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Properties

Label 378.2.k.a.215.2
Level $378$
Weight $2$
Character 378.215
Analytic conductor $3.018$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [378,2,Mod(215,378)] 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("378.215"); 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([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,0,2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 215.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 378.215
Dual form 378.2.k.a.269.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.500000 + 2.59808i) q^{7} +1.00000i q^{8} +(2.59808 - 1.50000i) q^{11} +3.46410i q^{13} +(-0.866025 + 2.50000i) q^{14} +(-0.500000 + 0.866025i) q^{16} +3.00000 q^{22} +(2.50000 + 4.33013i) q^{25} +(-1.73205 + 3.00000i) q^{26} +(-2.00000 + 1.73205i) q^{28} -9.00000i q^{29} +(-1.50000 + 0.866025i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(4.00000 - 6.92820i) q^{37} -10.3923 q^{41} +4.00000 q^{43} +(2.59808 + 1.50000i) q^{44} +(5.19615 - 9.00000i) q^{47} +(-6.50000 + 2.59808i) q^{49} +5.00000i q^{50} +(-3.00000 + 1.73205i) q^{52} +(-5.19615 + 3.00000i) q^{53} +(-2.59808 + 0.500000i) q^{56} +(4.50000 - 7.79423i) q^{58} +(2.59808 + 4.50000i) q^{59} +(-12.0000 - 6.92820i) q^{61} -1.73205 q^{62} -1.00000 q^{64} +(-1.00000 - 1.73205i) q^{67} -12.0000i q^{71} +(4.50000 - 2.59808i) q^{73} +(6.92820 - 4.00000i) q^{74} +(5.19615 + 6.00000i) q^{77} +(6.50000 - 11.2583i) q^{79} +(-9.00000 - 5.19615i) q^{82} +5.19615 q^{83} +(3.46410 + 2.00000i) q^{86} +(1.50000 + 2.59808i) q^{88} +(-5.19615 + 9.00000i) q^{89} +(-9.00000 + 1.73205i) q^{91} +(9.00000 - 5.19615i) q^{94} -8.66025i q^{97} +(-6.92820 - 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{7} - 2 q^{16} + 12 q^{22} + 10 q^{25} - 8 q^{28} - 6 q^{31} + 16 q^{37} + 16 q^{43} - 26 q^{49} - 12 q^{52} + 18 q^{58} - 48 q^{61} - 4 q^{64} - 4 q^{67} + 18 q^{73} + 26 q^{79} - 36 q^{82}+ \cdots + 36 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.59808 − 1.50000i 0.783349 − 0.452267i −0.0542666 − 0.998526i \(-0.517282\pi\)
0.837616 + 0.546259i \(0.183949\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) −0.866025 + 2.50000i −0.231455 + 0.668153i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) −1.73205 + 3.00000i −0.339683 + 0.588348i
\(27\) 0 0
\(28\) −2.00000 + 1.73205i −0.377964 + 0.327327i
\(29\) − 9.00000i − 1.67126i −0.549294 − 0.835629i \(-0.685103\pi\)
0.549294 − 0.835629i \(-0.314897\pi\)
\(30\) 0 0
\(31\) −1.50000 + 0.866025i −0.269408 + 0.155543i −0.628619 − 0.777714i \(-0.716379\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 − 6.92820i 0.657596 − 1.13899i −0.323640 − 0.946180i \(-0.604907\pi\)
0.981236 − 0.192809i \(-0.0617599\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.3923 −1.62301 −0.811503 − 0.584349i \(-0.801350\pi\)
−0.811503 + 0.584349i \(0.801350\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.59808 + 1.50000i 0.391675 + 0.226134i
\(45\) 0 0
\(46\) 0 0
\(47\) 5.19615 − 9.00000i 0.757937 − 1.31278i −0.185964 − 0.982556i \(-0.559541\pi\)
0.943901 − 0.330228i \(-0.107126\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 5.00000i 0.707107i
