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Properties

Label 68.6.e.a
Level $68$
Weight $6$
Character orbit 68.e
Analytic conductor $10.906$
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] = [68,6,Mod(13,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.13");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 68.e (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: \(10.9060997473\)
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: \( x^{16} + 1473 x^{14} + 868792 x^{12} + 259909217 x^{10} + 41026119309 x^{8} + 3204542941640 x^{6} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{21} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1 + 1) q^{3} + (\beta_{6} - 3 \beta_{5} - 3) q^{5} + ( - \beta_{10} + \beta_{7} + 8 \beta_{5} + \cdots - 8) q^{7}+ \cdots + ( - \beta_{8} + 127 \beta_{5} + \cdots + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1 + 1) q^{3} + (\beta_{6} - 3 \beta_{5} - 3) q^{5} + ( - \beta_{10} + \beta_{7} + 8 \beta_{5} + \cdots - 8) q^{7}+ \cdots + (90 \beta_{15} + 204 \beta_{12} + \cdots + 14440) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 22 q^{3} - 44 q^{5} - 118 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 22 q^{3} - 44 q^{5} - 118 q^{7} + 414 q^{11} + 252 q^{13} - 2784 q^{17} + 5068 q^{21} + 1102 q^{23} - 14612 q^{27} + 9672 q^{29} + 2162 q^{31} - 20108 q^{33} + 59228 q^{35} - 26148 q^{37} - 42488 q^{39} + 18032 q^{41} + 40544 q^{45} - 12144 q^{47} - 57046 q^{51} + 17284 q^{55} + 31700 q^{57} - 9724 q^{61} - 20022 q^{63} - 34212 q^{65} + 47080 q^{67} - 17796 q^{69} + 117038 q^{71} + 37912 q^{73} - 82670 q^{75} + 129822 q^{79} - 197196 q^{81} + 113848 q^{85} - 456228 q^{89} - 263080 q^{91} + 393180 q^{95} + 209900 q^{97} + 222358 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1473 x^{14} + 868792 x^{12} + 259909217 x^{10} + 41026119309 x^{8} + 3204542941640 x^{6} + \cdots + 12\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 444352193922880 \nu^{14} + \cdots + 61\!\cdots\!28 ) / 51\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 444352193922880 \nu^{14} + \cdots - 61\!\cdots\!28 ) / 51\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 444352193922880 \nu^{14} + \cdots + 96\!\cdots\!72 ) / 25\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!42 \nu^{14} + \cdots + 15\!\cdots\!88 ) / 23\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!37 \nu^{15} + \cdots + 18\!\cdots\!56 \nu ) / 18\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!99 \nu^{15} + \cdots - 69\!\cdots\!96 ) / 25\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\!\cdots\!99 \nu^{15} + \cdots - 69\!\cdots\!96 ) / 25\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 81\!\cdots\!77 \nu^{15} + \cdots + 17\!\cdots\!04 \nu ) / 47\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!42 \nu^{15} + \cdots + 12\!\cdots\!88 \nu ) / 15\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!07 \nu^{15} + \cdots - 58\!\cdots\!40 ) / 84\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!07 \nu^{15} + \cdots - 58\!\cdots\!40 ) / 84\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 39\!\cdots\!23 \nu^{15} + \cdots + 18\!\cdots\!92 ) / 63\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 39\!\cdots\!23 \nu^{15} + \cdots + 18\!\cdots\!92 ) / 63\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 54\!\cdots\!91 \nu^{15} + \cdots + 87\!\cdots\!44 ) / 63\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 54\!\cdots\!91 \nu^{15} + \cdots - 87\!