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Properties

Label 61.8.a.b
Level $61$
Weight $8$
Character orbit 61.a
Self dual yes
Analytic conductor $19.055$
Analytic rank $0$
Dimension $19$
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: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 4 x^{18} - 1853 x^{17} + 7710 x^{16} + 1430821 x^{15} - 6206876 x^{14} - 595839157 x^{13} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{13}\cdot 3\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + (\beta_{3} + 7) q^{3} + (\beta_{2} + 2 \beta_1 + 69) q^{4} + ( - \beta_{9} + \beta_1 + 22) q^{5} + (\beta_{14} - \beta_{9} - \beta_{3} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{17} + \beta_{16} + \cdots + 963) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + (\beta_{3} + 7) q^{3} + (\beta_{2} + 2 \beta_1 + 69) q^{4} + ( - \beta_{9} + \beta_1 + 22) q^{5} + (\beta_{14} - \beta_{9} - \beta_{3} + \cdots - 2) q^{6}+ \cdots + (4848 \beta_{18} - 907 \beta_{17} + \cdots + 1357618) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 23 q^{2} + 134 q^{3} + 1317 q^{4} + 429 q^{5} + q^{6} + 1017 q^{7} + 4479 q^{8} + 18249 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 23 q^{2} + 134 q^{3} + 1317 q^{4} + 429 q^{5} + q^{6} + 1017 q^{7} + 4479 q^{8} + 18249 q^{9} + 5029 q^{10} + 21895 q^{11} + 13639 q^{12} + 4293 q^{13} + 28184 q^{14} + 28006 q^{15} + 30149 q^{16} - 3512 q^{17} - 52668 q^{18} + 62964 q^{19} + 128631 q^{20} + 151802 q^{21} + 380355 q^{22} + 194883 q^{23} + 643935 q^{24} + 594054 q^{25} + 412571 q^{26} + 544388 q^{27} + 818132 q^{28} + 343584 q^{29} + 1287286 q^{30} + 376400 q^{31} + 1182835 q^{32} + 545950 q^{33} + 952570 q^{34} + 1101957 q^{35} + 1334624 q^{36} + 630086 q^{37} + 263947 q^{38} + 1043878 q^{39} + 78765 q^{40} + 1584539 q^{41} + 510094 q^{42} - 93664 q^{43} + 2646833 q^{44} + 206061 q^{45} - 559318 q^{46} + 1752508 q^{47} - 2563869 q^{48} + 2287550 q^{49} + 411602 q^{50} + 2278880 q^{51} - 3538347 q^{52} + 127548 q^{53} - 8008280 q^{54} - 2282221 q^{55} + 1035726 q^{56} - 2815944 q^{57} - 5418817 q^{58} + 5826729 q^{59} - 8433754 q^{60} - 4312639 q^{61} - 6139425 q^{62} - 10627967 q^{63} - 7157059 q^{64} - 10672623 q^{65} - 15073362 q^{66} - 10206319 q^{67} - 10901542 q^{68} - 11593480 q^{69} - 29382493 q^{70} + 4358512 q^{71} - 23635494 q^{72} - 15042353 q^{73} - 10739915 q^{74} + 9338192 q^{75} - 12059835 q^{76} + 6632395 q^{77} - 41176024 q^{78} + 3473903 q^{79} - 6140121 q^{80} + 21257391 q^{81} - 14428970 q^{82} + 19386052 q^{83} - 10704930 q^{84} - 1181128 q^{85} + 3163940 q^{86} + 23797004 q^{87} + 13465595 q^{88} + 10913876 q^{89} - 18268219 q^{90} + 17042997 q^{91} + 20668398 q^{92} + 23643110 q^{93} - 43306680 q^{94} + 27353940 q^{95} + 24794935 q^{96} + 55835432 q^{97} + 16445433 q^{98} + 26225127 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 4 x^{18} - 1853 x^{17} + 7710 x^{16} + 1430821 x^{15} - 6206876 x^{14} - 595839157 x^{13} + \cdots + 26\!\cdots\!12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 196 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!75 \nu^{18} + \cdots - 10\!\cdots\!20 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 75\!\cdots\!85 \nu^{18} + \cdots + 11\!\cdots\!96 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\!\cdots\!73 \nu^{18} + \cdots - 97\!\cdots\!72 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!93 \nu^{18} + \cdots - 15\!\cdots\!32 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!93 \nu^{18} + \cdots - 48\!\cdots\!08 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!55 \nu^{18} + \cdots - 11\!\cdots\!40 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33\!\cdots\!23 \nu^{18} + \cdots + 54\!\cdots\!40 ) / 30\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 76\!\cdots\!95 \nu^{18} + \cdots - 69\!\cdots\!76 ) / 61\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 56\!\cdots\!57 \nu^{18} + \cdots + 18\!\cdots\!08 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!61 \nu^{18} + \cdots + 22\!\cdots\!76 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!35 \nu^{18} + \cdots + 10\!\cdots\!20 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 42\!\cdots\!59 \nu^{18} + \cdots + 28\!\cdots\!20 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 65\!\cdots\!75 \nu^{18} + \cdots + 11\!\cdots\!60 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 16\!\cdots\!59 \nu^{18} + \cdots - 22\!\cdots\!36 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 19\!\cdots\!57 \nu^{18} + \cdots + 43\!\cdots\!64 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 35\!\cdots\!25 \nu^{18} + \cdots - 47\!