LMFDB - Embedded newform 2240.2.k.g.1791.16

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			1791.16
Root			$-0.00830917 - 1.38001i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.g.1791.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76002 q^{3} +1.00000i q^{5} +(2.62068 + 0.363381i) q^{7} +4.61769 q^{9} +O(q^{10})$	$q+2.76002 q^{3} +1.00000i q^{5} +(2.62068 + 0.363381i) q^{7} +4.61769 q^{9} -3.56590i q^{11} -0.710144i q^{13} +2.76002i q^{15} -5.25797i q^{17} +7.01944 q^{19} +(7.23312 + 1.00294i) q^{21} +1.77809i q^{23} -1.00000 q^{25} +4.46486 q^{27} -4.08118 q^{29} -1.18814 q^{31} -9.84194i q^{33} +(-0.363381 + 2.62068i) q^{35} -5.91299 q^{37} -1.96001i q^{39} +7.71105i q^{41} -6.89299i q^{43} +4.61769i q^{45} +6.90271 q^{47} +(6.73591 + 1.90461i) q^{49} -14.5121i q^{51} -0.578094 q^{53} +3.56590 q^{55} +19.3738 q^{57} -6.12636 q^{59} -8.34459i q^{61} +(12.1015 + 1.67798i) q^{63} +0.710144 q^{65} +11.2249i q^{67} +4.90755i q^{69} +15.9354i q^{71} +2.37777i q^{73} -2.76002 q^{75} +(1.29578 - 9.34508i) q^{77} -10.5062i q^{79} -1.52999 q^{81} -16.8366 q^{83} +5.25797 q^{85} -11.2641 q^{87} +14.5147i q^{89} +(0.258053 - 1.86106i) q^{91} -3.27930 q^{93} +7.01944i q^{95} +4.55712i q^{97} -16.4662i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.76002			1.59350			0.796748	−	0.604311i	$-0.206552\pi$
$3$	2.76002			1.59350			0.796748	+	0.604311i	$0.206552\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	2.62068	+	0.363381i	0.990523	+	0.137345i
$7$	2.62068	+	0.363381i	0.990523	+	0.137345i
$8$		0			0
$9$	4.61769			1.53923
$10$		0			0
$11$		−	3.56590i		−	1.07516i	−0.843213	−	0.537580i	$-0.819339\pi$
$11$		−	3.56590i		−	1.07516i	0.843213	−	0.537580i	$-0.180661\pi$
$12$		0			0
$13$		−	0.710144i		−	0.196959i	−0.995139	−	0.0984793i	$-0.968602\pi$
$13$		−	0.710144i		−	0.196959i	0.995139	−	0.0984793i	$-0.0313978\pi$
$14$		0			0
$15$			2.76002i			0.712633i
$16$		0			0
$17$		−	5.25797i		−	1.27525i	−0.770348	−	0.637623i	$-0.779918\pi$
$17$		−	5.25797i		−	1.27525i	0.770348	−	0.637623i	$-0.220082\pi$
$18$		0			0
$19$	7.01944			1.61037			0.805185	−	0.593023i	$-0.202066\pi$
$19$	7.01944			1.61037			0.805185	+	0.593023i	$0.202066\pi$
$20$		0			0
$21$	7.23312	+	1.00294i	1.57840	+	0.218859i
$22$		0			0
$23$			1.77809i			0.370757i	0.982667	+	0.185378i	$0.0593511\pi$
$23$			1.77809i			0.370757i	−0.982667	+	0.185378i	$0.940649\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	4.46486			0.859262
$28$		0			0
$29$	−4.08118			−0.757856			−0.378928	−	0.925426i	$-0.623707\pi$
$29$	−4.08118			−0.757856			−0.378928	+	0.925426i	$0.623707\pi$
$30$		0			0
$31$	−1.18814			−0.213397			−0.106699	−	0.994291i	$-0.534028\pi$
$31$	−1.18814			−0.213397			−0.106699	+	0.994291i	$0.534028\pi$
$32$		0			0
$33$		−	9.84194i		−	1.71326i
$34$		0			0
$35$	−0.363381	+	2.62068i	−0.0614226	+	0.442975i
$36$		0			0
$37$	−5.91299			−0.972089			−0.486045	−	0.873934i	$-0.661561\pi$
$37$	−5.91299			−0.972089			−0.486045	+	0.873934i	$0.661561\pi$
$38$		0			0
$39$		−	1.96001i		−	0.313853i
$40$		0			0
$41$			7.71105i			1.20426i	0.798397	+	0.602132i	$0.205682\pi$
$41$			7.71105i			1.20426i	−0.798397	+	0.602132i	$0.794318\pi$
$42$		0			0
$43$		−	6.89299i		−	1.05117i	−0.850741	−	0.525585i	$-0.823846\pi$
$43$		−	6.89299i		−	1.05117i	0.850741	−	0.525585i	$-0.176154\pi$
$44$		0			0
$45$			4.61769i			0.688365i
$46$		0			0
$47$	6.90271			1.00686			0.503432	−	0.864035i	$-0.332070\pi$
$47$	6.90271			1.00686			0.503432	+	0.864035i	$0.332070\pi$
$48$		0			0
$49$	6.73591	+	1.90461i	0.962273	+	0.272087i
$50$		0			0
$51$		−	14.5121i		−	2.03210i
$52$		0			0
$53$	−0.578094			−0.0794073			−0.0397036	−	0.999211i	$-0.512641\pi$
$53$	−0.578094			−0.0794073			−0.0397036	+	0.999211i	$0.512641\pi$
$54$		0			0
$55$	3.56590			0.480826
$56$		0			0
$57$	19.3738			2.56612
$58$		0			0
$59$	−6.12636			−0.797584			−0.398792	−	0.917041i	$-0.630571\pi$
$59$	−6.12636			−0.797584			−0.398792	+	0.917041i	$0.630571\pi$
$60$		0			0
$61$		−	8.34459i		−	1.06842i	−0.845353	−	0.534208i	$-0.820610\pi$
$61$		−	8.34459i		−	1.06842i	0.845353	−	0.534208i	$-0.179390\pi$
$62$		0			0
$63$	12.1015	+	1.67798i	1.52464	+	0.211406i
$64$		0			0
$65$	0.710144			0.0880825
$66$		0			0
$67$			11.2249i			1.37134i	0.727914	+	0.685668i	$0.240490\pi$
$67$			11.2249i			1.37134i	−0.727914	+	0.685668i	$0.759510\pi$
$68$		0			0
$69$			4.90755i			0.590799i
$70$		0			0
$71$			15.9354i			1.89119i	0.325349	+	0.945594i	$0.394518\pi$
$71$			15.9354i			1.89119i	−0.325349	+	0.945594i	$0.605482\pi$
$72$		0			0
$73$			2.37777i			0.278297i	0.990272	+	0.139149i	$0.0444366\pi$
$73$			2.37777i			0.278297i	−0.990272	+	0.139149i	$0.955563\pi$
$74$		0			0
$75$	−2.76002			−0.318699
$76$		0			0
$77$	1.29578	−	9.34508i	0.147668	−	1.06497i
$78$		0			0
$79$		−	10.5062i		−	1.18204i	−0.806656	−	0.591021i	$-0.798725\pi$
$79$		−	10.5062i		−	1.18204i	0.806656	−	0.591021i	$-0.201275\pi$
$80$		0			0
$81$	−1.52999			−0.169999
$82$		0			0
$83$	−16.8366			−1.84806			−0.924031	−	0.382318i	$-0.875126\pi$
$83$	−16.8366			−1.84806			−0.924031	+	0.382318i	$0.875126\pi$
$84$		0			0
$85$	5.25797			0.570307
$86$		0			0
$87$	−11.2641			−1.20764
$88$		0			0
$89$			14.5147i			1.53855i	0.638916	+	0.769277i	$0.279383\pi$
$89$			14.5147i			1.53855i	−0.638916	+	0.769277i	$0.720617\pi$
