↑

Properties

Label 245.3.g.c.148.2
Level $245$
Weight $3$
Character 245.148
Analytic conductor $6.676$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [245,3,Mod(148,245)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("245.148"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 10 x^{9} + 127 x^{8} - 160 x^{7} + 116 x^{6} + 288 x^{5} + 1471 x^{4} + \cdots + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 148.2
Root \(-1.28771 + 1.28771i\) of defining polynomial
Character \(\chi\) \(=\) 245.148
Dual form 245.3.g.c.197.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.28771 + 1.28771i) q^{2} +(0.143294 + 0.143294i) q^{3} +0.683608i q^{4} +(-3.65165 + 3.41547i) q^{5} -0.369042 q^{6} +(-6.03113 - 6.03113i) q^{8} -8.95893i q^{9} +(0.304133 - 9.10040i) q^{10} -13.3419 q^{11} +(-0.0979569 + 0.0979569i) q^{12} +(4.91251 + 4.91251i) q^{13} +(-1.01268 - 0.0338433i) q^{15} +12.7982 q^{16} +(14.2300 - 14.2300i) q^{17} +(11.5365 + 11.5365i) q^{18} +16.2253i q^{19} +(-2.33484 - 2.49630i) q^{20} +(17.1805 - 17.1805i) q^{22} +(-13.7245 - 13.7245i) q^{23} -1.72845i q^{24} +(1.66912 - 24.9442i) q^{25} -12.6518 q^{26} +(2.57341 - 2.57341i) q^{27} -29.5639i q^{29} +(1.34761 - 1.26045i) q^{30} -18.9874 q^{31} +(7.64408 - 7.64408i) q^{32} +(-1.91181 - 1.91181i) q^{33} +36.6482i q^{34} +6.12440 q^{36} +(5.25357 - 5.25357i) q^{37} +(-20.8935 - 20.8935i) q^{38} +1.40787i q^{39} +(42.6227 + 1.42444i) q^{40} -52.1667 q^{41} +(-11.3609 - 11.3609i) q^{43} -9.12062i q^{44} +(30.5990 + 32.7149i) q^{45} +35.3463 q^{46} +(39.0370 - 39.0370i) q^{47} +(1.83391 + 1.83391i) q^{48} +(29.9716 + 34.2703i) q^{50} +4.07814 q^{51} +(-3.35823 + 3.35823i) q^{52} +(-8.95323 - 8.95323i) q^{53} +6.62760i q^{54} +(48.7199 - 45.5688i) q^{55} +(-2.32499 + 2.32499i) q^{57} +(38.0697 + 38.0697i) q^{58} +13.8598i q^{59} +(0.0231356 - 0.692274i) q^{60} -114.718 q^{61} +(24.4502 - 24.4502i) q^{62} +70.8797i q^{64} +(-34.7173 - 1.16024i) q^{65} +4.92372 q^{66} +(-41.0675 + 41.0675i) q^{67} +(9.72773 + 9.72773i) q^{68} -3.93327i q^{69} -86.5580 q^{71} +(-54.0325 + 54.0325i) q^{72} +(17.7630 + 17.7630i) q^{73} +13.5301i q^{74} +(3.81353 - 3.33518i) q^{75} -11.0918 q^{76} +(-1.81292 - 1.81292i) q^{78} +26.6206i q^{79} +(-46.7347 + 43.7120i) q^{80} -79.8929 q^{81} +(67.1756 - 67.1756i) q^{82} +(32.6537 + 32.6537i) q^{83} +(-3.36086 + 100.565i) q^{85} +29.2590 q^{86} +(4.23632 - 