Properties

Label 161.4.c.b
Level $161$
Weight $4$
Character orbit 161.c
Analytic conductor $9.499$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,4,Mod(160,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.160");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 161.c (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: \(9.49930751092\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} - 2516x^{5} + 43236x^{4} + 20128x^{3} - 55968x^{2} - 6463604x + 23372755 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{2} q^{3} - 7 q^{4} + ( - \beta_{3} - \beta_1) q^{5} + \beta_{2} q^{6} - \beta_{4} q^{7} + 15 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{2} q^{3} - 7 q^{4} + ( - \beta_{3} - \beta_1) q^{5} + \beta_{2} q^{6} - \beta_{4} q^{7} + 15 q^{8} - 7 q^{9} + (\beta_{3} + \beta_1) q^{10} + ( - \beta_{4} + \beta_1) q^{11} + 7 \beta_{2} q^{12} + ( - \beta_{6} + \beta_{2}) q^{13} + \beta_{4} q^{14} + (\beta_{7} + 3 \beta_{4} - 2 \beta_{3}) q^{15} + 41 q^{16} + (\beta_{3} + \beta_1) q^{17} + 7 q^{18} + ( - \beta_{7} + 3 \beta_{4} - 6 \beta_{3}) q^{19} + (7 \beta_{3} + 7 \beta_1) q^{20} + (\beta_{4} - 7 \beta_{3}) q^{21} + (\beta_{4} - \beta_1) q^{22} + (\beta_{7} + \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_1 - 16) q^{23} - 15 \beta_{2} q^{24} + (\beta_{7} + 2 \beta_{5} + \beta_{4} + 121) q^{25} + (\beta_{6} - \beta_{2}) q^{26} - 20 \beta_{2} q^{27} + 7 \beta_{4} q^{28} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} + 40) q^{29} + ( - \beta_{7} - 3 \beta_{4} + 2 \beta_{3}) q^{30} + (7 \beta_{6} + 6 \beta_{2}) q^{31} - 161 q^{32} + ( - \beta_{7} + 3 \beta_{4} - 6 \beta_{3}) q^{33} + ( - \beta_{3} - \beta_1) q^{34} + ( - 7 \beta_{6} + 28 \beta_{2} + 112) q^{35} + 49 q^{36} + (2 \beta_{7} - 5 \beta_{4} - 4 \beta_{3} + 11 \beta_1) q^{37} + (\beta_{7} - 3 \beta_{4} + 6 \beta_{3}) q^{38} + (\beta_{7} + 2 \beta_{5} + \beta_{4} + 16) q^{39} + ( - 15 \beta_{3} - 15 \beta_1) q^{40} + ( - 8 \beta_{6} + 36 \beta_{2}) q^{41} + ( - \beta_{4} + 7 \beta_{3}) q^{42} + (\beta_{7} - 10 \beta_{4} - 2 \beta_{3} + 13 \beta_1) q^{43} + (7 \beta_{4} - 7 \beta_1) q^{44} + (7 \beta_{3} + 7 \beta_1) q^{45} + ( - \beta_{7} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_1 + 16) q^{46} + (7 \beta_{6} + 34 \beta_{2}) q^{47} - 41 \beta_{2} q^{48} + ( - \beta_{7} - 8 \beta_{6} - 2 \beta_{5} - \beta_{4} - 8 \beta_{2} - 95) q^{49} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} - 121) q^{50} + ( - \beta_{7} - 3 \beta_{4} + 2 \beta_{3}) q^{51} + (7 \beta_{6} - 7 \beta_{2}) q^{52} + (6 \beta_{7} + 9 \beta_{4} - 12 \beta_{3} + 9 \beta_1) q^{53} + 20 \beta_{2} q^{54} + ( - 8 \beta_{6} + 16 \beta_{2}) q^{55} - 15 \beta_{4} q^{56} + (34 \beta_{4} - 34 \beta_1) q^{57} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - 40) q^{58} + (8 \beta_{6} - 49 \beta_{2}) q^{59} + ( - 7 \beta_{7} - 21 \beta_{4} + 14 \beta_{3}) q^{60} + (2 \beta_{7} - 6 \beta_{4} - 5 \beta_{3} - 17 \beta_1) q^{61} + ( - 7 \beta_{6} - 6 \beta_{2}) q^{62} + 7 \beta_{4} q^{63} - 167 q^{64} + ( - 4 \beta_{7} - 28 \beta_{4} + 8 \beta_{3} + 16 \beta_1) q^{65} + (\beta_{7} - 3 \beta_{4} + 6 \beta_{3}) q^{66} + ( - 8 \beta_{7} - 19 \beta_{4} + 16 \beta_{3} - 5 \beta_1) q^{67} + ( - 7 \beta_{3} - 7 \beta_1) q^{68} + (\beta_{7} + 17 \beta_{6} - 3 \beta_{4} + 23 \beta_{3} + 7 \beta_{2} + 17 \beta_1) q^{69} + (7 \beta_{6} - 28 \beta_{2} - 112) q^{70} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} + 432) q^{71} - 105 q^{72} + ( - 14 \beta_{6} + 62 \beta_{2}) q^{73} + ( - 2 \beta_{7} + 5 \beta_{4} + 4 \beta_{3} - 11 \beta_1) q^{74} + (34 \beta_{6} - 139 \beta_{2}) q^{75} + (7 \beta_{7} - 21 \beta_{4} + 42 \beta_{3}) q^{76} + ( - 7 \beta_{6} - 21 \beta_{2} - 280) q^{77} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} - 16) q^{78} + ( - 2 \beta_{7} + 42 \beta_{4} + 4 \beta_{3} - 48 \beta_1) q^{79} + ( - 41 \beta_{3} - 41 \beta_1) q^{80} - 869 q^{81} + (8 \beta_{6} - 36 \beta_{2}) q^{82} + ( - 5 \beta_{7} + 15 \beta_{4} - 18 \beta_{3} + 12 \beta_1) q^{83} + ( - 7 \beta_{4} + 49 \beta_{3}) q^{84} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} - 246) q^{85} + ( - \beta_{7} + 10 \beta_{4} + 2 \beta_{3} - 13 \beta_1) q^{86} + ( - 34 \beta_{6} - 22 \beta_{2}) q^{87} + ( - 15 \beta_{4} + 15 \beta_1) q^{88} + (3 \beta_{7} - 9 \beta_{4} + 63 \beta_{3} + 45 \beta_1) q^{89} + ( - 7 \beta_{3} - 7 \beta_1) q^{90} + ( - 13 \beta_{4} + 28 \beta_{3} + 49 \beta_1) q^{91} + ( - 7 \beta_{7} - 7 \beta_{5} - 21 \beta_{4} + 7 \beta_{3} + 7 \beta_1 + 112) q^{92} + ( - 7 \beta_{7} - 14 \beta_{5} - 7 \beta_{4} + 330) q^{93} + ( - 7 \beta_{6} - 34 \beta_{2}) q^{94} + (8 \beta_{7} + 16 \beta_{5} + 8 \beta_{4} + 400) q^{95} + 161 \beta_{2} q^{96} + (7 \beta_{7} - 21 \beta_{4} - 7 \beta_{3} - 49 \beta_1) q^{97} + (\beta_{7} + 8 \beta_{6} + 2 \beta_{5} + \beta_{4} + 8 \beta_{2} + 95) q^{98} + (7 \beta_{4} - 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 56 q^{4} + 120 q^{8} - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 56 q^{4} + 120 q^{8} - 56 q^{9} + 328 q^{16} + 56 q^{18} - 124 q^{23} + 976 q^{25} + 312 q^{29} - 1288 q^{32} + 896 q^{35} + 392 q^{36} + 136 q^{39} + 124 q^{46} - 768 q^{49} - 976 q^{50} - 312 q^{58} - 1336 q^{64} - 896 q^{70} + 3448 q^{71} - 840 q^{72} - 2240 q^{77} - 136 q^{78} - 6952 q^{81} - 1976 q^{85} + 868 q^{92} + 2584 q^{93} + 3264 q^{95} + 768 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} - 2516x^{5} + 43236x^{4} + 20128x^{3} - 55968x^{2} - 6463604x + 23372755 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 312796779 \nu^{7} - 4501364056 \nu^{6} - 8102975775 \nu^{5} + 829696059414 \nu^{4} - 2127136655513 \nu^{3} + \cdots + 12\!\cdots\!90 ) / 201720841645470 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5190913 \nu^{7} + 15009457 \nu^{6} + 11579205 \nu^{5} - 13251233983 \nu^{4} + 182223722761 \nu^{3} + 424559789633 \nu^{2} + \cdots - 24605864501855 ) / 3240495448120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1901486953 \nu^{7} + 17906311497 \nu^{6} + 30216937765 \nu^{5} - 4804602683303 \nu^{4} + 40078148590801 \nu^{3} + \cdots - 86\!\cdots\!95 ) / 806883366581880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 389913038 \nu^{7} + 990048397 \nu^{6} - 9303870925 \nu^{5} - 964117175938 \nu^{4} + 14898947751346 \nu^{3} + \cdots - 20\!\cdots\!30 ) / 100860420822735 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5054316871 \nu^{7} + 5368063059 \nu^{6} - 302517171285 \nu^{5} - 16872010275461 \nu^{4} + 193879539447807 \nu^{3} + \cdots - 10\!\cdots\!05 ) / 806883366581880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 88808803 \nu^{7} + 205555327 \nu^{6} + 422480975 \nu^{5} - 222814087953 \nu^{4} + 3324397362451 \nu^{3} + \cdots - 438841756672305 ) / 9721486344360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7210546463 \nu^{7} - 20364939287 \nu^{6} + 145099265185 \nu^{5} + 18628750106953 \nu^{4} - 250873656237351 \nu^{3} + \cdots + 35\!\cdots\!25 ) / 403441683290940 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 4\beta_{4} - 9\beta_{3} + 7\beta_{2} - 8\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{7} + 21\beta_{6} - 9\beta_{4} - 13\beta_{3} - 140\beta_{2} - 17\beta _1 + 28 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 309\beta_{7} + 126\beta_{5} + 788\beta_{4} + 1153\beta_{3} + 203\beta_{2} + 874\beta _1 + 13146 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 39 \beta_{7} - 546 \beta_{6} - 1554 \beta_{5} + 5381 \beta_{4} - 11234 \beta_{3} + 3640 \beta_{2} - 8942 \beta _1 - 150101 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 41607 \beta_{7} + 132090 \beta_{6} + 1680 \beta_{5} - 269958 \beta_{4} + 335403 \beta_{3} - 807583 \beta_{2} + 266958 \beta _1 + 175280 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 530785 \beta_{7} - 967113 \beta_{6} + 139860 \beta_{5} + 1450916 \beta_{4} + 1932491 \beta_{3} + 5896268 \beta_{2} + 1394901 \beta _1 + 13216168 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 14120259 \beta_{7} - 924630 \beta_{6} - 13169730 \beta_{5} - 9511498 \beta_{4} - 100864787 \beta_{3} + 5541613 \beta_{2} - 76204388 \beta _1 - 1242867654 ) / 14 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).

