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Properties

Label 61.6.a.b
Level $61$
Weight $6$
Character orbit 61.a
Self dual yes
Analytic conductor $9.783$
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] = [61,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(9.78341300859\)
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: \( x^{14} - 3 x^{13} - 346 x^{12} + 760 x^{11} + 44872 x^{10} - 73466 x^{9} - 2694248 x^{8} + \cdots + 2545372552 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{5} + 2) q^{3} + (\beta_{2} - \beta_1 + 19) q^{4} + ( - \beta_{3} + 5) q^{5} + ( - \beta_{13} + \beta_{11} + \cdots + 15) q^{6}+ \cdots + ( - 2 \beta_{12} + 2 \beta_{10} + \cdots + 128) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{5} + 2) q^{3} + (\beta_{2} - \beta_1 + 19) q^{4} + ( - \beta_{3} + 5) q^{5} + ( - \beta_{13} + \beta_{11} + \cdots + 15) q^{6}+ \cdots + (925 \beta_{13} + 98 \beta_{12} + \cdots + 21436) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 11 q^{2} + 26 q^{3} + 261 q^{4} + 71 q^{5} + 205 q^{6} + 109 q^{7} + 543 q^{8} + 1728 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 11 q^{2} + 26 q^{3} + 261 q^{4} + 71 q^{5} + 205 q^{6} + 109 q^{7} + 543 q^{8} + 1728 q^{9} - 115 q^{10} + 1893 q^{11} + 295 q^{12} + 499 q^{13} + 2188 q^{14} + 862 q^{15} + 10397 q^{16} + 1780 q^{17} + 17370 q^{18} + 4016 q^{19} + 5607 q^{20} + 2500 q^{21} + 7355 q^{22} + 11017 q^{23} + 2699 q^{24} + 11855 q^{25} - 1759 q^{26} + 5624 q^{27} - 14000 q^{28} + 14816 q^{29} - 30416 q^{30} + 11462 q^{31} + 1437 q^{32} - 15928 q^{33} - 50510 q^{34} + 8039 q^{35} + 40916 q^{36} - 17702 q^{37} - 17021 q^{38} - 7506 q^{39} - 95013 q^{40} + 5801 q^{41} - 102978 q^{42} + 8194 q^{43} + 47645 q^{44} - 21789 q^{45} - 39138 q^{46} - 16308 q^{47} - 128033 q^{48} - 31195 q^{49} - 14152 q^{50} + 20616 q^{51} - 98667 q^{52} - 4394 q^{53} - 123344 q^{54} + 12343 q^{55} + 2422 q^{56} + 8136 q^{57} - 154349 q^{58} + 99977 q^{59} - 319822 q^{60} + 52094 q^{61} + 46935 q^{62} + 41153 q^{63} + 216381 q^{64} + 143953 q^{65} - 94758 q^{66} + 66109 q^{67} - 210960 q^{68} + 189258 q^{69} + 102851 q^{70} + 251686 q^{71} + 571496 q^{72} + 169601 q^{73} + 18665 q^{74} + 317916 q^{75} - 44867 q^{76} + 211299 q^{77} - 88738 q^{78} + 136045 q^{79} + 131243 q^{80} + 422594 q^{81} - 192210 q^{82} + 199016 q^{83} - 309736 q^{84} + 162282 q^{85} + 150228 q^{86} - 4380 q^{87} + 82907 q^{88} + 153370 q^{89} - 82467 q^{90} + 285993 q^{91} + 456392 q^{92} - 119274 q^{93} - 438812 q^{94} + 51324 q^{95} - 204513 q^{96} - 12398 q^{97} + 22799 q^{98} + 278641 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 346 x^{12} + 760 x^{11} + 44872 x^{10} - 73466 x^{9} - 2694248 x^{8} + \cdots + 2545372552 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22\!\cdots\!52 \nu^{13} + \cdots - 23\!\cdots\!72 ) / 58\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41\!\cdots\!97 \nu^{13} + \cdots - 59\!\cdots\!20 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!51 \nu^{13} + \cdots - 16\!\cdots\!08 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!21 \nu^{13} + \cdots + 47\!\cdots\!40 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 77\!\cdots\!41 \nu^{13} + \cdots + 79\!\cdots\!84 ) / 18\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 27\!\cdots\!89 \nu^{13} + \cdots + 43\!\cdots\!20 ) / 57\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!43 \nu^{13} + \cdots - 23\!\cdots\!92 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!29 \nu^{13} + \cdots + 11\!\cdots\!36 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 54\!\cdots\!37 \nu^{13} + \cdots - 43\!\cdots\!76 ) / 93\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!93 \nu^{13} + \cdots - 28\!\cdots\!52 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 53\!\cdots\!81 \nu^{13} + \cdots + 42\!