Properties

Label 15.11.c.a
Level $15$
Weight $11$
Character orbit 15.c
Analytic conductor $9.530$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 15.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.53035879011\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \( x^{14} + 11554 x^{12} + 52224391 x^{10} + 115670558124 x^{8} + 127683454012911 x^{6} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{20}\cdot 5^{21} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} + \beta_{2} + 3) q^{3} + (\beta_1 - 629) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} + 1563) q^{6} + (\beta_{7} + \beta_{6} + 3 \beta_{3} - 2 \beta_1 - 3610) q^{7} + ( - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{6} + 8 \beta_{5} + 29 \beta_{3} + 508 \beta_{2} + \cdots + 4) q^{8}+ \cdots + (\beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} - 10 \beta_{5} - 10 \beta_{3} - 135 \beta_{2} + \cdots + 8308) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{3} + \beta_{2} + 3) q^{3} + (\beta_1 - 629) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} + 1563) q^{6} + (\beta_{7} + \beta_{6} + 3 \beta_{3} - 2 \beta_1 - 3610) q^{7} + ( - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{6} + 8 \beta_{5} + 29 \beta_{3} + 508 \beta_{2} + \cdots + 4) q^{8}+ \cdots + ( - 176187 \beta_{13} - 3471 \beta_{12} + \cdots + 2592208497) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 44 q^{3} - 8802 q^{4} + 21886 q^{6} - 50548 q^{7} + 116362 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 44 q^{3} - 8802 q^{4} + 21886 q^{6} - 50548 q^{7} + 116362 q^{9} + 31250 q^{10} + 43756 q^{12} + 699408 q^{13} - 343750 q^{15} + 2871906 q^{16} - 3243880 q^{18} + 3814644 q^{19} - 2191008 q^{21} - 10493420 q^{22} + 9454542 q^{24} - 27343750 q^{25} + 13322636 q^{27} - 10989172 q^{28} + 20875000 q^{30} + 105444308 q^{31} - 187570700 q^{33} + 84960772 q^{34} + 80968490 q^{36} - 152902928 q^{37} - 262995952 q^{39} - 228656250 q^{40} + 1025108820 q^{42} - 82568592 q^{43} + 284500000 q^{45} + 302816052 q^{46} - 534917396 q^{48} + 1339929050 q^{49} - 519773324 q^{51} - 2117624528 q^{52} - 3171778694 q^{54} - 414437500 q^{55} + 2459677832 q^{57} + 2203542020 q^{58} + 918156250 q^{60} - 2372907732 q^{61} + 253855908 q^{63} + 5663115830 q^{64} + 915786920 q^{66} - 7807415008 q^{67} - 1032380604 q^{69} - 95812500 q^{70} + 2313658920 q^{72} + 10465834068 q^{73} - 85937500 q^{75} - 4927934540 q^{76} - 4082143640 q^{78} - 8333919076 q^{79} - 4284635426 q^{81} + 14404193720 q^{82} + 13837595568 q^{84} + 4711812500 q^{85} - 11735627260 q^{87} - 14973492180 q^{88} - 9226281250 q^{90} + 4013221984 q^{91} - 9561672552 q^{93} - 47501516708 q^{94} + 43132239458 q^{96} + 31262487532 q^{97} + 36258312560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 11554 x^{12} + 52224391 x^{10} + 115670558124 x^{8} + 127683454012911 x^{6} + \cdots + 62\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 570117271 \nu^{12} - 4883354422201 \nu^{10} + \cdots + 12\!\cdots\!60 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9510660222317 \nu^{13} + \cdots - 92\!\cdots\!00 \nu ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!99 \nu^{13} + \cdots - 48\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 84\!\cdots\!17 \nu^{13} + \cdots + 52\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 123638582890121 \nu^{13} + \cdots - 55\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!73 \nu^{13} + \cdots - 49\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!69 \nu^{13} + \cdots + 17\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73\!\cdots\!12 \nu^{13} + \cdots - 89\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53\!\cdots\!25 \nu^{13} + \cdots - 15\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!81 \nu^{13} + \cdots - 81\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!33 \nu^{13} + \cdots + 16\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!76 \nu^{13} + \cdots + 69\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 75\!\cdots\!77 \nu^{13} + \cdots + 45\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 624\beta_{2} ) / 625 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 2\beta_{5} + 2\beta_{4} - 54\beta_{3} + 4\beta_{2} + 601\beta _1 - 1031788 ) / 625 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 520 \beta_{13} - 505 \beta_{12} + 595 \beta_{11} + 45 \beta_{10} + 30 \beta_{9} - 75 \beta_{8} - 15 \beta_{7} + 700 \beta_{6} + 57 \beta_{5} + 19925 \beta_{3} + 1570612 \beta_{2} + 15 \beta _1 + 2770 ) / 625 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4050 \beta_{13} - 3675 \beta_{12} - 12275 \beta_{11} - 8475 \beta_{10} - 7350 \beta_{9} - 3675 \beta_{8} + 6125 \beta_{7} - 8094 \beta_{6} + 11794 \beta_{5} + 106 \beta_{4} + 873613 \beta_{3} + \cdots + 2594070886 ) / 625 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 387640 \beta_{13} + 409660 \beta_{12} - 433540 \beta_{11} - 53940 \beta_{10} + 20040 \beta_{9} + 75900 \beta_{8} + 79980 \beta_{7} - 759400 \beta_{6} - 1138659 \beta_{5} - 12739100 \beta_{3} + \cdots - 1683640 ) / 125 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18769200 \beta_{13} + 14105700 \beta_{12} + 67029100 \beta_{11} + 42201900 \beta_{10} + 28211400 \beta_{9} + 14105700 \beta_{8} - 28082500 \beta_{7} + \cdots - 7074626554004 ) / 625 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6383382580 \beta_{13} - 7114116145 \beta_{12} + 6666461755 \beta_{11} + 1090249305 \beta_{10} - 843371130 \beta_{9} - 1424531175 \beta_{8} + \cdots + 19953205330 ) / 625 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 69371017350 \beta_{13} - 43374086475 \beta_{12} - 266072905175 \beta_{11} - 164738965575 \beta_{10} - 86748172950 \beta_{9} - 43374086475 \beta_{8} + \cdots + 20\!