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Properties

Label 161.4.a.d
Level $161$
Weight $4$
Character orbit 161.a
Self dual yes
Analytic conductor $9.499$
Analytic rank $0$
Dimension $12$
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] = [161,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(9.49930751092\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 6) q^{4} + (\beta_{7} + \beta_1 + 1) q^{5} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 1) q^{6} + 7 q^{7} + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots - 5) q^{8}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 6) q^{4} + (\beta_{7} + \beta_1 + 1) q^{5} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 1) q^{6} + 7 q^{7} + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots - 5) q^{8}+ \cdots + (22 \beta_{11} + 31 \beta_{10} + \cdots + 217) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + q^{3} + 72 q^{4} + 16 q^{5} - 15 q^{6} + 84 q^{7} - 21 q^{8} + 203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + q^{3} + 72 q^{4} + 16 q^{5} - 15 q^{6} + 84 q^{7} - 21 q^{8} + 203 q^{9} + 106 q^{10} + 50 q^{11} - 139 q^{12} + 21 q^{13} + 28 q^{14} + 192 q^{15} + 516 q^{16} + 26 q^{17} + 251 q^{18} + 292 q^{19} - 12 q^{20} + 7 q^{21} + 206 q^{22} + 276 q^{23} + 42 q^{24} + 408 q^{25} + 409 q^{26} - 335 q^{27} + 504 q^{28} + 129 q^{29} + 168 q^{30} + 375 q^{31} - 54 q^{32} + 428 q^{33} + 674 q^{34} + 112 q^{35} + 2271 q^{36} + 958 q^{37} - 120 q^{38} + 89 q^{39} + 1424 q^{40} - 347 q^{41} - 105 q^{42} + 392 q^{43} - 1158 q^{44} - 1338 q^{45} + 92 q^{46} + 559 q^{47} - 4767 q^{48} + 588 q^{49} - 888 q^{50} + 204 q^{51} - 1765 q^{52} + 42 q^{53} - 1965 q^{54} + 78 q^{55} - 147 q^{56} + 156 q^{57} - 3695 q^{58} - 620 q^{59} - 1574 q^{60} - 1258 q^{61} + 489 q^{62} + 1421 q^{63} + 4335 q^{64} - 752 q^{65} - 7334 q^{66} + 2024 q^{67} - 5162 q^{68} + 23 q^{69} + 742 q^{70} + 1371 q^{71} - 2483 q^{72} - 1241 q^{73} - 4006 q^{74} + 1357 q^{75} + 4214 q^{76} + 350 q^{77} - 4787 q^{78} + 2660 q^{79} - 5734 q^{80} + 2400 q^{81} - 3551 q^{82} - 1476 q^{83} - 973 q^{84} + 408 q^{85} - 988 q^{86} + 1263 q^{87} + 3324 q^{88} + 1670 q^{89} + 1050 q^{90} + 147 q^{91} + 1656 q^{92} + 135 q^{93} - 3539 q^{94} + 2428 q^{95} - 3973 q^{96} + 1174 q^{97} + 196 q^{98} + 2400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1057087 \nu^{11} - 736072 \nu^{10} - 81046848 \nu^{9} + 59112285 \nu^{8} + 2160058053 \nu^{7} + \cdots + 27581677952 ) / 466611584 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1117105 \nu^{11} - 226867 \nu^{10} - 85952171 \nu^{9} + 21399060 \nu^{8} + 2305747495 \nu^{7} + \cdots + 19319250176 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1519271 \nu^{11} - 498841 \nu^{10} - 117032797 \nu^{9} + 43441900 \nu^{8} + 3145054901 \nu^{7} + \cdots + 27799261696 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1605177 \nu^{11} - 326483 \nu^{10} - 123035627 \nu^{9} + 30155740 \nu^{8} + 3282153271 \nu^{7} + \cdots + 31578353216 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99065 \nu^{11} - 21862 \nu^{10} - 7609238 \nu^{9} + 2030749 \nu^{8} + 203556131 \nu^{7} + \cdots + 1798578304 ) / 6964352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7097167 \nu^{11} - 2425124 \nu^{10} - 546101432 \nu^{9} + 210879061 \nu^{8} + \cdots + 160329330944 ) / 466611584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5914582 \nu^{11} + 1189665 \nu^{10} + 454454473 \nu^{9} - 112071589 \nu^{8} + \cdots - 101406339072 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7229129 \nu^{11} - 1769452 \nu^{10} - 555592220 \nu^{9} + 160203215 \nu^{8} + \cdots + 135239697088 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22127067 \nu^{11} - 4919124 \nu^{10} - 1700971564 \nu^{9} + 457650357 \nu^{8} + \cdots + 427632581376 ) / 466611584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 23\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{9} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} + 