Properties

Label 325.6.a.g
Level $325$
Weight $6$
Character orbit 325.a
Self dual yes
Analytic conductor $52.125$
Analytic rank $1$
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: \(1\)
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: \( x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} - \beta_1 - 6) q^{3} + (\beta_{3} + \beta_{2} + 22) q^{4} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + \cdots + 52) q^{6}+ \cdots + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots + 83) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} - \beta_1 - 6) q^{3} + (\beta_{3} + \beta_{2} + 22) q^{4} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + \cdots + 52) q^{6}+ \cdots + (1296 \beta_{5} - 325 \beta_{4} + \cdots - 4260) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 38 q^{3} + 134 q^{4} + 318 q^{6} - 220 q^{7} - 24 q^{8} + 518 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 38 q^{3} + 134 q^{4} + 318 q^{6} - 220 q^{7} - 24 q^{8} + 518 q^{9} - 170 q^{11} - 2238 q^{12} - 1014 q^{13} - 1440 q^{14} + 3506 q^{16} - 728 q^{17} - 7788 q^{18} + 1218 q^{19} - 396 q^{21} - 5154 q^{22} - 8954 q^{23} + 12618 q^{24} - 13112 q^{27} - 13212 q^{28} + 8364 q^{29} + 2862 q^{31} + 29672 q^{32} + 3772 q^{33} - 21484 q^{34} - 2138 q^{36} - 13840 q^{37} + 27706 q^{38} + 6422 q^{39} + 2248 q^{41} + 47028 q^{42} - 3822 q^{43} - 73482 q^{44} - 39886 q^{46} + 10860 q^{47} - 13962 q^{48} + 85902 q^{49} - 55028 q^{51} - 22646 q^{52} - 6584 q^{53} + 45504 q^{54} - 188800 q^{56} - 11692 q^{57} + 88500 q^{58} - 28874 q^{59} + 972 q^{61} - 123466 q^{62} + 125236 q^{63} + 14022 q^{64} - 158040 q^{66} - 60040 q^{67} - 3504 q^{68} + 153088 q^{69} - 74174 q^{71} - 298668 q^{72} - 48960 q^{73} + 226112 q^{74} - 93422 q^{76} - 231588 q^{77} - 53742 q^{78} + 214508 q^{79} + 116006 q^{81} - 58732 q^{82} - 123260 q^{83} - 235268 q^{84} - 304666 q^{86} - 228144 q^{87} - 127622 q^{88} + 94028 q^{89} + 37180 q^{91} - 430298 q^{92} - 192892 q^{93} - 125920 q^{94} - 385858 q^{96} - 246284 q^{97} + 4736 q^{98} - 32270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 18\nu^{4} + 851\nu^{3} - 1814\nu^{2} - 30420\nu - 12312 ) / 2016 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{5} - 18\nu^{4} - 851\nu^{3} + 3830\nu^{2} + 30420\nu - 96552 ) / 2016 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 30\nu^{4} - 103\nu^{3} - 3098\nu^{2} + 3060\nu + 29880 ) / 336 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{5} - 18\nu^{4} + 2705\nu^{3} - 1994\nu^{2} - 41148\nu + 80856 ) / 2016 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} - \beta_{4} - 4\beta_{3} + 4\beta_{2} + 89\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} + 12\beta_{4} + 111\beta_{3} + 107\beta_{2} - 88\beta _1 + 4742 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -326\beta_{5} - 127\beta_{4} - 644\beta_{3} + 300\beta_{2} + 8747\beta _1 - 4642 \) 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
9.61672
8.51599
1.34530
−2.75663
−5.93318
−10.7882
−9.61672 −28.9791 60.4813 0 278.684 189.995 −273.896 596.790 0
1.2 −8.51599 −11.2297 40.5221 0 95.6316 −229.647 −72.5738 −116.895 0
1.3 −1.34530 19.6439 −30.1902 0 −26.4269 −48.6530 83.6645 142.882 0
1.4 2.75663 −23.9621 −24.4010 0 −66.0547 −85.7300 −155.477 331.182 0
1.5 5.93318 7.05430 3.20258 0 41.8544 185.746 −170.860 −193.237 0
1.6 10.7882 −0.527294 84.3852 0 −5.68855 −231.710 565.142 −242.722 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.g 6
5.b even 2 1 65.6.a.d 6
5.c odd 4 2 325.6.b.g 12
15.d odd 2 1 585.6.a.m 6
20.d odd 2 1 1040.6.a.q 6
65.d even 2 1 845.6.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.d 6 5.b even 2 1
325.6.a.g 6 1.a even 1 1 trivial
325.6.b.g 12 5.c odd 4 2
585.6.a.m 6 15.d odd 2 1
845.6.a.h 6 65.d even 2 1
1040.6.a.q 6 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 163T_{2}^{4} + 8T_{2}^{3} + 6120T_{2}^{2} - 6624T_{2} - 19440 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 163 T^{4} + \cdots - 19440 \) Copy content Toggle raw display
$3$ \( T^{6} + 38 T^{5} + \cdots + 569784 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 7832693511488 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 57\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T + 169)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 60\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 60\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 38\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 18\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 25\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 86\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 41\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 91\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 13\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 10\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 66\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
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