LMFDB - Embedded newform 2240.4.a.bo.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2240,4,Mod(1,2240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2240.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,10,0,14,0,12,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$132.164278413$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.65685 q^{3} +5.00000 q^{5} +7.00000 q^{7} +17.3137 q^{9} +38.2548 q^{11} -19.3431 q^{13} +33.2843 q^{15} -87.2254 q^{17} -44.2254 q^{19} +46.5980 q^{21} -218.167 q^{23} +25.0000 q^{25} -64.4802 q^{27} +46.9411 q^{29} -194.558 q^{31} +254.657 q^{33} +35.0000 q^{35} -366.853 q^{37} -128.765 q^{39} -339.362 q^{41} -226.167 q^{43} +86.5685 q^{45} -11.6762 q^{47} +49.0000 q^{49} -580.647 q^{51} +209.019 q^{53} +191.274 q^{55} -294.402 q^{57} -616.000 q^{59} -320.735 q^{61} +121.196 q^{63} -96.7157 q^{65} +14.5097 q^{67} -1452.30 q^{69} +952.000 q^{71} +824.489 q^{73} +166.421 q^{75} +267.784 q^{77} -156.275 q^{79} -896.706 q^{81} -1036.53 q^{83} -436.127 q^{85} +312.480 q^{87} -170.225 q^{89} -135.402 q^{91} -1295.15 q^{93} -221.127 q^{95} +1059.87 q^{97} +662.333 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 10 q^{5} + 14 q^{7} + 12 q^{9} - 14 q^{11} - 50 q^{13} + 10 q^{15} - 50 q^{17} + 36 q^{19} + 14 q^{21} - 244 q^{23} + 50 q^{25} + 86 q^{27} + 26 q^{29} + 120 q^{31} + 498 q^{33} + 70 q^{35}+ \cdots + 940 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 10 * q^5 + 14 * q^7 + 12 * q^9 - 14 * q^11 - 50 * q^13 + 10 * q^15 - 50 * q^17 + 36 * q^19 + 14 * q^21 - 244 * q^23 + 50 * q^25 + 86 * q^27 + 26 * q^29 + 120 * q^31 + 498 * q^33 + 70 * q^35 - 564 * q^37 + 14 * q^39 - 328 * q^41 - 260 * q^43 + 60 * q^45 + 350 * q^47 + 98 * q^49 - 754 * q^51 + 56 * q^53 - 70 * q^55 - 668 * q^57 - 1232 * q^59 - 336 * q^61 + 84 * q^63 - 250 * q^65 - 152 * q^67 - 1332 * q^69 + 1904 * q^71 + 676 * q^73 + 50 * q^75 - 98 * q^77 - 1014 * q^79 - 1454 * q^81 - 376 * q^83 - 250 * q^85 + 410 * q^87 - 216 * q^89 - 350 * q^91 - 2760 * q^93 + 180 * q^95 + 2742 * q^97 + 940 * q^99

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
$3$	6.65685	1.28111	0.640556	−	0.767911i	$-0.278704\pi$
$3$	6.65685	1.28111	0.640556	+	0.767911i	$0.278704\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	17.3137	0.641248
$10$	0	0
$11$	38.2548	1.04857	0.524285	−	0.851543i	$-0.324333\pi$
$11$	38.2548	1.04857	0.524285	+	0.851543i	$0.324333\pi$
$12$	0	0
$13$	−19.3431	−0.412679	−0.206339	−	0.978480i	$-0.566155\pi$
$13$	−19.3431	−0.412679	−0.206339	+	0.978480i	$0.566155\pi$
$14$	0	0
$15$	33.2843	0.572931
$16$	0	0
$17$	−87.2254	−1.24443	−0.622214	−	0.782847i	$-0.713767\pi$
$17$	−87.2254	−1.24443	−0.622214	+	0.782847i	$0.713767\pi$
$18$	0	0
$19$	−44.2254	−0.534000	−0.267000	−	0.963697i	$-0.586032\pi$
$19$	−44.2254	−0.534000	−0.267000	+	0.963697i	$0.586032\pi$
$20$	0	0
$21$	46.5980	0.484215
$22$	0	0
$23$	−218.167	−1.97786	−0.988932	−	0.148371i	$-0.952597\pi$
$23$	−218.167	−1.97786	−0.988932	+	0.148371i	$0.952597\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−64.4802	−0.459601
$28$	0	0
$29$	46.9411	0.300578	0.150289	−	0.988642i	$-0.451980\pi$
$29$	46.9411	0.300578	0.150289	+	0.988642i	$0.451980\pi$
$30$	0	0
$31$	−194.558	−1.12722	−0.563609	−	0.826042i	$-0.690587\pi$
$31$	−194.558	−1.12722	−0.563609	+	0.826042i	$0.690587\pi$
$32$	0	0
$33$	254.657	1.34334
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−366.853	−1.63001	−0.815003	−	0.579457i	$-0.803265\pi$
$37$	−366.853	−1.63001	−0.815003	+	0.579457i	$0.803265\pi$
$38$	0	0
$39$	−128.765	−0.528688
$40$	0	0
$41$	−339.362	−1.29267	−0.646336	−	0.763053i	$-0.723699\pi$
$41$	−339.362	−1.29267	−0.646336	+	0.763053i	$0.723699\pi$
$42$	0	0
$43$	−226.167	−0.802095	−0.401047	−	0.916057i	$-0.631354\pi$
$43$	−226.167	−0.802095	−0.401047	+	0.916057i	$0.631354\pi$
$44$	0	0
$45$	86.5685	0.286775
$46$	0	0
$47$	−11.6762	−0.0362372	−0.0181186	−	0.999836i	$-0.505768\pi$
$47$	−11.6762	−0.0362372	−0.0181186	+	0.999836i	$0.505768\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−580.647	−1.59425
$52$	0	0
$53$	209.019	0.541717	0.270859	−	0.962619i	$-0.412692\pi$
$53$	209.019	0.541717	0.270859	+	0.962619i	$0.412692\pi$
$54$	0	0
$55$	191.274	0.468935
$56$	0	0
$57$	−294.402	−0.684114
$58$	0	0
$59$	−616.000	−1.35926	−0.679630	−	0.733555i	$-0.737860\pi$
$59$	−616.000	−1.35926	−0.679630	+	0.733555i	$0.737860\pi$
$60$	0	0
$61$	−320.735	−0.673212	−0.336606	−	0.941646i	$-0.609279\pi$
$61$	−320.735	−0.673212	−0.336606	+	0.941646i	$0.609279\pi$
$62$	0	0
$63$	121.196	0.242369
$64$	0	0
$65$	−96.7157	−0.184556
$66$	0	0
$67$	14.5097	0.0264573	0.0132286	−	0.999912i	$-0.495789\pi$
$67$	14.5097	0.0264573	0.0132286	+	0.999912i	$0.495789\pi$
$68$	0	0
$69$	−1452.30	−2.53387
$70$	0	0