$90$		0			0
$91$	0.258053	−	1.86106i	0.0270513	−	0.195092i
$92$		0			0
$93$	−3.27930			−0.340048
$94$		0			0
$95$			7.01944i			0.720180i
$96$		0			0
$97$			4.55712i			0.462706i	0.972870	+	0.231353i	$0.0743152\pi$
$97$			4.55712i			0.462706i	−0.972870	+	0.231353i	$0.925685\pi$
$98$		0			0
$99$		−	16.4662i		−	1.65492i
$100$		0			0
$101$		−	2.26039i		−	0.224917i	−0.993656	−	0.112459i	$-0.964127\pi$
$101$		−	2.26039i		−	0.224917i	0.993656	−	0.112459i	$-0.0358726\pi$
$102$		0			0
$103$	4.56924			0.450220			0.225110	−	0.974333i	$-0.427726\pi$
$103$	4.56924			0.450220			0.225110	+	0.974333i	$0.427726\pi$
$104$		0			0
$105$	−1.00294	+	7.23312i	−0.0978767	+	0.705880i
$106$		0			0
$107$			13.7076i			1.32516i	0.748990	+	0.662581i	$0.230539\pi$
$107$			13.7076i			1.32516i	−0.748990	+	0.662581i	$0.769461\pi$
$108$		0			0
$109$	12.5248			1.19966			0.599828	−	0.800129i	$-0.295236\pi$
$109$	12.5248			1.19966			0.599828	+	0.800129i	$0.295236\pi$
$110$		0			0
$111$	−16.3199			−1.54902
$112$		0			0
$113$	19.3182			1.81730			0.908651	−	0.417556i	$-0.137113\pi$
$113$	19.3182			1.81730			0.908651	+	0.417556i	$0.137113\pi$
$114$		0			0
$115$	−1.77809			−0.165807
$116$		0			0
$117$		−	3.27923i		−	0.303165i
$118$		0			0
$119$	1.91065	−	13.7795i	0.175149	−	1.26316i
$120$		0			0
$121$	−1.71565			−0.155968
$122$		0			0
$123$			21.2826i			1.91899i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			15.6456i			1.38832i	0.719820	+	0.694161i	$0.244225\pi$
$127$			15.6456i			1.38832i	−0.719820	+	0.694161i	$0.755775\pi$
$128$		0			0
$129$		−	19.0248i		−	1.67504i
$130$		0			0
$131$	−10.4368			−0.911870			−0.455935	−	0.890013i	$-0.650695\pi$
$131$	−10.4368			−0.911870			−0.455935	+	0.890013i	$0.650695\pi$
$132$		0			0
$133$	18.3957	+	2.55073i	1.59511	+	0.221177i
$134$		0			0
$135$			4.46486i			0.384274i
$136$		0			0
$137$	5.91901			0.505695			0.252848	−	0.967506i	$-0.418633\pi$
$137$	5.91901			0.505695			0.252848	+	0.967506i	$0.418633\pi$
$138$		0			0
$139$	−8.79248			−0.745768			−0.372884	−	0.927878i	$-0.621631\pi$
$139$	−8.79248			−0.745768			−0.372884	+	0.927878i	$0.621631\pi$
$140$		0			0
$141$	19.0516			1.60443
$142$		0			0
$143$	−2.53230			−0.211762
$144$		0			0
$145$		−	4.08118i		−	0.338923i
$146$		0			0
$147$	18.5912	+	5.25676i	1.53338	+	0.433570i
$148$		0			0
$149$	−11.0342			−0.903956			−0.451978	−	0.892029i	$-0.649281\pi$
$149$	−11.0342			−0.903956			−0.451978	+	0.892029i	$0.649281\pi$
$150$		0			0
$151$		−	18.4025i		−	1.49758i	−0.662809	−	0.748789i	$-0.730636\pi$
$151$		−	18.4025i		−	1.49758i	0.662809	−	0.748789i	$-0.269364\pi$
$152$		0			0
$153$		−	24.2797i		−	1.96290i
$154$		0			0
$155$		−	1.18814i		−	0.0954341i
$156$		0			0
$157$		−	2.62253i		−	0.209301i	−0.994509	−	0.104650i	$-0.966628\pi$
$157$		−	2.62253i		−	0.209301i	0.994509	−	0.104650i	$-0.0333723\pi$
$158$		0			0
$159$	−1.59555			−0.126535
$160$		0			0
$161$	−0.646123	+	4.65979i	−0.0509216	+	0.367243i
$162$		0			0
$163$		−	6.72228i		−	0.526529i	−0.964724	−	0.263265i	$-0.915201\pi$
$163$		−	6.72228i		−	0.526529i	0.964724	−	0.263265i	$-0.0847993\pi$
$164$		0			0
$165$	9.84194			0.766194
$166$		0			0
$167$	−23.8207			−1.84330			−0.921650	−	0.388023i	$-0.873158\pi$
$167$	−23.8207			−1.84330			−0.921650	+	0.388023i	$0.873158\pi$
$168$		0			0
$169$	12.4957			0.961207
$170$		0			0
$171$	32.4136			2.47873
$172$		0			0
$173$		−	2.58227i		−	0.196326i	−0.995170	−	0.0981632i	$-0.968703\pi$
$173$		−	2.58227i		−	0.196326i	0.995170	−	0.0981632i	$-0.0312967\pi$
$174$		0			0
$175$	−2.62068	−	0.363381i	−0.198105	−	0.0274690i
$176$		0			0
$177$	−16.9089			−1.27095
$178$		0			0
$179$		−	15.0289i		−	1.12331i	−0.827370	−	0.561657i	$-0.810164\pi$
$179$		−	15.0289i		−	1.12331i	0.827370	−	0.561657i	$-0.189836\pi$
$180$		0			0
$181$			22.9157i			1.70331i	0.524102	+	0.851656i	$0.324401\pi$
$181$			22.9157i			1.70331i	−0.524102	+	0.851656i	$0.675599\pi$
$182$		0			0
$183$		−	23.0312i		−	1.70252i
$184$		0			0
$185$		−	5.91299i		−	0.434732i
$186$		0			0
$187$	−18.7494			−1.37109
$188$		0			0
$189$	11.7010	+	1.62245i	0.851119	+	0.118016i
$190$		0			0
$191$			19.0445i			1.37801i	0.724755	+	0.689007i	$0.241953\pi$
$191$			19.0445i			1.37801i	−0.724755	+	0.689007i	$0.758047\pi$
$192$		0			0
$193$	1.53605			0.110567			0.0552837	−	0.998471i	$-0.482394\pi$
$193$	1.53605			0.110567			0.0552837	+	0.998471i	$0.482394\pi$
$194$		0			0
$195$	1.96001			0.140359
$196$		0			0
$197$	−5.80357			−0.413487			−0.206744	−	0.978395i	$-0.566287\pi$
$197$	−5.80357			−0.413487			−0.206744	+	0.978395i	$0.566287\pi$
$198$		0			0
$199$	−18.2990			−1.29718			−0.648592	−	0.761137i	$-0.724642\pi$
$199$	−18.2990			−1.29718			−0.648592	+	0.761137i	$0.724642\pi$
$200$		0			0
$201$			30.9808i			2.18522i
$202$		0			0
$203$	−10.6955	−	1.48302i	−0.750674	−	0.104088i
$204$		0			0
$205$	−7.71105			−0.538563
$206$		0			0
$207$			8.21066i			0.570680i
$208$		0			0
$209$		−	25.0306i		−	1.73141i
$210$		0			0
$211$			4.60086i			0.316736i	0.987380	+	0.158368i	$0.0506233\pi$
$211$			4.60086i			0.316736i	−0.987380	+	0.158368i	$0.949377\pi$
$212$		0			0
$213$			43.9821i			3.01360i
$214$		0			0
$215$	6.89299			0.470098
$216$		0			0
$217$	−3.11375	−	0.431749i	−0.211375	−	0.0293091i
$218$		0			0
$219$			6.56269i			0.443466i
