LMFDB - Embedded newform 2240.2.l.c.1569.13

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			1569.13
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.c.1569.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.236390 q^{3} +(1.55127 - 1.61046i) q^{5} -1.00000i q^{7} -2.94412 q^{9} +O(q^{10})$	$q+0.236390 q^{3} +(1.55127 - 1.61046i) q^{5} -1.00000i q^{7} -2.94412 q^{9} -3.36940i q^{11} -1.01585 q^{13} +(0.366703 - 0.380696i) q^{15} +0.988642i q^{17} -2.29607i q^{19} -0.236390i q^{21} +0.806457i q^{23} +(-0.187153 - 4.99650i) q^{25} -1.40513 q^{27} +3.17517i q^{29} -1.77004 q^{31} -0.796493i q^{33} +(-1.61046 - 1.55127i) q^{35} -9.11347 q^{37} -0.240136 q^{39} -8.66890 q^{41} +11.0221 q^{43} +(-4.56711 + 4.74138i) q^{45} -9.87647i q^{47} -1.00000 q^{49} +0.233705i q^{51} -4.36450 q^{53} +(-5.42629 - 5.22684i) q^{55} -0.542768i q^{57} -2.34114i q^{59} +10.2211i q^{61} +2.94412i q^{63} +(-1.57585 + 1.63598i) q^{65} -6.35033 q^{67} +0.190638i q^{69} -12.5115 q^{71} -13.0894i q^{73} +(-0.0442412 - 1.18112i) q^{75} -3.36940 q^{77} -7.51602 q^{79} +8.50020 q^{81} +14.5463 q^{83} +(1.59217 + 1.53365i) q^{85} +0.750577i q^{87} +0.147020 q^{89} +1.01585i q^{91} -0.418419 q^{93} +(-3.69773 - 3.56182i) q^{95} +9.75135i q^{97} +9.91993i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.236390			0.136480			0.0682399	−	0.997669i	$-0.478262\pi$
$3$	0.236390			0.136480			0.0682399	+	0.997669i	$0.478262\pi$
$4$		0			0
$5$	1.55127	−	1.61046i	0.693747	−	0.720219i
$5$	1.55127	−	1.61046i	0.693747	−	0.720219i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.94412			−0.981373
$10$		0			0
$11$		−	3.36940i		−	1.01591i	−0.861383	−	0.507957i	$-0.830401\pi$
$11$		−	3.36940i		−	1.01591i	0.861383	−	0.507957i	$-0.169599\pi$
$12$		0			0
$13$	−1.01585			−0.281745			−0.140872	−	0.990028i	$-0.544991\pi$
$13$	−1.01585			−0.281745			−0.140872	+	0.990028i	$0.544991\pi$
$14$		0			0
$15$	0.366703	−	0.380696i	0.0946824	−	0.0982953i
$16$		0			0
$17$			0.988642i			0.239781i	0.992787	+	0.119890i	$0.0382543\pi$
$17$			0.988642i			0.239781i	−0.992787	+	0.119890i	$0.961746\pi$
$18$		0			0
$19$		−	2.29607i		−	0.526755i	−0.964693	−	0.263378i	$-0.915163\pi$
$19$		−	2.29607i		−	0.526755i	0.964693	−	0.263378i	$-0.0848365\pi$
$20$		0			0
$21$		−	0.236390i		−	0.0515845i
$22$		0			0
$23$			0.806457i			0.168158i	0.996459	+	0.0840790i	$0.0267948\pi$
$23$			0.806457i			0.168158i	−0.996459	+	0.0840790i	$0.973205\pi$
$24$		0			0
$25$	−0.187153	−	4.99650i	−0.0374307	−	0.999299i
$26$		0			0
$27$	−1.40513			−0.270417
$28$		0			0
$29$			3.17517i			0.589613i	0.955557	+	0.294807i	$0.0952553\pi$
$29$			3.17517i			0.589613i	−0.955557	+	0.294807i	$0.904745\pi$
$30$		0			0
$31$	−1.77004			−0.317908			−0.158954	−	0.987286i	$-0.550812\pi$
$31$	−1.77004			−0.317908			−0.158954	+	0.987286i	$0.550812\pi$
$32$		0			0
$33$		−	0.796493i		−	0.138652i
$34$		0			0
$35$	−1.61046	−	1.55127i	−0.272217	−	0.262212i
$36$		0			0
$37$	−9.11347			−1.49825			−0.749123	−	0.662431i	$-0.769525\pi$
$37$	−9.11347			−1.49825			−0.749123	+	0.662431i	$0.769525\pi$
$38$		0			0
$39$	−0.240136			−0.0384525
$40$		0			0
$41$	−8.66890			−1.35385			−0.676927	−	0.736050i	$-0.736689\pi$
$41$	−8.66890			−1.35385			−0.676927	+	0.736050i	$0.736689\pi$
$42$		0			0
$43$	11.0221			1.68086			0.840430	−	0.541921i	$-0.182303\pi$
$43$	11.0221			1.68086			0.840430	+	0.541921i	$0.182303\pi$
$44$		0			0
$45$	−4.56711	+	4.74138i	−0.680825	+	0.706804i
$46$		0			0
$47$		−	9.87647i		−	1.44063i	−0.693646	−	0.720316i	$-0.743997\pi$
$47$		−	9.87647i		−	1.44063i	0.693646	−	0.720316i	$-0.256003\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			0.233705i			0.0327252i
$52$		0			0
$53$	−4.36450			−0.599510			−0.299755	−	0.954016i	$-0.596905\pi$
$53$	−4.36450			−0.599510			−0.299755	+	0.954016i	$0.596905\pi$
$54$		0			0
$55$	−5.42629	−	5.22684i	−0.731680	−	0.704787i
$56$		0			0
$57$		−	0.542768i		−	0.0718914i
$58$		0			0
$59$		−	2.34114i		−	0.304790i	−0.988320	−	0.152395i	$-0.951301\pi$
$59$		−	2.34114i		−	0.304790i	0.988320	−	0.152395i	$-0.0486986\pi$
$60$		0			0
$61$			10.2211i			1.30867i	0.756203	+	0.654337i	$0.227052\pi$
$61$			10.2211i			1.30867i	−0.756203	+	0.654337i	$0.772948\pi$
$62$		0			0
$63$			2.94412i			0.370924i
$64$		0			0
$65$	−1.57585	+	1.63598i	−0.195460	+	0.202918i
$66$		0			0
$67$	−6.35033			−0.775816			−0.387908	−	0.921698i	$-0.626802\pi$
$67$	−6.35033			−0.775816			−0.387908	+	0.921698i	$0.626802\pi$
$68$		0			0
$69$			0.190638i			0.0229502i
$70$		0			0
$71$	−12.5115			−1.48485			−0.742423	−	0.669931i	$-0.766323\pi$
$71$	−12.5115			−1.48485			−0.742423	+	0.669931i	$0.766323\pi$
$72$		0			0
$73$		−	13.0894i		−	1.53200i	−0.642841	−	0.766000i	$-0.722244\pi$
$73$		−	13.0894i		−	1.53200i	0.642841	−	0.766000i	$-0.277756\pi$
$74$		0			0
$75$	−0.0442412	−	1.18112i	−0.00510853	−	0.136384i
$76$		0			0
$77$	−3.36940			−0.383979
$78$		0			0
$79$	−7.51602			−0.845619			−0.422809	−	0.906219i	$-0.638956\pi$
$79$	−7.51602			−0.845619			−0.422809	+	0.906219i	$0.638956\pi$
$80$		0			0
$81$	8.50020			0.944467
$82$		0			0
$83$	14.5463			1.59666			0.798330	−	0.602220i	$-0.205717\pi$
$83$	14.5463			1.59666			0.798330	+	0.602220i	$0.205717\pi$
$84$		0			0
$85$	1.59217	+	1.53365i	0.172695	+	0.166347i
$86$		0			0
$87$			0.750577i			0.0804703i
$88$		0			0
$89$	0.147020			0.0155841			0.00779204	−	0.999970i	$-0.497520\pi$
