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Properties

Label 5.15.c.a
Level $5$
Weight $15$
Character orbit 5.c
Analytic conductor $6.216$
Analytic rank $0$
Dimension $12$
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] = [5,15,Mod(2,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.2");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 5.c (of order \(4\), degree \(2\), 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: \(6.21644840760\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 19558 x^{10} + 126669069 x^{8} + 319537876249 x^{6} + 246445182947944 x^{4} + \cdots + 494659590011136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{8}\cdot 5^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 11 \beta_{2} - 11) q^{2} + ( - \beta_{5} + \beta_{4} - 180 \beta_{2} - 180) q^{3} + (\beta_{8} - \beta_{6} + \cdots - 5748 \beta_{2}) q^{4}+ \cdots + (15 \beta_{11} + 4 \beta_{10} + \cdots + 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 11 \beta_{2} - 11) q^{2} + ( - \beta_{5} + \beta_{4} - 180 \beta_{2} - 180) q^{3} + (\beta_{8} - \beta_{6} + \cdots - 5748 \beta_{2}) q^{4}+ \cdots + ( - 248020455 \beta_{11} + \cdots - 248020455) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 130 q^{2} - 2160 q^{3} + 17240 q^{5} + 306744 q^{6} - 1646400 q^{7} + 5926980 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 130 q^{2} - 2160 q^{3} + 17240 q^{5} + 306744 q^{6} - 1646400 q^{7} + 5926980 q^{8} - 13200810 q^{10} + 29923504 q^{11} - 59937960 q^{12} + 55638180 q^{13} + 184224480 q^{15} - 1037647368 q^{16} + 850934660 q^{17} + 270137490 q^{18} + 478060020 q^{20} - 249957456 q^{21} - 2359887360 q^{22} + 8179357520 q^{23} - 16708599900 q^{25} - 12118305836 q^{26} + 14239650960 q^{27} + 56966840280 q^{28} - 125572575120 q^{30} - 86707686336 q^{31} + 185038230920 q^{32} + 220909657680 q^{33} - 332814062560 q^{35} - 524021029188 q^{36} + 459843960780 q^{37} + 989412055320 q^{38} - 1782654206100 q^{40} - 156617026256 q^{41} + 905682106560 q^{42} + 1654722058800 q^{43} - 2292625396620 q^{45} - 2435633641416 q^{46} + 1589991245840 q^{47} + 4481963789520 q^{48} - 1871993053150 q^{50} - 3547107536256 q^{51} + 3285360237900 q^{52} + 1133396613740 q^{53} - 3338422423920 q^{55} - 361136044080 q^{56} - 115055923680 q^{57} - 476843131920 q^{58} + 6677820506040 q^{60} + 6171807176304 q^{61} - 13406596533560 q^{62} - 11000614032240 q^{63} + 19687020330380 q^{65} + 26889794569248 q^{66} - 15697041941040 q^{67} - 52143555024860 q^{68} + 47152432127640 q^{70} + 40379406227584 q^{71} - 23059234580220 q^{72} - 62089383019380 q^{73} + 61270184515200 q^{75} - 6270244149360 q^{76} - 29708728137200 q^{77} - 43681118652600 q^{78} + 93124211032640 q^{80} + 35277674991012 q^{81} - 88245357269760 q^{82} - 41460816019840 q^{83} - 27610709184060 q^{85} + 66858814091224 q^{86} + 38039311006080 q^{87} + 159247556456160 q^{88} - 213819594380970 q^{90} - 191959976216736 q^{91} + 136323611740760 q^{92} + 181596579349680 q^{93} - 62631035996400 q^{95} - 294857825144736 q^{96} + 383795964738540 q^{97} + 331970262670270 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 19558 x^{10} + 126669069 x^{8} + 319537876249 x^{6} + 246445182947944 x^{4} + \cdots + 494659590011136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33\!\cdots\!13 \nu^{10} + \cdots - 13\!\cdots\!72 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!51 \nu^{11} + \cdots + 73\!\cdots\!84 \nu ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28\!\cdots\!21 \nu^{11} + \cdots + 19\!\cdots\!12 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!21 \nu^{11} + \cdots - 19\!\cdots\!12 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\!\cdots\!27 \nu^{11} + \cdots + 55\!\cdots\!44 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32\!\cdots\!27 \nu^{11} + \cdots - 55\!\cdots\!44 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!59 \nu^{11} + \cdots + 21\!\cdots\!60 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 36\!\cdots\!49 \nu^{11} + \cdots - 37\!\cdots\!56 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!05 \nu^{11} + \cdots - 47\!\cdots\!52 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41\!\cdots\!91 \nu^{11} + \cdots - 80\!\cdots\!04 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!27 \nu^{11} + \cdots + 48\!