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Properties

Label 245.3.m.b.128.2
Level $245$
Weight $3$
Character 245.128
Analytic conductor $6.676$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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(18,245)] 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("245.18"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 8])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.m (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 128.2
Character \(\chi\) \(=\) 245.128
Dual form 245.3.m.b.67.2

$q$-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+(-0.471334 - 1.75904i) q^{2} +(-0.0524492 + 0.195743i) q^{3} +(0.592022 - 0.341804i) q^{4} +(1.13206 + 4.87016i) q^{5} +0.369042 q^{6} +(-6.03113 - 6.03113i) q^{8} +(7.75866 + 4.47947i) q^{9} +(8.03325 - 4.28681i) q^{10} +(6.67094 + 11.5544i) q^{11} +(0.0358547 + 0.133812i) q^{12} +(-4.91251 - 4.91251i) q^{13} +(-1.01268 - 0.0338433i) q^{15} +(-6.39912 + 11.0836i) q^{16} +(19.4385 + 5.20854i) q^{17} +(4.22265 - 15.7592i) q^{18} +(14.0516 + 8.11267i) q^{19} +(2.33484 + 2.49630i) q^{20} +(17.1805 - 17.1805i) q^{22} +(18.7480 - 5.02351i) q^{23} +(1.49688 - 0.864224i) q^{24} +(-22.4369 + 11.0266i) q^{25} +(-6.32588 + 10.9567i) q^{26} +(-2.57341 + 2.57341i) q^{27} -29.5639i q^{29} +(0.417777 + 1.79729i) q^{30} +(-9.49369 - 16.4435i) q^{31} +(-10.4420 - 2.79793i) q^{32} +(-2.61158 + 0.699772i) q^{33} -36.6482i q^{34} +6.12440 q^{36} +(1.92294 + 7.17651i) q^{37} +(7.64756 - 28.5411i) q^{38} +(1.21925 - 0.703933i) q^{39} +(22.5450 - 36.2001i) q^{40} +52.1667 q^{41} +(-11.3609 - 11.3609i) q^{43} +(7.89869 + 4.56031i) q^{44} +(-13.0325 + 42.8569i) q^{45} +(-17.6732 - 30.6108i) q^{46} +(-14.2885 - 53.3255i) q^{47} +(-1.83391 - 1.83391i) q^{48} +(29.9716 + 34.2703i) q^{50} +(-2.03907 + 3.53178i) q^{51} +(-4.58742 - 1.22920i) q^{52} +(-3.27711 + 12.2303i) q^{53} +(5.73967 + 3.31380i) q^{54} +(-48.7199 + 45.5688i) q^{55} +(-2.32499 + 2.32499i) q^{57} +(-52.0041 + 13.9345i) q^{58} +(-12.0029 + 6.92989i) q^{59} +(-0.611094 + 0.326101i) q^{60} +(-57.3589 + 99.3486i) q^{61} +(-24.4502 + 24.4502i) q^{62} +70.8797i q^{64} +(18.3634 - 29.4859i) q^{65} +(2.46186 + 4.26406i) q^{66} +(56.0993 + 15.0318i) q^{67} +(13.2883 - 3.56060i) q^{68} +3.93327i q^{69} -86.5580 q^{71} +(-19.7773 - 73.8097i) q^{72} +(-6.50169 + 24.2646i) q^{73} +(11.7175 - 6.76507i) q^{74} +(-0.981586 - 4.97021i) q^{75} +11.0918 q^{76} +(-1.81292 - 1.81292i) q^{78} +(-23.0541 - 13.3103i) q^{79} +(-61.2231 - 18.6175i) q^{80} +(39.9464 + 69.1893i) q^{81} +(-24.5880 - 