LMFDB - Embedded newform 6040.2.a.p.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.a (trivial)

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:	yes
Analytic conductor:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 $ x^19 - 5x^18 - 29x^17 + 165x^16 + 325x^15 - 2208x^14 - 1891x^13 + 15895x^12 + 6652x^11 - 67665x^10 - 17345x^9 + 174105x^8 + 41499x^7 - 262172x^6 - 80919x^5 + 206783x^4 + 91643x^3 - 59750x^2 - 40224x - 5628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$-3.37325$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37325 q^{3} +1.00000 q^{5} -3.80572 q^{7} +8.37885 q^{9} +O(q^{10})$	$q+3.37325 q^{3} +1.00000 q^{5} -3.80572 q^{7} +8.37885 q^{9} -2.57642 q^{11} -4.49122 q^{13} +3.37325 q^{15} +4.43598 q^{17} -6.71504 q^{19} -12.8377 q^{21} -8.16971 q^{23} +1.00000 q^{25} +18.1442 q^{27} -8.65001 q^{29} -2.63378 q^{31} -8.69091 q^{33} -3.80572 q^{35} -2.00647 q^{37} -15.1500 q^{39} +0.367431 q^{41} -4.23046 q^{43} +8.37885 q^{45} -6.96940 q^{47} +7.48352 q^{49} +14.9637 q^{51} +12.5327 q^{53} -2.57642 q^{55} -22.6515 q^{57} -2.01138 q^{59} -6.13006 q^{61} -31.8876 q^{63} -4.49122 q^{65} -10.2416 q^{67} -27.5585 q^{69} +3.21081 q^{71} +0.138853 q^{73} +3.37325 q^{75} +9.80513 q^{77} +6.47365 q^{79} +36.0685 q^{81} -10.2321 q^{83} +4.43598 q^{85} -29.1787 q^{87} +16.5702 q^{89} +17.0923 q^{91} -8.88441 q^{93} -6.71504 q^{95} +7.03808 q^{97} -21.5874 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.37325	1.94755	0.973775	−	0.227515i	$-0.0730599\pi$
$3$	3.37325	1.94755	0.973775	+	0.227515i	$0.0730599\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.80572	−1.43843	−0.719214	−	0.694789i	$-0.755498\pi$
$7$	−3.80572	−1.43843	−0.719214	+	0.694789i	$0.755498\pi$
$8$	0	0
$9$	8.37885	2.79295
$10$	0	0
$11$	−2.57642	−0.776819	−0.388410	−	0.921487i	$-0.626975\pi$
$11$	−2.57642	−0.776819	−0.388410	+	0.921487i	$0.626975\pi$
$12$	0	0
$13$	−4.49122	−1.24564	−0.622821	−	0.782365i	$-0.714013\pi$
$13$	−4.49122	−1.24564	−0.622821	+	0.782365i	$0.714013\pi$
$14$	0	0
$15$	3.37325	0.870971
$16$	0	0
$17$	4.43598	1.07588	0.537941	−	0.842982i	$-0.319202\pi$
$17$	4.43598	1.07588	0.537941	+	0.842982i	$0.319202\pi$
$18$	0	0
$19$	−6.71504	−1.54054	−0.770268	−	0.637720i	$-0.779878\pi$
$19$	−6.71504	−1.54054	−0.770268	+	0.637720i	$0.779878\pi$
$20$	0	0
$21$	−12.8377	−2.80141
$22$	0	0
$23$	−8.16971	−1.70350	−0.851751	−	0.523947i	$-0.824459\pi$
$23$	−8.16971	−1.70350	−0.851751	+	0.523947i	$0.824459\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	18.1442	3.49186
$28$	0	0
$29$	−8.65001	−1.60627	−0.803133	−	0.595800i	$-0.796835\pi$
$29$	−8.65001	−1.60627	−0.803133	+	0.595800i	$0.796835\pi$
$30$	0	0
$31$	−2.63378	−0.473041	−0.236520	−	0.971627i	$-0.576007\pi$
$31$	−2.63378	−0.473041	−0.236520	+	0.971627i	$0.576007\pi$
$32$	0	0
$33$	−8.69091	−1.51289
$34$	0	0
$35$	−3.80572	−0.643284
$36$	0	0
$37$	−2.00647	−0.329862	−0.164931	−	0.986305i	$-0.552740\pi$
$37$	−2.00647	−0.329862	−0.164931	+	0.986305i	$0.552740\pi$
$38$	0	0
$39$	−15.1500	−2.42595
$40$	0	0
$41$	0.367431	0.0573831	0.0286916	−	0.999588i	$-0.490866\pi$
$41$	0.367431	0.0573831	0.0286916	+	0.999588i	$0.490866\pi$
$42$	0	0
$43$	−4.23046	−0.645139	−0.322570	−	0.946546i	$-0.604547\pi$
$43$	−4.23046	−0.645139	−0.322570	+	0.946546i	$0.604547\pi$
$44$	0	0
$45$	8.37885	1.24904
$46$	0	0
$47$	−6.96940	−1.01659	−0.508296	−	0.861183i	$-0.669724\pi$
$47$	−6.96940	−1.01659	−0.508296	+	0.861183i	$0.669724\pi$
$48$	0	0
$49$	7.48352	1.06907
$50$	0	0
$51$	14.9637	2.09533
$52$	0	0
$53$	12.5327	1.72150	0.860748	−	0.509031i	$-0.169996\pi$
$53$	12.5327	1.72150	0.860748	+	0.509031i	$0.169996\pi$
$54$	0	0
$55$	−2.57642	−0.347404
$56$	0	0
$57$	−22.6515	−3.00027
$58$	0	0
$59$	−2.01138	−0.261860	−0.130930	−	0.991392i	$-0.541796\pi$
$59$	−2.01138	−0.261860	−0.130930	+	0.991392i	$0.541796\pi$
$60$	0	0
$61$	−6.13006	−0.784874	−0.392437	−	0.919779i	$-0.628368\pi$
$61$	−6.13006	−0.784874	−0.392437	+	0.919779i	$0.628368\pi$
$62$	0	0
$63$	−31.8876	−4.01745
$64$	0	0
$65$	−4.49122	−0.557068
$66$	0	0
$67$	−10.2416	−1.25121	−0.625606	−	0.780139i	$-0.715148\pi$
$67$	−10.2416	−1.25121	−0.625606	+	0.780139i	$0.715148\pi$
$68$	0	0
$69$	−27.5585	−3.31765
$70$	0	0
$71$	3.21081	0.381053	0.190526	−	0.981682i	$-0.438981\pi$
$71$	3.21081	0.381053	0.190526	+	0.981682i	$0.438981\pi$
$72$	0	0
$73$	0.138853	0.0162515	0.00812573	−	0.999967i	$-0.497413\pi$
$73$	0.138853	0.0162515	0.00812573	+	0.999967i	$0.497413\pi$
