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Properties

Label 169.8.b.e
Level $169$
Weight $8$
Character orbit 169.b
Analytic conductor $52.793$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,8,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), 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: \(52.7930693068\)
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: \( x^{16} + 1591 x^{14} + 998837 x^{12} + 319862003 x^{10} + 57017400035 x^{8} + 5819167911653 x^{6} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 13^{8} \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{3} + 4) q^{3} + (\beta_1 - 72) q^{4} + (\beta_{11} - 2 \beta_{8} + 4 \beta_{5}) q^{5} + ( - \beta_{12} + 3 \beta_{8} - 2 \beta_{5}) q^{6} + (\beta_{10} - 2 \beta_{8} - 5 \beta_{5}) q^{7} + ( - \beta_{14} - \beta_{12} + \cdots - 82 \beta_{5}) q^{8}+ \cdots + ( - \beta_{4} + 12 \beta_{3} + \cdots + 756) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{3} + 4) q^{3} + (\beta_1 - 72) q^{4} + (\beta_{11} - 2 \beta_{8} + 4 \beta_{5}) q^{5} + ( - \beta_{12} + 3 \beta_{8} - 2 \beta_{5}) q^{6} + (\beta_{10} - 2 \beta_{8} - 5 \beta_{5}) q^{7} + ( - \beta_{14} - \beta_{12} + \cdots - 82 \beta_{5}) q^{8}+ \cdots + (2053 \beta_{15} + \cdots + 221880 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 56 q^{3} - 1154 q^{4} + 11976 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 56 q^{3} - 1154 q^{4} + 11976 q^{9} - 13626 q^{10} + 15816 q^{12} + 14484 q^{14} + 122754 q^{16} - 45648 q^{17} + 147452 q^{22} - 144936 q^{23} - 58488 q^{25} + 358616 q^{27} + 443544 q^{29} - 516232 q^{30} - 63120 q^{35} - 2325986 q^{36} - 2364492 q^{38} + 766030 q^{40} - 3121908 q^{42} + 782984 q^{43} - 9171424 q^{48} - 2217560 q^{49} - 8040712 q^{51} + 3329328 q^{53} + 836432 q^{55} - 7218312 q^{56} - 3527096 q^{61} - 25996776 q^{62} - 32227074 q^{64} + 16245108 q^{66} + 20236182 q^{68} - 9236984 q^{69} - 4412442 q^{74} - 43669976 q^{75} - 35263992 q^{77} + 12644416 q^{79} + 18632544 q^{81} - 18695502 q^{82} - 10399976 q^{87} - 73352808 q^{88} - 132263562 q^{90} + 64127424 q^{92} + 12052256 q^{94} - 32534160 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1591 x^{14} + 998837 x^{12} + 319862003 x^{10} + 57017400035 x^{8} + 5819167911653 x^{6} + \cdots + 11\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2095593961 \nu^{14} - 3169890760198 \nu^{12} + \cdots - 27\!\cdots\!56 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2095593961 \nu^{14} + 3169890760198 \nu^{12} + \cdots + 32\!\cdots\!32 ) / 40\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!73 \nu^{14} + \cdots - 28\!\cdots\!88 ) / 49\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 35\!\cdots\!81 \nu^{14} + \cdots + 69\!\cdots\!40 ) / 14\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 121102446247 \nu^{15} + 183841063433362 \nu^{13} + \cdots + 23\!\cdots\!08 \nu ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\!\cdots\!29 \nu^{14} + \cdots + 90\!\cdots\!44 ) / 73\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33\!\cdots\!59 \nu^{14} + \cdots - 26\!\cdots\!12 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 121102446247 \nu^{15} - 183841063433362 \nu^{13} + \cdots - 21\!\cdots\!28 \nu ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\!\cdots\!79 \nu^{14} + \cdots + 53\!\cdots\!92 ) / 73\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67\!\cdots\!51 \nu^{15} + \cdots + 66\!\cdots\!24 \nu ) / 62\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19\!\cdots\!07 \nu^{15} + \cdots - 11\!\cdots\!80 \nu ) / 12\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!77 \nu^{15} + \cdots + 15\!\cdots\!08 \nu ) / 31\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 80\!\cdots\!75 \nu^{15} + \cdots + 75\!\cdots\!00 \nu ) / 12\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 35\!\cdots\!67 \nu^{15} + \cdots - 47\!\cdots\!48 \nu ) / 51\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 33\!\cdots\!01 \nu^{15} + \cdots + 62\!