$89$	0.147020			0.0155841			0.00779204	+	0.999970i	$0.497520\pi$
$90$		0			0
$91$			1.01585i			0.106490i
$92$		0			0
$93$	−0.418419			−0.0433880
$94$		0			0
$95$	−3.69773	−	3.56182i	−0.379379	−	0.365435i
$96$		0			0
$97$			9.75135i			0.990099i	0.868865	+	0.495050i	$0.164850\pi$
$97$			9.75135i			0.990099i	−0.868865	+	0.495050i	$0.835150\pi$
$98$		0			0
$99$			9.91993i			0.996990i
$100$		0			0
$101$			1.47196i			0.146466i	0.997315	+	0.0732328i	$0.0233316\pi$
$101$			1.47196i			0.146466i	−0.997315	+	0.0732328i	$0.976668\pi$
$102$		0			0
$103$		−	10.2250i		−	1.00750i	−0.863850	−	0.503749i	$-0.831954\pi$
$103$		−	10.2250i		−	1.00750i	0.863850	−	0.503749i	$-0.168046\pi$
$104$		0			0
$105$	−0.380696	−	0.366703i	−0.0371521	−	0.0357866i
$106$		0			0
$107$	1.46681			0.141802			0.0709011	−	0.997483i	$-0.477413\pi$
$107$	1.46681			0.141802			0.0709011	+	0.997483i	$0.477413\pi$
$108$		0			0
$109$			15.9914i			1.53170i	0.643018	+	0.765851i	$0.277682\pi$
$109$			15.9914i			1.53170i	−0.643018	+	0.765851i	$0.722318\pi$
$110$		0			0
$111$	−2.15433			−0.204480
$112$		0			0
$113$			0.0774799i			0.00728869i	0.999993	+	0.00364435i	$0.00116003\pi$
$113$			0.0774799i			0.00728869i	−0.999993	+	0.00364435i	$0.998840\pi$
$114$		0			0
$115$	1.29877	+	1.25103i	0.121111	+	0.116659i
$116$		0			0
$117$	2.99077			0.276497
$118$		0			0
$119$	0.988642			0.0906287
$120$		0			0
$121$	−0.352885			−0.0320805
$122$		0			0
$123$	−2.04924			−0.184774
$124$		0			0
$125$	−8.33697	−	7.44949i	−0.745682	−	0.666302i
$126$		0			0
$127$		−	12.7130i		−	1.12810i	−0.825742	−	0.564049i	$-0.809243\pi$
$127$		−	12.7130i		−	1.12810i	0.825742	−	0.564049i	$-0.190757\pi$
$128$		0			0
$129$	2.60552			0.229403
$130$		0			0
$131$		−	11.5369i		−	1.00799i	−0.863707	−	0.503994i	$-0.831864\pi$
$131$		−	11.5369i		−	1.00799i	0.863707	−	0.503994i	$-0.168136\pi$
$132$		0			0
$133$	−2.29607			−0.199095
$134$		0			0
$135$	−2.17973	+	2.26290i	−0.187601	+	0.194760i
$136$		0			0
$137$		−	12.6090i		−	1.07726i	−0.842542	−	0.538631i	$-0.818942\pi$
$137$		−	12.6090i		−	1.07726i	0.842542	−	0.538631i	$-0.181058\pi$
$138$		0			0
$139$			0.753066i			0.0638742i	0.999490	+	0.0319371i	$0.0101676\pi$
$139$			0.753066i			0.0638742i	−0.999490	+	0.0319371i	$0.989832\pi$
$140$		0			0
$141$		−	2.33470i		−	0.196617i
$142$		0			0
$143$			3.42279i			0.286228i
$144$		0			0
$145$	5.11347	+	4.92552i	0.424651	+	0.409042i
$146$		0			0
$147$	−0.236390			−0.0194971
$148$		0			0
$149$			1.57940i			0.129389i	0.997905	+	0.0646946i	$0.0206073\pi$
$149$			1.57940i			0.129389i	−0.997905	+	0.0646946i	$0.979393\pi$
$150$		0			0
$151$	9.39158			0.764276			0.382138	−	0.924105i	$-0.375188\pi$
$151$	9.39158			0.764276			0.382138	+	0.924105i	$0.375188\pi$
$152$		0			0
$153$		−	2.91068i		−	0.235315i
$154$		0			0
$155$	−2.74579	+	2.85057i	−0.220548	+	0.228963i
$156$		0			0
$157$	−9.04730			−0.722053			−0.361027	−	0.932555i	$-0.617574\pi$
$157$	−9.04730			−0.722053			−0.361027	+	0.932555i	$0.617574\pi$
$158$		0			0
$159$	−1.03172			−0.0818210
$160$		0			0
$161$	0.806457			0.0635577
$162$		0			0
$163$	0.118681			0.00929585			0.00464793	−	0.999989i	$-0.498521\pi$
$163$	0.118681			0.00929585			0.00464793	+	0.999989i	$0.498521\pi$
$164$		0			0
$165$	−1.28272	−	1.23557i	−0.0998596	−	0.0961892i
$166$		0			0
$167$			2.27336i			0.175918i	0.996124	+	0.0879590i	$0.0280345\pi$
$167$			2.27336i			0.175918i	−0.996124	+	0.0879590i	$0.971966\pi$
$168$		0			0
$169$	−11.9681			−0.920620
$170$		0			0
$171$			6.75991i			0.516943i
$172$		0			0
$173$	17.2558			1.31193			0.655967	−	0.754789i	$-0.272261\pi$
$173$	17.2558			1.31193			0.655967	+	0.754789i	$0.272261\pi$
$174$		0			0
$175$	−4.99650	+	0.187153i	−0.377700	+	0.0141475i
$176$		0			0
$177$		−	0.553421i		−	0.0415977i
$178$		0			0
$179$		−	18.5755i		−	1.38839i	−0.719785	−	0.694197i	$-0.755759\pi$
$179$		−	18.5755i		−	1.38839i	0.719785	−	0.694197i	$-0.244241\pi$
$180$		0			0
$181$			9.04999i			0.672681i	0.941740	+	0.336340i	$0.109189\pi$
$181$			9.04999i			0.672681i	−0.941740	+	0.336340i	$0.890811\pi$
$182$		0			0
$183$			2.41616i			0.178608i
$184$		0			0
$185$	−14.1374	+	14.6769i	−1.03940	+	1.07906i
$186$		0			0
$187$	3.33113			0.243597
$188$		0			0
$189$			1.40513i			0.102208i
$190$		0			0
$191$	8.08031			0.584670			0.292335	−	0.956316i	$-0.405568\pi$
$191$	8.08031			0.584670			0.292335	+	0.956316i	$0.405568\pi$
$192$		0			0
$193$		−	23.5747i		−	1.69695i	−0.529237	−	0.848474i	$-0.677522\pi$
$193$		−	23.5747i		−	1.69695i	0.529237	−	0.848474i	$-0.322478\pi$
$194$		0			0
$195$	−0.372514	+	0.386728i	−0.0266763	+	0.0276942i
$196$		0			0
$197$	−3.47867			−0.247845			−0.123923	−	0.992292i	$-0.539547\pi$
$197$	−3.47867			−0.247845			−0.123923	+	0.992292i	$0.539547\pi$
$198$		0			0
$199$	−19.3657			−1.37280			−0.686400	−	0.727224i	$-0.740810\pi$
$199$	−19.3657			−1.37280			−0.686400	+	0.727224i	$0.740810\pi$
$200$		0			0
$201$	−1.50115			−0.105883
$202$		0			0
$203$	3.17517			0.222853
$204$		0			0
$205$	−13.4478	+	13.9609i	−0.939232	+	0.975072i
$206$		0			0
$207$		−	2.37431i		−	0.165026i
$208$		0			0
$209$	−7.73640			−0.535138
$210$		0			0
$211$		−	12.3617i		−	0.851011i	−0.904956	−	0.425506i	$-0.860096\pi$
$211$		−	12.3617i		−	0.851011i	0.904956	−	0.425506i	$-0.139904\pi$
$212$		0			0
$213$	−2.95760			−0.202651
$214$		0			0
$215$	17.0982	−	17.7507i	1.16609	−	1.21059i
$216$		0			0
$217$			1.77004i			0.120158i
$218$		0			0
