Properties

Label 570.2.m.b
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: \( x^{20} + 108x^{16} + 1318x^{12} + 4652x^{8} + 5057x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9} \)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{10} q^{3} + \beta_{2} q^{4} + (\beta_{6} + 1) q^{5} - q^{6} + (\beta_{16} - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{7} - \beta_{10} q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{10} q^{3} + \beta_{2} q^{4} + (\beta_{6} + 1) q^{5} - q^{6} + (\beta_{16} - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{7} - \beta_{10} q^{8} - \beta_{2} q^{9} + ( - \beta_{11} + \beta_{7}) q^{10} + (\beta_{16} + \beta_{9} + \beta_{6} + \beta_{4}) q^{11} - \beta_{7} q^{12} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_1) q^{13} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{5} - \beta_1) q^{14} + ( - \beta_{14} - \beta_{10}) q^{15} - q^{16} + ( - \beta_{17} - \beta_{15} + \beta_{9} + \beta_{6}) q^{17} + \beta_{10} q^{18} + (\beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} + \beta_{12} - \beta_{9} - \beta_{7}) q^{19} + (\beta_{8} + \beta_{2}) q^{20} + (\beta_{10} + \beta_{5} + \beta_1) q^{21} + ( - \beta_{14} - 2 \beta_{12} - \beta_{11} + \beta_{7} + \beta_{5}) q^{22} + ( - \beta_{9} + 2 \beta_{8} + \beta_{6} + 2 \beta_{4} + \beta_{2} + 1) q^{23} - \beta_{2} q^{24} + ( - \beta_{17} - \beta_{16} + \beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots - \beta_1) q^{25}+ \cdots + (\beta_{9} - \beta_{8} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 12 q^{5} - 20 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 12 q^{5} - 20 q^{6} - 4 q^{7} - 8 q^{11} - 20 q^{16} - 12 q^{17} - 4 q^{23} - 28 q^{25} + 24 q^{26} - 4 q^{28} - 12 q^{30} + 4 q^{35} + 20 q^{36} - 12 q^{38} + 4 q^{42} - 12 q^{43} - 44 q^{47} + 64 q^{55} + 12 q^{57} - 8 q^{58} - 24 q^{62} + 4 q^{63} + 8 q^{66} - 12 q^{68} - 4 q^{73} + 4 q^{76} + 88 q^{77} - 12 q^{80} - 20 q^{81} - 8 q^{82} + 76 q^{83} - 12 q^{85} + 8 q^{87} + 4 q^{92} - 24 q^{93} - 24 q^{95} + 20 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 108x^{16} + 1318x^{12} + 4652x^{8} + 5057x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1233\nu^{19} - 120336\nu^{15} - 289166\nu^{11} + 6031344\nu^{7} + 9961151\nu^{3} ) / 5112544 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1843\nu^{18} + 199306\nu^{14} + 2457754\nu^{10} + 8963426\nu^{6} + 11554103\nu^{2} ) / 2556272 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2595\nu^{18} + 273087\nu^{14} + 2661844\nu^{10} + 4366379\nu^{6} - 932561\nu^{2} ) / 1278136 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4296 \nu^{18} - 7207 \nu^{16} + 477582 \nu^{14} - 770625 \nu^{12} + 7084096 \nu^{10} - 8670841 \nu^{8} + 32921622 \nu^{6} - 24076543 \nu^{4} + 42067188 \nu^{2} + \cdots - 13013696 ) / 5112544 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3686 \nu^{19} + 4131 \nu^{17} - 398612 \nu^{15} + 450983 \nu^{13} - 4915508 \nu^{11} + 5923921 \nu^{9} - 17926852 \nu^{7} + 21144461 \nu^{5} + \cdots + 8864472 \nu ) / 5112544 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4296 \nu^{18} - 7207 \nu^{16} - 477582 \nu^{14} - 770625 \nu^{12} - 7084096 \nu^{10} - 8670841 \nu^{8} - 32921622 \nu^{6} - 24076543 \nu^{4} + \cdots - 13013696 ) / 5112544 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10283\nu^{17} - 1090267\nu^{13} - 11417761\nu^{9} - 27008625\nu^{5} - 12050376\nu ) / 5112544 