Properties

Label 8025.2.a.bf
Level $8025$
Weight $2$
Character orbit 8025.a
Self dual yes
Analytic conductor $64.080$
Analytic rank $1$
Dimension $12$
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] = [8025,2,Mod(1,8025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.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: \(64.0799476221\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} - 529 x^{3} + 267 x^{2} + 15 x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1605)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + (\beta_{5} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + (\beta_{5} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + \beta_{8} q^{11} + ( - \beta_{2} - 1) q^{12} + (\beta_{7} - 1) q^{13} + (\beta_{10} + \beta_{9} + \beta_{5} - \beta_{4} + \beta_1 - 1) q^{14} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - 2 \beta_{5} + 2) q^{16} + (\beta_{6} - \beta_{5} + \beta_{3} + 1) q^{17} - \beta_1 q^{18} + ( - \beta_{9} - \beta_{5} + \beta_{3} - \beta_{2} + 2) q^{19} + ( - \beta_{5} + 1) q^{21} + (\beta_{10} - \beta_{6} - \beta_{4} - 2) q^{22} + (\beta_{11} - \beta_{9} - \beta_{5} + \beta_1 - 1) q^{23} + (\beta_{3} + \beta_1) q^{24} + ( - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_1 - 1) q^{26} - q^{27} + ( - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{28} + ( - \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4}) q^{29} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{31} + ( - \beta_{10} - \beta_{8} - \beta_{3} + \beta_{2}) q^{32} - \beta_{8} q^{33} + ( - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{34} + (\beta_{2} + 1) q^{36} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 3) q^{37} + ( - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{38} + ( - \beta_{7} + 1) q^{39} + ( - \beta_{10} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 2) q^{41} + ( - \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{42} + ( - \beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{43} + ( - 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \cdots + 2) q^{44}+ \cdots + \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 12 q^{3} + 15 q^{4} + 3 q^{6} - 7 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 12 q^{3} + 15 q^{4} + 3 q^{6} - 7 q^{7} - 3 q^{8} + 12 q^{9} + 4 q^{11} - 15 q^{12} - 13 q^{13} + 4 q^{14} + 13 q^{16} + 4 q^{17} - 3 q^{18} + 14 q^{19} + 7 q^{21} - 15 q^{22} - 11 q^{23} + 3 q^{24} - 8 q^{26} - 12 q^{27} - 16 q^{28} - 7 q^{29} + 4 q^{31} - 4 q^{32} - 4 q^{33} + q^{34} + 15 q^{36} - 24 q^{37} + 11 q^{38} + 13 q^{39} + 13 q^{41} - 4 q^{42} - 25 q^{43} + 10 q^{44} - 22 q^{46} - 19 q^{47} - 13 q^{48} + 9 q^{49} - 4 q^{51} - 20 q^{52} - 11 q^{53} + 3 q^{54} - 37 q^{56} - 14 q^{57} + 2 q^{58} + 8 q^{59} + 7 q^{61} + 11 q^{62} - 7 q^{63} - 19 q^{64} + 15 q^{66} - 33 q^{67} + 24 q^{68} + 11 q^{69} - 3 q^{72} - 34 q^{73} - 27 q^{74} - 9 q^{76} + 29 q^{77} + 8 q^{78} + 12 q^{81} - q^{82} + 24 q^{83} + 16 q^{84} - 36 q^{86} + 7 q^{87} + 6 q^{88} - 10 q^{89} + 30 q^{91} + 28 q^{92} - 4 q^{93} - 8 q^{94} + 4 q^{96} - 16 q^{97} + 36 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} - 529 x^{3} + 267 x^{2} + 15 x - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 24 \nu^{10} - 4 \nu^{9} - 402 \nu^{8} - 101 \nu^{7} + 2336 \nu^{6} + 799 \nu^{5} - 5414 \nu^{4} - 1608 \nu^{3} + 4026 \nu^{2} + 336 \nu - 174 ) / 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} + 26 \nu^{10} + 61 \nu^{9} - 411 \nu^{8} - 481 \nu^{7} + 2204 \nu^{6} + 1817 \nu^{5} - 4583 \nu^{4} - 2918 \nu^{3} + 2916 \nu^{2} + 