$74$	0	0
$75$	3.37325	0.389510
$76$	0	0
$77$	9.80513	1.11740
$78$	0	0
$79$	6.47365	0.728342	0.364171	−	0.931332i	$-0.381352\pi$
$79$	6.47365	0.728342	0.364171	+	0.931332i	$0.381352\pi$
$80$	0	0
$81$	36.0685	4.00761
$82$	0	0
$83$	−10.2321	−1.12312	−0.561558	−	0.827437i	$-0.689798\pi$
$83$	−10.2321	−1.12312	−0.561558	+	0.827437i	$0.689798\pi$
$84$	0	0
$85$	4.43598	0.481149
$86$	0	0
$87$	−29.1787	−3.12828
$88$	0	0
$89$	16.5702	1.75644	0.878218	−	0.478260i	$-0.158732\pi$
$89$	16.5702	1.75644	0.878218	+	0.478260i	$0.158732\pi$
$90$	0	0
$91$	17.0923	1.79176
$92$	0	0
$93$	−8.88441	−0.921270
$94$	0	0
$95$	−6.71504	−0.688949
$96$	0	0
$97$	7.03808	0.714609	0.357305	−	0.933988i	$-0.383696\pi$
$97$	7.03808	0.714609	0.357305	+	0.933988i	$0.383696\pi$
$98$	0	0
$99$	−21.5874	−2.16962
$100$	0	0
$101$	2.70142	0.268802	0.134401	−	0.990927i	$-0.457089\pi$
$101$	2.70142	0.268802	0.134401	+	0.990927i	$0.457089\pi$
$102$	0	0
$103$	2.37032	0.233554	0.116777	−	0.993158i	$-0.462744\pi$
$103$	2.37032	0.233554	0.116777	+	0.993158i	$0.462744\pi$
$104$	0	0
$105$	−12.8377	−1.25283
$106$	0	0
$107$	−0.385224	−0.0372411	−0.0186205	−	0.999827i	$-0.505927\pi$
$107$	−0.385224	−0.0372411	−0.0186205	+	0.999827i	$0.505927\pi$
$108$	0	0
$109$	−5.71939	−0.547818	−0.273909	−	0.961756i	$-0.588317\pi$
$109$	−5.71939	−0.547818	−0.273909	+	0.961756i	$0.588317\pi$
$110$	0	0
$111$	−6.76834	−0.642423
$112$	0	0
$113$	15.3078	1.44004	0.720018	−	0.693955i	$-0.244134\pi$
$113$	15.3078	1.44004	0.720018	+	0.693955i	$0.244134\pi$
$114$	0	0
$115$	−8.16971	−0.761829
$116$	0	0
$117$	−37.6313	−3.47901
$118$	0	0
$119$	−16.8821	−1.54758
$120$	0	0
$121$	−4.36207	−0.396552
$122$	0	0
$123$	1.23944	0.111756
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.03938	0.624644	0.312322	−	0.949976i	$-0.398893\pi$
$127$	7.03938	0.624644	0.312322	+	0.949976i	$0.398893\pi$
$128$	0	0
$129$	−14.2704	−1.25644
$130$	0	0
$131$	11.9148	1.04100	0.520500	−	0.853862i	$-0.325746\pi$
$131$	11.9148	1.04100	0.520500	+	0.853862i	$0.325746\pi$
$132$	0	0
$133$	25.5556	2.21595
$134$	0	0
$135$	18.1442	1.56161
$136$	0	0
$137$	−8.94271	−0.764027	−0.382014	−	0.924157i	$-0.624769\pi$
$137$	−8.94271	−0.764027	−0.382014	+	0.924157i	$0.624769\pi$
$138$	0	0
$139$	−4.72445	−0.400723	−0.200361	−	0.979722i	$-0.564212\pi$
$139$	−4.72445	−0.400723	−0.200361	+	0.979722i	$0.564212\pi$
$140$	0	0
$141$	−23.5096	−1.97986
$142$	0	0
$143$	11.5713	0.967638
$144$	0	0
$145$	−8.65001	−0.718344
$146$	0	0
$147$	25.2438	2.08207
$148$	0	0
$149$	10.1001	0.827430	0.413715	−	0.910406i	$-0.364231\pi$
$149$	10.1001	0.827430	0.413715	+	0.910406i	$0.364231\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	37.1684	3.00488
$154$	0	0
$155$	−2.63378	−0.211550
$156$	0	0
$157$	9.52854	0.760460	0.380230	−	0.924892i	$-0.375845\pi$
$157$	9.52854	0.760460	0.380230	+	0.924892i	$0.375845\pi$
$158$	0	0
$159$	42.2759	3.35270
$160$	0	0
$161$	31.0916	2.45036
$162$	0	0
$163$	−16.7786	−1.31420	−0.657100	−	0.753804i	$-0.728217\pi$
$163$	−16.7786	−1.31420	−0.657100	+	0.753804i	$0.728217\pi$
$164$	0	0
$165$	−8.69091	−0.676587
$166$	0	0
$167$	21.4191	1.65746	0.828730	−	0.559648i	$-0.189064\pi$
$167$	21.4191	1.65746	0.828730	+	0.559648i	$0.189064\pi$
$168$	0	0
$169$	7.17108	0.551622
$170$	0	0
$171$	−56.2643	−4.30264
$172$	0	0
$173$	1.47999	0.112521	0.0562606	−	0.998416i	$-0.482082\pi$
$173$	1.47999	0.112521	0.0562606	+	0.998416i	$0.482082\pi$
$174$	0	0
$175$	−3.80572	−0.287686
$176$	0	0
$177$	−6.78490	−0.509984
$178$	0	0
$179$	6.08516	0.454826	0.227413	−	0.973798i	$-0.426973\pi$
$179$	6.08516	0.454826	0.227413	+	0.973798i	$0.426973\pi$
$180$	0	0
$181$	−25.4550	−1.89205	−0.946026	−	0.324091i	$-0.894942\pi$
$181$	−25.4550	−1.89205	−0.946026	+	0.324091i	$0.894942\pi$
$182$	0	0
$183$	−20.6783	−1.52858
$184$	0	0
$185$	−2.00647	−0.147519
$186$	0	0
$187$	−11.4289	−0.835766
$188$	0	0
$189$	−69.0518	−5.02278
$190$	0	0
$191$	−5.50059	−0.398009	−0.199004	−	0.979999i	$-0.563771\pi$
$191$	−5.50059	−0.398009	−0.199004	+	0.979999i	$0.563771\pi$
$192$	0	0
$193$	9.73741	0.700914	0.350457	−	0.936579i	$-0.386026\pi$
$193$	9.73741	0.700914	0.350457	+	0.936579i	$0.386026\pi$
$194$	0	0
$195$	−15.1500	−1.08492
$196$	0	0
$197$	−8.13384	−0.579512	−0.289756	−	0.957101i	$-0.593574\pi$
$197$	−8.13384	−0.579512	−0.289756	+	0.957101i	$0.593574\pi$
$198$	0	0
$199$	11.8288	0.838525	0.419262	−	0.907865i	$-0.362289\pi$
$199$	11.8288	0.838525	0.419262	+	0.907865i	$0.362289\pi$
