Properties

Label 675.4.b.k
Level $675$
Weight $4$
Character orbit 675.b
Analytic conductor $39.826$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2033649216.1
Defining polynomial: \(x^{6} + 47 x^{4} + 541 x^{2} + 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -7 - \beta_{2} + \beta_{4} ) q^{4} + ( 2 \beta_{1} + 14 \beta_{3} ) q^{7} + ( -8 \beta_{1} - 9 \beta_{3} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -7 - \beta_{2} + \beta_{4} ) q^{4} + ( 2 \beta_{1} + 14 \beta_{3} ) q^{7} + ( -8 \beta_{1} - 9 \beta_{3} + \beta_{5} ) q^{8} + ( -12 + 2 \beta_{2} + 4 \beta_{4} ) q^{11} + ( 10 \beta_{1} - 14 \beta_{3} - 4 \beta_{5} ) q^{13} + ( -30 - 16 \beta_{2} + 2 \beta_{4} ) q^{14} + ( 58 + 17 \beta_{2} ) q^{16} + ( 2 \beta_{1} + 3 \beta_{3} - 8 \beta_{5} ) q^{17} + ( -59 - 14 \beta_{2} - 4 \beta_{4} ) q^{19} + ( -42 \beta_{1} + 54 \beta_{3} - 2 \beta_{5} ) q^{22} + ( 32 \beta_{1} - 39 \beta_{3} - 4 \beta_{5} ) q^{23} + ( -126 - 28 \beta_{2} + 10 \beta_{4} ) q^{26} + ( -46 \beta_{1} - 116 \beta_{3} + 16 \beta_{5} ) q^{28} + ( 42 + 42 \beta_{2} + 8 \beta_{4} ) q^{29} + ( 83 - 30 \beta_{2} - 8 \beta_{4} ) q^{31} + ( 11 \beta_{1} + 183 \beta_{3} - 9 \beta_{5} ) q^{32} + ( 18 - 69 \beta_{2} + 2 \beta_{4} ) q^{34} + ( -64 \beta_{1} + 50 \beta_{3} + 8 \beta_{5} ) q^{37} + ( -41 \beta_{1} - 234 \beta_{3} + 14 \beta_{5} ) q^{38} + ( 132 - 42 \beta_{2} + 16 \beta_{4} ) q^{41} + ( -78 \beta_{1} + 16 \beta_{3} - 8 \beta_{5} ) q^{43} + ( 546 - 12 \beta_{2} - 10 \beta_{4} ) q^{44} + ( -456 - 25 \beta_{2} + 32 \beta_{4} ) q^{46} + ( -28 \beta_{1} + 168 \beta_{3} + 4 \beta_{5} ) q^{47} + ( 87 - 60 \beta_{2} + 4 \beta_{4} ) q^{49} + ( -154 \beta_{1} - 472 \beta_{3} - 4 \beta_{5} ) q^{52} + ( 14 \beta_{1} + 165 \beta_{3} - 12 \beta_{5} ) q^{53} + ( 354 + 162 \beta_{2} - 30 \beta_{4} ) q^{56} + ( 20 \beta_{1} + 678 \beta_{3} - 42 \beta_{5} ) q^{58} + ( 78 - 82 \beta_{2} + 12 \beta_{4} ) q^{59} + ( 173 + 60 \beta_{2} - 16 \beta_{4} ) q^{61} + ( 117 \beta_{1} - 498 \beta_{3} + 30 \beta_{5} ) q^{62} + ( 353 - 130 \beta_{2} + 11 \beta_{4} ) q^{64} + ( -52 \beta_{1} + 302 \beta_{3} - 24 \beta_{5} ) q^{67} + ( -51 \beta_{1} - 999 \beta_{3} + 5 \beta_{5} ) q^{68} + ( 192 - 34 \beta_{2} - 60 \beta_{4} ) q^{71} + ( -22 \beta_{1} - 404 \beta_{3} + 60 \beta_{5} ) q^{73} + ( 912 + 78 \beta_{2} - 64 \beta_{4} ) q^{74} + ( 59 + 275 \beta_{2} - 73 \beta_{4} ) q^{76} + ( -56 \beta_{1} - 60 \beta_{3} + 52 \beta_{5} ) q^{77} + ( -221 + 22 \beta_{2} - 12 \beta_{4} ) q^{79} + ( -38 \beta_{1} - 534 \beta_{3} + 42 \beta_{5} ) q^{82} + ( 120 \beta_{1} - 489 \beta_{3} - 60 \beta_{5} ) q^{83} + ( 1218 - 2 \beta_{2} - 78 \beta_{4} ) q^{86} + ( 278 \beta_{1} + 192 \beta_{3} - 4 \beta_{5} ) q^{88} + ( 756 - 66 \beta_{2} + 48 \beta_{4} ) q^{89} + ( -56 - 196 \beta_{2} + 76 \beta_{4} ) q^{91} + ( -481 \beta_{1} - 495 \beta_{3} - 7 \beta_{5} ) q^{92} + ( 396 - 108 \beta_{2} - 28 \beta_{4} ) q^{94} + ( 64 \beta_{1} + 440 \beta_{3} - 8 \beta_{5} ) q^{97} + ( -5 \beta_{1} - 876 \beta_{3} + 60 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 46 q^{4} + O(q^{10}) \) \( 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 47 x^{4} + 541 x^{2} + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 24 \nu^{2} + 6 \)\()/17\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 41 \nu^{3} + 397 \nu \)\()/102\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 41 \nu^{2} + 261 \)\()/17\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} + 157 \nu^{3} + 2007 \nu \)\()/34\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{2} - 15\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 9 \beta_{3} - 24 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-24 \beta_{4} + 41 \beta_{2} + 354\)
\(\nu^{5}\)\(=\)\(-41 \beta_{5} + 471 \beta_{3} + 587 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
5.20067i
4.45938i
0.258712i
0.258712i
4.45938i
5.20067i
5.20067i 0 −19.0470 0 0 24.4013i 57.4517i 0 0
649.2 4.45938i 0 −11.8861 0 0 5.08123i 17.3296i 0 0
649.3 0.258712i 0 7.93307 0 0 14.5174i 4.12208i 0 0
649.4 0.258712i 0 7.93307 0 0 14.5174i 4.12208i 0 0
649.5 4.45938i 0 −11.8861 0 0 5.08123i 17.3296i 0 0
649.6 5.20067i 0 −19.0470 0 0 24.4013i 57.4517i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.4.b.k 6
3.b odd 2 1 675.4.b.l 6
5.b even 2 1 inner 675.4.b.k 6
5.c odd 4 1 135.4.a.g yes 3
5.c odd 4 1 675.4.a.q 3
15.d odd 2 1 675.4.b.l 6
15.e even 4 1 135.4.a.f 3
15.e even 4 1 675.4.a.r 3
20.e even 4 1 2160.4.a.be 3
45.k odd 12 2 405.4.e.r 6
45.l even 12 2 405.4.e.t 6
60.l odd 4 1 2160.4.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 15.e even 4 1
135.4.a.g yes 3 5.c odd 4 1
405.4.e.r 6 45.k odd 12 2
405.4.e.t 6 45.l even 12 2
675.4.a.q 3 5.c odd 4 1
675.4.a.r 3 15.e even 4 1
675.4.b.k 6 1.a even 1 1 trivial
675.4.b.k 6 5.b even 2 1 inner
675.4.b.l 6 3.b odd 2 1
675.4.b.l 6 15.d odd 2 1
2160.4.a.be 3 20.e even 4 1
2160.4.a.bm 3 60.l odd 4 1

