LMFDB - Embedded newform 2240.2.l.d.1569.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1569,2240)]

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([0, 1, 1, 0]))

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

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

chi := DirichletCharacter("2240.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.3
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.d.1569.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41750 q^{3} +(1.94498 - 1.10320i) q^{5} -1.00000i q^{7} +2.84431 q^{9} +O(q^{10})$	$q-2.41750 q^{3} +(1.94498 - 1.10320i) q^{5} -1.00000i q^{7} +2.84431 q^{9} -2.55443i q^{11} +5.47711 q^{13} +(-4.70199 + 2.66698i) q^{15} +2.44523i q^{17} -4.99104i q^{19} +2.41750i q^{21} +1.10108i q^{23} +(2.56590 - 4.29140i) q^{25} +0.376386 q^{27} +1.96204i q^{29} +2.33842 q^{31} +6.17535i q^{33} +(-1.10320 - 1.94498i) q^{35} +6.16452 q^{37} -13.2409 q^{39} -9.40718 q^{41} +5.99845 q^{43} +(5.53213 - 3.13784i) q^{45} +8.05426i q^{47} -1.00000 q^{49} -5.91134i q^{51} -1.97706 q^{53} +(-2.81805 - 4.96833i) q^{55} +12.0658i q^{57} +1.44400i q^{59} -8.71679i q^{61} -2.84431i q^{63} +(10.6529 - 6.04234i) q^{65} -3.92408 q^{67} -2.66186i q^{69} +4.55130 q^{71} +7.55542i q^{73} +(-6.20307 + 10.3745i) q^{75} -2.55443 q^{77} -8.53342 q^{79} -9.44284 q^{81} -8.26470 q^{83} +(2.69757 + 4.75592i) q^{85} -4.74323i q^{87} +17.2784 q^{89} -5.47711i q^{91} -5.65314 q^{93} +(-5.50611 - 9.70748i) q^{95} -10.6168i q^{97} -7.26560i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{5} + 12 q^{9}+O(q^{10}) $ 24 * q + 4 * q^5 + 12 * q^9	$ 24 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{37} + 48 q^{45} - 24 q^{49} - 88 q^{53} + 36 q^{65} - 20 q^{77} + 16 q^{81} + 56 q^{85} - 40 q^{89} - 24 q^{93}+O(q^{100}) $ 24 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^37 + 48 * q^45 - 24 * q^49 - 88 * q^53 + 36 * q^65 - 20 * q^77 + 16 * q^81 + 56 * q^85 - 40 * q^89 - 24 * 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
$3$	−2.41750			−1.39574			−0.697872	−	0.716222i	$-0.745870\pi$
$3$	−2.41750			−1.39574			−0.697872	+	0.716222i	$0.745870\pi$
$4$		0			0
$5$	1.94498	−	1.10320i	0.869822	−	0.493366i
$5$	1.94498	−	1.10320i	0.869822	−	0.493366i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	2.84431			0.948103
$10$		0			0
$11$		−	2.55443i		−	0.770191i	−0.922877	−	0.385095i	$-0.874169\pi$
$11$		−	2.55443i		−	0.770191i	0.922877	−	0.385095i	$-0.125831\pi$
$12$		0			0
$13$	5.47711			1.51908			0.759538	−	0.650463i	$-0.225425\pi$
$13$	5.47711			1.51908			0.759538	+	0.650463i	$0.225425\pi$
$14$		0			0
$15$	−4.70199	+	2.66698i	−1.21405	+	0.688612i
$16$		0			0
$17$			2.44523i			0.593055i	0.955024	+	0.296528i	$0.0958287\pi$
$17$			2.44523i			0.593055i	−0.955024	+	0.296528i	$0.904171\pi$
$18$		0			0
$19$		−	4.99104i		−	1.14502i	−0.819896	−	0.572512i	$-0.805969\pi$
$19$		−	4.99104i		−	1.14502i	0.819896	−	0.572512i	$-0.194031\pi$
$20$		0			0
$21$			2.41750i			0.527542i
$22$		0			0
$23$			1.10108i			0.229591i	0.993389	+	0.114795i	$0.0366213\pi$
$23$			1.10108i			0.229591i	−0.993389	+	0.114795i	$0.963379\pi$
$24$		0			0
$25$	2.56590	−	4.29140i	0.513181	−	0.858280i
$26$		0			0
$27$	0.376386			0.0724355
$28$		0			0
$29$			1.96204i			0.364341i	0.983267	+	0.182171i	$0.0583123\pi$
$29$			1.96204i			0.364341i	−0.983267	+	0.182171i	$0.941688\pi$
$30$		0			0
$31$	2.33842			0.419993			0.209997	−	0.977702i	$-0.432655\pi$
$31$	2.33842			0.419993			0.209997	+	0.977702i	$0.432655\pi$
$32$		0			0
$33$			6.17535i			1.07499i
$34$		0			0
$35$	−1.10320	−	1.94498i	−0.186475	−	0.328762i
$36$		0			0
$37$	6.16452			1.01344			0.506720	−	0.862111i	$-0.330858\pi$
$37$	6.16452			1.01344			0.506720	+	0.862111i	$0.330858\pi$
$38$		0			0
$39$	−13.2409			−2.12024
$40$		0			0
$41$	−9.40718			−1.46915			−0.734577	−	0.678525i	$-0.762619\pi$
$41$	−9.40718			−1.46915			−0.734577	+	0.678525i	$0.762619\pi$
$42$		0			0
$43$	5.99845			0.914755			0.457377	−	0.889273i	$-0.348789\pi$
$43$	5.99845			0.914755			0.457377	+	0.889273i	$0.348789\pi$
$44$		0			0
$45$	5.53213	−	3.13784i	0.824681	−	0.467761i
$46$		0			0
$47$			8.05426i			1.17483i	0.809284	+	0.587417i	$0.199855\pi$
$47$			8.05426i			1.17483i	−0.809284	+	0.587417i	$0.800145\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	5.91134i		−	0.827753i
$52$		0			0
$53$	−1.97706			−0.271570			−0.135785	−	0.990738i	$-0.543356\pi$
$53$	−1.97706			−0.271570			−0.135785	+	0.990738i	$0.543356\pi$
$54$		0			0
$55$	−2.81805	−	4.96833i	−0.379986	−	0.669929i
$56$		0			0
$57$			12.0658i			1.59816i
$58$		0			0
$59$			1.44400i			0.187993i	0.995573	+	0.0939967i	$0.0299643\pi$
$59$			1.44400i			0.187993i	−0.995573	+	0.0939967i	$0.970036\pi$
$60$		0			0
$61$		−	8.71679i		−	1.11607i	−0.829817	−	0.558036i	$-0.811555\pi$
$61$		−	8.71679i		−	1.11607i	0.829817	−	0.558036i	$-0.188445\pi$
$62$		0			0
$63$		−	2.84431i		−	0.358349i
$64$		0			0
$65$	10.6529	−	6.04234i	1.32133	−	0.749460i
$66$		0			0
$67$	−3.92408			−0.479402			−0.239701	−	0.970847i	$-0.577049\pi$
$67$	−3.92408			−0.479402			−0.239701	+	0.970847i	$0.577049\pi$
$68$		0			0
$69$		−	2.66186i		−	0.320450i
$70$		0			0
$71$	4.55130			0.540140			0.270070	−	0.962841i	$-0.412953\pi$
$71$	4.55130			0.540140			0.270070	+	0.962841i	$0.412953\pi$
$72$		0			0
$73$			7.55542i			0.884296i	0.896942	+	0.442148i	$0.145783\pi$
$73$			7.55542i			0.884296i	−0.896942	+	0.442148i	$0.854217\pi$
$74$		0			0
$75$	−6.20307	+	10.3745i	−0.716269	+	1.19794i