$219$		−	3.09420i		−	0.209087i
$220$		0			0
$221$		−	1.00431i		−	0.0675570i
$222$		0			0
$223$			5.14747i			0.344700i	0.985036	+	0.172350i	$0.0551360\pi$
$223$			5.14747i			0.344700i	−0.985036	+	0.172350i	$0.944864\pi$
$224$		0			0
$225$	0.551002	+	14.7103i	0.0367335	+	0.980686i
$226$		0			0
$227$	5.48759			0.364224			0.182112	−	0.983278i	$-0.441707\pi$
$227$	5.48759			0.364224			0.182112	+	0.983278i	$0.441707\pi$
$228$		0			0
$229$			0.128516i			0.00849255i	0.999991	+	0.00424628i	$0.00135164\pi$
$229$			0.128516i			0.00849255i	−0.999991	+	0.00424628i	$0.998648\pi$
$230$		0			0
$231$	−0.796493			−0.0524054
$232$		0			0
$233$		−	22.6055i		−	1.48094i	−0.672091	−	0.740468i	$-0.734604\pi$
$233$		−	22.6055i		−	1.48094i	0.672091	−	0.740468i	$-0.265396\pi$
$234$		0			0
$235$	−15.9056	−	15.3210i	−1.03757	−	0.999433i
$236$		0			0
$237$	−1.77671			−0.115410
$238$		0			0
$239$	14.0951			0.911737			0.455869	−	0.890047i	$-0.349329\pi$
$239$	14.0951			0.911737			0.455869	+	0.890047i	$0.349329\pi$
$240$		0			0
$241$	−7.92029			−0.510191			−0.255095	−	0.966916i	$-0.582107\pi$
$241$	−7.92029			−0.510191			−0.255095	+	0.966916i	$0.582107\pi$
$242$		0			0
$243$	6.22475			0.399318
$244$		0			0
$245$	−1.55127	+	1.61046i	−0.0991067	+	0.102888i
$246$		0			0
$247$			2.33245i			0.148411i
$248$		0			0
$249$	3.43859			0.217912
$250$		0			0
$251$			11.1153i			0.701592i	0.936452	+	0.350796i	$0.114089\pi$
$251$			11.1153i			0.701592i	−0.936452	+	0.350796i	$0.885911\pi$
$252$		0			0
$253$	2.71728			0.170834
$254$		0			0
$255$	0.376372	+	0.362538i	0.0235693	+	0.0227030i
$256$		0			0
$257$			5.39563i			0.336570i	0.985738	+	0.168285i	$0.0538229\pi$
$257$			5.39563i			0.336570i	−0.985738	+	0.168285i	$0.946177\pi$
$258$		0			0
$259$			9.11347i			0.566284i
$260$		0			0
$261$		−	9.34807i		−	0.578631i
$262$		0			0
$263$			4.88914i			0.301478i	0.988574	+	0.150739i	$0.0481652\pi$
$263$			4.88914i			0.301478i	−0.988574	+	0.150739i	$0.951835\pi$
$264$		0			0
$265$	−6.77050	+	7.02885i	−0.415908	+	0.431779i
$266$		0			0
$267$	0.0347540			0.00212691
$268$		0			0
$269$		−	28.1206i		−	1.71454i	−0.514863	−	0.857272i	$-0.672157\pi$
$269$		−	28.1206i		−	1.71454i	0.514863	−	0.857272i	$-0.327843\pi$
$270$		0			0
$271$	−22.2213			−1.34985			−0.674924	−	0.737887i	$-0.735824\pi$
$271$	−22.2213			−1.34985			−0.674924	+	0.737887i	$0.735824\pi$
$272$		0			0
$273$			0.240136i			0.0145337i
$274$		0			0
$275$	−16.8352	+	0.630596i	−1.01520	+	0.0380264i
$276$		0			0
$277$	−4.70300			−0.282576			−0.141288	−	0.989969i	$-0.545124\pi$
$277$	−4.70300			−0.282576			−0.141288	+	0.989969i	$0.545124\pi$
$278$		0			0
$279$	5.21120			0.311986
$280$		0			0
$281$	−9.32935			−0.556542			−0.278271	−	0.960503i	$-0.589761\pi$
$281$	−9.32935			−0.556542			−0.278271	+	0.960503i	$0.589761\pi$
$282$		0			0
$283$	17.1601			1.02006			0.510031	−	0.860156i	$-0.329634\pi$
$283$	17.1601			1.02006			0.510031	+	0.860156i	$0.329634\pi$
$284$		0			0
$285$	−0.874106	−	0.841978i	−0.0517776	−	0.0498745i
$286$		0			0
$287$			8.66890i			0.511709i
$288$		0			0
$289$	16.0226			0.942505
$290$		0			0
$291$			2.30512i			0.135129i
$292$		0			0
$293$	11.1178			0.649509			0.324754	−	0.945798i	$-0.394718\pi$
$293$	11.1178			0.649509			0.324754	+	0.945798i	$0.394718\pi$
$294$		0			0
$295$	−3.77031	−	3.63173i	−0.219516	−	0.211447i
$296$		0			0
$297$			4.73445i			0.274721i
$298$		0			0
$299$		−	0.819236i		−	0.0473776i
$300$		0			0
$301$		−	11.0221i		−	0.635305i
$302$		0			0
$303$			0.347957i			0.0199896i
$304$		0			0
$305$	16.4606	+	15.8556i	0.942532	+	0.907889i
$306$		0			0
$307$	25.2534			1.44129			0.720643	−	0.693307i	$-0.243847\pi$
$307$	25.2534			1.44129			0.720643	+	0.693307i	$0.243847\pi$
$308$		0			0
$309$		−	2.41708i		−	0.137503i
$310$		0			0
$311$	20.7515			1.17671			0.588354	−	0.808603i	$-0.299776\pi$
$311$	20.7515			1.17671			0.588354	+	0.808603i	$0.299776\pi$
$312$		0			0
$313$		−	25.5714i		−	1.44538i	−0.691170	−	0.722692i	$-0.742905\pi$
$313$		−	25.5714i		−	1.44538i	0.691170	−	0.722692i	$-0.257095\pi$
$314$		0			0
$315$	4.74138	+	4.56711i	0.267147	+	0.257328i
$316$		0			0
$317$	22.3540			1.25552			0.627762	−	0.778405i	$-0.283971\pi$
$317$	22.3540			1.25552			0.627762	+	0.778405i	$0.283971\pi$
$318$		0			0
$319$	10.6984			0.598996
$320$		0			0
$321$	0.346740			0.0193531
$322$		0			0
$323$	2.26999			0.126306
$324$		0			0
$325$	0.190119	+	5.07567i	0.0105459	+	0.281547i
$326$		0			0
$327$			3.78021i			0.209046i
$328$		0			0
$329$	−9.87647			−0.544507
$330$		0			0
$331$		−	7.95850i		−	0.437439i	−0.975788	−	0.218719i	$-0.929812\pi$
$331$		−	7.95850i		−	0.437439i	0.975788	−	0.218719i	$-0.0701880\pi$
$332$		0			0
$333$	26.8312			1.47034
$334$		0			0
$335$	−9.85105	+	10.2269i	−0.538220	+	0.558758i
$336$		0			0
$337$			20.7896i			1.13248i	0.824240	+	0.566241i	$0.191603\pi$
$337$			20.7896i			1.13248i	−0.824240	+	0.566241i	$0.808397\pi$
$338$		0			0
$339$			0.0183155i			0.000994759i
$340$		0			0
$341$			5.96397i			0.322967i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	0.307015	+	0.295731i	0.0165291	+	0.0159216i
$346$		0			0
$347$	−27.6336			−1.48345			−0.741724	−	0.670705i	$-0.765992\pi$
$347$	−27.6336			−1.48345			−0.741724	+	0.670705i	$0.765992\pi$
$348$		0			0
$349$		−	15.4381i		−	0.826385i	−0.910644	−	0.413193i	$-0.864414\pi$
$349$		−	15.4381i		−	0.826385i	0.910644	−	0.413193i	$-0.135586\pi$
$350$		0			0
