Properties

Label 4025.2.a.w
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 3x^{7} - 6x^{6} + 19x^{5} + 12x^{4} - 34x^{3} - 12x^{2} + 17x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_{7} + \beta_{6}) q^{6} - q^{7} + (\beta_{6} + \beta_{5} + \beta_{2} + 1) q^{8} + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_{7} + \beta_{6}) q^{6} - q^{7} + (\beta_{6} + \beta_{5} + \beta_{2} + 1) q^{8} + \beta_{6} q^{9} + ( - \beta_{7} + \beta_{3} - 1) q^{11} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{12} + (2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{13} - \beta_1 q^{14} + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_1 - 1) q^{16} + ( - \beta_{5} - \beta_{4}) q^{17} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{18} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{2}) q^{19} - \beta_{3} q^{21} + (\beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_1) q^{22} - q^{23} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 1) q^{24} + ( - \beta_{6} + \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{26} + (\beta_{6} - 2 \beta_{3} + \beta_1) q^{27} + ( - \beta_{2} - \beta_1) q^{28} + (\beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{29} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{31} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{32} + (2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{33} + (\beta_{5} - \beta_{3} - \beta_{2}) q^{34} + (\beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{36} + ( - \beta_{7} + \beta_{6} - \beta_{2} + \beta_1 + 2) q^{37} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{38} + (3 \beta_{6} + 2 \beta_{3} - 3 \beta_{2} + 2) q^{39} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + ( - \beta_{7} - \beta_{6}) q^{42} + (\beta_{7} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{43} + ( - 2 \beta_{6} - \beta_{3} + \beta_{2}) q^{44} - \beta_1 q^{46} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + 4 \beta_{2} + 2 \beta_1 - 1) q^{47} + (\beta_{7} + 3 \beta_{6} + \beta_{4} - \beta_{2} + \beta_1 + 4) q^{48} + q^{49} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{51} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{52} + ( - \beta_{6} - 4 \beta_{4} + \beta_{2} - \beta_1 + 1) q^{53} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{54} + ( - \beta_{6} - \beta_{5} - \beta_{2} - 1) q^{56} + (2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{57} + (2 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} + \beta_{2} - \beta_1 + 3) q^{58} + (\beta_{7} + 2 \beta_{6} - 4 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{59} + ( - 2 \beta_{7} - \beta_{6} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 1) q^{61} + ( - 2 \beta_{7} + \beta_{6} + 4 \beta_{5} - \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{62} - \beta_{6} q^{63} + (2 \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{64} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 5) q^{66} + ( - 2 \beta_{6} - 4 \beta_{5} + \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{67} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{68} - \beta_{3} q^{69} + ( - \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{71} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 + 3) q^{72} + ( - 2 \beta_{7} - 4 \beta_{6} - \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{73}+ \cdots + ( - \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} + 2 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{2} + 2 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 12 q^{8} - 5 q^{11} + 3 q^{12} + 9 q^{13} - 3 q^{14} - q^{16} - q^{17} + 6 q^{18} - 4 q^{19} - 2 q^{21} - 5 q^{22} - 8 q^{23} + 16 q^{24} + 22 q^{26} - q^{27} - 5 q^{28} - 5 q^{29} - q^{31} + 2 q^{32} + 12 q^{33} - 2 q^{34} + 16 q^{36} + 18 q^{37} + 14 q^{38} + 14 q^{39} - q^{41} + q^{42} + 20 q^{43} - 3 q^{46} + 10 q^{47} + 31 q^{48} + 8 q^{49} - 4 q^{51} + 11 q^{52} + 11 q^{53} + 29 q^{54} - 12 q^{56} + 8 q^{57} + 24 q^{58} + 20 q^{59} - 6 q^{61} + 2 q^{62} + 8 q^{64} - 37 q^{66} + 23 q^{67} + 9 q^{68} - 2 q^{69} + 3 q^{71} + 29 q^{72} - 8 q^{73} + 35 q^{74} - 29 q^{76} + 5 q^{77} + 31 q^{78} + 4 q^{79} - 44 q^{81} + 27 q^{82} + 4 q^{83} - 3 q^{84} - 18 q^{86} + 27 q^{87} - 4 q^{88} - 17 q^{89} - 9 q^{91} - 5 q^{92} - 7 q^{93} + 13 q^{94} + 22 q^{96} + 41 q^{97} + 3 q^{98} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 6x^{6} + 19x^{5} + 