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Properties

Label 3900.2.by.k
Level $3900$
Weight $2$
Character orbit 3900.by
Analytic conductor $31.142$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3900,2,Mod(1849,3900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.by (of order \(6\), degree \(2\), not 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: \(31.1416567883\)
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: \( x^{16} - 26x^{14} + 471x^{12} - 4250x^{10} + 27661x^{8} - 93852x^{6} + 225180x^{4} - 174960x^{2} + 104976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{3} + (\beta_{11} + \beta_{4}) q^{7} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{3} + (\beta_{11} + \beta_{4}) q^{7} - \beta_{3} q^{9} + ( - \beta_{10} + \beta_{7} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{10} + \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} - 16 q^{19} - 8 q^{21} - 72 q^{31} + 8 q^{41} + 4 q^{49} - 16 q^{51} + 40 q^{59} + 28 q^{61} - 80 q^{79} - 8 q^{81} - 16 q^{89} - 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26x^{14} + 471x^{12} - 4250x^{10} + 27661x^{8} - 93852x^{6} + 225180x^{4} - 174960x^{2} + 104976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4391 \nu^{14} - 184276 \nu^{12} + 3768549 \nu^{10} - 46992688 \nu^{8} + 335323679 \nu^{6} + \cdots - 2288091888 ) / 1576825056 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17892881 \nu^{14} - 448232911 \nu^{12} + 8036427861 \nu^{10} - 69182444329 \nu^{8} + \cdots - 2762403269184 ) / 2059136420004 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12505 \nu^{15} - 314870 \nu^{13} + 5053179 \nu^{11} - 38145374 \nu^{9} + 143639161 \nu^{7} + \cdots - 19501525656 \nu ) / 8010905184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 522842861 \nu^{14} - 12576229576 \nu^{12} + 229133984055 \nu^{10} - 1944770746036 \nu^{8} + \cdots + 48106199023488 ) / 49419274080096 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5837011 \nu^{14} - 190429796 \nu^{12} + 3654766713 \nu^{10} - 40432554608 \nu^{8} + \cdots - 1859476594608 ) / 531390043872 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 212999305 \nu^{14} + 5071385750 \nu^{12} - 86071460619 \nu^{10} + 649735158014 \nu^{8} + \cdots - 12926325861576 ) / 16473091360032 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 377136448 \nu^{14} + 9201764333 \nu^{12} - 161326583904 \nu^{10} + 1325363591327 \nu^{8} + \cdots + 82275357025908 ) / 24709637040048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1514 \nu^{15} + 59137 \nu^{13} - 1199814 \nu^{11} + 15073663 \nu^{9} - 113952086 \nu^{7} + \cdots + 2049169428 \nu ) / 552681144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 440136845 \nu^{14} + 10335823678 \nu^{12} - 177856078551 \nu^{10} + \cdots - 32164935836328 ) / 16473091360032 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1882265105 \nu^{15} + 44279195254 \nu^{13} - 760609556259 \nu^{11} + \cdots - 304997970945672 \nu ) / 296515644480576 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2216448193 \nu^{15} + 61461640508 \nu^{13} - 1135232042403 \nu^{11} + \cdots + 470902844441232 \nu ) / 296515644480576 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17892881 \nu^{15} + 448232911 \nu^{13} - 8036427861 \nu^{11} + 69182444329 \nu^{9} + \cdots + 2762403269184 \nu ) / 2059136420004 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1106558695 \nu^{15} + 27218685500 \nu^{13} - 484354375221 \nu^{11} + \cdots + 159962754820464 \nu ) / 49419274080096 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2067967549 \nu^{15} + 51529366268 \nu^{13} - 914519014167 \nu^{11} + \cdots - 50352680221344 \nu ) / 74128911120144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} - 2\beta_{5} + 6\beta_{3} - \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} + 3\beta_{14} - 10\beta_{13} + 6\beta_{12} + 6\beta_{11} - \beta_{9} + 3\beta_{4} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 26\beta_{8} + 6\beta_{6} - 13\beta_{5} + 60\beta_{3} - 19\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -25\beta_{15} + 45\beta_{14} - 118\beta_{13} + 114\beta_{12} - 50\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 120\beta_{10} + 163\beta_{8} - 307\beta_{7} + 163\beta_{5} - 708 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 427\beta_{15} - 1842\beta_{11} - 427\beta_{9} - 609\beta_{4} - 1504\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1890 \beta_{10} - 2113 \beta_{8} - 4627 \beta_{7} - 1890 \beta_{6} + 4226 \beta_{5} - 9024 \beta_{3} + \cdots - 9024 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 13034 \beta_{15} - 8229 \beta_{14} + 19990 \beta_{13} - 27762 \beta_{12} - 27762 \beta_{11} + \cdots - 19990 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -56438\beta_{8} - 27780\beta_{6} + 28219\beta_{5} - 119940\beta_{3} + 