LMFDB - Embedded newform 2240.2.b.e.1121.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (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:	$6$
Coefficient field:	6.0.3534400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 $ x^6 - 2x^5 - 3x^4 + 16x^3 + x^2 - 12x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.2
Root			$0.627553 - 1.14620i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.e.1121.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29240i q^{3} +1.00000i q^{5} -1.00000 q^{7} -2.25511 q^{9} +O(q^{10})$	$q-2.29240i q^{3} +1.00000i q^{5} -1.00000 q^{7} -2.25511 q^{9} -0.744895i q^{11} -3.83991i q^{13} +2.29240 q^{15} +7.38741 q^{17} -4.58480i q^{19} +2.29240i q^{21} +2.51021 q^{23} -1.00000 q^{25} -1.70760i q^{27} +4.80261i q^{29} -3.09501 q^{31} -1.70760 q^{33} -1.00000i q^{35} +11.6798i q^{37} -8.80261 q^{39} -1.41520 q^{41} -2.00000i q^{43} -2.25511i q^{45} +1.83991 q^{47} +1.00000 q^{49} -16.9349i q^{51} -11.0950i q^{53} +0.744895 q^{55} -10.5102 q^{57} -11.0950i q^{59} -8.51021i q^{61} +2.25511 q^{63} +3.83991 q^{65} -10.5848i q^{67} -5.75441i q^{69} +3.41520 q^{71} -8.51021 q^{73} +2.29240i q^{75} +0.744895i q^{77} -8.36699 q^{79} -10.6798 q^{81} -4.58480i q^{83} +7.38741i q^{85} +11.0095 q^{87} -7.48979 q^{89} +3.83991i q^{91} +7.09501i q^{93} +4.58480 q^{95} -3.38741 q^{97} +1.67982i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 10 * q^9	$ 6 q - 6 q^{7} - 10 q^{9} - 4 q^{17} + 8 q^{23} - 6 q^{25} + 16 q^{31} - 24 q^{33} - 32 q^{39} - 36 q^{41} - 20 q^{47} + 6 q^{49} + 8 q^{55} - 56 q^{57} + 10 q^{63} - 8 q^{65} + 48 q^{71} - 44 q^{73} - 16 q^{79} - 2 q^{81} - 20 q^{87} - 52 q^{89} + 28 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 10 * q^9 - 4 * q^17 + 8 * q^23 - 6 * q^25 + 16 * q^31 - 24 * q^33 - 32 * q^39 - 36 * q^41 - 20 * q^47 + 6 * q^49 + 8 * q^55 - 56 * q^57 + 10 * q^63 - 8 * q^65 + 48 * q^71 - 44 * q^73 - 16 * q^79 - 2 * q^81 - 20 * q^87 - 52 * q^89 + 28 * q^97

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
$3$	− 2.29240i	− 1.32352i	−0.749716	−	0.661759i	$-0.769810\pi$
$3$	− 2.29240i	− 1.32352i	0.749716	−	0.661759i	$-0.230190\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.25511	−0.751702
$10$	0	0
$11$	− 0.744895i	− 0.224594i	−0.993675	−	0.112297i	$-0.964179\pi$
$11$	− 0.744895i	− 0.224594i	0.993675	−	0.112297i	$-0.0358208\pi$
$12$	0	0
$13$	− 3.83991i	− 1.06500i	−0.846430	−	0.532499i	$-0.821253\pi$
$13$	− 3.83991i	− 1.06500i	0.846430	−	0.532499i	$-0.178747\pi$
$14$	0	0
$15$	2.29240	0.591896
$16$	0	0
$17$	7.38741	1.79171	0.895856	−	0.444345i	$-0.146564\pi$
$17$	7.38741	1.79171	0.895856	+	0.444345i	$0.146564\pi$
$18$	0	0
$19$	− 4.58480i	− 1.05183i	−0.850538	−	0.525913i	$-0.823724\pi$
$19$	− 4.58480i	− 1.05183i	0.850538	−	0.525913i	$-0.176276\pi$
$20$	0	0
$21$	2.29240i	0.500243i
$22$	0	0
$23$	2.51021	0.523415	0.261707	−	0.965147i	$-0.415714\pi$
$23$	2.51021	0.523415	0.261707	+	0.965147i	$0.415714\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.70760i	− 0.328627i
$28$	0	0
$29$	4.80261i	0.891823i	0.895077	+	0.445911i	$0.147120\pi$
$29$	4.80261i	0.891823i	−0.895077	+	0.445911i	$0.852880\pi$
$30$	0	0
$31$	−3.09501	−0.555881	−0.277940	−	0.960598i	$-0.589652\pi$
$31$	−3.09501	−0.555881	−0.277940	+	0.960598i	$0.589652\pi$
$32$	0	0
$33$	−1.70760	−0.297255
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	11.6798i	1.92015i	0.279743	+	0.960075i	$0.409751\pi$
$37$	11.6798i	1.92015i	−0.279743	+	0.960075i	$0.590249\pi$
$38$	0	0
$39$	−8.80261	−1.40955
$40$	0	0
$41$	−1.41520	−0.221017	−0.110508	−	0.993875i	$-0.535248\pi$
$41$	−1.41520	−0.221017	−0.110508	+	0.993875i	$0.535248\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	0	0
$45$	− 2.25511i	− 0.336171i
$46$	0	0
$47$	1.83991	0.268378	0.134189	−	0.990956i	$-0.457157\pi$
$47$	1.83991	0.268378	0.134189	+	0.990956i	$0.457157\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 16.9349i	− 2.37136i
$52$	0	0
$53$	− 11.0950i	− 1.52402i	−0.647567	−	0.762009i	$-0.724213\pi$
$53$	− 11.0950i	− 1.52402i	0.647567	−	0.762009i	$-0.275787\pi$
$54$	0	0
$55$	0.744895	0.100442
$56$	0	0
$57$	−10.5102	−1.39211
$58$	0	0
$59$	− 11.0950i	− 1.44445i	−0.691659	−	0.722224i	$-0.743120\pi$
$59$	− 11.0950i	− 1.44445i	0.691659	−	0.722224i	$-0.256880\pi$
$60$	0	0
$61$	− 8.51021i	− 1.08962i	−0.838559	−	0.544810i	$-0.816602\pi$
$61$	− 8.51021i	− 1.08962i	0.838559	−	0.544810i	$-0.183398\pi$
$62$	0	0
$63$	2.25511	0.284117
$64$	0	0
$65$	3.83991	0.476282
$66$	0	0
$67$	− 10.5848i	− 1.29314i	−0.762855	−	0.646570i	$-0.776203\pi$
$67$	− 10.5848i	− 1.29314i	0.762855	−	0.646570i	$-0.223797\pi$
$68$	0	0
$69$	− 5.75441i	− 0.692749i
$70$	0	0
$71$	3.41520	0.405309	0.202655	−	0.979250i	$-0.435043\pi$
$71$	3.41520	0.405309	0.202655	+	0.979250i	$0.435043\pi$
$72$	0	0
