Properties

Label 75.9.d.b
Level $75$
Weight $9$
Character orbit 75.d
Analytic conductor $30.553$
Analytic rank $0$
Dimension $4$
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{14})\)
Defining polynomial: \( x^{4} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (3 \beta_{2} - 9 \beta_1) q^{3} + 248 q^{4} + ( - 9 \beta_{3} + 1512) q^{6} - 350 \beta_1 q^{7} - 8 \beta_{2} q^{8} + ( - 54 \beta_{3} + 2511) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (3 \beta_{2} - 9 \beta_1) q^{3} + 248 q^{4} + ( - 9 \beta_{3} + 1512) q^{6} - 350 \beta_1 q^{7} - 8 \beta_{2} q^{8} + ( - 54 \beta_{3} + 2511) q^{9} - 62 \beta_{3} q^{11} + (744 \beta_{2} - 2232 \beta_1) q^{12} - 5146 \beta_1 q^{13} - 350 \beta_{3} q^{14} - 67520 q^{16} + 3336 \beta_{2} q^{17} + (2511 \beta_{2} - 27216 \beta_1) q^{18} - 18938 q^{19} + ( - 1050 \beta_{3} - 78750) q^{21} - 31248 \beta_1 q^{22} + 20956 \beta_{2} q^{23} + (72 \beta_{3} - 12096) q^{24} - 5146 \beta_{3} q^{26} + ( - 4617 \beta_{2} - 104247 \beta_1) q^{27} - 86800 \beta_1 q^{28} + 4106 \beta_{3} q^{29} - 351478 q^{31} - 65472 \beta_{2} q^{32} + ( - 13950 \beta_{2} - 93744 \beta_1) q^{33} + 1681344 q^{34} + ( - 13392 \beta_{3} + 622728) q^{36} + 267034 \beta_1 q^{37} - 18938 \beta_{2} q^{38} + ( - 15438 \beta_{3} - 1157850) q^{39} - 16708 \beta_{3} q^{41} + ( - 78750 \beta_{2} - 529200 \beta_1) q^{42} + 705230 \beta_1 q^{43} - 15376 \beta_{3} q^{44} + 10561824 q^{46} - 181784 \beta_{2} q^{47} + ( - 202560 \beta_{2} + 607680 \beta_1) q^{48} + 2702301 q^{49} + ( - 30024 \beta_{3} + 5044032) q^{51} - 1276208 \beta_1 q^{52} + 294066 \beta_{2} q^{53} + ( - 104247 \beta_{3} - 2326968) q^{54} + 2800 \beta_{3} q^{56} + ( - 56814 \beta_{2} + 170442 \beta_1) q^{57} + 2069424 \beta_1 q^{58} + 122182 \beta_{3} q^{59} + 753602 q^{61} - 351478 \beta_{2} q^{62} + ( - 472500 \beta_{2} - 878850 \beta_1) q^{63} - 15712768 q^{64} + ( - 93744 \beta_{3} - 7030800) q^{66} + 453778 \beta_1 q^{67} + 827328 \beta_{2} q^{68} + ( - 188604 \beta_{3} + 31685472) q^{69} - 151644 \beta_{3} q^{71} + ( - 20088 \beta_{2} + 217728 \beta_1) q^{72} - 5534554 \beta_1 q^{73} + 267034 \beta_{3} q^{74} - 4696624 q^{76} - 542500 \beta_{2} q^{77} + ( - 1157850 \beta_{2} - 7780752 \beta_1) q^{78} + 22980982 q^{79} + ( - 271188 \beta_{3} - 30436479) q^{81} - 8420832 \beta_1 q^{82} + 2066606 \beta_{2} q^{83} + ( - 260400 \beta_{3} - 19530000) q^{84} + 705230 \beta_{3} q^{86} + (923850 \beta_{2} + 6208272 \beta_1) q^{87} + 249984 \beta_1 q^{88} + 646908 \beta_{3} q^{89} - 45027500 q^{91} + 5197088 \beta_{2} q^{92} + ( - 1054434 \beta_{2} + 3163302 \beta_1) q^{93} - 91619136 q^{94} + (589248 \beta_{3} - 98993664) q^{96} + 29454202 \beta_1 q^{97} + 2702301 \beta_{2} q^{98} + ( - 155682 \beta_{3} - 42184800) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 992 q^{4} + 6048 q^{6} + 10044 q^{9} - 270080 q^{16} - 75752 q^{19} - 315000 q^{21} - 48384 q^{24} - 1405912 q^{31} + 6725376 q^{34} + 2490912 q^{36} - 4631400 q^{39} + 42247296 q^{46} + 10809204 q^{49} + 20176128 q^{51} - 9307872 q^{54} + 3014408 q^{61} - 62851072 q^{64} - 28123200 q^{66} + 126741888 q^{69} - 18786496 q^{76} + 91923928 q^{79} - 121745916 q^{81} - 78120000 q^{84} - 180110000 q^{91} - 366476544 q^{94} - 395974656 q^{96} - 168739200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6\nu^{3} + 42\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\nu^{3} + 210\nu ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 5\beta_{2} ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} - 35\beta_{2} ) / 60 \) Copy content Toggle raw display

