Properties

Label 570.2.m.a
Level $570$
Weight $2$
Character orbit 570.m
Analytic conductor $4.551$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \( x^{20} + 153x^{16} + 6416x^{12} + 78648x^{8} + 19120x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{4} q^{4} + \beta_{6} q^{5} + q^{6} + ( - \beta_{9} + \beta_{6}) q^{7} - \beta_{11} q^{8} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{4} q^{4} + \beta_{6} q^{5} + q^{6} + ( - \beta_{9} + \beta_{6}) q^{7} - \beta_{11} q^{8} - \beta_{4} q^{9} - \beta_{13} q^{10} - \beta_1 q^{11} + \beta_{2} q^{12} + ( - \beta_{19} - \beta_{13} + \beta_{12}) q^{13} + ( - \beta_{15} - \beta_{13}) q^{14} + \beta_{14} q^{15} - q^{16} + ( - \beta_{19} - \beta_{17} - \beta_{16} - \beta_{7} - \beta_{5} + \beta_{3}) q^{17} + \beta_{11} q^{18} + (\beta_{17} - \beta_{4} - \beta_{3}) q^{19} - \beta_{7} q^{20} + (\beta_{14} + \beta_{12}) q^{21} - \beta_{3} q^{22} + ( - \beta_{9} - \beta_{6} + 2 \beta_{4} + 2) q^{23} + \beta_{4} q^{24} + (\beta_{17} - \beta_{16} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + \beta_1) q^{25} + ( - \beta_{9} - \beta_{7} - \beta_1) q^{26} - \beta_{2} q^{27} + ( - \beta_{7} + \beta_{5}) q^{28} + (\beta_{19} - \beta_{18} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} + \beta_{3}) q^{29} + \beta_{6} q^{30} + ( - \beta_{19} + \beta_{18} + \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{8} + \beta_{3}) q^{31} - \beta_{2} q^{32} - \beta_{19} q^{33} + (\beta_{18} - \beta_{14} + \beta_{12} + \beta_{10}) q^{34} + (\beta_{17} - \beta_{16} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - 4 \beta_{4} + \beta_1) q^{35} + q^{36} + (\beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{8} + \beta_{3}) q^{37} + ( - \beta_{18} + \beta_{11} - \beta_{10} - \beta_{8}) q^{38} + (\beta_{10} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_1) q^{39} - \beta_{14} q^{40} + ( - \beta_{19} - \beta_{14} - \beta_{12} - \beta_{11} + \beta_{3} + \beta_{2}) q^{41} + ( - \beta_{9} + \beta_{6}) q^{42} + ( - \beta_{17} + \beta_{16} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 3) q^{43} + ( - \beta_{10} - \beta_{8} + \beta_1) q^{44} + \beta_{7} q^{45} + ( - \beta_{15} + \beta_{13} - 2 \beta_{11} + 2 \beta_{2}) q^{46} + ( - \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{47} - \beta_{11} q^{48} + (\beta_{19} + 2 \beta_{17} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 2 \beta_1) q^{49}+ \cdots + (\beta_{10} + \beta_{8} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{5} + 20 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{5} + 20 q^{6} - 4 q^{7} - 8 q^{11} - 20 q^{16} + 4 q^{17} + 44 q^{23} + 4 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{30} + 4 q^{35} + 20 q^{36} - 4 q^{38} - 4 q^{42} + 52 q^{43} + 4 q^{47} + 16 q^{55} - 4 q^{57} + 8 q^{58} + 32 q^{61} - 8 q^{62} + 4 q^{63} - 8 q^{66} + 4 q^{68} - 20 q^{73} + 20 q^{76} - 24 q^{77} + 4 q^{80} - 20 q^{81} - 24 q^{82} - 116 q^{83} - 60 q^{85} + 8 q^{87} - 44 q^{92} + 8 q^{93} - 32 q^{95} - 20 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 153x^{16} + 6416x^{12} + 78648x^{8} + 19120x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -16765\nu^{16} - 2625903\nu^{12} - 115242314\nu^{8} - 1515721480\nu^{4} - 599186688 ) / 174141704 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 806911\nu^{17} + 123405389\nu^{13} + 5168924442\nu^{9} + 63121710948\nu^{5} + 12328816792\nu ) / 2089700448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -604825\nu^{17} - 92597906\nu^{13} - 3887923677\nu^{9} - 47719026672\nu^{5} - 10577293948\nu ) / 522425112 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 54129\nu^{18} + 8282406\nu^{14} + 347394797\nu^{10} + 4261561356\nu^{6} + 1096207364\nu^{2} ) / 31662128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11884471 \nu^{18} - 808344 \nu^{16} + 1818129965 \nu^{14} - 123229932 \nu^{12} + 76223127354 \nu^{10} - 5124003060 \nu^{8} + 933673009644 \nu^{6} + \cdots - 7006091376 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11884471 \nu^{18} - 808344 \nu^{16} - 1818129965 \nu^{14} - 123229932 \nu^{12} - 76223127354 \nu^{10} - 5124003060 \nu^{8} + \cdots - 7006091376 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13498293 \nu^{18} - 604298 \nu^{16} + 