Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [378,2,Mod(109,378)] 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("378.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,0] 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: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 378.109
Dual form 378.2.g.b.163.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 + 1.73205i) q^{7} +1.00000 q^{8} +(3.00000 - 5.19615i) q^{11} +5.00000 q^{13} +(2.50000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(2.00000 + 3.46410i) q^{19} -6.00000 q^{22} +(3.00000 + 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} +(-2.50000 - 4.33013i) q^{26} +(-0.500000 - 2.59808i) q^{28} -6.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(-0.500000 + 0.866025i) q^{32} -6.00000 q^{34} +(0.500000 + 0.866025i) q^{37} +(2.00000 - 3.46410i) q^{38} +6.00000 q^{41} -1.00000 q^{43} +(3.00000 + 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{46} +(-3.00000 - 5.19615i) q^{47} +(1.00000 - 6.92820i) q^{49} -5.00000 q^{50} +(-2.50000 + 4.33013i) q^{52} +(-3.00000 + 5.19615i) q^{53} +(-2.00000 + 1.73205i) q^{56} +(3.00000 + 5.19615i) q^{58} +(-3.00000 + 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} -1.00000 q^{62} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{67} +(3.00000 + 5.19615i) q^{68} -12.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(0.500000 - 0.866025i) q^{74} -4.00000 q^{76} +(3.00000 + 15.5885i) q^{77} +(0.500000 + 0.866025i) q^{79} +(-3.00000 - 5.19615i) q^{82} -6.00000 q^{83} +(0.500000 + 0.866025i) q^{86} +(3.00000 - 5.19615i) q^{88} +(-10.0000 + 8.66025i) q^{91} -6.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +17.0000 q^{97} +(-6.50000 + 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 4 q^{7} + 2 q^{8} + 6 q^{11} + 10 q^{13} + 5 q^{14} - q^{16} + 6 q^{17} + 4 q^{19} - 12 q^{22} + 6 q^{23} + 5 q^{25} - 5 q^{26} - q^{28} - 12 q^{29} + q^{31} - q^{32} - 12 q^{34}+ \cdots - 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 − 5.19615i 0.904534 − 1.56670i 0.0829925 − 0.996550i \(-0.473552\pi\)
0.821541 − 0.570149i \(-0.193114\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 2.50000 + 0.866025i 0.668153 + 0.231455i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 3.00000 − 5.19615i 0.727607 − 1.26025i −0.230285 − 0.973123i \(-0.573966\pi\)
0.957892 − 0.287129i \(-0.0927008\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 − 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) −2.50000 − 4.33013i −0.490290 − 0.849208i
\(27\) 0 0
\(28\) −0.500000 − 2.59808i −0.0944911 − 0.490990i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0.500000 − 0.866025i 0.0898027 − 0.155543i −0.817625 − 0.575751i \(-0.804710\pi\)
0.907428 + 0.420208i \(0.138043\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 2.00000 − 3.46410i 0.324443 − 0.561951i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 3.00000 + 5.19615i 0.452267 + 0.783349i
\(45\) 0 0
\(46\) 3.00000 − 5.19615i 0.442326 − 0.766131i
\(47\) −3.00000 − 5.19615i −0.437595 − 0.757937i 0.559908 − 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) 1.00000 − 6.92820i 0.142857 − 0.989743i
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) −2.50000 + 4.33013i −0.346688 + 0.600481i
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 + 1.73205i −0.267261 + 0.231455i
\(57\) 0 0
\(58\) 3.00000 + 5.19615i 0.393919 + 0.682288i
\(59\) −3.00000 + 5.19615i −0.390567 + 0.676481i −0.992524 − 0.122047i \(-0.961054\pi\)
0.601958 + 0.798528i \(0.294388\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.0640184 + 0.110883i 0.896258 − 0.443533i \(-0.146275\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.500000 − 0.866025i 0.0610847 − 0.105802i −0.833866 − 0.551967i \(-0.813877\pi\)
0.894951 + 0.446165i \(0.147211\pi\)
\(68\) 3.00000 + 5.19615i 0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 − 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0.500000 − 0.866025i 0.0581238 − 0.100673i
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 3.00000 + 15.5885i 0.341882 + 1.77647i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 − 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.00000 − 5.19615i −0.331295 − 0.573819i
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.500000 + 0.866025i 0.0539164 + 0.0933859i
\(87\) 0 0
\(88\) 3.00000 − 5.19615i 0.319801 − 0.553912i
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) −10.0000 + 8.66025i −1.04828 + 0.907841i
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −3.00000 + 5.19615i −0.309426 + 0.535942i
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 − 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) −6.50000 + 2.59808i −0.656599 + 0.262445i
\(99\) 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 378.2.g.b.109.1 ✓ 2
3.2 odd 2 378.2.g.e.109.1 yes 2
7.2 even 3 inner 378.2.g.b.163.1 yes 2
7.3 odd 6 2646.2.a.w.1.1 1
7.4 even 3 2646.2.a.x.1.1 1
9.2 odd 6 1134.2.h.n.109.1 2
9.4 even 3 1134.2.e.m.865.1 2
9.5 odd 6 1134.2.e.c.865.1 2
9.7 even 3 1134.2.h.d.109.1 2
21.2 odd 6 378.2.g.e.163.1 yes 2
21.11 odd 6 2646.2.a.h.1.1 1
21.17 even 6 2646.2.a.g.1.1 1
63.2 odd 6 1134.2.e.c.919.1 2
63.16 even 3 1134.2.e.m.919.1 2
63.23 odd 6 1134.2.h.n.541.1 2
63.58 even 3 1134.2.h.d.541.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.g.b.109.1 ✓ 2 1.1 even 1 trivial
378.2.g.b.163.1 yes 2 7.2 even 3 inner
378.2.g.e.109.1 yes 2 3.2 odd 2
378.2.g.e.163.1 yes 2 21.2 odd 6
1134.2.e.c.865.1 2 9.5 odd 6
1134.2.e.c.919.1 2 63.2 odd 6
1134.2.e.m.865.1 2 9.4 even 3
1134.2.e.m.919.1 2 63.16 even 3
1134.2.h.d.109.1 2 9.7 even 3
1134.2.h.d.541.1 2 63.58 even 3
1134.2.h.n.109.1 2 9.2 odd 6
1134.2.h.n.541.1 2 63.23 odd 6
2646.2.a.g.1.1 1 21.17 even 6
2646.2.a.h.1.1 1 21.11 odd 6
2646.2.a.w.1.1 1 7.3 odd 6
2646.2.a.x.1.1 1 7.4 even 3