$351$	1.42739			0.0761887
$352$		0			0
$353$			30.3576i			1.61577i	0.589340	+	0.807885i	$0.299388\pi$
$353$			30.3576i			1.61577i	−0.589340	+	0.807885i	$0.700612\pi$
$354$		0			0
$355$	−19.4087	+	20.1493i	−1.03011	+	1.06941i
$356$		0			0
$357$	0.233705			0.0123690
$358$		0			0
$359$	−2.44899			−0.129253			−0.0646263	−	0.997910i	$-0.520586\pi$
$359$	−2.44899			−0.129253			−0.0646263	+	0.997910i	$0.520586\pi$
$360$		0			0
$361$	13.7280			0.722529
$362$		0			0
$363$	−0.0834185			−0.00437833
$364$		0			0
$365$	−21.0799	−	20.3051i	−1.10337	−	1.06282i
$366$		0			0
$367$			6.02179i			0.314335i	0.987572	+	0.157167i	$0.0502362\pi$
$367$			6.02179i			0.314335i	−0.987572	+	0.157167i	$0.949764\pi$
$368$		0			0
$369$	25.5223			1.32864
$370$		0			0
$371$			4.36450i			0.226594i
$372$		0			0
$373$	16.5797			0.858466			0.429233	−	0.903194i	$-0.358784\pi$
$373$	16.5797			0.858466			0.429233	+	0.903194i	$0.358784\pi$
$374$		0			0
$375$	−1.97078	−	1.76098i	−0.101770	−	0.0909368i
$376$		0			0
$377$		−	3.22548i		−	0.166120i
$378$		0			0
$379$			3.55769i			0.182746i	0.995817	+	0.0913731i	$0.0291256\pi$
$379$			3.55769i			0.182746i	−0.995817	+	0.0913731i	$0.970874\pi$
$380$		0			0
$381$		−	3.00523i		−	0.153962i
$382$		0			0
$383$			29.0775i			1.48579i	0.669407	+	0.742896i	$0.266548\pi$
$383$			29.0775i			1.48579i	−0.669407	+	0.742896i	$0.733452\pi$
$384$		0			0
$385$	−5.22684	+	5.42629i	−0.266384	+	0.276549i
$386$		0			0
$387$	−32.4505			−1.64955
$388$		0			0
$389$			9.98022i			0.506017i	0.967464	+	0.253009i	$0.0814201\pi$
$389$			9.98022i			0.506017i	−0.967464	+	0.253009i	$0.918580\pi$
$390$		0			0
$391$	−0.797298			−0.0403211
$392$		0			0
$393$		−	2.72722i		−	0.137570i
$394$		0			0
$395$	−11.6593	+	12.1042i	−0.586645	+	0.609031i
$396$		0			0
$397$	4.52752			0.227230			0.113615	−	0.993525i	$-0.463757\pi$
$397$	4.52752			0.227230			0.113615	+	0.993525i	$0.463757\pi$
$398$		0			0
$399$	−0.542768			−0.0271724
$400$		0			0
$401$	35.6019			1.77787			0.888937	−	0.458029i	$-0.151444\pi$
$401$	35.6019			1.77787			0.888937	+	0.458029i	$0.151444\pi$
$402$		0			0
$403$	1.79808			0.0895688
$404$		0			0
$405$	13.1861	−	13.6892i	0.655221	−	0.680223i
$406$		0			0
$407$			30.7070i			1.52209i
$408$		0			0
$409$	−7.47440			−0.369586			−0.184793	−	0.982778i	$-0.559161\pi$
$409$	−7.47440			−0.369586			−0.184793	+	0.982778i	$0.559161\pi$
$410$		0			0
$411$		−	2.98064i		−	0.147024i
$412$		0			0
$413$	−2.34114			−0.115200
$414$		0			0
$415$	22.5651	−	23.4262i	1.10768	−	1.14995i
$416$		0			0
$417$			0.178017i			0.00871754i
$418$		0			0
$419$			29.0445i			1.41892i	0.704746	+	0.709459i	$0.251061\pi$
$419$			29.0445i			1.41892i	−0.704746	+	0.709459i	$0.748939\pi$
$420$		0			0
$421$			1.23277i			0.0600814i	0.999549	+	0.0300407i	$0.00956369\pi$
$421$			1.23277i			0.0600814i	−0.999549	+	0.0300407i	$0.990436\pi$
$422$		0			0
$423$			29.0775i			1.41380i
$424$		0			0
$425$	4.93975	−	0.185028i	0.239613	−	0.00897517i
$426$		0			0
$427$	10.2211			0.494633
$428$		0			0
$429$			0.809114i			0.0390644i
$430$		0			0
$431$	34.4116			1.65755			0.828775	−	0.559582i	$-0.189038\pi$
$431$	34.4116			1.65755			0.828775	+	0.559582i	$0.189038\pi$
$432$		0			0
$433$		−	0.864601i		−	0.0415501i	−0.999784	−	0.0207750i	$-0.993387\pi$
$433$		−	0.864601i		−	0.0415501i	0.999784	−	0.0207750i	$-0.00661338\pi$
$434$		0			0
$435$	1.20877	+	1.16434i	0.0579562	+	0.0558260i
$436$		0			0
$437$	1.85168			0.0885781
$438$		0			0
$439$	15.3822			0.734153			0.367076	−	0.930191i	$-0.380359\pi$
$439$	15.3822			0.734153			0.367076	+	0.930191i	$0.380359\pi$
$440$		0			0
$441$	2.94412			0.140196
$442$		0			0
$443$	22.5632			1.07201			0.536006	−	0.844214i	$-0.319933\pi$
$443$	22.5632			1.07201			0.536006	+	0.844214i	$0.319933\pi$
$444$		0			0
$445$	0.228067	−	0.236770i	0.0108114	−	0.0112240i
$446$		0			0
$447$			0.373354i			0.0176590i
$448$		0			0
$449$	6.45322			0.304546			0.152273	−	0.988338i	$-0.451341\pi$
$449$	6.45322			0.304546			0.152273	+	0.988338i	$0.451341\pi$
$450$		0			0
$451$			29.2090i			1.37540i
$452$		0			0
$453$	2.22008			0.104308
$454$		0			0
$455$	1.63598	+	1.57585i	0.0766958	+	0.0738768i
$456$		0			0
$457$			1.81364i			0.0848385i	0.999100	+	0.0424192i	$0.0135065\pi$
$457$			1.81364i			0.0848385i	−0.999100	+	0.0424192i	$0.986493\pi$
$458$		0			0
$459$		−	1.38917i		−	0.0648409i
$460$		0			0
$461$			0.374379i			0.0174366i	0.999962	+	0.00871828i	$0.00277515\pi$
$461$			0.374379i			0.0174366i	−0.999962	+	0.00871828i	$0.997225\pi$
$462$		0			0
$463$			4.82245i			0.224118i	0.993702	+	0.112059i	$0.0357446\pi$
$463$			4.82245i			0.224118i	−0.993702	+	0.112059i	$0.964255\pi$
$464$		0			0
$465$	−0.649078	+	0.673846i	−0.0301003	+	0.0312488i
$466$		0			0
$467$	−22.1830			−1.02651			−0.513255	−	0.858236i	$-0.671560\pi$
$467$	−22.1830			−1.02651			−0.513255	+	0.858236i	$0.671560\pi$
$468$		0			0
$469$			6.35033i			0.293231i
$470$		0			0
$471$	−2.13869			−0.0985457
$472$		0			0
$473$		−	37.1380i		−	1.70761i
$474$		0			0
$475$	−11.4723	+	0.429718i	−0.526386	+	0.0197168i
$476$		0			0
$477$	12.8496			0.588343
$478$		0			0
$479$	−22.5177			−1.02886			−0.514429	−	0.857533i	$-0.671996\pi$
$479$	−22.5177			−1.02886			−0.514429	+	0.857533i	$0.671996\pi$
$480$		0			0
$481$	9.25788			0.422123
$482$		0			0
$483$	0.190638			0.00867435
$484$		0			0
$485$	15.7041	+	15.1269i	0.713088	+	0.686878i
$486$		0			0
