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Properties

Label 245.4.a.l.1.3
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
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] = [245,4,Mod(1,245)] 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("245.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.4554679514\)
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 1.3
Root \(4.48565\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.48565 q^{2} -0.850238 q^{3} +4.14976 q^{4} -5.00000 q^{5} -2.96363 q^{6} -13.4206 q^{8} -26.2771 q^{9} -17.4283 q^{10} -6.90764 q^{11} -3.52829 q^{12} +22.1364 q^{13} +4.25119 q^{15} -79.9776 q^{16} -88.3030 q^{17} -91.5928 q^{18} -36.9560 q^{19} -20.7488 q^{20} -24.0776 q^{22} -95.5283 q^{23} +11.4107 q^{24} +25.0000 q^{25} +77.1598 q^{26} +45.2982 q^{27} +269.029 q^{29} +14.8182 q^{30} -197.114 q^{31} -171.409 q^{32} +5.87314 q^{33} -307.793 q^{34} -109.044 q^{36} +2.14546 q^{37} -128.816 q^{38} -18.8212 q^{39} +67.1029 q^{40} -174.127 q^{41} -17.0345 q^{43} -28.6650 q^{44} +131.385 q^{45} -332.978 q^{46} +528.029 q^{47} +68.0000 q^{48} +87.1413 q^{50} +75.0786 q^{51} +91.8608 q^{52} -641.114 q^{53} +157.894 q^{54} +34.5382 q^{55} +31.4214 q^{57} +937.742 q^{58} +642.975 q^{59} +17.6414 q^{60} -142.967 q^{61} -687.070 q^{62} +42.3480 q^{64} -110.682 q^{65} +20.4717 q^{66} +478.797 q^{67} -366.436 q^{68} +81.2218 q^{69} +105.550 q^{71} +352.654 q^{72} -986.512 q^{73} +7.47834 q^{74} -21.2560 q^{75} -153.358 q^{76} -65.6042 q^{78} -1099.86 q^{79} +399.888 q^{80} +670.967 q^{81} -606.947 q^{82} +1236.62 q^{83} +441.515 q^{85} -59.3763 q^{86} -228.739 q^{87} +92.7045 q^{88} +711.698 q^{89} +457.964 q^{90} -396.420 q^{92} +167.594 q^{93} +1840.52 q^{94} +184.780 q^{95} +145.739 q^{96} +636.553 q^{97} +181.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9} + 15 q^{10} - 74 q^{11} + 152 q^{12} - 44 q^{13} + 10 q^{15} - 79 q^{16} + 52 q^{17} - 411 q^{18} - 168 q^{19} - 65 q^{20}+ \cdots - 3488 q^{99}+O(q^{100}) \) 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 \(-0.288679\pi\)
0.616182 + 0.787604i \(0.288679\pi\)
\(3\) −0.850238 −0.163628 −0.0818142 − 0.996648i \(-0.526071\pi\)
−0.0818142 + 0.996648i \(0.526071\pi\)
\(4\) 4.14976 0.518720
\(5\) −5.00000 −0.447214
\(6\) −2.96363 −0.201650
\(7\) 0 0
\(8\) −13.4206 −0.593112
\(9\) −26.2771 −0.973226
\(10\) −17.4283 −0.551130
\(11\) −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 \(-0.424119\pi\)
0.236136 + 0.971720i \(0.424119\pi\)
\(14\) 0 0
\(15\) 4.25119 0.0731769
\(16\) −79.9776 −1.24965
\(17\) −88.3030 −1.25980 −0.629901 − 0.776676i \(-0.716904\pi\)
−0.629901 + 0.776676i \(0.716904\pi\)
\(18\) −91.5928 −1.19937
\(19\) −36.9560 −0.446225 −0.223113 − 0.974793i \(-0.571622\pi\)
−0.223113 + 0.974793i \(0.571622\pi\)