2064940743 \nu^{14} - 92070010 \nu^{12} + 86560976238 \nu^{10} - 3830938056 \nu^{8} + 1059916431540 \nu^{6} + \cdots - 6281901056 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12734103 \nu^{18} + 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} + 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} + \cdots - 7190240256 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13498293 \nu^{18} - 604298 \nu^{16} - 2064940743 \nu^{14} - 92070010 \nu^{12} - 86560976238 \nu^{10} - 3830938056 \nu^{8} + \cdots - 6281901056 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12734103 \nu^{18} - 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} - 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} + \cdots - 7190240256 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 45261895 \nu^{19} - 6924865889 \nu^{15} - 290369158398 \nu^{11} - 3558464452956 \nu^{7} - 850240722328 \nu^{3} ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21963356 \nu^{19} + 518651 \nu^{17} - 3360431092 \nu^{15} + 79345865 \nu^{13} - 140922857856 \nu^{11} + 3327761602 \nu^{9} + \cdots + 13651993928 \nu ) / 1393133632 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 21963356 \nu^{19} + 518651 \nu^{17} + 3360431092 \nu^{15} + 79345865 \nu^{13} + 140922857856 \nu^{11} + 3327761602 \nu^{9} + \cdots + 13651993928 \nu ) / 1393133632 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24345032 \nu^{19} - 518651 \nu^{17} + 3724856956 \nu^{15} - 79345865 \nu^{13} + 156208228924 \nu^{11} - 3327761602 \nu^{9} + \cdots - 12258860296 \nu ) / 1393133632 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 24345032 \nu^{19} - 518651 \nu^{17} - 3724856956 \nu^{15} - 79345865 \nu^{13} - 156208228924 \nu^{11} - 3327761602 \nu^{9} + \cdots - 12258860296 \nu ) / 1393133632 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 77025497 \nu^{19} - 2419300 \nu^{17} + 1726694 \nu^{16} - 11784791035 \nu^{15} - 370391624 \nu^{13} + 264708058 \nu^{12} + \cdots + 11065189808 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 77025497 \nu^{19} - 22270858 \nu^{18} - 2419300 \nu^{17} - 11784791035 \nu^{15} - 3406995518 \nu^{14} - 370391624 \nu^{13} + \cdots - 42309175792 \nu ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 113542400 \nu^{19} + 12734103 \nu^{18} + 201180 \nu^{16} + 17372793436 \nu^{15} + 1948517589 \nu^{14} + 31510836 \nu^{12} + 728609156748 \nu^{11} + \cdots + 7190240256 ) / 4179400896 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 77025497 \nu^{19} + 11784791035 \nu^{15} + 494177340558 \nu^{11} + 6057012474372 \nu^{7} + 1459603333880 \nu^{3} ) / 2089700448 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - 2 \beta_{17} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_{3} - 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{19} - 2 \beta_{18} - 7 \beta_{15} + 7 \beta_{14} - 7 \beta_{13} + 7 \beta_{12} + 4 \beta_{11} - \beta_{10} - \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{19} - 18\beta_{16} - 3\beta_{9} - 3\beta_{7} - 13\beta_{6} - 13\beta_{5} + 9\beta_{3} - 16\beta _1 - 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 51 \beta_{15} - 51 \beta_{14} - 59 \beta_{13} - 59 \beta_{12} - 13 \beta_{10} + 13 \beta_{8} + 28 \beta_{3} + 72 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 79 \beta_{19} + 158 \beta_{17} - 136 \beta_{10} - 143 \beta_{9} - 136 \beta_{8} + 143 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - 496 \beta_{4} - 79 \beta_{3} + 136 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 312 \beta_{19} + 294 \beta_{18} + 531 \beta_{15} - 531 \beta_{14} + 395 \beta_{13} - 395 \beta_{12} - 888 \beta_{11} + 147 \beta_{10} + 147 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 707 \beta_{19} + 1414 \beta_{16} - 343 \beta_{9} - 343 \beta_{7} + 1463 \beta_{6} + 1463 \beta_{5} - 707 \beta_{3} + 1220 \beta _1 + 4328 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3235 \beta_{15} + 3235 \beta_{14} + 4947 \beta_{13} + 4947 \beta_{12} + 1563 \beta_{10} - 1563 \beta_{8} - 3220 \beta_{3} - 9664 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6455 \beta_{19} - 12910 \beta_{17} + 11308 \beta_{10} + 14507 \beta_{9} + 11308 \beta_{8} - 14507 \beta_{7} - 4723 \beta_{6} + 4723 \beta_{5} + 39168 \beta_{4} + 6455 \beta_{3} - 