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Properties

Label 930.2.h.c
Level $930$
Weight $2$
Character orbit 930.h
Analytic conductor $7.426$
Analytic rank $0$
Dimension $16$
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] = [930,2,Mod(371,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.371");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.h (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: \(7.42608738798\)
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: \( x^{16} - 2x^{14} + 10x^{12} - 42x^{10} + 82x^{8} - 378x^{6} + 810x^{4} - 1458x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{5} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{4} - 1) q^{7} - \beta_{3} q^{8} + ( - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{5} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{4} - 1) q^{7} - \beta_{3} q^{8} + ( - \beta_{3} - \beta_{2}) q^{9} - q^{10} - \beta_{12} q^{11} + \beta_{5} q^{12} + (\beta_{14} - 2 \beta_{10} + \beta_1) q^{13} + ( - \beta_{7} - \beta_{3}) q^{14} + \beta_1 q^{15} + q^{16} + ( - \beta_{11} - \beta_{6} - \beta_{5}) q^{17} + (\beta_{9} + 1) q^{18} + (\beta_{15} + \beta_{4} + \beta_{2} - 1) q^{19} - \beta_{3} q^{20} + (\beta_{14} - \beta_{12} + \cdots + \beta_{5}) q^{21}+ \cdots + (3 \beta_{14} + 2 \beta_{11} + \cdots + \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 20 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 20 q^{7} - 4 q^{9} - 16 q^{10} + 16 q^{16} + 12 q^{18} - 4 q^{19} - 16 q^{25} + 20 q^{28} - 4 q^{31} - 16 q^{33} + 4 q^{36} - 4 q^{39} + 16 q^{40} + 12 q^{45} + 4 q^{49} + 52 q^{51} - 12 q^{63} - 16 q^{64} + 4 q^{66} + 112 q^{67} - 4 q^{69} + 20 q^{70} - 12 q^{72} + 4 q^{76} - 28 q^{78} - 32 q^{81} + 32 q^{82} - 24 q^{87} + 4 q^{90} + 16 q^{93} - 8 q^{94} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} + 10x^{12} - 42x^{10} + 82x^{8} - 378x^{6} + 810x^{4} - 1458x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23 \nu^{14} + 325 \nu^{12} - 2003 \nu^{10} - 375 \nu^{8} - 23567 \nu^{6} + 61461 \nu^{4} + \cdots + 449793 ) / 571536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23 \nu^{14} - 325 \nu^{12} + 2003 \nu^{10} + 375 \nu^{8} + 23567 \nu^{6} - 61461 \nu^{4} + \cdots - 449793 ) / 571536 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 52\nu^{12} + 179\nu^{10} - 660\nu^{8} + 2267\nu^{6} - 1620\nu^{4} + 10935\nu^{2} - 23328 ) / 20412 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23 \nu^{15} - 325 \nu^{13} + 2003 \nu^{11} + 375 \nu^{9} + 23567 \nu^{7} - 61461 \nu^{5} + \cdots - 449793 \nu ) / 571536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\nu^{15} - 265\nu^{13} + 569\nu^{11} - 1623\nu^{9} + 4763\nu^{7} - 2781\nu^{5} + 68661\nu^{3} - 84807\nu ) / 142884 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -40\nu^{14} + 431\nu^{12} - 454\nu^{10} + 249\nu^{8} + 1580\nu^{6} - 2997\nu^{4} + 13446\nu^{2} - 51759 ) / 142884 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49 \nu^{15} + 361 \nu^{13} - 347 \nu^{11} + 4557 \nu^{9} - 12263 \nu^{7} + 26649 \nu^{5} + \cdots + 79461 \nu ) / 244944 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\nu^{14} - 197\nu^{12} - 149\nu^{10} - 2409\nu^{8} + 5863\nu^{6} - 11205\nu^{4} + 46251\nu^{2} - 46737 ) / 63504 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35\nu^{15} - 169\nu^{13} - 181\nu^{11} - 273\nu^{9} + 467\nu^{7} - 7497\nu^{5} + 12555\nu^{3} + 40095\nu ) / 122472 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 617 \nu^{15} + 1027 \nu^{13} - 3245 \nu^{11} + 