$200$	0	0
$201$	−34.5476	−2.43680
$202$	0	0
$203$	32.9195	2.31050
$204$	0	0
$205$	0.367431	0.0256625
$206$	0	0
$207$	−68.4527	−4.75779
$208$	0	0
$209$	17.3007	1.19672
$210$	0	0
$211$	−20.3740	−1.40261	−0.701304	−	0.712863i	$-0.747398\pi$
$211$	−20.3740	−1.40261	−0.701304	+	0.712863i	$0.747398\pi$
$212$	0	0
$213$	10.8309	0.742119
$214$	0	0
$215$	−4.23046	−0.288515
$216$	0	0
$217$	10.0234	0.680435
$218$	0	0
$219$	0.468385	0.0316505
$220$	0	0
$221$	−19.9230	−1.34016
$222$	0	0
$223$	18.3631	1.22968	0.614842	−	0.788650i	$-0.289220\pi$
$223$	18.3631	1.22968	0.614842	+	0.788650i	$0.289220\pi$
$224$	0	0
$225$	8.37885	0.558590
$226$	0	0
$227$	−21.6600	−1.43762	−0.718812	−	0.695204i	$-0.755314\pi$
$227$	−21.6600	−1.43762	−0.718812	+	0.695204i	$0.755314\pi$
$228$	0	0
$229$	−30.1033	−1.98928	−0.994642	−	0.103381i	$-0.967034\pi$
$229$	−30.1033	−1.98928	−0.994642	+	0.103381i	$0.967034\pi$
$230$	0	0
$231$	33.0752	2.17619
$232$	0	0
$233$	−4.16170	−0.272642	−0.136321	−	0.990665i	$-0.543528\pi$
$233$	−4.16170	−0.272642	−0.136321	+	0.990665i	$0.543528\pi$
$234$	0	0
$235$	−6.96940	−0.454634
$236$	0	0
$237$	21.8373	1.41848
$238$	0	0
$239$	11.4316	0.739448	0.369724	−	0.929142i	$-0.379452\pi$
$239$	11.4316	0.739448	0.369724	+	0.929142i	$0.379452\pi$
$240$	0	0
$241$	10.4462	0.672900	0.336450	−	0.941701i	$-0.390774\pi$
$241$	10.4462	0.672900	0.336450	+	0.941701i	$0.390774\pi$
$242$	0	0
$243$	67.2356	4.31317
$244$	0	0
$245$	7.48352	0.478104
$246$	0	0
$247$	30.1587	1.91895
$248$	0	0
$249$	−34.5154	−2.18733
$250$	0	0
$251$	24.3107	1.53448	0.767239	−	0.641361i	$-0.221630\pi$
$251$	24.3107	1.53448	0.767239	+	0.641361i	$0.221630\pi$
$252$	0	0
$253$	21.0486	1.32331
$254$	0	0
$255$	14.9637	0.937062
$256$	0	0
$257$	−25.2368	−1.57423	−0.787115	−	0.616806i	$-0.788426\pi$
$257$	−25.2368	−1.57423	−0.787115	+	0.616806i	$0.788426\pi$
$258$	0	0
$259$	7.63608	0.474483
$260$	0	0
$261$	−72.4771	−4.48622
$262$	0	0
$263$	10.6038	0.653857	0.326928	−	0.945049i	$-0.393986\pi$
$263$	10.6038	0.653857	0.326928	+	0.945049i	$0.393986\pi$
$264$	0	0
$265$	12.5327	0.769877
$266$	0	0
$267$	55.8955	3.42075
$268$	0	0
$269$	−15.7916	−0.962828	−0.481414	−	0.876493i	$-0.659877\pi$
$269$	−15.7916	−0.962828	−0.481414	+	0.876493i	$0.659877\pi$
$270$	0	0
$271$	−31.6036	−1.91978	−0.959892	−	0.280369i	$-0.909543\pi$
$271$	−31.6036	−1.91978	−0.959892	+	0.280369i	$0.909543\pi$
$272$	0	0
$273$	57.6568	3.48955
$274$	0	0
$275$	−2.57642	−0.155364
$276$	0	0
$277$	0.318427	0.0191324	0.00956621	−	0.999954i	$-0.496955\pi$
$277$	0.318427	0.0191324	0.00956621	+	0.999954i	$0.496955\pi$
$278$	0	0
$279$	−22.0680	−1.32118
$280$	0	0
$281$	−19.2708	−1.14960	−0.574801	−	0.818294i	$-0.694920\pi$
$281$	−19.2708	−1.14960	−0.574801	+	0.818294i	$0.694920\pi$
$282$	0	0
$283$	−24.2315	−1.44041	−0.720207	−	0.693759i	$-0.755953\pi$
$283$	−24.2315	−1.44041	−0.720207	+	0.693759i	$0.755953\pi$
$284$	0	0
$285$	−22.6515	−1.34176
$286$	0	0
$287$	−1.39834	−0.0825414
$288$	0	0
$289$	2.67789	0.157523
$290$	0	0
$291$	23.7412	1.39174
$292$	0	0
$293$	25.3003	1.47806	0.739031	−	0.673671i	$-0.235284\pi$
$293$	25.3003	1.47806	0.739031	+	0.673671i	$0.235284\pi$
$294$	0	0
$295$	−2.01138	−0.117107
$296$	0	0
$297$	−46.7471	−2.71254
$298$	0	0
$299$	36.6920	2.12195
$300$	0	0
$301$	16.1000	0.927986
$302$	0	0
$303$	9.11259	0.523505
$304$	0	0
$305$	−6.13006	−0.351006
$306$	0	0
$307$	18.3536	1.04750	0.523748	−	0.851873i	$-0.324533\pi$
$307$	18.3536	1.04750	0.523748	+	0.851873i	$0.324533\pi$
$308$	0	0
$309$	7.99568	0.454859
$310$	0	0
$311$	14.7087	0.834053	0.417026	−	0.908894i	$-0.363072\pi$
$311$	14.7087	0.834053	0.417026	+	0.908894i	$0.363072\pi$
$312$	0	0
$313$	0.877216	0.0495832	0.0247916	−	0.999693i	$-0.492108\pi$
$313$	0.877216	0.0495832	0.0247916	+	0.999693i	$0.492108\pi$
$314$	0	0
$315$	−31.8876	−1.79666
$316$	0	0
$317$	−22.9407	−1.28848	−0.644240	−	0.764823i	$-0.722826\pi$
$317$	−22.9407	−1.28848	−0.644240	+	0.764823i	$0.722826\pi$
$318$	0	0
$319$	22.2860	1.24778
$320$	0	0
$321$	−1.29946	−0.0725288
$322$	0	0
$323$	−29.7878	−1.65744
$324$	0	0
$325$	−4.49122	−0.249128
$326$	0	0
$327$	−19.2930	−1.06690
$328$	0	0
$329$	26.5236	1.46229
$330$	0	0
$331$	4.67009	0.256692	0.128346	−	0.991729i	$-0.459033\pi$
$331$	4.67009	0.256692	0.128346	+	0.991729i	$0.459033\pi$
$332$	0	0
$333$	−16.8119	−0.921288
$334$	0	0
$335$	−10.2416	−0.559559
$336$	0	0
$337$	−20.0182	−1.09046	−0.545229	−	0.838287i	$-0.683557\pi$
