Properties

Label 1521.4.a.r
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta - 4) q^{4} + (\beta - 2) q^{5} + ( - 11 \beta + 10) q^{7} + ( - 11 \beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta - 4) q^{4} + (\beta - 2) q^{5} + ( - 11 \beta + 10) q^{7} + ( - 11 \beta + 4) q^{8} + ( - \beta + 4) q^{10} + (12 \beta + 34) q^{11} + ( - \beta - 44) q^{14} + ( - 15 \beta - 12) q^{16} + (17 \beta - 18) q^{17} + (32 \beta + 26) q^{19} + ( - 5 \beta + 12) q^{20} + (46 \beta + 48) q^{22} + (12 \beta - 104) q^{23} + ( - 3 \beta - 117) q^{25} + (43 \beta - 84) q^{28} + ( - 96 \beta + 70) q^{29} + (34 \beta + 26) q^{31} + (61 \beta - 92) q^{32} + ( - \beta + 68) q^{34} + (21 \beta - 64) q^{35} + ( - 5 \beta - 102) q^{37} + (58 \beta + 128) q^{38} + (15 \beta - 52) q^{40} + (22 \beta - 126) q^{41} + (143 \beta + 72) q^{43} + ( - 2 \beta - 88) q^{44} + ( - 92 \beta + 48) q^{46} + ( - 121 \beta + 278) q^{47} + ( - 99 \beta + 241) q^{49} + ( - 120 \beta - 12) q^{50} + ( - 30 \beta + 74) q^{53} + (22 \beta - 20) q^{55} + ( - 33 \beta + 524) q^{56} + ( - 26 \beta - 384) q^{58} + (124 \beta - 246) q^{59} + ( - 190 \beta - 434) q^{61} + (60 \beta + 136) q^{62} + (89 \beta + 340) q^{64} + (232 \beta - 150) q^{67} + ( - 69 \beta + 140) q^{68} + ( - 43 \beta + 84) q^{70} + ( - 231 \beta + 50) q^{71} + ( - 260 \beta - 98) q^{73} + ( - 107 \beta - 20) q^{74} + ( - 70 \beta + 24) q^{76} + ( - 386 \beta - 188) q^{77} + (40 \beta - 524) q^{79} + (3 \beta - 36) q^{80} + ( - 104 \beta + 88) q^{82} + ( - 182 \beta + 1070) q^{83} + ( - 35 \beta + 104) q^{85} + (215 \beta + 572) q^{86} + ( - 458 \beta - 392) q^{88} + ( - 388 \beta - 166) q^{89} + ( - 140 \beta + 464) q^{92} + (157 \beta - 484) q^{94} + ( - 6 \beta + 76) q^{95} + ( - 508 \beta + 718) q^{97} + (142 \beta - 396) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31} - 123 q^{32} + 135 q^{34} - 107 q^{35} - 209 q^{37} + 314 q^{38} - 89 q^{40} - 230 q^{41} + 287 q^{43} - 178 q^{44} + 4 q^{46} + 435 q^{47} + 383 q^{49} - 144 q^{50} + 118 q^{53} - 18 q^{55} + 1015 q^{56} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 769 q^{64} - 68 q^{67} + 211 q^{68} + 125 q^{70} - 131 q^{71} - 456 q^{73} - 147 q^{74} - 22 q^{76} - 762 q^{77} - 1008 q^{79} - 69 q^{80} + 72 q^{82} + 1958 q^{83} + 173 q^{85} + 1359 q^{86} - 1242 q^{88} - 720 q^{89} + 788 q^{92} - 811 q^{94} + 146 q^{95} + 928 q^{97} - 650 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−1.56155
2.56155
−1.56155 0 −5.56155 −3.56155 0 27.1771 21.1771 0 5.56155
1.2 2.56155 0 −1.43845 0.561553 0 −18.1771 −24.1771 0 1.43845
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)

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 1521.4.a.r 2
3.b odd 2 1 169.4.a.g 2
13.b even 2 1 117.4.a.d 2
39.d odd 2 1 13.4.a.b 2
39.f even 4 2 169.4.b.f 4
39.h odd 6 2 169.4.c.g 4
39.i odd 6 2 169.4.c.j 4
39.k even 12 4 169.4.e.f 8
52.b odd 2 1 1872.4.a.bb 2
156.h even 2 1 208.4.a.h 2
195.e odd 2 1 325.4.a.f 2
195.s even 4 2 325.4.b.e 4
273.g even 2 1 637.4.a.b 2
312.b odd 2 1 832.4.a.s 2
312.h even 2 1 832.4.a.z 2
429.e even 2 1 1573.4.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 39.d odd 2 1
117.4.a.d 2 13.b even 2 1
169.4.a.g 2 3.b odd 2 1
169.4.b.f 4 39.f even 4 2
169.4.c.g 4 39.h odd 6 2
169.4.c.j 4 39.i odd 6 2
169.4.e.f 8 39.k even 12 4
208.4.a.h 2 156.h even 2 1
325.4.a.f 2 195.e odd 2 1
325.4.b.e 4 195.s even 4 2
637.4.a.b 2 273.g even 2 1
832.4.a.s 2 312.b odd 2 1
832.4.a.z 2 312.h even 2 1
1521.4.a.r 2 1.a even 1 1 trivial
1573.4.a.b 2 429.e even 2 1
1872.4.a.bb 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 9T_{7} - 494 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 9T - 494 \) Copy content Toggle raw display
$11$ \( T^{2} - 80T + 988 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19T - 1138 \) Copy content Toggle raw display
$19$ \( T^{2} - 84T - 2588 \) Copy content Toggle raw display
$23$ \( T^{2} + 196T + 8992 \) Copy content Toggle raw display
$29$ \( T^{2} - 44T - 38684 \) Copy content Toggle raw display
$31$ \( T^{2} - 86T - 3064 \) Copy content Toggle raw display
$37$ \( T^{2} + 209T + 10814 \) Copy content Toggle raw display
$41$ \( T^{2} + 230T + 11168 \) Copy content Toggle raw display
$43$ \( T^{2} - 287T - 66316 \) Copy content Toggle raw display
$47$ \( T^{2} - 435T - 14918 \) Copy content Toggle raw display
$53$ \( T^{2} - 118T - 344 \) Copy content Toggle raw display
$59$ \( T^{2} + 368T - 31492 \) Copy content Toggle raw display
$61$ \( T^{2} + 1058 T + 126416 \) Copy content Toggle raw display
$67$ \( T^{2} + 68T - 227596 \) Copy content Toggle raw display
$71$ \( T^{2} + 131T - 222494 \) Copy content Toggle raw display
$73$ \( T^{2} + 456T - 235316 \) Copy content Toggle raw display
$79$ \( T^{2} + 1008 T + 247216 \) Copy content Toggle raw display
$83$ \( T^{2} - 1958 T + 817664 \) Copy content Toggle raw display
$89$ \( T^{2} + 720T - 510212 \) Copy content Toggle raw display
$97$ \( T^{2} - 928T - 881476 \) Copy content Toggle raw display
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