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Properties

Label 108.4.b.a
Level $108$
Weight $4$
Character orbit 108.b
Analytic conductor $6.372$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,4,Mod(107,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{4} - 1) q^{4} - \beta_{9} q^{5} + (\beta_{4} + \beta_{2}) q^{7} + (\beta_{10} - \beta_{9} - \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{4} - 1) q^{4} - \beta_{9} q^{5} + (\beta_{4} + \beta_{2}) q^{7} + (\beta_{10} - \beta_{9} - \beta_{5}) q^{8} + ( - \beta_{6} + \beta_{3} + 2) q^{10} + (\beta_{11} + \beta_{10} + \beta_{5}) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots + 3) q^{13}+ \cdots + (12 \beta_{11} + 4 \beta_{10} + \cdots - 42 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 24 q^{10} + 36 q^{13} + 24 q^{16} + 120 q^{22} - 132 q^{25} + 420 q^{28} - 360 q^{34} + 516 q^{37} - 1152 q^{40} - 696 q^{46} - 720 q^{49} + 204 q^{52} + 2832 q^{58} - 972 q^{61} + 2496 q^{64} - 1848 q^{70} + 660 q^{73} - 5004 q^{76} - 3888 q^{82} + 1056 q^{85} + 3168 q^{88} + 7608 q^{94} + 2532 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -34\nu^{11} + 383\nu^{9} - 2918\nu^{7} + 3680\nu^{5} + 28168\nu^{3} - 112656\nu ) / 3152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -149\nu^{10} + 1916\nu^{8} - 16832\nu^{6} + 54600\nu^{4} - 87232\nu^{2} - 1152 ) / 6304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -128\nu^{10} + 1523\nu^{8} - 13952\nu^{6} + 40368\nu^{4} - 69448\nu^{2} - 94640 ) / 3152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 283\nu^{10} - 3298\nu^{8} + 30256\nu^{6} - 92280\nu^{4} + 215120\nu^{2} - 17824 ) / 6304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 347\nu^{11} - 4158\nu^{9} + 38808\nu^{7} - 126648\nu^{5} + 315248\nu^{3} - 50880\nu ) / 12608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 252\nu^{10} - 2943\nu^{8} + 27468\nu^{6} - 85680\nu^{4} + 227592\nu^{2} - 34416 ) / 3152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -211\nu^{10} + 2626\nu^{8} - 24772\nu^{6} + 86712\nu^{4} - 213584\nu^{2} + 47616 ) / 1576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -1261\nu^{11} + 14002\nu^{9} - 128584\nu^{7} + 350728\nu^{5} - 853776\nu^{3} - 352704\nu ) / 12608 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -190\nu^{11} + 2233\nu^{9} - 20710\nu^{7} + 64600\nu^{5} - 159552\nu^{3} + 4560\nu ) / 1576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2105\nu^{11} + 25294\nu^{9} - 235552\nu^{7} + 773224\nu^{5} - 1928752\nu^{3} + 632064\nu ) / 12608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1453\nu^{11} - 17764\nu^{9} + 166060\nu^{7} - 567304\nu^{5} + 1428384\nu^{3} - 616416\nu ) / 6304 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{11} + 6\beta_{10} - 6\beta_{9} + \beta_{8} - 3\beta_{5} ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 6\beta_{6} - 8\beta_{4} + 2\beta_{3} - 4\beta_{2} + 72 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{10} - 16\beta_{9} + 5\beta_{8} - 39\beta_{5} + 2\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} + 26\beta_{6} - 46\beta_{4} - 8\beta_{3} - 14\beta_{2} - 240 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -21\beta_{11} - 28\beta_{10} - 28\beta_{9} + 20\beta_{8} - 42\beta_{5} + 5\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} + 32\beta_{6} - 104\beta_{4} - 64\beta_{3} + 32\beta_{2} - 1800 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -58\beta_{11} - 82\beta_{10} + 90\beta_{9} + 3\beta_{8} + 395\beta_{5} + 4\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -130\beta_{7} - 492\beta_{6} + 704\beta_{4} - 260\beta_{3} + 856\beta_{2} - 7104 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -32\beta_{10} + 1352\beta_{9} - 460\beta_{8} + 4044\beta_{5} - 32\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -676\beta_{7} - 6056\beta_{6} + 13544\beta_{4} + 2128\beta_{3} + 5032\beta_{2} + 57408 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11832\beta_{11} + 15344\beta_{10} + 15344\beta_{9} - 12496\beta_{8} + 15504\beta_{5} - 1576\beta_1 ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.61829 − 0.934317i
