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Properties

Label 1692.2.h.a
Level $1692$
Weight $2$
Character orbit 1692.h
Analytic conductor $13.511$
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] = [1692,2,Mod(845,1692)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1692, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1692.845");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1692 = 2^{2} \cdot 3^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1692.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: \(13.5106880220\)
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: \( x^{16} - 26 x^{14} - 16 x^{13} + 269 x^{12} + 288 x^{11} + 850 x^{10} + 2032 x^{9} + 6628 x^{8} + \cdots + 253609 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{47}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{5} + \beta_{11} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{5} + \beta_{11} q^{7} - \beta_{12} q^{11} - \beta_{13} q^{13} + \beta_{14} q^{17} - \beta_{3} q^{19} - \beta_{7} q^{23} + ( - \beta_{15} + 1) q^{25} + (\beta_{12} - \beta_{9} + \cdots + \beta_{7}) q^{29}+ \cdots + ( - \beta_{15} + 2 \beta_{11}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{25} + 8 q^{37} + 24 q^{49} + 8 q^{55} + 40 q^{61} + 32 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26 x^{14} - 16 x^{13} + 269 x^{12} + 288 x^{11} + 850 x^{10} + 2032 x^{9} + 6628 x^{8} + \cdots + 253609 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 67\!\cdots\!19 \nu^{15} + \cdots + 67\!\cdots\!83 ) / 25\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36\!\cdots\!81 \nu^{15} + \cdots + 63\!\cdots\!88 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22\!\cdots\!54 \nu^{15} + \cdots + 91\!\cdots\!25 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 78\!\cdots\!84 \nu^{15} + \cdots + 14\!\cdots\!63 ) / 63\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 40\!\cdots\!98 \nu^{15} + \cdots - 25\!\cdots\!45 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45\!\cdots\!57 \nu^{15} + \cdots + 16\!\cdots\!70 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 60\!\cdots\!27 \nu^{15} + \cdots - 60\!\cdots\!31 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32\!\cdots\!39 \nu^{15} + \cdots - 20\!\cdots\!58 ) / 63\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 80\!\cdots\!78 \nu^{15} + \cdots + 68\!\cdots\!91 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!55 \nu^{15} + \cdots + 60\!\cdots\!80 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!79 \nu^{15} + \cdots - 23\!\cdots\!80 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!99 \nu^{15} + \cdots + 96\!\cdots\!72 ) / 63\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 30\!\cdots\!92 \nu^{15} + \cdots - 11\!\cdots\!35 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17\!\cdots\!26 \nu^{15} + \cdots - 10\!\cdots\!51 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47\!\cdots\!45 \nu^{15} + \cdots - 56\!\cdots\!24 ) / 28\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + \beta_{11} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + \beta_{11} + \beta_{10} - 2\beta_{9} + \beta_{8} + 2\beta_{7} - 2\beta_{5} - 2\beta_{4} + 2\beta_{3} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} - 6 \beta_{14} + 6 \beta_{13} + 8 \beta_{12} + 10 \beta_{11} + 4 \beta_{9} + 4 \beta_{7} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11 \beta_{12} + 17 \beta_{11} + 9 \beta_{10} + 2 \beta_{9} + 11 \beta_{8} + 6 \beta_{7} + 16 \beta_{6} + \cdots + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4 \beta_{15} - 170 \beta_{14} + 190 \beta_{13} - 19 \beta_{12} - 29 \beta_{11} + 8 \beta_{10} + \cdots + 76 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{15} - 48 \beta_{14} + 48 \beta_{13} - 55 \beta_{12} - 75 \beta_{11} - 99 \beta_{10} + \cdots - 647 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 780 \beta_{15} - 2436 \beta_{14} + 2772 \beta_{13} - 3913 \beta_{12} - 4349 \beta_{11} - 8 \beta_{10} + \cdots - 2628 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 224 \beta_{15} - 3456 \beta_{14} + 3712 \beta_{13} - 11711 \beta_{12} - 15115 \beta_{11} + \cdots - 53746 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 22178 \beta_{15} - 12042 \beta_{14} + 13458 \beta_{13} - 119474 \beta_{12} - 134368 \beta_{11} + \cdots - 143494 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 28064 \beta_{15} - 41728 \beta_{14} + 47488 \beta_{13} - 375803 \beta_{12} - 463729 \beta_{11} + \cdots - 1257490 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 386212 \beta_{15} + 505582 \beta_{14} - 571626 \beta_{13} - 2341291 \beta_{12} - 2678569 \beta_{11} + \cdots - 3746060 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 417952 \beta_{15} + 412480 \beta_{14} - 429760 \beta_{13} - 3744144 \beta_{12} - 4480772 \beta_{11} + \cdots - 9849279 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3956312 \beta_{15} + 20437248 \beta_{14} - 22927216 \beta_{13} - 28456131 \beta_{12} + \cdots - 59778920 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 12373920 \beta_{15} + 54112320 \beta_{14} - 59071296 \beta_{13} - 78864459 \beta_{12} + \cdots - 137022858 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 19477810 \beta_{15} + 465046938 \beta_{14} - 519256714 \beta_{13} + 29247152 \beta_{12} + \cdots - 247575882 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1692\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(847\) \(1505\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
845.1
−0.590664 + 1.41421i
−0.590664 − 1.41421i
0.263277 − 1.41421i
0.263277 + 1.41421i
−2.18496 − 1.41421i
−2.18496 + 1.41421i
4.38784 + 1.41421i
4.38784 − 1.41421i
−0.227903 − 1.41421i
−0.227903 + 1.41421i
0.457675 + 1.41421i
0.457675 − 1.41421i
−3.79828 + 1.41421i
−3.79828 − 1.41421i
1.69301 − 1.41421i
1.69301 + 1.41421i
0 0 0 −3.93643 0 1.10235 0 0 0
845.2 0 0 0 −3.93643 0 1.10235 0 0 0
845.3 0 0 0 −2.84774 0 −3.53500 0 0 0
845.4 0 0 0 −2.84774 0 −3.53500 0 0 0
845.5 0 0 0 −0.591052 0 −1.72728 0 0 0
845.6 0 0 0 −0.591052 0 −1.72728 0 0 0
845.7 0 0 0 −0.213445 0 4.15994 0 0 0
845.8 0 0 0 −0.213445 0 4.15994 0 0 0
845.9 0 0 0 0.213445 0 4.15994 0 0 0
845.10 0 0 0 0.213445 0 4.15994 0 0 0
845.11 0 0 0 0.591052 0 −1.72728 0 0 0
845.12 0 0 0 0.591052 0 −1.72728 0 0 0
845.13 0 0 0 2.84774 0 −3.53500 0 0 0
845.14 0 0 0 2.84774 0 −3.53500 0 0 0
845.15 0 0 0 3.93643 0 1.10235 0 0 0
845.16 0 0 0 3.93643 0 1.10235 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 845.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1692.2.h.a ✓ 16
3.b odd 2 1 inner 1692.2.h.a ✓ 16
4.b odd 2 1 6768.2.o.d 16
12.b even 2 1 6768.2.o.d 16
47.b odd 2 1 inner 1692.2.h.a ✓ 16
141.c even 2 1 inner 1692.2.h.a ✓ 16
188.b even 2 1 6768.2.o.d 16
564.f odd 2 1 6768.2.o.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1692.2.h.a ✓ 16 1.a even 1 1 trivial
1692.2.h.a ✓ 16 3.b odd 2 1 inner
1692.2.h.a ✓ 16 47.b odd 2 1 inner
1692.2.h.a ✓ 16 141.c even 2 1 inner
6768.2.o.d 16 4.b odd 2 1
6768.2.o.d 16 12.b even 2 1
6768.2.o.d 16 188.b even 2 1
6768.2.o.d 16 564.f odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1692, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + 135 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 17 T^{2} + \cdots + 28)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 50 T^{6} + \cdots + 12800)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 42 T^{6} + \cdots + 3872)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 82 T^{6} + \cdots + 71824)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 50 T^{6} + 270 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 78 T^{6} + \cdots + 72200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 100 T^{6} + \cdots + 249218)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 68 T^{6} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 118)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 94 T^{6} + \cdots + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 144 T^{6} + \cdots + 456968)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23811286661761 \) Copy content Toggle raw display
$53$ \( (T^{8} + 266 T^{6} + \cdots + 234256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 174 T^{6} + \cdots + 141376)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 10 T^{3} + \cdots - 1880)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 464 T^{6} + \cdots + 6379592)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 332 T^{6} + \cdots + 9048064)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 354 T^{6} + \cdots + 70688)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + 7 T^{2} + \cdots - 40)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 106 T^{6} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 372 T^{6} + \cdots + 27709696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 125 T^{2} + \cdots + 1250)^{4} \) Copy content Toggle raw display
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