Properties

Label 93.2.p.a
Level $93$
Weight $2$
Character orbit 93.p
Analytic conductor $0.743$
Analytic rank $0$
Dimension $8$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 93.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.742608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \(x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{30}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{15}^{2} + 2 \zeta_{15}^{7} ) q^{3} + 2 \zeta_{15}^{6} q^{4} + ( -\zeta_{15} + \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{7} -3 \zeta_{15}^{4} q^{9} +O(q^{10})\) \( q + ( \zeta_{15}^{2} + 2 \zeta_{15}^{7} ) q^{3} + 2 \zeta_{15}^{6} q^{4} + ( -\zeta_{15} + \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{7} -3 \zeta_{15}^{4} q^{9} + ( 2 - 2 \zeta_{15} - 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{12} + ( -4 + 5 \zeta_{15} - 3 \zeta_{15}^{3} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 3 \zeta_{15}^{6} + 4 \zeta_{15}^{7} ) q^{13} + ( -4 \zeta_{15}^{2} - 4 \zeta_{15}^{7} ) q^{16} + ( 5 - 3 \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 5 \zeta_{15}^{6} - 8 \zeta_{15}^{7} ) q^{19} + ( -5 - 5 \zeta_{15} - \zeta_{15}^{5} - 4 \zeta_{15}^{6} ) q^{21} + 5 \zeta_{15}^{5} q^{25} + ( 6 \zeta_{15} + 3 \zeta_{15}^{6} ) q^{27} + ( -10 + 6 \zeta_{15}^{2} - 6 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 6 \zeta_{15}^{5} - 6 \zeta_{15}^{6} + 6 \zeta_{15}^{7} ) q^{28} + ( -5 \zeta_{15}^{2} + \zeta_{15}^{7} ) q^{31} + ( 6 + 6 \zeta_{15}^{5} ) q^{36} + ( 7 - 7 \zeta_{15} + 7 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 7 \zeta_{15}^{4} + 7 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{37} + ( -3 + 5 \zeta_{15} + 2 \zeta_{15}^{3} + 5 \zeta_{15}^{4} - 10 \zeta_{15}^{5} + 5 \zeta_{15}^{7} ) q^{39} + ( 7 - 6 \zeta_{15} + \zeta_{15}^{5} + \zeta_{15}^{6} ) q^{43} + ( 4 - 4 \zeta_{15}^{2} + 4 \zeta_{15}^{3} + 4 \zeta_{15}^{4} + 4 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{48} + ( -3 + 8 \zeta_{15} - 8 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 8 \zeta_{15}^{6} + 10 \zeta_{15}^{7} ) q^{49} + ( 6 + 6 \zeta_{15}^{3} - 8 \zeta_{15}^{4} + 6 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{52} + ( -7 - \zeta_{15} + 7 \zeta_{15}^{2} - 7 \zeta_{15}^{3} + 8 \zeta_{15}^{4} + 7 \zeta_{15}^{7} ) q^{57} + ( -5 - 4 \zeta_{15} + 5 \zeta_{15}^{2} - 5 \zeta_{15}^{3} - 4 \zeta_{15}^{4} - 14 \zeta_{15}^{6} + 5 \zeta_{15}^{7} ) q^{61} + ( 6 - 6 \zeta_{15} - 3 \zeta_{15}^{2} + 9 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 6 \zeta_{15}^{5} - 15 \zeta_{15}^{7} ) q^{63} + 8 \zeta_{15}^{3} q^{64} + ( -2 + 7 \zeta_{15} + 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 9 \zeta_{15}^{4} - 4 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{67} + ( 9 - 8 \zeta_{15} - \zeta_{15}^{2} - 17 \zeta_{15}^{4} + 8 \zeta_{15}^{5} + \zeta_{15}^{6} - 9 \zeta_{15}^{7} ) q^{73} + ( -10 \zeta_{15}^{2} - 5 \zeta_{15}^{7} ) q^{75} + ( -16 + 10 \zeta_{15} - 4 \zeta_{15}^{3} + 10 \zeta_{15}^{4} - 6 \zeta_{15}^{5} + 10 \zeta_{15}^{7} ) q^{76} + ( -7 - 3 \zeta_{15} - 7 \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 3 \zeta_{15}^{7} ) q^{79} + ( -9 + 9 \zeta_{15} - 9 \zeta_{15}^{3} + 9 \zeta_{15}^{4} - 9 \zeta_{15}^{5} + 9 \zeta_{15}^{7} ) q^{81} + ( 2 \zeta_{15} + 8 \zeta_{15}^{2} - 8 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{84} + ( 15 - 11 \zeta_{15} - 19 \zeta_{15}^{2} + 10 \zeta_{15}^{3} - \zeta_{15}^{4} + 11 \zeta_{15}^{5} + 5 \zeta_{15}^{6} - 20 \zeta_{15}^{7} ) q^{91} + ( 11 - 11 \zeta_{15}^{2} + 11 \zeta_{15}^{3} - 7 \zeta_{15}^{4} + 11 \zeta_{15}^{6} - 11 \zeta_{15}^{7} ) q^{93} + ( -3 + 3 \zeta_{15}^{2} - 11 \zeta_{15}^{5} - 8 \zeta_{15}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{3} - 4q^{4} - 4q^{7} - 3q^{9} + O(q^{10}) \) \( 8q + 3q^{3} - 4q^{4} - 4q^{7} - 3q^{9} + 6q^{12} + 9q^{13} - 8q^{16} + 7q^{19} - 33q^{21} - 20q^{25} - 18q^{28} - 4q^{31} + 24q^{36} + 9q^{37} + 27q^{39} + 44q^{43} + 12q^{48} + 9q^{49} + 18q^{52} - 21q^{57} - 24q^{63} - 16q^{64} + 16q^{67} + 3q^{73} - 15q^{75} - 66q^{76} - 35q^{79} + 9q^{81} + 24q^{84} - 5q^{91} + 15q^{93} + 15q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(34\)
\(\chi(n)\) \(-1\) \(-\zeta_{15}^{4}\)

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
0.913545 + 0.406737i
0.913545 0.406737i
−0.978148 + 0.207912i
−0.104528 + 0.994522i
0.669131 + 0.743145i
−0.978148 0.207912i
0.669131 0.743145i
−0.104528 0.994522i
0 −1.28716 + 1.15897i −1.61803 + 1.17557i 0 0 0.473579 + 4.50581i 0 0.313585 2.98357i 0
17.1 0 −1.28716 1.15897i −1.61803 1.17557i 0 0 0.473579 4.50581i 0 0.313585 + 2.98357i 0
44.1 0 0.704489 + 1.58231i 0.618034 1.90211i 0 0 0.802903 + 0.891714i 0 −2.00739 + 2.22943i 0
53.1 0 0.360114 1.69420i −1.61803 1.17557i 0 0 1.88052 0.837263i 0 −2.74064 1.22021i 0
65.1 0 1.72256 + 0.181049i 0.618034 1.90211i 0 0 −5.15701 + 1.09616i 0 2.93444 + 0.623735i 0
74.1 0 0.704489 1.58231i 0.618034 + 1.90211i 0 0 0.802903 0.891714i 0 −2.00739 2.22943i 0
83.1 0 1.72256 0.181049i 0.618034 + 1.90211i 0 0 −5.15701 1.09616i 0 2.93444 0.623735i 0
86.1 0 0.360114 + 1.69420i −1.61803 + 1.17557i 0 0 1.88052 + 0.837263i 0 −2.74064 + 1.22021i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
31.h odd 30 1 inner
93.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.2.p.a 8
3.b odd 2 1 CM 93.2.p.a 8
31.h odd 30 1 inner 93.2.p.a 8
93.p even 30 1 inner 93.2.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.p.a 8 1.a even 1 1 trivial
93.2.p.a 8 3.b odd 2 1 CM
93.2.p.a 8 31.h odd 30 1 inner
93.2.p.a 8 93.p even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(93, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 81 - 81 T + 54 T^{2} - 27 T^{3} + 9 T^{4} - 9 T^{5} + 6 T^{6} - 3 T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 3481 - 5841 T + 4655 T^{2} - 1304 T^{3} - 81 T^{4} + 76 T^{5} + 4 T^{7} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 32761 + 20453 T + 22336 T^{2} + 5724 T^{3} - 701 T^{4} + 82 T^{5} + 54 T^{6} - 9 T^{7} + T^{8} \)
$17$ \( T^{8} \)
$19$ \( 201601 + 196662 T + 83030 T^{2} + 14147 T^{3} - 216 T^{4} - 322 T^{5} - 7 T^{7} + T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( 923521 + 119164 T - 14415 T^{2} - 5704 T^{3} - 271 T^{4} - 184 T^{5} - 15 T^{6} + 4 T^{7} + T^{8} \)
$37$ \( 6651241 + 3667338 T + 266546 T^{2} - 224676 T^{3} + 18119 T^{4} + 1422 T^{5} - 131 T^{6} - 9 T^{7} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( 564001 - 485897 T + 719531 T^{2} - 283016 T^{3} + 64179 T^{4} - 9168 T^{5} + 834 T^{6} - 44 T^{7} + T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( 39300361 + 6392618 T^{2} + 103259 T^{4} + 562 T^{6} + T^{8} \)
$67$ \( 4932841 + 2807344 T + 1773155 T^{2} - 28784 T^{3} + 24244 T^{4} - 1264 T^{5} + 335 T^{6} - 16 T^{7} + T^{8} \)
$71$ \( T^{8} \)
$73$ \( 653262481 - 65533276 T + 9798844 T^{2} + 10923 T^{3} - 35396 T^{4} + 6196 T^{5} + 6 T^{6} - 3 T^{7} + T^{8} \)
$79$ \( 7317025 + 8304350 T + 4438700 T^{2} + 704900 T^{3} + 86445 T^{4} + 9960 T^{5} + 735 T^{6} + 35 T^{7} + T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( 1263447025 + 159063875 T + 7067250 T^{2} - 648175 T^{3} + 106215 T^{4} - 4975 T^{5} + 560 T^{6} - 15 T^{7} + T^{8} \)
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