LMFDB - Embedded newform 6040.2.a.p.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

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

Embedding invariants

Embedding label			1.7
Root			$1.68584$ 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)

$f(q)$	$=$	$q-1.68584 q^{3} +1.00000 q^{5} -4.78926 q^{7} -0.157936 q^{9} +O(q^{10})$	$q-1.68584 q^{3} +1.00000 q^{5} -4.78926 q^{7} -0.157936 q^{9} -1.17912 q^{11} +3.55016 q^{13} -1.68584 q^{15} +5.24056 q^{17} +2.24601 q^{19} +8.07394 q^{21} -8.41556 q^{23} +1.00000 q^{25} +5.32378 q^{27} -8.71732 q^{29} +3.38760 q^{31} +1.98782 q^{33} -4.78926 q^{35} +9.53075 q^{37} -5.98501 q^{39} -2.73536 q^{41} +4.70896 q^{43} -0.157936 q^{45} -11.7202 q^{47} +15.9370 q^{49} -8.83475 q^{51} -4.56229 q^{53} -1.17912 q^{55} -3.78642 q^{57} +2.95106 q^{59} +3.98085 q^{61} +0.756395 q^{63} +3.55016 q^{65} +13.8968 q^{67} +14.1873 q^{69} -3.61567 q^{71} +9.48202 q^{73} -1.68584 q^{75} +5.64713 q^{77} +1.13533 q^{79} -8.50125 q^{81} +15.4538 q^{83} +5.24056 q^{85} +14.6960 q^{87} -0.962116 q^{89} -17.0027 q^{91} -5.71097 q^{93} +2.24601 q^{95} +3.59211 q^{97} +0.186226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.68584	−0.973322	−0.486661	−	0.873591i	$-0.661785\pi$
$3$	−1.68584	−0.973322	−0.486661	+	0.873591i	$0.661785\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.78926	−1.81017	−0.905086	−	0.425229i	$-0.860194\pi$
$7$	−4.78926	−1.81017	−0.905086	+	0.425229i	$0.860194\pi$
$8$	0	0
$9$	−0.157936	−0.0526452
$10$	0	0
$11$	−1.17912	−0.355519	−0.177760	−	0.984074i	$-0.556885\pi$
$11$	−1.17912	−0.355519	−0.177760	+	0.984074i	$0.556885\pi$
$12$	0	0
$13$	3.55016	0.984637	0.492319	−	0.870415i	$-0.336149\pi$
$13$	3.55016	0.984637	0.492319	+	0.870415i	$0.336149\pi$
$14$	0	0
$15$	−1.68584	−0.435283
$16$	0	0
$17$	5.24056	1.27102	0.635511	−	0.772092i	$-0.280790\pi$
$17$	5.24056	1.27102	0.635511	+	0.772092i	$0.280790\pi$
$18$	0	0
$19$	2.24601	0.515270	0.257635	−	0.966242i	$-0.417057\pi$
$19$	2.24601	0.515270	0.257635	+	0.966242i	$0.417057\pi$
$20$	0	0
$21$	8.07394	1.76188
$22$	0	0
$23$	−8.41556	−1.75477	−0.877383	−	0.479790i	$-0.840713\pi$
$23$	−8.41556	−1.75477	−0.877383	+	0.479790i	$0.840713\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.32378	1.02456
$28$	0	0
$29$	−8.71732	−1.61877	−0.809383	−	0.587282i	$-0.800198\pi$
$29$	−8.71732	−1.61877	−0.809383	+	0.587282i	$0.800198\pi$
$30$	0	0
$31$	3.38760	0.608432	0.304216	−	0.952603i	$-0.401606\pi$
$31$	3.38760	0.608432	0.304216	+	0.952603i	$0.401606\pi$
$32$	0	0
$33$	1.98782	0.346034
$34$	0	0
$35$	−4.78926	−0.809533
$36$	0	0
$37$	9.53075	1.56685	0.783423	−	0.621489i	$-0.213472\pi$
$37$	9.53075	1.56685	0.783423	+	0.621489i	$0.213472\pi$
$38$	0	0
$39$	−5.98501	−0.958369
$40$	0	0
$41$	−2.73536	−0.427192	−0.213596	−	0.976922i	$-0.568518\pi$
$41$	−2.73536	−0.427192	−0.213596	+	0.976922i	$0.568518\pi$
$42$	0	0
$43$	4.70896	0.718110	0.359055	−	0.933316i	$-0.383099\pi$
$43$	4.70896	0.718110	0.359055	+	0.933316i	$0.383099\pi$
$44$	0	0
$45$	−0.157936	−0.0235436
$46$	0	0
$47$	−11.7202	−1.70956	−0.854781	−	0.518989i	$-0.826308\pi$
$47$	−11.7202	−1.70956	−0.854781	+	0.518989i	$0.826308\pi$
$48$	0	0
$49$	15.9370	2.27672
$50$	0	0
$51$	−8.83475	−1.23711
$52$	0	0
$53$	−4.56229	−0.626679	−0.313340	−	0.949641i	$-0.601448\pi$
$53$	−4.56229	−0.626679	−0.313340	+	0.949641i	$0.601448\pi$
$54$	0	0
$55$	−1.17912	−0.158993
$56$	0	0
$57$	−3.78642	−0.501524
$58$	0	0
$59$	2.95106	0.384195	0.192098	−	0.981376i	$-0.438471\pi$
$59$	2.95106	0.384195	0.192098	+	0.981376i	$0.438471\pi$
$60$	0	0
$61$	3.98085	0.509696	0.254848	−	0.966981i	$-0.417975\pi$
$61$	3.98085	0.509696	0.254848	+	0.966981i	$0.417975\pi$
$62$	0	0
$63$	0.756395	0.0952968
$64$	0	0
$65$	3.55016	0.440343
$66$	0	0
$67$	13.8968	1.69776	0.848881	−	0.528584i	$-0.177277\pi$
$67$	13.8968	1.69776	0.848881	+	0.528584i	$0.177277\pi$
$68$	0	0
$69$	14.1873	1.70795
$70$	0	0
$71$	−3.61567	−0.429101	−0.214550	−	0.976713i	$-0.568829\pi$
$71$	−3.61567	−0.429101	−0.214550	+	0.976713i	$0.568829\pi$
$72$	0	0
$73$	9.48202	1.10979	0.554893	−	0.831922i	$-0.312759\pi$
$73$	9.48202	1.10979	0.554893	+	0.831922i	$0.312759\pi$
$74$	0	0
$75$	−1.68584	−0.194664
$76$	0	0
$77$	5.64713	0.643550
$78$	0	0
$79$	1.13533	0.127734	0.0638671	−	0.997958i	$-0.479657\pi$
$79$	1.13533	0.127734	0.0638671	+	0.997958i	$0.479657\pi$
$80$	0	0
$81$	−8.50125	−0.944583
$82$	0	0
$83$	15.4538	1.69628	0.848138	−	0.529775i	$-0.177724\pi$