\cdots\!02 ) / 625 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20258138664560 \beta_{13} + 23560150483640 \beta_{12} - 19642754597660 \beta_{11} - 3847620857760 \beta_{10} + 3529875170160 \beta_{9} + \cdots - 40717501280060 ) / 625 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 47213902958400 \beta_{13} + 25082372159400 \beta_{12} + 187917643098200 \beta_{11} + 116559336715800 \beta_{10} + 50164744318800 \beta_{9} + \cdots - 11\!\cdots\!68 ) / 125 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 63\!\cdots\!40 \beta_{13} + \cdots + 66\!\cdots\!90 ) / 625 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 77\!\cdots\!50 \beta_{13} + \cdots + 16\!\cdots\!78 ) / 625 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 19\!\cdots\!20 \beta_{13} + \cdots - 46\!\cdots\!20 ) / 625 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
55.5349i
54.9539i
49.8576i
42.9372i
29.7613i
15.0833i
2.70449i
2.70449i
15.0833i
29.7613i
42.9372i
49.8576i
54.9539i
55.5349i
57.7709i −60.6159 + 235.318i −2313.48 1397.54i 13594.6 + 3501.84i 22792.7 74494.6i −51700.4 28528.1i 80737.3
11.2 52.7178i −210.660 121.125i −1755.17 1397.54i −6385.47 + 11105.5i −8585.72 38545.6i 29706.2 + 51032.6i −73675.4
11.3 52.0937i 196.615 142.800i −1689.75 1397.54i −7438.96 10242.4i −32323.0 34681.6i 18265.6 56152.9i 72803.2
11.4 40.7012i 230.652 + 76.4761i −632.586 1397.54i 3112.67 9387.81i 19744.7 15931.0i 47351.8 + 35278.8i −56881.6
11.5 27.5253i −80.1400 + 229.405i 266.360 1397.54i 6314.43 + 2205.87i −24115.7 35517.5i −46204.2 36769.0i −38467.7
11.6 17.3194i −236.663 + 55.1313i 724.039 1397.54i 954.839 + 4098.86i 2728.90 30275.0i 52970.1 26095.1i 24204.6
11.7 4.94055i 182.813 + 160.089i 999.591 1397.54i 790.929 903.195i −5515.83 9997.66i 7791.87 + 58532.7i 6904.63
11.8 4.94055i 182.813 160.089i 999.591 1397.54i 790.929 + 903.195i −5515.83 9997.66i 7791.87 58532.7i 6904.63
11.9 17.3194i −236.663 55.1313i 724.039 1397.54i 954.839 4098.86i 2728.90 30275.0i 52970.1 + 26095.1i 24204.6
11.10 27.5253i −80.1400 229.405i 266.360 1397.54i 6314.43 2205.87i −24115.7 35517.5i −46204.2 + 36769.0i −38467.7
11.11 40.7012i 230.652 76.4761i −632.586 1397.54i 3112.67 + 9387.81i 19744.7 15931.0i 47351.8 35278.8i −56881.6
11.12 52.0937i 196.615 + 142.800i −1689.75 1397.54i −7438.96 + 10242.4i −32323.0 34681.6i 18265.6 + 56152.9i 72803.2
11.13 52.7178i −210.660 + 121.125i −1755.17 1397.54i −6385.47 11105.5i −8585.72 38545.6i 29706.2 51032.6i −73675.4
11.14 57.7709i −60.6159 235.318i −2313.48 1397.54i 13594.6 3501.84i 22792.7 74494.6i −51700.4 + 28528.1i 80737.3
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.11.c.a 14
3.b odd 2 1 inner 15.11.c.a 14
4.b odd 2 1 240.11.l.b 14
5.b even 2 1 75.11.c.g 14
5.c odd 4 2 75.11.d.d 28
12.b even 2 1 240.11.l.b 14
15.d odd 2 1 75.11.c.g 14
15.e even 4 2 75.11.d.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.c.a 14 1.a even 1 1 trivial
15.11.c.a 14 3.b odd 2 1 inner
75.11.c.g 14 5.b even 2 1
75.11.c.g 14 15.d odd 2 1
75.11.d.d 28 5.c odd 4 2
75.11.d.d 28 15.e even 4 2
240.11.l.b 14 4.b odd 2 1
240.11.l.b 14 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 11569 T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{14} - 44 T^{13} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1953125)^{7} \) Copy content Toggle raw display
$7$ \( (T^{7} + 25274 T^{6} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + 212695941340 T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{7} - 349704 T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + 15608014678924 T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{7} - 1907322 T^{6} + \cdots + 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 277803738663444 T^{12} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} - 52722154 T^{6} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + 76451464 T^{6} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + 41284296 T^{6} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + 1186453866 T^{6} + \cdots - 71\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + 3903707504 T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} - 5232917034 T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + 4166959538 T^{6} + \cdots - 24\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} - 15631243766 T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
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