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Properties

Label 75.10.b.h
Level $75$
Weight $10$
Character orbit 75.b
Analytic conductor $38.628$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,10,Mod(49,75)]
 
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([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 1470x^{5} + 317749x^{4} + 221032x^{3} + 2888x^{2} + 10631640x + 19569212100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5^{4}\cdot 23^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - 81 \beta_1 q^{3} + (\beta_{4} - 6 \beta_{2} - 445) q^{4} - 81 \beta_{2} q^{6} + (\beta_{7} + \beta_{5} + \cdots + 3236 \beta_1) q^{7}+ \cdots - 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - 81 \beta_1 q^{3} + (\beta_{4} - 6 \beta_{2} - 445) q^{4} - 81 \beta_{2} q^{6} + (\beta_{7} + \beta_{5} + \cdots + 3236 \beta_1) q^{7}+ \cdots + (6561 \beta_{6} + 439587 \beta_{4} + \cdots - 171196173) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3584 q^{4} - 324 q^{6} - 52488 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3584 q^{4} - 324 q^{6} - 52488 q^{9} + 209392 q^{11} + 362124 q^{14} + 2639600 q^{16} - 786616 q^{19} + 2111832 q^{21} + 3991032 q^{24} - 12955004 q^{26} - 9853232 q^{29} - 195032 q^{31} - 33657288 q^{34} + 23514624 q^{36} - 22811544 q^{39} - 1993312 q^{41} - 413090432 q^{44} + 207835608 q^{46} - 161397840 q^{49} + 79275024 q^{51} + 2125764 q^{54} + 341308440 q^{56} - 451893424 q^{59} + 12590680 q^{61} - 1265766656 q^{64} - 128620872 q^{66} - 376891056 q^{69} + 45433376 q^{71} - 386381576 q^{74} + 1283407744 q^{76} - 2553924160 q^{79} + 344373768 q^{81} - 1635272064 q^{84} + 2852348932 q^{86} + 1795509504 q^{89} - 3675807352 q^{91} + 4381134456 q^{94} - 4734991728 q^{96} - 1373820912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 1470x^{5} + 317749x^{4} + 221032x^{3} + 2888x^{2} + 10631640x + 19569212100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1012807213 \nu^{7} - 649081591529 \nu^{6} + 1264526583365 \nu^{5} + 105101100915 \nu^{4} + \cdots + 22\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 140725171 \nu^{7} - 213007096 \nu^{6} - 47414508810 \nu^{5} + 1336144474140 \nu^{4} + \cdots + 15\!\cdots\!60 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3377112344123 \nu^{7} + 72830691175064 \nu^{6} + \cdots - 95\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1235491636 \nu^{7} + 1786830643 \nu^{6} + 480798622035 \nu^{5} - 17999640662355 \nu^{4} + \cdots - 22\!\cdots\!25 ) / 375183275969985 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 48338035252883 \nu^{7} + \cdots + 13\!\cdots\!00 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12595993203 \nu^{7} + 18245905298 \nu^{6} + 4879373130580 \nu^{5} + \cdots - 14\!\cdots\!80 ) / 500244367959980 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 302913050590799 \nu^{7} + \cdots - 84\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 349\beta_{3} - 349\beta_{2} - 405\beta _1 + 405 ) / 920 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} + 41\beta_{5} + 95\beta_{3} - 177975\beta_1 ) / 460 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 545 \beta_{7} + 545 \beta_{6} + 559 \beta_{5} - 559 \beta_{4} - 126725 \beta_{3} + 126725 \beta_{2} + \cdots - 571245 ) / 920 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2186\beta_{6} - 16145\beta_{4} + 20014\beta_{2} - 36890565 ) / 230 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 289759 \beta_{7} + 289759 \beta_{6} - 633461 \beta_{5} - 633461 \beta_{4} + 49224091 \beta_{3} + \cdots - 518953095 ) / 920 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2949155\beta_{7} - 19836899\beta_{5} + 88639735\beta_{3} + 32633136525\beta_1 ) / 460 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 153309695 \beta_{7} - 153309695 \beta_{6} - 448686601 \beta_{5} + 448686601 \beta_{4} + \cdots + 380898186555 ) / 920 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−13.2852 + 13.2852i
11.5143 + 11.5143i
16.5691 + 16.5691i
−13.7982 + 13.7982i
−13.7982 − 13.7982i
16.5691 − 16.5691i
11.5143 − 11.5143i
−13.2852 − 13.2852i
− 44.2736i − 81.0000i −1448.15 0 −3586.16 3879.20i 41447.0i −6561.00 0
49.2 − 38.7320i 81.0000i −988.165 0 3137.29 − 6373.44i 18442.8i −6561.00 0
49.3 − 15.4348i 81.0000i 273.767 0 1250.22 8162.66i − 12128.2i −6561.00 0
49.4 − 11.8931i − 81.0000i 370.553 0 −963.344 10946.0i − 10496.3i −6561.00 0
49.5 11.8931i 81.0000i 370.553 0 −963.344 − 10946.0i 10496.3i −6561.00 0
49.6 15.4348i − 81.0000i 273.767 0 1250.22 − 8162.66i 12128.2i −6561.00 0
49.7 38.7320i − 81.0000i −988.165 0 3137.29 6373.44i − 18442.8i −6561.00 0
49.8 44.2736i 81.0000i −1448.15 0 −3586.16 − 3879.20i − 41447.0i −6561.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.10.b.h 8
3.b odd 2 1 225.10.b.n 8
5.b even 2 1 inner 75.10.b.h 8
5.c odd 4 1 75.10.a.j ✓ 4
5.c odd 4 1 75.10.a.k yes 4
15.d odd 2 1 225.10.b.n 8
15.e even 4 1 225.10.a.r 4
15.e even 4 1 225.10.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.10.a.j ✓ 4 5.c odd 4 1
75.10.a.k yes 4 5.c odd 4 1
75.10.b.h 8 1.a even 1 1 trivial
75.10.b.h 8 5.b even 2 1 inner
225.10.a.r 4 15.e even 4 1
225.10.a.t 4 15.e even 4 1
225.10.b.n 8 3.b odd 2 1
225.10.b.n 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3840T_{2}^{6} + 4288068T_{2}^{4} + 1233074560T_{2}^{2} + 99088966656 \) acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 99088966656 \) Copy content Toggle raw display
$3$ \( (T^{2} + 6561)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 85\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 76\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 98\!\cdots\!25)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 28\!\cdots\!21)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 35\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
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