$487$		−	36.8362i		−	1.66921i	−0.550850	−	0.834604i	$-0.685696\pi$
$487$		−	36.8362i		−	1.66921i	0.550850	−	0.834604i	$-0.314304\pi$
$488$		0			0
$489$	0.0280551			0.00126870
$490$		0			0
$491$			21.5261i			0.971459i	0.874109	+	0.485729i	$0.161446\pi$
$491$			21.5261i			0.971459i	−0.874109	+	0.485729i	$0.838554\pi$
$492$		0			0
$493$	−3.13910			−0.141378
$494$		0			0
$495$	15.9756	+	15.3884i	0.718051	+	0.691659i
$496$		0			0
$497$			12.5115i			0.561219i
$498$		0			0
$499$		−	25.6995i		−	1.15047i	−0.817990	−	0.575233i	$-0.804911\pi$
$499$		−	25.6995i		−	1.15047i	0.817990	−	0.575233i	$-0.195089\pi$
$500$		0			0
$501$			0.537400i			0.0240093i
$502$		0			0
$503$			28.7371i			1.28132i	0.767824	+	0.640661i	$0.221340\pi$
$503$			28.7371i			1.28132i	−0.767824	+	0.640661i	$0.778660\pi$
$504$		0			0
$505$	2.37053	+	2.28340i	0.105487	+	0.101610i
$506$		0			0
$507$	−2.82913			−0.125646
$508$		0			0
$509$		−	7.89425i		−	0.349906i	−0.984577	−	0.174953i	$-0.944023\pi$
$509$		−	7.89425i		−	0.349906i	0.984577	−	0.174953i	$-0.0559774\pi$
$510$		0			0
$511$	−13.0894			−0.579041
$512$		0			0
$513$			3.22628i			0.142444i
$514$		0			0
$515$	−16.4669	−	15.8617i	−0.725619	−	0.698948i
$516$		0			0
$517$	−33.2778			−1.46356
$518$		0			0
$519$	4.07910			0.179053
$520$		0			0
$521$	−26.2583			−1.15040			−0.575198	−	0.818014i	$-0.695075\pi$
$521$	−26.2583			−1.15040			−0.575198	+	0.818014i	$0.695075\pi$
$522$		0			0
$523$	12.6053			0.551190			0.275595	−	0.961274i	$-0.411125\pi$
$523$	12.6053			0.551190			0.275595	+	0.961274i	$0.411125\pi$
$524$		0			0
$525$	−1.18112	+	0.0442412i	−0.0515484	+	0.00193084i
$526$		0			0
$527$		−	1.74993i		−	0.0762282i
$528$		0			0
$529$	22.3496			0.971723
$530$		0			0
$531$			6.89259i			0.299113i
$532$		0			0
$533$	8.80626			0.381441
$534$		0			0
$535$	2.27542	−	2.36224i	0.0983748	−	0.102129i
$536$		0			0
$537$		−	4.39105i		−	0.189488i
$538$		0			0
$539$			3.36940i			0.145131i
$540$		0			0
$541$		−	13.9643i		−	0.600371i	−0.953881	−	0.300186i	$-0.902951\pi$
$541$		−	13.9643i		−	0.600371i	0.953881	−	0.300186i	$-0.0970486\pi$
$542$		0			0
$543$			2.13933i			0.0918073i
$544$		0			0
$545$	25.7535	+	24.8069i	1.10316	+	1.06261i
$546$		0			0
$547$	−26.3936			−1.12851			−0.564253	−	0.825602i	$-0.690836\pi$
$547$	−26.3936			−1.12851			−0.564253	+	0.825602i	$0.690836\pi$
$548$		0			0
$549$		−	30.0921i		−	1.28430i
$550$		0			0
$551$	7.29041			0.310582
$552$		0			0
$553$			7.51602i			0.319614i
$554$		0			0
$555$	−3.34194	+	3.46946i	−0.141858	+	0.147271i
$556$		0			0
$557$	34.7328			1.47168			0.735839	−	0.677157i	$-0.236788\pi$
$557$	34.7328			1.47168			0.735839	+	0.677157i	$0.236788\pi$
$558$		0			0
$559$	−11.1968			−0.473573
$560$		0			0
$561$	0.787447			0.0332460
$562$		0			0
$563$	46.3924			1.95521			0.977604	−	0.210452i	$-0.0674934\pi$
$563$	46.3924			1.95521			0.977604	+	0.210452i	$0.0674934\pi$
$564$		0			0
$565$	0.124778	+	0.120192i	0.00524946	+	0.00505651i
$566$		0			0
$567$		−	8.50020i		−	0.356975i
$568$		0			0
$569$	−0.837791			−0.0351220			−0.0175610	−	0.999846i	$-0.505590\pi$
$569$	−0.837791			−0.0351220			−0.0175610	+	0.999846i	$0.505590\pi$
$570$		0			0
$571$		−	33.5095i		−	1.40233i	−0.712999	−	0.701165i	$-0.752664\pi$
$571$		−	33.5095i		−	1.40233i	0.712999	−	0.701165i	$-0.247336\pi$
$572$		0			0
$573$	1.91010			0.0797957
$574$		0			0
$575$	4.02946	−	0.150931i	0.168040	−	0.00629427i
$576$		0			0
$577$			25.3613i			1.05580i	0.849306	+	0.527902i	$0.177021\pi$
$577$			25.3613i			1.05580i	−0.849306	+	0.527902i	$0.822979\pi$
$578$		0			0
$579$		−	5.57283i		−	0.231599i
$580$		0			0
$581$		−	14.5463i		−	0.603481i
$582$		0			0
$583$			14.7058i			0.609051i
$584$		0			0
$585$	4.63948	−	4.81651i	0.191819	−	0.199138i
$586$		0			0
$587$	−24.2742			−1.00190			−0.500951	−	0.865476i	$-0.667016\pi$
$587$	−24.2742			−1.00190			−0.500951	+	0.865476i	$0.667016\pi$
$588$		0			0
$589$			4.06413i			0.167460i
$590$		0			0
$591$	−0.822323			−0.0338259
$592$		0			0
$593$		−	11.6582i		−	0.478745i	−0.970928	−	0.239373i	$-0.923058\pi$
$593$		−	11.6582i		−	0.478745i	0.970928	−	0.239373i	$-0.0769417\pi$
$594$		0			0
$595$	1.53365	−	1.59217i	0.0628733	−	0.0652725i
$596$		0			0
$597$	−4.57786			−0.187360
$598$		0			0
$599$	22.5455			0.921185			0.460593	−	0.887612i	$-0.347637\pi$
$599$	22.5455			0.921185			0.460593	+	0.887612i	$0.347637\pi$
$600$		0			0
$601$	−47.1404			−1.92290			−0.961449	−	0.274983i	$-0.911328\pi$
$601$	−47.1404			−1.92290			−0.961449	+	0.274983i	$0.911328\pi$
$602$		0			0
$603$	18.6961			0.761366
$604$		0			0
$605$	−0.547418	+	0.568307i	−0.0222557	+	0.0231050i
$606$		0			0
$607$			4.87557i			0.197893i	0.995093	+	0.0989466i	$0.0315473\pi$
$607$			4.87557i			0.197893i	−0.995093	+	0.0989466i	$0.968453\pi$
$608$		0			0
$609$	0.750577			0.0304149
$610$		0			0
$611$			10.0330i			0.405890i
$612$		0			0
$613$	45.8086			1.85019			0.925096	−	0.379732i	$-0.123984\pi$
$613$	45.8086			1.85019			0.925096	+	0.379732i	$0.123984\pi$
$614$		0			0
$615$	−3.17892	+	3.30022i	−0.128186	+	0.133078i
$616$		0			0
$617$		−	0.551269i		−	0.0221932i	−0.999938	−	0.0110966i	$-0.996468\pi$
$617$		−	0.551269i		−	0.0221932i	0.999938	−	0.0110966i	$-0.00353224\pi$
$618$		0			0
$619$			10.1227i			0.406866i	0.979089	+	0.203433i	$0.0652099\pi$
$619$			10.1227i			0.406866i	−0.979089	+	0.203433i	$0.934790\pi$
$620$		0			0
$621$		−	1.13318i		−	0.0454728i
$622$		0			0
$623$		−	0.147020i		−	0.00589023i