$83$	15.4538	1.69628	0.848138	+	0.529775i	$0.177724\pi$
$84$	0	0
$85$	5.24056	0.568418
$86$	0	0
$87$	14.6960	1.57558
$88$	0	0
$89$	−0.962116	−0.101984	−0.0509920	−	0.998699i	$-0.516238\pi$
$89$	−0.962116	−0.101984	−0.0509920	+	0.998699i	$0.516238\pi$
$90$	0	0
$91$	−17.0027	−1.78236
$92$	0	0
$93$	−5.71097	−0.592200
$94$	0	0
$95$	2.24601	0.230436
$96$	0	0
$97$	3.59211	0.364723	0.182362	−	0.983232i	$-0.441626\pi$
$97$	3.59211	0.364723	0.182362	+	0.983232i	$0.441626\pi$
$98$	0	0
$99$	0.186226	0.0187164
$100$	0	0
$101$	−11.2620	−1.12061	−0.560304	−	0.828287i	$-0.689316\pi$
$101$	−11.2620	−1.12061	−0.560304	+	0.828287i	$0.689316\pi$
$102$	0	0
$103$	−13.2954	−1.31004	−0.655019	−	0.755612i	$-0.727339\pi$
$103$	−13.2954	−1.31004	−0.655019	+	0.755612i	$0.727339\pi$
$104$	0	0
$105$	8.07394	0.787936
$106$	0	0
$107$	0.644033	0.0622610	0.0311305	−	0.999515i	$-0.490089\pi$
$107$	0.644033	0.0622610	0.0311305	+	0.999515i	$0.490089\pi$
$108$	0	0
$109$	14.0524	1.34598	0.672988	−	0.739653i	$-0.265010\pi$
$109$	14.0524	1.34598	0.672988	+	0.739653i	$0.265010\pi$
$110$	0	0
$111$	−16.0673	−1.52504
$112$	0	0
$113$	−6.81062	−0.640689	−0.320345	−	0.947301i	$-0.603799\pi$
$113$	−6.81062	−0.640689	−0.320345	+	0.947301i	$0.603799\pi$
$114$	0	0
$115$	−8.41556	−0.784755
$116$	0	0
$117$	−0.560697	−0.0518364
$118$	0	0
$119$	−25.0984	−2.30077
$120$	0	0
$121$	−9.60967	−0.873606
$122$	0	0
$123$	4.61139	0.415795
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.47889	0.486173	0.243086	−	0.970005i	$-0.421840\pi$
$127$	5.47889	0.486173	0.243086	+	0.970005i	$0.421840\pi$
$128$	0	0
$129$	−7.93857	−0.698952
$130$	0	0
$131$	−9.29141	−0.811795	−0.405897	−	0.913919i	$-0.633041\pi$
$131$	−9.29141	−0.811795	−0.405897	+	0.913919i	$0.633041\pi$
$132$	0	0
$133$	−10.7567	−0.932728
$134$	0	0
$135$	5.32378	0.458198
$136$	0	0
$137$	5.97746	0.510689	0.255344	−	0.966850i	$-0.417811\pi$
$137$	5.97746	0.510689	0.255344	+	0.966850i	$0.417811\pi$
$138$	0	0
$139$	−12.0766	−1.02433	−0.512164	−	0.858888i	$-0.671156\pi$
$139$	−12.0766	−1.02433	−0.512164	+	0.858888i	$0.671156\pi$
$140$	0	0
$141$	19.7583	1.66395
$142$	0	0
$143$	−4.18608	−0.350057
$144$	0	0
$145$	−8.71732	−0.723934
$146$	0	0
$147$	−26.8673	−2.21598
$148$	0	0
$149$	−12.0049	−0.983480	−0.491740	−	0.870742i	$-0.663639\pi$
$149$	−12.0049	−0.983480	−0.491740	+	0.870742i	$0.663639\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−0.827670	−0.0669132
$154$	0	0
$155$	3.38760	0.272099
$156$	0	0
$157$	−2.61722	−0.208877	−0.104438	−	0.994531i	$-0.533304\pi$
$157$	−2.61722	−0.208877	−0.104438	+	0.994531i	$0.533304\pi$
$158$	0	0
$159$	7.69131	0.609960
$160$	0	0
$161$	40.3044	3.17643
$162$	0	0
$163$	16.6073	1.30078	0.650391	−	0.759599i	$-0.274605\pi$
$163$	16.6073	1.30078	0.650391	+	0.759599i	$0.274605\pi$
$164$	0	0
$165$	1.98782	0.154751
$166$	0	0
$167$	−20.2987	−1.57076	−0.785379	−	0.619015i	$-0.787532\pi$
$167$	−20.2987	−1.57076	−0.785379	+	0.619015i	$0.787532\pi$
$168$	0	0
$169$	−0.396361	−0.0304893
$170$	0	0
$171$	−0.354725	−0.0271265
$172$	0	0
$173$	−19.2373	−1.46258	−0.731291	−	0.682066i	$-0.761082\pi$
$173$	−19.2373	−1.46258	−0.731291	+	0.682066i	$0.761082\pi$
$174$	0	0
$175$	−4.78926	−0.362034
$176$	0	0
$177$	−4.97502	−0.373945
$178$	0	0
$179$	23.5260	1.75841	0.879207	−	0.476439i	$-0.158073\pi$
$179$	23.5260	1.75841	0.879207	+	0.476439i	$0.158073\pi$
$180$	0	0
$181$	−2.07285	−0.154074	−0.0770369	−	0.997028i	$-0.524546\pi$
$181$	−2.07285	−0.154074	−0.0770369	+	0.997028i	$0.524546\pi$
$182$	0	0
$183$	−6.71109	−0.496098
$184$	0	0
$185$	9.53075	0.700715
$186$	0	0
$187$	−6.17926	−0.451872
$188$	0	0
$189$	−25.4970	−1.85463
$190$	0	0
$191$	−16.1010	−1.16503	−0.582513	−	0.812821i	$-0.697931\pi$
$191$	−16.1010	−1.16503	−0.582513	+	0.812821i	$0.697931\pi$
$192$	0	0
$193$	−20.6192	−1.48420	−0.742100	−	0.670290i	$-0.766170\pi$
$193$	−20.6192	−1.48420	−0.742100	+	0.670290i	$0.766170\pi$
$194$	0	0
$195$	−5.98501	−0.428596
$196$	0	0
$197$	−13.3013	−0.947677	−0.473839	−	0.880612i	$-0.657132\pi$
$197$	−13.3013	−0.947677	−0.473839	+	0.880612i	$0.657132\pi$
$198$	0	0
$199$	−22.8199	−1.61766	−0.808829	−	0.588043i	$-0.799898\pi$
$199$	−22.8199	−1.61766	−0.808829	+	0.588043i	$0.799898\pi$
$200$	0	0
$201$	−23.4278	−1.65247
$202$	0	0
$203$	41.7495	2.93024
$204$	0	0
$205$	−2.73536	−0.191046
$206$	0	0
$207$	1.32912	0.0923800
$208$	0	0
$209$	−2.64832	−0.183188
$210$	0	0
