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Properties

Label 68.3.d.c
Level $68$
Weight $3$
Character orbit 68.d
Analytic conductor $1.853$
Analytic rank $0$
Dimension $12$
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] = [68,3,Mod(67,68)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 68.d (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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 17x^{10} + 102x^{8} + 154x^{6} - 308x^{4} - 460x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{7} q^{3} + ( - \beta_{3} - 1) q^{4} - \beta_{6} q^{5} - \beta_{10} q^{6} + ( - \beta_{9} + \beta_{7}) q^{7} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{8} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{7} q^{3} + ( - \beta_{3} - 1) q^{4} - \beta_{6} q^{5} - \beta_{10} q^{6} + ( - \beta_{9} + \beta_{7}) q^{7} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{8} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{9} + (\beta_{11} - \beta_{9} - \beta_{8}) q^{10} + ( - \beta_{11} - \beta_{9}) q^{11} + (\beta_{11} + \beta_{8} + \cdots + \beta_{6}) q^{12}+ \cdots + ( - 4 \beta_{11} + 6 \beta_{9} - \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 14 q^{4} - 2 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 14 q^{4} - 2 q^{8} - 12 q^{9} + 8 q^{13} - 62 q^{16} + 44 q^{17} + 50 q^{18} + 64 q^{21} - 84 q^{25} - 12 q^{26} - 2 q^{32} - 32 q^{33} + 78 q^{34} - 98 q^{36} + 112 q^{38} - 96 q^{42} - 172 q^{49} + 110 q^{50} + 268 q^{52} - 200 q^{53} + 224 q^{60} - 254 q^{64} - 288 q^{66} + 2 q^{68} + 320 q^{69} + 160 q^{70} - 78 q^{72} + 272 q^{76} + 448 q^{77} - 532 q^{81} - 128 q^{84} + 64 q^{85} - 496 q^{86} + 328 q^{89} + 128 q^{93} - 544 q^{94} - 478 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 17x^{10} + 102x^{8} + 154x^{6} - 308x^{4} - 460x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 621\nu^{10} + 1073\nu^{8} - 65678\nu^{6} - 480824\nu^{4} - 301192\nu^{2} + 1017912 ) / 546446 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2169\nu^{10} - 31466\nu^{8} - 123021\nu^{6} + 293486\nu^{4} + 1653872\nu^{2} - 739931 ) / 273223 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2543\nu^{10} + 35632\nu^{8} + 149902\nu^{6} - 42778\nu^{4} - 263684\nu^{2} + 1160265 ) / 273223 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5931\nu^{10} - 89443\nu^{8} - 457700\nu^{6} - 460428\nu^{4} + 849208\nu^{2} + 898262 ) / 546446 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7621\nu^{10} + 125801\nu^{8} + 773492\nu^{6} + 1552396\nu^{4} + 692896\nu^{2} - 778494 ) / 546446 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7441\nu^{11} - 141329\nu^{9} - 1228102\nu^{7} - 5215946\nu^{5} - 8398764\nu^{3} + 11503532\nu ) / 4371568 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8909\nu^{11} + 87549\nu^{9} + 9870\nu^{7} - 2379486\nu^{5} + 27724\nu^{3} + 15843908\nu ) / 4371568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -8909\nu^{11} - 87549\nu^{9} - 9870\nu^{7} + 2379486\nu^{5} - 27724\nu^{3} + 1642364\nu ) / 4371568 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23897\nu^{11} + 324633\nu^{9} + 1317974\nu^{7} - 1239174\nu^{5} - 4749620\nu^{3} + 9176948\nu ) / 4371568 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -14367\nu^{11} - 236011\nu^{9} - 1373782\nu^{7} - 2167582\nu^{5} + 1197476\nu^{3} + 34628\nu ) / 1092892 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8833\nu^{11} + 130535\nu^{9} + 626830\nu^{7} + 199522\nu^{5} - 2268150\nu^{3} - 985548\nu ) / 546446 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 3\beta_{4} + 2\beta_{3} - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{11} - 2\beta_{10} + 4\beta_{9} - 5\beta_{8} - 4\beta_{7} + \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} - 10\beta_{4} - \beta_{3} + 3\beta_{2} - 5\beta _1 + 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{11} + 6\beta_{10} - 54\beta_{9} + 17\beta_{8} + 44\beta_{7} - 7\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{5} + 27\beta_{4} - 17\beta_{3} - 11\beta_{2} + 38\beta _1 - 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -111\beta_{11} + 18\beta_{10} + 488\beta_{9} + 15\beta_{8} - 320\beta_{7} - 27\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -162\beta_{5} - 77\beta_{4} + 252\beta_{3} + 55\beta_{2} - 427\beta _1 - 230 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 347\beta_{11} - 710\beta_{10} - 3516\beta_{9} - 1087\beta_{8} + 1716\beta_{7} + 843\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 898\beta_{5} - 519\beta_{4} - 2378\beta_{3} - 97\beta_{2} + 3701\beta _1 + 5074 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1797\beta_{11} + 8566\beta_{10} + 19576\beta_{9} + 13443\beta_{8} - 4560\beta_{7} - 10127\beta_{6} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.639662 − 2.66374i
0.639662 + 2.66374i
−0.639662 + 2.66374i
0.639662 − 2.66374i
−1.13191 + 0.280634i
1.13191 − 0.280634i
