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Properties

Label 195.2.bb.a.121.1
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 121.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 195.121
Dual form 195.2.bb.a.166.1

$q$-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+(-2.36603 - 1.36603i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(2.73205 + 4.73205i) q^{4} +1.00000i q^{5} +(2.36603 - 1.36603i) q^{6} +(-2.13397 + 1.23205i) q^{7} -9.46410i q^{8} +(-0.500000 - 0.866025i) q^{9} +(1.36603 - 2.36603i) q^{10} +(-3.00000 - 1.73205i) q^{11} -5.46410 q^{12} +(-0.866025 - 3.50000i) q^{13} +6.73205 q^{14} +(-0.866025 - 0.500000i) q^{15} +(-7.46410 + 12.9282i) q^{16} +(-1.63397 - 2.83013i) q^{17} +2.73205i q^{18} +(-1.26795 + 0.732051i) q^{19} +(-4.73205 + 2.73205i) q^{20} -2.46410i q^{21} +(4.73205 + 8.19615i) q^{22} +(-3.73205 + 6.46410i) q^{23} +(8.19615 + 4.73205i) q^{24} -1.00000 q^{25} +(-2.73205 + 9.46410i) q^{26} +1.00000 q^{27} +(-11.6603 - 6.73205i) q^{28} +(-0.366025 + 0.633975i) q^{29} +(1.36603 + 2.36603i) q^{30} -7.19615i q^{31} +(18.9282 - 10.9282i) q^{32} +(3.00000 - 1.73205i) q^{33} +8.92820i q^{34} +(-1.23205 - 2.13397i) q^{35} +(2.73205 - 4.73205i) q^{36} +(-3.46410 - 2.00000i) q^{37} +4.00000 q^{38} +(3.46410 + 1.00000i) q^{39} +9.46410 q^{40} +(-7.56218 - 4.36603i) q^{41} +(-3.36603 + 5.83013i) q^{42} +(-1.86603 - 3.23205i) q^{43} -18.9282i q^{44} +(0.866025 - 0.500000i) q^{45} +(17.6603 - 10.1962i) q^{46} +10.1962i q^{47} +(-7.46410 - 12.9282i) q^{48} +(-0.464102 + 0.803848i) q^{49} +(2.36603 + 1.36603i) q^{50} +3.26795 q^{51} +(14.1962 - 13.6603i) q^{52} +6.92820 q^{53} +(-2.36603 - 1.36603i) q^{54} +(1.73205 - 3.00000i) q^{55} +(11.6603 + 20.1962i) q^{56} -1.46410i q^{57} +(1.73205 - 1.00000i) q^{58} +(-9.29423 + 5.36603i) q^{59} -5.46410i q^{60} +(1.23205 + 2.13397i) q^{61} +(-9.83013 + 17.0263i) q^{62} +(2.13397 + 1.23205i) q^{63} -29.8564 q^{64} +(3.50000 - 0.866025i) q^{65} -9.46410 q^{66} +(4.79423 + 2.76795i) q^{67} +(8.92820 - 15.4641i) q^{68} +(-3.73205 - 6.46410i) q^{69} +6.73205i q^{70} +(8.02628 - 4.63397i) q^{71} +(-8.19615 + 4.73205i) q^{72} +5.39230i q^{73} +(5.46410 + 9.46410i) q^{74} +(0.500000 - 0.866025i) q^{75} +(-6.92820 - 4.00000i) q^{76} +8.53590 q^{77} +(-6.83013 - 7.09808i) q^{78} -11.9282 q^{79} +(-12.9282 - 7.46410i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(11.9282 + 20.6603i) q^{82} +9.46410i q^{83} +(11.6603 - 6.73205i) q^{84} +(2.83013 - 1.63397i) q^{85} +10.1962i q^{86} +(-0.366025 - 0.633975i) q^{87} +(-16.3923 + 28.3923i) q^{88} +(4.09808 + 2.36603i) q^{89} -2.73205 q^{90} +(6.16025 + 6.40192i) q^{91} -40.7846 q^{92} +(6.23205 + 3.59808i) q^{93} +(13.9282 - 24.1244i) q^{94} +(-0.732051 - 1.26795i) q^{95} +21.8564i q^{96} +(8.25833 - 4.76795i) q^{97} +(2.19615 - 1.26795i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.36603 − 1.36603i −1.67303 − 0.965926i −0.965926 − 0.258819i \(-0.916667\pi\)
−0.707107 − 0.707107i \(-0.750000\pi\)
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 2.73205 + 4.73205i 1.36603 + 2.36603i
\(5\) 1.00000i 0.447214i
\(6\) 2.36603 − 1.36603i 0.965926 − 0.557678i
