Properties

Label 536.1.ba.a
Level $536$
Weight $1$
Character orbit 536.ba
Analytic conductor $0.267$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 536.ba (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.267498846771\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Defining polynomial: \(x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{33}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{66}^{23} q^{2} + ( \zeta_{66}^{14} - \zeta_{66}^{31} ) q^{3} -\zeta_{66}^{13} q^{4} + ( \zeta_{66}^{4} - \zeta_{66}^{21} ) q^{6} -\zeta_{66}^{3} q^{8} + ( \zeta_{66}^{12} + \zeta_{66}^{28} - \zeta_{66}^{29} ) q^{9} +O(q^{10})\) \( q -\zeta_{66}^{23} q^{2} + ( \zeta_{66}^{14} - \zeta_{66}^{31} ) q^{3} -\zeta_{66}^{13} q^{4} + ( \zeta_{66}^{4} - \zeta_{66}^{21} ) q^{6} -\zeta_{66}^{3} q^{8} + ( \zeta_{66}^{12} + \zeta_{66}^{28} - \zeta_{66}^{29} ) q^{9} + ( -\zeta_{66} - \zeta_{66}^{7} ) q^{11} + ( -\zeta_{66}^{11} - \zeta_{66}^{27} ) q^{12} + \zeta_{66}^{26} q^{16} + ( \zeta_{66}^{18} + \zeta_{66}^{22} ) q^{17} + ( \zeta_{66}^{2} + \zeta_{66}^{18} - \zeta_{66}^{19} ) q^{18} + ( \zeta_{66}^{6} - \zeta_{66}^{25} ) q^{19} + ( \zeta_{66}^{24} + \zeta_{66}^{30} ) q^{22} + ( -\zeta_{66} - \zeta_{66}^{17} ) q^{24} + \zeta_{66}^{30} q^{25} + ( -\zeta_{66}^{9} + \zeta_{66}^{10} + \zeta_{66}^{26} - \zeta_{66}^{27} ) q^{27} + \zeta_{66}^{16} q^{32} + ( -\zeta_{66}^{5} - \zeta_{66}^{15} - \zeta_{66}^{21} + \zeta_{66}^{32} ) q^{33} + ( \zeta_{66}^{8} + \zeta_{66}^{12} ) q^{34} + ( \zeta_{66}^{8} - \zeta_{66}^{9} - \zeta_{66}^{25} ) q^{36} + ( -\zeta_{66}^{15} - \zeta_{66}^{29} ) q^{38} + ( \zeta_{66}^{10} - \zeta_{66}^{19} ) q^{41} + ( -\zeta_{66}^{5} - \zeta_{66}^{13} ) q^{43} + ( \zeta_{66}^{14} + \zeta_{66}^{20} ) q^{44} + ( -\zeta_{66}^{7} + \zeta_{66}^{24} ) q^{48} + \zeta_{66}^{2} q^{49} + \zeta_{66}^{20} q^{50} + ( -\zeta_{66}^{3} + \zeta_{66}^{16} + \zeta_{66}^{20} + \zeta_{66}^{32} ) q^{51} + ( 1 + \zeta_{66}^{16} - \zeta_{66}^{17} + \zeta_{66}^{32} ) q^{54} + ( \zeta_{66}^{4} + \zeta_{66}^{6} + \zeta_{66}^{20} - \zeta_{66}^{23} ) q^{57} + ( -\zeta_{66}^{17} + \zeta_{66}^{22} ) q^{59} + \zeta_{66}^{6} q^{64} + ( -\zeta_{66}^{5} - \zeta_{66}^{11} + \zeta_{66}^{22} + \zeta_{66}^{28} ) q^{66} -\zeta_{66}^{15} q^{67} + ( \zeta_{66}^{2} - \zeta_{66}^{31} ) q^{68} + ( -\zeta_{66}^{15} - \zeta_{66}^{31} + \zeta_{66}^{32} ) q^{72} + ( -\zeta_{66}^{9} + \zeta_{66}^{16} ) q^{73} + ( -\zeta_{66}^{11} + \zeta_{66}^{28} ) q^{75} + ( -\zeta_{66}^{5} - \zeta_{66}^{19} ) q^{76} + ( -\zeta_{66}^{7} + \zeta_{66}^{8} - \zeta_{66}^{23} + \zeta_{66}^{24} - \zeta_{66}^{25} ) q^{81} + ( 