$624$		0			0
$625$	−24.9299	+	1.87022i	−0.997198	+	0.0748089i
$626$		0			0
$627$	−1.82881			−0.0730355
$628$		0			0
$629$		−	9.00996i		−	0.359251i
$630$		0			0
$631$	−31.0973			−1.23796			−0.618982	−	0.785405i	$-0.712455\pi$
$631$	−31.0973			−1.23796			−0.618982	+	0.785405i	$0.712455\pi$
$632$		0			0
$633$		−	2.92217i		−	0.116146i
$634$		0			0
$635$	−20.4738	−	19.7213i	−0.812477	−	0.782614i
$636$		0			0
$637$	1.01585			0.0402492
$638$		0			0
$639$	36.8355			1.45719
$640$		0			0
$641$	29.2917			1.15695			0.578476	−	0.815699i	$-0.303648\pi$
$641$	29.2917			1.15695			0.578476	+	0.815699i	$0.303648\pi$
$642$		0			0
$643$	40.7053			1.60526			0.802630	−	0.596477i	$-0.203433\pi$
$643$	40.7053			1.60526			0.802630	+	0.596477i	$0.203433\pi$
$644$		0			0
$645$	4.04185	−	4.19608i	0.159148	−	0.165221i
$646$		0			0
$647$		−	24.5048i		−	0.963382i	−0.876341	−	0.481691i	$-0.840023\pi$
$647$		−	24.5048i		−	0.963382i	0.876341	−	0.481691i	$-0.159977\pi$
$648$		0			0
$649$	−7.88824			−0.309640
$650$		0			0
$651$			0.418419i			0.0163991i
$652$		0			0
$653$	13.4136			0.524915			0.262457	−	0.964944i	$-0.415467\pi$
$653$	13.4136			0.524915			0.262457	+	0.964944i	$0.415467\pi$
$654$		0			0
$655$	−18.5798	−	17.8969i	−0.725971	−	0.699288i
$656$		0			0
$657$			38.5368i			1.50346i
$658$		0			0
$659$			27.5200i			1.07203i	0.844210	+	0.536013i	$0.180070\pi$
$659$			27.5200i			1.07203i	−0.844210	+	0.536013i	$0.819930\pi$
$660$		0			0
$661$		−	5.66818i		−	0.220466i	−0.993906	−	0.110233i	$-0.964840\pi$
$661$		−	5.66818i		−	0.220466i	0.993906	−	0.110233i	$-0.0351598\pi$
$662$		0			0
$663$		−	0.237408i		−	0.00922017i
$664$		0			0
$665$	−3.56182	+	3.69773i	−0.138121	+	0.143392i
$666$		0			0
$667$	−2.56064			−0.0991482
$668$		0			0
$669$			1.21681i			0.0470446i
$670$		0			0
$671$	34.4389			1.32950
$672$		0			0
$673$		−	24.8165i		−	0.956605i	−0.878195	−	0.478302i	$-0.841252\pi$
$673$		−	24.8165i		−	0.956605i	0.878195	−	0.478302i	$-0.158748\pi$
$674$		0			0
$675$	0.262975	+	7.02073i	0.0101219	+	0.270228i
$676$		0			0
$677$	−20.9321			−0.804487			−0.402244	−	0.915533i	$-0.631770\pi$
$677$	−20.9321			−0.804487			−0.402244	+	0.915533i	$0.631770\pi$
$678$		0			0
$679$	9.75135			0.374222
$680$		0			0
$681$	1.29721			0.0497093
$682$		0			0
$683$	−26.0481			−0.996702			−0.498351	−	0.866975i	$-0.666061\pi$
$683$	−26.0481			−0.996702			−0.498351	+	0.866975i	$0.666061\pi$
$684$		0			0
$685$	−20.3063	−	19.5599i	−0.775864	−	0.747346i
$686$		0			0
$687$			0.0303798i			0.00115906i
$688$		0			0
$689$	4.43366			0.168909
$690$		0			0
$691$		−	42.7645i		−	1.62684i	−0.581678	−	0.813419i	$-0.697604\pi$
$691$		−	42.7645i		−	1.62684i	0.581678	−	0.813419i	$-0.302396\pi$
$692$		0			0
$693$	9.91993			0.376827
$694$		0			0
$695$	1.21278	+	1.16820i	0.0460034	+	0.0443125i
$696$		0			0
$697$		−	8.57044i		−	0.324629i
$698$		0			0
$699$		−	5.34372i		−	0.202118i
$700$		0			0
$701$		−	31.0993i		−	1.17460i	−0.809368	−	0.587302i	$-0.800190\pi$
$701$		−	31.0993i		−	1.17460i	0.809368	−	0.587302i	$-0.199810\pi$
$702$		0			0
$703$			20.9252i			0.789209i
$704$		0			0
$705$	−3.75993	−	3.62174i	−0.141607	−	0.136402i
$706$		0			0
$707$	1.47196			0.0553588
$708$		0			0
$709$			1.63578i			0.0614329i	0.999528	+	0.0307164i	$0.00977888\pi$
$709$			1.63578i			0.0614329i	−0.999528	+	0.0307164i	$0.990221\pi$
$710$		0			0
$711$	22.1281			0.829867
$712$		0			0
$713$		−	1.42746i		−	0.0534587i
$714$		0			0
$715$	5.51227	+	5.30966i	0.206147	+	0.198570i
$716$		0			0
$717$	3.33194			0.124434
$718$		0			0
$719$	14.5439			0.542397			0.271198	−	0.962523i	$-0.412580\pi$
$719$	14.5439			0.542397			0.271198	+	0.962523i	$0.412580\pi$
$720$		0			0
$721$	−10.2250			−0.380798
$722$		0			0
$723$	−1.87228			−0.0696307
$724$		0			0
$725$	15.8647	−	0.594243i	0.589200	−	0.0220696i
$726$		0			0
$727$			24.5867i			0.911870i	0.890013	+	0.455935i	$0.150695\pi$
$727$			24.5867i			0.911870i	−0.890013	+	0.455935i	$0.849305\pi$
$728$		0			0
$729$	−24.0291			−0.889968
$730$		0			0
$731$			10.8969i			0.403038i
$732$		0			0
$733$	−25.0758			−0.926196			−0.463098	−	0.886307i	$-0.653262\pi$
$733$	−25.0758			−0.926196			−0.463098	+	0.886307i	$0.653262\pi$
$734$		0			0
$735$	−0.366703	+	0.380696i	−0.0135261	+	0.0140422i
$736$		0			0
$737$			21.3968i			0.788162i
$738$		0			0
$739$		−	11.1764i		−	0.411131i	−0.978643	−	0.205566i	$-0.934097\pi$
$739$		−	11.1764i		−	0.411131i	0.978643	−	0.205566i	$-0.0659034\pi$
$740$		0			0
$741$			0.551369i			0.0202550i
$742$		0			0
$743$			49.7436i			1.82492i	0.409171	+	0.912458i	$0.365818\pi$
$743$			49.7436i			1.82492i	−0.409171	+	0.912458i	$0.634182\pi$
$744$		0			0
$745$	2.54355	+	2.45006i	0.0931886	+	0.0897634i
$746$		0			0
$747$	−42.8260			−1.56692
$748$		0			0
$749$		−	1.46681i		−	0.0535962i
$750$		0			0
$751$	−7.24769			−0.264472			−0.132236	−	0.991218i	$-0.542216\pi$
$751$	−7.24769			−0.264472			−0.132236	+	0.991218i	$0.542216\pi$
$752$		0			0
$753$			2.62755i			0.0957531i
$754$		0			0
$755$	14.5688	−	15.1248i	0.530214	−	0.550446i
$756$		0			0
$757$	−39.5805			−1.43858			−0.719289	−	0.694711i	$-0.755532\pi$
$757$	−39.5805			−1.43858			−0.719289	+	0.694711i	$0.755532\pi$
$758$		0			0
$759$	0.642338			0.0233154
$760$		0			0
$761$	18.9207			0.685874			0.342937	−	0.939358i	$-0.388578\pi$
$761$	18.9207			0.685874			0.342937	+	0.939358i	$0.388578\pi$
$762$		0			0
$763$	15.9914			0.578929
$764$		0			0