$76$		0			0
$77$	−2.55443			−0.291105
$78$		0			0
$79$	−8.53342			−0.960085			−0.480042	−	0.877245i	$-0.659379\pi$
$79$	−8.53342			−0.960085			−0.480042	+	0.877245i	$0.659379\pi$
$80$		0			0
$81$	−9.44284			−1.04920
$82$		0			0
$83$	−8.26470			−0.907169			−0.453584	−	0.891213i	$-0.649855\pi$
$83$	−8.26470			−0.907169			−0.453584	+	0.891213i	$0.649855\pi$
$84$		0			0
$85$	2.69757	+	4.75592i	0.292593	+	0.515852i
$86$		0			0
$87$		−	4.74323i		−	0.508527i
$88$		0			0
$89$	17.2784			1.83151			0.915754	−	0.401740i	$-0.131594\pi$
$89$	17.2784			1.83151			0.915754	+	0.401740i	$0.131594\pi$
$90$		0			0
$91$		−	5.47711i		−	0.574157i
$92$		0			0
$93$	−5.65314			−0.586203
$94$		0			0
$95$	−5.50611	−	9.70748i	−0.564915	−	0.995967i
$96$		0			0
$97$		−	10.6168i		−	1.07797i	−0.842314	−	0.538987i	$-0.818807\pi$
$97$		−	10.6168i		−	1.07797i	0.842314	−	0.538987i	$-0.181193\pi$
$98$		0			0
$99$		−	7.26560i		−	0.730220i
$100$		0			0
$101$		−	14.4284i		−	1.43568i	−0.696207	−	0.717841i	$-0.745130\pi$
$101$		−	14.4284i		−	1.43568i	0.696207	−	0.717841i	$-0.254870\pi$
$102$		0			0
$103$			1.19135i			0.117387i	0.998276	+	0.0586937i	$0.0186935\pi$
$103$			1.19135i			0.117387i	−0.998276	+	0.0586937i	$0.981306\pi$
$104$		0			0
$105$	2.66698	+	4.70199i	0.260271	+	0.458868i
$106$		0			0
$107$	18.8080			1.81824			0.909118	−	0.416539i	$-0.136757\pi$
$107$	18.8080			1.81824			0.909118	+	0.416539i	$0.136757\pi$
$108$		0			0
$109$		−	15.9149i		−	1.52437i	−0.647357	−	0.762187i	$-0.724126\pi$
$109$		−	15.9149i		−	1.52437i	0.647357	−	0.762187i	$-0.275874\pi$
$110$		0			0
$111$	−14.9027			−1.41450
$112$		0			0
$113$			4.74597i			0.446463i	0.974765	+	0.223232i	$0.0716606\pi$
$113$			4.74597i			0.446463i	−0.974765	+	0.223232i	$0.928339\pi$
$114$		0			0
$115$	1.21471	+	2.14158i	0.113272	+	0.199703i
$116$		0			0
$117$	15.5786			1.44024
$118$		0			0
$119$	2.44523			0.224154
$120$		0			0
$121$	4.47486			0.406806
$122$		0			0
$123$	22.7419			2.05056
$124$		0			0
$125$	0.256365	−	11.1774i	0.0229300	−	0.999737i
$126$		0			0
$127$		−	12.8858i		−	1.14343i	−0.820451	−	0.571717i	$-0.806278\pi$
$127$		−	12.8858i		−	1.14343i	0.820451	−	0.571717i	$-0.193722\pi$
$128$		0			0
$129$	−14.5013			−1.27676
$130$		0			0
$131$		−	14.3831i		−	1.25665i	−0.777949	−	0.628327i	$-0.783740\pi$
$131$		−	14.3831i		−	1.25665i	0.777949	−	0.628327i	$-0.216260\pi$
$132$		0			0
$133$	−4.99104			−0.432778
$134$		0			0
$135$	0.732063	−	0.415228i	0.0630060	−	0.0357372i
$136$		0			0
$137$		−	10.6481i		−	0.909727i	−0.890561	−	0.454864i	$-0.849688\pi$
$137$		−	10.6481i		−	0.909727i	0.890561	−	0.454864i	$-0.150312\pi$
$138$		0			0
$139$			19.7092i			1.67172i	0.548945	+	0.835858i	$0.315029\pi$
$139$			19.7092i			1.67172i	−0.548945	+	0.835858i	$0.684971\pi$
$140$		0			0
$141$		−	19.4712i		−	1.63977i
$142$		0			0
$143$		−	13.9909i		−	1.16998i
$144$		0			0
$145$	2.16452	+	3.81613i	0.179753	+	0.316912i
$146$		0			0
$147$	2.41750			0.199392
$148$		0			0
$149$			5.00028i			0.409639i	0.978800	+	0.204820i	$0.0656608\pi$
$149$			5.00028i			0.409639i	−0.978800	+	0.204820i	$0.934339\pi$
$150$		0			0
$151$	0.471590			0.0383774			0.0191887	−	0.999816i	$-0.493892\pi$
$151$	0.471590			0.0383774			0.0191887	+	0.999816i	$0.493892\pi$
$152$		0			0
$153$			6.95498i			0.562277i
$154$		0			0
$155$	4.54819	−	2.57975i	0.365320	−	0.207210i
$156$		0			0
$157$	−19.5755			−1.56229			−0.781146	−	0.624349i	$-0.785364\pi$
$157$	−19.5755			−1.56229			−0.781146	+	0.624349i	$0.785364\pi$
$158$		0			0
$159$	4.77954			0.379042
$160$		0			0
$161$	1.10108			0.0867772
$162$		0			0
$163$	15.1895			1.18974			0.594868	−	0.803823i	$-0.297204\pi$
$163$	15.1895			1.18974			0.594868	+	0.803823i	$0.297204\pi$
$164$		0			0
$165$	6.81264	+	12.0109i	0.530363	+	0.935050i
$166$		0			0
$167$		−	15.4112i		−	1.19255i	−0.802780	−	0.596276i	$-0.796646\pi$
$167$		−	15.4112i		−	1.19255i	0.802780	−	0.596276i	$-0.203354\pi$
$168$		0			0
$169$	16.9987			1.30759
$170$		0			0
$171$		−	14.1961i		−	1.08560i
$172$		0			0
$173$	1.11822			0.0850166			0.0425083	−	0.999096i	$-0.486465\pi$
$173$	1.11822			0.0850166			0.0425083	+	0.999096i	$0.486465\pi$
$174$		0			0
$175$	−4.29140	−	2.56590i	−0.324400	−	0.193964i
$176$		0			0
$177$		−	3.49088i		−	0.262391i
$178$		0			0
$179$			19.1919i			1.43447i	0.696830	+	0.717237i	$0.254593\pi$
$179$			19.1919i			1.43447i	−0.696830	+	0.717237i	$0.745407\pi$
$180$		0			0
$181$		−	22.3475i		−	1.66108i	−0.556962	−	0.830538i	$-0.688033\pi$
$181$		−	22.3475i		−	1.66108i	0.556962	−	0.830538i	$-0.311967\pi$
$182$		0			0
$183$			21.0729i			1.55775i
$184$		0			0
$185$	11.9899	−	6.80069i	0.881513	−	0.499997i
$186$		0			0
$187$	6.24618			0.456766
$188$		0			0
$189$		−	0.376386i		−	0.0273780i
$190$		0			0
$191$	12.8797			0.931943			0.465971	−	0.884800i	$-0.345705\pi$
$191$	12.8797			0.931943			0.465971	+	0.884800i	$0.345705\pi$
$192$		0			0
$193$		−	14.2001i		−	1.02214i	−0.859538	−	0.511072i	$-0.829249\pi$
$193$		−	14.2001i		−	1.02214i	0.859538	−	0.511072i	$-0.170751\pi$
$194$		0			0
$195$	−25.7533	+	14.6074i	−1.84423	+	1.04605i
$196$		0			0
$197$	−8.14660			−0.580421			−0.290211	−	0.956963i	$-0.593725\pi$
$197$	−8.14660			−0.580421			−0.290211	+	0.956963i	$0.593725\pi$
$198$		0			0
$199$	−24.4855			−1.73573			−0.867865	−	0.496800i	$-0.834508\pi$
$199$	−24.4855			−1.73573			−0.867865	+	0.496800i	$0.834508\pi$
$200$		0			0
$201$	9.48646			0.669123
$202$		0			0
$203$	1.96204			0.137708
$204$		0			0
$205$	−18.2968	+	10.3780i	−1.27790	+	0.724830i
$206$		0			0