91.7635i) q^{82} +(-32.6537 - 32.6537i) q^{83} +(-3.36086 + 100.565i) q^{85} +(-14.6295 + 25.3390i) q^{86} +(5.78692 + 1.55060i) q^{87} +(29.4528 - 109.919i) q^{88} +(62.6554 + 36.1741i) q^{89} +(81.5299 + 2.72471i) q^{90} +(9.38217 - 9.38217i) q^{92} +(3.71665 - 0.995873i) q^{93} +(-87.0673 + 50.2683i) q^{94} +(-23.6028 + 77.6173i) q^{95} +(1.09535 - 1.89720i) q^{96} +(-47.8969 + 47.8969i) q^{97} +119.529i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} + 2 q^{3} + 4 q^{5} - 36 q^{8} - 14 q^{10} - 24 q^{11} + 46 q^{12} + 8 q^{13} + 52 q^{15} + 20 q^{16} + 48 q^{17} - 4 q^{18} + 72 q^{20} + 104 q^{22} - 86 q^{23} - 16 q^{25} - 140 q^{26}+ \cdots + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.471334 − 1.75904i −0.235667 − 0.879522i −0.977847 − 0.209321i \(-0.932875\pi\)
0.742180 − 0.670201i \(-0.233792\pi\)
\(3\) −0.0524492 + 0.195743i −0.0174831 + 0.0652477i −0.974116 − 0.226048i \(-0.927420\pi\)
0.956633 + 0.291295i \(0.0940862\pi\)
\(4\) 0.592022 − 0.341804i 0.148005 − 0.0854510i
\(5\) 1.13206 + 4.87016i 0.226412 + 0.974032i
\(6\) 0.369042 0.0615070
\(7\) 0 0
\(8\) −6.03113 − 6.03113i −0.753891 − 0.753891i
\(9\) 7.75866 + 4.47947i 0.862074 + 0.497719i
\(10\) 8.03325 − 4.28681i 0.803325 − 0.428681i
\(11\) 6.67094 + 11.5544i 0.606449 + 1.05040i 0.991821 + 0.127639i \(0.0407400\pi\)
−0.385371 + 0.922762i \(0.625927\pi\)
\(12\) 0.0358547 + 0.133812i 0.00298789 + 0.0111510i
\(13\) −4.91251 − 4.91251i −0.377885 − 0.377885i 0.492454 − 0.870339i \(-0.336100\pi\)
−0.870339 + 0.492454i \(0.836100\pi\)
\(14\) 0 0
\(15\) −1.01268 − 0.0338433i −0.0675117 − 0.00225622i
\(16\) −6.39912 + 11.0836i −0.399945 + 0.692726i
\(17\) 19.4385 + 5.20854i 1.14344 + 0.306385i 0.780334 − 0.625362i \(-0.215049\pi\)
0.363108 + 0.931747i \(0.381716\pi\)
\(18\) 4.22265 − 15.7592i 0.234592 − 0.875509i
\(19\) 14.0516 + 8.11267i 0.739555 + 0.426983i 0.821908 − 0.569621i \(-0.192910\pi\)
−0.0823522 + 0.996603i \(0.526243\pi\)
\(20\) 2.33484 + 2.49630i 0.116742 + 0.124815i
\(21\) 0 0
\(22\) 17.1805 − 17.1805i 0.780931 − 0.780931i
\(23\) 18.7480 − 5.02351i 0.815131 − 0.218414i 0.172914 − 0.984937i \(-0.444682\pi\)
0.642217 + 0.766523i \(0.278015\pi\)
\(24\) 1.49688 − 0.864224i 0.0623700 − 0.0360093i
\(25\) −22.4369 + 11.0266i −0.897475 + 0.441064i
\(26\) −6.32588 + 10.9567i −0.243303 + 0.421413i
\(27\) −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 \(-0.170252\pi\)
\(30\) 0.417777 + 1.79729i 0.0139259 + 0.0599098i
\(31\) −9.49369 − 16.4435i −0.306248 − 0.530437i 0.671290 − 0.741194i \(-0.265740\pi\)
−0.977538 + 0.210757i \(0.932407\pi\)
