↑

Properties

Label 68.4.h.a
Level $68$
Weight $4$
Character orbit 68.h
Analytic conductor $4.012$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [68,4,Mod(9,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 68.h (of order \(8\), degree \(4\), 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: \(4.01212988039\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 386 x^{18} + 51207 x^{16} + 2935444 x^{14} + 86104695 x^{12} + 1377178386 x^{10} + \cdots + 4724087824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{14} + \beta_{10} - \beta_{9} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{18} + \beta_{15} - 12 \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + (\beta_{14} + \beta_{10} - \beta_{9} + \cdots - 1) q^{5}+ \cdots + ( - 7 \beta_{19} + 9 \beta_{18} + \cdots - 792) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 16 q^{5} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 16 q^{5} + 20 q^{9} - 156 q^{11} + 248 q^{15} + 72 q^{17} + 40 q^{19} - 272 q^{23} + 144 q^{25} + 660 q^{27} - 192 q^{29} - 672 q^{31} - 640 q^{33} + 64 q^{35} - 592 q^{37} - 1312 q^{39} - 304 q^{41} + 1260 q^{43} + 2312 q^{45} - 472 q^{49} + 1744 q^{51} + 2120 q^{53} - 992 q^{57} + 828 q^{59} - 24 q^{61} - 3512 q^{63} - 2808 q^{65} - 376 q^{67} - 80 q^{69} - 1064 q^{71} - 4016 q^{73} - 1788 q^{75} + 2992 q^{77} + 2744 q^{79} + 2636 q^{83} + 9256 q^{85} + 7832 q^{87} - 1696 q^{91} + 960 q^{93} + 3280 q^{95} - 3636 q^{97} - 16384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 386 x^{18} + 51207 x^{16} + 2935444 x^{14} + 86104695 x^{12} + 1377178386 x^{10} + \cdots + 4724087824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 66\!\cdots\!57 \nu^{18} + \cdots - 42\!\cdots\!48 ) / 22\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18\!\cdots\!67 \nu^{19} + \cdots + 30\!\cdots\!28 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18\!\cdots\!67 \nu^{19} + \cdots - 30\!\cdots\!28 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 73\!\cdots\!37 \nu^{19} + \cdots + 78\!\cdots\!40 ) / 85\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 73\!\cdots\!37 \nu^{19} + \cdots - 36\!\cdots\!20 ) / 85\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 73\!\cdots\!37 \nu^{19} + \cdots + 36\!\cdots\!20 ) / 85\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73\!\cdots\!37 \nu^{19} + \cdots - 78\!\cdots\!40 ) / 85\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 75425980514155 \nu^{19} + \cdots - 23\!\cdots\!52 \nu ) / 72\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28\!\cdots\!05 \nu^{19} + \cdots + 24\!\cdots\!92 ) / 14\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 28\!\cdots\!05 \nu^{19} + \cdots - 24\!\cdots\!92 ) / 14\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26\!\cdots\!43 \nu^{19} + \cdots - 41\!\cdots\!88 \nu ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 24\!\cdots\!72 \nu^{19} + \cdots + 14\!\cdots\!76 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 24\!\cdots\!72 \nu^{19} + \cdots - 14\!\cdots\!76 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41\!\cdots\!85 \nu^{19} + \cdots + 12\!\cdots\!52 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 41\!\cdots\!85 \nu^{19} + \cdots + 12\!\cdots\!52 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 91\!\cdots\!78 \nu^{19} + \cdots - 46\!\cdots\!12 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 91\!\cdots\!78 \nu^{19} + \cdots - 46\!\cdots\!12 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 26\!\cdots\!29 \nu^{19} + \cdots - 21\!\cdots\!60 ) / 30\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 26\!\cdots\!29 \nu^{19} + \cdots - 21\!\cdots\!60 ) / 30\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - \beta_{18} + \beta_{15} + \beta_{14} + 37 \beta_{10} - 37 \beta_{9} + 3 \beta_{7} + \cdots - 76 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{19} + 3 \beta_{18} + 3 \beta_{17} - 3 \beta_{16} + 10 \beta_{15} - 10 \beta_{14} + \cdots + 27 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 183 \beta_{19} + 183 \beta_{18} - 18 \beta_{17} - 18 \beta_{16} - 231 \beta_{15} - 231 \beta_{14} + \cdots + 9032 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 817 \beta_{19} - 817 \beta_{18} - 525 \beta_{17} + 525 \beta_{16} - 1920 \beta_{15} + \cdots - 4735 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 29047 \beta_{19} - 29047 \beta_{18} + 3240 \beta_{17} + 3240 \beta_{16} + 44227 \beta_{15} + \cdots - 1345720 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 178437 \beta_{19} + 178437 \beta_{18} + 91545 \beta_{17} - 91545 \beta_{16} + 323614 \beta_{15} + \cdots + 784125 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4611067 \beta_{19} + 4611067 \beta_{18} - 549906 \beta_{17} - 549906 \beta_{16} - 8069731 \beta_{15} + \cdots + 211386168 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 