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Properties

Label 975.2.bc.h.751.1
Level $975$
Weight $2$
Character 975.751
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(751,975)] 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("975.751"); 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, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,6,2,4,0,6,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 751.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 975.751
Dual form 975.2.bc.h.901.1

$q$-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+(0.633975 - 0.366025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.732051 + 1.26795i) q^{4} +(0.633975 + 0.366025i) q^{6} +(3.86603 + 2.23205i) q^{7} +2.53590i q^{8} +(-0.500000 + 0.866025i) q^{9} +(-3.00000 + 1.73205i) q^{11} -1.46410 q^{12} +(-0.866025 + 3.50000i) q^{13} +3.26795 q^{14} +(-0.535898 - 0.928203i) q^{16} +(3.36603 - 5.83013i) q^{17} +0.732051i q^{18} +(-4.73205 - 2.73205i) q^{19} +4.46410i q^{21} +(-1.26795 + 2.19615i) q^{22} +(0.267949 + 0.464102i) q^{23} +(-2.19615 + 1.26795i) q^{24} +(0.732051 + 2.53590i) q^{26} -1.00000 q^{27} +(-5.66025 + 3.26795i) q^{28} +(1.36603 + 2.36603i) q^{29} +3.19615i q^{31} +(-5.07180 - 2.92820i) q^{32} +(-3.00000 - 1.73205i) q^{33} -4.92820i q^{34} +(-0.732051 - 1.26795i) q^{36} +(-3.46410 + 2.00000i) q^{37} -4.00000 q^{38} +(-3.46410 + 1.00000i) q^{39} +(4.56218 - 2.63397i) q^{41} +(1.63397 + 2.83013i) q^{42} +(0.133975 - 0.232051i) q^{43} -5.07180i q^{44} +(0.339746 + 0.196152i) q^{46} +0.196152i q^{47} +(0.535898 - 0.928203i) q^{48} +(6.46410 + 11.1962i) q^{49} +6.73205 q^{51} +(-3.80385 - 3.66025i) q^{52} +6.92820 q^{53} +(-0.633975 + 0.366025i) q^{54} +(-5.66025 + 9.80385i) q^{56} -5.46410i q^{57} +(1.73205 + 1.00000i) q^{58} +(6.29423 + 3.63397i) q^{59} +(-2.23205 + 3.86603i) q^{61} +(1.16987 + 2.02628i) q^{62} +(-3.86603 + 2.23205i) q^{63} -2.14359 q^{64} -2.53590 q^{66} +(10.7942 - 6.23205i) q^{67} +(4.92820 + 8.53590i) q^{68} +(-0.267949 + 0.464102i) q^{69} +(-11.0263 - 6.36603i) q^{71} +(-2.19615 - 1.26795i) q^{72} +15.3923i q^{73} +(-1.46410 + 2.53590i) q^{74} +(6.92820 - 4.00000i) q^{76} -15.4641 q^{77} +(-1.83013 + 1.90192i) q^{78} +1.92820 q^{79} +(-0.500000 - 0.866025i) q^{81} +(1.92820 - 3.33975i) q^{82} -2.53590i q^{83} +(-5.66025 - 3.26795i) q^{84} -0.196152i q^{86} +(-1.36603 + 2.36603i) q^{87} +(-4.39230 - 7.60770i) q^{88} +(-1.09808 + 0.633975i) q^{89} +(-11.1603 + 11.5981i) q^{91} -0.784610 q^{92} +(-2.76795 + 1.59808i) q^{93} +(0.0717968 + 0.124356i) q^{94} -5.85641i q^{96} +(14.2583 + 8.23205i) q^{97} +(8.19615 + 4.73205i) q^{98} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{6} + 12 q^{7} - 2 q^{9} - 12 q^{11} + 8 q^{12} + 20 q^{14} - 16 q^{16} + 10 q^{17} - 12 q^{19} - 12 q^{22} + 8 q^{23} + 12 q^{24} - 4 q^{26} - 4 q^{27} + 12 q^{28}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.633975 − 0.366025i 0.448288 − 0.258819i −0.258819 − 0.965926i \(-0.583333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) −0.732051 + 1.26795i −0.366025 + 0.633975i
