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Properties

Label 148.4.e.a
Level $148$
Weight $4$
Character orbit 148.e
Analytic conductor $8.732$
Analytic rank $0$
Dimension $18$
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] = [148,4,Mod(121,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.121");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 148.e (of order \(3\), degree \(2\), 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: \(8.73228268085\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 158 x^{16} + 143 x^{15} + 16923 x^{14} + 19160 x^{13} + 989213 x^{12} + \cdots + 11209104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{3} + ( - \beta_{7} + \beta_{4}) q^{5} + ( - \beta_{14} - \beta_{9} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{12} + 9 \beta_{4} + \cdots - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{3} + ( - \beta_{7} + \beta_{4}) q^{5} + ( - \beta_{14} - \beta_{9} + \cdots - \beta_{2}) q^{7}+ \cdots + (11 \beta_{15} - 13 \beta_{12} + \cdots - 391) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 8 q^{3} + 13 q^{5} + 8 q^{7} - 79 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 8 q^{3} + 13 q^{5} + 8 q^{7} - 79 q^{9} - 16 q^{11} - 72 q^{13} - 32 q^{15} - 15 q^{17} - 20 q^{19} + 162 q^{21} + 288 q^{23} - 252 q^{25} - 136 q^{27} - 318 q^{29} - 80 q^{31} - 4 q^{33} - 256 q^{35} + 329 q^{37} + 572 q^{39} + 701 q^{41} - 104 q^{43} - 1078 q^{45} + 8 q^{47} - 363 q^{49} - 400 q^{51} + 608 q^{53} + 764 q^{55} + 410 q^{57} + 908 q^{59} + 181 q^{61} - 832 q^{63} + 844 q^{65} - 212 q^{67} - 404 q^{69} - 612 q^{71} + 2092 q^{73} - 3352 q^{75} + 548 q^{77} - 2276 q^{79} + 463 q^{81} + 572 q^{83} + 2586 q^{85} + 1152 q^{87} + 445 q^{89} - 2200 q^{91} - 2028 q^{93} + 884 q^{95} + 3786 q^{97} - 3688 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 158 x^{16} + 143 x^{15} + 16923 x^{14} + 19160 x^{13} + 989213 x^{12} + \cdots + 11209104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\!\cdots\!45 \nu^{17} + \cdots - 13\!\cdots\!16 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34\!\cdots\!11 \nu^{17} + \cdots - 60\!\cdots\!16 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!19 \nu^{17} + \cdots + 16\!\cdots\!36 ) / 31\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 78\!\cdots\!69 \nu^{17} + \cdots + 40\!\cdots\!48 ) / 65\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 76\!\cdots\!67 \nu^{17} + \cdots - 62\!\cdots\!04 ) / 13\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!60 \nu^{17} + \cdots + 53\!\cdots\!00 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26\!\cdots\!33 \nu^{17} + \cdots + 18\!\cdots\!64 ) / 21\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!53 \nu^{17} + \cdots - 92\!\cdots\!92 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 60\!\cdots\!71 \nu^{17} + \cdots + 41\!\cdots\!92 ) / 32\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26\!\cdots\!58 \nu^{17} + \cdots - 18\!\cdots\!88 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79\!\cdots\!89 \nu^{17} + \cdots - 55\!\cdots\!40 ) / 31\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 75\!\cdots\!91 \nu^{17} + \cdots - 11\!\cdots\!68 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 85\!\cdots\!28 \nu^{17} + \cdots - 79\!\cdots\!60 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!73 \nu^{17} + \cdots + 11\!\cdots\!36 ) / 13\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 16\!\cdots\!00 \nu^{17} + \cdots - 64\!\cdots\!48 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{17} + \cdots + 98\!\cdots\!80 ) / 43\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - 35\beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - 2\beta_{3} + 60\beta_{2} - 60\beta _1 - 44 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{15} - 68 \beta_{12} + 5 \beta_{11} + 20 \beta_{10} - 20 \beta_{9} + 5 \beta_{8} + \cdots - 2094 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 24 \beta_{17} - 21 \beta_{16} - 140 \beta_{14} - 104 \beta_{13} - 266 \beta_{12} + \cdots - 4048 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 762 \beta_{17} - 364 \beta_{16} + 762 \beta_{15} - 2192 \beta_{14} - 638 \beta_{13} - 2832 \beta_{10} + \cdots + 140129 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4254 \beta_{15} + 27114 \beta_{12} - 9159 \beta_{11} - 9547 \beta_{10} + 15489 \beta_{9} - 13587 \beta_{8} + \cdots + 609708 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 77268 \beta_{17} + 44636 \beta_{16} + 199056 \beta_{14} + 65175 \beta_{13} + 378656 \beta_{12} + \cdots + 1731308 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 525462 \beta_{17} + 386639 \beta_{16} - 525462 \beta_{15} + 1551860 \beta_{14} + 785330 \beta_{13} + \cdots - 58724465 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7308312 \beta_{15} - 30456517 \beta_{12} + 6200340 \beta_{11} + 29728848 \beta_{10} + \cdots - 