\cdots\!44 ) / 63\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 - 368 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{13} + 3 \beta_{12} + 8 \beta_{11} - 8 \beta_{10} - 4 \beta_{9} + 10 \beta_{8} + \cdots - 587 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{15} + 11 \beta_{14} + 17 \beta_{13} + 17 \beta_{12} + 93 \beta_{11} + 93 \beta_{10} + \cdots + 215009 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 27 \beta_{15} - 27 \beta_{14} + 789 \beta_{13} - 789 \beta_{12} - 1389 \beta_{11} + \cdots + 92621 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5532 \beta_{15} - 5532 \beta_{14} - 3057 \beta_{13} - 3057 \beta_{12} - 48676 \beta_{11} + \cdots - 67673908 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 52307 \beta_{15} + 52307 \beta_{14} - 655435 \beta_{13} + 655435 \beta_{12} + 722579 \beta_{11} + \cdots - 60211280 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1170973 \beta_{15} + 1170973 \beta_{14} - 497276 \beta_{13} - 497276 \beta_{12} + 10062205 \beta_{11} + \cdots + 10973436661 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 31538542 \beta_{15} - 31538542 \beta_{14} + 254562317 \beta_{13} - 254562317 \beta_{12} + \cdots + 19937984659 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 952069499 \beta_{15} - 952069499 \beta_{14} + 1193810863 \beta_{13} + 1193810863 \beta_{12} + \cdots - 7252643368889 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7835090447 \beta_{15} + 7835090447 \beta_{14} - 48415829035 \beta_{13} + 48415829035 \beta_{12} + \cdots - 3347653185035 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 382436790688 \beta_{15} + 382436790688 \beta_{14} - 724624796987 \beta_{13} - 724624796987 \beta_{12} + \cdots + 24\!\cdots\!60 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 7049953624819 \beta_{15} - 7049953624819 \beta_{14} + 36615179043599 \beta_{13} + \cdots + 22\!\cdots\!48 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 76341312947575 \beta_{15} - 76341312947575 \beta_{14} + 180333948855464 \beta_{13} + \cdots - 41\!\cdots\!63 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 29\!\cdots\!98 \beta_{15} + \cdots - 78\!\cdots\!79 \beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
− 17.1376i
− 16.7633i
− 11.9931i
1.10346i
4.75153i
5.68063i
18.1432i
19.2152i
17.1376i
16.7633i
11.9931i
− 1.10346i
− 4.75153i
− 5.68063i
− 18.1432i
− 19.2152i
0 −16.1376 − 16.1376i 0 −61.8583 − 61.8583i 0 −125.065 + 125.065i 0 277.844i 0
13.2 0 −15.7633 − 15.7633i 0 58.2322 + 58.2322i 0 5.21158 − 5.21158i 0 253.960i 0
13.3 0 −10.9931 − 10.9931i 0 −13.9092 − 13.9092i 0 92.3225 − 92.3225i 0 − 1.30311i 0
13.4 0 2.10346 + 2.10346i 0 −14.4897 − 14.4897i 0 −21.7489 + 21.7489i 0 − 234.151i 0
13.5 0 5.75153 + 5.75153i 0 50.9600 + 50.9600i 0 −125.564 + 125.564i 0 − 176.840i 0
13.6 0 6.68063 + 6.68063i 0 −20.9824 − 20.9824i 0 78.1871 − 78.1871i 0 − 153.738i 0
13.7 0 19.1432 + 19.1432i 0 47.8722 + 47.8722i 0 158.427 − 158.427i 0 489.921i 0
13.8 0 20.2152 + 20.2152i 0 −67.8249 − 67.8249i 0 −120.770 + 120.770i 0 574.306i 0
21.1 0 −16.1376 + 16.1376i 0 −61.8583 + 61.8583i 0 −125.065 − 125.065i 0 − 277.844i 0
21.2 0 −15.7633 + 15.7633i 0 58.2322 − 58.2322i 0 5.21158 + 5.21158i 0 − 253.960i 0
21.3 0 −10.9931 + 10.9931i 0 −13.9092 + 13.9092i 0 92.3225 + 92.3225i 0 1.30311i 0
21.4 0 2.10346 − 2.10346i 0 −14.4897 + 14.4897i 0 −21.7489 − 21.7489i 0 234.151i 0
21.5 0 5.75153 − 5.75153i 0 50.9600 − 50.9600i 0 −125.564 − 125.564i 0 176.840i 0
21.6 0 6.68063 − 6.68063i 0 −20.9824 + 20.9824i 0 78.1871 + 78.1871i 0 153.738i 0
21.7 0 19.1432 − 19.1432i 0 47.8722 − 47.8722i 0 158.427 + 158.427i 0 − 489.921i 0
21.8 0 20.2152 − 20.2152i 0 −67.8249 + 67.8249i 0 −120.770 − 120.770i 0 − 574.306i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.6.e.a ✓ 16
3.b odd 2 1 612.6.k.a 16
4.b odd 2 1 272.6.o.d 16
17.c even 4 1 inner 68.6.e.a ✓ 16
51.f odd 4 1 612.6.k.a 16
68.f odd 4 1 272.6.o.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.6.e.a ✓ 16 1.a even 1 1 trivial
68.6.e.a ✓ 16 17.c even 4 1 inner
272.6.o.d 16 4.b odd 2 1
272.6.o.d 16 68.f odd 4 1
612.6.k.a 16 3.b odd 2 1
612.6.k.a 16 51.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(68, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 62\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 67\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 65\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 73\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
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