$207$			3.13181i			0.217676i
$208$		0			0
$209$	−12.7493			−0.881887
$210$		0			0
$211$		−	25.6630i		−	1.76672i	−0.468698	−	0.883358i	$-0.655277\pi$
$211$		−	25.6630i		−	1.76672i	0.468698	−	0.883358i	$-0.344723\pi$
$212$		0			0
$213$	−11.0028			−0.753897
$214$		0			0
$215$	11.6669	−	6.61748i	0.795674	−	0.451308i
$216$		0			0
$217$		−	2.33842i		−	0.158743i
$218$		0			0
$219$		−	18.2652i		−	1.23425i
$220$		0			0
$221$			13.3928i			0.900896i
$222$		0			0
$223$		−	0.100136i		−	0.00670559i	−0.999994	−	0.00335280i	$-0.998933\pi$
$223$		−	0.100136i		−	0.00670559i	0.999994	−	0.00335280i	$-0.00106723\pi$
$224$		0			0
$225$	7.29822	−	12.2061i	0.486548	−	0.813738i
$226$		0			0
$227$	−29.9755			−1.98955			−0.994773	−	0.102114i	$-0.967439\pi$
$227$	−29.9755			−1.98955			−0.994773	+	0.102114i	$0.967439\pi$
$228$		0			0
$229$			5.13231i			0.339153i	0.985517	+	0.169576i	$0.0542399\pi$
$229$			5.13231i			0.339153i	−0.985517	+	0.169576i	$0.945760\pi$
$230$		0			0
$231$	6.17535			0.406308
$232$		0			0
$233$			19.5680i			1.28195i	0.767564	+	0.640973i	$0.221469\pi$
$233$			19.5680i			1.28195i	−0.767564	+	0.640973i	$0.778531\pi$
$234$		0			0
$235$	8.88545	+	15.6654i	0.579623	+	1.02190i
$236$		0			0
$237$	20.6295			1.34003
$238$		0			0
$239$	28.2504			1.82736			0.913682	−	0.406430i	$-0.133227\pi$
$239$	28.2504			1.82736			0.913682	+	0.406430i	$0.133227\pi$
$240$		0			0
$241$	−19.6708			−1.26711			−0.633554	−	0.773699i	$-0.718404\pi$
$241$	−19.6708			−1.26711			−0.633554	+	0.773699i	$0.718404\pi$
$242$		0			0
$243$	21.6989			1.39199
$244$		0			0
$245$	−1.94498	+	1.10320i	−0.124260	+	0.0704808i
$246$		0			0
$247$		−	27.3365i		−	1.73938i
$248$		0			0
$249$	19.9799			1.26618
$250$		0			0
$251$		−	9.03969i		−	0.570580i	−0.958441	−	0.285290i	$-0.907910\pi$
$251$		−	9.03969i		−	0.570580i	0.958441	−	0.285290i	$-0.0920899\pi$
$252$		0			0
$253$	2.81264			0.176829
$254$		0			0
$255$	−6.52139	−	11.4974i	−0.408385	−	0.719998i
$256$		0			0
$257$		−	23.1921i		−	1.44668i	−0.690491	−	0.723341i	$-0.742605\pi$
$257$		−	23.1921i		−	1.44668i	0.690491	−	0.723341i	$-0.257395\pi$
$258$		0			0
$259$		−	6.16452i		−	0.383045i
$260$		0			0
$261$			5.58064i			0.345433i
$262$		0			0
$263$		−	2.48532i		−	0.153251i	−0.997060	−	0.0766256i	$-0.975585\pi$
$263$		−	2.48532i		−	0.153251i	0.997060	−	0.0766256i	$-0.0244146\pi$
$264$		0			0
$265$	−3.84534	+	2.18109i	−0.236218	+	0.133983i
$266$		0			0
$267$	−41.7706			−2.55632
$268$		0			0
$269$			27.0597i			1.64986i	0.565235	+	0.824930i	$0.308786\pi$
$269$			27.0597i			1.64986i	−0.565235	+	0.824930i	$0.691214\pi$
$270$		0			0
$271$	1.98463			0.120558			0.0602788	−	0.998182i	$-0.480801\pi$
$271$	1.98463			0.120558			0.0602788	+	0.998182i	$0.480801\pi$
$272$		0			0
$273$			13.2409i			0.801376i
$274$		0			0
$275$	−10.9621	−	6.55443i	−0.661040	−	0.395247i
$276$		0			0
$277$	15.3192			0.920440			0.460220	−	0.887805i	$-0.347771\pi$
$277$	15.3192			0.920440			0.460220	+	0.887805i	$0.347771\pi$
$278$		0			0
$279$	6.65120			0.398197
$280$		0			0
$281$	8.20615			0.489538			0.244769	−	0.969581i	$-0.421288\pi$
$281$	8.20615			0.489538			0.244769	+	0.969581i	$0.421288\pi$
$282$		0			0
$283$	20.9508			1.24539			0.622697	−	0.782463i	$-0.286037\pi$
$283$	20.9508			1.24539			0.622697	+	0.782463i	$0.286037\pi$
$284$		0			0
$285$	13.3110	+	23.4678i	0.788477	+	1.39012i
$286$		0			0
$287$			9.40718i			0.555288i
$288$		0			0
$289$	11.0209			0.648286
$290$		0			0
$291$			25.6662i			1.50458i
$292$		0			0
$293$	−26.3205			−1.53766			−0.768829	−	0.639454i	$-0.779160\pi$
$293$	−26.3205			−1.53766			−0.768829	+	0.639454i	$0.779160\pi$
$294$		0			0
$295$	1.59302	+	2.80856i	0.0927495	+	0.163521i
$296$		0			0
$297$		−	0.961453i		−	0.0557891i
$298$		0			0
$299$			6.03073i			0.348766i
$300$		0			0
$301$		−	5.99845i		−	0.345745i
$302$		0			0
$303$			34.8807i			2.00385i
$304$		0			0
$305$	−9.61636	−	16.9540i	−0.550631	−	0.970783i
$306$		0			0
$307$	−22.3121			−1.27342			−0.636710	−	0.771103i	$-0.719705\pi$
$307$	−22.3121			−1.27342			−0.636710	+	0.771103i	$0.719705\pi$
$308$		0			0
$309$		−	2.88009i		−	0.163843i
$310$		0			0
$311$	4.90591			0.278189			0.139094	−	0.990279i	$-0.455581\pi$
$311$	4.90591			0.278189			0.139094	+	0.990279i	$0.455581\pi$
$312$		0			0
$313$			12.2324i			0.691414i	0.938343	+	0.345707i	$0.112361\pi$
$313$			12.2324i			0.691414i	−0.938343	+	0.345707i	$0.887639\pi$
$314$		0			0
$315$	−3.13784	−	5.53213i	−0.176797	−	0.311700i
$316$		0			0
$317$	13.6327			0.765689			0.382844	−	0.923813i	$-0.374945\pi$
$317$	13.6327			0.765689			0.382844	+	0.923813i	$0.374945\pi$
$318$		0			0
$319$	5.01190			0.280612
$320$		0			0
$321$	−45.4683			−2.53779
$322$		0			0
$323$	12.2042			0.679062
$324$		0			0
$325$	14.0537	−	23.5045i	0.779561	−	1.30379i
$326$		0			0
$327$			38.4743i			2.12764i
$328$		0			0
$329$	8.05426			0.444046
$330$		0			0
$331$		−	1.79947i		−	0.0989076i	−0.998776	−	0.0494538i	$-0.984252\pi$
$331$		−	1.79947i		−	0.0989076i	0.998776	−	0.0494538i	$-0.0157480\pi$
$332$		0			0
$333$	17.5338			0.960846
$334$		0			0
$335$	−7.63226	+	4.32904i	−0.416995	+	0.236521i
$336$		0			0
$337$			25.8851i			1.41005i	0.709183	+	0.705025i	$0.249064\pi$
$337$			25.8851i			1.41005i	−0.709183	+	0.705025i	$0.750936\pi$
$338$		0			0
$339$		−	11.4734i		−	0.623149i
$340$		0			0
$341$		−	5.97335i		−	0.323475i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−2.93656	−	5.17727i	−0.158099	−	0.278735i
$346$		0			0
$347$	6.06811			0.325753			0.162877	−	0.986646i	$-0.447923\pi$