67303\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 95083\beta_{15} - 112437\beta_{14} + 271900\beta_{13} - 403818\beta_{12} + 190166\beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -397686\beta_{10} - 384337\beta_{8} + 960967\beta_{7} - 384337\beta_{5} + 1631400 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -1358653\beta_{15} + 5765802\beta_{11} + 1358653\beta_{9} + 1550697\beta_{4} + 3745378\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5626656 \beta_{10} + 5296075 \beta_{8} + 13587139 \beta_{7} + 5626656 \beta_{6} - 10592150 \beta_{5} + \cdots + 22472268 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 38427590 \beta_{15} + 21514881 \beta_{14} - 51947632 \beta_{13} + 81522834 \beta_{12} + \cdots + 51947632 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.66001 + 0.958407i
−2.37225 − 1.36962i
3.23827 + 1.86962i
−0.793983 − 0.458407i
0.793983 + 0.458407i
−3.23827 − 1.86962i
2.37225 + 1.36962i
−1.66001 − 0.958407i
1.66001 − 0.958407i
−2.37225 + 1.36962i
3.23827 − 1.86962i
−0.793983 + 0.458407i
0.793983 − 0.458407i
−3.23827 + 1.86962i
2.37225 − 1.36962i
−1.66001 + 0.958407i
0 −0.866025 + 0.500000i 0 0 0 −3.14159 − 1.81380i 0 0.500000 − 0.866025i 0
1849.2 0 −0.866025 + 0.500000i 0 0 0 −0.116591 − 0.0673141i 0 0.500000 − 0.866025i 0
1849.3 0 −0.866025 + 0.500000i 0 0 0 2.20736 + 1.27442i 0 0.500000 − 0.866025i 0
1849.4 0 −0.866025 + 0.500000i 0 0 0 2.78287 + 1.60669i 0 0.500000 − 0.866025i 0
1849.5 0 0.866025 − 0.500000i 0 0 0 −2.78287 − 1.60669i 0 0.500000 − 0.866025i 0
1849.6 0 0.866025 − 0.500000i 0 0 0 −2.20736 − 1.27442i 0 0.500000 − 0.866025i 0
1849.7 0 0.866025 − 0.500000i 0 0 0 0.116591 + 0.0673141i 0 0.500000 − 0.866025i 0
1849.8 0 0.866025 − 0.500000i 0 0 0 3.14159 + 1.81380i 0 0.500000 − 0.866025i 0
3649.1 0 −0.866025 − 0.500000i 0 0 0 −3.14159 + 1.81380i 0 0.500000 + 0.866025i 0
3649.2 0 −0.866025 − 0.500000i 0 0 0 −0.116591 + 0.0673141i 0 0.500000 + 0.866025i 0
3649.3 0 −0.866025 − 0.500000i 0 0 0 2.20736 − 1.27442i 0 0.500000 + 0.866025i 0
3649.4 0 −0.866025 − 0.500000i 0 0 0 2.78287 − 1.60669i 0 0.500000 + 0.866025i 0
3649.5 0 0.866025 + 0.500000i 0 0 0 −2.78287 + 1.60669i 0 0.500000 + 0.866025i 0
3649.6 0 0.866025 + 0.500000i 0 0 0 −2.20736 + 1.27442i 0 0.500000 + 0.866025i 0
3649.7 0 0.866025 + 0.500000i 0 0 0 0.116591 − 0.0673141i 0 0.500000 + 0.866025i 0
3649.8 0 0.866025 + 0.500000i 0 0 0 3.14159 − 1.81380i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.2.by.k 16
5.b even 2 1 inner 3900.2.by.k 16
5.c odd 4 1 3900.2.q.o ✓ 8
5.c odd 4 1 3900.2.q.p yes 8
13.c even 3 1 inner 3900.2.by.k 16
65.n even 6 1 inner 3900.2.by.k 16
65.q odd 12 1 3900.2.q.o ✓ 8
65.q odd 12 1 3900.2.q.p yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3900.2.q.o ✓ 8 5.c odd 4 1
3900.2.q.o ✓ 8 65.q odd 12 1
3900.2.q.p yes 8 5.c odd 4 1
3900.2.q.p yes 8 65.q odd 12 1
3900.2.by.k 16 1.a even 1 1 trivial
3900.2.by.k 16 5.b even 2 1 inner
3900.2.by.k 16 13.c even 3 1 inner
3900.2.by.k 16 65.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{16} - 30 T_{7}^{14} + 611 T_{7}^{12} - 6894 T_{7}^{10} + 56865 T_{7}^{8} - 255672 T_{7}^{6} + \cdots + 256 \) Copy content Toggle raw display
\( T_{11}^{8} + 26T_{11}^{6} + 48T_{11}^{5} + 604T_{11}^{4} + 624T_{11}^{3} + 2448T_{11}^{2} - 1728T_{11} + 5184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 30 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{8} + 26 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 60 T^{14} + \cdots + 26873856 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{7} + \cdots + 323761)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 18 T^{2} + 324)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 50 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 18 T^{3} + \cdots - 776)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$41$ \( (T^{8} - 4 T^{7} + \cdots + 2742336)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2457068790016 \) Copy content Toggle raw display
$47$ \( (T^{8} + 356 T^{6} + \cdots + 23970816)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 144 T^{6} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 20 T^{7} + \cdots + 144576576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 14 T^{7} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 71511783383296 \) Copy content Toggle raw display
$71$ \( (T^{8} + 146 T^{6} + \cdots + 4981824)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 404 T^{6} + \cdots + 26967249)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 20 T^{3} + \cdots - 1297)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 244 T^{6} + \cdots + 1679616)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 8 T^{7} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 18339659776 \) Copy content Toggle raw display
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