Properties

Label 690.2.m.c
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \( x^{20} - 4 x^{19} - 3 x^{18} + 66 x^{17} - 163 x^{16} - 52 x^{15} + 1567 x^{14} - 6182 x^{13} + 17043 x^{12} - 35832 x^{11} + 60906 x^{10} - 87666 x^{9} + 106197 x^{8} - 102542 x^{7} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{16} q^{2} - \beta_{14} q^{3} - \beta_{12} q^{4} + \beta_{11} q^{5} - \beta_{15} q^{6} + (\beta_{17} + \beta_{15} + \beta_{14} - \beta_{10} + \beta_{9} + \beta_{5} + \beta_{4} + 1) q^{7} + \beta_{9} q^{8} + \beta_{17} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{16} q^{2} - \beta_{14} q^{3} - \beta_{12} q^{4} + \beta_{11} q^{5} - \beta_{15} q^{6} + (\beta_{17} + \beta_{15} + \beta_{14} - \beta_{10} + \beta_{9} + \beta_{5} + \beta_{4} + 1) q^{7} + \beta_{9} q^{8} + \beta_{17} q^{9} + \beta_{8} q^{10} + (\beta_{17} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{8} - \beta_{7} - 1) q^{11} + \beta_{10} q^{12} + (\beta_{19} - \beta_{17} + \beta_{16} - \beta_{15} - \beta_{13} - 2 \beta_{12} - \beta_{11} - \beta_{10} - 3 \beta_{8} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{17} + \beta_{15} + \beta_{13} + \beta_{11} - \beta_{9} + \beta_{8} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} - 2 q^{9} - 2 q^{10} - 24 q^{11} + 2 q^{12} - 4 q^{13} + 2 q^{14} + 2 q^{15} - 2 q^{16} + 16 q^{17} - 2 q^{18} + 14 q^{19} - 2 q^{20} - 13 q^{21} - 2 q^{22} - 2 q^{23} - 20 q^{24} - 2 q^{25} + 7 q^{26} + 2 q^{27} + 2 q^{28} + 18 q^{29} + 2 q^{30} + 22 q^{31} - 2 q^{32} + 2 q^{33} - 17 q^{34} + 13 q^{35} - 2 q^{36} - 16 q^{37} + 3 q^{38} + 4 q^{39} - 2 q^{40} + 29 q^{41} + 9 q^{42} - 22 q^{43} - 24 q^{44} + 20 q^{45} - 2 q^{46} - 94 q^{47} + 2 q^{48} - 22 q^{49} - 2 q^{50} - 5 q^{51} - 4 q^{52} + 58 q^{53} + 2 q^{54} + 9 q^{55} + 2 q^{56} - 25 q^{57} - 4 q^{58} + 45 q^{59} + 2 q^{60} + q^{61} - 9 q^{63} - 2 q^{64} - 4 q^{65} - 9 q^{66} + 16 q^{67} - 6 q^{68} + 24 q^{69} + 2 q^{70} + 59 q^{71} - 2 q^{72} + 3 q^{73} - 16 q^{74} + 2 q^{75} - 8 q^{76} - 19 q^{77} - 18 q^{78} - 20 q^{79} - 2 q^{80} - 2 q^{81} - 37 q^{82} + 13 q^{83} + 9 q^{84} + 5 q^{85} - 22 q^{86} + 4 q^{87} + 9 q^{88} - 97 q^{89} - 2 q^{90} - 18 q^{91} + 9 q^{92} + 22 q^{93} + 27 q^{94} + 3 q^{95} + 2 q^{96} - 17 q^{97} - 11 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 3 x^{18} + 66 x^{17} - 163 x^{16} - 52 x^{15} + 1567 x^{14} - 6182 x^{13} + 17043 x^{12} - 35832 x^{11} + 60906 x^{10} - 87666 x^{9} + 106197 x^{8} - 102542 x^{7} + \cdots + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 77\!\cdots\!35 \nu^{19} + \cdots - 13\!\cdots\!32 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 47\!\cdots\!30 \nu^{19} + \cdots - 57\!\cdots\!72 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 55\!\cdots\!96 \nu^{19} + \cdots - 18\!\cdots\!58 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 56\!\cdots\!12 \nu^{19} + \cdots + 20\!\cdots\!88 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 73\!\cdots\!18 \nu^{19} + \cdots - 63\!\cdots\!04 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 82\!\cdots\!88 \nu^{19} + \cdots + 25\!\cdots\!38 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 87\!\cdots\!48 \nu^{19} + \cdots - 10\!\cdots\!60 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!68 \nu^{19} + \cdots + 12\!\cdots\!95 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!76 \nu^{19} + \cdots + 12\!\cdots\!77 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!02 \nu^{19} + \cdots + 11\!\cdots\!84 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!08 \nu^{19} + \cdots + 10\!\cdots\!01 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31\!\cdots\!52 \nu^{19} + \cdots - 20\!\cdots\!18 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 34\!