Properties

Label 15.4.e.a
Level $15$
Weight $4$
Character orbit 15.e
Analytic conductor $0.885$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.885028650086\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.28356903014400.8
Defining polynomial: \( x^{8} + 209x^{4} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2}) q^{4} + (\beta_{7} - 2 \beta_{6} + \beta_{3} - 2 \beta_1) q^{5} + (2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 1) q^{6} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{4} + \beta_{2} - 1) q^{7} + (2 \beta_{7} - 2 \beta_{6} + 3 \beta_1) q^{8} + ( - 6 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2}) q^{4} + (\beta_{7} - 2 \beta_{6} + \beta_{3} - 2 \beta_1) q^{5} + (2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 1) q^{6} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{4} + \beta_{2} - 1) q^{7} + (2 \beta_{7} - 2 \beta_{6} + 3 \beta_1) q^{8} + ( - 6 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1) q^{9} + (5 \beta_{5} - 5 \beta_{4} - 10 \beta_{2} - 15) q^{10} + (2 \beta_{6} - 7 \beta_{3} - 7 \beta_1) q^{11} + (3 \beta_{7} + \beta_{6} + 2 \beta_{4} - 3 \beta_{3} - 17 \beta_{2} + 17) q^{12} + (\beta_{7} - \beta_{6} + 2 \beta_{5} + 8 \beta_{2} + 8) q^{13} + ( - 8 \beta_{7} - 7 \beta_{3} + 7 \beta_1) q^{14} + (3 \beta_{7} - 6 \beta_{6} + 5 \beta_{5} - 12 \beta_{3} + 25 \beta_{2} + 9 \beta_1 + 10) q^{15} + ( - 7 \beta_{5} + 7 \beta_{4} + 39) q^{16} + (7 \beta_{7} + 7 \beta_{6} + 20 \beta_{3}) q^{17} + (6 \beta_{7} - 6 \beta_{6} - 30 \beta_{2} - 21 \beta_1 - 30) q^{18} + ( - 6 \beta_{7} + 6 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 12 \beta_{2}) q^{19} + ( - 6 \beta_{7} + 2 \beta_{6} + 19 \beta_{3} + 2 \beta_1) q^{20} + ( - 11 \beta_{6} + \beta_{5} - \beta_{4} + 6 \beta_{3} + 6 \beta_1 - 62) q^{21} + ( - 5 \beta_{7} + 5 \beta_{6} - 10 \beta_{4} + 65 \beta_{2} - 65) q^{22} + ( - 9 \beta_{7} + 9 \beta_{6} + 14 \beta_1) q^{23} + (21 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + 3 \beta_{3} + 39 \beta_{2} - 3 \beta_1) q^{24} + (5 \beta_{7} - 5 \beta_{6} - 10 \beta_{5} + 20 \beta_{4} - 85 \beta_{2} - 20) q^{25} + (4 \beta_{6} - 4 \beta_{3} - 4 \beta_1) q^{26} + ( - 9 \beta_{7} - 6 \beta_{6} - 3 \beta_{4} - 18 \beta_{3} - 87 \beta_{2} + \cdots + 87) q^{27}+ \cdots + (66 \beta_{7} + 30 \beta_{6} - 30 \beta_{5} - 30 \beta_{4} + 129 \beta_{3} + \cdots - 129 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} - 12 q^{6} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} - 12 q^{6} - 16 q^{7} - 100 q^{10} + 132 q^{12} + 68 q^{13} + 90 q^{15} + 284 q^{16} - 240 q^{18} - 492 q^{21} - 500 q^{22} - 220 q^{25} + 702 q^{27} + 508 q^{28} + 660 q^{30} + 616 q^{31} - 240 q^{33} - 804 q^{36} - 1156 q^{37} - 600 q^{40} + 540 q^{42} + 548 q^{43} + 180 q^{45} + 736 q^{46} - 1116 q^{48} - 852 q^{51} + 224 q^{52} + 460 q^{55} + 684 q^{57} + 60 q^{58} + 540 q^{60} + 16 q^{61} + 1428 q^{63} + 2040 q^{66} + 404 q^{67} - 2220 q^{70} - 1800 q^{72} - 2512 q^{73} - 2910 q^{75} - 1488 q^{76} - 360 q^{78} + 288 q^{81} + 2800 q^{82} + 4940 q^{85} - 1680 q^{87} + 2460 q^{88} + 600 q^{90} - 1304 q^{91} + 3408 q^{93} + 4164 q^{96} + 1904 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 209x^{4} + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 249\nu^{2} ) / 680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 249\nu^{3} ) / 680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{6} - 20\nu^{5} - 40\nu^{4} - 1561\nu^{2} - 3620\nu - 4520 ) / 1360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{6} - 20\nu^{5} + 40\nu^{4} - 1561\nu^{2} - 3620\nu + 4520 ) / 1360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{7} - 40\nu^{5} - 2557\nu^{3} - 7240\nu ) / 2720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{7} + 40\nu^{5} - 2557\nu^{3} + 7240\nu ) / 2720 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 2\beta_{6} + 13\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 17\beta_{5} - 17\beta_{4} - 113 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34\beta_{7} - 34\beta_{6} - 181\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -249\beta_{7} + 249\beta_{6} - 249\beta_{5} - 249\beta_{4} + 1561\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -498\beta_{7} - 498\beta_{6} - 2557\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−2.66260 2.66260i