$211$	−9.47681	−0.652411	−0.326205	−	0.945299i	$-0.605770\pi$
$211$	−9.47681	−0.652411	−0.326205	+	0.945299i	$0.605770\pi$
$212$	0	0
$213$	6.09544	0.417653
$214$	0	0
$215$	4.70896	0.321149
$216$	0	0
$217$	−16.2241	−1.10137
$218$	0	0
$219$	−15.9852	−1.08018
$220$	0	0
$221$	18.6048	1.25150
$222$	0	0
$223$	19.6834	1.31810	0.659049	−	0.752100i	$-0.270959\pi$
$223$	19.6834	1.31810	0.659049	+	0.752100i	$0.270959\pi$
$224$	0	0
$225$	−0.157936	−0.0105290
$226$	0	0
$227$	−9.62635	−0.638924	−0.319462	−	0.947599i	$-0.603502\pi$
$227$	−9.62635	−0.638924	−0.319462	+	0.947599i	$0.603502\pi$
$228$	0	0
$229$	6.27899	0.414927	0.207464	−	0.978243i	$-0.433479\pi$
$229$	6.27899	0.414927	0.207464	+	0.978243i	$0.433479\pi$
$230$	0	0
$231$	−9.52017	−0.626382
$232$	0	0
$233$	6.72540	0.440596	0.220298	−	0.975433i	$-0.429297\pi$
$233$	6.72540	0.440596	0.220298	+	0.975433i	$0.429297\pi$
$234$	0	0
$235$	−11.7202	−0.764539
$236$	0	0
$237$	−1.91398	−0.124326
$238$	0	0
$239$	23.7929	1.53903	0.769517	−	0.638627i	$-0.220497\pi$
$239$	23.7929	1.53903	0.769517	+	0.638627i	$0.220497\pi$
$240$	0	0
$241$	−12.0919	−0.778908	−0.389454	−	0.921046i	$-0.627336\pi$
$241$	−12.0919	−0.778908	−0.389454	+	0.921046i	$0.627336\pi$
$242$	0	0
$243$	−1.63958	−0.105179
$244$	0	0
$245$	15.9370	1.01818
$246$	0	0
$247$	7.97370	0.507355
$248$	0	0
$249$	−26.0527	−1.65102
$250$	0	0
$251$	−22.3273	−1.40929	−0.704644	−	0.709561i	$-0.748893\pi$
$251$	−22.3273	−1.40929	−0.704644	+	0.709561i	$0.748893\pi$
$252$	0	0
$253$	9.92299	0.623853
$254$	0	0
$255$	−8.83475	−0.553254
$256$	0	0
$257$	−8.81643	−0.549954	−0.274977	−	0.961451i	$-0.588670\pi$
$257$	−8.81643	−0.549954	−0.274977	+	0.961451i	$0.588670\pi$
$258$	0	0
$259$	−45.6453	−2.83626
$260$	0	0
$261$	1.37677	0.0852202
$262$	0	0
$263$	2.64563	0.163137	0.0815683	−	0.996668i	$-0.474007\pi$
$263$	2.64563	0.163137	0.0815683	+	0.996668i	$0.474007\pi$
$264$	0	0
$265$	−4.56229	−0.280259
$266$	0	0
$267$	1.62198	0.0992633
$268$	0	0
$269$	−22.0993	−1.34742	−0.673709	−	0.738997i	$-0.735300\pi$
$269$	−22.0993	−1.34742	−0.673709	+	0.738997i	$0.735300\pi$
$270$	0	0
$271$	5.22290	0.317268	0.158634	−	0.987337i	$-0.449291\pi$
$271$	5.22290	0.317268	0.158634	+	0.987337i	$0.449291\pi$
$272$	0	0
$273$	28.6638	1.73481
$274$	0	0
$275$	−1.17912	−0.0711038
$276$	0	0
$277$	−7.22896	−0.434346	−0.217173	−	0.976133i	$-0.569684\pi$
$277$	−7.22896	−0.434346	−0.217173	+	0.976133i	$0.569684\pi$
$278$	0	0
$279$	−0.535023	−0.0320310
$280$	0	0
$281$	−25.5959	−1.52693	−0.763463	−	0.645852i	$-0.776502\pi$
$281$	−25.5959	−1.52693	−0.763463	+	0.645852i	$0.776502\pi$
$282$	0	0
$283$	4.62103	0.274691	0.137346	−	0.990523i	$-0.456143\pi$
$283$	4.62103	0.274691	0.137346	+	0.990523i	$0.456143\pi$
$284$	0	0
$285$	−3.78642	−0.224288
$286$	0	0
$287$	13.1004	0.773291
$288$	0	0
$289$	10.4634	0.615496
$290$	0	0
$291$	−6.05572	−0.354993
$292$	0	0
$293$	23.3315	1.36304	0.681522	−	0.731798i	$-0.261319\pi$
$293$	23.3315	1.36304	0.681522	+	0.731798i	$0.261319\pi$
$294$	0	0
$295$	2.95106	0.171817
$296$	0	0
$297$	−6.27739	−0.364251
$298$	0	0
$299$	−29.8766	−1.72781
$300$	0	0
$301$	−22.5525	−1.29990
$302$	0	0
$303$	18.9859	1.09071
$304$	0	0
$305$	3.98085	0.227943
$306$	0	0
$307$	26.4097	1.50728	0.753640	−	0.657288i	$-0.228296\pi$
$307$	26.4097	1.50728	0.753640	+	0.657288i	$0.228296\pi$
$308$	0	0
$309$	22.4140	1.27509
$310$	0	0
$311$	−14.7295	−0.835232	−0.417616	−	0.908624i	$-0.637134\pi$
$311$	−14.7295	−0.835232	−0.417616	+	0.908624i	$0.637134\pi$
$312$	0	0
$313$	−17.5665	−0.992917	−0.496459	−	0.868060i	$-0.665367\pi$
$313$	−17.5665	−0.992917	−0.496459	+	0.868060i	$0.665367\pi$
$314$	0	0
$315$	0.756395	0.0426180
$316$	0	0
$317$	32.7240	1.83796	0.918981	−	0.394301i	$-0.129013\pi$
$317$	32.7240	1.83796	0.918981	+	0.394301i	$0.129013\pi$
$318$	0	0
$319$	10.2788	0.575502
$320$	0	0
$321$	−1.08574	−0.0606000
$322$	0	0
$323$	11.7704	0.654920
$324$	0	0
$325$	3.55016	0.196927
$326$	0	0
$327$	−23.6901	−1.31007
$328$	0	0
$329$	56.1309	3.09460
$330$	0	0
$331$	−3.28802	−0.180726	−0.0903628	−	0.995909i	$-0.528803\pi$
$331$	−3.28802	−0.180726	−0.0903628	+	0.995909i	$0.528803\pi$
$332$	0	0
$333$	−1.50524	−0.0824869
$334$	0	0
$335$	13.8968	0.759262
$336$	0	0
$337$	0.972333	0.0529663	0.0264832	−	0.999649i	$-0.491569\pi$
$337$	0.972333	0.0529663	0.0264832	+	0.999649i	$0.491569\pi$
$338$	0	0
$339$	11.4816	0.623597
$340$	0	0
$341$	−3.99440	−0.216309
$342$	0	0
$343$	−42.8019	−2.31108
$344$	0	0
$345$	14.1873	0.763819