$71$	952.000	1.59129	0.795645	−	0.605763i	$-0.207132\pi$
$71$	952.000	1.59129	0.795645	+	0.605763i	$0.207132\pi$
$72$	0	0
$73$	824.489	1.32191	0.660953	−	0.750427i	$-0.270152\pi$
$73$	824.489	1.32191	0.660953	+	0.750427i	$0.270152\pi$
$74$	0	0
$75$	166.421	0.256222
$76$	0	0
$77$	267.784	0.396322
$78$	0	0
$79$	−156.275	−0.222561	−0.111280	−	0.993789i	$-0.535495\pi$
$79$	−156.275	−0.222561	−0.111280	+	0.993789i	$0.535495\pi$
$80$	0	0
$81$	−896.706	−1.23005
$82$	0	0
$83$	−1036.53	−1.37077	−0.685384	−	0.728182i	$-0.740366\pi$
$83$	−1036.53	−1.37077	−0.685384	+	0.728182i	$0.740366\pi$
$84$	0	0
$85$	−436.127	−0.556525
$86$	0	0
$87$	312.480	0.385074
$88$	0	0
$89$	−170.225	−0.202740	−0.101370	−	0.994849i	$-0.532323\pi$
$89$	−170.225	−0.202740	−0.101370	+	0.994849i	$0.532323\pi$
$90$	0	0
$91$	−135.402	−0.155978
$92$	0	0
$93$	−1295.15	−1.44409
$94$	0	0
$95$	−221.127	−0.238812
$96$	0	0
$97$	1059.87	1.10942	0.554710	−	0.832044i	$-0.312829\pi$
$97$	1059.87	1.10942	0.554710	+	0.832044i	$0.312829\pi$
$98$	0	0
$99$	662.333	0.672394
$100$	0	0
$101$	241.833	0.238251	0.119125	−	0.992879i	$-0.461991\pi$
$101$	241.833	0.238251	0.119125	+	0.992879i	$0.461991\pi$
$102$	0	0
$103$	1679.58	1.60673	0.803367	−	0.595484i	$-0.203040\pi$
$103$	1679.58	1.60673	0.803367	+	0.595484i	$0.203040\pi$
$104$	0	0
$105$	232.990	0.216547
$106$	0	0
$107$	1506.88	1.36146	0.680728	−	0.732537i	$-0.261664\pi$
$107$	1506.88	1.36146	0.680728	+	0.732537i	$0.261664\pi$
$108$	0	0
$109$	1252.41	1.10054	0.550271	−	0.834986i	$-0.314524\pi$
$109$	1252.41	1.10054	0.550271	+	0.834986i	$0.314524\pi$
$110$	0	0
$111$	−2442.09	−2.08822
$112$	0	0
$113$	1370.20	1.14069	0.570345	−	0.821405i	$-0.306810\pi$
$113$	1370.20	1.14069	0.570345	+	0.821405i	$0.306810\pi$
$114$	0	0
$115$	−1090.83	−0.884528
$116$	0	0
$117$	−334.902	−0.264630
$118$	0	0
$119$	−610.578	−0.470349
$120$	0	0
$121$	132.432	0.0994984
$122$	0	0
$123$	−2259.09	−1.65606
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1213.49	−0.847873	−0.423936	−	0.905692i	$-0.639352\pi$
$127$	−1213.49	−0.847873	−0.423936	+	0.905692i	$0.639352\pi$
$128$	0	0
$129$	−1505.56	−1.02757
$130$	0	0
$131$	−1982.42	−1.32217	−0.661087	−	0.750309i	$-0.729904\pi$
$131$	−1982.42	−1.32217	−0.661087	+	0.750309i	$0.729904\pi$
$132$	0	0
$133$	−309.578	−0.201833
$134$	0	0
$135$	−322.401	−0.205540
$136$	0	0
$137$	2210.95	1.37879	0.689394	−	0.724386i	$-0.257877\pi$
$137$	2210.95	1.37879	0.689394	+	0.724386i	$0.257877\pi$
$138$	0	0
$139$	528.039	0.322213	0.161107	−	0.986937i	$-0.448494\pi$
$139$	528.039	0.322213	0.161107	+	0.986937i	$0.448494\pi$
$140$	0	0
$141$	−77.7267	−0.0464239
$142$	0	0
$143$	−739.969	−0.432722
$144$	0	0
$145$	234.706	0.134422
$146$	0	0
$147$	326.186	0.183016
$148$	0	0
$149$	328.372	0.180545	0.0902727	−	0.995917i	$-0.471226\pi$
$149$	328.372	0.180545	0.0902727	+	0.995917i	$0.471226\pi$
$150$	0	0
$151$	−1029.43	−0.554793	−0.277396	−	0.960756i	$-0.589472\pi$
$151$	−1029.43	−0.554793	−0.277396	+	0.960756i	$0.589472\pi$
$152$	0	0
$153$	−1510.20	−0.797987
$154$	0	0
$155$	−972.792	−0.504107
$156$	0	0
$157$	−525.098	−0.266926	−0.133463	−	0.991054i	$-0.542610\pi$
$157$	−525.098	−0.266926	−0.133463	+	0.991054i	$0.542610\pi$
$158$	0	0
$159$	1391.41	0.694001
$160$	0	0
$161$	−1527.17	−0.747562
$162$	0	0
$163$	1002.63	0.481790	0.240895	−	0.970551i	$-0.422559\pi$
$163$	1002.63	0.481790	0.240895	+	0.970551i	$0.422559\pi$
$164$	0	0
$165$	1273.28	0.600758
$166$	0	0
$167$	359.422	0.166544	0.0832722	−	0.996527i	$-0.473463\pi$
$167$	359.422	0.166544	0.0832722	+	0.996527i	$0.473463\pi$
$168$	0	0
$169$	−1822.84	−0.829696
$170$	0	0
$171$	−765.706	−0.342427
$172$	0	0
$173$	−3293.65	−1.44747	−0.723733	−	0.690080i	$-0.757575\pi$
$173$	−3293.65	−1.44747	−0.723733	+	0.690080i	$0.757575\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−4100.62	−1.74137
$178$	0	0
$179$	2978.82	1.24384	0.621921	−	0.783080i	$-0.286353\pi$
$179$	2978.82	1.24384	0.621921	+	0.783080i	$0.286353\pi$
$180$	0	0
$181$	−1462.31	−0.600514	−0.300257	−	0.953858i	$-0.597072\pi$
$181$	−1462.31	−0.600514	−0.300257	+	0.953858i	$0.597072\pi$
$182$	0	0
$183$	−2135.09	−0.862460
$184$	0	0
$185$	−1834.26	−0.728961
$186$	0	0
$187$	−3336.79	−1.30487
$188$	0	0
$189$	−451.362	−0.173713
$190$	0	0
$191$	374.923	0.142034	0.0710169	−	0.997475i	$-0.477376\pi$
$191$	374.923	0.142034	0.0710169	+	0.997475i	$0.477376\pi$
$192$	0	0
$193$	733.028	0.273391	0.136696	−	0.990613i	$-0.456352\pi$
$193$	733.028	0.273391	0.136696	+	0.990613i	$0.456352\pi$
$194$	0	0
$195$	−643.823	−0.236436
$196$	0	0
$197$	2093.24	0.757043	0.378521	−	0.925593i	$-0.376433\pi$
$197$	2093.24	0.757043	0.378521	+	0.925593i	$0.376433\pi$
$198$	0	0
$199$	−2865.04	−1.02059	−0.510295	−	0.860000i	$-0.670464\pi$
$199$	−2865.04	−1.02059	−0.510295	+	0.860000i	$0.670464\pi$