$337$	−20.0182	−1.09046	−0.545229	+	0.838287i	$0.683557\pi$
$338$	0	0
$339$	51.6371	2.80454
$340$	0	0
$341$	6.78571	0.367467
$342$	0	0
$343$	−1.84013	−0.0993579
$344$	0	0
$345$	−27.5585	−1.48370
$346$	0	0
$347$	−31.9949	−1.71758	−0.858788	−	0.512332i	$-0.828782\pi$
$347$	−31.9949	−1.71758	−0.858788	+	0.512332i	$0.828782\pi$
$348$	0	0
$349$	−29.9764	−1.60460	−0.802300	−	0.596921i	$-0.796391\pi$
$349$	−29.9764	−1.60460	−0.802300	+	0.596921i	$0.796391\pi$
$350$	0	0
$351$	−81.4897	−4.34960
$352$	0	0
$353$	−17.4732	−0.930005	−0.465002	−	0.885309i	$-0.653947\pi$
$353$	−17.4732	−0.930005	−0.465002	+	0.885309i	$0.653947\pi$
$354$	0	0
$355$	3.21081	0.170412
$356$	0	0
$357$	−56.9476	−3.01399
$358$	0	0
$359$	23.8229	1.25732	0.628662	−	0.777678i	$-0.283603\pi$
$359$	23.8229	1.25732	0.628662	+	0.777678i	$0.283603\pi$
$360$	0	0
$361$	26.0918	1.37325
$362$	0	0
$363$	−14.7144	−0.772305
$364$	0	0
$365$	0.138853	0.00726787
$366$	0	0
$367$	−18.9346	−0.988380	−0.494190	−	0.869354i	$-0.664535\pi$
$367$	−18.9346	−0.988380	−0.494190	+	0.869354i	$0.664535\pi$
$368$	0	0
$369$	3.07865	0.160268
$370$	0	0
$371$	−47.6959	−2.47625
$372$	0	0
$373$	33.3904	1.72889	0.864445	−	0.502727i	$-0.167670\pi$
$373$	33.3904	1.72889	0.864445	+	0.502727i	$0.167670\pi$
$374$	0	0
$375$	3.37325	0.174194
$376$	0	0
$377$	38.8491	2.00083
$378$	0	0
$379$	−25.2769	−1.29839	−0.649194	−	0.760623i	$-0.724894\pi$
$379$	−25.2769	−1.29839	−0.649194	+	0.760623i	$0.724894\pi$
$380$	0	0
$381$	23.7456	1.21652
$382$	0	0
$383$	15.2268	0.778052	0.389026	−	0.921227i	$-0.372812\pi$
$383$	15.2268	0.778052	0.389026	+	0.921227i	$0.372812\pi$
$384$	0	0
$385$	9.80513	0.499716
$386$	0	0
$387$	−35.4464	−1.80184
$388$	0	0
$389$	2.61685	0.132680	0.0663398	−	0.997797i	$-0.478868\pi$
$389$	2.61685	0.132680	0.0663398	+	0.997797i	$0.478868\pi$
$390$	0	0
$391$	−36.2406	−1.83277
$392$	0	0
$393$	40.1916	2.02740
$394$	0	0
$395$	6.47365	0.325724
$396$	0	0
$397$	−36.2365	−1.81866	−0.909330	−	0.416076i	$-0.863405\pi$
$397$	−36.2365	−1.81866	−0.909330	+	0.416076i	$0.863405\pi$
$398$	0	0
$399$	86.2055	4.31567
$400$	0	0
$401$	0.212066	0.0105901	0.00529503	−	0.999986i	$-0.498315\pi$
$401$	0.212066	0.0105901	0.00529503	+	0.999986i	$0.498315\pi$
$402$	0	0
$403$	11.8289	0.589239
$404$	0	0
$405$	36.0685	1.79226
$406$	0	0
$407$	5.16951	0.256243
$408$	0	0
$409$	−27.0874	−1.33939	−0.669693	−	0.742638i	$-0.733574\pi$
$409$	−27.0874	−1.33939	−0.669693	+	0.742638i	$0.733574\pi$
$410$	0	0
$411$	−30.1660	−1.48798
$412$	0	0
$413$	7.65476	0.376666
$414$	0	0
$415$	−10.2321	−0.502273
$416$	0	0
$417$	−15.9368	−0.780427
$418$	0	0
$419$	26.0027	1.27032	0.635158	−	0.772382i	$-0.280935\pi$
$419$	26.0027	1.27032	0.635158	+	0.772382i	$0.280935\pi$
$420$	0	0
$421$	−13.5473	−0.660256	−0.330128	−	0.943936i	$-0.607092\pi$
$421$	−13.5473	−0.660256	−0.330128	+	0.943936i	$0.607092\pi$
$422$	0	0
$423$	−58.3955	−2.83929
$424$	0	0
$425$	4.43598	0.215176
$426$	0	0
$427$	23.3293	1.12898
$428$	0	0
$429$	39.0328	1.88452
$430$	0	0
$431$	−26.0281	−1.25373	−0.626864	−	0.779129i	$-0.715662\pi$
$431$	−26.0281	−1.25373	−0.626864	+	0.779129i	$0.715662\pi$
$432$	0	0
$433$	16.9097	0.812626	0.406313	−	0.913734i	$-0.366814\pi$
$433$	16.9097	0.812626	0.406313	+	0.913734i	$0.366814\pi$
$434$	0	0
$435$	−29.1787	−1.39901
$436$	0	0
$437$	54.8599	2.62431
$438$	0	0
$439$	−17.2413	−0.822882	−0.411441	−	0.911436i	$-0.634974\pi$
$439$	−17.2413	−0.822882	−0.411441	+	0.911436i	$0.634974\pi$
$440$	0	0
$441$	62.7032	2.98587
$442$	0	0
$443$	24.9806	1.18687	0.593433	−	0.804883i	$-0.297772\pi$
$443$	24.9806	1.18687	0.593433	+	0.804883i	$0.297772\pi$
$444$	0	0
$445$	16.5702	0.785502
$446$	0	0
$447$	34.0701	1.61146
$448$	0	0
$449$	39.3840	1.85864	0.929322	−	0.369269i	$-0.120392\pi$
$449$	39.3840	1.85864	0.929322	+	0.369269i	$0.120392\pi$
$450$	0	0
$451$	−0.946656	−0.0445763
$452$	0	0
$453$	3.37325	0.158489
$454$	0	0
$455$	17.0923	0.801301
$456$	0	0
$457$	−26.9955	−1.26280	−0.631399	−	0.775458i	$-0.717519\pi$
$457$	−26.9955	−1.26280	−0.631399	+	0.775458i	$0.717519\pi$
$458$	0	0
$459$	80.4873	3.75683
$460$	0	0
$461$	17.2448	0.803170	0.401585	−	0.915822i	$-0.368459\pi$
$461$	17.2448	0.803170	0.401585	+	0.915822i	$0.368459\pi$
$462$	0	0
$463$	10.9009	0.506610	0.253305	−	0.967387i	$-0.418482\pi$
$463$	10.9009	0.506610	0.253305	+	0.967387i	$0.418482\pi$
$464$	0	0
$465$	−8.88441	−0.412005
$466$	0	0
$467$	0.615217	0.0284688	0.0142344	−	0.999899i	$-0.495469\pi$