\cdots\!28 ) / 37\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} + 4 \beta_{7} + 120 \beta_{6} + 123 \beta_{5} + \cdots - 2 ) / 12000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 25 \beta_{11} + 8 \beta_{10} - 42 \beta_{9} - 8 \beta_{8} + 41 \beta_{7} - 1615 \beta_{6} + \cdots - 7824113 ) / 2400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10008 \beta_{11} + 12799 \beta_{10} + 7217 \beta_{9} + 56913 \beta_{8} - 35606 \beta_{7} + \cdots + 10008 ) / 12000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 132722 \beta_{11} + 1759 \beta_{10} + 267203 \beta_{9} - 1759 \beta_{8} - 129204 \beta_{7} + \cdots + 25850112620 ) / 1200 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 15544723 \beta_{11} - 31392724 \beta_{10} + 303278 \beta_{9} - 122306243 \beta_{8} + \cdots - 15544723 ) / 3000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2527121191 \beta_{11} - 403160934 \beta_{10} - 5457403316 \beta_{9} + 403160934 \beta_{8} + \cdots - 403426335073083 ) / 2400 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 227907549859 \beta_{11} + 597210412427 \beta_{10} - 141395312709 \beta_{9} + 2150364705749 \beta_{8} + \cdots + 227907549859 ) / 6000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 23713127852023 \beta_{11} + 5417141984206 \beta_{10} + 52843397688252 \beta_{9} + \cdots + 34\!\cdots\!55 ) / 2400 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 37\!\cdots\!32 \beta_{11} + \cdots - 37\!\cdots\!32 ) / 12000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 22\!\cdots\!47 \beta_{11} + \cdots - 30\!\cdots\!03 ) / 2400 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 16\!\cdots\!39 \beta_{11} + \cdots + 16\!\cdots\!39 ) / 6000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 0.908322i
97.0021i
− 63.9679i
1.56306i
36.8401i
− 68.5291i
0.908322i
− 97.0021i
63.9679i
− 1.56306i
− 36.8401i
68.5291i
−171.094 − 171.094i −1246.66 + 1246.66i 42162.0i −47605.5 − 61945.4i 426591. −263499. − 263499.i 4.41045e6 − 4.41045e6i 1.67464e6i −2.45346e6 + 1.87434e7i
2.2 −88.3818 − 88.3818i 1759.79 − 1759.79i − 761.322i 77798.3 + 7137.15i −311067. −1.04208e6 − 1.04208e6i −1.51533e6 + 1.51533e6i − 1.41078e6i −6.24516e6 − 7.50675e6i
2.3 −51.6522 − 51.6522i −332.943 + 332.943i − 11048.1i −48483.9 + 61260.4i 34394.5 918301. + 918301.i −1.41693e6 + 1.41693e6i 4.56127e6i 5.66853e6 − 659934.i
2.4 28.2979 + 28.2979i −2604.53 + 2604.53i − 14782.5i 49526.7 − 60420.4i −147406. −336380. − 336380.i 881946. − 881946.i − 8.78422e6i 3.11127e6 − 308268.i
2.5 76.5799 + 76.5799i 1769.89 − 1769.89i − 4655.03i −30740.1 − 71823.1i 271076. 378117. + 378117.i 1.61117e6 − 1.61117e6i − 1.48206e6i 3.14614e6 − 7.85428e6i
2.6 141.250 + 141.250i −425.547 + 425.547i 23518.9i 8124.40 + 77701.4i −120217. −477660. − 477660.i −1.00781e6 + 1.00781e6i 4.42079e6i −9.82773e6 + 1.21229e7i
3.1 −171.094 + 171.094i −1246.66 − 1246.66i − 42162.0i −47605.5 + 61945.4i 426591. −263499. + 263499.i 4.41045e6 + 4.41045e6i − 1.67464e6i −2.45346e6 − 1.87434e7i
3.2 −88.3818 + 88.3818i 1759.79 + 1759.79i 761.322i 77798.3 − 7137.15i −311067. −1.04208e6 + 1.04208e6i −1.51533e6 − 1.51533e6i 1.41078e6i −6.24516e6 + 7.50675e6i
3.3 −51.6522 + 51.6522i −332.943 − 332.943i 11048.1i −48483.9 − 61260.4i 34394.5 918301. − 918301.i −1.41693e6 − 1.41693e6i − 4.56127e6i 5.66853e6 + 659934.i
3.4 28.2979 − 28.2979i −2604.53 − 2604.53i 14782.5i 49526.7 + 60420.4i −147406. −336380. + 336380.i 881946. + 881946.i 8.78422e6i 3.11127e6 + 308268.i
3.5 76.5799 − 76.5799i 1769.89 + 1769.89i 4655.03i −30740.1 + 71823.1i 271076. 378117. − 378117.i 1.61117e6 + 1.61117e6i 1.48206e6i 3.14614e6 + 7.85428e6i
3.6 141.250 − 141.250i −425.547 − 425.547i − 23518.9i 8124.40 − 77701.4i −120217. −477660. + 477660.i −1.00781e6 − 1.00781e6i − 4.42079e6i −9.82773e6 − 1.21229e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.15.c.a ✓ 12
3.b odd 2 1 45.15.g.a 12
4.b odd 2 1 80.15.p.c 12
5.b even 2 1 25.15.c.b 12
5.c odd 4 1 inner 5.15.c.a ✓ 12
5.c odd 4 1 25.15.c.b 12
15.e even 4 1 45.15.g.a 12
20.e even 4 1 80.15.p.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.15.c.a ✓ 12 1.a even 1 1 trivial
5.15.c.a ✓ 12 5.c odd 4 1 inner
25.15.c.b 12 5.b even 2 1
25.15.c.b 12 5.c odd 4 1
45.15.g.a 12 3.b odd 2 1
45.15.g.a 12 15.e even 4 1
80.15.p.c 12 4.b odd 2 1
80.15.p.c 12 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 19\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 50\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 68\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 42\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 79\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
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