\(7\) −2.13397 + 1.23205i −0.806567 + 0.465671i −0.845762 − 0.533560i \(-0.820854\pi\)
0.0391956 + 0.999232i \(0.487520\pi\)
\(8\) − 9.46410i − 3.34607i
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 1.36603 − 2.36603i 0.431975 − 0.748203i
\(11\) −3.00000 − 1.73205i −0.904534 − 0.522233i −0.0258656 − 0.999665i \(-0.508234\pi\)
−0.878668 + 0.477432i \(0.841568\pi\)
\(12\) −5.46410 −1.57735
\(13\) −0.866025 − 3.50000i −0.240192 − 0.970725i
\(14\) 6.73205 1.79922
\(15\) −0.866025 − 0.500000i −0.223607 − 0.129099i
\(16\) −7.46410 + 12.9282i −1.86603 + 3.23205i
\(17\) −1.63397 − 2.83013i −0.396297 − 0.686407i 0.596969 − 0.802264i \(-0.296372\pi\)
−0.993266 + 0.115858i \(0.963038\pi\)
\(18\) 2.73205i 0.643951i
\(19\) −1.26795 + 0.732051i −0.290887 + 0.167944i −0.638342 − 0.769753i \(-0.720379\pi\)
0.347455 + 0.937697i \(0.387046\pi\)
\(20\) −4.73205 + 2.73205i −1.05812 + 0.610905i
\(21\) − 2.46410i − 0.537711i
\(22\) 4.73205 + 8.19615i 1.00888 + 1.74743i
\(23\) −3.73205 + 6.46410i −0.778186 + 1.34786i 0.154800 + 0.987946i \(0.450527\pi\)
−0.932986 + 0.359912i \(0.882807\pi\)
\(24\) 8.19615 + 4.73205i 1.67303 + 0.965926i
\(25\) −1.00000 −0.200000
\(26\) −2.73205 + 9.46410i −0.535799 + 1.85606i
\(27\) 1.00000 0.192450
\(28\) −11.6603 − 6.73205i −2.20358 − 1.27224i
\(29\) −0.366025 + 0.633975i −0.0679692 + 0.117726i −0.898007 − 0.439981i \(-0.854985\pi\)
0.830038 + 0.557707i \(0.188319\pi\)
\(30\) 1.36603 + 2.36603i 0.249401 + 0.431975i
\(31\) − 7.19615i − 1.29247i −0.763140 − 0.646234i \(-0.776343\pi\)
0.763140 − 0.646234i \(-0.223657\pi\)
\(32\) 18.9282 − 10.9282i 3.34607 − 1.93185i
\(33\) 3.00000 − 1.73205i 0.522233 − 0.301511i
\(34\) 8.92820i 1.53117i
\(35\) −1.23205 − 2.13397i −0.208255 − 0.360708i
\(36\) 2.73205 − 4.73205i 0.455342 − 0.788675i
\(37\) −3.46410 − 2.00000i −0.569495 − 0.328798i 0.187453 − 0.982274i \(-0.439977\pi\)
−0.756948 + 0.653476i \(0.773310\pi\)
\(38\) 4.00000 0.648886
\(39\) 3.46410 + 1.00000i 0.554700 + 0.160128i
\(40\) 9.46410 1.49641
\(41\) −7.56218 − 4.36603i −1.18101 − 0.681859i −0.224765 − 0.974413i \(-0.572161\pi\)
−0.956249 + 0.292554i \(0.905495\pi\)
\(42\) −3.36603 + 5.83013i −0.519389 + 0.899608i
\(43\) −1.86603 − 3.23205i −0.284566 − 0.492883i 0.687938 − 0.725770i \(-0.258516\pi\)
−0.972504 + 0.232887i \(0.925183\pi\)
\(44\) − 18.9282i − 2.85353i
\(45\) 0.866025 − 0.500000i 0.129099 − 0.0745356i
\(46\) 17.6603 − 10.1962i 2.60386 − 1.50334i
\(47\) 10.1962i 1.48726i 0.668590 + 0.743631i \(0.266898\pi\)
−0.668590 + 0.743631i \(0.733102\pi\)
\(48\) −7.46410 − 12.9282i −1.07735 − 1.86603i
\(49\) −0.464102 + 0.803848i −0.0663002 + 0.114835i
\(50\) 2.36603 + 1.36603i 0.334607 + 0.193185i
\(51\) 3.26795 0.457604
\(52\) 14.1962 − 13.6603i 1.96865 − 1.89434i
\(53\) 6.92820 0.951662 0.475831 − 0.879537i \(-0.342147\pi\)
0.475831 + 0.879537i \(0.342147\pi\)
\(54\) −2.36603 − 1.36603i −0.321975 − 0.185893i
\(55\) 1.73205 − 3.00000i 0.233550 − 0.404520i
\(56\) 11.6603 + 20.1962i 1.55817 + 2.69882i
\(57\) − 1.46410i − 0.193925i
\(58\) 1.73205 − 1.00000i 0.227429 − 0.131306i
\(59\) −9.29423 + 5.36603i −1.21001 + 0.698597i −0.962760 − 0.270356i \(-0.912859\pi\)
−0.247245 + 0.968953i \(0.579525\pi\)
