LMFDB - Embedded newform 6040.2.a.n.1.13

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 $ x^13 - 4x^12 - 14x^11 + 70x^10 + 41x^9 - 403x^8 + 109x^7 + 870x^6 - 444x^5 - 708x^4 + 373x^3 + 108x^2 - 82x + 11
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			1.13
Root			$-2.78181$ 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)

$f(q)$	$=$	$q+2.78181 q^{3} -1.00000 q^{5} -1.67394 q^{7} +4.73847 q^{9} +O(q^{10})$	$q+2.78181 q^{3} -1.00000 q^{5} -1.67394 q^{7} +4.73847 q^{9} -3.41127 q^{11} -0.294282 q^{13} -2.78181 q^{15} +1.48505 q^{17} -2.53267 q^{19} -4.65659 q^{21} +3.05047 q^{23} +1.00000 q^{25} +4.83608 q^{27} +1.45133 q^{29} -0.406937 q^{31} -9.48952 q^{33} +1.67394 q^{35} -7.08315 q^{37} -0.818636 q^{39} -8.70154 q^{41} -9.27859 q^{43} -4.73847 q^{45} -1.06067 q^{47} -4.19792 q^{49} +4.13113 q^{51} +9.86606 q^{53} +3.41127 q^{55} -7.04542 q^{57} -2.88487 q^{59} -8.14586 q^{61} -7.93192 q^{63} +0.294282 q^{65} +4.34761 q^{67} +8.48582 q^{69} +7.17770 q^{71} -9.44419 q^{73} +2.78181 q^{75} +5.71027 q^{77} +14.7552 q^{79} -0.762341 q^{81} -8.81009 q^{83} -1.48505 q^{85} +4.03731 q^{87} -15.2646 q^{89} +0.492611 q^{91} -1.13202 q^{93} +2.53267 q^{95} +7.86733 q^{97} -16.1642 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * 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
$3$	2.78181	1.60608	0.803039	−	0.595926i	$-0.203215\pi$
$3$	2.78181	1.60608	0.803039	+	0.595926i	$0.203215\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.67394	−0.632691	−0.316345	−	0.948644i	$-0.602456\pi$
$7$	−1.67394	−0.632691	−0.316345	+	0.948644i	$0.602456\pi$
$8$	0	0
$9$	4.73847	1.57949
$10$	0	0
$11$	−3.41127	−1.02854	−0.514269	−	0.857629i	$-0.671937\pi$
$11$	−3.41127	−1.02854	−0.514269	+	0.857629i	$0.671937\pi$
$12$	0	0
$13$	−0.294282	−0.0816191	−0.0408095	−	0.999167i	$-0.512994\pi$
$13$	−0.294282	−0.0816191	−0.0408095	+	0.999167i	$0.512994\pi$
$14$	0	0
$15$	−2.78181	−0.718260
$16$	0	0
$17$	1.48505	0.360178	0.180089	−	0.983650i	$-0.442361\pi$
$17$	1.48505	0.360178	0.180089	+	0.983650i	$0.442361\pi$
$18$	0	0
$19$	−2.53267	−0.581035	−0.290518	−	0.956870i	$-0.593828\pi$
$19$	−2.53267	−0.581035	−0.290518	+	0.956870i	$0.593828\pi$
$20$	0	0
$21$	−4.65659	−1.01615
$22$	0	0
$23$	3.05047	0.636066	0.318033	−	0.948080i	$-0.396978\pi$
$23$	3.05047	0.636066	0.318033	+	0.948080i	$0.396978\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.83608	0.930704
$28$	0	0
$29$	1.45133	0.269505	0.134752	−	0.990879i	$-0.456976\pi$
$29$	1.45133	0.269505	0.134752	+	0.990879i	$0.456976\pi$
$30$	0	0
$31$	−0.406937	−0.0730881	−0.0365440	−	0.999332i	$-0.511635\pi$
$31$	−0.406937	−0.0730881	−0.0365440	+	0.999332i	$0.511635\pi$
$32$	0	0
$33$	−9.48952	−1.65191
$34$	0	0
$35$	1.67394	0.282948
$36$	0	0
$37$	−7.08315	−1.16446	−0.582231	−	0.813023i	$-0.697820\pi$
$37$	−7.08315	−1.16446	−0.582231	+	0.813023i	$0.697820\pi$
$38$	0	0
$39$	−0.818636	−0.131087
$40$	0	0
$41$	−8.70154	−1.35895	−0.679476	−	0.733697i	$-0.737793\pi$
$41$	−8.70154	−1.35895	−0.679476	+	0.733697i	$0.737793\pi$
$42$	0	0
$43$	−9.27859	−1.41497	−0.707486	−	0.706727i	$-0.750171\pi$
$43$	−9.27859	−1.41497	−0.707486	+	0.706727i	$0.750171\pi$
$44$	0	0
$45$	−4.73847	−0.706369
$46$	0	0
$47$	−1.06067	−0.154715	−0.0773576	−	0.997003i	$-0.524648\pi$
$47$	−1.06067	−0.154715	−0.0773576	+	0.997003i	$0.524648\pi$
$48$	0	0
$49$	−4.19792	−0.599703
$50$	0	0
$51$	4.13113	0.578474
$52$	0	0
$53$	9.86606	1.35521	0.677604	−	0.735427i	$-0.263018\pi$
$53$	9.86606	1.35521	0.677604	+	0.735427i	$0.263018\pi$
$54$	0	0
$55$	3.41127	0.459976
$56$	0	0
$57$	−7.04542	−0.933188
$58$	0	0
$59$	−2.88487	−0.375578	−0.187789	−	0.982209i	$-0.560132\pi$
$59$	−2.88487	−0.375578	−0.187789	+	0.982209i	$0.560132\pi$
$60$	0	0
$61$	−8.14586	−1.04297	−0.521486	−	0.853260i	$-0.674622\pi$
$61$	−8.14586	−1.04297	−0.521486	+	0.853260i	$0.674622\pi$
$62$	0	0
$63$	−7.93192	−0.999327
$64$	0	0
$65$	0.294282	0.0365012
$66$	0	0
$67$	4.34761	0.531145	0.265572	−	0.964091i	$-0.414439\pi$
$67$	4.34761	0.531145	0.265572	+	0.964091i	$0.414439\pi$
$68$	0	0
$69$	8.48582	1.02157
$70$	0	0
$71$	7.17770	0.851836	0.425918	−	0.904762i	$-0.359951\pi$
$71$	7.17770	0.851836	0.425918	+	0.904762i	$0.359951\pi$
$72$	0	0
$73$	−9.44419	−1.10536	−0.552679	−	0.833394i	$-0.686395\pi$
$73$	−9.44419	−1.10536	−0.552679	+	0.833394i	$0.686395\pi$
$74$	0	0
$75$	2.78181	0.321216
$76$	0	0
$77$	5.71027	0.650746
$78$	0	0
$79$	14.7552	1.66009	0.830047	−	0.557693i	$-0.188313\pi$
