Properties

Label 15.5.f.a
Level $15$
Weight $5$
Character orbit 15.f
Analytic conductor $1.551$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 \) 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: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 12 \beta_{2} - \beta_1) q^{4} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 8 \beta_{2} + \cdots - 10) q^{5}+ \cdots + 27 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 12 \beta_{2} - \beta_1) q^{4} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 8 \beta_{2} + \cdots - 10) q^{5}+ \cdots + ( - 108 \beta_{6} + 432 \beta_{5} - 243 \beta_{4} - 324 \beta_{3} + \cdots + 243 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 84 q^{5} + 36 q^{6} + 20 q^{7} + 180 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 84 q^{5} + 36 q^{6} + 20 q^{7} + 180 q^{8} + 104 q^{10} - 288 q^{11} - 360 q^{12} - 340 q^{13} + 144 q^{15} + 620 q^{16} + 900 q^{17} + 564 q^{20} + 792 q^{21} - 1100 q^{22} - 1560 q^{23} - 1204 q^{25} - 3024 q^{26} + 3580 q^{28} - 2664 q^{30} - 512 q^{31} + 4980 q^{32} + 2700 q^{33} + 6600 q^{35} + 2484 q^{36} - 3820 q^{37} - 7680 q^{38} - 2952 q^{40} - 2712 q^{41} - 7380 q^{42} - 1240 q^{43} - 1944 q^{45} + 13528 q^{46} + 4800 q^{47} + 3600 q^{48} + 3744 q^{50} + 6264 q^{51} - 1240 q^{52} + 1020 q^{53} - 3644 q^{55} - 30720 q^{56} - 5400 q^{57} + 2340 q^{58} - 1044 q^{60} - 4760 q^{61} + 28680 q^{62} + 540 q^{63} - 1212 q^{65} + 10008 q^{66} - 8920 q^{67} - 1920 q^{68} + 7380 q^{70} + 7536 q^{71} - 4860 q^{72} + 11600 q^{73} - 5976 q^{75} + 4344 q^{76} - 360 q^{77} - 4680 q^{78} + 10644 q^{80} - 5832 q^{81} - 27200 q^{82} - 32400 q^{83} - 15628 q^{85} + 14592 q^{86} + 10620 q^{87} - 14340 q^{88} + 8964 q^{90} + 16528 q^{91} - 31800 q^{92} + 14040 q^{93} + 18864 q^{95} - 4068 q^{96} + 58640 q^{97} + 46440 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19857 \nu^{7} - 350053 \nu^{6} - 190938 \nu^{5} - 1295568 \nu^{4} + 60124233 \nu^{3} - 934582407 \nu^{2} + 716887878 \nu + 368125308 ) / 10440376750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1224717 \nu^{7} + 40833643 \nu^{6} + 212097928 \nu^{5} + 1138297958 \nu^{4} - 4285577073 \nu^{3} + 47520018367 \nu^{2} + \cdots + 621224634602 ) / 83523014000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1331 \nu^{7} - 726 \nu^{6} - 396 \nu^{5} + 79644 \nu^{4} - 3304389 \nu^{3} + 2589906 \nu^{2} - 983124 \nu - 20889564 ) / 39697250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6913292 \nu^{7} - 45272557 \nu^{6} - 24694122 \nu^{5} + 1350453158 \nu^{4} - 5477763248 \nu^{3} - 37379548933 \nu^{2} + \cdots + 599396712902 ) / 83523014000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 468490 \nu^{7} + 2002219 \nu^{6} + 16278122 \nu^{5} + 18734582 \nu^{4} + 693074870 \nu^{3} + 45879131 \nu^{2} + 15134603638 \nu + 30778570718 ) / 3340920560 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47399 \nu^{7} - 25854 \nu^{6} + 707666 \nu^{5} + 4673476 \nu^{4} - 77977231 \nu^{3} + 70577574 \nu^{2} + 999020754 \nu - 1154530656 ) / 317578000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} - 28\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{7} - 5\beta_{6} - 2\beta_{5} - 39\beta_{4} + 5\beta_{3} + 25\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -55\beta_{7} + 55\beta_{5} + 85\beta_{4} + 55\beta_{3} + 85\beta _1 - 959 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 305\beta_{7} + 305\beta_{6} - 475\beta_{3} - 2215\beta_{2} - 1679\beta _1 + 1910 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2799\beta_{6} - 2559\beta_{5} - 5319\beta_{4} + 5358\beta_{3} + 42542\beta_{2} + 5319\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15795\beta_{7} + 15795\beta_{6} + 11598\beta_{5} + 76931\beta_{4} - 15795\beta_{3} - 139095\beta_{2} - 123300 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−5.02811 5.02811i
