Properties

Label 480.2.m.b
Level $480$
Weight $2$
Character orbit 480.m
Analytic conductor $3.833$
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] = [480,2,Mod(239,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.m (of order \(2\), degree \(1\), 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: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{3} + (\beta_{10} + \beta_1) q^{5} - \beta_{4} q^{7} + ( - \beta_{12} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{3} + (\beta_{10} + \beta_1) q^{5} - \beta_{4} q^{7} + ( - \beta_{12} - 1) q^{9} + ( - \beta_{12} + \beta_{5}) q^{11} + ( - \beta_{11} + \beta_{8} - \beta_{7} - \beta_{4}) q^{13} + (\beta_{15} - \beta_{7} - \beta_{3} - \beta_1) q^{15} + ( - \beta_{14} - \beta_{13} - \beta_{2}) q^{17} + ( - \beta_{9} + 2) q^{19} + (2 \beta_{11} - \beta_{10} + \beta_{3}) q^{21} - \beta_{15} q^{23} + (\beta_{14} - \beta_{9} + \beta_{6} + 1) q^{25} + ( - 2 \beta_{14} - 2 \beta_{13} - \beta_{6} - \beta_{2}) q^{27} + ( - \beta_{8} - \beta_{7} - 2 \beta_{3} - \beta_1) q^{29} + (\beta_{15} + 2 \beta_{11} + \beta_{8} + \beta_{7}) q^{31} + ( - \beta_{14} - 2 \beta_{13} - \beta_{6} + 2 \beta_{2}) q^{33} + ( - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{9} - \beta_{5} + 2 \beta_{2}) q^{35} + (\beta_{11} - \beta_{8} + \beta_{7} - \beta_{4}) q^{37} + ( - 3 \beta_{15} + \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} + \beta_1) q^{39} + ( - \beta_{9} + 2 \beta_{5}) q^{41} + (\beta_{14} + \beta_{13} + 2 \beta_{6}) q^{43} + ( - 3 \beta_{15} - \beta_{11} - 2 \beta_{8} - 2 \beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{45} + \beta_{15} q^{47} + (\beta_{9} + 1) q^{49} + (\beta_{12} - \beta_{9} + \beta_{5} - 2) q^{51} + (3 \beta_{15} - 3 \beta_1) q^{53} + ( - \beta_{15} - 4 \beta_{11} - 2 \beta_{7} - \beta_{4}) q^{55} + (2 \beta_{14} + \beta_{13} - \beta_{6} - \beta_{2}) q^{57} + (\beta_{12} - \beta_{5}) q^{59} + (3 \beta_{15} + 3 \beta_{8} + 3 \beta_{7}) q^{61} + (\beta_{15} + \beta_{11} - \beta_{8} + \beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{63} + (3 \beta_{14} + 3 \beta_{13} + \beta_{9} - 2 \beta_{5} + \beta_{2}) q^{65} + ( - \beta_{14} + \beta_{13}) q^{67} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} + \beta_1) q^{69} + (2 \beta_{15} - 4 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + 4 \beta_{3}) q^{71} + ( - 2 \beta_{14} + 2 \beta_{13}) q^{73} + (\beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{9} - \beta_{6} - \beta_{5} - \beta_{2} - 2) q^{75} + ( - 4 \beta_{15} + 2 \beta_1) q^{77} + (3 \beta_{15} - 4 \beta_{11} + 3 \beta_{8} + 3 \beta_{7}) q^{79} + (\beta_{9} + 2 \beta_{5} - 3) q^{81} + (\beta_{14} + \beta_{13} - 4 \beta_{2}) q^{83} + (\beta_{15} - \beta_{11} + 2 \beta_{7} + \beta_{4}) q^{85} + ( - 3 \beta_{15} + \beta_{4} - 2 \beta_1) q^{87} + (2 \beta_{12} - 2 \beta_{5}) q^{89} + (4 \beta_{9} + 4) q^{91} + ( - \beta_{15} - \beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} + 3 \beta_1) q^{93} + ( - 3 \beta_{15} + 4 \beta_{10} - \beta_{8} - \beta_{7} - 2 \beta_{3} + 2 \beta_1) q^{95} + (2 \beta_{13} + 2 \beta_{6}) q^{97} + ( - \beta_{12} + 4 \beta_{9} - \beta_{5} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} + 32 q^{19} + 16 q^{25} + 16 q^{49} - 32 q^{51} - 32 q^{75} - 48 q^{81} + 64 q^{91} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{13} + 69\nu^{11} + 506\nu^{9} + 1488\nu^{7} + 1638\nu^{5} + 594\nu^{3} + 28\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{14} - 164\nu^{12} - 1250\nu^{10} - 3984\nu^{8} - 5334\nu^{6} - 3024\nu^{4} - 596\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{15} + 22 \nu^{14} + 62 \nu^{13} + 522 \nu^{12} + 344 \nu^{11} + 4080 \nu^{10} + 284 \nu^{9} + 13636 \nu^{8} - 2010 \nu^{7} + 20076 \nu^{6} - 3772 \nu^{5} + 13332 \nu^{4} - 2000 \nu^{3} + \cdots + 344 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{14} + 352\nu^{12} + 2692\nu^{10} + 8638\nu^{8} + 11726\nu^{6} + 6808\nu^{4} + 1496\nu^{2} + 76 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{15} + 12 \nu^{14} - 110 \nu^{13} + 282 \nu^{12} - 728 \nu^{11} + 2164 \nu^{10} - 1625 \nu^{9} + 7004 \nu^{8} - 118 \nu^{7} + 9752 \nu^{6} + 2276 \nu^{5} + 6092 \nu^{4} + 1648 \nu^{3} + \cdots + 136 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16 \nu^{15} + \nu^{14} + 376 \nu^{13} + 20 \nu^{12} + 2885 \nu^{11} + 100 \nu^{10} + 9330 \nu^{9} - 4 \nu^{8} + 12928 \nu^{7} - 918 \nu^{6} + 7872 \nu^{5} - 1440 \nu^{4} + 1834 \nu^{3} - 664 \nu^{2} + \cdots - 72 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8 \nu^{15} - 21 \nu^{14} + 196 \nu^{13} - 494 \nu^{12} + 1629 \nu^{11} - 3798 \nu^{10} + 6072 \nu^{9} - 12334 \nu^{8} + 10856 \nu^{7} - 17306 \nu^{6} + 9544 \nu^{5} - 11028 \nu^{4} + 3818 \nu^{3} + \cdots - 252 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14 \nu^{15} + 21 \nu^{14} - 326 \nu^{13} + 494 \nu^{12} - 2455 \nu^{11} + 3798 \nu^{10} - 7652 \nu^{9} + 12334 \nu^{8} - 9812 \nu^{7} + 17306 \nu^{6} - 5276 \nu^{5} + 11028 \nu^{4} + \cdots + 252 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6\nu^{14} + 141\nu^{12} + 1082\nu^{10} + 3502\nu^{8} + 4876\nu^{6} + 3046\nu^{4} + 804\nu^{2} + 68 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19 \nu^{15} + 6 \nu^{14} - 454 \nu^{13} + 138 \nu^{12} - 3598 \nu^{11} + 1012 \nu^{10} - 12338 \nu^{9} + 2976 \nu^{8} - 19070 \nu^{7} + 3276 \nu^{6} - 13604 \nu^{5} + 1188 \nu^{4} + \cdots + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -11\nu^{15} - 261\nu^{13} - 2042\nu^{11} - 6862\nu^{9} - 10334\nu^{7} - 7410\nu^{5} - 2412\nu^{3} - 252\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11 \nu^{15} + 12 \nu^{14} - 261 \nu^{13} + 282 \nu^{12} - 2040 \nu^{11} + 2164 \nu^{10} - 6817 \nu^{9} + 7004 \nu^{8} - 10018 \nu^{7} + 9752 \nu^{6} - 6554 \nu^{5} + 6092 \nu^{4} - 1640 \nu^{3} + \cdots + 136 ) / 8 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38 \nu^{15} - \nu^{14} + 896 \nu^{13} - 20 \nu^{12} + 6921 \nu^{11} - 100 \nu^{10} + 22674 \nu^{9} + 4 \nu^{8} + 32332 \nu^{7} + 918 \nu^{6} + 20960 \nu^{5} + 1440 \nu^{4} + 5842 \nu^{3} + 664 \nu^{2} + \cdots + 72 ) / 16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 38 \nu^{15} - \nu^{14} - 896 \nu^{13} - 20 \nu^{12} - 6921 \nu^{11} - 100 \nu^{10} - 22674 \nu^{9} + 4 \nu^{8} - 32332 \nu^{7} + 918 \nu^{6} - 20960 \nu^{5} + 1440 \nu^{4} - 5842 \nu^{3} + \cdots + 72 ) / 16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -19\nu^{15} - 451\nu^{13} - 3529\nu^{11} - 11832\nu^{9} - 17582\nu^{7} - 11966\nu^{5} - 3498\nu^{3} - 344\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - 2\beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} - \beta_{2} + \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{15} - 3 \beta_{13} + 8 \beta_{12} - 3 \beta_{11} - 3 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 9 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{15} - 20 \beta_{14} - 20 \beta_{13} - 12 \beta_{11} - 28 \beta_{10} - 13 \beta_{9} + 16 \beta_{8} - 8 \beta_{7} + 8 \beta_{4} + 8 \beta_{3} + 12 \beta_{2} - 10 \beta _1 + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 78 \beta_{15} - 10 \beta_{14} + 50 \beta_{13} - 83 \beta_{12} + 55 \beta_{11} + 27 \beta_{9} + 47 \beta_{8} + 47 \beta_{7} + 40 \beta_{6} + 29 \beta_{5} - 95 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 