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Properties

Label 900.3.p.e
Level $900$
Weight $3$
Character orbit 900.p
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(101,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (of order \(6\), 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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 10 x^{14} - 12 x^{13} + 6 x^{12} - 27 x^{11} - 585 x^{10} + 3618 x^{9} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{3} + \beta_{10} q^{7} + (\beta_{15} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{3} + \beta_{10} q^{7} + (\beta_{15} - \beta_{3} + 1) q^{9} + \beta_{11} q^{11} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots + \beta_1) q^{13}+ \cdots + ( - \beta_{15} + 8 \beta_{14} + \cdots + 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} - q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} - q^{7} + 14 q^{9} - 10 q^{13} + 2 q^{19} + q^{21} + 27 q^{23} - 16 q^{27} + 9 q^{29} + 8 q^{31} + 36 q^{33} - 22 q^{37} + 19 q^{39} + 54 q^{41} + 44 q^{43} - 108 q^{47} - 45 q^{49} + 90 q^{51} - 68 q^{57} + 9 q^{59} - 55 q^{61} - 107 q^{63} - 28 q^{67} - 147 q^{69} + 86 q^{73} + 342 q^{77} + 11 q^{79} - 130 q^{81} - 306 q^{83} + 375 q^{87} - 134 q^{91} - 83 q^{93} + 41 q^{97} + 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 10 x^{14} - 12 x^{13} + 6 x^{12} - 27 x^{11} - 585 x^{10} + 3618 x^{9} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4099 \nu^{15} + 12833 \nu^{14} - 9679 \nu^{13} - 10653 \nu^{12} + 361803 \nu^{11} + \cdots + 54664552701 ) / 36541883160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 515 \nu^{15} - 3479 \nu^{14} + 6649 \nu^{13} - 42933 \nu^{12} + 205755 \nu^{11} + \cdots - 19605389931 ) / 4060209240 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2405 \nu^{15} - 16133 \nu^{14} + 254923 \nu^{13} - 203007 \nu^{12} + 344697 \nu^{11} + \cdots + 504904556547 ) / 18270941580 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1817 \nu^{15} + 25193 \nu^{14} - 60625 \nu^{13} + 28569 \nu^{12} + 179409 \nu^{11} + \cdots + 140184038421 ) / 9135470790 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4759 \nu^{15} - 14899 \nu^{14} + 34049 \nu^{13} - 262149 \nu^{12} + 1530723 \nu^{11} + \cdots + 100045362573 ) / 18270941580 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 655 \nu^{15} - 1301 \nu^{14} + 10591 \nu^{13} - 9867 \nu^{12} - 49035 \nu^{11} + \cdots + 4565609631 ) / 2030104620 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6551 \nu^{15} - 50857 \nu^{14} + 172367 \nu^{13} + 206985 \nu^{12} - 535767 \nu^{11} + \cdots + 281616431751 ) / 18270941580 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10129 \nu^{15} + 32833 \nu^{14} - 98447 \nu^{13} - 6897 \nu^{12} - 298569 \nu^{11} + \cdots + 70544009781 ) / 18270941580 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27817 \nu^{15} - 106331 \nu^{14} + 379645 \nu^{13} - 604497 \nu^{12} - 150969 \nu^{11} + \cdots + 71366680449 ) / 36541883160 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10603 \nu^{15} - 25049 \nu^{14} + 120439 \nu^{13} - 320619 \nu^{12} - 840315 \nu^{11} + \cdots + 94768153443 ) / 12180627720 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31943 \nu^{15} - 51853 \nu^{14} - 346237 \nu^{13} + 1172577 \nu^{12} - 1145463 \nu^{11} + \cdots - 896524492329 ) / 36541883160 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11429 \nu^{15} - 14033 \nu^{14} + 1207 \nu^{13} + 50037 \nu^{12} - 164451 \nu^{11} + \cdots - 107787395061 ) / 12180627720 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18241 \nu^{15} - 46679 \nu^{14} + 216169 \nu^{13} - 945129 \nu^{12} + 2042403 \nu^{11} + \cdots - 90699441147 ) / 18270941580 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - 2\beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - 4\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{15} + 4 \beta_{14} + 9 \beta_{13} + 6 \beta_{12} + \beta_{11} - 3 \beta_{10} - 8 \beta_{9} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5 \beta_{15} - 42 \beta_{14} + 4 \beta_{13} + 20 \beta_{12} - 26 \beta_{11} + 6 \beta_{10} + \cdots + 103 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 56 \beta_{15} + 119 \beta_{14} + 23 \beta_{13} + 19 \beta_{12} - 38 \beta_{11} + 105 \beta_{10} + \cdots - 1416 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 159 \beta_{15} + 343 \beta_{14} + 93 \beta_{13} + 30 \beta_{12} + 154 \beta_{11} + 159 \beta_{10} + \cdots - 166 