Properties

Label 135.4.e.c
Level $135$
Weight $4$
Character orbit 135.e
Analytic conductor $7.965$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{13} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{10} - 5 \beta_{5} - \beta_{3} - 5) q^{4} + ( - 5 \beta_{5} - 5) q^{5} + ( - \beta_{12} - \beta_{7} + 3 \beta_{5}) q^{7} + (\beta_{4} + 7 \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{10} - 5 \beta_{5} - \beta_{3} - 5) q^{4} + ( - 5 \beta_{5} - 5) q^{5} + ( - \beta_{12} - \beta_{7} + 3 \beta_{5}) q^{7} + (\beta_{4} + 7 \beta_{2} + 1) q^{8} + 5 \beta_{2} q^{10} + ( - \beta_{11} + \beta_{9} + 2 \beta_{5} - 3 \beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - 15 \beta_{5} - \beta_{3} + 4 \beta_1 - 15) q^{13} + ( - \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 4 \beta_{5} + 2 \beta_{3} - 7 \beta_1 + 4) q^{14} + ( - 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots - 6 \beta_1) q^{16}+ \cdots + (34 \beta_{9} - 5 \beta_{8} + 9 \beta_{7} - 11 \beta_{4} + 25 \beta_{3} + \cdots + 188) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26} + 674 q^{28} + 296 q^{29} - 244 q^{31} + 314 q^{32} - 125 q^{34} + 220 q^{35} + 808 q^{37} - 305 q^{38} - 90 q^{40} + 47 q^{41} - 525 q^{43} + 110 q^{44} + 1434 q^{46} - 164 q^{47} - 1225 q^{49} - 50 q^{50} - 1682 q^{52} + 1012 q^{53} + 230 q^{55} + 981 q^{56} - 1183 q^{58} + 85 q^{59} - 828 q^{61} - 1572 q^{62} + 4472 q^{64} - 480 q^{65} - 1093 q^{67} - 2473 q^{68} + 105 q^{70} + 656 q^{71} + 4170 q^{73} + 1316 q^{74} - 2789 q^{76} - 24 q^{77} - 2110 q^{79} + 3240 q^{80} - 124 q^{82} - 1290 q^{83} - 805 q^{85} + 2569 q^{86} - 2271 q^{88} - 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 517 q^{94} - 1395 q^{95} - 1787 q^{97} + 2558 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20061126179 \nu^{13} + 306116042626 \nu^{12} - 1952416155384 \nu^{11} + 13978725434868 \nu^{10} - 67214635202751 \nu^{9} + \cdots + 81\!\cdots\!72 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22166149189 \nu^{13} - 82456841566 \nu^{12} + 1064588155344 \nu^{11} - 2918043973788 \nu^{10} + 34529619637641 \nu^{9} + \cdots - 36\!\cdots\!52 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 461405902117 \nu^{13} - 7040668980398 \nu^{12} + 44905571573832 \nu^{11} - 321510685001964 \nu^{10} + \cdots - 22\!\cdots\!56 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2939472898789 \nu^{13} - 5818762419041 \nu^{12} + 140176351013994 \nu^{11} - 170511125461188 \nu^{10} + \cdots - 93\!\cdots\!52 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4311956453557 \nu^{13} + 5840830212833 \nu^{12} - 197347051822122 \nu^{11} + 132817332275844 \nu^{10} + \cdots - 15\!\cdots\!24 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3624809887207 \nu^{13} - 20338354293458 \nu^{12} + 206799337599372 \nu^{11} - 852176307382044 \nu^{10} + \cdots - 12\!\cdots\!76 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 339722329867 \nu^{13} + 807058989248 \nu^{12} - 14594710056057 \nu^{11} + 24987551185914 \nu^{10} + \cdots + 10\!\cdots\!56 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 186796009849 \nu^{13} + 832866413286 \nu^{12} - 9801117027144 \nu^{11} + 34103389012188 \nu^{10} - 323151061131341 \nu^{9} + \cdots + 32\!\cdots\!72 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38213147684257 \nu^{13} + 75643911447533 \nu^{12} + \cdots + 12\!\cdots\!76 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21668969460439 \nu^{13} - 23844974735516 \nu^{12} + \cdots + 86\!