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Properties

Label 315.5.h.a
Level $315$
Weight $5$
Character orbit 315.h
Analytic conductor $32.562$
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] = [315,5,Mod(181,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.181");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(32.5615383714\)
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: \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 11) q^{4} - \beta_{5} q^{5} + ( - \beta_{10} - 4) q^{7} + (\beta_{9} + 2 \beta_{2} - 8 \beta_1 + 20) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 11) q^{4} - \beta_{5} q^{5} + ( - \beta_{10} - 4) q^{7} + (\beta_{9} + 2 \beta_{2} - 8 \beta_1 + 20) q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{10} + (\beta_{11} + 2 \beta_{10} - \beta_{9} + \cdots - 5) q^{11}+ \cdots + ( - 6 \beta_{11} + 28 \beta_{10} + \cdots - 5483) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 122 q^{4} - 50 q^{7} + 186 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 122 q^{4} - 50 q^{7} + 186 q^{8} - 126 q^{11} - 78 q^{14} + 578 q^{16} + 2264 q^{22} + 756 q^{23} - 1500 q^{25} + 1414 q^{28} + 2190 q^{29} + 8682 q^{32} + 150 q^{35} + 5564 q^{37} + 3944 q^{43} + 11196 q^{44} + 7844 q^{46} - 8796 q^{49} - 750 q^{50} - 11760 q^{53} + 6606 q^{56} - 18496 q^{58} + 20146 q^{64} + 750 q^{65} - 24096 q^{67} + 14400 q^{70} + 5664 q^{71} - 17604 q^{74} - 26904 q^{77} - 1590 q^{79} + 1050 q^{85} - 17604 q^{86} - 7268 q^{88} - 7182 q^{91} + 60252 q^{92} + 3000 q^{95} - 57714 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 545049904483 \nu^{11} + 3342865644154 \nu^{10} + 54129391887267 \nu^{9} + \cdots - 93\!\cdots\!00 ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 230060779665 \nu^{11} - 4634987015638 \nu^{10} - 6235493671465 \nu^{9} + \cdots - 88\!\cdots\!20 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22050672377839 \nu^{11} - 167967216649588 \nu^{10} + \cdots + 17\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8559661080213 \nu^{11} - 144006533906196 \nu^{10} - 392019044756793 \nu^{9} + \cdots + 11\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 545049904483 \nu^{11} - 3342865644154 \nu^{10} - 54129391887267 \nu^{9} + \cdots + 93\!\cdots\!00 ) / 32\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2017661537865 \nu^{11} + 30477299744326 \nu^{10} - 339396147765681 \nu^{9} + \cdots - 58\!\cdots\!80 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10105263298801 \nu^{11} - 7668759064702 \nu^{10} + \cdots + 10\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 85744361226611 \nu^{11} + 759213715715702 \nu^{10} + \cdots - 39\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 775009347434 \nu^{11} + 4971706132333 \nu^{10} + 90922162125716 \nu^{9} + \cdots - 17\!\cdots\!20 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19651585016711 \nu^{11} + 155617378754012 \nu^{10} + \cdots - 53\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15067388428199 \nu^{11} + 153255880811244 \nu^{10} + \cdots - 54\!