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Properties

Label 975.2.w.a.199.2
Level $975$
Weight $2$
Character 975.199
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [975,2,Mod(49,975)] 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("975.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,0,-4,0,6,-2,24,2,0,-12,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 199.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 975.199
Dual form 975.2.w.a.49.2

$q$-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+(0.366025 - 0.633975i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.732051 + 1.26795i) q^{4} +(0.633975 - 0.366025i) q^{6} +(-2.23205 - 3.86603i) q^{7} +2.53590 q^{8} +(0.500000 + 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} +1.46410i q^{12} +(3.50000 - 0.866025i) q^{13} -3.26795 q^{14} +(-0.535898 + 0.928203i) q^{16} +(5.83013 - 3.36603i) q^{17} +0.732051 q^{18} +(4.73205 - 2.73205i) q^{19} -4.46410i q^{21} +(-2.19615 + 1.26795i) q^{22} +(0.464102 + 0.267949i) q^{23} +(2.19615 + 1.26795i) q^{24} +(0.732051 - 2.53590i) q^{26} +1.00000i q^{27} +(3.26795 - 5.66025i) q^{28} +(-1.36603 + 2.36603i) q^{29} -3.19615i q^{31} +(2.92820 + 5.07180i) q^{32} +(-1.73205 - 3.00000i) q^{33} -4.92820i q^{34} +(-0.732051 + 1.26795i) q^{36} +(-2.00000 + 3.46410i) q^{37} -4.00000i q^{38} +(3.46410 + 1.00000i) q^{39} +(4.56218 + 2.63397i) q^{41} +(-2.83013 - 1.63397i) q^{42} +(-0.232051 + 0.133975i) q^{43} -5.07180i q^{44} +(0.339746 - 0.196152i) q^{46} -0.196152 q^{47} +(-0.928203 + 0.535898i) q^{48} +(-6.46410 + 11.1962i) q^{49} +6.73205 q^{51} +(3.66025 + 3.80385i) q^{52} +6.92820i q^{53} +(0.633975 + 0.366025i) q^{54} +(-5.66025 - 9.80385i) q^{56} +5.46410 q^{57} +(1.00000 + 1.73205i) q^{58} +(-6.29423 + 3.63397i) q^{59} +(-2.23205 - 3.86603i) q^{61} +(-2.02628 - 1.16987i) q^{62} +(2.23205 - 3.86603i) q^{63} +2.14359 q^{64} -2.53590 q^{66} +(6.23205 - 10.7942i) q^{67} +(8.53590 + 4.92820i) q^{68} +(0.267949 + 0.464102i) q^{69} +(-11.0263 + 6.36603i) q^{71} +(1.26795 + 2.19615i) q^{72} +15.3923 q^{73} +(1.46410 + 2.53590i) q^{74} +(6.92820 + 4.00000i) q^{76} +15.4641i q^{77} +(1.90192 - 1.83013i) q^{78} -1.92820 q^{79} +(-0.500000 + 0.866025i) q^{81} +(3.33975 - 1.92820i) q^{82} -2.53590 q^{83} +(5.66025 - 3.26795i) q^{84} +0.196152i q^{86} +(-2.36603 + 1.36603i) q^{87} +(-7.60770 - 4.39230i) q^{88} +(1.09808 + 0.633975i) q^{89} +(-11.1603 - 11.5981i) q^{91} +0.784610i q^{92} +(1.59808 - 2.76795i) q^{93} +(-0.0717968 + 0.124356i) q^{94} +5.85641i q^{96} +(-8.23205 - 14.2583i) q^{97} +(4.73205 + 8.19615i) q^{98} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 4 q^{4} + 6 q^{6} - 2 q^{7} + 24 q^{8} + 2 q^{9} - 12 q^{11} + 14 q^{13} - 20 q^{14} - 16 q^{16} + 6 q^{17} - 4 q^{18} + 12 q^{19} + 12 q^{22} - 12 q^{23} - 12 q^{24} - 4 q^{26} + 20 q^{28}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.366025 − 0.633975i 0.258819 − 0.448288i −0.707107 − 0.707107i \(-0.750000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(3\) 0.866025 + 0.500000i 0.500000 + 0.288675i