\cdots\!48 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 154567151645861 \nu^{13} - 238501344274534 \nu^{12} + \cdots + 52\!\cdots\!52 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 44692378643083 \nu^{13} + 78652198588502 \nu^{12} + \cdots - 14\!\cdots\!56 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 13\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - 23\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} - 2\beta_{12} + 2\beta_{11} - 31\beta_{10} + \beta_{6} - 294\beta_{5} - 31\beta_{3} + 6\beta _1 - 294 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{13} - 6 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 6 \beta_{7} - 36 \beta_{6} + 26 \beta_{5} + 36 \beta_{4} + 597 \beta_{2} - 597 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -92\beta_{9} + 76\beta_{8} - 84\beta_{7} - 38\beta_{4} + 897\beta_{3} - 328\beta_{2} + 7615 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 84 \beta_{13} + 284 \beta_{12} - 124 \beta_{11} - 128 \beta_{10} + 1103 \beta_{6} - 2457 \beta_{5} - 128 \beta_{3} + 16239 \beta _1 - 2457 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2366 \beta_{13} + 2862 \beta_{12} - 3190 \beta_{11} + 25711 \beta_{10} + 3190 \beta_{9} - 2366 \beta_{8} + 2862 \beta_{7} - 1191 \beta_{6} + 207304 \beta_{5} + 1191 \beta_{4} + 12966 \beta_{2} - 12966 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4990\beta_{9} + 2710\beta_{8} + 10050\beta_{7} - 32458\beta_{4} + 5658\beta_{3} - 451733\beta_{2} + 111860 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 69976 \beta_{13} - 89936 \beta_{12} + 100416 \beta_{11} - 736057 \beta_{10} + 35836 \beta_{6} - 5772441 \beta_{5} - 736057 \beta_{3} + 450816 \beta _1 - 5772441 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 82152 \beta_{13} - 319512 \beta_{12} + 169080 \beta_{11} + 216848 \beta_{10} - 169080 \beta_{9} - 82152 \beta_{8} - 319512 \beta_{7} - 942285 \beta_{6} + 4153523 \beta_{5} + \cdots - 12722871 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3026058 \beta_{9} + 2035002 \beta_{8} - 2711322 \beta_{7} - 1072205 \beta_{4} + 21074031 \beta_{3} - 14703286 \beta_{2} + 162674650 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2459146 \beta_{13} + 9645086 \beta_{12} - 5260346 \beta_{11} - 7736162 \beta_{10} + 27207296 \beta_{6} - 140767086 \beta_{5} - 7736162 \beta_{3} + 360996933 \beta _1 - 140767086 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(-1 - \beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
2.65775 + 4.60336i
2.13089 + 3.69081i
1.09722 + 1.90044i
0.112625 + 0.195072i
−0.785104 1.35984i
−1.52087 2.63422i
−2.69252 4.66357i
2.65775 4.60336i
2.13089 3.69081i
1.09722 1.90044i
0.112625 0.195072i
−0.785104 + 1.35984i
−1.52087 + 2.63422i
−2.69252 + 4.66357i
−2.65775 + 4.60336i 0 −10.1273 17.5410i −2.50000 4.33013i 0 6.71686 11.6339i 65.1396 0 26.5775
46.2 −2.13089 + 3.69081i 0 −5.08138 8.80120i −2.50000 4.33013i 0 −15.3820 + 26.6423i 9.21718 0 21.3089
46.3 −1.09722 + 1.90044i 0 1.59221 + 2.75778i −2.50000 4.33013i 0 −1.38302 + 2.39547i −24.5436 0 10.9722
46.4 −0.112625 + 0.195072i 0 3.97463 + 6.88426i −2.50000 4.33013i 0 15.5970 27.0148i −3.59257 0 1.12625
46.5 0.785104 1.35984i 0 2.76722 + 4.79297i −2.50000 4.33013i 0 −17.1199 + 29.6525i 21.2519 0 −7.85104
46.6 1.52087 2.63422i 0 −0.626094 1.08443i −2.50000 4.33013i 0 6.85611 11.8751i 20.5251 0 −15.2087
46.7 2.69252 4.66357i 0 −10.4993 18.1853i −2.50000 4.33013i 0 −6.28510 + 10.8861i −69.9976 0 −26.9252
91.1 −2.65775 4.60336i 0 −10.1273 + 17.5410i −2.50000 + 4.33013i 0 6.71686 + 11.6339i 65.1396 0 26.5775