Character values

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

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

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
−1.87083 1.87083i
−1.87083 + 1.87083i
1.87083 + 1.87083i
1.87083 1.87083i
−22.4499 −67.3498 45.0000i 248.000 0 1512.00 + 1010.25i 1750.00i 179.600 2511.00 + 6061.48i 0
74.2 −22.4499 −67.3498 + 45.0000i 248.000 0 1512.00 1010.25i 1750.00i 179.600 2511.00 6061.48i 0
74.3 22.4499 67.3498 45.0000i 248.000 0 1512.00 1010.25i 1750.00i −179.600 2511.00 6061.48i 0
74.4 22.4499 67.3498 + 45.0000i 248.000 0 1512.00 + 1010.25i 1750.00i −179.600 2511.00 + 6061.48i 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 inner
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 75.9.d.b 4
3.b odd 2 1 inner 75.9.d.b 4
5.b even 2 1 inner 75.9.d.b 4
5.c odd 4 1 3.9.b.a 2
5.c odd 4 1 75.9.c.c 2
15.d odd 2 1 inner 75.9.d.b 4
15.e even 4 1 3.9.b.a 2
15.e even 4 1 75.9.c.c 2
20.e even 4 1 48.9.e.b 2
40.i odd 4 1 192.9.e.e 2
40.k even 4 1 192.9.e.f 2
45.k odd 12 2 81.9.d.d 4
45.l even 12 2 81.9.d.d 4
60.l odd 4 1 48.9.e.b 2
120.q odd 4 1 192.9.e.f 2
120.w even 4 1 192.9.e.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 5.c odd 4 1
3.9.b.a 2 15.e even 4 1
48.9.e.b 2 20.e even 4 1
48.9.e.b 2 60.l odd 4 1
75.9.c.c 2 5.c odd 4 1
75.9.c.c 2 15.e even 4 1
75.9.d.b 4 1.a even 1 1 trivial
75.9.d.b 4 3.b odd 2 1 inner
75.9.d.b 4 5.b even 2 1 inner
75.9.d.b 4 15.d odd 2 1 inner
81.9.d.d 4 45.k odd 12 2
81.9.d.d 4 45.l even 12 2
192.9.e.e 2 40.i odd 4 1
192.9.e.e 2 120.w even 4 1
192.9.e.f 2 40.k even 4 1
192.9.e.f 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 504 \) acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 504)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 5022 T^{2} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 3062500)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 48434400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 662032900)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 5608963584)^{2} \) Copy content Toggle raw display
$19$ \( (T + 18938)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 221333583744)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 212426373600)^{2} \) Copy content Toggle raw display
$31$ \( (T + 351478)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1782678928900)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3517381526400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12433733822500)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 16654893018624)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 43583305427424)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 188098358162400)^{2} \) Copy content Toggle raw display
$61$ \( (T - 753602)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 5147861832100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 289748374473600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 765782199472900)^{2} \) Copy content Toggle raw display
$79$ \( (T - 22980982)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 21\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less