$200$	0	0
$201$	96.5887	0.0338948
$202$	0	0
$203$	328.588	0.113608
$204$	0	0
$205$	−1696.81	−0.578100
$206$	0	0
$207$	−3777.27	−1.26830
$208$	0	0
$209$	−1691.84	−0.559936
$210$	0	0
$211$	5643.65	1.84135	0.920674	−	0.390331i	$-0.127640\pi$
$211$	5643.65	1.84135	0.920674	+	0.390331i	$0.127640\pi$
$212$	0	0
$213$	6337.33	2.03862
$214$	0	0
$215$	−1130.83	−0.358708
$216$	0	0
$217$	−1361.91	−0.426048
$218$	0	0
$219$	5488.51	1.69351
$220$	0	0
$221$	1687.21	0.513549
$222$	0	0
$223$	6369.16	1.91260	0.956302	−	0.292381i	$-0.0944477\pi$
$223$	6369.16	1.91260	0.956302	+	0.292381i	$0.0944477\pi$
$224$	0	0
$225$	432.843	0.128250
$226$	0	0
$227$	−1015.67	−0.296972	−0.148486	−	0.988914i	$-0.547440\pi$
$227$	−1015.67	−0.296972	−0.148486	+	0.988914i	$0.547440\pi$
$228$	0	0
$229$	−4108.35	−1.18554	−0.592768	−	0.805373i	$-0.701965\pi$
$229$	−4108.35	−1.18554	−0.592768	+	0.805373i	$0.701965\pi$
$230$	0	0
$231$	1782.60	0.507733
$232$	0	0
$233$	608.431	0.171071	0.0855357	−	0.996335i	$-0.472740\pi$
$233$	608.431	0.171071	0.0855357	+	0.996335i	$0.472740\pi$
$234$	0	0
$235$	−58.3810	−0.0162058
$236$	0	0
$237$	−1040.30	−0.285126
$238$	0	0
$239$	5054.44	1.36797	0.683985	−	0.729496i	$-0.260245\pi$
$239$	5054.44	1.36797	0.683985	+	0.729496i	$0.260245\pi$
$240$	0	0
$241$	4.86782	0.00130109	0.000650547	−	1.00000i	$-0.499793\pi$
$241$	4.86782	0.00130109	0.000650547		1.00000i	$0.499793\pi$
$242$	0	0
$243$	−4228.27	−1.11623
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	855.458	0.220370
$248$	0	0
$249$	−6900.02	−1.75611
$250$	0	0
$251$	−547.921	−0.137787	−0.0688934	−	0.997624i	$-0.521947\pi$
$251$	−547.921	−0.137787	−0.0688934	+	0.997624i	$0.521947\pi$
$252$	0	0
$253$	−8345.92	−2.07393
$254$	0	0
$255$	−2903.23	−0.712971
$256$	0	0
$257$	−1774.61	−0.430729	−0.215364	−	0.976534i	$-0.569094\pi$
$257$	−1774.61	−0.430729	−0.215364	+	0.976534i	$0.569094\pi$
$258$	0	0
$259$	−2567.97	−0.616084
$260$	0	0
$261$	812.725	0.192745
$262$	0	0
$263$	1199.09	0.281138	0.140569	−	0.990071i	$-0.455107\pi$
$263$	1199.09	0.281138	0.140569	+	0.990071i	$0.455107\pi$
$264$	0	0
$265$	1045.10	0.242263
$266$	0	0
$267$	−1133.17	−0.259733
$268$	0	0
$269$	−3250.29	−0.736706	−0.368353	−	0.929686i	$-0.620078\pi$
$269$	−3250.29	−0.736706	−0.368353	+	0.929686i	$0.620078\pi$
$270$	0	0
$271$	896.143	0.200874	0.100437	−	0.994943i	$-0.467976\pi$
$271$	896.143	0.200874	0.100437	+	0.994943i	$0.467976\pi$
$272$	0	0
$273$	−901.352	−0.199825
$274$	0	0
$275$	956.371	0.209714
$276$	0	0
$277$	386.562	0.0838492	0.0419246	−	0.999121i	$-0.486651\pi$
$277$	386.562	0.0838492	0.0419246	+	0.999121i	$0.486651\pi$
$278$	0	0
$279$	−3368.53	−0.722826
$280$	0	0
$281$	−3335.10	−0.708025	−0.354013	−	0.935241i	$-0.615183\pi$
$281$	−3335.10	−0.708025	−0.354013	+	0.935241i	$0.615183\pi$
$282$	0	0
$283$	5412.26	1.13684	0.568419	−	0.822739i	$-0.307555\pi$
$283$	5412.26	1.13684	0.568419	+	0.822739i	$0.307555\pi$
$284$	0	0
$285$	−1472.01	−0.305945
$286$	0	0
$287$	−2375.54	−0.488584
$288$	0	0
$289$	2695.27	0.548600
$290$	0	0
$291$	7055.42	1.42129
$292$	0	0
$293$	−282.211	−0.0562695	−0.0281347	−	0.999604i	$-0.508957\pi$
$293$	−282.211	−0.0562695	−0.0281347	+	0.999604i	$0.508957\pi$
$294$	0	0
$295$	−3080.00	−0.607880
$296$	0	0
$297$	−2466.68	−0.481924
$298$	0	0
$299$	4220.03	0.816222
$300$	0	0
$301$	−1583.17	−0.303163
$302$	0	0
$303$	1609.85	0.305226
$304$	0	0
$305$	−1603.68	−0.301069
$306$	0	0
$307$	1919.67	0.356878	0.178439	−	0.983951i	$-0.442895\pi$
$307$	1919.67	0.356878	0.178439	+	0.983951i	$0.442895\pi$
$308$	0	0
$309$	11180.7	2.05841
$310$	0	0
$311$	−1213.31	−0.221223	−0.110612	−	0.993864i	$-0.535281\pi$
$311$	−1213.31	−0.221223	−0.110612	+	0.993864i	$0.535281\pi$
$312$	0	0
$313$	−1434.00	−0.258960	−0.129480	−	0.991582i	$-0.541331\pi$
$313$	−1434.00	−0.258960	−0.129480	+	0.991582i	$0.541331\pi$
$314$	0	0
$315$	605.980	0.108391
$316$	0	0
$317$	−6496.95	−1.15112	−0.575560	−	0.817760i	$-0.695216\pi$
$317$	−6496.95	−1.15112	−0.575560	+	0.817760i	$0.695216\pi$
$318$	0	0
$319$	1795.72	0.315176
$320$	0	0
$321$	10031.1	1.74418
$322$	0	0
$323$	3857.58	0.664524
$324$	0	0
$325$	−483.579	−0.0825357
$326$	0	0
$327$	8337.11	1.40992
$328$	0	0
$329$	−81.7333	−0.0136964
$330$	0	0
$331$	−9683.88	−1.60808	−0.804039	−	0.594576i	$-0.797320\pi$
$331$	−9683.88	−1.60808	−0.804039	+	0.594576i	$0.797320\pi$
$332$	0	0
$333$	−6351.58	−1.04524
$334$	0	0
$335$	72.5483	0.0118321
$336$	0	0
$337$	29.1319	0.00470895	0.00235447	−	0.999997i	$-0.499251\pi$
$337$	29.1319	0.00470895	0.00235447	+	0.999997i	$0.499251\pi$
$338$	0	0
$339$	9121.24	1.46135
$340$	0	0
$341$	−7442.80	−1.18197
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−7261.51	−1.13318
$346$	0	0
$347$	−7848.58	−1.21422	−0.607110	−	0.794618i	$-0.707671\pi$