$79$	14.7552	1.66009	0.830047	+	0.557693i	$0.188313\pi$
$80$	0	0
$81$	−0.762341	−0.0847045
$82$	0	0
$83$	−8.81009	−0.967033	−0.483516	−	0.875335i	$-0.660641\pi$
$83$	−8.81009	−0.967033	−0.483516	+	0.875335i	$0.660641\pi$
$84$	0	0
$85$	−1.48505	−0.161076
$86$	0	0
$87$	4.03731	0.432846
$88$	0	0
$89$	−15.2646	−1.61805	−0.809024	−	0.587776i	$-0.800004\pi$
$89$	−15.2646	−1.61805	−0.809024	+	0.587776i	$0.800004\pi$
$90$	0	0
$91$	0.492611	0.0516396
$92$	0	0
$93$	−1.13202	−0.117385
$94$	0	0
$95$	2.53267	0.259847
$96$	0	0
$97$	7.86733	0.798806	0.399403	−	0.916775i	$-0.369217\pi$
$97$	7.86733	0.798806	0.399403	+	0.916775i	$0.369217\pi$
$98$	0	0
$99$	−16.1642	−1.62456
$100$	0	0
$101$	−15.9421	−1.58630	−0.793151	−	0.609025i	$-0.791561\pi$
$101$	−15.9421	−1.58630	−0.793151	+	0.609025i	$0.791561\pi$
$102$	0	0
$103$	−15.4202	−1.51940	−0.759700	−	0.650274i	$-0.774654\pi$
$103$	−15.4202	−1.51940	−0.759700	+	0.650274i	$0.774654\pi$
$104$	0	0
$105$	4.65659	0.454436
$106$	0	0
$107$	17.8228	1.72299	0.861497	−	0.507762i	$-0.169527\pi$
$107$	17.8228	1.72299	0.861497	+	0.507762i	$0.169527\pi$
$108$	0	0
$109$	−6.83739	−0.654903	−0.327452	−	0.944868i	$-0.606190\pi$
$109$	−6.83739	−0.654903	−0.327452	+	0.944868i	$0.606190\pi$
$110$	0	0
$111$	−19.7040	−1.87022
$112$	0	0
$113$	9.74237	0.916485	0.458242	−	0.888827i	$-0.348479\pi$
$113$	9.74237	0.916485	0.458242	+	0.888827i	$0.348479\pi$
$114$	0	0
$115$	−3.05047	−0.284458
$116$	0	0
$117$	−1.39444	−0.128916
$118$	0	0
$119$	−2.48589	−0.227881
$120$	0	0
$121$	0.636791	0.0578901
$122$	0	0
$123$	−24.2060	−2.18258
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.2780	−1.08950	−0.544748	−	0.838600i	$-0.683375\pi$
$127$	−12.2780	−1.08950	−0.544748	+	0.838600i	$0.683375\pi$
$128$	0	0
$129$	−25.8113	−2.27256
$130$	0	0
$131$	0.171614	0.0149940	0.00749698	−	0.999972i	$-0.497614\pi$
$131$	0.171614	0.0149940	0.00749698	+	0.999972i	$0.497614\pi$
$132$	0	0
$133$	4.23955	0.367616
$134$	0	0
$135$	−4.83608	−0.416224
$136$	0	0
$137$	−10.0684	−0.860201	−0.430101	−	0.902781i	$-0.641522\pi$
$137$	−10.0684	−0.860201	−0.430101	+	0.902781i	$0.641522\pi$
$138$	0	0
$139$	4.83650	0.410226	0.205113	−	0.978738i	$-0.434244\pi$
$139$	4.83650	0.410226	0.205113	+	0.978738i	$0.434244\pi$
$140$	0	0
$141$	−2.95059	−0.248485
$142$	0	0
$143$	1.00388	0.0839483
$144$	0	0
$145$	−1.45133	−0.120526
$146$	0	0
$147$	−11.6778	−0.963170
$148$	0	0
$149$	−2.24631	−0.184025	−0.0920124	−	0.995758i	$-0.529330\pi$
$149$	−2.24631	−0.184025	−0.0920124	+	0.995758i	$0.529330\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	7.03686	0.568896
$154$	0	0
$155$	0.406937	0.0326860
$156$	0	0
$157$	19.5947	1.56383	0.781913	−	0.623388i	$-0.214244\pi$
$157$	19.5947	1.56383	0.781913	+	0.623388i	$0.214244\pi$
$158$	0	0
$159$	27.4455	2.17657
$160$	0	0
$161$	−5.10631	−0.402433
$162$	0	0
$163$	6.22056	0.487232	0.243616	−	0.969872i	$-0.421666\pi$
$163$	6.22056	0.487232	0.243616	+	0.969872i	$0.421666\pi$
$164$	0	0
$165$	9.48952	0.738758
$166$	0	0
$167$	−10.1137	−0.782625	−0.391313	−	0.920258i	$-0.627979\pi$
$167$	−10.1137	−0.782625	−0.391313	+	0.920258i	$0.627979\pi$
$168$	0	0
$169$	−12.9134	−0.993338
$170$	0	0
$171$	−12.0010	−0.917739
$172$	0	0
$173$	10.9544	0.832850	0.416425	−	0.909170i	$-0.363283\pi$
$173$	10.9544	0.832850	0.416425	+	0.909170i	$0.363283\pi$
$174$	0	0
$175$	−1.67394	−0.126538
$176$	0	0
$177$	−8.02516	−0.603208
$178$	0	0
$179$	−10.2384	−0.765256	−0.382628	−	0.923903i	$-0.624981\pi$
$179$	−10.2384	−0.765256	−0.382628	+	0.923903i	$0.624981\pi$
$180$	0	0
$181$	22.4447	1.66830	0.834151	−	0.551536i	$-0.185958\pi$
$181$	22.4447	1.66830	0.834151	+	0.551536i	$0.185958\pi$
$182$	0	0
$183$	−22.6602	−1.67509
$184$	0	0
$185$	7.08315	0.520763
$186$	0	0
$187$	−5.06591	−0.370456
$188$	0	0
$189$	−8.09532	−0.588848
$190$	0	0
$191$	−18.2452	−1.32018	−0.660088	−	0.751188i	$-0.729481\pi$
$191$	−18.2452	−1.32018	−0.660088	+	0.751188i	$0.729481\pi$
$192$	0	0
$193$	−23.1598	−1.66708	−0.833541	−	0.552457i	$-0.813690\pi$
$193$	−23.1598	−1.66708	−0.833541	+	0.552457i	$0.813690\pi$
$194$	0	0
$195$	0.818636	0.0586237
$196$	0	0
$197$	−0.381127	−0.0271542	−0.0135771	−	0.999908i	$-0.504322\pi$
$197$	−0.381127	−0.0271542	−0.0135771	+	0.999908i	$0.504322\pi$
$198$	0	0
$199$	12.4940	0.885678	0.442839	−	0.896601i	$-0.353971\pi$
$199$	12.4940	0.885678	0.442839	+	0.896601i	$0.353971\pi$
$200$	0	0
$201$	12.0942	0.853060
$202$	0	0
$203$	−2.42944	−0.170513
$204$	0	0
