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Properties

Label 325.6.a.h
Level $325$
Weight $6$
Character orbit 325.a
Self dual yes
Analytic conductor $52.125$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{4} - 1) q^{3} + ( - \beta_{4} + \beta_{3} - 2 \beta_1 + 11) q^{4} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots - 11) q^{6}+ \cdots + ( - \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 61) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + ( - \beta_{4} - 1) q^{3} + ( - \beta_{4} + \beta_{3} - 2 \beta_1 + 11) q^{4} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots - 11) q^{6}+ \cdots + (467 \beta_{8} + 536 \beta_{7} + \cdots - 83645) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21} + 4345 q^{22} + 1839 q^{23} - 8469 q^{24} - 845 q^{26} - 7649 q^{27} - 2836 q^{28} - 8737 q^{29} + 748 q^{31} - 423 q^{32} - 356 q^{33} - 17789 q^{34} + 512 q^{36} - 15486 q^{37} - 3425 q^{38} - 1859 q^{39} - 28676 q^{41} + 6876 q^{42} + 28665 q^{43} - 30599 q^{44} - 12056 q^{46} - 29452 q^{47} + 64759 q^{48} - 40907 q^{49} - 31006 q^{51} + 15379 q^{52} + 75977 q^{53} - 102761 q^{54} - 23002 q^{56} - 38038 q^{57} + 142384 q^{58} - 88142 q^{59} + 28165 q^{61} - 137308 q^{62} + 41492 q^{63} - 100845 q^{64} + 42577 q^{66} - 94754 q^{67} + 89267 q^{68} - 181747 q^{69} - 70562 q^{71} - 263778 q^{72} + 60602 q^{73} - 135676 q^{74} + 46373 q^{76} - 140292 q^{77} - 14027 q^{78} - 164073 q^{79} - 69935 q^{81} - 72887 q^{82} - 22458 q^{83} - 345656 q^{84} - 294920 q^{86} - 87031 q^{87} + 430607 q^{88} - 252698 q^{89} + 2028 q^{91} - 237824 q^{92} + 56556 q^{93} - 501606 q^{94} - 319181 q^{96} + 137986 q^{97} + 378699 q^{98} - 757776 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 603 \nu^{8} + 4139 \nu^{7} - 57736 \nu^{6} + 833832 \nu^{5} + 16685739 \nu^{4} + \cdots + 92692560 ) / 33188160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3059 \nu^{8} - 1477 \nu^{7} + 551488 \nu^{6} + 201192 \nu^{5} - 31471053 \nu^{4} + \cdots - 1760425200 ) / 33188160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3059 \nu^{8} - 1477 \nu^{7} + 551488 \nu^{6} + 201192 \nu^{5} - 31471053 \nu^{4} + \cdots - 366522480 ) / 33188160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 567 \nu^{8} - 280 \nu^{7} - 81415 \nu^{6} + 11598 \nu^{5} + 2823969 \nu^{4} + 266328 \nu^{3} + \cdots - 337308240 ) / 4148520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5761 \nu^{8} - 82657 \nu^{7} - 791792 \nu^{6} + 11905272 \nu^{5} + 29973327 \nu^{4} + \cdots + 2049237840 ) / 33188160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5327 \nu^{8} + 357 \nu^{7} - 877148 \nu^{6} - 154800 \nu^{5} + 42766929 \nu^{4} + \cdots - 119818320 ) / 16594080 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10807 \nu^{8} - 41615 \nu^{7} - 1966680 \nu^{6} + 5794488 \nu^{5} + 111516249 \nu^{4} + \cdots + 2036238960 ) / 33188160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 64\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{8} - \beta_{7} + 3\beta_{6} - 6\beta_{5} - 104\beta_{4} + 80\beta_{3} - 2\beta_{2} + 20\beta _1 + 2727 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5 \beta_{8} + 124 \beta_{7} + 3 \beta_{6} - 103 \beta_{5} + 160 \beta_{4} + 102 \beta_{3} + \cdots - 253 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 742 \beta_{8} - 112 \beta_{7} + 434 \beta_{6} - 666 \beta_{5} - 9493 \beta_{4} + 6361 \beta_{3} + \cdots + 197044 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1148 \beta_{8} + 12301 \beta_{7} + 268 \beta_{6} - 9129 \beta_{5} + 16577 \beta_{4} + 8449 \beta_{3} + \cdots + 21362 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 82105 \beta_{8} - 8033 \beta_{7} + 47447 \beta_{6} - 60362 \beta_{5} - 834948 \beta_{4} + \cdots + 15086387 \) Copy content Toggle raw display

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

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
−9.17271
−7.11691
−6.50844
−0.838151
0.603392
4.20685
5.30592
8.28147
9.23858
−10.1727 23.0461 71.4841 0 −234.441 −51.0548 −401.660 288.123 0
1.2 −8.11691 −3.22994 33.8842 0 26.2172 −20.7248 −15.2938 −232.567 0
1.3 −7.50844 −17.3551 24.3767 0 130.310 149.455 57.2388 58.2000 0
1.4 −1.83815 −11.9262 −28.6212 0 21.9221 −112.158 111.431 −100.766 0
1.5 −0.396608 11.4974 −31.8427 0 −4.55994 −8.04695 25.3205 −110.811 0
1.6 3.20685 −29.5845 −21.7161 0 −94.8732 −11.5734 −172.260 632.245 0
1.7 4.30592 19.9083 −13.4591 0 85.7235 125.783 −195.743 153.341 0
1.8 7.28147 −14.9935 21.0198 0 −109.175 135.391 −79.9521 −18.1955 0
1.9 8.23858 11.6375 35.8743 0 95.8765 −195.071 31.9185 −107.569 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.h ✓ 9
5.b even 2 1 325.6.a.i yes 9
5.c odd 4 2 325.6.b.h 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.6.a.h ✓ 9 1.a even 1 1 trivial
325.6.a.i yes 9 5.b even 2 1
325.6.b.h 18 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 5 T_{2}^{8} - 177 T_{2}^{7} - 607 T_{2}^{6} + 10684 T_{2}^{5} + 18202 T_{2}^{4} + \cdots + 374400 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 5 T^{8} + \cdots + 374400 \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots + 18204577776 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 54\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 97\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( (T - 169)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 85\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 55\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 70\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 65\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 68\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 13\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 25\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 15\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 43\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
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