\(32\) −10.4420 − 2.79793i −0.326313 − 0.0874352i
\(33\) −2.61158 + 0.699772i −0.0791389 + 0.0212052i
\(34\) − 36.6482i − 1.07789i
\(35\) 0 0
\(36\) 6.12440 0.170122
\(37\) 1.92294 + 7.17651i 0.0519714 + 0.193960i 0.987031 − 0.160531i \(-0.0513208\pi\)
−0.935059 + 0.354491i \(0.884654\pi\)
\(38\) 7.64756 − 28.5411i 0.201252 − 0.751081i
\(39\) 1.21925 − 0.703933i 0.0312627 − 0.0180496i
\(40\) 22.5450 − 36.2001i 0.563624 − 0.905003i
\(41\) 52.1667 1.27236 0.636179 − 0.771541i \(-0.280514\pi\)
0.636179 + 0.771541i \(0.280514\pi\)
\(42\) 0 0
\(43\) −11.3609 − 11.3609i −0.264206 − 0.264206i 0.562554 − 0.826760i \(-0.309819\pi\)
−0.826760 + 0.562554i \(0.809819\pi\)
\(44\) 7.89869 + 4.56031i 0.179516 + 0.103643i
\(45\) −13.0325 + 42.8569i −0.289610 + 0.952376i
\(46\) −17.6732 − 30.6108i −0.384199 − 0.665452i
\(47\) −14.2885 − 53.3255i −0.304011 − 1.13459i −0.933793 − 0.357814i \(-0.883522\pi\)
0.629782 − 0.776772i \(-0.283144\pi\)
\(48\) −1.83391 − 1.83391i −0.0382065 − 0.0382065i
\(49\) 0 0
\(50\) 29.9716 + 34.2703i 0.599431 + 0.685405i
\(51\) −2.03907 + 3.53178i −0.0399818 + 0.0692505i
\(52\) −4.58742 − 1.22920i −0.0882197 − 0.0236384i
\(53\) −3.27711 + 12.2303i −0.0618323 + 0.230761i −0.989926 − 0.141585i \(-0.954780\pi\)
0.928094 + 0.372347i \(0.121447\pi\)
\(54\) 5.73967 + 3.31380i 0.106290 + 0.0613667i
\(55\) −48.7199 + 45.5688i −0.885817 + 0.828524i
\(56\) 0 0
\(57\) −2.32499 + 2.32499i −0.0407894 + 0.0407894i
\(58\) −52.0041 + 13.9345i −0.896623 + 0.240249i
\(59\) −12.0029 + 6.92989i −0.203439 + 0.117456i −0.598259 − 0.801303i \(-0.704141\pi\)
0.394819 + 0.918759i \(0.370807\pi\)
\(60\) −0.611094 + 0.326101i −0.0101849 + 0.00543501i
\(61\) −57.3589 + 99.3486i −0.940311 + 1.62867i −0.175432 + 0.984492i \(0.556132\pi\)
−0.764879 + 0.644174i \(0.777201\pi\)
\(62\) −24.4502 + 24.4502i −0.394358 + 0.394358i
\(63\) 0 0
\(64\) 70.8797i 1.10750i
\(65\) 18.3634 − 29.4859i 0.282514 − 0.453630i
\(66\) 2.46186 + 4.26406i 0.0373009 + 0.0646070i
\(67\) 56.0993 + 15.0318i 0.837302 + 0.224355i 0.651897 − 0.758308i \(-0.273973\pi\)
0.185406 + 0.982662i \(0.440640\pi\)
\(68\) 13.2883 − 3.56060i 0.195417 − 0.0523617i
\(69\) 3.93327i 0.0570040i
\(70\) 0 0
\(71\) −86.5580 −1.21913 −0.609563 − 0.792737i \(-0.708655\pi\)
−0.609563 + 0.792737i \(0.708655\pi\)
\(72\) −19.7773 − 73.8097i −0.274684 − 1.02514i
\(73\) −6.50169 + 24.2646i −0.0890643 + 0.332392i −0.996053 − 0.0887630i \(-0.971709\pi\)
0.906989 + 0.421155i \(0.138375\pi\)
\(74\) 11.7175 − 6.76507i 0.158344 − 0.0914199i
\(75\) −0.981586 − 4.97021i −0.0130878 − 0.0662694i
\(76\) 11.0918 0.145944
\(77\) 0 0
