Properties

Label 325.6.a.e
Level $325$
Weight $6$
Character orbit 325.a
Self dual yes
Analytic conductor $52.125$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.1878612.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 16x^{2} + 16x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 2) q^{2} + (\beta_{3} + \beta_{2} + 1) q^{3} + (\beta_{3} - 4 \beta_{2} + 22) q^{4} + (4 \beta_{3} + 6 \beta_{2} - 20 \beta_1 - 18) q^{6} + ( - 2 \beta_{3} - 10 \beta_{2} + \cdots + 32) q^{7}+ \cdots + (5 \beta_{3} - 27 \beta_{2} + \cdots + 153) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 2) q^{2} + (\beta_{3} + \beta_{2} + 1) q^{3} + (\beta_{3} - 4 \beta_{2} + 22) q^{4} + (4 \beta_{3} + 6 \beta_{2} - 20 \beta_1 - 18) q^{6} + ( - 2 \beta_{3} - 10 \beta_{2} + \cdots + 32) q^{7}+ \cdots + (3955 \beta_{3} - 949 \beta_{2} + \cdots + 71441) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{2} + 4 q^{3} + 93 q^{4} - 74 q^{6} + 136 q^{7} + 447 q^{8} + 644 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 9 q^{2} + 4 q^{3} + 93 q^{4} - 74 q^{6} + 136 q^{7} + 447 q^{8} + 644 q^{9} - 516 q^{11} + 710 q^{12} + 676 q^{13} + 1604 q^{14} - 719 q^{16} - 344 q^{17} + 6905 q^{18} - 5012 q^{19} - 4448 q^{21} + 9318 q^{22} - 708 q^{23} - 2094 q^{24} + 1521 q^{26} + 4264 q^{27} + 7268 q^{28} + 5416 q^{29} - 10124 q^{31} - 9497 q^{32} + 16256 q^{33} + 3022 q^{34} + 36205 q^{36} - 1448 q^{37} - 11922 q^{38} + 676 q^{39} + 21576 q^{41} - 33128 q^{42} + 25740 q^{43} + 24710 q^{44} + 28754 q^{46} + 51096 q^{47} - 43282 q^{48} - 40412 q^{49} + 52984 q^{51} + 15717 q^{52} - 39448 q^{53} + 60340 q^{54} + 30188 q^{56} + 56160 q^{57} - 1374 q^{58} + 37764 q^{59} + 95016 q^{61} - 83102 q^{62} + 43592 q^{63} - 142095 q^{64} + 198500 q^{66} + 50592 q^{67} + 47110 q^{68} - 177648 q^{69} - 89308 q^{71} + 99183 q^{72} + 108312 q^{73} - 16766 q^{74} - 3442 q^{76} - 6208 q^{77} - 12506 q^{78} - 25464 q^{79} - 106348 q^{81} + 206426 q^{82} - 20584 q^{83} - 213448 q^{84} + 363666 q^{86} + 128312 q^{87} - 20558 q^{88} - 148328 q^{89} + 22984 q^{91} + 173474 q^{92} + 10496 q^{93} + 63916 q^{94} + 85154 q^{96} + 77816 q^{97} + 83505 q^{98} + 290668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 16x^{2} + 16x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 13\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 9\nu^{2} + \nu - 67 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 3\beta _1 + 36 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 9\beta_{2} + 29\beta _1 + 58 ) / 4 \) 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
4.24074
−1.52205
−3.13795
2.41925
−7.55727 14.4912 25.1123 0 −109.514 −26.5515 52.0524 −33.0063 0
1.2 −1.42789 −13.6012 −29.9611 0 19.4210 9.51473 88.4737 −58.0072 0
1.3 8.75180 26.2145 44.5940 0 229.424 −8.06989 110.220 444.199 0
1.4 9.23335 −23.1044 53.2548 0 −213.331 161.107 196.253 290.815 0
\(n\): e.g. 2-40 or 990-1000
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.e 4
5.b even 2 1 65.6.a.c 4
5.c odd 4 2 325.6.b.e 8
15.d odd 2 1 585.6.a.g 4
20.d odd 2 1 1040.6.a.o 4
65.d even 2 1 845.6.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.c 4 5.b even 2 1
325.6.a.e 4 1.a even 1 1 trivial
325.6.b.e 8 5.c odd 4 2
585.6.a.g 4 15.d odd 2 1
845.6.a.f 4 65.d even 2 1
1040.6.a.o 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 9T_{2}^{3} - 70T_{2}^{2} + 532T_{2} + 872 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9 T^{3} + \cdots + 872 \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 119376 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 136 T^{3} + \cdots + 328448 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 10689924208 \) Copy content Toggle raw display
$13$ \( (T - 169)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 100880603792 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 161727486480 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 22398353692848 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 170255904416240 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 7222593222000 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 6842070199824 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 25\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 34\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 76\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 48\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 54\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 96\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 68\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 49\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
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