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Properties

Label 304.7.d.a
Level $304$
Weight $7$
Character orbit 304.d
Analytic conductor $69.936$
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] = [304,7,Mod(191,304)]
 
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([1, 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("304.191");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.d (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: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 9260 x^{16} + 34907494 x^{14} + 69594229756 x^{12} + 79783654678785 x^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{37}\cdot 19^{10} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} - 20) q^{5} + (\beta_{11} + \beta_1) q^{7} + (\beta_{3} + \beta_{2} - 300) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} - 20) q^{5} + (\beta_{11} + \beta_1) q^{7} + (\beta_{3} + \beta_{2} - 300) q^{9} + (\beta_{13} + \beta_{11}) q^{11} + ( - \beta_{5} - 2 \beta_{2} + 126) q^{13} + ( - \beta_{12} + \beta_{11} - 12 \beta_1) q^{15} + ( - \beta_{7} - \beta_{4} - \beta_{3} + \cdots + 46) q^{17}+ \cdots + ( - 6 \beta_{17} - 8 \beta_{16} + \cdots + 3614 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 356 q^{5} - 5398 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 356 q^{5} - 5398 q^{9} + 2268 q^{13} + 820 q^{17} - 20784 q^{21} - 114 q^{25} + 38796 q^{29} - 1968 q^{33} + 69660 q^{37} + 54788 q^{41} - 42884 q^{45} - 84750 q^{49} - 53476 q^{53} + 26268 q^{61} + 589000 q^{65} - 585088 q^{69} - 393612 q^{73} - 1391672 q^{77} + 927922 q^{81} + 350640 q^{85} + 285588 q^{89} - 837872 q^{93} - 89628 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 9260 x^{16} + 34907494 x^{14} + 69594229756 x^{12} + 79783654678785 x^{10} + \cdots + 35\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\!\cdots\!67 \nu^{16} + \cdots + 50\!\cdots\!40 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 65\!\cdots\!67 \nu^{16} + \cdots + 32\!\cdots\!20 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!29 \nu^{16} + \cdots - 66\!\cdots\!80 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!69 \nu^{16} + \cdots - 11\!\cdots\!80 ) / 73\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!27 \nu^{16} + \cdots + 12\!\cdots\!20 ) / 73\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 66\!\cdots\!97 \nu^{16} + \cdots - 10\!\cdots\!80 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!47 \nu^{16} + \cdots - 69\!\cdots\!40 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!99 \nu^{16} + \cdots - 31\!\cdots\!80 ) / 73\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 66\!\cdots\!94 \nu^{17} + \cdots + 98\!\cdots\!20 \nu ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!29 \nu^{17} + \cdots + 14\!\cdots\!00 \nu ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!51 \nu^{17} + \cdots + 21\!\cdots\!60 \nu ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 54\!\cdots\!63 \nu^{17} + \cdots - 10\!\cdots\!00 \nu ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 61\!\cdots\!55 \nu^{17} + \cdots + 86\!\cdots\!60 \nu ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!67 \nu^{17} + \cdots + 60\!\cdots\!80 \nu ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 16\!\cdots\!73 \nu^{17} + \cdots + 10\!\cdots\!20 \nu ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 69\!\cdots\!93 \nu^{17} + \cdots + 61\!