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Properties

Label 1950.2.j.e
Level $1950$
Weight $2$
Character orbit 1950.j
Analytic conductor $15.571$
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] = [1950,2,Mod(1243,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.1243");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.j (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: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 390)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{8} q^{3} + q^{4} + \beta_{8} q^{6} + (\beta_{15} + \beta_{13} + \cdots - \beta_{2}) q^{7}+ \cdots - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{8} q^{3} + q^{4} + \beta_{8} q^{6} + (\beta_{15} + \beta_{13} + \cdots - \beta_{2}) q^{7}+ \cdots + (\beta_{15} + \beta_{14} + \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} + 4 q^{11} - 4 q^{13} + 16 q^{16} - 4 q^{17} + 4 q^{19} - 8 q^{21} - 4 q^{22} - 16 q^{23} + 4 q^{26} + 12 q^{31} - 16 q^{32} + 4 q^{34} - 4 q^{38} - 4 q^{39} + 4 q^{41} + 8 q^{42} + 16 q^{43} + 4 q^{44} + 16 q^{46} - 80 q^{49} - 4 q^{52} + 44 q^{53} + 16 q^{57} + 12 q^{59} - 32 q^{61} - 12 q^{62} + 16 q^{64} + 32 q^{67} - 4 q^{68} - 16 q^{69} + 16 q^{71} + 4 q^{76} + 32 q^{77} + 4 q^{78} - 16 q^{81} - 4 q^{82} - 8 q^{84} - 16 q^{86} - 28 q^{87} - 4 q^{88} + 4 q^{89} + 76 q^{91} - 16 q^{92} + 8 q^{97} + 80 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 217 \nu^{15} + 584 \nu^{14} - 5720 \nu^{13} + 17472 \nu^{12} - 49324 \nu^{11} + 194656 \nu^{10} + \cdots - 701696 ) / 113152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19 \nu^{15} + 342 \nu^{14} - 1352 \nu^{13} + 11024 \nu^{12} - 29156 \nu^{11} + 136680 \nu^{10} + \cdots + 303296 ) / 56576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12 \nu^{15} + 483 \nu^{14} - 208 \nu^{13} + 15080 \nu^{12} + 656 \nu^{11} + 179204 \nu^{10} + \cdots + 58464 ) / 56576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12 \nu^{15} + 557 \nu^{14} - 208 \nu^{13} + 15704 \nu^{12} + 656 \nu^{11} + 158588 \nu^{10} + \cdots + 68000 ) / 56576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 725 \nu^{15} - 22776 \nu^{13} - 273788 \nu^{11} - 1589116 \nu^{9} - 4674452 \nu^{7} + \cdots - 697504 \nu ) / 113152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 147 \nu^{14} + 4472 \nu^{12} + 51348 \nu^{10} + 279540 \nu^{8} + 759820 \nu^{6} + 1028936 \nu^{4} + \cdots + 86560 ) / 14144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 247 \nu^{15} + 527 \nu^{14} - 7384 \nu^{13} + 15912 \nu^{12} - 82420 \nu^{11} + 180404 \nu^{10} + \cdots + 147424 ) / 56576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 247 \nu^{15} - 527 \nu^{14} - 7384 \nu^{13} - 15912 \nu^{12} - 82420 \nu^{11} - 180404 \nu^{10} + \cdots - 147424 ) / 56576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 219 \nu^{14} + 6552 \nu^{12} + 72996 \nu^{10} + 374068 \nu^{8} + 890060 \nu^{6} + 889896 \nu^{4} + \cdots - 8544 ) / 14144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 410 \nu^{15} - 965 \nu^{14} + 12896 \nu^{13} - 29016 \nu^{12} + 154872 \nu^{11} - 326396 \nu^{10} + \cdots - 45472 ) / 56576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5\nu^{14} + 148\nu^{12} + 1624\nu^{10} + 8148\nu^{8} + 18932\nu^{6} + 19120\nu^{4} + 6900\nu^{2} + 608 ) / 272 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 199 \nu^{15} + 392 \nu^{14} + 6344 \nu^{13} + 12272 \nu^{12} + 77764 \nu^{11} + 146704 \nu^{10} + \cdots + 115456 ) / 28288 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 31 \nu^{15} - 936 \nu^{13} - 10612 \nu^{11} - 56340 \nu^{9} - 144828 \nu^{7} - 173864 \nu^{5} + \cdots - 8672 \nu ) / 1664 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2161 \nu^{15} + 966 \nu^{14} - 66040 \nu^{13} + 30160 \nu^{12} - 761740 \nu^{11} + 358408 \nu^{10} + \cdots + 116928 ) / 113152 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2693 \nu^{15} - 1156 \nu^{14} + 83512 \nu^{13} - 32032 \nu^{12} + 984892 \nu^{11} - 313008 \nu^{10} + \cdots + 497024 ) / 113152 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{9} + 2\beta_{8} + \beta_{6} - 2\beta_{5} - 2\beta_{3} + 2\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{14} - 3 \beta_{11} + 3 \beta_{9} - 8 \beta_{8} - \beta_{7} - 4 \beta_{6} + \cdots - 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{14} + 9 \beta_{12} + 2 \beta_{11} - 9 \beta_{10} - 2 \beta_{9} - 8 \beta_{7} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{15} - 10 \beta_{14} + 2 \beta_{13} + 23 \beta_{11} - 22 \beta_{9} + 65 \beta_{8} + \cdots + 72 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{15} - 11 \beta_{14} - 79 \beta_{12} - 38 \beta_{11} + 79 \beta_{10} + 34 \beta_{9} + \cdots - 172 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 162 \beta_{15} + 94 \beta_{14} - 40 \beta_{13} + 8 \beta_{12} - 197 \beta_{11} + 8 \beta_{10} + \cdots - 670 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 106 \beta_{15} + 106 \beta_{14} + 716 \beta_{12} + 496 \beta_{11} - 716 \beta_{10} - 420 \beta_{9} + \cdots + 1586 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1738 \beta_{15} - 898 \beta_{14} + 568 \beta_{13} - 160 \beta_{12} + 1772 \beta_{11} - 160 \beta_{10} + \cdots + 6320 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1036 \beta_{15} - 1036 \beta_{14} - 6676 \beta_{12} - 5656 \beta_{11} + 6676 \beta_{10} + \cdots - 14928 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 17948 \beta_{15} + 8788 \beta_{14} - 7116 \beta_{13} + 2236 \beta_{12} - 16380 \beta_{11} + \cdots - 59976 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10446 \beta_{15} + 10446 \beta_{14} + 63498 \beta_{12} + 60608 \beta_{11} - 63498 \beta_{10} + \cdots + 141580 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 181568 \beta_{15} - 87536 \beta_{14} + 83728 \beta_{13} - 27224 \beta_{12} + 153998 \beta_{11} + \cdots + 571068 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 107620 \beta_{15} - 107620 \beta_{14} - 611880 \beta_{12} - 628920 \beta_{11} + 611880 \beta_{10} + \cdots - 1347900 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1815852 \beta_{15} + 880988 \beta_{14} - 948768 \beta_{13} + 309968 \beta_{12} - 1463720 \beta_{11} + \cdots - 5450056 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1243.1
− 3.14581i
2.42515i
1.54171i
− 0.821039i
1.63314i
1.09662i
0.304860i
− 3.03462i
0.821039i
− 1.54171i
− 2.42515i
3.14581i
3.03462i
− 0.304860i
− 1.09662i
− 1.63314i
−1.00000 −0.707107 − 0.707107i 1.00000 0 0.707107 + 0.707107i − 4.05336i −1.00000 1.00000i 0
1243.2 −1.00000 −0.707107 − 0.707107i 1.00000 0 0.707107 + 0.707107i − 3.30195i −1.00000 1.00000i 0
1243.3 −1.00000 −0.707107 − 0.707107i 1.00000 0 0.707107 + 0.707107i 0.720033i −1.00000 1.00000i 0
1243.4 −1.00000 −0.707107 − 0.707107i 1.00000 0 0.707107 + 0.707107i 3.80685i −1.00000 1.00000i 0
1243.5 −1.00000 0.707107 + 0.707107i 1.00000 0 −0.707107 − 0.707107i − 4.24302i −1.00000 1.00000i 0
1243.6 −1.00000 0.707107 + 0.707107i 1.00000 0 −0.707107 − 0.707107i − 0.946681i −1.00000 1.00000i 0
1243.7 −1.00000 0.707107 + 0.707107i 1.00000 0 −0.707107 − 0.707107i 2.86620i −1.00000 1.00000i 0
