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Properties

Label 8.16.b.a
Level $8$
Weight $16$
Character orbit 8.b
Analytic conductor $11.415$
Analytic rank $0$
Dimension $14$
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] = [8,16,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(11.4154804080\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 6354 x^{12} + 136110 x^{11} + 41390651 x^{10} - 1368564777 x^{9} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{91}\cdot 3^{6}\cdot 5^{4}\cdot 31^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 6) q^{2} + \beta_{2} q^{3} + (\beta_{3} - 2 \beta_{2} + 5 \beta_1 + 3672) q^{4} + ( - \beta_{5} - 3 \beta_{2} + \cdots - 54) q^{5}+ \cdots + ( - \beta_{9} + \beta_{4} + \cdots - 4099555) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 6) q^{2} + \beta_{2} q^{3} + (\beta_{3} - 2 \beta_{2} + 5 \beta_1 + 3672) q^{4} + ( - \beta_{5} - 3 \beta_{2} + \cdots - 54) q^{5}+ \cdots + (1222248 \beta_{13} + \cdots + 1266125749320) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 90 q^{2} + 51444 q^{4} - 189428 q^{6} - 1647088 q^{7} + 1889640 q^{8} - 57395630 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 90 q^{2} + 51444 q^{4} - 189428 q^{6} - 1647088 q^{7} + 1889640 q^{8} - 57395630 q^{9} + 58467784 q^{10} + 399357832 q^{12} - 518960496 q^{14} + 712135312 q^{15} - 1435931120 q^{16} + 728554812 q^{17} + 526853306 q^{18} - 3449250768 q^{20} + 28367364252 q^{22} - 35548816080 q^{23} + 40155187088 q^{24} - 75899954794 q^{25} + 17666210712 q^{26} + 79863955680 q^{28} - 124878825712 q^{30} - 105758138816 q^{31} - 37651613280 q^{32} - 150458001384 q^{33} + 537472307308 q^{34} + 338679650892 q^{36} + 1649727781164 q^{38} - 2251546247120 q^{39} + 1251083710304 q^{40} - 53229185940 q^{41} - 2437011096800 q^{42} - 3416842360344 q^{44} - 3303531082064 q^{46} + 12527998446432 q^{47} - 6441543679584 q^{48} + 8427385380990 q^{49} + 1179755527374 q^{50} - 2436018627056 q^{52} + 3357642572216 q^{54} - 30557833792176 q^{55} + 7549064859072 q^{56} + 18277230892472 q^{57} - 8014960165320 q^{58} - 53574657402912 q^{60} + 77882578979904 q^{62} + 36142362113776 q^{63} + 76083381630528 q^{64} + 5437123965600 q^{65} - 134116957601160 q^{66} - 69772560247896 q^{68} + 133952399750848 q^{70} - 173249927708016 q^{71} + 163390222317848 q^{72} - 182057837882196 q^{73} + 2072780135688 q^{74} - 248503439494072 q^{76} + 636498768647600 q^{78} - 294370273271392 q^{79} + 766230078246336 q^{80} + 256903428263798 q^{81} - 10\!\cdots\!32 q^{82}+ \cdots - 51\!\cdots\!58 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 6354 x^{12} + 136110 x^{11} + 41390651 x^{10} - 1368564777 x^{9} + \cdots + 24\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57787459611 \nu^{13} + 1737913630882 \nu^{12} + \cdots - 83\!\cdots\!76 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 57787459611 \nu^{13} + 1737913630882 \nu^{12} + \cdots - 35\!\cdots\!08 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3186889963709 \nu^{13} + \cdots - 26\!\cdots\!32 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1396821254639 \nu^{13} + 221715839114138 \nu^{12} + \cdots - 30\!\cdots\!76 ) / 49\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1002455217403 \nu^{13} + 89631572299234 \nu^{12} + \cdots + 35\!\cdots\!68 ) / 47\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35879304688071 \nu^{13} + 430293662123350 \nu^{12} + \cdots - 62\!\cdots\!80 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26368160926857 \nu^{13} + \cdots - 21\!\cdots\!08 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 127928962719909 \nu^{13} + \cdots - 39\!\cdots\!84 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 72522157664001 \nu^{13} + \cdots + 50\!\cdots\!84 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 338295464497805 \nu^{13} + \cdots - 40\!\cdots\!36 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 269874431012467 \nu^{13} + \cdots - 21\!\cdots\!20 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 652797590290385 \nu^{13} + \cdots - 69\!\cdots\!08 ) / 74\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} - 7\beta _1 + 3636 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} - 8\beta_{3} + 189\beta_{2} + 3787\beta _1 - 202259 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{13} + 6 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} - 10 \beta_{9} - 134 \beta_{8} + \cdots - 49228208 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20 \beta_{13} - 198 \beta_{12} - 161 \beta_{11} - 878 \beta_{10} + 674 \beta_{9} - 302 \beta_{8} + \cdots + 718614158 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 16600 \beta_{13} + 1392 \beta_{12} + 16464 \beta_{11} + 51884 \beta_{10} + 11960 \beta_{9} + \cdots + 330007825365 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1129800 \beta_{13} + 2007260 \beta_{12} - 2573638 \beta_{11} + 671308 \beta_{10} + \cdots - 171680455469587 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 324650588 \beta_{13} + 287521870 \beta_{12} + 36361941 \beta_{11} - 153436126 \beta_{10} + \cdots + 14\!