$347$	6.06811			0.325753			0.162877	+	0.986646i	$0.447923\pi$
$348$		0			0
$349$		−	30.4463i		−	1.62975i	−0.579635	−	0.814876i	$-0.696805\pi$
$349$		−	30.4463i		−	1.62975i	0.579635	−	0.814876i	$-0.303195\pi$
$350$		0			0
$351$	2.06150			0.110035
$352$		0			0
$353$		−	0.771085i		−	0.0410407i	−0.999789	−	0.0205204i	$-0.993468\pi$
$353$		−	0.771085i		−	0.0410407i	0.999789	−	0.0205204i	$-0.00653229\pi$
$354$		0			0
$355$	8.85219	−	5.02099i	0.469825	−	0.266486i
$356$		0			0
$357$	−5.91134			−0.312861
$358$		0			0
$359$	−5.88246			−0.310465			−0.155232	−	0.987878i	$-0.549613\pi$
$359$	−5.88246			−0.310465			−0.155232	+	0.987878i	$0.549613\pi$
$360$		0			0
$361$	−5.91050			−0.311079
$362$		0			0
$363$	−10.8180			−0.567797
$364$		0			0
$365$	8.33514	+	14.6952i	0.436281	+	0.769180i
$366$		0			0
$367$			1.93231i			0.100866i	0.998727	+	0.0504329i	$0.0160601\pi$
$367$			1.93231i			0.100866i	−0.998727	+	0.0504329i	$0.983940\pi$
$368$		0			0
$369$	−26.7569			−1.39291
$370$		0			0
$371$			1.97706i			0.102644i
$372$		0			0
$373$	−11.4935			−0.595109			−0.297554	−	0.954705i	$-0.596171\pi$
$373$	−11.4935			−0.595109			−0.297554	+	0.954705i	$0.596171\pi$
$374$		0			0
$375$	−0.619764	+	27.0214i	−0.0320045	+	1.39538i
$376$		0			0
$377$			10.7463i			0.553462i
$378$		0			0
$379$			17.4833i			0.898059i	0.893517	+	0.449029i	$0.148230\pi$
$379$			17.4833i			0.898059i	−0.893517	+	0.449029i	$0.851770\pi$
$380$		0			0
$381$			31.1515i			1.59594i
$382$		0			0
$383$			22.9088i			1.17058i	0.810823	+	0.585292i	$0.199020\pi$
$383$			22.9088i			1.17058i	−0.810823	+	0.585292i	$0.800980\pi$
$384$		0			0
$385$	−4.96833	+	2.81805i	−0.253209	+	0.143621i
$386$		0			0
$387$	17.0614			0.867281
$388$		0			0
$389$			3.43009i			0.173912i	0.996212	+	0.0869562i	$0.0277140\pi$
$389$			3.43009i			0.173912i	−0.996212	+	0.0869562i	$0.972286\pi$
$390$		0			0
$391$	−2.69239			−0.136160
$392$		0			0
$393$			34.7711i			1.75397i
$394$		0			0
$395$	−16.5973	+	9.41406i	−0.835103	+	0.473673i
$396$		0			0
$397$	−27.6463			−1.38753			−0.693764	−	0.720203i	$-0.744049\pi$
$397$	−27.6463			−1.38753			−0.693764	+	0.720203i	$0.744049\pi$
$398$		0			0
$399$	12.0658			0.604048
$400$		0			0
$401$	−9.45990			−0.472405			−0.236202	−	0.971704i	$-0.575903\pi$
$401$	−9.45990			−0.472405			−0.236202	+	0.971704i	$0.575903\pi$
$402$		0			0
$403$	12.8078			0.638002
$404$		0			0
$405$	−18.3661	+	10.4173i	−0.912621	+	0.517641i
$406$		0			0
$407$		−	15.7469i		−	0.780543i
$408$		0			0
$409$	−1.55188			−0.0767354			−0.0383677	−	0.999264i	$-0.512216\pi$
$409$	−1.55188			−0.0767354			−0.0383677	+	0.999264i	$0.512216\pi$
$410$		0			0
$411$			25.7417i			1.26975i
$412$		0			0
$413$	1.44400			0.0710548
$414$		0			0
$415$	−16.0747	+	9.11761i	−0.789075	+	0.447566i
$416$		0			0
$417$		−	47.6471i		−	2.33329i
$418$		0			0
$419$			14.9384i			0.729790i	0.931049	+	0.364895i	$0.118895\pi$
$419$			14.9384i			0.729790i	−0.931049	+	0.364895i	$0.881105\pi$
$420$		0			0
$421$			32.8151i			1.59931i	0.600461	+	0.799654i	$0.294984\pi$
$421$			32.8151i			1.59931i	−0.600461	+	0.799654i	$0.705016\pi$
$422$		0			0
$423$			22.9088i			1.11386i
$424$		0			0
$425$	10.4935	+	6.27422i	0.509008	+	0.304345i
$426$		0			0
$427$	−8.71679			−0.421835
$428$		0			0
$429$			33.8230i			1.63299i
$430$		0			0
$431$	−20.5484			−0.989783			−0.494892	−	0.868955i	$-0.664792\pi$
$431$	−20.5484			−0.989783			−0.494892	+	0.868955i	$0.664792\pi$
$432$		0			0
$433$			35.7869i			1.71981i	0.510454	+	0.859905i	$0.329477\pi$
$433$			35.7869i			1.71981i	−0.510454	+	0.859905i	$0.670523\pi$
$434$		0			0
$435$	−5.23272	−	9.22549i	−0.250890	−	0.442328i
$436$		0			0
$437$	5.49553			0.262887
$438$		0			0
$439$	−32.4131			−1.54699			−0.773497	−	0.633800i	$-0.781494\pi$
$439$	−32.4131			−1.54699			−0.773497	+	0.633800i	$0.781494\pi$
$440$		0			0
$441$	−2.84431			−0.135443
$442$		0			0
$443$	−19.5723			−0.929907			−0.464953	−	0.885335i	$-0.653929\pi$
$443$	−19.5723			−0.929907			−0.464953	+	0.885335i	$0.653929\pi$
$444$		0			0
$445$	33.6062	−	19.0615i	1.59309	−	0.903603i
$446$		0			0
$447$		−	12.0882i		−	0.571752i
$448$		0			0
$449$	−20.6452			−0.974307			−0.487153	−	0.873316i	$-0.661965\pi$
$449$	−20.6452			−0.974307			−0.487153	+	0.873316i	$0.661965\pi$
$450$		0			0
$451$			24.0300i			1.13153i
$452$		0			0
$453$	−1.14007			−0.0535651
$454$		0			0
$455$	−6.04234	−	10.6529i	−0.283269	−	0.499414i
$456$		0			0
$457$			14.5940i			0.682680i	0.939940	+	0.341340i	$0.110881\pi$
$457$			14.5940i			0.682680i	−0.939940	+	0.341340i	$0.889119\pi$
$458$		0			0
$459$			0.920349i			0.0429582i
$460$		0			0
$461$			27.6406i			1.28735i	0.765298	+	0.643677i	$0.222592\pi$
$461$			27.6406i			1.28735i	−0.765298	+	0.643677i	$0.777408\pi$
$462$		0			0
$463$		−	7.73889i		−	0.359656i	−0.983698	−	0.179828i	$-0.942446\pi$
$463$		−	7.73889i		−	0.359656i	0.983698	−	0.179828i	$-0.0575542\pi$
$464$		0			0
$465$	−10.9953	+	6.23654i	−0.509893	+	0.289213i
$466$		0			0
$467$	19.9065			0.921165			0.460583	−	0.887617i	$-0.347641\pi$
$467$	19.9065			0.921165			0.460583	+	0.887617i	$0.347641\pi$
$468$		0			0
$469$			3.92408i			0.181197i
$470$		0			0
$471$	47.3237			2.18056
$472$		0			0
$473$		−	15.3226i		−	0.704536i
$474$		0			0
$475$	−21.4186	−	12.8065i	−0.982751	−	0.587604i
$476$		0			0
$477$	−5.62337			−0.257476
$478$		0			0
$479$	32.1124			1.46725			0.733627	−	0.679552i	$-0.237826\pi$
$479$	32.1124			1.46725			0.733627	+	0.679552i	$0.237826\pi$
$480$		0			0
$481$	33.7637			1.53949
$482$		0			0
$483$	−2.66186			−0.121119
