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Properties

Label 825.6.a.r
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
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] = [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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + \cdots + 1190400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 19) q^{4} - 9 \beta_1 q^{6} + ( - \beta_{4} - 2 \beta_1 + 7) q^{7} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 22) q^{8}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 19) q^{4} - 9 \beta_1 q^{6} + ( - \beta_{4} - 2 \beta_1 + 7) q^{7} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 22) q^{8}+ \cdots - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{2} + 81 q^{3} + 171 q^{4} - 9 q^{6} + 57 q^{7} + 177 q^{8} + 729 q^{9} - 1089 q^{11} + 1539 q^{12} - 723 q^{13} + 720 q^{14} + 4147 q^{16} + 2804 q^{17} - 81 q^{18} - 1601 q^{19} + 513 q^{21} + 121 q^{22} + 2392 q^{23} + 1593 q^{24} - 1987 q^{26} + 6561 q^{27} - 4436 q^{28} + 5966 q^{29} + 21575 q^{31} + 25493 q^{32} - 9801 q^{33} - 3098 q^{34} + 13851 q^{36} + 10228 q^{37} - 12765 q^{38} - 6507 q^{39} + 13304 q^{41} + 6480 q^{42} + 13829 q^{43} - 20691 q^{44} + 81283 q^{46} + 13998 q^{47} + 37323 q^{48} - 19468 q^{49} + 25236 q^{51} - 37131 q^{52} + 44166 q^{53} - 729 q^{54} + 79160 q^{56} - 14409 q^{57} + 7635 q^{58} + 11626 q^{59} + 49481 q^{61} + 94479 q^{62} + 4617 q^{63} + 109367 q^{64} + 1089 q^{66} - 26567 q^{67} + 119506 q^{68} + 21528 q^{69} + 78454 q^{71} + 14337 q^{72} + 100086 q^{73} + 162360 q^{74} + 194115 q^{76} - 6897 q^{77} - 17883 q^{78} - 8478 q^{79} + 59049 q^{81} + 52700 q^{82} + 157476 q^{83} - 39924 q^{84} + 251663 q^{86} + 53694 q^{87} - 21417 q^{88} - 65548 q^{89} - 106849 q^{91} + 350115 q^{92} + 194175 q^{93} - 35742 q^{94} + 229437 q^{96} - 116757 q^{97} + 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + \cdots + 1190400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - \nu^{7} - 213\nu^{6} + 243\nu^{5} + 13338\nu^{4} - 8432\nu^{3} - 242664\nu^{2} - 234544\nu + 288000 ) / 4800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{8} + \nu^{7} + 213 \nu^{6} - 243 \nu^{5} - 13338 \nu^{4} + 13232 \nu^{3} + 252264 \nu^{2} + \cdots - 672000 ) / 4800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{8} - \nu^{7} + 291 \nu^{6} + 219 \nu^{5} - 25980 \nu^{4} - 12448 \nu^{3} + 693568 \nu^{2} + \cdots - 1473600 ) / 4800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{8} + 15 \nu^{7} + 267 \nu^{6} - 2877 \nu^{5} - 22644 \nu^{4} + 145592 \nu^{3} + \cdots - 3888000 ) / 9600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 5 \nu^{7} - 247 \nu^{6} - 943 \nu^{5} + 20264 \nu^{4} + 51208 \nu^{3} - 581776 \nu^{2} + \cdots + 2483200 ) / 3200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23 \nu^{8} - 33 \nu^{7} - 5109 \nu^{6} + 9699 \nu^{5} + 348564 \nu^{4} - 728536 \nu^{3} + \cdots + 28147200 ) / 9600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 2\beta_{2} + 85\beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 2\beta_{3} + 117\beta_{2} - 62\beta _1 + 4339 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{8} - 6\beta_{7} + 10\beta_{6} - 2\beta_{5} + 183\beta_{4} + 149\beta_{3} - 368\beta_{2} + 8329\beta _1 - 5724 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{8} + 332 \beta_{7} - 320 \beta_{6} + 384 \beta_{5} - 168 \beta_{4} - 396 \beta_{3} + \cdots + 426279 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 768 \beta_{8} - 1080 \beta_{7} + 2472 \beta_{6} - 528 \beta_{5} + 25281 \beta_{4} + 18777 \beta_{3} + \cdots - 989250 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1056 \beta_{8} + 44418 \beta_{7} - 41442 \beta_{6} + 55074 \beta_{5} - 46540 \beta_{4} + \cdots + 45227887 \) 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
10.2706
8.36252
6.71211
3.27244
−0.624305
−1.92452
−5.77674
−8.18565
−11.1064
−10.2706 9.00000 73.4848 0 −92.4352 −129.851 −426.072 81.0000 0
1.2 −8.36252 9.00000 37.9318 0 −75.2627 15.4419 −49.6048 81.0000 0
1.3 −6.71211 9.00000 13.0525 0 −60.4090 177.172 127.178 81.0000 0
1.4 −3.27244 9.00000 −21.2911 0 −29.4519 −115.628 174.392 81.0000 0
1.5 0.624305 9.00000 −31.6102 0 5.61875 106.290 −39.7122 81.0000 0
1.6 1.92452 9.00000 −28.2962 0 17.3207 −59.1120 −116.041 81.0000 0
1.7 5.77674 9.00000 1.37069 0 51.9906 150.201 −176.937 81.0000 0
1.8 8.18565 9.00000 35.0048 0 73.6708 −163.661 24.5962 81.0000 0
1.9 11.1064 9.00000 91.3531 0 99.9580 76.1461 659.202 81.0000 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
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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.r ✓ 9
5.b even 2 1 825.6.a.s yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.r ✓ 9 1.a even 1 1 trivial
825.6.a.s yes 9 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + T_{2}^{8} - 229 T_{2}^{7} - 267 T_{2}^{6} + 16434 T_{2}^{5} + 16568 T_{2}^{4} + \cdots - 1190400 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + T^{8} + \cdots - 1190400 \) Copy content Toggle raw display
$3$ \( (T - 9)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 48\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T + 121)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 71\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 83\!\cdots\!77 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 55\!\cdots\!82 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 27\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 28\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 12\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 44\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 24\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 49\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 52\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 37\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
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