\(78\) −1.81292 − 1.81292i −0.0232426 − 0.0232426i
\(79\) −23.0541 − 13.3103i −0.291824 − 0.168485i 0.346940 − 0.937887i \(-0.387221\pi\)
−0.638764 + 0.769402i \(0.720554\pi\)
\(80\) −61.2231 − 18.6175i −0.765289 − 0.232718i
\(81\) 39.9464 + 69.1893i 0.493166 + 0.854189i
\(82\) −24.5880 − 91.7635i −0.299853 − 1.11907i
\(83\) −32.6537 − 32.6537i −0.393418 − 0.393418i 0.482486 − 0.875904i \(-0.339734\pi\)
−0.875904 + 0.482486i \(0.839734\pi\)
\(84\) 0 0
\(85\) −3.36086 + 100.565i −0.0395395 + 1.18312i
\(86\) −14.6295 + 25.3390i −0.170110 + 0.294640i
\(87\) 5.78692 + 1.55060i 0.0665164 + 0.0178230i
\(88\) 29.4528 − 109.919i 0.334691 − 1.24908i
\(89\) 62.6554 + 36.1741i 0.703993 + 0.406451i 0.808833 − 0.588038i \(-0.200100\pi\)
−0.104840 + 0.994489i \(0.533433\pi\)
\(90\) 81.5299 + 2.72471i 0.905888 + 0.0302745i
\(91\) 0 0
\(92\) 9.38217 − 9.38217i 0.101980 − 0.101980i
\(93\) 3.71665 − 0.995873i 0.0399640 − 0.0107083i
\(94\) −87.0673 + 50.2683i −0.926248 + 0.534769i
\(95\) −23.6028 + 77.6173i −0.248451 + 0.817024i
\(96\) 1.09535 − 1.89720i 0.0114099 − 0.0197625i
\(97\) −47.8969 + 47.8969i −0.493783 + 0.493783i −0.909496 − 0.415713i \(-0.863532\pi\)
0.415713 + 0.909496i \(0.363532\pi\)
\(98\) 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.m.b.128.2 24
5.2 odd 4 inner 245.3.m.b.177.5 24
7.2 even 3 245.3.g.b.148.2 12
7.3 odd 6 35.3.l.a.18.5 yes 24
7.4 even 3 inner 245.3.m.b.18.5 24
7.5 odd 6 245.3.g.c.148.2 12
7.6 odd 2 35.3.l.a.23.2 yes 24
21.17 even 6 315.3.ca.a.298.2 24
21.20 even 2 315.3.ca.a.163.5 24
35.2 odd 12 245.3.g.b.197.2 12
35.3 even 12 175.3.p.c.32.5 24
35.12 even 12 245.3.g.c.197.2 12
35.13 even 4 175.3.p.c.107.2 24
35.17 even 12 35.3.l.a.32.2 yes 24
35.24 odd 6 175.3.p.c.18.2 24
35.27 even 4 35.3.l.a.2.5 ✓ 24
35.32 odd 12 inner 245.3.m.b.67.2 24
35.34 odd 2 175.3.p.c.93.5 24
105.17 odd 12 315.3.ca.a.172.5 24
105.62 odd 4 315.3.ca.a.37.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.5 ✓ 24 35.27 even 4
35.3.l.a.18.5 yes 24 7.3 odd 6
35.3.l.a.23.2 yes 24 7.6 odd 2
35.3.l.a.32.2 yes 24 35.17 even 12
175.3.p.c.18.2 24 35.24 odd 6
175.3.p.c.32.5 24 35.3 even 12
175.3.p.c.93.5 24 35.34 odd 2
175.3.p.c.107.2 24 35.13 even 4
245.3.g.b.148.2 12 7.2 even 3
245.3.g.b.197.2 12 35.2 odd 12
245.3.g.c.148.2 12 7.5 odd 6
245.3.g.c.197.2 12 35.12 even 12
245.3.m.b.18.5 24 7.4 even 3 inner
245.3.m.b.67.2 24 35.32 odd 12 inner
245.3.m.b.128.2 24 1.1 even 1 trivial
245.3.m.b.177.5 24 5.2 odd 4 inner
315.3.ca.a.37.2 24 105.62 odd 4
315.3.ca.a.163.5 24 21.20 even 2
315.3.ca.a.172.5 24 105.17 odd 12
315.3.ca.a.298.2 24 21.17 even 6