4.23632i) q^{87} +(80.4666 + 80.4666i) q^{88} +72.3482i q^{89} +(-81.5299 - 2.72471i) q^{90} +(9.38217 - 9.38217i) q^{92} +(-2.72078 - 2.72078i) q^{93} +100.537i q^{94} +(-55.4172 - 59.2493i) q^{95} +2.19070 q^{96} +(47.8969 - 47.8969i) q^{97} +119.529i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 2 q^{3} + 4 q^{5} - 18 q^{8} - 14 q^{10} + 24 q^{11} + 46 q^{12} - 4 q^{13} + 26 q^{15} - 20 q^{16} + 48 q^{17} + 4 q^{18} - 36 q^{20} + 52 q^{22} + 86 q^{23} + 16 q^{25} - 140 q^{26} + 38 q^{27}+ \cdots - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28771 + 1.28771i −0.643855 + 0.643855i −0.951501 − 0.307646i \(-0.900459\pi\)
0.307646 + 0.951501i \(0.400459\pi\)
\(3\) 0.143294 + 0.143294i 0.0477647 + 0.0477647i 0.730586 − 0.682821i \(-0.239247\pi\)
−0.682821 + 0.730586i \(0.739247\pi\)
\(4\) 0.683608i 0.170902i
\(5\) −3.65165 + 3.41547i −0.730330 + 0.683094i
\(6\) −0.369042 −0.0615070
\(7\) 0 0
\(8\) −6.03113 − 6.03113i −0.753891 − 0.753891i
\(9\) − 8.95893i − 0.995437i
\(10\) 0.304133 − 9.10040i 0.0304133 − 0.910040i
\(11\) −13.3419 −1.21290 −0.606449 − 0.795122i \(-0.707407\pi\)
−0.606449 + 0.795122i \(0.707407\pi\)
\(12\) −0.0979569 + 0.0979569i −0.00816308 + 0.00816308i
\(13\) 4.91251 + 4.91251i 0.377885 + 0.377885i 0.870339 − 0.492454i \(-0.163900\pi\)
−0.492454 + 0.870339i \(0.663900\pi\)
\(14\) 0 0
\(15\) −1.01268 − 0.0338433i −0.0675117 − 0.00225622i
\(16\) 12.7982 0.799891
\(17\) 14.2300 − 14.2300i 0.837058 − 0.837058i −0.151413 − 0.988471i \(-0.548382\pi\)
0.988471 + 0.151413i \(0.0483822\pi\)
\(18\) 11.5365 + 11.5365i 0.640917 + 0.640917i
\(19\) 16.2253i 0.853965i 0.904260 + 0.426983i \(0.140423\pi\)
−0.904260 + 0.426983i \(0.859577\pi\)
\(20\) −2.33484 − 2.49630i −0.116742 − 0.124815i
\(21\) 0 0
\(22\) 17.1805 − 17.1805i 0.780931 − 0.780931i
\(23\) −13.7245 − 13.7245i −0.596717 − 0.596717i 0.342720 − 0.939437i \(-0.388652\pi\)
−0.939437 + 0.342720i \(0.888652\pi\)
\(24\) − 1.72845i − 0.0720187i
\(25\) 1.66912 − 24.9442i 0.0667648 − 0.997769i
\(26\) −12.6518 −0.486606
\(27\) 2.57341 − 2.57341i 0.0953114 − 0.0953114i
\(28\) 0 0
\(29\) − 29.5639i − 1.01944i −0.860339 − 0.509722i \(-0.829748\pi\)
0.860339 − 0.509722i \(-0.170252\pi\)
\(30\) 1.34761 − 1.26045i 0.0449204 − 0.0420151i
\(31\) −18.9874 −0.612496 −0.306248 − 0.951952i \(-0.599074\pi\)
−0.306248 + 0.951952i \(0.599074\pi\)
\(32\) 7.64408 − 7.64408i 0.238878 − 0.238878i