$220$		0			0
$221$	−3.73392			−0.251171
$222$		0			0
$223$	0.336784			0.0225527			0.0112763	−	0.999936i	$-0.496411\pi$
$223$	0.336784			0.0225527			0.0112763	+	0.999936i	$0.496411\pi$
$224$		0			0
$225$	−4.61769			−0.307846
$226$		0			0
$227$	6.22922			0.413448			0.206724	−	0.978399i	$-0.433720\pi$
$227$	6.22922			0.413448			0.206724	+	0.978399i	$0.433720\pi$
$228$		0			0
$229$		−	29.0736i		−	1.92124i	−0.277871	−	0.960618i	$-0.589629\pi$
$229$		−	29.0736i		−	1.92124i	0.277871	−	0.960618i	$-0.410371\pi$
$230$		0			0
$231$	3.57638	−	25.7926i	0.235308	−	1.69703i
$232$		0			0
$233$	−27.5560			−1.80525			−0.902627	−	0.430423i	$-0.858364\pi$
$233$	−27.5560			−1.80525			−0.902627	+	0.430423i	$0.858364\pi$
$234$		0			0
$235$			6.90271i			0.450283i
$236$		0			0
$237$		−	28.9973i		−	1.88358i
$238$		0			0
$239$		−	7.50740i		−	0.485614i	−0.970075	−	0.242807i	$-0.921932\pi$
$239$		−	7.50740i		−	0.485614i	0.970075	−	0.242807i	$-0.0780681\pi$
$240$		0			0
$241$			20.7284i			1.33523i	0.744505	+	0.667616i	$0.232685\pi$
$241$			20.7284i			1.33523i	−0.744505	+	0.667616i	$0.767315\pi$
$242$		0			0
$243$	−17.6174			−1.13016
$244$		0			0
$245$	−1.90461	+	6.73591i	−0.121681	+	0.430341i
$246$		0			0
$247$		−	4.98482i		−	0.317176i
$248$		0			0
$249$	−46.4694			−2.94488
$250$		0			0
$251$	8.92973			0.563640			0.281820	−	0.959467i	$-0.409062\pi$
$251$	8.92973			0.563640			0.281820	+	0.959467i	$0.409062\pi$
$252$		0			0
$253$	6.34048			0.398623
$254$		0			0
$255$	14.5121			0.908783
$256$		0			0
$257$		−	19.8217i		−	1.23644i	−0.786005	−	0.618221i	$-0.787854\pi$
$257$		−	19.8217i		−	1.23644i	0.786005	−	0.618221i	$-0.212146\pi$
$258$		0			0
$259$	−15.4960	−	2.14867i	−0.962877	−	0.133512i
$260$		0			0
$261$	−18.8456			−1.16651
$262$		0			0
$263$		−	14.3532i		−	0.885057i	−0.896754	−	0.442529i	$-0.854081\pi$
$263$		−	14.3532i		−	0.885057i	0.896754	−	0.442529i	$-0.145919\pi$
$264$		0			0
$265$		−	0.578094i		−	0.0355120i
$266$		0			0
$267$			40.0608i			2.45168i
$268$		0			0
$269$		−	24.0018i		−	1.46342i	−0.681618	−	0.731708i	$-0.738723\pi$
$269$		−	24.0018i		−	1.46342i	0.681618	−	0.731708i	$-0.261277\pi$
$270$		0			0
$271$	−16.9425			−1.02918			−0.514591	−	0.857436i	$-0.672056\pi$
$271$	−16.9425			−1.02918			−0.514591	+	0.857436i	$0.672056\pi$
$272$		0			0
$273$	0.712231	−	5.13655i	0.0431062	−	0.310878i
$274$		0			0
$275$			3.56590i			0.215032i
$276$		0			0
$277$	28.6804			1.72324			0.861619	−	0.507556i	$-0.169451\pi$
$277$	28.6804			1.72324			0.861619	+	0.507556i	$0.169451\pi$
$278$		0			0
$279$	−5.48649			−0.328467
$280$		0			0
$281$	10.8308			0.646112			0.323056	−	0.946380i	$-0.395290\pi$
$281$	10.8308			0.646112			0.323056	+	0.946380i	$0.395290\pi$
$282$		0			0
$283$	5.86572			0.348681			0.174340	−	0.984685i	$-0.444221\pi$
$283$	5.86572			0.348681			0.174340	+	0.984685i	$0.444221\pi$
$284$		0			0
$285$			19.3738i			1.14760i
$286$		0			0
$287$	−2.80205	+	20.2082i	−0.165400	+	1.19285i
$288$		0			0
$289$	−10.6463			−0.626253
$290$		0			0
$291$			12.5777i			0.737320i
$292$		0			0
$293$			6.29381i			0.367688i	0.982955	+	0.183844i	$0.0588542\pi$
$293$			6.29381i			0.367688i	−0.982955	+	0.183844i	$0.941146\pi$
$294$		0			0
$295$		−	6.12636i		−	0.356691i
$296$		0			0
$297$		−	15.9212i		−	0.923844i
$298$		0			0
$299$	1.26270			0.0730237
$300$		0			0
$301$	2.50478	−	18.0643i	0.144373	−	1.04121i
$302$		0			0
$303$		−	6.23872i		−	0.358405i
$304$		0			0
$305$	8.34459			0.477810
$306$		0			0
$307$	−1.98374			−0.113218			−0.0566091	−	0.998396i	$-0.518029\pi$
$307$	−1.98374			−0.113218			−0.0566091	+	0.998396i	$0.518029\pi$
$308$		0			0
$309$	12.6112			0.717424
$310$		0			0
$311$	4.80098			0.272238			0.136119	−	0.990692i	$-0.456537\pi$
$311$	4.80098			0.272238			0.136119	+	0.990692i	$0.456537\pi$
$312$		0			0
$313$			13.5434i			0.765516i	0.923849	+	0.382758i	$0.125026\pi$
$313$			13.5434i			0.765516i	−0.923849	+	0.382758i	$0.874974\pi$
$314$		0			0
$315$	−1.67798	+	12.1015i	−0.0945436	+	0.681841i
$316$		0			0
$317$	−8.95432			−0.502925			−0.251462	−	0.967867i	$-0.580911\pi$
$317$	−8.95432			−0.502925			−0.251462	+	0.967867i	$0.580911\pi$
$318$		0			0
$319$			14.5531i			0.814816i
$320$		0			0
$321$			37.8332i			2.11164i
$322$		0			0
$323$		−	36.9081i		−	2.05362i
$324$		0			0
$325$			0.710144i			0.0393917i
$326$		0			0
$327$	34.5686			1.91165
$328$		0			0
$329$	18.0898	+	2.50832i	0.997322	+	0.138288i
$330$		0			0
$331$			25.7007i			1.41264i	0.707894	+	0.706319i	$0.249645\pi$
$331$			25.7007i			1.41264i	−0.707894	+	0.706319i	$0.750355\pi$
$332$		0			0
$333$	−27.3044			−1.49627
$334$		0			0
$335$	−11.2249			−0.613280
$336$		0			0
$337$	−10.7849			−0.587490			−0.293745	−	0.955884i	$-0.594902\pi$
$337$	−10.7849			−0.587490			−0.293745	+	0.955884i	$0.594902\pi$
$338$		0			0
$339$	53.3185			2.89586
$340$		0			0
$341$			4.23681i			0.229436i
$342$		0			0
$343$	16.9605	+	7.43907i	0.915784	+	0.401672i
$344$		0			0
$345$	−4.90755			−0.264214
$346$		0			0
$347$		−	19.2996i		−	1.03606i	−0.855363	−	0.518029i	$-0.826666\pi$
$347$		−	19.2996i		−	1.03606i	0.855363	−	0.518029i	$-0.173334\pi$
$348$		0			0
$349$		−	14.4291i		−	0.772371i	−0.922421	−	0.386186i	$-0.873792\pi$
$349$		−	14.4291i		−	0.772371i	0.922421	−	0.386186i	$-0.126208\pi$
$350$		0			0
$351$		−	3.17069i		−	0.169239i