−2.08045 2.08045i
3.30519 + 3.30519i
3.80336 + 3.80336i
−5.02811 + 5.02811i
−2.08045 + 2.08045i
3.30519 3.30519i
3.80336 3.80336i
−5.02811 5.02811i −3.67423 + 3.67423i 34.5637i −23.8949 7.35070i 36.9489 −38.5593 38.5593i 93.3405 93.3405i 27.0000i 83.1861 + 157.106i
7.2 −2.08045 2.08045i 3.67423 3.67423i 7.34348i −8.43390 23.5344i −15.2881 65.1093 + 65.1093i −48.5649 + 48.5649i 27.0000i −31.4158 + 66.5084i
7.3 3.30519 + 3.30519i 3.67423 3.67423i 5.84858i −16.2403 + 19.0066i 24.2881 −33.1649 33.1649i 33.5524 33.5524i 27.0000i −116.498 + 9.14312i
7.4 3.80336 + 3.80336i −3.67423 + 3.67423i 12.9311i 6.56915 24.1215i −27.9489 16.6149 + 16.6149i 11.6720 11.6720i 27.0000i 116.728 66.7579i
13.1 −5.02811 + 5.02811i −3.67423 3.67423i 34.5637i −23.8949 + 7.35070i 36.9489 −38.5593 + 38.5593i 93.3405 + 93.3405i 27.0000i 83.1861 157.106i
13.2 −2.08045 + 2.08045i 3.67423 + 3.67423i 7.34348i −8.43390 + 23.5344i −15.2881 65.1093 65.1093i −48.5649 48.5649i 27.0000i −31.4158 66.5084i
13.3 3.30519 3.30519i 3.67423 + 3.67423i 5.84858i −16.2403 19.0066i 24.2881 −33.1649 + 33.1649i 33.5524 + 33.5524i 27.0000i −116.498 9.14312i
13.4 3.80336 3.80336i −3.67423 3.67423i 12.9311i 6.56915 + 24.1215i −27.9489 16.6149 16.6149i 11.6720 + 11.6720i 27.0000i 116.728 + 66.7579i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.f.a 8
3.b odd 2 1 45.5.g.e 8
4.b odd 2 1 240.5.bg.c 8
5.b even 2 1 75.5.f.e 8
5.c odd 4 1 inner 15.5.f.a 8
5.c odd 4 1 75.5.f.e 8
15.d odd 2 1 225.5.g.m 8
15.e even 4 1 45.5.g.e 8
15.e even 4 1 225.5.g.m 8
20.e even 4 1 240.5.bg.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.f.a 8 1.a even 1 1 trivial
15.5.f.a 8 5.c odd 4 1 inner
45.5.g.e 8 3.b odd 2 1
45.5.g.e 8 15.e even 4 1
75.5.f.e 8 5.b even 2 1
75.5.f.e 8 5.c odd 4 1
225.5.g.m 8 15.d odd 2 1
225.5.g.m 8 15.e even 4 1
240.5.bg.c 8 4.b odd 2 1
240.5.bg.c 8 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 60 T^{5} + 1973 T^{4} + \cdots + 276676 \) Copy content Toggle raw display
$3$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 84 T^{7} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{8} - 20 T^{7} + \cdots + 30620728960000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 144 T^{3} - 16742 T^{2} + \cdots + 27154144)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 340 T^{7} + \cdots + 158448579136 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 125636659418176 \) Copy content Toggle raw display
$19$ \( T^{8} + 567288 T^{6} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + 1560 T^{7} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + 1665012 T^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + 256 T^{3} + \cdots + 388673200000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 3820 T^{7} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + 1356 T^{3} + \cdots - 1195607024000)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 1240 T^{7} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} - 4800 T^{7} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} - 1020 T^{7} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + 28672428 T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2380 T^{3} + \cdots - 51045861284864)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 8920 T^{7} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} - 3768 T^{3} + \cdots - 4392786466304)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 11600 T^{7} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + 118621200 T^{6} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + 32400 T^{7} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + 128964168 T^{6} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} - 58640 T^{7} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
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