11308 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 32140 \beta_{19} - 32062 \beta_{18} - 46923 \beta_{15} + 46923 \beta_{14} - 27771 \beta_{13} + 27771 \beta_{12} + 99536 \beta_{11} - 16031 \beta_{10} - 16031 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 59911 \beta_{19} - 119822 \beta_{16} + 54059 \beta_{9} + 54059 \beta_{7} - 141819 \beta_{6} - 141819 \beta_{5} + 59911 \beta_{3} - 106756 \beta _1 - 363056 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 247291 \beta_{15} - 247291 \beta_{14} - 449099 \beta_{13} - 449099 \beta_{12} - 160815 \beta_{10} + 160815 \beta_{8} + 315700 \beta_{3} + 997024 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 562991 \beta_{19} + 1125982 \beta_{17} - 1018020 \beta_{10} - 1377203 \beta_{9} - 1018020 \beta_{8} + 1377203 \beta_{7} + 572851 \beta_{6} - 572851 \beta_{5} - 3416960 \beta_{4} + \cdots + 1018020 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 3076036 \beta_{19} + 3181742 \beta_{18} + 4317371 \beta_{15} - 4317371 \beta_{14} + 2261611 \beta_{13} - 2261611 \beta_{12} - 9836256 \beta_{11} + 1590871 \beta_{10} + \cdots + 1590871 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 5337647 \beta_{19} + 10675294 \beta_{16} - 5838259 \beta_{9} - 5838259 \beta_{7} + 13331811 \beta_{6} + 13331811 \beta_{5} - 5337647 \beta_{3} + 9760724 \beta _1 + 32468000 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 21071035 \beta_{15} + 21071035 \beta_{14} + 41593707 \beta_{13} + 41593707 \beta_{12} + 15598983 \beta_{10} - 15598983 \beta_{8} - 29845364 \beta_{3} - 96201728 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 50916399 \beta_{19} - 101832798 \beta_{17} + 93862708 \beta_{10} + 128862627 \beta_{9} + 93862708 \beta_{8} - 128862627 \beta_{7} - 58227795 \beta_{6} + \cdots - 93862708 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 288923220 \beta_{19} - 304181006 \beta_{18} - 401128059 \beta_{15} + 401128059 \beta_{14} - 198779851 \beta_{13} + 198779851 \beta_{12} + 936087104 \beta_{11} + \cdots - 152090503 \beta_{8} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.53190 + 1.53190i
−0.120370 + 0.120370i
−1.75036 + 1.75036i
2.19691 2.19691i
0.498616 0.498616i
1.53190 1.53190i
0.120370 0.120370i
1.75036 1.75036i
−2.19691 + 2.19691i
−0.498616 + 0.498616i
−1.53190 1.53190i
−0.120370 0.120370i
−1.75036 1.75036i
2.19691 + 2.19691i
0.498616 + 0.498616i
1.53190 + 1.53190i
0.120370 + 0.120370i
1.75036 + 1.75036i
−2.19691 2.19691i
−0.498616 0.498616i
−0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −2.23502 + 0.0685835i 1.00000 −2.16643 2.16643i 0.707107 + 0.707107i 1.00000i 1.53190 1.62889i
37.2 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.66396 + 1.49373i 1.00000 −0.170229 0.170229i 0.707107 + 0.707107i 1.00000i 0.120370 2.23283i
37.3 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −0.253765 2.22162i 1.00000 −2.47539 2.47539i 0.707107 + 0.707107i 1.00000i 1.75036 + 1.39149i
37.4 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.25884 + 1.84806i 1.00000 3.10690 + 3.10690i 0.707107 + 0.707107i 1.00000i −2.19691 0.416642i
37.5 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.89390 1.18875i 1.00000 0.705149 + 0.705149i 0.707107 + 0.707107i 1.00000i −0.498616 + 2.17977i
37.6 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −2.23502 + 0.0685835i 1.00000 −2.16643 2.16643i −0.707107 0.707107i 1.00000i −1.53190 + 1.62889i
37.7 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.66396 + 1.49373i 1.00000 −0.170229 0.170229i −0.707107 0.707107i 1.00000i −0.120370 + 2.23283i
37.8 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −0.253765 2.22162i 1.00000 −2.47539 2.47539i −0.707107 0.707107i 1.00000i −1.75036 1.39149i
37.9 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.25884 + 1.84806i 1.00000 3.10690 + 3.10690i −0.707107 0.707107i 1.00000i 2.19691 + 0.416642i
37.10 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.89390 1.18875i 1.00000 0.705149 + 0.705149i −0.707107 0.707107i 1.00000i 0.498616 2.17977i
493.1 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −2.23502 0.0685835i 1.00000 −2.16643 + 2.16643i 0.707107 0.707107i 1.00000i 1.53190 + 1.62889i
493.2 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −1.66396 1.49373i 1.00000 −0.170229 + 0.170229i 0.707107 0.707107i 1.00000i 0.120370 + 2.23283i