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14579 \nu^{18} - 5890 \nu^{16} - 1567849 \nu^{14} - 610604 \nu^{12} - 18501857 \nu^{10} - 5104050 \nu^{8} - 59930247 \nu^{6} - 3630112 \nu^{4} + \cdots + 9123648 ) / 5112544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14579 \nu^{18} - 5890 \nu^{16} + 1567849 \nu^{14} - 610604 \nu^{12} + 18501857 \nu^{10} - 5104050 \nu^{8} + 59930247 \nu^{6} - 3630112 \nu^{4} + 54117564 \nu^{2} + \cdots + 9123648 ) / 5112544 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30391\nu^{19} + 3256034\nu^{15} + 37292880\nu^{11} + 113829150\nu^{7} + 98273977\nu^{3} ) / 10225088 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14918 \nu^{19} + 1504 \nu^{17} - 1619593 \nu^{15} + 147562 \nu^{13} - 20551431 \nu^{11} + 408180 \nu^{9} - 78087767 \nu^{7} - 9194094 \nu^{5} + \cdots - 24973328 \nu ) / 5112544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14918 \nu^{19} + 1504 \nu^{17} + 1619593 \nu^{15} + 147562 \nu^{13} + 20551431 \nu^{11} + 408180 \nu^{9} + 78087767 \nu^{7} - 9194094 \nu^{5} + \cdots - 24973328 \nu ) / 5112544 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18604 \nu^{19} - 1504 \nu^{17} + 2018205 \nu^{15} - 147562 \nu^{13} + 25466939 \nu^{11} - 408180 \nu^{9} + 96014619 \nu^{7} + 9194094 \nu^{5} + \cdots + 19860784 \nu ) / 5112544 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18604 \nu^{19} - 1504 \nu^{17} - 2018205 \nu^{15} - 147562 \nu^{13} - 25466939 \nu^{11} - 408180 \nu^{9} - 96014619 \nu^{7} + 9194094 \nu^{5} + \cdots + 19860784 \nu ) / 5112544 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10893 \nu^{19} + 3076 \nu^{17} + 13621 \nu^{16} + 1169237 \nu^{15} + 319642 \nu^{13} + 1438589 \nu^{12} + 13586349 \nu^{11} + 2746920 \nu^{9} + 14554471 \nu^{8} + \cdots + 18284064 ) / 5112544 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 14579 \nu^{18} + 23836 \nu^{16} - 1567849 \nu^{14} + 2504338 \nu^{12} - 18501857 \nu^{10} + 24041672 \nu^{8} - 59930247 \nu^{6} + 37863114 \nu^{4} + \cdots - 8423808 ) / 5112544 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10893 \nu^{19} - 21341 \nu^{18} - 3076 \nu^{17} - 1169237 \nu^{15} - 2286103 \nu^{14} - 319642 \nu^{13} - 13586349 \nu^{11} - 26164285 \nu^{10} + \cdots + 963320 \nu ) / 5112544 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 11787 \nu^{18} - 17752 \nu^{17} - 8973 \nu^{16} - 1237829 \nu^{14} - 1889572 \nu^{13} - 946867 \nu^{12} - 11825941 \nu^{10} - 20478392 \nu^{9} - 9468811 \nu^{8} + \cdots - 349920 ) / 5112544 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 77979 \nu^{19} - 23574 \nu^{18} + 17946 \nu^{16} - 8384792 \nu^{15} - 2475658 \nu^{14} + 1893734 \nu^{12} - 98874134 \nu^{11} - 23651882 \nu^{10} + \cdots + 699840 ) / 10225088 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{17} + 2 \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 6 \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{19} - 2 \beta_{16} - 5 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} + 5 \beta_{11} + 8 \beta_{10} - 2 \beta_{8} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{16} + 18 \beta_{15} - 18 \beta_{14} + 9 \beta_{13} - 18 \beta_{12} + 9 \beta_{11} - 18 \beta_{10} + 15 \beta_{9} + 11 \beta_{8} + 9 \beta_{7} + 13 \beta_{6} + 9 \beta_{5} + 13 \beta_{4} - 9 \beta _1 - 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 48 \beta_{18} - 24 \beta_{16} + 59 \beta_{14} + 39 \beta_{13} + 63 \beta_{12} + 