−1.61829 + 0.934317i
0.456937 − 0.263813i
0.456937 + 0.263813i
2.48442 + 1.43438i
2.48442 − 1.43438i
−2.48442 + 1.43438i
−2.48442 − 1.43438i
−0.456937 − 0.263813i
−0.456937 + 0.263813i
1.61829 − 0.934317i
1.61829 + 0.934317i
−2.72087 − 0.772562i 0 6.80630 + 4.20408i − 3.33155i 0 16.9016i −15.2712 − 16.6971i 0 −2.57383 + 9.06473i
107.2 −2.72087 + 0.772562i 0 6.80630 − 4.20408i 3.33155i 0 − 16.9016i −15.2712 + 16.6971i 0 −2.57383 − 9.06473i
107.3 −1.51859 − 2.38619i 0 −3.38780 + 7.24726i − 13.1987i 0 − 4.49091i 22.4380 − 2.92167i 0 −31.4945 + 20.0433i
107.4 −1.51859 + 2.38619i 0 −3.38780 − 7.24726i 13.1987i 0 4.49091i 22.4380 + 2.92167i 0 −31.4945 − 20.0433i
107.5 −0.889241 − 2.68500i 0 −6.41850 + 4.77523i 14.9230i 0 − 30.0528i 18.5291 + 12.9874i 0 40.0683 − 13.2701i
107.6 −0.889241 + 2.68500i 0 −6.41850 − 4.77523i − 14.9230i 0 30.0528i 18.5291 − 12.9874i 0 40.0683 + 13.2701i
107.7 0.889241 − 2.68500i 0 −6.41850 − 4.77523i 14.9230i 0 30.0528i −18.5291 + 12.9874i 0 40.0683 + 13.2701i
107.8 0.889241 + 2.68500i 0 −6.41850 + 4.77523i − 14.9230i 0 − 30.0528i −18.5291 − 12.9874i 0 40.0683 − 13.2701i
107.9 1.51859 − 2.38619i 0 −3.38780 − 7.24726i − 13.1987i 0 4.49091i −22.4380 − 2.92167i 0 −31.4945 − 20.0433i
107.10 1.51859 + 2.38619i 0 −3.38780 + 7.24726i 13.1987i 0 − 4.49091i −22.4380 + 2.92167i 0 −31.4945 + 20.0433i
107.11 2.72087 − 0.772562i 0 6.80630 − 4.20408i − 3.33155i 0 − 16.9016i 15.2712 − 16.6971i 0 −2.57383 − 9.06473i
107.12 2.72087 + 0.772562i 0 6.80630 + 4.20408i 3.33155i 0 16.9016i 15.2712 + 16.6971i 0 −2.57383 + 9.06473i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.b.a ✓ 12
3.b odd 2 1 inner 108.4.b.a ✓ 12
4.b odd 2 1 inner 108.4.b.a ✓ 12
8.b even 2 1 1728.4.c.i 12
8.d odd 2 1 1728.4.c.i 12
12.b even 2 1 inner 108.4.b.a ✓ 12
24.f even 2 1 1728.4.c.i 12
24.h odd 2 1 1728.4.c.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.a ✓ 12 1.a even 1 1 trivial
108.4.b.a ✓ 12 3.b odd 2 1 inner
108.4.b.a ✓ 12 4.b odd 2 1 inner
108.4.b.a ✓ 12 12.b even 2 1 inner
1728.4.c.i 12 8.b even 2 1
1728.4.c.i 12 8.d odd 2 1
1728.4.c.i 12 24.f even 2 1
1728.4.c.i 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 408T_{5}^{4} + 43200T_{5}^{2} + 430592 \) acting on \(S_{4}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 408 T^{4} + \cdots + 430592)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 1209 T^{4} + \cdots + 5203467)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3912 T^{4} + \cdots - 442934784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 9 T^{2} + \cdots + 105973)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 16344 T^{4} + \cdots + 21293875712)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 22329 T^{4} + \cdots + 55289547)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 2278747067904)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 103008 T^{4} + \cdots + 469937979392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 24996 T^{4} + \cdots + 57669803712)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 129 T^{2} + \cdots + 20595901)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 8632577589248)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 4259903272128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 5276542957056)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 1446033784832)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 243 T^{2} + \cdots - 3664703)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 74359881698067)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 55\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 165 T^{2} + \cdots - 57176231)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 21\!\cdots\!47)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 632941911834624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 633 T^{2} + \cdots + 28667317)^{4} \) Copy content Toggle raw display
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