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Properties

Label 61.8.a.a
Level $61$
Weight $8$
Character orbit 61.a
Self dual yes
Analytic conductor $19.055$
Analytic rank $1$
Dimension $16$
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] = [61,8,Mod(1,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 61.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: \(19.0554865545\)
Analytic rank: \(1\)
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: \( x^{16} - x^{15} - 1385 x^{14} + 1645 x^{13} + 736192 x^{12} - 959218 x^{11} - 191200060 x^{10} + \cdots + 620354245761376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{5} - 5) q^{3} + (\beta_{2} + 2 \beta_1 + 46) q^{4} + ( - \beta_{8} + \beta_{5} + \beta_1 - 36) q^{5} + (\beta_{9} + \beta_{8} + 2 \beta_{5} + \cdots - 81) q^{6}+ \cdots + ( - \beta_{14} - 4 \beta_{13} + \cdots + 455) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{5} - 5) q^{3} + (\beta_{2} + 2 \beta_1 + 46) q^{4} + ( - \beta_{8} + \beta_{5} + \beta_1 - 36) q^{5} + (\beta_{9} + \beta_{8} + 2 \beta_{5} + \cdots - 81) q^{6}+ \cdots + (10738 \beta_{15} + 2233 \beta_{14} + \cdots - 4092528) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 17 q^{2} - 82 q^{3} + 741 q^{4} - 571 q^{5} - 1295 q^{6} - 355 q^{7} - 3201 q^{8} + 7314 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 17 q^{2} - 82 q^{3} + 741 q^{4} - 571 q^{5} - 1295 q^{6} - 355 q^{7} - 3201 q^{8} + 7314 q^{9} - 971 q^{10} - 18035 q^{11} - 14009 q^{12} - 4495 q^{13} - 32184 q^{14} - 25994 q^{15} + 95341 q^{16} - 29528 q^{17} + 5142 q^{18} - 56756 q^{19} - 170873 q^{20} - 121436 q^{21} - 232781 q^{22} - 340663 q^{23} - 596965 q^{24} - 40625 q^{25} - 566099 q^{26} - 424144 q^{27} - 907672 q^{28} - 670040 q^{29} - 935468 q^{30} - 497810 q^{31} - 754935 q^{32} - 569344 q^{33} - 661102 q^{34} - 1121485 q^{35} - 1309180 q^{36} - 109786 q^{37} - 1055353 q^{38} - 894234 q^{39} - 678945 q^{40} - 219237 q^{41} + 989598 q^{42} - 1455730 q^{43} - 1575123 q^{44} - 548823 q^{45} - 380526 q^{46} + 176228 q^{47} + 558487 q^{48} + 2263417 q^{49} + 373152 q^{50} - 1890624 q^{51} + 2591901 q^{52} - 1878614 q^{53} + 4064944 q^{54} - 898753 q^{55} - 1748314 q^{56} - 1688544 q^{57} + 6569855 q^{58} - 4956943 q^{59} + 16042082 q^{60} + 3631696 q^{61} + 6090523 q^{62} + 11609765 q^{63} + 13517229 q^{64} + 5917459 q^{65} + 21678498 q^{66} + 4562669 q^{67} + 16164052 q^{68} + 8065122 q^{69} + 16726003 q^{70} - 9512338 q^{71} + 8617724 q^{72} + 4179019 q^{73} + 7436853 q^{74} + 6938352 q^{75} + 22095341 q^{76} - 5208723 q^{77} + 15313466 q^{78} - 11499559 q^{79} - 5330077 q^{80} - 11519332 q^{81} + 13902258 q^{82} - 4419572 q^{83} + 24551540 q^{84} - 6476754 q^{85} - 21603212 q^{86} - 2072772 q^{87} + 11337643 q^{88} - 12894070 q^{89} + 711621 q^{90} - 28043859 q^{91} - 45341460 q^{92} - 26800482 q^{93} + 34842980 q^{94} - 30122012 q^{95} - 12974625 q^{96} - 77260834 q^{97} + 46572131 q^{98} - 65346383 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 1385 x^{14} + 1645 x^{13} + 736192 x^{12} - 959218 x^{11} - 191200060 x^{10} + \cdots + 620354245761376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 173 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50\!\cdots\!73 \nu^{15} + \cdots + 25\!\cdots\!52 ) / 89\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31\!\cdots\!19 \nu^{15} + \cdots - 12\!\cdots\!52 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 50\!\cdots\!45 \nu^{15} + \cdots + 14\!\cdots\!68 ) / 86\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\!\cdots\!15 \nu^{15} + \cdots - 61\!\cdots\!36 ) / 47\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!27 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!15 \nu^{15} + \cdots - 17\!\cdots\!72 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43\!\cdots\!05 \nu^{15} + \cdots - 61\!\cdots\!48 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 54\!\cdots\!11 \nu^{15} + \cdots - 70\!\cdots\!56 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!17 \nu^{15} + \cdots - 64\!\cdots\!40 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 58\!\cdots\!09 \nu^{15} + \cdots - 17\!\cdots\!28 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 41\!\cdots\!37 \nu^{15} + \cdots - 88\!\cdots\!28 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32\!\cdots\!77 \nu^{15} + \cdots - 29\!\cdots\!28 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 50\!\cdots\!21 \nu^{15} + \cdots - 97\!