43 \beta_{11} - 48 \beta_{9} + 24 \beta_{8} + 76 \beta_{7} - 24 \beta_{6} - 20 \beta_{5} + 24 \beta_{4} - 24 \beta_{3} + 48 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 166 \beta_{17} - 166 \beta_{14} + 83 \beta_{13} - 166 \beta_{12} + 83 \beta_{11} - 166 \beta_{10} - 131 \beta_{9} + 131 \beta_{8} + 83 \beta_{7} - 111 \beta_{6} + 83 \beta_{5} + 111 \beta_{4} + 52 \beta_{3} - 330 \beta_{2} - 83 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 492 \beta_{19} + 246 \beta_{16} + 411 \beta_{14} - 411 \beta_{13} + 359 \beta_{12} - 359 \beta_{11} - 976 \beta_{10} + 246 \beta_{8} - 246 \beta_{6} + 246 \beta_{4} - 246 \beta_{3} + 492 \beta_{2} - 186 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 544 \beta_{16} - 1582 \beta_{15} + 1582 \beta_{14} - 791 \beta_{13} + 1582 \beta_{12} - 791 \beta_{11} + 1582 \beta_{10} - 1637 \beta_{9} - 1093 \beta_{8} - 791 \beta_{7} - 1279 \beta_{6} - 791 \beta_{5} + \cdots + 3122 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4856 \beta_{18} + 2428 \beta_{16} - 5213 \beta_{14} - 3445 \beta_{13} - 5757 \beta_{12} - 3989 \beta_{11} + 4856 \beta_{9} - 2428 \beta_{8} - 7832 \beta_{7} + 2428 \beta_{6} + 1768 \beta_{5} - 2428 \beta_{4} + \cdots - 4856 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 15282 \beta_{17} + 15282 \beta_{14} - 7641 \beta_{13} + 15282 \beta_{12} - 7641 \beta_{11} + 15282 \beta_{10} + 12441 \beta_{9} - 12441 \beta_{8} - 7641 \beta_{7} + 10673 \beta_{6} - 7641 \beta_{5} + \cdots + 7641 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 47428 \beta_{19} - 23714 \beta_{16} - 38789 \beta_{14} + 38789 \beta_{13} - 33389 \beta_{12} + 33389 \beta_{11} + 93656 \beta_{10} - 23714 \beta_{8} + 23714 \beta_{6} - 23714 \beta_{4} + 23714 \beta_{3} + \cdots + 17050 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 52828 \beta_{16} + 148306 \beta_{15} - 148306 \beta_{14} + 74153 \beta_{13} - 148306 \beta_{12} + 74153 \beta_{11} - 148306 \beta_{10} + 156759 \beta_{9} + 103931 \beta_{8} + 74153 \beta_{7} + \cdots - 292662 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 461824 \beta_{18} - 230912 \beta_{16} + 489771 \beta_{14} + 324415 \beta_{13} + 542599 \beta_{12} + 377243 \beta_{11} - 461824 \beta_{9} + 230912 \beta_{8} + 746292 \beta_{7} + \cdots + 461824 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1441366 \beta_{17} - 1441366 \beta_{14} + 720683 \beta_{13} - 1441366 \beta_{12} + 720683 \beta_{11} - 1441366 \beta_{10} - 1176507 \beta_{9} + 1176507 \beta_{8} + 720683 \beta_{7} + \cdots - 720683 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4492972 \beta_{19} + 2246486 \beta_{16} + 3668827 \beta_{14} - 3668827 \beta_{13} + 3154175 \beta_{12} - 3154175 \beta_{11} - 8868288 \beta_{10} + 2246486 \beta_{8} + \cdots - 1606722 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 5007624 \beta_{16} - 14014766 \beta_{15} + 14014766 \beta_{14} - 7007383 \beta_{13} + 14014766 \beta_{12} - 7007383 \beta_{11} + 14014766 \beta_{10} - 14842429 \beta_{9} + \cdots + 27660770 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 43699624 \beta_{18} + 21849812 \beta_{16} - 46294061 \beta_{14} - 30672573 \beta_{13} - 51301685 \beta_{12} - 35680197 \beta_{11} + 43699624 \beta_{9} - 21849812 \beta_{8} + \cdots - 43699624 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 136287746 \beta_{17} + 136287746 \beta_{14} - 68143873 \beta_{13} + 136287746 \beta_{12} - 68143873 \beta_{11} + 136287746 \beta_{10} + 111270017 \beta_{9} + \cdots + 68143873 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 424999300 \beta_{19} - 212499650 \beta_{16} - 346996293 \beta_{14} + 346996293 \beta_{13} - 298289045 \beta_{12} + 298289045 \beta_{11} + 838836392 \beta_{10} + \cdots + 151909234 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\)

