Properties

Label 75.9.f.e
Level $75$
Weight $9$
Character orbit 75.f
Analytic conductor $30.553$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 4140 x^{13} + 1109893 x^{12} - 3063780 x^{11} + 8569800 x^{10} - 2336277960 x^{9} + 311176823556 x^{8} - 1727244961920 x^{7} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{20}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{6} q^{3} + (\beta_{9} + 3 \beta_{3} + 3 \beta_{2} - 158 \beta_1) q^{4} + ( - \beta_{7} + 3 \beta_{3} - 3 \beta_{2} - 142) q^{6} + (\beta_{15} - 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{9} + \beta_{8} - \beta_{7} - 7 \beta_{5} + \cdots - 285) q^{7}+ \cdots + 2187 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{6} q^{3} + (\beta_{9} + 3 \beta_{3} + 3 \beta_{2} - 158 \beta_1) q^{4} + ( - \beta_{7} + 3 \beta_{3} - 3 \beta_{2} - 142) q^{6} + (\beta_{15} - 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{9} + \beta_{8} - \beta_{7} - 7 \beta_{5} + \cdots - 285) q^{7}+ \cdots + (2187 \beta_{15} - 6561 \beta_{14} + 17496 \beta_{12} + \cdots - 3210516 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2268 q^{6} - 4540 q^{7} - 17460 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2268 q^{6} - 4540 q^{7} - 17460 q^{8} - 23616 q^{11} - 22680 q^{12} - 133420 q^{13} - 471380 q^{16} - 573300 q^{17} + 163944 q^{21} + 234700 q^{22} - 651480 q^{23} - 448848 q^{26} + 3567940 q^{28} + 1311776 q^{31} - 641460 q^{32} + 3815100 q^{33} + 5519988 q^{36} + 3607340 q^{37} - 8139840 q^{38} - 14740104 q^{41} + 9643860 q^{42} + 4805480 q^{43} + 14024216 q^{46} - 26529600 q^{47} - 3661200 q^{48} - 6168312 q^{51} + 15861080 q^{52} - 16612140 q^{53} + 10752000 q^{56} - 4714200 q^{57} - 63562980 q^{58} - 12550600 q^{61} + 35190840 q^{62} - 9928980 q^{63} + 46958616 q^{66} - 46836760 q^{67} + 197811840 q^{68} - 85681968 q^{71} + 38185020 q^{72} + 50835800 q^{73} + 101166648 q^{76} - 97175880 q^{77} - 131709240 q^{78} - 76527504 q^{81} + 181542400 q^{82} + 208234800 q^{83} - 187512576 q^{86} + 74298060 q^{87} - 138207420 q^{88} + 38623856 q^{91} - 652331400 q^{92} - 159787080 q^{93} + 531512604 q^{96} + 138370520 q^{97} + 50186520 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4140 x^{13} + 1109893 x^{12} - 3063780 x^{11} + 8569800 x^{10} - 2336277960 x^{9} + 311176823556 x^{8} - 1727244961920 x^{7} + \cdots + 66\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 96\!\cdots\!78 \nu^{15} + \cdots + 52\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 65\!\cdots\!74 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 39\!\cdots\!03 \nu^{15} + \cdots + 29\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!51 \nu^{15} + \cdots - 89\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59\!\cdots\!66 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!17 \nu^{15} + \cdots + 19\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!57 \nu^{15} + \cdots + 95\!\cdots\!00 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!79 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!91 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 48\!\cdots\!57 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!17 \nu^{15} + \cdots + 22\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72\!\cdots\!26 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38\!\cdots\!67 \nu^{15} + \cdots + 39\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!02 \nu^{15} + \cdots - 32\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 61\!\cdots\!43 \nu^{15} + \cdots + 33\!