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Properties

Label 1062.3.b.a
Level $1062$
Weight $3$
Character orbit 1062.b
Analytic conductor $28.937$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1062,3,Mod(827,1062)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1062, 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("1062.827");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1062 = 2 \cdot 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1062.b (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: \(28.9374040751\)
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: \( x^{16} + 92 x^{14} + 3060 x^{12} + 46232 x^{10} + 335608 x^{8} + 1151280 x^{6} + 1665360 x^{4} + \cdots + 104976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - 2 q^{4} + \beta_{2} q^{5} - \beta_{10} q^{7} + 2 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - 2 q^{4} + \beta_{2} q^{5} - \beta_{10} q^{7} + 2 \beta_{4} q^{8} + \beta_{9} q^{10} + (\beta_{14} + 2 \beta_{4}) q^{11} + ( - \beta_{7} + \beta_{6}) q^{13} - \beta_{12} q^{14} + 4 q^{16} + ( - \beta_{15} - 2 \beta_{13} + \cdots - \beta_1) q^{17}+ \cdots + (4 \beta_{15} + 2 \beta_{14} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} - 8 q^{10} + 64 q^{16} + 56 q^{22} - 72 q^{25} - 128 q^{31} - 56 q^{34} + 120 q^{37} + 16 q^{40} + 112 q^{43} - 96 q^{46} + 136 q^{49} - 112 q^{55} - 16 q^{58} - 128 q^{61} - 128 q^{64} + 40 q^{67} + 176 q^{70} + 280 q^{79} + 192 q^{82} + 16 q^{85} - 112 q^{88} - 128 q^{91} - 216 q^{94} - 464 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 92 x^{14} + 3060 x^{12} + 46232 x^{10} + 335608 x^{8} + 1151280 x^{6} + 1665360 x^{4} + \cdots + 104976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6565 \nu^{15} - 734732 \nu^{13} - 25000938 \nu^{11} - 100779476 \nu^{9} + \cdots + 641735702352 \nu ) / 10784394432 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18973 \nu^{15} - 1893548 \nu^{13} - 73516158 \nu^{11} - 1482988220 \nu^{9} + \cdots - 216755366256 \nu ) / 10784394432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{14} - 65 \nu^{12} - 738 \nu^{10} + 22456 \nu^{8} + 500528 \nu^{6} + 2886516 \nu^{4} + \cdots + 1574640 ) / 81648 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48113 \nu^{15} - 4509943 \nu^{13} - 154336356 \nu^{11} - 2430516718 \nu^{9} + \cdots - 31598586648 \nu ) / 5392197216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 83788 \nu^{14} + 7366019 \nu^{12} + 226595412 \nu^{10} + 2975046410 \nu^{8} + 16795461424 \nu^{6} + \cdots + 7808738904 ) / 1797399072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 90875 \nu^{14} + 8809411 \nu^{12} + 315495792 \nu^{10} + 5243615566 \nu^{8} + \cdots + 11142683352 ) / 1797399072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16405 \nu^{14} + 1539707 \nu^{12} + 52724943 \nu^{10} + 828113132 \nu^{8} + 6227787496 \nu^{6} + \cdots + 5934287448 ) / 299566512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26303 \nu^{15} + 2423422 \nu^{13} + 80812602 \nu^{11} + 1226898184 \nu^{9} + \cdots + 36917633088 \nu ) / 1540627776 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 78166 \nu^{14} + 7098491 \nu^{12} + 230732136 \nu^{10} + 3337358822 \nu^{8} + 22203067768 \nu^{6} + \cdots + 14905361448 ) / 898699536 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8783 \nu^{14} + 794464 \nu^{12} + 25687782 \nu^{10} + 369699616 \nu^{8} + 2472099212 \nu^{6} + \cdots + 1733946912 ) / 85590432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 520819 \nu^{14} + 47068241 \nu^{12} + 1518423264 \nu^{10} + 21717302978 \nu^{8} + \cdots + 38257972488 ) / 1797399072 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 188386 \nu^{15} - 16728296 \nu^{13} - 523460979 \nu^{11} - 7079171960 \nu^{9} + \cdots + 77206599576 \nu ) / 2696098608 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 64157 \nu^{15} - 5817277 \nu^{13} - 188827848 \nu^{11} - 2735288500 \nu^{9} + \cdots - 25581169872 \nu ) / 599133024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2251685 \nu^{15} - 204661930 \nu^{13} - 6667805934 \nu^{11} - 97081206208 \nu^{9} + \cdots - 596094202080 \nu ) / 10784394432 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 233887 \nu^{15} + 21241412 \nu^{13} + 690664446 \nu^{11} + 10002019046 \nu^{9} + \cdots + 19810924920 \nu ) / 770313888 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - 2\beta_{14} + \beta_{13} - \beta_{8} - 2\beta_{4} + 4\beta_{2} + \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{11} + 6\beta_{10} + 7\beta_{9} - 6\beta_{7} + 2\beta_{6} + 4\beta_{5} + 2\beta_{3} - 102 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17\beta_{15} + 37\beta_{14} - 17\beta_{13} - 3\beta_{12} + 68\beta_{8} + 94\beta_{4} - 107\beta_{2} - 38\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 98\beta_{11} - 180\beta_{10} - 200\beta_{9} + 300\beta_{7} - 70\beta_{6} - 152\beta_{5} - 136\beta_{3} + 2652 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 220 \beta_{15} - 722 \beta_{14} + 544 \beta_{13} + 