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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2916,2,Mod(973,2916)] 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("2916.973"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2916, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2916 = 2^{2} \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2916.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.2843772294\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 973.9
Character \(\chi\) \(=\) 2916.973
Dual form 2916.2.e.e.1945.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.16084 - 2.01063i) q^{5} +(-1.56011 - 2.70218i) q^{7} +(-1.56995 - 2.71923i) q^{11} +(1.75839 - 3.04563i) q^{13} +5.62105 q^{17} +5.22552 q^{19} +(1.01694 - 1.76139i) q^{23} +(-0.195102 - 0.337927i) q^{25} +(3.08920 + 5.35066i) q^{29} +(-2.41824 + 4.18851i) q^{31} -7.24414 q^{35} +1.75386 q^{37} +(5.89766 - 10.2150i) q^{41} +(-4.41147 - 7.64090i) q^{43} +(2.74404 + 4.75282i) q^{47} +(-1.36786 + 2.36921i) q^{49} -5.46571 q^{53} -7.28984 q^{55} +(-4.93094 + 8.54065i) q^{59} +(-5.42497 - 9.39632i) q^{61} +(-4.08243 - 7.07097i) q^{65} +(-3.28907 + 5.69683i) q^{67} +3.25445 q^{71} -5.61431 q^{73} +(-4.89857 + 8.48458i) q^{77} +(-7.87264 - 13.6358i) q^{79} +(-6.96069 - 12.0563i) q^{83} +(6.52514 - 11.3019i) q^{85} -7.16926 q^{89} -10.9731 q^{91} +(6.06600 - 10.5066i) q^{95} +(-6.76961 - 11.7253i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 12 q^{13} + 24 q^{19} - 30 q^{25} - 6 q^{31} + 36 q^{37} - 24 q^{43} - 48 q^{49} + 36 q^{55} - 60 q^{61} - 42 q^{67} + 120 q^{73} - 12 q^{79} - 72 q^{85} + 96 q^{91} - 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times\).

\(n\) \(1459\) \(2189\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.16084 − 2.01063i 0.519144 − 0.899183i −0.480609 − 0.876935i \(-0.659584\pi\)
0.999752 − 0.0222482i \(-0.00708240\pi\)
\(6\) 0 0
\(7\) −1.56011 − 2.70218i −0.589665 − 1.02133i −0.994276 − 0.106841i \(-0.965927\pi\)
0.404611 − 0.914489i \(-0.367407\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.56995 − 2.71923i −0.473357 − 0.819879i 0.526178 − 0.850375i \(-0.323625\pi\)
−0.999535 + 0.0304958i \(0.990291\pi\)
\(12\) 0 0
\(13\) 1.75839 − 3.04563i 0.487690 − 0.844704i −0.512209 − 0.858861i \(-0.671173\pi\)
0.999900 + 0.0141562i \(0.00450619\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.62105 1.36330 0.681652 − 0.731676i \(-0.261262\pi\)
0.681652 + 0.731676i \(0.261262\pi\)
\(18\) 0 0
\(19\) 5.22552 1.19882 0.599409 − 0.800443i \(-0.295402\pi\)
0.599409 + 0.800443i \(0.295402\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.01694 − 1.76139i 0.212047 − 0.367276i −0.740308 − 0.672268i \(-0.765320\pi\)
0.952355 + 0.304992i \(0.0986537\pi\)
\(24\) 0 0
\(25\) −0.195102 − 0.337927i −0.0390204 − 0.0675853i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.08920 + 5.35066i 0.573651 + 0.993592i 0.996187 + 0.0872460i \(0.0278066\pi\)
−0.422536 + 0.906346i \(0.638860\pi\)
\(30\) 0 0
\(31\) −2.41824 + 4.18851i −0.434328 + 0.752279i −0.997241 − 0.0742378i \(-0.976348\pi\)
