Properties

Label 575.2.a.k
Level $575$
Weight $2$
Character orbit 575.a
Self dual yes
Analytic conductor $4.591$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - x^{6} - 12x^{5} + 9x^{4} + 43x^{3} - 14x^{2} - 49x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{6} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{6} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{11} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{12} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{13} + (\beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 2) q^{14} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{16} + (2 \beta_{6} + 2) q^{17} + (2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - 2) q^{18} + ( - \beta_{6} + \beta_{4} + \beta_1 + 2) q^{19} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{21} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 4) q^{22} - q^{23} + ( - \beta_{6} - \beta_{5} - \beta_{3} + 2) q^{24} + ( - \beta_{6} + \beta_{5} - 2 \beta_{2} - 4) q^{26} + ( - \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1 - 1) q^{27} + ( - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{28} + ( - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2}) q^{29} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{31} + ( - \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 + 2) q^{32} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{33} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{34} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 2) q^{36} + (2 \beta_{3} + 2 \beta_{2}) q^{37} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_1 - 4) q^{38} + (3 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1 - 1) q^{39} + (\beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{41} + ( - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} + 6) q^{42} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{43} + (\beta_{6} + \beta_{4} - 2 \beta_{3} - 3 \beta_1) q^{44} + \beta_1 q^{46} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{47} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 2) q^{48} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 7) q^{49} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{51} + (2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 5 \beta_1 + 2) q^{52} + (2 \beta_{6} - 2 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{6} - 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{54} + (3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 7 \beta_1) q^{56} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{57} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{58} + ( - \beta_{6} + \beta_{4} + 2 \beta_{2} - 3 \beta_1 - 1) q^{59} + (2 \beta_{5} - 2 \beta_1 + 6) q^{61} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 3) q^{62} + ( - \beta_{6} - 3 \beta_{4} + 4 \beta_{2} + \beta_1 + 8) q^{63} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{64} + (2 \beta_{6} + 4 \beta_{4} - 6) q^{66} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{67} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 6) q^{68} - \beta_{5} q^{69} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{71} + (4 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 1) q^{72} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{73} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 6) q^{74} + (\beta_{6} + \beta_{4} + 2 \beta_{2} + 3 \beta_1 + 6) q^{76} + ( - \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{77} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{78} + ( - \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{79} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{3} + 4 \beta_1 + 5) q^{81} + (3 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{82} + (\beta_{6} + \beta_{4} + 2 \beta_{2} - \beta_1 + 4) q^{83} + (6 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 8) q^{84} + ( - \beta_{6} - 4 \beta_{5} - 3 \beta_{4} - \beta_1 - 6) q^{86} + (\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 7) q^{87} + (3 \beta_{6} - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 10) q^{88} + ( - 4 \beta_{5} - 2 \beta_{3} - 2 \beta_1 + 2) q^{89} + ( - \beta_{6} - 4 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 2) q^{91} + ( - \beta_{2} - 2) q^{92} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{93} + (2 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 3 \beta_1 + 2) q^{94} + (3 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{96} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 4) q^{97} + ( - \beta_{6} + 4 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - 6 \beta_1 - 10) q^{98} + (\beta_{6} + 4 \beta_{5} + 3 \beta_{4} - 2 \beta_{2} + 5 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 11 q^{4} + 5 q^{6} + 3 q^{7} - 6 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 11 q^{4} + 5 q^{6} + 3 q^{7} - 6 q^{8} + 15 q^{9} - q^{11} + 6 q^{12} - 3 q^{13} + 7 q^{14} + 7 q^{16} + 10 q^{17} - 24 q^{18} + 15 q^{19} + 2 q^{21} + 21 q^{22} - 7 q^{23} + 18 q^{24} - 20 q^{26} - 11 q^{28} + 3 q^{29} + 14 q^{31} + 17 q^{32} + 6 q^{33} + 20 q^{34} - 10 q^{37} - 31 q^{38} - 8 q^{39} + 19 q^{41} + 44 q^{42} + 5 q^{43} - 3 q^{44} + q^{46} - 14 q^{47} - 27 q^{48} + 40 q^{49} + 2 q^{51} + 16 q^{52} + 4 q^{53} - q^{54} - 9 q^{56} - 4 q^{57} - 13 q^{58} - 16 q^{59} + 40 q^{61} - 12 q^{62} + 53 q^{63} - 4 q^{64} - 54 q^{66} - 4 q^{67} + 20 q^{68} - 14 q^{71} - 6 q^{72} - 3 q^{73} - 18 q^{74} + 35 q^{76} - 17 q^{77} + 23 q^{78} - q^{79} + 47 q^{81} - 22 q^{82} + 17 q^{83} - 60 q^{84} - 35 q^{86} - 56 q^{87} + 57 q^{88} + 16 q^{89} + 25 q^{91} - 11 q^{92} + 14 q^{93} + 7 q^{94} - 19 q^{96} - 24 q^{97} - 46 q^{98} - 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 12x^{5} + 9x^{4} + 43x^{3} - 14x^{2} - 49x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{6} - \nu^{5} - 17\nu^{4} + 7\nu^{3} + 27\nu^{2} - 2\nu + 1 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{5} - 11\nu^{4} - 19\nu^{3} + 31\nu^{2} + 39\nu - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 11\nu^{4} - 26\nu^{3} - 31\nu^{2} + 41\nu + 22 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} + 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + \beta_{5} + 9\beta_{3} + 29\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{6} - 8\beta_{5} + 11\beta_{4} + \beta_{3} + 46\beta_{2} - 2\beta _1 + 132 \) 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
2.53289
2.27220
1.69496
−0.202227
−1.07994
−1.63662
−2.58128
−2.53289 0.345624 4.41555 0 −0.875428 −5.12894 −6.11832 −2.88054 0
1.2 −2.27220 −3.13672 3.16289 0 7.12726 4.34930 −2.64233 6.83902 0
1.3 −1.69496 3.30905 0.872898 0 −5.60872 0.852729 1.91040 7.94982 0
1.4 0.202227 −2.69619 −1.95910 0 −0.545243 −2.81698 −0.800639 4.26945 0
1.5 1.07994 −1.06928 −0.833738 0 −1.15476 2.95289 −3.06026 −1.85663 0
1.6 1.63662 2.46212 0.678510 0 4.02954 4.74965 −2.16277 3.06202 0
1.7 2.58128 0.785406 4.66299 0 2.02735 −1.95865 6.87392 −2.38314 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