7887 \nu^{9} - 53969 \nu^{7} + 21123 \nu^{5} + \cdots - 175689 \nu ) / 1714608 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 107 \nu^{15} - 199 \nu^{13} + 1499 \nu^{11} - 2103 \nu^{9} - 3061 \nu^{7} - 20343 \nu^{5} + \cdots - 10935 \nu ) / 285768 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 617 \nu^{14} - 1027 \nu^{12} + 3245 \nu^{10} - 7887 \nu^{8} + 53969 \nu^{6} - 21123 \nu^{4} + \cdots - 395847 ) / 571536 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} - 2\nu^{13} + 10\nu^{11} - 42\nu^{9} + 82\nu^{7} - 378\nu^{5} + 810\nu^{3} - 1458\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 761 \nu^{14} + 1243 \nu^{12} - 5837 \nu^{10} + 33303 \nu^{8} - 40721 \nu^{6} + 234891 \nu^{4} + \cdots + 693279 ) / 571536 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{6} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 2\beta_{13} - \beta_{9} + 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{14} - \beta_{12} - 5\beta_{11} - 3\beta_{10} + \beta_{8} + 4\beta_{6} - 4\beta_{5} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{15} + 5\beta_{13} + 3\beta_{9} + 6\beta_{7} + 2\beta_{4} + 5\beta_{3} - \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{14} - 8\beta_{12} - 21\beta_{11} - 5\beta_{10} - 6\beta_{8} - 5\beta_{6} + 7\beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9\beta_{15} + 7\beta_{13} - \beta_{9} - 8\beta_{7} - 16\beta_{4} + 56\beta_{3} + 9\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -27\beta_{14} + 5\beta_{11} + 23\beta_{10} + 26\beta_{8} + \beta_{6} + 57\beta_{5} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -25\beta_{15} + 3\beta_{13} - 83\beta_{9} - 22\beta_{7} + 50\beta_{4} + 19\beta_{3} + 12\beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -18\beta_{14} + 93\beta_{12} + 36\beta_{11} - 111\beta_{10} + 45\beta_{8} + 84\beta_{6} + 65\beta_{5} + 118\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -54\beta_{15} + 63\beta_{13} - 110\beta_{9} + 288\beta_{7} + 18\beta_{4} - 92\beta_{3} + 25\beta_{2} + 61 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 365 \beta_{14} - 203 \beta_{12} - 72 \beta_{11} - 416 \beta_{10} + 117 \beta_{8} - 245 \beta_{6} + \cdots + 133 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -159\beta_{15} + 736\beta_{13} - 56\beta_{9} - 32\beta_{7} - 544\beta_{4} - 25\beta_{3} + 336\beta_{2} - 256 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 725 \beta_{14} - 248 \beta_{12} - 1272 \beta_{11} + 520 \beta_{10} + 936 \beta_{8} - 176 \beta_{6} + \cdots - 600 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1
−0.687009 + 1.58997i
−1.06195 + 1.36831i
1.39665 + 1.02439i
−1.71746 + 0.224352i
1.71746 − 0.224352i
−1.39665 − 1.02439i
1.06195 − 1.36831i
0.687009 − 1.58997i
−0.687009 − 1.58997i
−1.06195 − 1.36831i
1.39665 − 1.02439i
−1.71746 − 0.224352i
1.71746 + 0.224352i
−1.39665 + 1.02439i
1.06195 + 1.36831i
0.687009 + 1.58997i
− 1.00000i −1.58997 − 0.687009i −1.00000 − 1.00000i −0.687009 + 1.58997i −4.71902 1.00000i 2.05604 + 2.18465i −1.00000
371.2 − 1.00000i −1.36831 − 1.06195i −1.00000 − 1.00000i −1.06195 + 1.36831i 1.28192 1.00000i 0.744540 + 2.90614i −1.00000
371.3 − 1.00000i −1.02439 + 1.39665i −1.00000 − 1.00000i 1.39665 + 1.02439i 0.608912 1.00000i −0.901245 − 2.86143i −1.00000
371.4 − 1.00000i −0.224352 − 1.71746i −1.00000 − 1.00000i −1.71746 + 0.224352i −2.17181 1.00000i −2.89933 + 0.770630i −1.00000
371.5 − 1.00000i 0.224352 + 1.71746i −1.00000 − 1.00000i 1.71746 − 0.224352i −2.17181 1.00000i −2.89933 + 0.770630i −1.00000