$347$	−7848.58	−1.21422	−0.607110	+	0.794618i	$0.707671\pi$
$348$	0	0
$349$	10269.6	1.57513	0.787567	−	0.616229i	$-0.211341\pi$
$349$	10269.6	1.57513	0.787567	+	0.616229i	$0.211341\pi$
$350$	0	0
$351$	1247.25	0.189668
$352$	0	0
$353$	2799.93	0.422168	0.211084	−	0.977468i	$-0.432301\pi$
$353$	2799.93	0.422168	0.211084	+	0.977468i	$0.432301\pi$
$354$	0	0
$355$	4760.00	0.711647
$356$	0	0
$357$	−4064.53	−0.602570
$358$	0	0
$359$	3163.29	0.465048	0.232524	−	0.972591i	$-0.425302\pi$
$359$	3163.29	0.465048	0.232524	+	0.972591i	$0.425302\pi$
$360$	0	0
$361$	−4903.11	−0.714844
$362$	0	0
$363$	881.583	0.127469
$364$	0	0
$365$	4122.45	0.591175
$366$	0	0
$367$	−3182.85	−0.452706	−0.226353	−	0.974045i	$-0.572680\pi$
$367$	−3182.85	−0.452706	−0.226353	+	0.974045i	$0.572680\pi$
$368$	0	0
$369$	−5875.62	−0.828923
$370$	0	0
$371$	1463.14	0.204750
$372$	0	0
$373$	2615.14	0.363021	0.181510	−	0.983389i	$-0.441901\pi$
$373$	2615.14	0.363021	0.181510	+	0.983389i	$0.441901\pi$
$374$	0	0
$375$	832.107	0.114586
$376$	0	0
$377$	−907.989	−0.124042
$378$	0	0
$379$	−672.434	−0.0911362	−0.0455681	−	0.998961i	$-0.514510\pi$
$379$	−672.434	−0.0911362	−0.0455681	+	0.998961i	$0.514510\pi$
$380$	0	0
$381$	−8078.03	−1.08622
$382$	0	0
$383$	−1169.86	−0.156075	−0.0780377	−	0.996950i	$-0.524865\pi$
$383$	−1169.86	−0.156075	−0.0780377	+	0.996950i	$0.524865\pi$
$384$	0	0
$385$	1338.92	0.177241
$386$	0	0
$387$	−3915.78	−0.514342
$388$	0	0
$389$	1122.22	0.146269	0.0731347	−	0.997322i	$-0.476700\pi$
$389$	1122.22	0.146269	0.0731347	+	0.997322i	$0.476700\pi$
$390$	0	0
$391$	19029.7	2.46131
$392$	0	0
$393$	−13196.7	−1.69385
$394$	0	0
$395$	−781.375	−0.0995323
$396$	0	0
$397$	1985.93	0.251060	0.125530	−	0.992090i	$-0.459937\pi$
$397$	1985.93	0.251060	0.125530	+	0.992090i	$0.459937\pi$
$398$	0	0
$399$	−2060.81	−0.258571
$400$	0	0
$401$	−4172.38	−0.519597	−0.259799	−	0.965663i	$-0.583656\pi$
$401$	−4172.38	−0.519597	−0.259799	+	0.965663i	$0.583656\pi$
$402$	0	0
$403$	3763.37	0.465178
$404$	0	0
$405$	−4483.53	−0.550095
$406$	0	0
$407$	−14033.9	−1.70918
$408$	0	0
$409$	−11700.8	−1.41459	−0.707295	−	0.706919i	$-0.750085\pi$
$409$	−11700.8	−1.41459	−0.707295	+	0.706919i	$0.750085\pi$
$410$	0	0
$411$	14718.0	1.76638
$412$	0	0
$413$	−4312.00	−0.513752
$414$	0	0
$415$	−5182.64	−0.613026
$416$	0	0
$417$	3515.08	0.412791
$418$	0	0
$419$	2733.20	0.318677	0.159339	−	0.987224i	$-0.449064\pi$
$419$	2733.20	0.318677	0.159339	+	0.987224i	$0.449064\pi$
$420$	0	0
$421$	−13549.4	−1.56854	−0.784272	−	0.620417i	$-0.786963\pi$
$421$	−13549.4	−1.56854	−0.784272	+	0.620417i	$0.786963\pi$
$422$	0	0
$423$	−202.158	−0.0232370
$424$	0	0
$425$	−2180.63	−0.248885
$426$	0	0
$427$	−2245.15	−0.254450
$428$	0	0
$429$	−4925.86	−0.554366
$430$	0	0
$431$	6429.25	0.718530	0.359265	−	0.933236i	$-0.383027\pi$
$431$	6429.25	0.718530	0.359265	+	0.933236i	$0.383027\pi$
$432$	0	0
$433$	8022.03	0.890333	0.445166	−	0.895448i	$-0.353145\pi$
$433$	8022.03	0.890333	0.445166	+	0.895448i	$0.353145\pi$
$434$	0	0
$435$	1562.40	0.172210
$436$	0	0
$437$	9648.50	1.05618
$438$	0	0
$439$	5569.88	0.605549	0.302774	−	0.953062i	$-0.402087\pi$
$439$	5569.88	0.605549	0.302774	+	0.953062i	$0.402087\pi$
$440$	0	0
$441$	848.372	0.0916069
$442$	0	0
$443$	−5486.21	−0.588392	−0.294196	−	0.955745i	$-0.595052\pi$
$443$	−5486.21	−0.588392	−0.294196	+	0.955745i	$0.595052\pi$
$444$	0	0
$445$	−851.127	−0.0906681
$446$	0	0
$447$	2185.92	0.231299
$448$	0	0
$449$	−7232.67	−0.760203	−0.380101	−	0.924945i	$-0.624111\pi$
$449$	−7232.67	−0.760203	−0.380101	+	0.924945i	$0.624111\pi$
$450$	0	0
$451$	−12982.3	−1.35546
$452$	0	0
$453$	−6852.76	−0.710752
$454$	0	0
$455$	−677.010	−0.0697554
$456$	0	0
$457$	−2900.51	−0.296893	−0.148446	−	0.988920i	$-0.547427\pi$
$457$	−2900.51	−0.296893	−0.148446	+	0.988920i	$0.547427\pi$
$458$	0	0
$459$	5624.31	0.571940
$460$	0	0
$461$	−6073.57	−0.613611	−0.306805	−	0.951772i	$-0.599260\pi$
$461$	−6073.57	−0.613611	−0.306805	+	0.951772i	$0.599260\pi$
$462$	0	0
$463$	18922.8	1.89939	0.949693	−	0.313183i	$-0.101395\pi$
$463$	18922.8	1.89939	0.949693	+	0.313183i	$0.101395\pi$
$464$	0	0
$465$	−6475.74	−0.645817
$466$	0	0
$467$	−6776.71	−0.671496	−0.335748	−	0.941952i	$-0.608989\pi$
$467$	−6776.71	−0.671496	−0.335748	+	0.941952i	$0.608989\pi$
$468$	0	0
$469$	101.568	0.00999991
$470$	0	0
$471$	−3495.50	−0.341962
$472$	0	0
$473$	−8651.96	−0.841052
$474$	0	0
$475$	−1105.63	−0.106800
$476$	0	0
$477$	3618.90	0.347375
$478$	0	0
$479$	−2397.32	−0.228677	−0.114338	−	0.993442i	$-0.536475\pi$
$479$	−2397.32	−0.228677	−0.114338	+	0.993442i	$0.536475\pi$
$480$	0	0
$481$	7096.09	0.672669
$482$	0	0
$483$	−10166.1	−0.957711
$484$	0	0
$485$	5299.37	0.496148
$486$	0	0
$487$	−5586.17	−0.519781	−0.259890	−	0.965638i	$-0.583686\pi$
