LMFDB - Embedded newform 6040.2.a.o.1.1

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.95828$ 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)

$f(q)$	$=$	$q-2.95828 q^{3} +1.00000 q^{5} +2.63281 q^{7} +5.75141 q^{9} +O(q^{10})$	$q-2.95828 q^{3} +1.00000 q^{5} +2.63281 q^{7} +5.75141 q^{9} +2.27282 q^{11} -1.87439 q^{13} -2.95828 q^{15} -3.68235 q^{17} -1.78425 q^{19} -7.78857 q^{21} +3.61300 q^{23} +1.00000 q^{25} -8.13943 q^{27} +8.68751 q^{29} +1.44348 q^{31} -6.72364 q^{33} +2.63281 q^{35} +3.63631 q^{37} +5.54495 q^{39} -4.60947 q^{41} +7.11075 q^{43} +5.75141 q^{45} +0.534101 q^{47} -0.0683312 q^{49} +10.8934 q^{51} -9.09126 q^{53} +2.27282 q^{55} +5.27832 q^{57} +10.5135 q^{59} -1.07040 q^{61} +15.1423 q^{63} -1.87439 q^{65} +12.4004 q^{67} -10.6883 q^{69} -0.0803696 q^{71} -9.28652 q^{73} -2.95828 q^{75} +5.98390 q^{77} +5.80748 q^{79} +6.82447 q^{81} +4.41680 q^{83} -3.68235 q^{85} -25.7001 q^{87} +6.03745 q^{89} -4.93490 q^{91} -4.27023 q^{93} -1.78425 q^{95} -0.00385206 q^{97} +13.0719 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * 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
$3$	−2.95828	−1.70796	−0.853981	−	0.520304i	$-0.825819\pi$
$3$	−2.95828	−1.70796	−0.853981	+	0.520304i	$0.825819\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.63281	0.995107	0.497554	−	0.867433i	$-0.334232\pi$
$7$	2.63281	0.995107	0.497554	+	0.867433i	$0.334232\pi$
$8$	0	0
$9$	5.75141	1.91714
$10$	0	0
$11$	2.27282	0.685282	0.342641	−	0.939466i	$-0.388679\pi$
$11$	2.27282	0.685282	0.342641	+	0.939466i	$0.388679\pi$
$12$	0	0
$13$	−1.87439	−0.519861	−0.259931	−	0.965627i	$-0.583700\pi$
$13$	−1.87439	−0.519861	−0.259931	+	0.965627i	$0.583700\pi$
$14$	0	0
$15$	−2.95828	−0.763824
$16$	0	0
$17$	−3.68235	−0.893101	−0.446551	−	0.894758i	$-0.647348\pi$
$17$	−3.68235	−0.893101	−0.446551	+	0.894758i	$0.647348\pi$
$18$	0	0
$19$	−1.78425	−0.409336	−0.204668	−	0.978831i	$-0.565611\pi$
$19$	−1.78425	−0.409336	−0.204668	+	0.978831i	$0.565611\pi$
$20$	0	0
$21$	−7.78857	−1.69961
$22$	0	0
$23$	3.61300	0.753363	0.376682	−	0.926343i	$-0.377065\pi$
$23$	3.61300	0.753363	0.376682	+	0.926343i	$0.377065\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−8.13943	−1.56643
$28$	0	0
$29$	8.68751	1.61323	0.806615	−	0.591077i	$-0.201297\pi$
$29$	8.68751	1.61323	0.806615	+	0.591077i	$0.201297\pi$
$30$	0	0
$31$	1.44348	0.259257	0.129629	−	0.991563i	$-0.458621\pi$
$31$	1.44348	0.259257	0.129629	+	0.991563i	$0.458621\pi$
$32$	0	0
$33$	−6.72364	−1.17044
$34$	0	0
$35$	2.63281	0.445025
$36$	0	0
$37$	3.63631	0.597806	0.298903	−	0.954283i	$-0.403379\pi$
$37$	3.63631	0.597806	0.298903	+	0.954283i	$0.403379\pi$
$38$	0	0
$39$	5.54495	0.887903
$40$	0	0
$41$	−4.60947	−0.719879	−0.359939	−	0.932976i	$-0.617203\pi$
$41$	−4.60947	−0.719879	−0.359939	+	0.932976i	$0.617203\pi$
$42$	0	0
$43$	7.11075	1.08438	0.542190	−	0.840256i	$-0.317595\pi$
$43$	7.11075	1.08438	0.542190	+	0.840256i	$0.317595\pi$
$44$	0	0
$45$	5.75141	0.857369
$46$	0	0
$47$	0.534101	0.0779066	0.0389533	−	0.999241i	$-0.487598\pi$
$47$	0.534101	0.0779066	0.0389533	+	0.999241i	$0.487598\pi$
$48$	0	0
$49$	−0.0683312	−0.00976160
$50$	0	0
$51$	10.8934	1.52538
$52$	0	0
$53$	−9.09126	−1.24878	−0.624390	−	0.781112i	$-0.714653\pi$
$53$	−9.09126	−1.24878	−0.624390	+	0.781112i	$0.714653\pi$
$54$	0	0
$55$	2.27282	0.306467
$56$	0	0
$57$	5.27832	0.699131
$58$	0	0
$59$	10.5135	1.36874	0.684368	−	0.729137i	$-0.260078\pi$
$59$	10.5135	1.36874	0.684368	+	0.729137i	$0.260078\pi$
$60$	0	0
$61$	−1.07040	−0.137051	−0.0685255	−	0.997649i	$-0.521829\pi$
$61$	−1.07040	−0.137051	−0.0685255	+	0.997649i	$0.521829\pi$
$62$	0	0
$63$	15.1423	1.90776
$64$	0	0
$65$	−1.87439	−0.232489
$66$	0	0
$67$	12.4004	1.51495	0.757473	−	0.652867i	$-0.226434\pi$
$67$	12.4004	1.51495	0.757473	+	0.652867i	$0.226434\pi$
$68$	0	0
$69$	−10.6883	−1.28672
$70$	0	0
$71$	−0.0803696	−0.00953812	−0.00476906	−	0.999989i	$-0.501518\pi$
$71$	−0.0803696	−0.00953812	−0.00476906	+	0.999989i	$0.501518\pi$
$72$	0	0
$73$	−9.28652	−1.08690	−0.543452	−	0.839440i	$-0.682883\pi$
$73$	−9.28652	−1.08690	−0.543452	+	0.839440i	$0.682883\pi$
$74$	0	0
$75$	−2.95828	−0.341592
$76$	0	0
$77$	5.98390	0.681929
$78$	0	0
$79$	5.80748	0.653393	0.326696	−	0.945129i	$-0.394065\pi$
$79$	5.80748	0.653393	0.326696	+	0.945129i	$0.394065\pi$
$80$	0	0
$81$	6.82447	0.758274
$82$	0	0
$83$	4.41680	0.484806	0.242403	−	0.970176i	$-0.422064\pi$