\(33\) −1.91181 − 1.91181i −0.0579337 − 0.0579337i
\(34\) 36.6482i 1.07789i
\(35\) 0 0
\(36\) 6.12440 0.170122
\(37\) 5.25357 − 5.25357i 0.141988 − 0.141988i −0.632540 − 0.774528i \(-0.717987\pi\)
0.774528 + 0.632540i \(0.217987\pi\)
\(38\) −20.8935 − 20.8935i −0.549829 − 0.549829i
\(39\) 1.40787i 0.0360991i
\(40\) 42.6227 + 1.42444i 1.06557 + 0.0356110i
\(41\) −52.1667 −1.27236 −0.636179 − 0.771541i \(-0.719486\pi\)
−0.636179 + 0.771541i \(0.719486\pi\)
\(42\) 0 0
\(43\) −11.3609 − 11.3609i −0.264206 − 0.264206i 0.562554 − 0.826760i \(-0.309819\pi\)
−0.826760 + 0.562554i \(0.809819\pi\)
\(44\) − 9.12062i − 0.207287i
\(45\) 30.5990 + 32.7149i 0.679977 + 0.726998i
\(46\) 35.3463 0.768398
\(47\) 39.0370 − 39.0370i 0.830574 − 0.830574i −0.157021 − 0.987595i \(-0.550189\pi\)
0.987595 + 0.157021i \(0.0501890\pi\)
\(48\) 1.83391 + 1.83391i 0.0382065 + 0.0382065i
\(49\) 0 0
\(50\) 29.9716 + 34.2703i 0.599431 + 0.685405i
\(51\) 4.07814 0.0799636
\(52\) −3.35823 + 3.35823i −0.0645813 + 0.0645813i
\(53\) −8.95323 − 8.95323i −0.168929 − 0.168929i 0.617580 − 0.786508i \(-0.288113\pi\)
−0.786508 + 0.617580i \(0.788113\pi\)
\(54\) 6.62760i 0.122733i
\(55\) 48.7199 − 45.5688i 0.885817 − 0.828524i
\(56\) 0 0
\(57\) −2.32499 + 2.32499i −0.0407894 + 0.0407894i
\(58\) 38.0697 + 38.0697i 0.656373 + 0.656373i
\(59\) 13.8598i 0.234912i 0.993078 + 0.117456i \(0.0374738\pi\)
−0.993078 + 0.117456i \(0.962526\pi\)
\(60\) 0.0231356 − 0.692274i 0.000385593 − 0.0115379i
\(61\) −114.718 −1.88062 −0.940311 − 0.340317i \(-0.889465\pi\)
−0.940311 + 0.340317i \(0.889465\pi\)
\(62\) 24.4502 − 24.4502i 0.394358 − 0.394358i
\(63\) 0 0
\(64\) 70.8797i 1.10750i
\(65\) −34.7173 − 1.16024i −0.534112 − 0.0178499i
\(66\) 4.92372 0.0746018
\(67\) −41.0675 + 41.0675i −0.612948 + 0.612948i −0.943713 − 0.330765i \(-0.892693\pi\)
0.330765 + 0.943713i \(0.392693\pi\)
\(68\) 9.72773 + 9.72773i 0.143055 + 0.143055i
\(69\) − 3.93327i − 0.0570040i
\(70\) 0 0
\(71\) −86.5580 −1.21913 −0.609563 − 0.792737i \(-0.708655\pi\)
−0.609563 + 0.792737i \(0.708655\pi\)
\(72\) −54.0325 + 54.0325i −0.750451 + 0.750451i
\(73\) 17.7630 + 17.7630i 0.243328 + 0.243328i 0.818226 − 0.574897i \(-0.194958\pi\)
−0.574897 + 0.818226i \(0.694958\pi\)
\(74\) 13.5301i 0.182840i
\(75\) 3.81353 − 3.33518i 0.0508471 − 0.0444691i
\(76\) −11.0918 −0.145944
\(77\) 0 0
\(78\) −1.81292 − 1.81292i −0.0232426 − 0.0232426i
