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Properties

Label 304.5.e.f
Level $304$
Weight $5$
Character orbit 304.e
Analytic conductor $31.424$
Analytic rank $0$
Dimension $20$
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] = [304,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(31.4244687775\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{50} \)
Twist minimal: no (minimal twist has level 152)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - 2) q^{7} + ( - \beta_{3} + \beta_{2} - 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - 2) q^{7} + ( - \beta_{3} + \beta_{2} - 19) q^{9} + ( - \beta_{6} - \beta_{3} + 1) q^{11} + \beta_{13} q^{13} + (\beta_{10} - 4 \beta_1) q^{15} + ( - \beta_{7} - \beta_{3} + 11) q^{17} + ( - \beta_{16} + \beta_{3} + 30) q^{19} + ( - \beta_{12} + 2 \beta_1) q^{21} + ( - \beta_{9} - 3 \beta_{3} - 29) q^{23} + ( - \beta_{8} + \beta_{6} + \beta_{4} + \cdots + 71) q^{25}+ \cdots + ( - 4 \beta_{16} - 4 \beta_{15} + \cdots - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 32 q^{7} - 372 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57} + 352 q^{61} - 6104 q^{63} + 1352 q^{73} + 9288 q^{77} - 4220 q^{81} + 16104 q^{83} + 10232 q^{85} - 2936 q^{87} + 36432 q^{93} - 14232 q^{95} - 760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + \cdots + 34\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 70\!\cdots\!53 \nu^{18} + \cdots + 76\!\cdots\!72 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70\!\cdots\!53 \nu^{18} + \cdots + 70\!\cdots\!72 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!71 \nu^{18} + \cdots + 58\!\cdots\!08 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64\!\cdots\!81 \nu^{18} + \cdots - 93\!\cdots\!96 ) / 30\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!99 \nu^{18} + \cdots - 39\!\cdots\!40 ) / 89\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!25 \nu^{18} + \cdots + 38\!\cdots\!16 ) / 28\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 67\!\cdots\!57 \nu^{18} + \cdots - 32\!\cdots\!60 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!95 \nu^{18} + \cdots - 24\!\cdots\!56 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70\!\cdots\!53 \nu^{19} + \cdots - 70\!\cdots\!44 \nu ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47\!\cdots\!39 \nu^{19} + \cdots - 33\!\cdots\!68 \nu ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64\!\cdots\!81 \nu^{19} + \cdots + 94\!\cdots\!08 \nu ) / 30\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!19 \nu^{19} + \cdots + 19\!\cdots\!12 \nu ) / 51\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!93 \nu^{19} + \cdots + 18\!\cdots\!40 \nu ) / 48\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!75 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 34\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!75 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 34\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 37\!\cdots\!53 \nu^{19} + \cdots - 60\!\cdots\!24 \nu ) / 26\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!41 \nu^{19} + \cdots + 15\!\cdots\!88 \nu ) / 58\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 15\!\cdots\!19 \nu^{19} + \cdots - 24\!\cdots\!12 \nu ) / 51\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} - \beta_{15} + \beta_{12} - 175\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{9} + 10 \beta_{7} - 22 \beta_{6} + 46 \beta_{5} - 6 \beta_{4} + \cdots + 17500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{19} + 28 \beta_{18} - 241 \beta_{17} - 291 \beta_{16} + 291 \beta_{15} - 182 \beta_{14} + \cdots + 35457 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 234 \beta_{16} + 234 \beta_{15} - 898 \beta_{9} + 146 \beta_{8} - 3504 \beta_{7} + 7428 \beta_{6} + \cdots - 3550438 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7980 \beta_{19} - 11004 \beta_{18} + 51293 \beta_{17} + 76025 \beta_{16} - 76025 \beta_{15} + \cdots - 7612981 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19822 \beta_{16} + 19822 \beta_{15} + 282914 \beta_{9} - 51140 \beta_{8} + 1007638 \beta_{7} + \cdots + 763911296 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3084394 \beta_{19} + 3297452 \beta_{18} - 