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Properties

Label 2960.2.a.u.1.3
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [2960,2,Mod(1,2960)] 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("2960.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-3,0,1,0,11,0,-11,0,0] 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: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.59774\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.34596 q^{3} -1.00000 q^{5} +2.59774 q^{7} +8.19547 q^{9} -4.74823 q^{11} +6.69193 q^{13} -3.34596 q^{15} -0.748228 q^{17} +3.34596 q^{19} +8.69193 q^{21} -1.49646 q^{23} +1.00000 q^{25} +17.3839 q^{27} +3.94370 q^{29} -7.79321 q^{31} -15.8874 q^{33} -2.59774 q^{35} -1.00000 q^{37} +22.3909 q^{39} -6.44724 q^{41} -1.94370 q^{43} -8.19547 q^{45} -1.84951 q^{47} -0.251772 q^{49} -2.50354 q^{51} +10.4472 q^{53} +4.74823 q^{55} +11.1955 q^{57} -5.84951 q^{59} +7.94370 q^{61} +21.2897 q^{63} -6.69193 q^{65} -1.84951 q^{67} -5.00709 q^{69} +3.88740 q^{71} -7.49646 q^{73} +3.34596 q^{75} -12.3346 q^{77} +16.5414 q^{79} +33.5793 q^{81} -15.2334 q^{83} +0.748228 q^{85} +13.1955 q^{87} +6.00000 q^{89} +17.3839 q^{91} -26.0758 q^{93} -3.34596 q^{95} -10.4472 q^{97} -38.9140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45}+ \cdots - 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0
\(3\) 3.34596 1.93179 0.965896 − 0.258929i \(-0.0833695\pi\)
0.965896 + 0.258929i \(0.0833695\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.59774 0.981852 0.490926 − 0.871201i \(-0.336659\pi\)
0.490926 + 0.871201i \(0.336659\pi\)
\(8\) 0 0
\(9\) 8.19547 2.73182
\(10\) 0 0
\(11\) −4.74823 −1.43164 −0.715822 − 0.698282i \(-0.753948\pi\)
−0.715822 + 0.698282i \(0.753948\pi\)
\(12\) 0 0
\(13\) 6.69193 1.85601 0.928003 − 0.372572i \(-0.121524\pi\)
0.928003 + 0.372572i \(0.121524\pi\)
\(14\) 0 0
\(15\) −3.34596 −0.863924
\(16\) 0 0
\(17\) −0.748228 −0.181472 −0.0907360 − 0.995875i \(-0.528922\pi\)
−0.0907360 + 0.995875i \(0.528922\pi\)
\(18\) 0 0
\(19\) 3.34596 0.767617 0.383808 − 0.923413i \(-0.374612\pi\)
0.383808 + 0.923413i \(0.374612\pi\)
\(20\) 0 0
\(21\) 8.69193 1.89673
\(22\) 0 0
\(23\) −1.49646 −0.312033 −0.156016 − 0.987754i \(-0.549865\pi\)
−0.156016 + 0.987754i \(0.549865\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 17.3839 3.34552
\(28\) 0 0
\(29\) 3.94370 0.732326 0.366163 − 0.930551i \(-0.380671\pi\)
0.366163 + 0.930551i \(0.380671\pi\)
\(30\) 0 0
\(31\) −7.79321 −1.39970 −0.699851 − 0.714289i \(-0.746750\pi\)
−0.699851 + 0.714289i \(0.746750\pi\)
\(32\) 0 0
\(33\) −15.8874 −2.76564
\(34\) 0 0
\(35\) −2.59774 −0.439097
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 22.3909 3.58542
\(40\) 0 0
\(41\) −6.44724 −1.00689 −0.503445 − 0.864027i \(-0.667934\pi\)
−0.503445 + 0.864027i \(0.667934\pi\)
\(42\) 0 0
\(43\) −1.94370 −0.296411 −0.148206 − 0.988957i \(-0.547350\pi\)
−0.148206 + 0.988957i \(0.547350\pi\)
\(44\) 0 0
\(45\) −8.19547 −1.22171
\(46\) 0 0
\(47\) −1.84951 −0.269778 −0.134889 − 0.990861i \(-0.543068\pi\)
−0.134889 + 0.990861i \(0.543068\pi\)
\(48\) 0 0
\(49\) −0.251772 −0.0359674
\(50\) 0 0
\(51\) −2.50354 −0.350566
\(52\) 0 0
\(53\) 10.4472 1.43504 0.717520 − 0.696538i \(-0.245277\pi\)
0.717520 + 0.696538i \(0.245277\pi\)
\(54\) 0 0
\(55\) 4.74823 0.640251
\(56\) 0 0
\(57\) 11.1955 1.48288
\(58\) 0 0
\(59\) −5.84951 −0.761541 −0.380770 − 0.924670i \(-0.624341\pi\)
−0.380770 + 0.924670i \(0.624341\pi\)
\(60\) 0 0
\(61\) 7.94370 1.01709 0.508543 − 0.861036i \(-0.330184\pi\)
0.508543 + 0.861036i \(0.330184\pi\)
\(62\) 0 0
\(63\) 21.2897 2.68225
\(64\) 0 0
\(65\) −6.69193 −0.830031
\(66\) 0 0
\(67\) −1.84951 −0.225953 −0.112977 − 0.993598i \(-0.536039\pi\)
−0.112977 + 0.993598i \(0.536039\pi\)
\(68\) 0 0
\(69\) −5.00709 −0.602783
\(70\) 0 0
\(71\) 3.88740 0.461349 0.230675 − 0.973031i \(-0.425907\pi\)
0.230675 + 0.973031i \(0.425907\pi\)
\(72\) 0 0
\(73\) −7.49646 −0.877394 −0.438697 − 0.898635i \(-0.644560\pi\)
−0.438697 + 0.898635i \(0.644560\pi\)
\(74\) 0 0
\(75\) 3.34596 0.386359
\(76\) 0 0
\(77\) −12.3346 −1.40566
\(78\) 0 0
\(79\) 16.5414 1.86106 0.930528 − 0.366220i \(-0.119348\pi\)
0.930528 + 0.366220i \(0.119348\pi\)
\(80\) 0 0
\(81\) 33.5793 3.73104
\(82\) 0 0
\(83\) −15.2334 −1.67208 −0.836039 − 0.548670i \(-0.815134\pi\)
−0.836039 + 0.548670i \(0.815134\pi\)
\(84\) 0 0
\(85\) 0.748228 0.0811567
\(86\) 0 0
\(87\) 13.1955 1.41470
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 17.3839 1.82232
\(92\) 0 0
\(93\) −26.0758 −2.70393
\(94\) 0 0
\(95\) −3.34596 −0.343289
\(96\) 0 0
\(97\) −10.4472 −1.06076 −0.530378 − 0.847761i \(-0.677950\pi\)
−0.530378 + 0.847761i \(0.677950\pi\)
\(98\) 0 0
\(99\) −38.9140 −3.91100
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 2960.2.a.u.1.3 3
4.3 odd 2 370.2.a.g.1.1 ✓ 3
12.11 even 2 3330.2.a.bg.1.1 3
20.3 even 4 1850.2.b.o.149.1 6
20.7 even 4 1850.2.b.o.149.6 6
20.19 odd 2 1850.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.1 ✓ 3 4.3 odd 2
1850.2.a.z.1.3 3 20.19 odd 2
1850.2.b.o.149.1 6 20.3 even 4
1850.2.b.o.149.6 6 20.7 even 4
2960.2.a.u.1.3 3 1.1 even 1 trivial
3330.2.a.bg.1.1 3 12.11 even 2