Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [14,12,Mod(9,14)] 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("14.9"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-128] 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: \(10.7568045278\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 9.3
Root \(-100.397 + 173.893i\) of defining polynomial
Character \(\chi\) \(=\) 14.9
Dual form 14.12.c.a.11.3

$q$-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+(-16.0000 + 27.7128i) q^{2} +(234.294 + 405.809i) q^{3} +(-512.000 - 886.810i) q^{4} +(-3326.37 + 5761.45i) q^{5} -14994.8 q^{6} +(12255.3 + 42745.0i) q^{7} +32768.0 q^{8} +(-21214.1 + 36743.8i) q^{9} +(-106444. - 184366. i) q^{10} +(-60907.1 - 105494. i) q^{11} +(239917. - 415549. i) q^{12} -1.81677e6 q^{13} +(-1.38067e6 - 344291. i) q^{14} -3.11740e6 q^{15} +(-524288. + 908093. i) q^{16} +(-2.70113e6 - 4.67849e6i) q^{17} +(-678850. - 1.17580e6i) q^{18} +(5.10442e6 - 8.84111e6i) q^{19} +6.81241e6 q^{20} +(-1.44750e7 + 1.49882e7i) q^{21} +3.89805e6 q^{22} +(-1.88132e7 + 3.25854e7i) q^{23} +(7.67735e6 + 1.32976e7i) q^{24} +(2.28454e6 + 3.95694e6i) q^{25} +(2.90684e7 - 5.03479e7i) q^{26} +6.31277e7 q^{27} +(3.16320e7 - 3.27536e7i) q^{28} -1.87020e7 q^{29} +(4.98784e7 - 8.63919e7i) q^{30} +(1.22797e8 + 2.12690e8i) q^{31} +(-1.67772e7 - 2.90590e7i) q^{32} +(2.85404e7 - 4.94333e7i) q^{33} +1.72872e8 q^{34} +(-2.87039e8 - 7.15776e7i) q^{35} +4.34464e7 q^{36} +(-3.25911e8 + 5.64495e8i) q^{37} +(1.63341e8 + 2.82916e8i) q^{38} +(-4.25659e8 - 7.37263e8i) q^{39} +(-1.08999e8 + 1.88791e8i) q^{40} -6.38428e8 q^{41} +(-1.83766e8 - 6.40954e8i) q^{42} +8.54732e8 q^{43} +(-6.23688e7 + 1.08026e8i) q^{44} +(-1.41132e8 - 2.44447e8i) q^{45} +(-6.02023e8 - 1.04273e9i) q^{46} +(4.23660e8 - 7.33801e8i) q^{47} -4.91351e8 q^{48} +(-1.67694e9 + 1.04770e9i) q^{49} -1.46210e8 q^{50} +(1.26572e9 - 2.19228e9i) q^{51} +(9.30187e8 + 1.61113e9i) q^{52} +(2.11267e9 + 3.65925e9i) q^{53} +(-1.01004e9 + 1.74945e9i) q^{54} +8.10399e8 q^{55} +(4.01581e8 + 1.40067e9i) q^{56} +4.78374e9 q^{57} +(2.99232e8 - 5.18285e8i) q^{58} +(1.10225e9 + 1.90916e9i) q^{59} +(1.59611e9 + 2.76454e9i) q^{60} +(4.84994e9 - 8.40035e9i) q^{61} -7.85899e9 q^{62} +(-1.83060e9 - 4.56488e8i) q^{63} +1.07374e9 q^{64} +(6.04326e9 - 1.04672e10i) q^{65} +(9.13291e8 + 1.58187e9i) q^{66} +(4.86446e9 + 8.42549e9i) q^{67} +(-2.76595e9 + 4.79077e9i) q^{68} -1.76313e10 q^{69} +(6.57624e9 - 6.80941e9i) q^{70} -2.83613e10 q^{71} +(-6.95142e8 + 1.20402e9i) q^{72} +(1.83417e9 + 3.17688e9i) q^{73} +(-1.04292e10 - 1.80638e10i) q^{74} +(-1.07051e9 + 1.85417e9i) q^{75} -1.04539e10 