$765$	−4.68753	−	4.51524i	−0.169478	−	0.163249i
$766$		0			0
$767$			2.37823i			0.0858730i
$768$		0			0
$769$	−44.4149			−1.60164			−0.800820	−	0.598905i	$-0.795603\pi$
$769$	−44.4149			−1.60164			−0.800820	+	0.598905i	$0.795603\pi$
$770$		0			0
$771$			1.27547i			0.0459350i
$772$		0			0
$773$	36.9670			1.32961			0.664805	−	0.747017i	$-0.268515\pi$
$773$	36.9670			1.32961			0.664805	+	0.747017i	$0.268515\pi$
$774$		0			0
$775$	0.331268	+	8.84398i	0.0118995	+	0.317685i
$776$		0			0
$777$			2.15433i			0.0772863i
$778$		0			0
$779$			19.9044i			0.713150i
$780$		0			0
$781$			42.1564i			1.50848i
$782$		0			0
$783$		−	4.46152i		−	0.159442i
$784$		0			0
$785$	−14.0348	+	14.5703i	−0.500922	+	0.520037i
$786$		0			0
$787$	4.67376			0.166602			0.0833008	−	0.996524i	$-0.473454\pi$
$787$	4.67376			0.166602			0.0833008	+	0.996524i	$0.473454\pi$
$788$		0			0
$789$			1.15574i			0.0411456i
$790$		0			0
$791$	0.0774799			0.00275487
$792$		0			0
$793$		−	10.3830i		−	0.368712i
$794$		0			0
$795$	−1.60048	+	1.66155i	−0.0567631	+	0.0589291i
$796$		0			0
$797$	27.2328			0.964634			0.482317	−	0.875997i	$-0.339795\pi$
$797$	27.2328			0.964634			0.482317	+	0.875997i	$0.339795\pi$
$798$		0			0
$799$	9.76429			0.345436
$800$		0			0
$801$	−0.432844			−0.0152938
$802$		0			0
$803$	−44.1035			−1.55638
$804$		0			0
$805$	1.25103	−	1.29877i	0.0440930	−	0.0457755i
$806$		0			0
$807$		−	6.64743i		−	0.234001i
$808$		0			0
$809$	31.7755			1.11717			0.558583	−	0.829449i	$-0.311345\pi$
$809$	31.7755			1.11717			0.558583	+	0.829449i	$0.311345\pi$
$810$		0			0
$811$			51.8625i			1.82114i	0.413355	+	0.910570i	$0.364357\pi$
$811$			51.8625i			1.82114i	−0.413355	+	0.910570i	$0.635643\pi$
$812$		0			0
$813$	−5.25289			−0.184227
$814$		0			0
$815$	0.184106	−	0.191132i	0.00644897	−	0.00669505i
$816$		0			0
$817$		−	25.3076i		−	0.885401i
$818$		0			0
$819$		−	2.99077i		−	0.104506i
$820$		0			0
$821$		−	11.1276i		−	0.388356i	−0.980966	−	0.194178i	$-0.937796\pi$
$821$		−	11.1276i		−	0.388356i	0.980966	−	0.194178i	$-0.0622039\pi$
$822$		0			0
$823$		−	44.7666i		−	1.56047i	−0.625488	−	0.780233i	$-0.715100\pi$
$823$		−	44.7666i		−	1.56047i	0.625488	−	0.780233i	$-0.284900\pi$
$824$		0			0
$825$	−3.97968	+	0.149066i	−0.138555	+	0.00518983i
$826$		0			0
$827$	48.7729			1.69600			0.848000	−	0.529996i	$-0.177807\pi$
$827$	48.7729			1.69600			0.848000	+	0.529996i	$0.177807\pi$
$828$		0			0
$829$			40.1625i			1.39490i	0.716632	+	0.697451i	$0.245683\pi$
$829$			40.1625i			1.39490i	−0.716632	+	0.697451i	$0.754317\pi$
$830$		0			0
$831$	−1.11174			−0.0385659
$832$		0			0
$833$		−	0.988642i		−	0.0342544i
$834$		0			0
$835$	3.66115	+	3.52659i	0.126699	+	0.122043i
$836$		0			0
$837$	2.48713			0.0859678
$838$		0			0
$839$	−42.5135			−1.46773			−0.733864	−	0.679297i	$-0.762285\pi$
$839$	−42.5135			−1.46773			−0.733864	+	0.679297i	$0.762285\pi$
$840$		0			0
$841$	18.9183			0.652356
$842$		0			0
$843$	−2.20536			−0.0759568
$844$		0			0
$845$	−18.5656	+	19.2741i	−0.638677	+	0.663048i
$846$		0			0
$847$			0.352885i			0.0121253i
$848$		0			0
$849$	4.05647			0.139218
$850$		0			0
$851$		−	7.34963i		−	0.251942i
$852$		0			0
$853$	−26.0098			−0.890560			−0.445280	−	0.895391i	$-0.646896\pi$
$853$	−26.0098			−0.890560			−0.445280	+	0.895391i	$0.646896\pi$
$854$		0			0
$855$	10.8866	+	10.4864i	0.372313	+	0.358628i
$856$		0			0
$857$			13.3029i			0.454418i	0.973846	+	0.227209i	$0.0729600\pi$
$857$			13.3029i			0.454418i	−0.973846	+	0.227209i	$0.927040\pi$
$858$		0			0
$859$		−	19.0863i		−	0.651217i	−0.945505	−	0.325609i	$-0.894431\pi$
$859$		−	19.0863i		−	0.651217i	0.945505	−	0.325609i	$-0.105569\pi$
$860$		0			0
$861$			2.04924i			0.0698379i
$862$		0			0
$863$		−	34.5461i		−	1.17596i	−0.808874	−	0.587982i	$-0.799923\pi$
$863$		−	34.5461i		−	1.17596i	0.808874	−	0.587982i	$-0.200077\pi$
$864$		0			0
$865$	26.7683	−	27.7898i	0.910151	−	0.944880i
$866$		0			0
$867$	3.78758			0.128633
$868$		0			0
$869$			25.3245i			0.859075i
$870$		0			0
$871$	6.45095			0.218582
$872$		0			0
$873$		−	28.7091i		−	0.971657i
$874$		0			0
$875$	−7.44949	+	8.33697i	−0.251839	+	0.281841i
$876$		0			0
$877$	7.13633			0.240977			0.120488	−	0.992715i	$-0.461554\pi$
$877$	7.13633			0.240977			0.120488	+	0.992715i	$0.461554\pi$
$878$		0			0
$879$	2.62814			0.0886448
$880$		0			0
$881$	−12.5504			−0.422835			−0.211418	−	0.977396i	$-0.567808\pi$
$881$	−12.5504			−0.422835			−0.211418	+	0.977396i	$0.567808\pi$
$882$		0			0
$883$	25.7331			0.865987			0.432993	−	0.901397i	$-0.357457\pi$
$883$	25.7331			0.865987			0.432993	+	0.901397i	$0.357457\pi$
$884$		0			0
$885$	−0.891262	−	0.858503i	−0.0299595	−	0.0288583i
$886$		0			0
$887$		−	15.8391i		−	0.531826i	−0.963997	−	0.265913i	$-0.914327\pi$
$887$		−	15.8391i		−	0.531826i	0.963997	−	0.265913i	$-0.0856735\pi$
$888$		0			0
$889$	−12.7130			−0.426381
$890$		0			0
$891$		−	28.6406i		−	0.959497i
$892$		0			0
$893$	−22.6771			−0.758860
$894$		0			0
$895$	−29.9150	−	28.8155i	−0.999948	−	0.963194i
$896$		0			0
$897$		−	0.193659i		−	0.00646609i
$898$		0			0
$899$		−	5.62016i		−	0.187443i
$900$		0			0
$901$		−	4.31493i		−	0.143751i
$902$		0			0
$903$		−	2.60552i		−	0.0867063i
$904$		0			0
$905$	14.5746	+	14.0389i	0.484477	+	0.466670i
$906$		0			0
$907$	−16.5440			−0.549333			−0.274667	−	0.961540i	$-0.588567\pi$
$907$	−16.5440			−0.549333			−0.274667	+	0.961540i	$0.588567\pi$
$908$		0			0