$467$	0.615217	0.0284688	0.0142344	+	0.999899i	$0.495469\pi$
$468$	0	0
$469$	38.9767	1.79978
$470$	0	0
$471$	32.1422	1.48103
$472$	0	0
$473$	10.8994	0.501156
$474$	0	0
$475$	−6.71504	−0.308107
$476$	0	0
$477$	105.009	4.80805
$478$	0	0
$479$	−11.2848	−0.515617	−0.257808	−	0.966196i	$-0.583000\pi$
$479$	−11.2848	−0.515617	−0.257808	+	0.966196i	$0.583000\pi$
$480$	0	0
$481$	9.01151	0.410890
$482$	0	0
$483$	104.880	4.77220
$484$	0	0
$485$	7.03808	0.319583
$486$	0	0
$487$	41.0311	1.85930	0.929648	−	0.368450i	$-0.120111\pi$
$487$	41.0311	1.85930	0.929648	+	0.368450i	$0.120111\pi$
$488$	0	0
$489$	−56.5984	−2.55947
$490$	0	0
$491$	−19.4719	−0.878756	−0.439378	−	0.898302i	$-0.644801\pi$
$491$	−19.4719	−0.878756	−0.439378	+	0.898302i	$0.644801\pi$
$492$	0	0
$493$	−38.3712	−1.72815
$494$	0	0
$495$	−21.5874	−0.970282
$496$	0	0
$497$	−12.2194	−0.548117
$498$	0	0
$499$	−3.46073	−0.154924	−0.0774618	−	0.996995i	$-0.524682\pi$
$499$	−3.46073	−0.154924	−0.0774618	+	0.996995i	$0.524682\pi$
$500$	0	0
$501$	72.2521	3.22799
$502$	0	0
$503$	−3.01552	−0.134455	−0.0672277	−	0.997738i	$-0.521415\pi$
$503$	−3.01552	−0.134455	−0.0672277	+	0.997738i	$0.521415\pi$
$504$	0	0
$505$	2.70142	0.120212
$506$	0	0
$507$	24.1899	1.07431
$508$	0	0
$509$	12.6552	0.560933	0.280467	−	0.959864i	$-0.409511\pi$
$509$	12.6552	0.560933	0.280467	+	0.959864i	$0.409511\pi$
$510$	0	0
$511$	−0.528434	−0.0233765
$512$	0	0
$513$	−121.839	−5.37933
$514$	0	0
$515$	2.37032	0.104449
$516$	0	0
$517$	17.9561	0.789708
$518$	0	0
$519$	4.99237	0.219141
$520$	0	0
$521$	−25.3687	−1.11142	−0.555712	−	0.831375i	$-0.687554\pi$
$521$	−25.3687	−1.11142	−0.555712	+	0.831375i	$0.687554\pi$
$522$	0	0
$523$	−9.03765	−0.395189	−0.197594	−	0.980284i	$-0.563313\pi$
$523$	−9.03765	−0.395189	−0.197594	+	0.980284i	$0.563313\pi$
$524$	0	0
$525$	−12.8377	−0.560282
$526$	0	0
$527$	−11.6834	−0.508936
$528$	0	0
$529$	43.7441	1.90192
$530$	0	0
$531$	−16.8531	−0.731360
$532$	0	0
$533$	−1.65022	−0.0714788
$534$	0	0
$535$	−0.385224	−0.0166547
$536$	0	0
$537$	20.5268	0.885796
$538$	0	0
$539$	−19.2807	−0.830477
$540$	0	0
$541$	16.8584	0.724798	0.362399	−	0.932023i	$-0.381958\pi$
$541$	16.8584	0.724798	0.362399	+	0.932023i	$0.381958\pi$
$542$	0	0
$543$	−85.8661	−3.68486
$544$	0	0
$545$	−5.71939	−0.244992
$546$	0	0
$547$	20.2498	0.865819	0.432910	−	0.901437i	$-0.357487\pi$
$547$	20.2498	0.865819	0.432910	+	0.901437i	$0.357487\pi$
$548$	0	0
$549$	−51.3628	−2.19211
$550$	0	0
$551$	58.0851	2.47451
$552$	0	0
$553$	−24.6369	−1.04767
$554$	0	0
$555$	−6.76834	−0.287300
$556$	0	0
$557$	29.8544	1.26497	0.632485	−	0.774573i	$-0.282035\pi$
$557$	29.8544	1.26497	0.632485	+	0.774573i	$0.282035\pi$
$558$	0	0
$559$	18.9999	0.803612
$560$	0	0
$561$	−38.5527	−1.62770
$562$	0	0
$563$	29.5901	1.24708	0.623538	−	0.781793i	$-0.285695\pi$
$563$	29.5901	1.24708	0.623538	+	0.781793i	$0.285695\pi$
$564$	0	0
$565$	15.3078	0.644004
$566$	0	0
$567$	−137.267	−5.76466
$568$	0	0
$569$	−29.6119	−1.24139	−0.620697	−	0.784051i	$-0.713150\pi$
$569$	−29.6119	−1.24139	−0.620697	+	0.784051i	$0.713150\pi$
$570$	0	0
$571$	11.7793	0.492946	0.246473	−	0.969150i	$-0.420728\pi$
$571$	11.7793	0.492946	0.246473	+	0.969150i	$0.420728\pi$
$572$	0	0
$573$	−18.5549	−0.775142
$574$	0	0
$575$	−8.16971	−0.340700
$576$	0	0
$577$	−17.4017	−0.724440	−0.362220	−	0.932093i	$-0.617981\pi$
$577$	−17.4017	−0.724440	−0.362220	+	0.932093i	$0.617981\pi$
$578$	0	0
$579$	32.8468	1.36507
$580$	0	0
$581$	38.9405	1.61552
$582$	0	0
$583$	−32.2894	−1.33729
$584$	0	0
$585$	−37.6313	−1.55586
$586$	0	0
$587$	−22.9410	−0.946877	−0.473438	−	0.880827i	$-0.656987\pi$
$587$	−22.9410	−0.946877	−0.473438	+	0.880827i	$0.656987\pi$
$588$	0	0
$589$	17.6859	0.728736
$590$	0	0
$591$	−27.4375	−1.12863
$592$	0	0
$593$	3.93575	0.161622	0.0808110	−	0.996729i	$-0.474249\pi$
$593$	3.93575	0.161622	0.0808110	+	0.996729i	$0.474249\pi$
$594$	0	0
$595$	−16.8821	−0.692098
$596$	0	0
$597$	39.9017	1.63307
$598$	0	0
$599$	31.4344	1.28437	0.642187	−	0.766548i	$-0.278027\pi$
$599$	31.4344	1.28437	0.642187	+	0.766548i	$0.278027\pi$
$600$	0	0
$601$	22.6787	0.925084	0.462542	−	0.886597i	$-0.346938\pi$
$601$	22.6787	0.925084	0.462542	+	0.886597i	$0.346938\pi$
$602$	0	0
$603$	−85.8129	−3.49457
$604$	0	0
$605$	−4.36207	−0.177344
$606$	0	0
$607$	−5.71008	−0.231765	−0.115883	−	0.993263i	$-0.536970\pi$
$607$	−5.71008	−0.231765	−0.115883	+	0.993263i	$0.536970\pi$
$608$	0	0
$609$	111.046	4.49981
$610$	0	0