\cdots\!02 \nu^{19} + \cdots - 18\!\cdots\!88 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 38\!\cdots\!72 \nu^{19} + \cdots + 92\!\cdots\!94 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 47\!\cdots\!22 \nu^{19} + \cdots + 55\!\cdots\!36 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 47\!\cdots\!34 \nu^{19} + \cdots - 78\!\cdots\!96 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 66\!\cdots\!69 \nu^{19} + \cdots + 20\!\cdots\!21 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 74\!\cdots\!15 \nu^{19} + \cdots + 17\!\cdots\!51 ) / 73\!\cdots\!31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} - \beta_{18} - 2 \beta_{17} - 2 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 4 \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{19} + 4 \beta_{17} + 4 \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{12} - \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + \beta_{7} - \beta_{3} - 6 \beta_{2} + 2 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{19} - 2 \beta_{18} - 49 \beta_{17} - 34 \beta_{16} - 38 \beta_{15} - 21 \beta_{14} - 13 \beta_{13} - \beta_{12} - 21 \beta_{11} + 18 \beta_{10} - 49 \beta_{9} - 47 \beta_{8} + 2 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} + 7 \beta_{2} - 13 \beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 71 \beta_{19} + 13 \beta_{18} + 134 \beta_{17} + 66 \beta_{16} + 114 \beta_{15} - 13 \beta_{14} + 48 \beta_{13} + 74 \beta_{12} + 48 \beta_{11} + 37 \beta_{10} + 123 \beta_{9} + 136 \beta_{8} - 13 \beta_{7} - 18 \beta_{6} - 18 \beta_{5} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 151 \beta_{19} - 4 \beta_{18} - 568 \beta_{17} - 238 \beta_{16} - 505 \beta_{15} - 151 \beta_{13} - 48 \beta_{12} - 94 \beta_{11} - 44 \beta_{10} - 501 \beta_{9} - 572 \beta_{8} + 43 \beta_{7} + 135 \beta_{6} + 43 \beta_{5} + 177 \beta_{4} + \cdots - 234 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 702 \beta_{19} + 2126 \beta_{17} + 408 \beta_{16} + 2126 \beta_{15} - 38 \beta_{14} + 877 \beta_{13} + 294 \beta_{12} + 664 \beta_{11} + 702 \beta_{10} + 1767 \beta_{9} + 2268 \beta_{8} - 290 \beta_{7} - 438 \beta_{6} + \cdots + 1065 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2390 \beta_{19} + 132 \beta_{18} - 6136 \beta_{17} - 7012 \beta_{15} + 2086 \beta_{14} - 2444 \beta_{13} - 383 \beta_{12} - 934 \beta_{11} - 3900 \beta_{10} - 4451 \beta_{9} - 7012 \beta_{8} + 1692 \beta_{7} + 2390 \beta_{6} + \cdots - 3746 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5111 \beta_{19} - 2040 \beta_{18} + 17477 \beta_{17} - 6362 \beta_{16} + 22925 \beta_{15} - 6791 \beta_{14} + 8259 \beta_{13} - 4107 \beta_{12} + 1897 \beta_{11} + 15050 \beta_{10} + 10299 \beta_{9} + \cdots + 17813 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 14021 \beta_{19} + 9229 \beta_{18} - 34089 \beta_{17} + 49307 \beta_{16} - 64960 \beta_{15} + 38468 \beta_{14} - 24860 \beta_{13} + 20295 \beta_{12} + 2741 \beta_{11} - 71749 \beta_{10} - 6488 \beta_{9} + \cdots - 60168 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 39811 \beta_{18} - 23464 \beta_{17} - 258961 \beta_{16} + 124682 \beta_{15} - 159691 \beta_{14} + 43390 \beta_{13} - 121103 \beta_{12} - 63275 \beta_{11} + 258961 \beta_{10} - 119880 \beta_{9} + \cdots + 194703 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 194262 \beta_{19} + 194262 \beta_{18} + 590249 \beta_{17} + 1222410 \beta_{16} + 590249 \beta_{14} + 612124 \beta_{12} + 374692 \beta_{11} - 907833 \beta_{10} + 907833 \beta_{9} + \cdots - 612124 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1291028 \beta_{19} - 672687 \beta_{18} - 4241226 \beta_{17} - 5090636 \beta_{16} - 1882408 \beta_{15} - 2176925 \beta_{14} - 672687 \beta_{13} - 2413435 \beta_{12} - 1882408 \beta_{11} + \cdots + 1291028 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7306375 \beta_{19} + 2462944 \beta_{18} + 22504950 \beta_{17} + 19715461 \beta_{16} + 14176470 \beta_{15} + 6870095 \beta_{14} + 5252433 \beta_{13} + 9769319 \beta_{12} + \cdots - 1185708 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 33609406 \beta_{19} - 8199524 \beta_{18} - 102255210 \beta_{17} - 69885095 \beta_{16} - 76091057 \beta_{15} - 18895885 \beta_{14} - 27803501 \beta_{13} - 35108969 \beta_{12} + \cdots - 10696361 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 140271748 \beta_{19} + 20954963 \beta_{18} + 427756633 \beta_{17} + 223899277 \beta_{16} + 353063932 \beta_{15} + 38521255 \beta_{14} + 129164655 \beta_{13} + \cdots + 100976707 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 546164663 \beta_{19} - 45887093 \beta_{18} - 1632307100 \beta_{17} - 619032881 \beta_{16} - 1487267752 \beta_{15} - 546164663 \beta_{13} - 324068924 \beta_{12} + \cdots - 573145788 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1926997917 \beta_{19} + 5726696450 \beta_{17} + 1243247780 \beta_{16} + 5726696450 \beta_{15} - 571490131 \beta_{14} + 2094830388 \beta_{13} + 683750137 \beta_{12} + \cdots + 2764584594 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 6182769136 \beta_{19} + 669331509 \beta_{18} - 18145320837 \beta_{17} - 20325168529 \beta_{15} + 4210658628 \beta_{14} - 7449454423 \beta_{13} - 286561532 \beta_{12} + \cdots - 11962551701 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{16}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.180562 1.25584i
−0.248715 + 1.72985i
−0.608732 + 1.33294i
0.706969 1.54805i
1.74971 + 2.01927i
−0.103821 0.119816i
−0.608732 1.33294i
0.706969 + 1.54805i
2.17335 + 0.638152i
−3.70312 1.08733i
2.21300 + 1.42221i
−0.359204 0.230846i
2.17335 0.638152i
−3.70312 + 1.08733i
1.74971 2.01927i
−0.103821 + 0.119816i
2.21300 1.42221i
−0.359204 + 0.230846i
0.180562 + 1.25584i
−0.248715 1.72985i
−0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i 0.959493 0.281733i −1.32376 0.850727i 0.415415 0.909632i −0.654861 + 0.755750i 0.841254 0.540641i
31.2 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i 0.959493 0.281733i 1.99973 + 1.28515i 0.415415 0.909632i −0.654861 + 0.755750i 0.841254 0.540641i
121.1 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.654861 0.755750i −0.668052 + 4.64640i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.142315 0.989821i
121.2 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.654861 0.755750i 0.0903198 0.628188i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.142315 0.989821i
151.1 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 0.142315 + 0.989821i 0.170792 + 0.0501489i 0.841254 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.2 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 0.142315 + 0.989821i 2.42713 + 0.712670i 0.841254 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
211.1 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.654861 + 0.755750i −0.668052 4.64640i −0.959493 0.281733i 0.841254 0.540641i −0.142315 + 0.989821i
211.2 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.654861 + 0.755750i 0.0903198 + 0.628188i −0.959493 0.281733i 0.841254 0.540641i −0.142315 + 0.989821i
271.1 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.841254 + 0.540641i −0.838100 1.83518i −0.654861 0.755750i −0.142315 0.989821i 0.415415 0.909632i
271.2 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.841254 + 0.540641i −0.113936 0.249486i −0.654861 0.755750i −0.142315 0.989821i 0.415415 0.909632i
301.1 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.415415 + 0.909632i −2.29325 + 2.64655i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.654861 0.755750i
