Properties

Label 980.2.m.a
Level $980$
Weight $2$
Character orbit 980.m
Analytic conductor $7.825$
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] = [980,2,Mod(97,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.m (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: \(7.82533939809\)
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: \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + 17766 x^{8} - 24680 x^{7} + 28682 x^{6} - 27248 x^{5} + 20861 x^{4} + \cdots + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 140)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{10} q^{5} + ( - 2 \beta_{7} - \beta_{4} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{10} q^{5} + ( - 2 \beta_{7} - \beta_{4} - \beta_{3}) q^{9} + \beta_{2} q^{11} + (\beta_{12} + \beta_{11} + \beta_{10} + \beta_{9}) q^{13} + (\beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{15} + ( - \beta_{15} + \beta_{12} - \beta_{10} + \beta_1) q^{17} + (\beta_{12} - \beta_{11} + \beta_{6} - \beta_{5} + 2 \beta_1) q^{19} + (2 \beta_{7} - \beta_{4} + 2) q^{23} + ( - 2 \beta_{14} - \beta_{8} - \beta_{4} - 1) q^{25} + (\beta_{15} - 2 \beta_{11} + 2 \beta_{9} - \beta_{6} - \beta_1) q^{27} + (2 \beta_{13} - \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2}) q^{29} + ( - \beta_{10} + \beta_{9} + 2 \beta_{6} + 2 \beta_{5}) q^{31} + ( - 2 \beta_{15} + \beta_{11} + \beta_{9} - 2 \beta_1) q^{33} + (\beta_{13} + \beta_{8} - 2 \beta_{7} + 2) q^{37} + (3 \beta_{14} - 2 \beta_{13} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{39} + ( - \beta_{15} + \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{5}) q^{41} + (3 \beta_{14} - \beta_{13} + \beta_{8} + 2 \beta_{7} + \beta_{4} - 3 \beta_{2} + 2) q^{43} + (3 \beta_{15} - \beta_{11} + \beta_{10} - 3 \beta_{6} + \beta_{5} - \beta_1) q^{45} + ( - \beta_{12} + \beta_{10} - 2 \beta_{6}) q^{47} + (\beta_{14} + 2 \beta_{8} + \beta_{4} - \beta_{3} - 4 \beta_{2} - 2) q^{51} + (\beta_{13} - \beta_{8} + \beta_{7} + 2 \beta_{4} + 1) q^{53} + ( - \beta_{15} + 2 \beta_{12} + \beta_{9} + 2 \beta_1) q^{55} + ( - 2 \beta_{14} + \beta_{13} + \beta_{8} - 3 \beta_{7} - 2 \beta_{3} - 2 \beta_{2} + 3) q^{57} + ( - \beta_{12} + \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{6} - \beta_{5}) q^{59} + (3 \beta_{15} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5}) q^{61} + (2 \beta_{14} + \beta_{13} + \beta_{8} + 4 \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{65} + ( - \beta_{14} + \beta_{7} - \beta_{3} - \beta_{2} - 1) q^{67} + (\beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{6} - \beta_{5} - \beta_1) q^{69} + (\beta_{14} + 2 \beta_{8} - \beta_{4} + \beta_{3} + \beta_{2}) q^{71} + ( - 3 \beta_{15} + \beta_{12} + \beta_{10} - 2 \beta_{5} - 3 \beta_1) q^{73} + (4 \beta_{15} - \beta_{12} + 2 \beta_{10} + 2 \beta_{9} - \beta_{6} + 2 \beta_{5} - 3 \beta_1) q^{75} + ( - 2 \beta_{14} + 2 \beta_{13} - 6 \beta_{7} + \beta_{2}) q^{79} + ( - 3 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 3) q^{81} + (2 \beta_{15} - \beta_{12} - \beta_{10} - \beta_{5} + 2 \beta_1) q^{83} + ( - 3 \beta_{14} + \beta_{13} + 4 \beta_{7} - \beta_{3}) q^{85} + ( - \beta_{15} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} - 