1 - \zeta_{66}^{9} ) q^{82} + ( \zeta_{66}^{8} - \zeta_{66}^{11} ) q^{83} + ( -\zeta_{66}^{3} + \zeta_{66}^{28} ) q^{86} + ( \zeta_{66}^{4} + \zeta_{66}^{10} ) q^{88} + ( \zeta_{66}^{4} - \zeta_{66}^{17} ) q^{89} + ( \zeta_{66}^{14} + \zeta_{66}^{30} ) q^{96} + ( \zeta_{66}^{24} - \zeta_{66}^{31} ) q^{97} -\zeta_{66}^{25} q^{98} + ( \zeta_{66}^{2} - \zeta_{66}^{3} - \zeta_{66}^{13} - \zeta_{66}^{19} - \zeta_{66}^{29} + \zeta_{66}^{30} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + q^{2} + 2q^{3} + q^{4} - q^{6} - 2q^{8} + O(q^{10}) \) \( 20q + q^{2} + 2q^{3} + q^{4} - q^{6} - 2q^{8} + 2q^{11} - 12q^{12} + q^{16} - 12q^{17} - q^{19} - 4q^{22} + 2q^{24} - 2q^{25} - 2q^{27} + q^{32} - 2q^{33} - q^{34} - q^{38} + 2q^{41} + 2q^{43} + 2q^{44} - q^{48} + q^{49} + q^{50} + q^{51} + 23q^{54} + q^{57} - 9q^{59} - 2q^{64} - 18q^{66} - 2q^{67} + 2q^{68} - q^{73} - 9q^{75} + 2q^{76} + 2q^{81} + 18q^{82} - 9q^{83} - q^{86} + 2q^{88} + 2q^{89} - q^{96} - q^{97} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/536\mathbb{Z}\right)^\times\).

\(n\) \(135\) \(269\) \(337\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{66}^{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.0475819 + 0.998867i
0.723734 + 0.690079i
−0.327068 0.945001i
0.981929 + 0.189251i
−0.327068 + 0.945001i
0.235759 0.971812i
−0.995472 + 0.0950560i
0.723734 0.690079i
−0.786053 0.618159i
0.235759 + 0.971812i
−0.786053 + 0.618159i
0.580057 0.814576i
0.928368 0.371662i
0.0475819 0.998867i
0.580057 + 0.814576i
−0.888835 0.458227i
−0.995472 0.0950560i
−0.888835 + 0.458227i
0.981929 0.189251i
0.928368 + 0.371662i
−0.888835 0.458227i −1.78153 + 0.523103i 0.580057 + 0.814576i 0 1.82318 + 0.351390i 0 −0.142315 0.989821i 2.05894 1.32320i 0
35.1 0.235759 0.971812i −0.279486 1.94387i −0.888835 0.458227i 0 −1.95496 0.186677i 0 −0.654861 + 0.755750i −2.74102 + 0.804835i 0
83.1 0.981929 + 0.189251i −0.738471 1.61703i 0.928368 + 0.371662i 0 −0.419102 1.72756i 0 0.841254 + 0.540641i −1.41457 + 1.63251i 0
123.1 −0.327068 0.945001i 0.0395325 + 0.0865641i −0.786053 + 0.618159i 0 0.0688733 0.0656706i 0 0.841254 + 0.540641i 0.648930 0.748905i 0
155.1 0.981929 0.189251i −0.738471 + 1.61703i 0.928368 0.371662i 0 −0.419102 + 1.72756i 0 0.841254 0.540641i −1.41457 1.63251i 0
211.1 0.723734 + 0.690079i 0.0930932 + 0.647478i 0.0475819 + 0.998867i 0 −0.379436 + 0.532843i 0 −0.654861 + 0.755750i 0.548932 0.161181i 0
