Properties

Label 675.4.a.q
Level $675$
Weight $4$
Character orbit 675.a
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 7) q^{4} + ( - 2 \beta_1 - 14) q^{7} + ( - \beta_{2} - 8 \beta_1 - 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 7) q^{4} + ( - 2 \beta_1 - 14) q^{7} + ( - \beta_{2} - 8 \beta_1 - 9) q^{8} + ( - 4 \beta_{2} + 2 \beta_1 - 12) q^{11} + (4 \beta_{2} + 10 \beta_1 - 14) q^{13} + (2 \beta_{2} + 16 \beta_1 + 30) q^{14} + (17 \beta_1 + 58) q^{16} + ( - 8 \beta_{2} - 2 \beta_1 - 3) q^{17} + ( - 4 \beta_{2} + 14 \beta_1 + 59) q^{19} + ( - 2 \beta_{2} + 42 \beta_1 - 54) q^{22} + (4 \beta_{2} + 32 \beta_1 - 39) q^{23} + ( - 10 \beta_{2} - 28 \beta_1 - 126) q^{26} + ( - 16 \beta_{2} - 46 \beta_1 - 116) q^{28} + (8 \beta_{2} - 42 \beta_1 - 42) q^{29} + (8 \beta_{2} - 30 \beta_1 + 83) q^{31} + ( - 9 \beta_{2} - 11 \beta_1 - 183) q^{32} + (2 \beta_{2} + 69 \beta_1 - 18) q^{34} + (8 \beta_{2} + 64 \beta_1 - 50) q^{37} + ( - 14 \beta_{2} - 41 \beta_1 - 234) q^{38} + ( - 16 \beta_{2} - 42 \beta_1 + 132) q^{41} + (8 \beta_{2} - 78 \beta_1 + 16) q^{43} + ( - 10 \beta_{2} + 12 \beta_1 - 546) q^{44} + ( - 32 \beta_{2} - 25 \beta_1 - 456) q^{46} + (4 \beta_{2} + 28 \beta_1 - 168) q^{47} + (4 \beta_{2} + 60 \beta_1 - 87) q^{49} + ( - 4 \beta_{2} + 154 \beta_1 + 472) q^{52} + (12 \beta_{2} + 14 \beta_1 + 165) q^{53} + (30 \beta_{2} + 162 \beta_1 + 354) q^{56} + (42 \beta_{2} + 20 \beta_1 + 678) q^{58} + (12 \beta_{2} + 82 \beta_1 - 78) q^{59} + (16 \beta_{2} + 60 \beta_1 + 173) q^{61} + (30 \beta_{2} - 117 \beta_1 + 498) q^{62} + (11 \beta_{2} + 130 \beta_1 - 353) q^{64} + ( - 24 \beta_{2} + 52 \beta_1 - 302) q^{67} + ( - 5 \beta_{2} - 51 \beta_1 - 999) q^{68} + (60 \beta_{2} - 34 \beta_1 + 192) q^{71} + ( - 60 \beta_{2} - 22 \beta_1 - 404) q^{73} + ( - 64 \beta_{2} - 78 \beta_1 - 912) q^{74} + (73 \beta_{2} + 275 \beta_1 + 59) q^{76} + (52 \beta_{2} + 56 \beta_1 + 60) q^{77} + ( - 12 \beta_{2} - 22 \beta_1 + 221) q^{79} + (42 \beta_{2} + 38 \beta_1 + 534) q^{82} + (60 \beta_{2} + 120 \beta_1 - 489) q^{83} + (78 \beta_{2} - 2 \beta_1 + 1218) q^{86} + (4 \beta_{2} + 278 \beta_1 + 192) q^{88} + (48 \beta_{2} + 66 \beta_1 - 756) q^{89} + ( - 76 \beta_{2} - 196 \beta_1 - 56) q^{91} + ( - 7 \beta_{2} + 481 \beta_1 + 495) q^{92} + ( - 28 \beta_{2} + 108 \beta_1 - 396) q^{94} + ( - 8 \beta_{2} - 64 \beta_1 - 440) q^{97} + ( - 60 \beta_{2} - 5 \beta_1 - 876) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 23 q^{4} - 44 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 23 q^{4} - 44 q^{7} - 36 q^{8} - 38 q^{11} - 28 q^{13} + 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} - 122 q^{22} - 81 q^{23} - 416 q^{26} - 410 q^{28} - 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} - 78 q^{37} - 757 q^{38} + 338 q^{41} - 22 q^{43} - 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} + 1566 q^{52} + 521 q^{53} + 1254 q^{56} + 2096 q^{58} - 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} - 878 q^{67} - 3053 q^{68} + 602 q^{71} - 1294 q^{73} - 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 1682 q^{82} - 1287 q^{83} + 3730 q^{86} + 858 q^{88} - 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} - 1392 q^{97} - 2693 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 23x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 15 \) 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
