Properties

Label 2475.4.a.t
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.029727264\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.47528.1
Defining polynomial: \(x^{3} - x^{2} - 26 x - 22\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 10 + \beta_{2} ) q^{4} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 3 + 9 \beta_{1} - 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 10 + \beta_{2} ) q^{4} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 3 + 9 \beta_{1} - 2 \beta_{2} ) q^{8} -11 q^{11} + ( -38 + 2 \beta_{2} ) q^{13} + ( 28 - 16 \beta_{1} + 6 \beta_{2} ) q^{14} + ( 60 - 2 \beta_{1} + 5 \beta_{2} ) q^{16} + ( -34 - 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -24 + 14 \beta_{1} - 8 \beta_{2} ) q^{19} + ( 11 - 11 \beta_{1} ) q^{22} + ( 48 - 24 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 48 - 24 \beta_{1} - 4 \beta_{2} ) q^{26} + ( -238 + 38 \beta_{1} - 12 \beta_{2} ) q^{28} + ( 62 + 34 \beta_{1} ) q^{29} + ( 96 - 40 \beta_{1} - 8 \beta_{2} ) q^{31} + ( -93 + 21 \beta_{1} + 4 \beta_{2} ) q^{32} + ( -10 - 50 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -286 + 20 \beta_{1} ) q^{37} + ( 222 - 66 \beta_{1} + 30 \beta_{2} ) q^{38} + ( -30 - 66 \beta_{1} ) q^{41} + ( -48 + 22 \beta_{1} + 14 \beta_{2} ) q^{43} + ( -110 - 11 \beta_{2} ) q^{44} + ( -436 + 52 \beta_{1} - 32 \beta_{2} ) q^{46} + ( 168 - 4 \beta_{2} ) q^{47} + ( 117 - 72 \beta_{1} ) q^{49} + ( -172 - 4 \beta_{1} - 32 \beta_{2} ) q^{52} + ( 126 - 96 \beta_{1} + 16 \beta_{2} ) q^{53} + ( 600 - 156 \beta_{1} + 14 \beta_{2} ) q^{56} + ( 516 + 96 \beta_{1} + 34 \beta_{2} ) q^{58} + ( -172 - 32 \beta_{1} - 8 \beta_{2} ) q^{59} + ( 134 + 12 \beta_{1} + 68 \beta_{2} ) q^{61} + ( -816 - 24 \beta_{2} ) q^{62} + ( -10 - 28 \beta_{1} - 27 \beta_{2} ) q^{64} + ( 176 - 100 \beta_{1} + 16 \beta_{2} ) q^{67} + ( -558 - 30 \beta_{1} - 38 \beta_{2} ) q^{68} + ( 324 - 60 \beta_{1} + 28 \beta_{2} ) q^{71} + ( -194 - 36 \beta_{1} + 38 \beta_{2} ) q^{73} + ( 626 - 266 \beta_{1} + 20 \beta_{2} ) q^{74} + ( -1002 + 254 \beta_{1} - 62 \beta_{2} ) q^{76} + ( 44 - 22 \beta_{1} + 22 \beta_{2} ) q^{77} + ( -212 + 94 \beta_{1} + 8 \beta_{2} ) q^{79} + ( -1092 - 96 \beta_{1} - 66 \beta_{2} ) q^{82} + ( 60 - 180 \beta_{1} + 54 \beta_{2} ) q^{83} + ( 492 + 72 \beta_{1} - 6 \beta_{2} ) q^{86} + ( -33 - 99 \beta_{1} + 22 \beta_{2} ) q^{88} + ( -254 - 28 \beta_{1} - 120 \beta_{2} ) q^{89} + ( -244 - 40 \beta_{1} + 92 \beta_{2} ) q^{91} + ( 776 - 416 \beta_{1} + 84 \beta_{2} ) q^{92} + ( -188 + 140 \beta_{1} + 8 \beta_{2} ) q^{94} + ( -658 - 100 \beta_{1} + 44 \beta_{2} ) q^{97} + ( -1341 + 45 \beta_{1} - 72 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} + 30q^{4} - 10q^{7} + 18q^{8} + O(q^{10}) \) \( 3q - 2q^{2} + 30q^{4} - 10q^{7} + 