LMFDB - Embedded newform 6040.2.a.s.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:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+2.40154 q^{3} +1.00000 q^{5} +4.37326 q^{7} +2.76742 q^{9} +O(q^{10})$	$q+2.40154 q^{3} +1.00000 q^{5} +4.37326 q^{7} +2.76742 q^{9} +1.38559 q^{11} -2.27210 q^{13} +2.40154 q^{15} +5.99857 q^{17} -4.33791 q^{19} +10.5026 q^{21} -2.89546 q^{23} +1.00000 q^{25} -0.558557 q^{27} -8.25947 q^{29} +1.81268 q^{31} +3.32755 q^{33} +4.37326 q^{35} -1.37234 q^{37} -5.45654 q^{39} +8.97470 q^{41} +8.79073 q^{43} +2.76742 q^{45} -3.33325 q^{47} +12.1254 q^{49} +14.4058 q^{51} +10.2665 q^{53} +1.38559 q^{55} -10.4177 q^{57} +6.09212 q^{59} -1.58311 q^{61} +12.1026 q^{63} -2.27210 q^{65} +2.09277 q^{67} -6.95359 q^{69} +12.7702 q^{71} +15.1595 q^{73} +2.40154 q^{75} +6.05953 q^{77} +13.0724 q^{79} -9.64365 q^{81} -1.04119 q^{83} +5.99857 q^{85} -19.8355 q^{87} -14.9386 q^{89} -9.93648 q^{91} +4.35324 q^{93} -4.33791 q^{95} -12.1406 q^{97} +3.83449 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * 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
$3$	2.40154	1.38653	0.693266	−	0.720682i	$-0.256171\pi$
$3$	2.40154	1.38653	0.693266	+	0.720682i	$0.256171\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.37326	1.65294	0.826469	−	0.562982i	$-0.190346\pi$
$7$	4.37326	1.65294	0.826469	+	0.562982i	$0.190346\pi$
$8$	0	0
$9$	2.76742	0.922473
$10$	0	0
$11$	1.38559	0.417770	0.208885	−	0.977940i	$-0.433017\pi$
$11$	1.38559	0.417770	0.208885	+	0.977940i	$0.433017\pi$
$12$	0	0
$13$	−2.27210	−0.630166	−0.315083	−	0.949064i	$-0.602032\pi$
$13$	−2.27210	−0.630166	−0.315083	+	0.949064i	$0.602032\pi$
$14$	0	0
$15$	2.40154	0.620076
$16$	0	0
$17$	5.99857	1.45487	0.727433	−	0.686179i	$-0.240713\pi$
$17$	5.99857	1.45487	0.727433	+	0.686179i	$0.240713\pi$
$18$	0	0
$19$	−4.33791	−0.995185	−0.497593	−	0.867411i	$-0.665783\pi$
$19$	−4.33791	−0.995185	−0.497593	+	0.867411i	$0.665783\pi$
$20$	0	0
$21$	10.5026	2.29185
$22$	0	0
$23$	−2.89546	−0.603746	−0.301873	−	0.953348i	$-0.597612\pi$
$23$	−2.89546	−0.603746	−0.301873	+	0.953348i	$0.597612\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.558557	−0.107494
$28$	0	0
$29$	−8.25947	−1.53375	−0.766873	−	0.641799i	$-0.778188\pi$
$29$	−8.25947	−1.53375	−0.766873	+	0.641799i	$0.778188\pi$
$30$	0	0
$31$	1.81268	0.325568	0.162784	−	0.986662i	$-0.447953\pi$
$31$	1.81268	0.325568	0.162784	+	0.986662i	$0.447953\pi$
$32$	0	0
$33$	3.32755	0.579251
$34$	0	0
$35$	4.37326	0.739216
$36$	0	0
$37$	−1.37234	−0.225611	−0.112806	−	0.993617i	$-0.535984\pi$
$37$	−1.37234	−0.225611	−0.112806	+	0.993617i	$0.535984\pi$
$38$	0	0
$39$	−5.45654	−0.873746
$40$	0	0
$41$	8.97470	1.40161	0.700807	−	0.713351i	$-0.252824\pi$
$41$	8.97470	1.40161	0.700807	+	0.713351i	$0.252824\pi$
$42$	0	0
$43$	8.79073	1.34057	0.670287	−	0.742102i	$-0.266171\pi$
$43$	8.79073	1.34057	0.670287	+	0.742102i	$0.266171\pi$
$44$	0	0
$45$	2.76742	0.412542
$46$	0	0
$47$	−3.33325	−0.486205	−0.243103	−	0.970001i	$-0.578165\pi$
$47$	−3.33325	−0.486205	−0.243103	+	0.970001i	$0.578165\pi$
$48$	0	0
$49$	12.1254	1.73220
$50$	0	0
$51$	14.4058	2.01722
$52$	0	0
$53$	10.2665	1.41022	0.705109	−	0.709099i	$-0.250898\pi$
$53$	10.2665	1.41022	0.705109	+	0.709099i	$0.250898\pi$
$54$	0	0
$55$	1.38559	0.186832
$56$	0	0
$57$	−10.4177	−1.37986
$58$	0	0
$59$	6.09212	0.793127	0.396563	−	0.918007i	$-0.370203\pi$
$59$	6.09212	0.793127	0.396563	+	0.918007i	$0.370203\pi$
$60$	0	0
$61$	−1.58311	−0.202696	−0.101348	−	0.994851i	$-0.532316\pi$
$61$	−1.58311	−0.202696	−0.101348	+	0.994851i	$0.532316\pi$
$62$	0	0
$63$	12.1026	1.52479
$64$	0	0
$65$	−2.27210	−0.281819
$66$	0	0
$67$	2.09277	0.255672	0.127836	−	0.991795i	$-0.459197\pi$
$67$	2.09277	0.255672	0.127836	+	0.991795i	$0.459197\pi$
$68$	0	0
$69$	−6.95359	−0.837113
$70$	0	0
$71$	12.7702	1.51555	0.757774	−	0.652517i	$-0.226287\pi$
$71$	12.7702	1.51555	0.757774	+	0.652517i	$0.226287\pi$
$72$	0	0
$73$	15.1595	1.77428	0.887142	−	0.461496i	$-0.152687\pi$
$73$	15.1595	1.77428	0.887142	+	0.461496i	$0.152687\pi$
$74$	0	0
$75$	2.40154	0.277307
$76$	0	0
$77$	6.05953	0.690547
$78$	0	0
$79$	13.0724	1.47076	0.735378	−	0.677657i	$-0.237005\pi$
$79$	13.0724	1.47076	0.735378	+	0.677657i	$0.237005\pi$
$80$	0	0
$81$	−9.64365	−1.07152
$82$	0	0
$83$	−1.04119	−0.114285	−0.0571426	−	0.998366i	$-0.518199\pi$
$83$	−1.04119	−0.114285	−0.0571426	+	0.998366i	$0.518199\pi$