$611$	31.3011	1.26631
$612$	0	0
$613$	7.85985	0.317456	0.158728	−	0.987322i	$-0.449261\pi$
$613$	7.85985	0.317456	0.158728	+	0.987322i	$0.449261\pi$
$614$	0	0
$615$	1.23944	0.0499790
$616$	0	0
$617$	−12.7505	−0.513317	−0.256658	−	0.966502i	$-0.582622\pi$
$617$	−12.7505	−0.513317	−0.256658	+	0.966502i	$0.582622\pi$
$618$	0	0
$619$	−46.4014	−1.86503	−0.932515	−	0.361130i	$-0.882391\pi$
$619$	−46.4014	−1.86503	−0.932515	+	0.361130i	$0.882391\pi$
$620$	0	0
$621$	−148.233	−5.94838
$622$	0	0
$623$	−63.0615	−2.52651
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	58.3598	2.33067
$628$	0	0
$629$	−8.90066	−0.354893
$630$	0	0
$631$	18.8448	0.750199	0.375099	−	0.926985i	$-0.377609\pi$
$631$	18.8448	0.750199	0.375099	+	0.926985i	$0.377609\pi$
$632$	0	0
$633$	−68.7268	−2.73165
$634$	0	0
$635$	7.03938	0.279349
$636$	0	0
$637$	−33.6101	−1.33168
$638$	0	0
$639$	26.9029	1.06426
$640$	0	0
$641$	5.20934	0.205757	0.102878	−	0.994694i	$-0.467195\pi$
$641$	5.20934	0.205757	0.102878	+	0.994694i	$0.467195\pi$
$642$	0	0
$643$	39.8396	1.57112	0.785560	−	0.618785i	$-0.212375\pi$
$643$	39.8396	1.57112	0.785560	+	0.618785i	$0.212375\pi$
$644$	0	0
$645$	−14.2704	−0.561897
$646$	0	0
$647$	1.37749	0.0541547	0.0270773	−	0.999633i	$-0.491380\pi$
$647$	1.37749	0.0541547	0.0270773	+	0.999633i	$0.491380\pi$
$648$	0	0
$649$	5.18216	0.203417
$650$	0	0
$651$	33.8116	1.32518
$652$	0	0
$653$	−8.05089	−0.315056	−0.157528	−	0.987515i	$-0.550352\pi$
$653$	−8.05089	−0.315056	−0.157528	+	0.987515i	$0.550352\pi$
$654$	0	0
$655$	11.9148	0.465549
$656$	0	0
$657$	1.16342	0.0453895
$658$	0	0
$659$	−18.1094	−0.705443	−0.352721	−	0.935728i	$-0.614744\pi$
$659$	−18.1094	−0.705443	−0.352721	+	0.935728i	$0.614744\pi$
$660$	0	0
$661$	7.59373	0.295362	0.147681	−	0.989035i	$-0.452819\pi$
$661$	7.59373	0.295362	0.147681	+	0.989035i	$0.452819\pi$
$662$	0	0
$663$	−67.2052	−2.61003
$664$	0	0
$665$	25.5556	0.991003
$666$	0	0
$667$	70.6680	2.73628
$668$	0	0
$669$	61.9434	2.39487
$670$	0	0
$671$	15.7936	0.609705
$672$	0	0
$673$	24.3801	0.939785	0.469893	−	0.882724i	$-0.344293\pi$
$673$	24.3801	0.939785	0.469893	+	0.882724i	$0.344293\pi$
$674$	0	0
$675$	18.1442	0.698371
$676$	0	0
$677$	8.67232	0.333304	0.166652	−	0.986016i	$-0.446704\pi$
$677$	8.67232	0.333304	0.166652	+	0.986016i	$0.446704\pi$
$678$	0	0
$679$	−26.7850	−1.02791
$680$	0	0
$681$	−73.0647	−2.79985
$682$	0	0
$683$	−24.8358	−0.950317	−0.475158	−	0.879900i	$-0.657609\pi$
$683$	−24.8358	−0.950317	−0.475158	+	0.879900i	$0.657609\pi$
$684$	0	0
$685$	−8.94271	−0.341683
$686$	0	0
$687$	−101.546	−3.87423
$688$	0	0
$689$	−56.2871	−2.14437
$690$	0	0
$691$	4.34267	0.165203	0.0826014	−	0.996583i	$-0.473677\pi$
$691$	4.34267	0.165203	0.0826014	+	0.996583i	$0.473677\pi$
$692$	0	0
$693$	82.1556	3.12083
$694$	0	0
$695$	−4.72445	−0.179209
$696$	0	0
$697$	1.62992	0.0617375
$698$	0	0
$699$	−14.0385	−0.530983
$700$	0	0
$701$	3.56255	0.134556	0.0672779	−	0.997734i	$-0.478569\pi$
$701$	3.56255	0.134556	0.0672779	+	0.997734i	$0.478569\pi$
$702$	0	0
$703$	13.4735	0.508164
$704$	0	0
$705$	−23.5096	−0.885421
$706$	0	0
$707$	−10.2809	−0.386652
$708$	0	0
$709$	32.3494	1.21491	0.607454	−	0.794355i	$-0.292191\pi$
$709$	32.3494	1.21491	0.607454	+	0.794355i	$0.292191\pi$
$710$	0	0
$711$	54.2417	2.03422
$712$	0	0
$713$	21.5172	0.805826
$714$	0	0
$715$	11.5713	0.432741
$716$	0	0
$717$	38.5617	1.44011
$718$	0	0
$719$	−19.1705	−0.714939	−0.357470	−	0.933925i	$-0.616360\pi$
$719$	−19.1705	−0.714939	−0.357470	+	0.933925i	$0.616360\pi$
$720$	0	0
$721$	−9.02077	−0.335951
$722$	0	0
$723$	35.2377	1.31051
$724$	0	0
$725$	−8.65001	−0.321253
$726$	0	0
$727$	16.3944	0.608035	0.304017	−	0.952666i	$-0.401672\pi$
$727$	16.3944	0.608035	0.304017	+	0.952666i	$0.401672\pi$
$728$	0	0
$729$	118.597	4.39250
$730$	0	0
$731$	−18.7662	−0.694094
$732$	0	0
$733$	19.1591	0.707657	0.353828	−	0.935310i	$-0.384880\pi$
$733$	19.1591	0.707657	0.353828	+	0.935310i	$0.384880\pi$
$734$	0	0
$735$	25.2438	0.931132
$736$	0	0
$737$	26.3867	0.971966
$738$	0	0
$739$	17.8584	0.656932	0.328466	−	0.944516i	$-0.393468\pi$
$739$	17.8584	0.656932	0.328466	+	0.944516i	$0.393468\pi$
$740$	0	0
$741$	101.733	3.73726
$742$	0	0
$743$	−30.0882	−1.10383	−0.551915	−	0.833901i	$-0.686103\pi$
$743$	−30.0882	−1.10383	−0.551915	+	0.833901i	$0.686103\pi$
$744$	0	0
$745$	10.1001	0.370038
$746$	0	0
$747$	−85.7330	−3.13681
$748$	0	0
$749$	1.46606	0.0535686
$750$	0	0
$751$	−38.0985	−1.39023	−0.695116	−	0.718897i	$-0.744647\pi$