$346$	0	0
$347$	−10.3706	−0.556722	−0.278361	−	0.960477i	$-0.589791\pi$
$347$	−10.3706	−0.556722	−0.278361	+	0.960477i	$0.589791\pi$
$348$	0	0
$349$	−11.6433	−0.623252	−0.311626	−	0.950205i	$-0.600874\pi$
$349$	−11.6433	−0.623252	−0.311626	+	0.950205i	$0.600874\pi$
$350$	0	0
$351$	18.9003	1.00882
$352$	0	0
$353$	15.5832	0.829412	0.414706	−	0.909955i	$-0.363884\pi$
$353$	15.5832	0.829412	0.414706	+	0.909955i	$0.363884\pi$
$354$	0	0
$355$	−3.61567	−0.191900
$356$	0	0
$357$	42.3120	2.23939
$358$	0	0
$359$	−2.08463	−0.110023	−0.0550113	−	0.998486i	$-0.517519\pi$
$359$	−2.08463	−0.110023	−0.0550113	+	0.998486i	$0.517519\pi$
$360$	0	0
$361$	−13.9554	−0.734496
$362$	0	0
$363$	16.2004	0.850300
$364$	0	0
$365$	9.48202	0.496311
$366$	0	0
$367$	−27.8275	−1.45258	−0.726291	−	0.687387i	$-0.758758\pi$
$367$	−27.8275	−1.45258	−0.726291	+	0.687387i	$0.758758\pi$
$368$	0	0
$369$	0.432011	0.0224896
$370$	0	0
$371$	21.8500	1.13440
$372$	0	0
$373$	−23.5229	−1.21797	−0.608984	−	0.793182i	$-0.708423\pi$
$373$	−23.5229	−1.21797	−0.608984	+	0.793182i	$0.708423\pi$
$374$	0	0
$375$	−1.68584	−0.0870565
$376$	0	0
$377$	−30.9479	−1.59390
$378$	0	0
$379$	−22.7298	−1.16755	−0.583777	−	0.811914i	$-0.698426\pi$
$379$	−22.7298	−1.16755	−0.583777	+	0.811914i	$0.698426\pi$
$380$	0	0
$381$	−9.23654	−0.473202
$382$	0	0
$383$	−24.4353	−1.24858	−0.624291	−	0.781191i	$-0.714612\pi$
$383$	−24.4353	−1.24858	−0.624291	+	0.781191i	$0.714612\pi$
$384$	0	0
$385$	5.64713	0.287805
$386$	0	0
$387$	−0.743713	−0.0378050
$388$	0	0
$389$	−6.54041	−0.331612	−0.165806	−	0.986158i	$-0.553023\pi$
$389$	−6.54041	−0.331612	−0.165806	+	0.986158i	$0.553023\pi$
$390$	0	0
$391$	−44.1022	−2.23035
$392$	0	0
$393$	15.6639	0.790137
$394$	0	0
$395$	1.13533	0.0571244
$396$	0	0
$397$	24.1816	1.21364	0.606820	−	0.794839i	$-0.292445\pi$
$397$	24.1816	1.21364	0.606820	+	0.794839i	$0.292445\pi$
$398$	0	0
$399$	18.1342	0.907844
$400$	0	0
$401$	18.4253	0.920115	0.460057	−	0.887889i	$-0.347829\pi$
$401$	18.4253	0.920115	0.460057	+	0.887889i	$0.347829\pi$
$402$	0	0
$403$	12.0265	0.599084
$404$	0	0
$405$	−8.50125	−0.422430
$406$	0	0
$407$	−11.2379	−0.557043
$408$	0	0
$409$	6.95826	0.344064	0.172032	−	0.985091i	$-0.444967\pi$
$409$	6.95826	0.344064	0.172032	+	0.985091i	$0.444967\pi$
$410$	0	0
$411$	−10.0771	−0.497065
$412$	0	0
$413$	−14.1334	−0.695459
$414$	0	0
$415$	15.4538	0.758598
$416$	0	0
$417$	20.3593	0.997000
$418$	0	0
$419$	−25.2342	−1.23277	−0.616387	−	0.787444i	$-0.711404\pi$
$419$	−25.2342	−1.23277	−0.616387	+	0.787444i	$0.711404\pi$
$420$	0	0
$421$	−7.85171	−0.382669	−0.191335	−	0.981525i	$-0.561282\pi$
$421$	−7.85171	−0.382669	−0.191335	+	0.981525i	$0.561282\pi$
$422$	0	0
$423$	1.85103	0.0900002
$424$	0	0
$425$	5.24056	0.254204
$426$	0	0
$427$	−19.0653	−0.922637
$428$	0	0
$429$	7.05707	0.340718
$430$	0	0
$431$	28.4386	1.36984	0.684920	−	0.728619i	$-0.259837\pi$
$431$	28.4386	1.36984	0.684920	+	0.728619i	$0.259837\pi$
$432$	0	0
$433$	2.45235	0.117853	0.0589263	−	0.998262i	$-0.481232\pi$
$433$	2.45235	0.117853	0.0589263	+	0.998262i	$0.481232\pi$
$434$	0	0
$435$	14.6960	0.704620
$436$	0	0
$437$	−18.9015	−0.904179
$438$	0	0
$439$	−12.6622	−0.604333	−0.302167	−	0.953255i	$-0.597710\pi$
$439$	−12.6622	−0.604333	−0.302167	+	0.953255i	$0.597710\pi$
$440$	0	0
$441$	−2.51703	−0.119858
$442$	0	0
$443$	3.19950	0.152013	0.0760064	−	0.997107i	$-0.475783\pi$
$443$	3.19950	0.152013	0.0760064	+	0.997107i	$0.475783\pi$
$444$	0	0
$445$	−0.962116	−0.0456087
$446$	0	0
$447$	20.2384	0.957243
$448$	0	0
$449$	−15.5437	−0.733550	−0.366775	−	0.930310i	$-0.619538\pi$
$449$	−15.5437	−0.733550	−0.366775	+	0.930310i	$0.619538\pi$
$450$	0	0
$451$	3.22533	0.151875
$452$	0	0
$453$	−1.68584	−0.0792078
$454$	0	0
$455$	−17.0027	−0.797097
$456$	0	0
$457$	−2.88086	−0.134761	−0.0673806	−	0.997727i	$-0.521464\pi$
$457$	−2.88086	−0.134761	−0.0673806	+	0.997727i	$0.521464\pi$
$458$	0	0
$459$	27.8996	1.30224
$460$	0	0
$461$	−13.2623	−0.617686	−0.308843	−	0.951113i	$-0.599942\pi$
$461$	−13.2623	−0.617686	−0.308843	+	0.951113i	$0.599942\pi$
$462$	0	0
$463$	−37.1104	−1.72467	−0.862333	−	0.506342i	$-0.830998\pi$
$463$	−37.1104	−1.72467	−0.862333	+	0.506342i	$0.830998\pi$
$464$	0	0
$465$	−5.71097	−0.264840
$466$	0	0
$467$	29.3945	1.36021	0.680107	−	0.733112i	$-0.261933\pi$
$467$	29.3945	1.36021	0.680107	+	0.733112i	$0.261933\pi$
$468$	0	0
$469$	−66.5553	−3.07324
$470$	0	0
$471$	4.41222	0.203304
$472$	0	0
