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Properties

Label 161.6.a.c
Level $161$
Weight $6$
Character orbit 161.a
Self dual yes
Analytic conductor $25.822$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,6,Mod(1,161)]
 
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([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 161.a (trivial)

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: yes
Analytic conductor: \(25.8217949899\)
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: \( x^{14} - 352 x^{12} - 26 x^{11} + 47103 x^{10} + 4686 x^{9} - 3009340 x^{8} - 392976 x^{7} + \cdots - 2194301952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1 + 2) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 19) q^{4} + (\beta_{5} - \beta_{2} + \beta_1 + 8) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots - 26) q^{6}+ \cdots + ( - \beta_{10} - \beta_{9} - \beta_{6} + \cdots + 80) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1 + 2) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 19) q^{4} + (\beta_{5} - \beta_{2} + \beta_1 + 8) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots - 26) q^{6}+ \cdots + ( - 97 \beta_{13} + 11 \beta_{12} + \cdots - 11522) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 24 q^{3} + 256 q^{4} + 108 q^{5} - 356 q^{6} - 686 q^{7} - 78 q^{8} + 1112 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 24 q^{3} + 256 q^{4} + 108 q^{5} - 356 q^{6} - 686 q^{7} - 78 q^{8} + 1112 q^{9} - 298 q^{10} + 458 q^{11} + 3148 q^{12} + 1484 q^{13} + 3766 q^{15} + 6148 q^{16} + 2178 q^{17} - 3738 q^{18} + 5520 q^{19} + 14152 q^{20} - 1176 q^{21} + 5558 q^{22} + 7406 q^{23} - 8190 q^{24} + 6926 q^{25} + 7288 q^{26} - 8772 q^{27} - 12544 q^{28} - 8194 q^{29} - 1696 q^{30} + 14222 q^{31} - 12346 q^{32} + 21698 q^{33} + 34760 q^{34} - 5292 q^{35} + 58140 q^{36} - 8880 q^{37} + 9468 q^{38} + 33156 q^{39} - 13838 q^{40} + 46764 q^{41} + 17444 q^{42} + 18978 q^{43} + 80286 q^{44} + 99404 q^{45} + 22094 q^{47} + 238392 q^{48} + 33614 q^{49} + 31528 q^{50} + 45212 q^{51} + 111290 q^{52} + 48684 q^{53} + 25532 q^{54} + 154902 q^{55} + 3822 q^{56} + 44942 q^{57} - 57174 q^{58} + 134150 q^{59} + 154584 q^{60} + 3846 q^{61} + 26916 q^{62} - 54488 q^{63} + 168212 q^{64} - 26274 q^{65} - 115298 q^{66} + 78784 q^{67} - 11306 q^{68} + 12696 q^{69} + 14602 q^{70} - 14584 q^{71} - 176210 q^{72} - 41068 q^{73} - 60718 q^{74} + 127514 q^{75} + 131040 q^{76} - 22442 q^{77} - 473954 q^{78} + 84494 q^{79} + 430056 q^{80} - 66986 q^{81} - 219390 q^{82} + 93722 q^{83} - 154252 q^{84} - 154620 q^{85} + 94970 q^{86} + 354496 q^{87} - 77882 q^{88} - 94102 q^{89} - 371532 q^{90} - 72716 q^{91} + 135424 q^{92} - 380154 q^{93} + 116916 q^{94} + 148200 q^{95} - 814078 q^{96} + 384190 q^{97} - 174402 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 352 x^{12} - 26 x^{11} + 47103 x^{10} + 4686 x^{9} - 3009340 x^{8} - 392976 x^{7} + \cdots - 2194301952 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{13} + \cdots + 99\!\cdots\!60 ) / 72\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{13} + \cdots - 27\!\cdots\!32 ) / 72\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!95 \nu^{13} + \cdots - 92\!\cdots\!52 ) / 36\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 80\!\cdots\!29 \nu^{13} + \cdots - 78\!\cdots\!76 ) / 36\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!39 \nu^{13} + \cdots - 20\!\cdots\!64 ) / 72\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!29 \nu^{13} + \cdots + 34\!\cdots\!44 ) / 72\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 72\!\cdots\!77 \nu^{13} + \cdots + 34\!\cdots\!08 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!01 \nu^{13} + \cdots + 22\!\cdots\!24 ) / 25\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46\!\cdots\!57 \nu^{13} + \cdots + 10\!\cdots\!32 ) / 72\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!11 \nu^{13} + \cdots - 20\!\cdots\!00 ) / 36\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 40\!\cdots\!29 \nu^{13} + \cdots + 20\!\cdots\!12 ) / 51\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31\!\cdots\!41 \nu^{13} + \cdots + 23\!