181 \beta_{15} + 280 \beta_{14} + 280 \beta_{13} + 144 \beta_{11} + 362 \beta_{10} + 161 \beta_{9} - 211 \beta_{8} + 77 \beta_{7} - 74 \beta_{4} - 134 \beta_{3} - 134 \beta_{2} + 114 \beta _1 - 504 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 984 \beta_{15} + 175 \beta_{14} - 672 \beta_{13} + 956 \beta_{12} - 756 \beta_{11} - 292 \beta_{9} - 571 \beta_{8} - 571 \beta_{7} - 497 \beta_{6} - 372 \beta_{5} + 1091 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1130 \beta_{15} - 1792 \beta_{14} - 1792 \beta_{13} - 876 \beta_{11} - 2260 \beta_{10} - 990 \beta_{9} + 1326 \beta_{8} - 426 \beta_{7} + 396 \beta_{4} + 900 \beta_{3} + 784 \beta_{2} - 680 \beta _1 + 2910 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 12191 \beta_{15} - 2388 \beta_{14} + 8496 \beta_{13} - 11446 \beta_{12} + 9624 \beta_{11} + 3407 \beta_{9} + 6971 \beta_{8} + 6971 \beta_{7} + 6108 \beta_{6} + 4632 \beta_{5} - 13034 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 13931 \beta_{15} + 22287 \beta_{14} + 22287 \beta_{13} + 10707 \beta_{11} + 27862 \beta_{10} + 12147 \beta_{9} - 16395 \beta_{8} + 5019 \beta_{7} - 4587 \beta_{4} - 11376 \beta_{3} - 9423 \beta_{2} + \cdots - 34858 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 74968 \beta_{15} + 15180 \beta_{14} - 52613 \beta_{13} + 69508 \beta_{12} - 59741 \beta_{11} - 20483 \beta_{9} - 42622 \beta_{8} - 42622 \beta_{7} - 37433 \beta_{6} - 28542 \beta_{5} + \cdots + 79079 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 170986 \beta_{15} - 274428 \beta_{14} - 274428 \beta_{13} - 131044 \beta_{11} - 341972 \beta_{10} - 148879 \beta_{9} + 201492 \beta_{8} - 60596 \beta_{7} + 54976 \beta_{4} + 140896 \beta_{3} + \cdots + 423384 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 919486 \beta_{15} - 188422 \beta_{14} + 646906 \beta_{13} - 848621 \beta_{12} + 735189 \beta_{11} + 249125 \beta_{9} + 521589 \beta_{8} + 521589 \beta_{7} + 458484 \beta_{6} + 350371 \beta_{5} + \cdots - 965133 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2095431 \beta_{15} + 3367136 \beta_{14} + 3367136 \beta_{13} + 1604464 \beta_{11} + 4190862 \beta_{10} + 1823791 \beta_{9} - 2470697 \beta_{8} + 738231 \beta_{7} - 667774 \beta_{4} + \cdots - 5168936 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 11266000 \beta_{15} + 2318785 \beta_{14} - 7933264 \beta_{13} + 10380392 \beta_{12} - 9018824 \beta_{11} - 3042952 \beta_{9} - 6385193 \beta_{8} - 6385193 \beta_{7} + \cdots + 11804009 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
3.49930i
0.528036i
3.49930i
0.528036i
1.05636i
0.724535i
1.05636i
0.724535i
2.13875i
0.357857i
2.13875i
0.357857i
0.886177i
2.08509i
0.886177i
2.08509i
0 −1.30656 1.13705i 0 −2.10100 0.765367i 0 −2.27411 0 0.414214 + 2.97127i 0
239.2 0 −1.30656 1.13705i 0 2.10100 + 0.765367i 0 2.27411 0 0.414214 + 2.97127i 0
239.3 0 −1.30656 + 1.13705i 0 −2.10100 + 0.765367i 0 −2.27411 0 0.414214 2.97127i 0
239.4 0 −1.30656 + 1.13705i 0 2.10100 0.765367i 0 2.27411 0 0.414214 2.97127i 0
239.5 0 −0.541196 1.64533i 0 −1.25928 + 1.84776i 0 3.29066 0 −2.41421 + 1.78089i 0
239.6 0 −0.541196 1.64533i 0 1.25928 1.84776i 0 −3.29066 0 −2.41421 + 1.78089i 0
239.7 0 −0.541196 + 1.64533i 0 −1.25928 1.84776i 0 3.29066 0 −2.41421 1.78089i 0
239.8 0 −0.541196 + 1.64533i 0 1.25928 + 1.84776i 0 −3.29066 0 −2.41421 1.78089i 0
239.9 0 0.541196 1.64533i 0 −1.25928 + 1.84776i 0 −3.29066 0 −2.41421 1.78089i 0
239.10 0 0.541196 1.64533i 0 1.25928 1.84776i 0 3.29066 0 −2.41421 1.78089i 0
239.11 0 0.541196 + 1.64533i 0 −1.25928 1.84776i 0 −3.29066 0 −2.41421 + 1.78089i 0