1001 \nu - 66 ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27 \nu^{11} - 38 \nu^{10} - 451 \nu^{9} + 612 \nu^{8} + 2663 \nu^{7} - 3372 \nu^{6} - 6651 \nu^{5} + 7486 \nu^{4} + 6172 \nu^{3} - 5566 \nu^{2} - 581 \nu + 55 ) / 49 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24 \nu^{11} - 61 \nu^{10} - 390 \nu^{9} + 985 \nu^{8} + 2182 \nu^{7} - 5480 \nu^{6} - 4834 \nu^{5} + 12360 \nu^{4} + 2813 \nu^{3} - 9510 \nu^{2} + 1939 \nu + 283 ) / 49 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{11} - 6 \nu^{10} - 83 \nu^{9} + 97 \nu^{8} + 482 \nu^{7} - 542 \nu^{6} - 1150 \nu^{5} + 1252 \nu^{4} + 913 \nu^{3} - 1017 \nu^{2} + 91 \nu + 12 ) / 7 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22 \nu^{11} - 60 \nu^{10} - 382 \nu^{9} + 956 \nu^{8} + 2384 \nu^{7} - 5203 \nu^{6} - 6481 \nu^{5} + 11232 \nu^{4} + 6813 \nu^{3} - 7860 \nu^{2} - 840 \nu + 190 ) / 49 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5 \nu^{11} + 6 \nu^{10} + 83 \nu^{9} - 97 \nu^{8} - 482 \nu^{7} + 542 \nu^{6} + 1157 \nu^{5} - 1252 \nu^{4} - 976 \nu^{3} + 1024 \nu^{2} + 28 \nu - 33 ) / 7 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43 \nu^{11} + 95 \nu^{10} + 711 \nu^{9} - 1530 \nu^{8} - 4134 \nu^{7} + 8479 \nu^{6} + 10086 \nu^{5} - 19009 \nu^{4} - 8668 \nu^{3} + 14356 \nu^{2} - 581 \nu - 358 ) / 49 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - 2\beta_{5} + 6\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + \beta_{8} + 9\beta_{3} - \beta_{2} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{11} - 9 \beta_{10} - 9 \beta_{9} - \beta_{8} + 10 \beta_{7} + 2 \beta_{6} - 18 \beta_{5} - 2 \beta_{3} + 37 \beta_{2} - \beta _1 + 94 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{11} + 12\beta_{10} - \beta_{9} + 12\beta_{8} - \beta_{7} - \beta_{5} + 68\beta_{3} - 14\beta_{2} + 167\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79 \beta_{11} - 67 \beta_{10} - 67 \beta_{9} - 14 \beta_{8} + 79 \beta_{7} + 27 \beta_{6} - 131 \beta_{5} - 2 \beta_{4} - 28 \beta_{3} + 234 \beta_{2} - 18 \beta _1 + 577 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18 \beta_{11} + 108 \beta_{10} - 14 \beta_{9} + 104 \beta_{8} - 16 \beta_{7} + \beta_{6} - 10 \beta_{5} + \beta_{4} + 486 \beta_{3} - 140 \beta_{2} + 1035 \beta _1 - 85 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 577 \beta_{11} - 473 \beta_{10} - 473 \beta_{9} - 136 \beta_{8} + 576 \beta_{7} + 257 \beta_{6} - 897 \beta_{5} - 32 \beta_{4} - 281 \beta_{3} + 1506 \beta_{2} - 214 \beta _1 + 3634 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 209 \beta_{11} + 873 \beta_{10} - 129 \beta_{9} + 801 \beta_{8} - 177 \beta_{7} + 18 \beta_{6} - 55 \beta_{5} + 17 \beta_{4} + 3389 \beta_{3} - 1225 \beta_{2} + 6575 \beta _1 - 971 \) 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
2.51437
2.43355
2.36308
1.79305
0.921208
0.851790
0.142275
−0.155707
−1.37578
−1.73551
−2.11086
−2.64148
−2.51437 −1.00000 4.32208 0 2.51437 0.976551 −5.83856 1.00000 0
1.2 −2.43355 −1.00000 3.92219 0 2.43355 −4.14842 −4.67775 1.00000 0
1.3 −2.36308 −1.00000 3.58415 0 2.36308 1.44739 −3.74348 1.00000 0
1.4 −1.79305 −1.00000 1.21504 0 1.79305 −0.475581 1.40748 1.00000 0
1.5 −0.921208 −1.00000 −1.15138 0 0.921208 −3.92138 2.90307 1.00000 0
1.6 −0.851790 −1.00000 −1.27445 0 0.851790 0.870792 2.78915 1.00000 0
1.7 −0.142275 −1.00000 −1.97976 0 0.142275 1.55684 0.566220 1.00000 0
1.8 0.155707 −1.00000 −1.97576 0 −0.155707 −3.91791 −0.619053 1.00000 0
1.9 1.37578 −1.00000 −0.107224 0 −1.37578 1.29286 −2.89908 1.00000 0
1.10 1.73551 −1.00000 1.01200 0 −1.73551 4.69063 −1.71469 1.00000 0