−1.13191 − 0.280634i
1.13191 + 0.280634i
0.244153 + 1.75378i
−0.244153 − 1.75378i
0.244153 − 1.75378i
−0.244153 + 1.75378i
−1.57067 − 1.23814i −0.995325 0.933996 + 3.88943i − 6.78371i 1.56332 + 1.23236i −9.60090 3.34868 − 7.26542i −8.00933 −8.39921 + 10.6549i
67.2 −1.57067 − 1.23814i 0.995325 0.933996 + 3.88943i 6.78371i −1.56332 − 1.23236i 9.60090 3.34868 − 7.26542i −8.00933 8.39921 − 10.6549i
67.3 −1.57067 + 1.23814i −0.995325 0.933996 − 3.88943i 6.78371i 1.56332 − 1.23236i −9.60090 3.34868 + 7.26542i −8.00933 −8.39921 − 10.6549i
67.4 −1.57067 + 1.23814i 0.995325 0.933996 − 3.88943i − 6.78371i −1.56332 + 1.23236i 9.60090 3.34868 + 7.26542i −8.00933 8.39921 + 10.6549i
67.5 −0.242431 − 1.98525i −3.64418 −3.88245 + 0.962573i 4.63032i 0.883462 + 7.23462i −3.07874 2.85218 + 7.47429i 4.28005 9.19236 − 1.12253i
67.6 −0.242431 − 1.98525i 3.64418 −3.88245 + 0.962573i − 4.63032i −0.883462 − 7.23462i 3.07874 2.85218 + 7.47429i 4.28005 −9.19236 + 1.12253i
67.7 −0.242431 + 1.98525i −3.64418 −3.88245 − 0.962573i − 4.63032i 0.883462 − 7.23462i −3.07874 2.85218 − 7.47429i 4.28005 9.19236 + 1.12253i
67.8 −0.242431 + 1.98525i 3.64418 −3.88245 − 0.962573i 4.63032i −0.883462 + 7.23462i 3.07874 2.85218 − 7.47429i 4.28005 −9.19236 − 1.12253i
67.9 1.31310 − 1.50857i −3.11918 −0.551542 − 3.96179i − 5.34242i −4.09579 + 4.70549i 1.53101 −6.70085 − 4.37019i 0.729282 −8.05939 − 7.01512i
67.10 1.31310 − 1.50857i 3.11918 −0.551542 − 3.96179i 5.34242i 4.09579 − 4.70549i −1.53101 −6.70085 − 4.37019i 0.729282 8.05939 + 7.01512i
67.11 1.31310 + 1.50857i −3.11918 −0.551542 + 3.96179i 5.34242i −4.09579 − 4.70549i 1.53101 −6.70085 + 4.37019i 0.729282 −8.05939 + 7.01512i
67.12 1.31310 + 1.50857i 3.11918 −0.551542 + 3.96179i − 5.34242i 4.09579 + 4.70549i −1.53101 −6.70085 + 4.37019i 0.729282 8.05939 − 7.01512i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.b even 2 1 inner
68.d 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.d.c ✓ 12
3.b odd 2 1 612.3.e.g 12
4.b odd 2 1 inner 68.3.d.c ✓ 12
8.b even 2 1 1088.3.g.k 12
8.d odd 2 1 1088.3.g.k 12
12.b even 2 1 612.3.e.g 12
17.b even 2 1 inner 68.3.d.c ✓ 12
51.c odd 2 1 612.3.e.g 12
68.d odd 2 1 inner 68.3.d.c ✓ 12
136.e odd 2 1 1088.3.g.k 12
136.h even 2 1 1088.3.g.k 12
204.h even 2 1 612.3.e.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.3.d.c ✓ 12 1.a even 1 1 trivial
68.3.d.c ✓ 12 4.b odd 2 1 inner
68.3.d.c ✓ 12 17.b even 2 1 inner
68.3.d.c ✓ 12 68.d odd 2 1 inner
612.3.e.g 12 3.b odd 2 1
612.3.e.g 12 12.b even 2 1
612.3.e.g 12 51.c odd 2 1
612.3.e.g 12 204.h even 2 1
1088.3.g.k 12 8.b even 2 1
1088.3.g.k 12 8.d odd 2 1
1088.3.g.k 12 136.e odd 2 1
1088.3.g.k 12 136.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 24T_{3}^{4} + 152T_{3}^{2} - 128 \) acting on \(S_{3}^{\mathrm{new}}(68, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + 4 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 24 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 96 T^{4} + \cdots + 28160)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 104 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 280 T^{4} + \cdots - 476288)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 2 T^{2} + \cdots + 1208)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} - 22 T^{5} + \cdots + 24137569)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 768 T^{4} + \cdots + 3604480)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 2408 T^{4} + \cdots - 1438208)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 4512 T^{4} + \cdots + 2160688640)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 3336 T^{4} + \cdots - 1145864192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 4960 T^{4} + \cdots + 1451056640)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 4480 T^{4} + \cdots + 1126400000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 7456 T^{4} + \cdots + 3136798720)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 9312 T^{4} + \cdots + 3690987520)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 50 T^{2} + \cdots - 206504)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + 5920 T^{4} + \cdots + 563200000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 11680 T^{4} + \cdots + 17600000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 13952 T^{4} + \cdots + 12547194880)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 16584 T^{4} + \cdots - 26416890368)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 3200 T^{4} + \cdots + 720896000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 4072 T^{4} + \cdots - 1602232832)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 32032 T^{4} + \cdots + 329832448000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 82 T^{2} + \cdots + 50392)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + 21888 T^{4} + \cdots + 8090255360)^{2} \) Copy content Toggle raw display
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