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Properties

Label 195.2.bb.a.121.2
Level $195$
Weight $2$
Character 195.121
Analytic conductor $1.557$
Analytic rank $1$
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] = [195,2,Mod(121,195)] 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("195.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.55708283941\)
Analytic rank: \(1\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 121.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 195.121
Dual form 195.2.bb.a.166.2

$q$-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+(-0.633975 - 0.366025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.732051 - 1.26795i) q^{4} -1.00000i q^{5} +(0.633975 - 0.366025i) q^{6} +(-3.86603 + 2.23205i) q^{7} +2.53590i q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.366025 + 0.633975i) q^{10} +(-3.00000 - 1.73205i) q^{11} +1.46410 q^{12} +(0.866025 + 3.50000i) q^{13} +3.26795 q^{14} +(0.866025 + 0.500000i) q^{15} +(-0.535898 + 0.928203i) q^{16} +(-3.36603 - 5.83013i) q^{17} +0.732051i q^{18} +(-4.73205 + 2.73205i) q^{19} +(-1.26795 + 0.732051i) q^{20} -4.46410i q^{21} +(1.26795 + 2.19615i) q^{22} +(-0.267949 + 0.464102i) q^{23} +(-2.19615 - 1.26795i) q^{24} -1.00000 q^{25} +(0.732051 - 2.53590i) q^{26} +1.00000 q^{27} +(5.66025 + 3.26795i) q^{28} +(1.36603 - 2.36603i) q^{29} +(-0.366025 - 0.633975i) q^{30} -3.19615i q^{31} +(5.07180 - 2.92820i) q^{32} +(3.00000 - 1.73205i) q^{33} +4.92820i q^{34} +(2.23205 + 3.86603i) q^{35} +(-0.732051 + 1.26795i) q^{36} +(3.46410 + 2.00000i) q^{37} +4.00000 q^{38} +(-3.46410 - 1.00000i) q^{39} +2.53590 q^{40} +(4.56218 + 2.63397i) q^{41} +(-1.63397 + 2.83013i) q^{42} +(-0.133975 - 0.232051i) q^{43} +5.07180i q^{44} +(-0.866025 + 0.500000i) q^{45} +(0.339746 - 0.196152i) q^{46} +0.196152i q^{47} +(-0.535898 - 0.928203i) q^{48} +(6.46410 - 11.1962i) q^{49} +(0.633975 + 0.366025i) q^{50} +6.73205 q^{51} +(3.80385 - 3.66025i) q^{52} -6.92820 q^{53} +(-0.633975 - 0.366025i) q^{54} +(-1.73205 + 3.00000i) q^{55} +(-5.66025 - 9.80385i) q^{56} -5.46410i q^{57} +(-1.73205 + 1.00000i) q^{58} +(6.29423 - 3.63397i) q^{59} -1.46410i q^{60} +(-2.23205 - 3.86603i) q^{61} +(-1.16987 + 2.02628i) q^{62} +(3.86603 + 2.23205i) q^{63} -2.14359 q^{64} +(3.50000 - 0.866025i) q^{65} -2.53590 q^{66} +(-10.7942 - 6.23205i) q^{67} +(-4.92820 + 8.53590i) q^{68} +(-0.267949 - 0.464102i) q^{69} -3.26795i q^{70} +(-11.0263 + 6.36603i) q^{71} +(2.19615 - 1.26795i) q^{72} +15.3923i q^{73} +(-1.46410 - 2.53590i) q^{74} +(0.500000 - 0.866025i) q^{75} +(6.92820 + 4.00000i) q^{76} +15.4641 q^{77} +(1.83013 + 1.90192i) q^{78} +1.92820 q^{79} +(0.928203 + 0.535898i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-1.92820 - 3.33975i) q^{82} -2.53590i q^{83} +(-5.66025 + 3.26795i) q^{84} +(-5.83013 + 3.36603i) q^{85} +0.196152i q^{86} +(1.36603 + 2.36603i) q^{87} +(4.39230 - 7.60770i) q^{88} +(-1.09808 - 0.633975i) q^{89} +0.732051 q^{90} +(-11.1603 - 11.5981i) q^{91} +0.784610 q^{92} +(2.76795 + 1.59808i) q^{93} +(0.0717968 - 0.124356i) q^{94} +(2.73205 + 4.73205i) q^{95} +5.85641i q^{96} +(-14.2583 + 8.23205i) q^{97} +(-8.19615 + 4.73205i) q^{98} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} - 2 q^{3} + 4 q^{4} + 6 q^{6} - 12 q^{7} - 2 q^{9} + 2 q^{10} - 12 q^{11} - 8 q^{12} + 20 q^{14} - 16 q^{16} - 10 q^{17} - 12 q^{19} - 12 q^{20} + 12 q^{22} - 8 q^{23} + 12 q^{24} - 4 q^{25}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).

