Properties

Label 15.5.d.c
Level $15$
Weight $5$
Character orbit 15.d
Analytic conductor $1.551$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 15.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{10}, \sqrt{-26})\)
Defining polynomial: \( x^{4} + 8x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 6 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (\beta_{3} + 15) q^{6} + (\beta_{2} - 2 \beta_1) q^{7} - 22 \beta_{2} q^{8} + (3 \beta_{3} - 36) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 6 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (\beta_{3} + 15) q^{6} + (\beta_{2} - 2 \beta_1) q^{7} - 22 \beta_{2} q^{8} + (3 \beta_{3} - 36) q^{9} + (5 \beta_{2} - 10 \beta_1 + 20) q^{10} - 4 \beta_{3} q^{11} + ( - 6 \beta_{2} - 6 \beta_1) q^{12} + ( - 16 \beta_{2} + 32 \beta_1) q^{13} - 2 \beta_{3} q^{14} + (2 \beta_{3} + 66 \beta_{2} - 15 \beta_1 + 30) q^{15} - 124 q^{16} + 88 \beta_{2} q^{17} + ( - 51 \beta_{2} + 30 \beta_1) q^{18} + 308 q^{19} + (6 \beta_{3} - 12 \beta_{2}) q^{20} + ( - 3 \beta_{3} + 117) q^{21} + (20 \beta_{2} - 40 \beta_1) q^{22} - 131 \beta_{2} q^{23} + ( - 22 \beta_{3} - 330) q^{24} + (20 \beta_{2} - 40 \beta_1 - 545) q^{25} + 32 \beta_{3} q^{26} + ( - 234 \beta_{2} + 9 \beta_1) q^{27} + ( - 6 \beta_{2} + 12 \beta_1) q^{28} + 8 \beta_{3} q^{29} + ( - 15 \beta_{3} + 20 \beta_{2} + 20 \beta_1 + 585) q^{30} + 32 q^{31} + 228 \beta_{2} q^{32} + (264 \beta_{2} - 60 \beta_1) q^{33} + 880 q^{34} + ( - 4 \beta_{3} - 117 \beta_{2}) q^{35} + ( - 18 \beta_{3} + 216) q^{36} + ( - 84 \beta_{2} + 168 \beta_1) q^{37} + 308 \beta_{2} q^{38} + (48 \beta_{3} - 1872) q^{39} + ( - 110 \beta_{2} + 220 \beta_1 - 440) q^{40} - 86 \beta_{3} q^{41} + (132 \beta_{2} - 30 \beta_1) q^{42} + (169 \beta_{2} - 338 \beta_1) q^{43} + 24 \beta_{3} q^{44} + (36 \beta_{3} - 102 \beta_{2} + 60 \beta_1 + 1755) q^{45} - 1310 q^{46} - 773 \beta_{2} q^{47} + ( - 124 \beta_{2} - 124 \beta_1) q^{48} + 2167 q^{49} + ( - 40 \beta_{3} - 545 \beta_{2}) q^{50} + (88 \beta_{3} + 1320) q^{51} + (96 \beta_{2} - 192 \beta_1) q^{52} + 448 \beta_{2} q^{53} + (9 \beta_{3} - 2295) q^{54} + (40 \beta_{2} - 80 \beta_1 - 2340) q^{55} + 44 \beta_{3} q^{56} + (308 \beta_{2} + 308 \beta_1) q^{57} + ( - 40 \beta_{2} + 80 \beta_1) q^{58} - 164 \beta_{3} q^{59} + ( - 12 \beta_{3} - 396 \beta_{2} + 90 \beta_1 - 180) q^{60} - 928 q^{61} + 32 \beta_{2} q^{62} + (315 \beta_{2} + 72 \beta_1) q^{63} + 4264 q^{64} + (64 \beta_{3} + 1872 \beta_{2}) q^{65} + ( - 60 \beta_{3} + 2340) q^{66} + ( - 169 \beta_{2} + 338 \beta_1) q^{67} - 528 \beta_{2} q^{68} + ( - 131 \beta_{3} - 1965) q^{69} + (20 \beta_{2} - 40 \beta_1 - 1170) q^{70} + 192 \beta_{3} q^{71} + (1122 \beta_{2} - 660 \beta_1) q^{72} + ( - 276 \beta_{2} + 552 \beta_1) q^{73} + 168 \beta_{3} q^{74} + ( - 60 \beta_{3} - 545 \beta_{2} - 545 \beta_1 + 2340) q^{75} - 1848 q^{76} - 468 \beta_{2} q^{77} + ( - 2112 \beta_{2} + 480 \beta_1) q^{78} + 8 q^{79} + (124 \beta_{3} - 248 \beta_{2}) q^{80} + ( - 216 \beta_{3} - 3969) q^{81} + (430 \beta_{2} - 860 \beta_1) q^{82} - 1403 \beta_{2} q^{83} + (18 \beta_{3} - 702) q^{84} + (440 \beta_{2} - 880 \beta_1 + 1760) q^{85} - 338 \beta_{3} q^{86} + ( - 528 \beta_{2} + 120 \beta_1) q^{87} + ( - 440 \beta_{2} + 880 \beta_1) q^{88} + 384 \beta_{3} q^{89} + (60 \beta_{3} + 1575 \beta_{2} + 360 \beta_1 - 720) q^{90} + 3744 q^{91} + 786 \beta_{2} q^{92} + (32 \beta_{2} + 32 \beta_1) q^{93} - 7730 q^{94} + ( - 308 \beta_{3} + 616 \beta_{2}) q^{95} + (228 \beta_{3} + 3420) q^{96} + ( - 164 \beta_{2} + 328 \beta_1) q^{97} + 2167 \beta_{2} q^{98} + (144 \beta_{3} + 7020) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{4} + 