$484$		0			0
$485$	−11.7125	−	20.6495i	−0.531836	−	0.937646i
$486$		0			0
$487$			31.9597i			1.44823i	0.689679	+	0.724115i	$0.257752\pi$
$487$			31.9597i			1.44823i	−0.689679	+	0.724115i	$0.742248\pi$
$488$		0			0
$489$	−36.7207			−1.66057
$490$		0			0
$491$		−	7.88545i		−	0.355865i	−0.984043	−	0.177933i	$-0.943059\pi$
$491$		−	7.88545i		−	0.355865i	0.984043	−	0.177933i	$-0.0569409\pi$
$492$		0			0
$493$	−4.79763			−0.216075
$494$		0			0
$495$	−8.01540	−	14.1315i	−0.360265	−	0.635162i
$496$		0			0
$497$		−	4.55130i		−	0.204154i
$498$		0			0
$499$		−	10.8487i		−	0.485654i	−0.970070	−	0.242827i	$-0.921925\pi$
$499$		−	10.8487i		−	0.485654i	0.970070	−	0.242827i	$-0.0780747\pi$
$500$		0			0
$501$			37.2565i			1.66450i
$502$		0			0
$503$			0.0921625i			0.00410932i	0.999998	+	0.00205466i	$0.000654019\pi$
$503$			0.0921625i			0.00410932i	−0.999998	+	0.00205466i	$0.999346\pi$
$504$		0			0
$505$	−15.9174	−	28.0630i	−0.708316	−	1.24879i
$506$		0			0
$507$	−41.0944			−1.82506
$508$		0			0
$509$			28.3558i			1.25685i	0.777872	+	0.628423i	$0.216299\pi$
$509$			28.3558i			1.25685i	−0.777872	+	0.628423i	$0.783701\pi$
$510$		0			0
$511$	7.55542			0.334232
$512$		0			0
$513$		−	1.87856i		−	0.0829403i
$514$		0			0
$515$	1.31430	+	2.31716i	0.0579149	+	0.102106i
$516$		0			0
$517$	20.5741			0.904847
$518$		0			0
$519$	−2.70329			−0.118661
$520$		0			0
$521$	−14.8781			−0.651819			−0.325910	−	0.945401i	$-0.605671\pi$
$521$	−14.8781			−0.651819			−0.325910	+	0.945401i	$0.605671\pi$
$522$		0			0
$523$	28.4064			1.24212			0.621062	−	0.783762i	$-0.286702\pi$
$523$	28.4064			1.24212			0.621062	+	0.783762i	$0.286702\pi$
$524$		0			0
$525$	10.3745	+	6.20307i	0.452779	+	0.270724i
$526$		0			0
$527$			5.71798i			0.249079i
$528$		0			0
$529$	21.7876			0.947288
$530$		0			0
$531$			4.10719i			0.178237i
$532$		0			0
$533$	−51.5241			−2.23176
$534$		0			0
$535$	36.5812	−	20.7489i	1.58154	−	0.897055i
$536$		0			0
$537$		−	46.3965i		−	2.00216i
$538$		0			0
$539$			2.55443i			0.110027i
$540$		0			0
$541$		−	6.30660i		−	0.271142i	−0.990768	−	0.135571i	$-0.956713\pi$
$541$		−	6.30660i		−	0.271142i	0.990768	−	0.135571i	$-0.0432868\pi$
$542$		0			0
$543$			54.0251i			2.31844i
$544$		0			0
$545$	−17.5573	−	30.9542i	−0.752073	−	1.32593i
$546$		0			0
$547$	13.7648			0.588542			0.294271	−	0.955722i	$-0.404923\pi$
$547$	13.7648			0.588542			0.294271	+	0.955722i	$0.404923\pi$
$548$		0			0
$549$		−	24.7932i		−	1.05815i
$550$		0			0
$551$	9.79262			0.417179
$552$		0			0
$553$			8.53342i			0.362878i
$554$		0			0
$555$	−28.9855	+	16.4407i	−1.23037	+	0.697868i
$556$		0			0
$557$	−25.2978			−1.07190			−0.535950	−	0.844249i	$-0.680047\pi$
$557$	−25.2978			−1.07190			−0.535950	+	0.844249i	$0.680047\pi$
$558$		0			0
$559$	32.8541			1.38958
$560$		0			0
$561$	−15.1001			−0.637528
$562$		0			0
$563$	−0.0730290			−0.00307781			−0.00153890	−	0.999999i	$-0.500490\pi$
$563$	−0.0730290			−0.00307781			−0.00153890	+	0.999999i	$0.500490\pi$
$564$		0			0
$565$	5.23575	+	9.23082i	0.220270	+	0.388344i
$566$		0			0
$567$			9.44284i			0.396562i
$568$		0			0
$569$	4.34990			0.182357			0.0911787	−	0.995835i	$-0.470937\pi$
$569$	4.34990			0.182357			0.0911787	+	0.995835i	$0.470937\pi$
$570$		0			0
$571$			7.45790i			0.312103i	0.987749	+	0.156052i	$0.0498766\pi$
$571$			7.45790i			0.312103i	−0.987749	+	0.156052i	$0.950123\pi$
$572$		0			0
$573$	−31.1367			−1.30075
$574$		0			0
$575$	4.72517	+	2.82526i	0.197053	+	0.117822i
$576$		0			0
$577$			3.44590i			0.143455i	0.997424	+	0.0717273i	$0.0228511\pi$
$577$			3.44590i			0.143455i	−0.997424	+	0.0717273i	$0.977149\pi$
$578$		0			0
$579$			34.3287i			1.42665i
$580$		0			0
$581$			8.26470i			0.342878i
$582$		0			0
$583$			5.05027i			0.209161i
$584$		0			0
$585$	30.3000	−	17.1863i	1.25275	−	0.710565i
$586$		0			0
$587$	21.3571			0.881502			0.440751	−	0.897629i	$-0.354712\pi$
$587$	21.3571			0.881502			0.440751	+	0.897629i	$0.354712\pi$
$588$		0			0
$589$		−	11.6712i		−	0.480902i
$590$		0			0
$591$	19.6944			0.810120
$592$		0			0
$593$		−	9.05876i		−	0.371999i	−0.982550	−	0.185999i	$-0.940448\pi$
$593$		−	9.05876i		−	0.371999i	0.982550	−	0.185999i	$-0.0595523\pi$
$594$		0			0
$595$	4.75592	−	2.69757i	0.194974	−	0.110590i
$596$		0			0
$597$	59.1937			2.42264
$598$		0			0
$599$	6.73982			0.275381			0.137691	−	0.990475i	$-0.456032\pi$
$599$	6.73982			0.275381			0.137691	+	0.990475i	$0.456032\pi$
$600$		0			0
$601$	25.1963			1.02778			0.513890	−	0.857856i	$-0.328204\pi$
$601$	25.1963			1.02778			0.513890	+	0.857856i	$0.328204\pi$
$602$		0			0
$603$	−11.1613			−0.454523
$604$		0			0
$605$	8.70353	−	4.93667i	0.353849	−	0.200704i
$606$		0			0
$607$		−	17.2576i		−	0.700463i	−0.936663	−	0.350231i	$-0.886103\pi$
$607$		−	17.2576i		−	0.700463i	0.936663	−	0.350231i	$-0.113897\pi$
$608$		0			0
$609$	−4.74323			−0.192205
$610$		0			0
$611$			44.1140i			1.78466i
$612$		0			0
$613$	1.55561			0.0628305			0.0314153	−	0.999506i	$-0.489999\pi$
$613$	1.55561			0.0628305			0.0314153	+	0.999506i	$0.489999\pi$
$614$		0			0
$615$	44.2325	−	25.0888i	1.78363	−	1.01168i
$616$		0			0
$617$			4.19511i			0.168889i	0.996428	+	0.0844443i	$0.0269115\pi$
$617$			4.19511i			0.168889i	−0.996428	+	0.0844443i	$0.973088\pi$
$618$		0			0
$619$		−	25.7431i		−	1.03470i	−0.855773	−	0.517351i	$-0.826918\pi$
$619$		−	25.7431i		−	1.03470i	0.855773	−	0.517351i	$-0.173082\pi$
$620$		0			0
$621$			0.414431i			0.0166305i
$622$		0			0
$623$		−	17.2784i		−	0.692245i
$624$		0			0