\(n\) \(106\) \(131\) \(157\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.633975 − 0.366025i −0.448288 − 0.258819i 0.258819 − 0.965926i \(-0.416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) −0.732051 − 1.26795i −0.366025 − 0.633975i
\(5\) − 1.00000i − 0.447214i
\(6\) 0.633975 − 0.366025i 0.258819 − 0.149429i
\(7\) −3.86603 + 2.23205i −1.46122 + 0.843636i −0.999068 − 0.0431647i \(-0.986256\pi\)
−0.462152 + 0.886801i \(0.652923\pi\)
\(8\) 2.53590i 0.896575i
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −0.366025 + 0.633975i −0.115747 + 0.200480i
\(11\) −3.00000 − 1.73205i −0.904534 − 0.522233i −0.0258656 − 0.999665i \(-0.508234\pi\)
−0.878668 + 0.477432i \(0.841568\pi\)
\(12\) 1.46410 0.422650
\(13\) 0.866025 + 3.50000i 0.240192 + 0.970725i
\(14\) 3.26795 0.873396
\(15\) 0.866025 + 0.500000i 0.223607 + 0.129099i
\(16\) −0.535898 + 0.928203i −0.133975 + 0.232051i
\(17\) −3.36603 − 5.83013i −0.816381 − 1.41401i −0.908332 − 0.418250i \(-0.862644\pi\)
0.0919509 − 0.995764i \(-0.470690\pi\)
\(18\) 0.732051i 0.172546i
\(19\) −4.73205 + 2.73205i −1.08561 + 0.626775i −0.932403 − 0.361419i \(-0.882292\pi\)
−0.153203 + 0.988195i \(0.548959\pi\)
\(20\) −1.26795 + 0.732051i −0.283522 + 0.163692i
\(21\) − 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 \(-0.851127\pi\)
0.836737 + 0.547605i \(0.184460\pi\)
\(24\) −2.19615 − 1.26795i −0.448288 − 0.258819i
\(25\) −1.00000 −0.200000
\(26\) 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 −0.710867 − 0.703326i \(-0.751697\pi\)
0.964532 + 0.263966i \(0.0850307\pi\)
\(30\) −0.366025 − 0.633975i −0.0668268 − 0.115747i
\(31\) − 3.19615i − 0.574046i −0.957924 − 0.287023i \(-0.907334\pi\)
0.957924 − 0.287023i \(-0.0926656\pi\)
\(32\) 5.07180 − 2.92820i 0.896575 − 0.517638i
\(33\) 3.00000 − 1.73205i 0.522233 − 0.301511i
\(34\) 4.92820i 0.845180i
\(35\) 2.23205 + 3.86603i 0.377285 + 0.653478i
\(36\) −0.732051 + 1.26795i −0.122008 + 0.211325i
\(37\) 3.46410 + 2.00000i 0.569495 + 0.328798i 0.756948 − 0.653476i \(-0.226690\pi\)
−0.187453 + 0.982274i \(0.560023\pi\)
\(38\) 4.00000 0.648886
\(39\) −3.46410 − 1.00000i −0.554700 − 0.160128i
\(40\) 2.53590 0.400961
\(41\) 4.56218 + 2.63397i 0.712492 + 0.411358i 0.811983 − 0.583681i \(-0.198388\pi\)
−0.0994908 + 0.995038i \(0.531721\pi\)
\(42\) −1.63397 + 2.83013i −0.252128 + 0.436698i
\(43\) −0.133975 − 0.232051i −0.0204309 − 0.0353874i 0.855629 − 0.517589i \(-0.173170\pi\)
−0.876060 + 0.482202i \(0.839837\pi\)
\(44\) 5.07180i 0.764602i
\(45\) −0.866025 + 0.500000i −0.129099 + 0.0745356i
\(46\) 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
\(50\) 0.633975 + 0.366025i 0.0896575 + 0.0517638i
\(51\) 6.73205 0.942676
\(52\) 3.80385 − 3.66025i 0.527499 − 0.507586i
\(53\) −6.92820 −0.951662 −0.475831 − 0.879537i \(-0.657853\pi\)
