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Properties

Label 75.5.c.i
Level $75$
Weight $5$
Character orbit 75.c
Analytic conductor $7.753$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,5,Mod(26,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 73x^{4} + 1096x^{2} + 180 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{3} - \beta_{2} - 8) q^{4} + (\beta_{5} + \beta_{4} - 4 \beta_1) q^{6} + (\beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 13) q^{7} + (\beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{3} - \beta_{2} - 8) q^{4} + (\beta_{5} + \beta_{4} - 4 \beta_1) q^{6} + (\beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 13) q^{7} + (\beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \cdots + 1) q^{8}+ \cdots + (64 \beta_{5} + 188 \beta_{4} + \cdots + 1424) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} - 50 q^{4} - 2 q^{6} - 76 q^{7} + 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{3} - 50 q^{4} - 2 q^{6} - 76 q^{7} + 118 q^{9} + 452 q^{12} + 424 q^{13} + 802 q^{16} - 1160 q^{18} - 244 q^{19} - 876 q^{21} - 340 q^{22} - 786 q^{24} + 352 q^{27} + 3764 q^{28} + 3772 q^{31} - 4420 q^{33} + 3124 q^{34} - 7606 q^{36} - 1896 q^{37} - 1336 q^{39} + 1980 q^{42} + 7384 q^{43} + 8196 q^{46} - 14668 q^{48} - 1318 q^{49} - 8492 q^{51} - 8976 q^{52} - 278 q^{54} + 11584 q^{57} + 23740 q^{58} + 6452 q^{61} - 14796 q^{63} + 3174 q^{64} - 12760 q^{66} - 13816 q^{67} + 5472 q^{69} + 2040 q^{72} - 596 q^{73} + 21348 q^{76} + 1400 q^{78} - 16124 q^{79} + 5086 q^{81} + 31240 q^{82} - 14736 q^{84} + 4900 q^{87} - 15660 q^{88} - 11632 q^{91} + 8184 q^{93} + 34924 q^{94} + 14354 q^{96} - 9756 q^{97} + 9680 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 73x^{4} + 1096x^{2} + 180 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} + 71\nu^{3} + 47\nu^{2} + 1002\nu + 114 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} + 71\nu^{3} + 95\nu^{2} + 1002\nu + 1266 ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 79\nu^{3} + 1346\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{5} + \nu^{4} - 142\nu^{3} + 47\nu^{2} - 2004\nu + 90 ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} - 43\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} - 47\beta_{3} + 79\beta_{2} + 1022 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -79\beta_{5} - 142\beta_{4} + 158\beta_{2} + 2051\beta _1 - 79 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 7.20990i
− 4.56632i
− 0.407512i
0.407512i
4.56632i
7.20990i
− 7.20990i −8.77108 + 2.01697i −35.9827 0 14.5422 + 63.2386i −23.3388 144.073i 72.8637 − 35.3820i 0
26.2 − 4.56632i 7.98405 − 4.15391i −4.85128 0 −18.9681 − 36.4577i −61.6068 − 50.9086i 46.4900 − 66.3301i 0
26.3 − 0.407512i −3.21297 + 8.40695i 15.8339 0 3.42594 + 1.30932i 46.9457 − 12.9727i −60.3537 − 54.0225i 0
26.4 0.407512i −3.21297 − 8.40695i 15.8339 0 3.42594 − 1.30932i 46.9457 12.9727i −60.3537 + 54.0225i 0
26.5 4.56632i 7.98405 + 4.15391i −4.85128 0 −18.9681 + 36.4577i −61.6068 50.9086i 46.4900 + 66.3301i 0
26.6 7.20990i −8.77108 − 2.01697i −35.9827 0 14.5422 − 63.2386i −23.3388 − 144.073i 72.8637 + 35.3820i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.5.c.i 6
3.b odd 2 1 inner 75.5.c.i 6
5.b even 2 1 15.5.c.a ✓ 6
5.c odd 4 2 75.5.d.d 12
15.d odd 2 1 15.5.c.a ✓ 6
15.e even 4 2 75.5.d.d 12
20.d odd 2 1 240.5.l.d 6
60.h even 2 1 240.5.l.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.c.a ✓ 6 5.b even 2 1
15.5.c.a ✓ 6 15.d odd 2 1
75.5.c.i 6 1.a even 1 1 trivial
75.5.c.i 6 3.b odd 2 1 inner
75.5.d.d 12 5.c odd 4 2
75.5.d.d 12 15.e even 4 2
240.5.l.d 6 20.d odd 2 1
240.5.l.d 6 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{6} + 73T_{2}^{4} + 1096T_{2}^{2} + 180 \) Copy content Toggle raw display
\( T_{7}^{3} + 38T_{7}^{2} - 2550T_{7} - 67500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 73 T^{4} + \cdots + 180 \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{5} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} + 38 T^{2} + \cdots - 67500)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 552448800000 \) Copy content Toggle raw display
$13$ \( (T^{3} - 212 T^{2} + \cdots - 179200)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 345773292505920 \) Copy content Toggle raw display
$19$ \( (T^{3} + 122 T^{2} + \cdots - 6584528)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 512545320524880 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} - 1886 T^{2} + \cdots - 171289728)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 948 T^{2} + \cdots - 148979200)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} - 3692 T^{2} + \cdots + 9836480000)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} - 3226 T^{2} + \cdots + 8122222912)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 6908 T^{2} + \cdots - 28105138800)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + 298 T^{2} + \cdots + 62842083800)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 8062 T^{2} + \cdots - 94953979728)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 77\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} + 4878 T^{2} + \cdots - 501103547800)^{2} \) Copy content Toggle raw display
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