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Properties

Label 750.3.b.b
Level $750$
Weight $3$
Character orbit 750.b
Analytic conductor $20.436$
Analytic rank $0$
Dimension $16$
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] = [750,3,Mod(749,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.749");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 750.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: \(20.4360198270\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 19 x^{14} + 4 x^{13} + 4 x^{12} - 114 x^{11} + 1528 x^{10} + 2092 x^{9} + \cdots + 13393856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{2} + 1) q^{3} + 2 q^{4} + (\beta_{14} + \beta_{5}) q^{6} + ( - \beta_{14} - \beta_{13} - \beta_{12} + \cdots + 1) q^{7}+ \cdots + (\beta_{15} + \beta_{10} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{2} + 1) q^{3} + 2 q^{4} + (\beta_{14} + \beta_{5}) q^{6} + ( - \beta_{14} - \beta_{13} - \beta_{12} + \cdots + 1) q^{7}+ \cdots + ( - 10 \beta_{15} + 6 \beta_{14} + \cdots - 22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{3} + 32 q^{4} + 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{3} + 32 q^{4} + 4 q^{6} + 2 q^{9} + 20 q^{12} + 64 q^{16} - 60 q^{17} + 40 q^{18} - 8 q^{19} + 62 q^{21} - 100 q^{23} + 8 q^{24} + 130 q^{27} + 48 q^{31} + 150 q^{33} - 16 q^{34} + 4 q^{36} - 200 q^{38} + 50 q^{39} + 200 q^{42} - 56 q^{46} - 80 q^{47} + 40 q^{48} - 80 q^{49} - 2 q^{51} - 160 q^{53} + 20 q^{54} + 250 q^{57} + 64 q^{61} - 160 q^{62} + 90 q^{63} + 128 q^{64} + 72 q^{66} - 120 q^{68} - 132 q^{69} + 80 q^{72} - 16 q^{76} + 120 q^{77} + 300 q^{78} + 148 q^{79} - 298 q^{81} - 220 q^{83} + 124 q^{84} + 30 q^{87} - 300 q^{91} - 200 q^{92} + 120 q^{93} + 200 q^{94} + 16 q^{96} - 560 q^{98} - 362 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 19 x^{14} + 4 x^{13} + 4 x^{12} - 114 x^{11} + 1528 x^{10} + 2092 x^{9} + \cdots + 13393856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2249 \nu^{15} + 2375 \nu^{14} - 141659 \nu^{13} + 977853 \nu^{12} - 3068200 \nu^{11} + \cdots - 11727644320 ) / 21159854856 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 5 \nu^{14} + 14 \nu^{13} + 18 \nu^{12} + 22 \nu^{11} - 92 \nu^{10} + 1436 \nu^{9} + \cdots + 24869896 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15035 \nu^{15} + 26408 \nu^{14} - 184097 \nu^{13} - 617202 \nu^{12} + 13679822 \nu^{11} + \cdots + 106975620584 ) / 63479564568 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31957 \nu^{15} + 414755 \nu^{14} - 2265938 \nu^{13} + 6163065 \nu^{12} - 7981975 \nu^{11} + \cdots - 46801886656 ) / 63479564568 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29003 \nu^{15} - 163945 \nu^{14} + 361315 \nu^{13} + 1447395 \nu^{12} - 4977640 \nu^{11} + \cdots + 314537466656 ) / 42319709712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13837 \nu^{15} + 108728 \nu^{14} - 441557 \nu^{13} + 624831 \nu^{12} - 381592 \nu^{11} + \cdots - 21791049280 ) / 21159854856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25747 \nu^{15} + 250433 \nu^{14} - 1153673 \nu^{13} + 2828997 \nu^{12} - 7042186 \nu^{11} + \cdots + 2179746464 ) / 42319709712 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41339 \nu^{15} - 167566 \nu^{14} + 28975 \nu^{13} + 4441164 \nu^{12} - 10412554 \nu^{11} + \cdots + 716803866800 ) / 42319709712 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23674 \nu^{15} - 102992 \nu^{14} + 152879 \nu^{13} + 2126121 \nu^{12} - 6247139 \nu^{11} + \cdots + 258404471752 ) / 21159854856 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18811 \nu^{15} + 176953 \nu^{14} - 822421 \nu^{13} + 1483685 \nu^{12} + 425866 \nu^{11} + \cdots + 38434979520 ) / 14106569904 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25898 \nu^{15} - 189265 \nu^{14} + 722578 \nu^{13} - 180105 \nu^{12} - 3007390 \nu^{11} + \cdots + 128135681744 ) / 14106569904 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24356 \nu^{15} - 265375 \nu^{14} + 1464004 \nu^{13} - 4396221 \nu^{12} + 8068958 \nu^{11} + \cdots - 57511541920 ) / 14106569904 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 34090 \nu^{15} + 231225 \nu^{14} - 737640 \nu^{13} - 648605 \nu^{12} + 5942184 \nu^{11} + \cdots - 206787783104 ) / 14106569904 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 331031 \nu^{15} - 2381446 \nu^{14} + 8412049 \nu^{13} - 2929536 \nu^{12} + \cdots + 1949731637504 ) / 126959129136 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 63292 \nu^{15} - 512735 \nu^{14} + 2100488 \nu^{13} - 2598123 \nu^{12} - 1778090 \nu^{11} + \cdots + 267219234688 ) / 21159854856 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - \beta_{10} + \beta_{8} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{15} - \beta_{13} + 2 \beta_{12} - 2 \beta_{10} - \beta_{9} + \beta_{8} + 5 \beta_{6} + \cdots - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{15} + \beta_{13} + 13 \beta_{12} - \beta_{10} + \beta_{9} + 2 \beta_{8} + 13 \beta_{6} + \cdots - 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 33 \beta_{15} + 9 \beta_{14} + 19 \beta_{13} + 38 \beta_{12} - 9 \beta_{11} + 2 \beta_{10} + \cdots - 143 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 87 \beta_{15} + 144 \beta_{14} + 99 \beta_{13} + 65 \beta_{12} - 36 \beta_{11} + 12 \beta_{10} + \cdots - 285 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 57 \beta_{15} + 675 \beta_{14} + 251 \beta_{13} - 105 \beta_{12} - 162 \beta_{11} + 142 \beta_{10} + \cdots - 229 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 543 \beta_{15} + 1980 \beta_{14} - 336 \beta_{13} - 1343 \beta_{12} + 45 \beta_{11} + 681 \beta_{10} + \cdots + 330 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1557 \beta_{15} + 282 \beta_{14} - 2074 \beta_{13} - 2492 \beta_{12} + 672 \beta_{11} + 835 \beta_{10} + \cdots + 2739 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 23733 \beta_{15} - 25506 \beta_{14} - 30221 \beta_{13} - 32592 \beta_{12} + 10602 \beta_{11} + \cdots + 43192 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 100896 \beta_{15} - 159300 \beta_{14} - 100585 \beta_{13} - 112612 \beta_{12} + 32346 \beta_{11} + \cdots + 77207 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 349398 \beta_{15} - 590184 \beta_{14} - 243572 \beta_{13} - 260633 \beta_{12} + 48240 \beta_{11} + \cdots - 176240 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 918399 \beta_{15} - 1503981 \beta_{14} - 224192 \beta_{13} - 110194 \beta_{12} - 41499 \beta_{11} + \cdots - 1125074 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1088877 \beta_{15} - 1993950 \beta_{14} + 1952265 \beta_{13} + 2682170 \beta_{12} - 501822 \beta_{11} + \cdots + 1511907 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 6870504 \beta_{15} + 4781268 \beta_{14} + 15893021 \beta_{13} + 18252003 \beta_{12} - 2041713 \beta_{11} + \cdots + 37275425 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 65811822 \beta_{15} + 49625730 \beta_{14} + 74726385 \beta_{13} + 85417279 \beta_{12} + \cdots + 185969445 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
2.47438 − 2.61270i
2.47438 + 2.61270i
2.35016 − 2.67900i
2.35016 + 2.67900i
−1.13927 − 2.10322i
−1.13927 + 2.10322i
−1.47816 − 1.69078i