\cdots\!12 ) / 46\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{13} - 2 \beta_{12} + 9 \beta_{11} + 15 \beta_{10} + 6 \beta_{9} + 14 \beta_{8} + 4 \beta_{6} + \cdots + 4480 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 149 \beta_{13} - 139 \beta_{12} + 257 \beta_{11} + 257 \beta_{10} - 55 \beta_{9} + 291 \beta_{8} + \cdots + 11191 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 811 \beta_{13} - 590 \beta_{12} + 2525 \beta_{11} + 3395 \beta_{10} + 1342 \beta_{9} + 3668 \beta_{8} + \cdots + 473572 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 20775 \beta_{13} - 18333 \beta_{12} + 46495 \beta_{11} + 47271 \beta_{10} + 3947 \beta_{9} + \cdots + 2055133 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 154779 \beta_{13} - 114424 \beta_{12} + 487301 \beta_{11} + 602583 \beta_{10} + 237768 \beta_{9} + \cdots + 54761602 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2929835 \beta_{13} - 2472619 \beta_{12} + 7531475 \beta_{11} + 7809103 \beta_{10} + 1763273 \beta_{9} + \cdots + 336948847 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 26085799 \beta_{13} - 19487262 \beta_{12} + 82606833 \beta_{11} + 97825223 \beta_{10} + \cdots + 6777544580 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 418607955 \beta_{13} - 341797605 \beta_{12} + 1164810907 \beta_{11} + 1227976075 \beta_{10} + \cdots + 52730037749 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4153703975 \beta_{13} - 3129730436 \beta_{12} + 13189027737 \beta_{11} + 15223930091 \beta_{10} + \cdots + 885329657910 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 60388917723 \beta_{13} - 48187998831 \beta_{12} + 176200875699 \beta_{11} + 188146483399 \beta_{10} + \cdots + 8063139819979 \) 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
12.1602
9.46450
9.34749
4.84161
3.49718
3.41922
1.00213
0.661834
−2.68640
−4.83560
−5.83267
−8.68557
−9.56513
−9.78883
−11.1602 −8.26149 92.5507 59.9263 92.2001 −86.5340 −675.760 −174.748 −668.792
1.2 −8.46450 −7.61614 39.6477 −44.1841 64.4668 −161.647 −64.7342 −184.994 373.996
1.3 −8.34749 15.6781 37.6806 70.7102 −130.873 107.126 −47.4184 2.80335 −590.253
1.4 −3.84161 0.815702 −17.2421 −106.804 −3.13361 96.9015 189.169 −242.335 410.301
1.5 −2.49718 −11.5502 −25.7641 1.70766 28.8429 −142.436 144.247 −109.593 −4.26435
1.6 −2.41922 28.1310 −26.1474 −22.9420 −68.0553 117.139 140.671 548.356 55.5019
1.7 −0.00213054 −25.7472 −32.0000 −62.6827 0.0548556 −62.4396 0.136355 419.920 0.133548
1.8 0.338166 7.33909 −31.8856 105.715 2.48183 −19.5266 −21.6039 −189.138 35.7492
1.9 3.68640 −20.0449 −18.4104 15.5243 −73.8936 184.412 −185.833 158.798 57.2288
1.10 5.83560 27.9774 2.05426 75.8923 163.265 −83.7417 −174.751 539.735 442.877
1.11 6.83267 16.7283 14.6853 −8.80653 114.299 220.590 −118.305 36.8358 −60.1721
1.12 9.68557 5.57423 61.8103 42.8925 53.9896 15.7016 288.729 −211.928 415.439
1.13 10.5651 26.9109 79.6219 −93.3093 284.317 −134.380 503.132 481.195 −985.825
1.14 10.7888 −29.9348 84.3988 37.3608 −322.962 57.8349 565.322 653.093 403.080
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 61.6.a.b ✓ 14
3.b odd 2 1 549.6.a.e 14
4.b odd 2 1 976.6.a.h 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.6.a.b ✓ 14 1.a even 1 1 trivial
549.6.a.e 14 3.b odd 2 1
976.6.a.h 14 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 11 T_{2}^{13} - 294 T_{2}^{12} + 3262 T_{2}^{11} + 30539 T_{2}^{10} - 340791 T_{2}^{9} + \cdots - 2139648 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(61))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 11 T^{13} + \cdots - 2139648 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 20\!\cdots\!60 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 80\!\cdots\!92 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 12\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 34\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 28\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 25\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 46\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 38\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 17\!\cdots\!78 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 18\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 12\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T - 3721)^{14} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 27\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 31\!\cdots\!50 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 62\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 50\!\cdots\!32 \) Copy content Toggle raw display
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