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Properties

Label 64.6.e.a
Level $64$
Weight $6$
Character orbit 64.e
Analytic conductor $10.265$
Analytic rank $0$
Dimension $18$
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] = [64,6,Mod(17,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 64.e (of order \(4\), degree \(2\), not 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: \(10.2645644680\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{16} - 30 x^{15} - 42 x^{14} - 344 x^{13} + 2904 x^{12} + 5344 x^{11} + 16576 x^{10} + \cdots + 68719476736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{72} \)
Twist minimal: no (minimal twist has level 16)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} - \beta_{5} q^{5} + (\beta_{4} - 11 \beta_1) q^{7} + (\beta_{6} - \beta_{3} + \cdots + 63 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} - \beta_{5} q^{5} + (\beta_{4} - 11 \beta_1) q^{7} + (\beta_{6} - \beta_{3} + \cdots + 63 \beta_1) q^{9}+ \cdots + ( - 16 \beta_{17} - 110 \beta_{15} + \cdots + 16471) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{3} - 2 q^{5} + 606 q^{11} - 2 q^{13} - 1796 q^{15} - 4 q^{17} + 2362 q^{19} + 484 q^{21} - 4216 q^{27} + 4070 q^{29} + 11536 q^{31} - 4 q^{33} - 8636 q^{35} - 10650 q^{37} + 15382 q^{43} + 5762 q^{45} - 44176 q^{47} - 14410 q^{49} + 2748 q^{51} + 24726 q^{53} + 29734 q^{59} - 48082 q^{61} + 12156 q^{63} + 27684 q^{65} + 75210 q^{67} + 22804 q^{69} - 154726 q^{75} + 41060 q^{77} + 52864 q^{79} - 13126 q^{81} - 227838 q^{83} - 138652 q^{85} + 231164 q^{91} + 180688 q^{93} + 250380 q^{95} - 4 q^{97} + 296770 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{16} - 30 x^{15} - 42 x^{14} - 344 x^{13} + 2904 x^{12} + 5344 x^{11} + 16576 x^{10} + \cdots + 68719476736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13201861 \nu^{17} + 91238416 \nu^{16} + 259875289 \nu^{15} - 1089436474 \nu^{14} + \cdots + 37\!\cdots\!84 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 181944863 \nu^{17} + 3770692432 \nu^{16} + 1392584293 \nu^{15} - 76307038258 \nu^{14} + \cdots + 13\!\cdots\!48 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 68229307 \nu^{17} - 778072432 \nu^{16} - 1243260583 \nu^{15} + 21646856518 \nu^{14} + \cdots - 56\!\cdots\!08 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20028053 \nu^{17} - 75584528 \nu^{16} - 255116777 \nu^{15} + 65939162 \nu^{14} + \cdots - 13\!\cdots\!52 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 71113927 \nu^{17} + 1558513488 \nu^{16} - 855026403 \nu^{15} - 33026286562 \nu^{14} + \cdots + 62\!\cdots\!32 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 784516657 \nu^{17} - 1545631312 \nu^{16} - 15933947893 \nu^{15} - 6288779822 \nu^{14} + \cdots - 10\!\cdots\!88 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3401231 \nu^{17} + 57739504 \nu^{16} - 342078389 \nu^{15} - 1266122478 \nu^{14} + \cdots + 12\!\cdots\!44 ) / 901447063437312 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 339823113 \nu^{17} - 513251408 \nu^{16} - 8617678637 \nu^{15} - 3874526078 \nu^{14} + \cdots - 35\!\cdots\!52 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 318373 \nu^{17} - 423232 \nu^{16} + 5937593 \nu^{15} + 37697430 \nu^{14} + \cdots - 15\!\cdots\!56 ) / 21463025319936 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3677220871 \nu^{17} + 178650695504 \nu^{16} + 180631925981 \nu^{15} - 3295448180066 \nu^{14} + \cdots + 71\!\cdots\!56 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5352592453 \nu^{17} - 36219942928 \nu^{16} + 51573678503 \nu^{15} - 483366905798 \nu^{14} + \cdots + 19\!