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Properties

Label 308.3.h.a
Level $308$
Weight $3$
Character orbit 308.h
Analytic conductor $8.392$
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] = [308,3,Mod(197,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 308.h (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: \(8.39239214230\)
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: \( x^{12} - 4 x^{11} - 47 x^{10} + 200 x^{9} + 869 x^{8} - 3952 x^{7} - 5397 x^{6} + 27740 x^{5} + \cdots + 647488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} + (\beta_{4} - 1) q^{5} + \beta_{5} q^{7} + (\beta_{8} + \beta_{4} - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} + (\beta_{4} - 1) q^{5} + \beta_{5} q^{7} + (\beta_{8} + \beta_{4} - \beta_{3} + 2) q^{9} + ( - \beta_{6} + \beta_{3} + 1) q^{11} + (\beta_{11} + \beta_{5}) q^{13} + (\beta_{9} + \beta_{8} + \beta_{6} + \cdots + 2) q^{15}+ \cdots + ( - 4 \beta_{11} - \beta_{10} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{3} - 6 q^{5} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{3} - 6 q^{5} + 30 q^{9} + 20 q^{11} + 26 q^{15} + 2 q^{23} - 34 q^{25} - 70 q^{27} + 6 q^{31} + 54 q^{33} + 18 q^{37} + 156 q^{45} - 164 q^{47} - 84 q^{49} - 88 q^{53} + 70 q^{55} + 134 q^{59} + 114 q^{67} + 98 q^{69} + 50 q^{71} - 8 q^{75} - 28 q^{77} - 112 q^{81} - 46 q^{89} - 84 q^{91} + 90 q^{93} - 270 q^{97} - 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 47 x^{10} + 200 x^{9} + 869 x^{8} - 3952 x^{7} - 5397 x^{6} + 27740 x^{5} + \cdots + 647488 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9723393847957 \nu^{11} + 350750792675157 \nu^{10} + \cdots + 62\!\cdots\!00 ) / 89\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7980295481731 \nu^{11} + 48766747768459 \nu^{10} - 573744199280756 \nu^{9} + \cdots - 34\!\cdots\!72 ) / 29\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58036717715583 \nu^{11} - 183177773300213 \nu^{10} + \cdots + 14\!\cdots\!72 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 189845301460613 \nu^{11} + 657161203335757 \nu^{10} + \cdots - 15\!\cdots\!08 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 58036717715583 \nu^{11} - 183177773300213 \nu^{10} + \cdots - 59\!\cdots\!68 ) / 74\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 367816299424599 \nu^{11} + \cdots + 27\!\cdots\!04 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81096861252973 \nu^{11} - 250973384954927 \nu^{10} + \cdots - 42\!\cdots\!80 ) / 89\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 483659886833327 \nu^{11} - 839129099962003 \nu^{10} + \cdots - 29\!\cdots\!48 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 84310614959657 \nu^{11} + 338254073152417 \nu^{10} + \cdots - 45\!\cdots\!28 ) / 74\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!43 \nu^{11} + \cdots - 30\!\cdots\!08 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 357234048193799 \nu^{11} + \cdots + 30\!\cdots\!44 ) / 74\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} + \beta_{5} + 2\beta_{4} + 2\beta_{2} + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} - 3 \beta_{10} - 4 \beta_{9} - \beta_{8} + 3 \beta_{7} - 2 \beta_{6} + 32 \beta_{5} + \cdots + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 24 \beta_{8} - 2 \beta_{7} + 14 \beta_{6} + 31 \beta_{5} + \cdots + 155 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 85 \beta_{11} - 115 \beta_{10} - 46 \beta_{9} - 17 \beta_{8} + 125 \beta_{7} - 6 \beta_{6} + 844 \beta_{5} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 240 \beta_{11} + 134 \beta_{10} - 26 \beta_{9} - 108 \beta_{8} + 40 \beta_{7} + 382 \beta_{6} + \cdots - 811 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1953 \beta_{11} - 2625 \beta_{10} + 1008 \beta_{9} + 205 \beta_{8} + 3339 \beta_{7} + 1298 \beta_{6} + \cdots - 2653 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5010 \beta_{11} + 3482 \beta_{10} + 2220 \beta_{9} - 17080 \beta_{8} + 1826 \beta_{7} + 7486 \beta_{6} + \cdots - 114985 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 32169 \beta_{11} - 41673 \beta_{10} + 69572 \beta_{9} + 23335 \beta_{8} + 58167 \beta_{7} + 54542 \beta_{6} + \cdots - 79567 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 58800 \beta_{11} + 49872 \beta_{10} + 105332 \beta_{9} - 641704 \beta_{8} + 27110 \beta_{7} + \cdots - 4370707 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 177749 \beta_{11} - 223751 \beta_{10} + 2240434 \beta_{9} + 850885 \beta_{8} + 359337 \beta_{7} + \cdots - 1708277 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\) \(155\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
4.71781 − 1.32288i
4.71781 + 1.32288i
3.28429 − 1.32288i
3.28429 + 1.32288i
1.86354 − 1.32288i
1.86354 + 1.32288i
−0.689084 − 1.32288i
−0.689084 + 1.32288i
−2.41425 − 1.32288i
−2.41425 + 1.32288i
−4.76230 − 1.32288i
−4.76230 + 1.32288i
0 −5.21781 0 2.57895 0 − 2.64575i 0 18.2256 0
197.2 0 −5.21781 0 2.57895 0 2.64575i 0 18.2256 0
197.3 0 −3.78429 0 −7.35091 0 − 2.64575i 0 5.32083 0
197.4 0 −3.78429 0 −7.35091 0 2.64575i 0 5.32083 0
197.5 0 −2.36354 0 4.17768 0 − 2.64575i 0 −3.41370 0
197.6 0 −2.36354 0 4.17768 0 2.64575i 0 −3.41370 0
197.7 0 0.189084 0 −1.35011 0 − 2.64575i 0 −8.96425 0
197.8 0 0.189084 0 −1.35011 0 2.64575i 0 −8.96425 0
197.9 0 1.91425 0 −5.65031 0 − 2.64575i 0 −5.33565 0
197.10 0 1.91425 0 −5.65031 0 2.64575i 0 −5.33565 0
197.11 0 4.26230 0 4.59470 0 − 2.64575i 0 9.16721 0
197.12 0 4.26230 0 4.59470 0 2.64575i 0 9.16721 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.3.h.a ✓ 12
3.b odd 2 1 2772.3.j.a 12
4.b odd 2 1 1232.3.n.d 12
7.b odd 2 1 2156.3.h.e 12
11.b odd 2 1 inner 308.3.h.a ✓ 12
33.d even 2 1 2772.3.j.a 12
44.c even 2 1 1232.3.n.d 12
77.b even 2 1 2156.3.h.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.3.h.a ✓ 12 1.a even 1 1 trivial
308.3.h.a ✓ 12 11.b odd 2 1 inner
1232.3.n.d 12 4.b odd 2 1
1232.3.n.d 12 44.c even 2 1
2156.3.h.e 12 7.b odd 2 1
2156.3.h.e 12 77.b even 2 1
2772.3.j.a 12 3.b odd 2 1
2772.3.j.a 12 33.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(308, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 5 T^{5} - 22 T^{4} + \cdots - 72)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 3 T^{5} + \cdots - 2776)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 304633348096 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 59708136226816 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 624069146853376 \) Copy content Toggle raw display
$23$ \( (T^{6} - T^{5} + \cdots + 11116512)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 73\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} + \cdots - 356426392)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 9 T^{5} + \cdots - 131584)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{6} + 82 T^{5} + \cdots - 175587408)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 44 T^{5} + \cdots + 6370708736)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 67 T^{5} + \cdots - 178483176)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{6} - 57 T^{5} + \cdots - 960443552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 25 T^{5} + \cdots + 34761056)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{6} + 23 T^{5} + \cdots + 51226216928)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 135 T^{5} + \cdots - 1708727328)^{2} \) Copy content Toggle raw display
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