3 \beta_{6} + \beta_1) q^{87} + ( - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{6} + 2 \beta_{5} + \beta_1) q^{89} + (\beta_{14} - \beta_{13} + \beta_{8} + 8 \beta_{7} + 2 \beta_{4} - \beta_{2} + 8) q^{93} + ( - \beta_{14} + \beta_{8} + 2 \beta_{7} - 3 \beta_{3} + \beta_{2}) q^{95} + ( - \beta_{15} + 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{6} + \cdots + \beta_1) q^{97}+ \cdots + (3 \beta_{14} - 2 \beta_{13} - 2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} - \beta_{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 \( 16 q - 20 q^{15} + 32 q^{23} - 12 q^{25} + 28 q^{37} + 28 q^{43} - 40 q^{51} + 20 q^{53} + 44 q^{57} - 68 q^{65} - 16 q^{67} - 8 q^{71} - 48 q^{81} + 124 q^{93} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + 17766 x^{8} - 24680 x^{7} + 28682 x^{6} - 27248 x^{5} + 20861 x^{4} + \cdots + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 103 \nu^{14} + 721 \nu^{13} - 4446 \nu^{12} + 17303 \nu^{11} - 57239 \nu^{10} + 144768 \nu^{9} - 310061 \nu^{8} + 532511 \nu^{7} - 766976 \nu^{6} + 885763 \nu^{5} + \cdots - 24976 ) / 2156 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24 \nu^{14} + 168 \nu^{13} - 1045 \nu^{12} + 4086 \nu^{11} - 13636 \nu^{10} + 34729 \nu^{9} - 74976 \nu^{8} + 129612 \nu^{7} - 187089 \nu^{6} + 215748 \nu^{5} - 199886 \nu^{4} + \cdots - 3892 ) / 98 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 921 \nu^{15} + 4134 \nu^{14} - 27981 \nu^{13} + 258387 \nu^{12} - 993848 \nu^{11} + 3534057 \nu^{10} - 8600435 \nu^{9} + 18275426 \nu^{8} - 28971201 \nu^{7} + \cdots - 627760 ) / 84868 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 921 \nu^{15} - 17949 \nu^{14} + 126600 \nu^{13} - 689883 \nu^{12} + 2686219 \nu^{11} - 8556602 \nu^{10} + 21803093 \nu^{9} - 46150211 \nu^{8} + 80475178 \nu^{7} + \cdots - 3476900 ) / 84868 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2249 \nu^{15} + 5826 \nu^{14} - 25703 \nu^{13} - 60501 \nu^{12} + 480220 \nu^{11} - 2553517 \nu^{10} + 7821237 \nu^{9} - 19570106 \nu^{8} + 37195221 \nu^{7} + \cdots - 1543192 ) / 84868 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2249 \nu^{15} + 27909 \nu^{14} - 180284 \nu^{13} + 887769 \nu^{12} - 3199847 \nu^{11} + 9537142 \nu^{10} - 22582291 \nu^{9} + 44855531 \nu^{8} - 72251158 \nu^{7} + \cdots + 796740 ) / 84868 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4498 \nu^{15} - 33735 \nu^{14} + 205987 \nu^{13} - 827268 \nu^{12} + 2719627 \nu^{11} - 6983625 \nu^{10} + 14761054 \nu^{9} - 25285425 \nu^{8} + 35055937 \nu^{7} + \cdots + 661584 ) / 84868 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6283 \nu^{15} - 51669 \nu^{14} + 332274 \nu^{13} - 1453425 \nu^{12} + 5182367 \nu^{11} - 14821924 \nu^{10} + 35371231 \nu^{9} - 70552899 \nu^{8} + 117927236 \nu^{7} + \cdots - 4705876 ) / 84868 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37258 \nu^{15} - 235269 \nu^{14} + 1410813 \nu^{13} - 5023983 \nu^{12} + 15617521 \nu^{11} - 35160459 \nu^{10} + 66702361 \nu^{9} - 93259339 \nu^{8} + \cdots - 838334 ) / 466774 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37258 \nu^{15} + 323601 \nu^{14} - 2029137 \nu^{13} + 8859497 \nu^{12} - 30592393 \nu^{11} + 84923417 \nu^{10} - 192984213 \nu^{9} + 363819389 \nu^{8} + \cdots + 6124398 ) / 466774 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 110763 \nu^{15} + 827042 \nu^{14} - 5299391 \nu^{13} + 21834951 \nu^{12} - 76233328 \nu^{11} + 206125891 \nu^{10} - 474572071 \nu^{9} + 888153026 \nu^{8} + \cdots + 29355256 ) / 933548 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 110763 \nu^{15} + 834403 \nu^{14} - 5350918 \nu^{13} + 22193475 \nu^{12} - 77714621 \nu^{11} + 211712024 \nu^{10} - 490152277 \nu^{9} + 926997889 \nu^{8} + \cdots + 36641780 ) / 933548 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5508 \nu^{15} + 46506 \nu^{14} - 302999 \nu^{13} + 1332783 \nu^{12} - 4790008 \nu^{11} + 13638265 \nu^{10} - 32542271 \nu^{9} + 63996516 \nu^{8} + \cdots + 3098746 ) / 42434 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12566 \nu^{15} + 94245 \nu^{14} - 600897 \nu^{13} + 2476448 \nu^{12} - 8609785 \nu^{11} + 23257531 \nu^{10} - 53380894 \nu^{9} + 99905415 \nu^{8} + \cdots + 4065068 ) / 84868 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 852 \nu^{15} + 6390 \nu^{14} - 40676 \nu^{13} + 167479 \nu^{12} - 580296 \nu^{11} + 1562572 \nu^{10} - 3560985 \nu^{9} + 6611661 \nu^{8} - 10337168 \nu^{7} + \cdots + 170743 ) / 4763 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{4} + \beta_{3} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 7\beta_{7} - 5\beta_{6} - 2\beta_{5} - 3\beta_{4} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} - 4 \beta_{12} - 4 \beta_{10} - 15 \beta_{7} - 2 \beta_{6} - 14 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + 2 \beta _1 + 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 10 \beta_{15} + 5 \beta_{14} + 2 \beta_{12} + 12 \beta_{11} - 10 \beta_{9} + 36 \beta_{7} + 32 \beta_{6} - 3 \beta_{5} + 30 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 5 \beta _1 + 61 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 35 \beta_{15} + 14 \beta_{14} + 36 \beta_{12} + 16 \beta_{11} + 36 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + 146 \beta_{7} + 35 \beta_{6} + 93 \beta_{5} + 16 \beta_{4} + 74 \beta_{3} + 18 \beta_{2} - 25 \beta _1 - 70 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 58 \beta_{15} - 49 \beta_{14} + 7 \beta_{13} + 19 \beta_{12} - 86 \beta_{11} + 60 \beta_{10} + 87 \beta_{9} - 7 \beta_{8} - 127 \beta_{7} - 211 \beta_{6} + 118 \beta_{5} - 224 \beta_{4} + 105 \beta_{3} + 84 \beta_{2} - 105 \beta _1 - 468 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 400 \beta_{15} - 242 \beta_{14} + 28 \beta_{13} - 272 \beta_{12} - 248 \beta_{11} - 216 \beta_{10} + 88 \beta_{9} + 20 \beta_{8} - 1225 \beta_{7} - 440 \beta_{6} - 568 \beta_{5} - 385 \beta_{4} - 525 \beta_{3} - 70 \beta_{2} + 120 \beta _1 + 89 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 78 \beta_{15} + 192 \beta_{14} - 48 \beta_{13} - 371 \beta_{12} + 409 \beta_{11} - 807 \beta_{10} - 693 \beta_{9} + 132 \beta_{8} - 224 \beta_{7} + 1246 \beta_{6} - 1457 \beta_{5} + 1386 \beta_{4} - 1371 \beta_{3} + \cdots + 3481 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3645 \beta_{15} + 2608 \beta_{14} - 450 \beta_{13} + 1922 \beta_{12} + 2552 \beta_{11} + 722 \beta_{10} - 1288 \beta_{9} - 34 \beta_{8} + 9166 \beta_{7} + 4575 \beta_{6} + 