$352$		0			0
$353$		−	3.26839i		−	0.173959i	−0.996210	−	0.0869794i	$-0.972279\pi$
$353$		−	3.26839i		−	0.173959i	0.996210	−	0.0869794i	$-0.0277214\pi$
$354$		0			0
$355$	−15.9354			−0.845765
$356$		0			0
$357$	5.27342	−	38.0315i	0.279099	−	2.01284i
$358$		0			0
$359$		−	7.97539i		−	0.420925i	−0.977602	−	0.210462i	$-0.932503\pi$
$359$		−	7.97539i		−	0.420925i	0.977602	−	0.210462i	$-0.0674969\pi$
$360$		0			0
$361$	30.2726			1.59329
$362$		0			0
$363$	−4.73521			−0.248534
$364$		0			0
$365$	−2.37777			−0.124458
$366$		0			0
$367$	21.1841			1.10580			0.552900	−	0.833248i	$-0.313521\pi$
$367$	21.1841			1.10580			0.552900	+	0.833248i	$0.313521\pi$
$368$		0			0
$369$			35.6073i			1.85364i
$370$		0			0
$371$	−1.51500	−	0.210068i	−0.0786548	−	0.0109062i
$372$		0			0
$373$	18.5804			0.962057			0.481029	−	0.876705i	$-0.340263\pi$
$373$	18.5804			0.962057			0.481029	+	0.876705i	$0.340263\pi$
$374$		0			0
$375$		−	2.76002i		−	0.142527i
$376$		0			0
$377$			2.89822i			0.149266i
$378$		0			0
$379$			25.3000i			1.29957i	0.760117	+	0.649786i	$0.225142\pi$
$379$			25.3000i			1.29957i	−0.760117	+	0.649786i	$0.774858\pi$
$380$		0			0
$381$			43.1821i			2.21229i
$382$		0			0
$383$	−3.72136			−0.190153			−0.0950763	−	0.995470i	$-0.530309\pi$
$383$	−3.72136			−0.190153			−0.0950763	+	0.995470i	$0.530309\pi$
$384$		0			0
$385$	9.34508	+	1.29578i	0.476269	+	0.0660391i
$386$		0			0
$387$		−	31.8297i		−	1.61799i
$388$		0			0
$389$	−32.2204			−1.63364			−0.816820	−	0.576893i	$-0.804265\pi$
$389$	−32.2204			−1.63364			−0.816820	+	0.576893i	$0.804265\pi$
$390$		0			0
$391$	9.34913			0.472806
$392$		0			0
$393$	−28.8058			−1.45306
$394$		0			0
$395$	10.5062			0.528625
$396$		0			0
$397$			16.1716i			0.811632i	0.913955	+	0.405816i	$0.133013\pi$
$397$			16.1716i			0.811632i	−0.913955	+	0.405816i	$0.866987\pi$
$398$		0			0
$399$	50.7724	+	7.04007i	2.54180	+	0.352444i
$400$		0			0
$401$	−16.4693			−0.822437			−0.411219	−	0.911537i	$-0.634897\pi$
$401$	−16.4693			−0.822437			−0.411219	+	0.911537i	$0.634897\pi$
$402$		0			0
$403$			0.843754i			0.0420304i
$404$		0			0
$405$		−	1.52999i		−	0.0760260i
$406$		0			0
$407$			21.0851i			1.04515i
$408$		0			0
$409$		−	29.4223i		−	1.45484i	−0.686194	−	0.727419i	$-0.740720\pi$
$409$		−	29.4223i		−	1.45484i	0.686194	−	0.727419i	$-0.259280\pi$
$410$		0			0
$411$	16.3366			0.805823
$412$		0			0
$413$	−16.0552	−	2.22620i	−0.790026	−	0.109544i
$414$		0			0
$415$		−	16.8366i		−	0.826478i
$416$		0			0
$417$	−24.2674			−1.18838
$418$		0			0
$419$	−21.0972			−1.03066			−0.515332	−	0.856991i	$-0.672331\pi$
$419$	−21.0972			−1.03066			−0.515332	+	0.856991i	$0.672331\pi$
$420$		0			0
$421$	−0.808118			−0.0393853			−0.0196926	−	0.999806i	$-0.506269\pi$
$421$	−0.808118			−0.0393853			−0.0196926	+	0.999806i	$0.506269\pi$
$422$		0			0
$423$	31.8746			1.54980
$424$		0			0
$425$			5.25797i			0.255049i
$426$		0			0
$427$	3.03227	−	21.8685i	0.146742	−	1.05829i
$428$		0			0
$429$	−6.98920			−0.337442
$430$		0			0
$431$		−	0.250132i		−	0.0120484i	−0.999982	−	0.00602422i	$-0.998082\pi$
$431$		−	0.250132i		−	0.0120484i	0.999982	−	0.00602422i	$-0.00191758\pi$
$432$		0			0
$433$		−	2.35341i		−	0.113098i	−0.998400	−	0.0565489i	$-0.981990\pi$
$433$		−	2.35341i		−	0.113098i	0.998400	−	0.0565489i	$-0.0180097\pi$
$434$		0			0
$435$		−	11.2641i		−	0.540073i
$436$		0			0
$437$			12.4812i			0.597056i
$438$		0			0
$439$	−31.9950			−1.52704			−0.763520	−	0.645784i	$-0.776531\pi$
$439$	−31.9950			−1.52704			−0.763520	+	0.645784i	$0.776531\pi$
$440$		0			0
$441$	31.1044	+	8.79490i	1.48116	+	0.418805i
$442$		0			0
$443$		−	18.1663i		−	0.863106i	−0.902088	−	0.431553i	$-0.857966\pi$
$443$		−	18.1663i		−	0.863106i	0.902088	−	0.431553i	$-0.142034\pi$
$444$		0			0
$445$	−14.5147			−0.688062
$446$		0			0
$447$	−30.4546			−1.44045
$448$		0			0
$449$	0.348007			0.0164235			0.00821173	−	0.999966i	$-0.497386\pi$
$449$	0.348007			0.0164235			0.00821173	+	0.999966i	$0.497386\pi$
$450$		0			0
$451$	27.4968			1.29478
$452$		0			0
$453$		−	50.7913i		−	2.38638i
$454$		0			0
$455$	1.86106	+	0.258053i	0.0872478	+	0.0120977i
$456$		0			0
$457$	−32.6756			−1.52850			−0.764251	−	0.644919i	$-0.776891\pi$
$457$	−32.6756			−1.52850			−0.764251	+	0.644919i	$0.776891\pi$
$458$		0			0
$459$		−	23.4761i		−	1.09577i
$460$		0			0
$461$			10.4378i			0.486137i	0.970009	+	0.243068i	$0.0781539\pi$
$461$			10.4378i			0.486137i	−0.970009	+	0.243068i	$0.921846\pi$
$462$		0			0
$463$		−	28.6675i		−	1.33229i	−0.745822	−	0.666145i	$-0.767943\pi$
$463$		−	28.6675i		−	1.33229i	0.745822	−	0.666145i	$-0.232057\pi$
$464$		0			0
$465$		−	3.27930i		−	0.152074i
$466$		0			0
$467$	−6.80444			−0.314872			−0.157436	−	0.987529i	$-0.550323\pi$
$467$	−6.80444			−0.314872			−0.157436	+	0.987529i	$0.550323\pi$
$468$		0			0
$469$	−4.07891	+	29.4168i	−0.188346	+	1.35834i
$470$		0			0
$471$		−	7.23822i		−	0.333520i
$472$		0			0
$473$	−24.5797			−1.13018
$474$		0			0
$475$	−7.01944			−0.322074
$476$		0			0
$477$	−2.66946			−0.122226
$478$		0			0
$479$	12.2106			0.557918			0.278959	−	0.960303i	$-0.410011\pi$
$479$	12.2106			0.557918			0.278959	+	0.960303i	$0.410011\pi$
$480$		0			0
$481$			4.19907i			0.191461i
$482$		0			0
$483$	−1.78331	+	12.8611i	−0.0811434	+	0.585201i
$484$		0			0