$473$	−5.55245	−0.255302
$474$	0	0
$475$	2.24601	0.103054
$476$	0	0
$477$	0.720548	0.0329916
$478$	0	0
$479$	29.2253	1.33534	0.667670	−	0.744457i	$-0.267292\pi$
$479$	29.2253	1.33534	0.667670	+	0.744457i	$0.267292\pi$
$480$	0	0
$481$	33.8357	1.54277
$482$	0	0
$483$	−67.9468	−3.09169
$484$	0	0
$485$	3.59211	0.163109
$486$	0	0
$487$	33.6969	1.52695	0.763476	−	0.645836i	$-0.223491\pi$
$487$	33.6969	1.52695	0.763476	+	0.645836i	$0.223491\pi$
$488$	0	0
$489$	−27.9973	−1.26608
$490$	0	0
$491$	−34.9759	−1.57844	−0.789220	−	0.614111i	$-0.789515\pi$
$491$	−34.9759	−1.57844	−0.789220	+	0.614111i	$0.789515\pi$
$492$	0	0
$493$	−45.6836	−2.05749
$494$	0	0
$495$	0.186226	0.00837021
$496$	0	0
$497$	17.3164	0.776746
$498$	0	0
$499$	−19.9030	−0.890981	−0.445491	−	0.895287i	$-0.646971\pi$
$499$	−19.9030	−0.890981	−0.445491	+	0.895287i	$0.646971\pi$
$500$	0	0
$501$	34.2204	1.52885
$502$	0	0
$503$	−21.4091	−0.954585	−0.477293	−	0.878744i	$-0.658382\pi$
$503$	−21.4091	−0.954585	−0.477293	+	0.878744i	$0.658382\pi$
$504$	0	0
$505$	−11.2620	−0.501151
$506$	0	0
$507$	0.668202	0.0296759
$508$	0	0
$509$	20.1679	0.893928	0.446964	−	0.894552i	$-0.352505\pi$
$509$	20.1679	0.893928	0.446964	+	0.894552i	$0.352505\pi$
$510$	0	0
$511$	−45.4119	−2.00890
$512$	0	0
$513$	11.9573	0.527927
$514$	0	0
$515$	−13.2954	−0.585867
$516$	0	0
$517$	13.8195	0.607782
$518$	0	0
$519$	32.4310	1.42356
$520$	0	0
$521$	−27.0068	−1.18319	−0.591596	−	0.806235i	$-0.701502\pi$
$521$	−27.0068	−1.18319	−0.591596	+	0.806235i	$0.701502\pi$
$522$	0	0
$523$	37.5250	1.64085	0.820427	−	0.571751i	$-0.193736\pi$
$523$	37.5250	1.64085	0.820427	+	0.571751i	$0.193736\pi$
$524$	0	0
$525$	8.07394	0.352376
$526$	0	0
$527$	17.7529	0.773330
$528$	0	0
$529$	47.8217	2.07921
$530$	0	0
$531$	−0.466077	−0.0202260
$532$	0	0
$533$	−9.71098	−0.420629
$534$	0	0
$535$	0.644033	0.0278440
$536$	0	0
$537$	−39.6611	−1.71150
$538$	0	0
$539$	−18.7917	−0.809418
$540$	0	0
$541$	9.04100	0.388703	0.194352	−	0.980932i	$-0.437740\pi$
$541$	9.04100	0.388703	0.194352	+	0.980932i	$0.437740\pi$
$542$	0	0
$543$	3.49450	0.149963
$544$	0	0
$545$	14.0524	0.601939
$546$	0	0
$547$	−1.72613	−0.0738039	−0.0369020	−	0.999319i	$-0.511749\pi$
$547$	−1.72613	−0.0738039	−0.0369020	+	0.999319i	$0.511749\pi$
$548$	0	0
$549$	−0.628718	−0.0268330
$550$	0	0
$551$	−19.5792	−0.834102
$552$	0	0
$553$	−5.43738	−0.231221
$554$	0	0
$555$	−16.0673	−0.682021
$556$	0	0
$557$	−10.8157	−0.458276	−0.229138	−	0.973394i	$-0.573591\pi$
$557$	−10.8157	−0.458276	−0.229138	+	0.973394i	$0.573591\pi$
$558$	0	0
$559$	16.7176	0.707078
$560$	0	0
$561$	10.4173	0.439817
$562$	0	0
$563$	−28.3778	−1.19598	−0.597991	−	0.801503i	$-0.704034\pi$
$563$	−28.3778	−1.19598	−0.597991	+	0.801503i	$0.704034\pi$
$564$	0	0
$565$	−6.81062	−0.286525
$566$	0	0
$567$	40.7147	1.70986
$568$	0	0
$569$	−9.33414	−0.391307	−0.195654	−	0.980673i	$-0.562683\pi$
$569$	−9.33414	−0.391307	−0.195654	+	0.980673i	$0.562683\pi$
$570$	0	0
$571$	19.2264	0.804599	0.402300	−	0.915508i	$-0.368211\pi$
$571$	19.2264	0.804599	0.402300	+	0.915508i	$0.368211\pi$
$572$	0	0
$573$	27.1437	1.13395
$574$	0	0
$575$	−8.41556	−0.350953
$576$	0	0
$577$	30.6039	1.27406	0.637028	−	0.770841i	$-0.280164\pi$
$577$	30.6039	1.27406	0.637028	+	0.770841i	$0.280164\pi$
$578$	0	0
$579$	34.7606	1.44460
$580$	0	0
$581$	−74.0124	−3.07055
$582$	0	0
$583$	5.37951	0.222796
$584$	0	0
$585$	−0.560697	−0.0231820
$586$	0	0
$587$	−14.9701	−0.617884	−0.308942	−	0.951081i	$-0.599975\pi$
$587$	−14.9701	−0.617884	−0.308942	+	0.951081i	$0.599975\pi$
$588$	0	0
$589$	7.60860	0.313507
$590$	0	0
$591$	22.4239	0.922395
$592$	0	0
$593$	−20.1585	−0.827810	−0.413905	−	0.910320i	$-0.635835\pi$
$593$	−20.1585	−0.827810	−0.413905	+	0.910320i	$0.635835\pi$
$594$	0	0
$595$	−25.0984	−1.02893
$596$	0	0
$597$	38.4707	1.57450
$598$	0	0
$599$	44.0047	1.79798	0.898992	−	0.437964i	$-0.144300\pi$
$599$	44.0047	1.79798	0.898992	+	0.437964i	$0.144300\pi$
$600$	0	0
$601$	−1.87877	−0.0766366	−0.0383183	−	0.999266i	$-0.512200\pi$
$601$	−1.87877	−0.0766366	−0.0383183	+	0.999266i	$0.512200\pi$
$602$	0	0
$603$	−2.19480	−0.0893790
$604$	0	0
$605$	−9.60967	−0.390689
$606$	0	0
$607$	31.4087	1.27484	0.637420	−	0.770517i	$-0.280002\pi$
$607$	31.4087	1.27484	0.637420	+	0.770517i	$0.280002\pi$
$608$	0	0
$609$	−70.3831	−2.85207
$610$	0	0
$611$	−41.6085	−1.68330
$612$	0	0
$613$	−38.5568	−1.55729	−0.778647	−	0.627462i	$-0.784094\pi$