$84$	0	0
$85$	5.99857	0.650636
$86$	0	0
$87$	−19.8355	−2.12659
$88$	0	0
$89$	−14.9386	−1.58348	−0.791742	−	0.610855i	$-0.790826\pi$
$89$	−14.9386	−1.58348	−0.791742	+	0.610855i	$0.790826\pi$
$90$	0	0
$91$	−9.93648	−1.04163
$92$	0	0
$93$	4.35324	0.451410
$94$	0	0
$95$	−4.33791	−0.445060
$96$	0	0
$97$	−12.1406	−1.23269	−0.616345	−	0.787476i	$-0.711387\pi$
$97$	−12.1406	−1.23269	−0.616345	+	0.787476i	$0.711387\pi$
$98$	0	0
$99$	3.83449	0.385381
$100$	0	0
$101$	−11.4175	−1.13609	−0.568044	−	0.822998i	$-0.692300\pi$
$101$	−11.4175	−1.13609	−0.568044	+	0.822998i	$0.692300\pi$
$102$	0	0
$103$	2.28203	0.224855	0.112428	−	0.993660i	$-0.464137\pi$
$103$	2.28203	0.224855	0.112428	+	0.993660i	$0.464137\pi$
$104$	0	0
$105$	10.5026	1.02495
$106$	0	0
$107$	−6.88872	−0.665957	−0.332979	−	0.942934i	$-0.608054\pi$
$107$	−6.88872	−0.665957	−0.332979	+	0.942934i	$0.608054\pi$
$108$	0	0
$109$	−19.8503	−1.90131	−0.950656	−	0.310247i	$-0.899588\pi$
$109$	−19.8503	−1.90131	−0.950656	+	0.310247i	$0.899588\pi$
$110$	0	0
$111$	−3.29573	−0.312817
$112$	0	0
$113$	10.8344	1.01922	0.509609	−	0.860406i	$-0.329790\pi$
$113$	10.8344	1.01922	0.509609	+	0.860406i	$0.329790\pi$
$114$	0	0
$115$	−2.89546	−0.270003
$116$	0	0
$117$	−6.28784	−0.581311
$118$	0	0
$119$	26.2333	2.40480
$120$	0	0
$121$	−9.08015	−0.825468
$122$	0	0
$123$	21.5532	1.94338
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.3421	1.09518	0.547591	−	0.836746i	$-0.315545\pi$
$127$	12.3421	1.09518	0.547591	+	0.836746i	$0.315545\pi$
$128$	0	0
$129$	21.1113	1.85875
$130$	0	0
$131$	−0.104911	−0.00916614	−0.00458307	−	0.999989i	$-0.501459\pi$
$131$	−0.104911	−0.00916614	−0.00458307	+	0.999989i	$0.501459\pi$
$132$	0	0
$133$	−18.9708	−1.64498
$134$	0	0
$135$	−0.558557	−0.0480729
$136$	0	0
$137$	7.66366	0.654751	0.327375	−	0.944894i	$-0.393836\pi$
$137$	7.66366	0.654751	0.327375	+	0.944894i	$0.393836\pi$
$138$	0	0
$139$	−16.5039	−1.39984	−0.699922	−	0.714219i	$-0.746782\pi$
$139$	−16.5039	−1.39984	−0.699922	+	0.714219i	$0.746782\pi$
$140$	0	0
$141$	−8.00496	−0.674139
$142$	0	0
$143$	−3.14818	−0.263264
$144$	0	0
$145$	−8.25947	−0.685912
$146$	0	0
$147$	29.1198	2.40176
$148$	0	0
$149$	9.19861	0.753579	0.376790	−	0.926299i	$-0.377028\pi$
$149$	9.19861	0.753579	0.376790	+	0.926299i	$0.377028\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	16.6005	1.34207
$154$	0	0
$155$	1.81268	0.145598
$156$	0	0
$157$	7.93216	0.633055	0.316528	−	0.948583i	$-0.397483\pi$
$157$	7.93216	0.633055	0.316528	+	0.948583i	$0.397483\pi$
$158$	0	0
$159$	24.6556	1.95531
$160$	0	0
$161$	−12.6626	−0.997955
$162$	0	0
$163$	−4.45116	−0.348642	−0.174321	−	0.984689i	$-0.555773\pi$
$163$	−4.45116	−0.348642	−0.174321	+	0.984689i	$0.555773\pi$
$164$	0	0
$165$	3.32755	0.259049
$166$	0	0
$167$	−23.0131	−1.78081	−0.890405	−	0.455170i	$-0.849578\pi$
$167$	−23.0131	−1.78081	−0.890405	+	0.455170i	$0.849578\pi$
$168$	0	0
$169$	−7.83758	−0.602891
$170$	0	0
$171$	−12.0048	−0.918031
$172$	0	0
$173$	−14.3308	−1.08955	−0.544774	−	0.838583i	$-0.683385\pi$
$173$	−14.3308	−1.08955	−0.544774	+	0.838583i	$0.683385\pi$
$174$	0	0
$175$	4.37326	0.330588
$176$	0	0
$177$	14.6305	1.09970
$178$	0	0
$179$	−3.41792	−0.255468	−0.127734	−	0.991808i	$-0.540770\pi$
$179$	−3.41792	−0.255468	−0.127734	+	0.991808i	$0.540770\pi$
$180$	0	0
$181$	9.18457	0.682684	0.341342	−	0.939939i	$-0.389119\pi$
$181$	9.18457	0.682684	0.341342	+	0.939939i	$0.389119\pi$
$182$	0	0
$183$	−3.80191	−0.281045
$184$	0	0
$185$	−1.37234	−0.100896
$186$	0	0
$187$	8.31153	0.607799
$188$	0	0
$189$	−2.44272	−0.177681
$190$	0	0
$191$	4.84230	0.350377	0.175188	−	0.984535i	$-0.443947\pi$
$191$	4.84230	0.350377	0.175188	+	0.984535i	$0.443947\pi$
$192$	0	0
$193$	17.0375	1.22639	0.613194	−	0.789932i	$-0.289884\pi$
$193$	17.0375	1.22639	0.613194	+	0.789932i	$0.289884\pi$
$194$	0	0
$195$	−5.45654	−0.390751
$196$	0	0
$197$	−4.16488	−0.296735	−0.148368	−	0.988932i	$-0.547402\pi$
$197$	−4.16488	−0.296735	−0.148368	+	0.988932i	$0.547402\pi$
$198$	0	0
$199$	−24.2391	−1.71827	−0.859133	−	0.511752i	$-0.828997\pi$
$199$	−24.2391	−1.71827	−0.859133	+	0.511752i	$0.828997\pi$
$200$	0	0
$201$	5.02588	0.354498
$202$	0	0
$203$	−36.1208	−2.53519
$204$	0	0
$205$	8.97470	0.626820
$206$	0	0
$207$	−8.01296	−0.556939
$208$	0	0
$209$	−6.01055	−0.415758
$210$	0	0
$211$	23.1103	1.59098	0.795491	−	0.605965i	$-0.207213\pi$
$211$	23.1103	1.59098	0.795491	+	0.605965i	$0.207213\pi$