$487$	−5586.17	−0.519781	−0.259890	+	0.965638i	$0.583686\pi$
$488$	0	0
$489$	6674.33	0.617227
$490$	0	0
$491$	537.392	0.0493934	0.0246967	−	0.999695i	$-0.492138\pi$
$491$	537.392	0.0493934	0.0246967	+	0.999695i	$0.492138\pi$
$492$	0	0
$493$	−4094.46	−0.374047
$494$	0	0
$495$	3311.67	0.300704
$496$	0	0
$497$	6664.00	0.601451
$498$	0	0
$499$	598.965	0.0537342	0.0268671	−	0.999639i	$-0.491447\pi$
$499$	598.965	0.0537342	0.0268671	+	0.999639i	$0.491447\pi$
$500$	0	0
$501$	2392.62	0.213362
$502$	0	0
$503$	−4426.76	−0.392405	−0.196202	−	0.980563i	$-0.562861\pi$
$503$	−4426.76	−0.392405	−0.196202	+	0.980563i	$0.562861\pi$
$504$	0	0
$505$	1209.17	0.106549
$506$	0	0
$507$	−12134.4	−1.06293
$508$	0	0
$509$	−17727.7	−1.54374	−0.771872	−	0.635779i	$-0.780679\pi$
$509$	−17727.7	−1.54374	−0.771872	+	0.635779i	$0.780679\pi$
$510$	0	0
$511$	5771.43	0.499634
$512$	0	0
$513$	2851.66	0.245427
$514$	0	0
$515$	8397.88	0.718553
$516$	0	0
$517$	−446.671	−0.0379972
$518$	0	0
$519$	−21925.4	−1.85437
$520$	0	0
$521$	8662.79	0.728453	0.364226	−	0.931310i	$-0.381333\pi$
$521$	8662.79	0.728453	0.364226	+	0.931310i	$0.381333\pi$
$522$	0	0
$523$	−7770.40	−0.649667	−0.324833	−	0.945771i	$-0.605308\pi$
$523$	−7770.40	−0.649667	−0.324833	+	0.945771i	$0.605308\pi$
$524$	0	0
$525$	1164.95	0.0968430
$526$	0	0
$527$	16970.4	1.40274
$528$	0	0
$529$	35429.6	2.91194
$530$	0	0
$531$	−10665.2	−0.871624
$532$	0	0
$533$	6564.34	0.533458
$534$	0	0
$535$	7534.41	0.608861
$536$	0	0
$537$	19829.6	1.59350
$538$	0	0
$539$	1874.49	0.149796
$540$	0	0
$541$	−21641.0	−1.71981	−0.859906	−	0.510453i	$-0.829478\pi$
$541$	−21641.0	−1.71981	−0.859906	+	0.510453i	$0.829478\pi$
$542$	0	0
$543$	−9734.41	−0.769325
$544$	0	0
$545$	6262.05	0.492177
$546$	0	0
$547$	7489.29	0.585409	0.292705	−	0.956203i	$-0.405445\pi$
$547$	7489.29	0.585409	0.292705	+	0.956203i	$0.405445\pi$
$548$	0	0
$549$	−5553.11	−0.431696
$550$	0	0
$551$	−2075.99	−0.160508
$552$	0	0
$553$	−1093.93	−0.0841201
$554$	0	0
$555$	−12210.4	−0.933881
$556$	0	0
$557$	−25297.9	−1.92443	−0.962214	−	0.272295i	$-0.912217\pi$
$557$	−25297.9	−1.92443	−0.962214	+	0.272295i	$0.912217\pi$
$558$	0	0
$559$	4374.77	0.331007
$560$	0	0
$561$	−22212.5	−1.67168
$562$	0	0
$563$	15661.3	1.17237	0.586186	−	0.810177i	$-0.300629\pi$
$563$	15661.3	1.17237	0.586186	+	0.810177i	$0.300629\pi$
$564$	0	0
$565$	6851.02	0.510132
$566$	0	0
$567$	−6276.94	−0.464915
$568$	0	0
$569$	−9982.75	−0.735498	−0.367749	−	0.929925i	$-0.619871\pi$
$569$	−9982.75	−0.735498	−0.367749	+	0.929925i	$0.619871\pi$
$570$	0	0
$571$	−11583.6	−0.848966	−0.424483	−	0.905436i	$-0.639544\pi$
$571$	−11583.6	−0.848966	−0.424483	+	0.905436i	$0.639544\pi$
$572$	0	0
$573$	2495.81	0.181961
$574$	0	0
$575$	−5454.16	−0.395573
$576$	0	0
$577$	−595.378	−0.0429565	−0.0214783	−	0.999769i	$-0.506837\pi$
$577$	−595.378	−0.0429565	−0.0214783	+	0.999769i	$0.506837\pi$
$578$	0	0
$579$	4879.66	0.350245
$580$	0	0
$581$	−7255.70	−0.518102
$582$	0	0
$583$	7996.00	0.568028
$584$	0	0
$585$	−1674.51	−0.118346
$586$	0	0
$587$	−15750.3	−1.10747	−0.553736	−	0.832693i	$-0.686798\pi$
$587$	−15750.3	−1.10747	−0.553736	+	0.832693i	$0.686798\pi$
$588$	0	0
$589$	8604.42	0.601934
$590$	0	0
$591$	13934.4	0.969856
$592$	0	0
$593$	−417.878	−0.0289379	−0.0144690	−	0.999895i	$-0.504606\pi$
$593$	−417.878	−0.0289379	−0.0144690	+	0.999895i	$0.504606\pi$
$594$	0	0
$595$	−3052.89	−0.210347
$596$	0	0
$597$	−19072.2	−1.30749
$598$	0	0
$599$	19997.3	1.36406	0.682028	−	0.731326i	$-0.261098\pi$
$599$	19997.3	1.36406	0.682028	+	0.731326i	$0.261098\pi$
$600$	0	0
$601$	−15992.6	−1.08545	−0.542723	−	0.839912i	$-0.682607\pi$
$601$	−15992.6	−1.08545	−0.542723	+	0.839912i	$0.682607\pi$
$602$	0	0
$603$	251.216	0.0169657
$604$	0	0
$605$	662.162	0.0444970
$606$	0	0
$607$	−14159.2	−0.946793	−0.473396	−	0.880850i	$-0.656972\pi$
$607$	−14159.2	−0.946793	−0.473396	+	0.880850i	$0.656972\pi$
$608$	0	0
$609$	2187.36	0.145544
$610$	0	0
$611$	225.854	0.0149543
$612$	0	0
$613$	4629.41	0.305025	0.152512	−	0.988302i	$-0.451264\pi$
$613$	4629.41	0.305025	0.152512	+	0.988302i	$0.451264\pi$
$614$	0	0
$615$	−11295.4	−0.740611
$616$	0	0
$617$	−23165.3	−1.51151	−0.755753	−	0.654857i	$-0.772729\pi$
$617$	−23165.3	−1.51151	−0.755753	+	0.654857i	$0.772729\pi$
$618$	0	0
$619$	12370.6	0.803258	0.401629	−	0.915803i	$-0.368444\pi$
$619$	12370.6	0.803258	0.401629	+	0.915803i	$0.368444\pi$
$620$	0	0
$621$	14067.4	0.909028
$622$	0	0
$623$	−1191.58	−0.0766285
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−11262.3	−0.717341
$628$	0	0
$629$	31998.9	2.02842
$630$	0	0
$631$	−13980.2	−0.882002	−0.441001	−	0.897507i	$-0.645377\pi$
$631$	−13980.2	−0.882002	−0.441001	+	0.897507i	$0.645377\pi$
$632$	0	0
$633$	37568.9	2.35897
$634$	0	0
$635$	−6067.45	−0.379180
$636$	0	0