1243.8 −1.00000 0.707107 + 0.707107i 1.00000 0 −0.707107 − 0.707107i 5.15192i −1.00000 1.00000i 0
1357.1 −1.00000 −0.707107 + 0.707107i 1.00000 0 0.707107 − 0.707107i − 3.80685i −1.00000 − 1.00000i 0
1357.2 −1.00000 −0.707107 + 0.707107i 1.00000 0 0.707107 − 0.707107i − 0.720033i −1.00000 − 1.00000i 0
1357.3 −1.00000 −0.707107 + 0.707107i 1.00000 0 0.707107 − 0.707107i 3.30195i −1.00000 − 1.00000i 0
1357.4 −1.00000 −0.707107 + 0.707107i 1.00000 0 0.707107 − 0.707107i 4.05336i −1.00000 − 1.00000i 0
1357.5 −1.00000 0.707107 − 0.707107i 1.00000 0 −0.707107 + 0.707107i − 5.15192i −1.00000 − 1.00000i 0
1357.6 −1.00000 0.707107 − 0.707107i 1.00000 0 −0.707107 + 0.707107i − 2.86620i −1.00000 − 1.00000i 0
1357.7 −1.00000 0.707107 − 0.707107i 1.00000 0 −0.707107 + 0.707107i 0.946681i −1.00000 − 1.00000i 0
1357.8 −1.00000 0.707107 − 0.707107i 1.00000 0 −0.707107 + 0.707107i 4.24302i −1.00000 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1243.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.j.e 16
5.b even 2 1 390.2.j.b ✓ 16
5.c odd 4 1 390.2.t.b yes 16
5.c odd 4 1 1950.2.t.e 16
13.d odd 4 1 1950.2.t.e 16
15.d odd 2 1 1170.2.m.i 16
15.e even 4 1 1170.2.w.i 16
65.f even 4 1 390.2.j.b ✓ 16
65.g odd 4 1 390.2.t.b yes 16
65.k even 4 1 inner 1950.2.j.e 16
195.n even 4 1 1170.2.w.i 16
195.u odd 4 1 1170.2.m.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.b ✓ 16 5.b even 2 1
390.2.j.b ✓ 16 65.f even 4 1
390.2.t.b yes 16 5.c odd 4 1
390.2.t.b yes 16 65.g odd 4 1
1170.2.m.i 16 15.d odd 2 1
1170.2.m.i 16 195.u odd 4 1
1170.2.w.i 16 15.e even 4 1
1170.2.w.i 16 195.n even 4 1
1950.2.j.e 16 1.a even 1 1 trivial
1950.2.j.e 16 65.k even 4 1 inner
1950.2.t.e 16 5.c odd 4 1
1950.2.t.e 16 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7}^{16} + 96 T_{7}^{14} + 3760 T_{7}^{12} + 77336 T_{7}^{10} + 890304 T_{7}^{8} + 5594816 T_{7}^{6} + \cdots + 4734976 \) Copy content Toggle raw display
\( T_{17}^{16} + 4 T_{17}^{15} + 8 T_{17}^{14} - 72 T_{17}^{13} + 940 T_{17}^{12} + 2240 T_{17}^{11} + \cdots + 2534464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 96 T^{14} + \cdots + 4734976 \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{15} + \cdots + 58003456 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 4 T^{15} + \cdots + 2534464 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 77275104256 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21484523776 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 837825347584 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1092810981376 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 79370665984 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 349241344 \) Copy content Toggle raw display
$47$ \( T^{16} + 176 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 50182602625024 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 48729781571584 \) Copy content Toggle raw display
$61$ \( (T^{8} + 16 T^{7} + \cdots - 3982336)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 16 T^{7} + \cdots + 220928)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 846046756864 \) Copy content Toggle raw display
$73$ \( (T^{8} - 384 T^{6} + \cdots + 2967568)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 604860841984 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 509407657984 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 17644340224 \) Copy content Toggle raw display
$97$ \( (T^{8} - 4 T^{7} + \cdots + 12563008)^{2} \) Copy content Toggle raw display
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