$212$	0	0
$213$	30.6683	2.10136
$214$	0	0
$215$	8.79073	0.599523
$216$	0	0
$217$	7.92734	0.538143
$218$	0	0
$219$	36.4062	2.46010
$220$	0	0
$221$	−13.6293	−0.916808
$222$	0	0
$223$	4.18603	0.280317	0.140159	−	0.990129i	$-0.455239\pi$
$223$	4.18603	0.280317	0.140159	+	0.990129i	$0.455239\pi$
$224$	0	0
$225$	2.76742	0.184495
$226$	0	0
$227$	−6.43784	−0.427294	−0.213647	−	0.976911i	$-0.568534\pi$
$227$	−6.43784	−0.427294	−0.213647	+	0.976911i	$0.568534\pi$
$228$	0	0
$229$	−22.4370	−1.48268	−0.741340	−	0.671129i	$-0.765809\pi$
$229$	−22.4370	−1.48268	−0.741340	+	0.671129i	$0.765809\pi$
$230$	0	0
$231$	14.5522	0.957467
$232$	0	0
$233$	−12.1980	−0.799119	−0.399559	−	0.916707i	$-0.630837\pi$
$233$	−12.1980	−0.799119	−0.399559	+	0.916707i	$0.630837\pi$
$234$	0	0
$235$	−3.33325	−0.217437
$236$	0	0
$237$	31.3939	2.03925
$238$	0	0
$239$	−1.60613	−0.103892	−0.0519459	−	0.998650i	$-0.516542\pi$
$239$	−1.60613	−0.103892	−0.0519459	+	0.998650i	$0.516542\pi$
$240$	0	0
$241$	−1.61794	−0.104220	−0.0521102	−	0.998641i	$-0.516595\pi$
$241$	−1.61794	−0.104220	−0.0521102	+	0.998641i	$0.516595\pi$
$242$	0	0
$243$	−21.4840	−1.37820
$244$	0	0
$245$	12.1254	0.774665
$246$	0	0
$247$	9.85616	0.627132
$248$	0	0
$249$	−2.50046	−0.158460
$250$	0	0
$251$	−28.6208	−1.80653	−0.903264	−	0.429086i	$-0.858836\pi$
$251$	−28.6208	−1.80653	−0.903264	+	0.429086i	$0.858836\pi$
$252$	0	0
$253$	−4.01191	−0.252227
$254$	0	0
$255$	14.4058	0.902128
$256$	0	0
$257$	17.9824	1.12171	0.560855	−	0.827914i	$-0.310473\pi$
$257$	17.9824	1.12171	0.560855	+	0.827914i	$0.310473\pi$
$258$	0	0
$259$	−6.00160	−0.372921
$260$	0	0
$261$	−22.8574	−1.41484
$262$	0	0
$263$	8.50226	0.524272	0.262136	−	0.965031i	$-0.415573\pi$
$263$	8.50226	0.524272	0.262136	+	0.965031i	$0.415573\pi$
$264$	0	0
$265$	10.2665	0.630668
$266$	0	0
$267$	−35.8756	−2.19555
$268$	0	0
$269$	−14.4836	−0.883082	−0.441541	−	0.897241i	$-0.645568\pi$
$269$	−14.4836	−0.883082	−0.441541	+	0.897241i	$0.645568\pi$
$270$	0	0
$271$	23.2895	1.41474	0.707369	−	0.706844i	$-0.249882\pi$
$271$	23.2895	1.41474	0.707369	+	0.706844i	$0.249882\pi$
$272$	0	0
$273$	−23.8629	−1.44425
$274$	0	0
$275$	1.38559	0.0835540
$276$	0	0
$277$	−22.6702	−1.36212	−0.681059	−	0.732229i	$-0.738480\pi$
$277$	−22.6702	−1.36212	−0.681059	+	0.732229i	$0.738480\pi$
$278$	0	0
$279$	5.01645	0.300327
$280$	0	0
$281$	10.9152	0.651145	0.325572	−	0.945517i	$-0.394443\pi$
$281$	10.9152	0.651145	0.325572	+	0.945517i	$0.394443\pi$
$282$	0	0
$283$	2.51430	0.149460	0.0747298	−	0.997204i	$-0.476191\pi$
$283$	2.51430	0.149460	0.0747298	+	0.997204i	$0.476191\pi$
$284$	0	0
$285$	−10.4177	−0.617091
$286$	0	0
$287$	39.2487	2.31678
$288$	0	0
$289$	18.9828	1.11664
$290$	0	0
$291$	−29.1562	−1.70916
$292$	0	0
$293$	0.999784	0.0584080	0.0292040	−	0.999573i	$-0.490703\pi$
$293$	0.999784	0.0584080	0.0292040	+	0.999573i	$0.490703\pi$
$294$	0	0
$295$	6.09212	0.354697
$296$	0	0
$297$	−0.773928	−0.0449079
$298$	0	0
$299$	6.57877	0.380460
$300$	0	0
$301$	38.4442	2.21588
$302$	0	0
$303$	−27.4197	−1.57522
$304$	0	0
$305$	−1.58311	−0.0906485
$306$	0	0
$307$	−21.1706	−1.20827	−0.604134	−	0.796883i	$-0.706481\pi$
$307$	−21.1706	−1.20827	−0.604134	+	0.796883i	$0.706481\pi$
$308$	0	0
$309$	5.48040	0.311769
$310$	0	0
$311$	−34.1035	−1.93383	−0.966915	−	0.255099i	$-0.917892\pi$
$311$	−34.1035	−1.93383	−0.966915	+	0.255099i	$0.917892\pi$
$312$	0	0
$313$	−14.2795	−0.807127	−0.403564	−	0.914952i	$-0.632229\pi$
$313$	−14.2795	−0.807127	−0.403564	+	0.914952i	$0.632229\pi$
$314$	0	0
$315$	12.1026	0.681907
$316$	0	0
$317$	22.9499	1.28899	0.644497	−	0.764607i	$-0.277067\pi$
$317$	22.9499	1.28899	0.644497	+	0.764607i	$0.277067\pi$
$318$	0	0
$319$	−11.4442	−0.640753
$320$	0	0
$321$	−16.5436	−0.923372
$322$	0	0
$323$	−26.0213	−1.44786
$324$	0	0
$325$	−2.27210	−0.126033
$326$	0	0
$327$	−47.6713	−2.63623
$328$	0	0
$329$	−14.5772	−0.803667
$330$	0	0
$331$	−27.3526	−1.50344	−0.751718	−	0.659484i	$-0.770775\pi$
$331$	−27.3526	−1.50344	−0.751718	+	0.659484i	$0.770775\pi$
$332$	0	0
$333$	−3.79784	−0.208120
$334$	0	0
$335$	2.09277	0.114340
$336$	0	0
$337$	−22.7562	−1.23961	−0.619803	−	0.784757i	$-0.712788\pi$
$337$	−22.7562	−1.23961	−0.619803	+	0.784757i	$0.712788\pi$
$338$	0	0
$339$	26.0194	1.41318
$340$	0	0
$341$	2.51163	0.136012
$342$	0	0
$343$	22.4148	1.21029
$344$	0	0
$345$	−6.95359	−0.374369
$346$	0	0
$347$	26.8193	1.43974	0.719868	−	0.694111i	$-0.244202\pi$
