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Properties

Label 169.5.d.c
Level $169$
Weight $5$
Character orbit 169.d
Analytic conductor $17.470$
Analytic rank $0$
Dimension $16$
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] = [169,5,Mod(70,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.70");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 169.d (of order \(4\), degree \(2\), 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: \(17.4695237612\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 152 x^{14} + 9190 x^{12} + 285720 x^{10} + 4862025 x^{8} + 43573680 x^{6} + 169417008 x^{4} + \cdots + 3779136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 13)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + (\beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{10} - \beta_{9} + \cdots + 8 \beta_{3}) q^{4}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots + 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + (\beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{10} - \beta_{9} + \cdots + 8 \beta_{3}) q^{4}+ \cdots + (14 \beta_{14} - 112 \beta_{13} + \cdots + 3922) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 4 q^{3} - 8 q^{5} + 128 q^{6} - 56 q^{7} - 90 q^{8} + 328 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 4 q^{3} - 8 q^{5} + 128 q^{6} - 56 q^{7} - 90 q^{8} + 328 q^{9} - 500 q^{11} + 808 q^{14} - 844 q^{15} - 460 q^{16} - 2434 q^{18} - 1712 q^{19} - 838 q^{20} + 1076 q^{21} + 3048 q^{22} - 5772 q^{24} + 3352 q^{27} - 2980 q^{28} - 1628 q^{29} - 4060 q^{31} - 3662 q^{32} + 860 q^{33} + 2502 q^{34} + 9784 q^{35} - 10468 q^{37} + 18816 q^{40} - 3440 q^{41} + 2408 q^{42} - 10736 q^{44} + 1004 q^{45} - 11436 q^{46} - 1484 q^{47} + 6004 q^{48} - 21616 q^{50} + 7204 q^{53} - 6760 q^{54} - 13872 q^{55} - 12736 q^{57} - 7974 q^{58} + 4840 q^{59} + 6472 q^{60} + 324 q^{61} + 13988 q^{63} - 23872 q^{66} - 2216 q^{67} + 12888 q^{68} + 34524 q^{70} + 1240 q^{71} + 6606 q^{72} + 15448 q^{73} + 5764 q^{74} + 39688 q^{76} - 17064 q^{79} + 37630 q^{80} - 4256 q^{81} - 12788 q^{83} + 7736 q^{84} + 22884 q^{85} - 67260 q^{86} - 58684 q^{87} + 37624 q^{89} - 49884 q^{92} + 2492 q^{93} + 61212 q^{94} + 94664 q^{96} + 57056 q^{97} - 27950 q^{98} + 21632 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 152 x^{14} + 9190 x^{12} + 285720 x^{10} + 4862025 x^{8} + 43573680 x^{6} + 169417008 x^{4} + \cdots + 3779136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2495 \nu^{14} + 3822073 \nu^{12} + 342888545 \nu^{10} + 9265559079 \nu^{8} + \cdots + 13453755092928 ) / 1848457158912 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 759103 \nu^{15} + 12268161 \nu^{14} - 141903785 \nu^{13} + 1670382423 \nu^{12} + \cdots + 19280394785088 ) / 33272228860416 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2297 \nu^{15} - 204802 \nu^{13} - 1715276 \nu^{11} + 329438598 \nu^{9} + \cdots + 461609215200 \nu ) / 84878134848 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53905 \nu^{15} - 1946495 \nu^{13} + 403691033 \nu^{11} + 33967214799 \nu^{9} + \cdots + 32386072836672 \nu ) / 1848457158912 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 759103 \nu^{15} + 12268161 \nu^{14} + 141903785 \nu^{13} + 1670382423 \nu^{12} + \cdots + 52552623645504 ) / 33272228860416 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 958933 \nu^{14} + 127064675 \nu^{12} + 6269679835 \nu^{10} + 142169221341 \nu^{8} + \cdots - 17722730943936 ) / 616152386304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1159493 \nu^{14} - 158929651 \nu^{12} - 8274255659 \nu^{10} - 206871004557 \nu^{8} + \cdots + 1613002009536 ) / 616152386304 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 215356 \nu^{15} + 55893 \nu^{14} - 30842309 \nu^{13} + 9623157 \nu^{12} + \cdots + 2257820868672 ) / 462114289728 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 