$205$	8.70154	0.607742
$206$	0	0
$207$	14.4545	1.00466
$208$	0	0
$209$	8.63965	0.597617
$210$	0	0
$211$	−14.2386	−0.980228	−0.490114	−	0.871658i	$-0.663045\pi$
$211$	−14.2386	−0.980228	−0.490114	+	0.871658i	$0.663045\pi$
$212$	0	0
$213$	19.9670	1.36812
$214$	0	0
$215$	9.27859	0.632795
$216$	0	0
$217$	0.681189	0.0462421
$218$	0	0
$219$	−26.2719	−1.77529
$220$	0	0
$221$	−0.437023	−0.0293974
$222$	0	0
$223$	−2.12708	−0.142440	−0.0712199	−	0.997461i	$-0.522689\pi$
$223$	−2.12708	−0.142440	−0.0712199	+	0.997461i	$0.522689\pi$
$224$	0	0
$225$	4.73847	0.315898
$226$	0	0
$227$	−14.6909	−0.975072	−0.487536	−	0.873103i	$-0.662104\pi$
$227$	−14.6909	−0.975072	−0.487536	+	0.873103i	$0.662104\pi$
$228$	0	0
$229$	22.9497	1.51656	0.758279	−	0.651930i	$-0.226040\pi$
$229$	22.9497	1.51656	0.758279	+	0.651930i	$0.226040\pi$
$230$	0	0
$231$	15.8849	1.04515
$232$	0	0
$233$	−17.6978	−1.15942	−0.579710	−	0.814823i	$-0.696834\pi$
$233$	−17.6978	−1.15942	−0.579710	+	0.814823i	$0.696834\pi$
$234$	0	0
$235$	1.06067	0.0691907
$236$	0	0
$237$	41.0463	2.66624
$238$	0	0
$239$	9.47461	0.612862	0.306431	−	0.951893i	$-0.400865\pi$
$239$	9.47461	0.612862	0.306431	+	0.951893i	$0.400865\pi$
$240$	0	0
$241$	8.46067	0.545000	0.272500	−	0.962156i	$-0.412150\pi$
$241$	8.46067	0.545000	0.272500	+	0.962156i	$0.412150\pi$
$242$	0	0
$243$	−16.6289	−1.06675
$244$	0	0
$245$	4.19792	0.268195
$246$	0	0
$247$	0.745320	0.0474236
$248$	0	0
$249$	−24.5080	−1.55313
$250$	0	0
$251$	6.05619	0.382264	0.191132	−	0.981564i	$-0.438784\pi$
$251$	6.05619	0.382264	0.191132	+	0.981564i	$0.438784\pi$
$252$	0	0
$253$	−10.4060	−0.654218
$254$	0	0
$255$	−4.13113	−0.258701
$256$	0	0
$257$	14.4430	0.900932	0.450466	−	0.892794i	$-0.351258\pi$
$257$	14.4430	0.900932	0.450466	+	0.892794i	$0.351258\pi$
$258$	0	0
$259$	11.8568	0.736744
$260$	0	0
$261$	6.87706	0.425679
$262$	0	0
$263$	−14.5631	−0.898001	−0.449001	−	0.893531i	$-0.648220\pi$
$263$	−14.5631	−0.898001	−0.449001	+	0.893531i	$0.648220\pi$
$264$	0	0
$265$	−9.86606	−0.606067
$266$	0	0
$267$	−42.4633	−2.59871
$268$	0	0
$269$	2.02685	0.123579	0.0617897	−	0.998089i	$-0.480319\pi$
$269$	2.02685	0.123579	0.0617897	+	0.998089i	$0.480319\pi$
$270$	0	0
$271$	5.73756	0.348532	0.174266	−	0.984699i	$-0.444245\pi$
$271$	5.73756	0.348532	0.174266	+	0.984699i	$0.444245\pi$
$272$	0	0
$273$	1.37035	0.0829373
$274$	0	0
$275$	−3.41127	−0.205708
$276$	0	0
$277$	3.32687	0.199892	0.0999462	−	0.994993i	$-0.468133\pi$
$277$	3.32687	0.199892	0.0999462	+	0.994993i	$0.468133\pi$
$278$	0	0
$279$	−1.92826	−0.115442
$280$	0	0
$281$	−23.1191	−1.37917	−0.689584	−	0.724206i	$-0.742207\pi$
$281$	−23.1191	−1.37917	−0.689584	+	0.724206i	$0.742207\pi$
$282$	0	0
$283$	−6.49216	−0.385919	−0.192959	−	0.981207i	$-0.561809\pi$
$283$	−6.49216	−0.385919	−0.192959	+	0.981207i	$0.561809\pi$
$284$	0	0
$285$	7.04542	0.417335
$286$	0	0
$287$	14.5659	0.859796
$288$	0	0
$289$	−14.7946	−0.870272
$290$	0	0
$291$	21.8854	1.28295
$292$	0	0
$293$	−13.0045	−0.759729	−0.379865	−	0.925042i	$-0.624029\pi$
$293$	−13.0045	−0.759729	−0.379865	+	0.925042i	$0.624029\pi$
$294$	0	0
$295$	2.88487	0.167964
$296$	0	0
$297$	−16.4972	−0.957264
$298$	0	0
$299$	−0.897697	−0.0519152
$300$	0	0
$301$	15.5318	0.895240
$302$	0	0
$303$	−44.3480	−2.54773
$304$	0	0
$305$	8.14586	0.466431
$306$	0	0
$307$	24.8426	1.41784	0.708922	−	0.705287i	$-0.249182\pi$
$307$	24.8426	1.41784	0.708922	+	0.705287i	$0.249182\pi$
$308$	0	0
$309$	−42.8961	−2.44028
$310$	0	0
$311$	−2.99643	−0.169912	−0.0849560	−	0.996385i	$-0.527075\pi$
$311$	−2.99643	−0.169912	−0.0849560	+	0.996385i	$0.527075\pi$
$312$	0	0
$313$	−12.3770	−0.699588	−0.349794	−	0.936827i	$-0.613748\pi$
$313$	−12.3770	−0.699588	−0.349794	+	0.936827i	$0.613748\pi$
$314$	0	0
$315$	7.93192	0.446913
$316$	0	0
$317$	−6.70337	−0.376499	−0.188249	−	0.982121i	$-0.560281\pi$
$317$	−6.70337	−0.376499	−0.188249	+	0.982121i	$0.560281\pi$
$318$	0	0
$319$	−4.95087	−0.277196
$320$	0	0
$321$	49.5796	2.76727
$322$	0	0
$323$	−3.76115	−0.209276
$324$	0	0
$325$	−0.294282	−0.0163238
$326$	0	0
$327$	−19.0203	−1.05183
$328$	0	0
$329$	1.77551	0.0978868
$330$	0	0
$331$	7.63666	0.419749	0.209874	−	0.977728i	$-0.432695\pi$
$331$	7.63666	0.419749	0.209874	+	0.977728i	$0.432695\pi$
$332$	0	0
$333$	−33.5632	−1.83925
$334$	0	0
$335$	−4.34761	−0.237535
$336$	0	0
$337$	−16.6635	−0.907718	−0.453859	−	0.891073i	$-0.649953\pi$
$337$	−16.6635	−0.907718	−0.453859	+	0.891073i	$0.649953\pi$
$338$	0	0
