Properties

Label 75.9.f.b
Level $75$
Weight $9$
Character orbit 75.f
Analytic conductor $30.553$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1485512441856.2
Defining polynomial: \( x^{8} + 119x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 2 \beta_1) q^{2} + 27 \beta_{3} q^{3} + ( - 4 \beta_{6} - 106 \beta_{2}) q^{4} + (27 \beta_{7} + 162) q^{6} + (126 \beta_{5} - 49 \beta_1) q^{7} + (338 \beta_{4} + 172 \beta_{3}) q^{8} - 2187 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 2 \beta_1) q^{2} + 27 \beta_{3} q^{3} + ( - 4 \beta_{6} - 106 \beta_{2}) q^{4} + (27 \beta_{7} + 162) q^{6} + (126 \beta_{5} - 49 \beta_1) q^{7} + (338 \beta_{4} + 172 \beta_{3}) q^{8} - 2187 \beta_{2} q^{9} + ( - 674 \beta_{7} + 6594) q^{11} + ( - 324 \beta_{5} + 2862 \beta_1) q^{12} + ( - 3588 \beta_{4} + 6741 \beta_{3}) q^{13} + (203 \beta_{6} - 17094 \beta_{2}) q^{14} + (1872 \beta_{7} + 20540) q^{16} + (6550 \beta_{5} + 15782 \beta_1) q^{17} + (2187 \beta_{4} + 4374 \beta_{3}) q^{18} + (3742 \beta_{6} - 42379 \beta_{2}) q^{19} + ( - 3402 \beta_{7} + 3969) q^{21} + ( - 2550 \beta_{5} + 79824 \beta_1) q^{22} + ( - 17806 \beta_{4} + 26254 \beta_{3}) q^{23} + (9126 \beta_{6} - 13932 \beta_{2}) q^{24} + ( - 435 \beta_{7} - 454698) q^{26} + 59049 \beta_1 q^{27} + ( - 13944 \beta_{4} + 74746 \beta_{3}) q^{28} + (14714 \beta_{6} - 172578 \beta_{2}) q^{29} + ( - 12218 \beta_{7} - 855269) q^{31} + (54756 \beta_{5} - 255384 \beta_1) q^{32} + ( - 54594 \beta_{4} + 178038 \beta_{3}) q^{33} + (28882 \beta_{6} - 998592 \beta_{2}) q^{34} + (8748 \beta_{7} - 231822) q^{36} + ( - 28284 \beta_{5} - 165284 \beta_1) q^{37} + (64831 \beta_{4} + 601154 \beta_{3}) q^{38} + ( - 96876 \beta_{6} - 546021 \beta_{2}) q^{39} + (51002 \beta_{7} - 3628644) q^{41} + (16443 \beta_{5} + 461538 \beta_1) q^{42} + (120858 \beta_{4} + 2878445 \beta_{3}) q^{43} + ( - 97820 \beta_{6} - 1815108 \beta_{2}) q^{44} + ( - 9358 \beta_{7} - 2299704) q^{46} + ( - 185028 \beta_{5} + 390930 \beta_1) q^{47} + (151632 \beta_{4} + 554580 \beta_{3}) q^{48} + (12348 \beta_{6} - 3566710 \beta_{2}) q^{49} + ( - 176850 \beta_{7} - 1278342) q^{51} + ( - 461220 \beta_{5} + 2695122 \beta_1) q^{52} + (186994 \beta_{4} + 6908924 \beta_{3}) q^{53} + (59049 \beta_{6} - 354294 \beta_{2}) q^{54} + ( - 5110 \beta_{7} - 5851860) q^{56} + (303102 \beta_{5} + 1144233 \beta_1) q^{57} + (260862 \beta_{4} + 2375688 \beta_{3}) q^{58} + ( - 231792 \beta_{6} + 1623114 \beta_{2}) q^{59} + (861704 \beta_{7} - 9247765) q^{61} + (928577 \beta_{5} + 3396622 \beta_1) q^{62} + ( - 275562 \beta_{4} + 107163 \beta_{3}) q^{63} + (333360 \beta_{6} - 11282264 \beta_{2}) q^{64} + (68850 \beta_{7} - 6465744) q^{66} + ( - 1725330 \beta_{5} - 7266903 \beta_1) q^{67} + ( - 504916 \beta_{4} + 1942708 \beta_{3}) q^{68} + ( - 480762 \beta_{6} - 2126574 \beta_{2}) q^{69} + ( - 1788530 \beta_{7} + 4137612) q^{71} + (739206 \beta_{5} + 376164 \beta_1) q^{72} + ( - 827556 \beta_{4} - 6571876 \beta_{3}) q^{73} + ( - 221852 \beta_{6} + 4894896 \beta_{2}) q^{74} + ( - 227136 \beta_{7} + 1704578) q^{76} + (929922 \beta_{5} - 12042618 \beta_1) q^{77} + ( - 35235 \beta_{4} - 12276846 \beta_{3}) q^{78} + (3134420 \beta_{6} - 7983770 \beta_{2}) q^{79} - 4782969 q^{81} + (3322632 \beta_{5} + 219012 \beta_1) q^{82} + ( - 3359044 \beta_{4} + 2620042 \beta_{3}) q^{83} + ( - 376488 \beta_{6} - 6054426 \beta_{2}) q^{84} + (3120161 \beta_{7} + 33949074) q^{86} + (1191834 \beta_{5} + 4659606 \beta_1) q^{87} + (1880988 \beta_{4} - 30303888 \beta_{3}) q^{88} + ( - 793148 \beta_{6} + 69916476 \beta_{2}) q^{89} + ( - 1025178 \beta_{7} + 63379071) q^{91} + ( - 2202484 \beta_{5} + 12611836 \beta_1) q^{92} + ( - 989658 \beta_{4} - 23092263 \beta_{3}) q^{93} + (20874 \beta_{6} + 23188284 \beta_{2}) q^{94} + ( - 1478412 \beta_{7} + 20686104) q^{96} + ( - 9891360 \beta_{5} + 7687027 \beta_1) q^{97} + (3640798 \beta_{4} + 8837444 \beta_{3}) q^{98} + ( - 1474038 \beta_{6} - 14421078 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1296 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1296 q^{6} + 52752 q^{11} + 164320 q^{16} + 31752 q^{21} - 3637584 q^{26} - 6842152 q^{31} - 1854576 q^{36} - 29029152 q^{41} - 18397632 q^{46} - 10226736 q^{51} - 46814880 q^{56} - 73982120 q^{61} - 51725952 q^{66} + 33100896 q^{71} + 13636624 q^{76} - 38263752 q^{81} + 271592592 q^{86} + 507032568 q^{91} + 165488832 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 119x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 79\nu ) / 65 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 144\nu^{2} ) / 325 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{7} + 827\nu^{3} ) / 1625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{6} + 50\nu^{4} - 1222\nu^{2} + 2975 ) / 325 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{6} - 50\nu^{4} - 1222\nu^{2} - 2975 ) / 325 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 54\nu^{7} + 75\nu^{5} + 6801\nu^{3} + 15675\nu ) / 1625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54\nu^{7} - 75\nu^{5} + 6801\nu^{3} - 15675\nu ) / 1625 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + 26\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 27\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{5} + 13\beta_{4} - 238 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 79\beta_{7} - 79\beta_{6} + 1254\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -36\beta_{5} - 36\beta_{4} - 611\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -827\beta_{7} - 827\beta_{6} + 13602\beta_{3} ) / 12 \) Copy content Toggle raw display