$613$	−38.5568	−1.55729	−0.778647	+	0.627462i	$0.784094\pi$
$614$	0	0
$615$	4.61139	0.185949
$616$	0	0
$617$	10.6786	0.429903	0.214951	−	0.976625i	$-0.431041\pi$
$617$	10.6786	0.429903	0.214951	+	0.976625i	$0.431041\pi$
$618$	0	0
$619$	−15.0106	−0.603328	−0.301664	−	0.953414i	$-0.597542\pi$
$619$	−15.0106	−0.603328	−0.301664	+	0.953414i	$0.597542\pi$
$620$	0	0
$621$	−44.8026	−1.79787
$622$	0	0
$623$	4.60783	0.184609
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.46466	0.178301
$628$	0	0
$629$	49.9464	1.99149
$630$	0	0
$631$	−12.9588	−0.515882	−0.257941	−	0.966161i	$-0.583044\pi$
$631$	−12.9588	−0.515882	−0.257941	+	0.966161i	$0.583044\pi$
$632$	0	0
$633$	15.9764	0.635005
$634$	0	0
$635$	5.47889	0.217423
$636$	0	0
$637$	56.5791	2.24174
$638$	0	0
$639$	0.571042	0.0225901
$640$	0	0
$641$	−29.9531	−1.18308	−0.591539	−	0.806277i	$-0.701479\pi$
$641$	−29.9531	−1.18308	−0.591539	+	0.806277i	$0.701479\pi$
$642$	0	0
$643$	−40.4514	−1.59525	−0.797624	−	0.603155i	$-0.793910\pi$
$643$	−40.4514	−1.59525	−0.797624	+	0.603155i	$0.793910\pi$
$644$	0	0
$645$	−7.93857	−0.312581
$646$	0	0
$647$	16.0943	0.632732	0.316366	−	0.948637i	$-0.397537\pi$
$647$	16.0943	0.632732	0.316366	+	0.948637i	$0.397537\pi$
$648$	0	0
$649$	−3.47966	−0.136589
$650$	0	0
$651$	27.3513	1.07198
$652$	0	0
$653$	−33.9810	−1.32978	−0.664889	−	0.746942i	$-0.731521\pi$
$653$	−33.9810	−1.32978	−0.664889	+	0.746942i	$0.731521\pi$
$654$	0	0
$655$	−9.29141	−0.363046
$656$	0	0
$657$	−1.49755	−0.0584249
$658$	0	0
$659$	5.84598	0.227727	0.113863	−	0.993496i	$-0.463677\pi$
$659$	5.84598	0.227727	0.113863	+	0.993496i	$0.463677\pi$
$660$	0	0
$661$	10.5250	0.409375	0.204688	−	0.978827i	$-0.434382\pi$
$661$	10.5250	0.409375	0.204688	+	0.978827i	$0.434382\pi$
$662$	0	0
$663$	−31.3648	−1.21811
$664$	0	0
$665$	−10.7567	−0.417129
$666$	0	0
$667$	73.3612	2.84056
$668$	0	0
$669$	−33.1831	−1.28293
$670$	0	0
$671$	−4.69391	−0.181207
$672$	0	0
$673$	−3.99228	−0.153891	−0.0769454	−	0.997035i	$-0.524517\pi$
$673$	−3.99228	−0.153891	−0.0769454	+	0.997035i	$0.524517\pi$
$674$	0	0
$675$	5.32378	0.204912
$676$	0	0
$677$	−1.59882	−0.0614478	−0.0307239	−	0.999528i	$-0.509781\pi$
$677$	−1.59882	−0.0614478	−0.0307239	+	0.999528i	$0.509781\pi$
$678$	0	0
$679$	−17.2035	−0.660211
$680$	0	0
$681$	16.2285	0.621878
$682$	0	0
$683$	1.23339	0.0471944	0.0235972	−	0.999722i	$-0.492488\pi$
$683$	1.23339	0.0471944	0.0235972	+	0.999722i	$0.492488\pi$
$684$	0	0
$685$	5.97746	0.228387
$686$	0	0
$687$	−10.5854	−0.403858
$688$	0	0
$689$	−16.1969	−0.617052
$690$	0	0
$691$	−1.55289	−0.0590746	−0.0295373	−	0.999564i	$-0.509403\pi$
$691$	−1.55289	−0.0590746	−0.0295373	+	0.999564i	$0.509403\pi$
$692$	0	0
$693$	−0.891883	−0.0338798
$694$	0	0
$695$	−12.0766	−0.458093
$696$	0	0
$697$	−14.3348	−0.542970
$698$	0	0
$699$	−11.3380	−0.428841
$700$	0	0
$701$	−13.4906	−0.509532	−0.254766	−	0.967003i	$-0.581998\pi$
$701$	−13.4906	−0.509532	−0.254766	+	0.967003i	$0.581998\pi$
$702$	0	0
$703$	21.4062	0.807349
$704$	0	0
$705$	19.7583	0.744142
$706$	0	0
$707$	53.9365	2.02849
$708$	0	0
$709$	15.1429	0.568703	0.284351	−	0.958720i	$-0.408222\pi$
$709$	15.1429	0.568703	0.284351	+	0.958720i	$0.408222\pi$
$710$	0	0
$711$	−0.179308	−0.00672459
$712$	0	0
$713$	−28.5086	−1.06766
$714$	0	0
$715$	−4.18608	−0.156550
$716$	0	0
$717$	−40.1110	−1.49797
$718$	0	0
$719$	−27.5067	−1.02583	−0.512913	−	0.858441i	$-0.671434\pi$
$719$	−27.5067	−1.02583	−0.512913	+	0.858441i	$0.671434\pi$
$720$	0	0
$721$	63.6753	2.37139
$722$	0	0
$723$	20.3851	0.758128
$724$	0	0
$725$	−8.71732	−0.323753
$726$	0	0
$727$	−7.40922	−0.274793	−0.137396	−	0.990516i	$-0.543873\pi$
$727$	−7.40922	−0.274793	−0.137396	+	0.990516i	$0.543873\pi$
$728$	0	0
$729$	28.2678	1.04696
$730$	0	0
$731$	24.6776	0.912734
$732$	0	0
$733$	15.9022	0.587361	0.293680	−	0.955904i	$-0.405120\pi$
$733$	15.9022	0.587361	0.293680	+	0.955904i	$0.405120\pi$
$734$	0	0
$735$	−26.8673	−0.991017
$736$	0	0
$737$	−16.3860	−0.603587
$738$	0	0
$739$	33.4462	1.23034	0.615169	−	0.788395i	$-0.289088\pi$
$739$	33.4462	1.23034	0.615169	+	0.788395i	$0.289088\pi$
$740$	0	0
$741$	−13.4424	−0.493819
$742$	0	0
$743$	46.9482	1.72236	0.861182	−	0.508297i	$-0.169725\pi$
$743$	46.9482	1.72236	0.861182	+	0.508297i	$0.169725\pi$
$744$	0	0
$745$	−12.0049	−0.439826
$746$	0	0
$747$	−2.44071	−0.0893008
$748$	0	0
$749$	−3.08444	−0.112703
$750$	0	0
$751$	−39.3367	−1.43542	−0.717708	−	0.696345i	$-0.754809\pi$