\cdots\!64 \nu ) / 31\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + 13\beta_{5} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + 13\beta _1 - 2583 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{15} - 13\beta_{14} - 10\beta_{12} + 6\beta_{11} - 325\beta_{8} - 4349\beta_{5} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{9} + 52\beta_{7} - 18\beta_{6} + 67\beta_{4} - 1563\beta_{3} - 1503\beta_{2} - 6723\beta _1 + 864902 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2498 \beta_{15} + 7046 \beta_{14} - 1920 \beta_{13} + 5692 \beta_{12} - 8636 \beta_{11} + \cdots + 1787549 \beta_{5} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 614 \beta_{9} - 35232 \beta_{7} + 8892 \beta_{6} - 50290 \beta_{4} + 1138178 \beta_{3} + \cdots - 355199513 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1565725 \beta_{15} - 3357835 \beta_{14} + 1345984 \beta_{13} - 2650006 \beta_{12} + \cdots - 789907117 \beta_{5} ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 77041 \beta_{9} + 18533444 \beta_{7} - 3480702 \beta_{6} + 28700653 \beta_{4} - 607070229 \beta_{3} + \cdots + 156793010544 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 886481140 \beta_{15} + 1575202876 \beta_{14} - 709523200 \beta_{13} + 1137492760 \beta_{12} + \cdots + 359691594733 \beta_{5} ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 350132260 \beta_{9} - 8931498272 \beta_{7} + 1303760088 \beta_{6} - 15009735188 \beta_{4} + \cdots - 71332230924843 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 478095208231 \beta_{15} - 740641622201 \beta_{14} + 338842305664 \beta_{13} + \cdots - 166326435048989 \beta_{5} ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 348642011475 \beta_{9} + 4132838704692 \beta_{7} - 485859842634 \beta_{6} + 7574372804679 \beta_{4} + \cdots + 32\!\cdots\!30 ) / 13 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 251000441048214 \beta_{15} + 349852894181154 \beta_{14} - 154716697982080 \beta_{13} + \cdots + 77\!\cdots\!57 \beta_{5} ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 258518026429598 \beta_{9} + \cdots - 15\!\cdots\!17 ) / 13 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 12\!\cdots\!33 \beta_{15} + \cdots - 36\!\cdots\!93 \beta_{5} ) / 13 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
168.1
− 20.7945i
− 22.0918i
− 16.2034i
− 9.24529i
− 11.1141i
− 10.7397i
− 5.78595i
− 7.08346i
7.08346i
5.78595i
10.7397i
11.1141i
9.24529i
16.2034i
22.0918i
20.7945i
− 21.7945i −67.8320 −346.998 − 157.804i 1478.36i 842.108i 4772.94i 2414.17 −3439.24
168.2 − 21.0918i 5.55059 −316.864 126.804i − 117.072i 29.3160i 3983.49i −2156.19 2674.54
168.3 − 17.2034i 86.0173 −167.957 − 289.904i − 1479.79i − 968.548i 687.398i 5211.97 −4987.34
168.4 − 10.2453i 9.69718 23.0341 294.418i − 99.3504i 1430.82i − 1547.39i −2092.96 3016.39
168.5 − 10.1141i −70.4106 25.7042 − 163.930i 712.143i − 246.938i − 1554.59i 2770.65 −1658.01
168.6 − 9.73969i 69.5263 33.1385 − 459.709i − 677.164i 861.089i − 1569.44i 2646.91 −4477.42
168.7 − 6.78595i −30.1769 81.9509 − 93.2351i 204.779i − 1618.45i − 1424.72i −1276.35 −632.688
168.8 − 6.08346i 25.6281 90.9916 442.310i − 155.908i − 761.546i − 1332.23i −1530.20 2690.77
168.9 6.08346i 25.6281 90.9916 − 442.310i 155.908i 761.546i 1332.23i −1530.20 2690.77
168.10 6.78595i −30.1769 81.9509 93.2351i − 204.779i 1618.45i 1424.72i −1276.35 −632.688
168.11 9.73969i 69.5263 33.1385 459.709i 677.164i − 861.089i 1569.44i 2646.91 −4477.42
168.12 10.1141i −70.4106 25.7042 163.930i − 712.143i 246.938i 1554.59i 2770.65 −1658.01
168.13 10.2453i 9.69718 23.0341 − 294.418i 99.3504i − 1430.82i 1547.39i −2092.96 3016.39
168.14 17.2034i 86.0173 −167.957 289.904i 1479.79i 968.548i − 687.398i 5211.97 −4987.34
168.15 21.0918i 5.55059 −316.864 − 126.804i 117.072i − 29.3160i − 3983.49i −2156.19 2674.54
168.16 21.7945i −67.8320 −346.998 157.804i − 1478.36i − 842.108i − 4772.94i 2414.17 −3439.24
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 168.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.8.b.e 16
13.b even 2 1 inner 169.8.b.e 16
13.d odd 4 1 169.8.a.e 8
13.d odd 4 1 169.8.a.f 8
13.f odd 12 2 13.8.c.a ✓ 16
39.k even 12 2 117.8.g.d 16
52.l even 12 2 208.8.i.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.c.a ✓ 16 13.f odd 12 2
117.8.g.d 16 39.k even 12 2
169.8.a.e 8 13.d odd 4 1
169.8.a.f 8 13.d odd 4 1
169.8.b.e 16 1.a even 1 1 trivial
169.8.b.e 16 13.b even 2 1 inner
208.8.i.d 16 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 1601 T_{2}^{14} + 1009056 T_{2}^{12} + 322391392 T_{2}^{10} + 56816563968 T_{2}^{8} + \cdots + 10\!\cdots\!04 \) acting on \(S_{8}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( (T^{8} + \cdots - 1189005516288)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 18\!\cdots\!33)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 21\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 50\!\cdots\!43)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 85\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 81\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 12\!\cdots\!75)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
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