Character values

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

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

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
−2.30795 2.30795i
2.30795 + 2.30795i
1.08321 + 1.08321i
−1.08321 1.08321i
−2.30795 + 2.30795i
2.30795 2.30795i
1.08321 1.08321i
−1.08321 + 1.08321i
−10.7561 10.7561i −33.0681 + 33.0681i 24.6120i 0 711.369 986.622 + 986.622i −3018.29 + 3018.29i 2187.00i 0
7.2 −5.85713 5.85713i 33.0681 33.0681i 187.388i 0 −387.369 1106.65 + 1106.65i −2596.98 + 2596.98i 2187.00i 0
7.3 5.85713 + 5.85713i −33.0681 + 33.0681i 187.388i 0 −387.369 −1106.65 1106.65i 2596.98 2596.98i 2187.00i 0
7.4 10.7561 + 10.7561i 33.0681 33.0681i 24.6120i 0 711.369 −986.622 986.622i 3018.29 3018.29i 2187.00i 0
43.1 −10.7561 + 10.7561i −33.0681 33.0681i 24.6120i 0 711.369 986.622 986.622i −3018.29 3018.29i 2187.00i 0
43.2 −5.85713 + 5.85713i 33.0681 + 33.0681i 187.388i 0 −387.369 1106.65 1106.65i −2596.98 2596.98i 2187.00i 0
43.3 5.85713 5.85713i −33.0681 33.0681i 187.388i 0 −387.369 −1106.65 + 1106.65i 2596.98 + 2596.98i 2187.00i 0
43.4 10.7561 10.7561i 33.0681 + 33.0681i 24.6120i 0 711.369 −986.622 + 986.622i 3018.29 + 3018.29i 2187.00i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
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.b 8
5.b even 2 1 inner 75.9.f.b 8
5.c odd 4 2 inner 75.9.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.f.b 8 1.a even 1 1 trivial
75.9.f.b 8 5.b even 2 1 inner
75.9.f.b 8 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 58248T_{2}^{4} + 252047376 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 58248 T^{4} + \cdots + 252047376 \) Copy content Toggle raw display
$3$ \( (T^{4} + 4782969)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 9789455818674 T^{4} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( (T^{2} - 13188 T - 144589428)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 72\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + 15186082274 T^{2} + \cdots + 16\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + 238829819256 T^{2} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1710538 T + 669683339425)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + 7257288 T + 12090158821080)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 27\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + 49755394000584 T^{2} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 18495530 T - 221887828921799)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 40\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8275224 T - 13\!\cdots\!56)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 31\!\cdots\!61 \) Copy content Toggle raw display
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