q^{76} +(3.76291e9 - 3.89633e9i) q^{77} +2.72422e10 q^{78} +(-6.34771e9 + 1.09946e10i) q^{79} +(-3.48796e9 - 6.04132e9i) q^{80} +(1.85485e10 + 3.21269e10i) q^{81} +(1.02148e10 - 1.76926e10i) q^{82} -4.03358e10 q^{83} +(2.07029e10 + 5.16259e9i) q^{84} +3.59398e10 q^{85} +(-1.36757e10 + 2.36870e10i) q^{86} +(-4.38177e9 - 7.58944e9i) q^{87} +(-1.99580e9 - 3.45683e9i) q^{88} +(1.64368e10 - 2.84694e10i) q^{89} +9.03243e9 q^{90} +(-2.22651e10 - 7.76579e10i) q^{91} +3.85295e10 q^{92} +(-5.75412e10 + 9.96642e10i) q^{93} +(1.35571e10 + 2.34816e10i) q^{94} +(3.39584e10 + 5.88177e10i) q^{95} +(7.86161e9 - 1.36167e10i) q^{96} +1.02279e11 q^{97} +(-2.20377e9 - 6.32361e10i) q^{98} +5.16834e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} + 266 q^{3} - 4096 q^{4} + 7504 q^{5} - 17024 q^{6} - 42224 q^{7} + 262144 q^{8} - 123520 q^{9} + 240128 q^{10} + 213026 q^{11} + 272384 q^{12} - 2609712 q^{13} + 1257536 q^{14} + 2275500 q^{15}+ \cdots + 393415805736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 + 27.7128i −0.353553 + 0.612372i
\(3\) 234.294 + 405.809i 0.556666 + 0.964174i 0.997772 + 0.0667186i \(0.0212530\pi\)
−0.441106 + 0.897455i \(0.645414\pi\)
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) −3326.37 + 5761.45i −0.476032 + 0.824511i −0.999623 − 0.0274583i \(-0.991259\pi\)
0.523591 + 0.851970i \(0.324592\pi\)
\(6\) −14994.8 −0.787245
\(7\) 12255.3 + 42745.0i 0.275603 + 0.961271i
\(8\) 32768.0 0.353553
\(9\) −21214.1 + 36743.8i −0.119754 + 0.207420i
\(10\) −106444. − 184366.i −0.336605 − 0.583018i
\(11\) −60907.1 − 105494.i −0.114027 − 0.197501i 0.803363 − 0.595489i \(-0.203042\pi\)
−0.917390 + 0.397988i \(0.869708\pi\)
\(12\) 239917. − 415549.i 0.278333 − 0.482087i
\(13\) −1.81677e6 −1.35710 −0.678550 − 0.734554i \(-0.737391\pi\)
−0.678550 + 0.734554i \(0.737391\pi\)
\(14\) −1.38067e6 − 344291.i −0.686097 − 0.171089i
\(15\) −3.11740e6 −1.05996
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) −2.70113e6 − 4.67849e6i −0.461398 − 0.799165i 0.537633 − 0.843179i \(-0.319319\pi\)
−0.999031 + 0.0440143i \(0.985985\pi\)
\(18\) −678850. − 1.17580e6i −0.0846788 − 0.146668i
\(19\) 5.10442e6 − 8.84111e6i 0.472935 − 0.819148i −0.526585 − 0.850122i \(-0.676528\pi\)
0.999520 + 0.0309748i \(0.00986115\pi\)
\(20\) 6.81241e6 0.476032
\(21\) −1.44750e7 + 1.49882e7i −0.773414 + 0.800837i
\(22\) 3.89805e6 0.161259
\(23\) −1.88132e7 + 3.25854e7i −0.609481 + 1.05565i 0.381845 + 0.924226i \(0.375289\pi\)
−0.991326 + 0.131425i \(0.958045\pi\)
\(24\) 7.67735e6 + 1.32976e7i 0.196811 + 0.340887i
\(25\) 2.28454e6 + 3.95694e6i 0.0467873 + 0.0810381i
