pi$
$984$		0			0
$985$		−	27.9221i		−	0.889671i
$986$		0			0
$987$	−27.8153	+	10.3107i	−0.885371	+	0.328194i
$988$		0			0
$989$	−22.3966			−0.712170
$990$		0			0
$991$			13.0915i			0.415865i	0.978143	+	0.207933i	$0.0666735\pi$
$991$			13.0915i			0.415865i	−0.978143	+	0.207933i	$0.933327\pi$
$992$		0			0
$993$		−	24.6937i		−	0.783632i
$994$		0			0
$995$			21.4173i			0.678974i
$996$		0			0
$997$		−	45.8336i		−	1.45157i	−0.687924	−	0.725783i	$-0.741478\pi$
$997$		−	45.8336i		−	1.45157i	0.687924	−	0.725783i	$-0.258522\pi$
$998$		0			0
$999$	−63.5727			−2.01135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.f.1791.5		16
4.3	odd	2		2240.2.k.g.1791.11		16
7.6	odd	2		2240.2.k.g.1791.12		16
8.3	odd	2		1120.2.k.b.671.6	yes	16
8.5	even	2		1120.2.k.a.671.12	yes	16
28.27	even	2	inner	2240.2.k.f.1791.6		16
56.13	odd	2		1120.2.k.b.671.5	yes	16
56.27	even	2		1120.2.k.a.671.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.k.a.671.11	✓	16	56.27	even	2
1120.2.k.a.671.12	yes	16	8.5	even	2
1120.2.k.b.671.5	yes	16	56.13	odd	2
1120.2.k.b.671.6	yes	16	8.3	odd	2
2240.2.k.f.1791.5		16	1.1	even	1	trivial
2240.2.k.f.1791.6		16	28.27	even	2	inner
2240.2.k.g.1791.11		16	4.3	odd	2
2240.2.k.g.1791.12		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)

\(f(q)\)	\(=\)	\(q-1.44498 q^{3} -1.00000i q^{5} +(2.48080 - 0.919594i) q^{7} -0.912023 q^{9} +O(q^{10})\)	\(q-1.44498 q^{3} -1.00000i q^{5} +(2.48080 - 0.919594i) q^{7} -0.912023 q^{9} -5.66807i q^{11} -5.14588i q^{13} +1.44498i q^{15} -1.65490i q^{17} +1.33951 q^{19} +(-3.58471 + 1.32880i) q^{21} +3.62209i q^{23} -1.00000 q^{25} +5.65281 q^{27} +0.0482240 q^{29} -3.08555 q^{31} +8.19027i q^{33} +(-0.919594 - 2.48080i) q^{35} -11.2462 q^{37} +7.43570i q^{39} -2.37889i q^{41} +6.18334i q^{43} +0.912023i q^{45} +7.75943 q^{47} +(5.30869 - 4.56265i) q^{49} +2.39131i q^{51} -10.5028 q^{53} -5.66807 q^{55} -1.93556 q^{57} +5.32378 q^{59} +2.30924i q^{61} +(-2.26254 + 0.838692i) q^{63} -5.14588 q^{65} -0.207825i q^{67} -5.23385i q^{69} +6.67465i q^{71} -3.42442i q^{73} +1.44498 q^{75} +(-5.21233 - 14.0613i) q^{77} +1.73371i q^{79} -5.43214 q^{81} +12.2384 q^{83} -1.65490 q^{85} -0.0696828 q^{87} +4.28261i q^{89} +(-4.73212 - 12.7659i) q^{91} +4.45856 q^{93} -1.33951i q^{95} -12.1226i q^{97} +5.16942i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more