\cdots\!20 ) / 10\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 5\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 2\beta_{3} + 5\beta_{2} + 5\beta _1 + 105 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{10} - 5 \beta_{9} + 6 \beta_{8} - 3 \beta_{6} + 70 \beta_{5} - 6 \beta_{4} - 21 \beta_{3} + \cdots + 135 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 52 \beta_{10} - 5 \beta_{9} - 36 \beta_{8} - 12 \beta_{6} + 120 \beta_{5} - 116 \beta_{4} + 188 \beta_{3} + \cdots + 1530 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 49 \beta_{8} + 31 \beta_{7} - 52 \beta_{6} + 760 \beta_{5} + \cdots + 594 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 120 \beta_{11} + 2540 \beta_{10} - 205 \beta_{9} - 3060 \beta_{8} - 1080 \beta_{7} - 1020 \beta_{6} + \cdots - 102240 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 880 \beta_{11} + 10007 \beta_{10} + 10840 \beta_{9} + 49 \beta_{8} + 3595 \beta_{7} - 13762 \beta_{6} + \cdots - 288660 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3480 \beta_{11} + 5364 \beta_{10} + 26265 \beta_{9} - 147552 \beta_{8} - 132300 \beta_{7} + \cdots - 13655640 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 258240 \beta_{11} - 224427 \beta_{10} + 1164820 \beta_{9} - 480129 \beta_{8} - 593715 \beta_{7} + \cdots - 43435800 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 268872 \beta_{11} - 1718332 \beta_{10} + 860303 \beta_{9} - 552840 \beta_{8} - 2006412 \beta_{7} + \cdots - 191284272 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 20929760 \beta_{11} - 64688969 \beta_{10} + 75805660 \beta_{9} - 23187043 \beta_{8} - 71344585 \beta_{7} + \cdots - 3588162480 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
7.50299 + 2.23607i
7.50299 − 2.23607i
5.81769 + 2.23607i
5.81769 − 2.23607i
2.17355 + 2.23607i
2.17355 − 2.23607i
−2.23331 + 2.23607i
−2.23331 − 2.23607i
−3.62973 + 2.23607i
−3.62973 − 2.23607i
−6.63119 + 2.23607i
−6.63119 − 2.23607i
−6.50299 0 26.2889 − 11.1803i 0 −19.9894 − 44.7373i −66.9085 0 72.7056i
181.2 −6.50299 0 26.2889 11.1803i 0 −19.9894 + 44.7373i −66.9085 0 − 72.7056i
181.3 −4.81769 0 7.21016 − 11.1803i 0 44.8989 − 19.6236i 42.3467 0 53.8634i
181.4 −4.81769 0 7.21016 11.1803i 0 44.8989 + 19.6236i 42.3467 0 − 53.8634i
181.5 −1.17355 0 −14.6228 − 11.1803i 0 −40.5002 + 27.5814i 35.9373 0 13.1207i
181.6 −1.17355 0 −14.6228 11.1803i 0 −40.5002 − 27.5814i 35.9373 0 − 13.1207i
181.7 3.23331 0 −5.54568 − 11.1803i 0 −26.6804 + 41.0994i −69.6640 0 − 36.1495i
181.8 3.23331 0 −5.54568 11.1803i 0 −26.6804 − 41.0994i −69.6640 0 36.1495i
181.9 4.62973 0 5.43443 − 11.1803i 0 15.2405 − 46.5696i −48.9158 0 − 51.7620i
181.10 4.62973 0 5.43443 11.1803i 0 15.2405 + 46.5696i −48.9158 0 51.7620i
181.11 7.63119 0 42.2350 − 11.1803i 0 2.03056 + 48.9579i 200.204 0 − 85.3193i
181.12 7.63119 0 42.2350 11.1803i 0 2.03056 − 48.9579i 200.204 0 85.3193i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.h.a 12
3.b odd 2 1 35.5.d.a ✓ 12
7.b odd 2 1 inner 315.5.h.a 12
12.b even 2 1 560.5.f.b 12
15.d odd 2 1 175.5.d.i 12
15.e even 4 2 175.5.c.d 24
21.c even 2 1 35.5.d.a ✓ 12
84.h odd 2 1 560.5.f.b 12
105.g even 2 1 175.5.d.i 12
105.k odd 4 2 175.5.c.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.d.a ✓ 12 3.b odd 2 1
35.5.d.a ✓ 12 21.c even 2 1
175.5.c.d 24 15.e even 4 2
175.5.c.d 24 105.k odd 4 2
175.5.d.i 12 15.d odd 2 1
175.5.d.i 12 105.g even 2 1
315.5.h.a 12 1.a even 1 1 trivial
315.5.h.a 12 7.b odd 2 1 inner
560.5.f.b 12 12.b even 2 1
560.5.f.b 12 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} - 74T_{2}^{4} + 168T_{2}^{3} + 1348T_{2}^{2} - 2340T_{2} - 4200 \) acting on \(S_{5}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 3 T^{5} + \cdots - 4200)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 5924759078688)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 449297249971200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 58\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
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