$339$	27.1014	1.47195
$340$	0	0
$341$	1.38817	0.0751739
$342$	0	0
$343$	18.7447	1.01212
$344$	0	0
$345$	−8.48582	−0.456861
$346$	0	0
$347$	35.7180	1.91744	0.958721	−	0.284349i	$-0.0917774\pi$
$347$	35.7180	1.91744	0.958721	+	0.284349i	$0.0917774\pi$
$348$	0	0
$349$	18.2771	0.978351	0.489176	−	0.872185i	$-0.337298\pi$
$349$	18.2771	0.978351	0.489176	+	0.872185i	$0.337298\pi$
$350$	0	0
$351$	−1.42317	−0.0759632
$352$	0	0
$353$	3.91623	0.208440	0.104220	−	0.994554i	$-0.466765\pi$
$353$	3.91623	0.208440	0.104220	+	0.994554i	$0.466765\pi$
$354$	0	0
$355$	−7.17770	−0.380953
$356$	0	0
$357$	−6.91527	−0.365995
$358$	0	0
$359$	−35.8478	−1.89198	−0.945988	−	0.324201i	$-0.894905\pi$
$359$	−35.8478	−1.89198	−0.945988	+	0.324201i	$0.894905\pi$
$360$	0	0
$361$	−12.5856	−0.662398
$362$	0	0
$363$	1.77143	0.0929760
$364$	0	0
$365$	9.44419	0.494331
$366$	0	0
$367$	34.4451	1.79802	0.899011	−	0.437926i	$-0.144287\pi$
$367$	34.4451	1.79802	0.899011	+	0.437926i	$0.144287\pi$
$368$	0	0
$369$	−41.2320	−2.14645
$370$	0	0
$371$	−16.5152	−0.857427
$372$	0	0
$373$	20.7875	1.07634	0.538169	−	0.842837i	$-0.319116\pi$
$373$	20.7875	1.07634	0.538169	+	0.842837i	$0.319116\pi$
$374$	0	0
$375$	−2.78181	−0.143652
$376$	0	0
$377$	−0.427099	−0.0219967
$378$	0	0
$379$	−3.44363	−0.176887	−0.0884437	−	0.996081i	$-0.528189\pi$
$379$	−3.44363	−0.176887	−0.0884437	+	0.996081i	$0.528189\pi$
$380$	0	0
$381$	−34.1550	−1.74981
$382$	0	0
$383$	−4.20901	−0.215070	−0.107535	−	0.994201i	$-0.534296\pi$
$383$	−4.20901	−0.215070	−0.107535	+	0.994201i	$0.534296\pi$
$384$	0	0
$385$	−5.71027	−0.291023
$386$	0	0
$387$	−43.9663	−2.23493
$388$	0	0
$389$	−22.8921	−1.16067	−0.580337	−	0.814376i	$-0.697079\pi$
$389$	−22.8921	−1.16067	−0.580337	+	0.814376i	$0.697079\pi$
$390$	0	0
$391$	4.53010	0.229097
$392$	0	0
$393$	0.477397	0.0240815
$394$	0	0
$395$	−14.7552	−0.742417
$396$	0	0
$397$	−10.0808	−0.505939	−0.252969	−	0.967474i	$-0.581407\pi$
$397$	−10.0808	−0.505939	−0.252969	+	0.967474i	$0.581407\pi$
$398$	0	0
$399$	11.7936	0.590419
$400$	0	0
$401$	35.7492	1.78523	0.892615	−	0.450819i	$-0.148868\pi$
$401$	35.7492	1.78523	0.892615	+	0.450819i	$0.148868\pi$
$402$	0	0
$403$	0.119754	0.00596538
$404$	0	0
$405$	0.762341	0.0378810
$406$	0	0
$407$	24.1626	1.19769
$408$	0	0
$409$	0.426187	0.0210736	0.0105368	−	0.999944i	$-0.496646\pi$
$409$	0.426187	0.0210736	0.0105368	+	0.999944i	$0.496646\pi$
$410$	0	0
$411$	−28.0084	−1.38155
$412$	0	0
$413$	4.82910	0.237625
$414$	0	0
$415$	8.81009	0.432470
$416$	0	0
$417$	13.4542	0.658856
$418$	0	0
$419$	−6.28123	−0.306858	−0.153429	−	0.988160i	$-0.549032\pi$
$419$	−6.28123	−0.306858	−0.153429	+	0.988160i	$0.549032\pi$
$420$	0	0
$421$	−6.13086	−0.298800	−0.149400	−	0.988777i	$-0.547734\pi$
$421$	−6.13086	−0.298800	−0.149400	+	0.988777i	$0.547734\pi$
$422$	0	0
$423$	−5.02597	−0.244371
$424$	0	0
$425$	1.48505	0.0720355
$426$	0	0
$427$	13.6357	0.659878
$428$	0	0
$429$	2.79259	0.134828
$430$	0	0
$431$	25.3208	1.21966	0.609829	−	0.792533i	$-0.291238\pi$
$431$	25.3208	1.21966	0.609829	+	0.792533i	$0.291238\pi$
$432$	0	0
$433$	29.9814	1.44081	0.720407	−	0.693552i	$-0.243955\pi$
$433$	29.9814	1.44081	0.720407	+	0.693552i	$0.243955\pi$
$434$	0	0
$435$	−4.03731	−0.193574
$436$	0	0
$437$	−7.72584	−0.369577
$438$	0	0
$439$	16.2520	0.775667	0.387834	−	0.921729i	$-0.373224\pi$
$439$	16.2520	0.775667	0.387834	+	0.921729i	$0.373224\pi$
$440$	0	0
$441$	−19.8917	−0.947224
$442$	0	0
$443$	17.5817	0.835333	0.417667	−	0.908600i	$-0.362848\pi$
$443$	17.5817	0.835333	0.417667	+	0.908600i	$0.362848\pi$
$444$	0	0
$445$	15.2646	0.723613
$446$	0	0
$447$	−6.24880	−0.295558
$448$	0	0
$449$	−22.0957	−1.04276	−0.521381	−	0.853324i	$-0.674583\pi$
$449$	−22.0957	−1.04276	−0.521381	+	0.853324i	$0.674583\pi$
$450$	0	0
$451$	29.6833	1.39773
$452$	0	0
$453$	2.78181	0.130701
$454$	0	0
$455$	−0.492611	−0.0230939
$456$	0	0
$457$	−9.64916	−0.451369	−0.225684	−	0.974200i	$-0.572462\pi$
$457$	−9.64916	−0.451369	−0.225684	+	0.974200i	$0.572462\pi$
$458$	0	0
$459$	7.18182	0.335219
$460$	0	0
$461$	34.6855	1.61546	0.807731	−	0.589551i	$-0.200695\pi$
$461$	34.6855	1.61546	0.807731	+	0.589551i	$0.200695\pi$
$462$	0	0
$463$	32.6710	1.51835	0.759175	−	0.650886i	$-0.225602\pi$
$463$	32.6710	1.51835	0.759175	+	0.650886i	$0.225602\pi$
$464$	0	0
$465$	1.13202	0.0524963
$466$	0	0
$467$	−4.86911	−0.225316	−0.112658	−	0.993634i	$-0.535936\pi$
$467$	−4.86911	−0.225316	−0.112658	+	0.993634i	$0.535936\pi$
$468$	0	0
