Properties

Label 304.7.e.e
Level $304$
Weight $7$
Character orbit 304.e
Analytic conductor $69.936$
Analytic rank $0$
Dimension $10$
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] = [304,7,Mod(113,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not 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: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 38)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + ( - \beta_1 - 11) q^{5} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 23) q^{7} + ( - 3 \beta_{4} - 4 \beta_{3} + \beta_{2} - \beta_1 - 288) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + ( - \beta_1 - 11) q^{5} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 23) q^{7} + ( - 3 \beta_{4} - 4 \beta_{3} + \beta_{2} - \beta_1 - 288) q^{9} + ( - \beta_{4} + 4 \beta_{3} - 6 \beta_{2} - 6 \beta_1 - 364) q^{11} + ( - 2 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + 8 \beta_{5}) q^{13} + ( - \beta_{9} + 3 \beta_{8} - 27 \beta_{6} + 42 \beta_{5}) q^{15} + ( - 15 \beta_{4} + 12 \beta_{3} + \beta_{2} + 18 \beta_1 - 1048) q^{17} + ( - \beta_{9} + 5 \beta_{8} + 3 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + \cdots + 1719) q^{19}+ \cdots + (69 \beta_{4} - 8650 \beta_{3} - 548 \beta_{2} + 13226 \beta_1 - 107058) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9} - 3644 q^{11} - 10420 q^{17} + 17230 q^{19} - 37712 q^{23} - 52078 q^{25} + 161720 q^{35} + 78876 q^{39} - 6308 q^{43} + 309808 q^{45} - 322220 q^{47} - 235770 q^{49} + 377880 q^{55} + 24228 q^{57} + 426304 q^{61} + 517916 q^{63} - 786076 q^{73} + 2303716 q^{77} + 5261090 q^{81} + 101500 q^{83} - 1261380 q^{85} + 2460732 q^{87} - 2827032 q^{93} - 3106292 q^{95} - 1061428 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1003849 \nu^{8} - 4834130050 \nu^{6} - 6581721953521 \nu^{4} + \cdots - 10\!\cdots\!20 ) / 486640529977344 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3036701 \nu^{8} - 14052901482 \nu^{6} - 16755790470373 \nu^{4} + \cdots + 32\!\cdots\!16 ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 80563009 \nu^{8} - 413005498610 \nu^{6} - 584839838800809 \nu^{4} + \cdots - 57\!\cdots\!20 ) / 27\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 94415663 \nu^{8} + 484279228654 \nu^{6} + 700898949271623 \nu^{4} + \cdots + 29\!\cdots\!84 ) / 27\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 128881 \nu^{9} - 758922410 \nu^{7} - 1584249446409 \nu^{5} + \cdots - 46\!\cdots\!84 \nu ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 128881 \nu^{9} - 758922410 \nu^{7} - 1584249446409 \nu^{5} + \cdots - 20\!\cdots\!24 \nu ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 124220604299 \nu^{9} - 619692963258790 \nu^{7} + \cdots - 57\!\cdots\!96 \nu ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50824099611 \nu^{9} - 266986461529670 \nu^{7} + \cdots - 16\!\cdots\!64 \nu ) / 86\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16319424587 \nu^{9} - 80878018999270 \nu^{7} + \cdots - 17\!\cdots\!68 \nu ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - 24\beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -3\beta_{4} - 3\beta_{3} + \beta_{2} - 2\beta _1 - 1009 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{9} - 87\beta_{7} - 413\beta_{6} + 12807\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7515\beta_{4} + 7854\beta_{3} - 1885\beta_{2} + 3743\beta _1 + 2156385 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -24879\beta_{9} + 21540\beta_{8} + 221829\beta_{7} + 585421\beta_{6} - 29424777\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17515695\beta_{4} - 19389633\beta_{3} + 4436769\beta_{2} - 7253208\beta _1 - 4953548237 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 66728061\beta_{9} - 124185672\beta_{8} - 523404171\beta_{7} - 728121863\beta_{6} + 68153767635\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40590401331\beta_{4} + 47391878700\beta_{3} - 10844673557\beta_{2} + 13578744841\beta _1 + 11468377737353 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 177182073387 \beta_{9} + 466496662860 \beta_{8} + 1225986883473 \beta_{7} + 322329106801 \beta_{6} - 158100274392957 \beta_{5} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
48.7374i
47.4187i
11.5377i
16.6843i
3.82791i
3.82791i
16.6843i
11.5377i
47.4187i
48.7374i
0 51.5658i 0 39.9185 0 25.1565 0 −1930.03 0
113.2 0 44.5903i 0 −145.013 0 −126.579 0 −1259.29 0
113.3 0 14.3661i 0 −88.1981 0 −443.109 0 522.615 0
113.4 0 13.8558i 0 −9.64927 0 472.908 0 537.016 0
113.5 0 6.65634i 0 146.942 0 183.624 0 684.693 0
113.6 0 6.65634i 0 146.942 0 183.624 0 684.693 0
113.7 0 13.8558i 0 −9.64927 0 472.908 0 537.016 0
113.8 0 14.3661i 0 −88.1981 0 −443.109 0 522.615 0
113.9 0 44.5903i 0 −145.013 0 −126.579 0 −1259.29 0
113.10 0 51.5658i 0 39.9185 0 25.1565 0 −1930.03 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.7.e.e 10
4.b odd 2 1 38.7.b.a 10
12.b even 2 1 342.7.d.a 10
19.b odd 2 1 inner 304.7.e.e 10
76.d even 2 1 38.7.b.a 10
228.b odd 2 1 342.7.d.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.7.b.a 10 4.b odd 2 1
38.7.b.a 10 76.d even 2 1
304.7.e.e 10 1.a even 1 1 trivial
304.7.e.e 10 19.b odd 2 1 inner
342.7.d.a 10 12.b even 2 1
342.7.d.a 10 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{10} + 5090T_{3}^{8} + 7401465T_{3}^{6} + 2608321032T_{3}^{4} + 310957885440T_{3}^{2} + 9281499955200 \) Copy content Toggle raw display
\( T_{5}^{5} + 56T_{5}^{4} - 24475T_{5}^{3} - 1262450T_{5}^{2} + 65160000T_{5} + 723900000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 5090 T^{8} + \cdots + 9281499955200 \) Copy content Toggle raw display
$5$ \( (T^{5} + 56 T^{4} - 24475 T^{3} + \cdots + 723900000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} - 112 T^{4} + \cdots - 122526279250)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 1822 T^{4} + \cdots + 73\!\cdots\!80)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 30601962 T^{8} + \cdots + 25\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( (T^{5} + 5210 T^{4} + \cdots + 12\!\cdots\!70)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 17230 T^{9} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{5} + 18856 T^{4} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 3652502562 T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + 6466714608 T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + 28355255280 T^{8} + \cdots + 78\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{10} + 33767398128 T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + 3154 T^{4} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} + 161110 T^{4} + \cdots + 52\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 93419714370 T^{8} + \cdots + 55\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{10} + 193990727946 T^{8} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} - 213152 T^{4} + \cdots + 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 508295939058 T^{8} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} + 574923916200 T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + 393038 T^{4} + \cdots + 13\!\cdots\!50)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 867503738544 T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{5} - 50750 T^{4} + \cdots - 54\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 1573209139080 T^{8} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + 2670627879816 T^{8} + \cdots + 71\!\cdots\!08 \) Copy content Toggle raw display
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