$909$		−	4.33363i		−	0.143738i
$910$		0			0
$911$	−38.4342			−1.27338			−0.636691	−	0.771119i	$-0.719697\pi$
$911$	−38.4342			−1.27338			−0.636691	+	0.771119i	$0.719697\pi$
$912$		0			0
$913$		−	49.0123i		−	1.62207i
$914$		0			0
$915$	3.89112	+	3.74810i	0.128637	+	0.123908i
$916$		0			0
$917$	−11.5369			−0.380983
$918$		0			0
$919$	−44.8577			−1.47972			−0.739860	−	0.672761i	$-0.765108\pi$
$919$	−44.8577			−1.47972			−0.739860	+	0.672761i	$0.765108\pi$
$920$		0			0
$921$	5.96964			0.196706
$922$		0			0
$923$	12.7098			0.418348
$924$		0			0
$925$	1.70562	+	45.5354i	0.0560804	+	1.49720i
$926$		0			0
$927$			30.1036i			0.988731i
$928$		0			0
$929$	19.4332			0.637581			0.318791	−	0.947825i	$-0.396723\pi$
$929$	19.4332			0.637581			0.318791	+	0.947825i	$0.396723\pi$
$930$		0			0
$931$			2.29607i			0.0752507i
$932$		0			0
$933$	4.90544			0.160597
$934$		0			0
$935$	5.16747	−	5.36465i	0.168994	−	0.175443i
$936$		0			0
$937$			36.3664i			1.18804i	0.804451	+	0.594020i	$0.202460\pi$
$937$			36.3664i			1.18804i	−0.804451	+	0.594020i	$0.797540\pi$
$938$		0			0
$939$		−	6.04483i		−	0.197266i
$940$		0			0
$941$			12.6304i			0.411740i	0.978579	+	0.205870i	$0.0660024\pi$
$941$			12.6304i			0.411740i	−0.978579	+	0.205870i	$0.933998\pi$
$942$		0			0
$943$		−	6.99110i		−	0.227661i
$944$		0			0
$945$	2.26290	+	2.17973i	0.0736123	+	0.0709066i
$946$		0			0
$947$	36.6341			1.19045			0.595224	−	0.803560i	$-0.297063\pi$
$947$	36.6341			1.19045			0.595224	+	0.803560i	$0.297063\pi$
$948$		0			0
$949$			13.2968i			0.431633i
$950$		0			0
$951$	5.28426			0.171354
$952$		0			0
$953$			6.97886i			0.226068i	0.993591	+	0.113034i	$0.0360568\pi$
$953$			6.97886i			0.226068i	−0.993591	+	0.113034i	$0.963943\pi$
$954$		0			0
$955$	12.5347	−	13.0130i	0.405613	−	0.421091i
$956$		0			0
$957$	2.52900			0.0817509
$958$		0			0
$959$	−12.6090			−0.407166
$960$		0			0
$961$	−27.8670			−0.898935
$962$		0			0
$963$	−4.31848			−0.139161
$964$		0			0
$965$	−37.9662	−	36.5707i	−1.22217	−	1.17725i
$966$		0			0
$967$		−	35.7173i		−	1.14859i	−0.818648	−	0.574296i	$-0.805276\pi$
$967$		−	35.7173i		−	1.14859i	0.818648	−	0.574296i	$-0.194724\pi$
$968$		0			0
$969$	0.536604			0.0172382
$970$		0			0
$971$			12.6075i			0.404592i	0.979324	+	0.202296i	$0.0648404\pi$
$971$			12.6075i			0.404592i	−0.979324	+	0.202296i	$0.935160\pi$
$972$		0			0
$973$	0.753066			0.0241422
$974$		0			0
$975$	0.0449422	+	1.19984i	0.00143930	+	0.0384255i
$976$		0			0
$977$		−	42.3742i		−	1.35567i	−0.735213	−	0.677836i	$-0.762918\pi$
$977$		−	42.3742i		−	1.35567i	0.735213	−	0.677836i	$-0.237082\pi$
$978$		0			0
$979$		−	0.495370i		−	0.0158321i
$980$		0			0
$981$		−	47.0807i		−	1.50317i
$982$		0			0
$983$		−	51.8265i		−	1.65301i	−0.562930	−	0.826505i	$-0.690326\pi$
$983$		−	51.8265i		−	1.65301i	0.562930	−	0.826505i	$-0.309674\pi$
$984$		0			0
$985$	−5.39634	+	5.60226i	−0.171942	+	0.178503i
$986$		0			0
$987$	−2.33470			−0.0743143
$988$		0			0
$989$			8.88888i			0.282650i
$990$		0			0
$991$	−20.2175			−0.642230			−0.321115	−	0.947040i	$-0.604058\pi$
$991$	−20.2175			−0.642230			−0.321115	+	0.947040i	$0.604058\pi$
$992$		0			0
$993$		−	1.88131i		−	0.0597016i
$994$		0			0
$995$	−30.0414	+	31.1877i	−0.952376	+	0.988717i
$996$		0			0
$997$	−61.2159			−1.93873			−0.969363	−	0.245632i	$-0.921004\pi$
$997$	−61.2159			−1.93873			−0.969363	+	0.245632i	$0.921004\pi$
$998$		0			0
$999$	12.8056			0.405152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.c.1569.13	yes	24
4.3	odd	2	inner	2240.2.l.c.1569.12	yes	24
5.4	even	2		2240.2.l.d.1569.12	yes	24
8.3	odd	2		2240.2.l.d.1569.14	yes	24
8.5	even	2		2240.2.l.d.1569.11	yes	24
20.19	odd	2		2240.2.l.d.1569.13	yes	24
40.19	odd	2	inner	2240.2.l.c.1569.11	✓	24
40.29	even	2	inner	2240.2.l.c.1569.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.c.1569.11	✓	24	40.19	odd	2	inner
2240.2.l.c.1569.12	yes	24	4.3	odd	2	inner
2240.2.l.c.1569.13	yes	24	1.1	even	1	trivial
2240.2.l.c.1569.14	yes	24	40.29	even	2	inner
2240.2.l.d.1569.11	yes	24	8.5	even	2
2240.2.l.d.1569.12	yes	24	5.4	even	2
2240.2.l.d.1569.13	yes	24	20.19	odd	2
2240.2.l.d.1569.14	yes	24	8.3	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.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.236390 q^{3} +(1.55127 - 1.61046i) q^{5} -1.00000i q^{7} -2.94412 q^{9} +O(q^{10})\)	\(q+0.236390 q^{3} +(1.55127 - 1.61046i) q^{5} -1.00000i q^{7} -2.94412 q^{9} -3.36940i q^{11} -1.01585 q^{13} +(0.366703 - 0.380696i) q^{15} +0.988642i q^{17} -2.29607i q^{19} -0.236390i q^{21} +0.806457i q^{23} +(-0.187153 - 4.99650i) q^{25} -1.40513 q^{27} +3.17517i q^{29} -1.77004 q^{31} -0.796493i q^{33} +(-1.61046 - 1.55127i) q^{35} -9.11347 q^{37} -0.240136 q^{39} -8.66890 q^{41} +11.0221 q^{43} +(-4.56711 + 4.74138i) q^{45} -9.87647i q^{47} -1.00000 q^{49} +0.233705i q^{51} -4.36450 q^{53} +(-5.42629 - 5.22684i) q^{55} -0.542768i q^{57} -2.34114i q^{59} +10.2211i q^{61} +2.94412i q^{63} +(-1.57585 + 1.63598i) q^{65} -6.35033 q^{67} +0.190638i q^{69} -12.5115 q^{71} -13.0894i q^{73} +(-0.0442412 - 1.18112i) q^{75} -3.36940 q^{77} -7.51602 q^{79} +8.50020 q^{81} +14.5463 q^{83} +(1.59217 + 1.53365i) q^{85} +0.750577i q^{87} +0.147020 q^{89} +1.01585i q^{91} -0.418419 q^{93} +(-3.69773 - 3.56182i) q^{95} +9.75135i q^{97} +9.91993i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} + 12 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more