$347$	26.8193	1.43974	0.719868	+	0.694111i	$0.244202\pi$
$348$	0	0
$349$	−19.0337	−1.01885	−0.509424	−	0.860516i	$-0.670142\pi$
$349$	−19.0337	−1.01885	−0.509424	+	0.860516i	$0.670142\pi$
$350$	0	0
$351$	1.26910	0.0677393
$352$	0	0
$353$	31.9301	1.69947	0.849733	−	0.527213i	$-0.176763\pi$
$353$	31.9301	1.69947	0.849733	+	0.527213i	$0.176763\pi$
$354$	0	0
$355$	12.7702	0.677774
$356$	0	0
$357$	63.0005	3.33434
$358$	0	0
$359$	−28.7814	−1.51903	−0.759513	−	0.650492i	$-0.774563\pi$
$359$	−28.7814	−1.51903	−0.759513	+	0.650492i	$0.774563\pi$
$360$	0	0
$361$	−0.182515	−0.00960607
$362$	0	0
$363$	−21.8064	−1.14454
$364$	0	0
$365$	15.1595	0.793484
$366$	0	0
$367$	20.8361	1.08764	0.543819	−	0.839203i	$-0.316978\pi$
$367$	20.8361	1.08764	0.543819	+	0.839203i	$0.316978\pi$
$368$	0	0
$369$	24.8368	1.29295
$370$	0	0
$371$	44.8983	2.33100
$372$	0	0
$373$	−22.1765	−1.14826	−0.574128	−	0.818766i	$-0.694659\pi$
$373$	−22.1765	−1.14826	−0.574128	+	0.818766i	$0.694659\pi$
$374$	0	0
$375$	2.40154	0.124015
$376$	0	0
$377$	18.7663	0.966515
$378$	0	0
$379$	−25.4780	−1.30872	−0.654358	−	0.756185i	$-0.727061\pi$
$379$	−25.4780	−1.30872	−0.654358	+	0.756185i	$0.727061\pi$
$380$	0	0
$381$	29.6401	1.51851
$382$	0	0
$383$	−6.34548	−0.324239	−0.162119	−	0.986771i	$-0.551833\pi$
$383$	−6.34548	−0.324239	−0.162119	+	0.986771i	$0.551833\pi$
$384$	0	0
$385$	6.05953	0.308822
$386$	0	0
$387$	24.3276	1.23664
$388$	0	0
$389$	28.5911	1.44962	0.724812	−	0.688947i	$-0.241927\pi$
$389$	28.5911	1.44962	0.724812	+	0.688947i	$0.241927\pi$
$390$	0	0
$391$	−17.3686	−0.878370
$392$	0	0
$393$	−0.251949	−0.0127091
$394$	0	0
$395$	13.0724	0.657742
$396$	0	0
$397$	18.1488	0.910864	0.455432	−	0.890271i	$-0.349485\pi$
$397$	18.1488	0.910864	0.455432	+	0.890271i	$0.349485\pi$
$398$	0	0
$399$	−45.5593	−2.28082
$400$	0	0
$401$	26.4923	1.32296	0.661480	−	0.749963i	$-0.269929\pi$
$401$	26.4923	1.32296	0.661480	+	0.749963i	$0.269929\pi$
$402$	0	0
$403$	−4.11859	−0.205162
$404$	0	0
$405$	−9.64365	−0.479197
$406$	0	0
$407$	−1.90149	−0.0942535
$408$	0	0
$409$	22.8384	1.12929	0.564643	−	0.825335i	$-0.309014\pi$
$409$	22.8384	1.12929	0.564643	+	0.825335i	$0.309014\pi$
$410$	0	0
$411$	18.4046	0.907833
$412$	0	0
$413$	26.6425	1.31099
$414$	0	0
$415$	−1.04119	−0.0511099
$416$	0	0
$417$	−39.6349	−1.94093
$418$	0	0
$419$	9.19823	0.449363	0.224682	−	0.974432i	$-0.427866\pi$
$419$	9.19823	0.449363	0.224682	+	0.974432i	$0.427866\pi$
$420$	0	0
$421$	15.7106	0.765687	0.382843	−	0.923813i	$-0.374945\pi$
$421$	15.7106	0.765687	0.382843	+	0.923813i	$0.374945\pi$
$422$	0	0
$423$	−9.22451	−0.448511
$424$	0	0
$425$	5.99857	0.290973
$426$	0	0
$427$	−6.92335	−0.335044
$428$	0	0
$429$	−7.56051	−0.365025
$430$	0	0
$431$	−21.0157	−1.01229	−0.506146	−	0.862448i	$-0.668930\pi$
$431$	−21.0157	−1.01229	−0.506146	+	0.862448i	$0.668930\pi$
$432$	0	0
$433$	22.3069	1.07200	0.536001	−	0.844217i	$-0.319934\pi$
$433$	22.3069	1.07200	0.536001	+	0.844217i	$0.319934\pi$
$434$	0	0
$435$	−19.8355	−0.951039
$436$	0	0
$437$	12.5603	0.600839
$438$	0	0
$439$	−18.4527	−0.880699	−0.440350	−	0.897826i	$-0.645146\pi$
$439$	−18.4527	−0.880699	−0.440350	+	0.897826i	$0.645146\pi$
$440$	0	0
$441$	33.5561	1.59791
$442$	0	0
$443$	33.7222	1.60219	0.801095	−	0.598537i	$-0.204251\pi$
$443$	33.7222	1.60219	0.801095	+	0.598537i	$0.204251\pi$
$444$	0	0
$445$	−14.9386	−0.708156
$446$	0	0
$447$	22.0909	1.04486
$448$	0	0
$449$	−33.2412	−1.56875	−0.784374	−	0.620289i	$-0.787015\pi$
$449$	−33.2412	−1.56875	−0.784374	+	0.620289i	$0.787015\pi$
$450$	0	0
$451$	12.4352	0.585552
$452$	0	0
$453$	2.40154	0.112834
$454$	0	0
$455$	−9.93648	−0.465829
$456$	0	0
$457$	−35.3399	−1.65313	−0.826566	−	0.562841i	$-0.809709\pi$
$457$	−35.3399	−1.65313	−0.826566	+	0.562841i	$0.809709\pi$
$458$	0	0
$459$	−3.35054	−0.156390
$460$	0	0
$461$	−16.8161	−0.783205	−0.391602	−	0.920135i	$-0.628079\pi$
$461$	−16.8161	−0.783205	−0.391602	+	0.920135i	$0.628079\pi$
$462$	0	0
$463$	1.34280	0.0624053	0.0312026	−	0.999513i	$-0.490066\pi$
$463$	1.34280	0.0624053	0.0312026	+	0.999513i	$0.490066\pi$
$464$	0	0
$465$	4.35324	0.201877
$466$	0	0
$467$	5.25285	0.243073	0.121537	−	0.992587i	$-0.461218\pi$
$467$	5.25285	0.243073	0.121537	+	0.992587i	$0.461218\pi$
$468$	0	0
$469$	9.15223	0.422611
$470$	0	0
$471$	19.0494	0.877752
$472$	0	0
$473$	12.1803	0.560051
$474$	0	0
$475$	−4.33791	−0.199037
$476$	0	0
$477$	28.4118	1.30089
