Properties

Label 75.7.d.a
Level 75
Weight 7
Character orbit 75.d
Analytic conductor 17.254
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.2540562715\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -27 i q^{3} -64 q^{4} + 286 i q^{7} -729 q^{9} +O(q^{10})\) \( q -27 i q^{3} -64 q^{4} + 286 i q^{7} -729 q^{9} + 1728 i q^{12} + 506 i q^{13} + 4096 q^{16} + 10582 q^{19} + 7722 q^{21} + 19683 i q^{27} -18304 i q^{28} + 35282 q^{31} + 46656 q^{36} + 89206 i q^{37} + 13662 q^{39} + 111386 i q^{43} -110592 i q^{48} + 35853 q^{49} -32384 i q^{52} -285714 i q^{57} -420838 q^{61} -208494 i q^{63} -262144 q^{64} -172874 i q^{67} + 638066 i q^{73} -677248 q^{76} + 204622 q^{79} + 531441 q^{81} -494208 q^{84} -144716 q^{91} -952614 i q^{93} + 56446 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 128q^{4} - 1458q^{9} + O(q^{10}) \) \( 2q - 128q^{4} - 1458q^{9} + 8192q^{16} + 21164q^{19} + 15444q^{21} + 70564q^{31} + 93312q^{36} + 27324q^{39} + 71706q^{49} - 841676q^{61} - 524288q^{64} - 1354496q^{76} + 409244q^{79} + 1062882q^{81} - 988416q^{84} - 289432q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
1.00000i
1.00000i
0 27.0000i −64.0000 0 0 286.000i 0 −729.000 0
74.2 0 27.0000i −64.0000 0 0 286.000i 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.7.d.a 2
3.b odd 2 1 CM 75.7.d.a 2
5.b even 2 1 inner 75.7.d.a 2
5.c odd 4 1 3.7.b.a 1
5.c odd 4 1 75.7.c.a 1
15.d odd 2 1 inner 75.7.d.a 2
15.e even 4 1 3.7.b.a 1
15.e even 4 1 75.7.c.a 1
20.e even 4 1 48.7.e.a 1
35.f even 4 1 147.7.b.a 1
40.i odd 4 1 192.7.e.b 1
40.k even 4 1 192.7.e.a 1
45.k odd 12 2 81.7.d.a 2
45.l even 12 2 81.7.d.a 2
60.l odd 4 1 48.7.e.a 1
105.k odd 4 1 147.7.b.a 1
120.q odd 4 1 192.7.e.a 1
120.w even 4 1 192.7.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.7.b.a 1 5.c odd 4 1
3.7.b.a 1 15.e even 4 1
48.7.e.a 1 20.e even 4 1
48.7.e.a 1 60.l odd 4 1
75.7.c.a 1 5.c odd 4 1
75.7.c.a 1 15.e even 4 1
75.7.d.a 2 1.a even 1 1 trivial
75.7.d.a 2 3.b odd 2 1 CM
75.7.d.a 2 5.b even 2 1 inner
75.7.d.a 2 15.d odd 2 1 inner
81.7.d.a 2 45.k odd 12 2
81.7.d.a 2 45.l even 12 2
147.7.b.a 1 35.f even 4 1
147.7.b.a 1 105.k odd 4 1
192.7.e.a 1 40.k even 4 1
192.7.e.a 1 120.q odd 4 1
192.7.e.b 1 40.i odd 4 1
192.7.e.b 1 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{7}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 64 T^{2} )^{2} \)
$3$ \( 1 + 729 T^{2} \)
$5$ 1
$7$ \( 1 - 153502 T^{2} + 13841287201 T^{4} \)
$11$ \( ( 1 - 1331 T )^{2}( 1 + 1331 T )^{2} \)
$13$ \( 1 - 9397582 T^{2} + 23298085122481 T^{4} \)
$17$ \( ( 1 + 24137569 T^{2} )^{2} \)
$19$ \( ( 1 - 10582 T + 47045881 T^{2} )^{2} \)
$23$ \( ( 1 + 148035889 T^{2} )^{2} \)
$29$ \( ( 1 - 24389 T )^{2}( 1 + 24389 T )^{2} \)
$31$ \( ( 1 - 35282 T + 887503681 T^{2} )^{2} \)
$37$ \( 1 + 2826257618 T^{2} + 6582952005840035281 T^{4} \)
$41$ \( ( 1 - 68921 T )^{2}( 1 + 68921 T )^{2} \)
$43$ \( 1 - 235885102 T^{2} + 39959630797262576401 T^{4} \)
$47$ \( ( 1 + 10779215329 T^{2} )^{2} \)
$53$ \( ( 1 + 22164361129 T^{2} )^{2} \)
$59$ \( ( 1 - 205379 T )^{2}( 1 + 205379 T )^{2} \)
$61$ \( ( 1 + 420838 T + 51520374361 T^{2} )^{2} \)
$67$ \( 1 - 151031344462 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \)
$71$ \( ( 1 - 357911 T )^{2}( 1 + 357911 T )^{2} \)
$73$ \( 1 + 104459767778 T^{2} + \)\(22\!\cdots\!21\)\( T^{4} \)
$79$ \( ( 1 - 204622 T + 243087455521 T^{2} )^{2} \)
$83$ \( ( 1 + 326940373369 T^{2} )^{2} \)
$89$ \( ( 1 - 704969 T )^{2}( 1 + 704969 T )^{2} \)
$97$ \( 1 - 1662757858942 T^{2} + \)\(69\!\cdots\!41\)\( T^{4} \)
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