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Properties

Label 75.9.f.a
Level $75$
Weight $9$
Character orbit 75.f
Analytic conductor $30.553$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \beta_1 q^{2} - 27 \beta_{3} q^{3} - 148 \beta_{2} q^{4} + 486 q^{6} + 2345 \beta_1 q^{7} - 2424 \beta_{3} q^{8} - 2187 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 \beta_1 q^{2} - 27 \beta_{3} q^{3} - 148 \beta_{2} q^{4} + 486 q^{6} + 2345 \beta_1 q^{7} - 2424 \beta_{3} q^{8} - 2187 \beta_{2} q^{9} - 234 q^{11} - 3996 \beta_1 q^{12} + 9741 \beta_{3} q^{13} + 42210 \beta_{2} q^{14} + 5744 q^{16} + 72678 \beta_1 q^{17} - 13122 \beta_{3} q^{18} - 181693 \beta_{2} q^{19} + 189945 q^{21} - 1404 \beta_1 q^{22} - 219990 \beta_{3} q^{23} - 196344 \beta_{2} q^{24} - 175338 q^{26} - 59049 \beta_1 q^{27} - 347060 \beta_{3} q^{28} + 240174 \beta_{2} q^{29} + 836725 q^{31} - 586080 \beta_1 q^{32} + 6318 \beta_{3} q^{33} + 1308204 \beta_{2} q^{34} - 323676 q^{36} + 496660 \beta_1 q^{37} - 1090158 \beta_{3} q^{38} + 789021 \beta_{2} q^{39} + 2822220 q^{41} + 1139670 \beta_1 q^{42} - 2287837 \beta_{3} q^{43} + 34632 \beta_{2} q^{44} + 3959820 q^{46} - 4403538 \beta_1 q^{47} - 155088 \beta_{3} q^{48} + 10732274 \beta_{2} q^{49} + 5886918 q^{51} + 1441668 \beta_1 q^{52} + 973740 \beta_{3} q^{53} - 1062882 \beta_{2} q^{54} + 17052840 q^{56} - 4905711 \beta_1 q^{57} + 1441044 \beta_{3} q^{58} + 12785958 \beta_{2} q^{59} + 517403 q^{61} + 5020350 \beta_1 q^{62} - 5128515 \beta_{3} q^{63} - 12019904 \beta_{2} q^{64} - 113724 q^{66} + 1687023 \beta_1 q^{67} - 10756344 \beta_{3} q^{68} - 17819190 \beta_{2} q^{69} - 20828628 q^{71} - 5301288 \beta_1 q^{72} - 23574700 \beta_{3} q^{73} + 8939880 \beta_{2} q^{74} - 26890564 q^{76} - 548730 \beta_1 q^{77} + 4734126 \beta_{3} q^{78} + 42187930 \beta_{2} q^{79} - 4782969 q^{81} + 16933320 \beta_1 q^{82} + 54440934 \beta_{3} q^{83} - 28111860 \beta_{2} q^{84} + 41181066 q^{86} + 6484698 \beta_1 q^{87} + 567216 \beta_{3} q^{88} - 95161428 \beta_{2} q^{89} - 68527935 q^{91} - 32558520 \beta_1 q^{92} - 22591575 \beta_{3} q^{93} - 79263684 \beta_{2} q^{94} - 47472480 q^{96} + 57863323 \beta_1 q^{97} + 64393644 \beta_{3} q^{98} + 511758 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1944 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 1944 q^{6} - 936 q^{11} + 22976 q^{16} + 759780 q^{21} - 701352 q^{26} + 3346900 q^{31} - 1294704 q^{36} + 11288880 q^{41} + 15839280 q^{46} + 23547672 q^{51} + 68211360 q^{56} + 2069612 q^{61} - 454896 q^{66} - 83314512 q^{71} - 107562256 q^{76} - 19131876 q^{81} + 164724264 q^{86} - 274111740 q^{91} - 189889920 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
−1.22474 − 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 − 1.22474i
−7.34847 − 7.34847i −33.0681 + 33.0681i − 148.000i 0 486.000 −2872.03 − 2872.03i −2968.78 + 2968.78i − 2187.00i 0
7.2 7.34847 + 7.34847i 33.0681 − 33.0681i − 148.000i 0 486.000 2872.03 + 2872.03i 2968.78 − 2968.78i − 2187.00i 0
43.1 −7.34847 + 7.34847i −33.0681 − 33.0681i 148.000i 0 486.000 −2872.03 + 2872.03i −2968.78 − 2968.78i 2187.00i 0
43.2 7.34847 − 7.34847i 33.0681 + 33.0681i 148.000i 0 486.000 2872.03 − 2872.03i 2968.78 + 2968.78i 2187.00i 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
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.f.a ✓ 4
5.b even 2 1 inner 75.9.f.a ✓ 4
5.c odd 4 2 inner 75.9.f.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.f.a ✓ 4 1.a even 1 1 trivial
75.9.f.a ✓ 4 5.b even 2 1 inner
75.9.f.a ✓ 4 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 11664 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 11664 \) Copy content Toggle raw display
$3$ \( T^{4} + 4782969 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 272153483555625 \) Copy content Toggle raw display
$11$ \( (T + 234)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 81\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{4} + 25\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{2} + 33012346249)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + 57683550276)^{2} \) Copy content Toggle raw display
$31$ \( (T - 836725)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 54\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T - 2822220)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 24\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{4} + 33\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + 80\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{2} + 163480721977764)^{2} \) Copy content Toggle raw display
$61$ \( (T - 517403)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 72\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T + 20828628)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 79\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{2} + 90\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 10\!\cdots\!69 \) Copy content Toggle raw display
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