$73$	−8.51021	−0.996045	−0.498022	−	0.867164i	$-0.665940\pi$
$73$	−8.51021	−0.996045	−0.498022	+	0.867164i	$0.665940\pi$
$74$	0	0
$75$	2.29240i	0.264704i
$76$	0	0
$77$	0.744895i	0.0848887i
$78$	0	0
$79$	−8.36699	−0.941360	−0.470680	−	0.882304i	$-0.655991\pi$
$79$	−8.36699	−0.941360	−0.470680	+	0.882304i	$0.655991\pi$
$80$	0	0
$81$	−10.6798	−1.18665
$82$	0	0
$83$	− 4.58480i	− 0.503248i	−0.967825	−	0.251624i	$-0.919035\pi$
$83$	− 4.58480i	− 0.503248i	0.967825	−	0.251624i	$-0.0809645\pi$
$84$	0	0
$85$	7.38741i	0.801278i
$86$	0	0
$87$	11.0095	1.18034
$88$	0	0
$89$	−7.48979	−0.793916	−0.396958	−	0.917837i	$-0.629934\pi$
$89$	−7.48979	−0.793916	−0.396958	+	0.917837i	$0.629934\pi$
$90$	0	0
$91$	3.83991i	0.402532i
$92$	0	0
$93$	7.09501i	0.735719i
$94$	0	0
$95$	4.58480	0.470391
$96$	0	0
$97$	−3.38741	−0.343940	−0.171970	−	0.985102i	$-0.555013\pi$
$97$	−3.38741	−0.343940	−0.171970	+	0.985102i	$0.555013\pi$
$98$	0	0
$99$	1.67982i	0.168828i
$100$	0	0
$101$	− 11.1696i	− 1.11142i	−0.831377	−	0.555709i	$-0.812447\pi$
$101$	− 11.1696i	− 1.11142i	0.831377	−	0.555709i	$-0.187553\pi$
$102$	0	0
$103$	−13.8399	−1.36369	−0.681843	−	0.731498i	$-0.738821\pi$
$103$	−13.8399	−1.36369	−0.681843	+	0.731498i	$0.738821\pi$
$104$	0	0
$105$	−2.29240	−0.223715
$106$	0	0
$107$	8.19003i	0.791760i	0.918302	+	0.395880i	$0.129560\pi$
$107$	8.19003i	0.791760i	−0.918302	+	0.395880i	$0.870440\pi$
$108$	0	0
$109$	− 12.3670i	− 1.18454i	−0.805738	−	0.592272i	$-0.798231\pi$
$109$	− 12.3670i	− 1.18454i	0.805738	−	0.592272i	$-0.201769\pi$
$110$	0	0
$111$	26.7748	2.54135
$112$	0	0
$113$	−4.07459	−0.383305	−0.191653	−	0.981463i	$-0.561385\pi$
$113$	−4.07459	−0.383305	−0.191653	+	0.981463i	$0.561385\pi$
$114$	0	0
$115$	2.51021i	0.234078i
$116$	0	0
$117$	8.65940i	0.800561i
$118$	0	0
$119$	−7.38741	−0.677203
$120$	0	0
$121$	10.4451	0.949557
$122$	0	0
$123$	3.24420i	0.292520i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−7.09501	−0.629581	−0.314790	−	0.949161i	$-0.601934\pi$
$127$	−7.09501	−0.629581	−0.314790	+	0.949161i	$0.601934\pi$
$128$	0	0
$129$	−4.58480	−0.403669
$130$	0	0
$131$	3.67982i	0.321507i	0.986995	+	0.160754i	$0.0513924\pi$
$131$	3.67982i	0.321507i	−0.986995	+	0.160754i	$0.948608\pi$
$132$	0	0
$133$	4.58480i	0.397553i
$134$	0	0
$135$	1.70760	0.146967
$136$	0	0
$137$	−3.48979	−0.298153	−0.149076	−	0.988826i	$-0.547630\pi$
$137$	−3.48979	−0.298153	−0.149076	+	0.988826i	$0.547630\pi$
$138$	0	0
$139$	11.6798i	0.990669i	0.868702	+	0.495335i	$0.164955\pi$
$139$	11.6798i	0.990669i	−0.868702	+	0.495335i	$0.835045\pi$
$140$	0	0
$141$	− 4.21781i	− 0.355204i
$142$	0	0
$143$	−2.86033	−0.239193
$144$	0	0
$145$	−4.80261	−0.398835
$146$	0	0
$147$	− 2.29240i	− 0.189074i
$148$	0	0
$149$	− 6.19003i	− 0.507107i	−0.967321	−	0.253553i	$-0.918401\pi$
$149$	− 6.19003i	− 0.507107i	0.967321	−	0.253553i	$-0.0815993\pi$
$150$	0	0
$151$	9.97222	0.811528	0.405764	−	0.913978i	$-0.367006\pi$
$151$	9.97222	0.811528	0.405764	+	0.913978i	$0.367006\pi$
$152$	0	0
$153$	−16.6594	−1.34683
$154$	0	0
$155$	− 3.09501i	− 0.248597i
$156$	0	0
$157$	23.1696i	1.84914i	0.381017	+	0.924568i	$0.375574\pi$
$157$	23.1696i	1.84914i	−0.381017	+	0.924568i	$0.624426\pi$
$158$	0	0
$159$	−25.4342	−2.01707
$160$	0	0
$161$	−2.51021	−0.197832
$162$	0	0
$163$	− 4.07459i	− 0.319147i	−0.987186	−	0.159573i	$-0.948988\pi$
$163$	− 4.07459i	− 0.319147i	0.987186	−	0.159573i	$-0.0510119\pi$
$164$	0	0
$165$	− 1.70760i	− 0.132936i
$166$	0	0
$167$	24.1791	1.87104	0.935518	−	0.353278i	$-0.114933\pi$
$167$	24.1791	1.87104	0.935518	+	0.353278i	$0.114933\pi$
$168$	0	0
$169$	−1.74489	−0.134223
$170$	0	0
$171$	10.3392i	0.790659i
$172$	0	0
$173$	14.0299i	1.06668i	0.845902	+	0.533338i	$0.179063\pi$
$173$	14.0299i	1.06668i	−0.845902	+	0.533338i	$0.820937\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−25.4342	−1.91175
$178$	0	0
$179$	− 5.67982i	− 0.424529i	−0.977212	−	0.212265i	$-0.931916\pi$
$179$	− 5.67982i	− 0.424529i	0.977212	−	0.212265i	$-0.0680839\pi$
$180$	0	0
$181$	− 5.56438i	− 0.413597i	−0.978384	−	0.206799i	$-0.933695\pi$
$181$	− 5.56438i	− 0.413597i	0.978384	−	0.206799i	$-0.0663045\pi$
$182$	0	0
$183$	−19.5088	−1.44213
$184$	0	0
$185$	−11.6798	−0.858717
$186$	0	0
$187$	− 5.50285i	− 0.402408i
$188$	0	0
$189$	1.70760i	0.124210i
$190$	0	0
$191$	−14.5570	−1.05331	−0.526655	−	0.850079i	$-0.676554\pi$
$191$	−14.5570	−1.05331	−0.526655	+	0.850079i	$0.676554\pi$
$192$	0	0
$193$	26.8494	1.93266	0.966332	−	0.257299i	$-0.0828325\pi$
$193$	26.8494	1.93266	0.966332	+	0.257299i	$0.0828325\pi$
$194$	0	0
$195$	− 8.80261i	− 0.630368i
$196$	0	0
$197$	23.2442i	1.65608i	0.560669	+	0.828040i	$0.310544\pi$
$197$	23.2442i	1.65608i	−0.560669	+	0.828040i	$0.689456\pi$
$198$	0	0
$199$	12.1154	0.858840	0.429420	−	0.903105i	$-0.358718\pi$
