Properties

Label 1050.2.b.f
Level $1050$
Weight $2$
Character orbit 1050.b
Analytic conductor $8.384$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(251,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.b (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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.101415451701035401216.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} - \beta_{7} q^{3} - q^{4} - \beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{10} q^{8} + (\beta_{15} - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} - \beta_{7} q^{3} - q^{4} - \beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{10} q^{8} + (\beta_{15} - \beta_{6}) q^{9} + \beta_{15} q^{11} + \beta_{7} q^{12} + (\beta_{7} - \beta_{4}) q^{13} + \beta_{11} q^{14} + q^{16} + (\beta_{12} + \beta_{7} + \beta_{4}) q^{17} - \beta_{5} q^{18} + ( - \beta_{14} + 2 \beta_{3} - 2 \beta_{2}) q^{19} + (\beta_{15} - \beta_{14} - \beta_{13} + \beta_{3} - 1) q^{21} + \beta_1 q^{22} + ( - 2 \beta_{10} + 2 \beta_{5} + \beta_1) q^{23} + \beta_{2} q^{24} + (\beta_{3} + \beta_{2}) q^{26} + (\beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{4}) q^{27} - \beta_{9} q^{28} + (2 \beta_{15} - \beta_{13} - \beta_{11}) q^{29} + ( - \beta_{14} - \beta_{3} + \beta_{2}) q^{31} - \beta_{10} q^{32} + (\beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4}) q^{33} + (\beta_{14} - \beta_{3} + \beta_{2}) q^{34} + ( - \beta_{15} + \beta_{6}) q^{36} + (\beta_{9} - \beta_{8} + 3 \beta_1) q^{37} + (\beta_{12} + 2 \beta_{7} + 2 \beta_{4}) q^{38} + ( - \beta_{15} + \beta_{6} + 3) q^{39} + (2 \beta_{13} - 2 \beta_{11} - 2 \beta_{3} - 2 \beta_{2}) q^{41} + (\beta_{12} + \beta_{10} - \beta_{8} + \beta_{4} + \beta_1) q^{42} + (2 \beta_{9} - 2 \beta_{8}) q^{43} - \beta_{15} q^{44} + (\beta_{15} - 2 \beta_{6} - 2) q^{46} + 2 \beta_{12} q^{47} - \beta_{7} q^{48} + ( - \beta_{15} - \beta_{14} + 2 \beta_{6} - \beta_{3} + \beta_{2} + 1) q^{49} + ( - \beta_{15} - \beta_{13} - \beta_{11} - 2) q^{51} + ( - \beta_{7} + \beta_{4}) q^{52} + ( - 2 \beta_{10} - 4 \beta_{5} - 2 \beta_1) q^{53} + (\beta_{14} - \beta_{13} + \beta_{11} + 2 \beta_{3} + \beta_{2}) q^{54} - \beta_{11} q^{56} + ( - 5 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{5} + \beta_1) q^{57} + (\beta_{9} - \beta_{8} + 2 \beta_1) q^{58} + (\beta_{13} - \beta_{11} - \beta_{3} - \beta_{2}) q^{59} + (3 \beta_{3} - 3 \beta_{2}) q^{61} + (\beta_{12} - \beta_{7} - \beta_{4}) q^{62} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \beta_{5} + 2 \beta_1) q^{63} - q^{64} + (\beta_{14} - \beta_{13} + \beta_{11} - \beta_{3} + \beta_{2}) q^{66} + (2 \beta_{9} - 2 \beta_{8}) q^{67} + ( - \beta_{12} - \beta_{7} - \beta_{4}) q^{68} + ( - \beta_{14} + \beta_{13} - \beta_{11} - 5 \beta_{3} - 3 \beta_{2}) q^{69} + ( - 2 \beta_{15} + 3 \beta_{13} + 3 \beta_{11}) q^{71} + \beta_{5} q^{72} + (\beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{4}) q^{73} + ( - 3 \beta_{15} + \beta_{13} + \beta_{11}) q^{74} + (\beta_{14} - 2 \beta_{3} + 2 \beta_{2}) q^{76} + ( - 2 \beta_{10} + 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_1) q^{77} + ( - 3 \beta_{10} + \beta_{5}) q^{78} + (\beta_{15} - 2 \beta_{6} + 2) q^{79} + (\beta_{15} - 2 \beta_{13} - 2 \beta_{11} + 5) q^{81} + (2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{4}) q^{82} + (2 \beta_{12} - \beta_{7} - \beta_{4}) q^{83} + ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{3} + 1) q^{84} + (2 \beta_{13} + 2 \beta_{11}) q^{86} + (3 \beta_{9} + 3 \beta_{8} + 2 \beta_{7}) q^{87} - \beta_1 q^{88} + (2 \beta_{13} - 2 \beta_{11} + 4 \beta_{3} + 4 \beta_{2}) q^{89} + (2 \beta_{14} - \beta_{3} + \beta_{2} + 2) q^{91} + (2 \beta_{10} - 2 \beta_{5} - \beta_1) q^{92} + (4 \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{5} + \beta_1) q^{93} + 2 \beta_{14} q^{94} - \beta_{2} q^{96} + ( - \beta_{9} - \beta_{8} + 4 \beta_{7} - 4 \beta_{4}) q^{97} + (\beta_{12} - \beta_{10} - \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_1) q^{98} + (\beta_{15} - 2 \beta_{13} - 2 \beta_{11} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{21} + 48 q^{39} - 32 q^{46} + 16 q^{49} - 32 q^{51} - 16 q^{64} + 32 q^{79} + 80 q^{81} + 16 q^{84} + 32 q^{91} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{12} + 10\nu^{8} - 5\nu^{4} - 508 ) / 180 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{13} - 130\nu^{9} + 1055\nu^{5} - 764\nu ) / 240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49\nu^{15} - 910\nu^{11} + 7625\nu^{7} - 7268\nu^{3} ) / 1440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{13} + 194\nu^{9} - 1531\nu^{5} + 412\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{12} - 190\nu^{8} + 1495\nu^{4} - 212 ) / 120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} - 16\nu^{10} + 113\nu^{6} + 154\nu^{2} ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{15} - 158\nu^{11} + 1233\nu^{7} - 52\nu^{3} ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35 \nu^{15} + 51 \nu^{14} - 30 \nu^{13} - 2 \nu^{12} - 650 \nu^{11} - 930 \nu^{10} + 540 \nu^{9} + 20 \nu^{8} + 5395 \nu^{7} + 7635 \nu^{6} - 4470 \nu^{5} - 10 \nu^{4} - 5140 \nu^{3} - 6252 \nu^{2} + \cdots - 1016 ) / 1440 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35 \nu^{15} - 51 \nu^{14} - 30 \nu^{13} + 2 \nu^{12} - 650 \nu^{11} + 930 \nu^{10} + 540 \nu^{9} - 20 \nu^{8} + 5395 \nu^{7} - 7635 \nu^{6} - 4470 \nu^{5} + 10 \nu^{4} - 5140 \nu^{3} + 6252 \nu^{2} + \cdots + 1016 ) / 1440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -13\nu^{14} + 230\nu^{10} - 1805\nu^{6} + 316\nu^{2} ) / 240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 96 \nu^{15} + 55 \nu^{14} - 76 \nu^{13} + 90 \nu^{12} - 1680 \nu^{11} - 970 \nu^{10} + 1360 \nu^{9} - 1620 \nu^{8} + 13080 \nu^{7} + 7655 \nu^{6} - 10820 \nu^{5} + 12690 \nu^{4} - 552 \nu^{3} + \cdots - 3240 ) / 1440 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -41\nu^{15} + 10\nu^{13} + 734\nu^{11} - 172\nu^{9} - 5857\nu^{7} + 1274\nu^{5} + 2212\nu^{3} + 472\nu ) / 288 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 96 \nu^{15} + 55 \nu^{14} + 76 \nu^{13} + 90 \nu^{12} + 1680 \nu^{11} - 970 \nu^{10} - 1360 \nu^{9} - 1620 \nu^{8} - 13080 \nu^{7} + 7655 \nu^{6} + 10820 \nu^{5} + 12690 \nu^{4} + \cdots - 3240 ) / 1440 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 241\nu^{15} + 110\nu^{13} - 4270\nu^{11} - 1940\nu^{9} + 33785\nu^{7} + 15310\nu^{5} - 8372\nu^{3} - 1240\nu ) / 1440 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\nu^{14} - 194\nu^{10} + 1531\nu^{6} - 268\nu^{2} ) / 72 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_{11} + 2\beta_{4} - \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + 2\beta_{10} + 2\beta_{9} - 2\beta_{8} - 2\beta_{6} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{14} + 2 \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{9} - \beta_{8} + 7 \beta_{7} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{15} - 4\beta_{13} - 4\beta_{11} + 6\beta_{5} + 9\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8 \beta_{14} + 8 \beta_{13} - 9 \beta_{12} - 8 \beta_{11} - 9 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + 13 \beta_{4} - 8 \beta_{3} - 18 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 32\beta_{15} + 70\beta_{10} + 10\beta_{9} - 10\beta_{8} - 14\beta_{6} + 5\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{14} + 2 \beta_{13} + 29 \beta_{12} - 2 \beta_{11} - 29 \beta_{9} - 29 \beta_{8} + 35 \beta_{7} + 29 \beta_{4} + 84 \beta_{3} - 2 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 36\beta_{15} - 72\beta_{13} - 72\beta_{11} + 102\beta_{5} + 63\beta _1 + 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 22 \beta_{14} + 22 \beta_{13} - 111 \beta_{12} - 22 \beta_{11} - 111 \beta_{9} - 111 \beta_{8} - 111 \beta_{7} - 49 \beta_{4} - 22 \beta_{3} - 292 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 278\beta_{15} + 902\beta_{10} - 58\beta_{9} + 58\beta_{8} + 82\beta_{6} - 29\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 244 \beta_{14} - 244 \beta_{13} + 391 \beta_{12} + 244 \beta_{11} - 391 \beta_{9} - 391 \beta_{8} - 299 \beta_{7} + 391 \beta_{4} + 862 \beta_{3} + 244 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 350\beta_{15} - 700\beta_{13} - 700\beta_{11} + 990\beta_{5} - 135\beta _1 - 1782 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 688 \beta_{14} - 688 \beta_{13} - 705 \beta_{12} + 688 \beta_{11} - 705 \beta_{9} - 705 \beta_{8} - 705 \beta_{7} - 2651 \beta_{4} + 688 \beta_{3} - 2682 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 524\beta_{15} + 6214\beta_{10} - 2366\beta_{9} + 2366\beta_{8} + 3346\beta_{6} - 1183\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 4546 \beta_{14} - 4546 \beta_{13} + 2897 \beta_{12} + 4546 \beta_{11} - 2897 \beta_{9} - 2897 \beta_{8} - 9961 \beta_{7} + 2897 \beta_{4} + 3648 \beta_{3} + 4546 \beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.81768 + 0.332046i
−0.137538 + 0.752908i
−0.332046 1.81768i
0.752908 0.137538i
−0.752908 + 0.137538i
0.332046 + 1.81768i
0.137538 0.752908i
−1.81768 0.332046i
−0.137538 0.752908i
1.81768 0.332046i
0.752908 + 0.137538i
−0.332046 + 1.81768i
0.332046 1.81768i
−0.752908 0.137538i
−1.81768 + 0.332046i
0.137538 + 0.752908i
1.00000i −1.68014 0.420861i −1.00000 0 −0.420861 + 1.68014i 1.16372 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.2 1.00000i −1.68014 + 0.420861i −1.00000 0 0.420861 + 1.68014i −1.16372 + 2.37608i 1.00000i 2.64575 1.41421i 0
251.3 1.00000i −0.420861 1.68014i −1.00000 0 −1.68014 + 0.420861i −2.57794 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.4 1.00000i −0.420861 + 1.68014i −1.00000 0 1.68014 + 0.420861i 2.57794 + 0.595188i 1.00000i −2.64575 1.41421i 0
251.5 1.00000i 0.420861 1.68014i −1.00000 0 −1.68014 0.420861i 2.57794 0.595188i 1.00000i −2.64575 1.41421i 0
251.6 1.00000i 0.420861 + 1.68014i −1.00000 0 1.68014 0.420861i −2.57794 + 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.7 1.00000i 1.68014 0.420861i −1.00000 0 −0.420861 1.68014i −1.16372 2.37608i 1.00000i 2.64575 1.41421i 0
251.8 1.00000i 1.68014 + 0.420861i −1.00000 0 0.420861 1.68014i 1.16372 + 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.9 1.00000i −1.68014 0.420861i −1.00000 0 0.420861 1.68014i −1.16372 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.10 1.00000i −1.68014 + 0.420861i −1.00000 0 −0.420861 1.68014i 1.16372 + 2.37608i 1.00000i 2.64575 1.41421i 0
251.11 1.00000i −0.420861 1.68014i −1.00000 0 1.68014 0.420861i 2.57794 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.12 1.00000i −0.420861 + 1.68014i −1.00000 0 −1.68014 0.420861i −2.57794 + 0.595188i 1.00000i −2.64575 1.41421i 0
251.13 1.00000i 0.420861 1.68014i −1.00000 0 1.68014 + 0.420861i −2.57794 0.595188i 1.00000i −2.64575 1.41421i 0
