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Properties

Label 75.5.d.d
Level $75$
Weight $5$
Character orbit 75.d
Analytic conductor $7.753$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,5,Mod(74,75)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.74");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 145x^{10} + 7978x^{8} + 203285x^{6} + 2280046x^{4} + 7912765x^{2} + 6027025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 8) q^{4} + ( - \beta_{11} - \beta_{8} + \beta_{2} - 1) q^{6} + ( - \beta_{9} + 2 \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{4} + \cdots - 11 \beta_1) q^{8}+ \cdots + ( - \beta_{10} - \beta_{8} - 2 \beta_{2} - 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 8) q^{4} + ( - \beta_{11} - \beta_{8} + \beta_{2} - 1) q^{6} + ( - \beta_{9} + 2 \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{4} + \cdots - 11 \beta_1) q^{8}+ \cdots + (165 \beta_{11} + 16 \beta_{10} + \cdots - 1709) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 100 q^{4} - 4 q^{6} - 236 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 100 q^{4} - 4 q^{6} - 236 q^{9} + 1604 q^{16} + 488 q^{19} - 1752 q^{21} + 1572 q^{24} + 7544 q^{31} - 6248 q^{34} - 15212 q^{36} + 2672 q^{39} + 16392 q^{46} + 2636 q^{49} - 16984 q^{51} + 556 q^{54} + 12904 q^{61} - 6348 q^{64} - 25520 q^{66} - 10944 q^{69} + 42696 q^{76} + 32248 q^{79} + 10172 q^{81} + 29472 q^{84} - 23264 q^{91} - 69848 q^{94} + 28708 q^{96} - 19360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 145x^{10} + 7978x^{8} + 203285x^{6} + 2280046x^{4} + 7912765x^{2} + 6027025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 56551 \nu^{10} + 6249405 \nu^{8} + 233239477 \nu^{6} + 3241784077 \nu^{4} + 11258013783 \nu^{2} + 7270876567 ) / 3061271556 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1890\nu^{10} - 210557\nu^{8} - 8021356\nu^{6} - 115447651\nu^{4} - 435323452\nu^{2} - 725264825 ) / 16197204 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7736157 \nu^{11} - 1058105 \nu^{10} + 875781225 \nu^{9} - 10929660 \nu^{8} + \cdots + 34193330329270 ) / 2147263334280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54153099 \nu^{11} + 2982797995 \nu^{10} - 6130468575 \nu^{9} + 388687933920 \nu^{8} + \cdots + 25\!\cdots\!70 ) / 15030843339960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11525891 \nu^{11} - 228725655 \nu^{9} + 78013282297 \nu^{7} + 4750819444105 \nu^{5} + \cdots + 536694293717605 \nu ) / 2505140556660 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12592917 \nu^{11} - 1058105 \nu^{10} + 3143011725 \nu^{9} - 10929660 \nu^{8} + \cdots + 34193330329270 ) / 2147263334280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7736157 \nu^{11} + 2116210 \nu^{10} + 875781225 \nu^{9} + 21859320 \nu^{8} + \cdots - 68386660658540 ) / 1073631667140 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11533121 \nu^{11} - 109657485 \nu^{10} - 1246762665 \nu^{9} - 12326937980 \nu^{8} + \cdots - 17945675224610 ) / 835046852220 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37271052 \nu^{11} - 1058105 \nu^{10} + 4665843450 \nu^{9} - 10929660 \nu^{8} + \cdots + 34193330329270 ) / 1073631667140 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 303268108 \nu^{11} + 2083801545 \nu^{10} + 35020927650 \nu^{9} + 235129930245 \nu^{8} + \cdots + 140498556918615 ) / 7515421669980 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 407066197 \nu^{11} + 1096884180 \nu^{10} - 46241791635 \nu^{9} + 124187488425 \nu^{8} + \cdots - 21012520102875 ) / 7515421669980 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 15\beta_{10} + 45\beta_{8} - 3\beta_{6} - 5\beta_{5} + 3\beta_{3} - 15\beta_{2} + 15 ) / 300 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{11} - 6\beta_{10} + 6\beta_{8} + \beta_{4} + \beta_{3} - 36\beta_{2} - 58\beta _1 - 1434 ) / 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 87 \beta_{11} - 459 \beta_{10} - 180 \beta_{9} - 1551 \beta_{8} - 90 \beta_{7} + 156 \beta_{6} + \cdots - 546 ) / 300 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 64 \beta_{11} + 64 \beta_{10} - 64 \beta_{8} - 20 \beta_{7} - 9 \beta_{4} + 31 \beta_{3} + 504 \beta_{2} + \cdots + 16686 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5751 \beta_{11} + 15852 \beta_{10} + 11550 \beta_{9} + 59058 \beta_{8} + 6900 \beta_{7} - 