Properties

Label 675.4.b.l
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: \( x^{6} + 47x^{4} + 541x^{2} + 36 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} - \beta_{2} - 7) q^{4} + ( - 14 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{5} - 9 \beta_{3} - 8 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} - \beta_{2} - 7) q^{4} + ( - 14 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{5} - 9 \beta_{3} - 8 \beta_1) q^{8} + ( - 4 \beta_{4} - 2 \beta_{2} + 12) q^{11} + (4 \beta_{5} + 14 \beta_{3} - 10 \beta_1) q^{13} + ( - 2 \beta_{4} + 16 \beta_{2} + 30) q^{14} + (17 \beta_{2} + 58) q^{16} + ( - 8 \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{17} + ( - 4 \beta_{4} - 14 \beta_{2} - 59) q^{19} + (2 \beta_{5} - 54 \beta_{3} + 42 \beta_1) q^{22} + ( - 4 \beta_{5} - 39 \beta_{3} + 32 \beta_1) q^{23} + ( - 10 \beta_{4} + 28 \beta_{2} + 126) q^{26} + ( - 16 \beta_{5} + 116 \beta_{3} + 46 \beta_1) q^{28} + ( - 8 \beta_{4} - 42 \beta_{2} - 42) q^{29} + ( - 8 \beta_{4} - 30 \beta_{2} + 83) q^{31} + ( - 9 \beta_{5} + 183 \beta_{3} + 11 \beta_1) q^{32} + (2 \beta_{4} - 69 \beta_{2} + 18) q^{34} + ( - 8 \beta_{5} - 50 \beta_{3} + 64 \beta_1) q^{37} + (14 \beta_{5} - 234 \beta_{3} - 41 \beta_1) q^{38} + ( - 16 \beta_{4} + 42 \beta_{2} - 132) q^{41} + (8 \beta_{5} - 16 \beta_{3} + 78 \beta_1) q^{43} + (10 \beta_{4} + 12 \beta_{2} - 546) q^{44} + (32 \beta_{4} - 25 \beta_{2} - 456) q^{46} + (4 \beta_{5} + 168 \beta_{3} - 28 \beta_1) q^{47} + (4 \beta_{4} - 60 \beta_{2} + 87) q^{49} + (4 \beta_{5} + 472 \beta_{3} + 154 \beta_1) q^{52} + ( - 12 \beta_{5} + 165 \beta_{3} + 14 \beta_1) q^{53} + (30 \beta_{4} - 162 \beta_{2} - 354) q^{56} + (42 \beta_{5} - 678 \beta_{3} - 20 \beta_1) q^{58} + ( - 12 \beta_{4} + 82 \beta_{2} - 78) q^{59} + ( - 16 \beta_{4} + 60 \beta_{2} + 173) q^{61} + (30 \beta_{5} - 498 \beta_{3} + 117 \beta_1) q^{62} + (11 \beta_{4} - 130 \beta_{2} + 353) q^{64} + (24 \beta_{5} - 302 \beta_{3} + 52 \beta_1) q^{67} + (5 \beta_{5} - 999 \beta_{3} - 51 \beta_1) q^{68} + (60 \beta_{4} + 34 \beta_{2} - 192) q^{71} + ( - 60 \beta_{5} + 404 \beta_{3} + 22 \beta_1) q^{73} + (64 \beta_{4} - 78 \beta_{2} - 912) q^{74} + ( - 73 \beta_{4} + 275 \beta_{2} + 59) q^{76} + (52 \beta_{5} - 60 \beta_{3} - 56 \beta_1) q^{77} + ( - 12 \beta_{4} + 22 \beta_{2} - 221) q^{79} + ( - 42 \beta_{5} + 534 \beta_{3} + 38 \beta_1) q^{82} + ( - 60 \beta_{5} - 489 \beta_{3} + 120 \beta_1) q^{83} + (78 \beta_{4} + 2 \beta_{2} - 1218) q^{86} + (4 \beta_{5} - 192 \beta_{3} - 278 \beta_1) q^{88} + ( - 48 \beta_{4} + 66 \beta_{2} - 756) q^{89} + (76 \beta_{4} - 196 \beta_{2} - 56) q^{91} + ( - 7 \beta_{5} - 495 \beta_{3} - 481 \beta_1) q^{92} + ( - 28 \beta_{4} - 108 \beta_{2} + 396) q^{94} + (8 \beta_{5} - 440 \beta_{3} - 64 \beta_1) q^{97} + (60 \beta_{5} - 876 \beta_{3} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 46 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 46 q^{4} + 76 q^{11} + 216 q^{14} + 382 q^{16} - 374 q^{19} + 832 q^{26} - 320 q^{29} + 454 q^{31} - 34 q^{34} - 676 q^{41} - 3272 q^{44} - 2850 q^{46} + 394 q^{49} - 2508 q^{56} - 280 q^{59} + 1190 q^{61} + 1836 q^{64} - 1204 q^{71} - 5756 q^{74} + 1050 q^{76} - 1258 q^{79} - 7460 q^{86} - 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 24\nu^{2} + 6 ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 41\nu^{3} + 397\nu ) / 102 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 41\nu^{2} + 261 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} + 157\nu^{3} + 2007\nu ) / 34 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 9\beta_{3} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -24\beta_{4} + 41\beta_{2} + 354 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -41\beta_{5} + 471\beta_{3} + 587\beta_1 \) Copy content Toggle raw display

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 47 T^{4} + 541 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 832 T^{4} + 146304 T^{2} + \cdots + 3240000 \) Copy content Toggle raw display
$11$ \( (T^{3} - 38 T^{2} - 2612 T + 83280)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 9936 T^{4} + \cdots + 10024014400 \) Copy content Toggle raw display
$17$ \( T^{6} + 23315 T^{4} + \cdots + 306790808769 \) Copy content Toggle raw display
$19$ \( (T^{3} + 187 T^{2} + 3587 T - 525871)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 53163 T^{4} + \cdots + 4177858328361 \) Copy content Toggle raw display
$29$ \( (T^{3} + 160 T^{2} - 47768 T - 7892760)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 227 T^{2} - 17973 T - 246321)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 185969950926400 \) Copy content Toggle raw display
$41$ \( (T^{3} + 338 T^{2} - 42812 T - 12116640)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 340939975993600 \) Copy content Toggle raw display
$47$ \( T^{6} + 114368 T^{4} + \cdots + 80202240000 \) Copy content Toggle raw display
$53$ \( T^{6} + 148667 T^{4} + \cdots + 883203364521 \) Copy content Toggle raw display
$59$ \( (T^{3} + 140 T^{2} - 166448 T - 34131480)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 595 T^{2} - 2749 T + 1782607)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 127577025000000 \) Copy content Toggle raw display
$71$ \( (T^{3} + 602 T^{2} - 583652 T - 280550880)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 1866252 T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} + 629 T^{2} + 97059 T + 2010303)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 2388123 T^{4} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2154 T^{2} + 1057572 T + 74325600)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 849792 T^{4} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
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