\cdots\!00 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{5} - 27\beta_{2} ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 27\beta_{9} + 4\beta_{8} + 69\beta_{3} + 69\beta_{2} - 10934\beta_1 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 85 \beta_{6} + \beta_{4} - 686 \beta_{3} + 777 \beta _1 + 776 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 252 \beta_{14} - 54 \beta_{13} - 333 \beta_{12} + 333 \beta_{11} + 252 \beta_{10} + 4492 \beta_{7} - 12249 \beta_{6} + 12249 \beta_{5} - 21339 \beta_{4} + 77613 \beta_{3} - 77613 \beta_{2} + \cdots - 7495850 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23787 \beta_{15} + 13815 \beta_{14} - 23787 \beta_{13} - 35649 \beta_{12} + 56979 \beta_{9} + 27477 \beta_{8} - 27477 \beta_{7} - 2225773 \beta_{5} + 56979 \beta_{4} + 13818366 \beta_{2} + \cdots + 25855038 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2538 \beta_{15} - 6274 \beta_{14} - 11011 \beta_{12} - 11011 \beta_{11} - 6274 \beta_{10} - 600949 \beta_{9} - 159616 \beta_{8} + 521963 \beta_{6} + 521963 \beta_{5} - 2838803 \beta_{3} + \cdots + 206761164 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 18229455 \beta_{15} + 18229455 \beta_{13} - 36089793 \beta_{11} + 2015847 \beta_{10} + 69651333 \beta_{9} + 26251425 \beta_{8} + 26251425 \beta_{7} - 2062510117 \beta_{6} + \cdots - 26691404094 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 63307764 \beta_{14} + 69910938 \beta_{13} + 205029171 \beta_{12} - 205029171 \beta_{11} - 63307764 \beta_{10} - 3918798500 \beta_{7} + 14419287063 \beta_{6} + \cdots + 4287483101386 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 505840601 \beta_{15} + 193668291 \beta_{14} + 505840601 \beta_{13} + 1245732943 \beta_{12} - 2681309833 \beta_{9} - 944521371 \beta_{8} + 944521371 \beta_{7} + \cdots - 948332962462 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 65295387198 \beta_{15} - 20529981126 \beta_{14} + 118354382001 \beta_{12} + 118354382001 \beta_{11} - 20529981126 \beta_{10} + 9695336611527 \beta_{9} + \cdots - 33\!\cdots\!20 \beta_1 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10217224989423 \beta_{15} - 10217224989423 \beta_{13} + 30136591664217 \beta_{11} + 9122606167761 \beta_{10} - 69938627778657 \beta_{9} + \cdots + 23\!\cdots\!70 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2925541320320 \beta_{14} - 2149201323270 \beta_{13} - 1815338707165 \beta_{12} + 1815338707165 \beta_{11} - 2925541320320 \beta_{10} + \cdots - 98\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 76\!\cdots\!79 \beta_{15} + \cdots + 21\!\cdots\!34 ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 49\!\cdots\!18 \beta_{15} + \cdots + 21\!\cdots\!52 \beta_1 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 21\!\cdots\!57 \beta_{15} + \cdots - 71\!\cdots\!78 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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
16.9634 + 16.9634i
19.6124 + 19.6124i
9.06979 + 9.06979i
1.64191 + 1.64191i
0.963769 + 0.963769i
−8.66090 8.66090i
−20.7939 20.7939i
−18.7966 18.7966i
16.9634 16.9634i
19.6124 19.6124i
9.06979 9.06979i
1.64191 1.64191i
0.963769 0.963769i
−8.66090 + 8.66090i
−20.7939 + 20.7939i
−18.7966 + 18.7966i
−19.4129 19.4129i 33.0681 33.0681i 497.724i 0 −1283.90 −1858.12 1858.12i 4692.57 4692.57i 2187.00i 0
7.2 −17.1630 17.1630i −33.0681 + 33.0681i 333.134i 0 1135.09 −1268.31 1268.31i 1323.85 1323.85i 2187.00i 0
7.3 −6.62030 6.62030i −33.0681 + 33.0681i 168.343i 0 437.842 216.619 + 216.619i −2809.28 + 2809.28i 2187.00i 0
7.4 −4.09140 4.09140i 33.0681 33.0681i 222.521i 0 −270.590 2635.24 + 2635.24i −1957.82 + 1957.82i 2187.00i 0
7.5 −3.41326 3.41326i 33.0681 33.0681i 232.699i 0 −225.740 −2986.31 2986.31i −1668.06 + 1668.06i 2187.00i 0
