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Properties

Label 3870.2.c.f.2321.4
Level $3870$
Weight $2$
Character 3870.2321
Analytic conductor $30.902$
Analytic rank $0$
Dimension $12$
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] = [3870,2,Mod(2321,3870)] 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("3870.2321"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3870, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3870.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,12,0,12,-12,0,0,12,0,-12,0,0,8,0,0,12,0,0,0,-12,0,0,0,0,12, 8,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.9021055822\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 6 x^{10} + 48 x^{9} + 53 x^{8} - 516 x^{7} + 540 x^{6} + 1520 x^{5} - 2672 x^{4} + \cdots + 31812 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2321.4
Root \(0.522916 + 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 3870.2321
Dual form 3870.2.c.f.2321.9

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.92901i q^{7} +1.00000 q^{8} -1.00000 q^{10} +2.15373i q^{11} +2.99451 q^{13} -1.92901i q^{14} +1.00000 q^{16} +7.71964i q^{17} +4.37850i q^{19} -1.00000 q^{20} +2.15373i q^{22} -6.55103i q^{23} +1.00000 q^{25} +2.99451 q^{26} -1.92901i q^{28} -7.88724 q^{29} +0.611707 q^{31} +1.00000 q^{32} +7.71964i q^{34} +1.92901i q^{35} +6.19045i q^{37} +4.37850i q^{38} -1.00000 q^{40} +0.159904i q^{41} +(6.55289 + 0.244114i) q^{43} +2.15373i q^{44} -6.55103i q^{46} +7.35906i q^{47} +3.27892 q^{49} +1.00000 q^{50} +2.99451 q^{52} +3.69351i q^{53} -2.15373i q^{55} -1.92901i q^{56} -7.88724 q^{58} -6.73290i q^{59} -2.00442i q^{61} +0.611707 q^{62} +1.00000 q^{64} -2.99451 q^{65} -3.37731 q^{67} +7.71964i q^{68} +1.92901i q^{70} +12.9092 q^{71} +1.50560i q^{73} +6.19045i q^{74} +4.37850i q^{76} +4.15456 q^{77} +2.90649 q^{79} -1.00000 q^{80} +0.159904i q^{82} +12.8111i q^{83} -7.71964i q^{85} +(6.55289 + 0.244114i) q^{86} +2.15373i q^{88} -3.11887 q^{89} -5.77643i q^{91} -6.55103i q^{92} +7.35906i q^{94} -4.37850i q^{95} -3.54772 q^{97} +3.27892 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} - 12 q^{5} + 12 q^{8} - 12 q^{10} + 8 q^{13} + 12 q^{16} - 12 q^{20} + 12 q^{25} + 8 q^{26} - 12 q^{29} - 12 q^{31} + 12 q^{32} - 12 q^{40} + 20 q^{43} - 36 q^{49} + 12 q^{50}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3870\mathbb{Z}\right)^\times\).

