Properties

Label 675.4.b.n
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: 6.0.12559936.1
Defining polynomial: \(x^{6} - 6 x^{3} + 49 x^{2} - 42 x + 18\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{2} - \beta_{4} ) q^{2} + ( -7 + 3 \beta_{1} - \beta_{3} ) q^{4} + ( -2 \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{7} + ( -29 \beta_{2} + 7 \beta_{4} + 5 \beta_{5} ) q^{8} +O(q^{10})\) \( q + ( 2 \beta_{2} - \beta_{4} ) q^{2} + ( -7 + 3 \beta_{1} - \beta_{3} ) q^{4} + ( -2 \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{7} + ( -29 \beta_{2} + 7 \beta_{4} + 5 \beta_{5} ) q^{8} + ( -3 + 9 \beta_{1} - 5 \beta_{3} ) q^{11} + ( -5 \beta_{2} + 6 \beta_{4} + 2 \beta_{5} ) q^{13} + ( -13 - 16 \beta_{1} + 5 \beta_{3} ) q^{14} + ( 69 - 37 \beta_{1} + 9 \beta_{3} ) q^{16} + ( 51 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} ) q^{17} + ( 28 - 33 \beta_{1} + \beta_{3} ) q^{19} + ( -95 \beta_{2} + 37 \beta_{4} + 19 \beta_{5} ) q^{22} + ( -85 \beta_{2} - 25 \beta_{4} - 5 \beta_{5} ) q^{23} + ( 72 - 21 \beta_{1} + 10 \beta_{3} ) q^{26} + ( 124 \beta_{2} - 36 \beta_{4} - 2 \beta_{5} ) q^{28} + ( -47 + \beta_{1} - 25 \beta_{3} ) q^{29} + ( -39 + 19 \beta_{1} + 17 \beta_{3} ) q^{31} + ( 295 \beta_{2} - 95 \beta_{4} - 15 \beta_{5} ) q^{32} + ( -145 + 29 \beta_{1} + 7 \beta_{3} ) q^{34} + ( -138 \beta_{2} + 55 \beta_{4} - 25 \beta_{5} ) q^{37} + ( 417 \beta_{2} - 66 \beta_{4} - 35 \beta_{5} ) q^{38} + ( 188 + 26 \beta_{1} + 10 \beta_{3} ) q^{41} + ( 281 \beta_{2} - 17 \beta_{4} - 29 \beta_{5} ) q^{43} + ( 535 - 155 \beta_{1} + 35 \beta_{3} ) q^{44} + ( -95 - 35 \beta_{1} - 35 \beta_{3} ) q^{46} + ( -9 \beta_{2} - 123 \beta_{4} + 5 \beta_{5} ) q^{47} + ( -161 - 53 \beta_{1} + 41 \beta_{3} ) q^{49} + ( 315 \beta_{2} - 95 \beta_{4} - 25 \beta_{5} ) q^{52} + ( 136 \beta_{2} - 38 \beta_{4} + 30 \beta_{5} ) q^{53} + ( -744 + 42 \beta_{1} ) q^{56} + ( -55 \beta_{2} + 173 \beta_{4} + 51 \beta_{5} ) q^{58} + ( -104 - 68 \beta_{1} ) q^{59} + ( -26 - 43 \beta_{1} + 31 \beta_{3} ) q^{61} + ( -321 \beta_{2} - 27 \beta_{4} - 15 \beta_{5} ) q^{62} + ( -1053 + 169 \beta_{1} - 53 \beta_{3} ) q^{64} + ( 40 \beta_{2} - 121 \beta_{4} + 3 \beta_{5} ) q^{67} + ( -215 \beta_{2} + 115 \beta_{4} + 55 \beta_{5} ) q^{68} + ( 64 - 182 \beta_{1} - 30 \beta_{3} ) q^{71} + ( 306 \beta_{2} + 59 \beta_{4} + 3 \beta_{5} ) q^{73} + ( 931 - 68 \beta_{1} + 5 \beta_{3} ) q^{74} + ( -1266 + 394 \beta_{1} - 128 \beta_{3} ) q^{76} + ( 658 \beta_{2} - 104 \beta_{4} + 80 \beta_{5} ) q^{77} + ( -343 - 38 \beta_{1} - 54 \beta_{3} ) q^{79} + ( 70 \beta_{2} - 212 \beta_{4} + 6 \beta_{5} ) q^{82} + ( -102 \beta_{2} - 24 \beta_{4} + 60 \beta_{5} ) q^{83} + ( -691 + 443 \beta_{1} - 75 \beta_{3} ) q^{86} + ( 1945 \beta_{2} - 569 \beta_{4} - 73 \beta_{5} ) q^{88} + ( -360 - 60 \beta_{3} ) q^{89} + ( -230 + 81 \beta_{1} + 23 \beta_{3} ) q^{91} + ( -415 \beta_{2} + 35 \beta_{4} - 5 \beta_{5} ) q^{92} + ( -1345 + 89 \beta_{1} - 113 \beta_{3} ) q^{94} + ( 202 \beta_{2} + 113 \beta_{4} + \beta_{5} ) q^{97} + ( 179 \beta_{2} - 97 \beta_{4} - 135 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 34 q^{4} + O(q^{10}) \) \( 6 q - 34 q^{4} + 10 q^{11} - 120 q^{14} + 322 q^{16} + 100 q^{19} + 370 q^{26} - 230 q^{29} - 230 q^{31} - 826 q^{34} + 1160 q^{41} + 2830 q^{44} - 570 q^{46} - 1154 q^{49} - 4380 q^{56} - 760 q^{59} - 304 q^{61} - 5874 q^{64} + 80 q^{71} + 5440 q^{74} - 6552 q^{76} - 2026 q^{79} - 3110 q^{86} - 2040 q^{89} - 1264 q^{91} - 7666 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 6 x^{3} + 49 x^{2} - 42 x + 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{5} + 49 \nu^{4} + 21 \nu^{3} - 9 \nu^{2} + 1544 \)\()/334\)
\(\beta_{2}\)\(=\)\((\)\( -49 \nu^{5} - 21 \nu^{4} - 9 \nu^{3} + 147 \nu^{2} - 2338 \nu + 1056 \)\()/1002\)
\(\beta_{3}\)\(=\)\((\)\( 27 \nu^{5} + 107 \nu^{4} + 189 \nu^{3} - 81 \nu^{2} + 1872 \)\()/334\)
\(\beta_{4}\)\(=\)\((\)\( -245 \nu^{5} - 105 \nu^{4} - 45 \nu^{3} + 1737 \nu^{2} - 11690 \nu + 5280 \)\()/1002\)
\(\beta_{5}\)\(=\)\((\)\( 63 \nu^{5} + 27 \nu^{4} - 60 \nu^{3} - 523 \nu^{2} + 2505 \nu - 1143 \)\()/167\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} - 1\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 5 \beta_{2}\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{5} - 14 \beta_{4} + 7 \beta_{3} + 25 \beta_{2} - 14 \beta_{1} + 25\)\()/6\)
\(\nu^{4}\)\(=\)\(-\beta_{3} + 9 \beta_{1} - 36\)
\(\nu^{5}\)\(=\)\((\)\(49 \beta_{5} + 116 \beta_{4} + 49 \beta_{3} - 265 \beta_{2} - 116 \beta_{1} + 265\)\()/6\)

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
−2.05655 2.05655i
0.455589 0.455589i
1.60096 + 1.60096i
1.60096 1.60096i
0.455589 + 0.455589i
−2.05655 + 2.05655i
5.45876i 0 −21.7980 0 0 11.8065i 75.3201i 0 0
649.2 2.58488i 0 1.31841 0 0 22.8935i 24.0869i 0 0
649.3 2.12612i 0 3.47962 0 0 30.7000i 24.4070i 0 0
649.4 2.12612i 0 3.47962 0 0 30.7000i 24.4070i 0 0
649.5 2.58488i 0 1.31841 0 0 22.8935i 24.0869i 0 0
649.6 5.45876i 0 −21.7980 0 0 11.8065i 75.3201i 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.n 6