215356 \nu^{15} + 55893 \nu^{14} + 30842309 \nu^{13} + 9623157 \nu^{12} + \cdots + 2257820868672 ) / 462114289728 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3060949 \nu^{15} + 538883915 \nu^{13} + 37885091971 \nu^{11} + 1360934568357 \nu^{9} + \cdots + 765185781986496 \nu ) / 5545371476736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1329607 \nu^{15} + 3102951 \nu^{14} - 195461153 \nu^{13} + 427193457 \nu^{12} + \cdots - 16283407247424 ) / 1584391850496 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1329607 \nu^{15} - 3102951 \nu^{14} - 195461153 \nu^{13} - 427193457 \nu^{12} + \cdots + 16283407247424 ) / 1584391850496 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6779555 \nu^{15} - 975858589 \nu^{13} - 54738814373 \nu^{11} - 1537765043619 \nu^{9} + \cdots - 220416880660416 \nu ) / 5545371476736 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1350115 \nu^{15} + 2814993 \nu^{14} - 195044105 \nu^{13} + 372172023 \nu^{12} + \cdots + 7337428110144 ) / 792195925248 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4114661 \nu^{15} + 8444979 \nu^{14} + 590866771 \nu^{13} + 1116516069 \nu^{12} + \cdots + 22012284330432 ) / 2376587775744 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{15} + 3 \beta_{14} - 6 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots + 10 ) / 26 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{15} + 4 \beta_{14} - 7 \beta_{12} + 7 \beta_{11} + 6 \beta_{9} + 6 \beta_{8} + 18 \beta_{7} + \cdots - 237 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 46 \beta_{15} - 46 \beta_{14} + 105 \beta_{13} + 35 \beta_{12} + 35 \beta_{11} + 61 \beta_{10} + \cdots + 137 ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 175 \beta_{15} - 175 \beta_{14} + 381 \beta_{12} - 381 \beta_{11} - 126 \beta_{9} - 126 \beta_{8} + \cdots + 7317 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1738 \beta_{15} + 1738 \beta_{14} - 3697 \beta_{13} - 1891 \beta_{12} - 1891 \beta_{11} + \cdots - 13919 ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8533 \beta_{15} + 8533 \beta_{14} - 18797 \beta_{12} + 18797 \beta_{11} + 3810 \beta_{9} + \cdots - 277557 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5734 \beta_{15} - 5734 \beta_{14} + 10949 \beta_{13} + 8055 \beta_{12} + 8055 \beta_{11} + 12751 \beta_{10} + \cdots + 62363 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 430867 \beta_{15} - 430867 \beta_{14} + 918997 \beta_{12} - 918997 \beta_{11} - 169350 \beta_{9} + \cdots + 12044181 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3450646 \beta_{15} + 3450646 \beta_{14} - 5961249 \beta_{13} - 5621987 \beta_{12} - 5621987 \beta_{11} + \cdots - 42409007 ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21896869 \beta_{15} + 21896869 \beta_{14} - 45236733 \beta_{12} + 45236733 \beta_{11} + \cdots - 563039973 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 166473118 \beta_{15} - 166473118 \beta_{14} + 266960641 \beta_{13} + 294126187 \beta_{12} + \cdots + 2153831855 ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1111784059 \beta_{15} - 1111784059 \beta_{14} + 2244399605 \beta_{12} - 2244399605 \beta_{11} + \cdots + 27332655477 ) / 13 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8204329606 \beta_{15} + 8204329606 \beta_{14} - 12528591137 \beta_{13} - 15142173123 \beta_{12} + \cdots - 108512590607 ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4333135633 \beta_{15} + 4333135633 \beta_{14} - 8616856273 \beta_{12} + 8616856273 \beta_{11} + \cdots - 103986087897 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 408752692174 \beta_{15} - 408752692174 \beta_{14} + 605785013889 \beta_{13} + 772345024907 \beta_{12} + \cdots + 5458332935375 ) / 13 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1
− 4.89748i
− 0.200628i
5.33868i
3.68702i
− 7.09996i
0.816521i
− 3.98977i
4.34562i
4.89748i
