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Properties

Label 60.5.c.a
Level $60$
Weight $5$
Character orbit 60.c
Analytic conductor $6.202$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,5,Mod(31,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 60.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(6.20219778503\)
Analytic rank: \(0\)
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: \( x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + \cdots + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + \beta_{2} q^{3} + (\beta_{7} + \beta_{3} + 2) q^{4} - \beta_{8} q^{5} + ( - \beta_{9} - \beta_{2} + 1) q^{6} + ( - \beta_{15} + \beta_{11} - \beta_{10} + \cdots - 1) q^{7}+ \cdots - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{2} + \beta_{2} q^{3} + (\beta_{7} + \beta_{3} + 2) q^{4} - \beta_{8} q^{5} + ( - \beta_{9} - \beta_{2} + 1) q^{6} + ( - \beta_{15} + \beta_{11} - \beta_{10} + \cdots - 1) q^{7}+ \cdots + ( - 54 \beta_{14} - 108 \beta_{13} + \cdots - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{2} + 26 q^{4} + 18 q^{6} - 180 q^{8} - 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{2} + 26 q^{4} + 18 q^{6} - 180 q^{8} - 432 q^{9} + 50 q^{10} - 352 q^{13} - 804 q^{14} - 190 q^{16} + 324 q^{18} + 600 q^{20} + 288 q^{21} + 436 q^{22} - 1998 q^{24} + 2000 q^{25} - 852 q^{26} - 1156 q^{28} - 3456 q^{29} + 7668 q^{32} + 4772 q^{34} - 702 q^{36} + 9376 q^{37} - 1320 q^{38} + 550 q^{40} + 1248 q^{41} - 324 q^{42} - 6420 q^{44} - 1112 q^{46} - 4176 q^{48} - 3952 q^{49} - 1500 q^{50} + 12704 q^{52} - 5184 q^{53} - 486 q^{54} - 2604 q^{56} - 11232 q^{57} + 12708 q^{58} + 3150 q^{60} - 3808 q^{61} - 16152 q^{62} - 11902 q^{64} + 2400 q^{65} - 2916 q^{66} - 12312 q^{68} + 9792 q^{69} - 17100 q^{70} + 4860 q^{72} + 11040 q^{73} + 30516 q^{74} - 5160 q^{76} - 27456 q^{77} - 3600 q^{78} + 10800 q^{80} + 11664 q^{81} - 54040 q^{82} - 2052 q^{84} - 11200 q^{85} + 39768 q^{86} - 7220 q^{88} + 7584 q^{89} - 1350 q^{90} + 28848 q^{92} + 19872 q^{93} + 49776 q^{94} + 18882 q^{96} - 14496 q^{97} + 23940 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + \cdots + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11071 \nu^{15} + 226436 \nu^{14} + 593367 \nu^{13} - 554294 \nu^{12} - 21258985 \nu^{11} + \cdots + 741363154944 ) / 15454961664 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6617 \nu^{15} + 26278 \nu^{14} + 88297 \nu^{13} + 60460 \nu^{12} - 2080027 \nu^{11} + \cdots + 64036536320 ) / 2575826944 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37951 \nu^{15} - 50364 \nu^{14} - 476215 \nu^{13} - 676818 \nu^{12} + 8411705 \nu^{11} + \cdots - 214580068352 ) / 7727480832 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40727 \nu^{15} - 80592 \nu^{14} - 189281 \nu^{13} + 3024494 \nu^{12} + 5984551 \nu^{11} + \cdots - 583508426752 ) / 7727480832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 99391 \nu^{15} - 371968 \nu^{14} - 502471 \nu^{13} + 212754 \nu^{12} + 14496273 \nu^{11} + \cdots - 357198462976 ) / 15454961664 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18263 \nu^{15} + 54973 \nu^{14} + 350339 \nu^{13} - 167015 \nu^{12} - 5995975 \nu^{11} + \cdots + 215238705152 ) / 965935104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 158297 \nu^{15} - 321188 \nu^{14} - 2321441 \nu^{13} - 1304702 \nu^{12} + 40098543 \nu^{11} + \cdots - 1177881673728 ) / 7727480832 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9755 \nu^{15} + 13120 \nu^{14} + 127395 \nu^{13} + 144550 \nu^{12} - 2087125 \nu^{11} + \cdots + 48431104000 ) / 454557696 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28268 \nu^{15} - 40905 \nu^{14} - 260496 \nu^{13} - 675047 \nu^{12} + 5667738 \nu^{11} + \cdots - 150021603328 ) / 1287913472 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 388523 \nu^{15} - 954528 \nu^{14} - 7727251 \nu^{13} - 4008678 \nu^{12} + 134476773 \nu^{11} + \cdots - 4911664201728 ) / 15454961664 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 155836 \nu^{15} + 5199 \nu^{14} - 2013264 \nu^{13} - 5482783 \nu^{12} + 30058250 \nu^{11} + \cdots - 756971995136 ) / 3863740416 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 39828 \nu^{15} - 39001 \nu^{14} - 561192 \nu^{13} - 949223 \nu^{12} + 8771394 \nu^{11} + \cdots - 246523756544 ) / 965935104 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 322639 \nu^{15} - 355100 \nu^{14} - 4508775 \nu^{13} - 7378706 \nu^{12} + 75683209 \nu^{11} + \cdots - 2320671178752 ) / 7727480832 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 742535 \nu^{15} + 988864 \nu^{14} + 9060815 \nu^{13} + 15925822 \nu^{12} + \cdots + 3457620639744 ) / 15454961664 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 383855 \nu^{15} - 709602 \nu^{14} - 4960223 \nu^{13} - 4710044 \nu^{12} + 88176749 \nu^{11} + \cdots - 2323994116096 ) / 7727480832 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 5 \beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 3 \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 40 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 10 \beta_{15} + 4 \beta_{14} + 9 \beta_{13} + 23 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + \cdots + 253 ) / 120 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10 \beta_{15} + 5 \beta_{14} + 21 \beta_{13} - 10 \beta_{12} - 10 \beta_{11} + 9 \beta_{10} + \cdots + 288 ) / 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{15} - 59 \beta_{14} + 72 \beta_{13} + 17 \beta_{12} - 105 \beta_{11} - 15 \beta_{10} + \cdots - 2415 ) / 120 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 24 \beta_{15} - 30 \beta_{14} + 103 \beta_{13} - 15 \beta_{12} - 45 \beta_{11} - 30 \beta_{10} + \cdots + 231 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 130 \beta_{15} - 135 \beta_{14} + 159 \beta_{13} - 140 \beta_{12} - 120 \beta_{11} + 5 \beta_{10} + \cdots - 2150 ) / 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2130 \beta_{15} - 561 \beta_{14} + 1342 \beta_{13} + 873 \beta_{12} - 225 \beta_{11} - 2445 \beta_{10} + \cdots + 8765 ) / 120 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2660 \beta_{15} + 2668 \beta_{14} + 9183 \beta_{13} - 3349 \beta_{12} + 1305 \beta_{11} - 5292 \beta_{10} + \cdots - 59759 ) / 120 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 10170 \beta_{15} + 7745 \beta_{14} - 531 \beta_{13} + 7090 \beta_{12} - 2870 \beta_{11} + 2821 \beta_{10} + \cdots - 54628 ) / 40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14382 \beta_{15} + 16053 \beta_{14} + 10384 \beta_{13} - 207 \beta_{12} - 2121 \beta_{11} + \cdots - 47583 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 83800 \beta_{15} + 168842 \beta_{14} + 46919 \beta_{13} - 70151 \beta_{12} - 16005 \beta_{11} + \cdots - 4468481 ) / 120 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 208250 \beta_{15} + 175845 \beta_{14} - 49381 \beta_{13} + 70700 \beta_{12} - 83200 \beta_{11} + \cdots - 535350 ) / 40 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 565350 \beta_{15} + 969483 \beta_{14} + 46014 \beta_{13} - 974619 \beta_{12} + 197715 \beta_{11} + \cdots - 6725815 ) / 120 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 780740 \beta_{15} + 820372 \beta_{14} - 1525773 \beta_{13} - 1778281 \beta_{12} + 3214605 \beta_{11} + \cdots - 5561451 ) / 120 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 34870 \beta_{15} + 344889 \beta_{14} - 111235 \beta_{13} + 157722 \beta_{12} + 30306 \beta_{11} + \cdots - 4002364 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
