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Properties

Label 2550.2.a.bi.1.1
Level $2550$
Weight $2$
Character 2550.1
Self dual yes
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2550,2,Mod(1,2550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2550.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,2,2,0,-2,0,-2,2,0,0,2,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 2550.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.44949 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.89898 q^{11} +1.00000 q^{12} +6.00000 q^{13} +2.44949 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.89898 q^{19} -2.44949 q^{21} +4.89898 q^{22} +6.44949 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} -2.44949 q^{28} -9.34847 q^{29} -6.44949 q^{31} -1.00000 q^{32} -4.89898 q^{33} -1.00000 q^{34} +1.00000 q^{36} -0.449490 q^{37} -6.89898 q^{38} +6.00000 q^{39} -1.10102 q^{41} +2.44949 q^{42} -2.89898 q^{43} -4.89898 q^{44} -6.44949 q^{46} +4.89898 q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000 q^{51} +6.00000 q^{52} -1.10102 q^{53} -1.00000 q^{54} +2.44949 q^{56} +6.89898 q^{57} +9.34847 q^{58} +5.79796 q^{59} +13.3485 q^{61} +6.44949 q^{62} -2.44949 q^{63} +1.00000 q^{64} +4.89898 q^{66} +4.00000 q^{67} +1.00000 q^{68} +6.44949 q^{69} +2.44949 q^{71} -1.00000 q^{72} +14.8990 q^{73} +0.449490 q^{74} +6.89898 q^{76} +12.0000 q^{77} -6.00000 q^{78} -1.55051 q^{79} +1.00000 q^{81} +1.10102 q^{82} +2.89898 q^{83} -2.44949 q^{84} +2.89898 q^{86} -9.34847 q^{87} +4.89898 q^{88} -1.79796 q^{89} -14.6969 q^{91} +6.44949 q^{92} -6.44949 q^{93} -4.89898 q^{94} -1.00000 q^{96} -3.79796 q^{97} +1.00000 q^{98} -4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 12 q^{13} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 4 q^{19} + 8 q^{23} - 2 q^{24} - 12 q^{26} + 2 q^{27} - 4 q^{29} - 8 q^{31} - 2 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −2.44949 −0.925820 −0.462910 − 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 − 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 2.44949 0.654654
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 6.89898 1.58273 0.791367 − 0.611341i \(-0.209370\pi\)
0.791367 + 0.611341i \(0.209370\pi\)
\(20\) 0 0
\(21\) −2.44949 −0.534522
\(22\) 4.89898 1.04447
\(23\) 6.44949 1.34481 0.672406 − 0.740183i \(-0.265261\pi\)
0.672406 + 0.740183i \(0.265261\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) −2.44949 −0.462910
\(29\) −9.34847 −1.73597 −0.867984 − 0.496593i \(-0.834584\pi\)
−0.867984 + 0.496593i \(0.834584\pi\)
\(30\) 0 0
\(31\) −6.44949 −1.15836 −0.579181 − 0.815199i \(-0.696628\pi\)
−0.579181 + 0.815199i \(0.696628\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.89898 −0.852803
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.449490 −0.0738957 −0.0369478 − 0.999317i \(-0.511764\pi\)
−0.0369478 + 0.999317i \(0.511764\pi\)
\(38\) −6.89898 −1.11916
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −1.10102 −0.171951 −0.0859753 − 0.996297i \(-0.527401\pi\)
−0.0859753 + 0.996297i \(0.527401\pi\)
\(42\) 2.44949 0.377964
\(43\) −2.89898 −0.442090 −0.221045 − 0.975264i \(-0.570947\pi\)
−0.221045 + 0.975264i \(0.570947\pi\)
\(44\) −4.89898 −0.738549
\(45\) 0 0
\(46\) −6.44949 −0.950925
\(47\) 4.89898 0.714590 0.357295 − 0.933992i \(-0.383699\pi\)
0.357295 + 0.933992i \(0.383699\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 6.00000 0.832050
\(53\) −1.10102 −0.151237 −0.0756184 − 0.997137i \(-0.524093\pi\)
−0.0756184 + 0.997137i \(0.524093\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.44949 0.327327
\(57\) 6.89898 0.913792
\(58\) 9.34847 1.22751
\(59\) 5.79796 0.754830 0.377415 − 0.926044i \(-0.376813\pi\)
0.377415 + 0.926044i \(0.376813\pi\)
\(60\) 0 0
\(61\) 13.3485 1.70910 0.854548 − 0.519372i \(-0.173834\pi\)
0.854548 + 0.519372i \(0.173834\pi\)
\(62\) 6.44949 0.819086
\(63\) −2.44949 −0.308607
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.89898 0.603023
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 1.00000 0.121268
\(69\) 6.44949 0.776427
\(70\) 0 0
\(71\) 2.44949 0.290701 0.145350 − 0.989380i \(-0.453569\pi\)
0.145350 + 0.989380i \(0.453569\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.8990 1.74379 0.871897 − 0.489690i \(-0.162890\pi\)
0.871897 + 0.489690i \(0.162890\pi\)
\(74\) 0.449490 0.0522521
\(75\) 0 0
\(76\) 6.89898 0.791367
\(77\) 12.0000 1.36753
\(78\) −6.00000 −0.679366
\(79\) −1.55051 −0.174446 −0.0872230 − 0.996189i \(-0.527799\pi\)
−0.0872230 + 0.996189i \(0.527799\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.10102 0.121587
\(83\) 2.89898 0.318204 0.159102 − 0.987262i \(-0.449140\pi\)
0.159102 + 0.987262i \(0.449140\pi\)
\(84\) −2.44949 −0.267261
\(85\) 0 0
\(86\) 2.89898 0.312605
\(87\) −9.34847 −1.00226
\(88\) 4.89898 0.522233
\(89\) −1.79796 −0.190583 −0.0952916 − 0.995449i \(-0.530378\pi\)
−0.0952916 + 0.995449i \(0.530378\pi\)
\(90\) 0 0
\(91\) −14.6969 −1.54066
\(92\) 6.44949 0.672406
\(93\) −6.44949 −0.668781
\(94\) −4.89898 −0.505291
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −3.79796 −0.385624 −0.192812 − 0.981236i \(-0.561761\pi\)
−0.192812 + 0.981236i \(0.561761\pi\)
\(98\) 1.00000 0.101015
\(99\) −4.89898 −0.492366
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2550.2.a.bi.1.1 2
3.2 odd 2 7650.2.a.dg.1.1 2
5.2 odd 4 510.2.d.c.409.1 ✓ 4
5.3 odd 4 510.2.d.c.409.3 yes 4
5.4 even 2 2550.2.a.bj.1.2 2
15.2 even 4 1530.2.d.e.919.4 4
15.8 even 4 1530.2.d.e.919.2 4
15.14 odd 2 7650.2.a.ct.1.2 2
20.3 even 4 4080.2.m.o.2449.1 4
20.7 even 4 4080.2.m.o.2449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.d.c.409.1 ✓ 4 5.2 odd 4
510.2.d.c.409.3 yes 4 5.3 odd 4
1530.2.d.e.919.2 4 15.8 even 4
1530.2.d.e.919.4 4 15.2 even 4
2550.2.a.bi.1.1 2 1.1 even 1 trivial
2550.2.a.bj.1.2 2 5.4 even 2
4080.2.m.o.2449.1 4 20.3 even 4
4080.2.m.o.2449.3 4 20.7 even 4
7650.2.a.ct.1.2 2 15.14 odd 2
7650.2.a.dg.1.1 2 3.2 odd 2