8283 \beta_{6} + \cdots + 21603 ) / 300 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2829 \beta_{11} - 2829 \beta_{10} + 2829 \beta_{8} + 1140 \beta_{7} + 199 \beta_{4} - 2081 \beta_{3} + \cdots - 928176 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 61578 \beta_{11} - 113529 \beta_{10} - 118734 \beta_{9} - 463743 \beta_{8} - 82362 \beta_{7} + \cdots - 175107 ) / 60 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 53208 \beta_{11} + 53208 \beta_{10} - 53208 \beta_{8} - 17620 \beta_{7} + 8397 \beta_{4} + \cdots + 23629382 ) / 20 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15258009 \beta_{11} + 20840373 \beta_{10} + 28242990 \beta_{9} + 93037137 \beta_{8} + 22131720 \beta_{7} + \cdots + 36098382 ) / 300 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 823188 \beta_{11} - 823188 \beta_{10} + 823188 \beta_{8} - 24528 \beta_{7} - 615331 \beta_{4} + \cdots - 553849278 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 728479503 \beta_{11} - 780794976 \beta_{10} - 1291533210 \beta_{9} - 3799343934 \beta_{8} + \cdots - 1509274479 ) / 300 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
− 6.59187i
6.59187i
6.18435i
− 6.18435i
− 2.02555i
2.02555i
1.02555i
− 1.02555i
− 5.18435i
5.18435i
5.59187i
− 5.59187i
−7.20990 −2.01697 − 8.77108i 35.9827 0 14.5422 + 63.2386i 23.3388i −144.073 −72.8637 + 35.3820i 0
74.2 −7.20990 −2.01697 + 8.77108i 35.9827 0 14.5422 − 63.2386i − 23.3388i −144.073 −72.8637 − 35.3820i 0
74.3 −4.56632 4.15391 − 7.98405i 4.85128 0 −18.9681 + 36.4577i − 61.6068i 50.9086 −46.4900 − 66.3301i 0
74.4 −4.56632 4.15391 + 7.98405i 4.85128 0 −18.9681 − 36.4577i 61.6068i 50.9086 −46.4900 + 66.3301i 0
74.5 −0.407512 −8.40695 − 3.21297i −15.8339 0 3.42594 + 1.30932i − 46.9457i 12.9727 60.3537 + 54.0225i 0
74.6 −0.407512 −8.40695 + 3.21297i −15.8339 0 3.42594 − 1.30932i 46.9457i 12.9727 60.3537 − 54.0225i 0
74.7 0.407512 8.40695 − 3.21297i −15.8339 0 3.42594 − 1.30932i − 46.9457i −12.9727 60.3537 − 54.0225i 0
74.8 0.407512 8.40695 + 3.21297i −15.8339 0 3.42594 + 1.30932i 46.9457i −12.9727 60.3537 + 54.0225i 0
74.9 4.56632 −4.15391 − 7.98405i 4.85128 0 −18.9681 − 36.4577i − 61.6068i −50.9086 −46.4900 + 66.3301i 0
74.10 4.56632 −4.15391 + 7.98405i 4.85128 0 −18.9681 + 36.4577i 61.6068i −50.9086 −46.4900 − 66.3301i 0
74.11 7.20990 2.01697 − 8.77108i 35.9827 0 14.5422 − 63.2386i 23.3388i 144.073 −72.8637 − 35.3820i 0
74.12 7.20990 2.01697 + 8.77108i 35.9827 0 14.5422 + 63.2386i − 23.3388i 144.073 −72.8637 + 35.3820i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.12
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 75.5.d.d 12
3.b odd 2 1 inner 75.5.d.d 12
5.b even 2 1 inner 75.5.d.d 12
5.c odd 4 1 15.5.c.a ✓ 6
5.c odd 4 1 75.5.c.i 6
15.d odd 2 1 inner 75.5.d.d 12
15.e even 4 1 15.5.c.a ✓ 6
15.e even 4 1 75.5.c.i 6
20.e even 4 1 240.5.l.d 6
60.l odd 4 1 240.5.l.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.c.a ✓ 6 5.c odd 4 1
15.5.c.a ✓ 6 15.e even 4 1
75.5.c.i 6 5.c odd 4 1
75.5.c.i 6 15.e even 4 1
75.5.d.d 12 1.a even 1 1 trivial
75.5.d.d 12 3.b odd 2 1 inner
75.5.d.d 12 5.b even 2 1 inner
75.5.d.d 12 15.d odd 2 1 inner
240.5.l.d 6 20.e even 4 1
240.5.l.d 6 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 73T_{2}^{4} + 1096T_{2}^{2} - 180 \) acting on \(S_{5}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 73 T^{4} + \cdots - 180)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 6544 T^{4} + \cdots + 4556250000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 53380 T^{4} + \cdots + 552448800000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 20424 T^{4} + \cdots + 32112640000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 345773292505920)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 122 T^{2} + \cdots + 6584528)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 512545320524880)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 1886 T^{2} + \cdots - 171289728)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 17\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 3226 T^{2} + \cdots + 8122222912)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 8062 T^{2} + \cdots + 94953979728)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 77\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
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