Properties

Label 75.9.f.c
Level $75$
Weight $9$
Character orbit 75.f
Analytic conductor $30.553$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: 8.0.151613669376.2
Defining polynomial: \( x^{8} - 7x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 2 \beta_{3}) q^{2} + 27 \beta_{2} q^{3} + ( - 4 \beta_{4} + 224 \beta_1) q^{4} + ( - 27 \beta_{6} + 162) q^{6} + (96 \beta_{5} - 1471 \beta_{3}) q^{7} + (8 \beta_{7} - 1808 \beta_{2}) q^{8} - 2187 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + 2 \beta_{3}) q^{2} + 27 \beta_{2} q^{3} + ( - 4 \beta_{4} + 224 \beta_1) q^{4} + ( - 27 \beta_{6} + 162) q^{6} + (96 \beta_{5} - 1471 \beta_{3}) q^{7} + (8 \beta_{7} - 1808 \beta_{2}) q^{8} - 2187 \beta_1 q^{9} + ( - 116 \beta_{6} + 13494) q^{11} + ( - 324 \beta_{5} + 6048 \beta_{3}) q^{12} + (1092 \beta_{7} + 3531 \beta_{2}) q^{13} + (1663 \beta_{4} - 53754 \beta_1) q^{14} + (768 \beta_{6} + 50240) q^{16} + ( - 620 \beta_{5} + 1318 \beta_{3}) q^{17} + (2187 \beta_{7} + 4374 \beta_{2}) q^{18} + (2332 \beta_{4} + 91571 \beta_1) q^{19} + (2592 \beta_{6} - 119151) q^{21} + ( - 14190 \beta_{5} + 81276 \beta_{3}) q^{22} + (13724 \beta_{7} - 12986 \beta_{2}) q^{23} + (216 \beta_{4} + 146448 \beta_1) q^{24} + ( - 5715 \beta_{6} + 532242) q^{26} - 59049 \beta_{3} q^{27} + (39156 \beta_{7} + 509216 \beta_{2}) q^{28} + (23404 \beta_{4} + 148062 \beta_1) q^{29} + (20168 \beta_{6} - 363539) q^{31} + ( - 43584 \beta_{5} + 203904 \beta_{3}) q^{32} + (9396 \beta_{7} + 364338 \beta_{2}) q^{33} + ( - 2558 \beta_{4} + 298068 \beta_1) q^{34} + ( - 8748 \beta_{6} + 489888) q^{36} + (32016 \beta_{5} - 1456796 \beta_{3}) q^{37} + ( - 77579 \beta_{7} + 908234 \beta_{2}) q^{38} + (29484 \beta_{4} - 286011 \beta_1) q^{39} + ( - 27652 \beta_{6} + 4239276) q^{41} + (134703 \beta_{5} - 1451358 \beta_{3}) q^{42} + ( - 121332 \beta_{7} + 444685 \beta_{2}) q^{43} + ( - 79960 \beta_{4} + 3674112 \beta_1) q^{44} + ( - 14462 \beta_{6} + 6344916) q^{46} + (161712 \beta_{5} + 1614270 \beta_{3}) q^{47} + ( - 62208 \beta_{7} + 1356480 \beta_{2}) q^{48} + ( - 282432 \beta_{4} + 5039810 \beta_1) q^{49} + ( - 16740 \beta_{6} + 106758) q^{51} + ( - 286980 \beta_{5} + 2835168 \beta_{3}) q^{52} + ( - 168356 \beta_{7} + 1243604 \beta_{2}) q^{53} + (59049 \beta_{4} - 354294 \beta_1) q^{54} + ( - 161800 \beta_{6} + 7619280) q^{56} + (188892 \beta_{5} + 2472417 \beta_{3}) q^{57} + ( - 7638 \beta_{7} + 10656948 \beta_{2}) q^{58} + (148848 \beta_{4} + 7828614 \beta_1) q^{59} + (538636 \beta_{6} + 3705875) q^{61} + (484547 \beta_{5} - 10165702 \beta_{3}) q^{62} + ( - 209952 \beta_{7} - 3217077 \beta_{2}) q^{63} + ( - 487680 \beta_{4} + 8759296 \beta_1) q^{64} + ( - 383130 \beta_{6} + 6583356) q^{66} + ( - 1122900 \beta_{5} + 379023 \beta_{3}) q^{67} + ( - 154696 \beta_{7} - 1455872 \beta_{2}) q^{68} + (370548 \beta_{4} + 1051866 \beta_1) q^{69} + (101260 \beta_{6} - 3530868) q^{71} + (17496 \beta_{5} + 3954096 \beta_{3}) q^{72} + (1523544 \beta_{7} - 5871476 \beta_{2}) q^{73} + (1520828 \beta_{4} - 23724264 \beta_1) q^{74} + ( - 156084 \beta_{6} - 7415392) q^{76} + (1807332 \beta_{5} - 25061322 \beta_{3}) q^{77} + (462915 \beta_{7} + 14370534 \beta_{2}) q^{78} + ( - 232600 \beta_{4} - 24785990 \beta_1) q^{79} - 4782969 q^{81} + ( - 4405188 \beta_{5} + 21419688 \beta_{3}) q^{82} + (3756056 \beta_{7} + 6252442 \beta_{2}) q^{83} + (1057212 \beta_{4} - 41246496 \beta_1) q^{84} + ( - 202021 \beta_{6} - 54115266) q^{86} + (1895724 \beta_{5} + 3997674 \beta_{3}) q^{87} + ( - 521232 \beta_{7} - 23962848 \beta_{2}) q^{88} + ( - 535288 \beta_{4} + 85933356 \beta_1) q^{89} + (1945308 \beta_{6} - 64643679) q^{91} + ( - 2918344 \beta_{5} + 22782464 \beta_{3}) q^{92} + ( - 1633608 \beta_{7} - 9815553 \beta_{2}) q^{93} + ( - 1290846 \beta_{4} - 65995596 \beta_1) q^{94} + ( - 1176768 \beta_{6} + 16516224) q^{96} + (3498360 \beta_{5} + 23773483 \beta_{3}) q^{97} + ( - 6734402 \beta_{7} - 142257796 \beta_{2}) q^{98} + (253692 \beta_{4} - 29511378 \beta_1) 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 \( 8 q + 1296 q^{6} + 107952 q^{11} + 401920 q^{16} - 953208 q^{21} + 4257936 q^{26} - 2908312 q^{31} + 3919104 q^{36} + 33914208 q^{41} + 50759328 q^{46} + 854064 q^{51} + 60954240 q^{56} + 29647000 q^{61} + 52666848 q^{66} - 28246944 q^{71} - 59323136 q^{76} - 38263752 q^{81} - 432922128 q^{86} - 517149432 q^{91} + 132129792 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 9\nu^{2} ) / 80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 11\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 71\nu^{3} ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{4} - 42 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} + 87\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 69\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27\nu^{7} + 3\nu^{3} ) / 160 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 6\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 30\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 54\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{4} + 42 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{5} - 174\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{6} + 230\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 71\beta_{7} - 6\beta_{3} ) / 12 \) Copy content Toggle raw display