$469$	−7.27764	−0.336050
$470$	0	0
$471$	54.5087	2.51163
$472$	0	0
$473$	31.6518	1.45535
$474$	0	0
$475$	−2.53267	−0.116207
$476$	0	0
$477$	46.7500	2.14054
$478$	0	0
$479$	12.3217	0.562992	0.281496	−	0.959562i	$-0.409169\pi$
$479$	12.3217	0.562992	0.281496	+	0.959562i	$0.409169\pi$
$480$	0	0
$481$	2.08444	0.0950423
$482$	0	0
$483$	−14.2048	−0.646339
$484$	0	0
$485$	−7.86733	−0.357237
$486$	0	0
$487$	−2.25883	−0.102357	−0.0511787	−	0.998690i	$-0.516298\pi$
$487$	−2.25883	−0.102357	−0.0511787	+	0.998690i	$0.516298\pi$
$488$	0	0
$489$	17.3044	0.782533
$490$	0	0
$491$	4.03311	0.182012	0.0910058	−	0.995850i	$-0.470992\pi$
$491$	4.03311	0.182012	0.0910058	+	0.995850i	$0.470992\pi$
$492$	0	0
$493$	2.15529	0.0970695
$494$	0	0
$495$	16.1642	0.726527
$496$	0	0
$497$	−12.0150	−0.538948
$498$	0	0
$499$	25.6356	1.14761	0.573804	−	0.818993i	$-0.305467\pi$
$499$	25.6356	1.14761	0.573804	+	0.818993i	$0.305467\pi$
$500$	0	0
$501$	−28.1345	−1.25696
$502$	0	0
$503$	−10.7105	−0.477557	−0.238778	−	0.971074i	$-0.576747\pi$
$503$	−10.7105	−0.477557	−0.238778	+	0.971074i	$0.576747\pi$
$504$	0	0
$505$	15.9421	0.709416
$506$	0	0
$507$	−35.9226	−1.59538
$508$	0	0
$509$	−30.3812	−1.34662	−0.673311	−	0.739359i	$-0.735129\pi$
$509$	−30.3812	−1.34662	−0.673311	+	0.739359i	$0.735129\pi$
$510$	0	0
$511$	15.8090	0.699350
$512$	0	0
$513$	−12.2482	−0.540772
$514$	0	0
$515$	15.4202	0.679496
$516$	0	0
$517$	3.61825	0.159130
$518$	0	0
$519$	30.4731	1.33762
$520$	0	0
$521$	20.3721	0.892516	0.446258	−	0.894904i	$-0.352756\pi$
$521$	20.3721	0.892516	0.446258	+	0.894904i	$0.352756\pi$
$522$	0	0
$523$	24.6572	1.07819	0.539093	−	0.842246i	$-0.318767\pi$
$523$	24.6572	1.07819	0.539093	+	0.842246i	$0.318767\pi$
$524$	0	0
$525$	−4.65659	−0.203230
$526$	0	0
$527$	−0.604322	−0.0263247
$528$	0	0
$529$	−13.6946	−0.595419
$530$	0	0
$531$	−13.6699	−0.593221
$532$	0	0
$533$	2.56071	0.110916
$534$	0	0
$535$	−17.8228	−0.770547
$536$	0	0
$537$	−28.4813	−1.22906
$538$	0	0
$539$	14.3203	0.616817
$540$	0	0
$541$	−2.11835	−0.0910749	−0.0455374	−	0.998963i	$-0.514500\pi$
$541$	−2.11835	−0.0910749	−0.0455374	+	0.998963i	$0.514500\pi$
$542$	0	0
$543$	62.4369	2.67942
$544$	0	0
$545$	6.83739	0.292882
$546$	0	0
$547$	24.9630	1.06734	0.533671	−	0.845692i	$-0.320812\pi$
$547$	24.9630	1.06734	0.533671	+	0.845692i	$0.320812\pi$
$548$	0	0
$549$	−38.5989	−1.64736
$550$	0	0
$551$	−3.67574	−0.156592
$552$	0	0
$553$	−24.6994	−1.05033
$554$	0	0
$555$	19.7040	0.836387
$556$	0	0
$557$	−0.252010	−0.0106780	−0.00533900	−	0.999986i	$-0.501699\pi$
$557$	−0.252010	−0.0106780	−0.00533900	+	0.999986i	$0.501699\pi$
$558$	0	0
$559$	2.73052	0.115489
$560$	0	0
$561$	−14.0924	−0.594982
$562$	0	0
$563$	−7.46310	−0.314532	−0.157266	−	0.987556i	$-0.550268\pi$
$563$	−7.46310	−0.314532	−0.157266	+	0.987556i	$0.550268\pi$
$564$	0	0
$565$	−9.74237	−0.409865
$566$	0	0
$567$	1.27611	0.0535918
$568$	0	0
$569$	0.905755	0.0379712	0.0189856	−	0.999820i	$-0.493956\pi$
$569$	0.905755	0.0379712	0.0189856	+	0.999820i	$0.493956\pi$
$570$	0	0
$571$	−3.02201	−0.126467	−0.0632336	−	0.997999i	$-0.520141\pi$
$571$	−3.02201	−0.126467	−0.0632336	+	0.997999i	$0.520141\pi$
$572$	0	0
$573$	−50.7547	−2.12031
$574$	0	0
$575$	3.05047	0.127213
$576$	0	0
$577$	−14.9909	−0.624079	−0.312039	−	0.950069i	$-0.601012\pi$
$577$	−14.9909	−0.624079	−0.312039	+	0.950069i	$0.601012\pi$
$578$	0	0
$579$	−64.4263	−2.67747
$580$	0	0
$581$	14.7476	0.611833
$582$	0	0
$583$	−33.6558	−1.39388
$584$	0	0
$585$	1.39444	0.0576532
$586$	0	0
$587$	2.34664	0.0968562	0.0484281	−	0.998827i	$-0.484579\pi$
$587$	2.34664	0.0968562	0.0484281	+	0.998827i	$0.484579\pi$
$588$	0	0
$589$	1.03064	0.0424668
$590$	0	0
$591$	−1.06022	−0.0436118
$592$	0	0
$593$	20.2828	0.832913	0.416457	−	0.909156i	$-0.363272\pi$
$593$	20.2828	0.832913	0.416457	+	0.909156i	$0.363272\pi$
$594$	0	0
$595$	2.48589	0.101911
$596$	0	0
$597$	34.7560	1.42247
$598$	0	0
$599$	−23.3855	−0.955507	−0.477753	−	0.878494i	$-0.658549\pi$
$599$	−23.3855	−0.955507	−0.477753	+	0.878494i	$0.658549\pi$
$600$	0	0
$601$	11.6040	0.473339	0.236669	−	0.971590i	$-0.423944\pi$
$601$	11.6040	0.473339	0.236669	+	0.971590i	$0.423944\pi$
$602$	0	0
$603$	20.6010	0.838937
$604$	0	0
$605$	−0.636791	−0.0258892
$606$	0	0
$607$	7.09663	0.288043	0.144022	−	0.989575i	$-0.453996\pi$
$607$	7.09663	0.288043	0.144022	+	0.989575i	$0.453996\pi$
$608$	0	0
$609$	−6.75823	−0.273857
$610$	0	0
$611$	0.312137	0.0126277
$612$	0	0