\cdots\!88 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 854506111 \nu^{17} + 4609185584 \nu^{16} - 13728641659 \nu^{15} - 82161030866 \nu^{14} + \cdots + 81\!\cdots\!16 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 315720403 \nu^{17} + 1287782128 \nu^{16} + 5285488927 \nu^{15} - 10886047222 \nu^{14} + \cdots + 55\!\cdots\!52 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1230660089 \nu^{17} - 866116816 \nu^{16} + 21994602461 \nu^{15} + 66752621374 \nu^{14} + \cdots - 17\!\cdots\!84 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26723135 \nu^{17} + 59722832 \nu^{16} - 438730555 \nu^{15} - 2644002546 \nu^{14} + \cdots + 10\!\cdots\!08 ) / 300482354479104 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 30192590489 \nu^{17} + 79531824464 \nu^{16} + 808472927741 \nu^{15} + \cdots + 56\!\cdots\!96 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 13563385 \nu^{17} - 30520744 \nu^{16} + 250193021 \nu^{15} + 1242908022 \nu^{14} + \cdots - 71\!\cdots\!64 ) / 37560294309888 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - 2\beta_{8} + \beta_{7} - 8\beta_{6} + \beta_{3} + 4\beta_{2} + 30\beta _1 + 1 ) / 256 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{17} + 4 \beta_{16} - 2 \beta_{15} + 4 \beta_{14} - \beta_{13} + 4 \beta_{12} - 4 \beta_{10} + \cdots + 139 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{17} + 4 \beta_{16} - 10 \beta_{15} + 12 \beta_{14} + \beta_{13} - 12 \beta_{12} + \cdots + 1261 ) / 256 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 18 \beta_{17} + 4 \beta_{16} + 38 \beta_{15} + 76 \beta_{14} - 5 \beta_{13} - 12 \beta_{12} + \cdots + 3443 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 118 \beta_{17} + 28 \beta_{16} - 318 \beta_{15} - 92 \beta_{14} - 91 \beta_{13} - 180 \beta_{12} + \cdots + 35957 ) / 256 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 310 \beta_{17} - 108 \beta_{16} + 62 \beta_{15} + 844 \beta_{14} - 213 \beta_{13} - 156 \beta_{12} + \cdots - 189565 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 626 \beta_{17} + 4300 \beta_{16} + 4170 \beta_{15} - 3532 \beta_{14} + 333 \beta_{13} - 68 \beta_{12} + \cdots - 158667 ) / 256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3782 \beta_{17} - 2860 \beta_{16} - 20242 \beta_{15} - 2580 \beta_{14} + 5419 \beta_{13} + \cdots - 1624909 ) / 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 17778 \beta_{17} - 40852 \beta_{16} + 62154 \beta_{15} + 55636 \beta_{14} - 8147 \beta_{13} + \cdots + 29869381 ) / 256 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 83702 \beta_{17} - 24556 \beta_{16} + 132958 \beta_{15} + 29420 \beta_{14} - 261573 \beta_{13} + \cdots - 64313277 ) / 256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 434814 \beta_{17} + 399148 \beta_{16} - 1275174 \beta_{15} + 1276692 \beta_{14} - 121283 \beta_{13} + \cdots - 285994891 ) / 256 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 937978 \beta_{17} + 237460 \beta_{16} + 510318 \beta_{15} + 175724 \beta_{14} + 2596491 \beta_{13} + \cdots - 927982765 ) / 256 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1923890 \beta_{17} - 7623636 \beta_{16} + 16413962 \beta_{15} + 14211220 \beta_{14} + \cdots + 2805783141 ) / 256 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 22983530 \beta_{17} - 1869036 \beta_{16} - 90803586 \beta_{15} - 73143188 \beta_{14} + \cdots + 17914392803 ) / 256 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 70188706 \beta_{17} + 104645804 \beta_{16} + 240542458 \beta_{15} - 31556716 \beta_{14} + \cdots - 98054897707 ) / 256 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 236301222 \beta_{17} + 660741524 \beta_{16} + 668848910 \beta_{15} - 365025172 \beta_{14} + \cdots + 344735904883 ) / 256 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 378271122 \beta_{17} - 18981844 \beta_{16} - 2714152342 \beta_{15} - 2117654636 \beta_{14} + \cdots + 115394170437 ) / 256 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.592893 + 3.95582i