7.6 11.1104 + 11.1104i −33.0681 + 33.0681i 9.11845i 0 −734.799 1159.26 + 1159.26i 2945.57 2945.57i 2187.00i 0
7.7 18.3444 + 18.3444i 33.0681 33.0681i 417.032i 0 1213.23 1693.91 + 1693.91i −2954.03 + 2954.03i 2187.00i 0
7.8 21.2461 + 21.2461i −33.0681 + 33.0681i 646.792i 0 −1405.14 −1862.28 1862.28i −8302.80 + 8302.80i 2187.00i 0
43.1 −19.4129 + 19.4129i 33.0681 + 33.0681i 497.724i 0 −1283.90 −1858.12 + 1858.12i 4692.57 + 4692.57i 2187.00i 0
43.2 −17.1630 + 17.1630i −33.0681 33.0681i 333.134i 0 1135.09 −1268.31 + 1268.31i 1323.85 + 1323.85i 2187.00i 0
43.3 −6.62030 + 6.62030i −33.0681 33.0681i 168.343i 0 437.842 216.619 216.619i −2809.28 2809.28i 2187.00i 0
43.4 −4.09140 + 4.09140i 33.0681 + 33.0681i 222.521i 0 −270.590 2635.24 2635.24i −1957.82 1957.82i 2187.00i 0
43.5 −3.41326 + 3.41326i 33.0681 + 33.0681i 232.699i 0 −225.740 −2986.31 + 2986.31i −1668.06 1668.06i 2187.00i 0
43.6 11.1104 11.1104i −33.0681 33.0681i 9.11845i 0 −734.799 1159.26 1159.26i 2945.57 + 2945.57i 2187.00i 0
43.7 18.3444 18.3444i 33.0681 + 33.0681i 417.032i 0 1213.23 1693.91 1693.91i −2954.03 2954.03i 2187.00i 0
43.8 21.2461 21.2461i −33.0681 33.0681i 646.792i 0 −1405.14 −1862.28 + 1862.28i −8302.80 8302.80i 2187.00i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
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 75.9.f.e 16
5.b even 2 1 15.9.f.a 16
5.c odd 4 1 15.9.f.a 16
5.c odd 4 1 inner 75.9.f.e 16
15.d odd 2 1 45.9.g.c 16
15.e even 4 1 45.9.g.c 16
20.d odd 2 1 240.9.bg.d 16
20.e even 4 1 240.9.bg.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.f.a 16 5.b even 2 1
15.9.f.a 16 5.c odd 4 1
45.9.g.c 16 15.d odd 2 1
45.9.g.c 16 15.e even 4 1
75.9.f.e 16 1.a even 1 1 trivial
75.9.f.e 16 5.c odd 4 1 inner
240.9.bg.d 16 20.d odd 2 1
240.9.bg.d 16 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 5820 T_{2}^{13} + 1126741 T_{2}^{12} + 3704100 T_{2}^{11} + 16936200 T_{2}^{10} + 3824426040 T_{2}^{9} + 327220620996 T_{2}^{8} + 2076332044800 T_{2}^{7} + \cdots + 45\!\cdots\!56 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 5820 T^{13} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( (T^{4} + 4782969)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 4540 T^{15} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + 11808 T^{7} + \cdots + 79\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 133420 T^{15} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + 573300 T^{15} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + 140583592296 T^{14} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + 651480 T^{15} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + 4670365028124 T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} - 655888 T^{7} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 3607340 T^{15} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + 7370052 T^{7} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 4805480 T^{15} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + 26529600 T^{15} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + 16612140 T^{15} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6275300 T^{7} + \cdots + 86\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 46836760 T^{15} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} + 42840984 T^{7} + \cdots + 36\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 50835800 T^{15} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} - 208234800 T^{15} + \cdots + 78\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} - 138370520 T^{15} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
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