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 598 \beta_{15} + 417 \beta_{14} + 247 \beta_{13} + 425 \beta_{12} + 37 \beta_{11} - 966 \beta_{10} + \cdots - 392 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 398 \beta_{15} - 826 \beta_{14} - 283 \beta_{13} - 1079 \beta_{12} + 1168 \beta_{11} - 1542 \beta_{10} + \cdots + 4299 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8079 \beta_{15} - 674 \beta_{14} + 5475 \beta_{13} - 4353 \beta_{12} - 4391 \beta_{11} - 8886 \beta_{10} + \cdots + 21488 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2812 \beta_{15} - 21111 \beta_{14} + 679 \beta_{13} - 26791 \beta_{12} - 13094 \beta_{11} + 9591 \beta_{10} + \cdots + 85504 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 24662 \beta_{15} + 16967 \beta_{14} + 6404 \beta_{13} - 46376 \beta_{12} + 30967 \beta_{11} + \cdots - 1055946 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 31551 \beta_{15} - 325664 \beta_{14} - 63420 \beta_{13} - 134610 \beta_{12} - 120851 \beta_{11} + \cdots - 138784 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 229072 \beta_{15} - 680946 \beta_{14} + 140782 \beta_{13} - 165328 \beta_{12} - 714482 \beta_{11} + \cdots - 1621616 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.844479 − 2.87869i
2.47109 − 1.70110i
−1.69925 − 2.47235i
2.94519 + 0.570848i
−2.99949 + 0.0553819i
1.13333 + 2.77769i
0.127146 + 2.99730i
−1.82249 + 2.38297i
0.844479 + 2.87869i
2.47109 + 1.70110i
−1.69925 + 2.47235i
2.94519 − 0.570848i
−2.99949 − 0.0553819i
1.13333 − 2.77769i
0.127146 − 2.99730i
−1.82249 − 2.38297i
0 −2.91526 + 0.708005i 0 0 0 4.96248 − 8.59526i 0 7.99746 − 4.12804i 0
101.2 0 −2.70874 − 1.28947i 0 0 0 −5.40503 + 9.36178i 0 5.67451 + 6.98569i 0
101.3 0 −1.29149 + 2.70777i 0 0 0 −1.73252 + 3.00081i 0 −5.66408 − 6.99415i 0
101.4 0 −0.978225 − 2.83603i 0 0 0 2.71262 − 4.69840i 0 −7.08615 + 5.54856i 0
101.5 0 1.54771 + 2.56994i 0 0 0 0.725042 − 1.25581i 0 −4.20921 + 7.95503i 0
101.6 0 1.83888 − 2.37034i 0 0 0 −4.13058 + 7.15437i 0 −2.23701 − 8.71756i 0
101.7 0 2.53217 − 1.60876i 0 0 0 4.69530 − 8.13249i 0 3.82376 − 8.14732i 0
101.8 0 2.97496 + 0.386835i 0 0 0 −2.32731 + 4.03103i 0 8.70072 + 2.30163i 0
401.1 0 −2.91526 − 0.708005i 0 0 0 4.96248 + 8.59526i 0 7.99746 + 4.12804i 0
401.2 0 −2.70874 + 1.28947i 0 0 0 −5.40503 − 9.36178i 0 5.67451 − 6.98569i 0
401.3 0 −1.29149 − 2.70777i 0 0 0 −1.73252 − 3.00081i 0 −5.66408 + 6.99415i 0
401.4 0 −0.978225 + 2.83603i 0 0 0 2.71262 + 4.69840i 0 −7.08615 − 5.54856i 0
401.5 0 1.54771 − 2.56994i 0 0 0 0.725042 + 1.25581i 0 −4.20921 − 7.95503i 0
401.6 0 1.83888 + 2.37034i 0 0 0 −4.13058 − 7.15437i 0 −2.23701 + 8.71756i 0
401.7 0 2.53217 + 1.60876i 0 0 0 4.69530 + 8.13249i 0 3.82376 + 8.14732i 0
401.8 0 2.97496 − 0.386835i 0 0 0 −2.32731 − 4.03103i 0 8.70072 − 2.30163i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.p.e yes 16
3.b odd 2 1 2700.3.p.d 16
5.b even 2 1 900.3.p.d ✓ 16
5.c odd 4 2 900.3.u.d 32
9.c even 3 1 2700.3.p.d 16
9.d odd 6 1 inner 900.3.p.e yes 16
15.d odd 2 1 2700.3.p.e 16
15.e even 4 2 2700.3.u.d 32
45.h odd 6 1 900.3.p.d ✓ 16
45.j even 6 1 2700.3.p.e 16
45.k odd 12 2 2700.3.u.d 32
45.l even 12 2 900.3.u.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.p.d ✓ 16 5.b even 2 1
900.3.p.d ✓ 16 45.h odd 6 1
900.3.p.e yes 16 1.a even 1 1 trivial
900.3.p.e yes 16 9.d odd 6 1 inner
900.3.u.d 32 5.c odd 4 2
900.3.u.d 32 45.l even 12 2
2700.3.p.d 16 3.b odd 2 1
2700.3.p.d 16 9.c even 3 1
2700.3.p.e 16 15.d odd 2 1
2700.3.p.e 16 45.j even 6 1
2700.3.u.d 32 15.e even 4 2
2700.3.u.d 32 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + T_{7}^{15} + 219 T_{7}^{14} + 218 T_{7}^{13} + 33299 T_{7}^{12} + 36030 T_{7}^{11} + \cdots + 1115296517776 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 1115296517776 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 318893306317056 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 206194665870400 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} - T^{7} + \cdots + 261665566)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 74\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{8} + 11 T^{7} + \cdots + 38426783044)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 68\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 17\!\cdots\!46)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 24\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 29\!\cdots\!21 \) Copy content Toggle raw display
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