91.2 −2.13089 3.69081i 0 −5.08138 + 8.80120i −2.50000 + 4.33013i 0 −15.3820 26.6423i 9.21718 0 21.3089
91.3 −1.09722 1.90044i 0 1.59221 2.75778i −2.50000 + 4.33013i 0 −1.38302 2.39547i −24.5436 0 10.9722
91.4 −0.112625 0.195072i 0 3.97463 6.88426i −2.50000 + 4.33013i 0 15.5970 + 27.0148i −3.59257 0 1.12625
91.5 0.785104 + 1.35984i 0 2.76722 4.79297i −2.50000 + 4.33013i 0 −17.1199 29.6525i 21.2519 0 −7.85104
91.6 1.52087 + 2.63422i 0 −0.626094 + 1.08443i −2.50000 + 4.33013i 0 6.85611 + 11.8751i 20.5251 0 −15.2087
91.7 2.69252 + 4.66357i 0 −10.4993 + 18.1853i −2.50000 + 4.33013i 0 −6.28510 10.8861i −69.9976 0 −26.9252
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.4.e.c 14
3.b odd 2 1 45.4.e.c 14
9.c even 3 1 inner 135.4.e.c 14
9.c even 3 1 405.4.a.n 7
9.d odd 6 1 45.4.e.c 14
9.d odd 6 1 405.4.a.m 7
15.d odd 2 1 225.4.e.d 14
15.e even 4 2 225.4.k.d 28
45.h odd 6 1 225.4.e.d 14
45.h odd 6 1 2025.4.a.bb 7
45.j even 6 1 2025.4.a.ba 7
45.l even 12 2 225.4.k.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.c 14 3.b odd 2 1
45.4.e.c 14 9.d odd 6 1
135.4.e.c 14 1.a even 1 1 trivial
135.4.e.c 14 9.c even 3 1 inner
225.4.e.d 14 15.d odd 2 1
225.4.e.d 14 45.h odd 6 1
225.4.k.d 28 15.e even 4 2
225.4.k.d 28 45.l even 12 2
405.4.a.m 7 9.d odd 6 1
405.4.a.n 7 9.c even 3 1
2025.4.a.ba 7 45.j even 6 1
2025.4.a.bb 7 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 2 T_{2}^{13} + 48 T_{2}^{12} + 60 T_{2}^{11} + 1605 T_{2}^{10} + 1800 T_{2}^{9} + 23232 T_{2}^{8} + 2346 T_{2}^{7} + 209529 T_{2}^{6} + 55412 T_{2}^{5} + 765088 T_{2}^{4} - 276096 T_{2}^{3} + 1572480 T_{2}^{2} + \cdots + 82944 \) acting on \(S_{4}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + 48 T^{12} + \cdots + 82944 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + 22 T^{13} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{14} + 23 T^{13} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{14} + 96 T^{13} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{7} - 161 T^{6} + \cdots + 60588009792)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} - 279 T^{6} + \cdots - 1377989598400)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 96 T^{13} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{14} - 296 T^{13} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{14} + 244 T^{13} + \cdots + 86\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{7} - 404 T^{6} + \cdots - 83646911884544)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} - 47 T^{13} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{14} + 525 T^{13} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{14} + 164 T^{13} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{7} - 506 T^{6} + \cdots + 11\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} - 85 T^{13} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{14} + 828 T^{13} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{14} + 1093 T^{13} + \cdots + 85\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{7} - 328 T^{6} + \cdots + 32\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} - 2085 T^{6} + \cdots + 82\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + 2110 T^{13} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{14} + 1290 T^{13} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{7} + 3048 T^{6} + \cdots + 32\!\cdots\!50)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 1787 T^{13} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
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