493.3 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −0.253765 + 2.22162i 1.00000 −2.47539 + 2.47539i 0.707107 0.707107i 1.00000i 1.75036 1.39149i
493.4 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.25884 1.84806i 1.00000 3.10690 3.10690i 0.707107 0.707107i 1.00000i −2.19691 + 0.416642i
493.5 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.89390 + 1.18875i 1.00000 0.705149 0.705149i 0.707107 0.707107i 1.00000i −0.498616 2.17977i
493.6 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −2.23502 0.0685835i 1.00000 −2.16643 + 2.16643i −0.707107 + 0.707107i 1.00000i −1.53190 1.62889i
493.7 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.66396 1.49373i 1.00000 −0.170229 + 0.170229i −0.707107 + 0.707107i 1.00000i −0.120370 2.23283i
493.8 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −0.253765 + 2.22162i 1.00000 −2.47539 + 2.47539i −0.707107 + 0.707107i 1.00000i −1.75036 + 1.39149i
493.9 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.25884 1.84806i 1.00000 3.10690 3.10690i −0.707107 + 0.707107i 1.00000i 2.19691 0.416642i
493.10 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.89390 + 1.18875i 1.00000 0.705149 0.705149i −0.707107 + 0.707107i 1.00000i 0.498616 + 2.17977i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.m.a 20
3.b odd 2 1 1710.2.p.d 20
5.c odd 4 1 inner 570.2.m.a 20
15.e even 4 1 1710.2.p.d 20
19.b odd 2 1 inner 570.2.m.a 20
57.d even 2 1 1710.2.p.d 20
95.g even 4 1 inner 570.2.m.a 20
285.j odd 4 1 1710.2.p.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.m.a 20 1.a even 1 1 trivial
570.2.m.a 20 5.c odd 4 1 inner
570.2.m.a 20 19.b odd 2 1 inner
570.2.m.a 20 95.g even 4 1 inner
1710.2.p.d 20 3.b odd 2 1
1710.2.p.d 20 15.e even 4 1
1710.2.p.d 20 57.d even 2 1
1710.2.p.d 20 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 2 T_{7}^{9} + 2 T_{7}^{8} + 8 T_{7}^{7} + 320 T_{7}^{6} + 864 T_{7}^{5} + 1120 T_{7}^{4} - 1600 T_{7}^{3} + 1600 T_{7}^{2} + 640 T_{7} + 128 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T^{10} + 2 T^{9} + T^{8} + 8 T^{7} + 22 T^{6} + \cdots + 3125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 2 T^{9} + 2 T^{8} + 8 T^{7} + \cdots + 128)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 2 T^{4} - 32 T^{3} - 8 T^{2} + \cdots - 272)^{4} \) Copy content Toggle raw display
$13$ \( T^{20} + 2656 T^{16} + 1634400 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( (T^{10} - 2 T^{9} + 2 T^{8} + 16 T^{7} + \cdots + 86528)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 54 T^{18} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} - 22 T^{9} + 242 T^{8} - 1464 T^{7} + \cdots + 128)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 136 T^{8} + 7088 T^{6} + \cdots - 9193472)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 136 T^{8} + 7088 T^{6} + \cdots + 9193472)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 21472 T^{16} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{10} + 216 T^{8} + 13864 T^{6} + \cdots + 700928)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 26 T^{9} + 338 T^{8} + \cdots + 12500000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 2 T^{9} + 2 T^{8} - 968 T^{7} + \cdots + 1520768)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 33488 T^{16} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{10} - 144 T^{8} + 6008 T^{6} + \cdots - 2196608)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 8 T^{4} - 16 T^{3} + 112 T^{2} + \cdots - 320)^{4} \) Copy content Toggle raw display
$67$ \( T^{20} + 39168 T^{16} + \cdots + 17592186044416 \) Copy content Toggle raw display
$71$ \( (T^{10} + 232 T^{8} + 19456 T^{6} + \cdots + 22151168)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 10 T^{9} + 50 T^{8} + \cdots + 22957088)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 184 T^{8} + 9968 T^{6} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 58 T^{9} + 1682 T^{8} + \cdots + 14623232)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 472 T^{8} + 63560 T^{6} + \cdots - 62720000)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 94128 T^{16} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
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