$83$	4.41680	0.484806	0.242403	+	0.970176i	$0.422064\pi$
$84$	0	0
$85$	−3.68235	−0.399407
$86$	0	0
$87$	−25.7001	−2.75534
$88$	0	0
$89$	6.03745	0.639969	0.319984	−	0.947423i	$-0.396322\pi$
$89$	6.03745	0.639969	0.319984	+	0.947423i	$0.396322\pi$
$90$	0	0
$91$	−4.93490	−0.517318
$92$	0	0
$93$	−4.27023	−0.442802
$94$	0	0
$95$	−1.78425	−0.183061
$96$	0	0
$97$	−0.00385206	−0.000391117 0	−0.000195559	−	1.00000i	$-0.500062\pi$
$97$	−0.00385206	−0.000391117 0	−0.000195559		1.00000i	$0.500062\pi$
$98$	0	0
$99$	13.0719	1.31378
$100$	0	0
$101$	4.77354	0.474985	0.237492	−	0.971389i	$-0.423675\pi$
$101$	4.77354	0.474985	0.237492	+	0.971389i	$0.423675\pi$
$102$	0	0
$103$	−15.8678	−1.56350	−0.781749	−	0.623593i	$-0.785672\pi$
$103$	−15.8678	−1.56350	−0.781749	+	0.623593i	$0.785672\pi$
$104$	0	0
$105$	−7.78857	−0.760087
$106$	0	0
$107$	−6.68945	−0.646694	−0.323347	−	0.946281i	$-0.604808\pi$
$107$	−6.68945	−0.646694	−0.323347	+	0.946281i	$0.604808\pi$
$108$	0	0
$109$	16.4266	1.57338	0.786690	−	0.617348i	$-0.211793\pi$
$109$	16.4266	1.57338	0.786690	+	0.617348i	$0.211793\pi$
$110$	0	0
$111$	−10.7572	−1.02103
$112$	0	0
$113$	−4.22914	−0.397844	−0.198922	−	0.980015i	$-0.563744\pi$
$113$	−4.22914	−0.397844	−0.198922	+	0.980015i	$0.563744\pi$
$114$	0	0
$115$	3.61300	0.336914
$116$	0	0
$117$	−10.7804	−0.996644
$118$	0	0
$119$	−9.69492	−0.888732
$120$	0	0
$121$	−5.83428	−0.530389
$122$	0	0
$123$	13.6361	1.22953
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.855024	−0.0758712	−0.0379356	−	0.999280i	$-0.512078\pi$
$127$	−0.855024	−0.0758712	−0.0379356	+	0.999280i	$0.512078\pi$
$128$	0	0
$129$	−21.0356	−1.85208
$130$	0	0
$131$	−0.154714	−0.0135174	−0.00675871	−	0.999977i	$-0.502151\pi$
$131$	−0.154714	−0.0135174	−0.00675871	+	0.999977i	$0.502151\pi$
$132$	0	0
$133$	−4.69760	−0.407333
$134$	0	0
$135$	−8.13943	−0.700530
$136$	0	0
$137$	−8.56170	−0.731475	−0.365738	−	0.930718i	$-0.619183\pi$
$137$	−8.56170	−0.731475	−0.365738	+	0.930718i	$0.619183\pi$
$138$	0	0
$139$	2.67491	0.226883	0.113441	−	0.993545i	$-0.463813\pi$
$139$	2.67491	0.226883	0.113441	+	0.993545i	$0.463813\pi$
$140$	0	0
$141$	−1.58002	−0.133062
$142$	0	0
$143$	−4.26015	−0.356251
$144$	0	0
$145$	8.68751	0.721458
$146$	0	0
$147$	0.202143	0.0166724
$148$	0	0
$149$	2.04323	0.167388	0.0836942	−	0.996491i	$-0.473328\pi$
$149$	2.04323	0.167388	0.0836942	+	0.996491i	$0.473328\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−21.1787	−1.71220
$154$	0	0
$155$	1.44348	0.115943
$156$	0	0
$157$	−13.8173	−1.10274	−0.551371	−	0.834260i	$-0.685895\pi$
$157$	−13.8173	−1.10274	−0.551371	+	0.834260i	$0.685895\pi$
$158$	0	0
$159$	26.8945	2.13287
$160$	0	0
$161$	9.51234	0.749677
$162$	0	0
$163$	11.7983	0.924118	0.462059	−	0.886849i	$-0.347111\pi$
$163$	11.7983	0.924118	0.462059	+	0.886849i	$0.347111\pi$
$164$	0	0
$165$	−6.72364	−0.523435
$166$	0	0
$167$	−11.9011	−0.920934	−0.460467	−	0.887677i	$-0.652318\pi$
$167$	−11.9011	−0.920934	−0.460467	+	0.887677i	$0.652318\pi$
$168$	0	0
$169$	−9.48668	−0.729744
$170$	0	0
$171$	−10.2620	−0.784753
$172$	0	0
$173$	−20.7055	−1.57421	−0.787104	−	0.616821i	$-0.788420\pi$
$173$	−20.7055	−1.57421	−0.787104	+	0.616821i	$0.788420\pi$
$174$	0	0
$175$	2.63281	0.199021
$176$	0	0
$177$	−31.1017	−2.33775
$178$	0	0
$179$	6.98372	0.521988	0.260994	−	0.965340i	$-0.415950\pi$
$179$	6.98372	0.521988	0.260994	+	0.965340i	$0.415950\pi$
$180$	0	0
$181$	21.8490	1.62402	0.812011	−	0.583642i	$-0.198373\pi$
$181$	21.8490	1.62402	0.812011	+	0.583642i	$0.198373\pi$
$182$	0	0
$183$	3.16655	0.234078
$184$	0	0
$185$	3.63631	0.267347
$186$	0	0
$187$	−8.36933	−0.612026
$188$	0	0
$189$	−21.4295	−1.55877
$190$	0	0
$191$	11.7927	0.853292	0.426646	−	0.904419i	$-0.359695\pi$
$191$	11.7927	0.853292	0.426646	+	0.904419i	$0.359695\pi$
$192$	0	0
$193$	9.70501	0.698582	0.349291	−	0.937014i	$-0.386422\pi$
$193$	9.70501	0.698582	0.349291	+	0.937014i	$0.386422\pi$
$194$	0	0
$195$	5.54495	0.397082
$196$	0	0
$197$	−1.93479	−0.137848	−0.0689239	−	0.997622i	$-0.521957\pi$
$197$	−1.93479	−0.137848	−0.0689239	+	0.997622i	$0.521957\pi$
$198$	0	0
$199$	−6.20343	−0.439749	−0.219875	−	0.975528i	$-0.570565\pi$
$199$	−6.20343	−0.439749	−0.219875	+	0.975528i	$0.570565\pi$
$200$	0	0
$201$	−36.6837	−2.58747
$202$	0	0
$203$	22.8725	1.60534
$204$	0	0
$205$	−4.60947	−0.321940
$206$	0	0
$207$	20.7799	1.44430
$208$	0	0
$209$	−4.05529	−0.280510
$210$	0	0
$211$	−10.8855	−0.749387	−0.374693	−	0.927149i	$-0.622252\pi$
