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Properties

Label 75.13.f.b
Level $75$
Weight $13$
Character orbit 75.f
Analytic conductor $68.550$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.7");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(68.5495362957\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1803691x^{12} + 837302586771x^{8} + 28116579901499881x^{4} + 244891489442452934656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{44}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{13} + \beta_{11}) q^{3} + (\beta_{9} + 933 \beta_{5}) q^{4} + ( - \beta_{2} + 3 \beta_1 - 6015) q^{6} + (2 \beta_{8} - 2 \beta_{7} + \cdots + 804 \beta_{3}) q^{7}+ \cdots - 177147 \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{13} + \beta_{11}) q^{3} + (\beta_{9} + 933 \beta_{5}) q^{4} + ( - \beta_{2} + 3 \beta_1 - 6015) q^{6} + (2 \beta_{8} - 2 \beta_{7} + \cdots + 804 \beta_{3}) q^{7}+ \cdots + (1240029 \beta_{12} + \cdots - 46492761591 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 96228 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 96228 q^{6} + 4199904 q^{11} + 51792860 q^{16} - 330254496 q^{21} - 2073579048 q^{26} - 2473998944 q^{31} + 2643741828 q^{36} + 10857619296 q^{41} + 27162102816 q^{46} + 30572650368 q^{51} + 152759259720 q^{56} + 571432480160 q^{61} + 183001354056 q^{66} - 556384405248 q^{71} - 2085818116072 q^{76} - 502096953744 q^{81} - 3148168178016 q^{86} + 1472124684096 q^{91} + 56136992004 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1803691x^{12} + 837302586771x^{8} + 28116579901499881x^{4} + 244891489442452934656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 188672499 \nu^{12} - 333097413127878 \nu^{8} + \cdots - 22\!\cdots\!76 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 576187371 \nu^{12} + \cdots + 80\!\cdots\!04 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7339986173 \nu^{13} + \cdots - 10\!\cdots\!52 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1310212593 \nu^{12} + \cdots - 18\!\cdots\!72 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22766781 \nu^{14} + 41343964028407 \nu^{10} + \cdots + 98\!\cdots\!69 \nu^{2} ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 95419820249 \nu^{13} + \cdots - 13\!\cdots\!76 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2954534982601 \nu^{13} + \cdots - 59\!\cdots\!24 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 350686277457 \nu^{13} + \cdots - 48\!\cdots\!08 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!09 \nu^{14} + \cdots + 22\!\cdots\!91 \nu^{2} ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 211560980472513 \nu^{14} + \cdots + 30\!\cdots\!12 \nu^{2} ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 44\!\cdots\!47 \nu^{15} + \cdots + 66\!\cdots\!03 \nu^{3} ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!77 \nu^{14} + \cdots - 19\!\cdots\!73 \nu^{2} ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27\!\cdots\!07 \nu^{15} + \cdots + 38\!\cdots\!43 \nu^{3} ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!89 \nu^{15} + \cdots - 20\!\cdots\!61 \nu^{3} ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!23 \nu^{15} + \cdots - 17\!\cdots\!27 \nu^{3} ) / 55\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 26\beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} + 21\beta_{9} + 130314\beta_{5} ) / 243 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} + 3\beta_{14} - 325\beta_{13} + 7235\beta_{11} ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{4} + 2348\beta_{2} + 21972\beta _1 - 109579719 ) / 243 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8940\beta_{8} - 9666\beta_{7} - 1029953\beta_{6} + 19991320\beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5718\beta_{12} + 822034\beta_{10} - 6816765\beta_{9} - 33883141428\beta_{5} ) / 81 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11159586\beta_{15} - 9271431\beta_{14} + 1089031943\beta_{13} - 18730029283\beta_{11} ) / 81 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -24207327\beta_{4} - 2541972664\beta_{2} - 18765038346\beta _1 + 95910470241549 ) / 243 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -4347330320\beta_{8} + 2926949076\beta_{7} + 380395685331\beta_{6} - 5875845142818\beta_{3} ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -30345125652\beta_{12} - 2601540044074\beta_{10} + 17210511636009\beta_{9} + 90896259134533758\beta_{5} ) / 243 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 14669687217342 \beta_{15} + 8318147172309 \beta_{14} + \cdots + 16\!