371.6 − 1.00000i 1.02439 − 1.39665i −1.00000 − 1.00000i −1.39665 − 1.02439i 0.608912 1.00000i −0.901245 − 2.86143i −1.00000
371.7 − 1.00000i 1.36831 + 1.06195i −1.00000 − 1.00000i 1.06195 − 1.36831i 1.28192 1.00000i 0.744540 + 2.90614i −1.00000
371.8 − 1.00000i 1.58997 + 0.687009i −1.00000 − 1.00000i 0.687009 − 1.58997i −4.71902 1.00000i 2.05604 + 2.18465i −1.00000
371.9 1.00000i −1.58997 + 0.687009i −1.00000 1.00000i −0.687009 − 1.58997i −4.71902 − 1.00000i 2.05604 − 2.18465i −1.00000
371.10 1.00000i −1.36831 + 1.06195i −1.00000 1.00000i −1.06195 − 1.36831i 1.28192 − 1.00000i 0.744540 − 2.90614i −1.00000
371.11 1.00000i −1.02439 − 1.39665i −1.00000 1.00000i 1.39665 − 1.02439i 0.608912 − 1.00000i −0.901245 + 2.86143i −1.00000
371.12 1.00000i −0.224352 + 1.71746i −1.00000 1.00000i −1.71746 − 0.224352i −2.17181 − 1.00000i −2.89933 − 0.770630i −1.00000
371.13 1.00000i 0.224352 − 1.71746i −1.00000 1.00000i 1.71746 + 0.224352i −2.17181 − 1.00000i −2.89933 − 0.770630i −1.00000
371.14 1.00000i 1.02439 + 1.39665i −1.00000 1.00000i −1.39665 + 1.02439i 0.608912 − 1.00000i −0.901245 + 2.86143i −1.00000
371.15 1.00000i 1.36831 − 1.06195i −1.00000 1.00000i 1.06195 + 1.36831i 1.28192 − 1.00000i 0.744540 − 2.90614i −1.00000
371.16 1.00000i 1.58997 − 0.687009i −1.00000 1.00000i 0.687009 + 1.58997i −4.71902 − 1.00000i 2.05604 − 2.18465i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 371.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.h.c ✓ 16
3.b odd 2 1 inner 930.2.h.c ✓ 16
31.b odd 2 1 inner 930.2.h.c ✓ 16
93.c even 2 1 inner 930.2.h.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.c ✓ 16 1.a even 1 1 trivial
930.2.h.c ✓ 16 3.b odd 2 1 inner
930.2.h.c ✓ 16 31.b odd 2 1 inner
930.2.h.c ✓ 16 93.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 5T_{7}^{3} - 2T_{7}^{2} - 14T_{7} + 8 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 5 T^{3} - 2 T^{2} + \cdots + 8)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 31 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 86 T^{6} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 50 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} - 30 T^{2} + \cdots - 32)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 61 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 136 T^{6} + \cdots + 123904)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 2 T^{7} + \cdots + 923521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 26 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 196 T^{6} + \cdots + 3444736)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 245 T^{6} + \cdots + 988036)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 176 T^{6} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 277 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 324 T^{6} + \cdots + 14258176)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 224 T^{6} + \cdots + 4528384)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 28 T^{3} + \cdots - 2816)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 209 T^{6} + \cdots + 3564544)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 461 T^{6} + \cdots + 62980096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 223 T^{6} + \cdots + 1149184)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 22 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 335 T^{6} + \cdots + 19219456)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + \cdots - 1744)^{4} \) Copy content Toggle raw display
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