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

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 575.2.a.k 7
3.b odd 2 1 5175.2.a.cg 7
4.b odd 2 1 9200.2.a.da 7
5.b even 2 1 575.2.a.l yes 7
5.c odd 4 2 575.2.b.f 14
15.d odd 2 1 5175.2.a.cb 7
20.d odd 2 1 9200.2.a.db 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.a.k 7 1.a even 1 1 trivial
575.2.a.l yes 7 5.b even 2 1
575.2.b.f 14 5.c odd 4 2
5175.2.a.cb 7 15.d odd 2 1
5175.2.a.cg 7 3.b odd 2 1
9200.2.a.da 7 4.b odd 2 1
9200.2.a.db 7 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(575))\):

\( T_{2}^{7} + T_{2}^{6} - 12T_{2}^{5} - 9T_{2}^{4} + 43T_{2}^{3} + 14T_{2}^{2} - 49T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{7} - 18T_{3}^{5} + 85T_{3}^{3} - 8T_{3}^{2} - 65T_{3} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + T^{6} - 12 T^{5} - 9 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{7} - 18 T^{5} + 85 T^{3} - 8 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 3 T^{6} - 40 T^{5} + \cdots + 1472 \) Copy content Toggle raw display
$11$ \( T^{7} + T^{6} - 58 T^{5} - 84 T^{4} + \cdots - 4800 \) Copy content Toggle raw display
$13$ \( T^{7} + 3 T^{6} - 46 T^{5} + \cdots - 1637 \) Copy content Toggle raw display
$17$ \( T^{7} - 10 T^{6} - 36 T^{5} + \cdots + 46080 \) Copy content Toggle raw display
$19$ \( T^{7} - 15 T^{6} + 30 T^{5} + \cdots - 1600 \) Copy content Toggle raw display
$23$ \( (T + 1)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} - 3 T^{6} - 122 T^{5} + \cdots - 3375 \) Copy content Toggle raw display
$31$ \( T^{7} - 14 T^{6} + 14 T^{5} + \cdots - 24350 \) Copy content Toggle raw display
$37$ \( T^{7} + 10 T^{6} - 116 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$41$ \( T^{7} - 19 T^{6} + 22 T^{5} + \cdots + 14217 \) Copy content Toggle raw display
$43$ \( T^{7} - 5 T^{6} - 144 T^{5} + \cdots - 14400 \) Copy content Toggle raw display
$47$ \( T^{7} + 14 T^{6} - 18 T^{5} + \cdots + 31170 \) Copy content Toggle raw display
$53$ \( T^{7} - 4 T^{6} - 124 T^{5} + \cdots - 3456 \) Copy content Toggle raw display
$59$ \( T^{7} + 16 T^{6} - 91 T^{5} + \cdots + 149340 \) Copy content Toggle raw display
$61$ \( T^{7} - 40 T^{6} + 544 T^{5} + \cdots + 79616 \) Copy content Toggle raw display
$67$ \( T^{7} + 4 T^{6} - 224 T^{5} + \cdots + 46080 \) Copy content Toggle raw display
$71$ \( T^{7} + 14 T^{6} - 18 T^{5} + \cdots + 31170 \) Copy content Toggle raw display
$73$ \( T^{7} + 3 T^{6} - 186 T^{5} + \cdots - 603981 \) Copy content Toggle raw display
$79$ \( T^{7} + T^{6} - 232 T^{5} + \cdots + 22720 \) Copy content Toggle raw display
$83$ \( T^{7} - 17 T^{6} - 40 T^{5} + \cdots + 192 \) Copy content Toggle raw display
$89$ \( T^{7} - 16 T^{6} - 244 T^{5} + \cdots + 2478720 \) Copy content Toggle raw display
$97$ \( T^{7} + 24 T^{6} + 4 T^{5} - 2232 T^{4} + \cdots - 896 \) Copy content Toggle raw display
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