251.14 1.00000i 0.420861 + 1.68014i −1.00000 0 −1.68014 + 0.420861i 2.57794 + 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.15 1.00000i 1.68014 0.420861i −1.00000 0 0.420861 + 1.68014i 1.16372 2.37608i 1.00000i 2.64575 1.41421i 0
251.16 1.00000i 1.68014 + 0.420861i −1.00000 0 −0.420861 + 1.68014i −1.16372 + 2.37608i 1.00000i 2.64575 + 1.41421i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.b.f 16
3.b odd 2 1 inner 1050.2.b.f 16
5.b even 2 1 inner 1050.2.b.f 16
5.c odd 4 1 210.2.d.a 8
5.c odd 4 1 210.2.d.b yes 8
7.b odd 2 1 inner 1050.2.b.f 16
15.d odd 2 1 inner 1050.2.b.f 16
15.e even 4 1 210.2.d.a 8
15.e even 4 1 210.2.d.b yes 8
20.e even 4 1 1680.2.k.e 8
20.e even 4 1 1680.2.k.f 8
21.c even 2 1 inner 1050.2.b.f 16
35.c odd 2 1 inner 1050.2.b.f 16
35.f even 4 1 210.2.d.a 8
35.f even 4 1 210.2.d.b yes 8
60.l odd 4 1 1680.2.k.e 8
60.l odd 4 1 1680.2.k.f 8
105.g even 2 1 inner 1050.2.b.f 16
105.k odd 4 1 210.2.d.a 8
105.k odd 4 1 210.2.d.b yes 8
140.j odd 4 1 1680.2.k.e 8
140.j odd 4 1 1680.2.k.f 8
420.w even 4 1 1680.2.k.e 8
420.w even 4 1 1680.2.k.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.d.a 8 5.c odd 4 1
210.2.d.a 8 15.e even 4 1
210.2.d.a 8 35.f even 4 1
210.2.d.a 8 105.k odd 4 1
210.2.d.b yes 8 5.c odd 4 1
210.2.d.b yes 8 15.e even 4 1
210.2.d.b yes 8 35.f even 4 1
210.2.d.b yes 8 105.k odd 4 1
1050.2.b.f 16 1.a even 1 1 trivial
1050.2.b.f 16 3.b odd 2 1 inner
1050.2.b.f 16 5.b even 2 1 inner
1050.2.b.f 16 7.b odd 2 1 inner
1050.2.b.f 16 15.d odd 2 1 inner
1050.2.b.f 16 21.c even 2 1 inner
1050.2.b.f 16 35.c odd 2 1 inner
1050.2.b.f 16 105.g even 2 1 inner
1680.2.k.e 8 20.e even 4 1
1680.2.k.e 8 60.l odd 4 1
1680.2.k.e 8 140.j odd 4 1
1680.2.k.e 8 420.w even 4 1
1680.2.k.f 8 20.e even 4 1
1680.2.k.f 8 60.l odd 4 1
1680.2.k.f 8 140.j odd 4 1
1680.2.k.f 8 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{17}^{4} - 24T_{17}^{2} + 32 \) Copy content Toggle raw display
\( T_{37}^{4} - 128T_{37}^{2} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} - 10 T^{4} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{6} - 10 T^{4} - 196 T^{2} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 12 T^{2} + 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{2} + 32)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 52 T^{2} + 648)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 64 T^{2} + 576)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 64 T^{2} + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 40 T^{2} + 288)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 128 T^{2} + 1296)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 1152)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 128 T^{2} + 2304)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 80 T^{2} + 1152)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 232 T^{2} + 11664)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{2} + 72)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 108 T^{2} + 648)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 128 T^{2} + 2304)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 256 T^{2} + 15376)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 40 T^{2} + 288)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 24)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 108 T^{2} + 2888)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 416 T^{2} + 41472)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 280 T^{2} + 14112)^{4} \) Copy content Toggle raw display
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