$485$	−4.55712			−0.206928
$486$		0			0
$487$		−	3.96998i		−	0.179897i	−0.995946	−	0.0899484i	$-0.971330\pi$
$487$		−	3.96998i		−	0.179897i	0.995946	−	0.0899484i	$-0.0286702\pi$
$488$		0			0
$489$		−	18.5536i		−	0.839023i
$490$		0			0
$491$			14.2188i			0.641687i	0.947132	+	0.320843i	$0.103966\pi$
$491$			14.2188i			0.641687i	−0.947132	+	0.320843i	$0.896034\pi$
$492$		0			0
$493$			21.4587i			0.966453i
$494$		0			0
$495$	16.4662			0.740102
$496$		0			0
$497$	−5.79064	+	41.7616i	−0.259746	+	1.87327i
$498$		0			0
$499$			7.97405i			0.356968i	0.983943	+	0.178484i	$0.0571192\pi$
$499$			7.97405i			0.356968i	−0.983943	+	0.178484i	$0.942881\pi$
$500$		0			0
$501$	−65.7455			−2.93729
$502$		0			0
$503$	7.83272			0.349244			0.174622	−	0.984636i	$-0.444130\pi$
$503$	7.83272			0.349244			0.174622	+	0.984636i	$0.444130\pi$
$504$		0			0
$505$	2.26039			0.100586
$506$		0			0
$507$	34.4883			1.53168
$508$		0			0
$509$			34.4801i			1.52830i	0.645036	+	0.764152i	$0.276842\pi$
$509$			34.4801i			1.52830i	−0.645036	+	0.764152i	$0.723158\pi$
$510$		0			0
$511$	−0.864038	+	6.23138i	−0.0382228	+	0.275660i
$512$		0			0
$513$	31.3408			1.38373
$514$		0			0
$515$			4.56924i			0.201345i
$516$		0			0
$517$		−	24.6144i		−	1.08254i
$518$		0			0
$519$		−	7.12711i		−	0.312845i
$520$		0			0
$521$		−	4.62219i		−	0.202502i	−0.994861	−	0.101251i	$-0.967716\pi$
$521$		−	4.62219i		−	0.202502i	0.994861	−	0.101251i	$-0.0322845\pi$
$522$		0			0
$523$	−0.841598			−0.0368005			−0.0184003	−	0.999831i	$-0.505857\pi$
$523$	−0.841598			−0.0368005			−0.0184003	+	0.999831i	$0.505857\pi$
$524$		0			0
$525$	−7.23312	−	1.00294i	−0.315679	−	0.0437718i
$526$		0			0
$527$			6.24724i			0.272134i
$528$		0			0
$529$	19.8384			0.862540
$530$		0			0
$531$	−28.2897			−1.22767
$532$		0			0
$533$	5.47596			0.237190
$534$		0			0
$535$	−13.7076			−0.592631
$536$		0			0
$537$		−	41.4801i		−	1.79000i
$538$		0			0
$539$	6.79165	−	24.0196i	0.292537	−	1.03460i
$540$		0			0
$541$	19.7127			0.847514			0.423757	−	0.905776i	$-0.360711\pi$
$541$	19.7127			0.847514			0.423757	+	0.905776i	$0.360711\pi$
$542$		0			0
$543$			63.2477i			2.71422i
$544$		0			0
$545$			12.5248i			0.536503i
$546$		0			0
$547$			4.70337i			0.201102i	0.994932	+	0.100551i	$0.0320605\pi$
$547$			4.70337i			0.201102i	−0.994932	+	0.100551i	$0.967940\pi$
$548$		0			0
$549$		−	38.5328i		−	1.64454i
$550$		0			0
$551$	−28.6476			−1.22043
$552$		0			0
$553$	3.81776	−	27.5334i	0.162348	−	1.17084i
$554$		0			0
$555$		−	16.3199i		−	0.692743i
$556$		0			0
$557$	−11.6257			−0.492598			−0.246299	−	0.969194i	$-0.579215\pi$
$557$	−11.6257			−0.492598			−0.246299	+	0.969194i	$0.579215\pi$
$558$		0			0
$559$	−4.89501			−0.207037
$560$		0			0
$561$	−51.7487			−2.18483
$562$		0			0
$563$	1.67687			0.0706717			0.0353358	−	0.999375i	$-0.488750\pi$
$563$	1.67687			0.0706717			0.0353358	+	0.999375i	$0.488750\pi$
$564$		0			0
$565$			19.3182i			0.812722i
$566$		0			0
$567$	−4.00962	−	0.555971i	−0.168388	−	0.0233486i
$568$		0			0
$569$	−1.40487			−0.0588952			−0.0294476	−	0.999566i	$-0.509375\pi$
$569$	−1.40487			−0.0588952			−0.0294476	+	0.999566i	$0.509375\pi$
$570$		0			0
$571$		−	3.33515i		−	0.139572i	−0.997562	−	0.0697859i	$-0.977768\pi$
$571$		−	3.33515i		−	0.139572i	0.997562	−	0.0697859i	$-0.0222316\pi$
$572$		0			0
$573$			52.5632i			2.19586i
$574$		0			0
$575$		−	1.77809i		−	0.0741513i
$576$		0			0
$577$		−	32.9986i		−	1.37375i	−0.726775	−	0.686875i	$-0.758982\pi$
$577$		−	32.9986i		−	1.37375i	0.726775	−	0.686875i	$-0.241018\pi$
$578$		0			0
$579$	4.23953			0.176189
$580$		0			0
$581$	−44.1234	−	6.11812i	−1.83055	−	0.253822i
$582$		0			0
$583$			2.06143i			0.0853755i
$584$		0			0
$585$	3.27923			0.135579
$586$		0			0
$587$	25.5667			1.05525			0.527625	−	0.849477i	$-0.323082\pi$
$587$	25.5667			1.05525			0.527625	+	0.849477i	$0.323082\pi$
$588$		0			0
$589$	−8.34011			−0.343648
$590$		0			0
$591$	−16.0180			−0.658891
$592$		0			0
$593$		−	25.9625i		−	1.06615i	−0.846067	−	0.533077i	$-0.821036\pi$
$593$		−	25.9625i		−	1.06615i	0.846067	−	0.533077i	$-0.178964\pi$
$594$		0			0
$595$	13.7795	+	1.91065i	0.564903	+	0.0783290i
$596$		0			0
$597$	−50.5056			−2.06706
$598$		0			0
$599$			9.59648i			0.392102i	0.980594	+	0.196051i	$0.0628118\pi$
$599$			9.59648i			0.392102i	−0.980594	+	0.196051i	$0.937188\pi$
$600$		0			0
$601$		−	9.00881i		−	0.367477i	−0.982975	−	0.183738i	$-0.941180\pi$
$601$		−	9.00881i		−	0.367477i	0.982975	−	0.183738i	$-0.0588199\pi$
$602$		0			0
$603$			51.8330i			2.11080i
$604$		0			0
$605$		−	1.71565i		−	0.0697509i
$606$		0			0
$607$	2.04773			0.0831148			0.0415574	−	0.999136i	$-0.486768\pi$
$607$	2.04773			0.0831148			0.0415574	+	0.999136i	$0.486768\pi$
$608$		0			0
$609$	−29.5196	−	4.09317i	−1.19620	−	0.165864i
$610$		0			0
$611$		−	4.90192i		−	0.198310i
$612$		0			0
$613$	9.01716			0.364199			0.182100	−	0.983280i	$-0.441711\pi$
$613$	9.01716			0.364199			0.182100	+	0.983280i	$0.441711\pi$
$614$		0			0
$615$	−21.2826			−0.858198
$616$		0			0
$617$	13.3988			0.539414			0.269707	−	0.962942i	$-0.413073\pi$
$617$	13.3988			0.539414			0.269707	+	0.962942i	$0.413073\pi$
$618$		0			0
$619$	11.4821			0.461504			0.230752	−	0.973013i	$-0.425881\pi$
$619$	11.4821			0.461504			0.230752	+	0.973013i	$0.425881\pi$
$620$		0			0