$625$	−11.8323	−	22.0227i	−0.473291	−	0.880906i
$626$		0			0
$627$	30.8214			1.23089
$628$		0			0
$629$			15.0737i			0.601026i
$630$		0			0
$631$	4.14801			0.165130			0.0825648	−	0.996586i	$-0.473689\pi$
$631$	4.14801			0.165130			0.0825648	+	0.996586i	$0.473689\pi$
$632$		0			0
$633$			62.0404i			2.46588i
$634$		0			0
$635$	−14.2156	−	25.0627i	−0.564131	−	0.994584i
$636$		0			0
$637$	−5.47711			−0.217011
$638$		0			0
$639$	12.9453			0.512108
$640$		0			0
$641$	12.3082			0.486144			0.243072	−	0.970008i	$-0.421845\pi$
$641$	12.3082			0.486144			0.243072	+	0.970008i	$0.421845\pi$
$642$		0			0
$643$	−14.9655			−0.590181			−0.295090	−	0.955469i	$-0.595350\pi$
$643$	−14.9655			−0.590181			−0.295090	+	0.955469i	$0.595350\pi$
$644$		0			0
$645$	−28.2047	+	15.9978i	−1.11056	+	0.629911i
$646$		0			0
$647$			40.1417i			1.57813i	0.614307	+	0.789067i	$0.289436\pi$
$647$			40.1417i			1.57813i	−0.614307	+	0.789067i	$0.710564\pi$
$648$		0			0
$649$	3.68862			0.144791
$650$		0			0
$651$			5.65314i			0.221564i
$652$		0			0
$653$	−23.2472			−0.909735			−0.454867	−	0.890559i	$-0.650313\pi$
$653$	−23.2472			−0.909735			−0.454867	+	0.890559i	$0.650313\pi$
$654$		0			0
$655$	−15.8674	−	27.9748i	−0.619990	−	1.09307i
$656$		0			0
$657$			21.4900i			0.838403i
$658$		0			0
$659$		−	14.7905i		−	0.576158i	−0.957607	−	0.288079i	$-0.906983\pi$
$659$		−	14.7905i		−	0.576158i	0.957607	−	0.288079i	$-0.0930165\pi$
$660$		0			0
$661$		−	37.3069i		−	1.45107i	−0.688184	−	0.725536i	$-0.741592\pi$
$661$		−	37.3069i		−	1.45107i	0.688184	−	0.725536i	$-0.258408\pi$
$662$		0			0
$663$		−	32.3771i		−	1.25742i
$664$		0			0
$665$	−9.70748	+	5.50611i	−0.376440	+	0.213518i
$666$		0			0
$667$	−2.16036			−0.0836495
$668$		0			0
$669$			0.242078i			0.00935929i
$670$		0			0
$671$	−22.2665			−0.859588
$672$		0			0
$673$		−	20.8085i		−	0.802110i	−0.916054	−	0.401055i	$-0.868644\pi$
$673$		−	20.8085i		−	0.802110i	0.916054	−	0.401055i	$-0.131356\pi$
$674$		0			0
$675$	0.965770	−	1.61522i	0.0371725	−	0.0621699i
$676$		0			0
$677$	−22.9322			−0.881357			−0.440678	−	0.897665i	$-0.645262\pi$
$677$	−22.9322			−0.881357			−0.440678	+	0.897665i	$0.645262\pi$
$678$		0			0
$679$	−10.6168			−0.407436
$680$		0			0
$681$	72.4658			2.77690
$682$		0			0
$683$	40.0656			1.53307			0.766534	−	0.642204i	$-0.221980\pi$
$683$	40.0656			1.53307			0.766534	+	0.642204i	$0.221980\pi$
$684$		0			0
$685$	−11.7470	−	20.7103i	−0.448828	−	0.791301i
$686$		0			0
$687$		−	12.4074i		−	0.473370i
$688$		0			0
$689$	−10.8286			−0.412536
$690$		0			0
$691$		−	14.3188i		−	0.544712i	−0.962197	−	0.272356i	$-0.912197\pi$
$691$		−	14.3188i		−	0.544712i	0.962197	−	0.272356i	$-0.0878029\pi$
$692$		0			0
$693$	−7.26560			−0.275997
$694$		0			0
$695$	21.7432	+	38.3341i	0.824767	+	1.45410i
$696$		0			0
$697$		−	23.0027i		−	0.871289i
$698$		0			0
$699$		−	47.3058i		−	1.78927i
$700$		0			0
$701$		−	20.6746i		−	0.780870i	−0.920631	−	0.390435i	$-0.872325\pi$
$701$		−	20.6746i		−	0.780870i	0.920631	−	0.390435i	$-0.127675\pi$
$702$		0			0
$703$		−	30.7674i		−	1.16041i
$704$		0			0
$705$	−21.4806	−	37.8711i	−0.809005	−	1.42631i
$706$		0			0
$707$	−14.4284			−0.542637
$708$		0			0
$709$		−	23.9451i		−	0.899276i	−0.893211	−	0.449638i	$-0.851553\pi$
$709$		−	23.9451i		−	0.899276i	0.893211	−	0.449638i	$-0.148447\pi$
$710$		0			0
$711$	−24.2717			−0.910259
$712$		0			0
$713$			2.57479i			0.0964267i
$714$		0			0
$715$	−15.4348	−	27.2121i	−0.577227	−	1.01767i
$716$		0			0
$717$	−68.2952			−2.55053
$718$		0			0
$719$	19.8543			0.740440			0.370220	−	0.928944i	$-0.379282\pi$
$719$	19.8543			0.740440			0.370220	+	0.928944i	$0.379282\pi$
$720$		0			0
$721$	1.19135			0.0443683
$722$		0			0
$723$	47.5542			1.76856
$724$		0			0
$725$	8.41990	+	5.03440i	0.312707	+	0.186973i
$726$		0			0
$727$		−	44.0636i		−	1.63423i	−0.576476	−	0.817114i	$-0.695572\pi$
$727$		−	44.0636i		−	1.63423i	0.576476	−	0.817114i	$-0.304428\pi$
$728$		0			0
$729$	−24.1286			−0.893652
$730$		0			0
$731$			14.6676i			0.542500i
$732$		0			0
$733$	−9.56797			−0.353401			−0.176700	−	0.984265i	$-0.556542\pi$
$733$	−9.56797			−0.353401			−0.176700	+	0.984265i	$0.556542\pi$
$734$		0			0
$735$	4.70199	−	2.66698i	0.173436	−	0.0983732i
$736$		0			0
$737$			10.0238i			0.369231i
$738$		0			0
$739$			26.4892i			0.974421i	0.873285	+	0.487211i	$0.161986\pi$
$739$			26.4892i			0.974421i	−0.873285	+	0.487211i	$0.838014\pi$
$740$		0			0
$741$			66.0859i			2.42773i
$742$		0			0
$743$		−	20.0864i		−	0.736897i	−0.929648	−	0.368448i	$-0.879889\pi$
$743$		−	20.0864i		−	0.736897i	0.929648	−	0.368448i	$-0.120111\pi$
$744$		0			0
$745$	5.51631	+	9.72546i	0.202102	+	0.356313i
$746$		0			0
$747$	−23.5074			−0.860089
$748$		0			0
$749$		−	18.8080i		−	0.687228i
$750$		0			0
$751$	−34.0342			−1.24192			−0.620962	−	0.783841i	$-0.713258\pi$
$751$	−34.0342			−1.24192			−0.620962	+	0.783841i	$0.713258\pi$
$752$		0			0
$753$			21.8534i			0.796384i
$754$		0			0
$755$	0.917233	−	0.520257i	0.0333815	−	0.0189341i
$756$		0			0
$757$	13.0130			0.472965			0.236483	−	0.971636i	$-0.424005\pi$
$757$	13.0130			0.472965			0.236483	+	0.971636i	$0.424005\pi$
$758$		0			0
$759$	−6.79955			−0.246808
$760$		0			0
$761$	−5.21488			−0.189039			−0.0945196	−	0.995523i	$-0.530132\pi$
$761$	−5.21488			−0.189039			−0.0945196	+	0.995523i	$0.530132\pi$
$762$		0			0
$763$	−15.9149			−0.576159
$764$		0			0
$765$	7.67273	+	13.5273i	0.277408	+	0.489081i
$766$		0			0
$767$			7.90897i			0.285576i
$768$		0			0
$769$	44.2904			1.59715			0.798577	−	0.601893i	$-0.205587\pi$
