Properties

Label 640.2.l.b
Level $640$
Weight $2$
Character orbit 640.l
Analytic conductor $5.110$
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] = [640,2,Mod(161,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.l (of order \(4\), 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: \(5.11042572936\)
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: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{5}) q^{3} - \beta_1 q^{5} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{4} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{5}) q^{3} - \beta_1 q^{5} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{4} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{4}) q^{9} + (\beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{11} + (\beta_{14} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - 1) q^{13} + (\beta_{14} - 1) q^{15} + (\beta_{14} - \beta_{13} + \beta_{12} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 1) q^{17} + ( - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{19} + (\beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} + \cdots - 1) q^{21}+ \cdots + ( - 2 \beta_{14} - 2 \beta_{11} + \beta_{8} - \beta_{7} - \beta_{6} - 4 \beta_{5} - \beta_{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{11} - 8 q^{15} + 8 q^{19} - 24 q^{27} + 16 q^{29} + 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} + 32 q^{51} - 16 q^{53} + 8 q^{59} - 16 q^{61} + 40 q^{63} - 40 q^{67} - 16 q^{69} - 16 q^{77} + 16 q^{79} - 16 q^{81} - 40 q^{83} + 16 q^{85} - 32 q^{91} + 48 q^{93} + 32 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} - 632 \nu^{9} + 3284 \nu^{8} + 6101 \nu^{7} - 1962 \nu^{6} - 10212 \nu^{5} - 4496 \nu^{4} + \cdots - 7296 ) / 2688 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9 \nu^{15} - 80 \nu^{14} + 152 \nu^{13} + 168 \nu^{12} + 65 \nu^{11} - 740 \nu^{10} - 1008 \nu^{9} + 1260 \nu^{8} + 2359 \nu^{7} + 1444 \nu^{6} - 5092 \nu^{5} - 4728 \nu^{4} + 2944 \nu^{3} + \cdots - 12288 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41 \nu^{15} + 80 \nu^{14} - 264 \nu^{13} - 312 \nu^{12} + 47 \nu^{11} + 1380 \nu^{10} + 1312 \nu^{9} - 2204 \nu^{8} - 4071 \nu^{7} - 1316 \nu^{6} + 8052 \nu^{5} + 6376 \nu^{4} - 4208 \nu^{3} + \cdots + 15488 ) / 384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 303 \nu^{15} - 1892 \nu^{14} + 1724 \nu^{13} + 1784 \nu^{12} + 3497 \nu^{11} - 6864 \nu^{10} - 19332 \nu^{9} + 11948 \nu^{8} + 27583 \nu^{7} + 39328 \nu^{6} - 63120 \nu^{5} + \cdots - 209536 ) / 2688 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 397 \nu^{15} - 502 \nu^{14} - 772 \nu^{13} - 664 \nu^{12} + 2523 \nu^{11} + 5226 \nu^{10} - 3292 \nu^{9} - 9932 \nu^{8} - 10139 \nu^{7} + 16854 \nu^{6} + 24480 \nu^{5} - 4720 \nu^{4} + \cdots + 8832 ) / 2688 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 655 \nu^{15} + 60 \nu^{14} + 2516 \nu^{13} + 2480 \nu^{12} - 3641 \nu^{11} - 14664 \nu^{10} - 3308 \nu^{9} + 27372 \nu^{8} + 36625 \nu^{7} - 15880 \nu^{6} - 84768 \nu^{5} + \cdots - 123008 ) / 2688 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 715 \nu^{15} + 2724 \nu^{14} - 1240 \nu^{13} - 1648 \nu^{12} - 7037 \nu^{11} + 2592 \nu^{10} + 27040 \nu^{9} - 2292 \nu^{8} - 22475 \nu^{7} - 67264 \nu^{6} + 45252 \nu^{5} + \cdots + 227968 ) / 2688 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} - 11700 \nu^{9} - 2228 \nu^{8} + 6320 \nu^{7} + 32015 \nu^{6} - 9900 \nu^{5} - 42864 \nu^{4} + \cdots - 83072 ) / 1344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 137 \nu^{15} + 466 \nu^{14} - 144 \nu^{13} - 264 \nu^{12} - 1303 \nu^{11} + 106 \nu^{10} + 4672 \nu^{9} + 492 \nu^{8} - 3289 \nu^{7} - 12506 \nu^{6} + 4972 \nu^{5} + 17552 \nu^{4} + \cdots + 32000 ) / 448 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 935 \nu^{15} + 3056 \nu^{14} - 676 \nu^{13} - 1552 \nu^{12} - 8961 \nu^{11} - 636 \nu^{10} + 30236 \nu^{9} + 5644 \nu^{8} - 17975 \nu^{7} - 81876 \nu^{6} + 25272 \nu^{5} + \cdots + 203136 ) / 2688 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1011 \nu^{15} + 2684 \nu^{14} - 164 \nu^{13} - 632 \nu^{12} - 8357 \nu^{11} - 3480 \nu^{10} + 25260 \nu^{9} + 9028 \nu^{8} - 7939 \nu^{7} - 73048 \nu^{6} + 8664 \nu^{5} + \cdots + 166912 ) / 2688 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + 31596 \nu^{9} + 17228 \nu^{8} - 5477 \nu^{7} - 95888 \nu^{6} - 4752 \nu^{5} + \cdots + 176384 ) / 2688 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1583 \nu^{15} + 4222 \nu^{14} - 120 \nu^{13} - 1032 \nu^{12} - 13337 \nu^{11} - 5850 \nu^{10} + 40424 \nu^{9} + 15908 \nu^{8} - 12087 \nu^{7} - 117622 \nu^{6} + 9804 \nu^{5} + \cdots + 252160 ) / 2688 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 239 \nu^{15} - 554 \nu^{14} - 56 \nu^{13} + 16 \nu^{12} + 1833 \nu^{11} + 1254 \nu^{10} - 5096 \nu^{9} - 2764 \nu^{8} + 311 \nu^{7} + 15690 \nu^{6} + 1044 \nu^{5} - 17384 \nu^{4} + \cdots - 29952 ) / 384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 356 \nu^{15} + 1000 \nu^{14} - 139 \nu^{13} - 312 \nu^{12} - 3008 \nu^{11} - 856 \nu^{10} + 9619 \nu^{9} + 2572 \nu^{8} - 4104 \nu^{7} - 27068 \nu^{6} + 5701 \nu^{5} + 34644 \nu^{4} + \cdots + 65248 ) / 224 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - \beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{14} - \beta_{13} + \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 2\beta_{13} + \beta_{12} - 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} + \beta_{14} - 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{2} - 3 \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} + \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 12 \beta _1 - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - 4 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \beta_{2} - 2 \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7 \beta_{15} + 4 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 19 \beta_{8} + 9 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 12 \beta _1 - 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 17 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 14 \beta_{6} - 11 \beta_{5} + 2 \beta_{4} + 11 \beta_{3} + 14 \beta_{2} + 11 \beta _1 - 32 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} - 16 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} - 2 \beta_{5} + 10 \beta_{4} - \beta_{3} - 3 \beta_{2} - 17 \beta _1 - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4 \beta_{15} - 13 \beta_{14} + 17 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 13 \beta_{10} + 3 \beta_{9} + 36 \beta_{8} - 10 \beta_{7} + 10 \beta_{6} + 7 \beta_{5} - 30 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 51 \beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 31 \beta_{15} + 4 \beta_{14} + 84 \beta_{13} - 43 \beta_{12} + 30 \beta_{11} - 56 \beta_{10} + 35 \beta_{9} - 41 \beta_{8} + 69 \beta_{7} - 26 \beta_{6} - \beta_{5} + 37 \beta_{4} + 9 \beta_{3} + 31 \beta_{2} - 20 \beta _1 + 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 46 \beta_{15} - 44 \beta_{14} + 49 \beta_{13} - 52 \beta_{12} + 26 \beta_{11} - 14 \beta_{10} + 24 \beta_{9} - 88 \beta_{8} + \beta_{7} + 33 \beta_{6} + 6 \beta_{5} + \beta_{4} - 7 \beta_{3} - 28 \beta_{2} - 47 \beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 5 \beta_{15} - 38 \beta_{14} + 76 \beta_{13} - 17 \beta_{12} - 84 \beta_{11} + 88 \beta_{10} - 3 \beta_{9} + 139 \beta_{8} - 75 \beta_{7} + 36 \beta_{6} + 55 \beta_{5} - 75 \beta_{4} - 37 \beta_{3} - 33 \beta_{2} - 6 \beta _1 + 139 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 32 \beta_{15} + 9 \beta_{14} - 23 \beta_{13} - 81 \beta_{12} + 27 \beta_{11} - 9 \beta_{10} + 95 \beta_{9} - 178 \beta_{8} + 50 \beta_{7} - 18 \beta_{6} - 61 \beta_{5} - 14 \beta_{4} - 33 \beta_{3} + 66 \beta_{2} - 131 \beta _1 + 68 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 45 \beta_{15} + 4 \beta_{14} + 82 \beta_{13} - 85 \beta_{12} - 58 \beta_{11} + 11 \beta_{10} + 77 \beta_{9} - 108 \beta_{8} + 125 \beta_{6} + 24 \beta_{5} - 4 \beta_{4} - 32 \beta_{3} - 89 \beta_{2} + 32 \beta _1 + 63 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.21331 + 0.726558i
−1.39563 0.228522i
−0.530822 1.31081i
−0.966675 + 1.03225i
−0.296075 + 1.38287i
1.26868 + 0.624862i
1.32070 0.505727i
1.38652 + 0.278517i
1.21331 0.726558i
−1.39563 + 0.228522i
−0.530822 + 1.31081i
−0.966675 1.03225i
−0.296075 1.38287i
1.26868 0.624862i
1.32070 + 0.505727i
1.38652 0.278517i
0 −1.82762 1.82762i 0 0.707107 0.707107i 0 4.50961i 0 3.68037i 0
161.2 0 −1.42313 1.42313i 0 −0.707107 + 0.707107i 0 0.690576i 0 1.05061i 0
161.3 0 −1.37027 1.37027i 0 0.707107 0.707107i 0 2.73482i 0 0.755274i 0
161.4 0 −0.209571 0.209571i 0 −0.707107 + 0.707107i 0 1.73696i 0 2.91216i 0
161.5 0 0.120009 + 0.120009i 0 0.707107 0.707107i 0 2.66881i 0 2.97120i 0
161.6 0 0.720673 + 0.720673i 0 −0.707107 + 0.707107i 0 4.02840i 0 1.96126i 0
161.7 0 1.66366 + 1.66366i 0 0.707107 0.707107i 0 2.89402i 0 2.53555i 0
161.8 0 2.32624 + 2.32624i 0 −0.707107 + 0.707107i 0 0.982011i 0 7.82281i 0
481.1 0 −1.82762 + 1.82762i 0 0.707107 + 0.707107i 0 4.50961i 0 3.68037i 0
481.2 0 −1.42313 + 1.42313i 0 −0.707107 0.707107i 0 0.690576i 0 1.05061i 0
481.3 0 −1.37027 + 1.37027i 0 0.707107 + 0.707107i 0 2.73482i 0 0.755274i 0
481.4 0 −0.209571 + 0.209571i 0 −0.707107 0.707107i 0 1.73696i 0 2.91216i 0
481.5 0 0.120009 0.120009i 0 0.707107 + 0.707107i 0 2.66881i 0 2.97120i 0
481.6 0 0.720673 0.720673i 0 −0.707107 0.707107i 0 4.02840i 0 1.96126i 0
481.7 0 1.66366 1.66366i 0 0.707107 + 0.707107i 0 2.89402i 0 2.53555i 0
481.8 0 2.32624 2.32624i 0 −0.707107 0.707107i 0 0.982011i 0 7.82281i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.l.b 16
4.b odd 2 1 640.2.l.a 16
8.b even 2 1 80.2.l.a 16
8.d odd 2 1 320.2.l.a 16
16.e even 4 1 80.2.l.a 16
16.e even 4 1 inner 640.2.l.b 16
16.f odd 4 1 320.2.l.a 16
16.f odd 4 1 640.2.l.a 16
24.f even 2 1 2880.2.t.c 16
24.h odd 2 1 720.2.t.c 16
32.g even 8 1 5120.2.a.s 8
32.g even 8 1 5120.2.a.v 8
32.h odd 8 1 5120.2.a.t 8
32.h odd 8 1 5120.2.a.u 8
40.e odd 2 1 1600.2.l.i 16
40.f even 2 1 400.2.l.h 16
40.i odd 4 1 400.2.q.g 16
40.i odd 4 1 400.2.q.h 16
40.k even 4 1 1600.2.q.g 16
40.k even 4 1 1600.2.q.h 16
48.i odd 4 1 720.2.t.c 16
48.k even 4 1 2880.2.t.c 16
80.i odd 4 1 400.2.q.h 16
