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Properties

Label 14.14.a.a.1.1
Level $14$
Weight $14$
Character 14.1
Self dual yes
Analytic conductor $15.012$
Analytic rank $1$
Dimension $1$
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] = [14,14,Mod(1,14)] 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("14.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-64] 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: yes
Analytic conductor: \(15.0123300533\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 14.1

$q$-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)
 
\(f(q)\) \(=\) \(q-64.0000 q^{2} +1626.00 q^{3} +4096.00 q^{4} -36400.0 q^{5} -104064. q^{6} -117649. q^{7} -262144. q^{8} +1.04955e6 q^{9} +2.32960e6 q^{10} +2.60529e6 q^{11} +6.66010e6 q^{12} -1.26245e7 q^{13} +7.52954e6 q^{14} -5.91864e7 q^{15} +1.67772e7 q^{16} -1.30752e8 q^{17} -6.71714e7 q^{18} -2.49436e8 q^{19} -1.49094e8 q^{20} -1.91297e8 q^{21} -1.66738e8 q^{22} +4.89054e8 q^{23} -4.26246e8 q^{24} +1.04257e8 q^{25} +8.07966e8 q^{26} -8.85796e8 q^{27} -4.81890e8 q^{28} -1.12116e8 q^{29} +3.78793e9 q^{30} -9.10307e9 q^{31} -1.07374e9 q^{32} +4.23620e9 q^{33} +8.36815e9 q^{34} +4.28242e9 q^{35} +4.29897e9 q^{36} +1.83082e10 q^{37} +1.59639e10 q^{38} -2.05274e10 q^{39} +9.54204e9 q^{40} +1.30824e10 q^{41} +1.22430e10 q^{42} -6.71235e10 q^{43} +1.06713e10 q^{44} -3.82037e10 q^{45} -3.12995e10 q^{46} +1.05240e11 q^{47} +2.72798e10 q^{48} +1.38413e10 q^{49} -6.67244e9 q^{50} -2.12603e11 q^{51} -5.17098e10 q^{52} -2.52217e10 q^{53} +5.66909e10 q^{54} -9.48325e10 q^{55} +3.08410e10 q^{56} -4.05583e11 q^{57} +7.17542e9 q^{58} -2.76775e11 q^{59} -2.42427e11 q^{60} +7.59389e11 q^{61} +5.82596e11 q^{62} -1.23479e11 q^{63} +6.87195e10 q^{64} +4.59531e11 q^{65} -2.71117e11 q^{66} +1.03966e12 q^{67} -5.35562e11 q^{68} +7.95202e11 q^{69} -2.74075e11 q^{70} +1.81709e12 q^{71} -2.75134e11 q^{72} +4.00342e11 q^{73} -1.17172e12 q^{74} +1.69522e11 q^{75} -1.02169e12 q^{76} -3.06510e11 q^{77} +1.31375e12 q^{78} -3.59780e12 q^{79} -6.10691e11 q^{80} -3.11363e12 q^{81} -8.37272e11 q^{82} -1.30903e12 q^{83} -7.83554e11 q^{84} +4.75939e12 q^{85} +4.29590e12 q^{86} -1.82300e11 q^{87} -6.82961e11 q^{88} +1.65329e12 q^{89} +2.44504e12 q^{90} +1.48526e12 q^{91} +2.00317e12 q^{92} -1.48016e13 q^{93} -6.73536e12 q^{94} +9.07947e12 q^{95} -1.74590e12 q^{96} -1.27369e13 q^{97} -8.85842e11 q^{98} +2.73439e12 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −64.0000 −0.707107