120 \beta_{12} - 2530 \beta_{8} + \cdots + 1600 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2488 \beta_{11} + 5088 \beta_{10} + 6718 \beta_{9} - 11904 \beta_{7} + 1832 \beta_{6} + 5404 \beta_{5} + \cdots - 83604 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 338 \beta_{15} + 13294 \beta_{14} - 19562 \beta_{13} - 2142 \beta_{12} + 90104 \beta_{8} + \cdots - 63824 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 65408 \beta_{11} - 148440 \beta_{10} - 242552 \beta_{9} + 456456 \beta_{7} - 44584 \beta_{6} + \cdots + 2827500 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 149732 \beta_{15} - 177056 \beta_{14} + 697852 \beta_{13} - 11424 \beta_{12} - 3213580 \beta_{8} + \cdots + 2469220 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1767592 \beta_{11} + 4531608 \beta_{10} + 8969524 \beta_{9} - 17328840 \beta_{7} + 1019936 \beta_{6} + \cdots - 98968656 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9217804 \beta_{15} - 1281548 \beta_{14} - 24800900 \beta_{13} + 3187308 \beta_{12} + \cdots - 94066832 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 48974216 \beta_{11} - 144182688 \beta_{10} - 334152032 \beta_{9} + 654294432 \beta_{7} + \cdots + 3534529920 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 430166768 \beta_{15} + 248067640 \beta_{14} + 886008928 \beta_{13} - 195901776 \beta_{12} + \cdots + 3554312512 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1391991952 \beta_{11} + 4750569600 \beta_{10} + 12473025640 \beta_{9} - 24611885376 \beta_{7} + \cdots - 127885870272 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 18178809976 \beta_{15} - 14405197832 \beta_{14} - 31918909832 \beta_{13} + 9383722584 \beta_{12} + \cdots - 133669018352 \beta_1 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1062\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
827.1
− 6.10540i
− 3.73892i
2.51963i
− 2.39409i
5.11215i
0.991854i
− 0.393353i
1.17970i
− 1.17970i
0.393353i
− 0.991854i
− 5.11215i
2.39409i
− 2.51963i
3.73892i
6.10540i
− 1.41421i 0 −2.00000 − 8.04351i 0 −0.518401 2.82843i 0 −11.3752
827.2 − 1.41421i 0 −2.00000 − 5.18281i 0 3.97564 2.82843i 0 −7.32960
827.3 − 1.41421i 0 −2.00000 − 4.21334i 0 −12.7312 2.82843i 0 −5.95857
827.4 − 1.41421i 0 −2.00000 − 3.18168i 0 −4.94820 2.82843i 0 −4.49957
827.5 − 1.41421i 0 −2.00000 0.885622i 0 9.02400 2.82843i 0 1.25246
827.6 − 1.41421i 0 −2.00000 2.24174i 0 10.0398 2.82843i 0 3.17029
827.7 − 1.41421i 0 −2.00000 6.06465i 0 −8.05144 2.82843i 0 8.57671
827.8 − 1.41421i 0 −2.00000 8.60090i 0 3.20979 2.82843i 0 12.1635
827.9 1.41421i 0 −2.00000 − 8.60090i 0 3.20979 − 2.82843i 0 12.1635
827.10 1.41421i 0 −2.00000 − 6.06465i 0 −8.05144 − 2.82843i 0 8.57671
827.11 1.41421i 0 −2.00000 − 2.24174i 0 10.0398 − 2.82843i 0 3.17029
827.12 1.41421i 0 −2.00000 − 0.885622i 0 9.02400 − 2.82843i 0 1.25246
827.13 1.41421i 0 −2.00000 3.18168i 0 −4.94820 − 2.82843i 0 −4.49957
827.14 1.41421i 0 −2.00000 4.21334i 0 −12.7312 − 2.82843i 0 −5.95857
827.15 1.41421i 0 −2.00000 5.18281i 0 3.97564 − 2.82843i 0 −7.32960
827.16 1.41421i 0 −2.00000 8.04351i 0 −0.518401 − 2.82843i 0 −11.3752
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 827.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1062.3.b.a ✓ 16
3.b odd 2 1 inner 1062.3.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1062.3.b.a ✓ 16 1.a even 1 1 trivial
1062.3.b.a ✓ 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 236 T_{5}^{14} + 21760 T_{5}^{12} + 1004452 T_{5}^{10} + 24883342 T_{5}^{8} + \cdots + 3349284129 \) acting on \(S_{3}^{\mathrm{new}}(1062, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 3349284129 \) Copy content Toggle raw display
$7$ \( (T^{8} - 230 T^{6} + \cdots + 303993)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 8044834249 \) Copy content Toggle raw display
$13$ \( (T^{8} - 508 T^{6} + \cdots - 12120597)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 84\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( (T^{8} - 1542 T^{6} + \cdots + 1596022737)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 93\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( (T^{8} + 64 T^{7} + \cdots + 30072692123)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 60 T^{7} + \cdots + 39301359267)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 40\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 1459067073021)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$59$ \( (T^{2} + 59)^{8} \) Copy content Toggle raw display
$61$ \( (T^{8} + 64 T^{7} + \cdots - 14063694237)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 1610273928037)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 58\!\cdots\!41 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 236817558966531)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 140 T^{7} + \cdots + 4596312249)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 298872918486381)^{2} \) Copy content Toggle raw display
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