18q^{8} - 33q^{11} - 114q^{13} + 68q^{14} + 178q^{16} - 104q^{17} - 58q^{19} + 22q^{22} + 120q^{23} + 120q^{26} - 676q^{28} + 220q^{29} + 248q^{31} - 258q^{32} - 80q^{34} - 838q^{37} + 600q^{38} - 156q^{41} - 122q^{43} - 330q^{44} - 1256q^{46} + 504q^{47} + 279q^{49} - 520q^{52} + 282q^{53} + 1644q^{56} + 1644q^{58} - 548q^{59} + 414q^{61} - 2448q^{62} - 58q^{64} + 428q^{67} - 1704q^{68} + 912q^{71} - 618q^{73} + 1612q^{74} - 2752q^{76} + 110q^{77} - 542q^{79} - 3372q^{82} + 1548q^{86} - 198q^{88} - 790q^{89} - 772q^{91} + 1912q^{92} - 424q^{94} - 2074q^{97} - 3978q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 26 x - 22\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 17 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 17\)

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
−4.06484
−0.906392
5.97123
−5.06484 0 17.6526 0 0 −27.4348 −48.8887 0 0
1.2 −1.90639 0 −4.36567 0 0 22.9186 23.5738 0 0
1.3 4.97123 0 16.7131 0 0 −5.48376 43.3148 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 2475.4.a.t 3
3.b odd 2 1 825.4.a.r 3
5.b even 2 1 495.4.a.k 3
15.d odd 2 1 165.4.a.e 3
15.e even 4 2 825.4.c.k 6
165.d even 2 1 1815.4.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.e 3 15.d odd 2 1
495.4.a.k 3 5.b even 2 1
825.4.a.r 3 3.b odd 2 1
825.4.c.k 6 15.e even 4 2
1815.4.a.r 3 165.d even 2 1
2475.4.a.t 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{3} + 2 T_{2}^{2} - 25 T_{2} - 48 \)
\( T_{7}^{3} + 10 T_{7}^{2} - 604 T_{7} - 3448 \)
\( T_{29}^{3} - 220 T_{29}^{2} - 14308 T_{29} + 629760 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -48 - 25 T + 2 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( -3448 - 604 T + 10 T^{2} + T^{3} \)
$11$ \( ( 11 + T )^{3} \)
$13$ \( 37216 + 3712 T + 114 T^{2} + T^{3} \)
$17$ \( 8448 + 2792 T + 104 T^{2} + T^{3} \)
$19$ \( 65520 - 11496 T + 58 T^{2} + T^{3} \)
$23$ \( 148224 - 10736 T - 120 T^{2} + T^{3} \)
$29$ \( 629760 - 14308 T - 220 T^{2} + T^{3} \)
$31$ \( 9589248 - 38592 T - 248 T^{2} + T^{3} \)
$37$ \( 18607336 + 223548 T + 838 T^{2} + T^{3} \)
$41$ \( 3013632 - 106596 T + 156 T^{2} + T^{3} \)
$43$ \( -1445400 - 44940 T + 122 T^{2} + T^{3} \)
$47$ \( -4372224 + 82192 T - 504 T^{2} + T^{3} \)
$53$ \( -3654264 - 222068 T - 282 T^{2} + T^{3} \)
$59$ \( 1206720 + 57584 T + 548 T^{2} + T^{3} \)
$61$ \( 342344792 - 681332 T - 414 T^{2} + T^{3} \)
$67$ \( 8135552 - 206752 T - 428 T^{2} + T^{3} \)
$71$ \( 2867712 + 97888 T - 912 T^{2} + T^{3} \)
$73$ \( -26458592 - 100544 T + 618 T^{2} + T^{3} \)
$79$ \( -88503440 - 161224 T + 542 T^{2} + T^{3} \)
$83$ \( -434328048 - 1091340 T + T^{3} \)
$89$ \( -1941629400 - 2118532 T + 790 T^{2} + T^{3} \)
$97$ \( -98075336 + 967212 T + 2074 T^{2} + T^{3} \)
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