\(3\) 1626.00 1.28775 0.643876 − 0.765130i \(-0.277325\pi\)
0.643876 + 0.765130i \(0.277325\pi\)
\(4\) 4096.00 0.500000
\(5\) −36400.0 −1.04183 −0.520914 − 0.853609i \(-0.674409\pi\)
−0.520914 + 0.853609i \(0.674409\pi\)
\(6\) −104064. −0.910578
\(7\) −117649. −0.377964
\(8\) −262144. −0.353553
\(9\) 1.04955e6 0.658306
\(10\) 2.32960e6 0.736684
\(11\) 2.60529e6 0.443408 0.221704 − 0.975114i \(-0.428838\pi\)
0.221704 + 0.975114i \(0.428838\pi\)
\(12\) 6.66010e6 0.643876
\(13\) −1.26245e7 −0.725406 −0.362703 − 0.931905i \(-0.618146\pi\)
−0.362703 + 0.931905i \(0.618146\pi\)
\(14\) 7.52954e6 0.267261
\(15\) −5.91864e7 −1.34162
\(16\) 1.67772e7 0.250000
\(17\) −1.30752e8 −1.31381 −0.656903 − 0.753975i \(-0.728134\pi\)
−0.656903 + 0.753975i \(0.728134\pi\)
\(18\) −6.71714e7 −0.465493
\(19\) −2.49436e8 −1.21636 −0.608178 − 0.793801i \(-0.708099\pi\)
−0.608178 + 0.793801i \(0.708099\pi\)
\(20\) −1.49094e8 −0.520914
\(21\) −1.91297e8 −0.486725
\(22\) −1.66738e8 −0.313537
\(23\) 4.89054e8 0.688852 0.344426 − 0.938813i \(-0.388074\pi\)
0.344426 + 0.938813i \(0.388074\pi\)
\(24\) −4.26246e8 −0.455289
\(25\) 1.04257e8 0.0854072
\(26\) 8.07966e8 0.512940
\(27\) −8.85796e8 −0.440017
\(28\) −4.81890e8 −0.188982
\(29\) −1.12116e8 −0.0350010 −0.0175005 − 0.999847i \(-0.505571\pi\)
−0.0175005 + 0.999847i \(0.505571\pi\)
\(30\) 3.78793e9 0.948667
\(31\) −9.10307e9 −1.84220 −0.921100 − 0.389326i \(-0.872708\pi\)
−0.921100 + 0.389326i \(0.872708\pi\)
\(32\) −1.07374e9 −0.176777
\(33\) 4.23620e9 0.570999
\(34\) 8.36815e9 0.929002
\(35\) 4.28242e9 0.393774
\(36\) 4.29897e9 0.329153
\(37\) 1.83082e10 1.17310 0.586548 − 0.809914i \(-0.300486\pi\)
0.586548 + 0.809914i \(0.300486\pi\)
\(38\) 1.59639e10 0.860094
\(39\) −2.05274e10 −0.934144
\(40\) 9.54204e9 0.368342
\(41\) 1.30824e10 0.430122 0.215061 − 0.976601i \(-0.431005\pi\)
0.215061 + 0.976601i \(0.431005\pi\)
\(42\) 1.22430e10 0.344166
\(43\) −6.71235e10 −1.61931 −0.809654 − 0.586908i \(-0.800345\pi\)
−0.809654 + 0.586908i \(0.800345\pi\)
\(44\) 1.06713e10 0.221704
\(45\) −3.82037e10 −0.685843
\(46\) −3.12995e10 −0.487092
\(47\) 1.05240e11 1.42411 0.712057 − 0.702122i \(-0.247764\pi\)
0.712057 + 0.702122i \(0.247764\pi\)
\(48\) 2.72798e10 0.321938
\(49\) 1.38413e10 0.142857
\(50\) −6.67244e9 −0.0603920
\(51\) −2.12603e11 −1.69186
\(52\) −5.17098e10 −0.362703
\(53\) −2.52217e10 −0.156308 −0.0781541 − 0.996941i \(-0.524903\pi\)
−0.0781541 + 0.996941i \(0.524903\pi\)
\(54\) 5.66909e10 0.311139