Hecke kernels

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

\( T_{2}^{6} + 47 T_{2}^{4} + 541 T_{2}^{2} + 36 \)
\( T_{7}^{6} + 832 T_{7}^{4} + 146304 T_{7}^{2} + 3240000 \)
\( T_{11}^{3} + 38 T_{11}^{2} - 2612 T_{11} - 83280 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 36 + 541 T^{2} + 47 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( T^{6} \)
$7$ \( 3240000 + 146304 T^{2} + 832 T^{4} + T^{6} \)
$11$ \( ( -83280 - 2612 T + 38 T^{2} + T^{3} )^{2} \)
$13$ \( 10024014400 + 26546496 T^{2} + 9936 T^{4} + T^{6} \)
$17$ \( 306790808769 + 152769235 T^{2} + 23315 T^{4} + T^{6} \)
$19$ \( ( -525871 + 3587 T + 187 T^{2} + T^{3} )^{2} \)
$23$ \( 4177858328361 + 874061523 T^{2} + 53163 T^{4} + T^{6} \)
$29$ \( ( 7892760 - 47768 T - 160 T^{2} + T^{3} )^{2} \)
$31$ \( ( -246321 - 17973 T - 227 T^{2} + T^{3} )^{2} \)
$37$ \( 185969950926400 + 12112190256 T^{2} + 205932 T^{4} + T^{6} \)
$41$ \( ( 12116640 - 42812 T - 338 T^{2} + T^{3} )^{2} \)
$43$ \( 340939975993600 + 26385567696 T^{2} + 320316 T^{4} + T^{6} \)
$47$ \( 80202240000 + 3205848064 T^{2} + 114368 T^{4} + T^{6} \)
$53$ \( 883203364521 + 4747623907 T^{2} + 148667 T^{4} + T^{6} \)
$59$ \( ( 34131480 - 166448 T - 140 T^{2} + T^{3} )^{2} \)
$61$ \( ( 1782607 - 2749 T - 595 T^{2} + T^{3} )^{2} \)
$67$ \( 127577025000000 + 25602118704 T^{2} + 618988 T^{4} + T^{6} \)
$71$ \( ( 280550880 - 583652 T - 602 T^{2} + T^{3} )^{2} \)
$73$ \( 163289762572038400 + 1054986577104 T^{2} + 1866252 T^{4} + T^{6} \)
$79$ \( ( 2010303 + 97059 T + 629 T^{2} + T^{3} )^{2} \)
$83$ \( 119996015960256921 + 1025510933043 T^{2} + 2388123 T^{4} + T^{6} \)
$89$ \( ( -74325600 + 1057572 T - 2154 T^{2} + T^{3} )^{2} \)
$97$ \( 4044390164070400 + 118816444416 T^{2} + 849792 T^{4} + T^{6} \)
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