0.562912 + 0.826517i \(0.309681\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.24414 −1.22448
\(36\) 0 0
\(37\) 1.75386 0.288332 0.144166 − 0.989553i \(-0.453950\pi\)
0.144166 + 0.989553i \(0.453950\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.89766 − 10.2150i 0.921060 − 1.59532i 0.123281 − 0.992372i \(-0.460658\pi\)
0.797779 − 0.602951i \(-0.206008\pi\)
\(42\) 0 0
\(43\) −4.41147 − 7.64090i −0.672743 − 1.16523i −0.977123 − 0.212675i \(-0.931782\pi\)
0.304379 − 0.952551i \(-0.401551\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.74404 + 4.75282i 0.400260 + 0.693270i 0.993757 − 0.111566i \(-0.0355866\pi\)
−0.593497 + 0.804836i \(0.702253\pi\)
\(48\) 0 0
\(49\) −1.36786 + 2.36921i −0.195409 + 0.338458i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.46571 −0.750773 −0.375387 − 0.926868i \(-0.622490\pi\)
−0.375387 + 0.926868i \(0.622490\pi\)
\(54\) 0 0
\(55\) −7.28984 −0.982962
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.93094 + 8.54065i −0.641954 + 1.11190i 0.343042 + 0.939320i \(0.388543\pi\)
−0.984996 + 0.172577i \(0.944791\pi\)
\(60\) 0 0
\(61\) −5.42497 − 9.39632i −0.694596 − 1.20308i −0.970317 − 0.241838i \(-0.922250\pi\)
0.275721 − 0.961238i \(-0.411083\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.08243 − 7.07097i −0.506363 − 0.877046i
\(66\) 0 0
\(67\) −3.28907 + 5.69683i −0.401824 + 0.695979i −0.993946 − 0.109869i \(-0.964957\pi\)
0.592123 + 0.805848i \(0.298290\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.25445 0.386232 0.193116 − 0.981176i \(-0.438141\pi\)
0.193116 + 0.981176i \(0.438141\pi\)
\(72\) 0 0
\(73\) −5.61431 −0.657105 −0.328552 − 0.944486i \(-0.606561\pi\)
−0.328552 + 0.944486i \(0.606561\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.89857 + 8.48458i −0.558244 + 0.966907i
\(78\) 0 0
\(79\) −7.87264 − 13.6358i −0.885741 − 1.53415i −0.844862 − 0.534984i \(-0.820318\pi\)
−0.0408789 − 0.999164i \(-0.513016\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.96069 − 12.0563i −0.764035 − 1.32335i −0.940755 − 0.339087i \(-0.889882\pi\)
0.176720 − 0.984261i \(-0.443451\pi\)
\(84\) 0 0
\(85\) 6.52514 − 11.3019i 0.707751 − 1.22586i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.16926 −0.759940 −0.379970 − 0.924999i \(-0.624066\pi\)
−0.379970 + 0.924999i \(0.624066\pi\)
\(90\) 0 0
\(91\) −10.9731 −1.15030
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.06600 − 10.5066i 0.622358 − 1.07796i
\(96\) 0 0
\(97\) −6.76961 − 11.7253i −0.687349 − 1.19052i −0.972692 − 0.232099i \(-0.925441\pi\)
0.285343 − 0.958426i \(-0.407893\pi\)
\(98\) 0 0
\(99\) 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 2916.2.e.e.973.9 24
3.2 odd 2 inner 2916.2.e.e.973.4 24
9.2 odd 6 inner 2916.2.e.e.1945.4 24
9.4 even 3 2916.2.a.e.1.4 ✓ 12
9.5 odd 6 2916.2.a.e.1.9 yes 12
9.7 even 3 inner 2916.2.e.e.1945.9 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2916.2.a.e.1.4 ✓ 12 9.4 even 3
2916.2.a.e.1.9 yes 12 9.5 odd 6
2916.2.e.e.973.4 24 3.2 odd 2 inner
2916.2.e.e.973.9 24 1.1 even 1 trivial
2916.2.e.e.1945.4 24 9.2 odd 6 inner
2916.2.e.e.1945.9 24 9.7 even 3 inner