$199$	12.1154	0.858840	0.429420	+	0.903105i	$0.358718\pi$
$200$	0	0
$201$	−24.2646	−1.71149
$202$	0	0
$203$	− 4.80261i	− 0.337077i
$204$	0	0
$205$	− 1.41520i	− 0.0988416i
$206$	0	0
$207$	−5.66079	−0.393452
$208$	0	0
$209$	−3.41520	−0.236234
$210$	0	0
$211$	2.08550i	0.143572i	0.997420	+	0.0717858i	$0.0228698\pi$
$211$	2.08550i	0.143572i	−0.997420	+	0.0717858i	$0.977130\pi$
$212$	0	0
$213$	− 7.82900i	− 0.536434i
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	3.09501	0.210103
$218$	0	0
$219$	19.5088i	1.31828i
$220$	0	0
$221$	− 28.3670i	− 1.90817i
$222$	0	0
$223$	24.1791	1.61915	0.809577	−	0.587014i	$-0.199697\pi$
$223$	24.1791	1.61915	0.809577	+	0.587014i	$0.199697\pi$
$224$	0	0
$225$	2.25511	0.150340
$226$	0	0
$227$	− 23.8976i	− 1.58614i	−0.609130	−	0.793071i	$-0.708481\pi$
$227$	− 23.8976i	− 1.58614i	0.609130	−	0.793071i	$-0.291519\pi$
$228$	0	0
$229$	− 16.7748i	− 1.10851i	−0.832346	−	0.554256i	$-0.813003\pi$
$229$	− 16.7748i	− 1.10851i	0.832346	−	0.554256i	$-0.186997\pi$
$230$	0	0
$231$	1.70760	0.112352
$232$	0	0
$233$	−23.8698	−1.56377	−0.781883	−	0.623426i	$-0.785740\pi$
$233$	−23.8698	−1.56377	−0.781883	+	0.623426i	$0.785740\pi$
$234$	0	0
$235$	1.83991i	0.120022i
$236$	0	0
$237$	19.1805i	1.24591i
$238$	0	0
$239$	−7.19739	−0.465560	−0.232780	−	0.972529i	$-0.574782\pi$
$239$	−7.19739	−0.465560	−0.232780	+	0.972529i	$0.574782\pi$
$240$	0	0
$241$	4.83039	0.311153	0.155577	−	0.987824i	$-0.450276\pi$
$241$	4.83039	0.311153	0.155577	+	0.987824i	$0.450276\pi$
$242$	0	0
$243$	19.3596i	1.24192i
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	−17.6052	−1.12019
$248$	0	0
$249$	−10.5102	−0.666057
$250$	0	0
$251$	− 11.6798i	− 0.737223i	−0.929583	−	0.368612i	$-0.879833\pi$
$251$	− 11.6798i	− 0.737223i	0.929583	−	0.368612i	$-0.120167\pi$
$252$	0	0
$253$	− 1.86984i	− 0.117556i
$254$	0	0
$255$	16.9349	1.06051
$256$	0	0
$257$	12.6594	0.789671	0.394836	−	0.918752i	$-0.370802\pi$
$257$	12.6594	0.789671	0.394836	+	0.918752i	$0.370802\pi$
$258$	0	0
$259$	− 11.6798i	− 0.725748i
$260$	0	0
$261$	− 10.8304i	− 0.670385i
$262$	0	0
$263$	−20.5848	−1.26931	−0.634657	−	0.772794i	$-0.718858\pi$
$263$	−20.5848	−1.26931	−0.634657	+	0.772794i	$0.718858\pi$
$264$	0	0
$265$	11.0950	0.681561
$266$	0	0
$267$	17.1696i	1.05076i
$268$	0	0
$269$	8.07459i	0.492317i	0.969230	+	0.246158i	$0.0791683\pi$
$269$	8.07459i	0.492317i	−0.969230	+	0.246158i	$0.920832\pi$
$270$	0	0
$271$	−1.16961	−0.0710485	−0.0355243	−	0.999369i	$-0.511310\pi$
$271$	−1.16961	−0.0710485	−0.0355243	+	0.999369i	$0.511310\pi$
$272$	0	0
$273$	8.80261	0.532758
$274$	0	0
$275$	0.744895i	0.0449189i
$276$	0	0
$277$	− 6.51021i	− 0.391161i	−0.980688	−	0.195580i	$-0.937341\pi$
$277$	− 6.51021i	− 0.391161i	0.980688	−	0.195580i	$-0.0626590\pi$
$278$	0	0
$279$	6.97958	0.417857
$280$	0	0
$281$	−2.08550	−0.124410	−0.0622052	−	0.998063i	$-0.519813\pi$
$281$	−2.08550	−0.124410	−0.0622052	+	0.998063i	$0.519813\pi$
$282$	0	0
$283$	29.5028i	1.75376i	0.480707	+	0.876881i	$0.340380\pi$
$283$	29.5028i	1.75376i	−0.480707	+	0.876881i	$0.659620\pi$
$284$	0	0
$285$	− 10.5102i	− 0.622571i
$286$	0	0
$287$	1.41520	0.0835364
$288$	0	0
$289$	37.5739	2.21023
$290$	0	0
$291$	7.76532i	0.455211i
$292$	0	0
$293$	− 1.32970i	− 0.0776818i	−0.999245	−	0.0388409i	$-0.987633\pi$
$293$	− 1.32970i	− 0.0776818i	0.999245	−	0.0388409i	$-0.0123666\pi$
$294$	0	0
$295$	11.0950	0.645977
$296$	0	0
$297$	−1.27198	−0.0738079
$298$	0	0
$299$	− 9.63898i	− 0.557436i
$300$	0	0
$301$	2.00000i	0.115278i
$302$	0	0
$303$	−25.6052	−1.47098
$304$	0	0
$305$	8.51021	0.487293
$306$	0	0
$307$	15.8976i	0.907325i	0.891173	+	0.453663i	$0.149883\pi$
$307$	15.8976i	0.907325i	−0.891173	+	0.453663i	$0.850117\pi$
$308$	0	0
$309$	31.7266i	1.80486i
$310$	0	0
$311$	31.6798	1.79640	0.898199	−	0.439590i	$-0.144876\pi$
$311$	31.6798	1.79640	0.898199	+	0.439590i	$0.144876\pi$
$312$	0	0
$313$	25.1418	1.42110	0.710550	−	0.703647i	$-0.248446\pi$
$313$	25.1418	1.42110	0.710550	+	0.703647i	$0.248446\pi$
$314$	0	0
$315$	2.25511i	0.127061i
$316$	0	0
$317$	12.7340i	0.715212i	0.933873	+	0.357606i	$0.116407\pi$
$317$	12.7340i	0.715212i	−0.933873	+	0.357606i	$0.883593\pi$
$318$	0	0
$319$	3.57744	0.200298
$320$	0	0
$321$	18.7748	1.04791
$322$	0	0
$323$	− 33.8698i	− 1.88457i
$324$	0	0
$325$	3.83991i	0.213000i
$326$	0	0
$327$	−28.3501	−1.56777
$328$	0	0
$329$	−1.83991	−0.101437
$330$	0	0
$331$	− 17.0394i	− 0.936573i	−0.883577	−	0.468286i	$-0.844872\pi$
$331$	− 17.0394i	− 0.936573i	0.883577	−	0.468286i	$-0.155128\pi$
$332$	0	0
$333$	− 26.3392i	− 1.44338i
$334$	0	0
$335$	10.5848	0.578310
$336$	0	0
$337$	16.0746	0.875639	0.437819	−	0.899063i	$-0.355751\pi$
$337$	16.0746	0.875639	0.437819	+	0.899063i	$0.355751\pi$
$338$	0	0
$339$	9.34060i	0.507312i
$340$	0	0
$341$	2.30546i	0.124848i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.75441	0.309807