$478$	0	0
$479$	1.87667	0.0857474	0.0428737	−	0.999080i	$-0.486349\pi$
$479$	1.87667	0.0857474	0.0428737	+	0.999080i	$0.486349\pi$
$480$	0	0
$481$	3.11809	0.142173
$482$	0	0
$483$	−30.4099	−1.38370
$484$	0	0
$485$	−12.1406	−0.551276
$486$	0	0
$487$	14.9815	0.678878	0.339439	−	0.940628i	$-0.389763\pi$
$487$	14.9815	0.678878	0.339439	+	0.940628i	$0.389763\pi$
$488$	0	0
$489$	−10.6897	−0.483404
$490$	0	0
$491$	1.64656	0.0743081	0.0371541	−	0.999310i	$-0.488171\pi$
$491$	1.64656	0.0743081	0.0371541	+	0.999310i	$0.488171\pi$
$492$	0	0
$493$	−49.5450	−2.23139
$494$	0	0
$495$	3.83449	0.172348
$496$	0	0
$497$	55.8476	2.50511
$498$	0	0
$499$	0.398432	0.0178363	0.00891814	−	0.999960i	$-0.497161\pi$
$499$	0.398432	0.0178363	0.00891814	+	0.999960i	$0.497161\pi$
$500$	0	0
$501$	−55.2671	−2.46915
$502$	0	0
$503$	−26.4136	−1.17773	−0.588863	−	0.808233i	$-0.700424\pi$
$503$	−26.4136	−1.17773	−0.588863	+	0.808233i	$0.700424\pi$
$504$	0	0
$505$	−11.4175	−0.508074
$506$	0	0
$507$	−18.8223	−0.835927
$508$	0	0
$509$	−11.5686	−0.512768	−0.256384	−	0.966575i	$-0.582531\pi$
$509$	−11.5686	−0.512768	−0.256384	+	0.966575i	$0.582531\pi$
$510$	0	0
$511$	66.2964	2.93278
$512$	0	0
$513$	2.42297	0.106977
$514$	0	0
$515$	2.28203	0.100558
$516$	0	0
$517$	−4.61851	−0.203122
$518$	0	0
$519$	−34.4160	−1.51069
$520$	0	0
$521$	15.7824	0.691439	0.345719	−	0.938338i	$-0.387635\pi$
$521$	15.7824	0.691439	0.345719	+	0.938338i	$0.387635\pi$
$522$	0	0
$523$	−20.9953	−0.918061	−0.459030	−	0.888421i	$-0.651803\pi$
$523$	−20.9953	−0.918061	−0.459030	+	0.888421i	$0.651803\pi$
$524$	0	0
$525$	10.5026	0.458370
$526$	0	0
$527$	10.8735	0.473657
$528$	0	0
$529$	−14.6163	−0.635491
$530$	0	0
$531$	16.8594	0.731638
$532$	0	0
$533$	−20.3914	−0.883249
$534$	0	0
$535$	−6.88872	−0.297825
$536$	0	0
$537$	−8.20830	−0.354214
$538$	0	0
$539$	16.8008	0.723662
$540$	0	0
$541$	−16.9595	−0.729145	−0.364572	−	0.931175i	$-0.618785\pi$
$541$	−16.9595	−0.729145	−0.364572	+	0.931175i	$0.618785\pi$
$542$	0	0
$543$	22.0572	0.946564
$544$	0	0
$545$	−19.8503	−0.850293
$546$	0	0
$547$	34.2131	1.46285	0.731424	−	0.681923i	$-0.238856\pi$
$547$	34.2131	1.46285	0.731424	+	0.681923i	$0.238856\pi$
$548$	0	0
$549$	−4.38112	−0.186982
$550$	0	0
$551$	35.8289	1.52636
$552$	0	0
$553$	57.1689	2.43107
$554$	0	0
$555$	−3.29573	−0.139896
$556$	0	0
$557$	−11.6313	−0.492832	−0.246416	−	0.969164i	$-0.579253\pi$
$557$	−11.6313	−0.492832	−0.246416	+	0.969164i	$0.579253\pi$
$558$	0	0
$559$	−19.9734	−0.844784
$560$	0	0
$561$	19.9605	0.842733
$562$	0	0
$563$	33.3024	1.40353	0.701765	−	0.712409i	$-0.252396\pi$
$563$	33.3024	1.40353	0.701765	+	0.712409i	$0.252396\pi$
$564$	0	0
$565$	10.8344	0.455808
$566$	0	0
$567$	−42.1742	−1.77115
$568$	0	0
$569$	−7.58629	−0.318034	−0.159017	−	0.987276i	$-0.550832\pi$
$569$	−7.58629	−0.318034	−0.159017	+	0.987276i	$0.550832\pi$
$570$	0	0
$571$	−4.56985	−0.191242	−0.0956212	−	0.995418i	$-0.530484\pi$
$571$	−4.56985	−0.191242	−0.0956212	+	0.995418i	$0.530484\pi$
$572$	0	0
$573$	11.6290	0.485808
$574$	0	0
$575$	−2.89546	−0.120749
$576$	0	0
$577$	3.19106	0.132846	0.0664228	−	0.997792i	$-0.478841\pi$
$577$	3.19106	0.132846	0.0664228	+	0.997792i	$0.478841\pi$
$578$	0	0
$579$	40.9164	1.70043
$580$	0	0
$581$	−4.55339	−0.188906
$582$	0	0
$583$	14.2252	0.589146
$584$	0	0
$585$	−6.28784	−0.259970
$586$	0	0
$587$	3.38721	0.139805	0.0699026	−	0.997554i	$-0.477731\pi$
$587$	3.38721	0.139805	0.0699026	+	0.997554i	$0.477731\pi$
$588$	0	0
$589$	−7.86326	−0.324000
$590$	0	0
$591$	−10.0021	−0.411433
$592$	0	0
$593$	36.8050	1.51140	0.755700	−	0.654918i	$-0.227297\pi$
$593$	36.8050	1.51140	0.755700	+	0.654918i	$0.227297\pi$
$594$	0	0
$595$	26.2333	1.07546
$596$	0	0
$597$	−58.2114	−2.38243
$598$	0	0
$599$	−19.2796	−0.787741	−0.393871	−	0.919166i	$-0.628864\pi$
$599$	−19.2796	−0.787741	−0.393871	+	0.919166i	$0.628864\pi$
$600$	0	0
$601$	−37.7132	−1.53835	−0.769177	−	0.639036i	$-0.779333\pi$
$601$	−37.7132	−1.53835	−0.769177	+	0.639036i	$0.779333\pi$
$602$	0	0
$603$	5.79157	0.235851
$604$	0	0
$605$	−9.08015	−0.369161
$606$	0	0
$607$	4.63660	0.188194	0.0940969	−	0.995563i	$-0.470004\pi$
$607$	4.63660	0.188194	0.0940969	+	0.995563i	$0.470004\pi$
$608$	0	0
$609$	−86.7458	−3.51512
$610$	0	0
$611$	7.57347	0.306390
$612$	0	0
$613$	19.4641	0.786148	0.393074	−	0.919507i	$-0.371412\pi$
$613$	19.4641	0.786148	0.393074	+	0.919507i	$0.371412\pi$
$614$	0	0