2909 \beta_{5} + 4570 \beta_{4} + \cdots + 2494 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3305 \beta_{15} + 1309 \beta_{14} - 176 \beta_{13} + 4638 \beta_{12} - 312 \beta_{11} + 7536 \beta_{10} + 4828 \beta_{9} - 1474 \beta_{8} + 10625 \beta_{7} - 5493 \beta_{6} + 14142 \beta_{5} - 6413 \beta_{4} + \cdots - 24056 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 27668 \beta_{15} - 21904 \beta_{14} + 4356 \beta_{13} - 11876 \beta_{12} - 21172 \beta_{11} + 3216 \beta_{10} + 15040 \beta_{9} - 1634 \beta_{8} - 60753 \beta_{7} - 41348 \beta_{6} - 8804 \beta_{5} + \cdots - 41671 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 57555 \beta_{15} - 35867 \beta_{14} + 7475 \beta_{13} - 48811 \beta_{12} - 21576 \beta_{11} - 57400 \beta_{10} - 26853 \beta_{9} + 11973 \beta_{8} - 141868 \beta_{7} + 3476 \beta_{6} - 119673 \beta_{5} + \cdots + 146745 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 170623 \beta_{15} + 148774 \beta_{14} - 30758 \beta_{13} + 53306 \beta_{12} + 147184 \beta_{11} - 91176 \beta_{10} - 149788 \beta_{9} + 29644 \beta_{8} + 337546 \beta_{7} + 331357 \beta_{6} + \cdots + 465346 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 657499 \beta_{15} + 463584 \beta_{14} - 103744 \beta_{13} + 451091 \beta_{12} + 335591 \beta_{11} + 361643 \beta_{10} + 80673 \beta_{9} - 71496 \beta_{8} + 1444309 \beta_{7} + \cdots - 702512 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.500000 + 1.78727i
0.500000 + 1.27536i
0.500000 + 0.617773i
0.500000 + 0.105864i
0.500000 1.10586i
0.500000 1.61777i
0.500000 2.27536i
0.500000 2.78727i
0.500000 1.78727i
0.500000 1.27536i
0.500000 0.617773i
0.500000 0.105864i
0.500000 + 1.10586i
0.500000 + 1.61777i
0.500000 + 2.27536i
0.500000 + 2.78727i
0 −2.28727 2.28727i 0 −1.22200 1.87262i 0 0 0 7.46321i 0
97.2 0 −1.77536 1.77536i 0 1.45225 1.70029i 0 0 0 3.30382i 0
97.3 0 −1.11777 1.11777i 0 −0.524151 + 2.17377i 0 0 0 0.501168i 0
97.4 0 −0.605864 0.605864i 0 2.15010 + 0.614051i 0 0 0 2.26586i 0
97.5 0 0.605864 + 0.605864i 0 −2.15010 0.614051i 0 0 0 2.26586i 0
97.6 0 1.11777 + 1.11777i 0 0.524151 2.17377i 0 0 0 0.501168i 0
97.7 0 1.77536 + 1.77536i 0 −1.45225 + 1.70029i 0 0 0 3.30382i 0
97.8 0 2.28727 + 2.28727i 0 1.22200 + 1.87262i 0 0 0 7.46321i 0
293.1 0 −2.28727 + 2.28727i 0 −1.22200 + 1.87262i 0 0 0 7.46321i 0
293.2 0 −1.77536 + 1.77536i 0 1.45225 + 1.70029i 0 0 0 3.30382i 0
293.3 0 −1.11777 + 1.11777i 0 −0.524151 2.17377i 0 0 0 0.501168i 0
293.4 0 −0.605864 + 0.605864i 0 2.15010 0.614051i 0 0 0 2.26586i 0
293.5 0 0.605864 0.605864i 0 −2.15010 + 0.614051i 0 0 0 2.26586i 0
293.6 0 1.11777 1.11777i 0 0.524151 + 2.17377i 0 0 0 0.501168i 0
293.7 0 1.77536 1.77536i 0 −1.45225 1.70029i 0 0 0 3.30382i 0
293.8 0 2.28727 2.28727i 0 1.22200 1.87262i 0 0 0 7.46321i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.m.a 16
5.c odd 4 1 inner 980.2.m.a 16
7.b odd 2 1 inner 980.2.m.a 16
7.c even 3 1 140.2.u.a 16
7.c even 3 1 980.2.v.a 16
7.d odd 6 1 140.2.u.a 16
7.d odd 6 1 980.2.v.a 16
21.g even 6 1 1260.2.dq.a 16
21.h odd 6 1 1260.2.dq.a 16
28.f even 6 1 560.2.ci.d 16
28.g odd 6 1 560.2.ci.d 16
35.f even 4 1 inner 980.2.m.a 16
35.i odd 6 1 700.2.bc.b 16
35.j even 6 1 700.2.bc.b 16
35.k even 12 1 140.2.u.a 16