227.1 0.580057 + 0.814576i 1.21769 0.782560i −0.327068 + 0.945001i 0 1.34378 + 0.537970i 0 −0.959493 + 0.281733i 0.454947 0.996196i 0
291.1 0.235759 + 0.971812i −0.279486 + 1.94387i −0.888835 + 0.458227i 0 −1.95496 + 0.186677i 0 −0.654861 0.755750i −2.74102 0.804835i 0
307.1 0.928368 0.371662i −0.759713 0.876756i 0.723734 0.690079i 0 −1.03115 0.531595i 0 0.415415 0.909632i −0.0492216 + 0.342344i 0
315.1 0.723734 0.690079i 0.0930932 0.647478i 0.0475819 0.998867i 0 −0.379436 0.532843i 0 −0.654861 0.755750i 0.548932 + 0.161181i 0
323.1 0.928368 + 0.371662i −0.759713 + 0.876756i 0.723734 + 0.690079i 0 −1.03115 + 0.531595i 0 0.415415 + 0.909632i −0.0492216 0.342344i 0
339.1 −0.995472 0.0950560i 0.396666 + 0.254922i 0.981929 + 0.189251i 0 −0.370638 0.291473i 0 −0.959493 0.281733i −0.323056 0.707394i 0
371.1 −0.786053 0.618159i 1.30379 + 1.50465i 0.235759 + 0.971812i 0 −0.0947329 1.98869i 0 0.415415 0.909632i −0.421801 + 2.93369i 0
395.1 −0.888835 + 0.458227i −1.78153 0.523103i 0.580057 0.814576i 0 1.82318 0.351390i 0 −0.142315 + 0.989821i 2.05894 + 1.32320i 0
419.1 −0.995472 + 0.0950560i 0.396666 0.254922i 0.981929 0.189251i 0 −0.370638 + 0.291473i 0 −0.959493 + 0.281733i −0.323056 + 0.707394i 0
435.1 0.0475819 + 0.998867i 1.50842 0.442913i −0.995472 + 0.0950560i 0 0.514186 + 1.48564i 0 −0.142315 0.989821i 1.23792 0.795563i 0
451.1 0.580057 0.814576i 1.21769 + 0.782560i −0.327068 0.945001i 0 1.34378 0.537970i 0 −0.959493 0.281733i 0.454947 + 0.996196i 0
467.1 0.0475819 0.998867i 1.50842 + 0.442913i −0.995472 0.0950560i 0 0.514186 1.48564i 0 −0.142315 + 0.989821i 1.23792 + 0.795563i 0
475.1 −0.327068 + 0.945001i 0.0395325 0.0865641i −0.786053 0.618159i 0 0.0688733 + 0.0656706i 0 0.841254 0.540641i 0.648930 + 0.748905i 0
523.1 −0.786053 + 0.618159i 1.30379 1.50465i 0.235759 0.971812i 0 −0.0947329 + 1.98869i 0 0.415415 + 0.909632i −0.421801 2.93369i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 523.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
67.g even 33 1 inner
536.ba odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.1.ba.a 20
4.b odd 2 1 2144.1.bu.a 20
8.b even 2 1 2144.1.bu.a 20
8.d odd 2 1 CM 536.1.ba.a 20
67.g even 33 1 inner 536.1.ba.a 20
268.o odd 66 1 2144.1.bu.a 20
536.ba odd 66 1 inner 536.1.ba.a 20
536.bf even 66 1 2144.1.bu.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.1.ba.a 20 1.a even 1 1 trivial
536.1.ba.a 20 8.d odd 2 1 CM