3.b odd 2 1 675.4.b.m 6
5.b even 2 1 inner 675.4.b.n 6
5.c odd 4 1 135.4.a.h yes 3
5.c odd 4 1 675.4.a.p 3
15.d odd 2 1 675.4.b.m 6
15.e even 4 1 135.4.a.e 3
15.e even 4 1 675.4.a.s 3
20.e even 4 1 2160.4.a.bq 3
45.k odd 12 2 405.4.e.q 6
45.l even 12 2 405.4.e.v 6
60.l odd 4 1 2160.4.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.e 3 15.e even 4 1
135.4.a.h yes 3 5.c odd 4 1
405.4.e.q 6 45.k odd 12 2
405.4.e.v 6 45.l even 12 2
675.4.a.p 3 5.c odd 4 1
675.4.a.s 3 15.e even 4 1
675.4.b.m 6 3.b odd 2 1
675.4.b.m 6 15.d odd 2 1
675.4.b.n 6 1.a even 1 1 trivial
675.4.b.n 6 5.b even 2 1 inner
2160.4.a.bi 3 60.l odd 4 1
2160.4.a.bq 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 \(S_{4}^{\mathrm{new}}(675, [\chi])\):

\( T_{2}^{6} + 41 T_{2}^{4} + 364 T_{2}^{2} + 900 \)
\( T_{7}^{6} + 1606 T_{7}^{4} + 698409 T_{7}^{2} + 68856804 \)
\( T_{11}^{3} - 5 T_{11}^{2} - 2888 T_{11} + 31260 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 900 + 364 T^{2} + 41 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( T^{6} \)
$7$ \( 68856804 + 698409 T^{2} + 1606 T^{4} + T^{6} \)
$11$ \( ( 31260 - 2888 T - 5 T^{2} + T^{3} )^{2} \)
$13$ \( 41280625 + 501411 T^{2} + 1587 T^{4} + T^{6} \)
$17$ \( 1743897600 + 18504064 T^{2} + 12809 T^{4} + T^{6} \)
$19$ \( ( 368012 - 16663 T - 50 T^{2} + T^{3} )^{2} \)
$23$ \( 306362250000 + 577935000 T^{2} + 48825 T^{4} + T^{6} \)
$29$ \( ( -6440340 - 47408 T + 115 T^{2} + T^{3} )^{2} \)
$31$ \( ( -938304 - 29232 T + 115 T^{2} + T^{3} )^{2} \)
$37$ \( 513801865171204 + 22021941993 T^{2} + 283302 T^{4} + T^{6} \)
$41$ \( ( -3917280 + 89812 T - 580 T^{2} + T^{3} )^{2} \)
$43$ \( 28707478180096 + 28852500384 T^{2} + 350169 T^{4} + T^{6} \)
$47$ \( 207021450297600 + 57904129504 T^{2} + 519329 T^{4} + T^{6} \)
$53$ \( 160233065222400 + 13510637584 T^{2} + 276344 T^{4} + T^{6} \)
$59$ \( ( -5205120 - 27392 T + 380 T^{2} + T^{3} )^{2} \)
$61$ \( ( -5069066 - 87475 T + 152 T^{2} + T^{3} )^{2} \)
$67$ \( 431363822490000 + 59618589201 T^{2} + 488002 T^{4} + T^{6} \)
$71$ \( ( 216071280 - 677372 T - 40 T^{2} + T^{3} )^{2} \)
$73$ \( 270535190786116 + 37739718249 T^{2} + 431334 T^{4} + T^{6} \)
$79$ \( ( -90596925 + 52215 T + 1013 T^{2} + T^{3} )^{2} \)
$83$ \( 7146869573505600 + 136509985584 T^{2} + 675756 T^{4} + T^{6} \)
$89$ \( ( -125064000 + 46800 T + 1020 T^{2} + T^{3} )^{2} \)
$97$ \( 752435512191364 + 40694604489 T^{2} + 587526 T^{4} + T^{6} \)
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