$346$	0	0
$347$	0.945827i	0.0507747i	0.999678	+	0.0253873i	$0.00808191\pi$
$347$	0.945827i	0.0507747i	−0.999678	+	0.0253873i	$0.991918\pi$
$348$	0	0
$349$	− 26.5292i	− 1.42008i	−0.704163	−	0.710039i	$-0.748677\pi$
$349$	− 26.5292i	− 1.42008i	0.704163	−	0.710039i	$-0.251323\pi$
$350$	0	0
$351$	−6.55702	−0.349988
$352$	0	0
$353$	13.9314	0.741492	0.370746	−	0.928734i	$-0.379102\pi$
$353$	13.9314	0.741492	0.370746	+	0.928734i	$0.379102\pi$
$354$	0	0
$355$	3.41520i	0.181260i
$356$	0	0
$357$	16.9349i	0.896291i
$358$	0	0
$359$	−27.9444	−1.47485	−0.737425	−	0.675429i	$-0.763959\pi$
$359$	−27.9444	−1.47485	−0.737425	+	0.675429i	$0.763959\pi$
$360$	0	0
$361$	−2.02042	−0.106338
$362$	0	0
$363$	− 23.9444i	− 1.25676i
$364$	0	0
$365$	− 8.51021i	− 0.445445i
$366$	0	0
$367$	−7.00951	−0.365894	−0.182947	−	0.983123i	$-0.558564\pi$
$367$	−7.00951	−0.365894	−0.182947	+	0.983123i	$0.558564\pi$
$368$	0	0
$369$	3.19142	0.166139
$370$	0	0
$371$	11.0950i	0.576024i
$372$	0	0
$373$	− 2.83039i	− 0.146552i	−0.997312	−	0.0732761i	$-0.976655\pi$
$373$	− 2.83039i	− 0.146552i	0.997312	−	0.0732761i	$-0.0233454\pi$
$374$	0	0
$375$	−2.29240	−0.118379
$376$	0	0
$377$	18.4416	0.949790
$378$	0	0
$379$	− 10.8494i	− 0.557297i	−0.960393	−	0.278649i	$-0.910113\pi$
$379$	− 10.8494i	− 0.557297i	0.960393	−	0.278649i	$-0.0898865\pi$
$380$	0	0
$381$	16.2646i	0.833262i
$382$	0	0
$383$	37.5497	1.91870	0.959349	−	0.282222i	$-0.0910716\pi$
$383$	37.5497	1.91870	0.959349	+	0.282222i	$0.0910716\pi$
$384$	0	0
$385$	−0.744895	−0.0379634
$386$	0	0
$387$	4.51021i	0.229267i
$388$	0	0
$389$	9.38741i	0.475961i	0.971270	+	0.237980i	$0.0764854\pi$
$389$	9.38741i	0.475961i	−0.971270	+	0.237980i	$0.923515\pi$
$390$	0	0
$391$	18.5440	0.937809
$392$	0	0
$393$	8.43562	0.425521
$394$	0	0
$395$	− 8.36699i	− 0.420989i
$396$	0	0
$397$	15.3705i	0.771425i	0.922619	+	0.385713i	$0.126044\pi$
$397$	15.3705i	0.771425i	−0.922619	+	0.385713i	$0.873956\pi$
$398$	0	0
$399$	10.5102	0.526169
$400$	0	0
$401$	13.7653	0.687407	0.343704	−	0.939078i	$-0.388318\pi$
$401$	13.7653	0.687407	0.343704	+	0.939078i	$0.388318\pi$
$402$	0	0
$403$	11.8846i	0.592012i
$404$	0	0
$405$	− 10.6798i	− 0.530684i
$406$	0	0
$407$	8.70024	0.431255
$408$	0	0
$409$	−0.0745931	−0.00368839	−0.00184420	−	0.999998i	$-0.500587\pi$
$409$	−0.0745931	−0.00368839	−0.00184420	+	0.999998i	$0.500587\pi$
$410$	0	0
$411$	8.00000i	0.394611i
$412$	0	0
$413$	11.0950i	0.545950i
$414$	0	0
$415$	4.58480	0.225059
$416$	0	0
$417$	26.7748	1.31117
$418$	0	0
$419$	22.0746i	1.07841i	0.842173	+	0.539207i	$0.181276\pi$
$419$	22.0746i	1.07841i	−0.842173	+	0.539207i	$0.818724\pi$
$420$	0	0
$421$	− 5.38741i	− 0.262567i	−0.991345	−	0.131283i	$-0.958090\pi$
$421$	− 5.38741i	− 0.262567i	0.991345	−	0.131283i	$-0.0419097\pi$
$422$	0	0
$423$	−4.14919	−0.201740
$424$	0	0
$425$	−7.38741	−0.358342
$426$	0	0
$427$	8.51021i	0.411838i
$428$	0	0
$429$	6.55702i	0.316576i
$430$	0	0
$431$	−29.9167	−1.44103	−0.720517	−	0.693437i	$-0.756096\pi$
$431$	−29.9167	−1.44103	−0.720517	+	0.693437i	$0.756096\pi$
$432$	0	0
$433$	5.15058	0.247521	0.123760	−	0.992312i	$-0.460505\pi$
$433$	5.15058	0.247521	0.123760	+	0.992312i	$0.460505\pi$
$434$	0	0
$435$	11.0095i	0.527866i
$436$	0	0
$437$	− 11.5088i	− 0.550541i
$438$	0	0
$439$	−15.4152	−0.735727	−0.367864	−	0.929880i	$-0.619911\pi$
$439$	−15.4152	−0.735727	−0.367864	+	0.929880i	$0.619911\pi$
$440$	0	0
$441$	−2.25511	−0.107386
$442$	0	0
$443$	− 25.5088i	− 1.21196i	−0.795480	−	0.605980i	$-0.792781\pi$
$443$	− 25.5088i	− 1.21196i	0.795480	−	0.605980i	$-0.207219\pi$
$444$	0	0
$445$	− 7.48979i	− 0.355050i
$446$	0	0
$447$	−14.1900	−0.671165
$448$	0	0
$449$	23.0841	1.08941	0.544703	−	0.838629i	$-0.316642\pi$
$449$	23.0841	1.08941	0.544703	+	0.838629i	$0.316642\pi$
$450$	0	0
$451$	1.05417i	0.0496391i
$452$	0	0
$453$	− 22.8603i	− 1.07407i
$454$	0	0
$455$	−3.83991	−0.180018
$456$	0	0
$457$	0.774830	0.0362450	0.0181225	−	0.999836i	$-0.494231\pi$
$457$	0.774830	0.0362450	0.0181225	+	0.999836i	$0.494231\pi$
$458$	0	0
$459$	− 12.6147i	− 0.588806i
$460$	0	0
$461$	− 34.0599i	− 1.58633i	−0.609009	−	0.793163i	$-0.708433\pi$
$461$	− 34.0599i	− 1.58633i	0.609009	−	0.793163i	$-0.291567\pi$
$462$	0	0
$463$	12.1492	0.564621	0.282310	−	0.959323i	$-0.408899\pi$
$463$	12.1492	0.564621	0.282310	+	0.959323i	$0.408899\pi$
$464$	0	0
$465$	−7.09501	−0.329023
$466$	0	0
$467$	18.8772i	0.873533i	0.899575	+	0.436766i	$0.143876\pi$
$467$	18.8772i	0.873533i	−0.899575	+	0.436766i	$0.856124\pi$
$468$	0	0
$469$	10.5848i	0.488761i
$470$	0	0
$471$	53.1140	2.44737
$472$	0	0
$473$	−1.48979	−0.0685006
$474$	0	0
$475$	4.58480i	0.210365i
$476$	0	0
$477$	25.0204i	1.14561i
$478$	0	0
$479$	3.56438	0.162861	0.0814304	−	0.996679i	$-0.474051\pi$
$479$	3.56438	0.162861	0.0814304	+	0.996679i	$0.474051\pi$