\(60\) − 5.46410i − 0.705412i
\(61\) 1.23205 + 2.13397i 0.157748 + 0.273227i 0.934056 − 0.357126i \(-0.116243\pi\)
−0.776308 + 0.630353i \(0.782910\pi\)
\(62\) −9.83013 + 17.0263i −1.24843 + 2.16234i
\(63\) 2.13397 + 1.23205i 0.268856 + 0.155224i
\(64\) −29.8564 −3.73205
\(65\) 3.50000 − 0.866025i 0.434122 − 0.107417i
\(66\) −9.46410 −1.16495
\(67\) 4.79423 + 2.76795i 0.585708 + 0.338159i 0.763399 − 0.645928i \(-0.223529\pi\)
−0.177690 + 0.984086i \(0.556863\pi\)
\(68\) 8.92820 − 15.4641i 1.08270 − 1.87530i
\(69\) −3.73205 − 6.46410i −0.449286 − 0.778186i
\(70\) 6.73205i 0.804634i
\(71\) 8.02628 − 4.63397i 0.952544 − 0.549952i 0.0586738 − 0.998277i \(-0.481313\pi\)
0.893870 + 0.448326i \(0.147979\pi\)
\(72\) −8.19615 + 4.73205i −0.965926 + 0.557678i
\(73\) 5.39230i 0.631122i 0.948905 + 0.315561i \(0.102193\pi\)
−0.948905 + 0.315561i \(0.897807\pi\)
\(74\) 5.46410 + 9.46410i 0.635189 + 1.10018i
\(75\) 0.500000 − 0.866025i 0.0577350 − 0.100000i
\(76\) −6.92820 − 4.00000i −0.794719 − 0.458831i
\(77\) 8.53590 0.972756
\(78\) −6.83013 − 7.09808i −0.773360 − 0.803699i
\(79\) −11.9282 −1.34203 −0.671014 − 0.741445i \(-0.734141\pi\)
−0.671014 + 0.741445i \(0.734141\pi\)
\(80\) −12.9282 − 7.46410i −1.44542 − 0.834512i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 11.9282 + 20.6603i 1.31725 + 2.28154i
\(83\) 9.46410i 1.03882i 0.854525 + 0.519410i \(0.173848\pi\)
−0.854525 + 0.519410i \(0.826152\pi\)
\(84\) 11.6603 − 6.73205i 1.27224 − 0.734527i
\(85\) 2.83013 − 1.63397i 0.306970 − 0.177229i
\(86\) 10.1962i 1.09948i
\(87\) −0.366025 − 0.633975i −0.0392420 − 0.0679692i
\(88\) −16.3923 + 28.3923i −1.74743 + 3.02663i
\(89\) 4.09808 + 2.36603i 0.434395 + 0.250798i 0.701217 − 0.712948i \(-0.252640\pi\)
−0.266822 + 0.963746i \(0.585974\pi\)
\(90\) −2.73205 −0.287983
\(91\) 6.16025 + 6.40192i 0.645770 + 0.671104i
\(92\) −40.7846 −4.25209
\(93\) 6.23205 + 3.59808i 0.646234 + 0.373103i
\(94\) 13.9282 − 24.1244i 1.43658 − 2.48824i
\(95\) −0.732051 − 1.26795i −0.0751068 − 0.130089i
\(96\) 21.8564i 2.23071i
\(97\) 8.25833 − 4.76795i 0.838506 − 0.484112i −0.0182499 − 0.999833i \(-0.505809\pi\)
0.856756 + 0.515722i \(0.172476\pi\)
\(98\) 2.19615 − 1.26795i 0.221845 − 0.128082i
\(99\) 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.121.1 ✓ 4
3.2 odd 2 585.2.bu.a.316.2 4
5.2 odd 4 975.2.w.f.199.2 4
5.3 odd 4 975.2.w.a.199.1 4
5.4 even 2 975.2.bc.h.901.2 4
13.6 odd 12 2535.2.a.s.1.2 2
13.7 odd 12 2535.2.a.n.1.1 2
13.10 even 6 inner 195.2.bb.a.166.1 yes 4
39.20 even 12 7605.2.a.bk.1.2 2
39.23 odd 6 585.2.bu.a.361.2 4
39.32 even 12 7605.2.a.y.1.1 2
65.23 odd 12 975.2.w.f.49.2 4
65.49 even 6 975.2.bc.h.751.2 4
65.62 odd 12 975.2.w.a.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.bb.a.121.1 ✓ 4 1.1 even 1 trivial
195.2.bb.a.166.1 yes 4 13.10 even 6 inner
585.2.bu.a.316.2 4 3.2 odd 2
585.2.bu.a.361.2 4 39.23 odd 6
975.2.w.a.49.1 4 65.62 odd 12
975.2.w.a.199.1 4 5.3 odd 4
975.2.w.f.49.2 4 65.23 odd 12
975.2.w.f.199.2 4 5.2 odd 4
975.2.bc.h.751.2 4 65.49 even 6
975.2.bc.h.901.2 4 5.4 even 2
2535.2.a.n.1.1 2 13.7 odd 12
2535.2.a.s.1.2 2 13.6 odd 12
7605.2.a.y.1.1 2 39.32 even 12
7605.2.a.bk.1.2 2 39.20 even 12