−1.47816 + 1.69078i
3.42729 − 1.76303i
3.42729 + 1.76303i
0.997224 − 3.00000i
0.997224 + 3.00000i
−1.68126 − 1.34568i
−1.68126 + 1.34568i
−1.95036 − 0.543464i
−1.95036 + 0.543464i
−1.41421 −1.47438 − 2.61270i 2.00000 0 2.08508 + 3.69492i − 7.39353i −2.82843 −4.65243 + 7.70421i 0
749.2 −1.41421 −1.47438 + 2.61270i 2.00000 0 2.08508 − 3.69492i 7.39353i −2.82843 −4.65243 − 7.70421i 0
749.3 −1.41421 −1.35016 − 2.67900i 2.00000 0 1.90941 + 3.78869i − 0.207413i −2.82843 −5.35413 + 7.23417i 0
749.4 −1.41421 −1.35016 + 2.67900i 2.00000 0 1.90941 − 3.78869i 0.207413i −2.82843 −5.35413 − 7.23417i 0
749.5 −1.41421 2.13927 − 2.10322i 2.00000 0 −3.02538 + 2.97440i 4.95030i −2.82843 0.152952 − 8.99870i 0
749.6 −1.41421 2.13927 + 2.10322i 2.00000 0 −3.02538 − 2.97440i − 4.95030i −2.82843 0.152952 + 8.99870i 0
749.7 −1.41421 2.47816 − 1.69078i 2.00000 0 −3.50465 + 2.39112i − 6.14755i −2.82843 3.28255 − 8.38003i 0
749.8 −1.41421 2.47816 + 1.69078i 2.00000 0 −3.50465 − 2.39112i 6.14755i −2.82843 3.28255 + 8.38003i 0
749.9 1.41421 −2.42729 − 1.76303i 2.00000 0 −3.43271 − 2.49330i 10.5743i 2.82843 2.78348 + 8.55875i 0
749.10 1.41421 −2.42729 + 1.76303i 2.00000 0 −3.43271 + 2.49330i − 10.5743i 2.82843 2.78348 − 8.55875i 0
749.11 1.41421 0.00277587 − 3.00000i 2.00000 0 0.00392567 − 4.24264i 5.72578i 2.82843 −8.99998 − 0.0166552i 0
749.12 1.41421 0.00277587 + 3.00000i 2.00000 0 0.00392567 + 4.24264i − 5.72578i 2.82843 −8.99998 + 0.0166552i 0
749.13 1.41421 2.68126 − 1.34568i 2.00000 0 3.79187 − 1.90308i 6.63242i 2.82843 5.37828 − 7.21624i 0
749.14 1.41421 2.68126 + 1.34568i 2.00000 0 3.79187 + 1.90308i − 6.63242i 2.82843 5.37828 + 7.21624i 0
749.15 1.41421 2.95036 − 0.543464i 2.00000 0 4.17244 − 0.768574i 11.2431i 2.82843 8.40929 − 3.20683i 0
749.16 1.41421 2.95036 + 0.543464i 2.00000 0 4.17244 + 0.768574i − 11.2431i 2.82843 8.40929 + 3.20683i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 750.3.b.b 16
3.b odd 2 1 750.3.b.a 16
5.b even 2 1 750.3.b.a 16
5.c odd 4 2 750.3.d.b ✓ 32
15.d odd 2 1 inner 750.3.b.b 16
15.e even 4 2 750.3.d.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.3.b.a 16 3.b odd 2 1
750.3.b.a 16 5.b even 2 1
750.3.b.b 16 1.a even 1 1 trivial
750.3.b.b 16 15.d odd 2 1 inner
750.3.d.b ✓ 32 5.c odd 4 2
750.3.d.b ✓ 32 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(750, [\chi])\):

\( T_{7}^{16} + 432 T_{7}^{14} + 75058 T_{7}^{12} + 6807924 T_{7}^{10} + 350415105 T_{7}^{8} + \cdots + 44394015856 \) Copy content Toggle raw display
\( T_{17}^{8} + 30 T_{17}^{7} - 327 T_{17}^{6} - 15250 T_{17}^{5} - 22356 T_{17}^{4} + 1887550 T_{17}^{3} + \cdots - 66810599 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} - 10 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 44394015856 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 75082443750000 \) Copy content Toggle raw display
$17$ \( (T^{8} + 30 T^{7} + \cdots - 66810599)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots + 2454161551)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 50 T^{7} + \cdots - 1643010624)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} - 24 T^{7} + \cdots - 11958077609)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 3120689355625)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 5540803798041)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 2353010577039)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 242321950359729)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 386900061553904)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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