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Properties

Label 1225.4.a.y.1.1
Level $1225$
Weight $4$
Character 1225.1
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $3$
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] = [1225,4,Mod(1,1225)] 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("1225.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,2,13,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.14360.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.48565\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.48565 q^{2} +0.850238 q^{3} +4.14976 q^{4} -2.96363 q^{6} +13.4206 q^{8} -26.2771 q^{9} -6.90764 q^{11} +3.52829 q^{12} -22.1364 q^{13} -79.9776 q^{16} +88.3030 q^{17} +91.5928 q^{18} -36.9560 q^{19} +24.0776 q^{22} +95.5283 q^{23} +11.4107 q^{24} +77.1598 q^{26} -45.2982 q^{27} +269.029 q^{29} -197.114 q^{31} +171.409 q^{32} -5.87314 q^{33} -307.793 q^{34} -109.044 q^{36} -2.14546 q^{37} +128.816 q^{38} -18.8212 q^{39} -174.127 q^{41} +17.0345 q^{43} -28.6650 q^{44} -332.978 q^{46} -528.029 q^{47} -68.0000 q^{48} +75.0786 q^{51} -91.8608 q^{52} +641.114 q^{53} +157.894 q^{54} -31.4214 q^{57} -937.742 q^{58} +642.975 q^{59} -142.967 q^{61} +687.070 q^{62} +42.3480 q^{64} +20.4717 q^{66} -478.797 q^{67} +366.436 q^{68} +81.2218 q^{69} +105.550 q^{71} -352.654 q^{72} +986.512 q^{73} +7.47834 q^{74} -153.358 q^{76} +65.6042 q^{78} -1099.86 q^{79} +670.967 q^{81} +606.947 q^{82} -1236.62 q^{83} -59.3763 q^{86} +228.739 q^{87} -92.7045 q^{88} +711.698 q^{89} +396.420 q^{92} -167.594 q^{93} +1840.52 q^{94} +145.739 q^{96} -636.553 q^{97} +181.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 2 q^{3} + 13 q^{4} - 24 q^{6} + 15 q^{8} + 81 q^{9} - 74 q^{11} - 152 q^{12} + 44 q^{13} - 79 q^{16} - 52 q^{17} + 411 q^{18} - 168 q^{19} - 184 q^{22} + 124 q^{23} - 420 q^{24} + 446 q^{26}+ \cdots - 3488 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.48565 −1.23236 −0.616182 − 0.787604i \(-0.711321\pi\)
−0.616182 + 0.787604i \(0.711321\pi\)
\(3\) 0.850238 0.163628 0.0818142 − 0.996648i \(-0.473929\pi\)
0.0818142 + 0.996648i \(0.473929\pi\)
\(4\) 4.14976 0.518720
\(5\) 0 0
\(6\) −2.96363 −0.201650
\(7\) 0 0
\(8\) 13.4206 0.593112
\(9\) −26.2771 −0.973226
\(10\) 0 0
\(11\) −6.90764 −0.189339 −0.0946696 − 0.995509i \(-0.530179\pi\)
−0.0946696 + 0.995509i \(0.530179\pi\)
\(12\) 3.52829 0.0848774
\(13\) −22.1364 −0.472272 −0.236136 − 0.971720i \(-0.575881\pi\)
−0.236136 + 0.971720i \(0.575881\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −79.9776 −1.24965
\(17\) 88.3030 1.25980 0.629901 − 0.776676i \(-0.283096\pi\)
0.629901 + 0.776676i \(0.283096\pi\)
\(18\) 91.5928 1.19937
\(19\) −36.9560 −0.446225 −0.223113 − 0.974793i \(-0.571622\pi\)
−0.223113 + 0.974793i \(0.571622\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 24.0776 0.233335
\(23\) 95.5283 0.866045 0.433022 − 0.901383i \(-0.357447\pi\)
0.433022 + 0.901383i \(0.357447\pi\)
\(24\) 11.4107 0.0970500
\(25\) 0 0
\(26\) 77.1598 0.582010
\(27\) −45.2982 −0.322876
\(28\) 0 0
\(29\) 269.029 1.72267 0.861336 − 0.508035i \(-0.169628\pi\)
0.861336 + 0.508035i \(0.169628\pi\)
\(30\) 0 0
\(31\) −197.114 −1.14202 −0.571012 − 0.820942i \(-0.693449\pi\)
−0.571012 + 0.820942i \(0.693449\pi\)
\(32\) 171.409 0.946911
\(33\) −5.87314 −0.0309813
\(34\) −307.793 −1.55253
\(35\) 0 0
\(36\) −109.044 −0.504832
\(37\) −2.14546 −0.00953276 −0.00476638 − 0.999989i \(-0.501517\pi\)