$637$	−947.814	−0.0589541
$638$	0	0
$639$	16482.7	1.02041
$640$	0	0
$641$	−16060.9	−0.989655	−0.494828	−	0.868991i	$-0.664769\pi$
$641$	−16060.9	−0.989655	−0.494828	+	0.868991i	$0.664769\pi$
$642$	0	0
$643$	−4502.17	−0.276125	−0.138063	−	0.990424i	$-0.544087\pi$
$643$	−4502.17	−0.276125	−0.138063	+	0.990424i	$0.544087\pi$
$644$	0	0
$645$	−7527.79	−0.459545
$646$	0	0
$647$	−29414.8	−1.78735	−0.893675	−	0.448715i	$-0.851882\pi$
$647$	−29414.8	−1.78735	−0.893675	+	0.448715i	$0.851882\pi$
$648$	0	0
$649$	−23565.0	−1.42528
$650$	0	0
$651$	−9066.03	−0.545815
$652$	0	0
$653$	−13013.6	−0.779882	−0.389941	−	0.920840i	$-0.627505\pi$
$653$	−13013.6	−0.779882	−0.389941	+	0.920840i	$0.627505\pi$
$654$	0	0
$655$	−9912.09	−0.591294
$656$	0	0
$657$	14275.0	0.847671
$658$	0	0
$659$	23474.2	1.38759	0.693797	−	0.720171i	$-0.255937\pi$
$659$	23474.2	1.38759	0.693797	+	0.720171i	$0.255937\pi$
$660$	0	0
$661$	9266.36	0.545264	0.272632	−	0.962118i	$-0.412106\pi$
$661$	9266.36	0.545264	0.272632	+	0.962118i	$0.412106\pi$
$662$	0	0
$663$	11231.5	0.657914
$664$	0	0
$665$	−1547.89	−0.0902625
$666$	0	0
$667$	−10241.0	−0.594501
$668$	0	0
$669$	42398.6	2.45026
$670$	0	0
$671$	−12269.7	−0.705909
$672$	0	0
$673$	−25067.2	−1.43576	−0.717882	−	0.696164i	$-0.754888\pi$
$673$	−25067.2	−1.43576	−0.717882	+	0.696164i	$0.754888\pi$
$674$	0	0
$675$	−1612.01	−0.0919202
$676$	0	0
$677$	22409.6	1.27219	0.636093	−	0.771613i	$-0.280550\pi$
$677$	22409.6	1.27219	0.636093	+	0.771613i	$0.280550\pi$
$678$	0	0
$679$	7419.11	0.419322
$680$	0	0
$681$	−6761.20	−0.380455
$682$	0	0
$683$	−8757.53	−0.490626	−0.245313	−	0.969444i	$-0.578891\pi$
$683$	−8757.53	−0.490626	−0.245313	+	0.969444i	$0.578891\pi$
$684$	0	0
$685$	11054.7	0.616613
$686$	0	0
$687$	−27348.7	−1.51880
$688$	0	0
$689$	−4043.09	−0.223555
$690$	0	0
$691$	8468.42	0.466214	0.233107	−	0.972451i	$-0.425111\pi$
$691$	8468.42	0.466214	0.233107	+	0.972451i	$0.425111\pi$
$692$	0	0
$693$	4636.33	0.254141
$694$	0	0
$695$	2640.19	0.144098
$696$	0	0
$697$	29601.0	1.60864
$698$	0	0
$699$	4050.23	0.219162
$700$	0	0
$701$	−15996.9	−0.861906	−0.430953	−	0.902374i	$-0.641823\pi$
$701$	−15996.9	−0.861906	−0.430953	+	0.902374i	$0.641823\pi$
$702$	0	0
$703$	16224.2	0.870423
$704$	0	0
$705$	−388.633	−0.0207614
$706$	0	0
$707$	1692.83	0.0900503
$708$	0	0
$709$	−19903.0	−1.05426	−0.527131	−	0.849784i	$-0.676732\pi$
$709$	−19903.0	−1.05426	−0.527131	+	0.849784i	$0.676732\pi$
$710$	0	0
$711$	−2705.70	−0.142717
$712$	0	0
$713$	42446.1	2.22948
$714$	0	0
$715$	−3699.84	−0.193519
$716$	0	0
$717$	33646.7	1.75252
$718$	0	0
$719$	11073.1	0.574347	0.287174	−	0.957879i	$-0.407284\pi$
$719$	11073.1	0.574347	0.287174	+	0.957879i	$0.407284\pi$
$720$	0	0
$721$	11757.0	0.607288
$722$	0	0
$723$	32.4043	0.00166685
$724$	0	0
$725$	1173.53	0.0601155
$726$	0	0
$727$	−31652.7	−1.61476	−0.807382	−	0.590029i	$-0.799116\pi$
$727$	−31652.7	−1.61476	−0.807382	+	0.590029i	$0.799116\pi$
$728$	0	0
$729$	−3935.94	−0.199967
$730$	0	0
$731$	19727.5	0.998149
$732$	0	0
$733$	−16958.3	−0.854528	−0.427264	−	0.904127i	$-0.640523\pi$
$733$	−16958.3	−0.854528	−0.427264	+	0.904127i	$0.640523\pi$
$734$	0	0
$735$	1630.93	0.0818473
$736$	0	0
$737$	555.065	0.0277423
$738$	0	0
$739$	−11616.6	−0.578245	−0.289123	−	0.957292i	$-0.593364\pi$
$739$	−11616.6	−0.578245	−0.289123	+	0.957292i	$0.593364\pi$
$740$	0	0
$741$	5694.66	0.282319
$742$	0	0
$743$	−15928.0	−0.786464	−0.393232	−	0.919439i	$-0.628643\pi$
$743$	−15928.0	−0.786464	−0.393232	+	0.919439i	$0.628643\pi$
$744$	0	0
$745$	1641.86	0.0807423
$746$	0	0
$747$	−17946.1	−0.879003
$748$	0	0
$749$	10548.2	0.514582
$750$	0	0
$751$	−25571.9	−1.24252	−0.621260	−	0.783604i	$-0.713379\pi$
$751$	−25571.9	−1.24252	−0.621260	+	0.783604i	$0.713379\pi$
$752$	0	0
$753$	−3647.43	−0.176520
$754$	0	0
$755$	−5147.14	−0.248111
$756$	0	0
$757$	−6202.41	−0.297794	−0.148897	−	0.988853i	$-0.547572\pi$
$757$	−6202.41	−0.297794	−0.148897	+	0.988853i	$0.547572\pi$
$758$	0	0
$759$	−55557.6	−2.65693
$760$	0	0
$761$	−29199.1	−1.39089	−0.695444	−	0.718580i	$-0.744792\pi$
$761$	−29199.1	−1.39089	−0.695444	+	0.718580i	$0.744792\pi$
$762$	0	0
$763$	8766.87	0.415966
$764$	0	0
$765$	−7550.98	−0.356871
$766$	0	0
$767$	11915.4	0.560938
$768$	0	0
$769$	21838.2	1.02407	0.512033	−	0.858966i	$-0.328892\pi$
$769$	21838.2	1.02407	0.512033	+	0.858966i	$0.328892\pi$
$770$	0	0
$771$	−11813.3	−0.551812
$772$	0	0
$773$	25544.8	1.18859	0.594296	−	0.804246i	$-0.297431\pi$
$773$	25544.8	1.18859	0.594296	+	0.804246i	$0.297431\pi$
$774$	0	0
$775$	−4863.96	−0.225443
$776$	0	0
$777$	−17094.6	−0.789273
$778$	0	0
$779$	15008.4	0.690286
$780$	0	0
$781$	36418.6	1.66858
$782$	0	0
$783$	−3026.77	−0.138146
$784$	0	0
$785$	−2625.49	−0.119373
$786$	0	0
