Properties

Label 1300.2.bs.d
Level $1300$
Weight $2$
Character orbit 1300.bs
Analytic conductor $10.381$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \( x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9 \) 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 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{3} + 1) q^{3} + (\beta_{8} + \beta_{7} + \beta_{5} - \beta_{2}) q^{7} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{3} + 1) q^{3} + (\beta_{8} + \beta_{7} + \beta_{5} - \beta_{2}) q^{7} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3}) q^{9} + (\beta_{19} + \beta_{18} + 2 \beta_{14} + \beta_{12} + \beta_{9} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 1) q^{11}+ \cdots + ( - 2 \beta_{19} - 3 \beta_{18} - \beta_{17} + 4 \beta_{16} + 2 \beta_{15} + \beta_{14} + 2 \beta_{13} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} - 6 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} - 6 q^{7} - 12 q^{9} - 8 q^{13} + 20 q^{19} - 12 q^{21} - 6 q^{23} + 20 q^{27} + 24 q^{29} + 8 q^{31} + 10 q^{33} + 4 q^{39} + 6 q^{41} - 38 q^{43} + 14 q^{49} - 30 q^{53} + 76 q^{57} - 24 q^{59} - 32 q^{61} + 24 q^{63} - 22 q^{67} - 16 q^{69} + 44 q^{73} + 12 q^{77} + 2 q^{81} - 38 q^{87} - 30 q^{89} - 72 q^{91} + 48 q^{93} - 46 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3047 \nu^{18} - 80534 \nu^{16} - 834504 \nu^{14} - 4296019 \nu^{12} - 11423658 \nu^{10} - 14767004 \nu^{8} - 8552045 \nu^{6} - 4645944 \nu^{4} + \cdots + 99723 ) / 1134948 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19367 \nu^{19} + 69462 \nu^{18} + 805551 \nu^{17} + 1878198 \nu^{16} + 13356712 \nu^{15} + 19986924 \nu^{14} + 115467750 \nu^{13} + 105446208 \nu^{12} + \cdots - 217278 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15705 \nu^{19} + 7449 \nu^{18} + 402397 \nu^{17} + 223669 \nu^{16} + 3930438 \nu^{15} + 2731870 \nu^{14} + 17694998 \nu^{13} + 17453738 \nu^{12} + 30393363 \nu^{11} + \cdots + 661521 ) / 4539792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 66482 \nu^{19} - 47115 \nu^{18} - 2012742 \nu^{17} - 1207191 \nu^{16} - 25148026 \nu^{15} - 11791314 \nu^{14} - 168552744 \nu^{13} - 53084994 \nu^{12} + \cdots + 15821217 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 66482 \nu^{19} + 47115 \nu^{18} - 2012742 \nu^{17} + 1207191 \nu^{16} - 25148026 \nu^{15} + 11791314 \nu^{14} - 168552744 \nu^{13} + 53084994 \nu^{12} + \cdots - 15821217 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 66482 \nu^{19} - 83679 \nu^{18} + 2012742 \nu^{17} - 2173599 \nu^{16} + 25148026 \nu^{15} - 21805362 \nu^{14} + 168552744 \nu^{13} - 104637222 \nu^{12} + \cdots + 17017893 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 165961 \nu^{19} - 181245 \nu^{18} - 5106855 \nu^{17} - 5590593 \nu^{16} - 65333654 \nu^{15} - 71485608 \nu^{14} - 453746592 \nu^{13} - 492632442 \nu^{12} + \cdots - 45388899 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27888 \nu^{19} + 17400 \nu^{18} + 876466 \nu^{17} + 537244 \nu^{16} + 11480801 \nu^{15} + 6830496 \nu^{14} + 81710792 \nu^{13} + 46336394 \nu^{12} + \cdots + 4472424 ) / 2269896 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 84623 \nu^{19} - 64035 \nu^{18} - 2581149 \nu^{17} - 1764837 \nu^{16} - 32554318 \nu^{15} - 19533108 \nu^{14} - 220915596 \nu^{13} - 112032558 \nu^{12} + \cdots + 12376215 ) / 6809688 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27888 \nu^{19} - 17400 \nu^{18} + 876466 \nu^{17} - 537244 \nu^{16} + 11480801 \nu^{15} - 6830496 \nu^{14} + 81710792 \nu^{13} - 46336394 \nu^{12} + \cdots - 4472424 ) / 2269896 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 84623 \nu^{19} + 64035 \nu^{18} - 2581149 \nu^{17} + 1764837 \nu^{16} - 32554318 \nu^{15} + 19533108 \nu^{14} - 220915596 \nu^{13} + 