12x^{4} - 34x^{3} - 12x^{2} + 17x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 11\nu^{3} + 15\nu^{2} - 13\nu - 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} + 3\nu^{6} + 5\nu^{5} - 17\nu^{4} - 5\nu^{3} + 23\nu^{2} - 2\nu - 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 3\nu^{6} - 5\nu^{5} + 17\nu^{4} + 6\nu^{3} - 24\nu^{2} - \nu + 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{7} + 3\nu^{6} + 6\nu^{5} - 18\nu^{4} - 11\nu^{3} + 26\nu^{2} + 5\nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 7\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 7\beta_{6} + 6\beta_{5} + \beta_{3} + 9\beta_{2} + 21\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 10\beta_{6} + 8\beta_{5} + \beta_{4} + 9\beta_{3} + 34\beta_{2} + 45\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{7} + 43\beta_{6} + 31\beta_{5} + 3\beta_{4} + 15\beta_{3} + 63\beta_{2} + 122\beta _1 + 56 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.86556
−1.34053
−0.673262
−0.468649
1.07669
1.38128
2.33027
2.55976
−1.86556 1.47264 1.48032 0 −2.74730 −1.00000 0.969501 −0.831331 0
1.2 −1.34053 −2.02792 −0.202970 0 2.71850 −1.00000 2.95316 1.11246 0
1.3 −0.673262 0.897707 −1.54672 0 −0.604392 −1.00000 2.38787 −2.19412 0
1.4 −0.468649 2.11571 −1.78037 0 −0.991526 −1.00000 1.77166 1.47624 0
1.5 1.07669 0.452781 −0.840734 0 0.487505 −1.00000 −3.05860 −2.79499 0
1.6 1.38128 −1.77230 −0.0920619 0 −2.44804 −1.00000 −2.88973 0.141045 0
1.7 2.33027 −1.65734 3.43017 0 −3.86204 −1.00000 3.33268 −0.253237 0
1.8 2.55976 2.51872 4.55237 0 6.44731 −1.00000 6.53345 3.34393 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(23\) \(1\)

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 4025.2.a.w yes 8
5.b even 2 1 4025.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.s 8 5.b even 2 1
4025.2.a.w yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{8} - 3T_{2}^{7} - 6T_{2}^{6} + 19T_{2}^{5} + 12T_{2}^{4} - 34T_{2}^{3} - 12T_{2}^{2} + 17T_{2} + 7 \) Copy content Toggle raw display
\( T_{3}^{8} - 2T_{3}^{7} - 10T_{3}^{6} + 19T_{3}^{5} + 31T_{3}^{4} - 59T_{3}^{3} - 23T_{3}^{2} + 61T_{3} - 19 \) Copy content Toggle raw display
\( T_{11}^{8} + 5T_{11}^{7} - 18T_{11}^{6} - 97T_{11}^{5} + 61T_{11}^{4} + 444T_{11}^{3} + 98T_{11}^{2} - 384T_{11} - 71 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} - 6 T^{6} + 19 T^{5} + \cdots + 7 \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} - 10 T^{6} + 19 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} - 18 T^{6} - 97 T^{5} + \cdots - 71 \) Copy content Toggle raw display
$13$ \( T^{8} - 9 T^{7} - 14 T^{6} + 267 T^{5} + \cdots - 111 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} - 33 T^{6} + 40 T^{5} + \cdots - 7 \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{7} - 61 T^{6} - 192 T^{5} + \cdots - 243 \) Copy content Toggle raw display
$23$ \( (T + 1)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 5 T^{7} - 139 T^{6} + \cdots + 159651 \) Copy content Toggle raw display
$31$ \( T^{8} + T^{7} - 148 T^{6} + \cdots - 195513 \) Copy content Toggle raw display
$37$ \( T^{8} - 18 T^{7} + 80 T^{6} + \cdots + 3375 \) Copy content Toggle raw display
$41$ \( T^{8} + T^{7} - 110 T^{6} + \cdots - 13569 \) Copy content Toggle raw display
$43$ \( T^{8} - 20 T^{7} + 26 T^{6} + \cdots - 650071 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} - 173 T^{6} + \cdots + 352131 \) Copy content Toggle raw display
$53$ \( T^{8} - 11 T^{7} - 239 T^{6} + \cdots + 1118241 \) Copy content Toggle raw display
$59$ \( T^{8} - 20 T^{7} + 19 T^{6} + \cdots - 71079 \) Copy content Toggle raw display
$61$ \( T^{8} + 6 T^{7} - 332 T^{6} + \cdots + 5400047 \) Copy content Toggle raw display
$67$ \( T^{8} - 23 T^{7} - 125 T^{6} + \cdots - 21960131 \) Copy content Toggle raw display
$71$ \( T^{8} - 3 T^{7} - 255 T^{6} + \cdots - 461939 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} - 342 T^{6} + \cdots + 7171111 \) Copy content Toggle raw display
$79$ \( T^{8} - 4 T^{7} - 72 T^{6} + 70 T^{5} + \cdots + 243 \) Copy content Toggle raw display
$83$ \( T^{8} - 4 T^{7} - 393 T^{6} + \cdots + 18949233 \) Copy content Toggle raw display
$89$ \( T^{8} + 17 T^{7} + 8 T^{6} + \cdots + 3313 \) Copy content Toggle raw display
$97$ \( T^{8} - 41 T^{7} + 458 T^{6} + \cdots - 12386987 \) Copy content Toggle raw display
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