$480$	0	0
$481$	44.8494	2.04496
$482$	0	0
$483$	5.75441i	0.261835i
$484$	0	0
$485$	− 3.38741i	− 0.153815i
$486$	0	0
$487$	−30.6594	−1.38931	−0.694655	−	0.719343i	$-0.744443\pi$
$487$	−30.6594	−1.38931	−0.694655	+	0.719343i	$0.744443\pi$
$488$	0	0
$489$	−9.34060	−0.422397
$490$	0	0
$491$	9.44513i	0.426253i	0.977025	+	0.213126i	$0.0683646\pi$
$491$	9.44513i	0.426253i	−0.977025	+	0.213126i	$0.931635\pi$
$492$	0	0
$493$	35.4789i	1.59789i
$494$	0	0
$495$	−1.67982	−0.0755021
$496$	0	0
$497$	−3.41520	−0.153193
$498$	0	0
$499$	36.2537i	1.62294i	0.584395	+	0.811470i	$0.301332\pi$
$499$	36.2537i	1.62294i	−0.584395	+	0.811470i	$0.698668\pi$
$500$	0	0
$501$	− 55.4283i	− 2.47635i
$502$	0	0
$503$	16.6703	0.743292	0.371646	−	0.928375i	$-0.378794\pi$
$503$	16.6703	0.743292	0.371646	+	0.928375i	$0.378794\pi$
$504$	0	0
$505$	11.1696	0.497041
$506$	0	0
$507$	4.00000i	0.177646i
$508$	0	0
$509$	− 4.07459i	− 0.180603i	−0.995914	−	0.0903016i	$-0.971217\pi$
$509$	− 4.07459i	− 0.180603i	0.995914	−	0.0903016i	$-0.0287831\pi$
$510$	0	0
$511$	8.51021	0.376470
$512$	0	0
$513$	−7.82900	−0.345659
$514$	0	0
$515$	− 13.8399i	− 0.609859i
$516$	0	0
$517$	− 1.37054i	− 0.0602762i
$518$	0	0
$519$	32.1622	1.41177
$520$	0	0
$521$	24.4546	1.07138	0.535689	−	0.844416i	$-0.320052\pi$
$521$	24.4546	1.07138	0.535689	+	0.844416i	$0.320052\pi$
$522$	0	0
$523$	− 0.584803i	− 0.0255717i	−0.999918	−	0.0127858i	$-0.995930\pi$
$523$	− 0.584803i	− 0.0255717i	0.999918	−	0.0127858i	$-0.00406997\pi$
$524$	0	0
$525$	− 2.29240i	− 0.100049i
$526$	0	0
$527$	−22.8641	−0.995978
$528$	0	0
$529$	−16.6988	−0.726037
$530$	0	0
$531$	25.0204i	1.08579i
$532$	0	0
$533$	5.43423i	0.235382i
$534$	0	0
$535$	−8.19003	−0.354086
$536$	0	0
$537$	−13.0204	−0.561873
$538$	0	0
$539$	− 0.744895i	− 0.0320849i
$540$	0	0
$541$	− 1.44298i	− 0.0620385i	−0.999519	−	0.0310193i	$-0.990125\pi$
$541$	− 1.44298i	− 0.0620385i	0.999519	−	0.0310193i	$-0.00987532\pi$
$542$	0	0
$543$	−12.7558	−0.547404
$544$	0	0
$545$	12.3670	0.529744
$546$	0	0
$547$	24.5102i	1.04798i	0.851724	+	0.523990i	$0.175557\pi$
$547$	24.5102i	1.04798i	−0.851724	+	0.523990i	$0.824443\pi$
$548$	0	0
$549$	19.1914i	0.819070i
$550$	0	0
$551$	22.0190	0.938042
$552$	0	0
$553$	8.36699	0.355801
$554$	0	0
$555$	26.7748i	1.13653i
$556$	0	0
$557$	− 16.1492i	− 0.684263i	−0.939652	−	0.342131i	$-0.888851\pi$
$557$	− 16.1492i	− 0.684263i	0.939652	−	0.342131i	$-0.111149\pi$
$558$	0	0
$559$	−7.67982	−0.324822
$560$	0	0
$561$	−12.6147	−0.532595
$562$	0	0
$563$	33.7544i	1.42258i	0.702899	+	0.711289i	$0.251888\pi$
$563$	33.7544i	1.42258i	−0.702899	+	0.711289i	$0.748112\pi$
$564$	0	0
$565$	− 4.07459i	− 0.171419i
$566$	0	0
$567$	10.6798	0.448510
$568$	0	0
$569$	33.5088	1.40476	0.702381	−	0.711801i	$-0.252120\pi$
$569$	33.5088	1.40476	0.702381	+	0.711801i	$0.252120\pi$
$570$	0	0
$571$	21.6798i	0.907272i	0.891187	+	0.453636i	$0.149873\pi$
$571$	21.6798i	0.907272i	−0.891187	+	0.453636i	$0.850127\pi$
$572$	0	0
$573$	33.3705i	1.39407i
$574$	0	0
$575$	−2.51021	−0.104683
$576$	0	0
$577$	39.9722	1.66407	0.832033	−	0.554727i	$-0.187177\pi$
$577$	39.9722	1.66407	0.832033	+	0.554727i	$0.187177\pi$
$578$	0	0
$579$	− 61.5497i	− 2.55792i
$580$	0	0
$581$	4.58480i	0.190210i
$582$	0	0
$583$	−8.26462	−0.342286
$584$	0	0
$585$	−8.65940	−0.358022
$586$	0	0
$587$	− 41.6052i	− 1.71723i	−0.512620	−	0.858616i	$-0.671325\pi$
$587$	− 41.6052i	− 1.71723i	0.512620	−	0.858616i	$-0.328675\pi$
$588$	0	0
$589$	14.1900i	0.584690i
$590$	0	0
$591$	53.2850	2.19185
$592$	0	0
$593$	−2.80261	−0.115089	−0.0575447	−	0.998343i	$-0.518327\pi$
$593$	−2.80261	−0.115089	−0.0575447	+	0.998343i	$0.518327\pi$
$594$	0	0
$595$	− 7.38741i	− 0.302854i
$596$	0	0
$597$	− 27.7734i	− 1.13669i
$598$	0	0
$599$	−40.8026	−1.66715	−0.833575	−	0.552407i	$-0.813710\pi$
$599$	−40.8026	−1.66715	−0.833575	+	0.552407i	$0.813710\pi$
$600$	0	0
$601$	−39.8143	−1.62406	−0.812029	−	0.583617i	$-0.801637\pi$
$601$	−39.8143	−1.62406	−0.812029	+	0.583617i	$0.801637\pi$
$602$	0	0
$603$	23.8698i	0.972055i
$604$	0	0
$605$	10.4451i	0.424655i
$606$	0	0
$607$	−14.3093	−0.580796	−0.290398	−	0.956906i	$-0.593788\pi$
$607$	−14.3093	−0.580796	−0.290398	+	0.956906i	$0.593788\pi$
$608$	0	0
$609$	−11.0095	−0.446128
$610$	0	0
$611$	− 7.06508i	− 0.285822i
$612$	0	0
$613$	18.1345i	0.732444i	0.930528	+	0.366222i	$0.119349\pi$
$613$	18.1345i	0.732444i	−0.930528	+	0.366222i	$0.880651\pi$
$614$	0	0
$615$	−3.24420	−0.130819
$616$	0	0
$617$	−36.6038	−1.47361	−0.736807	−	0.676103i	$-0.763668\pi$
$617$	−36.6038	−1.47361	−0.736807	+	0.676103i	$0.763668\pi$
$618$	0	0
$619$	− 38.3991i	− 1.54339i	−0.635993	−	0.771695i	$-0.719409\pi$
$619$	− 38.3991i	− 1.54339i	0.635993	−	0.771695i	$-0.280591\pi$
$620$	0	0
$621$	− 4.28643i	− 0.172009i
$622$	0	0
$623$	7.48979	0.300072