$613$	6.01247	0.242841	0.121421	−	0.992601i	$-0.461255\pi$
$613$	6.01247	0.242841	0.121421	+	0.992601i	$0.461255\pi$
$614$	0	0
$615$	24.2060	0.976082
$616$	0	0
$617$	−8.30614	−0.334393	−0.167196	−	0.985924i	$-0.553471\pi$
$617$	−8.30614	−0.334393	−0.167196	+	0.985924i	$0.553471\pi$
$618$	0	0
$619$	45.5582	1.83114	0.915569	−	0.402161i	$-0.131741\pi$
$619$	45.5582	1.83114	0.915569	+	0.402161i	$0.131741\pi$
$620$	0	0
$621$	14.7523	0.591990
$622$	0	0
$623$	25.5521	1.02372
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.0339	0.959820
$628$	0	0
$629$	−10.5188	−0.419413
$630$	0	0
$631$	2.96427	0.118006	0.0590029	−	0.998258i	$-0.481208\pi$
$631$	2.96427	0.118006	0.0590029	+	0.998258i	$0.481208\pi$
$632$	0	0
$633$	−39.6092	−1.57432
$634$	0	0
$635$	12.2780	0.487237
$636$	0	0
$637$	1.23537	0.0489472
$638$	0	0
$639$	34.0113	1.34546
$640$	0	0
$641$	16.0060	0.632201	0.316100	−	0.948726i	$-0.397626\pi$
$641$	16.0060	0.632201	0.316100	+	0.948726i	$0.397626\pi$
$642$	0	0
$643$	−32.0435	−1.26367	−0.631837	−	0.775102i	$-0.717699\pi$
$643$	−32.0435	−1.26367	−0.631837	+	0.775102i	$0.717699\pi$
$644$	0	0
$645$	25.8113	1.01632
$646$	0	0
$647$	−39.9964	−1.57242	−0.786209	−	0.617960i	$-0.787959\pi$
$647$	−39.9964	−1.57242	−0.786209	+	0.617960i	$0.787959\pi$
$648$	0	0
$649$	9.84108	0.386296
$650$	0	0
$651$	1.89494	0.0742685
$652$	0	0
$653$	−5.90149	−0.230943	−0.115472	−	0.993311i	$-0.536838\pi$
$653$	−5.90149	−0.230943	−0.115472	+	0.993311i	$0.536838\pi$
$654$	0	0
$655$	−0.171614	−0.00670551
$656$	0	0
$657$	−44.7510	−1.74590
$658$	0	0
$659$	−29.4593	−1.14757	−0.573785	−	0.819006i	$-0.694525\pi$
$659$	−29.4593	−1.14757	−0.573785	+	0.819006i	$0.694525\pi$
$660$	0	0
$661$	−7.95360	−0.309359	−0.154680	−	0.987965i	$-0.549434\pi$
$661$	−7.95360	−0.309359	−0.154680	+	0.987965i	$0.549434\pi$
$662$	0	0
$663$	−1.21572	−0.0472145
$664$	0	0
$665$	−4.23955	−0.164403
$666$	0	0
$667$	4.42722	0.171423
$668$	0	0
$669$	−5.91714	−0.228770
$670$	0	0
$671$	27.7878	1.07274
$672$	0	0
$673$	−13.2248	−0.509778	−0.254889	−	0.966970i	$-0.582039\pi$
$673$	−13.2248	−0.509778	−0.254889	+	0.966970i	$0.582039\pi$
$674$	0	0
$675$	4.83608	0.186141
$676$	0	0
$677$	−33.1605	−1.27446	−0.637230	−	0.770673i	$-0.719920\pi$
$677$	−33.1605	−1.27446	−0.637230	+	0.770673i	$0.719920\pi$
$678$	0	0
$679$	−13.1694	−0.505397
$680$	0	0
$681$	−40.8674	−1.56604
$682$	0	0
$683$	35.6919	1.36571	0.682857	−	0.730552i	$-0.260737\pi$
$683$	35.6919	1.36571	0.682857	+	0.730552i	$0.260737\pi$
$684$	0	0
$685$	10.0684	0.384694
$686$	0	0
$687$	63.8417	2.43571
$688$	0	0
$689$	−2.90340	−0.110611
$690$	0	0
$691$	−24.3807	−0.927484	−0.463742	−	0.885970i	$-0.653493\pi$
$691$	−24.3807	−0.927484	−0.463742	+	0.885970i	$0.653493\pi$
$692$	0	0
$693$	27.0579	1.02785
$694$	0	0
$695$	−4.83650	−0.183459
$696$	0	0
$697$	−12.9222	−0.489464
$698$	0	0
$699$	−49.2318	−1.86212
$700$	0	0
$701$	33.2140	1.25448	0.627238	−	0.778828i	$-0.284185\pi$
$701$	33.2140	1.25448	0.627238	+	0.778828i	$0.284185\pi$
$702$	0	0
$703$	17.9393	0.676594
$704$	0	0
$705$	2.95059	0.111126
$706$	0	0
$707$	26.6862	1.00364
$708$	0	0
$709$	−2.19138	−0.0822988	−0.0411494	−	0.999153i	$-0.513102\pi$
$709$	−2.19138	−0.0822988	−0.0411494	+	0.999153i	$0.513102\pi$
$710$	0	0
$711$	69.9172	2.62210
$712$	0	0
$713$	−1.24135	−0.0464889
$714$	0	0
$715$	−1.00388	−0.0375428
$716$	0	0
$717$	26.3566	0.984304
$718$	0	0
$719$	39.8153	1.48486	0.742430	−	0.669924i	$-0.233673\pi$
$719$	39.8153	1.48486	0.742430	+	0.669924i	$0.233673\pi$
$720$	0	0
$721$	25.8126	0.961310
$722$	0	0
$723$	23.5360	0.875312
$724$	0	0
$725$	1.45133	0.0539009
$726$	0	0
$727$	9.46250	0.350945	0.175472	−	0.984484i	$-0.443855\pi$
$727$	9.46250	0.350945	0.175472	+	0.984484i	$0.443855\pi$
$728$	0	0
$729$	−43.9715	−1.62857
$730$	0	0
$731$	−13.7792	−0.509641
$732$	0	0
$733$	−1.77867	−0.0656965	−0.0328483	−	0.999460i	$-0.510458\pi$
$733$	−1.77867	−0.0656965	−0.0328483	+	0.999460i	$0.510458\pi$
$734$	0	0
$735$	11.6778	0.430743
$736$	0	0
$737$	−14.8309	−0.546302
$738$	0	0
$739$	−17.8904	−0.658109	−0.329055	−	0.944311i	$-0.606730\pi$
$739$	−17.8904	−0.658109	−0.329055	+	0.944311i	$0.606730\pi$
$740$	0	0
$741$	2.07334	0.0761660
$742$	0	0
$743$	−2.71843	−0.0997293	−0.0498647	−	0.998756i	$-0.515879\pi$
$743$	−2.71843	−0.0997293	−0.0498647	+	0.998756i	$0.515879\pi$
$744$	0	0
$745$	2.24631	0.0822984
$746$	0	0
$747$	−41.7463	−1.52742
$748$	0	0
$749$	−29.8343	−1.09012
$750$	0	0
$751$	18.7959	0.685874	0.342937	−	0.939358i	$-0.388578\pi$