−1.18766 1.18766i
1.18766 + 1.18766i
2.66260 + 2.66260i
−2.66260 + 2.66260i
−1.18766 + 1.18766i
1.18766 1.18766i
2.66260 2.66260i
−2.66260 + 2.66260i −4.37420 + 2.80471i 6.17891i 9.55729 + 5.80157i 4.17891 19.1146i 9.35782 + 9.35782i −4.84884 4.84884i 11.2672 24.5367i −40.8945 + 10.0000i
2.2 −1.18766 + 1.18766i 5.11173 + 0.932827i 5.17891i −2.48157 10.9015i −7.17891 + 4.96314i −13.3578 13.3578i −15.6521 15.6521i 25.2597 + 9.53673i 15.8945 + 10.0000i
2.3 1.18766 1.18766i −0.932827 5.11173i 5.17891i 2.48157 + 10.9015i −7.17891 4.96314i −13.3578 13.3578i 15.6521 + 15.6521i −25.2597 + 9.53673i 15.8945 + 10.0000i
2.4 2.66260 2.66260i −2.80471 + 4.37420i 6.17891i −9.55729 5.80157i 4.17891 + 19.1146i 9.35782 + 9.35782i 4.84884 + 4.84884i −11.2672 24.5367i −40.8945 + 10.0000i
8.1 −2.66260 2.66260i −4.37420 2.80471i 6.17891i 9.55729 5.80157i 4.17891 + 19.1146i 9.35782 9.35782i −4.84884 + 4.84884i 11.2672 + 24.5367i −40.8945 10.0000i
8.2 −1.18766 1.18766i 5.11173 0.932827i 5.17891i −2.48157 + 10.9015i −7.17891 4.96314i −13.3578 + 13.3578i −15.6521 + 15.6521i 25.2597 9.53673i 15.8945 10.0000i
8.3 1.18766 + 1.18766i −0.932827 + 5.11173i 5.17891i 2.48157 10.9015i −7.17891 + 4.96314i −13.3578 + 13.3578i 15.6521 15.6521i −25.2597 9.53673i 15.8945 10.0000i
8.4 2.66260 + 2.66260i −2.80471 4.37420i 6.17891i −9.55729 + 5.80157i 4.17891 19.1146i 9.35782 9.35782i 4.84884 4.84884i −11.2672 + 24.5367i −40.8945 10.0000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.4.e.a 8
3.b odd 2 1 inner 15.4.e.a 8
4.b odd 2 1 240.4.v.c 8
5.b even 2 1 75.4.e.c 8
5.c odd 4 1 inner 15.4.e.a 8
5.c odd 4 1 75.4.e.c 8
12.b even 2 1 240.4.v.c 8
15.d odd 2 1 75.4.e.c 8
15.e even 4 1 inner 15.4.e.a 8
15.e even 4 1 75.4.e.c 8
20.e even 4 1 240.4.v.c 8
60.l odd 4 1 240.4.v.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.e.a 8 1.a even 1 1 trivial
15.4.e.a 8 3.b odd 2 1 inner
15.4.e.a 8 5.c odd 4 1 inner
15.4.e.a 8 15.e even 4 1 inner
75.4.e.c 8 5.b even 2 1
75.4.e.c 8 5.c odd 4 1
75.4.e.c 8 15.d odd 2 1
75.4.e.c 8 15.e even 4 1
240.4.v.c 8 4.b odd 2 1
240.4.v.c 8 12.b even 2 1
240.4.v.c 8 20.e even 4 1
240.4.v.c 8 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 209T^{4} + 1600 \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{8} + 110 T^{6} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 2000 T + 62500)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1990 T^{2} + 961000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 34 T^{3} + 578 T^{2} - 2720 T + 6400)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 68589044 T^{4} + \cdots + 32321044225600 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2772 T^{2} + 876096)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 227401574425600 \) Copy content Toggle raw display
$29$ \( (T^{4} - 18390 T^{2} + 40401000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 154 T - 23096)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 578 T^{3} + 167042 T^{2} + \cdots + 1695792400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 160240 T^{2} + \cdots + 1201216000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 274 T^{3} + 37538 T^{2} + \cdots + 193210000)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 971492324 T^{4} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + 59035541924 T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} - 709410 T^{2} + \cdots + 94361796000)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 202 T^{3} + \cdots + 80906113600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 568060 T^{2} + \cdots + 15888196000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1256 T^{3} + \cdots + 37974316900)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 348072 T^{2} + \cdots + 797271696)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 10775280164 T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} - 3626460 T^{2} + \cdots + 3249684036000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 952 T^{3} + \cdots + 615926736100)^{2} \) Copy content Toggle raw display
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