Properties

Label 2475.4.a.q
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( -4 + 2 \beta ) q^{4} + ( -10 + 4 \beta ) q^{7} + ( -6 - 10 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( -4 + 2 \beta ) q^{4} + ( -10 + 4 \beta ) q^{7} + ( -6 - 10 \beta ) q^{8} + 11 q^{11} + ( -40 + 20 \beta ) q^{13} + ( 2 - 6 \beta ) q^{14} + ( -4 - 32 \beta ) q^{16} + ( -62 + 12 \beta ) q^{17} + ( 36 + 60 \beta ) q^{19} + ( 11 + 11 \beta ) q^{22} + ( -49 - 36 \beta ) q^{23} + ( 20 - 20 \beta ) q^{26} + ( 64 - 36 \beta ) q^{28} + ( -72 + 56 \beta ) q^{29} + ( -17 + 28 \beta ) q^{31} + ( -52 + 44 \beta ) q^{32} + ( -26 - 50 \beta ) q^{34} + ( -27 + 8 \beta ) q^{37} + ( 216 + 96 \beta ) q^{38} + ( -268 + 4 \beta ) q^{41} + ( 30 + 16 \beta ) q^{43} + ( -44 + 22 \beta ) q^{44} + ( -157 - 85 \beta ) q^{46} + ( -136 - 120 \beta ) q^{47} + ( -195 - 80 \beta ) q^{49} + ( 280 - 160 \beta ) q^{52} + ( -246 - 56 \beta ) q^{53} + ( -60 + 76 \beta ) q^{56} + ( 96 - 16 \beta ) q^{58} + ( -317 + 132 \beta ) q^{59} + ( 420 + 184 \beta ) q^{61} + ( 67 + 11 \beta ) q^{62} + ( 112 + 248 \beta ) q^{64} + ( -377 + 20 \beta ) q^{67} + ( 320 - 172 \beta ) q^{68} + ( 339 - 76 \beta ) q^{71} + ( 200 + 468 \beta ) q^{73} + ( -3 - 19 \beta ) q^{74} + ( 216 - 168 \beta ) q^{76} + ( -110 + 44 \beta ) q^{77} + ( 158 + 656 \beta ) q^{79} + ( -256 - 264 \beta ) q^{82} + ( 234 + 120 \beta ) q^{83} + ( 78 + 46 \beta ) q^{86} + ( -66 - 110 \beta ) q^{88} + ( 921 + 328 \beta ) q^{89} + ( 640 - 360 \beta ) q^{91} + ( -20 + 46 \beta ) q^{92} + ( -496 - 256 \beta ) q^{94} + ( -1097 - 144 \beta ) q^{97} + ( -435 - 275 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 8q^{4} - 20q^{7} - 12q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 8q^{4} - 20q^{7} - 12q^{8} + 22q^{11} - 80q^{13} + 4q^{14} - 8q^{16} - 124q^{17} + 72q^{19} + 22q^{22} - 98q^{23} + 40q^{26} + 128q^{28} - 144q^{29} - 34q^{31} - 104q^{32} - 52q^{34} - 54q^{37} + 432q^{38} - 536q^{41} + 60q^{43} - 88q^{44} - 314q^{46} - 272q^{47} - 390q^{49} + 560q^{52} - 492q^{53} - 120q^{56} + 192q^{58} - 634q^{59} + 840q^{61} + 134q^{62} + 224q^{64} - 754q^{67} + 640q^{68} + 678q^{71} + 400q^{73} - 6q^{74} + 432q^{76} - 220q^{77} + 316q^{79} - 512q^{82} + 468q^{83} + 156q^{86} - 132q^{88} + 1842q^{89} + 1280q^{91} - 40q^{92} - 992q^{94} - 2194q^{97} - 870q^{98} + O(q^{100}) \)

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
−1.73205
1.73205
−0.732051 0 −7.46410 0 0 −16.9282 11.3205 0 0
1.2 2.73205 0 −0.535898 0 0 −3.07180 −23.3205 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.q 2
3.b odd 2 1 275.4.a.b 2
5.b even 2 1 99.4.a.c 2
15.d odd 2 1 11.4.a.a 2
15.e even 4 2 275.4.b.c 4
20.d odd 2 1 1584.4.a.bc 2
55.d odd 2 1 1089.4.a.v 2
60.h even 2 1 176.4.a.i 2
105.g even 2 1 539.4.a.e 2
120.i odd 2 1 704.4.a.p 2
120.m even 2 1 704.4.a.n 2
165.d even 2 1 121.4.a.c 2
165.o odd 10 4 121.4.c.c 8
165.r even 10 4 121.4.c.f 8
195.e odd 2 1 1859.4.a.a 2
660.g odd 2 1 1936.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 15.d odd 2 1
99.4.a.c 2 5.b even 2 1
121.4.a.c 2 165.d even 2 1
121.4.c.c 8 165.o odd 10 4
121.4.c.f 8 165.r even 10 4
176.4.a.i 2 60.h even 2 1
275.4.a.b 2 3.b odd 2 1
275.4.b.c 4 15.e even 4 2
539.4.a.e 2 105.g even 2 1
704.4.a.n 2 120.m even 2 1
704.4.a.p 2 120.i odd 2 1
1089.4.a.v 2 55.d odd 2 1
1584.4.a.bc 2 20.d odd 2 1
1859.4.a.a 2 195.e odd 2 1
1936.4.a.w 2 660.g odd 2 1
2475.4.a.q 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} - 2 T_{2} - 2 \)
\( T_{7}^{2} + 20 T_{7} + 52 \)
\( T_{29}^{2} + 144 T_{29} - 4224 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 52 + 20 T + T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 400 + 80 T + T^{2} \)
$17$ \( 3412 + 124 T + T^{2} \)
$19$ \( -9504 - 72 T + T^{2} \)
$23$ \( -1487 + 98 T + T^{2} \)
$29$ \( -4224 + 144 T + T^{2} \)
$31$ \( -2063 + 34 T + T^{2} \)
$37$ \( 537 + 54 T + T^{2} \)
$41$ \( 71776 + 536 T + T^{2} \)
$43$ \( 132 - 60 T + T^{2} \)
$47$ \( -24704 + 272 T + T^{2} \)
$53$ \( 51108 + 492 T + T^{2} \)
$59$ \( 48217 + 634 T + T^{2} \)
$61$ \( 74832 - 840 T + T^{2} \)
$67$ \( 140929 + 754 T + T^{2} \)
$71$ \( 97593 - 678 T + T^{2} \)
$73$ \( -617072 - 400 T + T^{2} \)
$79$ \( -1266044 - 316 T + T^{2} \)
$83$ \( 11556 - 468 T + T^{2} \)
$89$ \( 525489 - 1842 T + T^{2} \)
$97$ \( 1141201 + 2194 T + T^{2} \)
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