112032558 \nu^{12} + \cdots - 12376215 ) / 6809688 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 165961 \nu^{19} - 320169 \nu^{18} + 5106855 \nu^{17} - 9346989 \nu^{16} + 65333654 \nu^{15} - 111459456 \nu^{14} + 453746592 \nu^{13} - 703524858 \nu^{12} + \cdots - 31334967 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 282319 \nu^{19} + 174918 \nu^{18} + 8264211 \nu^{17} + 5018946 \nu^{16} + 98942810 \nu^{15} + 58624560 \nu^{14} + 628204794 \nu^{13} + 361205580 \nu^{12} + \cdots + 42156486 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 353 \nu^{19} - 10585 \nu^{17} - 130732 \nu^{15} - 864458 \nu^{13} - 3326801 \nu^{11} - 7574149 \nu^{9} - 9886332 \nu^{7} - 6763380 \nu^{5} - 1984833 \nu^{3} + \cdots - 8376 ) / 16752 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 397382 \nu^{19} + 50979 \nu^{18} + 11805666 \nu^{17} + 1600971 \nu^{16} + 144498028 \nu^{15} + 20826990 \nu^{14} + 949107816 \nu^{13} + 145593894 \nu^{12} + \cdots + 24710427 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 210017 \nu^{19} + 54111 \nu^{18} + 6333330 \nu^{17} + 1652277 \nu^{16} + 78814861 \nu^{15} + 20796687 \nu^{14} + 526330083 \nu^{13} + 139985934 \nu^{12} + \cdots + 22008474 ) / 6809688 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 67147 \nu^{19} - 27215 \nu^{18} + 2045559 \nu^{17} - 784089 \nu^{16} + 25759345 \nu^{15} - 9210622 \nu^{14} + 174392772 \nu^{13} - 57243966 \nu^{12} + \cdots + 7414695 ) / 2269896 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 938081 \nu^{19} + 181527 \nu^{18} + 28105911 \nu^{17} + 5420187 \nu^{16} + 346829194 \nu^{15} + 66687072 \nu^{14} + 2291248896 \nu^{13} + 439458966 \nu^{12} + \cdots - 35447103 ) / 13619376 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 1080005 \nu^{19} - 98094 \nu^{18} - 32065821 \nu^{17} - 2808162 \nu^{16} - 391463830 \nu^{15} - 32618304 \nu^{14} - 2554928178 \nu^{13} + \cdots - 15698898 ) / 13619376 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{19} + \beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 3 \beta_{12} + 3 \beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 5 \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9 \beta_{19} + 7 \beta_{18} - 13 \beta_{16} + 15 \beta_{15} + 9 \beta_{14} + 11 \beta_{13} + 5 \beta_{12} - 6 \beta_{11} - 9 \beta_{10} - 11 \beta_{9} + 9 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 5 \beta_{5} - 4 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - 6 \beta _1 + 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13 \beta_{19} - 11 \beta_{18} - 22 \beta_{17} + 11 \beta_{16} + 9 \beta_{15} + 17 \beta_{14} + 11 \beta_{13} + 29 \beta_{12} - 4 \beta_{11} - 29 \beta_{10} - 15 \beta_{9} - 29 \beta_{8} - 7 \beta_{7} + 3 \beta_{6} - 15 \beta_{5} + 30 \beta_{4} - 24 \beta_{3} + \cdots - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 38 \beta_{19} - 29 \beta_{18} + 58 \beta_{16} - 67 \beta_{15} - 38 \beta_{14} - 47 \beta_{13} - 16 \beta_{12} + 34 \beta_{11} + 35 \beta_{10} + 42 \beta_{9} - 35 \beta_{8} - 45 \beta_{7} - 40 \beta_{6} - 13 \beta_{5} + 20 \beta_{4} + 38 \beta_{3} + \cdots - 119 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 118 \beta_{19} + 106 \beta_{18} + 212 \beta_{17} - 106 \beta_{16} - 94 \beta_{15} - 166 \beta_{14} - 106 \beta_{13} - 242 \beta_{12} + 74 \beta_{11} + 246 \beta_{10} + 180 \beta_{9} + 246 \beta_{8} + 30 \beta_{7} - 119 \beta_{6} + \cdots + 352 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 637 \beta_{19} + 499 \beta_{18} - 951 \beta_{16} + 1089 \beta_{15} + 637 \beta_{14} + 775 \beta_{13} + 247 \beta_{12} - 624 \beta_{11} - 537 \beta_{10} - 603 \beta_{9} + 537 \beta_{8} + 699 \beta_{7} + 541 \beta_{6} + 151 \beta_{5} + \cdots + 1830 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 975 \beta_{19} - 