\(n\) \(24\) \(120\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1
−10.4466 + 12.7456i
−3.79050 + 4.28219i
3.79050 + 1.54876i
10.4466 6.91469i
−10.4466 12.7456i
−3.79050 4.28219i
3.79050 1.54876i
10.4466 + 6.91469i
−1.00000 5.83095i −7.00000 −20.8933 5.83095i −5.36058 17.7275i 15.0000 −7.00000 20.8933
160.2 −1.00000 5.83095i −7.00000 −7.58101 5.83095i −14.7738 + 11.1685i 15.0000 −7.00000 7.58101
160.3 −1.00000 5.83095i −7.00000 7.58101 5.83095i 14.7738 11.1685i 15.0000 −7.00000 −7.58101
160.4 −1.00000 5.83095i −7.00000 20.8933 5.83095i 5.36058 + 17.7275i 15.0000 −7.00000 −20.8933
160.5 −1.00000 5.83095i −7.00000 −20.8933 5.83095i −5.36058 + 17.7275i 15.0000 −7.00000 20.8933
160.6 −1.00000 5.83095i −7.00000 −7.58101 5.83095i −14.7738 11.1685i 15.0000 −7.00000 7.58101
160.7 −1.00000 5.83095i −7.00000 7.58101 5.83095i 14.7738 + 11.1685i 15.0000 −7.00000 −7.58101
160.8 −1.00000 5.83095i −7.00000 20.8933 5.83095i 5.36058 17.7275i 15.0000 −7.00000 −20.8933
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.c.b 8
7.b odd 2 1 inner 161.4.c.b 8
23.b odd 2 1 inner 161.4.c.b 8
161.c even 2 1 inner 161.4.c.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.c.b 8 1.a even 1 1 trivial
161.4.c.b 8 7.b odd 2 1 inner
161.4.c.b 8 23.b odd 2 1 inner
161.4.c.b 8 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 34)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 494 T^{2} + 25088)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 384 T^{6} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{4} + 878 T^{2} + 156800)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2130 T^{2} + 1098304)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 494 T^{2} + 25088)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 29852 T^{2} + \cdots + 181260800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 62 T^{3} - 10626 T^{2} + \cdots + 148035889)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 78 T - 34400)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 109674 T^{2} + \cdots + 2371690000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 137518 T^{2} + \cdots + 648288032)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 204864 T^{2} + \cdots + 1075840000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 143962 T^{2} + \cdots + 955194632)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 199162 T^{2} + \cdots + 15649936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 587646 T^{2} + \cdots + 83535828768)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 272932 T^{2} + 1522756)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 206686 T^{2} + \cdots + 5188137248)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1031614 T^{2} + \cdots + 184044871808)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 862 T + 149840)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 619848 T^{2} + \cdots + 10862642176)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1933424 T^{2} + \cdots + 441709764608)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 719516 T^{2} + \cdots + 29725071488)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 1489338 T^{2} + \cdots + 14041528200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2089066 T^{2} + \cdots + 279557553800)^{2} \) Copy content Toggle raw display
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