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Properties

Label 148.2.n.b
Level $148$
Weight $2$
Character orbit 148.n
Analytic conductor $1.182$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [148,2,Mod(21,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 148.n (of order \(18\), degree \(6\), 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: \(1.18178594991\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 198x^{8} + 734x^{6} + 1329x^{4} + 1134x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} - 1) q^{3} + ( - \beta_{8} - \beta_{5}) q^{5} + (\beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{9} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} - 1) q^{3} + ( - \beta_{8} - \beta_{5}) q^{5} + (\beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 6 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 6 q^{7} + 12 q^{9} - 6 q^{11} + 6 q^{13} + 9 q^{15} + 9 q^{17} - 9 q^{19} - 6 q^{21} - 27 q^{23} - 18 q^{25} - 6 q^{27} + 3 q^{33} - 18 q^{35} + 12 q^{37} - 6 q^{39} + 15 q^{41} + 9 q^{45} - 12 q^{47} - 42 q^{49} - 9 q^{51} + 33 q^{53} + 27 q^{55} - 9 q^{57} + 9 q^{61} + 3 q^{63} + 9 q^{65} + 6 q^{67} + 27 q^{69} - 24 q^{71} + 36 q^{75} - 51 q^{77} + 15 q^{79} + 42 q^{81} + 33 q^{83} - 18 q^{85} - 9 q^{87} + 36 q^{89} + 48 q^{91} + 15 q^{93} + 27 q^{95} - 54 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 198x^{8} + 734x^{6} + 1329x^{4} + 1134x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 63 \nu^{11} - 57 \nu^{10} + 1341 \nu^{9} - 1444 \nu^{8} + 8788 \nu^{7} - 12673 \nu^{6} + 21466 \nu^{5} + \cdots - 38912 ) / 2584 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 63 \nu^{11} - 57 \nu^{10} - 1341 \nu^{9} - 1444 \nu^{8} - 8788 \nu^{7} - 12673 \nu^{6} + \cdots - 38912 ) / 2584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -45\nu^{11} - 1004\nu^{9} - 7200\nu^{7} - 20547\nu^{5} - 22907\nu^{3} - 7311\nu + 646 ) / 1292 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6 \nu^{11} - 456 \nu^{10} + 103 \nu^{9} - 10260 \nu^{8} + 3885 \nu^{7} - 75221 \nu^{6} + \cdots - 141075 ) / 2584 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6 \nu^{11} + 456 \nu^{10} + 103 \nu^{9} + 10260 \nu^{8} + 3885 \nu^{7} + 75221 \nu^{6} + \cdots + 141075 ) / 2584 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 155 \nu^{11} + 171 \nu^{10} - 3530 \nu^{9} + 3686 \nu^{8} - 26415 \nu^{7} + 24776 \nu^{6} + \cdots + 18867 ) / 2584 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{11} - 20 \nu^{10} + 194 \nu^{9} - 433 \nu^{8} + 1304 \nu^{7} - 2945 \nu^{6} + 3341 \nu^{5} + \cdots - 3765 ) / 136 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 120 \nu^{11} - 608 \nu^{10} + 2785 \nu^{9} - 13680 \nu^{8} + 21461 \nu^{7} - 100187 \nu^{6} + \cdots - 173565 ) / 2584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 218 \nu^{11} + 228 \nu^{10} + 4871 \nu^{9} + 5130 \nu^{8} + 35203 \nu^{7} + 37449 \nu^{6} + \cdots + 57779 ) / 2584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 120 \nu^{11} + 608 \nu^{10} + 2785 \nu^{9} + 13680 \nu^{8} + 21461 \nu^{7} + 100187 \nu^{6} + \cdots + 173565 ) / 2584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 261 \nu^{11} - 380 \nu^{10} - 5694 \nu^{9} - 8227 \nu^{8} - 39176 \nu^{7} - 55955 \nu^{6} + \cdots - 70243 ) / 2584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{8} - \beta_{5} - \beta_{4} + 2\beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} - 3 \beta_{9} - \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \cdots - 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} - 7 \beta_{10} - 7 \beta_{8} - 6 \beta_{7} + 7 \beta_{5} + 7 \beta_{4} - 12 \beta_{3} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 22 \beta_{11} - 20 \beta_{10} + 39 \beta_{9} + 20 \beta_{8} + 22 \beta_{7} + 39 \beta_{6} + 31 \beta_{5} + \cdots + 101 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 90 \beta_{11} + 76 \beta_{10} - 9 \beta_{9} + 76 \beta_{8} + 90 \beta_{7} + 9 \beta_{6} - 61 \beta_{5} + \cdots - 60 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 78 \beta_{11} + 93 \beta_{10} - 161 \beta_{9} - 93 \beta_{8} - 78 \beta_{7} - 161 \beta_{6} + \cdots - 334 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1146 \beta_{11} - 892 \beta_{10} + 198 \beta_{9} - 892 \beta_{8} - 1146 \beta_{7} - 198 \beta_{6} + \cdots + 813 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2576 \beta_{11} - 3493 \beta_{10} + 5820 \beta_{9} + 3493 \beta_{8} + 2576 \beta_{7} + 5820 \beta_{6} + \cdots + 10858 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 13848 \beta_{11} + 10489 \beta_{10} - 2943 \beta_{9} + 10489 \beta_{8} + 13848 \beta_{7} + \cdots - 9987 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 29056 \beta_{11} + 41927 \beta_{10} - 68826 \beta_{9} - 41927 \beta_{8} - 29056 \beta_{7} + \cdots - 121925 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 163644 \beta_{11} - 122602 \beta_{10} + 38091 \beta_{9} - 122602 \beta_{8} - 163644 \beta_{7} + \cdots + 118653 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/148\mathbb{Z}\right)^\times\).