2.44021 + 1.43016i
2.44021 − 1.43016i
−1.14149 + 2.58786i
−1.14149 − 2.58786i
1.85226 − 2.13755i
1.85226 + 2.13755i
2.70166 + 0.837276i
2.70166 − 0.837276i
2.77114 + 0.566380i
2.77114 − 0.566380i
−1.85197 + 2.13780i
−1.85197 − 2.13780i
−2.48191 − 1.35651i
−2.48191 + 1.35651i
−2.28990 + 1.66022i
−2.28990 − 1.66022i
−3.95975 − 0.566024i 5.19615i 15.3592 + 4.48263i 11.1803 2.94115 − 20.5755i − 24.1355i −58.2814 − 26.4438i −27.0000 −44.2713 − 6.32834i
31.2 −3.95975 + 0.566024i − 5.19615i 15.3592 − 4.48263i 11.1803 2.94115 + 20.5755i 24.1355i −58.2814 + 26.4438i −27.0000 −44.2713 + 6.32834i
31.3 −3.40825 − 2.09376i − 5.19615i 7.23235 + 14.2721i −11.1803 −10.8795 + 17.7098i 61.3317i 5.23271 − 63.7857i −27.0000 38.1054 + 23.4089i
31.4 −3.40825 + 2.09376i 5.19615i 7.23235 − 14.2721i −11.1803 −10.8795 − 17.7098i − 61.3317i 5.23271 + 63.7857i −27.0000 38.1054 − 23.4089i
31.5 −3.34902 − 2.18726i 5.19615i 6.43181 + 14.6503i −11.1803 11.3653 − 17.4020i − 1.39605i 10.5038 − 63.1322i −27.0000 37.4431 + 24.4543i
31.6 −3.34902 + 2.18726i − 5.19615i 6.43181 − 14.6503i −11.1803 11.3653 + 17.4020i 1.39605i 10.5038 + 63.1322i −27.0000 37.4431 − 24.4543i
31.7 −3.08838 − 2.54203i − 5.19615i 3.07620 + 15.7015i 11.1803 −13.2088 + 16.0477i − 86.5709i 30.4132 − 56.3120i −27.0000 −34.5292 − 28.4207i
31.8 −3.08838 + 2.54203i 5.19615i 3.07620 − 15.7015i 11.1803 −13.2088 − 16.0477i 86.5709i 30.4132 + 56.3120i −27.0000 −34.5292 + 28.4207i
31.9 0.264404 − 3.99125i 5.19615i −15.8602 − 2.11060i −11.1803 20.7392 + 1.37388i 29.5855i −12.6174 + 62.7439i −27.0000 −2.95612 + 44.6236i
31.10 0.264404 + 3.99125i − 5.19615i −15.8602 + 2.11060i −11.1803 20.7392 − 1.37388i − 29.5855i −12.6174 − 62.7439i −27.0000 −2.95612 − 44.6236i
31.11 1.28004 − 3.78966i 5.19615i −12.7230 − 9.70180i 11.1803 19.6916 + 6.65126i − 89.0673i −53.0524 + 35.7972i −27.0000 14.3112 − 42.3697i
31.12 1.28004 + 3.78966i − 5.19615i −12.7230 + 9.70180i 11.1803 19.6916 − 6.65126i 89.0673i −53.0524 − 35.7972i −27.0000 14.3112 + 42.3697i
31.13 2.37483 − 3.21872i − 5.19615i −4.72038 − 15.2878i −11.1803 −16.7250 − 12.3400i − 19.2859i −60.4174 − 21.1124i −27.0000 −26.5514 + 35.9864i
31.14 2.37483 + 3.21872i 5.19615i −4.72038 + 15.2878i −11.1803 −16.7250 + 12.3400i 19.2859i −60.4174 + 21.1124i −27.0000 −26.5514 − 35.9864i
31.15 3.88613 − 0.947630i − 5.19615i 14.2040 − 7.36522i 11.1803 −4.92403 − 20.1929i − 12.7755i 48.2191 − 42.0823i −27.0000 43.4482 − 10.5948i
31.16 3.88613 + 0.947630i 5.19615i 14.2040 + 7.36522i 11.1803 −4.92403 + 20.1929i 12.7755i 48.2191 + 42.0823i −27.0000 43.4482 + 10.5948i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.5.c.a ✓ 16
3.b odd 2 1 180.5.c.c 16
4.b odd 2 1 inner 60.5.c.a ✓ 16
5.b even 2 1 300.5.c.d 16
5.c odd 4 2 300.5.f.b 32
8.b even 2 1 960.5.e.f 16
8.d odd 2 1 960.5.e.f 16
12.b even 2 1 180.5.c.c 16
20.d odd 2 1 300.5.c.d 16
20.e even 4 2 300.5.f.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.c.a ✓ 16 1.a even 1 1 trivial
60.5.c.a ✓ 16 4.b odd 2 1 inner
180.5.c.c 16 3.b odd 2 1
180.5.c.c 16 12.b even 2 1
300.5.c.d 16 5.b even 2 1
300.5.c.d 16 20.d odd 2 1
300.5.f.b 32 5.c odd 4 2
300.5.f.b 32 20.e even 4 2
960.5.e.f 16 8.b even 2 1
960.5.e.f 16 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( (T^{2} + 27)^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 236865200916224)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 81\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 27\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 86\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 68\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 61\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
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