Properties

Label 325.6.a.f
Level $325$
Weight $6$
Character orbit 325.a
Self dual yes
Analytic conductor $52.125$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)

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: yes
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 161x^{4} + 328x^{3} + 6584x^{2} - 10688x - 47440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 3) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 23) q^{4} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots - 10) q^{6}+ \cdots + ( - 4 \beta_{5} - 3 \beta_{4} + \cdots + 167) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - 3) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 23) q^{4} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots - 10) q^{6}+ \cdots + ( - 1219 \beta_{5} - 3106 \beta_{4} + \cdots + 46690) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9} + 800 q^{11} + 832 q^{12} + 1014 q^{13} + 2108 q^{14} + 322 q^{16} - 4396 q^{17} - 6142 q^{18} + 5304 q^{19} + 1072 q^{21} - 8008 q^{22} + 1140 q^{23} + 4756 q^{24} - 338 q^{26} - 800 q^{27} - 524 q^{28} - 3664 q^{29} + 16664 q^{31} + 9894 q^{32} + 13092 q^{33} + 1380 q^{34} + 1818 q^{36} - 4488 q^{37} + 30112 q^{38} - 3380 q^{39} + 12436 q^{41} + 68336 q^{42} - 1516 q^{43} - 35124 q^{44} - 4556 q^{46} - 212 q^{47} + 16712 q^{48} - 27842 q^{49} - 27664 q^{51} + 22646 q^{52} + 15612 q^{53} - 132912 q^{54} + 44420 q^{56} + 86796 q^{57} + 39664 q^{58} - 11896 q^{59} + 57232 q^{61} + 95488 q^{62} - 207956 q^{63} - 39914 q^{64} + 9908 q^{66} + 25428 q^{67} + 16236 q^{68} + 72404 q^{69} + 214104 q^{71} + 130806 q^{72} - 60808 q^{73} - 182104 q^{74} + 366636 q^{76} + 5192 q^{77} - 8788 q^{78} + 36024 q^{79} + 350514 q^{81} + 1124 q^{82} - 34684 q^{83} + 6024 q^{84} + 29204 q^{86} + 8640 q^{87} + 40520 q^{88} - 52028 q^{89} - 29068 q^{91} + 283232 q^{92} - 26508 q^{93} + 157572 q^{94} + 547908 q^{96} - 129316 q^{97} + 113942 q^{98} + 301264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 161x^{4} + 328x^{3} + 6584x^{2} - 10688x - 47440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 2\nu^{4} + 125\nu^{3} + 212\nu^{2} - 2956\nu - 4800 ) / 240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 125\nu^{3} + 28\nu^{2} + 3196\nu - 8400 ) / 240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 10\nu^{3} + 125\nu^{2} - 808\nu - 2260 ) / 60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 2\nu^{3} - 101\nu^{2} - 116\nu + 1288 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - \beta _1 + 55 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 5\beta_{4} - 2\beta_{3} - 2\beta_{2} + 79\beta _1 - 29 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{5} - 10\beta_{4} + 105\beta_{3} + 105\beta_{2} - 143\beta _1 + 4325 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 105\beta_{5} + 645\beta_{4} - 248\beta_{3} - 488\beta_{2} + 6993\beta _1 - 5415 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.62003
7.62628
4.15349
−2.16876
−7.05260
−10.1784
−9.62003 −10.7175 60.5450 0 103.103 −18.4289 −274.604 −128.135 0
1.2 −7.62628 29.7832 26.1601 0 −227.135 −171.199 44.5366 644.042 0
1.3 −4.15349 −29.2293 −14.7485 0 121.404 −42.5171 194.170 611.349 0
1.4 2.16876 2.56930 −27.2965 0 5.57220 75.5966 −128.600 −236.399 0
1.5 7.05260 −22.8190 17.7392 0 −160.933 −141.347 −100.576 277.708 0
1.6 10.1784 10.4132 71.6007 0 105.990 125.896 403.073 −134.565 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.6.a.f 6
5.b even 2 1 65.6.a.e 6
5.c odd 4 2 325.6.b.f 12
15.d odd 2 1 585.6.a.k 6
20.d odd 2 1 1040.6.a.r 6
65.d even 2 1 845.6.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.e 6 5.b even 2 1
325.6.a.f 6 1.a even 1 1 trivial
325.6.b.f 12 5.c odd 4 2
585.6.a.k 6 15.d odd 2 1
845.6.a.g 6 65.d even 2 1
1040.6.a.r 6 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2T_{2}^{5} - 161T_{2}^{4} - 328T_{2}^{3} + 6584T_{2}^{2} + 10688T_{2} - 47440 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots - 47440 \) Copy content Toggle raw display
$3$ \( T^{6} + 20 T^{5} + \cdots - 5696136 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 180452981952 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 674919089487832 \) Copy content Toggle raw display
$13$ \( (T - 169)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 58\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 24\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 47\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 39\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 78\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 34\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 51\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 29\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 31\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 14\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 20\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 22\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 22\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
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