10737017 \beta_{17} - 19356063 \beta_{16} + \cdots + 1682376793 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20655942 \beta_{16} - 20655942 \beta_{15} - 78374274 \beta_{9} + 12438198 \beta_{8} + \cdots - 169264445898 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1002420040 \beta_{19} - 904682252 \beta_{18} + 2263300229 \beta_{17} + 4867692981 \beta_{16} + \cdots - 378339049933 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8124839966 \beta_{16} + 8124839966 \beta_{15} + 20445235362 \beta_{9} - 2525244936 \beta_{8} + \cdots + 38178353707492 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 296492631078 \beta_{19} + 239102205084 \beta_{18} - 483370031553 \beta_{17} + \cdots + 86120707333457 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2588688905206 \beta_{16} - 2588688905206 \beta_{15} - 5162871005442 \beta_{9} + 434472103386 \beta_{8} + \cdots - 87\!\cdots\!06 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 82864254888292 \beta_{19} - 62001344303324 \beta_{18} + 104708928870253 \beta_{17} + \cdots - 19\!\cdots\!17 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 753426470023374 \beta_{16} + 753426470023374 \beta_{15} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 22\!\cdots\!26 \beta_{19} + \cdots + 45\!\cdots\!89 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 20\!\cdots\!02 \beta_{16} + \cdots - 46\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 58\!\cdots\!48 \beta_{19} + \cdots - 10\!\cdots\!09 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
− 15.6226i
− 14.8343i
− 13.8615i
− 11.4302i
− 9.35107i
− 6.69459i
− 6.40102i
− 5.59979i
− 1.51452i
− 1.48358i
1.48358i
1.51452i
5.59979i
6.40102i
6.69459i
9.35107i
11.4302i
13.8615i
14.8343i
15.6226i
0 − 15.6226i 0 6.30407 0 28.3216 0 −163.066 0
113.2 0 − 14.8343i 0 −45.4605 0 2.86483 0 −139.057 0
113.3 0 − 13.8615i 0 16.6305 0 34.1005 0 −111.141 0
113.4 0 − 11.4302i 0 −9.10929 0 −75.9168 0 −49.6505 0
113.5 0 − 9.35107i 0 21.0112 0 9.13929 0 −6.44248 0
113.6 0 − 6.69459i 0 2.80752 0 −75.3813 0 36.1825 0
113.7 0 − 6.40102i 0 47.0652 0 −32.2009 0 40.0269 0
113.8 0 − 5.59979i 0 −33.0135 0 43.4389 0 49.6423 0
113.9 0 − 1.51452i 0 15.8064 0 74.7269 0 78.7062 0
113.10 0 − 1.48358i 0 −22.0416 0 −25.0930 0 78.7990 0
113.11 0 1.48358i 0 −22.0416 0 −25.0930 0 78.7990 0
113.12 0 1.51452i 0 15.8064 0 74.7269 0 78.7062 0
113.13 0 5.59979i 0 −33.0135 0 43.4389 0 49.6423 0
113.14 0 6.40102i 0 47.0652 0 −32.2009 0 40.0269 0
113.15 0 6.69459i 0 2.80752 0 −75.3813 0 36.1825 0
113.16 0 9.35107i 0 21.0112 0 9.13929 0 −6.44248 0
113.17 0 11.4302i 0 −9.10929 0 −75.9168 0 −49.6505 0
113.18 0 13.8615i 0 16.6305 0 34.1005 0 −111.141 0
113.19 0 14.8343i 0 −45.4605 0 2.86483 0 −139.057 0
113.20 0 15.6226i 0 6.30407 0 28.3216 0 −163.066 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.20
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 304.5.e.f 20
4.b odd 2 1 152.5.e.a ✓ 20
12.b even 2 1 1368.5.o.a 20
19.b odd 2 1 inner 304.5.e.f 20
76.d even 2 1 152.5.e.a ✓ 20
228.b odd 2 1 1368.5.o.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.5.e.a ✓ 20 4.b odd 2 1
152.5.e.a ✓ 20 76.d even 2 1
304.5.e.f 20 1.a even 1 1 trivial
304.5.e.f 20 19.b odd 2 1 inner
1368.5.o.a 20 12.b even 2 1
1368.5.o.a 20 228.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 \(S_{5}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{20} + 996 T_{3}^{18} + 408854 T_{3}^{16} + 89661524 T_{3}^{14} + 11414409521 T_{3}^{12} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
\( T_{5}^{10} - 3478 T_{5}^{8} + 6348 T_{5}^{7} + 3312217 T_{5}^{6} - 21809988 T_{5}^{5} + \cdots + 1386394827776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 1386394827776)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 379548178285972)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 92\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 92\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 33\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 62\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 33\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 80\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
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