1.11 2.11086 −1.00000 2.45572 0 −2.11086 −2.18333 0.961964 1.00000 0
1.12 2.64148 −1.00000 4.97740 0 −2.64148 −3.18844 7.86473 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(107\) \(1\)

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 8025.2.a.bf 12
5.b even 2 1 1605.2.a.n 12
15.d odd 2 1 4815.2.a.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1605.2.a.n 12 5.b even 2 1
4815.2.a.u 12 15.d odd 2 1
8025.2.a.bf 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\( T_{2}^{12} + 3 T_{2}^{11} - 15 T_{2}^{10} - 49 T_{2}^{9} + 71 T_{2}^{8} + 278 T_{2}^{7} - 92 T_{2}^{6} - 649 T_{2}^{5} - 127 T_{2}^{4} + 529 T_{2}^{3} + 267 T_{2}^{2} - 15 T_{2} - 6 \) Copy content Toggle raw display
\( T_{7}^{12} + 7 T_{7}^{11} - 22 T_{7}^{10} - 228 T_{7}^{9} - 24 T_{7}^{8} + 1958 T_{7}^{7} + 496 T_{7}^{6} - 7966 T_{7}^{5} + 1487 T_{7}^{4} + 13857 T_{7}^{3} - 9891 T_{7}^{2} - 2125 T_{7} + 2452 \) Copy content Toggle raw display
\( T_{11}^{12} - 4 T_{11}^{11} - 61 T_{11}^{10} + 259 T_{11}^{9} + 1231 T_{11}^{8} - 5905 T_{11}^{7} - 8630 T_{11}^{6} + 57130 T_{11}^{5} + 1170 T_{11}^{4} - 219209 T_{11}^{3} + 127197 T_{11}^{2} + 283848 T_{11} - 241932 \) Copy content Toggle raw display
\( T_{13}^{12} + 13 T_{13}^{11} + 2 T_{13}^{10} - 606 T_{13}^{9} - 2459 T_{13}^{8} + 600 T_{13}^{7} + 14736 T_{13}^{6} + 4272 T_{13}^{5} - 34752 T_{13}^{4} + 1600 T_{13}^{3} + 30720 T_{13}^{2} - 15744 T_{13} + 1536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} - 15 T^{10} - 49 T^{9} + \cdots - 6 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 7 T^{11} - 22 T^{10} + \cdots + 2452 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} - 61 T^{10} + \cdots - 241932 \) Copy content Toggle raw display
$13$ \( T^{12} + 13 T^{11} + 2 T^{10} + \cdots + 1536 \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} - 78 T^{10} + \cdots + 14528 \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{11} - 6 T^{10} + \cdots - 2744 \) Copy content Toggle raw display
$23$ \( T^{12} + 11 T^{11} - 98 T^{10} + \cdots - 163584 \) Copy content Toggle raw display
$29$ \( T^{12} + 7 T^{11} - 110 T^{10} + \cdots + 2900296 \) Copy content Toggle raw display
$31$ \( T^{12} - 4 T^{11} - 156 T^{10} + \cdots + 1036544 \) Copy content Toggle raw display
$37$ \( T^{12} + 24 T^{11} + \cdots + 2560288768 \) Copy content Toggle raw display
$41$ \( T^{12} - 13 T^{11} - 170 T^{10} + \cdots - 2011068 \) Copy content Toggle raw display
$43$ \( T^{12} + 25 T^{11} + \cdots - 449959160 \) Copy content Toggle raw display
$47$ \( T^{12} + 19 T^{11} + \cdots - 243892224 \) Copy content Toggle raw display
$53$ \( T^{12} + 11 T^{11} + \cdots - 6817093632 \) Copy content Toggle raw display
$59$ \( T^{12} - 8 T^{11} - 377 T^{10} + \cdots - 541472768 \) Copy content Toggle raw display
$61$ \( T^{12} - 7 T^{11} - 139 T^{10} + 770 T^{9} + \cdots - 14 \) Copy content Toggle raw display
$67$ \( T^{12} + 33 T^{11} + \cdots + 4253150328 \) Copy content Toggle raw display
$71$ \( T^{12} - 477 T^{10} + \cdots + 1745519872 \) Copy content Toggle raw display
$73$ \( T^{12} + 34 T^{11} + \cdots - 133064496 \) Copy content Toggle raw display
$79$ \( T^{12} - 498 T^{10} + \cdots + 4997668664 \) Copy content Toggle raw display
$83$ \( T^{12} - 24 T^{11} + \cdots - 4744520192 \) Copy content Toggle raw display
$89$ \( T^{12} + 10 T^{11} - 391 T^{10} + \cdots - 4477728 \) Copy content Toggle raw display
$97$ \( T^{12} + 16 T^{11} + \cdots + 164588990820 \) Copy content Toggle raw display
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