\(26\) 2.90684e7 − 5.03479e7i 0.479807 − 0.831051i
\(27\) 6.31277e7 0.846680
\(28\) 3.16320e7 − 3.27536e7i 0.347342 − 0.359658i
\(29\) −1.87020e7 −0.169316 −0.0846581 − 0.996410i \(-0.526980\pi\)
−0.0846581 + 0.996410i \(0.526980\pi\)
\(30\) 4.98784e7 − 8.63919e7i 0.374753 − 0.649092i
\(31\) 1.22797e8 + 2.12690e8i 0.770367 + 1.33431i 0.937362 + 0.348357i \(0.113260\pi\)
−0.166995 + 0.985958i \(0.553406\pi\)
\(32\) −1.67772e7 − 2.90590e7i −0.0883883 − 0.153093i
\(33\) 2.85404e7 − 4.94333e7i 0.126950 − 0.219884i
\(34\) 1.72872e8 0.652515
\(35\) −2.87039e8 − 7.15776e7i −0.923775 − 0.230358i
\(36\) 4.34464e7 0.119754
\(37\) −3.25911e8 + 5.64495e8i −0.772663 + 1.33829i 0.163436 + 0.986554i \(0.447742\pi\)
−0.936099 + 0.351737i \(0.885591\pi\)
\(38\) 1.63341e8 + 2.82916e8i 0.334416 + 0.579225i
\(39\) −4.25659e8 − 7.37263e8i −0.755452 − 1.30848i
\(40\) −1.08999e8 + 1.88791e8i −0.168303 + 0.291509i
\(41\) −6.38428e8 −0.860598 −0.430299 − 0.902686i \(-0.641592\pi\)
−0.430299 + 0.902686i \(0.641592\pi\)
\(42\) −1.83766e8 − 6.40954e8i −0.216967 − 0.756756i
\(43\) 8.54732e8 0.886652 0.443326 − 0.896360i \(-0.353798\pi\)
0.443326 + 0.896360i \(0.353798\pi\)
\(44\) −6.23688e7 + 1.08026e8i −0.0570135 + 0.0987503i
\(45\) −1.41132e8 − 2.44447e8i −0.114013 − 0.197477i
\(46\) −6.02023e8 − 1.04273e9i −0.430968 − 0.746458i
\(47\) 4.23660e8 − 7.33801e8i 0.269451 − 0.466702i −0.699269 − 0.714858i \(-0.746491\pi\)
0.968720 + 0.248156i \(0.0798245\pi\)
\(48\) −4.91351e8 −0.278333
\(49\) −1.67694e9 + 1.04770e9i −0.848086 + 0.529859i
\(50\) −1.46210e8 −0.0661673
\(51\) 1.26572e9 − 2.19228e9i 0.513689 − 0.889736i
\(52\) 9.30187e8 + 1.61113e9i 0.339275 + 0.587642i
\(53\) 2.11267e9 + 3.65925e9i 0.693927 + 1.20192i 0.970541 + 0.240936i \(0.0774544\pi\)
−0.276614 + 0.960981i \(0.589212\pi\)
\(54\) −1.01004e9 + 1.74945e9i −0.299347 + 0.518484i
\(55\) 8.10399e8 0.217122
\(56\) 4.01581e8 + 1.40067e9i 0.0974405 + 0.339861i
\(57\) 4.78374e9 1.05307
\(58\) 2.99232e8 − 5.18285e8i 0.0598623 − 0.103685i
\(59\) 1.10225e9 + 1.90916e9i 0.200722 + 0.347661i 0.948761 − 0.315994i \(-0.102338\pi\)
−0.748039 + 0.663655i \(0.769005\pi\)
\(60\) 1.59611e9 + 2.76454e9i 0.264991 + 0.458977i
\(61\) 4.84994e9 − 8.40035e9i 0.735229 − 1.27345i −0.219394 − 0.975636i \(-0.570408\pi\)
0.954623 − 0.297817i \(-0.0962587\pi\)
\(62\) −7.85899e9 −1.08946
\(63\) −1.83060e9 − 4.56488e8i −0.232391 − 0.0579504i
\(64\) 1.07374e9 0.125000
\(65\) 6.04326e9 − 1.04672e10i 0.646023 − 1.11894i
\(66\) 9.13291e8 + 1.58187e9i 0.0897672 + 0.155481i
\(67\) 4.86446e9 + 8.42549e9i 0.440173 + 0.762401i 0.997702 − 0.0677559i \(-0.0215839\pi\)