$751$	18.7959	0.685874	0.342937	+	0.939358i	$0.388578\pi$
$752$	0	0
$753$	16.8472	0.613945
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−2.46092	−0.0894436	−0.0447218	−	0.998999i	$-0.514240\pi$
$757$	−2.46092	−0.0894436	−0.0447218	+	0.998999i	$0.514240\pi$
$758$	0	0
$759$	−28.9475	−1.05073
$760$	0	0
$761$	45.9503	1.66570	0.832848	−	0.553501i	$-0.186709\pi$
$761$	45.9503	1.66570	0.832848	+	0.553501i	$0.186709\pi$
$762$	0	0
$763$	11.4454	0.414351
$764$	0	0
$765$	−7.03686	−0.254418
$766$	0	0
$767$	0.848965	0.0306543
$768$	0	0
$769$	5.40397	0.194872	0.0974361	−	0.995242i	$-0.468936\pi$
$769$	5.40397	0.194872	0.0974361	+	0.995242i	$0.468936\pi$
$770$	0	0
$771$	40.1778	1.44697
$772$	0	0
$773$	0.639572	0.0230038	0.0115019	−	0.999934i	$-0.496339\pi$
$773$	0.639572	0.0230038	0.0115019	+	0.999934i	$0.496339\pi$
$774$	0	0
$775$	−0.406937	−0.0146176
$776$	0	0
$777$	32.9833	1.18327
$778$	0	0
$779$	22.0382	0.789599
$780$	0	0
$781$	−24.4851	−0.876145
$782$	0	0
$783$	7.01873	0.250829
$784$	0	0
$785$	−19.5947	−0.699364
$786$	0	0
$787$	15.0163	0.535272	0.267636	−	0.963520i	$-0.413757\pi$
$787$	15.0163	0.535272	0.267636	+	0.963520i	$0.413757\pi$
$788$	0	0
$789$	−40.5119	−1.44226
$790$	0	0
$791$	−16.3082	−0.579851
$792$	0	0
$793$	2.39718	0.0851264
$794$	0	0
$795$	−27.4455	−0.973392
$796$	0	0
$797$	18.9827	0.672403	0.336201	−	0.941790i	$-0.390858\pi$
$797$	18.9827	0.672403	0.336201	+	0.941790i	$0.390858\pi$
$798$	0	0
$799$	−1.57515	−0.0557250
$800$	0	0
$801$	−72.3309	−2.55569
$802$	0	0
$803$	32.2167	1.13690
$804$	0	0
$805$	5.10631	0.179974
$806$	0	0
$807$	5.63832	0.198478
$808$	0	0
$809$	−45.6266	−1.60415	−0.802073	−	0.597226i	$-0.796270\pi$
$809$	−45.6266	−1.60415	−0.802073	+	0.597226i	$0.796270\pi$
$810$	0	0
$811$	4.31580	0.151548	0.0757741	−	0.997125i	$-0.475857\pi$
$811$	4.31580	0.151548	0.0757741	+	0.997125i	$0.475857\pi$
$812$	0	0
$813$	15.9608	0.559770
$814$	0	0
$815$	−6.22056	−0.217897
$816$	0	0
$817$	23.4997	0.822149
$818$	0	0
$819$	2.33422	0.0815642
$820$	0	0
$821$	−40.4931	−1.41322	−0.706609	−	0.707604i	$-0.749776\pi$
$821$	−40.4931	−1.41322	−0.706609	+	0.707604i	$0.749776\pi$
$822$	0	0
$823$	17.0340	0.593769	0.296885	−	0.954913i	$-0.404052\pi$
$823$	17.0340	0.593769	0.296885	+	0.954913i	$0.404052\pi$
$824$	0	0
$825$	−9.48952	−0.330383
$826$	0	0
$827$	12.0916	0.420465	0.210233	−	0.977651i	$-0.432578\pi$
$827$	12.0916	0.420465	0.210233	+	0.977651i	$0.432578\pi$
$828$	0	0
$829$	−11.3081	−0.392747	−0.196374	−	0.980529i	$-0.562917\pi$
$829$	−11.3081	−0.392747	−0.196374	+	0.980529i	$0.562917\pi$
$830$	0	0
$831$	9.25472	0.321043
$832$	0	0
$833$	−6.23412	−0.215999
$834$	0	0
$835$	10.1137	0.350001
$836$	0	0
$837$	−1.96798	−0.0680234
$838$	0	0
$839$	38.4492	1.32741	0.663707	−	0.747993i	$-0.268982\pi$
$839$	38.4492	1.32741	0.663707	+	0.747993i	$0.268982\pi$
$840$	0	0
$841$	−26.8937	−0.927367
$842$	0	0
$843$	−64.3129	−2.21505
$844$	0	0
$845$	12.9134	0.444234
$846$	0	0
$847$	−1.06595	−0.0366265
$848$	0	0
$849$	−18.0599	−0.619816
$850$	0	0
$851$	−21.6069	−0.740675
$852$	0	0
$853$	2.53284	0.0867228	0.0433614	−	0.999059i	$-0.486193\pi$
$853$	2.53284	0.0867228	0.0433614	+	0.999059i	$0.486193\pi$
$854$	0	0
$855$	12.0010	0.410425
$856$	0	0
$857$	31.4141	1.07309	0.536543	−	0.843873i	$-0.319730\pi$
$857$	31.4141	1.07309	0.536543	+	0.843873i	$0.319730\pi$
$858$	0	0
$859$	−7.78722	−0.265697	−0.132848	−	0.991136i	$-0.542412\pi$
$859$	−7.78722	−0.265697	−0.132848	+	0.991136i	$0.542412\pi$
$860$	0	0
$861$	40.5195	1.38090
$862$	0	0
$863$	−37.7126	−1.28375	−0.641877	−	0.766808i	$-0.721844\pi$
$863$	−37.7126	−1.28375	−0.641877	+	0.766808i	$0.721844\pi$
$864$	0	0
$865$	−10.9544	−0.372462
$866$	0	0
$867$	−41.1558	−1.39773
$868$	0	0
$869$	−50.3342	−1.70747
$870$	0	0
$871$	−1.27942	−0.0433515
$872$	0	0
$873$	37.2791	1.26170
$874$	0	0
$875$	1.67394	0.0565896
$876$	0	0
$877$	14.9651	0.505335	0.252668	−	0.967553i	$-0.418692\pi$
$877$	14.9651	0.505335	0.252668	+	0.967553i	$0.418692\pi$
$878$	0	0
$879$	−36.1760	−1.22018
$880$	0	0
$881$	−40.4677	−1.36339	−0.681697	−	0.731635i	$-0.738758\pi$
$881$	−40.4677	−1.36339	−0.681697	+	0.731635i	$0.738758\pi$
$882$	0	0
$883$	11.9557	0.402342	0.201171	−	0.979556i	$-0.435525\pi$
$883$	11.9557	0.402342	0.201171	+	0.979556i	$0.435525\pi$
$884$	0	0
$885$	8.02516	0.269763
$886$	0	0
$887$	14.9045	0.500445	0.250222	−	0.968188i	$-0.419496\pi$
$887$	14.9045	0.500445	0.250222	+	0.968188i	$0.419496\pi$
$888$	0	0
