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Properties

Label 336.7.m.a
Level $336$
Weight $7$
Character orbit 336.m
Analytic conductor $77.298$
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] = [336,7,Mod(127,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.127");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.m (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: \(77.2981720963\)
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: \( x^{12} - x^{11} + 759 x^{10} + 1504 x^{9} + 438301 x^{8} + 1012329 x^{7} + 103515094 x^{6} + \cdots + 1869265225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2}\cdot 5^{2}\cdot 7^{6} \)
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.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + ( - \beta_{2} - 31) q^{5} + \beta_{7} q^{7} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + ( - \beta_{2} - 31) q^{5} + \beta_{7} q^{7} - 243 q^{9} + (3 \beta_{11} + \beta_{10} + \cdots - 71 \beta_1) q^{11}+ \cdots + ( - 729 \beta_{11} + \cdots + 17253 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 376 q^{5} - 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 376 q^{5} - 2916 q^{9} - 2040 q^{13} - 2792 q^{17} - 1932 q^{25} - 25720 q^{29} + 22464 q^{33} + 74664 q^{37} + 176056 q^{41} + 91368 q^{45} - 201684 q^{49} + 95912 q^{53} + 445824 q^{57} - 359928 q^{61} - 1155920 q^{65} - 196128 q^{69} - 275400 q^{73} - 208544 q^{77} + 708588 q^{81} + 360480 q^{85} - 1913000 q^{89} + 778896 q^{93} - 1430856 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 759 x^{10} + 1504 x^{9} + 438301 x^{8} + 1012329 x^{7} + 103515094 x^{6} + \cdots + 1869265225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 62\!\cdots\!21 \nu^{11} + \cdots - 60\!\cdots\!25 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!05 \nu^{11} + \cdots + 12\!\cdots\!35 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 85\!\cdots\!11 \nu^{11} + \cdots + 29\!\cdots\!75 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!89 \nu^{11} + \cdots - 40\!\cdots\!69 ) / 18\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28\!\cdots\!25 \nu^{11} + \cdots - 63\!\cdots\!95 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!19 \nu^{11} + \cdots - 30\!\cdots\!75 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59851042068771 \nu^{11} + 59847562714852 \nu^{10} + \cdots + 59\!\cdots\!75 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 92\!\cdots\!03 \nu^{11} + \cdots - 85\!\cdots\!75 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!09 \nu^{11} + \cdots - 20\!\cdots\!25 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!05 \nu^{11} + \cdots - 28\!\cdots\!25 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 59\!\cdots\!35 \nu^{11} + \cdots - 58\!\cdots\!75 ) / 18\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 168 \beta_{11} - 357 \beta_{10} + 63 \beta_{9} - 42 \beta_{8} + 33 \beta_{7} + 63 \beta_{6} + \cdots + 2044 ) / 23520 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3948 \beta_{11} - 1533 \beta_{10} - 15813 \beta_{9} - 7098 \beta_{8} - 14883 \beta_{7} + \cdots - 2976736 ) / 23520 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6279\beta_{6} - 2212\beta_{5} - 7768\beta_{4} - 35091\beta_{3} - 31283\beta_{2} - 1663592 ) / 2940 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 526428 \beta_{11} + 221613 \beta_{10} + 2692613 \beta_{9} + 855498 \beta_{8} + 2775603 \beta_{7} + \cdots - 399280336 ) / 7840 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6228768 \beta_{11} + 11655147 \beta_{10} - 13368873 \beta_{9} + 3269742 \beta_{8} + \cdots + 517198276 ) / 4704 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 19269621 \beta_{6} + 20580763 \beta_{5} - 69842648 \beta_{4} + 15176259 \beta_{3} - 202344583 \beta_{2} + 31755973088 ) / 735 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2187932712 \beta_{11} - 3557621823 \beta_{10} + 5200224477 \beta_{9} - 1068954558 \beta_{8} + \cdots + 92586234916 ) / 3360 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 101814376356 \beta_{11} - 25386144651 \beta_{10} - 614693172051 \beta_{9} - 133928334246 \beta_{8} + \cdots - 73685578386112 ) / 7840 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 273590547363 \beta_{6} - 253889820464 \beta_{5} - 2317983889376 \beta_{4} - 2706519370527 \beta_{3} + \cdots - 581674226644 ) / 2940 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 27980618628228 \beta_{11} + 3639106065063 \beta_{10} + 173366195167263 \beta_{9} + \cdots - 19\!\cdots\!00 ) / 4704 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 36\!\cdots\!12 \beta_{11} + \cdots - 13\!\cdots\!64 ) / 23520 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.0459650 − 0.0796137i
10.0063 − 17.3314i
−7.05681 + 12.2227i
−1.84604 + 3.19744i
−10.9479 + 18.9623i
10.2985 − 17.8375i
0.0459650 + 0.0796137i
10.0063 + 17.3314i
−7.05681 − 12.2227i
−1.84604 − 3.19744i
−10.9479 − 18.9623i
10.2985 + 17.8375i
0 − 15.5885i 0 −146.777 0 129.642i 0 −243.000 0
127.2 0 − 15.5885i 0 −145.699 0 129.642i 0 −243.000 0
127.3 0 − 15.5885i 0 −127.553 0 − 129.642i 0 −243.000 0
127.4 0 − 15.5885i 0 −25.2014 0 − 129.642i 0 −243.000 0
127.5 0 − 15.5885i 0 125.154 0 129.642i 0 −243.000 0
127.6 0 − 15.5885i 0 132.076 0 − 129.642i 0 −243.000 0
127.7 0 15.5885i 0 −146.777 0 − 129.642i 0 −243.000 0
127.8 0 15.5885i 0 −145.699 0 − 129.642i 0 −243.000 0
127.9 0 15.5885i 0 −127.553 0 129.642i 0 −243.000 0
127.10 0 15.5885i 0 −25.2014 0 129.642i 0 −243.000 0
127.11 0 15.5885i 0 125.154 0 − 129.642i 0 −243.000 0
127.12 0 15.5885i 0 132.076 0 129.642i 0 −243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.m.a ✓ 12
4.b odd 2 1 inner 336.7.m.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.7.m.a ✓ 12 1.a even 1 1 trivial
336.7.m.a ✓ 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 188T_{5}^{5} - 28720T_{5}^{4} - 6253600T_{5}^{3} + 131746400T_{5}^{2} + 51855776000T_{5} + 1136311960000 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 243)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 1136311960000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16807)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 56\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 32\!\cdots\!60)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 92\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 16\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 76\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
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