$621$			7.93890i			0.318577i
$622$		0			0
$623$	−5.27436	+	38.0383i	−0.211313	+	1.52397i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	69.0850i		−	2.75899i
$628$		0			0
$629$			31.0903i			1.23965i
$630$		0			0
$631$			46.8147i			1.86366i	0.362892	+	0.931831i	$0.381789\pi$
$631$			46.8147i			1.86366i	−0.362892	+	0.931831i	$0.618211\pi$
$632$		0			0
$633$			12.6984i			0.504718i
$634$		0			0
$635$	−15.6456			−0.620877
$636$		0			0
$637$	1.35255	−	4.78346i	0.0535899	−	0.189528i
$638$		0			0
$639$			73.5849i			2.91098i
$640$		0			0
$641$	0.572206			0.0226008			0.0113004	−	0.999936i	$-0.496403\pi$
$641$	0.572206			0.0226008			0.0113004	+	0.999936i	$0.496403\pi$
$642$		0			0
$643$	−22.6282			−0.892370			−0.446185	−	0.894941i	$-0.647218\pi$
$643$	−22.6282			−0.892370			−0.446185	+	0.894941i	$0.647218\pi$
$644$		0			0
$645$	19.0248			0.749099
$646$		0			0
$647$	−8.65956			−0.340442			−0.170221	−	0.985406i	$-0.554448\pi$
$647$	−8.65956			−0.340442			−0.170221	+	0.985406i	$0.554448\pi$
$648$		0			0
$649$			21.8460i			0.857530i
$650$		0			0
$651$	−8.59399	−	1.19164i	−0.336825	−	0.0467039i
$652$		0			0
$653$	−41.4029			−1.62022			−0.810110	−	0.586278i	$-0.800593\pi$
$653$	−41.4029			−1.62022			−0.810110	+	0.586278i	$0.800593\pi$
$654$		0			0
$655$		−	10.4368i		−	0.407801i
$656$		0			0
$657$			10.9798i			0.428364i
$658$		0			0
$659$		−	6.69789i		−	0.260913i	−0.991454	−	0.130456i	$-0.958356\pi$
$659$		−	6.69789i		−	0.260913i	0.991454	−	0.130456i	$-0.0416443\pi$
$660$		0			0
$661$			38.1593i			1.48423i	0.670275	+	0.742113i	$0.266176\pi$
$661$			38.1593i			1.48423i	−0.670275	+	0.742113i	$0.733824\pi$
$662$		0			0
$663$	−10.3057			−0.400239
$664$		0			0
$665$	−2.55073	+	18.3957i	−0.0989132	+	0.713355i
$666$		0			0
$667$		−	7.25669i		−	0.280980i
$668$		0			0
$669$	0.929528			0.0359376
$670$		0			0
$671$	−29.7560			−1.14872
$672$		0			0
$673$	11.6699			0.449840			0.224920	−	0.974377i	$-0.427788\pi$
$673$	11.6699			0.449840			0.224920	+	0.974377i	$0.427788\pi$
$674$		0			0
$675$	−4.46486			−0.171852
$676$		0			0
$677$		−	33.0741i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$677$		−	33.0741i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$678$		0			0
$679$	−1.65597	+	11.9428i	−0.0635504	+	0.458321i
$680$		0			0
$681$	17.1928			0.658828
$682$		0			0
$683$		−	33.1196i		−	1.26729i	−0.773626	−	0.633643i	$-0.781559\pi$
$683$		−	33.1196i		−	1.26729i	0.773626	−	0.633643i	$-0.218441\pi$
$684$		0			0
$685$			5.91901i			0.226154i
$686$		0			0
$687$		−	80.2436i		−	3.06148i
$688$		0			0
$689$			0.410530i			0.0156399i
$690$		0			0
$691$	−22.2428			−0.846157			−0.423078	−	0.906093i	$-0.639051\pi$
$691$	−22.2428			−0.846157			−0.423078	+	0.906093i	$0.639051\pi$
$692$		0			0
$693$	5.98352	−	43.1527i	0.227295	−	1.63924i
$694$		0			0
$695$		−	8.79248i		−	0.333518i
$696$		0			0
$697$	40.5445			1.53573
$698$		0			0
$699$	−76.0551			−2.87667
$700$		0			0
$701$	16.3324			0.616868			0.308434	−	0.951246i	$-0.400195\pi$
$701$	16.3324			0.616868			0.308434	+	0.951246i	$0.400195\pi$
$702$		0			0
$703$	−41.5059			−1.56542
$704$		0			0
$705$			19.0516i			0.717525i
$706$		0			0
$707$	0.821384	−	5.92376i	0.0308913	−	0.222786i
$708$		0			0
$709$	24.0935			0.904850			0.452425	−	0.891802i	$-0.350559\pi$
$709$	24.0935			0.904850			0.452425	+	0.891802i	$0.350559\pi$
$710$		0			0
$711$		−	48.5145i		−	1.81944i
$712$		0			0
$713$		−	2.11262i		−	0.0791184i
$714$		0			0
$715$		−	2.53230i		−	0.0947028i
$716$		0			0
$717$		−	20.7206i		−	0.773823i
$718$		0			0
$719$	−4.47809			−0.167004			−0.0835022	−	0.996508i	$-0.526611\pi$
$719$	−4.47809			−0.167004			−0.0835022	+	0.996508i	$0.526611\pi$
$720$		0			0
$721$	11.9745	+	1.66037i	0.445954	+	0.0618356i
$722$		0			0
$723$			57.2107i			2.12769i
$724$		0			0
$725$	4.08118			0.151571
$726$		0			0
$727$	−30.8397			−1.14378			−0.571890	−	0.820331i	$-0.693789\pi$
$727$	−30.8397			−1.14378			−0.571890	+	0.820331i	$0.693789\pi$
$728$		0			0
$729$	−44.0343			−1.63090
$730$		0			0
$731$	−36.2431			−1.34050
$732$		0			0
$733$		−	47.5794i		−	1.75738i	−0.477389	−	0.878692i	$-0.658417\pi$
$733$		−	47.5794i		−	1.75738i	0.477389	−	0.878692i	$-0.341583\pi$
$734$		0			0
$735$	−5.25676	+	18.5912i	−0.193898	+	0.685747i
$736$		0			0
$737$	40.0268			1.47441
$738$		0			0
$739$			50.9114i			1.87281i	0.350925	+	0.936404i	$0.385867\pi$
$739$			50.9114i			1.87281i	−0.350925	+	0.936404i	$0.614133\pi$
$740$		0			0
$741$		−	13.7582i		−	0.505419i
$742$		0			0
$743$			3.89710i			0.142971i	0.997442	+	0.0714854i	$0.0227739\pi$
$743$			3.89710i			0.142971i	−0.997442	+	0.0714854i	$0.977226\pi$
$744$		0			0
$745$		−	11.0342i		−	0.404261i
$746$		0			0
$747$	−77.7464			−2.84459
$748$		0			0
$749$	−4.98108	+	35.9232i	−0.182005	+	1.31260i
$750$		0			0
$751$		−	3.41437i		−	0.124592i	−0.998058	−	0.0622961i	$-0.980158\pi$
$751$		−	3.41437i		−	0.124592i	0.998058	−	0.0622961i	$-0.0198423\pi$
$752$		0			0
$753$	24.6462			0.898158
$754$		0			0
$755$	18.4025			0.669737
$756$		0			0
$757$	−0.000925711	0		−3.36455e−5	0		−1.68228e−5	−	1.00000i	$-0.500005\pi$
$757$	−0.000925711	0		−3.36455e−5	0		−1.68228e−5		1.00000i	$0.500005\pi$
$758$		0			0
$759$	17.4998			0.635204
$760$		0			0
$761$			14.4178i			0.522644i	0.965252	+	0.261322i	$0.0841584\pi$