6\beta_{4} + \beta_{3} + 31\beta_{2} - 2\beta _1 + 315 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} - 6 \beta_{10} + 36 \beta_{9} + 40 \beta_{7} + 44 \beta_{6} + 44 \beta_{5} + 14 \beta_{4} + \cdots - 185 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27 \beta_{11} - 10 \beta_{10} + 75 \beta_{9} + 16 \beta_{8} - 65 \beta_{7} + 83 \beta_{6} - 109 \beta_{5} + \cdots + 8130 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{11} - 354 \beta_{10} + 1026 \beta_{9} + 48 \beta_{8} + 1242 \beta_{7} + 1570 \beta_{6} + \cdots - 6210 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 476 \beta_{11} - 748 \beta_{10} + 2172 \beta_{9} + 1184 \beta_{8} - 1532 \beta_{7} + 2692 \beta_{6} + \cdots + 224802 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2552 \beta_{11} - 15264 \beta_{10} + 26977 \beta_{9} + 3408 \beta_{8} + 36409 \beta_{7} + \cdots - 202781 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3329 \beta_{11} - 36960 \beta_{10} + 56722 \beta_{9} + 57632 \beta_{8} - 29538 \beta_{7} + \cdots + 6459459 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 100129 \beta_{11} - 579662 \beta_{10} + 683620 \beta_{9} + 162544 \beta_{8} + 1056368 \beta_{7} + \cdots - 6540689 \) 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
−5.60949
−4.89764
−3.54240
−3.06154
−0.718333
0.285899
1.40034
2.11879
3.76511
3.76781
4.95487
5.53660
−5.60949 −7.98435 23.4664 −5.27021 44.7882 7.00000 −86.7589 36.7498 29.5632
1.2 −4.89764 7.57994 15.9869 −14.0191 −37.1238 7.00000 −39.1171 30.4555 68.6607
1.3 −3.54240 −2.15594 4.54860 6.95559 7.63719 7.00000 12.2262 −22.3519 −24.6395
1.4 −3.06154 5.71887 1.37302 21.1978 −17.5086 7.00000 20.2888 5.70552 −64.8978
1.5 −0.718333 −2.25536 −7.48400 −17.9779 1.62010 7.00000 11.1227 −21.9133 12.9141
1.6 0.285899 8.01264 −7.91826 1.18183 2.29081 7.00000 −4.55102 37.2024 0.337884
1.7 1.40034 −4.22464 −6.03905 13.7756 −5.91594 7.00000 −19.6594 −9.15238 19.2904
1.8 2.11879 −9.53944 −3.51074 −13.8523 −20.2120 7.00000 −24.3888 64.0009 −29.3501
1.9 3.76511 9.31158 6.17602 1.25334 35.0591 7.00000 −6.86747 59.7056 4.71897
1.10 3.76781 2.58468 6.19640 21.4417 9.73858 7.00000 −6.79562 −20.3194 80.7883
1.11 4.95487 3.24611 16.5508 −2.30377 16.0841 7.00000 42.3679 −16.4628 −11.4149
1.12 5.53660 −9.29409 22.6539 3.61751 −51.4577 7.00000 81.1328 59.3802 20.0287
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 161.4.a.d ✓ 12
3.b odd 2 1 1449.4.a.o 12
7.b odd 2 1 1127.4.a.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.a.d ✓ 12 1.a even 1 1 trivial
1127.4.a.h 12 7.b odd 2 1
1449.4.a.o 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 4 T_{2}^{11} - 76 T_{2}^{10} + 311 T_{2}^{9} + 1979 T_{2}^{8} - 8466 T_{2}^{7} + \cdots - 70656 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(161))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{11} + \cdots - 70656 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 394600512 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 9891773184 \) Copy content Toggle raw display
$7$ \( (T - 7)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 23\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 14\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 63\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( (T - 23)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 52\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 33\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 28\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 32\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 75\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 36\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 39\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 60\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 26\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 81\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 22\!\cdots\!52 \) Copy content Toggle raw display
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