\cdots\!12 ) / 71\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 173 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - 2\beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} + 2\beta_{6} - 13\beta_{5} + 2\beta_{2} + 317\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 17 \beta_{15} - \beta_{14} + 18 \beta_{13} + 16 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} + \cdots + 55592 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 294 \beta_{15} - 59 \beta_{14} - 816 \beta_{13} - 1247 \beta_{12} + 91 \beta_{11} + 685 \beta_{10} + \cdots - 49749 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11091 \beta_{15} - 1424 \beta_{14} + 11177 \beta_{13} + 11091 \beta_{12} - 1328 \beta_{11} + \cdots + 21364244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 246672 \beta_{15} - 45516 \beta_{14} - 517397 \beta_{13} - 634274 \beta_{12} + 77286 \beta_{11} + \cdots - 34841479 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6093027 \beta_{15} - 1074617 \beta_{14} + 6221874 \beta_{13} + 6276294 \beta_{12} - 122621 \beta_{11} + \cdots + 8861792736 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 153617560 \beta_{15} - 24577553 \beta_{14} - 291230572 \beta_{13} - 307480919 \beta_{12} + \cdots - 24036074667 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3206190875 \beta_{15} - 655053778 \beta_{14} + 3372332375 \beta_{13} + 3377448481 \beta_{12} + \cdots + 3836718410648 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 85801024712 \beta_{15} - 11552833386 \beta_{14} - 154131787315 \beta_{13} - 147391130528 \beta_{12} + \cdots - 15542147799611 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1657732512607 \beta_{15} - 359777318619 \beta_{14} + 1798433678456 \beta_{13} + 1784869166280 \beta_{12} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 45511308767376 \beta_{15} - 5067728632299 \beta_{14} - 78872989623114 \beta_{13} - 70640637548313 \beta_{12} + \cdots - 94\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 848729730531339 \beta_{15} - 186312030934140 \beta_{14} + 947657439960273 \beta_{13} + \cdots + 77\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 23\!\cdots\!12 \beta_{15} + \cdots - 54\!\cdots\!95 \) 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
20.7549
19.9673
15.9593
9.87718
8.86740
8.59875
6.94773
0.457125
−2.88930
−2.93652
−6.00242
−9.07147
−13.2436
−15.8260
−18.1614
−22.2991
−21.7549 52.1383 345.276 128.976 −1134.26 −6.40684 −4726.83 531.400 −2805.87
1.2 −20.9673 −32.0998 311.629 −420.671 673.047 20.5229 −3850.21 −1156.60 8820.36
1.3 −16.9593 24.1663 159.618 249.508 −409.843 −1320.93 −536.226 −1602.99 −4231.48
1.4 −10.8772 −68.6919 −9.68694 11.7008 747.174 −806.002 1497.65 2531.58 −127.272
1.5 −9.86740 22.2122 −30.6344 −187.599 −219.177 1243.50 1565.31 −1693.62 1851.12
1.6 −9.59875 −71.9070 −35.8640 −346.448 690.217 1190.10 1572.89 2983.62 3325.46
1.7 −7.94773 45.2379 −64.8336 −7.24731 −359.538 252.374 1532.59 −140.534 57.5997
1.8 −1.45713 −30.4607 −125.877 218.948 44.3851 721.241 369.930 −1259.15 −319.034
1.9 1.88930 −85.4731 −124.431 366.599 −161.484 1002.97 −476.917 5118.66 692.614
1.10 1.93652 68.1342 −124.250 83.7015 131.944 −1156.02 −488.488 2455.27 162.090
1.11 5.00242 72.4549 −102.976 −537.233 362.450 1322.42 −1155.44 3062.72 −2687.46
1.12 8.07147 10.3434 −62.8514 381.592 83.4867 −1301.24 −1540.45 −2080.01 3080.01
1.13 12.2436 −5.20379 21.9047 −0.245446 −63.7129 267.939 −1298.98 −2159.92 −3.00513
1.14 14.8260 28.9491 91.8099 −362.141 429.199 −818.461 −536.554 −1348.95 −5369.09
1.15 17.1614 −65.8229 166.514 51.4287 −1129.61 635.670 660.958 2145.66 882.589
1.16 21.2991 −45.9770 325.651 −201.869 −979.268 −1602.67 4209.78 −73.1142 −4299.62
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(61\) \(-1\)

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 61.8.a.a ✓ 16
3.b odd 2 1 549.8.a.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.8.a.a ✓ 16 1.a even 1 1 trivial
549.8.a.b 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 17 T_{2}^{15} - 1250 T_{2}^{14} - 20370 T_{2}^{13} + 591047 T_{2}^{12} + \cdots + 904656683470848 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(61))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 904656683470848 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots - 85\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 35\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 55\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 53\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 43\!\cdots\!18 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 24\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( (T - 226981)^{16} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 18\!\cdots\!50 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 10\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 56\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
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