\cdots\!89 \beta_{11} ) / 81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 11896971493239\beta_{4} + 883138253161388\beta_{2} + 5269981228558032\beta _1 - 28788602268823184001 ) / 81 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 16\!\cdots\!00 \beta_{8} + \cdots + 15\!\cdots\!52 \beta_{3} ) / 81 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 40\!\cdots\!82 \beta_{12} + \cdots - 82\!\cdots\!60 \beta_{5} ) / 243 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 57\!\cdots\!70 \beta_{15} + \cdots - 49\!\cdots\!93 \beta_{11} ) / 27 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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
−21.1307 − 21.1307i
−22.1946 − 22.1946i
−8.08604 − 8.08604i
−8.24683 − 8.24683i
8.24683 + 8.24683i
8.08604 + 8.08604i
22.1946 + 22.1946i
21.1307 + 21.1307i
−21.1307 + 21.1307i
−22.1946 + 22.1946i
−8.08604 + 8.08604i
−8.24683 + 8.24683i
8.24683 − 8.24683i
8.08604 − 8.08604i
22.1946 − 22.1946i
21.1307 − 21.1307i
−74.4147 − 74.4147i 297.613 − 297.613i 6979.09i 0 −44293.6 46958.0 + 46958.0i 214544. − 214544.i − 177147.i 0
7.2 −55.5611 − 55.5611i −297.613 + 297.613i 2078.08i 0 33071.4 154903. + 154903.i −112118. + 112118.i − 177147.i 0
7.3 −35.2808 − 35.2808i 297.613 − 297.613i − 1606.53i 0 −21000.1 −96257.5 − 96257.5i −201190. + 201190.i − 177147.i 0
7.4 −13.7178 − 13.7178i −297.613 + 297.613i − 3719.65i 0 8165.18 −65492.6 − 65492.6i −107213. + 107213.i − 177147.i 0
7.5 13.7178 + 13.7178i 297.613 − 297.613i − 3719.65i 0 8165.18 65492.6 + 65492.6i 107213. − 107213.i − 177147.i 0
7.6 35.2808 + 35.2808i −297.613 + 297.613i − 1606.53i 0 −21000.1 96257.5 + 96257.5i 201190. − 201190.i − 177147.i 0
7.7 55.5611 + 55.5611i 297.613 − 297.613i 2078.08i 0 33071.4 −154903. − 154903.i 112118. − 112118.i − 177147.i 0
7.8 74.4147 + 74.4147i −297.613 + 297.613i 6979.09i 0 −44293.6 −46958.0 − 46958.0i −214544. + 214544.i − 177147.i 0
43.1 −74.4147 + 74.4147i 297.613 + 297.613i − 6979.09i 0 −44293.6 46958.0 − 46958.0i 214544. + 214544.i 177147.i 0
43.2 −55.5611 + 55.5611i −297.613 − 297.613i − 2078.08i 0 33071.4 154903. − 154903.i −112118. − 112118.i 177147.i 0
43.3 −35.2808 + 35.2808i 297.613 + 297.613i 1606.53i 0 −21000.1 −96257.5 + 96257.5i −201190. − 201190.i 177147.i 0
43.4 −13.7178 + 13.7178i −297.613 − 297.613i 3719.65i 0 8165.18 −65492.6 + 65492.6i −107213. − 107213.i 177147.i 0
43.5 13.7178 − 13.7178i 297.613 + 297.613i 3719.65i 0 8165.18 65492.6 − 65492.6i 107213. + 107213.i 177147.i 0
43.6 35.2808 − 35.2808i −297.613 − 297.613i 1606.53i 0 −21000.1 96257.5 − 96257.5i 201190. + 201190.i 177147.i 0
43.7 55.5611 − 55.5611i 297.613 + 297.613i − 2078.08i 0 33071.4 −154903. + 154903.i 112118. + 112118.i 177147.i 0
43.8 74.4147 − 74.4147i −297.613 − 297.613i − 6979.09i 0 −44293.6 −46958.0 + 46958.0i −214544. − 214544.i 177147.i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
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 75.13.f.b ✓ 16
5.b even 2 1 inner 75.13.f.b ✓ 16
5.c odd 4 2 inner 75.13.f.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.13.f.b ✓ 16 1.a even 1 1 trivial
75.13.f.b ✓ 16 5.b even 2 1 inner
75.13.f.b ✓ 16 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 167116041 T_{2}^{12} + \cdots + 41\!\cdots\!56 \) acting on \(S_{13}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( (T^{4} + 31381059609)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 12\!\cdots\!24)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 59\!\cdots\!00)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 43\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 89\!\cdots\!44)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 35\!\cdots\!44)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
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