239.12 0 0.541196 + 1.64533i 0 1.25928 + 1.84776i 0 3.29066 0 −2.41421 + 1.78089i 0
239.13 0 1.30656 1.13705i 0 −2.10100 0.765367i 0 2.27411 0 0.414214 2.97127i 0
239.14 0 1.30656 1.13705i 0 2.10100 + 0.765367i 0 −2.27411 0 0.414214 2.97127i 0
239.15 0 1.30656 + 1.13705i 0 −2.10100 + 0.765367i 0 2.27411 0 0.414214 + 2.97127i 0
239.16 0 1.30656 + 1.13705i 0 2.10100 0.765367i 0 −2.27411 0 0.414214 + 2.97127i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.16
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
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.m.b 16
3.b odd 2 1 inner 480.2.m.b 16
4.b odd 2 1 120.2.m.b 16
5.b even 2 1 inner 480.2.m.b 16
5.c odd 4 2 2400.2.b.i 16
8.b even 2 1 120.2.m.b 16
8.d odd 2 1 inner 480.2.m.b 16
12.b even 2 1 120.2.m.b 16
15.d odd 2 1 inner 480.2.m.b 16
15.e even 4 2 2400.2.b.i 16
20.d odd 2 1 120.2.m.b 16
20.e even 4 2 600.2.b.i 16
24.f even 2 1 inner 480.2.m.b 16
24.h odd 2 1 120.2.m.b 16
40.e odd 2 1 inner 480.2.m.b 16
40.f even 2 1 120.2.m.b 16
40.i odd 4 2 600.2.b.i 16
40.k even 4 2 2400.2.b.i 16
60.h even 2 1 120.2.m.b 16
60.l odd 4 2 600.2.b.i 16
120.i odd 2 1 120.2.m.b 16
120.m even 2 1 inner 480.2.m.b 16
120.q odd 4 2 2400.2.b.i 16
120.w even 4 2 600.2.b.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.m.b 16 4.b odd 2 1
120.2.m.b 16 8.b even 2 1
120.2.m.b 16 12.b even 2 1
120.2.m.b 16 20.d odd 2 1
120.2.m.b 16 24.h odd 2 1
120.2.m.b 16 40.f even 2 1
120.2.m.b 16 60.h even 2 1
120.2.m.b 16 120.i odd 2 1
480.2.m.b 16 1.a even 1 1 trivial
480.2.m.b 16 3.b odd 2 1 inner
480.2.m.b 16 5.b even 2 1 inner
480.2.m.b 16 8.d odd 2 1 inner
480.2.m.b 16 15.d odd 2 1 inner
480.2.m.b 16 24.f even 2 1 inner
480.2.m.b 16 40.e odd 2 1 inner
480.2.m.b 16 120.m even 2 1 inner
600.2.b.i 16 20.e even 4 2
600.2.b.i 16 40.i odd 4 2
600.2.b.i 16 60.l odd 4 2
600.2.b.i 16 120.w even 4 2
2400.2.b.i 16 5.c odd 4 2
2400.2.b.i 16 15.e even 4 2
2400.2.b.i 16 40.k even 4 2
2400.2.b.i 16 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 16T_{7}^{2} + 56 \) acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 4 T^{6} + 14 T^{4} + 36 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{6} + 22 T^{4} - 100 T^{2} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 56)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{2} + 112)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 32 T^{2} + 224)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16 T^{2} + 32)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 4)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 40 T^{2} + 112)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{2} + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{2} + 224)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 80 T^{2} + 448)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 112 T^{2} + 2744)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{2} + 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 144 T^{2} + 2592)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 24 T^{2} + 112)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 72)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 16 T^{2} + 56)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 192 T^{2} + 7168)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 64 T^{2} + 896)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 272 T^{2} + 64)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 136 T^{2} + 4232)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 96 T^{2} + 1792)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 128 T^{2} + 896)^{4} \) Copy content Toggle raw display
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