$615$	21.5532	0.869107
$616$	0	0
$617$	8.20227	0.330211	0.165105	−	0.986276i	$-0.447204\pi$
$617$	8.20227	0.330211	0.165105	+	0.986276i	$0.447204\pi$
$618$	0	0
$619$	10.8319	0.435369	0.217685	−	0.976019i	$-0.430150\pi$
$619$	10.8319	0.435369	0.217685	+	0.976019i	$0.430150\pi$
$620$	0	0
$621$	1.61728	0.0648993
$622$	0	0
$623$	−65.3303	−2.61740
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−14.4346	−0.576463
$628$	0	0
$629$	−8.23207	−0.328234
$630$	0	0
$631$	−24.6828	−0.982606	−0.491303	−	0.870989i	$-0.663479\pi$
$631$	−24.6828	−0.982606	−0.491303	+	0.870989i	$0.663479\pi$
$632$	0	0
$633$	55.5005	2.20595
$634$	0	0
$635$	12.3421	0.489780
$636$	0	0
$637$	−27.5501	−1.09158
$638$	0	0
$639$	35.3406	1.39805
$640$	0	0
$641$	2.42837	0.0959149	0.0479574	−	0.998849i	$-0.484729\pi$
$641$	2.42837	0.0959149	0.0479574	+	0.998849i	$0.484729\pi$
$642$	0	0
$643$	−34.0822	−1.34407	−0.672035	−	0.740519i	$-0.734580\pi$
$643$	−34.0822	−1.34407	−0.672035	+	0.740519i	$0.734580\pi$
$644$	0	0
$645$	21.1113	0.831258
$646$	0	0
$647$	−27.2783	−1.07242	−0.536210	−	0.844084i	$-0.680145\pi$
$647$	−27.2783	−1.07242	−0.536210	+	0.844084i	$0.680145\pi$
$648$	0	0
$649$	8.44116	0.331344
$650$	0	0
$651$	19.0379	0.746153
$652$	0	0
$653$	−6.02497	−0.235775	−0.117888	−	0.993027i	$-0.537612\pi$
$653$	−6.02497	−0.235775	−0.117888	+	0.993027i	$0.537612\pi$
$654$	0	0
$655$	−0.104911	−0.00409922
$656$	0	0
$657$	41.9526	1.63673
$658$	0	0
$659$	15.9762	0.622346	0.311173	−	0.950353i	$-0.399278\pi$
$659$	15.9762	0.622346	0.311173	+	0.950353i	$0.399278\pi$
$660$	0	0
$661$	−22.3865	−0.870734	−0.435367	−	0.900253i	$-0.643381\pi$
$661$	−22.3865	−0.870734	−0.435367	+	0.900253i	$0.643381\pi$
$662$	0	0
$663$	−32.7314	−1.27118
$664$	0	0
$665$	−18.9708	−0.735657
$666$	0	0
$667$	23.9150	0.925993
$668$	0	0
$669$	10.0529	0.388669
$670$	0	0
$671$	−2.19353	−0.0846804
$672$	0	0
$673$	14.9141	0.574896	0.287448	−	0.957796i	$-0.407193\pi$
$673$	14.9141	0.574896	0.287448	+	0.957796i	$0.407193\pi$
$674$	0	0
$675$	−0.558557	−0.0214989
$676$	0	0
$677$	12.2676	0.471481	0.235741	−	0.971816i	$-0.424248\pi$
$677$	12.2676	0.471481	0.235741	+	0.971816i	$0.424248\pi$
$678$	0	0
$679$	−53.0940	−2.03756
$680$	0	0
$681$	−15.4608	−0.592458
$682$	0	0
$683$	4.59217	0.175715	0.0878573	−	0.996133i	$-0.471998\pi$
$683$	4.59217	0.175715	0.0878573	+	0.996133i	$0.471998\pi$
$684$	0	0
$685$	7.66366	0.292813
$686$	0	0
$687$	−53.8835	−2.05578
$688$	0	0
$689$	−23.3266	−0.888671
$690$	0	0
$691$	−28.2987	−1.07653	−0.538266	−	0.842775i	$-0.680921\pi$
$691$	−28.2987	−1.07653	−0.538266	+	0.842775i	$0.680921\pi$
$692$	0	0
$693$	16.7692	0.637011
$694$	0	0
$695$	−16.5039	−0.626029
$696$	0	0
$697$	53.8354	2.03916
$698$	0	0
$699$	−29.2941	−1.10800
$700$	0	0
$701$	10.8022	0.407993	0.203996	−	0.978972i	$-0.434607\pi$
$701$	10.8022	0.407993	0.203996	+	0.978972i	$0.434607\pi$
$702$	0	0
$703$	5.95309	0.224525
$704$	0	0
$705$	−8.00496	−0.301484
$706$	0	0
$707$	−49.9319	−1.87788
$708$	0	0
$709$	1.30779	0.0491152	0.0245576	−	0.999698i	$-0.492182\pi$
$709$	1.30779	0.0491152	0.0245576	+	0.999698i	$0.492182\pi$
$710$	0	0
$711$	36.1767	1.35673
$712$	0	0
$713$	−5.24856	−0.196560
$714$	0	0
$715$	−3.14818	−0.117735
$716$	0	0
$717$	−3.85719	−0.144049
$718$	0	0
$719$	−30.2231	−1.12713	−0.563565	−	0.826072i	$-0.690571\pi$
$719$	−30.2231	−1.12713	−0.563565	+	0.826072i	$0.690571\pi$
$720$	0	0
$721$	9.97992	0.371672
$722$	0	0
$723$	−3.88555	−0.144505
$724$	0	0
$725$	−8.25947	−0.306749
$726$	0	0
$727$	−37.1286	−1.37702	−0.688512	−	0.725225i	$-0.741736\pi$
$727$	−37.1286	−1.37702	−0.688512	+	0.725225i	$0.741736\pi$
$728$	0	0
$729$	−22.6638	−0.839401
$730$	0	0
$731$	52.7318	1.95036
$732$	0	0
$733$	−35.5123	−1.31168	−0.655838	−	0.754902i	$-0.727685\pi$
$733$	−35.5123	−1.31168	−0.655838	+	0.754902i	$0.727685\pi$
$734$	0	0
$735$	29.1198	1.07410
$736$	0	0
$737$	2.89971	0.106812
$738$	0	0
$739$	23.0758	0.848856	0.424428	−	0.905462i	$-0.360475\pi$
$739$	23.0758	0.848856	0.424428	+	0.905462i	$0.360475\pi$
$740$	0	0
$741$	23.6700	0.869539
$742$	0	0
$743$	−40.2073	−1.47506	−0.737531	−	0.675313i	$-0.764009\pi$
$743$	−40.2073	−1.47506	−0.737531	+	0.675313i	$0.764009\pi$
$744$	0	0
$745$	9.19861	0.337011
$746$	0	0
$747$	−2.88140	−0.105425
$748$	0	0
$749$	−30.1262	−1.10079
$750$	0	0
$751$	−6.83278	−0.249332	−0.124666	−	0.992199i	$-0.539786\pi$
$751$	−6.83278	−0.249332	−0.124666	+	0.992199i	$0.539786\pi$