$211$	−10.8855	−0.749387	−0.374693	+	0.927149i	$0.622252\pi$
$212$	0	0
$213$	0.237756	0.0162908
$214$	0	0
$215$	7.11075	0.484949
$216$	0	0
$217$	3.80041	0.257989
$218$	0	0
$219$	27.4721	1.85639
$220$	0	0
$221$	6.90215	0.464289
$222$	0	0
$223$	5.59841	0.374897	0.187449	−	0.982274i	$-0.439978\pi$
$223$	5.59841	0.374897	0.187449	+	0.982274i	$0.439978\pi$
$224$	0	0
$225$	5.75141	0.383427
$226$	0	0
$227$	−10.9148	−0.724439	−0.362219	−	0.932093i	$-0.617981\pi$
$227$	−10.9148	−0.724439	−0.362219	+	0.932093i	$0.617981\pi$
$228$	0	0
$229$	−9.68430	−0.639957	−0.319978	−	0.947425i	$-0.603676\pi$
$229$	−9.68430	−0.639957	−0.319978	+	0.947425i	$0.603676\pi$
$230$	0	0
$231$	−17.7020	−1.16471
$232$	0	0
$233$	21.5485	1.41169	0.705844	−	0.708367i	$-0.250568\pi$
$233$	21.5485	1.41169	0.705844	+	0.708367i	$0.250568\pi$
$234$	0	0
$235$	0.534101	0.0348409
$236$	0	0
$237$	−17.1801	−1.11597
$238$	0	0
$239$	0.203994	0.0131953	0.00659763	−	0.999978i	$-0.497900\pi$
$239$	0.203994	0.0131953	0.00659763	+	0.999978i	$0.497900\pi$
$240$	0	0
$241$	−2.01705	−0.129929	−0.0649647	−	0.997888i	$-0.520693\pi$
$241$	−2.01705	−0.129929	−0.0649647	+	0.997888i	$0.520693\pi$
$242$	0	0
$243$	4.22962	0.271330
$244$	0	0
$245$	−0.0683312	−0.00436552
$246$	0	0
$247$	3.34438	0.212798
$248$	0	0
$249$	−13.0661	−0.828031
$250$	0	0
$251$	−12.6240	−0.796819	−0.398410	−	0.917208i	$-0.630438\pi$
$251$	−12.6240	−0.796819	−0.398410	+	0.917208i	$0.630438\pi$
$252$	0	0
$253$	8.21171	0.516266
$254$	0	0
$255$	10.8934	0.682172
$256$	0	0
$257$	−15.8520	−0.988824	−0.494412	−	0.869228i	$-0.664616\pi$
$257$	−15.8520	−0.988824	−0.494412	+	0.869228i	$0.664616\pi$
$258$	0	0
$259$	9.57371	0.594881
$260$	0	0
$261$	49.9654	3.09278
$262$	0	0
$263$	19.2195	1.18513	0.592563	−	0.805524i	$-0.298116\pi$
$263$	19.2195	1.18513	0.592563	+	0.805524i	$0.298116\pi$
$264$	0	0
$265$	−9.09126	−0.558472
$266$	0	0
$267$	−17.8605	−1.09304
$268$	0	0
$269$	20.3887	1.24312	0.621561	−	0.783366i	$-0.286499\pi$
$269$	20.3887	1.24312	0.621561	+	0.783366i	$0.286499\pi$
$270$	0	0
$271$	10.6334	0.645933	0.322967	−	0.946410i	$-0.395320\pi$
$271$	10.6334	0.645933	0.322967	+	0.946410i	$0.395320\pi$
$272$	0	0
$273$	14.5988	0.883559
$274$	0	0
$275$	2.27282	0.137056
$276$	0	0
$277$	0.589020	0.0353908	0.0176954	−	0.999843i	$-0.494367\pi$
$277$	0.589020	0.0353908	0.0176954	+	0.999843i	$0.494367\pi$
$278$	0	0
$279$	8.30206	0.497032
$280$	0	0
$281$	23.2769	1.38858	0.694292	−	0.719694i	$-0.255718\pi$
$281$	23.2769	1.38858	0.694292	+	0.719694i	$0.255718\pi$
$282$	0	0
$283$	29.9437	1.77997	0.889983	−	0.455994i	$-0.150716\pi$
$283$	29.9437	1.77997	0.889983	+	0.455994i	$0.150716\pi$
$284$	0	0
$285$	5.27832	0.312661
$286$	0	0
$287$	−12.1359	−0.716357
$288$	0	0
$289$	−3.44029	−0.202370
$290$	0	0
$291$	0.0113955	0.000668014 0
$292$	0	0
$293$	6.06023	0.354043	0.177021	−	0.984207i	$-0.443354\pi$
$293$	6.06023	0.354043	0.177021	+	0.984207i	$0.443354\pi$
$294$	0	0
$295$	10.5135	0.612117
$296$	0	0
$297$	−18.4995	−1.07345
$298$	0	0
$299$	−6.77216	−0.391644
$300$	0	0
$301$	18.7212	1.07907
$302$	0	0
$303$	−14.1215	−0.811256
$304$	0	0
$305$	−1.07040	−0.0612911
$306$	0	0
$307$	28.1236	1.60510	0.802550	−	0.596585i	$-0.203476\pi$
$307$	28.1236	1.60510	0.802550	+	0.596585i	$0.203476\pi$
$308$	0	0
$309$	46.9413	2.67040
$310$	0	0
$311$	28.0424	1.59014	0.795070	−	0.606517i	$-0.207434\pi$
$311$	28.0424	1.59014	0.795070	+	0.606517i	$0.207434\pi$
$312$	0	0
$313$	7.04309	0.398099	0.199049	−	0.979989i	$-0.436215\pi$
$313$	7.04309	0.398099	0.199049	+	0.979989i	$0.436215\pi$
$314$	0	0
$315$	15.1423	0.853174
$316$	0	0
$317$	−3.83743	−0.215532	−0.107766	−	0.994176i	$-0.534370\pi$
$317$	−3.83743	−0.215532	−0.107766	+	0.994176i	$0.534370\pi$
$318$	0	0
$319$	19.7452	1.10552
$320$	0	0
$321$	19.7893	1.10453
$322$	0	0
$323$	6.57025	0.365579
$324$	0	0
$325$	−1.87439	−0.103972
$326$	0	0
$327$	−48.5943	−2.68727
$328$	0	0
$329$	1.40618	0.0775254
$330$	0	0
$331$	−4.26989	−0.234694	−0.117347	−	0.993091i	$-0.537439\pi$
$331$	−4.26989	−0.234694	−0.117347	+	0.993091i	$0.537439\pi$
$332$	0	0
$333$	20.9139	1.14608
$334$	0	0
$335$	12.4004	0.677504
$336$	0	0
$337$	−2.95661	−0.161057	−0.0805285	−	0.996752i	$-0.525661\pi$
$337$	−2.95661	−0.161057	−0.0805285	+	0.996752i	$0.525661\pi$
$338$	0	0
$339$	12.5110	0.679502
$340$	0	0
$341$	3.28078	0.177664
$342$	0	0
$343$	−18.6095	−1.00482
$344$	0	0
$345$	−10.6883	−0.575437
$346$	0	0