80.j even 4 1 1600.2.q.h 16
80.k odd 4 1 1600.2.l.i 16
80.q even 4 1 400.2.l.h 16
80.s even 4 1 1600.2.q.g 16
80.t odd 4 1 400.2.q.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.l.a 16 8.b even 2 1
80.2.l.a 16 16.e even 4 1
320.2.l.a 16 8.d odd 2 1
320.2.l.a 16 16.f odd 4 1
400.2.l.h 16 40.f even 2 1
400.2.l.h 16 80.q even 4 1
400.2.q.g 16 40.i odd 4 1
400.2.q.g 16 80.t odd 4 1
400.2.q.h 16 40.i odd 4 1
400.2.q.h 16 80.i odd 4 1
640.2.l.a 16 4.b odd 2 1
640.2.l.a 16 16.f odd 4 1
640.2.l.b 16 1.a even 1 1 trivial
640.2.l.b 16 16.e even 4 1 inner
720.2.t.c 16 24.h odd 2 1
720.2.t.c 16 48.i odd 4 1
1600.2.l.i 16 40.e odd 2 1
1600.2.l.i 16 80.k odd 4 1
1600.2.q.g 16 40.k even 4 1
1600.2.q.g 16 80.s even 4 1
1600.2.q.h 16 40.k even 4 1
1600.2.q.h 16 80.j even 4 1
2880.2.t.c 16 24.f even 2 1
2880.2.t.c 16 48.k even 4 1
5120.2.a.s 8 32.g even 8 1
5120.2.a.t 8 32.h odd 8 1
5120.2.a.u 8 32.h odd 8 1
5120.2.a.v 8 32.g even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8 T_{3}^{13} + 112 T_{3}^{12} + 80 T_{3}^{11} + 32 T_{3}^{10} + 176 T_{3}^{9} + 2632 T_{3}^{8} + 2560 T_{3}^{7} + 1024 T_{3}^{6} - 2656 T_{3}^{5} + 5824 T_{3}^{4} + 1088 T_{3}^{3} + 128 T_{3}^{2} - 64 T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(640, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{13} + 112 T^{12} + 80 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 64 T^{14} + 1616 T^{12} + \cdots + 204304 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + 32 T^{14} + \cdots + 1290496 \) Copy content Toggle raw display
$13$ \( T^{16} - 128 T^{13} + \cdots + 20647936 \) Copy content Toggle raw display
$17$ \( (T^{8} - 72 T^{6} - 64 T^{5} + 1408 T^{4} + \cdots + 13888)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} - 8 T^{15} + 32 T^{14} + \cdots + 614656 \) Copy content Toggle raw display
$23$ \( T^{16} + 128 T^{14} + 4784 T^{12} + \cdots + 1731856 \) Copy content Toggle raw display
$29$ \( T^{16} - 16 T^{15} + \cdots + 3017085184 \) Copy content Toggle raw display
$31$ \( (T^{8} - 96 T^{6} + 64 T^{5} + 2848 T^{4} + \cdots - 20224)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 16 T^{15} + 128 T^{14} + \cdots + 18939904 \) Copy content Toggle raw display
$41$ \( T^{16} + 384 T^{14} + \cdots + 110660014336 \) Copy content Toggle raw display
$43$ \( T^{16} + 8 T^{15} + 32 T^{14} + \cdots + 53640976 \) Copy content Toggle raw display
$47$ \( (T^{8} + 20 T^{7} - 8 T^{6} - 2392 T^{5} + \cdots + 575044)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 16 T^{15} + \cdots + 383725735936 \) Copy content Toggle raw display
$59$ \( T^{16} - 8 T^{15} + \cdots + 12227051776 \) Copy content Toggle raw display
$61$ \( T^{16} + 16 T^{15} + \cdots + 1393986371584 \) Copy content Toggle raw display
$67$ \( T^{16} + 40 T^{15} + \cdots + 46120451769616 \) Copy content Toggle raw display
$71$ \( T^{16} + 640 T^{14} + \cdots + 3333516427264 \) Copy content Toggle raw display
$73$ \( T^{16} + 560 T^{14} + \cdots + 15847788544 \) Copy content Toggle raw display
$79$ \( (T^{8} - 8 T^{7} - 160 T^{6} + 352 T^{5} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 40 T^{15} + \cdots + 2050640656 \) Copy content Toggle raw display
$89$ \( T^{16} + 416 T^{14} + \cdots + 684153962496 \) Copy content Toggle raw display
$97$ \( (T^{8} - 440 T^{6} - 416 T^{5} + \cdots - 8549312)^{2} \) Copy content Toggle raw display
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