60 q^{6} - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{4} + 60 q^{6} - 144 q^{9} + 80 q^{10} + 120 q^{15} - 496 q^{16} + 1232 q^{19} + 468 q^{21} - 1320 q^{24} - 2180 q^{25} + 2340 q^{30} + 128 q^{31} + 3520 q^{34} + 864 q^{36} - 7488 q^{39} - 1760 q^{40} + 7020 q^{45} - 5240 q^{46} + 8668 q^{49} + 5280 q^{51} - 9180 q^{54} - 9360 q^{55} - 720 q^{60} - 3712 q^{61} + 17056 q^{64} + 9360 q^{66} - 7860 q^{69} - 4680 q^{70} + 9360 q^{75} - 7392 q^{76} + 32 q^{79} - 15876 q^{81} - 2808 q^{84} + 7040 q^{85} - 2880 q^{90} + 14976 q^{91} - 30920 q^{94} + 13680 q^{96} + 28080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 8x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 26\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -26\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
−1.58114 2.54951i
−1.58114 + 2.54951i
1.58114 2.54951i
1.58114 + 2.54951i
−3.16228 −4.74342 7.64853i −6.00000 −6.32456 24.1868i 15.0000 + 24.1868i 15.2971i 69.5701 −36.0000 + 72.5603i 20.0000 + 76.4853i
14.2 −3.16228 −4.74342 + 7.64853i −6.00000 −6.32456 + 24.1868i 15.0000 24.1868i 15.2971i 69.5701 −36.0000 72.5603i 20.0000 76.4853i
14.3 3.16228 4.74342 7.64853i −6.00000 6.32456 + 24.1868i 15.0000 24.1868i 15.2971i −69.5701 −36.0000 72.5603i 20.0000 + 76.4853i
14.4 3.16228 4.74342 + 7.64853i −6.00000 6.32456 24.1868i 15.0000 + 24.1868i 15.2971i −69.5701 −36.0000 + 72.5603i 20.0000 76.4853i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.d.c 4
3.b odd 2 1 inner 15.5.d.c 4
4.b odd 2 1 240.5.c.c 4
5.b even 2 1 inner 15.5.d.c 4
5.c odd 4 2 75.5.c.h 4
12.b even 2 1 240.5.c.c 4
15.d odd 2 1 inner 15.5.d.c 4
15.e even 4 2 75.5.c.h 4
20.d odd 2 1 240.5.c.c 4
60.h even 2 1 240.5.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.d.c 4 1.a even 1 1 trivial
15.5.d.c 4 3.b odd 2 1 inner
15.5.d.c 4 5.b even 2 1 inner
15.5.d.c 4 15.d odd 2 1 inner
75.5.c.h 4 5.c odd 4 2
75.5.c.h 4 15.e even 4 2
240.5.c.c 4 4.b odd 2 1
240.5.c.c 4 12.b even 2 1
240.5.c.c 4 20.d odd 2 1
240.5.c.c 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 10 \) acting on \(S_{5}^{\mathrm{new}}(15, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 72T^{2} + 6561 \) Copy content Toggle raw display
$5$ \( T^{4} + 1090 T^{2} + 390625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 234)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9360)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 59904)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 77440)^{2} \) Copy content Toggle raw display
$19$ \( (T - 308)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 171610)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 37440)^{2} \) Copy content Toggle raw display
$31$ \( (T - 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1651104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4326660)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6683274)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 5975290)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2007040)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 15734160)^{2} \) Copy content Toggle raw display
$61$ \( (T + 928)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6683274)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 21565440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 17825184)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 19684090)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 86261760)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6293664)^{2} \) Copy content Toggle raw display
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