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Properties

Label 315.5.h.c
Level $315$
Weight $5$
Character orbit 315.h
Analytic conductor $32.562$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 168 x^{18} + 13470 x^{16} - 617180 x^{14} + 17491185 x^{12} - 288192948 x^{10} + \cdots + 11797576257600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{8}\cdot 5^{7} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + (\beta_1 + 6) q^{4} - \beta_{11} q^{5} + ( - \beta_{6} + 1) q^{7} + ( - \beta_{13} - 3 \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + (\beta_1 + 6) q^{4} - \beta_{11} q^{5} + ( - \beta_{6} + 1) q^{7} + ( - \beta_{13} - 3 \beta_{10}) q^{8} + \beta_{2} q^{10} + ( - \beta_{14} - \beta_{13} + 2 \beta_{10}) q^{11} + ( - \beta_{9} + \beta_{6} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - 64 \beta_{18} + \cdots - 23 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 116 q^{4} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 116 q^{4} + 24 q^{7} - 660 q^{16} - 848 q^{22} - 2500 q^{25} + 1984 q^{28} - 2160 q^{37} - 9480 q^{43} + 11104 q^{46} + 7340 q^{49} - 2792 q^{58} - 40380 q^{64} + 21160 q^{67} + 3300 q^{70} - 6776 q^{79} - 6600 q^{85} + 54272 q^{88} + 46320 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 168 x^{18} + 13470 x^{16} - 617180 x^{14} + 17491185 x^{12} - 288192948 x^{10} + \cdots + 11797576257600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 52\!\cdots\!26 \nu^{18} + \cdots - 44\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 52\!\cdots\!26 \nu^{18} + \cdots + 65\!\cdots\!00 ) / 89\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!39 \nu^{18} + \cdots - 28\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!10 \nu^{18} + \cdots - 38\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!11 \nu^{18} + \cdots - 25\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!57 \nu^{18} + \cdots + 22\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35\!\cdots\!01 \nu^{18} + \cdots - 93\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 54\!\cdots\!13 \nu^{18} + \cdots - 37\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!09 \nu^{18} + \cdots + 27\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 54\!\cdots\!53 \nu^{19} + \cdots + 27\!\cdots\!00 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 54\!\cdots\!53 \nu^{19} + \cdots + 15\!\cdots\!00 \nu ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!14 \nu^{19} + \cdots - 63\!\cdots\!80 \nu ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 22\!\cdots\!54 \nu^{19} + \cdots - 14\!\cdots\!00 \nu ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 52\!\cdots\!49 \nu^{19} + \cdots - 15\!\cdots\!00 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!57 \nu^{19} + \cdots - 21\!\cdots\!00 \nu ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 70\!\cdots\!39 \nu^{19} + \cdots + 59\!\cdots\!00 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 76\!\cdots\!13 \nu^{19} + \cdots - 15\!\cdots\!00 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 20\!\cdots\!53 \nu^{19} + \cdots + 11\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 43\!\cdots\!47 \nu^{19} + \cdots - 33\!\cdots\!00 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 5\beta_{10} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + 5\beta _1 + 85 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{13} + 15\beta_{12} + 64\beta_{11} + 100\beta_{10} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{8} - 5\beta_{7} - 25\beta_{6} + 20\beta_{5} - 15\beta_{4} + 112\beta_{2} + 70\beta _1 + 655 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{18} - 5 \beta_{17} + 15 \beta_{16} - 12 \beta_{15} + 2 \beta_{14} + 5 \beta_{13} + \cdots - 279 \beta_{10} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 150 \beta_{9} + 5 \beta_{8} + 15 \beta_{7} - 1975 \beta_{6} + 1090 \beta_{5} - 1825 \beta_{4} + \cdots - 98925 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 280 \beta_{19} + 2115 \beta_{18} - 1785 \beta_{17} + 5495 \beta_{16} - 1030 \beta_{15} + \cdots - 286655 \beta_{10} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 9000 \beta_{9} - 22715 \beta_{8} + 21335 \beta_{7} - 54095 \beta_{6} + 35960 \beta_{5} + \cdots - 8327945 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 12360 \beta_{19} + 137495 \beta_{18} - 55365 \beta_{17} + 163275 \beta_{16} + 160720 \beta_{15} + \cdots - 18500515 \beta_{10} ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 31050 \beta_{9} - 380987 \beta_{8} + 352855 \beta_{7} + 384489 \beta_{6} + 870 \beta_{5} + \cdots - 84399569 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 106920 \beta_{19} + 4277855 \beta_{18} + 1260435 \beta_{17} - 4208765 \beta_{16} + \cdots - 809606915 \beta_{10} ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 17663700 \beta_{9} - 88351075 \beta_{8} + 82709935 \beta_{7} + 314747745 \beta_{6} - 98247340 \beta_{5} + \cdots - 14719611065 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 47074040 \beta_{19} - 70794265 \beta_{18} + 275381275 \beta_{17} - 827827845 \beta_{16} + \cdots - 21570435715 \beta_{10} ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1961131050 \beta_{9} - 1849744535 \beta_{8} + 1782570435 \beta_{7} + 20261328125 \beta_{6} + \cdots - 194171744485 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 698356200 \beta_{19} - 3732499381 \beta_{18} + 3854906255 \beta_{17} - 11237796865 \beta_{16} + \cdots + 46819256433 \beta_{10} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 111226546800 \beta_{9} + 90020125565 \beta_{8} - 82273698705 \beta_{7} + 797704278625 \beta_{6} + \cdots + 20622432747975 ) / 5 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 152509316760 \beta_{19} - 1365451381505 \beta_{18} + 827219837395 \beta_{17} + \cdots + 74699689681685 \beta_{10} ) / 5 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 3679822737750 \beta_{9} + 12333913941305 \beta_{8} - 11606133995245 \beta_{7} + 11065704413965 \beta_{6} + \cdots + 22\!\cdots\!15 ) / 5 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 2953920110120 \beta_{19} - 62332770393665 \beta_{18} + 15732113189555 \beta_{17} + \cdots + 56\!\cdots\!05 \beta_{10} ) / 5 \) 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
6.81708 + 2.23607i
6.81708 − 2.23607i
5.38548 + 2.23607i
5.38548 − 2.23607i
5.25461 + 2.23607i
5.25461 − 2.23607i
2.17381 + 2.23607i
2.17381 − 2.23607i
1.08979 + 2.23607i
1.08979 − 2.23607i
−1.08979 + 2.23607i
−1.08979 − 2.23607i
−2.17381 + 2.23607i
−2.17381 − 2.23607i
−5.25461 + 2.23607i
−5.25461 − 2.23607i
−5.38548 + 2.23607i
−5.38548 − 2.23607i
−6.81708 + 2.23607i
−6.81708 − 2.23607i
−6.81708 0 30.4725 − 11.1803i 0 34.6367 + 34.6597i −98.6603 0 76.2172i
181.2 −6.81708 0 30.4725 11.1803i 0 34.6367 − 34.6597i −98.6603 0 − 76.2172i
181.3 −5.38548 0 13.0034 − 11.1803i 0 −47.5947 − 11.6511i 16.1380 0 60.2115i
181.4 −5.38548 0 13.0034 11.1803i 0 −47.5947 + 11.6511i 16.1380 0 − 60.2115i
181.5 −5.25461 0 11.6110 − 11.1803i 0 6.58023 − 48.5562i 23.0627 0 58.7484i
181.6 −5.25461 0 11.6110 11.1803i 0 6.58023 + 48.5562i 23.0627 0 − 58.7484i
181.7 −2.17381 0 −11.2746 − 11.1803i 0 47.0248 − 13.7722i 59.2897 0 24.3039i
181.8 −2.17381 0 −11.2746 11.1803i 0 47.0248 + 13.7722i 59.2897 0 − 24.3039i
181.9 −1.08979 0 −14.8124 − 11.1803i 0 −34.6471 + 34.6494i 33.5790 0 12.1842i
181.10 −1.08979 0 −14.8124 11.1803i 0 −34.6471 − 34.6494i 33.5790 0 − 12.1842i
181.11 1.08979 0 −14.8124 − 11.1803i 0 −34.6471 − 34.6494i −33.5790 0 − 12.1842i
181.12 1.08979 0 −14.8124 11.1803i 0 −34.6471 + 34.6494i −33.5790 0 12.1842i
181.13 2.17381 0 −11.2746 − 11.1803i 0 47.0248 + 13.7722i −59.2897 0 − 24.3039i
181.14 2.17381 0 −11.2746 11.1803i 0 47.0248 − 13.7722i −59.2897 0 24.3039i
181.15 5.25461 0 11.6110 − 11.1803i 0 6.58023 + 48.5562i −23.0627 0 − 58.7484i
181.16 5.25461 0 11.6110 11.1803i 0 6.58023 − 48.5562i −23.0627 0 58.7484i
181.17 5.38548 0 13.0034 − 11.1803i 0 −47.5947 + 11.6511i −16.1380 0 − 60.2115i
181.18 5.38548 0 13.0034 11.1803i 0 −47.5947 − 11.6511i −16.1380 0 60.2115i
181.19 6.81708 0 30.4725 − 11.1803i 0 34.6367 − 34.6597i 98.6603 0 − 76.2172i
181.20 6.81708 0 30.4725 11.1803i 0 34.6367 + 34.6597i 98.6603 0 76.2172i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 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.c ✓ 20
3.b odd 2 1 inner 315.5.h.c ✓ 20
7.b odd 2 1 inner 315.5.h.c ✓ 20
21.c even 2 1 inner 315.5.h.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.5.h.c ✓ 20 1.a even 1 1 trivial
315.5.h.c ✓ 20 3.b odd 2 1 inner
315.5.h.c ✓ 20 7.b odd 2 1 inner
315.5.h.c ✓ 20 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 109T_{2}^{8} + 4047T_{2}^{6} - 58087T_{2}^{4} + 239320T_{2}^{2} - 208860 \) acting on \(S_{5}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 109 T^{8} + \cdots - 208860)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 79\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 26\!\cdots\!60)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 16\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 7807717738336)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 748772772797056)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 79\!\cdots\!44)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 70\!\cdots\!60)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 80\!\cdots\!36)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
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