$751$	−38.0985	−1.39023	−0.695116	+	0.718897i	$0.744647\pi$
$752$	0	0
$753$	82.0062	2.98847
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−8.54920	−0.310726	−0.155363	−	0.987857i	$-0.549655\pi$
$757$	−8.54920	−0.310726	−0.155363	+	0.987857i	$0.549655\pi$
$758$	0	0
$759$	71.0022	2.57722
$760$	0	0
$761$	−13.3327	−0.483309	−0.241655	−	0.970362i	$-0.577690\pi$
$761$	−13.3327	−0.483309	−0.241655	+	0.970362i	$0.577690\pi$
$762$	0	0
$763$	21.7664	0.787997
$764$	0	0
$765$	37.1684	1.34382
$766$	0	0
$767$	9.03356	0.326183
$768$	0	0
$769$	−3.18700	−0.114926	−0.0574632	−	0.998348i	$-0.518301\pi$
$769$	−3.18700	−0.114926	−0.0574632	+	0.998348i	$0.518301\pi$
$770$	0	0
$771$	−85.1303	−3.06589
$772$	0	0
$773$	5.99035	0.215458	0.107729	−	0.994180i	$-0.465642\pi$
$773$	5.99035	0.215458	0.107729	+	0.994180i	$0.465642\pi$
$774$	0	0
$775$	−2.63378	−0.0946081
$776$	0	0
$777$	25.7584	0.924078
$778$	0	0
$779$	−2.46732	−0.0884007
$780$	0	0
$781$	−8.27238	−0.296009
$782$	0	0
$783$	−156.948	−5.60885
$784$	0	0
$785$	9.52854	0.340088
$786$	0	0
$787$	−4.33648	−0.154579	−0.0772894	−	0.997009i	$-0.524627\pi$
$787$	−4.33648	−0.154579	−0.0772894	+	0.997009i	$0.524627\pi$
$788$	0	0
$789$	35.7692	1.27342
$790$	0	0
$791$	−58.2572	−2.07139
$792$	0	0
$793$	27.5315	0.977671
$794$	0	0
$795$	42.2759	1.49937
$796$	0	0
$797$	−48.3726	−1.71345	−0.856723	−	0.515777i	$-0.827503\pi$
$797$	−48.3726	−1.71345	−0.856723	+	0.515777i	$0.827503\pi$
$798$	0	0
$799$	−30.9161	−1.09373
$800$	0	0
$801$	138.839	4.90564
$802$	0	0
$803$	−0.357742	−0.0126244
$804$	0	0
$805$	31.0916	1.09584
$806$	0	0
$807$	−53.2689	−1.87515
$808$	0	0
$809$	−39.1516	−1.37650	−0.688248	−	0.725476i	$-0.741620\pi$
$809$	−39.1516	−1.37650	−0.688248	+	0.725476i	$0.741620\pi$
$810$	0	0
$811$	−1.41593	−0.0497201	−0.0248601	−	0.999691i	$-0.507914\pi$
$811$	−1.41593	−0.0497201	−0.0248601	+	0.999691i	$0.507914\pi$
$812$	0	0
$813$	−106.607	−3.73887
$814$	0	0
$815$	−16.7786	−0.587728
$816$	0	0
$817$	28.4077	0.993860
$818$	0	0
$819$	143.214	5.00431
$820$	0	0
$821$	−8.35788	−0.291692	−0.145846	−	0.989307i	$-0.546590\pi$
$821$	−8.35788	−0.291692	−0.145846	+	0.989307i	$0.546590\pi$
$822$	0	0
$823$	−10.1203	−0.352770	−0.176385	−	0.984321i	$-0.556440\pi$
$823$	−10.1203	−0.352770	−0.176385	+	0.984321i	$0.556440\pi$
$824$	0	0
$825$	−8.69091	−0.302579
$826$	0	0
$827$	−45.3023	−1.57531	−0.787657	−	0.616114i	$-0.788706\pi$
$827$	−45.3023	−1.57531	−0.787657	+	0.616114i	$0.788706\pi$
$828$	0	0
$829$	−43.4382	−1.50867	−0.754336	−	0.656488i	$-0.772041\pi$
$829$	−43.4382	−1.50867	−0.754336	+	0.656488i	$0.772041\pi$
$830$	0	0
$831$	1.07413	0.0372613
$832$	0	0
$833$	33.1967	1.15020
$834$	0	0
$835$	21.4191	0.741239
$836$	0	0
$837$	−47.7879	−1.65179
$838$	0	0
$839$	−37.9516	−1.31023	−0.655117	−	0.755528i	$-0.727381\pi$
$839$	−37.9516	−1.31023	−0.655117	+	0.755528i	$0.727381\pi$
$840$	0	0
$841$	45.8226	1.58009
$842$	0	0
$843$	−65.0054	−2.23890
$844$	0	0
$845$	7.17108	0.246693
$846$	0	0
$847$	16.6008	0.570412
$848$	0	0
$849$	−81.7391	−2.80528
$850$	0	0
$851$	16.3923	0.561921
$852$	0	0
$853$	−19.9158	−0.681903	−0.340952	−	0.940081i	$-0.610749\pi$
$853$	−19.9158	−0.681903	−0.340952	+	0.940081i	$0.610749\pi$
$854$	0	0
$855$	−56.2643	−1.92420
$856$	0	0
$857$	10.1274	0.345944	0.172972	−	0.984927i	$-0.444663\pi$
$857$	10.1274	0.345944	0.172972	+	0.984927i	$0.444663\pi$
$858$	0	0
$859$	20.5785	0.702131	0.351065	−	0.936351i	$-0.385819\pi$
$859$	20.5785	0.702131	0.351065	+	0.936351i	$0.385819\pi$
$860$	0	0
$861$	−4.71696	−0.160754
$862$	0	0
$863$	−19.8318	−0.675083	−0.337542	−	0.941311i	$-0.609595\pi$
$863$	−19.8318	−0.675083	−0.337542	+	0.941311i	$0.609595\pi$
$864$	0	0
$865$	1.47999	0.0503210
$866$	0	0
$867$	9.03319	0.306783
$868$	0	0
$869$	−16.6788	−0.565790
$870$	0	0
$871$	45.9974	1.55856
$872$	0	0
$873$	58.9710	1.99587
$874$	0	0
$875$	−3.80572	−0.128657
$876$	0	0
$877$	−6.14680	−0.207563	−0.103781	−	0.994600i	$-0.533094\pi$
$877$	−6.14680	−0.207563	−0.103781	+	0.994600i	$0.533094\pi$
$878$	0	0
$879$	85.3445	2.87860
$880$	0	0
$881$	−44.3491	−1.49416	−0.747080	−	0.664734i	$-0.768545\pi$
$881$	−44.3491	−1.49416	−0.747080	+	0.664734i	$0.768545\pi$
$882$	0	0
$883$	−27.9200	−0.939582	−0.469791	−	0.882778i	$-0.655671\pi$
$883$	−27.9200	−0.939582	−0.469791	+	0.882778i	$0.655671\pi$
$884$	0	0
$885$	−6.78490	−0.228072
$886$	0	0
$887$	6.93478	0.232847	0.116424	−	0.993200i	$-0.462857\pi$