\(n\) \(1721\) \(1981\) \(3097\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) − 1.92901i − 0.729097i −0.931184 − 0.364548i \(-0.881223\pi\)
0.931184 − 0.364548i \(-0.118777\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 2.15373i 0.649373i 0.945822 + 0.324687i \(0.105259\pi\)
−0.945822 + 0.324687i \(0.894741\pi\)
\(12\) 0 0
\(13\) 2.99451 0.830527 0.415264 − 0.909701i \(-0.363689\pi\)
0.415264 + 0.909701i \(0.363689\pi\)
\(14\) − 1.92901i − 0.515549i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.71964i 1.87229i 0.351617 + 0.936144i \(0.385632\pi\)
−0.351617 + 0.936144i \(0.614368\pi\)
\(18\) 0 0
\(19\) 4.37850i 1.00450i 0.864723 + 0.502248i \(0.167494\pi\)
−0.864723 + 0.502248i \(0.832506\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.15373i 0.459176i
\(23\) − 6.55103i − 1.36598i −0.730426 − 0.682992i \(-0.760678\pi\)
0.730426 − 0.682992i \(-0.239322\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.99451 0.587271
\(27\) 0 0
\(28\) − 1.92901i − 0.364548i
\(29\) −7.88724 −1.46462 −0.732312 − 0.680969i \(-0.761559\pi\)
−0.732312 + 0.680969i \(0.761559\pi\)
\(30\) 0 0
\(31\) 0.611707 0.109866 0.0549329 − 0.998490i \(-0.482505\pi\)
0.0549329 + 0.998490i \(0.482505\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.71964i 1.32391i
\(35\) 1.92901i 0.326062i
\(36\) 0 0
\(37\) 6.19045i 1.01770i 0.860854 + 0.508852i \(0.169930\pi\)
−0.860854 + 0.508852i \(0.830070\pi\)
\(38\) 4.37850i 0.710286i
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0.159904i 0.0249727i 0.999922 + 0.0124864i \(0.00397464\pi\)
−0.999922 + 0.0124864i \(0.996025\pi\)
\(42\) 0 0
\(43\) 6.55289 + 0.244114i 0.999307 + 0.0372271i
\(44\) 2.15373i 0.324687i
\(45\) 0 0
\(46\) − 6.55103i − 0.965896i
\(47\) 7.35906i 1.07343i 0.843764 + 0.536715i \(0.180335\pi\)
−0.843764 + 0.536715i \(0.819665\pi\)
\(48\) 0 0
\(49\) 3.27892 0.468418
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.99451 0.415264
\(53\) 3.69351i 0.507343i 0.967290 + 0.253671i \(0.0816382\pi\)
−0.967290 + 0.253671i \(0.918362\pi\)
\(54\) 0 0
\(55\) − 2.15373i − 0.290409i
\(56\) − 1.92901i − 0.257775i
\(57\) 0 0
\(58\) −7.88724 −1.03565
\(59\) − 6.73290i − 0.876549i −0.898841 − 0.438274i \(-0.855590\pi\)
0.898841 − 0.438274i \(-0.144410\pi\)
\(60\) 0 0
\(61\) − 2.00442i − 0.256639i −0.991733 − 0.128320i \(-0.959042\pi\)
0.991733 − 0.128320i \(-0.0409584\pi\)
\(62\) 0.611707 0.0776869
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.99451 −0.371423
\(66\) 0 0
\(67\) −3.37731 −0.412604 −0.206302 − 0.978488i \(-0.566143\pi\)
−0.206302 + 0.978488i \(0.566143\pi\)
\(68\) 7.71964i 0.936144i
\(69\) 0 0
\(70\) 1.92901i 0.230561i
\(71\) 12.9092 1.53204 0.766019 − 0.642818i \(-0.222235\pi\)
0.766019 + 0.642818i \(0.222235\pi\)
\(72\) 0 0
\(73\) 1.50560i 0.176217i 0.996111 + 0.0881084i \(0.0280822\pi\)
−0.996111 + 0.0881084i \(0.971918\pi\)
\(74\) 6.19045i 0.719625i
\(75\) 0 0
\(76\) 4.37850i 0.502248i
\(77\) 4.15456 0.473456
\(78\) 0 0
\(79\) 2.90649 0.327005 0.163503 − 0.986543i \(-0.447721\pi\)
0.163503 + 0.986543i \(0.447721\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 0.159904i 0.0176584i
\(83\) 12.8111i 1.40620i 0.711091 + 0.703100i \(0.248201\pi\)
−0.711091 + 0.703100i \(0.751799\pi\)
\(84\) 0 0
\(85\) − 7.71964i − 0.837313i
\(86\) 6.55289 + 0.244114i 0.706617 + 0.0263235i
\(87\) 0 0
\(88\) 2.15373i 0.229588i
\(89\) −3.11887 −0.330600 −0.165300 − 0.986243i \(-0.552859\pi\)
−0.165300 + 0.986243i \(0.552859\pi\)
\(90\) 0 0
\(91\) − 5.77643i − 0.605535i
\(92\) − 6.55103i − 0.682992i
\(93\) 0 0
\(94\) 7.35906i 0.759029i
\(95\) − 4.37850i − 0.449224i
\(96\) 0 0
\(97\) −3.54772 −0.360217 −0.180108 − 0.983647i \(-0.557645\pi\)
−0.180108 + 0.983647i \(0.557645\pi\)
\(98\) 3.27892 0.331221
\(99\) 0 0
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 3870.2.c.f.2321.4 yes 12
3.2 odd 2 3870.2.c.e.2321.4 ✓ 12
43.42 odd 2 3870.2.c.e.2321.9 yes 12
129.128 even 2 inner 3870.2.c.f.2321.9 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3870.2.c.e.2321.4 ✓ 12 3.2 odd 2
3870.2.c.e.2321.9 yes 12 43.42 odd 2
3870.2.c.f.2321.4 yes 12 1.1 even 1 trivial
3870.2.c.f.2321.9 yes 12 129.128 even 2 inner