971 \beta_{18} - 1942 \beta_{17} + 971 \beta_{16} + 967 \beta_{15} + 1443 \beta_{14} + 971 \beta_{13} + 1969 \beta_{12} - 912 \beta_{11} - 2041 \beta_{10} - 1883 \beta_{9} - 2041 \beta_{8} - 27 \beta_{7} + \cdots - 2992 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5317 \beta_{19} - 4287 \beta_{18} + 7635 \beta_{16} - 8665 \beta_{15} - 5317 \beta_{14} - 6347 \beta_{13} - 2055 \beta_{12} + 5386 \beta_{11} + 4159 \beta_{10} + 4309 \beta_{9} - 4159 \beta_{8} - 5403 \beta_{7} + \cdots - 14638 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7845 \beta_{19} + 8595 \beta_{18} + 17190 \beta_{17} - 8615 \beta_{16} - 9365 \beta_{15} - 12173 \beta_{14} - 8595 \beta_{13} - 15989 \beta_{12} + 9474 \beta_{11} + 16869 \beta_{10} + 18089 \beta_{9} + \cdots + 25044 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 22087 \beta_{19} + 18229 \beta_{18} - 30559 \beta_{16} + 34417 \beta_{15} + 22087 \beta_{14} + 25945 \beta_{13} + 8707 \beta_{12} - 22676 \beta_{11} - 16317 \beta_{10} - 15599 \beta_{9} + 16317 \beta_{8} + \cdots + 59485 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 62818 \beta_{19} - 74282 \beta_{18} - 148564 \beta_{17} + 74742 \beta_{16} + 86206 \beta_{15} + 101842 \beta_{14} + 74282 \beta_{13} + 130194 \beta_{12} - 90060 \beta_{11} - 139346 \beta_{10} + \cdots - 207904 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 365529 \beta_{19} - 307025 \beta_{18} + 490501 \beta_{16} - 549005 \beta_{15} - 365529 \beta_{14} - 424033 \beta_{13} - 147445 \beta_{12} + 377384 \beta_{11} + 259089 \beta_{10} + \cdots - 973110 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 504149 \beta_{19} + 631589 \beta_{18} + 1263178 \beta_{17} - 638233 \beta_{16} - 765673 \beta_{15} - 849325 \beta_{14} - 631589 \beta_{13} - 1063299 \beta_{12} + 813496 \beta_{11} + \cdots + 1717868 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 3015847 \beta_{19} + 2566165 \beta_{18} - 3952079 \beta_{16} + 4401761 \beta_{15} + 3015847 \beta_{14} + 3465529 \beta_{13} + 1241503 \beta_{12} - 3121006 \beta_{11} - 2076279 \beta_{10} + \cdots + 7979906 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 4062775 \beta_{19} - 5311317 \beta_{18} - 10622634 \beta_{17} + 5389093 \beta_{16} + 6637635 \beta_{15} + 7067203 \beta_{14} + 5311317 \beta_{13} + 8703539 \beta_{12} + \cdots - 14156328 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 12417331 \beta_{19} - 10666792 \beta_{18} + 15983777 \beta_{16} - 17734316 \beta_{15} - 12417331 \beta_{14} - 14167870 \beta_{13} - 5196791 \beta_{12} + 12862622 \beta_{11} + \cdots - 32752987 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 32875304 \beta_{19} + 44334764 \beta_{18} + 88669528 \beta_{17} - 45144368 \beta_{16} - 56603828 \beta_{15} - 58682048 \beta_{14} - 44334764 \beta_{13} - 71348356 \beta_{12} + \cdots + 116477984 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(\beta_{4} + \beta_{5}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
2.44766i
1.14923i
2.86589i
1.70974i
0.125665i
0.402430i
0.676406i
2.27790i
1.86950i
1.49418i
2.44766i
1.14923i
2.86589i
1.70974i
0.125665i
0.402430i
0.676406i
2.27790i
1.86950i
1.49418i
0 −0.877081 + 3.27331i 0 0 0 −2.27273 1.31216i 0 −7.34722 4.24192i 0
193.2 0 −0.387600 + 1.44654i 0 0 0 −2.10545 1.21558i 0 0.655821 + 0.378639i 0
193.3 0 −0.355132 + 1.32537i 0 0 0 3.40883 + 1.96809i 0 0.967585 + 0.558635i 0
193.4 0 0.563138 2.10166i 0 0 0 −1.25530 0.724750i 0 −1.50178 0.867051i 0
193.5 0 0.690650 2.57754i 0 0 0 −1.87342 1.08162i 0 −3.56864 2.06036i 0
293.1 0 −2.49316 + 0.668040i 0 0 0 0.749297 0.432607i 0 3.17149 1.83106i 0
293.2 0 −0.834228 + 0.223531i 0 0 0 −2.07295 + 1.19682i 0 −1.95211 + 1.12705i 0
293.3 0 −0.243028 + 0.0651192i 0 0 0 4.30824 2.48736i 0 −2.54325 + 1.46835i 0