\cdots\!82 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6962854628 \beta_{13} - 4578219874 \beta_{12} - 1344782523 \beta_{11} - 10220322330 \beta_{10} + \cdots - 16\!\cdots\!56 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1241250818976 \beta_{13} + 187901689480 \beta_{12} - 1587647469956 \beta_{11} + 1670154050368 \beta_{10} + \cdots - 36\!\cdots\!50 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 201599671400976 \beta_{13} - 146754756519032 \beta_{12} + 214591658187884 \beta_{11} + \cdots - 51\!\cdots\!71 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 48\!\cdots\!36 \beta_{13} + \cdots + 46\!\cdots\!24 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22\!\cdots\!00 \beta_{13} + \cdots + 54\!\cdots\!10 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
85.4255 + 19.3112i
85.4255 − 19.3112i
73.6274 + 48.1688i
73.6274 − 48.1688i
48.9348 + 74.1268i
48.9348 − 74.1268i
11.1806 + 89.3919i
11.1806 − 89.3919i
−53.8860 + 74.8506i
−53.8860 − 74.8506i
−70.4661 + 60.3351i
−70.4661 − 60.3351i
−93.3164 + 5.91227i
−93.3164 − 5.91227i
−176.851 − 38.6225i 4900.64i 29784.6 + 13660.8i 174283.i 189275. − 866683.i 3.39868e6 −4.73983e6 − 3.56629e6i −9.66732e6 6.73124e6 − 3.08221e7i
5.2 −176.851 + 38.6225i − 4900.64i 29784.6 − 13660.8i − 174283.i 189275. + 866683.i 3.39868e6 −4.73983e6 + 3.56629e6i −9.66732e6 6.73124e6 + 3.08221e7i
5.3 −153.255 − 96.3377i 27.2769i 14206.1 + 29528.4i − 212327.i 2627.80 − 4180.32i −3.74922e6 667546. − 5.89396e6i 1.43482e7 −2.04551e7 + 3.25401e7i
5.4 −153.255 + 96.3377i − 27.2769i 14206.1 − 29528.4i 212327.i 2627.80 + 4180.32i −3.74922e6 667546. + 5.89396e6i 1.43482e7 −2.04551e7 − 3.25401e7i
5.5 −103.870 − 148.254i − 6357.96i −11190.2 + 30798.1i 267219.i −942590. + 660399.i 120583. 5.72825e6 − 1.53999e6i −2.60748e7 3.96161e7 − 2.77559e7i
5.6 −103.870 + 148.254i 6357.96i −11190.2 − 30798.1i − 267219.i −942590. − 660399.i 120583. 5.72825e6 + 1.53999e6i −2.60748e7 3.96161e7 + 2.77559e7i
5.7 −28.3613 − 178.784i 2380.16i −31159.3 + 10141.1i − 9943.39i 425534. − 67504.4i 1.24365e6 2.69677e6 + 5.28316e6i 8.68373e6 −1.77772e6 + 282007.i
5.8 −28.3613 + 178.784i − 2380.16i −31159.3 − 10141.1i 9943.39i 425534. + 67504.4i 1.24365e6 2.69677e6 − 5.28316e6i 8.68373e6 −1.77772e6 − 282007.i
5.9 101.772 − 149.701i − 3583.46i −12052.9 − 30470.8i − 82632.2i −536448. − 364695.i −153730. −5.78817e6 − 1.29673e6i 1.50775e6 −1.23701e7 − 8.40964e6i
5.10 101.772 + 149.701i 3583.46i −12052.9 + 30470.8i 82632.2i −536448. + 364695.i −153730. −5.78817e6 + 1.29673e6i 1.50775e6 −1.23701e7 + 8.40964e6i
5.11 134.932 − 120.670i 6537.39i 3645.36 − 32564.6i 116500.i 788869. + 882104.i −2.94378e6 −3.43770e6 − 4.83390e6i −2.83885e7 1.40581e7 + 1.57196e7i
5.12 134.932 + 120.670i − 6537.39i 3645.36 + 32564.6i − 116500.i 788869. − 882104.i −2.94378e6 −3.43770e6 + 4.83390e6i −2.83885e7 1.40581e7 − 1.57196e7i
5.13 180.633 − 11.8245i − 1858.96i 32488.4 − 4271.80i 290191.i −21981.3 − 335789.i 1.26027e6 5.81795e6 − 1.15579e6i 1.08932e7 3.43137e6 + 5.24180e7i
5.14 180.633 + 11.8245i 1858.96i 32488.4 + 4271.80i − 290191.i −21981.3 + 335789.i 1.26027e6 5.81795e6 + 1.15579e6i 1.08932e7 3.43137e6 − 5.24180e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.16.b.a ✓ 14
3.b odd 2 1 72.16.d.b 14
4.b odd 2 1 32.16.b.a 14
8.b even 2 1 inner 8.16.b.a ✓ 14
8.d odd 2 1 32.16.b.a 14
24.h odd 2 1 72.16.d.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.b.a ✓ 14 1.a even 1 1 trivial
8.16.b.a ✓ 14 8.b even 2 1 inner
32.16.b.a 14 4.b odd 2 1
32.16.b.a 14 8.d odd 2 1
72.16.d.b 14 3.b odd 2 1
72.16.d.b 14 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 40\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 77\!\cdots\!68 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 21\!\cdots\!52)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 28\!\cdots\!48)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 25\!\cdots\!52)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 69\!\cdots\!68)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 64\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 52\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 48\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 29\!\cdots\!92)^{2} \) Copy content Toggle raw display
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