$752$	0	0
$753$	−68.7341	−2.50481
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−10.8682	−0.395012	−0.197506	−	0.980302i	$-0.563284\pi$
$757$	−10.8682	−0.395012	−0.197506	+	0.980302i	$0.563284\pi$
$758$	0	0
$759$	−9.63479	−0.349721
$760$	0	0
$761$	30.2643	1.09708	0.548541	−	0.836124i	$-0.315184\pi$
$761$	30.2643	1.09708	0.548541	+	0.836124i	$0.315184\pi$
$762$	0	0
$763$	−86.8105	−3.14275
$764$	0	0
$765$	16.6005	0.600194
$766$	0	0
$767$	−13.8419	−0.499802
$768$	0	0
$769$	−11.7965	−0.425392	−0.212696	−	0.977118i	$-0.568224\pi$
$769$	−11.7965	−0.425392	−0.212696	+	0.977118i	$0.568224\pi$
$770$	0	0
$771$	43.1855	1.55529
$772$	0	0
$773$	−15.4018	−0.553963	−0.276982	−	0.960875i	$-0.589334\pi$
$773$	−15.4018	−0.553963	−0.276982	+	0.960875i	$0.589334\pi$
$774$	0	0
$775$	1.81268	0.0651135
$776$	0	0
$777$	−14.4131	−0.517067
$778$	0	0
$779$	−38.9315	−1.39486
$780$	0	0
$781$	17.6943	0.633150
$782$	0	0
$783$	4.61339	0.164869
$784$	0	0
$785$	7.93216	0.283111
$786$	0	0
$787$	39.8048	1.41889	0.709444	−	0.704762i	$-0.248946\pi$
$787$	39.8048	1.41889	0.709444	+	0.704762i	$0.248946\pi$
$788$	0	0
$789$	20.4186	0.726920
$790$	0	0
$791$	47.3819	1.68471
$792$	0	0
$793$	3.59698	0.127732
$794$	0	0
$795$	24.6556	0.874442
$796$	0	0
$797$	27.6880	0.980758	0.490379	−	0.871509i	$-0.336858\pi$
$797$	27.6880	0.980758	0.490379	+	0.871509i	$0.336858\pi$
$798$	0	0
$799$	−19.9947	−0.707363
$800$	0	0
$801$	−41.3413	−1.46072
$802$	0	0
$803$	21.0048	0.741242
$804$	0	0
$805$	−12.6626	−0.446299
$806$	0	0
$807$	−34.7831	−1.22442
$808$	0	0
$809$	39.8055	1.39949	0.699744	−	0.714394i	$-0.253297\pi$
$809$	39.8055	1.39949	0.699744	+	0.714394i	$0.253297\pi$
$810$	0	0
$811$	−53.2636	−1.87034	−0.935168	−	0.354203i	$-0.884752\pi$
$811$	−53.2636	−1.87034	−0.935168	+	0.354203i	$0.884752\pi$
$812$	0	0
$813$	55.9308	1.96158
$814$	0	0
$815$	−4.45116	−0.155917
$816$	0	0
$817$	−38.1334	−1.33412
$818$	0	0
$819$	−27.4984	−0.960871
$820$	0	0
$821$	23.3740	0.815757	0.407879	−	0.913036i	$-0.366269\pi$
$821$	23.3740	0.815757	0.407879	+	0.913036i	$0.366269\pi$
$822$	0	0
$823$	−49.5519	−1.72727	−0.863636	−	0.504116i	$-0.831818\pi$
$823$	−49.5519	−1.72727	−0.863636	+	0.504116i	$0.831818\pi$
$824$	0	0
$825$	3.32755	0.115850
$826$	0	0
$827$	−30.8188	−1.07167	−0.535837	−	0.844321i	$-0.680004\pi$
$827$	−30.8188	−1.07167	−0.535837	+	0.844321i	$0.680004\pi$
$828$	0	0
$829$	−23.2316	−0.806866	−0.403433	−	0.915009i	$-0.632183\pi$
$829$	−23.2316	−0.806866	−0.403433	+	0.915009i	$0.632183\pi$
$830$	0	0
$831$	−54.4434	−1.88862
$832$	0	0
$833$	72.7352	2.52012
$834$	0	0
$835$	−23.0131	−0.796402
$836$	0	0
$837$	−1.01249	−0.0349967
$838$	0	0
$839$	55.6651	1.92177	0.960886	−	0.276945i	$-0.0893219\pi$
$839$	55.6651	1.92177	0.960886	+	0.276945i	$0.0893219\pi$
$840$	0	0
$841$	39.2189	1.35238
$842$	0	0
$843$	26.2133	0.902834
$844$	0	0
$845$	−7.83758	−0.269621
$846$	0	0
$847$	−39.7099	−1.36445
$848$	0	0
$849$	6.03820	0.207230
$850$	0	0
$851$	3.97356	0.136212
$852$	0	0
$853$	−9.69378	−0.331909	−0.165954	−	0.986133i	$-0.553070\pi$
$853$	−9.69378	−0.331909	−0.165954	+	0.986133i	$0.553070\pi$
$854$	0	0
$855$	−12.0048	−0.410556
$856$	0	0
$857$	5.57015	0.190273	0.0951364	−	0.995464i	$-0.469671\pi$
$857$	5.57015	0.190273	0.0951364	+	0.995464i	$0.469671\pi$
$858$	0	0
$859$	7.56622	0.258156	0.129078	−	0.991634i	$-0.458798\pi$
$859$	7.56622	0.258156	0.129078	+	0.991634i	$0.458798\pi$
$860$	0	0
$861$	94.2576	3.21229
$862$	0	0
$863$	45.5745	1.55138	0.775688	−	0.631117i	$-0.217403\pi$
$863$	45.5745	1.55138	0.775688	+	0.631117i	$0.217403\pi$
$864$	0	0
$865$	−14.3308	−0.487261
$866$	0	0
$867$	45.5881	1.54825
$868$	0	0
$869$	18.1129	0.614437
$870$	0	0
$871$	−4.75497	−0.161116
$872$	0	0
$873$	−33.5981	−1.13712
$874$	0	0
$875$	4.37326	0.147843
$876$	0	0
$877$	58.4983	1.97535	0.987673	−	0.156528i	$-0.0500302\pi$
$877$	58.4983	1.97535	0.987673	+	0.156528i	$0.0500302\pi$
$878$	0	0
$879$	2.40103	0.0809846
$880$	0	0
$881$	58.3766	1.96676	0.983378	−	0.181568i	$-0.0581174\pi$
$881$	58.3766	1.96676	0.983378	+	0.181568i	$0.0581174\pi$
$882$	0	0
$883$	−41.4346	−1.39439	−0.697193	−	0.716883i	$-0.745568\pi$
$883$	−41.4346	−1.39439	−0.697193	+	0.716883i	$0.745568\pi$
$884$	0	0
$885$	14.6305	0.491799
$886$	0	0
$887$	−49.4283	−1.65964	−0.829819	−	0.558032i	$-0.811557\pi$
$887$	−49.4283	−1.65964	−0.829819	+	0.558032i	$0.811557\pi$
$888$	0	0
$889$	53.9752	1.81027
$890$	0	0
$891$	−13.3621	−0.447647