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
0.339574 0.339574i
2.20512 2.20512i
−1.20277 + 1.20277i
0.922947 0.922947i
−0.850665 + 0.850665i
−0.339574 + 0.339574i
−2.20512 + 2.20512i
1.20277 1.20277i
−0.922947 + 0.922947i
0.850665 0.850665i
0.339574 + 0.339574i
2.20512 + 2.20512i
−1.20277 1.20277i
0.922947 + 0.922947i
−0.850665 0.850665i
−0.339574 0.339574i
−2.20512 2.20512i
1.20277 + 1.20277i
−0.922947 0.922947i
0.850665 + 0.850665i
−0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −1.29975 + 1.81952i −1.00000 −0.728588 0.728588i 0.707107 + 0.707107i 1.00000i −0.367533 2.20566i
37.2 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.114611 2.23313i −1.00000 −1.40368 1.40368i 0.707107 + 0.707107i 1.00000i 1.49802 + 1.66010i
37.3 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.528178 + 2.17279i −1.00000 0.904140 + 0.904140i 0.707107 + 0.707107i 1.00000i −1.90987 1.16292i
37.4 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 1.42113 1.72638i −1.00000 3.40461 + 3.40461i 0.707107 + 0.707107i 1.00000i 0.215841 + 2.22563i
37.5 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 2.23583 0.0328054i −1.00000 −3.17648 3.17648i 0.707107 + 0.707107i 1.00000i −1.55777 + 1.60417i
37.6 0.707107 0.707107i −0.707107 0.707107i 1.00000i −1.29975 + 1.81952i −1.00000 −0.728588 0.728588i −0.707107 0.707107i 1.00000i 0.367533 + 2.20566i
37.7 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0.114611 2.23313i −1.00000 −1.40368 1.40368i −0.707107 0.707107i 1.00000i −1.49802 1.66010i
37.8 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0.528178 + 2.17279i −1.00000 0.904140 + 0.904140i −0.707107 0.707107i 1.00000i 1.90987 + 1.16292i
37.9 0.707107 0.707107i −0.707107 0.707107i 1.00000i 1.42113 1.72638i −1.00000 3.40461 + 3.40461i −0.707107 0.707107i 1.00000i −0.215841 2.22563i
37.10 0.707107 0.707107i −0.707107 0.707107i 1.00000i 2.23583 0.0328054i −1.00000 −3.17648 3.17648i −0.707107 0.707107i 1.00000i 1.55777 1.60417i
493.1 −0.707107 0.707107i 0.707107 0.707107i 1.00000i −1.29975 1.81952i −1.00000 −0.728588 + 0.728588i 0.707107 0.707107i 1.00000i −0.367533 + 2.20566i
493.2 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.114611 + 2.23313i −1.00000 −1.40368 + 1.40368i 0.707107 0.707107i 1.00000i 1.49802 1.66010i
493.3 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.528178 2.17279i −1.00000 0.904140 0.904140i 0.707107 0.707107i 1.00000i −1.90987 + 1.16292i
493.4 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 1.42113 + 1.72638i −1.00000 3.40461 3.40461i 0.707107 0.707107i 1.00000i 0.215841 2.22563i
493.5 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 2.23583 + 0.0328054i −1.00000 −3.17648 + 3.17648i 0.707107 0.707107i 1.00000i −1.55777 1.60417i
493.6 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −1.29975 1.81952i −1.00000 −0.728588 + 0.728588i −0.707107 + 0.707107i 1.00000i 0.367533 2.20566i
493.7 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.114611 + 2.23313i −1.00000 −1.40368 + 1.40368i −0.707107 + 0.707107i 1.00000i −1.49802 + 1.66010i