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.82900i	0.312660i
$628$	0	0
$629$	86.2836i	3.44035i
$630$	0	0
$631$	31.1974	1.24195	0.620974	−	0.783831i	$-0.286737\pi$
$631$	31.1974	1.24195	0.620974	+	0.783831i	$0.286737\pi$
$632$	0	0
$633$	4.78080	0.190020
$634$	0	0
$635$	− 7.09501i	− 0.281557i
$636$	0	0
$637$	− 3.83991i	− 0.152143i
$638$	0	0
$639$	−7.70163	−0.304672
$640$	0	0
$641$	0.339213	0.0133981	0.00669905	−	0.999978i	$-0.497868\pi$
$641$	0.339213	0.0133981	0.00669905	+	0.999978i	$0.497868\pi$
$642$	0	0
$643$	34.8216i	1.37323i	0.727021	+	0.686616i	$0.240904\pi$
$643$	34.8216i	1.37323i	−0.727021	+	0.686616i	$0.759096\pi$
$644$	0	0
$645$	− 4.58480i	− 0.180526i
$646$	0	0
$647$	−41.5497	−1.63349	−0.816743	−	0.577002i	$-0.804223\pi$
$647$	−41.5497	−1.63349	−0.816743	+	0.577002i	$0.804223\pi$
$648$	0	0
$649$	−8.26462	−0.324415
$650$	0	0
$651$	− 7.09501i	− 0.278075i
$652$	0	0
$653$	16.7340i	0.654852i	0.944877	+	0.327426i	$0.106181\pi$
$653$	16.7340i	0.654852i	−0.944877	+	0.327426i	$0.893819\pi$
$654$	0	0
$655$	−3.67982	−0.143782
$656$	0	0
$657$	19.1914	0.748729
$658$	0	0
$659$	47.1440i	1.83647i	0.396038	+	0.918234i	$0.370385\pi$
$659$	47.1440i	1.83647i	−0.396038	+	0.918234i	$0.629615\pi$
$660$	0	0
$661$	− 27.8698i	− 1.08401i	−0.840375	−	0.542006i	$-0.817665\pi$
$661$	− 27.8698i	− 1.08401i	0.840375	−	0.542006i	$-0.182335\pi$
$662$	0	0
$663$	−65.0285	−2.52550
$664$	0	0
$665$	−4.58480	−0.177791
$666$	0	0
$667$	12.0556i	0.466793i
$668$	0	0
$669$	− 55.4283i	− 2.14298i
$670$	0	0
$671$	−6.33921	−0.244723
$672$	0	0
$673$	19.3406	0.745525	0.372763	−	0.927927i	$-0.378411\pi$
$673$	19.3406	0.745525	0.372763	+	0.927927i	$0.378411\pi$
$674$	0	0
$675$	1.70760i	0.0657255i
$676$	0	0
$677$	− 28.6893i	− 1.10262i	−0.834300	−	0.551310i	$-0.814128\pi$
$677$	− 28.6893i	− 1.10262i	0.834300	−	0.551310i	$-0.185872\pi$
$678$	0	0
$679$	3.38741	0.129997
$680$	0	0
$681$	−54.7830	−2.09929
$682$	0	0
$683$	13.6798i	0.523444i	0.965143	+	0.261722i	$0.0842903\pi$
$683$	13.6798i	0.523444i	−0.965143	+	0.261722i	$0.915710\pi$
$684$	0	0
$685$	− 3.48979i	− 0.133338i
$686$	0	0
$687$	−38.4546	−1.46714
$688$	0	0
$689$	−42.6038	−1.62308
$690$	0	0
$691$	12.1492i	0.462177i	0.972933	+	0.231088i	$0.0742287\pi$
$691$	12.1492i	0.462177i	−0.972933	+	0.231088i	$0.925771\pi$
$692$	0	0
$693$	− 1.67982i	− 0.0638109i
$694$	0	0
$695$	−11.6798	−0.443041
$696$	0	0
$697$	−10.4546	−0.395998
$698$	0	0
$699$	54.7193i	2.06967i
$700$	0	0
$701$	− 16.5162i	− 0.623808i	−0.950114	−	0.311904i	$-0.899033\pi$
$701$	− 16.5162i	− 0.623808i	0.950114	−	0.311904i	$-0.100967\pi$
$702$	0	0
$703$	53.5497	2.01966
$704$	0	0
$705$	4.21781	0.158852
$706$	0	0
$707$	11.1696i	0.420076i
$708$	0	0
$709$	14.8435i	0.557458i	0.960370	+	0.278729i	$0.0899130\pi$
$709$	14.8435i	0.557458i	−0.960370	+	0.278729i	$0.910087\pi$
$710$	0	0
$711$	18.8685	0.707622
$712$	0	0
$713$	−7.76913	−0.290956
$714$	0	0
$715$	− 2.86033i	− 0.106970i
$716$	0	0
$717$	16.4993i	0.616178i
$718$	0	0
$719$	10.9240	0.407397	0.203699	−	0.979034i	$-0.434704\pi$
$719$	10.9240	0.407397	0.203699	+	0.979034i	$0.434704\pi$
$720$	0	0
$721$	13.8399	0.515425
$722$	0	0
$723$	− 11.0732i	− 0.411817i
$724$	0	0
$725$	− 4.80261i	− 0.178365i
$726$	0	0
$727$	−3.12877	−0.116040	−0.0580198	−	0.998315i	$-0.518479\pi$
$727$	−3.12877	−0.116040	−0.0580198	+	0.998315i	$0.518479\pi$
$728$	0	0
$729$	12.3406	0.457059
$730$	0	0
$731$	− 14.7748i	− 0.546467i
$732$	0	0
$733$	− 15.3705i	− 0.567724i	−0.958865	−	0.283862i	$-0.908384\pi$
$733$	− 15.3705i	− 0.567724i	0.958865	−	0.283862i	$-0.0916157\pi$
$734$	0	0
$735$	2.29240	0.0845565
$736$	0	0
$737$	−7.88457	−0.290432
$738$	0	0
$739$	49.8252i	1.83285i	0.400208	+	0.916425i	$0.368938\pi$
$739$	49.8252i	1.83285i	−0.400208	+	0.916425i	$0.631062\pi$
$740$	0	0
$741$	40.3582i	1.48260i
$742$	0	0
$743$	14.2238	0.521820	0.260910	−	0.965363i	$-0.415977\pi$
$743$	14.2238	0.521820	0.260910	+	0.965363i	$0.415977\pi$
$744$	0	0
$745$	6.19003	0.226785
$746$	0	0
$747$	10.3392i	0.378292i
$748$	0	0
$749$	− 8.19003i	− 0.299257i
$750$	0	0
$751$	−15.9314	−0.581344	−0.290672	−	0.956823i	$-0.593879\pi$
$751$	−15.9314	−0.581344	−0.290672	+	0.956823i	$0.593879\pi$
$752$	0	0
$753$	−26.7748	−0.975729
$754$	0	0
$755$	9.97222i	0.362926i
$756$	0	0
$757$	− 11.5088i	− 0.418295i	−0.977884	−	0.209148i	$-0.932931\pi$
$757$	− 11.5088i	− 0.418295i	0.977884	−	0.209148i	$-0.0670689\pi$
$758$	0	0
$759$	−4.28643	−0.155588
$760$	0	0
$761$	38.3801	1.39128	0.695638	−	0.718393i	$-0.255122\pi$
$761$	38.3801	1.39128	0.695638	+	0.718393i	$0.255122\pi$
$762$	0	0
$763$	12.3670i	0.447715i
$764$	0	0
$765$	− 16.6594i	− 0.602322i
$766$	0	0
$767$	−42.6038	−1.53834
$768$	0	0
$769$	30.7939	1.11045	0.555227	−	0.831699i	$-0.312631\pi$
$769$	30.7939	1.11045	0.555227	+	0.831699i	$0.312631\pi$
$770$	0	0