\(4\) 0.732051 + 1.26795i 0.366025 + 0.633975i
\(5\) 0 0
\(6\) 0.633975 − 0.366025i 0.258819 − 0.149429i
\(7\) −2.23205 − 3.86603i −0.843636 − 1.46122i −0.886801 − 0.462152i \(-0.847077\pi\)
0.0431647 − 0.999068i \(-0.486256\pi\)
\(8\) 2.53590 0.896575
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) −3.00000 − 1.73205i −0.904534 − 0.522233i −0.0258656 − 0.999665i \(-0.508234\pi\)
−0.878668 + 0.477432i \(0.841568\pi\)
\(12\) 1.46410i 0.422650i
\(13\) 3.50000 − 0.866025i 0.970725 − 0.240192i
\(14\) −3.26795 −0.873396
\(15\) 0 0
\(16\) −0.535898 + 0.928203i −0.133975 + 0.232051i
\(17\) 5.83013 − 3.36603i 1.41401 − 0.816381i 0.418250 − 0.908332i \(-0.362644\pi\)
0.995764 + 0.0919509i \(0.0293103\pi\)
\(18\) 0.732051 0.172546
\(19\) 4.73205 − 2.73205i 1.08561 − 0.626775i 0.153203 − 0.988195i \(-0.451041\pi\)
0.932403 + 0.361419i \(0.117708\pi\)
\(20\) 0 0
\(21\) − 4.46410i − 0.974147i
\(22\) −2.19615 + 1.26795i −0.468221 + 0.270328i
\(23\) 0.464102 + 0.267949i 0.0967719 + 0.0558713i 0.547605 − 0.836737i \(-0.315540\pi\)
−0.450833 + 0.892608i \(0.648873\pi\)
\(24\) 2.19615 + 1.26795i 0.448288 + 0.258819i
\(25\) 0 0
\(26\) 0.732051 − 2.53590i 0.143567 − 0.497331i
\(27\) 1.00000i 0.192450i
\(28\) 3.26795 − 5.66025i 0.617584 − 1.06969i
\(29\) −1.36603 + 2.36603i −0.253665 + 0.439360i −0.964532 − 0.263966i \(-0.914969\pi\)
0.710867 + 0.703326i \(0.248303\pi\)
\(30\) 0 0
\(31\) − 3.19615i − 0.574046i −0.957924 − 0.287023i \(-0.907334\pi\)
0.957924 − 0.287023i \(-0.0926656\pi\)
\(32\) 2.92820 + 5.07180i 0.517638 + 0.896575i
\(33\) −1.73205 − 3.00000i −0.301511 − 0.522233i
\(34\) − 4.92820i − 0.845180i
\(35\) 0 0
\(36\) −0.732051 + 1.26795i −0.122008 + 0.211325i
\(37\) −2.00000 + 3.46410i −0.328798 + 0.569495i −0.982274 − 0.187453i \(-0.939977\pi\)
0.653476 + 0.756948i \(0.273310\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 3.46410 + 1.00000i 0.554700 + 0.160128i
\(40\) 0 0
\(41\) 4.56218 + 2.63397i 0.712492 + 0.411358i 0.811983 − 0.583681i \(-0.198388\pi\)
−0.0994908 + 0.995038i \(0.531721\pi\)
\(42\) −2.83013 − 1.63397i −0.436698 − 0.252128i
\(43\) −0.232051 + 0.133975i −0.0353874 + 0.0204309i −0.517589 − 0.855629i \(-0.673170\pi\)
0.482202 + 0.876060i \(0.339837\pi\)
\(44\) − 5.07180i − 0.764602i
\(45\) 0 0
\(46\) 0.339746 − 0.196152i 0.0500928 − 0.0289211i
\(47\) −0.196152 −0.0286118 −0.0143059 − 0.999898i \(-0.504554\pi\)
−0.0143059 + 0.999898i \(0.504554\pi\)
\(48\) −0.928203 + 0.535898i −0.133975 + 0.0773503i
\(49\) −6.46410 + 11.1962i −0.923443 + 1.59945i
\(50\) 0 0
\(51\) 6.73205 0.942676
\(52\) 3.66025 + 3.80385i 0.507586 + 0.527499i
\(53\) 6.92820i 0.951662i 0.879537 + 0.475831i \(0.157853\pi\)
−0.879537 + 0.475831i \(0.842147\pi\)
\(54\) 0.633975 + 0.366025i 0.0862730 + 0.0498097i
\(55\) 0 0
\(56\) −5.66025 − 9.80385i −0.756383 − 1.31009i
\(57\) 5.46410 0.723738
\(58\) 1.00000 + 1.73205i 0.131306 + 0.227429i
\(59\) −6.29423 + 3.63397i −0.819439 + 0.473103i −0.850223 − 0.526423i \(-0.823533\pi\)