\cdots\!44 ) / 36\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + \beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 84\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{13} + 4 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} + \cdots + 4339 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{13} - 10 \beta_{11} - 123 \beta_{10} - 28 \beta_{9} + 165 \beta_{8} - 20 \beta_{7} + \cdots + 1744 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 352 \beta_{13} + 682 \beta_{12} + 872 \beta_{11} + 286 \beta_{10} - 166 \beta_{9} + 332 \beta_{8} + \cdots + 426289 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2436 \beta_{13} + 232 \beta_{12} - 1720 \beta_{11} - 13613 \beta_{10} - 6008 \beta_{9} + \cdots + 397458 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 51858 \beta_{13} + 95564 \beta_{12} + 141360 \beta_{11} + 33810 \beta_{10} - 40624 \beta_{9} + \cdots + 44802011 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 453868 \beta_{13} + 79148 \beta_{12} - 198962 \beta_{11} - 1506567 \beta_{10} - 969656 \beta_{9} + \cdots + 71846424 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7241616 \beta_{13} + 12565386 \beta_{12} + 20302388 \beta_{11} + 3743406 \beta_{10} + \cdots + 4902861433 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 70695576 \beta_{13} + 17249684 \beta_{12} - 17008992 \beta_{11} - 168847889 \beta_{10} + \cdots + 11441628242 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 982558162 \beta_{13} + 1603742508 \beta_{12} + 2734409836 \beta_{11} + 398747562 \beta_{10} + \cdots + 551755529963 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10113167360 \beta_{13} + 3100652168 \beta_{12} - 731364266 \beta_{11} - 19145720227 \beta_{10} + \cdots + 1693898928688 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
11.2204
9.63041
7.38942
6.95375
4.44555
3.69414
0.274238
−1.38940
−2.54612
−4.49209
−6.11177
−9.02350
−9.45431
−10.5907
−11.2204 25.2646 93.8964 87.5136 −283.478 −49.0000 −694.499 395.301 −981.934
1.2 −9.63041 14.7462 60.7448 −33.9515 −142.012 −49.0000 −276.825 −25.5502 326.967
1.3 −7.38942 −1.95430 22.6035 34.2443 14.4411 −49.0000 69.4344 −239.181 −253.046
1.4 −6.95375 −28.9053 16.3547 25.9151 201.000 −49.0000 108.794 592.517 −180.207
1.5 −4.44555 6.11723 −12.2371 −72.2150 −27.1945 −49.0000 196.658 −205.579 321.036
1.6 −3.69414 25.8952 −18.3534 38.4835 −95.6605 −49.0000 186.012 427.563 −142.163
1.7 −0.274238 −12.6540 −31.9248 26.0126 3.47020 −49.0000 17.5306 −82.8771 −7.13365
1.8 1.38940 6.35929 −30.0696 −90.6657 8.83558 −49.0000 −86.2393 −202.559 −125.971
1.9 2.54612 14.0011 −25.5173 76.9446 35.6484 −49.0000 −146.446 −46.9703 195.910
1.10 4.49209 −13.8540 −11.8212 −34.9371 −62.2332 −49.0000 −196.848 −51.0678 −156.941
1.11 6.11177 −22.5174 5.35368 −81.7218 −137.621 −49.0000 −162.856 264.033 −499.464
1.12 9.02350 −20.5877 49.4236 76.3940 −185.773 −49.0000 157.222 180.854 689.342
1.13 9.45431 22.2358 57.3840 68.0220 210.224 −49.0000 239.988 251.432 643.101
1.14 10.5907 9.85319 80.1625 −12.0386 104.352 −49.0000 510.074 −145.915 −127.497
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.6.a.c ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.6.a.c ✓ 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 352 T_{2}^{12} + 26 T_{2}^{11} + 47103 T_{2}^{10} - 4686 T_{2}^{9} - 3009340 T_{2}^{8} + \cdots - 2194301952 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(161))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 2194301952 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 52\!\cdots\!40 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( (T + 49)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 72\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 39\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 42\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T - 529)^{14} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 47\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 42\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 59\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 45\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 70\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 11\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 63\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 17\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 59\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 28\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 40\!\cdots\!40 \) Copy content Toggle raw display
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