293.4 0 2.41732 0.647720i 0 0 0 −3.34088 + 1.92886i 0 2.82584 1.63150i 0
293.5 0 2.51912 0.674996i 0 0 0 1.45437 0.839682i 0 3.29226 1.90079i 0
357.1 0 −0.877081 3.27331i 0 0 0 −2.27273 + 1.31216i 0 −7.34722 + 4.24192i 0
357.2 0 −0.387600 1.44654i 0 0 0 −2.10545 + 1.21558i 0 0.655821 0.378639i 0
357.3 0 −0.355132 1.32537i 0 0 0 3.40883 1.96809i 0 0.967585 0.558635i 0
357.4 0 0.563138 + 2.10166i 0 0 0 −1.25530 + 0.724750i 0 −1.50178 + 0.867051i 0
357.5 0 0.690650 + 2.57754i 0 0 0 −1.87342 + 1.08162i 0 −3.56864 + 2.06036i 0
457.1 0 −2.49316 0.668040i 0 0 0 0.749297 + 0.432607i 0 3.17149 + 1.83106i 0
457.2 0 −0.834228 0.223531i 0 0 0 −2.07295 1.19682i 0 −1.95211 1.12705i 0
457.3 0 −0.243028 0.0651192i 0 0 0 4.30824 + 2.48736i 0 −2.54325 1.46835i 0
457.4 0 2.41732 + 0.647720i 0 0 0 −3.34088 1.92886i 0 2.82584 + 1.63150i 0
457.5 0 2.51912 + 0.674996i 0 0 0 1.45437 + 0.839682i 0 3.29226 + 1.90079i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 457.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bs.d 20
5.b even 2 1 260.2.bk.c yes 20
5.c odd 4 1 260.2.bf.c 20
5.c odd 4 1 1300.2.bn.d 20
13.f odd 12 1 1300.2.bn.d 20
65.o even 12 1 inner 1300.2.bs.d 20
65.s odd 12 1 260.2.bf.c 20
65.t even 12 1 260.2.bk.c yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bf.c 20 5.c odd 4 1
260.2.bf.c 20 65.s odd 12 1
260.2.bk.c yes 20 5.b even 2 1
260.2.bk.c yes 20 65.t even 12 1
1300.2.bn.d 20 5.c odd 4 1
1300.2.bn.d 20 13.f odd 12 1
1300.2.bs.d 20 1.a even 1 1 trivial
1300.2.bs.d 20 65.o even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 2 T_{3}^{19} + 8 T_{3}^{18} - 28 T_{3}^{17} - 26 T_{3}^{16} + 106 T_{3}^{15} - 500 T_{3}^{14} + 1366 T_{3}^{13} + 2867 T_{3}^{12} - 5410 T_{3}^{11} + 14776 T_{3}^{10} - 43502 T_{3}^{9} - 9730 T_{3}^{8} - 76600 T_{3}^{7} + \cdots + 21904 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 2 T^{19} + 8 T^{18} + \cdots + 21904 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 6 T^{19} - 24 T^{18} + \cdots + 27625536 \) Copy content Toggle raw display
$11$ \( T^{20} - 12 T^{18} + 128 T^{17} + \cdots + 8202496 \) Copy content Toggle raw display
$13$ \( T^{20} + 8 T^{19} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} + 93 T^{18} + \cdots + 109980446689 \) Copy content Toggle raw display
$19$ \( T^{20} - 20 T^{19} + \cdots + 472279824 \) Copy content Toggle raw display
$23$ \( T^{20} + 6 T^{19} + 60 T^{18} + \cdots + 3139984 \) Copy content Toggle raw display
$29$ \( T^{20} - 24 T^{19} + \cdots + 341103961 \) Copy content Toggle raw display
$31$ \( T^{20} - 8 T^{19} + \cdots + 54838398976 \) Copy content Toggle raw display
$37$ \( T^{20} - 285 T^{18} + \cdots + 52\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( T^{20} - 6 T^{19} + \cdots + 10017978284161 \) Copy content Toggle raw display
$43$ \( T^{20} + 38 T^{19} + \cdots + 319577657344 \) Copy content Toggle raw display
$47$ \( T^{20} + 224 T^{18} + \cdots + 5858983936 \) Copy content Toggle raw display
$53$ \( T^{20} + 30 T^{19} + \cdots + 90\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{20} + 24 T^{19} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{20} + 32 T^{19} + \cdots + 43\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{20} + 22 T^{19} + \cdots + 11546104977936 \) Copy content Toggle raw display
$71$ \( T^{20} - 24 T^{18} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{10} - 22 T^{9} - 123 T^{8} + \cdots - 119575728)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 207459544743936 \) Copy content Toggle raw display
$83$ \( T^{20} + 896 T^{18} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 237919334033956 \) Copy content Toggle raw display
$97$ \( T^{20} + 46 T^{19} + \cdots + 963004643584 \) Copy content Toggle raw display
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