$761$			14.4178i			0.522644i	−0.965252	+	0.261322i	$0.915842\pi$
$762$		0			0
$763$	32.8234	+	4.55127i	1.18829	+	0.164767i
$764$		0			0
$765$	24.2797			0.877835
$766$		0			0
$767$			4.35060i			0.157091i
$768$		0			0
$769$		−	5.13991i		−	0.185350i	−0.995696	−	0.0926750i	$-0.970458\pi$
$769$		−	5.13991i		−	0.185350i	0.995696	−	0.0926750i	$-0.0295418\pi$
$770$		0			0
$771$		−	54.7081i		−	1.97026i
$772$		0			0
$773$		−	16.7753i		−	0.603366i	−0.953408	−	0.301683i	$-0.902452\pi$
$773$		−	16.7753i		−	0.603366i	0.953408	−	0.301683i	$-0.0975484\pi$
$774$		0			0
$775$	1.18814			0.0426794
$776$		0			0
$777$	−42.7693	−	5.93036i	−1.53434	−	0.212751i
$778$		0			0
$779$			54.1273i			1.93931i
$780$		0			0
$781$	56.8242			2.03333
$782$		0			0
$783$	−18.2219			−0.651197
$784$		0			0
$785$	2.62253			0.0936021
$786$		0			0
$787$	−36.3429			−1.29549			−0.647743	−	0.761859i	$-0.724287\pi$
$787$	−36.3429			−1.29549			−0.647743	+	0.761859i	$0.724287\pi$
$788$		0			0
$789$		−	39.6151i		−	1.41034i
$790$		0			0
$791$	50.6267	+	7.01987i	1.80008	+	0.249598i
$792$		0			0
$793$	−5.92586			−0.210434
$794$		0			0
$795$		−	1.59555i		−	0.0565883i
$796$		0			0
$797$		−	0.523868i		−	0.0185564i	−0.999957	−	0.00927818i	$-0.997047\pi$
$797$		−	0.523868i		−	0.0185564i	0.999957	−	0.00927818i	$-0.00295338\pi$
$798$		0			0
$799$		−	36.2943i		−	1.28400i
$800$		0			0
$801$			67.0243i			2.36819i
$802$		0			0
$803$	8.47890			0.299214
$804$		0			0
$805$	−4.65979	−	0.646123i	−0.164236	−	0.0227728i
$806$		0			0
$807$		−	66.2454i		−	2.33195i
$808$		0			0
$809$	42.8403			1.50618			0.753092	−	0.657915i	$-0.228561\pi$
$809$	42.8403			1.50618			0.753092	+	0.657915i	$0.228561\pi$
$810$		0			0
$811$	−33.1175			−1.16291			−0.581456	−	0.813578i	$-0.697517\pi$
$811$	−33.1175			−1.16291			−0.581456	+	0.813578i	$0.697517\pi$
$812$		0			0
$813$	−46.7615			−1.64000
$814$		0			0
$815$	6.72228			0.235471
$816$		0			0
$817$		−	48.3849i		−	1.69277i
$818$		0			0
$819$	1.19161	−	8.59380i	0.0416382	−	0.300292i
$820$		0			0
$821$	25.5033			0.890073			0.445036	−	0.895512i	$-0.353191\pi$
$821$	25.5033			0.890073			0.445036	+	0.895512i	$0.353191\pi$
$822$		0			0
$823$			13.2075i			0.460385i	0.973145	+	0.230193i	$0.0739356\pi$
$823$			13.2075i			0.460385i	−0.973145	+	0.230193i	$0.926064\pi$
$824$		0			0
$825$			9.84194i			0.342653i
$826$		0			0
$827$			31.0372i			1.07927i	0.841900	+	0.539634i	$0.181437\pi$
$827$			31.0372i			1.07927i	−0.841900	+	0.539634i	$0.818563\pi$
$828$		0			0
$829$		−	42.6771i		−	1.48224i	−0.671375	−	0.741118i	$-0.734296\pi$
$829$		−	42.6771i		−	1.48224i	0.671375	−	0.741118i	$-0.265704\pi$
$830$		0			0
$831$	79.1584			2.74597
$832$		0			0
$833$	10.0144	−	35.4172i	0.346978	−	1.22713i
$834$		0			0
$835$		−	23.8207i		−	0.824349i
$836$		0			0
$837$	−5.30490			−0.183364
$838$		0			0
$839$	−20.3306			−0.701890			−0.350945	−	0.936396i	$-0.614140\pi$
$839$	−20.3306			−0.701890			−0.350945	+	0.936396i	$0.614140\pi$
$840$		0			0
$841$	−12.3440			−0.425655
$842$		0			0
$843$	29.8932			1.02958
$844$		0			0
$845$			12.4957i			0.429865i
$846$		0			0
$847$	−4.49616	−	0.623433i	−0.154490	−	0.0214214i
$848$		0			0
$849$	16.1895			0.555622
$850$		0			0
$851$		−	10.5138i		−	0.360409i
$852$		0			0
$853$		−	5.97835i		−	0.204695i	−0.994749	−	0.102347i	$-0.967365\pi$
$853$		−	5.97835i		−	0.204695i	0.994749	−	0.102347i	$-0.0326353\pi$
$854$		0			0
$855$			32.4136i			1.10852i
$856$		0			0
$857$		−	14.3976i		−	0.491812i	−0.969294	−	0.245906i	$-0.920915\pi$
$857$		−	14.3976i		−	0.491812i	0.969294	−	0.245906i	$-0.0790854\pi$
$858$		0			0
$859$	19.7597			0.674193			0.337097	−	0.941470i	$-0.390555\pi$
$859$	19.7597			0.674193			0.337097	+	0.941470i	$0.390555\pi$
$860$		0			0
$861$	−7.73371	+	55.7749i	−0.263564	+	1.90080i
$862$		0			0
$863$		−	4.55913i		−	0.155195i	−0.996985	−	0.0775973i	$-0.975275\pi$
$863$		−	4.55913i		−	0.155195i	0.996985	−	0.0775973i	$-0.0247248\pi$
$864$		0			0
$865$	2.58227			0.0877998
$866$		0			0
$867$	−29.3840			−0.997932
$868$		0			0
$869$	−37.4641			−1.27088
$870$		0			0
$871$	7.97128			0.270096
$872$		0			0
$873$			21.0434i			0.712211i
$874$		0			0
$875$	0.363381	−	2.62068i	0.0122845	−	0.0885951i
$876$		0			0
$877$	57.4704			1.94064			0.970318	−	0.241833i	$-0.0777486\pi$
$877$	57.4704			1.94064			0.970318	+	0.241833i	$0.0777486\pi$
$878$		0			0
$879$			17.3710i			0.585910i
$880$		0			0
$881$		−	31.2401i		−	1.05251i	−0.850328	−	0.526253i	$-0.823597\pi$
$881$		−	31.2401i		−	1.05251i	0.850328	−	0.526253i	$-0.176403\pi$
$882$		0			0
$883$		−	47.3199i		−	1.59244i	−0.605005	−	0.796221i	$-0.706829\pi$
$883$		−	47.3199i		−	1.59244i	0.605005	−	0.796221i	$-0.293171\pi$
$884$		0			0
$885$		−	16.9089i		−	0.568385i
$886$		0			0
$887$	41.2127			1.38379			0.691894	−	0.721999i	$-0.256777\pi$
$887$	41.2127			1.38379			0.691894	+	0.721999i	$0.256777\pi$
$888$		0			0
$889$	−5.68532	+	41.0021i	−0.190679	+	1.37517i
$890$		0			0
$891$			5.45581i			0.182776i
$892$		0			0
$893$	48.4532			1.62142
$894$		0			0
$895$	15.0289			0.502362
$896$		0			0
$897$	3.48507			0.116363
$898$		0			0
$899$	4.84903			0.161724
$900$		0			0
$901$			3.03960i			0.101264i
$902$		0			0
$903$	6.91324	−	49.8578i	0.230058	−	1.65916i
$904$		0			0
$905$	−22.9157			−0.761744
$906$		0			0
$907$		−	17.7972i		−	0.590948i	−0.955351	−	0.295474i	$-0.904523\pi$