$892$	0	0
$893$	14.4594	0.483864
$894$	0	0
$895$	−3.41792	−0.114249
$896$	0	0
$897$	15.7992	0.527521
$898$	0	0
$899$	−14.9718	−0.499338
$900$	0	0
$901$	61.5845	2.05168
$902$	0	0
$903$	92.3254	3.07240
$904$	0	0
$905$	9.18457	0.305306
$906$	0	0
$907$	52.0408	1.72798	0.863992	−	0.503505i	$-0.167956\pi$
$907$	52.0408	1.72798	0.863992	+	0.503505i	$0.167956\pi$
$908$	0	0
$909$	−31.5971	−1.04801
$910$	0	0
$911$	−1.76859	−0.0585961	−0.0292980	−	0.999571i	$-0.509327\pi$
$911$	−1.76859	−0.0585961	−0.0292980	+	0.999571i	$0.509327\pi$
$912$	0	0
$913$	−1.44265	−0.0477449
$914$	0	0
$915$	−3.80191	−0.125687
$916$	0	0
$917$	−0.458804	−0.0151511
$918$	0	0
$919$	4.58484	0.151240	0.0756199	−	0.997137i	$-0.475906\pi$
$919$	4.58484	0.151240	0.0756199	+	0.997137i	$0.475906\pi$
$920$	0	0
$921$	−50.8421	−1.67530
$922$	0	0
$923$	−29.0152	−0.955048
$924$	0	0
$925$	−1.37234	−0.0451222
$926$	0	0
$927$	6.31533	0.207423
$928$	0	0
$929$	48.6461	1.59603	0.798013	−	0.602640i	$-0.205885\pi$
$929$	48.6461	1.59603	0.798013	+	0.602640i	$0.205885\pi$
$930$	0	0
$931$	−52.5990	−1.72386
$932$	0	0
$933$	−81.9010	−2.68132
$934$	0	0
$935$	8.31153	0.271816
$936$	0	0
$937$	28.7413	0.938939	0.469469	−	0.882949i	$-0.344445\pi$
$937$	28.7413	0.938939	0.469469	+	0.882949i	$0.344445\pi$
$938$	0	0
$939$	−34.2930	−1.11911
$940$	0	0
$941$	48.5675	1.58326	0.791628	−	0.611004i	$-0.209234\pi$
$941$	48.5675	1.58326	0.791628	+	0.611004i	$0.209234\pi$
$942$	0	0
$943$	−25.9859	−0.846218
$944$	0	0
$945$	−2.44272	−0.0794616
$946$	0	0
$947$	−3.17038	−0.103024	−0.0515118	−	0.998672i	$-0.516404\pi$
$947$	−3.17038	−0.103024	−0.0515118	+	0.998672i	$0.516404\pi$
$948$	0	0
$949$	−34.4438	−1.11809
$950$	0	0
$951$	55.1152	1.78723
$952$	0	0
$953$	9.35544	0.303053	0.151526	−	0.988453i	$-0.451581\pi$
$953$	9.35544	0.303053	0.151526	+	0.988453i	$0.451581\pi$
$954$	0	0
$955$	4.84230	0.156693
$956$	0	0
$957$	−27.4838	−0.888424
$958$	0	0
$959$	33.5152	1.08226
$960$	0	0
$961$	−27.7142	−0.894006
$962$	0	0
$963$	−19.0640	−0.614327
$964$	0	0
$965$	17.0375	0.548457
$966$	0	0
$967$	−32.5980	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$967$	−32.5980	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$968$	0	0
$969$	−62.4912	−2.00751
$970$	0	0
$971$	22.6826	0.727920	0.363960	−	0.931415i	$-0.381424\pi$
$971$	22.6826	0.727920	0.363960	+	0.931415i	$0.381424\pi$
$972$	0	0
$973$	−72.1760	−2.31386
$974$	0	0
$975$	−5.45654	−0.174749
$976$	0	0
$977$	12.3366	0.394684	0.197342	−	0.980335i	$-0.436769\pi$
$977$	12.3366	0.394684	0.197342	+	0.980335i	$0.436769\pi$
$978$	0	0
$979$	−20.6987	−0.661532
$980$	0	0
$981$	−54.9340	−1.75391
$982$	0	0
$983$	24.8597	0.792902	0.396451	−	0.918056i	$-0.370242\pi$
$983$	24.8597	0.792902	0.396451	+	0.918056i	$0.370242\pi$
$984$	0	0
$985$	−4.16488	−0.132704
$986$	0	0
$987$	−35.0078	−1.11431
$988$	0	0
$989$	−25.4532	−0.809366
$990$	0	0
$991$	8.59575	0.273053	0.136527	−	0.990636i	$-0.456406\pi$
$991$	8.59575	0.273053	0.136527	+	0.990636i	$0.456406\pi$
$992$	0	0
$993$	−65.6886	−2.08456
$994$	0	0
$995$	−24.2391	−0.768432
$996$	0	0
$997$	60.7232	1.92312	0.961561	−	0.274590i	$-0.0885421\pi$
$997$	60.7232	1.92312	0.961561	+	0.274590i	$0.0885421\pi$
$998$	0	0
$999$	0.766529	0.0242519

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.s.1.19	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.s.1.19	✓	24	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)

\(f(q)\)	\(=\)	\(q+2.40154 q^{3} +1.00000 q^{5} +4.37326 q^{7} +2.76742 q^{9} +O(q^{10})\)	\(q+2.40154 q^{3} +1.00000 q^{5} +4.37326 q^{7} +2.76742 q^{9} +1.38559 q^{11} -2.27210 q^{13} +2.40154 q^{15} +5.99857 q^{17} -4.33791 q^{19} +10.5026 q^{21} -2.89546 q^{23} +1.00000 q^{25} -0.558557 q^{27} -8.25947 q^{29} +1.81268 q^{31} +3.32755 q^{33} +4.37326 q^{35} -1.37234 q^{37} -5.45654 q^{39} +8.97470 q^{41} +8.79073 q^{43} +2.76742 q^{45} -3.33325 q^{47} +12.1254 q^{49} +14.4058 q^{51} +10.2665 q^{53} +1.38559 q^{55} -10.4177 q^{57} +6.09212 q^{59} -1.58311 q^{61} +12.1026 q^{63} -2.27210 q^{65} +2.09277 q^{67} -6.95359 q^{69} +12.7702 q^{71} +15.1595 q^{73} +2.40154 q^{75} +6.05953 q^{77} +13.0724 q^{79} -9.64365 q^{81} -1.04119 q^{83} +5.99857 q^{85} -19.8355 q^{87} -14.9386 q^{89} -9.93648 q^{91} +4.35324 q^{93} -4.33791 q^{95} -12.1406 q^{97} +3.83449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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