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Properties

Label 825.6.a.x
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $13$
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] = [825,6,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("825.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 318 x^{11} + 776 x^{10} + 37929 x^{9} - 75673 x^{8} - 2114192 x^{7} + \cdots + 1037920000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 165)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + \beta_1 + 17) q^{4} - 9 \beta_1 q^{6} + (\beta_{6} - 4 \beta_1 + 23) q^{7} + (\beta_{7} - \beta_{6} + 2 \beta_{2} + \cdots + 23) q^{8}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + \beta_1 + 17) q^{4} - 9 \beta_1 q^{6} + (\beta_{6} - 4 \beta_1 + 23) q^{7} + (\beta_{7} - \beta_{6} + 2 \beta_{2} + \cdots + 23) q^{8}+ \cdots - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 3 q^{2} - 117 q^{3} + 229 q^{4} - 27 q^{6} + 284 q^{7} + 369 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 3 q^{2} - 117 q^{3} + 229 q^{4} - 27 q^{6} + 284 q^{7} + 369 q^{8} + 1053 q^{9} - 1573 q^{11} - 2061 q^{12} + 366 q^{13} - 2758 q^{14} + 4141 q^{16} + 2056 q^{17} + 243 q^{18} - 310 q^{19} - 2556 q^{21} - 363 q^{22} + 3612 q^{23} - 3321 q^{24} + 2280 q^{26} - 9477 q^{27} + 7896 q^{28} - 4848 q^{29} - 24 q^{31} + 38111 q^{32} + 14157 q^{33} + 5518 q^{34} + 18549 q^{36} - 8420 q^{37} + 474 q^{38} - 3294 q^{39} - 15120 q^{41} + 24822 q^{42} + 35492 q^{43} - 27709 q^{44} - 20280 q^{46} + 46544 q^{47} - 37269 q^{48} + 81837 q^{49} - 18504 q^{51} + 107194 q^{52} + 42256 q^{53} - 2187 q^{54} - 196602 q^{56} + 2790 q^{57} + 114160 q^{58} - 65592 q^{59} - 52042 q^{61} + 94972 q^{62} + 23004 q^{63} + 185977 q^{64} + 3267 q^{66} + 80580 q^{67} + 61108 q^{68} - 32508 q^{69} - 77820 q^{71} + 29889 q^{72} + 103050 q^{73} - 240028 q^{74} - 271174 q^{76} - 34364 q^{77} - 20520 q^{78} - 112258 q^{79} + 85293 q^{81} + 64060 q^{82} + 292150 q^{83} - 71064 q^{84} - 319250 q^{86} + 43632 q^{87} - 44649 q^{88} - 295810 q^{89} - 24200 q^{91} + 121328 q^{92} + 216 q^{93} - 358144 q^{94} - 342999 q^{96} + 49072 q^{97} + 101815 q^{98} - 127413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 3 x^{12} - 318 x^{11} + 776 x^{10} + 37929 x^{9} - 75673 x^{8} - 2114192 x^{7} + \cdots + 1037920000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 98460050695 \nu^{12} + 380358084197 \nu^{11} - 32301835849092 \nu^{10} + \cdots - 51\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27841265635 \nu^{12} - 85089357991 \nu^{11} - 8273830228724 \nu^{10} + \cdots + 33\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57164094459 \nu^{12} - 31884388543 \nu^{11} - 17639107494692 \nu^{10} + \cdots + 16\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 144095905415 \nu^{12} + 130660652197 \nu^{11} - 45786876578692 \nu^{10} + \cdots - 21\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 144095905415 \nu^{12} + 130660652197 \nu^{11} - 45786876578692 \nu^{10} + \cdots - 17\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 253626793841 \nu^{12} - 89938014403 \nu^{11} + 79341363823868 \nu^{10} + \cdots + 56\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 275021907439 \nu^{12} - 460470051581 \nu^{11} + 87482514802756 \nu^{10} + \cdots + 89\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29222637581 \nu^{12} - 31772032279 \nu^{11} + 9048605735004 \nu^{10} + \cdots + 12\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 88381788501 \nu^{12} + 68117355599 \nu^{11} - 27536799447084 \nu^{10} + \cdots - 36\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 131423379819 \nu^{12} + 92251209617 \nu^{11} - 41444081106252 \nu^{10} + \cdots - 