$347$	6.79433	0.364739	0.182369	−	0.983230i	$-0.441623\pi$
$347$	6.79433	0.364739	0.182369	+	0.983230i	$0.441623\pi$
$348$	0	0
$349$	−1.12790	−0.0603752	−0.0301876	−	0.999544i	$-0.509610\pi$
$349$	−1.12790	−0.0603752	−0.0301876	+	0.999544i	$0.509610\pi$
$350$	0	0
$351$	15.2564	0.814328
$352$	0	0
$353$	17.2788	0.919658	0.459829	−	0.888007i	$-0.347911\pi$
$353$	17.2788	0.919658	0.459829	+	0.888007i	$0.347911\pi$
$354$	0	0
$355$	−0.0803696	−0.00426558
$356$	0	0
$357$	28.6803	1.51792
$358$	0	0
$359$	3.68240	0.194350	0.0971748	−	0.995267i	$-0.469019\pi$
$359$	3.68240	0.194350	0.0971748	+	0.995267i	$0.469019\pi$
$360$	0	0
$361$	−15.8164	−0.832444
$362$	0	0
$363$	17.2594	0.905885
$364$	0	0
$365$	−9.28652	−0.486079
$366$	0	0
$367$	23.9260	1.24893	0.624463	−	0.781054i	$-0.285318\pi$
$367$	23.9260	1.24893	0.624463	+	0.781054i	$0.285318\pi$
$368$	0	0
$369$	−26.5110	−1.38011
$370$	0	0
$371$	−23.9355	−1.24267
$372$	0	0
$373$	21.0283	1.08880	0.544402	−	0.838824i	$-0.316756\pi$
$373$	21.0283	1.08880	0.544402	+	0.838824i	$0.316756\pi$
$374$	0	0
$375$	−2.95828	−0.152765
$376$	0	0
$377$	−16.2837	−0.838656
$378$	0	0
$379$	30.1152	1.54691	0.773457	−	0.633849i	$-0.218526\pi$
$379$	30.1152	1.54691	0.773457	+	0.633849i	$0.218526\pi$
$380$	0	0
$381$	2.52940	0.129585
$382$	0	0
$383$	25.8288	1.31979	0.659894	−	0.751358i	$-0.270601\pi$
$383$	25.8288	1.31979	0.659894	+	0.751358i	$0.270601\pi$
$384$	0	0
$385$	5.98390	0.304968
$386$	0	0
$387$	40.8968	2.07890
$388$	0	0
$389$	−8.84236	−0.448325	−0.224163	−	0.974552i	$-0.571965\pi$
$389$	−8.84236	−0.448325	−0.224163	+	0.974552i	$0.571965\pi$
$390$	0	0
$391$	−13.3044	−0.672830
$392$	0	0
$393$	0.457687	0.0230873
$394$	0	0
$395$	5.80748	0.292206
$396$	0	0
$397$	−13.2416	−0.664576	−0.332288	−	0.943178i	$-0.607821\pi$
$397$	−13.2416	−0.664576	−0.332288	+	0.943178i	$0.607821\pi$
$398$	0	0
$399$	13.8968	0.695710
$400$	0	0
$401$	14.5279	0.725490	0.362745	−	0.931888i	$-0.381840\pi$
$401$	14.5279	0.725490	0.362745	+	0.931888i	$0.381840\pi$
$402$	0	0
$403$	−2.70565	−0.134778
$404$	0	0
$405$	6.82447	0.339110
$406$	0	0
$407$	8.26469	0.409666
$408$	0	0
$409$	14.8855	0.736041	0.368021	−	0.929818i	$-0.380036\pi$
$409$	14.8855	0.736041	0.368021	+	0.929818i	$0.380036\pi$
$410$	0	0
$411$	25.3279	1.24933
$412$	0	0
$413$	27.6799	1.36204
$414$	0	0
$415$	4.41680	0.216812
$416$	0	0
$417$	−7.91313	−0.387508
$418$	0	0
$419$	16.7556	0.818564	0.409282	−	0.912408i	$-0.365779\pi$
$419$	16.7556	0.818564	0.409282	+	0.912408i	$0.365779\pi$
$420$	0	0
$421$	26.6622	1.29944	0.649719	−	0.760175i	$-0.274887\pi$
$421$	26.6622	1.29944	0.649719	+	0.760175i	$0.274887\pi$
$422$	0	0
$423$	3.07183	0.149358
$424$	0	0
$425$	−3.68235	−0.178620
$426$	0	0
$427$	−2.81816	−0.136380
$428$	0	0
$429$	12.6027	0.608464
$430$	0	0
$431$	−13.8811	−0.668631	−0.334315	−	0.942461i	$-0.608505\pi$
$431$	−13.8811	−0.668631	−0.334315	+	0.942461i	$0.608505\pi$
$432$	0	0
$433$	−12.2923	−0.590732	−0.295366	−	0.955384i	$-0.595442\pi$
$433$	−12.2923	−0.590732	−0.295366	+	0.955384i	$0.595442\pi$
$434$	0	0
$435$	−25.7001	−1.23222
$436$	0	0
$437$	−6.44652	−0.308379
$438$	0	0
$439$	−11.1793	−0.533561	−0.266780	−	0.963757i	$-0.585960\pi$
$439$	−11.1793	−0.533561	−0.266780	+	0.963757i	$0.585960\pi$
$440$	0	0
$441$	−0.393001	−0.0187143
$442$	0	0
$443$	−8.27722	−0.393263	−0.196631	−	0.980477i	$-0.563000\pi$
$443$	−8.27722	−0.393263	−0.196631	+	0.980477i	$0.563000\pi$
$444$	0	0
$445$	6.03745	0.286203
$446$	0	0
$447$	−6.04446	−0.285893
$448$	0	0
$449$	33.6636	1.58868	0.794342	−	0.607471i	$-0.207816\pi$
$449$	33.6636	1.58868	0.794342	+	0.607471i	$0.207816\pi$
$450$	0	0
$451$	−10.4765	−0.493320
$452$	0	0
$453$	2.95828	0.138992
$454$	0	0
$455$	−4.93490	−0.231351
$456$	0	0
$457$	5.27795	0.246892	0.123446	−	0.992351i	$-0.460605\pi$
$457$	5.27795	0.246892	0.123446	+	0.992351i	$0.460605\pi$
$458$	0	0
$459$	29.9722	1.39898
$460$	0	0
$461$	−13.2778	−0.618409	−0.309204	−	0.950996i	$-0.600063\pi$
$461$	−13.2778	−0.618409	−0.309204	+	0.950996i	$0.600063\pi$
$462$	0	0
$463$	−7.18073	−0.333717	−0.166859	−	0.985981i	$-0.553362\pi$
$463$	−7.18073	−0.333717	−0.166859	+	0.985981i	$0.553362\pi$
$464$	0	0
$465$	−4.27023	−0.198027
$466$	0	0
$467$	13.2462	0.612960	0.306480	−	0.951877i	$-0.400849\pi$
$467$	13.2462	0.612960	0.306480	+	0.951877i	$0.400849\pi$
$468$	0	0
$469$	32.6478	1.50753
$470$	0	0
$471$	40.8755	1.88344
$472$	0	0
$473$	16.1615	0.743105
$474$	0	0
$475$	−1.78425	−0.0818672