$787$	−37223.2	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$787$	−37223.2	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$788$	0	0
$789$	7982.19	0.360169
$790$	0	0
$791$	9591.42	0.431140
$792$	0	0
$793$	6204.03	0.277820
$794$	0	0
$795$	6957.06	0.310366
$796$	0	0
$797$	40384.6	1.79485	0.897425	−	0.441168i	$-0.145436\pi$
$797$	40384.6	1.79485	0.897425	+	0.441168i	$0.145436\pi$
$798$	0	0
$799$	1018.46	0.0450945
$800$	0	0
$801$	−2947.23	−0.130007
$802$	0	0
$803$	31540.7	1.38611
$804$	0	0
$805$	−7635.83	−0.334320
$806$	0	0
$807$	−21636.7	−0.943803
$808$	0	0
$809$	−1955.76	−0.0849948	−0.0424974	−	0.999097i	$-0.513531\pi$
$809$	−1955.76	−0.0849948	−0.0424974	+	0.999097i	$0.513531\pi$
$810$	0	0
$811$	−34301.8	−1.48520	−0.742600	−	0.669735i	$-0.766408\pi$
$811$	−34301.8	−1.48520	−0.742600	+	0.669735i	$0.766408\pi$
$812$	0	0
$813$	5965.49	0.257342
$814$	0	0
$815$	5013.13	0.215463
$816$	0	0
$817$	10002.3	0.428319
$818$	0	0
$819$	−2344.31	−0.100021
$820$	0	0
$821$	13665.6	0.580918	0.290459	−	0.956887i	$-0.406192\pi$
$821$	13665.6	0.580918	0.290459	+	0.956887i	$0.406192\pi$
$822$	0	0
$823$	21519.5	0.911449	0.455724	−	0.890121i	$-0.349380\pi$
$823$	21519.5	0.911449	0.455724	+	0.890121i	$0.349380\pi$
$824$	0	0
$825$	6366.42	0.268667
$826$	0	0
$827$	35220.6	1.48094	0.740471	−	0.672088i	$-0.234602\pi$
$827$	35220.6	1.48094	0.740471	+	0.672088i	$0.234602\pi$
$828$	0	0
$829$	−31365.5	−1.31408	−0.657039	−	0.753857i	$-0.728191\pi$
$829$	−31365.5	−1.31408	−0.657039	+	0.753857i	$0.728191\pi$
$830$	0	0
$831$	2573.28	0.107420
$832$	0	0
$833$	−4274.04	−0.177775
$834$	0	0
$835$	1797.11	0.0744810
$836$	0	0
$837$	12545.2	0.518070
$838$	0	0
$839$	28287.1	1.16398	0.581990	−	0.813196i	$-0.302274\pi$
$839$	28287.1	1.16398	0.581990	+	0.813196i	$0.302274\pi$
$840$	0	0
$841$	−22185.5	−0.909653
$842$	0	0
$843$	−22201.2	−0.907060
$844$	0	0
$845$	−9114.21	−0.371051
$846$	0	0
$847$	927.026	0.0376068
$848$	0	0
$849$	36028.6	1.45642
$850$	0	0
$851$	80035.0	3.22393
$852$	0	0
$853$	−9405.41	−0.377533	−0.188766	−	0.982022i	$-0.560449\pi$
$853$	−9405.41	−0.377533	−0.188766	+	0.982022i	$0.560449\pi$
$854$	0	0
$855$	−3828.53	−0.153138
$856$	0	0
$857$	−27966.9	−1.11474	−0.557369	−	0.830265i	$-0.688189\pi$
$857$	−27966.9	−1.11474	−0.557369	+	0.830265i	$0.688189\pi$
$858$	0	0
$859$	6281.11	0.249486	0.124743	−	0.992189i	$-0.460189\pi$
$859$	6281.11	0.249486	0.124743	+	0.992189i	$0.460189\pi$
$860$	0	0
$861$	−15813.6	−0.625931
$862$	0	0
$863$	4757.13	0.187642	0.0938208	−	0.995589i	$-0.470092\pi$
$863$	4757.13	0.187642	0.0938208	+	0.995589i	$0.470092\pi$
$864$	0	0
$865$	−16468.3	−0.647327
$866$	0	0
$867$	17942.0	0.702818
$868$	0	0
$869$	−5978.28	−0.233371
$870$	0	0
$871$	−280.663	−0.0109184
$872$	0	0
$873$	18350.3	0.711414
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	30240.5	1.16437	0.582184	−	0.813057i	$-0.302198\pi$
$877$	30240.5	1.16437	0.582184	+	0.813057i	$0.302198\pi$
$878$	0	0
$879$	−1878.64	−0.0720875
$880$	0	0
$881$	44875.5	1.71611	0.858056	−	0.513556i	$-0.171672\pi$
$881$	44875.5	1.71611	0.858056	+	0.513556i	$0.171672\pi$
$882$	0	0
$883$	4892.13	0.186448	0.0932238	−	0.995645i	$-0.470283\pi$
$883$	4892.13	0.186448	0.0932238	+	0.995645i	$0.470283\pi$
$884$	0	0
$885$	−20503.1	−0.778762
$886$	0	0
$887$	−1761.40	−0.0666765	−0.0333382	−	0.999444i	$-0.510614\pi$
$887$	−1761.40	−0.0666765	−0.0333382	+	0.999444i	$0.510614\pi$
$888$	0	0
$889$	−8494.43	−0.320466
$890$	0	0
$891$	−34303.3	−1.28979
$892$	0	0
$893$	516.384	0.0193507
$894$	0	0
$895$	14894.1	0.556263
$896$	0	0
$897$	28092.1	1.04567
$898$	0	0
$899$	−9132.79	−0.338816
$900$	0	0
$901$	−18231.8	−0.674128
$902$	0	0
$903$	−10538.9	−0.388386
$904$	0	0
$905$	−7311.57	−0.268558
$906$	0	0
$907$	23689.1	0.867238	0.433619	−	0.901096i	$-0.357236\pi$
$907$	23689.1	0.867238	0.433619	+	0.901096i	$0.357236\pi$
$908$	0	0
$909$	4187.03	0.152778
$910$	0	0
$911$	13877.3	0.504692	0.252346	−	0.967637i	$-0.418798\pi$
$911$	13877.3	0.504692	0.252346	+	0.967637i	$0.418798\pi$
$912$	0	0
$913$	−39652.2	−1.43735
$914$	0	0
$915$	−10675.4	−0.385704
$916$	0	0
$917$	−13876.9	−0.499735
$918$	0	0
$919$	−14331.6	−0.514426	−0.257213	−	0.966355i	$-0.582804\pi$
$919$	−14331.6	−0.514426	−0.257213	+	0.966355i	$0.582804\pi$
$920$	0	0
$921$	12779.0	0.457201
$922$	0	0
$923$	−18414.7	−0.656692
$924$	0	0
$925$	−9171.32	−0.326001
$926$	0	0
$927$	29079.7	1.03032
$928$	0	0
$929$	16668.4	0.588668	0.294334	−	0.955703i	$-0.404902\pi$
$929$	16668.4	0.588668	0.294334	+	0.955703i	$0.404902\pi$
$930$	0	0
$931$	−2167.04	−0.0762857
$932$	0	0
$933$	−8076.82	−0.283412
$934$	0	0
$935$	−16684.0	−0.583555
$936$	0	0
$937$	30384.9	1.05937	0.529685	−	0.848194i	$-0.322310\pi$
$937$	30384.9	1.05937	0.529685	+	0.848194i	$0.322310\pi$
$938$	0	0
$939$	−9545.94	−0.331757
$940$	0	0