30\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + 2\beta_{2} + 82\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{7} + \beta_{6} + 2 \beta_{4} + \cdots + 3971 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6 \beta_{12} - \beta_{11} - 23 \beta_{9} + 22 \beta_{8} + 146 \beta_{7} - 129 \beta_{6} - 8 \beta_{5} + \cdots + 5723 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 136 \beta_{12} - 162 \beta_{11} - 150 \beta_{10} + 114 \beta_{9} + 112 \beta_{8} + 316 \beta_{7} + \cdots + 380121 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1374 \beta_{12} - 300 \beta_{11} + 94 \beta_{10} - 4780 \beta_{9} + 4272 \beta_{8} + 17937 \beta_{7} + \cdots + 976475 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 16581 \beta_{12} - 20891 \beta_{11} - 17069 \beta_{10} + 6027 \beta_{9} + 27348 \beta_{8} + \cdots + 39756651 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 215814 \beta_{12} - 62729 \beta_{11} + 22916 \beta_{10} - 735359 \beta_{9} + 635622 \beta_{8} + \cdots + 143791499 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2029958 \beta_{12} - 2561932 \beta_{11} - 1789092 \beta_{10} - 529332 \beta_{9} + 4773968 \beta_{8} + \cdots + 4388166261 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 29661480 \beta_{12} - 10853408 \beta_{11} + 3788912 \beta_{10} - 101806760 \beta_{9} + \cdots + 19889101103 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 253208285 \beta_{12} - 312587401 \beta_{11} - 182161957 \beta_{10} - 226530071 \beta_{9} + \cdots + 501417262347 \) 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
−10.3606
−8.30687
−7.80204
−6.98295
−3.62063
−1.48806
−0.570908
3.81085
3.87461
5.95684
7.38043
9.81979
11.2895
−10.3606 −9.00000 75.3416 0 93.2452 248.805 −449.044 81.0000 0
1.2 −8.30687 −9.00000 37.0041 0 74.7618 54.6353 −41.5686 81.0000 0
1.3 −7.80204 −9.00000 28.8719 0 70.2184 −85.0125 24.4055 81.0000 0
1.4 −6.98295 −9.00000 16.7616 0 62.8465 45.4926 106.409 81.0000 0
1.5 −3.62063 −9.00000 −18.8910 0 32.5857 −187.526 184.258 81.0000 0
1.6 −1.48806 −9.00000 −29.7857 0 13.3926 −7.94259 91.9409 81.0000 0
1.7 −0.570908 −9.00000 −31.6741 0 5.13817 245.179 36.3520 81.0000 0
1.8 3.81085 −9.00000 −17.4775 0 −34.2976 −89.4064 −188.551 81.0000 0
1.9 3.87461 −9.00000 −16.9874 0 −34.8714 134.363 −189.807 81.0000 0
1.10 5.95684 −9.00000 3.48390 0 −53.6115 79.6742 −169.866 81.0000 0
1.11 7.38043 −9.00000 22.4707 0 −66.4239 −191.451 −70.3301 81.0000 0
1.12 9.81979 −9.00000 64.4282 0 −88.3781 193.324 318.438 81.0000 0
1.13 11.2895 −9.00000 95.4537 0 −101.606 −156.135 716.362 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

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 825.6.a.x 13
5.b even 2 1 825.6.a.w 13
5.c odd 4 2 165.6.c.a ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.c.a ✓ 26 5.c odd 4 2
825.6.a.w 13 5.b even 2 1
825.6.a.x 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 3 T_{2}^{12} - 318 T_{2}^{11} + 776 T_{2}^{10} + 37929 T_{2}^{9} - 75673 T_{2}^{8} + \cdots + 1037920000 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + \cdots + 1037920000 \) Copy content Toggle raw display
$3$ \( (T + 9)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T + 121)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots - 39\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 60\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 63\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 86\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 51\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 81\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 82\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
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