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: \(x^{3} - x^{2} - 23 x + 6\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( -14 - 2 \beta_{1} ) q^{7} + ( -9 - 8 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( -14 - 2 \beta_{1} ) q^{7} + ( -9 - 8 \beta_{1} - \beta_{2} ) q^{8} + ( -12 + 2 \beta_{1} - 4 \beta_{2} ) q^{11} + ( -14 + 10 \beta_{1} + 4 \beta_{2} ) q^{13} + ( 30 + 16 \beta_{1} + 2 \beta_{2} ) q^{14} + ( 58 + 17 \beta_{1} ) q^{16} + ( -3 - 2 \beta_{1} - 8 \beta_{2} ) q^{17} + ( 59 + 14 \beta_{1} - 4 \beta_{2} ) q^{19} + ( -54 + 42 \beta_{1} - 2 \beta_{2} ) q^{22} + ( -39 + 32 \beta_{1} + 4 \beta_{2} ) q^{23} + ( -126 - 28 \beta_{1} - 10 \beta_{2} ) q^{26} + ( -116 - 46 \beta_{1} - 16 \beta_{2} ) q^{28} + ( -42 - 42 \beta_{1} + 8 \beta_{2} ) q^{29} + ( 83 - 30 \beta_{1} + 8 \beta_{2} ) q^{31} + ( -183 - 11 \beta_{1} - 9 \beta_{2} ) q^{32} + ( -18 + 69 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -50 + 64 \beta_{1} + 8 \beta_{2} ) q^{37} + ( -234 - 41 \beta_{1} - 14 \beta_{2} ) q^{38} + ( 132 - 42 \beta_{1} - 16 \beta_{2} ) q^{41} + ( 16 - 78 \beta_{1} + 8 \beta_{2} ) q^{43} + ( -546 + 12 \beta_{1} - 10 \beta_{2} ) q^{44} + ( -456 - 25 \beta_{1} - 32 \beta_{2} ) q^{46} + ( -168 + 28 \beta_{1} + 4 \beta_{2} ) q^{47} + ( -87 + 60 \beta_{1} + 4 \beta_{2} ) q^{49} + ( 472 + 154 \beta_{1} - 4 \beta_{2} ) q^{52} + ( 165 + 14 \beta_{1} + 12 \beta_{2} ) q^{53} + ( 354 + 162 \beta_{1} + 30 \beta_{2} ) q^{56} + ( 678 + 20 \beta_{1} + 42 \beta_{2} ) q^{58} + ( -78 + 82 \beta_{1} + 12 \beta_{2} ) q^{59} + ( 173 + 60 \beta_{1} + 16 \beta_{2} ) q^{61} + ( 498 - 117 \beta_{1} + 30 \beta_{2} ) q^{62} + ( -353 + 130 \beta_{1} + 11 \beta_{2} ) q^{64} + ( -302 + 52 \beta_{1} - 24 \beta_{2} ) q^{67} + ( -999 - 51 \beta_{1} - 5 \beta_{2} ) q^{68} + ( 192 - 34 \beta_{1} + 60 \beta_{2} ) q^{71} + ( -404 - 22 \beta_{1} - 60 \beta_{2} ) q^{73} + ( -912 - 78 \beta_{1} - 64 \beta_{2} ) q^{74} + ( 59 + 275 \beta_{1} + 73 \beta_{2} ) q^{76} + ( 60 + 56 \beta_{1} + 52 \beta_{2} ) q^{77} + ( 221 - 22 \beta_{1} - 12 \beta_{2} ) q^{79} + ( 534 + 38 \beta_{1} + 42 \beta_{2} ) q^{82} + ( -489 + 120 \beta_{1} + 60 \beta_{2} ) q^{83} + ( 1218 - 2 \beta_{1} + 78 \beta_{2} ) q^{86} + ( 192 + 278 \beta_{1} + 4 \beta_{2} ) q^{88} + ( -756 + 66 \beta_{1} + 48 \beta_{2} ) q^{89} + ( -56 - 196 \beta_{1} - 76 \beta_{2} ) q^{91} + ( 495 + 481 \beta_{1} - 7 \beta_{2} ) q^{92} + ( -396 + 108 \beta_{1} - 28 \beta_{2} ) q^{94} + ( -440 - 64 \beta_{1} - 8 \beta_{2} ) q^{97} + ( -876 - 5 \beta_{1} - 60 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 23 q^{4} - 44 q^{7} - 36 q^{8} + O(q^{10}) \) \( 3 q - q^{2} + 23 q^{4} - 44 q^{7} - 36 q^{8} - 38 q^{11} - 28 q^{13} + 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} - 122 q^{22} - 81 q^{23} - 416 q^{26} - 410 q^{28} - 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} - 78 q^{37} - 757 q^{38} + 338 q^{41} - 22 q^{43} - 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} + 1566 q^{52} + 521 q^{53} + 1254 q^{56} + 2096 q^{58} - 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} - 878 q^{67} - 3053 q^{68} + 602 q^{71} - 1294 q^{73} - 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 1682 q^{82} - 1287 q^{83} + 3730 q^{86} + 858 q^{88} - 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} - 1392 q^{97} - 2693 q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 15\)

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):

\( T_{2}^{3} + T_{2}^{2} - 23 T_{2} - 6 \)
\( T_{7}^{3} + 44 T_{7}^{2} + 552 T_{7} + 1800 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 - 23 T + T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( 1800 + 552 T + 44 T^{2} + T^{3} \)
$11$ \( -83280 - 2612 T + 38 T^{2} + T^{3} \)
$13$ \( -100120 - 4576 T + 28 T^{2} + T^{3} \)
$17$ \( -553887 - 11477 T + 19 T^{2} + T^{3} \)
$19$ \( 525871 + 3587 T - 187 T^{2} + T^{3} \)
$23$ \( -2043981 - 23301 T + 81 T^{2} + T^{3} \)
$29$ \( -7892760 - 47768 T + 160 T^{2} + T^{3} \)
$31$ \( -246321 - 17973 T - 227 T^{2} + T^{3} \)
$37$ \( -13637080 - 99924 T + 78 T^{2} + T^{3} \)
$41$ \( 12116640 - 42812 T - 338 T^{2} + T^{3} \)
$43$ \( -18464560 - 159916 T + 22 T^{2} + T^{3} \)
$47$ \( -283200 + 54208 T + 472 T^{2} + T^{3} \)
$53$ \( 939789 + 61387 T - 521 T^{2} + T^{3} \)
$59$ \( -34131480 - 166448 T + 140 T^{2} + T^{3} \)
$61$ \( 1782607 - 2749 T - 595 T^{2} + T^{3} \)
$67$ \( -11295000 + 75948 T + 878 T^{2} + T^{3} \)
$71$ \( 280550880 - 583652 T - 602 T^{2} + T^{3} \)
$73$ \( -404091280 - 95908 T + 1294 T^{2} + T^{3} \)
$79$ \( -2010303 + 97059 T - 629 T^{2} + T^{3} \)
$83$ \( -346404411 - 365877 T + 1287 T^{2} + T^{3} \)
$89$ \( 74325600 + 1057572 T + 2154 T^{2} + T^{3} \)
$97$ \( 63595520 + 543936 T + 1392 T^{2} + T^{3} \)
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