−0.00476638 + 0.999989i \(0.501517\pi\)
\(38\) 128.816 0.549912
\(39\) −18.8212 −0.0772771
\(40\) 0 0
\(41\) −174.127 −0.663271 −0.331636 − 0.943408i \(-0.607600\pi\)
−0.331636 + 0.943408i \(0.607600\pi\)
\(42\) 0 0
\(43\) 17.0345 0.0604125 0.0302062 − 0.999544i \(-0.490384\pi\)
0.0302062 + 0.999544i \(0.490384\pi\)
\(44\) −28.6650 −0.0982140
\(45\) 0 0
\(46\) −332.978 −1.06728
\(47\) −528.029 −1.63874 −0.819371 − 0.573264i \(-0.805677\pi\)
−0.819371 + 0.573264i \(0.805677\pi\)
\(48\) −68.0000 −0.204478
\(49\) 0 0
\(50\) 0 0
\(51\) 75.0786 0.206139
\(52\) −91.8608 −0.244977
\(53\) 641.114 1.66158 0.830790 − 0.556586i \(-0.187889\pi\)
0.830790 + 0.556586i \(0.187889\pi\)
\(54\) 157.894 0.397900
\(55\) 0 0
\(56\) 0 0
\(57\) −31.4214 −0.0730151
\(58\) −937.742 −2.12296
\(59\) 642.975 1.41878 0.709391 − 0.704815i \(-0.248970\pi\)
0.709391 + 0.704815i \(0.248970\pi\)
\(60\) 0 0
\(61\) −142.967 −0.300083 −0.150042 − 0.988680i \(-0.547941\pi\)
−0.150042 + 0.988680i \(0.547941\pi\)
\(62\) 687.070 1.40739
\(63\) 0 0
\(64\) 42.3480 0.0827109
\(65\) 0 0
\(66\) 20.4717 0.0381802
\(67\) −478.797 −0.873050 −0.436525 − 0.899692i \(-0.643791\pi\)
−0.436525 + 0.899692i \(0.643791\pi\)
\(68\) 366.436 0.653484
\(69\) 81.2218 0.141710
\(70\) 0 0
\(71\) 105.550 0.176430 0.0882150 − 0.996101i \(-0.471884\pi\)
0.0882150 + 0.996101i \(0.471884\pi\)
\(72\) −352.654 −0.577232
\(73\) 986.512 1.58168 0.790839 − 0.612024i \(-0.209644\pi\)
0.790839 + 0.612024i \(0.209644\pi\)
\(74\) 7.47834 0.0117478
\(75\) 0 0
\(76\) −153.358 −0.231466
\(77\) 0 0
\(78\) 65.6042 0.0952335
\(79\) −1099.86 −1.56638 −0.783190 − 0.621783i \(-0.786409\pi\)
−0.783190 + 0.621783i \(0.786409\pi\)
\(80\) 0 0
\(81\) 670.967 0.920394
\(82\) 606.947 0.817391
\(83\) −1236.62 −1.63538 −0.817691 − 0.575657i \(-0.804746\pi\)
−0.817691 + 0.575657i \(0.804746\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −59.3763 −0.0744501
\(87\) 228.739 0.281878
\(88\) −92.7045 −0.112299
\(89\) 711.698 0.847638 0.423819 − 0.905747i \(-0.360689\pi\)
0.423819 + 0.905747i \(0.360689\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 396.420 0.449235
\(93\) −167.594 −0.186867
\(94\) 1840.52 2.01953
\(95\) 0 0
\(96\) 145.739 0.154942
\(97\) −636.553 −0.666311 −0.333156 − 0.942872i \(-0.608113\pi\)
−0.333156 + 0.942872i \(0.608113\pi\)
\(98\) 0 0
\(99\) 181.513 0.184270
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 1225.4.a.y.1.1 3
5.4 even 2 245.4.a.l.1.3 3
7.6 odd 2 175.4.a.f.1.1 3
15.14 odd 2 2205.4.a.bm.1.1 3
21.20 even 2 1575.4.a.ba.1.3 3
35.4 even 6 245.4.e.n.226.1 6
35.9 even 6 245.4.e.n.116.1 6
35.13 even 4 175.4.b.e.99.5 6
35.19 odd 6 245.4.e.m.116.1 6
35.24 odd 6 245.4.e.m.226.1 6
35.27 even 4 175.4.b.e.99.2 6
35.34 odd 2 35.4.a.c.1.3 ✓ 3
105.104 even 2 315.4.a.p.1.1 3
140.139 even 2 560.4.a.u.1.2 3
280.69 odd 2 2240.4.a.bt.1.2 3
280.139 even 2 2240.4.a.bv.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.3 ✓ 3 35.34 odd 2
175.4.a.f.1.1 3 7.6 odd 2
175.4.b.e.99.2 6 35.27 even 4
175.4.b.e.99.5 6 35.13 even 4
245.4.a.l.1.3 3 5.4 even 2
245.4.e.m.116.1 6 35.19 odd 6
245.4.e.m.226.1 6 35.24 odd 6
245.4.e.n.116.1 6 35.9 even 6
245.4.e.n.226.1 6 35.4 even 6
315.4.a.p.1.1 3 105.104 even 2
560.4.a.u.1.2 3 140.139 even 2
1225.4.a.y.1.1 3 1.1 even 1 trivial
1575.4.a.ba.1.3 3 21.20 even 2
2205.4.a.bm.1.1 3 15.14 odd 2
2240.4.a.bt.1.2 3 280.69 odd 2
2240.4.a.bv.1.2 3 280.139 even 2