\(5\) 0 0
\(6\) 0.633975 + 0.366025i 0.258819 + 0.149429i
\(7\) 3.86603 + 2.23205i 1.46122 + 0.843636i 0.999068 − 0.0431647i \(-0.0137440\pi\)
0.462152 + 0.886801i \(0.347077\pi\)
\(8\) 2.53590i 0.896575i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −3.00000 + 1.73205i −0.904534 + 0.522233i −0.878668 − 0.477432i \(-0.841568\pi\)
−0.0258656 + 0.999665i \(0.508234\pi\)
\(12\) −1.46410 −0.422650
\(13\) −0.866025 + 3.50000i −0.240192 + 0.970725i
\(14\) 3.26795 0.873396
\(15\) 0 0
\(16\) −0.535898 − 0.928203i −0.133975 − 0.232051i
\(17\) 3.36603 − 5.83013i 0.816381 − 1.41401i −0.0919509 − 0.995764i \(-0.529310\pi\)
0.908332 − 0.418250i \(-0.137356\pi\)
\(18\) 0.732051i 0.172546i
\(19\) −4.73205 − 2.73205i −1.08561 − 0.626775i −0.153203 − 0.988195i \(-0.548959\pi\)
−0.932403 + 0.361419i \(0.882292\pi\)
\(20\) 0 0
\(21\) 4.46410i 0.974147i
\(22\) −1.26795 + 2.19615i −0.270328 + 0.468221i
\(23\) 0.267949 + 0.464102i 0.0558713 + 0.0967719i 0.892608 − 0.450833i \(-0.148873\pi\)
−0.836737 + 0.547605i \(0.815540\pi\)
\(24\) −2.19615 + 1.26795i −0.448288 + 0.258819i
\(25\) 0 0
\(26\) 0.732051 + 2.53590i 0.143567 + 0.497331i
\(27\) −1.00000 −0.192450
\(28\) −5.66025 + 3.26795i −1.06969 + 0.617584i
\(29\) 1.36603 + 2.36603i 0.253665 + 0.439360i 0.964532 − 0.263966i \(-0.0850307\pi\)
−0.710867 + 0.703326i \(0.751697\pi\)
\(30\) 0 0
\(31\) 3.19615i 0.574046i 0.957924 + 0.287023i \(0.0926656\pi\)
−0.957924 + 0.287023i \(0.907334\pi\)
\(32\) −5.07180 − 2.92820i −0.896575 − 0.517638i
\(33\) −3.00000 − 1.73205i −0.522233 − 0.301511i
\(34\) − 4.92820i − 0.845180i
\(35\) 0 0
\(36\) −0.732051 − 1.26795i −0.122008 − 0.211325i
\(37\) −3.46410 + 2.00000i −0.569495 + 0.328798i −0.756948 − 0.653476i \(-0.773310\pi\)
0.187453 + 0.982274i \(0.439977\pi\)
\(38\) −4.00000 −0.648886
\(39\) −3.46410 + 1.00000i −0.554700 + 0.160128i
\(40\) 0 0
\(41\) 4.56218 − 2.63397i 0.712492 − 0.411358i −0.0994908 − 0.995038i \(-0.531721\pi\)
0.811983 + 0.583681i \(0.198388\pi\)
\(42\) 1.63397 + 2.83013i 0.252128 + 0.436698i
\(43\) 0.133975 − 0.232051i 0.0204309 − 0.0353874i −0.855629 − 0.517589i \(-0.826830\pi\)
0.876060 + 0.482202i \(0.160163\pi\)
\(44\) − 5.07180i − 0.764602i
\(45\) 0 0
\(46\) 0.339746 + 0.196152i 0.0500928 + 0.0289211i
\(47\) 0.196152i 0.0286118i 0.999898 + 0.0143059i \(0.00455386\pi\)
−0.999898 + 0.0143059i \(0.995446\pi\)
\(48\) 0.535898 − 0.928203i 0.0773503 − 0.133975i
\(49\) 6.46410 + 11.1962i 0.923443 + 1.59945i
\(50\) 0 0
\(51\) 6.73205 0.942676
\(52\) −3.80385 − 3.66025i −0.527499 − 0.507586i
\(53\) 6.92820 0.951662 0.475831 − 0.879537i \(-0.342147\pi\)
0.475831 + 0.879537i \(0.342147\pi\)
\(54\) −0.633975 + 0.366025i −0.0862730 + 0.0498097i
\(55\) 0 0
\(56\) −5.66025 + 9.80385i −0.756383 + 1.31009i
\(57\) − 5.46410i − 0.723738i
\(58\) 1.73205 + 1.00000i 0.227429 + 0.131306i
\(59\) 6.29423 + 3.63397i 0.819439 + 0.473103i 0.850223 − 0.526423i \(-0.176467\pi\)