$476$	0	0
$477$	−52.2876	−2.39408
$478$	0	0
$479$	2.24047	0.102370	0.0511849	−	0.998689i	$-0.483700\pi$
$479$	2.24047	0.102370	0.0511849	+	0.998689i	$0.483700\pi$
$480$	0	0
$481$	−6.81585	−0.310776
$482$	0	0
$483$	−28.1401	−1.28042
$484$	0	0
$485$	−0.00385206	−0.000174913 0
$486$	0	0
$487$	−13.2937	−0.602394	−0.301197	−	0.953562i	$-0.597386\pi$
$487$	−13.2937	−0.602394	−0.301197	+	0.953562i	$0.597386\pi$
$488$	0	0
$489$	−34.9028	−1.57836
$490$	0	0
$491$	36.1062	1.62945	0.814724	−	0.579848i	$-0.196888\pi$
$491$	36.1062	1.62945	0.814724	+	0.579848i	$0.196888\pi$
$492$	0	0
$493$	−31.9905	−1.44078
$494$	0	0
$495$	13.0719	0.587539
$496$	0	0
$497$	−0.211598	−0.00949145
$498$	0	0
$499$	−24.6056	−1.10150	−0.550749	−	0.834671i	$-0.685658\pi$
$499$	−24.6056	−1.10150	−0.550749	+	0.834671i	$0.685658\pi$
$500$	0	0
$501$	35.2067	1.57292
$502$	0	0
$503$	−26.5686	−1.18464	−0.592318	−	0.805704i	$-0.701787\pi$
$503$	−26.5686	−1.18464	−0.592318	+	0.805704i	$0.701787\pi$
$504$	0	0
$505$	4.77354	0.212420
$506$	0	0
$507$	28.0642	1.24638
$508$	0	0
$509$	21.5028	0.953094	0.476547	−	0.879149i	$-0.341888\pi$
$509$	21.5028	0.953094	0.476547	+	0.879149i	$0.341888\pi$
$510$	0	0
$511$	−24.4496	−1.08159
$512$	0	0
$513$	14.5228	0.641198
$514$	0	0
$515$	−15.8678	−0.699218
$516$	0	0
$517$	1.21392	0.0533880
$518$	0	0
$519$	61.2525	2.68869
$520$	0	0
$521$	28.4415	1.24604	0.623022	−	0.782204i	$-0.285905\pi$
$521$	28.4415	1.24604	0.623022	+	0.782204i	$0.285905\pi$
$522$	0	0
$523$	15.8777	0.694282	0.347141	−	0.937813i	$-0.387153\pi$
$523$	15.8777	0.694282	0.347141	+	0.937813i	$0.387153\pi$
$524$	0	0
$525$	−7.78857	−0.339921
$526$	0	0
$527$	−5.31541	−0.231543
$528$	0	0
$529$	−9.94620	−0.432444
$530$	0	0
$531$	60.4672	2.62405
$532$	0	0
$533$	8.63994	0.374237
$534$	0	0
$535$	−6.68945	−0.289210
$536$	0	0
$537$	−20.6598	−0.891536
$538$	0	0
$539$	−0.155305	−0.00668945
$540$	0	0
$541$	−36.5312	−1.57060	−0.785299	−	0.619116i	$-0.787491\pi$
$541$	−36.5312	−1.57060	−0.785299	+	0.619116i	$0.787491\pi$
$542$	0	0
$543$	−64.6353	−2.77377
$544$	0	0
$545$	16.4266	0.703637
$546$	0	0
$547$	−39.0731	−1.67064	−0.835322	−	0.549761i	$-0.814719\pi$
$547$	−39.0731	−1.67064	−0.835322	+	0.549761i	$0.814719\pi$
$548$	0	0
$549$	−6.15632	−0.262745
$550$	0	0
$551$	−15.5007	−0.660353
$552$	0	0
$553$	15.2900	0.650196
$554$	0	0
$555$	−10.7572	−0.456619
$556$	0	0
$557$	−26.5209	−1.12373	−0.561863	−	0.827230i	$-0.689915\pi$
$557$	−26.5209	−1.12373	−0.561863	+	0.827230i	$0.689915\pi$
$558$	0	0
$559$	−13.3283	−0.563727
$560$	0	0
$561$	24.7588	1.04532
$562$	0	0
$563$	17.2778	0.728172	0.364086	−	0.931365i	$-0.381381\pi$
$563$	17.2778	0.728172	0.364086	+	0.931365i	$0.381381\pi$
$564$	0	0
$565$	−4.22914	−0.177921
$566$	0	0
$567$	17.9675	0.754564
$568$	0	0
$569$	−0.881311	−0.0369465	−0.0184732	−	0.999829i	$-0.505881\pi$
$569$	−0.881311	−0.0369465	−0.0184732	+	0.999829i	$0.505881\pi$
$570$	0	0
$571$	1.82820	0.0765077	0.0382538	−	0.999268i	$-0.487820\pi$
$571$	1.82820	0.0765077	0.0382538	+	0.999268i	$0.487820\pi$
$572$	0	0
$573$	−34.8862	−1.45739
$574$	0	0
$575$	3.61300	0.150673
$576$	0	0
$577$	22.0500	0.917953	0.458976	−	0.888449i	$-0.348216\pi$
$577$	22.0500	0.917953	0.458976	+	0.888449i	$0.348216\pi$
$578$	0	0
$579$	−28.7101	−1.19315
$580$	0	0
$581$	11.6286	0.482434
$582$	0	0
$583$	−20.6628	−0.855767
$584$	0	0
$585$	−10.7804	−0.445713
$586$	0	0
$587$	−12.2611	−0.506070	−0.253035	−	0.967457i	$-0.581429\pi$
$587$	−12.2611	−0.506070	−0.253035	+	0.967457i	$0.581429\pi$
$588$	0	0
$589$	−2.57554	−0.106123
$590$	0	0
$591$	5.72363	0.235439
$592$	0	0
$593$	−7.47214	−0.306844	−0.153422	−	0.988161i	$-0.549029\pi$
$593$	−7.47214	−0.306844	−0.153422	+	0.988161i	$0.549029\pi$
$594$	0	0
$595$	−9.69492	−0.397453
$596$	0	0
$597$	18.3515	0.751076
$598$	0	0
$599$	25.3561	1.03602	0.518011	−	0.855374i	$-0.326672\pi$
$599$	25.3561	1.03602	0.518011	+	0.855374i	$0.326672\pi$
$600$	0	0
$601$	−6.54158	−0.266837	−0.133418	−	0.991060i	$-0.542595\pi$
$601$	−6.54158	−0.266837	−0.133418	+	0.991060i	$0.542595\pi$
$602$	0	0
$603$	71.3196	2.90436
$604$	0	0
$605$	−5.83428	−0.237197
$606$	0	0
$607$	8.53095	0.346260	0.173130	−	0.984899i	$-0.444612\pi$
$607$	8.53095	0.346260	0.173130	+	0.984899i	$0.444612\pi$
$608$	0	0
$609$	−67.6633	−2.74186
$610$	0	0
$611$	−1.00111	−0.0405006
$612$	0	0
$613$	12.6239	0.509874	0.254937	−	0.966958i	$-0.417945\pi$
$613$	12.6239	0.509874	0.254937	+	0.966958i	$0.417945\pi$
