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Properties

Label 832.2.bk.c
Level $832$
Weight $2$
Character orbit 832.bk
Analytic conductor $6.644$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [832,2,Mod(223,832)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("832.223"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 6, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 832.bk (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,-2,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.64355344817\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 42 x^{18} + 739 x^{16} + 7108 x^{14} + 40843 x^{12} + 143746 x^{10} + 305029 x^{8} + \cdots + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{19} - \beta_{4}) q^{3} - \beta_{15} q^{5} - \beta_{12} q^{7} + ( - \beta_{19} - \beta_{18} + \beta_{17} + \cdots - 1) q^{9} + (\beta_{19} + \beta_{18} + \beta_{16} + \cdots + 1) q^{11} + (\beta_{19} - \beta_{17} + \cdots + \beta_{2}) q^{13}+ \cdots + (3 \beta_{19} - \beta_{18} + 2 \beta_{15} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3} - 2 q^{7} - 12 q^{9} + 18 q^{11} + 6 q^{13} + 4 q^{15} - 6 q^{17} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 16 q^{27} - 4 q^{31} + 6 q^{33} - 18 q^{35} - 10 q^{37} - 22 q^{39} + 4 q^{41} - 24 q^{43}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 42 x^{18} + 739 x^{16} + 7108 x^{14} + 40843 x^{12} + 143746 x^{10} + 305029 x^{8} + \cdots + 2704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{18} - 29 \nu^{16} - 297 \nu^{14} - 1128 \nu^{12} + 757 \nu^{10} + 16791 \nu^{8} + 42045 \nu^{6} + \cdots + 832 ) / 624 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{18} - 29 \nu^{16} - 297 \nu^{14} - 1128 \nu^{12} + 757 \nu^{10} + 16791 \nu^{8} + 42045 \nu^{6} + \cdots + 832 ) / 624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 521 \nu^{19} + 5655 \nu^{18} + 127 \nu^{17} + 149461 \nu^{16} + 308336 \nu^{15} + \cdots - 9672208 ) / 4104672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25 \nu^{18} + 881 \nu^{16} + 12508 \nu^{14} + 92654 \nu^{12} + 386337 \nu^{10} + 909345 \nu^{8} + \cdots + 8944 ) / 6864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43 \nu^{19} + 845 \nu^{18} + 1481 \nu^{17} + 31265 \nu^{16} + 20324 \nu^{15} + 471848 \nu^{14} + \cdots + 873392 ) / 178464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4 \nu^{19} + 181 \nu^{17} + 3333 \nu^{15} + 32293 \nu^{13} + 178036 \nu^{11} + 565143 \nu^{9} + \cdots - 4056 ) / 8112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43 \nu^{19} - 845 \nu^{18} + 1481 \nu^{17} - 31265 \nu^{16} + 20324 \nu^{15} - 471848 \nu^{14} + \cdots - 873392 ) / 178464 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2529 \nu^{19} + 819 \nu^{18} - 64709 \nu^{17} - 4303 \nu^{16} - 406600 \nu^{15} + \cdots + 6308432 ) / 4104672 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2529 \nu^{19} - 819 \nu^{18} - 64709 \nu^{17} + 4303 \nu^{16} - 406600 \nu^{15} + \cdots - 6308432 ) / 4104672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2020 \nu^{19} + 11037 \nu^{18} + 63520 \nu^{17} + 371618 \nu^{16} + 780874 \nu^{15} + \cdots - 4572464 ) / 2052336 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3519 \nu^{19} - 24713 \nu^{18} + 127167 \nu^{17} - 852891 \nu^{16} + 1870084 \nu^{15} + \cdots - 9509968 ) / 4104672 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3050 \nu^{19} - 650 \nu^{18} - 107339 \nu^{17} - 28483 \nu^{16} - 1509245 \nu^{15} + \cdots + 12497888 ) / 2052336 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3050 \nu^{19} - 650 \nu^{18} + 107339 \nu^{17} - 28483 \nu^{16} + 1509245 \nu^{15} + \cdots + 12497888 ) / 2052336 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8453 \nu^{19} - 16809 \nu^{18} - 327193 \nu^{17} - 636519 \nu^{16} - 5222952 \nu^{15} + \cdots - 13141440 ) / 4104672 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7016 \nu^{19} - 6474 \nu^{18} - 270999 \nu^{17} - 273429 \nu^{16} - 4274889 \nu^{15} + \cdots - 2451176 ) / 2052336 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14553 \nu^{19} + 18603 \nu^{18} + 541871 \nu^{17} + 696319 \nu^{16} + 8241442 \nu^{15} + \cdots - 4769856 ) / 4104672 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 7016 \nu^{19} - 6474 \nu^{18} + 270999 \nu^{17} - 273429 \nu^{16} + 4274889 \nu^{15} + \cdots - 2451176 ) / 2052336 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 17203 \nu^{19} + 7475 \nu^{18} - 621815 \nu^{17} + 263419 \nu^{16} - 9106258 \nu^{15} + \cdots + 10883600 ) / 4104672 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 65 \nu^{19} + 25 \nu^{18} + 2405 \nu^{17} + 881 \nu^{16} + 36296 \nu^{15} + 12508 \nu^{14} + \cdots + 8944 ) / 13728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{9} - \beta_{8} + \beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{19} - 2 \beta_{17} - 2 \beta_{16} + 2 \beta_{13} - 2 \beta_{12} + \beta_{9} + \beta_{8} + \cdots - 7 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{16} - 11 \beta_{15} + 8 \beta_{14} - 8 \beta_{13} - 9 \beta_{9} + 9 \beta_{8} + \beta_{7} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 29 \beta_{19} + 28 \beta_{17} + 25 \beta_{16} - 3 \beta_{15} - 25 \beta_{13} + 25 \beta_{12} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{19} + 8 \beta_{17} + 50 \beta_{16} + 114 \beta_{15} - 62 \beta_{14} + 63 \beta_{13} + \cdots - 195 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 368 \beta_{19} + 16 \beta_{18} - 334 \beta_{17} - 270 \beta_{16} + 56 \beta_{15} + 8 \beta_{14} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 60 \beta_{19} - 166 \beta_{17} - 649 \beta_{16} - 1191 \beta_{15} + 496 \beta_{14} - 514 \beta_{13} + \cdots + 1596 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4379 \beta_{19} - 344 \beta_{18} + 3790 \beta_{17} + 2857 \beta_{16} - 761 \beta_{15} - 172 \beta_{14} + \cdots - 1169 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 849 \beta_{19} + 2414 \beta_{17} + 7724 \beta_{16} + 12592 \beta_{15} - 4152 \beta_{14} + \cdots - 13991 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 50178 \beta_{19} + 5104 \beta_{18} - 42084 \beta_{17} - 30302 \beta_{16} + 9230 \beta_{15} + \cdots + 14072 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10554 \beta_{19} - 30572 \beta_{17} - 88235 \beta_{16} - 134257 \beta_{15} + 36558 \beta_{14} + \cdots + 129784 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 562121 \beta_{19} - 65440 \beta_{18} + 462140 \beta_{17} + 323193 \beta_{16} - 106227 \beta_{15} + \cdots - 161821 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 123411 \beta_{19} + 361900 \beta_{17} + 985738 \beta_{16} + 1439010 \beta_{15} - 338194 \beta_{14} + \cdots - 1259551 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6211392 \beta_{19} + 780800 \beta_{18} - 5043330 \beta_{17} - 3463202 \beta_{16} + 1189728 \beta_{15} + \cdots + 1815908 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1397100 \beta_{19} - 4129850 \beta_{17} - 10870721 \beta_{16} - 15473151 \beta_{15} + 3266780 \beta_{14} + \cdots + 12649496 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 68060923 \beta_{19} - 8956648 \beta_{18} + 54834330 \beta_{17} + 37229797 \beta_{16} + \cdots - 20090949 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 15529113 \beta_{19} + 46143018 \beta_{17} + 118954260 \beta_{16} + 166700600 \beta_{15} + \cdots - 130201203 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 741912090 \beta_{19} + 100421232 \beta_{18} - 594847784 \beta_{17} - 401051778 \beta_{16} + \cdots + 220380300 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{7}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
3.28743i
1.64554i
0.617361i
− 1.94717i
− 2.60317i
− 2.62715i
− 1.69752i
− 0.264811i
1.00733i
2.58216i
− 3.28743i
− 1.64554i
− 0.617361i
1.94717i
2.60317i
2.62715i
1.69752i
0.264811i
− 1.00733i
− 2.58216i
0 −1.64372 − 2.84700i 0 1.86794 + 1.86794i 0 0.261804 + 0.977067i 0 −3.90361 + 6.76124i 0
223.2 0 −0.822771 − 1.42508i 0 −2.97055 − 2.97055i 0 −0.488718 − 1.82392i 0 0.146095 − 0.253045i 0
223.3 0 −0.308681 − 0.534651i 0 0.0694235 + 0.0694235i 0 0.428124 + 1.59778i 0 1.30943 − 2.26800i 0
223.4 0 0.973584 + 1.68630i 0 −0.923557 − 0.923557i 0 −1.13446 − 4.23387i 0 −0.395732 + 0.685428i 0