$769$	44.2904			1.59715			0.798577	+	0.601893i	$0.205587\pi$
$770$		0			0
$771$			56.0669i			2.01920i
$772$		0			0
$773$	36.3166			1.30622			0.653109	−	0.757264i	$-0.273464\pi$
$773$	36.3166			1.30622			0.653109	+	0.757264i	$0.273464\pi$
$774$		0			0
$775$	6.00017	−	10.0351i	0.215533	−	0.360472i
$776$		0			0
$777$			14.9027i			0.534632i
$778$		0			0
$779$			46.9516i			1.68222i
$780$		0			0
$781$		−	11.6260i		−	0.416011i
$782$		0			0
$783$			0.738483i			0.0263912i
$784$		0			0
$785$	−38.0739	+	21.5956i	−1.35892	+	0.770781i
$786$		0			0
$787$	−7.81506			−0.278577			−0.139288	−	0.990252i	$-0.544482\pi$
$787$	−7.81506			−0.278577			−0.139288	+	0.990252i	$0.544482\pi$
$788$		0			0
$789$			6.00825i			0.213899i
$790$		0			0
$791$	4.74597			0.168747
$792$		0			0
$793$		−	47.7428i		−	1.69540i
$794$		0			0
$795$	9.29612	−	5.27279i	0.329699	−	0.187006i
$796$		0			0
$797$	−36.2395			−1.28367			−0.641834	−	0.766844i	$-0.721826\pi$
$797$	−36.2395			−1.28367			−0.641834	+	0.766844i	$0.721826\pi$
$798$		0			0
$799$	−19.6945			−0.696741
$800$		0			0
$801$	49.1451			1.73646
$802$		0			0
$803$	19.2998			0.681076
$804$		0			0
$805$	2.14158	−	1.21471i	0.0754807	−	0.0428129i
$806$		0			0
$807$		−	65.4169i		−	2.30278i
$808$		0			0
$809$	−11.8407			−0.416298			−0.208149	−	0.978097i	$-0.566744\pi$
$809$	−11.8407			−0.416298			−0.208149	+	0.978097i	$0.566744\pi$
$810$		0			0
$811$		−	43.4442i		−	1.52553i	−0.646675	−	0.762766i	$-0.723841\pi$
$811$		−	43.4442i		−	1.52553i	0.646675	−	0.762766i	$-0.276159\pi$
$812$		0			0
$813$	−4.79784			−0.168268
$814$		0			0
$815$	29.5434	−	16.7571i	1.03486	−	0.586975i
$816$		0			0
$817$		−	29.9385i		−	1.04742i
$818$		0			0
$819$		−	15.5786i		−	0.544360i
$820$		0			0
$821$		−	35.3166i		−	1.23256i	−0.787528	−	0.616279i	$-0.788639\pi$
$821$		−	35.3166i		−	1.23256i	0.787528	−	0.616279i	$-0.211361\pi$
$822$		0			0
$823$			6.15779i			0.214647i	0.994224	+	0.107324i	$0.0342281\pi$
$823$			6.15779i			0.214647i	−0.994224	+	0.107324i	$0.965772\pi$
$824$		0			0
$825$	26.5009	+	15.8453i	0.922643	+	0.551664i
$826$		0			0
$827$	19.6425			0.683037			0.341518	−	0.939875i	$-0.389059\pi$
$827$	19.6425			0.683037			0.341518	+	0.939875i	$0.389059\pi$
$828$		0			0
$829$			11.4645i			0.398178i	0.979981	+	0.199089i	$0.0637982\pi$
$829$			11.4645i			0.398178i	−0.979981	+	0.199089i	$0.936202\pi$
$830$		0			0
$831$	−37.0341			−1.28470
$832$		0			0
$833$		−	2.44523i		−	0.0847222i
$834$		0			0
$835$	−17.0016	−	29.9744i	−0.588364	−	1.03731i
$836$		0			0
$837$	0.880149			0.0304224
$838$		0			0
$839$	−8.74339			−0.301855			−0.150928	−	0.988545i	$-0.548226\pi$
$839$	−8.74339			−0.301855			−0.150928	+	0.988545i	$0.548226\pi$
$840$		0			0
$841$	25.1504			0.867255
$842$		0			0
$843$	−19.8384			−0.683269
$844$		0			0
$845$	33.0622	−	18.7529i	1.13737	−	0.645121i
$846$		0			0
$847$		−	4.47486i		−	0.153758i
$848$		0			0
$849$	−50.6485			−1.73825
$850$		0			0
$851$			6.78762i			0.232677i
$852$		0			0
$853$	−2.58567			−0.0885318			−0.0442659	−	0.999020i	$-0.514095\pi$
$853$	−2.58567			−0.0885318			−0.0442659	+	0.999020i	$0.514095\pi$
$854$		0			0
$855$	−15.6611	−	27.6111i	−0.535598	−	0.944279i
$856$		0			0
$857$			9.13080i			0.311902i	0.987765	+	0.155951i	$0.0498442\pi$
$857$			9.13080i			0.311902i	−0.987765	+	0.155951i	$0.950156\pi$
$858$		0			0
$859$		−	49.7731i		−	1.69824i	−0.528203	−	0.849118i	$-0.677134\pi$
$859$		−	49.7731i		−	1.69824i	0.528203	−	0.849118i	$-0.322866\pi$
$860$		0			0
$861$		−	22.7419i		−	0.775040i
$862$		0			0
$863$			46.0261i			1.56675i	0.621552	+	0.783373i	$0.286502\pi$
$863$			46.0261i			1.56675i	−0.621552	+	0.783373i	$0.713498\pi$
$864$		0			0
$865$	2.17492	−	1.23362i	0.0739493	−	0.0419443i
$866$		0			0
$867$	−26.6429			−0.904841
$868$		0			0
$869$			21.7981i			0.739449i
$870$		0			0
$871$	−21.4926			−0.728249
$872$		0			0
$873$		−	30.1975i		−	1.02203i
$874$		0			0
$875$	−11.1774	−	0.256365i	−0.377865	−	0.00866674i
$876$		0			0
$877$	55.8392			1.88556			0.942778	−	0.333423i	$-0.108204\pi$
$877$	55.8392			1.88556			0.942778	+	0.333423i	$0.108204\pi$
$878$		0			0
$879$	63.6297			2.14618
$880$		0			0
$881$	0.502969			0.0169455			0.00847273	−	0.999964i	$-0.497303\pi$
$881$	0.502969			0.0169455			0.00847273	+	0.999964i	$0.497303\pi$
$882$		0			0
$883$	16.6391			0.559952			0.279976	−	0.960007i	$-0.409673\pi$
$883$	16.6391			0.559952			0.279976	+	0.960007i	$0.409673\pi$
$884$		0			0
$885$	−3.85114	−	6.78970i	−0.129455	−	0.228233i
$886$		0			0
$887$			18.2792i			0.613755i	0.951749	+	0.306878i	$0.0992842\pi$
$887$			18.2792i			0.613755i	−0.951749	+	0.306878i	$0.900716\pi$
$888$		0			0
$889$	−12.8858			−0.432177
$890$		0			0
$891$			24.1211i			0.808087i
$892$		0			0
$893$	40.1991			1.34521
$894$		0			0
$895$	21.1725	+	37.3280i	0.707720	+	1.24774i
$896$		0			0
$897$		−	14.5793i		−	0.486788i
$898$		0			0
$899$			4.58808i			0.153021i
$900$		0			0
$901$		−	4.83436i		−	0.161056i
$902$		0			0
$903$			14.5013i			0.482571i
$904$		0			0
$905$	−24.6537	−	43.4655i	−0.819518	−	1.44484i
$906$		0			0
$907$	3.94455			0.130976			0.0654882	−	0.997853i	$-0.479140\pi$
$907$	3.94455			0.130976			0.0654882	+	0.997853i	$0.479140\pi$
$908$		0			0
$909$		−	41.0389i		−	1.36117i
$910$		0			0
$911$	−4.30622			−0.142672			−0.0713358	−	0.997452i	$-0.522726\pi$
$911$	−4.30622			−0.142672			−0.0713358	+	0.997452i	$0.522726\pi$
$912$		0			0
$913$			21.1116i			0.698693i
$914$		0			0
$915$	23.2475	+	40.9863i	0.768540	+	1.35497i
$916$		0			0
$917$	−14.3831			−0.474971