5.20067
0.258712
−4.45938
−5.20067 0 19.0470 0 0 −24.4013 −57.4517 0 0
1.2 −0.258712 0 −7.93307 0 0 −14.5174 4.12208 0 0
1.3 4.45938 0 11.8861 0 0 −5.08123 17.3296 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 675.4.a.q 3
3.b odd 2 1 675.4.a.r 3
5.b even 2 1 135.4.a.g yes 3
5.c odd 4 2 675.4.b.k 6
15.d odd 2 1 135.4.a.f 3
15.e even 4 2 675.4.b.l 6
20.d odd 2 1 2160.4.a.be 3
45.h odd 6 2 405.4.e.t 6
45.j even 6 2 405.4.e.r 6
60.h even 2 1 2160.4.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 15.d odd 2 1
135.4.a.g yes 3 5.b even 2 1
405.4.e.r 6 45.j even 6 2
405.4.e.t 6 45.h odd 6 2
675.4.a.q 3 1.a even 1 1 trivial
675.4.a.r 3 3.b odd 2 1
675.4.b.k 6 5.c odd 4 2
675.4.b.l 6 15.e even 4 2
2160.4.a.be 3 20.d odd 2 1
2160.4.a.bm 3 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2}^{3} + T_{2}^{2} - 23T_{2} - 6 \) Copy content Toggle raw display
\( T_{7}^{3} + 44T_{7}^{2} + 552T_{7} + 1800 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 23T - 6 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 44 T^{2} + 552 T + 1800 \) Copy content Toggle raw display
$11$ \( T^{3} + 38 T^{2} - 2612 T - 83280 \) Copy content Toggle raw display
$13$ \( T^{3} + 28 T^{2} - 4576 T - 100120 \) Copy content Toggle raw display
$17$ \( T^{3} + 19 T^{2} - 11477 T - 553887 \) Copy content Toggle raw display
$19$ \( T^{3} - 187 T^{2} + 3587 T + 525871 \) Copy content Toggle raw display
$23$ \( T^{3} + 81 T^{2} - 23301 T - 2043981 \) Copy content Toggle raw display
$29$ \( T^{3} + 160 T^{2} - 47768 T - 7892760 \) Copy content Toggle raw display
$31$ \( T^{3} - 227 T^{2} - 17973 T - 246321 \) Copy content Toggle raw display
$37$ \( T^{3} + 78 T^{2} - 99924 T - 13637080 \) Copy content Toggle raw display
$41$ \( T^{3} - 338 T^{2} + \cdots + 12116640 \) Copy content Toggle raw display
$43$ \( T^{3} + 22 T^{2} - 159916 T - 18464560 \) Copy content Toggle raw display
$47$ \( T^{3} + 472 T^{2} + 54208 T - 283200 \) Copy content Toggle raw display
$53$ \( T^{3} - 521 T^{2} + 61387 T + 939789 \) Copy content Toggle raw display
$59$ \( T^{3} + 140 T^{2} + \cdots - 34131480 \) Copy content Toggle raw display
$61$ \( T^{3} - 595 T^{2} - 2749 T + 1782607 \) Copy content Toggle raw display
$67$ \( T^{3} + 878 T^{2} + \cdots - 11295000 \) Copy content Toggle raw display
$71$ \( T^{3} - 602 T^{2} + \cdots + 280550880 \) Copy content Toggle raw display
$73$ \( T^{3} + 1294 T^{2} + \cdots - 404091280 \) Copy content Toggle raw display
$79$ \( T^{3} - 629 T^{2} + 97059 T - 2010303 \) Copy content Toggle raw display
$83$ \( T^{3} + 1287 T^{2} + \cdots - 346404411 \) Copy content Toggle raw display
$89$ \( T^{3} + 2154 T^{2} + \cdots + 74325600 \) Copy content Toggle raw display
$97$ \( T^{3} + 1392 T^{2} + \cdots + 63595520 \) Copy content Toggle raw display
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