Properties

Label 75.5.f.b
Level $75$
Weight $5$
Character orbit 75.f
Analytic conductor $7.753$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,5,Mod(7,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.75274723129\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{3} q^{3} - 13 \beta_{2} q^{4} - 9 q^{6} + 30 \beta_1 q^{7} - 29 \beta_{3} q^{8} - 27 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 \beta_{3} q^{3} - 13 \beta_{2} q^{4} - 9 q^{6} + 30 \beta_1 q^{7} - 29 \beta_{3} q^{8} - 27 \beta_{2} q^{9} + 126 q^{11} + 39 \beta_1 q^{12} - 114 \beta_{3} q^{13} + 90 \beta_{2} q^{14} - 121 q^{16} + 308 \beta_1 q^{17} - 27 \beta_{3} q^{18} + 62 \beta_{2} q^{19} - 270 q^{21} + 126 \beta_1 q^{22} + 40 \beta_{3} q^{23} + 261 \beta_{2} q^{24} + 342 q^{26} + 81 \beta_1 q^{27} - 390 \beta_{3} q^{28} - 1296 \beta_{2} q^{29} - 650 q^{31} - 585 \beta_1 q^{32} + 378 \beta_{3} q^{33} + 924 \beta_{2} q^{34} - 351 q^{36} - 690 \beta_1 q^{37} + 62 \beta_{3} q^{38} + 1026 \beta_{2} q^{39} + 3150 q^{41} - 270 \beta_1 q^{42} + 1608 \beta_{3} q^{43} - 1638 \beta_{2} q^{44} - 120 q^{46} - 468 \beta_1 q^{47} - 363 \beta_{3} q^{48} + 299 \beta_{2} q^{49} - 2772 q^{51} - 1482 \beta_1 q^{52} + 1820 \beta_{3} q^{53} + 243 \beta_{2} q^{54} + 2610 q^{56} - 186 \beta_1 q^{57} - 1296 \beta_{3} q^{58} + 198 \beta_{2} q^{59} - 5782 q^{61} - 650 \beta_1 q^{62} - 810 \beta_{3} q^{63} + 181 \beta_{2} q^{64} - 1134 q^{66} + 4068 \beta_1 q^{67} - 4004 \beta_{3} q^{68} - 360 \beta_{2} q^{69} + 5112 q^{71} - 783 \beta_1 q^{72} + 2340 \beta_{3} q^{73} - 2070 \beta_{2} q^{74} + 806 q^{76} + 3780 \beta_1 q^{77} + 1026 \beta_{3} q^{78} + 2470 \beta_{2} q^{79} - 729 q^{81} + 3150 \beta_1 q^{82} - 1196 \beta_{3} q^{83} + 3510 \beta_{2} q^{84} - 4824 q^{86} + 3888 \beta_1 q^{87} - 3654 \beta_{3} q^{88} + 1062 \beta_{2} q^{89} + 10260 q^{91} + 520 \beta_1 q^{92} - 1950 \beta_{3} q^{93} - 1404 \beta_{2} q^{94} + 5265 q^{96} - 3012 \beta_1 q^{97} + 299 \beta_{3} q^{98} - 3402 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{6} + 504 q^{11} - 484 q^{16} - 1080 q^{21} + 1368 q^{26} - 2600 q^{31} - 1404 q^{36} + 12600 q^{41} - 480 q^{46} - 11088 q^{51} + 10440 q^{56} - 23128 q^{61} - 4536 q^{66} + 20448 q^{71} + 3224 q^{76} - 2916 q^{81} - 19296 q^{86} + 41040 q^{91} + 21060 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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
\(\nu\)\(=\) \( \beta_1 \) 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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
−1.22474 1.22474i 3.67423 3.67423i 13.0000i 0 −9.00000 −36.7423 36.7423i −35.5176 + 35.5176i 27.0000i 0
7.2 1.22474 + 1.22474i −3.67423 + 3.67423i 13.0000i 0 −9.00000 36.7423 + 36.7423i 35.5176 35.5176i 27.0000i 0
43.1 −1.22474 + 1.22474i 3.67423 + 3.67423i 13.0000i 0 −9.00000 −36.7423 + 36.7423i −35.5176 35.5176i 27.0000i 0
43.2 1.22474 1.22474i −3.67423 3.67423i 13.0000i 0 −9.00000 36.7423 36.7423i 35.5176 + 35.5176i 27.0000i 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 75.5.f.b 4
3.b odd 2 1 225.5.g.e 4
5.b even 2 1 inner 75.5.f.b 4
5.c odd 4 2 inner 75.5.f.b 4
15.d odd 2 1 225.5.g.e 4
15.e even 4 2 225.5.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.5.f.b 4 1.a even 1 1 trivial
75.5.f.b 4 5.b even 2 1 inner
75.5.f.b 4 5.c odd 4 2 inner
225.5.g.e 4 3.b odd 2 1
225.5.g.e 4 15.d odd 2 1
225.5.g.e 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9 \) acting on \(S_{5}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} + 729 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 7290000 \) Copy content Toggle raw display
$11$ \( (T - 126)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 1520064144 \) Copy content Toggle raw display
$17$ \( T^{4} + 80992606464 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3844)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 23040000 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1679616)^{2} \) Copy content Toggle raw display
$31$ \( (T + 650)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 2040040890000 \) Copy content Toggle raw display
$41$ \( (T - 3150)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 60170924888064 \) Copy content Toggle raw display
$47$ \( T^{4} + 431743613184 \) Copy content Toggle raw display
$53$ \( T^{4} + 98747943840000 \) Copy content Toggle raw display
$59$ \( (T^{2} + 39204)^{2} \) Copy content Toggle raw display
$61$ \( (T + 5782)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 24\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T - 5112)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 269839758240000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 6100900)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 18414809397504 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1127844)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 740734170810624 \) Copy content Toggle raw display
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