$887$	6.93478	0.232847	0.116424	+	0.993200i	$0.462857\pi$
$888$	0	0
$889$	−26.7899	−0.898505
$890$	0	0
$891$	−92.9276	−3.11319
$892$	0	0
$893$	46.7998	1.56610
$894$	0	0
$895$	6.08516	0.203404
$896$	0	0
$897$	123.771	4.13261
$898$	0	0
$899$	22.7822	0.759829
$900$	0	0
$901$	55.5947	1.85213
$902$	0	0
$903$	54.3092	1.80730
$904$	0	0
$905$	−25.4550	−0.846151
$906$	0	0
$907$	−22.3509	−0.742149	−0.371075	−	0.928603i	$-0.621011\pi$
$907$	−22.3509	−0.742149	−0.371075	+	0.928603i	$0.621011\pi$
$908$	0	0
$909$	22.6348	0.750749
$910$	0	0
$911$	35.4959	1.17603	0.588015	−	0.808850i	$-0.299909\pi$
$911$	35.4959	1.17603	0.588015	+	0.808850i	$0.299909\pi$
$912$	0	0
$913$	26.3621	0.872459
$914$	0	0
$915$	−20.6783	−0.683602
$916$	0	0
$917$	−45.3443	−1.49740
$918$	0	0
$919$	−44.9349	−1.48226	−0.741132	−	0.671359i	$-0.765711\pi$
$919$	−44.9349	−1.48226	−0.741132	+	0.671359i	$0.765711\pi$
$920$	0	0
$921$	61.9114	2.04005
$922$	0	0
$923$	−14.4205	−0.474655
$924$	0	0
$925$	−2.00647	−0.0659724
$926$	0	0
$927$	19.8605	0.652305
$928$	0	0
$929$	21.6642	0.710779	0.355390	−	0.934718i	$-0.384348\pi$
$929$	21.6642	0.710779	0.355390	+	0.934718i	$0.384348\pi$
$930$	0	0
$931$	−50.2521	−1.64695
$932$	0	0
$933$	49.6161	1.62436
$934$	0	0
$935$	−11.4289	−0.373766
$936$	0	0
$937$	13.3315	0.435521	0.217761	−	0.976002i	$-0.430125\pi$
$937$	13.3315	0.435521	0.217761	+	0.976002i	$0.430125\pi$
$938$	0	0
$939$	2.95907	0.0965657
$940$	0	0
$941$	−6.62779	−0.216060	−0.108030	−	0.994148i	$-0.534454\pi$
$941$	−6.62779	−0.216060	−0.108030	+	0.994148i	$0.534454\pi$
$942$	0	0
$943$	−3.00181	−0.0977522
$944$	0	0
$945$	−69.0518	−2.24626
$946$	0	0
$947$	−20.0496	−0.651523	−0.325762	−	0.945452i	$-0.605621\pi$
$947$	−20.0496	−0.651523	−0.325762	+	0.945452i	$0.605621\pi$
$948$	0	0
$949$	−0.623618	−0.0202435
$950$	0	0
$951$	−77.3849	−2.50938
$952$	0	0
$953$	−13.5161	−0.437831	−0.218915	−	0.975744i	$-0.570252\pi$
$953$	−13.5161	−0.437831	−0.218915	+	0.975744i	$0.570252\pi$
$954$	0	0
$955$	−5.50059	−0.177995
$956$	0	0
$957$	75.1764	2.43011
$958$	0	0
$959$	34.0335	1.09900
$960$	0	0
$961$	−24.0632	−0.776233
$962$	0	0
$963$	−3.22774	−0.104012
$964$	0	0
$965$	9.73741	0.313458
$966$	0	0
$967$	−26.7288	−0.859541	−0.429770	−	0.902938i	$-0.641406\pi$
$967$	−26.7288	−0.859541	−0.429770	+	0.902938i	$0.641406\pi$
$968$	0	0
$969$	−100.482	−3.22794
$970$	0	0
$971$	50.9996	1.63665	0.818327	−	0.574753i	$-0.194902\pi$
$971$	50.9996	1.63665	0.818327	+	0.574753i	$0.194902\pi$
$972$	0	0
$973$	17.9799	0.576410
$974$	0	0
$975$	−15.1500	−0.485189
$976$	0	0
$977$	38.8774	1.24380	0.621899	−	0.783098i	$-0.286362\pi$
$977$	38.8774	1.24380	0.621899	+	0.783098i	$0.286362\pi$
$978$	0	0
$979$	−42.6917	−1.36443
$980$	0	0
$981$	−47.9219	−1.53003
$982$	0	0
$983$	6.39114	0.203846	0.101923	−	0.994792i	$-0.467501\pi$
$983$	6.39114	0.203846	0.101923	+	0.994792i	$0.467501\pi$
$984$	0	0
$985$	−8.13384	−0.259166
$986$	0	0
$987$	89.4709	2.84789
$988$	0	0
$989$	34.5616	1.09900
$990$	0	0
$991$	−42.5343	−1.35115	−0.675573	−	0.737293i	$-0.736104\pi$
$991$	−42.5343	−1.35115	−0.675573	+	0.737293i	$0.736104\pi$
$992$	0	0
$993$	15.7534	0.499920
$994$	0	0
$995$	11.8288	0.375000
$996$	0	0
$997$	9.04748	0.286537	0.143268	−	0.989684i	$-0.454239\pi$
$997$	9.04748	0.286537	0.143268	+	0.989684i	$0.454239\pi$
$998$	0	0
$999$	−36.4059	−1.15183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.19	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.19	✓	19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37325 q^{3} +1.00000 q^{5} -3.80572 q^{7} +8.37885 q^{9} +O(q^{10})\)	\(q+3.37325 q^{3} +1.00000 q^{5} -3.80572 q^{7} +8.37885 q^{9} -2.57642 q^{11} -4.49122 q^{13} +3.37325 q^{15} +4.43598 q^{17} -6.71504 q^{19} -12.8377 q^{21} -8.16971 q^{23} +1.00000 q^{25} +18.1442 q^{27} -8.65001 q^{29} -2.63378 q^{31} -8.69091 q^{33} -3.80572 q^{35} -2.00647 q^{37} -15.1500 q^{39} +0.367431 q^{41} -4.23046 q^{43} +8.37885 q^{45} -6.96940 q^{47} +7.48352 q^{49} +14.9637 q^{51} +12.5327 q^{53} -2.57642 q^{55} -22.6515 q^{57} -2.01138 q^{59} -6.13006 q^{61} -31.8876 q^{63} -4.49122 q^{65} -10.2416 q^{67} -27.5585 q^{69} +3.21081 q^{71} +0.138853 q^{73} +3.37325 q^{75} +9.80513 q^{77} +6.47365 q^{79} +36.0685 q^{81} -10.2321 q^{83} +4.43598 q^{85} -29.1787 q^{87} +16.5702 q^{89} +17.0923 q^{91} -8.88441 q^{93} -6.71504 q^{95} +7.03808 q^{97} -21.5874 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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