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Properties

Label 68.3.c.a
Level $68$
Weight $3$
Character orbit 68.c
Analytic conductor $1.853$
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] = [68,3,Mod(35,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 68.c (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: \(1.85286579765\)
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: \( x^{16} - 2 x^{15} + 5 x^{14} - 8 x^{13} - 16 x^{11} + 64 x^{9} + 256 x^{7} - 1024 x^{5} - 8192 x^{3} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{12} q^{3} + \beta_{10} q^{4} - \beta_{2} q^{5} + (\beta_{11} + \beta_{9}) q^{6} - \beta_{4} q^{7} + (\beta_{13} - \beta_{11}) q^{8} + (\beta_{9} - \beta_{8} + \beta_{5} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{12} q^{3} + \beta_{10} q^{4} - \beta_{2} q^{5} + (\beta_{11} + \beta_{9}) q^{6} - \beta_{4} q^{7} + (\beta_{13} - \beta_{11}) q^{8} + (\beta_{9} - \beta_{8} + \beta_{5} + \cdots - 3) q^{9}+ \cdots + (8 \beta_{15} + 6 \beta_{14} + \cdots - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 6 q^{4} - 6 q^{6} - 2 q^{8} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} - 6 q^{4} - 6 q^{6} - 2 q^{8} - 48 q^{9} + 10 q^{10} + 6 q^{12} - 16 q^{13} - 16 q^{14} + 50 q^{16} + 58 q^{18} - 74 q^{20} + 32 q^{21} + 18 q^{22} + 86 q^{24} + 96 q^{25} - 56 q^{26} - 16 q^{28} + 16 q^{29} - 64 q^{30} - 122 q^{32} - 48 q^{33} - 18 q^{36} + 16 q^{37} + 22 q^{40} + 16 q^{41} + 8 q^{42} + 118 q^{44} - 128 q^{45} + 52 q^{46} - 106 q^{48} - 32 q^{49} - 146 q^{50} - 72 q^{52} - 16 q^{53} + 356 q^{54} + 88 q^{56} - 112 q^{57} - 214 q^{58} + 320 q^{60} + 112 q^{61} - 112 q^{62} + 42 q^{64} - 16 q^{65} + 364 q^{66} - 144 q^{69} + 136 q^{70} - 190 q^{72} + 128 q^{73} + 62 q^{74} - 112 q^{76} + 272 q^{77} - 556 q^{78} - 370 q^{80} + 384 q^{81} - 88 q^{82} + 312 q^{84} + 172 q^{86} - 250 q^{88} - 336 q^{89} - 326 q^{90} + 164 q^{92} + 16 q^{93} - 352 q^{94} + 150 q^{96} - 304 q^{97} + 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 5 x^{14} - 8 x^{13} - 16 x^{11} + 64 x^{9} + 256 x^{7} - 1024 x^{5} - 8192 x^{3} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 22 \nu^{14} + 13 \nu^{13} - 12 \nu^{12} + 64 \nu^{11} + 16 \nu^{10} + 192 \nu^{9} + \cdots - 245760 ) / 32768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 8 \nu^{12} + 16 \nu^{10} - 64 \nu^{8} - 256 \nu^{6} + \cdots + 32768 ) / 16384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - 2 \nu^{13} + \nu^{12} - 4 \nu^{11} - 12 \nu^{10} - 4 \nu^{9} + 32 \nu^{8} + 128 \nu^{7} + \cdots + 24576 ) / 2048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} - 3 \nu^{13} - 24 \nu^{11} - 32 \nu^{10} - 160 \nu^{9} + 256 \nu^{8} + \cdots + 98304 ) / 16384 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{15} + 2 \nu^{14} + 9 \nu^{13} + 36 \nu^{12} - 16 \nu^{11} - 16 \nu^{10} - 448 \nu^{9} + \cdots - 16384 ) / 32768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} - 3 \nu^{13} - 24 \nu^{11} - 32 \nu^{10} - 160 \nu^{9} + 256 \nu^{8} + \cdots + 114688 ) / 16384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} + 34 \nu^{14} - 39 \nu^{13} + 4 \nu^{12} - 64 \nu^{11} - 176 \nu^{10} - 128 \nu^{9} + \cdots + 507904 ) / 32768 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} + 11 \nu^{13} + 20 \nu^{12} + 40 \nu^{11} - 256 \nu^{8} - 640 \nu^{7} + \cdots - 98304 ) / 16384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} - \nu^{13} - 2 \nu^{12} + 16 \nu^{11} + 16 \nu^{10} + 32 \nu^{9} - 64 \nu^{8} + \cdots - 8192 ) / 8192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{15} - 2 \nu^{14} - \nu^{13} + 12 \nu^{12} + 32 \nu^{11} + 112 \nu^{10} - 64 \nu^{9} + \cdots - 49152 ) / 32768 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{15} + 