\cdots\!20 ) / 58\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 196 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 10\beta_{3} + \beta_{2} + 307\beta _1 - 109 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{17} - 14 \beta_{16} - 7 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} - 6 \beta_{11} + \cdots + 60027 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{18} - 52 \beta_{17} + 77 \beta_{16} + 581 \beta_{15} + 24 \beta_{14} + 24 \beta_{13} + \cdots - 41809 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 535 \beta_{18} + 7064 \beta_{17} - 7343 \beta_{16} - 4429 \beta_{15} + 3631 \beta_{14} - 1725 \beta_{13} + \cdots + 20583821 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 17404 \beta_{18} - 35232 \beta_{17} + 67636 \beta_{16} + 276651 \beta_{15} + 15638 \beta_{14} + \cdots - 15051963 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 381186 \beta_{18} + 3357780 \beta_{17} - 3114484 \beta_{16} - 2249119 \beta_{15} + 1912481 \beta_{14} + \cdots + 7441913683 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12791031 \beta_{18} - 17645820 \beta_{17} + 41254367 \beta_{16} + 124100421 \beta_{15} + \cdots - 6131600329 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 185116085 \beta_{18} + 1497599520 \beta_{17} - 1251290017 \beta_{16} - 1066422489 \beta_{15} + \cdots + 2776778417777 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7418717370 \beta_{18} - 7889169448 \beta_{17} + 21598765298 \beta_{16} + 54140830711 \beta_{15} + \cdots - 2814361853999 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 75216110080 \beta_{18} + 651429700572 \beta_{17} - 496711650774 \beta_{16} - 490874369883 \beta_{15} + \cdots + 10\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3818275448613 \beta_{18} - 3353723507876 \beta_{17} + 10453620294509 \beta_{16} + 23265226989473 \beta_{15} + \cdots - 13\!\cdots\!49 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 26978266675635 \beta_{18} + 280160115848264 \beta_{17} - 197815944868579 \beta_{16} + \cdots + 41\!\cdots\!13 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 18\!\cdots\!12 \beta_{18} + \cdots - 67\!\cdots\!19 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 85\!\cdots\!38 \beta_{18} + \cdots + 16\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 84\!\cdots\!31 \beta_{18} + \cdots - 33\!\cdots\!13 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 22\!\cdots\!37 \beta_{18} + \cdots + 64\!\cdots\!21 \) 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
−20.7471
−18.6752
−18.6242
−16.2381
−15.2682
−12.3106
−8.21907
−4.98040
−0.340895
0.203745
3.46085
7.84802
8.64765
12.0308
12.1179
17.1672
18.3992
19.1995
20.3290
−19.7471 −75.9843 261.949 175.748 1500.47 −559.590 −2645.12 3586.61 −3470.52
1.2 −17.6752 58.0023 184.414 −178.214 −1025.20 1114.90 −997.126 1177.27 3149.97
1.3 −17.6242 −31.4831 182.612 331.348 554.864 1426.26 −962.486 −1195.81 −5839.73
1.4 −15.2381 −9.62494 104.201 −286.418 146.666 −399.849 362.655 −2094.36 4364.48
1.5 −14.2682 89.9172 75.5824 −399.144 −1282.96 −1169.60 747.907 5898.11 5695.08
1.6 −11.3106 85.1141 −0.0696797 523.158 −962.694 503.209 1448.55 5057.41 −5917.24
1.7 −7.21907 −15.5721 −75.8851 401.113 112.416 −419.478 1471.86 −1944.51 −2895.66
1.8 −3.98040 15.7313 −112.156 −453.796 −62.6170 −1288.57 955.919 −1939.53 1806.29
1.9 0.659105 52.6301 −127.566 68.7263 34.6888 248.822 −168.445 582.927 45.2978
1.10 1.20374 −50.3987 −126.551 −248.593 −60.6672 −1088.75 −306.414 353.030 −299.242
1.11 4.46085 −23.9207 −108.101 −252.056 −106.707 859.955 −1053.21 −1614.80 −1124.38
1.12 8.84802 −70.0008 −49.7125 231.918 −619.368 −1321.92 −1572.40 2713.11 2052.02
1.13 9.64765 43.7399 −34.9228 464.886 421.988 1723.63 −1571.82 −273.817 4485.06
1.14 13.0308 −81.1172 41.8005 −450.096 −1057.02 189.744 −1123.24 4393.00 −5865.10
1.15 13.1179 84.5313 44.0782 21.2653 1108.87 −233.177 −1100.87 4958.55 278.955
1.16 18.1672 −42.1380 202.049 368.351 −765.530 354.471 1345.26 −411.392 6691.92
1.17 19.3992 64.8335 248.329 4.75307 1257.72 427.245 2334.28 2016.38 92.2057
1.18 20.1995 36.7696 280.022 427.056 742.730 −981.337 3070.77 −834.994 8626.33
1.19 21.3290 2.97031 326.928 −321.005 63.3539 1631.04 4242.95 −2178.18 −6846.73
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
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.b ✓ 19
3.b odd 2 1 549.8.a.e 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.8.a.b ✓ 19 1.a even 1 1 trivial
549.8.a.e 19 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{19} - 23 T_{2}^{18} - 1610 T_{2}^{17} + 37630 T_{2}^{16} + 1062593 T_{2}^{15} + \cdots - 34\!\cdots\!68 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(61))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} + \cdots - 34\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( T^{19} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{19} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{19} + \cdots + 64\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots + 37\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 74\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 29\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots - 15\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots - 61\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 18\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots - 36\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T + 226981)^{19} \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots - 75\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots + 83\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 12\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 41\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 67\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 81\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 22\!\cdots\!08 \) Copy content Toggle raw display
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