493.8 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0.528178 2.17279i −1.00000 0.904140 0.904140i −0.707107 + 0.707107i 1.00000i 1.90987 1.16292i
493.9 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 1.42113 + 1.72638i −1.00000 3.40461 3.40461i −0.707107 + 0.707107i 1.00000i −0.215841 + 2.22563i
493.10 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 2.23583 + 0.0328054i −1.00000 −3.17648 + 3.17648i −0.707107 + 0.707107i 1.00000i 1.55777 + 1.60417i
\(n\): 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.b 20
3.b odd 2 1 1710.2.p.c 20
5.c odd 4 1 inner 570.2.m.b 20
15.e even 4 1 1710.2.p.c 20
19.b odd 2 1 inner 570.2.m.b 20
57.d even 2 1 1710.2.p.c 20
95.g even 4 1 inner 570.2.m.b 20
285.j odd 4 1 1710.2.p.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.m.b 20 1.a even 1 1 trivial
570.2.m.b 20 5.c odd 4 1 inner
570.2.m.b 20 19.b odd 2 1 inner
570.2.m.b 20 95.g even 4 1 inner
1710.2.p.c 20 3.b odd 2 1
1710.2.p.c 20 15.e even 4 1
1710.2.p.c 20 57.d even 2 1
1710.2.p.c 20 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 2 T_{7}^{9} + 2 T_{7}^{8} + 8 T_{7}^{7} + 496 T_{7}^{6} + 1184 T_{7}^{5} + 1408 T_{7}^{4} - 320 T_{7}^{3} + 1600 T_{7}^{2} + 3200 T_{7} + 3200 \) 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
$5$ \( (T^{10} - 6 T^{9} + 25 T^{8} - 80 T^{7} + \cdots + 3125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 2 T^{9} + 2 T^{8} + 8 T^{7} + \cdots + 3200)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 2 T^{4} - 36 T^{3} - 16 T^{2} + \cdots - 400)^{4} \) Copy content Toggle raw display
$13$ \( T^{20} + 1248 T^{16} + 203872 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( (T^{10} + 6 T^{9} + 18 T^{8} - 64 T^{7} + \cdots + 12800)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} - 58 T^{18} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} + 2 T^{9} + 2 T^{8} - 24 T^{7} + \cdots + 156800)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 112 T^{8} + 4320 T^{6} + \cdots - 51200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 192 T^{8} + 12544 T^{6} + \cdots + 204800)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 8928 T^{16} + \cdots + 362615934976 \) Copy content Toggle raw display
$41$ \( (T^{10} + 64 T^{8} + 1320 T^{6} + \cdots + 51200)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 6 T^{9} + 18 T^{8} + 224 T^{7} + \cdots + 768800)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 22 T^{9} + 242 T^{8} + \cdots + 2000000)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 30928 T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{10} - 160 T^{8} + 1944 T^{6} + \cdots - 3200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 248 T^{3} - 816 T^{2} + \cdots + 56000)^{4} \) Copy content Toggle raw display
$67$ \( T^{20} + 85248 T^{16} + \cdots + 91\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{10} + 200 T^{8} + 12288 T^{6} + \cdots + 3276800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 2 T^{9} + 2 T^{8} + \cdots + 1468820000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 384 T^{8} + 28256 T^{6} + \cdots - 2508800)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 38 T^{9} + 722 T^{8} + \cdots + 356979200)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 192 T^{8} + 12680 T^{6} + \cdots - 3699200)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 113680 T^{16} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
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