536.1.ba.a 20 67.g even 33 1 inner
536.1.ba.a 20 536.ba odd 66 1 inner
2144.1.bu.a 20 4.b odd 2 1
2144.1.bu.a 20 8.b even 2 1
2144.1.bu.a 20 268.o odd 66 1
2144.1.bu.a 20 536.bf even 66 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(536, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
$3$ \( 1 - 13 T + 157 T^{2} - 565 T^{3} + 1149 T^{4} - 1491 T^{5} + 1613 T^{6} - 767 T^{7} + 768 T^{8} - 1011 T^{9} + 528 T^{10} - 43 T^{11} + 31 T^{12} - 8 T^{13} - 37 T^{14} + 16 T^{15} + 5 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$5$ \( T^{20} \)
$7$ \( T^{20} \)
$11$ \( 1 + 5 T + 11 T^{2} + 62 T^{3} + 178 T^{4} + 77 T^{5} + 49 T^{6} - 19 T^{7} - 154 T^{8} - 130 T^{9} + 164 T^{10} - 49 T^{11} + 11 T^{12} - 40 T^{13} + 42 T^{14} - 11 T^{15} - 5 T^{16} + 8 T^{17} - 2 T^{19} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( 1 + 23 T + 198 T^{2} + 615 T^{3} + 967 T^{4} + 1705 T^{5} + 5193 T^{6} + 12860 T^{7} + 22352 T^{8} + 29402 T^{9} + 31107 T^{10} + 27356 T^{11} + 20317 T^{12} + 12816 T^{13} + 6854 T^{14} + 3080 T^{15} + 1143 T^{16} + 340 T^{17} + 77 T^{18} + 12 T^{19} + T^{20} \)
$19$ \( 1 + 12 T + 44 T^{2} - 375 T^{3} + 725 T^{4} - 1100 T^{5} + 1563 T^{6} - 604 T^{7} + 11 T^{8} + 10 T^{9} + 120 T^{10} + 109 T^{11} - 33 T^{12} - 98 T^{13} + 12 T^{14} - T^{16} - T^{17} + T^{19} + T^{20} \)
$23$ \( T^{20} \)
$29$ \( T^{20} \)
$31$ \( T^{20} \)
$37$ \( T^{20} \)
$41$ \( 1 + 5 T + 22 T^{2} + 29 T^{3} + 24 T^{4} - 231 T^{5} + 71 T^{6} - 8 T^{7} + 198 T^{8} + 24 T^{9} - 243 T^{10} + 116 T^{11} + 121 T^{12} - 95 T^{13} + 20 T^{14} + 22 T^{15} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20} \)
$43$ \( 1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$47$ \( T^{20} \)
$53$ \( T^{20} \)
$59$ \( 1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$71$ \( T^{20} \)
$73$ \( 1 - 10 T + 88 T^{2} + 197 T^{3} + 384 T^{4} + 792 T^{5} + 188 T^{6} - 1451 T^{7} - 792 T^{8} + 417 T^{9} + 483 T^{10} + 65 T^{11} + 88 T^{12} + 34 T^{13} - 43 T^{14} - 44 T^{15} - T^{16} - T^{17} + T^{19} + T^{20} \)
$79$ \( T^{20} \)
$83$ \( 1 + 5 T - 15 T^{3} + 145 T^{4} + 616 T^{5} + 1116 T^{6} + 1631 T^{7} + 2453 T^{8} + 3302 T^{9} + 3750 T^{10} + 3713 T^{11} + 3223 T^{12} + 2424 T^{13} + 1571 T^{14} + 869 T^{15} + 402 T^{16} + 151 T^{17} + 44 T^{18} + 9 T^{19} + T^{20} \)
$89$ \( 1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20} \)
$97$ \( 1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20} \)
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