−0.0307841 + 0.999526i \(0.509800\pi\)
\(60\) 0 0
\(61\) −2.23205 + 3.86603i −0.285785 + 0.494994i −0.972799 − 0.231650i \(-0.925588\pi\)
0.687014 + 0.726644i \(0.258921\pi\)
\(62\) 1.16987 + 2.02628i 0.148574 + 0.257338i
\(63\) −3.86603 + 2.23205i −0.487073 + 0.281212i
\(64\) −2.14359 −0.267949
\(65\) 0 0
\(66\) −2.53590 −0.312148
\(67\) 10.7942 − 6.23205i 1.31872 − 0.761366i 0.335201 − 0.942146i \(-0.391196\pi\)
0.983524 + 0.180780i \(0.0578623\pi\)
\(68\) 4.92820 + 8.53590i 0.597632 + 1.03513i
\(69\) −0.267949 + 0.464102i −0.0322573 + 0.0558713i
\(70\) 0 0
\(71\) −11.0263 − 6.36603i −1.30858 − 0.755508i −0.326720 − 0.945121i \(-0.605943\pi\)
−0.981859 + 0.189613i \(0.939277\pi\)
\(72\) −2.19615 − 1.26795i −0.258819 − 0.149429i
\(73\) 15.3923i 1.80153i 0.434304 + 0.900767i \(0.356994\pi\)
−0.434304 + 0.900767i \(0.643006\pi\)
\(74\) −1.46410 + 2.53590i −0.170198 + 0.294792i
\(75\) 0 0
\(76\) 6.92820 − 4.00000i 0.794719 − 0.458831i
\(77\) −15.4641 −1.76230
\(78\) −1.83013 + 1.90192i −0.207221 + 0.215350i
\(79\) 1.92820 0.216940 0.108470 − 0.994100i \(-0.465405\pi\)
0.108470 + 0.994100i \(0.465405\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 1.92820 − 3.33975i 0.212934 − 0.368813i
\(83\) − 2.53590i − 0.278351i −0.990268 − 0.139176i \(-0.955555\pi\)
0.990268 − 0.139176i \(-0.0444452\pi\)
\(84\) −5.66025 − 3.26795i −0.617584 − 0.356562i
\(85\) 0 0
\(86\) − 0.196152i − 0.0211517i
\(87\) −1.36603 + 2.36603i −0.146453 + 0.253665i
\(88\) −4.39230 − 7.60770i −0.468221 − 0.810983i
\(89\) −1.09808 + 0.633975i −0.116396 + 0.0672012i −0.557068 − 0.830467i \(-0.688074\pi\)
0.440672 + 0.897668i \(0.354740\pi\)
\(90\) 0 0
\(91\) −11.1603 + 11.5981i −1.16991 + 1.21581i
\(92\) −0.784610 −0.0818012
\(93\) −2.76795 + 1.59808i −0.287023 + 0.165713i
\(94\) 0.0717968 + 0.124356i 0.00740527 + 0.0128263i
\(95\) 0 0
\(96\) − 5.85641i − 0.597717i
\(97\) 14.2583 + 8.23205i 1.44771 + 0.835838i 0.998345 − 0.0575081i \(-0.0183155\pi\)
0.449369 + 0.893346i \(0.351649\pi\)
\(98\) 8.19615 + 4.73205i 0.827936 + 0.478009i
\(99\) − 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.bc.h.751.1 4
5.2 odd 4 975.2.w.a.49.2 4
5.3 odd 4 975.2.w.f.49.1 4
5.4 even 2 195.2.bb.a.166.2 yes 4
13.4 even 6 inner 975.2.bc.h.901.1 4
15.14 odd 2 585.2.bu.a.361.1 4
65.4 even 6 195.2.bb.a.121.2 ✓ 4
65.17 odd 12 975.2.w.f.199.1 4
65.24 odd 12 2535.2.a.n.1.2 2
65.43 odd 12 975.2.w.a.199.2 4
65.54 odd 12 2535.2.a.s.1.1 2
195.89 even 12 7605.2.a.bk.1.1 2
195.119 even 12 7605.2.a.y.1.2 2
195.134 odd 6 585.2.bu.a.316.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.bb.a.121.2 ✓ 4 65.4 even 6
195.2.bb.a.166.2 yes 4 5.4 even 2
585.2.bu.a.316.1 4 195.134 odd 6
585.2.bu.a.361.1 4 15.14 odd 2
975.2.w.a.49.2 4 5.2 odd 4
975.2.w.a.199.2 4 65.43 odd 12
975.2.w.f.49.1 4 5.3 odd 4
975.2.w.f.199.1 4 65.17 odd 12
975.2.bc.h.751.1 4 1.1 even 1 trivial
975.2.bc.h.901.1 4 13.4 even 6 inner
2535.2.a.n.1.2 2 65.24 odd 12
2535.2.a.s.1.1 2 65.54 odd 12
7605.2.a.y.1.2 2 195.119 even 12
7605.2.a.bk.1.1 2 195.89 even 12