$918$		0			0
$919$	−29.2170			−0.963780			−0.481890	−	0.876232i	$-0.660050\pi$
$919$	−29.2170			−0.963780			−0.481890	+	0.876232i	$0.660050\pi$
$920$		0			0
$921$	53.9396			1.77737
$922$		0			0
$923$	24.9279			0.820513
$924$		0			0
$925$	15.8176	−	26.4544i	0.520078	−	0.869816i
$926$		0			0
$927$			3.38857i			0.111295i
$928$		0			0
$929$	13.0429			0.427922			0.213961	−	0.976842i	$-0.431363\pi$
$929$	13.0429			0.427922			0.213961	+	0.976842i	$0.431363\pi$
$930$		0			0
$931$			4.99104i			0.163575i
$932$		0			0
$933$	−11.8600			−0.388280
$934$		0			0
$935$	12.1487	−	6.89078i	0.397305	−	0.225352i
$936$		0			0
$937$			43.7965i			1.43077i	0.698731	+	0.715385i	$0.253749\pi$
$937$			43.7965i			1.43077i	−0.698731	+	0.715385i	$0.746251\pi$
$938$		0			0
$939$		−	29.5717i		−	0.965037i
$940$		0			0
$941$			12.3386i			0.402227i	0.979568	+	0.201114i	$0.0644560\pi$
$941$			12.3386i			0.402227i	−0.979568	+	0.201114i	$0.935544\pi$
$942$		0			0
$943$		−	10.3580i		−	0.337304i
$944$		0			0
$945$	−0.415228	−	0.732063i	−0.0135074	−	0.0238140i
$946$		0			0
$947$	44.3160			1.44008			0.720039	−	0.693934i	$-0.244124\pi$
$947$	44.3160			1.44008			0.720039	+	0.693934i	$0.244124\pi$
$948$		0			0
$949$			41.3819i			1.34331i
$950$		0			0
$951$	−32.9571			−1.06871
$952$		0			0
$953$			29.5353i			0.956744i	0.878157	+	0.478372i	$0.158773\pi$
$953$			29.5353i			0.956744i	−0.878157	+	0.478372i	$0.841227\pi$
$954$		0			0
$955$	25.0508	−	14.2089i	0.810624	−	0.459788i
$956$		0			0
$957$	−12.1163			−0.391663
$958$		0			0
$959$	−10.6481			−0.343845
$960$		0			0
$961$	−25.5318			−0.823606
$962$		0			0
$963$	53.4957			1.72387
$964$		0			0
$965$	−15.6655	−	27.6189i	−0.504290	−	0.889083i
$966$		0			0
$967$		−	11.0409i		−	0.355053i	−0.984116	−	0.177527i	$-0.943190\pi$
$967$		−	11.0409i		−	0.355053i	0.984116	−	0.177527i	$-0.0568095\pi$
$968$		0			0
$969$	−29.5038			−0.947797
$970$		0			0
$971$			51.1639i			1.64193i	0.570981	+	0.820963i	$0.306563\pi$
$971$			51.1639i			1.64193i	−0.570981	+	0.820963i	$0.693437\pi$
$972$		0			0
$973$	19.7092			0.631849
$974$		0			0
$975$	−33.9749	+	56.8221i	−1.08807	+	1.81976i
$976$		0			0
$977$			0.412964i			0.0132119i	0.999978	+	0.00660595i	$0.00210275\pi$
$977$			0.412964i			0.0132119i	−0.999978	+	0.00660595i	$0.997897\pi$
$978$		0			0
$979$		−	44.1366i		−	1.41061i
$980$		0			0
$981$		−	45.2670i		−	1.44526i
$982$		0			0
$983$			23.4442i			0.747754i	0.927478	+	0.373877i	$0.121972\pi$
$983$			23.4442i			0.747754i	−0.927478	+	0.373877i	$0.878028\pi$
$984$		0			0
$985$	−15.8450	+	8.98732i	−0.504863	+	0.286360i
$986$		0			0
$987$	−19.4712			−0.619774
$988$		0			0
$989$			6.60477i			0.210019i
$990$		0			0
$991$	−9.55300			−0.303461			−0.151731	−	0.988422i	$-0.548485\pi$
$991$	−9.55300			−0.303461			−0.151731	+	0.988422i	$0.548485\pi$
$992$		0			0
$993$			4.35021i			0.138050i
$994$		0			0
$995$	−47.6238	+	27.0124i	−1.50978	+	0.856349i
$996$		0			0
$997$	10.6904			0.338568			0.169284	−	0.985567i	$-0.445855\pi$
$997$	10.6904			0.338568			0.169284	+	0.985567i	$0.445855\pi$
$998$		0			0
$999$	2.32024			0.0734090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.d.1569.3	yes	24
4.3	odd	2	inner	2240.2.l.d.1569.22	yes	24
5.4	even	2		2240.2.l.c.1569.22	yes	24
8.3	odd	2		2240.2.l.c.1569.4	yes	24
8.5	even	2		2240.2.l.c.1569.21	yes	24
20.19	odd	2		2240.2.l.c.1569.3	✓	24
40.19	odd	2	inner	2240.2.l.d.1569.21	yes	24
40.29	even	2	inner	2240.2.l.d.1569.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.c.1569.3	✓	24	20.19	odd	2
2240.2.l.c.1569.4	yes	24	8.3	odd	2
2240.2.l.c.1569.21	yes	24	8.5	even	2
2240.2.l.c.1569.22	yes	24	5.4	even	2
2240.2.l.d.1569.3	yes	24	1.1	even	1	trivial
2240.2.l.d.1569.4	yes	24	40.29	even	2	inner
2240.2.l.d.1569.21	yes	24	40.19	odd	2	inner
2240.2.l.d.1569.22	yes	24	4.3	odd	2	inner

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.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.41750 q^{3} +(1.94498 - 1.10320i) q^{5} -1.00000i q^{7} +2.84431 q^{9} +O(q^{10})\)	\(q-2.41750 q^{3} +(1.94498 - 1.10320i) q^{5} -1.00000i q^{7} +2.84431 q^{9} -2.55443i q^{11} +5.47711 q^{13} +(-4.70199 + 2.66698i) q^{15} +2.44523i q^{17} -4.99104i q^{19} +2.41750i q^{21} +1.10108i q^{23} +(2.56590 - 4.29140i) q^{25} +0.376386 q^{27} +1.96204i q^{29} +2.33842 q^{31} +6.17535i q^{33} +(-1.10320 - 1.94498i) q^{35} +6.16452 q^{37} -13.2409 q^{39} -9.40718 q^{41} +5.99845 q^{43} +(5.53213 - 3.13784i) q^{45} +8.05426i q^{47} -1.00000 q^{49} -5.91134i q^{51} -1.97706 q^{53} +(-2.81805 - 4.96833i) q^{55} +12.0658i q^{57} +1.44400i q^{59} -8.71679i q^{61} -2.84431i q^{63} +(10.6529 - 6.04234i) q^{65} -3.92408 q^{67} -2.66186i q^{69} +4.55130 q^{71} +7.55542i q^{73} +(-6.20307 + 10.3745i) q^{75} -2.55443 q^{77} -8.53342 q^{79} -9.44284 q^{81} -8.26470 q^{83} +(2.69757 + 4.75592i) q^{85} -4.74323i q^{87} +17.2784 q^{89} -5.47711i q^{91} -5.65314 q^{93} +(-5.50611 - 9.70748i) q^{95} -10.6168i q^{97} -7.26560i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} + 12 q^{9}+O(q^{10}) \) 24 * q + 4 * q^5 + 12 * q^9	\( 24 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{37} + 48 q^{45} - 24 q^{49} - 88 q^{53} + 36 q^{65} - 20 q^{77} + 16 q^{81} + 56 q^{85} - 40 q^{89} - 24 q^{93}+O(q^{100}) \) 24 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^37 + 48 * q^45 - 24 * q^49 - 88 * q^53 + 36 * q^65 - 20 * q^77 + 16 * q^81 + 56 * q^85 - 40 * q^89 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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