−0.557529 + 0.830157i \(0.688251\pi\)
\(68\) −2.76595e9 + 4.79077e9i −0.230699 + 0.399582i
\(69\) −1.76313e10 −1.35711
\(70\) 6.57624e9 − 6.80941e9i 0.467669 − 0.484251i
\(71\) −2.83613e10 −1.86554 −0.932770 − 0.360472i \(-0.882616\pi\)
−0.932770 + 0.360472i \(0.882616\pi\)
\(72\) −6.95142e8 + 1.20402e9i −0.0423394 + 0.0733340i
\(73\) 1.83417e9 + 3.17688e9i 0.103553 + 0.179360i 0.913146 − 0.407632i \(-0.133645\pi\)
−0.809593 + 0.586992i \(0.800312\pi\)
\(74\) −1.04292e10 − 1.80638e10i −0.546355 − 0.946315i
\(75\) −1.07051e9 + 1.85417e9i −0.0520898 + 0.0902223i
\(76\) −1.04539e10 −0.472935
\(77\) 3.76291e9 − 3.89633e9i 0.158426 − 0.164043i
\(78\) 2.72422e10 1.06837
\(79\) −6.34771e9 + 1.09946e10i −0.232096 + 0.402002i −0.958425 − 0.285345i \(-0.907892\pi\)
0.726329 + 0.687348i \(0.241225\pi\)
\(80\) −3.48796e9 − 6.04132e9i −0.119008 − 0.206128i
\(81\) 1.85485e10 + 3.21269e10i 0.591072 + 1.02377i
\(82\) 1.02148e10 − 1.76926e10i 0.304267 − 0.527007i
\(83\) −4.03358e10 −1.12399 −0.561994 − 0.827141i \(-0.689966\pi\)
−0.561994 + 0.827141i \(0.689966\pi\)
\(84\) 2.07029e10 + 5.16259e9i 0.540126 + 0.134689i
\(85\) 3.59398e10 0.878561
\(86\) −1.36757e10 + 2.36870e10i −0.313479 + 0.542961i
\(87\) −4.38177e9 − 7.58944e9i −0.0942526 − 0.163250i
\(88\) −1.99580e9 − 3.45683e9i −0.0403147 − 0.0698270i
\(89\) 1.64368e10 − 2.84694e10i 0.312013 − 0.540423i −0.666785 − 0.745250i \(-0.732330\pi\)
0.978798 + 0.204827i \(0.0656633\pi\)
\(90\) 9.03243e9 0.161239
\(91\) −2.22651e10 − 7.76579e10i −0.374021 − 1.30454i
\(92\) 3.85295e10 0.609481
\(93\) −5.75412e10 + 9.96642e10i −0.857674 + 1.48554i
\(94\) 1.35571e10 + 2.34816e10i 0.190530 + 0.330008i
\(95\) 3.39584e10 + 5.88177e10i 0.450264 + 0.779881i
\(96\) 7.86161e9 − 1.36167e10i 0.0984056 − 0.170443i
\(97\) 1.02279e11 1.20932 0.604661 − 0.796483i \(-0.293308\pi\)
0.604661 + 0.796483i \(0.293308\pi\)
\(98\) −2.20377e9 − 6.32361e10i −0.0246276 − 0.706678i
\(99\) 5.16834e9 0.0546208
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 14.12.c.a.9.3 ✓ 8
3.2 odd 2 126.12.g.e.37.4 8
4.3 odd 2 112.12.i.a.65.2 8
7.2 even 3 98.12.a.j.1.2 4
7.3 odd 6 98.12.c.l.67.2 8
7.4 even 3 inner 14.12.c.a.11.3 yes 8
7.5 odd 6 98.12.a.l.1.3 4
7.6 odd 2 98.12.c.l.79.2 8
21.11 odd 6 126.12.g.e.109.4 8
28.11 odd 6 112.12.i.a.81.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.3 ✓ 8 1.1 even 1 trivial
14.12.c.a.11.3 yes 8 7.4 even 3 inner
98.12.a.j.1.2 4 7.2 even 3
98.12.a.l.1.3 4 7.5 odd 6
98.12.c.l.67.2 8 7.3 odd 6
98.12.c.l.79.2 8 7.6 odd 2
112.12.i.a.65.2 8 4.3 odd 2
112.12.i.a.81.2 8 28.11 odd 6
126.12.g.e.37.4 8 3.2 odd 2
126.12.g.e.109.4 8 21.11 odd 6