$941$	1196.35	0.0414452	0.0207226	−	0.999785i	$-0.493403\pi$
$941$	1196.35	0.0414452	0.0207226	+	0.999785i	$0.493403\pi$
$942$	0	0
$943$	74037.5	2.55673
$944$	0	0
$945$	−2256.81	−0.0776867
$946$	0	0
$947$	1788.41	0.0613681	0.0306840	−	0.999529i	$-0.490231\pi$
$947$	1788.41	0.0613681	0.0306840	+	0.999529i	$0.490231\pi$
$948$	0	0
$949$	−15948.2	−0.545523
$950$	0	0
$951$	−43249.2	−1.47471
$952$	0	0
$953$	8578.60	0.291593	0.145796	−	0.989315i	$-0.453426\pi$
$953$	8578.60	0.291593	0.145796	+	0.989315i	$0.453426\pi$
$954$	0	0
$955$	1874.61	0.0635194
$956$	0	0
$957$	11953.9	0.403776
$958$	0	0
$959$	15476.6	0.521133
$960$	0	0
$961$	8061.99	0.270618
$962$	0	0
$963$	26089.7	0.873031
$964$	0	0
$965$	3665.14	0.122264
$966$	0	0
$967$	55459.3	1.84431	0.922156	−	0.386818i	$-0.126426\pi$
$967$	55459.3	1.84431	0.922156	+	0.386818i	$0.126426\pi$
$968$	0	0
$969$	25679.3	0.851330
$970$	0	0
$971$	22047.3	0.728662	0.364331	−	0.931270i	$-0.381298\pi$
$971$	22047.3	0.728662	0.364331	+	0.931270i	$0.381298\pi$
$972$	0	0
$973$	3696.27	0.121785
$974$	0	0
$975$	−3219.11	−0.105738
$976$	0	0
$977$	14402.3	0.471617	0.235809	−	0.971800i	$-0.424226\pi$
$977$	14402.3	0.471617	0.235809	+	0.971800i	$0.424226\pi$
$978$	0	0
$979$	−6511.94	−0.212587
$980$	0	0
$981$	21683.9	0.705721
$982$	0	0
$983$	−7817.11	−0.253639	−0.126819	−	0.991926i	$-0.540477\pi$
$983$	−7817.11	−0.253639	−0.126819	+	0.991926i	$0.540477\pi$
$984$	0	0
$985$	10466.2	0.338560
$986$	0	0
$987$	−544.087	−0.0175466
$988$	0	0
$989$	49342.0	1.58643
$990$	0	0
$991$	−24501.6	−0.785386	−0.392693	−	0.919670i	$-0.628457\pi$
$991$	−24501.6	−0.785386	−0.392693	+	0.919670i	$0.628457\pi$
$992$	0	0
$993$	−64464.2	−2.06013
$994$	0	0
$995$	−14325.2	−0.456421
$996$	0	0
$997$	50696.0	1.61039	0.805195	−	0.593010i	$-0.202060\pi$
$997$	50696.0	1.61039	0.805195	+	0.593010i	$0.202060\pi$
$998$	0	0
$999$	23654.8	0.749152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.bo.1.2		2
4.3	odd	2		2240.4.a.bn.1.1		2
8.3	odd	2		35.4.a.b.1.1	✓	2
8.5	even	2		560.4.a.r.1.1		2
24.11	even	2		315.4.a.f.1.2		2
40.3	even	4		175.4.b.c.99.2		4
40.19	odd	2		175.4.a.c.1.2		2
40.27	even	4		175.4.b.c.99.3		4
56.3	even	6		245.4.e.i.226.2		4
56.11	odd	6		245.4.e.h.226.2		4
56.19	even	6		245.4.e.i.116.2		4
56.27	even	2		245.4.a.k.1.1		2
56.51	odd	6		245.4.e.h.116.2		4
120.59	even	2		1575.4.a.z.1.1		2
168.83	odd	2		2205.4.a.u.1.2		2
280.139	even	2		1225.4.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.1	✓	2	8.3	odd	2
175.4.a.c.1.2		2	40.19	odd	2
175.4.b.c.99.2		4	40.3	even	4
175.4.b.c.99.3		4	40.27	even	4
245.4.a.k.1.1		2	56.27	even	2
245.4.e.h.116.2		4	56.51	odd	6
245.4.e.h.226.2		4	56.11	odd	6
245.4.e.i.116.2		4	56.19	even	6
245.4.e.i.226.2		4	56.3	even	6
315.4.a.f.1.2		2	24.11	even	2
560.4.a.r.1.1		2	8.5	even	2
1225.4.a.m.1.2		2	280.139	even	2
1575.4.a.z.1.1		2	120.59	even	2
2205.4.a.u.1.2		2	168.83	odd	2
2240.4.a.bn.1.1		2	4.3	odd	2
2240.4.a.bo.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q+6.65685 q^{3} +5.00000 q^{5} +7.00000 q^{7} +17.3137 q^{9} +38.2548 q^{11} -19.3431 q^{13} +33.2843 q^{15} -87.2254 q^{17} -44.2254 q^{19} +46.5980 q^{21} -218.167 q^{23} +25.0000 q^{25} -64.4802 q^{27} +46.9411 q^{29} -194.558 q^{31} +254.657 q^{33} +35.0000 q^{35} -366.853 q^{37} -128.765 q^{39} -339.362 q^{41} -226.167 q^{43} +86.5685 q^{45} -11.6762 q^{47} +49.0000 q^{49} -580.647 q^{51} +209.019 q^{53} +191.274 q^{55} -294.402 q^{57} -616.000 q^{59} -320.735 q^{61} +121.196 q^{63} -96.7157 q^{65} +14.5097 q^{67} -1452.30 q^{69} +952.000 q^{71} +824.489 q^{73} +166.421 q^{75} +267.784 q^{77} -156.275 q^{79} -896.706 q^{81} -1036.53 q^{83} -436.127 q^{85} +312.480 q^{87} -170.225 q^{89} -135.402 q^{91} -1295.15 q^{93} -221.127 q^{95} +1059.87 q^{97} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 10 q^{5} + 14 q^{7} + 12 q^{9} - 14 q^{11} - 50 q^{13} + 10 q^{15} - 50 q^{17} + 36 q^{19} + 14 q^{21} - 244 q^{23} + 50 q^{25} + 86 q^{27} + 26 q^{29} + 120 q^{31} + 498 q^{33} + 70 q^{35}+ \cdots + 940 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 10 * q^5 + 14 * q^7 + 12 * q^9 - 14 * q^11 - 50 * q^13 + 10 * q^15 - 50 * q^17 + 36 * q^19 + 14 * q^21 - 244 * q^23 + 50 * q^25 + 86 * q^27 + 26 * q^29 + 120 * q^31 + 498 * q^33 + 70 * q^35 - 564 * q^37 + 14 * q^39 - 328 * q^41 - 260 * q^43 + 60 * q^45 + 350 * q^47 + 98 * q^49 - 754 * q^51 + 56 * q^53 - 70 * q^55 - 668 * q^57 - 1232 * q^59 - 336 * q^61 + 84 * q^63 - 250 * q^65 - 152 * q^67 - 1332 * q^69 + 1904 * q^71 + 676 * q^73 + 50 * q^75 - 98 * q^77 - 1014 * q^79 - 1454 * q^81 - 376 * q^83 - 250 * q^85 + 410 * q^87 - 216 * q^89 - 350 * q^91 - 2760 * q^93 + 180 * q^95 + 2742 * q^97 + 940 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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