Properties

Label 2475.4.a.l
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - \beta ) q^{2} + ( 5 + 7 \beta ) q^{4} + ( 17 - 9 \beta ) q^{7} + ( -19 - 25 \beta ) q^{8} +O(q^{10})\) \( q + ( -3 - \beta ) q^{2} + ( 5 + 7 \beta ) q^{4} + ( 17 - 9 \beta ) q^{7} + ( -19 - 25 \beta ) q^{8} + 11 q^{11} + ( 30 - 10 \beta ) q^{13} + ( -15 + 19 \beta ) q^{14} + ( 117 + 63 \beta ) q^{16} + ( -67 - 17 \beta ) q^{17} + ( 21 - 45 \beta ) q^{19} + ( -33 - 11 \beta ) q^{22} + ( 26 - 4 \beta ) q^{23} + ( -50 + 10 \beta ) q^{26} + ( -167 + 11 \beta ) q^{28} + ( 75 + 71 \beta ) q^{29} + ( 129 - 117 \beta ) q^{31} + ( -451 - 169 \beta ) q^{32} + ( 269 + 135 \beta ) q^{34} + ( 301 - 43 \beta ) q^{37} + ( 117 + 159 \beta ) q^{38} + ( 150 - 156 \beta ) q^{41} + ( 108 - 156 \beta ) q^{43} + ( 55 + 77 \beta ) q^{44} + ( -62 - 10 \beta ) q^{46} + ( -74 + 100 \beta ) q^{47} + ( 270 - 225 \beta ) q^{49} + ( -130 + 90 \beta ) q^{52} + ( 143 - 169 \beta ) q^{53} + ( 577 - 29 \beta ) q^{56} + ( -509 - 359 \beta ) q^{58} + ( 122 - 158 \beta ) q^{59} + ( -137 + 119 \beta ) q^{61} + ( 81 + 339 \beta ) q^{62} + ( 1093 + 623 \beta ) q^{64} + ( -208 + 150 \beta ) q^{67} + ( -811 - 673 \beta ) q^{68} + ( -763 - 61 \beta ) q^{71} + ( -6 - 58 \beta ) q^{73} + ( -731 - 129 \beta ) q^{74} + ( -1155 - 393 \beta ) q^{76} + ( 187 - 99 \beta ) q^{77} + ( -590 - 114 \beta ) q^{79} + ( 174 + 474 \beta ) q^{82} + ( -414 + 270 \beta ) q^{83} + ( 300 + 516 \beta ) q^{86} + ( -209 - 275 \beta ) q^{88} + ( 867 + 43 \beta ) q^{89} + ( 870 - 350 \beta ) q^{91} + ( 18 + 134 \beta ) q^{92} + ( -178 - 326 \beta ) q^{94} + ( -60 + 454 \beta ) q^{97} + ( 90 + 630 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 7q^{2} + 17q^{4} + 25q^{7} - 63q^{8} + O(q^{10}) \) \( 2q - 7q^{2} + 17q^{4} + 25q^{7} - 63q^{8} + 22q^{11} + 50q^{13} - 11q^{14} + 297q^{16} - 151q^{17} - 3q^{19} - 77q^{22} + 48q^{23} - 90q^{26} - 323q^{28} + 221q^{29} + 141q^{31} - 1071q^{32} + 673q^{34} + 559q^{37} + 393q^{38} + 144q^{41} + 60q^{43} + 187q^{44} - 134q^{46} - 48q^{47} + 315q^{49} - 170q^{52} + 117q^{53} + 1125q^{56} - 1377q^{58} + 86q^{59} - 155q^{61} + 501q^{62} + 2809q^{64} - 266q^{67} - 2295q^{68} - 1587q^{71} - 70q^{73} - 1591q^{74} - 2703q^{76} + 275q^{77} - 1294q^{79} + 822q^{82} - 558q^{83} + 1116q^{86} - 693q^{88} + 1777q^{89} + 1390q^{91} + 170q^{92} - 682q^{94} + 334q^{97} + 810q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
−5.56155 0 22.9309 0 0 −6.05398 −83.0388 0 0
1.2 −1.43845 0 −5.93087 0 0 31.0540 20.0388 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.l 2
3.b odd 2 1 275.4.a.c 2
5.b even 2 1 495.4.a.e 2
15.d odd 2 1 55.4.a.b 2
15.e even 4 2 275.4.b.b 4
60.h even 2 1 880.4.a.r 2
165.d even 2 1 605.4.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.b 2 15.d odd 2 1
275.4.a.c 2 3.b odd 2 1
275.4.b.b 4 15.e even 4 2
495.4.a.e 2 5.b even 2 1
605.4.a.g 2 165.d even 2 1
880.4.a.r 2 60.h even 2 1
2475.4.a.l 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} + 7 T_{2} + 8 \)
\( T_{7}^{2} - 25 T_{7} - 188 \)
\( T_{29}^{2} - 221 T_{29} - 9214 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + 7 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -188 - 25 T + T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 200 - 50 T + T^{2} \)
$17$ \( 4472 + 151 T + T^{2} \)
$19$ \( -8604 + 3 T + T^{2} \)
$23$ \( 508 - 48 T + T^{2} \)
$29$ \( -9214 - 221 T + T^{2} \)
$31$ \( -53208 - 141 T + T^{2} \)
$37$ \( 70262 - 559 T + T^{2} \)
$41$ \( -98244 - 144 T + T^{2} \)
$43$ \( -102528 - 60 T + T^{2} \)
$47$ \( -41924 + 48 T + T^{2} \)
$53$ \( -117962 - 117 T + T^{2} \)
$59$ \( -104248 - 86 T + T^{2} \)
$61$ \( -54178 + 155 T + T^{2} \)
$67$ \( -77936 + 266 T + T^{2} \)
$71$ \( 613828 + 1587 T + T^{2} \)
$73$ \( -13072 + 70 T + T^{2} \)
$79$ \( 363376 + 1294 T + T^{2} \)
$83$ \( -231984 + 558 T + T^{2} \)
$89$ \( 781574 - 1777 T + T^{2} \)
$97$ \( -848104 - 334 T + T^{2} \)
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