\(55\) −9.48325e10 −0.461955
\(56\) 3.08410e10 0.133631
\(57\) −4.05583e11 −1.56637
\(58\) 7.17542e9 0.0247494
\(59\) −2.76775e11 −0.854256 −0.427128 − 0.904191i \(-0.640475\pi\)
−0.427128 + 0.904191i \(0.640475\pi\)
\(60\) −2.42427e11 −0.670809
\(61\) 7.59389e11 1.88721 0.943605 − 0.331074i \(-0.107411\pi\)
0.943605 + 0.331074i \(0.107411\pi\)
\(62\) 5.82596e11 1.30263
\(63\) −1.23479e11 −0.248816
\(64\) 6.87195e10 0.125000
\(65\) 4.59531e11 0.755749
\(66\) −2.71117e11 −0.403758
\(67\) 1.03966e12 1.40413 0.702065 − 0.712113i \(-0.252262\pi\)
0.702065 + 0.712113i \(0.252262\pi\)
\(68\) −5.35562e11 −0.656903
\(69\) 7.95202e11 0.887071
\(70\) −2.74075e11 −0.278440
\(71\) 1.81709e12 1.68343 0.841717 − 0.539918i \(-0.181545\pi\)
0.841717 + 0.539918i \(0.181545\pi\)
\(72\) −2.75134e11 −0.232746
\(73\) 4.00342e11 0.309623 0.154811 − 0.987944i \(-0.450523\pi\)
0.154811 + 0.987944i \(0.450523\pi\)
\(74\) −1.17172e12 −0.829504
\(75\) 1.69522e11 0.109983
\(76\) −1.02169e12 −0.608178
\(77\) −3.06510e11 −0.167592
\(78\) 1.31375e12 0.660539
\(79\) −3.59780e12 −1.66518 −0.832589 − 0.553891i \(-0.813142\pi\)
−0.832589 + 0.553891i \(0.813142\pi\)
\(80\) −6.10691e11 −0.260457
\(81\) −3.11363e12 −1.22494
\(82\) −8.37272e11 −0.304142
\(83\) −1.30903e12 −0.439483 −0.219742 − 0.975558i \(-0.570521\pi\)
−0.219742 + 0.975558i \(0.570521\pi\)
\(84\) −7.83554e11 −0.243362
\(85\) 4.75939e12 1.36876
\(86\) 4.29590e12 1.14502
\(87\) −1.82300e11 −0.0450726
\(88\) −6.82961e11 −0.156768
\(89\) 1.65329e12 0.352625 0.176313 − 0.984334i \(-0.443583\pi\)
0.176313 + 0.984334i \(0.443583\pi\)
\(90\) 2.44504e12 0.484964
\(91\) 1.48526e12 0.274178
\(92\) 2.00317e12 0.344426
\(93\) −1.48016e13 −2.37230
\(94\) −6.73536e12 −1.00700
\(95\) 9.07947e12 1.26723
\(96\) −1.74590e12 −0.227645
\(97\) −1.27369e13 −1.55256 −0.776279 − 0.630390i \(-0.782895\pi\)
−0.776279 + 0.630390i \(0.782895\pi\)
\(98\) −8.85842e11 −0.101015
\(99\) 2.73439e12 0.291898
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 14.14.a.a.1.1 ✓ 1
3.2 odd 2 126.14.a.e.1.1 1
4.3 odd 2 112.14.a.a.1.1 1
7.2 even 3 98.14.c.f.67.1 2
7.3 odd 6 98.14.c.g.79.1 2
7.4 even 3 98.14.c.f.79.1 2
7.5 odd 6 98.14.c.g.67.1 2
7.6 odd 2 98.14.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.14.a.a.1.1 ✓ 1 1.1 even 1 trivial
98.14.a.a.1.1 1 7.6 odd 2
98.14.c.f.67.1 2 7.2 even 3
98.14.c.f.79.1 2 7.4 even 3
98.14.c.g.67.1 2 7.5 odd 6
98.14.c.g.79.1 2 7.3 odd 6
112.14.a.a.1.1 1 4.3 odd 2
126.14.a.e.1.1 1 3.2 odd 2