14 \nu^{14} - \nu^{13} + 28 \nu^{12} - 16 \nu^{10} - 128 \nu^{9} - 448 \nu^{8} + \cdots + 114688 ) / 32768 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7 \nu^{15} + 18 \nu^{14} - 11 \nu^{13} + 44 \nu^{12} + 64 \nu^{11} - 112 \nu^{10} - 320 \nu^{9} + \cdots + 81920 ) / 32768 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5 \nu^{15} - 14 \nu^{14} + 9 \nu^{13} - 28 \nu^{12} + 8 \nu^{11} - 32 \nu^{10} + 256 \nu^{9} + \cdots - 180224 ) / 16384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11 \nu^{15} - 10 \nu^{14} - 33 \nu^{13} - 60 \nu^{12} - 32 \nu^{11} + 48 \nu^{10} + 384 \nu^{9} + \cdots + 245760 ) / 32768 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{5} + \beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{14} + 4\beta_{12} + 4\beta_{11} + 2\beta_{7} + 2\beta_{5} + 4\beta_{3} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{14} + 4\beta_{8} - 2\beta_{7} + 4\beta_{6} + 2\beta_{5} - 4\beta_{4} + 8\beta_{3} - 3\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4 \beta_{14} + 8 \beta_{13} + 4 \beta_{12} - 4 \beta_{11} - 16 \beta_{10} - 8 \beta_{9} + 6 \beta_{7} + \cdots + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4 \beta_{14} - 24 \beta_{12} + 40 \beta_{11} - 16 \beta_{10} + 16 \beta_{9} + 12 \beta_{8} + 6 \beta_{7} + \cdots + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 32 \beta_{15} + 28 \beta_{14} - 8 \beta_{13} + 92 \beta_{12} + 36 \beta_{11} + 16 \beta_{10} + \cdots - 146 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4 \beta_{14} - 16 \beta_{13} + 80 \beta_{12} + 32 \beta_{11} - 80 \beta_{10} + 52 \beta_{8} + 46 \beta_{7} + \cdots - 174 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 32 \beta_{15} + 92 \beta_{14} - 120 \beta_{13} + 148 \beta_{12} - 52 \beta_{11} + 16 \beta_{10} + \cdots + 430 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 192 \beta_{15} - 188 \beta_{14} - 96 \beta_{13} - 72 \beta_{12} + 216 \beta_{11} - 208 \beta_{10} + \cdots + 146 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 544 \beta_{15} - 484 \beta_{14} + 792 \beta_{13} - 116 \beta_{12} - 460 \beta_{11} + 656 \beta_{10} + \cdots + 238 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 384 \beta_{15} + 388 \beta_{14} - 560 \beta_{13} + 1504 \beta_{12} + 464 \beta_{11} - 1360 \beta_{10} + \cdots - 3630 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 416 \beta_{15} + 476 \beta_{14} - 2008 \beta_{13} + 5828 \beta_{12} - 1188 \beta_{11} + 272 \beta_{10} + \cdots + 14894 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 960 \beta_{15} - 60 \beta_{14} + 1664 \beta_{13} + 2760 \beta_{12} - 6392 \beta_{11} - 464 \beta_{10} + \cdots - 17390 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 6304 \beta_{15} + 412 \beta_{14} + 8376 \beta_{13} - 11716 \beta_{12} - 11964 \beta_{11} - 7280 \beta_{10} + \cdots + 3694 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
1.98158 − 0.270838i
1.98158 + 0.270838i
1.79176 − 0.888585i
1.79176 + 0.888585i
0.769455 − 1.84606i
0.769455 + 1.84606i
0.485397 − 1.94020i
0.485397 + 1.94020i
0.166129 − 1.99309i
0.166129 + 1.99309i
−0.811096 − 1.82815i
−0.811096 + 1.82815i
−1.43994 − 1.38801i
−1.43994 + 1.38801i
−1.94329 − 0.472891i
−1.94329 + 0.472891i
−1.98158 − 0.270838i 5.07846i 3.85329 + 1.07337i −6.88513 1.37544 − 10.0634i − 5.70996i −7.34489 − 3.17059i −16.7907 13.6434 + 1.86476i
35.2 −1.98158 + 0.270838i − 5.07846i 3.85329 − 1.07337i −6.88513 1.37544 + 10.0634i 5.70996i −7.34489 + 3.17059i −16.7907 13.6434 − 1.86476i
35.3 −1.79176 − 0.888585i − 0.561065i 2.42083 + 3.18427i 3.63253 −0.498554 + 1.00530i − 4.58728i −1.50807 − 7.85657i 8.68521 −6.50863 − 3.22781i
35.4 −1.79176 + 0.888585i 0.561065i 2.42083 − 3.18427i 3.63253 −0.498554 − 1.00530i 4.58728i −1.50807 + 7.85657i 8.68521 −6.50863 + 3.22781i