$751$	−39.3367	−1.43542	−0.717708	+	0.696345i	$0.754809\pi$
$752$	0	0
$753$	37.6404	1.37169
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	38.7559	1.40861	0.704304	−	0.709899i	$-0.251259\pi$
$757$	38.7559	1.40861	0.704304	+	0.709899i	$0.251259\pi$
$758$	0	0
$759$	−16.7286	−0.607209
$760$	0	0
$761$	−19.3171	−0.700245	−0.350123	−	0.936704i	$-0.613860\pi$
$761$	−19.3171	−0.700245	−0.350123	+	0.936704i	$0.613860\pi$
$762$	0	0
$763$	−67.3007	−2.43645
$764$	0	0
$765$	−0.827670	−0.0299245
$766$	0	0
$767$	10.4767	0.378293
$768$	0	0
$769$	−11.2909	−0.407161	−0.203581	−	0.979058i	$-0.565258\pi$
$769$	−11.2909	−0.407161	−0.203581	+	0.979058i	$0.565258\pi$
$770$	0	0
$771$	14.8631	0.535282
$772$	0	0
$773$	7.11477	0.255900	0.127950	−	0.991781i	$-0.459160\pi$
$773$	7.11477	0.255900	0.127950	+	0.991781i	$0.459160\pi$
$774$	0	0
$775$	3.38760	0.121686
$776$	0	0
$777$	76.9507	2.76059
$778$	0	0
$779$	−6.14366	−0.220119
$780$	0	0
$781$	4.26332	0.152553
$782$	0	0
$783$	−46.4091	−1.65853
$784$	0	0
$785$	−2.61722	−0.0934125
$786$	0	0
$787$	−17.0445	−0.607572	−0.303786	−	0.952740i	$-0.598251\pi$
$787$	−17.0445	−0.607572	−0.303786	+	0.952740i	$0.598251\pi$
$788$	0	0
$789$	−4.46012	−0.158784
$790$	0	0
$791$	32.6179	1.15976
$792$	0	0
$793$	14.1327	0.501865
$794$	0	0
$795$	7.69131	0.272783
$796$	0	0
$797$	−44.7936	−1.58667	−0.793335	−	0.608785i	$-0.791657\pi$
$797$	−44.7936	−1.58667	−0.793335	+	0.608785i	$0.791657\pi$
$798$	0	0
$799$	−61.4202	−2.17289
$800$	0	0
$801$	0.151952	0.00536897
$802$	0	0
$803$	−11.1805	−0.394550
$804$	0	0
$805$	40.3044	1.42054
$806$	0	0
$807$	37.2559	1.31147
$808$	0	0
$809$	8.60627	0.302580	0.151290	−	0.988489i	$-0.451657\pi$
$809$	8.60627	0.302580	0.151290	+	0.988489i	$0.451657\pi$
$810$	0	0
$811$	−53.1530	−1.86646	−0.933228	−	0.359286i	$-0.883020\pi$
$811$	−53.1530	−1.86646	−0.933228	+	0.359286i	$0.883020\pi$
$812$	0	0
$813$	−8.80498	−0.308804
$814$	0	0
$815$	16.6073	0.581728
$816$	0	0
$817$	10.5764	0.370021
$818$	0	0
$819$	2.68532	0.0938328
$820$	0	0
$821$	14.5588	0.508104	0.254052	−	0.967191i	$-0.418236\pi$
$821$	14.5588	0.508104	0.254052	+	0.967191i	$0.418236\pi$
$822$	0	0
$823$	−45.4404	−1.58395	−0.791977	−	0.610551i	$-0.790948\pi$
$823$	−45.4404	−1.58395	−0.791977	+	0.610551i	$0.790948\pi$
$824$	0	0
$825$	1.98782	0.0692069
$826$	0	0
$827$	−36.4041	−1.26589	−0.632947	−	0.774195i	$-0.718155\pi$
$827$	−36.4041	−1.26589	−0.632947	+	0.774195i	$0.718155\pi$
$828$	0	0
$829$	−25.3060	−0.878913	−0.439457	−	0.898264i	$-0.644829\pi$
$829$	−25.3060	−0.878913	−0.439457	+	0.898264i	$0.644829\pi$
$830$	0	0
$831$	12.1869	0.422758
$832$	0	0
$833$	83.5190	2.89376
$834$	0	0
$835$	−20.2987	−0.702465
$836$	0	0
$837$	18.0349	0.623376
$838$	0	0
$839$	29.8613	1.03093	0.515463	−	0.856912i	$-0.327620\pi$
$839$	29.8613	1.03093	0.515463	+	0.856912i	$0.327620\pi$
$840$	0	0
$841$	46.9916	1.62040
$842$	0	0
$843$	43.1507	1.48619
$844$	0	0
$845$	−0.396361	−0.0136352
$846$	0	0
$847$	46.0232	1.58138
$848$	0	0
$849$	−7.79032	−0.267363
$850$	0	0
$851$	−80.2066	−2.74945
$852$	0	0
$853$	−0.462410	−0.0158326	−0.00791631	−	0.999969i	$-0.502520\pi$
$853$	−0.462410	−0.0158326	−0.00791631	+	0.999969i	$0.502520\pi$
$854$	0	0
$855$	−0.354725	−0.0121313
$856$	0	0
$857$	10.9521	0.374118	0.187059	−	0.982349i	$-0.440105\pi$
$857$	10.9521	0.374118	0.187059	+	0.982349i	$0.440105\pi$
$858$	0	0
$859$	−30.6640	−1.04624	−0.523121	−	0.852259i	$-0.675232\pi$
$859$	−30.6640	−1.04624	−0.523121	+	0.852259i	$0.675232\pi$
$860$	0	0
$861$	−22.0852	−0.752660
$862$	0	0
$863$	−18.5123	−0.630166	−0.315083	−	0.949064i	$-0.602032\pi$
$863$	−18.5123	−0.630166	−0.315083	+	0.949064i	$0.602032\pi$
$864$	0	0
$865$	−19.2373	−0.654086
$866$	0	0
$867$	−17.6397	−0.599076
$868$	0	0
$869$	−1.33869	−0.0454119
$870$	0	0
$871$	49.3358	1.67168
$872$	0	0
$873$	−0.567321	−0.0192009
$874$	0	0
$875$	−4.78926	−0.161907
$876$	0	0
$877$	21.7724	0.735202	0.367601	−	0.929984i	$-0.380179\pi$
$877$	21.7724	0.735202	0.367601	+	0.929984i	$0.380179\pi$
$878$	0	0
$879$	−39.3333	−1.32668
$880$	0	0
$881$	39.6883	1.33713	0.668566	−	0.743653i	$-0.266908\pi$
$881$	39.6883	1.33713	0.668566	+	0.743653i	$0.266908\pi$
$882$	0	0
$883$	3.71364	0.124974	0.0624869	−	0.998046i	$-0.480097\pi$
$883$	3.71364	0.124974	0.0624869	+	0.998046i	$0.480097\pi$
$884$	0	0
$885$	−4.97502	−0.167233
$886$	0	0
$887$	−1.81200	−0.0608410	−0.0304205	−	0.999537i	$-0.509685\pi$
$887$	−1.81200	−0.0608410	−0.0304205	+	0.999537i	$0.509685\pi$