\(n\) \(75\) \(113\)
\(\chi(n)\) \(1\) \(\beta_{1} - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
3.40333i
− 1.43371i
0.916904i
− 1.60094i
− 1.10818i
2.39375i
1.10818i
− 2.39375i
− 0.916904i
1.60094i
− 3.40333i
1.43371i
0 −1.76604 + 0.642788i 0 −2.94737 − 0.519700i 0 0.757565 − 4.29637i 0 0.407604 − 0.342020i 0
21.2 0 −1.76604 + 0.642788i 0 1.24163 + 0.218933i 0 −0.697258 + 3.95435i 0 0.407604 − 0.342020i 0
25.1 0 −1.17365 − 0.984808i 0 −0.794062 − 2.18167i 0 −1.16600 + 0.424388i 0 −0.113341 − 0.642788i 0
25.2 0 −1.17365 − 0.984808i 0 1.38646 + 3.80926i 0 2.93204 − 1.06718i 0 −0.113341 − 0.642788i 0
41.1 0 −0.0603074 − 0.342020i 0 −0.959710 − 1.14374i 0 2.91005 − 2.44183i 0 2.70574 − 0.984808i 0
41.2 0 −0.0603074 − 0.342020i 0 2.07305 + 2.47057i 0 −1.73641 + 1.45702i 0 2.70574 − 0.984808i 0
65.1 0 −0.0603074 + 0.342020i 0 −0.959710 + 1.14374i 0 2.91005 + 2.44183i 0 2.70574 + 0.984808i 0
65.2 0 −0.0603074 + 0.342020i 0 2.07305 − 2.47057i 0 −1.73641 − 1.45702i 0 2.70574 + 0.984808i 0
77.1 0 −1.17365 + 0.984808i 0 −0.794062 + 2.18167i 0 −1.16600 − 0.424388i 0 −0.113341 + 0.642788i 0
77.2 0 −1.17365 + 0.984808i 0 1.38646 − 3.80926i 0 2.93204 + 1.06718i 0 −0.113341 + 0.642788i 0
141.1 0 −1.76604 − 0.642788i 0 −2.94737 + 0.519700i 0 0.757565 + 4.29637i 0 0.407604 + 0.342020i 0
141.2 0 −1.76604 − 0.642788i 0 1.24163 − 0.218933i 0 −0.697258 − 3.95435i 0 0.407604 + 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.2.n.b ✓ 12
3.b odd 2 1 1332.2.ct.c 12
4.b odd 2 1 592.2.bq.c 12
37.h even 18 1 inner 148.2.n.b ✓ 12
37.i odd 36 2 5476.2.a.i 12
111.n odd 18 1 1332.2.ct.c 12
148.o odd 18 1 592.2.bq.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.2.n.b ✓ 12 1.a even 1 1 trivial
148.2.n.b ✓ 12 37.h even 18 1 inner
592.2.bq.c 12 4.b odd 2 1
592.2.bq.c 12 148.o odd 18 1
1332.2.ct.c 12 3.b odd 2 1
1332.2.ct.c 12 111.n odd 18 1
5476.2.a.i 12 37.i odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 6T_{3}^{5} + 15T_{3}^{4} + 19T_{3}^{3} + 12T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(148, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 6 T^{5} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{10} + \cdots + 29241 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots + 341056 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 23409 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 2601 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + \cdots + 29241 \) Copy content Toggle raw display
$19$ \( T^{12} + 9 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{12} + 27 T^{11} + \cdots + 1498176 \) Copy content Toggle raw display
$29$ \( T^{12} - 108 T^{10} + \cdots + 927369 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 145733184 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} - 15 T^{11} + \cdots + 8450649 \) Copy content Toggle raw display
$43$ \( T^{12} + 291 T^{10} + \cdots + 18455616 \) Copy content Toggle raw display
$47$ \( T^{12} + 12 T^{11} + \cdots + 82464561 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 6354162369 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 31012266609 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 17434297521 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 24180561001 \) Copy content Toggle raw display
$71$ \( T^{12} + 24 T^{11} + \cdots + 23707161 \) Copy content Toggle raw display
$73$ \( (T^{6} - 123 T^{4} + \cdots - 14984)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 9606548169 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2300257521 \) Copy content Toggle raw display
$89$ \( (T^{6} - 18 T^{5} + \cdots + 7803)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 18723269889 \) Copy content Toggle raw display
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