35.5 −0.769455 − 1.84606i 3.37544i −2.81588 + 2.84092i 0.849076 6.23128 − 2.59725i 11.7244i 7.41120 + 3.01232i −2.39363 −0.653325 − 1.56744i
35.6 −0.769455 + 1.84606i − 3.37544i −2.81588 − 2.84092i 0.849076 6.23128 + 2.59725i − 11.7244i 7.41120 − 3.01232i −2.39363 −0.653325 + 1.56744i
35.7 −0.485397 − 1.94020i − 5.75074i −3.52878 + 1.88354i 6.91078 −11.1576 + 2.79139i 1.37485i 5.36730 + 5.93229i −24.0710 −3.35447 − 13.4083i
35.8 −0.485397 + 1.94020i 5.75074i −3.52878 − 1.88354i 6.91078 −11.1576 − 2.79139i − 1.37485i 5.36730 − 5.93229i −24.0710 −3.35447 + 13.4083i
35.9 −0.166129 − 1.99309i − 0.954378i −3.94480 + 0.662220i −8.73270 −1.90216 + 0.158550i − 9.85804i 1.97521 + 7.75233i 8.08916 1.45076 + 17.4050i
35.10 −0.166129 + 1.99309i 0.954378i −3.94480 − 0.662220i −8.73270 −1.90216 − 0.158550i 9.85804i 1.97521 − 7.75233i 8.08916 1.45076 − 17.4050i
35.11 0.811096 − 1.82815i 2.38010i −2.68425 − 2.96561i 7.55480 4.35118 + 1.93049i − 8.84753i −7.59875 + 2.50181i 3.33511 6.12767 − 13.8113i
35.12 0.811096 + 1.82815i − 2.38010i −2.68425 + 2.96561i 7.55480 4.35118 − 1.93049i 8.84753i −7.59875 − 2.50181i 3.33511 6.12767 + 13.8113i
35.13 1.43994 − 1.38801i − 2.26952i 0.146832 − 3.99730i −1.51874 −3.15013 − 3.26797i 6.17390i −5.33689 − 5.95967i 3.84927 −2.18689 + 2.10804i
35.14 1.43994 + 1.38801i 2.26952i 0.146832 + 3.99730i −1.51874 −3.15013 + 3.26797i − 6.17390i −5.33689 + 5.95967i 3.84927 −2.18689 − 2.10804i
35.15 1.94329 − 0.472891i 3.70181i 3.55275 − 1.83793i −1.81060 1.75055 + 7.19368i 1.19314i 6.03488 − 5.25169i −4.70338 −3.51853 + 0.856218i
35.16 1.94329 + 0.472891i − 3.70181i 3.55275 + 1.83793i −1.81060 1.75055 − 7.19368i − 1.19314i 6.03488 + 5.25169i −4.70338 −3.51853 − 0.856218i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.3.c.a ✓ 16
3.b odd 2 1 612.3.f.a 16
4.b odd 2 1 inner 68.3.c.a ✓ 16
8.b even 2 1 1088.3.d.d 16
8.d odd 2 1 1088.3.d.d 16
12.b even 2 1 612.3.f.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.3.c.a ✓ 16 1.a even 1 1 trivial
68.3.c.a ✓ 16 4.b odd 2 1 inner
612.3.f.a 16 3.b odd 2 1
612.3.f.a 16 12.b even 2 1
1088.3.d.d 16 8.b even 2 1
1088.3.d.d 16 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(68, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{16} + 96 T^{14} + \cdots + 1114112 \) Copy content Toggle raw display
$5$ \( (T^{8} - 124 T^{6} + \cdots + 26624)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 73585983488 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 80797915676672 \) Copy content Toggle raw display
$13$ \( (T^{8} + 8 T^{7} + \cdots + 7402496)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 17)^{8} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 51\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + \cdots - 1219608576)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots - 1682997248)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 8 T^{7} + \cdots - 421375744)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 25\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 13718565354752)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 34\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 114674126139392)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 18\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 60\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 11640921052928)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 18\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 38\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 325689299096576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 418057423028992)^{2} \) Copy content Toggle raw display
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