\cdots\!40 \nu ) / 50\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 1029 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 2\beta_{14} + 3\beta_{13} - 8\beta_{11} - 1720\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 40 \beta_{9} - 136 \beta_{8} - 94 \beta_{7} + 22 \beta_{6} + 6 \beta_{5} - 18 \beta_{4} + \cdots + 1770675 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 366 \beta_{17} - 114 \beta_{16} - 3093 \beta_{15} - 8422 \beta_{14} - 11645 \beta_{13} + \cdots + 3566360 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 137048 \beta_{9} + 563820 \beta_{8} + 398146 \beta_{7} - 79044 \beta_{6} - 208912 \beta_{5} + \cdots - 3675732999 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1662822 \beta_{17} + 689012 \beta_{16} + 8315187 \beta_{15} + 28831150 \beta_{14} + \cdots - 8123345730 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 385368600 \beta_{9} - 1780158272 \beta_{8} - 1254765214 \beta_{7} + 255750338 \beta_{6} + \cdots + 8380482118143 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5515821894 \beta_{17} - 2605520086 \beta_{16} - 21706941253 \beta_{15} - 90079402398 \beta_{14} + \cdots + 19454342148068 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1034699305976 \beta_{9} + 5139483786548 \beta_{8} + 3581027312450 \beta_{7} - 790882080872 \beta_{6} + \cdots - 20\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 16360318553070 \beta_{17} + 8373528509232 \beta_{16} + 56311125024371 \beta_{15} + \cdots - 47\!\cdots\!54 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 27\!\cdots\!72 \beta_{9} + \cdots + 49\!\cdots\!67 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 46\!\cdots\!54 \beta_{17} + \cdots + 12\!\cdots\!32 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 72\!\cdots\!80 \beta_{9} + \cdots - 12\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 12\!\cdots\!94 \beta_{17} + \cdots - 30\!\cdots\!22 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 19\!\cdots\!28 \beta_{9} + \cdots + 31\!\cdots\!15 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 34\!\cdots\!78 \beta_{17} + \cdots + 78\!\cdots\!36 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
− 51.4224i
− 46.2127i
− 37.2032i
− 32.5617i
− 29.4723i
− 26.1005i
− 16.9079i
− 14.0987i
− 1.13088i
1.13088i
14.0987i
16.9079i
26.1005i
29.4723i
32.5617i
37.2032i
46.2127i
51.4224i
0 − 51.4224i 0 86.8235 0 − 407.754i 0 −1915.26 0
191.2 0 − 46.2127i 0 −207.467 0 100.036i 0 −1406.61 0
191.3 0 − 37.2032i 0 151.911 0 223.398i 0 −655.075 0
191.4 0 − 32.5617i 0 22.3004 0 493.464i 0 −331.265 0
191.5 0 − 29.4723i 0 −144.356 0 − 460.478i 0 −139.619 0
191.6 0 − 26.1005i 0 −59.6866 0 254.555i 0 47.7648 0
191.7 0 − 16.9079i 0 127.458 0 − 413.383i 0 443.122 0
191.8 0 − 14.0987i 0 5.44164 0 − 294.894i 0 530.227 0
191.9 0 − 1.13088i 0 −160.426 0 − 311.200i 0 727.721 0
191.10 0 1.13088i 0 −160.426 0 311.200i 0 727.721 0
191.11 0 14.0987i 0 5.44164 0 294.894i 0 530.227 0
191.12 0 16.9079i 0 127.458 0 413.383i 0 443.122 0
191.13 0 26.1005i 0 −59.6866 0 − 254.555i 0 47.7648 0
191.14 0 29.4723i 0 −144.356 0 460.478i 0 −139.619 0
191.15 0 32.5617i 0 22.3004 0 − 493.464i 0 −331.265 0
191.16 0 37.2032i 0 151.911 0 − 223.398i 0 −655.075 0
191.17 0 46.2127i 0 −207.467 0 − 100.036i 0 −1406.61 0
191.18 0 51.4224i 0 86.8235 0 407.754i 0 −1915.26 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.18
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 304.7.d.a ✓ 18
4.b odd 2 1 inner 304.7.d.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.7.d.a ✓ 18 1.a even 1 1 trivial
304.7.d.a ✓ 18 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 9260 T_{3}^{16} + 34907494 T_{3}^{14} + 69594229756 T_{3}^{12} + 79783654678785 T_{3}^{10} + \cdots + 35\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{9} + \cdots - 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{9} + \cdots + 21\!\cdots\!20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 86\!\cdots\!30)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2476099)^{9} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 73\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots + 27\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 58\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 57\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 28\!\cdots\!30)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots - 15\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
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