$889$	20.5526	0.689313
$890$	0	0
$891$	2.60055	0.0871218
$892$	0	0
$893$	2.68634	0.0898950
$894$	0	0
$895$	10.2384	0.342233
$896$	0	0
$897$	−2.49722	−0.0833798
$898$	0	0
$899$	−0.590599	−0.0196976
$900$	0	0
$901$	14.6516	0.488116
$902$	0	0
$903$	43.2066	1.43783
$904$	0	0
$905$	−22.4447	−0.746087
$906$	0	0
$907$	−0.748645	−0.0248583	−0.0124292	−	0.999923i	$-0.503956\pi$
$907$	−0.748645	−0.0248583	−0.0124292	+	0.999923i	$0.503956\pi$
$908$	0	0
$909$	−75.5413	−2.50555
$910$	0	0
$911$	40.6611	1.34716	0.673582	−	0.739113i	$-0.264755\pi$
$911$	40.6611	1.34716	0.673582	+	0.739113i	$0.264755\pi$
$912$	0	0
$913$	30.0536	0.994630
$914$	0	0
$915$	22.6602	0.749125
$916$	0	0
$917$	−0.287271	−0.00948654
$918$	0	0
$919$	−16.8577	−0.556085	−0.278042	−	0.960569i	$-0.589686\pi$
$919$	−16.8577	−0.556085	−0.278042	+	0.960569i	$0.589686\pi$
$920$	0	0
$921$	69.1075	2.27717
$922$	0	0
$923$	−2.11227	−0.0695261
$924$	0	0
$925$	−7.08315	−0.232892
$926$	0	0
$927$	−73.0682	−2.39988
$928$	0	0
$929$	−32.9331	−1.08050	−0.540251	−	0.841504i	$-0.681671\pi$
$929$	−32.9331	−1.08050	−0.540251	+	0.841504i	$0.681671\pi$
$930$	0	0
$931$	10.6320	0.348448
$932$	0	0
$933$	−8.33550	−0.272892
$934$	0	0
$935$	5.06591	0.165673
$936$	0	0
$937$	9.78337	0.319609	0.159804	−	0.987149i	$-0.448914\pi$
$937$	9.78337	0.319609	0.159804	+	0.987149i	$0.448914\pi$
$938$	0	0
$939$	−34.4304	−1.12359
$940$	0	0
$941$	−38.0148	−1.23925	−0.619624	−	0.784899i	$-0.712715\pi$
$941$	−38.0148	−1.23925	−0.619624	+	0.784899i	$0.712715\pi$
$942$	0	0
$943$	−26.5438	−0.864384
$944$	0	0
$945$	8.09532	0.263341
$946$	0	0
$947$	2.13393	0.0693433	0.0346716	−	0.999399i	$-0.488961\pi$
$947$	2.13393	0.0693433	0.0346716	+	0.999399i	$0.488961\pi$
$948$	0	0
$949$	2.77925	0.0902184
$950$	0	0
$951$	−18.6475	−0.604686
$952$	0	0
$953$	−30.0346	−0.972917	−0.486459	−	0.873704i	$-0.661712\pi$
$953$	−30.0346	−0.972917	−0.486459	+	0.873704i	$0.661712\pi$
$954$	0	0
$955$	18.2452	0.590401
$956$	0	0
$957$	−13.7724	−0.445198
$958$	0	0
$959$	16.8539	0.544241
$960$	0	0
$961$	−30.8344	−0.994658
$962$	0	0
$963$	84.4527	2.72145
$964$	0	0
$965$	23.1598	0.745542
$966$	0	0
$967$	−24.3222	−0.782149	−0.391074	−	0.920359i	$-0.627896\pi$
$967$	−24.3222	−0.782149	−0.391074	+	0.920359i	$0.627896\pi$
$968$	0	0
$969$	−10.4628	−0.336114
$970$	0	0
$971$	49.6376	1.59295	0.796473	−	0.604674i	$-0.206697\pi$
$971$	49.6376	1.59295	0.796473	+	0.604674i	$0.206697\pi$
$972$	0	0
$973$	−8.09601	−0.259546
$974$	0	0
$975$	−0.818636	−0.0262173
$976$	0	0
$977$	1.02878	0.0329134	0.0164567	−	0.999865i	$-0.494761\pi$
$977$	1.02878	0.0329134	0.0164567	+	0.999865i	$0.494761\pi$
$978$	0	0
$979$	52.0719	1.66422
$980$	0	0
$981$	−32.3987	−1.03441
$982$	0	0
$983$	−50.6737	−1.61624	−0.808119	−	0.589019i	$-0.799514\pi$
$983$	−50.6737	−1.61624	−0.808119	+	0.589019i	$0.799514\pi$
$984$	0	0
$985$	0.381127	0.0121437
$986$	0	0
$987$	4.93912	0.157214
$988$	0	0
$989$	−28.3041	−0.900016
$990$	0	0
$991$	−32.2187	−1.02346	−0.511731	−	0.859146i	$-0.670995\pi$
$991$	−32.2187	−1.02346	−0.511731	+	0.859146i	$0.670995\pi$
$992$	0	0
$993$	21.2437	0.674149
$994$	0	0
$995$	−12.4940	−0.396087
$996$	0	0
$997$	17.3631	0.549894	0.274947	−	0.961459i	$-0.411340\pi$
$997$	17.3631	0.549894	0.274947	+	0.961459i	$0.411340\pi$
$998$	0	0
$999$	−34.2547	−1.08377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.n.1.13	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.n.1.13	✓	13	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+2.78181 q^{3} -1.00000 q^{5} -1.67394 q^{7} +4.73847 q^{9} +O(q^{10})\)	\(q+2.78181 q^{3} -1.00000 q^{5} -1.67394 q^{7} +4.73847 q^{9} -3.41127 q^{11} -0.294282 q^{13} -2.78181 q^{15} +1.48505 q^{17} -2.53267 q^{19} -4.65659 q^{21} +3.05047 q^{23} +1.00000 q^{25} +4.83608 q^{27} +1.45133 q^{29} -0.406937 q^{31} -9.48952 q^{33} +1.67394 q^{35} -7.08315 q^{37} -0.818636 q^{39} -8.70154 q^{41} -9.27859 q^{43} -4.73847 q^{45} -1.06067 q^{47} -4.19792 q^{49} +4.13113 q^{51} +9.86606 q^{53} +3.41127 q^{55} -7.04542 q^{57} -2.88487 q^{59} -8.14586 q^{61} -7.93192 q^{63} +0.294282 q^{65} +4.34761 q^{67} +8.48582 q^{69} +7.17770 q^{71} -9.44419 q^{73} +2.78181 q^{75} +5.71027 q^{77} +14.7552 q^{79} -0.762341 q^{81} -8.81009 q^{83} -1.48505 q^{85} +4.03731 q^{87} -15.2646 q^{89} +0.492611 q^{91} -1.13202 q^{93} +2.53267 q^{95} +7.86733 q^{97} -16.1642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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