0.200628i
− 5.33868i
− 3.68702i
7.09996i
− 0.816521i
3.98977i
− 4.34562i
−4.64224 + 4.64224i −14.5736 − 27.1007i 20.2793 − 20.2793i 67.6539 − 67.6539i 15.6160 + 15.6160i 51.5321 + 51.5321i 131.389 188.283i
70.2 −4.36504 + 4.36504i −5.52024 − 22.1071i 5.37551 − 5.37551i 24.0961 − 24.0961i −63.9230 − 63.9230i 26.6579 + 26.6579i −50.5269 46.9287i
70.3 −2.50360 + 2.50360i 10.3300 3.46397i −8.65321 + 8.65321i −25.8622 + 25.8622i −2.15860 − 2.15860i −48.7300 − 48.7300i 25.7089 − 43.3284i
70.4 −1.47633 + 1.47633i 16.4234 11.6409i −17.2508 + 17.2508i −24.2464 + 24.2464i 8.05998 + 8.05998i −40.8071 − 40.8071i 188.729 − 50.9358i
70.5 1.05833 − 1.05833i −6.58616 13.7599i 10.5663 − 10.5663i −6.97034 + 6.97034i 39.0712 + 39.0712i 31.4958 + 31.4958i −37.6226 − 22.3652i
70.6 1.74699 − 1.74699i −8.51417 9.89603i −16.9748 + 16.9748i −14.8742 + 14.8742i −36.7630 − 36.7630i 45.2402 + 45.2402i −8.50884 59.3096i
70.7 4.03282 − 4.03282i 8.56159 − 16.5272i 29.3294 − 29.3294i 34.5273 − 34.5273i 33.5543 + 33.5543i −2.12627 − 2.12627i −7.69920 − 236.560i
70.8 5.14906 − 5.14906i 1.87913 − 37.0257i −26.6717 + 26.6717i 9.67578 − 9.67578i −21.4569 − 21.4569i −108.263 − 108.263i −77.4689 274.669i
99.1 −4.64224 − 4.64224i −14.5736 27.1007i 20.2793 + 20.2793i 67.6539 + 67.6539i 15.6160 − 15.6160i 51.5321 − 51.5321i 131.389 − 188.283i
99.2 −4.36504 − 4.36504i −5.52024 22.1071i 5.37551 + 5.37551i 24.0961 + 24.0961i −63.9230 + 63.9230i 26.6579 − 26.6579i −50.5269 − 46.9287i
99.3 −2.50360 − 2.50360i 10.3300 − 3.46397i −8.65321 − 8.65321i −25.8622 − 25.8622i −2.15860 + 2.15860i −48.7300 + 48.7300i 25.7089 43.3284i
99.4 −1.47633 − 1.47633i 16.4234 − 11.6409i −17.2508 − 17.2508i −24.2464 − 24.2464i 8.05998 − 8.05998i −40.8071 + 40.8071i 188.729 50.9358i
99.5 1.05833 + 1.05833i −6.58616 − 13.7599i 10.5663 + 10.5663i −6.97034 − 6.97034i 39.0712 − 39.0712i 31.4958 − 31.4958i −37.6226 22.3652i
99.6 1.74699 + 1.74699i −8.51417 − 9.89603i −16.9748 − 16.9748i −14.8742 − 14.8742i −36.7630 + 36.7630i 45.2402 − 45.2402i −8.50884 − 59.3096i
99.7 4.03282 + 4.03282i 8.56159 16.5272i 29.3294 + 29.3294i 34.5273 + 34.5273i 33.5543 − 33.5543i −2.12627 + 2.12627i −7.69920 236.560i
99.8 5.14906 + 5.14906i 1.87913 37.0257i −26.6717 − 26.6717i 9.67578 + 9.67578i −21.4569 + 21.4569i −108.263 + 108.263i −77.4689 − 274.669i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.5.d.c 16
13.b even 2 1 169.5.d.d 16
13.d odd 4 1 inner 169.5.d.c 16
13.d odd 4 1 169.5.d.d 16
13.e even 6 1 13.5.f.a ✓ 16
13.f odd 12 1 13.5.f.a ✓ 16
39.h odd 6 1 117.5.bd.c 16
39.k even 12 1 117.5.bd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.5.f.a ✓ 16 13.e even 6 1
13.5.f.a ✓ 16 13.f odd 12 1
117.5.bd.c 16 39.h odd 6 1
117.5.bd.c 16 39.k even 12 1
169.5.d.c 16 1.a even 1 1 trivial
169.5.d.c 16 13.d odd 4 1 inner
169.5.d.d 16 13.b even 2 1
169.5.d.d 16 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 2 T_{2}^{15} + 2 T_{2}^{14} + 10 T_{2}^{13} + 3717 T_{2}^{12} + 8788 T_{2}^{11} + \cdots + 2116736064 \) acting on \(S_{5}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 2116736064 \) Copy content Toggle raw display
$3$ \( (T^{8} - 2 T^{7} + \cdots + 12313296)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 82\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 91\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 45\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 19\!\cdots\!71)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 94\!\cdots\!29 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 87\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 20\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 24\!\cdots\!17)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 77\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 37\!\cdots\!24 \) Copy content Toggle raw display
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