$614$	0	0
$615$	13.6361	0.549861
$616$	0	0
$617$	−7.31605	−0.294533	−0.147266	−	0.989097i	$-0.547047\pi$
$617$	−7.31605	−0.294533	−0.147266	+	0.989097i	$0.547047\pi$
$618$	0	0
$619$	−34.9395	−1.40434	−0.702169	−	0.712011i	$-0.747785\pi$
$619$	−34.9395	−1.40434	−0.702169	+	0.712011i	$0.747785\pi$
$620$	0	0
$621$	−29.4078	−1.18009
$622$	0	0
$623$	15.8954	0.636838
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	11.9967	0.479101
$628$	0	0
$629$	−13.3902	−0.533902
$630$	0	0
$631$	−36.2550	−1.44329	−0.721644	−	0.692265i	$-0.756613\pi$
$631$	−36.2550	−1.44329	−0.721644	+	0.692265i	$0.756613\pi$
$632$	0	0
$633$	32.2022	1.27992
$634$	0	0
$635$	−0.855024	−0.0339306
$636$	0	0
$637$	0.128079	0.00507468
$638$	0	0
$639$	−0.462238	−0.0182859
$640$	0	0
$641$	27.0375	1.06792	0.533959	−	0.845510i	$-0.320704\pi$
$641$	27.0375	1.06792	0.533959	+	0.845510i	$0.320704\pi$
$642$	0	0
$643$	−14.0511	−0.554123	−0.277061	−	0.960852i	$-0.589361\pi$
$643$	−14.0511	−0.554123	−0.277061	+	0.960852i	$0.589361\pi$
$644$	0	0
$645$	−21.0356	−0.828275
$646$	0	0
$647$	3.17714	0.124906	0.0624531	−	0.998048i	$-0.480108\pi$
$647$	3.17714	0.124906	0.0624531	+	0.998048i	$0.480108\pi$
$648$	0	0
$649$	23.8952	0.937969
$650$	0	0
$651$	−11.2427	−0.440635
$652$	0	0
$653$	−26.3977	−1.03302	−0.516510	−	0.856281i	$-0.672769\pi$
$653$	−26.3977	−1.03302	−0.516510	+	0.856281i	$0.672769\pi$
$654$	0	0
$655$	−0.154714	−0.00604518
$656$	0	0
$657$	−53.4105	−2.08374
$658$	0	0
$659$	37.7136	1.46911	0.734557	−	0.678547i	$-0.237390\pi$
$659$	37.7136	1.46911	0.734557	+	0.678547i	$0.237390\pi$
$660$	0	0
$661$	−11.8757	−0.461912	−0.230956	−	0.972964i	$-0.574185\pi$
$661$	−11.8757	−0.461912	−0.230956	+	0.972964i	$0.574185\pi$
$662$	0	0
$663$	−20.4185	−0.792988
$664$	0	0
$665$	−4.69760	−0.182165
$666$	0	0
$667$	31.3880	1.21535
$668$	0	0
$669$	−16.5617	−0.640311
$670$	0	0
$671$	−2.43283	−0.0939185
$672$	0	0
$673$	−19.2637	−0.742561	−0.371280	−	0.928521i	$-0.621081\pi$
$673$	−19.2637	−0.742561	−0.371280	+	0.928521i	$0.621081\pi$
$674$	0	0
$675$	−8.13943	−0.313287
$676$	0	0
$677$	−36.4765	−1.40191	−0.700954	−	0.713207i	$-0.747242\pi$
$677$	−36.4765	−1.40191	−0.700954	+	0.713207i	$0.747242\pi$
$678$	0	0
$679$	−0.0101417	−0.000389204 0
$680$	0	0
$681$	32.2889	1.23731
$682$	0	0
$683$	41.3434	1.58196	0.790981	−	0.611840i	$-0.209571\pi$
$683$	41.3434	1.58196	0.790981	+	0.611840i	$0.209571\pi$
$684$	0	0
$685$	−8.56170	−0.327126
$686$	0	0
$687$	28.6489	1.09302
$688$	0	0
$689$	17.0405	0.649193
$690$	0	0
$691$	−31.5265	−1.19932	−0.599662	−	0.800253i	$-0.704698\pi$
$691$	−31.5265	−1.19932	−0.599662	+	0.800253i	$0.704698\pi$
$692$	0	0
$693$	34.4158	1.30735
$694$	0	0
$695$	2.67491	0.101465
$696$	0	0
$697$	16.9737	0.642925
$698$	0	0
$699$	−63.7464	−2.41111
$700$	0	0
$701$	27.3836	1.03427	0.517133	−	0.855905i	$-0.326999\pi$
$701$	27.3836	1.03427	0.517133	+	0.855905i	$0.326999\pi$
$702$	0	0
$703$	−6.48811	−0.244704
$704$	0	0
$705$	−1.58002	−0.0595069
$706$	0	0
$707$	12.5678	0.472661
$708$	0	0
$709$	−31.6921	−1.19022	−0.595110	−	0.803644i	$-0.702892\pi$
$709$	−31.6921	−1.19022	−0.595110	+	0.803644i	$0.702892\pi$
$710$	0	0
$711$	33.4012	1.25264
$712$	0	0
$713$	5.21531	0.195315
$714$	0	0
$715$	−4.26015	−0.159320
$716$	0	0
$717$	−0.603470	−0.0225370
$718$	0	0
$719$	−26.5223	−0.989117	−0.494558	−	0.869144i	$-0.664670\pi$
$719$	−26.5223	−0.989117	−0.494558	+	0.869144i	$0.664670\pi$
$720$	0	0
$721$	−41.7768	−1.55585
$722$	0	0
$723$	5.96699	0.221915
$724$	0	0
$725$	8.68751	0.322646
$726$	0	0
$727$	4.37175	0.162139	0.0810697	−	0.996708i	$-0.474166\pi$
$727$	4.37175	0.162139	0.0810697	+	0.996708i	$0.474166\pi$
$728$	0	0
$729$	−32.9858	−1.22170
$730$	0	0
$731$	−26.1843	−0.968461
$732$	0	0
$733$	19.3012	0.712906	0.356453	−	0.934313i	$-0.383986\pi$
$733$	19.3012	0.712906	0.356453	+	0.934313i	$0.383986\pi$
$734$	0	0
$735$	0.202143	0.00745615
$736$	0	0
$737$	28.1838	1.03816
$738$	0	0
$739$	19.8083	0.728661	0.364330	−	0.931270i	$-0.381298\pi$
$739$	19.8083	0.728661	0.364330	+	0.931270i	$0.381298\pi$
$740$	0	0
$741$	−9.89361	−0.363451
$742$	0	0
$743$	−13.5992	−0.498907	−0.249454	−	0.968387i	$-0.580251\pi$
$743$	−13.5992	−0.498907	−0.249454	+	0.968387i	$0.580251\pi$
$744$	0	0
$745$	2.04323	0.0748583
$746$	0	0
$747$	25.4028	0.929440
$748$	0	0
$749$	−17.6120	−0.643529
$750$	0	0
$751$	26.0789	0.951634	0.475817	−	0.879544i	$-0.342153\pi$
$751$	26.0789	0.951634	0.475817	+	0.879544i	$0.342153\pi$