Character values

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

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

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
0.662382 + 1.88713i
1.88713 + 0.662382i
−1.88713 0.662382i
−0.662382 1.88713i
0.662382 1.88713i
1.88713 0.662382i
−1.88713 + 0.662382i
−0.662382 + 1.88713i
−17.7465 17.7465i −33.0681 + 33.0681i 373.880i 0 1173.69 3270.12 + 3270.12i 2091.96 2091.96i 2187.00i 0
7.2 −12.8476 12.8476i 33.0681 33.0681i 74.1200i 0 −849.690 −333.082 333.082i −2336.72 + 2336.72i 2187.00i 0
7.3 12.8476 + 12.8476i −33.0681 + 33.0681i 74.1200i 0 −849.690 333.082 + 333.082i 2336.72 2336.72i 2187.00i 0
7.4 17.7465 + 17.7465i 33.0681 33.0681i 373.880i 0 1173.69 −3270.12 3270.12i −2091.96 + 2091.96i 2187.00i 0
43.1 −17.7465 + 17.7465i −33.0681 33.0681i 373.880i 0 1173.69 3270.12 3270.12i 2091.96 + 2091.96i 2187.00i 0
43.2 −12.8476 + 12.8476i 33.0681 + 33.0681i 74.1200i 0 −849.690 −333.082 + 333.082i −2336.72 2336.72i 2187.00i 0
43.3 12.8476 12.8476i −33.0681 33.0681i 74.1200i 0 −849.690 333.082 333.082i 2336.72 + 2336.72i 2187.00i 0
43.4 17.7465 17.7465i 33.0681 + 33.0681i 373.880i 0 1173.69 −3270.12 + 3270.12i −2091.96 2091.96i 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.c 8
5.b even 2 1 inner 75.9.f.c 8
5.c odd 4 2 inner 75.9.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.f.c 8 1.a even 1 1 trivial
75.9.f.c 8 5.b even 2 1 inner
75.9.f.c 8 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 505728T_{2}^{4} + 43237380096 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 505728 T^{4} + \cdots + 43237380096 \) Copy content Toggle raw display
$3$ \( (T^{4} + 4782969)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 457467317346834 T^{4} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( (T^{2} - 26988 T + 163195812)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 73\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + 32041029074 T^{2} + \cdots + 56\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1581918894216 T^{2} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 727078 T - 438913901975)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8478552 T + 16897916126160)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + 184787676030024 T^{2} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 7411750 T - 393607242140759)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{2} + 7061736 T - 1929008156976)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
show more
show less