790118057 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 56100036 \beta_{17} - 40510308 \beta_{16} - 147627409 \beta_{14} - 67640581 \beta_{13} + \cdots - 1905732108 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 672051738 \beta_{17} - 456555662 \beta_{16} + 672051738 \beta_{15} - 1533534980 \beta_{14} + \cdots + 64012021686 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5563458684 \beta_{15} + 20969526170 \beta_{12} - 5883102764 \beta_{11} - 20948264560 \beta_{10} + \cdots + 494022323997 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 60980981934 \beta_{17} + 42780736936 \beta_{16} + 135320579960 \beta_{14} + \cdots + 1337994771827 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 529384266306 \beta_{17} + 380406257798 \beta_{16} - 529384266306 \beta_{15} + 1240692103761 \beta_{14} + \cdots - 44445780070836 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5494174918056 \beta_{15} - 18668245134056 \beta_{12} + 4678300367979 \beta_{11} + \cdots - 457229912113138 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 49152876883722 \beta_{17} - 35404976934635 \beta_{16} - 111782623528676 \beta_{14} + \cdots - 11\!\cdots\!96 \beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(75\) \(113\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
4.71895 − 8.17347i
3.82793 − 6.63017i
2.58479 − 4.47698i
0.0560431 − 0.0970695i
−0.250111 + 0.433206i
−0.261942 + 0.453696i
−2.83381 + 4.90831i
−3.54624 + 6.14226i
−3.79561 + 6.57420i
4.71895 + 8.17347i
3.82793 + 6.63017i
2.58479 + 4.47698i
0.0560431 + 0.0970695i
−0.250111 − 0.433206i
−0.261942 − 0.453696i
−2.83381 − 4.90831i
−3.54624 − 6.14226i
−3.79561 − 6.57420i
0 −4.21895 − 7.30744i 0 5.02695 + 8.70694i 0 11.5901 + 20.0747i 0 −22.0991 + 38.2768i 0
121.2 0 −3.32793 − 5.76415i 0 −7.11475 − 12.3231i 0 −2.13422 − 3.69658i 0 −8.65027 + 14.9827i 0
121.3 0 −2.08479 − 3.61096i 0 5.08518 + 8.80779i 0 −9.76342 − 16.9107i 0 4.80732 − 8.32652i 0
121.4 0 0.443957 + 0.768956i 0 −6.22194 − 10.7767i 0 10.4932 + 18.1748i 0 13.1058 − 22.6999i 0
121.5 0 0.750111 + 1.29923i 0 0.678675 + 1.17550i 0 −4.90671 − 8.49867i 0 12.3747 − 21.4336i 0
121.6 0 0.761942 + 1.31972i 0 7.47780 + 12.9519i 0 8.33682 + 14.4398i 0 12.3389 − 21.3716i 0
121.7 0 3.33381 + 5.77433i 0 −5.27903 − 9.14355i 0 −18.0575 − 31.2765i 0 −8.72861 + 15.1184i 0
121.8 0 4.04624 + 7.00829i 0 −3.50828 − 6.07652i 0 9.82889 + 17.0241i 0 −19.2441 + 33.3317i 0
121.9 0 4.29561 + 7.44022i 0 10.3554 + 17.9361i 0 −1.38721 − 2.40272i 0 −23.4046 + 40.5380i 0
137.1 0 −4.21895 + 7.30744i 0 5.02695 − 8.70694i 0 11.5901 − 20.0747i 0 −22.0991 − 38.2768i 0
137.2 0 −3.32793 + 5.76415i 0 −7.11475 + 12.3231i 0 −2.13422 + 3.69658i 0 −8.65027 − 14.9827i 0
137.3 0 −2.08479 + 3.61096i 0 5.08518 − 8.80779i 0 −9.76342 + 16.9107i 0 4.80732 + 8.32652i 0
137.4 0 0.443957 − 0.768956i 0 −6.22194 + 10.7767i 0 10.4932 − 18.1748i 0 13.1058 + 22.6999i 0
137.5 0 0.750111 − 1.29923i 0 0.678675 − 1.17550i 0 −4.90671 + 8.49867i 0 12.3747 + 21.4336i 0
137.6 0 0.761942 − 1.31972i 0 7.47780 − 12.9519i 0 8.33682 − 14.4398i 0 12.3389 + 21.3716i 0
137.7 0 3.33381 − 5.77433i 0 −5.27903 + 9.14355i 0 −18.0575 + 31.2765i 0 −8.72861 − 15.1184i 0
137.8 0 4.04624 − 7.00829i 0 −3.50828 + 6.07652i 0 9.82889 − 17.0241i 0 −19.2441 − 33.3317i 0
137.9 0 4.29561 − 7.44022i 0 10.3554 − 17.9361i 0 −1.38721 + 2.40272i 0 −23.4046 − 40.5380i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.4.e.a ✓ 18
3.b odd 2 1 1332.4.j.a 18
37.c even 3 1 inner 148.4.e.a ✓ 18
111.i odd 6 1 1332.4.j.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.4.e.a ✓ 18 1.a even 1 1 trivial
148.4.e.a ✓ 18 37.c even 3 1 inner
1332.4.j.a 18 3.b odd 2 1
1332.4.j.a 18 111.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(148, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 48555003904 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 31\!\cdots\!01 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{9} + 8 T^{8} + \cdots + 474677710848)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 76\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots + 38\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots - 24\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 71\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 21\!\cdots\!33 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 15\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 88\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots + 26\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 94\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 77\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 18\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 75\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 12\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 44\!\cdots\!76)^{2} \) Copy content Toggle raw display
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