\(51\) 0 0
\(52\) −3.00000 + 1.73205i −0.416025 + 0.240192i
\(53\) −5.19615 + 3.00000i −0.713746 + 0.412082i −0.812447 − 0.583036i \(-0.801865\pi\)
0.0987002 + 0.995117i \(0.468532\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.59808 + 0.500000i −0.347183 + 0.0668153i
\(57\) 0 0
\(58\) 4.50000 − 7.79423i 0.590879 − 1.02343i
\(59\) 2.59808 + 4.50000i 0.338241 + 0.585850i 0.984102 − 0.177605i \(-0.0568349\pi\)
−0.645861 + 0.763455i \(0.723502\pi\)
\(60\) 0 0
\(61\) −12.0000 − 6.92820i −1.53644 − 0.887066i −0.999043 − 0.0437377i \(-0.986073\pi\)
−0.537400 − 0.843328i \(-0.680593\pi\)
\(62\) −1.73205 −0.219971
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 − 1.73205i −0.122169 − 0.211604i 0.798454 − 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 12.0000i − 1.42414i −0.702109 − 0.712069i \(-0.747758\pi\)
0.702109 − 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 4.50000 − 2.59808i 0.526685 − 0.304082i −0.212980 − 0.977056i \(-0.568317\pi\)
0.739666 + 0.672975i \(0.234984\pi\)
\(74\) 6.92820 − 4.00000i 0.805387 − 0.464991i
\(75\) 0 0
\(76\) 0 0
\(77\) 5.19615 + 6.00000i 0.592157 + 0.683763i
\(78\) 0 0
\(79\) 6.50000 − 11.2583i 0.731307 − 1.26666i −0.225018 − 0.974355i \(-0.572244\pi\)
0.956325 − 0.292306i \(-0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −9.00000 − 5.19615i −0.993884 − 0.573819i
\(83\) 5.19615 0.570352 0.285176 − 0.958475i \(-0.407948\pi\)
0.285176 + 0.958475i \(0.407948\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.46410 + 2.00000i 0.373544 + 0.215666i
\(87\) 0 0
\(88\) 1.50000 + 2.59808i 0.159901 + 0.276956i
\(89\) −5.19615 + 9.00000i −0.550791 + 0.953998i 0.447427 + 0.894321i \(0.352341\pi\)
−0.998218 + 0.0596775i \(0.980993\pi\)
\(90\) 0 0
\(91\) −9.00000 + 1.73205i −0.943456 + 0.181568i
\(92\) 0 0
\(93\) 0 0
\(94\) 9.00000 − 5.19615i 0.928279 − 0.535942i
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.66025i − 0.879316i −0.898165 − 0.439658i \(-0.855100\pi\)
0.898165 − 0.439658i \(-0.144900\pi\)
\(98\) −6.92820 − 1.00000i −0.699854 − 0.101015i
\(99\) 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.k.a.215.2 yes 4
3.2 odd 2 inner 378.2.k.a.215.1 ✓ 4
7.2 even 3 2646.2.d.c.2645.2 4
7.3 odd 6 inner 378.2.k.a.269.1 yes 4
7.5 odd 6 2646.2.d.c.2645.1 4
9.2 odd 6 1134.2.l.d.215.2 4
9.4 even 3 1134.2.t.a.593.1 4
9.5 odd 6 1134.2.t.a.593.2 4
9.7 even 3 1134.2.l.d.215.1 4
21.2 odd 6 2646.2.d.c.2645.4 4
21.5 even 6 2646.2.d.c.2645.3 4
21.17 even 6 inner 378.2.k.a.269.2 yes 4
63.31 odd 6 1134.2.l.d.269.1 4
63.38 even 6 1134.2.t.a.1025.1 4
63.52 odd 6 1134.2.t.a.1025.2 4
63.59 even 6 1134.2.l.d.269.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.a.215.1 ✓ 4 3.2 odd 2 inner
378.2.k.a.215.2 yes 4 1.1 even 1 trivial
378.2.k.a.269.1 yes 4 7.3 odd 6 inner
378.2.k.a.269.2 yes 4 21.17 even 6 inner
1134.2.l.d.215.1 4 9.7 even 3
1134.2.l.d.215.2 4 9.2 odd 6
1134.2.l.d.269.1 4 63.31 odd 6
1134.2.l.d.269.2 4 63.59 even 6
1134.2.t.a.593.1 4 9.4 even 3
1134.2.t.a.593.2 4 9.5 odd 6
1134.2.t.a.1025.1 4 63.38 even 6
1134.2.t.a.1025.2 4 63.52 odd 6
2646.2.d.c.2645.1 4 7.5 odd 6
2646.2.d.c.2645.2 4 7.2 even 3
2646.2.d.c.2645.3 4 21.5 even 6
2646.2.d.c.2645.4 4 21.2 odd 6