$752$	0	0
$753$	37.3453	1.36094
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	4.00724	0.145646	0.0728229	−	0.997345i	$-0.476799\pi$
$757$	4.00724	0.145646	0.0728229	+	0.997345i	$0.476799\pi$
$758$	0	0
$759$	−24.2925	−0.881763
$760$	0	0
$761$	−8.19040	−0.296902	−0.148451	−	0.988920i	$-0.547429\pi$
$761$	−8.19040	−0.296902	−0.148451	+	0.988920i	$0.547429\pi$
$762$	0	0
$763$	43.2480	1.56568
$764$	0	0
$765$	−21.1787	−0.765718
$766$	0	0
$767$	−19.7063	−0.711552
$768$	0	0
$769$	32.6768	1.17836	0.589179	−	0.808003i	$-0.299451\pi$
$769$	32.6768	1.17836	0.589179	+	0.808003i	$0.299451\pi$
$770$	0	0
$771$	46.8948	1.68887
$772$	0	0
$773$	22.6483	0.814604	0.407302	−	0.913294i	$-0.366470\pi$
$773$	22.6483	0.814604	0.407302	+	0.913294i	$0.366470\pi$
$774$	0	0
$775$	1.44348	0.0518515
$776$	0	0
$777$	−28.3217	−1.01603
$778$	0	0
$779$	8.22447	0.294672
$780$	0	0
$781$	−0.182666	−0.00653630
$782$	0	0
$783$	−70.7114	−2.52702
$784$	0	0
$785$	−13.8173	−0.493161
$786$	0	0
$787$	−7.98409	−0.284602	−0.142301	−	0.989823i	$-0.545450\pi$
$787$	−7.98409	−0.284602	−0.142301	+	0.989823i	$0.545450\pi$
$788$	0	0
$789$	−56.8566	−2.02415
$790$	0	0
$791$	−11.1345	−0.395897
$792$	0	0
$793$	2.00635	0.0712475
$794$	0	0
$795$	26.8945	0.953849
$796$	0	0
$797$	25.4458	0.901337	0.450668	−	0.892691i	$-0.351186\pi$
$797$	25.4458	0.901337	0.450668	+	0.892691i	$0.351186\pi$
$798$	0	0
$799$	−1.96675	−0.0695785
$800$	0	0
$801$	34.7239	1.22691
$802$	0	0
$803$	−21.1066	−0.744836
$804$	0	0
$805$	9.51234	0.335266
$806$	0	0
$807$	−60.3155	−2.12321
$808$	0	0
$809$	18.2938	0.643176	0.321588	−	0.946880i	$-0.395783\pi$
$809$	18.2938	0.643176	0.321588	+	0.946880i	$0.395783\pi$
$810$	0	0
$811$	5.33827	0.187452	0.0937260	−	0.995598i	$-0.470122\pi$
$811$	5.33827	0.187452	0.0937260	+	0.995598i	$0.470122\pi$
$812$	0	0
$813$	−31.4566	−1.10323
$814$	0	0
$815$	11.7983	0.413278
$816$	0	0
$817$	−12.6874	−0.443876
$818$	0	0
$819$	−28.3826	−0.991768
$820$	0	0
$821$	−19.8205	−0.691739	−0.345869	−	0.938283i	$-0.612416\pi$
$821$	−19.8205	−0.691739	−0.345869	+	0.938283i	$0.612416\pi$
$822$	0	0
$823$	−28.0579	−0.978039	−0.489019	−	0.872273i	$-0.662645\pi$
$823$	−28.0579	−0.978039	−0.489019	+	0.872273i	$0.662645\pi$
$824$	0	0
$825$	−6.72364	−0.234087
$826$	0	0
$827$	−37.5004	−1.30402	−0.652008	−	0.758212i	$-0.726073\pi$
$827$	−37.5004	−1.30402	−0.652008	+	0.758212i	$0.726073\pi$
$828$	0	0
$829$	2.51068	0.0871994	0.0435997	−	0.999049i	$-0.486117\pi$
$829$	2.51068	0.0871994	0.0435997	+	0.999049i	$0.486117\pi$
$830$	0	0
$831$	−1.74248	−0.0604461
$832$	0	0
$833$	0.251620	0.00871810
$834$	0	0
$835$	−11.9011	−0.411854
$836$	0	0
$837$	−11.7491	−0.406109
$838$	0	0
$839$	−2.51030	−0.0866653	−0.0433326	−	0.999061i	$-0.513798\pi$
$839$	−2.51030	−0.0866653	−0.0433326	+	0.999061i	$0.513798\pi$
$840$	0	0
$841$	46.4728	1.60251
$842$	0	0
$843$	−68.8595	−2.37165
$844$	0	0
$845$	−9.48668	−0.326352
$846$	0	0
$847$	−15.3605	−0.527794
$848$	0	0
$849$	−88.5816	−3.04011
$850$	0	0
$851$	13.1380	0.450365
$852$	0	0
$853$	44.6364	1.52832	0.764160	−	0.645026i	$-0.223154\pi$
$853$	44.6364	1.52832	0.764160	+	0.645026i	$0.223154\pi$
$854$	0	0
$855$	−10.2620	−0.350952
$856$	0	0
$857$	49.3063	1.68427	0.842136	−	0.539266i	$-0.181298\pi$
$857$	49.3063	1.68427	0.842136	+	0.539266i	$0.181298\pi$
$858$	0	0
$859$	40.7004	1.38868	0.694340	−	0.719647i	$-0.255696\pi$
$859$	40.7004	1.38868	0.694340	+	0.719647i	$0.255696\pi$
$860$	0	0
$861$	35.9012	1.22351
$862$	0	0
$863$	23.7892	0.809795	0.404897	−	0.914362i	$-0.367307\pi$
$863$	23.7892	0.809795	0.404897	+	0.914362i	$0.367307\pi$
$864$	0	0
$865$	−20.7055	−0.704007
$866$	0	0
$867$	10.1773	0.345640
$868$	0	0
$869$	13.1994	0.447758
$870$	0	0
$871$	−23.2431	−0.787562
$872$	0	0
$873$	−0.0221548	−0.000749825 0
$874$	0	0
$875$	2.63281	0.0890051
$876$	0	0
$877$	−12.0265	−0.406105	−0.203053	−	0.979168i	$-0.565086\pi$
$877$	−12.0265	−0.406105	−0.203053	+	0.979168i	$0.565086\pi$
$878$	0	0
$879$	−17.9279	−0.604691
$880$	0	0
$881$	35.1023	1.18263	0.591313	−	0.806442i	$-0.298610\pi$
$881$	35.1023	1.18263	0.591313	+	0.806442i	$0.298610\pi$
$882$	0	0
$883$	4.47638	0.150642	0.0753210	−	0.997159i	$-0.476002\pi$
$883$	4.47638	0.150642	0.0753210	+	0.997159i	$0.476002\pi$
$884$	0	0
$885$	−31.1017	−1.04547
$886$	0	0
$887$	−8.76454	−0.294284	−0.147142	−	0.989115i	$-0.547008\pi$
$887$	−8.76454	−0.294284	−0.147142	+	0.989115i	$0.547008\pi$
$888$	0	0
$889$	−2.25111	−0.0754999