0.0307841 + 0.999526i \(0.490200\pi\)
\(60\) 0 0
\(61\) −2.23205 − 3.86603i −0.285785 − 0.494994i 0.687014 − 0.726644i \(-0.258921\pi\)
−0.972799 + 0.231650i \(0.925588\pi\)
\(62\) −2.02628 − 1.16987i −0.257338 − 0.148574i
\(63\) 2.23205 − 3.86603i 0.281212 − 0.487073i
\(64\) 2.14359 0.267949
\(65\) 0 0
\(66\) −2.53590 −0.312148
\(67\) 6.23205 − 10.7942i 0.761366 − 1.31872i −0.180780 − 0.983524i \(-0.557862\pi\)
0.942146 − 0.335201i \(-0.108804\pi\)
\(68\) 8.53590 + 4.92820i 1.03513 + 0.597632i
\(69\) 0.267949 + 0.464102i 0.0322573 + 0.0558713i
\(70\) 0 0
\(71\) −11.0263 + 6.36603i −1.30858 + 0.755508i −0.981859 − 0.189613i \(-0.939277\pi\)
−0.326720 + 0.945121i \(0.605943\pi\)
\(72\) 1.26795 + 2.19615i 0.149429 + 0.258819i
\(73\) 15.3923 1.80153 0.900767 − 0.434304i \(-0.143006\pi\)
0.900767 + 0.434304i \(0.143006\pi\)
\(74\) 1.46410 + 2.53590i 0.170198 + 0.294792i
\(75\) 0 0
\(76\) 6.92820 + 4.00000i 0.794719 + 0.458831i
\(77\) 15.4641i 1.76230i
\(78\) 1.90192 − 1.83013i 0.215350 − 0.207221i
\(79\) −1.92820 −0.216940 −0.108470 − 0.994100i \(-0.534595\pi\)
−0.108470 + 0.994100i \(0.534595\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 3.33975 − 1.92820i 0.368813 − 0.212934i
\(83\) −2.53590 −0.278351 −0.139176 − 0.990268i \(-0.544445\pi\)
−0.139176 + 0.990268i \(0.544445\pi\)
\(84\) 5.66025 − 3.26795i 0.617584 − 0.356562i
\(85\) 0 0
\(86\) 0.196152i 0.0211517i
\(87\) −2.36603 + 1.36603i −0.253665 + 0.146453i
\(88\) −7.60770 − 4.39230i −0.810983 − 0.468221i
\(89\) 1.09808 + 0.633975i 0.116396 + 0.0672012i 0.557068 − 0.830467i \(-0.311926\pi\)
−0.440672 + 0.897668i \(0.645260\pi\)
\(90\) 0 0
\(91\) −11.1603 − 11.5981i −1.16991 − 1.21581i
\(92\) 0.784610i 0.0818012i
\(93\) 1.59808 − 2.76795i 0.165713 − 0.287023i
\(94\) −0.0717968 + 0.124356i −0.00740527 + 0.0128263i
\(95\) 0 0
\(96\) 5.85641i 0.597717i
\(97\) −8.23205 − 14.2583i −0.835838 − 1.44771i −0.893346 − 0.449369i \(-0.851649\pi\)
0.0575081 − 0.998345i \(-0.481685\pi\)
\(98\) 4.73205 + 8.19615i 0.478009 + 0.827936i
\(99\) − 3.46410i − 0.348155i
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 975.2.w.a.199.2 4
5.2 odd 4 975.2.bc.h.901.1 4
5.3 odd 4 195.2.bb.a.121.2 ✓ 4
5.4 even 2 975.2.w.f.199.1 4
13.10 even 6 975.2.w.f.49.1 4
15.8 even 4 585.2.bu.a.316.1 4
65.23 odd 12 195.2.bb.a.166.2 yes 4
65.33 even 12 2535.2.a.s.1.1 2
65.49 even 6 inner 975.2.w.a.49.2 4
65.58 even 12 2535.2.a.n.1.2 2
65.62 odd 12 975.2.bc.h.751.1 4
195.23 even 12 585.2.bu.a.361.1 4
195.98 odd 12 7605.2.a.y.1.2 2
195.188 odd 12 7605.2.a.bk.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.bb.a.121.2 ✓ 4 5.3 odd 4
195.2.bb.a.166.2 yes 4 65.23 odd 12
585.2.bu.a.316.1 4 15.8 even 4
585.2.bu.a.361.1 4 195.23 even 12
975.2.w.a.49.2 4 65.49 even 6 inner
975.2.w.a.199.2 4 1.1 even 1 trivial
975.2.w.f.49.1 4 13.10 even 6
975.2.w.f.199.1 4 5.4 even 2
975.2.bc.h.751.1 4 65.62 odd 12
975.2.bc.h.901.1 4 5.2 odd 4
2535.2.a.n.1.2 2 65.58 even 12
2535.2.a.s.1.1 2 65.33 even 12
7605.2.a.y.1.2 2 195.98 odd 12
7605.2.a.bk.1.1 2 195.188 odd 12