3.95408 − 0.604357i
−3.62794 − 1.68466i
0.562915 − 3.96019i
−1.58716 + 3.67164i
−3.75808 + 1.36997i
2.81847 + 2.83835i
3.67701 − 1.57467i
−2.63220 − 3.01190i
0.592893 − 3.95582i
3.95408 + 0.604357i
−3.62794 + 1.68466i
0.562915 + 3.96019i
−1.58716 − 3.67164i
−3.75808 − 1.36997i
2.81847 − 2.83835i
3.67701 + 1.57467i
−2.63220 + 3.01190i
0 −16.8936 − 16.8936i 0 66.0049 − 66.0049i 0 − 75.3048i 0 327.790i 0
17.2 0 −14.5768 − 14.5768i 0 −33.9573 + 33.9573i 0 141.886i 0 181.969i 0
17.3 0 −13.2148 − 13.2148i 0 −26.0955 + 26.0955i 0 − 106.802i 0 106.262i 0
17.4 0 −3.84023 − 3.84023i 0 8.73514 − 8.73514i 0 28.0117i 0 − 213.505i 0
17.5 0 3.18000 + 3.18000i 0 −67.3647 + 67.3647i 0 − 148.379i 0 − 222.775i 0
17.6 0 4.57839 + 4.57839i 0 48.5981 − 48.5981i 0 106.338i 0 − 201.077i 0
17.7 0 8.39316 + 8.39316i 0 −8.40645 + 8.40645i 0 149.265i 0 − 102.110i 0
17.8 0 12.7302 + 12.7302i 0 39.8132 − 39.8132i 0 − 248.565i 0 81.1158i 0
17.9 0 20.6438 + 20.6438i 0 −28.3274 + 28.3274i 0 55.5494i 0 609.330i 0
49.1 0 −16.8936 + 16.8936i 0 66.0049 + 66.0049i 0 75.3048i 0 − 327.790i 0
49.2 0 −14.5768 + 14.5768i 0 −33.9573 − 33.9573i 0 − 141.886i 0 − 181.969i 0
49.3 0 −13.2148 + 13.2148i 0 −26.0955 − 26.0955i 0 106.802i 0 − 106.262i 0
49.4 0 −3.84023 + 3.84023i 0 8.73514 + 8.73514i 0 − 28.0117i 0 213.505i 0
49.5 0 3.18000 − 3.18000i 0 −67.3647 − 67.3647i 0 148.379i 0 222.775i 0
49.6 0 4.57839 − 4.57839i 0 48.5981 + 48.5981i 0 − 106.338i 0 201.077i 0
49.7 0 8.39316 − 8.39316i 0 −8.40645 − 8.40645i 0 − 149.265i 0 102.110i 0
49.8 0 12.7302 − 12.7302i 0 39.8132 + 39.8132i 0 248.565i 0 − 81.1158i 0
49.9 0 20.6438 − 20.6438i 0 −28.3274 − 28.3274i 0 − 55.5494i 0 − 609.330i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.6.e.a 18
3.b odd 2 1 576.6.k.a 18
4.b odd 2 1 16.6.e.a ✓ 18
8.b even 2 1 128.6.e.a 18
8.d odd 2 1 128.6.e.b 18
12.b even 2 1 144.6.k.a 18
16.e even 4 1 inner 64.6.e.a 18
16.e even 4 1 128.6.e.a 18
16.f odd 4 1 16.6.e.a ✓ 18
16.f odd 4 1 128.6.e.b 18
32.g even 8 2 1024.6.a.l 18
32.h odd 8 2 1024.6.a.k 18
48.i odd 4 1 576.6.k.a 18
48.k even 4 1 144.6.k.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.6.e.a ✓ 18 4.b odd 2 1
16.6.e.a ✓ 18 16.f odd 4 1
64.6.e.a 18 1.a even 1 1 trivial
64.6.e.a 18 16.e even 4 1 inner
128.6.e.a 18 8.b even 2 1
128.6.e.a 18 16.e even 4 1
128.6.e.b 18 8.d odd 2 1
128.6.e.b 18 16.f odd 4 1
144.6.k.a 18 12.b even 2 1
144.6.k.a 18 48.k even 4 1
576.6.k.a 18 3.b odd 2 1
576.6.k.a 18 48.i odd 4 1
1024.6.a.k 18 32.h odd 8 2
1024.6.a.l 18 32.g even 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(64, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 82\!\cdots\!92 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 88\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots + 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 33\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 99\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 68\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 78\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots + 16\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 46\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 14\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 28\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 33\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 84\!\cdots\!16)^{2} \) Copy content Toggle raw display
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