$771$	− 29.0204i	− 1.04514i
$772$	0	0
$773$	− 21.1805i	− 0.761810i	−0.924614	−	0.380905i	$-0.875612\pi$
$773$	− 21.1805i	− 0.761810i	0.924614	−	0.380905i	$-0.124388\pi$
$774$	0	0
$775$	3.09501	0.111176
$776$	0	0
$777$	−26.7748	−0.960542
$778$	0	0
$779$	6.48840i	0.232471i
$780$	0	0
$781$	− 2.54396i	− 0.0910302i
$782$	0	0
$783$	8.20093	0.293077
$784$	0	0
$785$	−23.1696	−0.826959
$786$	0	0
$787$	34.2368i	1.22041i	0.792243	+	0.610206i	$0.208913\pi$
$787$	34.2368i	1.22041i	−0.792243	+	0.610206i	$0.791087\pi$
$788$	0	0
$789$	47.1886i	1.67996i
$790$	0	0
$791$	4.07459	0.144876
$792$	0	0
$793$	−32.6784	−1.16044
$794$	0	0
$795$	− 25.4342i	− 0.902059i
$796$	0	0
$797$	10.9905i	0.389303i	0.980872	+	0.194651i	$0.0623576\pi$
$797$	10.9905i	0.389303i	−0.980872	+	0.194651i	$0.937642\pi$
$798$	0	0
$799$	13.5922	0.480856
$800$	0	0
$801$	16.8903	0.596788
$802$	0	0
$803$	6.33921i	0.223706i
$804$	0	0
$805$	− 2.51021i	− 0.0884733i
$806$	0	0
$807$	18.5102	0.651590
$808$	0	0
$809$	−14.5549	−0.511722	−0.255861	−	0.966714i	$-0.582359\pi$
$809$	−14.5549	−0.511722	−0.255861	+	0.966714i	$0.582359\pi$
$810$	0	0
$811$	− 5.63898i	− 0.198011i	−0.995087	−	0.0990056i	$-0.968434\pi$
$811$	− 5.63898i	− 0.198011i	0.995087	−	0.0990056i	$-0.0315662\pi$
$812$	0	0
$813$	2.68121i	0.0940341i
$814$	0	0
$815$	4.07459	0.142727
$816$	0	0
$817$	−9.16961	−0.320804
$818$	0	0
$819$	− 8.65940i	− 0.302584i
$820$	0	0
$821$	− 36.3114i	− 1.26728i	−0.773629	−	0.633639i	$-0.781561\pi$
$821$	− 36.3114i	− 1.26728i	0.773629	−	0.633639i	$-0.218439\pi$
$822$	0	0
$823$	−18.0746	−0.630041	−0.315020	−	0.949085i	$-0.602011\pi$
$823$	−18.0746	−0.630041	−0.315020	+	0.949085i	$0.602011\pi$
$824$	0	0
$825$	1.70760	0.0594509
$826$	0	0
$827$	− 29.6243i	− 1.03014i	−0.857149	−	0.515068i	$-0.827767\pi$
$827$	− 29.6243i	− 1.03014i	0.857149	−	0.515068i	$-0.172233\pi$
$828$	0	0
$829$	40.2836i	1.39911i	0.714579	+	0.699554i	$0.246618\pi$
$829$	40.2836i	1.39911i	−0.714579	+	0.699554i	$0.753382\pi$
$830$	0	0
$831$	−14.9240	−0.517708
$832$	0	0
$833$	7.38741	0.255959
$834$	0	0
$835$	24.1791i	0.836753i
$836$	0	0
$837$	5.28504i	0.182678i
$838$	0	0
$839$	47.1549	1.62797	0.813984	−	0.580888i	$-0.197294\pi$
$839$	47.1549	1.62797	0.813984	+	0.580888i	$0.197294\pi$
$840$	0	0
$841$	5.93492	0.204652
$842$	0	0
$843$	4.78080i	0.164660i
$844$	0	0
$845$	− 1.74489i	− 0.0600262i
$846$	0	0
$847$	−10.4451	−0.358899
$848$	0	0
$849$	67.6324	2.32114
$850$	0	0
$851$	29.3188i	1.00504i
$852$	0	0
$853$	46.6784i	1.59824i	0.601172	+	0.799119i	$0.294701\pi$
$853$	46.6784i	1.59824i	−0.601172	+	0.799119i	$0.705299\pi$
$854$	0	0
$855$	−10.3392	−0.353594
$856$	0	0
$857$	−16.0190	−0.547200	−0.273600	−	0.961844i	$-0.588214\pi$
$857$	−16.0190	−0.547200	−0.273600	+	0.961844i	$0.588214\pi$
$858$	0	0
$859$	38.4209i	1.31090i	0.755237	+	0.655452i	$0.227522\pi$
$859$	38.4209i	1.31090i	−0.755237	+	0.655452i	$0.772478\pi$
$860$	0	0
$861$	− 3.24420i	− 0.110562i
$862$	0	0
$863$	45.8143	1.55954	0.779768	−	0.626068i	$-0.215337\pi$
$863$	45.8143	1.55954	0.779768	+	0.626068i	$0.215337\pi$
$864$	0	0
$865$	−14.0299	−0.477032
$866$	0	0
$867$	− 86.1345i	− 2.92528i
$868$	0	0
$869$	6.23253i	0.211424i
$870$	0	0
$871$	−40.6447	−1.37719
$872$	0	0
$873$	7.63898	0.258540
$874$	0	0
$875$	1.00000i	0.0338062i
$876$	0	0
$877$	− 14.9240i	− 0.503948i	−0.967734	−	0.251974i	$-0.918920\pi$
$877$	− 14.9240i	− 0.503948i	0.967734	−	0.251974i	$-0.0810798\pi$
$878$	0	0
$879$	−3.04820	−0.102813
$880$	0	0
$881$	−44.0190	−1.48304	−0.741519	−	0.670931i	$-0.765894\pi$
$881$	−44.0190	−1.48304	−0.741519	+	0.670931i	$0.765894\pi$
$882$	0	0
$883$	− 36.8903i	− 1.24146i	−0.784026	−	0.620728i	$-0.786837\pi$
$883$	− 36.8903i	− 1.24146i	0.784026	−	0.620728i	$-0.213163\pi$
$884$	0	0
$885$	− 25.4342i	− 0.854962i
$886$	0	0
$887$	−41.5497	−1.39510	−0.697551	−	0.716536i	$-0.745727\pi$
$887$	−41.5497	−1.39510	−0.697551	+	0.716536i	$0.745727\pi$
$888$	0	0
$889$	7.09501	0.237959
$890$	0	0
$891$	7.95534i	0.266514i
$892$	0	0
$893$	− 8.43562i	− 0.282287i
$894$	0	0
$895$	5.67982	0.189855
$896$	0	0
$897$	−22.0964	−0.737777
$898$	0	0
$899$	− 14.8641i	− 0.495747i
$900$	0	0
$901$	− 81.9635i	− 2.73060i
$902$	0	0
$903$	4.58480	0.152573
$904$	0	0
$905$	5.56438	0.184966
$906$	0	0
$907$	− 6.90499i	− 0.229276i	−0.993407	−	0.114638i	$-0.963429\pi$
$907$	− 6.90499i	− 0.229276i	0.993407	−	0.114638i	$-0.0365709\pi$
$908$	0	0
$909$	25.1886i	0.835454i
$910$	0	0
$911$	37.1140	1.22964	0.614822	−	0.788666i	$-0.289228\pi$
$911$	37.1140	1.22964	0.614822	+	0.788666i	$0.289228\pi$
$912$	0	0
$913$	−3.41520	−0.113027
$914$	0	0
$915$	− 19.5088i	− 0.644942i
$916$	0	0
$917$	− 3.67982i	− 0.121518i
$918$	0	0
$919$	53.9167	1.77855	0.889273	−	0.457376i	$-0.151211\pi$
$919$	53.9167	1.77855	0.889273	+	0.457376i	$0.151211\pi$
$920$	0	0
$921$	36.4437	1.20086