35435733 \beta_{19} - 35435733 \beta_{18} - 16016109 \beta_{17} + 16016109 \beta_{16} + \cdots - 130732575 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 744231207 \beta_{19} - 744231207 \beta_{18} + 93607440 \beta_{17} + 93607440 \beta_{16} + \cdots - 34029614240 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6685937981 \beta_{19} + 6685937981 \beta_{18} + 2794092669 \beta_{17} - 2794092669 \beta_{16} + \cdots + 21990700169 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 122048526995 \beta_{19} + 122048526995 \beta_{18} - 15997873434 \beta_{17} - 15997873434 \beta_{16} + \cdots + 5573913424880 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1223765115749 \beta_{19} - 1223765115749 \beta_{18} - 485940245145 \beta_{17} + \cdots - 3724640172803 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 20275361190559 \beta_{19} - 20275361190559 \beta_{18} + 2741571129624 \beta_{17} + \cdots - 925220823807416 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 219751261461613 \beta_{19} + 219751261461613 \beta_{18} + 84315128513913 \beta_{17} + \cdots + 634075012355485 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 34\!\cdots\!95 \beta_{19} + \cdots + 15\!\cdots\!16 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 38\!\cdots\!85 \beta_{19} + \cdots - 10\!\cdots\!19 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 57\!\cdots\!99 \beta_{19} + \cdots - 26\!\cdots\!48 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 68\!\cdots\!41 \beta_{19} + \cdots + 18\!\cdots\!13 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(-\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
− 4.89250i
− 1.72030i
− 0.687153i
2.64433i
5.06984i
− 13.1344i
− 3.88661i
− 0.399124i
3.92723i
11.0786i
13.1344i
3.88661i
0.399124i
− 3.92723i
− 11.0786i
4.89250i
1.72030i
0.687153i
− 2.64433i
− 5.06984i
0 −8.35203 − 3.45952i 0 1.10263 − 2.66200i 0 12.6032 + 30.4268i 0 38.6962 + 38.6962i 0
9.2 0 −2.93673 − 1.21643i 0 −3.25287 + 7.85313i 0 −11.4951 − 27.7517i 0 −11.9472 − 11.9472i 0
9.3 0 −1.17304 − 0.485891i 0 4.23880 − 10.2334i 0 −1.42338 − 3.43635i 0 −17.9519 − 17.9519i 0
9.4 0 4.51415 + 1.86982i 0 −5.52643 + 13.3420i 0 6.22170 + 15.0205i 0 −2.21055 − 2.21055i 0
9.5 0 8.65475 + 3.58492i 0 6.50893 − 15.7140i 0 −4.49217 − 10.8451i 0 42.9613 + 42.9613i 0
25.1 0 −3.84696 + 9.28739i 0 −11.2802 − 4.67240i 0 11.6591 − 4.82935i 0 −52.3647 − 52.3647i 0
25.2 0 −1.13836 + 2.74825i 0 10.4791 + 4.34058i 0 −21.3631 + 8.84890i 0 12.8349 + 12.8349i 0
25.3 0 −0.116901 + 0.282223i 0 1.12545 + 0.466177i 0 26.0449 − 10.7882i 0 19.0259 + 19.0259i 0
25.4 0 1.15026 − 2.77697i 0 −19.5194 − 8.08522i 0 −21.8743 + 9.06062i 0 12.7034 + 12.7034i 0
25.5 0 3.24486 − 7.83379i 0 8.12399 + 3.36507i 0 4.11921 − 1.70623i 0 −31.7472 − 31.7472i 0
49.1 0 −3.84696 − 9.28739i 0 −11.2802 + 4.67240i 0 11.6591 + 4.82935i 0 −52.3647 + 52.3647i 0
49.2 0 −1.13836 − 2.74825i 0 10.4791 − 4.34058i 0 −21.3631 − 8.84890i 0 12.8349 − 12.8349i 0
49.3 0 −0.116901 − 0.282223i 0 1.12545 − 0.466177i 0 26.0449 + 10.7882i 0 19.0259 − 19.0259i 0
49.4 0 1.15026 + 2.77697i 0 −19.5194 + 8.08522i 0 −21.8743 − 9.06062i 0 12.7034 − 12.7034i 0
49.5 0 3.24486 + 7.83379i 0 8.12399 − 3.36507i 0 4.11921 + 1.70623i 0 −31.7472 + 31.7472i 0
53.1 0 −8.35203 + 3.45952i 0 1.10263 + 2.66200i 0 12.6032 − 30.4268i 0 38.6962 − 38.6962i 0
53.2 0 −2.93673 + 1.21643i 0 −3.25287 − 7.85313i 0 −11.4951 + 27.7517i 0 −11.9472 + 11.9472i 0
53.3 0 −1.17304 + 0.485891i 0 4.23880 + 10.2334i 0 −1.42338 + 3.43635i 0 −17.9519 + 17.9519i 0
53.4 0 4.51415 − 1.86982i 0 −5.52643 − 13.3420i 0 6.22170 − 15.0205i 0 −2.21055 + 2.21055i 0
53.5 0 8.65475 − 3.58492i 0 6.50893 + 15.7140i 0 −4.49217 + 10.8451i 0 42.9613 − 42.9613i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.4.h.a ✓ 20
3.b odd 2 1 612.4.w.a 20
17.d even 8 1 inner 68.4.h.a ✓ 20
17.e odd 16 2 1156.4.a.l 20
17.e odd 16 2 1156.4.b.g 20
51.g odd 8 1 612.4.w.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.4.h.a ✓ 20 1.a even 1 1 trivial
68.4.h.a ✓ 20 17.d even 8 1 inner
612.4.w.a 20 3.b odd 2 1
612.4.w.a 20 51.g odd 8 1
1156.4.a.l 20 17.e odd 16 2
1156.4.b.g 20 17.e odd 16 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(68, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 151170810368 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 43\!\cdots\!48 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 37\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 81\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 34\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 40\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 68\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 50\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 49\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 92\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
show more
show less