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Properties

Label 304.5.r.a
Level $304$
Weight $5$
Character orbit 304.r
Analytic conductor $31.424$
Analytic rank $0$
Dimension $10$
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(65,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
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("304.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 304.r (of order \(6\), degree \(2\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 109x^{8} + 4107x^{6} + 61507x^{4} + 300520x^{2} + 108300 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3}) q^{3} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \cdots + 2) q^{5}+ \cdots + (2 \beta_{9} + \beta_{7} + 2 \beta_{6} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3}) q^{3} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \cdots + 2) q^{5}+ \cdots + ( - 106 \beta_{9} + 55 \beta_{7} + \cdots - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 9 q^{3} + 8 q^{5} + 24 q^{7} - 58 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 9 q^{3} + 8 q^{5} + 24 q^{7} - 58 q^{9} - 50 q^{11} - 624 q^{13} - 504 q^{15} - 292 q^{17} - 305 q^{19} + 1158 q^{21} - 98 q^{23} - 681 q^{25} + 2598 q^{29} - 3441 q^{33} - 694 q^{35} + 5552 q^{39} - 1407 q^{41} - 5424 q^{43} + 9572 q^{45} + 2416 q^{47} - 17826 q^{49} + 3342 q^{51} + 1122 q^{53} - 11424 q^{55} - 7906 q^{57} - 15387 q^{59} + 860 q^{61} - 5318 q^{63} - 14763 q^{67} + 27264 q^{71} + 1561 q^{73} - 18392 q^{77} - 24750 q^{79} + 14311 q^{81} - 6002 q^{83} - 14944 q^{85} + 31996 q^{87} - 22566 q^{89} - 8724 q^{91} + 12476 q^{93} + 7312 q^{95} + 46287 q^{97} + 2048 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 109x^{8} + 4107x^{6} + 61507x^{4} + 300520x^{2} + 108300 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 56\nu^{4} + 649\nu^{2} + 300\nu + 570 ) / 600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 56\nu^{4} - 649\nu^{2} + 300\nu - 570 ) / 600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 109\nu^{7} + 3917\nu^{5} + 50867\nu^{3} + 177210\nu + 57000 ) / 114000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{9} - 204 \nu^{7} - 285 \nu^{6} - 12087 \nu^{5} - 24510 \nu^{4} - 263572 \nu^{3} + \cdots - 1786950 ) / 171000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{9} - 204 \nu^{7} + 285 \nu^{6} - 12087 \nu^{5} + 24510 \nu^{4} - 263572 \nu^{3} + \cdots + 1786950 ) / 171000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23 \nu^{9} + 95 \nu^{8} + 2127 \nu^{7} + 7980 \nu^{6} + 65961 \nu^{5} + 202065 \nu^{4} + \cdots + 1259700 ) / 171000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23 \nu^{9} - 95 \nu^{8} + 2127 \nu^{7} - 7980 \nu^{6} + 65961 \nu^{5} - 202065 \nu^{4} + \cdots - 1259700 ) / 171000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\nu^{9} + 1104\nu^{7} + 37767\nu^{5} + 497882\nu^{3} - 28500\nu^{2} + 1895160\nu - 627000 ) / 57000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 12\beta_{3} - 33\beta_{2} - 33\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{6} - 10\beta_{5} - 43\beta_{4} + 10\beta_{2} - 10\beta _1 + 756 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 126 \beta_{9} - 53 \beta_{8} - 53 \beta_{7} + 23 \beta_{6} + 23 \beta_{5} + 63 \beta_{4} - 1116 \beta_{3} + \cdots + 558 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -560\beta_{6} + 560\beta_{5} + 1759\beta_{4} - 860\beta_{2} + 860\beta _1 - 28628 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6358 \beta_{9} + 2319 \beta_{8} + 2319 \beta_{7} - 639 \beta_{6} - 639 \beta_{5} - 3179 \beta_{4} + \cdots - 33954 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 900 \beta_{8} + 900 \beta_{7} + 25770 \beta_{6} - 25770 \beta_{5} - 72067 \beta_{4} + 50970 \beta_{2} + \cdots + 1130464 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 301214 \beta_{9} - 96037 \beta_{8} - 96037 \beta_{7} + 30427 \beta_{6} + 30427 \beta_{5} + \cdots + 1763502 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
4.58432i
− 2.91518i
6.00126i
− 6.56352i
0.625165i
− 4.58432i
2.91518i
− 6.00126i
6.56352i
− 0.625165i
0 −10.1306 − 5.84893i 0 20.8352 − 36.0877i 0 −24.1856 0 27.9200 + 48.3589i 0
65.2 0 −8.18489 − 4.72555i 0 −7.00036 + 12.1250i 0 −19.8223 0 4.16163 + 7.20815i 0
65.3 0 1.61920 + 0.934844i 0 −20.3151 + 35.1867i 0 11.8590 0 −38.7521 − 67.1207i 0
65.4 0 3.48758 + 2.01356i 0 5.86031 − 10.1504i 0 44.4232 0 −32.3912 − 56.1032i 0
65.5 0 8.70876 + 5.02800i 0 4.61985 − 8.00181i 0 −0.274467 0 10.0617 + 17.4273i 0
145.1 0 −10.1306 + 5.84893i 0 20.8352 + 36.0877i 0 −24.1856 0 27.9200 − 48.3589i 0
145.2 0 −8.18489 + 4.72555i 0 −7.00036 − 12.1250i 0 −19.8223 0 4.16163 − 7.20815i 0
145.3 0 1.61920 − 0.934844i 0 −20.3151 − 35.1867i 0 11.8590 0 −38.7521 + 67.1207i 0
145.4 0 3.48758 − 2.01356i 0 5.86031 + 10.1504i 0 44.4232 0 −32.3912 + 56.1032i 0
145.5 0 8.70876 − 5.02800i 0 4.61985 + 8.00181i 0 −0.274467 0 10.0617 − 17.4273i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.5.r.a 10
4.b odd 2 1 19.5.d.a ✓ 10
12.b even 2 1 171.5.p.a 10
19.d odd 6 1 inner 304.5.r.a 10
76.f even 6 1 19.5.d.a ✓ 10
228.n odd 6 1 171.5.p.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.5.d.a ✓ 10 4.b odd 2 1
19.5.d.a ✓ 10 76.f even 6 1
171.5.p.a 10 12.b even 2 1
171.5.p.a 10 228.n odd 6 1
304.5.r.a 10 1.a even 1 1 trivial
304.5.r.a 10 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 9 T_{3}^{9} - 133 T_{3}^{8} - 1440 T_{3}^{7} + 17599 T_{3}^{6} + 132579 T_{3}^{5} + \cdots + 70073667 \) acting on \(S_{5}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 9 T^{9} + \cdots + 70073667 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 6589766238916 \) Copy content Toggle raw display
$7$ \( (T^{5} - 12 T^{4} + \cdots + 69320)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 25 T^{4} + \cdots + 8646509620)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 61\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 27\!\cdots\!75 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 42\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 35\!\cdots\!63 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 91\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 56\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 11\!\cdots\!47 \) Copy content Toggle raw display
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