301.2 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.415415 + 0.909632i 1.54913 1.78779i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.654861 0.755750i
331.1 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.841254 0.540641i −0.838100 + 1.83518i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
331.2 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.841254 0.540641i −0.113936 + 0.249486i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
361.1 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 0.142315 0.989821i 0.170792 0.0501489i 0.841254 + 0.540641i 0.415415 0.909632i −0.959493 0.281733i
361.2 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 0.142315 0.989821i 2.42713 0.712670i 0.841254 + 0.540641i 0.415415 0.909632i −0.959493 0.281733i
541.1 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.415415 0.909632i −2.29325 2.64655i −0.142315 0.989821i −0.959493 0.281733i −0.654861 + 0.755750i
541.2 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.415415 0.909632i 1.54913 + 1.78779i −0.142315 0.989821i −0.959493 0.281733i −0.654861 + 0.755750i
601.1 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.959493 + 0.281733i −1.32376 + 0.850727i 0.415415 + 0.909632i −0.654861 0.755750i 0.841254 + 0.540641i
601.2 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.959493 + 0.281733i 1.99973 1.28515i 0.415415 + 0.909632i −0.654861 0.755750i 0.841254 + 0.540641i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.m.c 20
23.c even 11 1 inner 690.2.m.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.c 20 1.a even 1 1 trivial
690.2.m.c 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} - 2 T_{7}^{19} + 20 T_{7}^{18} - 94 T_{7}^{17} + 200 T_{7}^{16} - 705 T_{7}^{15} + 3796 T_{7}^{14} - 9544 T_{7}^{13} + 11360 T_{7}^{12} - 17376 T_{7}^{11} + 61478 T_{7}^{10} - 66922 T_{7}^{9} - 30534 T_{7}^{8} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} + 20 T^{18} - 94 T^{17} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{20} + 24 T^{19} + 273 T^{18} + \cdots + 4489 \) Copy content Toggle raw display
$13$ \( T^{20} + 4 T^{19} + 23 T^{18} + 197 T^{17} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{20} - 16 T^{19} + \cdots + 3500852224 \) Copy content Toggle raw display
$19$ \( T^{20} - 14 T^{19} + 147 T^{18} + \cdots + 59274601 \) Copy content Toggle raw display
$23$ \( T^{20} + 2 T^{19} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} - 18 T^{19} + \cdots + 59067463607296 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 152659665224704 \) Copy content Toggle raw display
$37$ \( T^{20} + 16 T^{19} + \cdots + 42213400681 \) Copy content Toggle raw display
$41$ \( T^{20} - 29 T^{19} + \cdots + 35\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{20} + 22 T^{19} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( (T^{10} + 47 T^{9} + 812 T^{8} + \cdots + 1159181)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} - 58 T^{19} + \cdots + 43\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{20} - 45 T^{19} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 554879105680384 \) Copy content Toggle raw display
$67$ \( T^{20} - 16 T^{19} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{20} - 59 T^{19} + \cdots + 59348755456 \) Copy content Toggle raw display
$73$ \( T^{20} - 3 T^{19} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{20} + 20 T^{19} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{20} - 13 T^{19} + \cdots + 10300915926016 \) Copy content Toggle raw display
$89$ \( T^{20} + 97 T^{19} + \cdots + 10\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{20} + 17 T^{19} + \cdots + 61671996335104 \) Copy content Toggle raw display
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