$890$	0	0
$891$	15.5108	0.519631
$892$	0	0
$893$	−0.952971	−0.0318900
$894$	0	0
$895$	6.98372	0.233440
$896$	0	0
$897$	20.0339	0.668914
$898$	0	0
$899$	12.5403	0.418242
$900$	0	0
$901$	33.4772	1.11529
$902$	0	0
$903$	−55.3826	−1.84302
$904$	0	0
$905$	21.8490	0.726285
$906$	0	0
$907$	30.8891	1.02566	0.512829	−	0.858491i	$-0.328598\pi$
$907$	30.8891	1.02566	0.512829	+	0.858491i	$0.328598\pi$
$908$	0	0
$909$	27.4546	0.910610
$910$	0	0
$911$	−44.6420	−1.47906	−0.739528	−	0.673126i	$-0.764951\pi$
$911$	−44.6420	−1.47906	−0.739528	+	0.673126i	$0.764951\pi$
$912$	0	0
$913$	10.0386	0.332229
$914$	0	0
$915$	3.16655	0.104683
$916$	0	0
$917$	−0.407332	−0.0134513
$918$	0	0
$919$	−9.92665	−0.327450	−0.163725	−	0.986506i	$-0.552351\pi$
$919$	−9.92665	−0.327450	−0.163725	+	0.986506i	$0.552351\pi$
$920$	0	0
$921$	−83.1974	−2.74145
$922$	0	0
$923$	0.150644	0.00495850
$924$	0	0
$925$	3.63631	0.119561
$926$	0	0
$927$	−91.2621	−2.99744
$928$	0	0
$929$	10.5436	0.345925	0.172963	−	0.984928i	$-0.444666\pi$
$929$	10.5436	0.345925	0.172963	+	0.984928i	$0.444666\pi$
$930$	0	0
$931$	0.121920	0.00399577
$932$	0	0
$933$	−82.9573	−2.71590
$934$	0	0
$935$	−8.36933	−0.273706
$936$	0	0
$937$	−1.20308	−0.0393030	−0.0196515	−	0.999807i	$-0.506256\pi$
$937$	−1.20308	−0.0393030	−0.0196515	+	0.999807i	$0.506256\pi$
$938$	0	0
$939$	−20.8354	−0.679938
$940$	0	0
$941$	−4.58892	−0.149594	−0.0747972	−	0.997199i	$-0.523831\pi$
$941$	−4.58892	−0.149594	−0.0747972	+	0.997199i	$0.523831\pi$
$942$	0	0
$943$	−16.6540	−0.542330
$944$	0	0
$945$	−21.4295	−0.697103
$946$	0	0
$947$	−14.7675	−0.479880	−0.239940	−	0.970788i	$-0.577128\pi$
$947$	−14.7675	−0.479880	−0.239940	+	0.970788i	$0.577128\pi$
$948$	0	0
$949$	17.4065	0.565040
$950$	0	0
$951$	11.3522	0.368120
$952$	0	0
$953$	24.6034	0.796981	0.398491	−	0.917172i	$-0.369534\pi$
$953$	24.6034	0.796981	0.398491	+	0.917172i	$0.369534\pi$
$954$	0	0
$955$	11.7927	0.381604
$956$	0	0
$957$	−58.4117	−1.88818
$958$	0	0
$959$	−22.5413	−0.727896
$960$	0	0
$961$	−28.9164	−0.932786
$962$	0	0
$963$	−38.4738	−1.23980
$964$	0	0
$965$	9.70501	0.312415
$966$	0	0
$967$	−51.8786	−1.66830	−0.834151	−	0.551536i	$-0.814042\pi$
$967$	−51.8786	−1.66830	−0.834151	+	0.551536i	$0.814042\pi$
$968$	0	0
$969$	−19.4366	−0.624394
$970$	0	0
$971$	15.2065	0.487999	0.244000	−	0.969775i	$-0.421540\pi$
$971$	15.2065	0.487999	0.244000	+	0.969775i	$0.421540\pi$
$972$	0	0
$973$	7.04252	0.225773
$974$	0	0
$975$	5.54495	0.177581
$976$	0	0
$977$	21.1635	0.677081	0.338540	−	0.940952i	$-0.390067\pi$
$977$	21.1635	0.677081	0.338540	+	0.940952i	$0.390067\pi$
$978$	0	0
$979$	13.7221	0.438559
$980$	0	0
$981$	94.4759	3.01638
$982$	0	0
$983$	−10.9519	−0.349310	−0.174655	−	0.984630i	$-0.555881\pi$
$983$	−10.9519	−0.349310	−0.174655	+	0.984630i	$0.555881\pi$
$984$	0	0
$985$	−1.93479	−0.0616474
$986$	0	0
$987$	−4.15988	−0.132411
$988$	0	0
$989$	25.6912	0.816932
$990$	0	0
$991$	48.1855	1.53066	0.765331	−	0.643637i	$-0.222575\pi$
$991$	48.1855	1.53066	0.765331	+	0.643637i	$0.222575\pi$
$992$	0	0
$993$	12.6315	0.400849
$994$	0	0
$995$	−6.20343	−0.196662
$996$	0	0
$997$	0.299205	0.00947592	0.00473796	−	0.999989i	$-0.498492\pi$
$997$	0.299205	0.00947592	0.00473796	+	0.999989i	$0.498492\pi$
$998$	0	0
$999$	−29.5975	−0.936424

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.1	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.1	✓	15	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)

\(f(q)\)	\(=\)	\(q-2.95828 q^{3} +1.00000 q^{5} +2.63281 q^{7} +5.75141 q^{9} +O(q^{10})\)	\(q-2.95828 q^{3} +1.00000 q^{5} +2.63281 q^{7} +5.75141 q^{9} +2.27282 q^{11} -1.87439 q^{13} -2.95828 q^{15} -3.68235 q^{17} -1.78425 q^{19} -7.78857 q^{21} +3.61300 q^{23} +1.00000 q^{25} -8.13943 q^{27} +8.68751 q^{29} +1.44348 q^{31} -6.72364 q^{33} +2.63281 q^{35} +3.63631 q^{37} +5.54495 q^{39} -4.60947 q^{41} +7.11075 q^{43} +5.75141 q^{45} +0.534101 q^{47} -0.0683312 q^{49} +10.8934 q^{51} -9.09126 q^{53} +2.27282 q^{55} +5.27832 q^{57} +10.5135 q^{59} -1.07040 q^{61} +15.1423 q^{63} -1.87439 q^{65} +12.4004 q^{67} -10.6883 q^{69} -0.0803696 q^{71} -9.28652 q^{73} -2.95828 q^{75} +5.98390 q^{77} +5.80748 q^{79} +6.82447 q^{81} +4.41680 q^{83} -3.68235 q^{85} -25.7001 q^{87} +6.03745 q^{89} -4.93490 q^{91} -4.27023 q^{93} -1.78425 q^{95} -0.00385206 q^{97} +13.0719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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