\(79\) 26.6206i 0.336970i 0.985704 + 0.168485i \(0.0538874\pi\)
−0.985704 + 0.168485i \(0.946113\pi\)
\(80\) −46.7347 + 43.7120i −0.584184 + 0.546401i
\(81\) −79.8929 −0.986332
\(82\) 67.1756 − 67.1756i 0.819214 − 0.819214i
\(83\) 32.6537 + 32.6537i 0.393418 + 0.393418i 0.875904 − 0.482486i \(-0.160266\pi\)
−0.482486 + 0.875904i \(0.660266\pi\)
\(84\) 0 0
\(85\) −3.36086 + 100.565i −0.0395395 + 1.18312i
\(86\) 29.2590 0.340220
\(87\) 4.23632 − 4.23632i 0.0486934 − 0.0486934i
\(88\) 80.4666 + 80.4666i 0.914393 + 0.914393i
\(89\) 72.3482i 0.812901i 0.913673 + 0.406451i \(0.133234\pi\)
−0.913673 + 0.406451i \(0.866766\pi\)
\(90\) −81.5299 − 2.72471i −0.905888 − 0.0302745i
\(91\) 0 0
\(92\) 9.38217 − 9.38217i 0.101980 − 0.101980i
\(93\) −2.72078 − 2.72078i −0.0292557 − 0.0292557i
\(94\) 100.537i 1.06954i
\(95\) −55.4172 − 59.2493i −0.583338 − 0.623677i
\(96\) 2.19070 0.0228198
\(97\) 47.8969 − 47.8969i 0.493783 − 0.493783i −0.415713 − 0.909496i \(-0.636468\pi\)
0.909496 + 0.415713i \(0.136468\pi\)
\(98\) 0 0
\(99\) 119.529i 1.20736i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 245.3.g.c.148.2 12
5.2 odd 4 inner 245.3.g.c.197.2 12
7.2 even 3 35.3.l.a.18.5 yes 24
7.3 odd 6 245.3.m.b.128.2 24
7.4 even 3 35.3.l.a.23.2 yes 24
7.5 odd 6 245.3.m.b.18.5 24
7.6 odd 2 245.3.g.b.148.2 12
21.2 odd 6 315.3.ca.a.298.2 24
21.11 odd 6 315.3.ca.a.163.5 24
35.2 odd 12 35.3.l.a.32.2 yes 24
35.4 even 6 175.3.p.c.93.5 24
35.9 even 6 175.3.p.c.18.2 24
35.12 even 12 245.3.m.b.67.2 24
35.17 even 12 245.3.m.b.177.5 24
35.18 odd 12 175.3.p.c.107.2 24
35.23 odd 12 175.3.p.c.32.5 24
35.27 even 4 245.3.g.b.197.2 12
35.32 odd 12 35.3.l.a.2.5 ✓ 24
105.2 even 12 315.3.ca.a.172.5 24
105.32 even 12 315.3.ca.a.37.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.5 ✓ 24 35.32 odd 12
35.3.l.a.18.5 yes 24 7.2 even 3
35.3.l.a.23.2 yes 24 7.4 even 3
35.3.l.a.32.2 yes 24 35.2 odd 12
175.3.p.c.18.2 24 35.9 even 6
175.3.p.c.32.5 24 35.23 odd 12
175.3.p.c.93.5 24 35.4 even 6
175.3.p.c.107.2 24 35.18 odd 12
245.3.g.b.148.2 12 7.6 odd 2
245.3.g.b.197.2 12 35.27 even 4
245.3.g.c.148.2 12 1.1 even 1 trivial
245.3.g.c.197.2 12 5.2 odd 4 inner
245.3.m.b.18.5 24 7.5 odd 6
245.3.m.b.67.2 24 35.12 even 12
245.3.m.b.128.2 24 7.3 odd 6
245.3.m.b.177.5 24 35.17 even 12
315.3.ca.a.37.2 24 105.32 even 12
315.3.ca.a.163.5 24 21.11 odd 6
315.3.ca.a.172.5 24 105.2 even 12
315.3.ca.a.298.2 24 21.2 odd 6