$907$		−	17.7972i		−	0.590948i	0.955351	−	0.295474i	$-0.0954775\pi$
$908$		0			0
$909$		−	10.4378i		−	0.346200i
$910$		0			0
$911$			30.0894i			0.996907i	0.866916	+	0.498454i	$0.166099\pi$
$911$			30.0894i			0.996907i	−0.866916	+	0.498454i	$0.833901\pi$
$912$		0			0
$913$			60.0378i			1.98696i
$914$		0			0
$915$	23.0312			0.761388
$916$		0			0
$917$	−27.3516	−	3.79255i	−0.903228	−	0.125241i
$918$		0			0
$919$			40.8706i			1.34820i	0.738642	+	0.674098i	$0.235468\pi$
$919$			40.8706i			1.34820i	−0.738642	+	0.674098i	$0.764532\pi$
$920$		0			0
$921$	−5.47516			−0.180413
$922$		0			0
$923$	11.3165			0.372486
$924$		0			0
$925$	5.91299			0.194418
$926$		0			0
$927$	21.0993			0.692993
$928$		0			0
$929$			53.3160i			1.74924i	0.484808	+	0.874621i	$0.338890\pi$
$929$			53.3160i			1.74924i	−0.484808	+	0.874621i	$0.661110\pi$
$930$		0			0
$931$	47.2823	+	13.3693i	1.54962	+	0.438161i
$932$		0			0
$933$	13.2508			0.433811
$934$		0			0
$935$		−	18.7494i		−	0.613171i
$936$		0			0
$937$		−	4.39190i		−	0.143477i	−0.997423	−	0.0717385i	$-0.977145\pi$
$937$		−	4.39190i		−	0.143477i	0.997423	−	0.0717385i	$-0.0228547\pi$
$938$		0			0
$939$			37.3799i			1.21985i
$940$		0			0
$941$			7.37371i			0.240376i	0.992751	+	0.120188i	$0.0383497\pi$
$941$			7.37371i			0.240376i	−0.992751	+	0.120188i	$0.961650\pi$
$942$		0			0
$943$	−13.7109			−0.446489
$944$		0			0
$945$	−1.62245	+	11.7010i	−0.0527782	+	0.380632i
$946$		0			0
$947$		−	7.10034i		−	0.230730i	−0.993323	−	0.115365i	$-0.963196\pi$
$947$		−	7.10034i		−	0.230730i	0.993323	−	0.115365i	$-0.0368038\pi$
$948$		0			0
$949$	1.68856			0.0548130
$950$		0			0
$951$	−24.7141			−0.801409
$952$		0			0
$953$	−37.6237			−1.21875			−0.609376	−	0.792881i	$-0.708580\pi$
$953$	−37.6237			−1.21875			−0.609376	+	0.792881i	$0.708580\pi$
$954$		0			0
$955$	−19.0445			−0.616266
$956$		0			0
$957$			40.1667i			1.29841i
$958$		0			0
$959$	15.5118	+	2.15086i	0.500903	+	0.0694548i
$960$		0			0
$961$	−29.5883			−0.954462
$962$		0			0
$963$			63.2974i			2.03973i
$964$		0			0
$965$			1.53605i			0.0494472i
$966$		0			0
$967$		−	34.2478i		−	1.10134i	−0.834724	−	0.550668i	$-0.814373\pi$
$967$		−	34.2478i		−	1.10134i	0.834724	−	0.550668i	$-0.185627\pi$
$968$		0			0
$969$		−	101.867i		−	3.27243i
$970$		0			0
$971$	38.1291			1.22362			0.611811	−	0.791004i	$-0.290441\pi$
$971$	38.1291			1.22362			0.611811	+	0.791004i	$0.290441\pi$
$972$		0			0
$973$	−23.0423	−	3.19502i	−0.738701	−	0.102428i
$974$		0			0
$975$			1.96001i			0.0627705i
$976$		0			0
$977$	−53.0027			−1.69571			−0.847853	−	0.530231i	$-0.822105\pi$
$977$	−53.0027			−1.69571			−0.847853	+	0.530231i	$0.822105\pi$
$978$		0			0
$979$	51.7579			1.65419
$980$		0			0
$981$	57.8356			1.84655
$982$		0			0
$983$	10.7474			0.342789			0.171395	−	0.985202i	$-0.445173\pi$
$983$	10.7474			0.342789			0.171395	+	0.985202i	$0.445173\pi$
$984$		0			0
$985$		−	5.80357i		−	0.184917i
$986$		0			0
$987$	49.9281	+	6.92299i	1.58923	+	0.220361i
$988$		0			0
$989$	12.2563			0.389728
$990$		0			0
$991$			35.9302i			1.14136i	0.821172	+	0.570680i	$0.193320\pi$
$991$			35.9302i			1.14136i	−0.821172	+	0.570680i	$0.806680\pi$
$992$		0			0
$993$			70.9343i			2.25103i
$994$		0			0
$995$		−	18.2990i		−	0.580118i
$996$		0			0
$997$			56.8192i			1.79948i	0.436425	+	0.899741i	$0.356245\pi$
$997$			56.8192i			1.79948i	−0.436425	+	0.899741i	$0.643755\pi$
$998$		0			0
$999$	−26.4007			−0.835280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.g.1791.16		16
4.3	odd	2		2240.2.k.f.1791.2		16
7.6	odd	2		2240.2.k.f.1791.1		16
8.3	odd	2		1120.2.k.a.671.15	✓	16
8.5	even	2		1120.2.k.b.671.1	yes	16
28.27	even	2	inner	2240.2.k.g.1791.15		16
56.13	odd	2		1120.2.k.a.671.16	yes	16
56.27	even	2		1120.2.k.b.671.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.k.a.671.15	✓	16	8.3	odd	2
1120.2.k.a.671.16	yes	16	56.13	odd	2
1120.2.k.b.671.1	yes	16	8.5	even	2
1120.2.k.b.671.2	yes	16	56.27	even	2
2240.2.k.f.1791.1		16	7.6	odd	2
2240.2.k.f.1791.2		16	4.3	odd	2
2240.2.k.g.1791.15		16	28.27	even	2	inner
2240.2.k.g.1791.16		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.76002 q^{3} +1.00000i q^{5} +(2.62068 + 0.363381i) q^{7} +4.61769 q^{9} +O(q^{10})\)	\(q+2.76002 q^{3} +1.00000i q^{5} +(2.62068 + 0.363381i) q^{7} +4.61769 q^{9} -3.56590i q^{11} -0.710144i q^{13} +2.76002i q^{15} -5.25797i q^{17} +7.01944 q^{19} +(7.23312 + 1.00294i) q^{21} +1.77809i q^{23} -1.00000 q^{25} +4.46486 q^{27} -4.08118 q^{29} -1.18814 q^{31} -9.84194i q^{33} +(-0.363381 + 2.62068i) q^{35} -5.91299 q^{37} -1.96001i q^{39} +7.71105i q^{41} -6.89299i q^{43} +4.61769i q^{45} +6.90271 q^{47} +(6.73591 + 1.90461i) q^{49} -14.5121i q^{51} -0.578094 q^{53} +3.56590 q^{55} +19.3738 q^{57} -6.12636 q^{59} -8.34459i q^{61} +(12.1015 + 1.67798i) q^{63} +0.710144 q^{65} +11.2249i q^{67} +4.90755i q^{69} +15.9354i q^{71} +2.37777i q^{73} -2.76002 q^{75} +(1.29578 - 9.34508i) q^{77} -10.5062i q^{79} -1.52999 q^{81} -16.8366 q^{83} +5.25797 q^{85} -11.2641 q^{87} +14.5147i q^{89} +(0.258053 - 1.86106i) q^{91} -3.27930 q^{93} +7.01944i q^{95} +4.55712i q^{97} -16.4662i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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

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