$922$	0	0
$923$	− 13.1140i	− 0.431654i
$924$	0	0
$925$	− 11.6798i	− 0.384030i
$926$	0	0
$927$	31.2104	1.02509
$928$	0	0
$929$	−53.3041	−1.74885	−0.874425	−	0.485161i	$-0.838761\pi$
$929$	−53.3041	−1.74885	−0.874425	+	0.485161i	$0.838761\pi$
$930$	0	0
$931$	− 4.58480i	− 0.150261i
$932$	0	0
$933$	− 72.6229i	− 2.37757i
$934$	0	0
$935$	5.50285	0.179962
$936$	0	0
$937$	38.9518	1.27250	0.636250	−	0.771483i	$-0.280485\pi$
$937$	38.9518	1.27250	0.636250	+	0.771483i	$0.280485\pi$
$938$	0	0
$939$	− 57.6352i	− 1.88085i
$940$	0	0
$941$	33.1886i	1.08192i	0.841049	+	0.540959i	$0.181939\pi$
$941$	33.1886i	1.08192i	−0.841049	+	0.540959i	$0.818061\pi$
$942$	0	0
$943$	−3.55244	−0.115683
$944$	0	0
$945$	−1.70760	−0.0555482
$946$	0	0
$947$	− 22.8157i	− 0.741410i	−0.928751	−	0.370705i	$-0.879116\pi$
$947$	− 22.8157i	− 0.741410i	0.928751	−	0.370705i	$-0.120884\pi$
$948$	0	0
$949$	32.6784i	1.06079i
$950$	0	0
$951$	29.1914	0.946597
$952$	0	0
$953$	−14.4356	−0.467616	−0.233808	−	0.972283i	$-0.575119\pi$
$953$	−14.4356	−0.467616	−0.233808	+	0.972283i	$0.575119\pi$
$954$	0	0
$955$	− 14.5570i	− 0.471054i
$956$	0	0
$957$	− 8.20093i	− 0.265098i
$958$	0	0
$959$	3.48979	0.112691
$960$	0	0
$961$	−21.4209	−0.690997
$962$	0	0
$963$	− 18.4694i	− 0.595167i
$964$	0	0
$965$	26.8494i	0.864313i
$966$	0	0
$967$	37.7544	1.21410	0.607050	−	0.794664i	$-0.292353\pi$
$967$	37.7544	1.21410	0.607050	+	0.794664i	$0.292353\pi$
$968$	0	0
$969$	−77.6433	−2.49426
$970$	0	0
$971$	− 32.1154i	− 1.03063i	−0.857000	−	0.515317i	$-0.827674\pi$
$971$	− 32.1154i	− 1.03063i	0.857000	−	0.515317i	$-0.172326\pi$
$972$	0	0
$973$	− 11.6798i	− 0.374438i
$974$	0	0
$975$	8.80261	0.281909
$976$	0	0
$977$	−8.04084	−0.257249	−0.128625	−	0.991693i	$-0.541056\pi$
$977$	−8.04084	−0.257249	−0.128625	+	0.991693i	$0.541056\pi$
$978$	0	0
$979$	5.57911i	0.178309i
$980$	0	0
$981$	27.8889i	0.890423i
$982$	0	0
$983$	2.16009	0.0688962	0.0344481	−	0.999406i	$-0.489033\pi$
$983$	2.16009	0.0688962	0.0344481	+	0.999406i	$0.489033\pi$
$984$	0	0
$985$	−23.2442	−0.740622
$986$	0	0
$987$	4.21781i	0.134254i
$988$	0	0
$989$	− 5.02042i	− 0.159640i
$990$	0	0
$991$	−45.1140	−1.43309	−0.716547	−	0.697538i	$-0.754279\pi$
$991$	−45.1140	−1.43309	−0.716547	+	0.697538i	$0.754279\pi$
$992$	0	0
$993$	−39.0613	−1.23957
$994$	0	0
$995$	12.1154i	0.384085i
$996$	0	0
$997$	22.6703i	0.717976i	0.933342	+	0.358988i	$0.116878\pi$
$997$	22.6703i	0.717976i	−0.933342	+	0.358988i	$0.883122\pi$
$998$	0	0
$999$	19.9444	0.631014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.e.1121.2	✓	6
4.3	odd	2		2240.2.b.f.1121.5	yes	6
8.3	odd	2		2240.2.b.f.1121.2	yes	6
8.5	even	2	inner	2240.2.b.e.1121.5	yes	6
16.3	odd	4		8960.2.a.bn.1.1		3
16.5	even	4		8960.2.a.bk.1.1		3
16.11	odd	4		8960.2.a.bh.1.3		3
16.13	even	4		8960.2.a.bq.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.e.1121.2	✓	6	1.1	even	1	trivial
2240.2.b.e.1121.5	yes	6	8.5	even	2	inner
2240.2.b.f.1121.2	yes	6	8.3	odd	2
2240.2.b.f.1121.5	yes	6	4.3	odd	2
8960.2.a.bh.1.3		3	16.11	odd	4
8960.2.a.bk.1.1		3	16.5	even	4
8960.2.a.bn.1.1		3	16.3	odd	4
8960.2.a.bq.1.3		3	16.13	even	4

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

\(f(q)\)	\(=\)	\(q-2.29240i q^{3} +1.00000i q^{5} -1.00000 q^{7} -2.25511 q^{9} +O(q^{10})\)	\(q-2.29240i q^{3} +1.00000i q^{5} -1.00000 q^{7} -2.25511 q^{9} -0.744895i q^{11} -3.83991i q^{13} +2.29240 q^{15} +7.38741 q^{17} -4.58480i q^{19} +2.29240i q^{21} +2.51021 q^{23} -1.00000 q^{25} -1.70760i q^{27} +4.80261i q^{29} -3.09501 q^{31} -1.70760 q^{33} -1.00000i q^{35} +11.6798i q^{37} -8.80261 q^{39} -1.41520 q^{41} -2.00000i q^{43} -2.25511i q^{45} +1.83991 q^{47} +1.00000 q^{49} -16.9349i q^{51} -11.0950i q^{53} +0.744895 q^{55} -10.5102 q^{57} -11.0950i q^{59} -8.51021i q^{61} +2.25511 q^{63} +3.83991 q^{65} -10.5848i q^{67} -5.75441i q^{69} +3.41520 q^{71} -8.51021 q^{73} +2.29240i q^{75} +0.744895i q^{77} -8.36699 q^{79} -10.6798 q^{81} -4.58480i q^{83} +7.38741i q^{85} +11.0095 q^{87} -7.48979 q^{89} +3.83991i q^{91} +7.09501i q^{93} +4.58480 q^{95} -3.38741 q^{97} +1.67982i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 10 * q^9	\( 6 q - 6 q^{7} - 10 q^{9} - 4 q^{17} + 8 q^{23} - 6 q^{25} + 16 q^{31} - 24 q^{33} - 32 q^{39} - 36 q^{41} - 20 q^{47} + 6 q^{49} + 8 q^{55} - 56 q^{57} + 10 q^{63} - 8 q^{65} + 48 q^{71} - 44 q^{73} - 16 q^{79} - 2 q^{81} - 20 q^{87} - 52 q^{89} + 28 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 10 * q^9 - 4 * q^17 + 8 * q^23 - 6 * q^25 + 16 * q^31 - 24 * q^33 - 32 * q^39 - 36 * q^41 - 20 * q^47 + 6 * q^49 + 8 * q^55 - 56 * q^57 + 10 * q^63 - 8 * q^65 + 48 * q^71 - 44 * q^73 - 16 * q^79 - 2 * q^81 - 20 * q^87 - 52 * q^89 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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