35.k even 12 1 700.2.bc.b 16
35.k even 12 1 980.2.v.a 16
35.l odd 12 1 140.2.u.a 16
35.l odd 12 1 700.2.bc.b 16
35.l odd 12 1 980.2.v.a 16
105.w odd 12 1 1260.2.dq.a 16
105.x even 12 1 1260.2.dq.a 16
140.w even 12 1 560.2.ci.d 16
140.x odd 12 1 560.2.ci.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.u.a 16 7.c even 3 1
140.2.u.a 16 7.d odd 6 1
140.2.u.a 16 35.k even 12 1
140.2.u.a 16 35.l odd 12 1
560.2.ci.d 16 28.f even 6 1
560.2.ci.d 16 28.g odd 6 1
560.2.ci.d 16 140.w even 12 1
560.2.ci.d 16 140.x odd 12 1
700.2.bc.b 16 35.i odd 6 1
700.2.bc.b 16 35.j even 6 1
700.2.bc.b 16 35.k even 12 1
700.2.bc.b 16 35.l odd 12 1
980.2.m.a 16 1.a even 1 1 trivial
980.2.m.a 16 5.c odd 4 1 inner
980.2.m.a 16 7.b odd 2 1 inner
980.2.m.a 16 35.f even 4 1 inner
980.2.v.a 16 7.c even 3 1
980.2.v.a 16 7.d odd 6 1
980.2.v.a 16 35.k even 12 1
980.2.v.a 16 35.l odd 12 1
1260.2.dq.a 16 21.g even 6 1
1260.2.dq.a 16 21.h odd 6 1
1260.2.dq.a 16 105.w odd 12 1
1260.2.dq.a 16 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 156T_{3}^{12} + 5366T_{3}^{8} + 30012T_{3}^{4} + 14641 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 156 T^{12} + 5366 T^{8} + \cdots + 14641 \) Copy content Toggle raw display
$5$ \( T^{16} + 6 T^{14} + 33 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} - 13 T^{2} - 6 T + 10)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 1800 T^{12} + \cdots + 268435456 \) Copy content Toggle raw display
$17$ \( T^{16} + 2286 T^{12} + 20417 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{8} - 74 T^{6} + 1853 T^{4} + \cdots + 68644)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 16 T^{7} + 128 T^{6} - 540 T^{5} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 162 T^{6} + 7073 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 134 T^{6} + 5345 T^{4} + \cdots + 414736)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 14 T^{7} + 98 T^{6} - 114 T^{5} + \cdots + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 170 T^{6} + 7001 T^{4} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 14 T^{7} + 98 T^{6} + \cdots + 10432900)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 3318 T^{12} + 2247521 T^{8} + \cdots + 10000 \) Copy content Toggle raw display
$53$ \( (T^{8} - 10 T^{7} + 50 T^{6} + \cdots + 188356)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 266 T^{6} + 19181 T^{4} + \cdots + 2502724)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 224 T^{6} + 16718 T^{4} + \cdots + 4092529)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 8 T^{7} + 32 T^{6} - 4 T^{5} + \cdots + 49)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} - 142 T^{2} - 560 T + 448)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 48846 T^{12} + \cdots + 3841600000000 \) Copy content Toggle raw display
$79$ \( (T^{8} + 270 T^{6} + 15989 T^{4} + \cdots + 1909924)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 25542 T^{12} + \cdots + 5006411536 \) Copy content Toggle raw display
$89$ \( (T^{8} - 184 T^{6} + 7142 T^{4} + \cdots + 182329)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 68112 T^{12} + \cdots + 47698139955456 \) Copy content Toggle raw display
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