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Properties

Label 315.3.h.d
Level $315$
Weight $3$
Character orbit 315.h
Analytic conductor $8.583$
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] = [315,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.181");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(8.58312832735\)
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: \( x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_1 + 4) q^{4} - \beta_{8} q^{5} + (\beta_{9} - \beta_{8} - 1) q^{7} + ( - \beta_{5} - \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_1 + 4) q^{4} - \beta_{8} q^{5} + (\beta_{9} - \beta_{8} - 1) q^{7} + ( - \beta_{5} - \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{8}+ \cdots + ( - 8 \beta_{11} - \beta_{10} + \cdots - 79) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 44 q^{4} - 8 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 44 q^{4} - 8 q^{7} - 4 q^{8} + 16 q^{11} + 40 q^{14} + 92 q^{16} - 88 q^{22} + 64 q^{23} - 60 q^{25} + 88 q^{28} - 104 q^{29} + 228 q^{32} - 60 q^{35} + 32 q^{37} + 152 q^{43} - 192 q^{44} + 200 q^{46} + 60 q^{49} - 20 q^{50} - 176 q^{53} + 368 q^{56} - 400 q^{58} - 20 q^{64} + 240 q^{65} + 168 q^{67} - 60 q^{70} - 32 q^{71} - 184 q^{74} - 8 q^{77} + 120 q^{79} + 120 q^{85} - 400 q^{86} - 536 q^{88} + 24 q^{91} - 192 q^{92} - 884 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1163071964 \nu^{11} + 179172244 \nu^{10} + 20630949354 \nu^{9} + 16671982336 \nu^{8} + \cdots + 2151653525805 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5593516534 \nu^{11} - 2605315656 \nu^{10} + 102990651177 \nu^{9} + 27516350446 \nu^{8} + \cdots + 10342332760284 ) / 2402029152369 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 42289552636 \nu^{11} - 136320072447 \nu^{10} + 936265293410 \nu^{9} + \cdots - 203704095121695 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15660232032 \nu^{11} - 28994320136 \nu^{10} + 329223217160 \nu^{9} - 365904134124 \nu^{8} + \cdots - 16499782051746 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2752740836 \nu^{11} + 4168730751 \nu^{10} + 43893742358 \nu^{9} + 122141383813 \nu^{8} + \cdots + 17534758495653 ) / 731052350721 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 79026253784 \nu^{11} + 146272115177 \nu^{10} - 1671226017089 \nu^{9} + \cdots + 86851646987175 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 82515469676 \nu^{11} - 145734598445 \nu^{10} + 1733118865151 \nu^{9} + \cdots - 97210890476343 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5888 \nu^{11} + 11357 \nu^{10} - 121362 \nu^{9} + 142983 \nu^{8} - 1697439 \nu^{7} + \cdots + 4306806 ) / 1130283 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 91626156422 \nu^{11} - 196270735236 \nu^{10} + 1900808184921 \nu^{9} - 2547333298184 \nu^{8} + \cdots - 2426520744774 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 161873480282 \nu^{11} + 275300339232 \nu^{10} - 3313587375429 \nu^{9} + \cdots + 207714123345456 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 722764520 \nu^{11} - 1406324924 \nu^{10} + 14880207305 \nu^{9} - 17611281862 \nu^{8} + \cdots - 498205455783 ) / 72788762193 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} + 3 \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} + 8 \beta_{4} + \cdots - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} - 6\beta_{2} + 11\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 23 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} - 59 \beta_{8} + 10 \beta_{7} - 27 \beta_{6} + 15 \beta_{5} + \cdots - 138 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 96 \beta_{11} - 48 \beta_{10} + 54 \beta_{9} - 16 \beta_{8} - 137 \beta_{7} - 145 \beta_{6} + \cdots + 117 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13 \beta_{11} - 144 \beta_{10} - 144 \beta_{9} - 13 \beta_{8} + 13 \beta_{6} - 213 \beta_{5} + \cdots + 1644 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1022 \beta_{11} + 762 \beta_{10} - 672 \beta_{9} + 1082 \beta_{8} + 1748 \beta_{7} + 1374 \beta_{6} + \cdots + 1644 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5220 \beta_{11} + 1416 \beta_{10} + 2142 \beta_{9} + 13222 \beta_{8} - 793 \beta_{7} + 4423 \beta_{6} + \cdots - 20703 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3194 \beta_{11} - 429 \beta_{10} - 429 \beta_{9} - 3194 \beta_{8} + 3194 \beta_{6} + 5598 \beta_{5} + \cdots - 24291 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 77615 \beta_{11} + 24843 \beta_{10} + 20103 \beta_{9} - 179477 \beta_{8} + 4867 \beta_{7} + \cdots - 268665 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 312327 \beta_{11} - 124221 \beta_{10} + 128205 \beta_{9} - 218377 \beta_{8} - 292652 \beta_{7} + \cdots + 366186 ) / 4 \) 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
1.31896 − 2.28450i
1.31896 + 2.28450i
0.378061 + 0.654821i
0.378061 − 0.654821i
−1.74681 − 3.02556i
−1.74681 + 3.02556i
−1.01714 + 1.76174i
−1.01714 − 1.76174i
0.198184 − 0.343264i
0.198184 + 0.343264i
1.86875 + 3.23677i
1.86875 − 3.23677i
−3.50369 0 8.27584 − 2.23607i 0 −6.69736 − 2.03600i −14.9812 0 7.83449i
181.2 −3.50369 0 8.27584 2.23607i 0 −6.69736 + 2.03600i −14.9812 0 − 7.83449i
181.3 −2.91758 0 4.51225 − 2.23607i 0 6.13981 − 3.36195i −1.49451 0 6.52390i
181.4 −2.91758 0 4.51225 2.23607i 0 6.13981 + 3.36195i −1.49451 0 − 6.52390i
181.5 0.112974 0 −3.98724 − 2.23607i 0 −6.71303 − 1.98374i −0.902349 0 − 0.252617i
181.6 0.112974 0 −3.98724 2.23607i 0 −6.71303 + 1.98374i −0.902349 0 0.252617i
181.7 1.71214 0 −1.06857 − 2.23607i 0 −3.33344 + 6.15534i −8.67811 0 − 3.82847i
181.8 1.71214 0 −1.06857 2.23607i 0 −3.33344 − 6.15534i −8.67811 0 3.82847i
181.9 2.79155 0 3.79273 − 2.23607i 0 4.15782 − 5.63139i −0.578591 0 − 6.24209i
181.10 2.79155 0 3.79273 2.23607i 0 4.15782 + 5.63139i −0.578591 0 6.24209i
181.11 3.80460 0 10.4750 − 2.23607i 0 2.44621 − 6.55866i 24.6348 0 − 8.50735i
181.12 3.80460 0 10.4750 2.23607i 0 2.44621 + 6.55866i 24.6348 0 8.50735i
\(n\): 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 315.3.h.d 12
3.b odd 2 1 105.3.h.a ✓ 12
7.b odd 2 1 inner 315.3.h.d 12
12.b even 2 1 1680.3.s.c 12
15.d odd 2 1 525.3.h.d 12
15.e even 4 2 525.3.e.c 24
21.c even 2 1 105.3.h.a ✓ 12
84.h odd 2 1 1680.3.s.c 12
105.g even 2 1 525.3.h.d 12
105.k odd 4 2 525.3.e.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.h.a ✓ 12 3.b odd 2 1
105.3.h.a ✓ 12 21.c even 2 1
315.3.h.d 12 1.a even 1 1 trivial
315.3.h.d 12 7.b odd 2 1 inner
525.3.e.c 24 15.e even 4 2
525.3.e.c 24 105.k odd 4 2
525.3.h.d 12 15.d odd 2 1
525.3.h.d 12 105.g even 2 1
1680.3.s.c 12 12.b even 2 1
1680.3.s.c 12 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 2T_{2}^{5} - 21T_{2}^{4} + 40T_{2}^{3} + 103T_{2}^{2} - 198T_{2} + 21 \) acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{5} - 21 T^{4} + \cdots + 21)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} - 8 T^{5} + \cdots + 388416)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1316818944 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 867491057664 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} - 32 T^{5} + \cdots - 11126976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 52 T^{5} + \cdots + 172225344)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{6} - 16 T^{5} + \cdots + 82379584)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{6} - 76 T^{5} + \cdots + 44197696)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} + 88 T^{5} + \cdots + 19593854784)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{6} - 84 T^{5} + \cdots + 35588736064)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 16 T^{5} + \cdots - 12730697664)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{6} - 60 T^{5} + \cdots - 595422656)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 74\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
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