223.5 0 1.30158 + 2.25441i 0 1.95675 + 1.95675i 0 0.433253 + 1.61692i 0 −1.88824 + 3.27053i 0
479.1 0 −1.31358 − 2.27518i 0 −1.26617 + 1.26617i 0 2.62760 − 0.704063i 0 −1.95097 + 3.37918i 0
479.2 0 −0.848762 − 1.47010i 0 2.81710 − 2.81710i 0 0.0793086 − 0.0212507i 0 0.0592058 − 0.102547i 0
479.3 0 −0.132405 − 0.229333i 0 −2.13851 + 2.13851i 0 −4.85609 + 1.30119i 0 1.46494 − 2.53735i 0
479.4 0 0.503667 + 0.872377i 0 −0.871309 + 0.871309i 0 3.24146 − 0.868547i 0 0.992639 − 1.71930i 0
479.5 0 1.29108 + 2.23621i 0 1.45889 − 1.45889i 0 −1.59228 + 0.426650i 0 −1.83376 + 3.17617i 0
735.1 0 −1.64372 + 2.84700i 0 1.86794 − 1.86794i 0 0.261804 − 0.977067i 0 −3.90361 − 6.76124i 0
735.2 0 −0.822771 + 1.42508i 0 −2.97055 + 2.97055i 0 −0.488718 + 1.82392i 0 0.146095 + 0.253045i 0
735.3 0 −0.308681 + 0.534651i 0 0.0694235 − 0.0694235i 0 0.428124 − 1.59778i 0 1.30943 + 2.26800i 0
735.4 0 0.973584 − 1.68630i 0 −0.923557 + 0.923557i 0 −1.13446 + 4.23387i 0 −0.395732 − 0.685428i 0
735.5 0 1.30158 − 2.25441i 0 1.95675 − 1.95675i 0 0.433253 − 1.61692i 0 −1.88824 − 3.27053i 0
799.1 0 −1.31358 + 2.27518i 0 −1.26617 − 1.26617i 0 2.62760 + 0.704063i 0 −1.95097 − 3.37918i 0
799.2 0 −0.848762 + 1.47010i 0 2.81710 + 2.81710i 0 0.0793086 + 0.0212507i 0 0.0592058 + 0.102547i 0
799.3 0 −0.132405 + 0.229333i 0 −2.13851 − 2.13851i 0 −4.85609 − 1.30119i 0 1.46494 + 2.53735i 0
799.4 0 0.503667 − 0.872377i 0 −0.871309 − 0.871309i 0 3.24146 + 0.868547i 0 0.992639 + 1.71930i 0
799.5 0 1.29108 − 2.23621i 0 1.45889 + 1.45889i 0 −1.59228 − 0.426650i 0 −1.83376 − 3.17617i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.bk.c ✓ 20
4.b odd 2 1 832.2.bk.f yes 20
8.b even 2 1 832.2.bk.e yes 20
8.d odd 2 1 832.2.bk.d yes 20
13.f odd 12 1 832.2.bk.d yes 20
52.l even 12 1 832.2.bk.e yes 20
104.u even 12 1 inner 832.2.bk.c ✓ 20
104.x odd 12 1 832.2.bk.f yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
832.2.bk.c ✓ 20 1.a even 1 1 trivial
832.2.bk.c ✓ 20 104.u even 12 1 inner
832.2.bk.d yes 20 8.d odd 2 1
832.2.bk.d yes 20 13.f odd 12 1
832.2.bk.e yes 20 8.b even 2 1
832.2.bk.e yes 20 52.l even 12 1
832.2.bk.f yes 20 4.b odd 2 1
832.2.bk.f yes 20 104.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(832, [\chi])\):

\( T_{3}^{20} + 2 T_{3}^{19} + 23 T_{3}^{18} + 30 T_{3}^{17} + 308 T_{3}^{16} + 360 T_{3}^{15} + \cdots + 2704 \) Copy content Toggle raw display
\( T_{5}^{20} + 394 T_{5}^{16} - 72 T_{5}^{15} + 1464 T_{5}^{13} + 34185 T_{5}^{12} - 3288 T_{5}^{11} + \cdots + 46656 \) Copy content Toggle raw display
\( T_{7}^{20} + 2 T_{7}^{19} - 13 T_{7}^{18} - 32 T_{7}^{17} - 230 T_{7}^{16} + 534 T_{7}^{15} + \cdots + 20736 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} + \cdots + 2704 \) Copy content Toggle raw display
$5$ \( T^{20} + 394 T^{16} + \cdots + 46656 \) Copy content Toggle raw display
$7$ \( T^{20} + 2 T^{19} + \cdots + 20736 \) Copy content Toggle raw display
$11$ \( T^{20} - 18 T^{19} + \cdots + 1679616 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 18692084961 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 74859148816 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 16172655282576 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 3978241669809 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 5737153536 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 72731077969 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 204391697409 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 1056751104256 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 39516008702976 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 681536104704 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 50029917751489 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 196741925136 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 28665198864 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 86\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 94549330427904 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 101437154064 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 84630358674576 \) Copy content Toggle raw display
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