−0.475831 + 0.879537i \(0.657853\pi\)
\(54\) −0.633975 − 0.366025i −0.0862730 − 0.0498097i
\(55\) −1.73205 + 3.00000i −0.233550 + 0.404520i
\(56\) −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 −0.0307841 − 0.999526i \(-0.509800\pi\)
0.850223 + 0.526423i \(0.176467\pi\)
\(60\) − 1.46410i − 0.189015i
\(61\) −2.23205 − 3.86603i −0.285785 − 0.494994i 0.687014 − 0.726644i \(-0.258921\pi\)
−0.972799 + 0.231650i \(0.925588\pi\)
\(62\) −1.16987 + 2.02628i −0.148574 + 0.257338i
\(63\) 3.86603 + 2.23205i 0.487073 + 0.281212i
\(64\) −2.14359 −0.267949
\(65\) 3.50000 − 0.866025i 0.434122 − 0.107417i
\(66\) −2.53590 −0.312148
\(67\) −10.7942 − 6.23205i −1.31872 − 0.761366i −0.335201 − 0.942146i \(-0.608804\pi\)
−0.983524 + 0.180780i \(0.942138\pi\)
\(68\) −4.92820 + 8.53590i −0.597632 + 1.03513i
\(69\) −0.267949 − 0.464102i −0.0322573 − 0.0558713i
\(70\) − 3.26795i − 0.390595i
\(71\) −11.0263 + 6.36603i −1.30858 + 0.755508i −0.981859 − 0.189613i \(-0.939277\pi\)
−0.326720 + 0.945121i \(0.605943\pi\)
\(72\) 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
\(75\) 0.500000 − 0.866025i 0.0577350 − 0.100000i
\(76\) 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 \(0.465405\pi\)
\(80\) 0.928203 + 0.535898i 0.103776 + 0.0599153i
\(81\) −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
\(85\) −5.83013 + 3.36603i −0.632366 + 0.365097i
\(86\) 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 0.440672 − 0.897668i \(-0.354740\pi\)
−0.557068 + 0.830467i \(0.688074\pi\)
\(90\) 0.732051 0.0771649
\(91\) −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
\(95\) 2.73205 + 4.73205i 0.280302 + 0.485498i
\(96\) 5.85641i 0.597717i
\(97\) −14.2583 + 8.23205i −1.44771 + 0.835838i −0.998345 − 0.0575081i \(-0.981685\pi\)
−0.449369 + 0.893346i \(0.648351\pi\)
\(98\) −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 195.2.bb.a.121.2 ✓ 4
3.2 odd 2 585.2.bu.a.316.1 4
5.2 odd 4 975.2.w.a.199.2 4
5.3 odd 4 975.2.w.f.199.1 4
5.4 even 2 975.2.bc.h.901.1 4
13.6 odd 12 2535.2.a.n.1.2 2
13.7 odd 12 2535.2.a.s.1.1 2
13.10 even 6 inner 195.2.bb.a.166.2 yes 4
39.20 even 12 7605.2.a.y.1.2 2
39.23 odd 6 585.2.bu.a.361.1 4
39.32 even 12 7605.2.a.bk.1.1 2
65.23 odd 12 975.2.w.a.49.2 4
65.49 even 6 975.2.bc.h.751.1 4
65.62 odd 12 975.2.w.f.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.bb.a.121.2 ✓ 4 1.1 even 1 trivial
195.2.bb.a.166.2 yes 4 13.10 even 6 inner
585.2.bu.a.316.1 4 3.2 odd 2
585.2.bu.a.361.1 4 39.23 odd 6
975.2.w.a.49.2 4 65.23 odd 12
975.2.w.a.199.2 4 5.2 odd 4
975.2.w.f.49.1 4 65.62 odd 12
975.2.w.f.199.1 4 5.3 odd 4
975.2.bc.h.751.1 4 65.49 even 6
975.2.bc.h.901.1 4 5.4 even 2
2535.2.a.n.1.2 2 13.6 odd 12
2535.2.a.s.1.1 2 13.7 odd 12
7605.2.a.y.1.2 2 39.20 even 12
7605.2.a.bk.1.1 2 39.32 even 12