$888$	0	0
$889$	−26.2398	−0.880056
$890$	0	0
$891$	10.0240	0.335817
$892$	0	0
$893$	−26.3236	−0.880886
$894$	0	0
$895$	23.5260	0.786387
$896$	0	0
$897$	50.3672	1.68171
$898$	0	0
$899$	−29.5308	−0.984908
$900$	0	0
$901$	−23.9090	−0.796523
$902$	0	0
$903$	38.0199	1.26522
$904$	0	0
$905$	−2.07285	−0.0689039
$906$	0	0
$907$	47.9592	1.59246	0.796230	−	0.604994i	$-0.206824\pi$
$907$	47.9592	1.59246	0.796230	+	0.604994i	$0.206824\pi$
$908$	0	0
$909$	1.77867	0.0589946
$910$	0	0
$911$	−3.77083	−0.124933	−0.0624666	−	0.998047i	$-0.519897\pi$
$911$	−3.77083	−0.124933	−0.0624666	+	0.998047i	$0.519897\pi$
$912$	0	0
$913$	−18.2219	−0.603058
$914$	0	0
$915$	−6.71109	−0.221862
$916$	0	0
$917$	44.4990	1.46949
$918$	0	0
$919$	−15.1021	−0.498174	−0.249087	−	0.968481i	$-0.580130\pi$
$919$	−15.1021	−0.498174	−0.249087	+	0.968481i	$0.580130\pi$
$920$	0	0
$921$	−44.5225	−1.46707
$922$	0	0
$923$	−12.8362	−0.422508
$924$	0	0
$925$	9.53075	0.313369
$926$	0	0
$927$	2.09982	0.0689672
$928$	0	0
$929$	25.5880	0.839517	0.419758	−	0.907636i	$-0.362115\pi$
$929$	25.5880	0.839517	0.419758	+	0.907636i	$0.362115\pi$
$930$	0	0
$931$	35.7948	1.17313
$932$	0	0
$933$	24.8316	0.812950
$934$	0	0
$935$	−6.17926	−0.202084
$936$	0	0
$937$	44.0737	1.43983	0.719913	−	0.694064i	$-0.244182\pi$
$937$	44.0737	1.43983	0.719913	+	0.694064i	$0.244182\pi$
$938$	0	0
$939$	29.6143	0.966428
$940$	0	0
$941$	2.47251	0.0806016	0.0403008	−	0.999188i	$-0.487168\pi$
$941$	2.47251	0.0806016	0.0403008	+	0.999188i	$0.487168\pi$
$942$	0	0
$943$	23.0196	0.749622
$944$	0	0
$945$	−25.4970	−0.829417
$946$	0	0
$947$	11.1441	0.362135	0.181067	−	0.983471i	$-0.442045\pi$
$947$	11.1441	0.362135	0.181067	+	0.983471i	$0.442045\pi$
$948$	0	0
$949$	33.6627	1.09274
$950$	0	0
$951$	−55.1675	−1.78893
$952$	0	0
$953$	34.8520	1.12897	0.564483	−	0.825444i	$-0.309075\pi$
$953$	34.8520	1.12897	0.564483	+	0.825444i	$0.309075\pi$
$954$	0	0
$955$	−16.1010	−0.521016
$956$	0	0
$957$	−17.3284	−0.560148
$958$	0	0
$959$	−28.6276	−0.924435
$960$	0	0
$961$	−19.5241	−0.629811
$962$	0	0
$963$	−0.101716	−0.00327774
$964$	0	0
$965$	−20.6192	−0.663754
$966$	0	0
$967$	−8.05008	−0.258873	−0.129437	−	0.991588i	$-0.541317\pi$
$967$	−8.05008	−0.258873	−0.129437	+	0.991588i	$0.541317\pi$
$968$	0	0
$969$	−19.8430	−0.637448
$970$	0	0
$971$	15.0168	0.481911	0.240956	−	0.970536i	$-0.422539\pi$
$971$	15.0168	0.481911	0.240956	+	0.970536i	$0.422539\pi$
$972$	0	0
$973$	57.8382	1.85421
$974$	0	0
$975$	−5.98501	−0.191674
$976$	0	0
$977$	−56.4400	−1.80567	−0.902837	−	0.429983i	$-0.858520\pi$
$977$	−56.4400	−1.80567	−0.902837	+	0.429983i	$0.858520\pi$
$978$	0	0
$979$	1.13445	0.0362573
$980$	0	0
$981$	−2.21938	−0.0708592
$982$	0	0
$983$	46.5036	1.48324	0.741618	−	0.670823i	$-0.234059\pi$
$983$	46.5036	1.48324	0.741618	+	0.670823i	$0.234059\pi$
$984$	0	0
$985$	−13.3013	−0.423814
$986$	0	0
$987$	−94.6279	−3.01204
$988$	0	0
$989$	−39.6286	−1.26012
$990$	0	0
$991$	−12.6757	−0.402656	−0.201328	−	0.979524i	$-0.564526\pi$
$991$	−12.6757	−0.402656	−0.201328	+	0.979524i	$0.564526\pi$
$992$	0	0
$993$	5.54308	0.175904
$994$	0	0
$995$	−22.8199	−0.723439
$996$	0	0
$997$	−45.8365	−1.45166	−0.725829	−	0.687876i	$-0.758543\pi$
$997$	−45.8365	−1.45166	−0.725829	+	0.687876i	$0.758543\pi$
$998$	0	0
$999$	50.7396	1.60533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.7	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.7	✓	19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.68584 q^{3} +1.00000 q^{5} -4.78926 q^{7} -0.157936 q^{9} +O(q^{10})\)	\(q-1.68584 q^{3} +1.00000 q^{5} -4.78926 q^{7} -0.157936 q^{9} -1.17912 q^{11} +3.55016 q^{13} -1.68584 q^{15} +5.24056 q^{17} +2.24601 q^{19} +8.07394 q^{21} -8.41556 q^{23} +1.00000 q^{25} +5.32378 q^{27} -8.71732 q^{29} +3.38760 q^{31} +1.98782 q^{33} -4.78926 q^{35} +9.53075 q^{37} -5.98501 q^{39} -2.73536 q^{41} +4.70896 q^{43} -0.157936 q^{45} -11.7202 q^{47} +15.9370 q^{49} -8.83475 q^{51} -4.56229 q^{53} -1.17912 q^{55} -3.78642 q^{57} +2.95106 q^{59} +3.98085 q^{61} +0.756395 q^{63} +3.55016 q^{65} +13.8968 q^{67} +14.1873 q^{69} -3.61567 q^{71} +9.48202 q^{73} -1.68584 q^{75} +5.64713 q^{77} +1.13533 q^{79} -8.50125 q^{81} +15.4538 q^{83} +5.24056 q^{85} +14.6960 q^{87} -0.962116 q^{89} -17.0027 q^{91} -5.71097 q^{93} +2.24601 q^{95} +3.59211 q^{97} +0.186226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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