\(20\) −20.7488 −0.231979
\(21\) 0 0
\(22\) −24.0776 −0.233335
\(23\) −95.5283 −0.866045 −0.433022 − 0.901383i \(-0.642553\pi\)
−0.433022 + 0.901383i \(0.642553\pi\)
\(24\) 11.4107 0.0970500
\(25\) 25.0000 0.200000
\(26\) 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 \(0.169628\pi\)
\(30\) 14.8182 0.0901805
\(31\) −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 \(-0.498483\pi\)
0.00476638 + 0.999989i \(0.498483\pi\)
\(38\) −128.816 −0.549912
\(39\) −18.8212 −0.0772771
\(40\) 67.1029 0.265248
\(41\) −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 \(-0.509616\pi\)
−0.0302062 + 0.999544i \(0.509616\pi\)
\(44\) −28.6650 −0.0982140
\(45\) 131.385 0.435240
\(46\) −332.978 −1.06728
\(47\) 528.029 1.63874 0.819371 − 0.573264i \(-0.194323\pi\)
0.819371 + 0.573264i \(0.194323\pi\)
\(48\) 68.0000 0.204478
\(49\) 0 0
\(50\) 87.1413 0.246473
\(51\) 75.0786 0.206139
\(52\) 91.8608 0.244977
\(53\) −641.114 −1.66158 −0.830790 − 0.556586i \(-0.812111\pi\)
−0.830790 + 0.556586i \(0.812111\pi\)
\(54\) 157.894 0.397900
\(55\) 34.5382 0.0846750
\(56\) 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 \(0.248970\pi\)
\(60\) 17.6414 0.0379583
\(61\) −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
\(65\) −110.682 −0.211206
\(66\) 20.4717 0.0381802
\(67\) 478.797 0.873050 0.436525 − 0.899692i \(-0.356209\pi\)
0.436525 + 0.899692i \(0.356209\pi\)
\(68\) −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 \(-0.790356\pi\)
−0.790839 + 0.612024i \(0.790356\pi\)
\(74\) 7.47834 0.0117478
\(75\) −21.2560 −0.0327257
\(76\) −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 \(0.786409\pi\)
\(80\) 399.888 0.558860
\(81\) 670.967 0.920394
\(82\) −606.947 −0.817391
\(83\) 1236.62 1.63538 0.817691 − 0.575657i \(-0.195254\pi\)
0.817691 + 0.575657i \(0.195254\pi\)
\(84\) 0 0
\(85\) 441.515 0.563400
\(86\) −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 \(0.360689\pi\)
\(90\) 457.964 0.536374
\(91\) 0 0
\(92\) −396.420 −0.449235
\(93\) 167.594 0.186867
\(94\) 1840.52 2.01953
\(95\) 184.780 0.199558
\(96\) 145.739 0.154942
\(97\) 636.553 0.666311 0.333156 − 0.942872i \(-0.391887\pi\)
0.333156 + 0.942872i \(0.391887\pi\)
\(98\) 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 245.4.a.l.1.3 3
3.2 odd 2 2205.4.a.bm.1.1 3
5.4 even 2 1225.4.a.y.1.1 3
7.2 even 3 245.4.e.n.116.1 6
7.3 odd 6 245.4.e.m.226.1 6
7.4 even 3 245.4.e.n.226.1 6
7.5 odd 6 245.4.e.m.116.1 6
7.6 odd 2 35.4.a.c.1.3 ✓ 3
21.20 even 2 315.4.a.p.1.1 3
28.27 even 2 560.4.a.u.1.2 3
35.13 even 4 175.4.b.e.99.2 6
35.27 even 4 175.4.b.e.99.5 6
35.34 odd 2 175.4.a.f.1.1 3
56.13 odd 2 2240.4.a.bt.1.2 3
56.27 even 2 2240.4.a.bv.1.2 3
105.104 even 2 1575.4.a.ba.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.3 ✓ 3 7.6 odd 2
175.4.a.f.1.1 3 35.34 odd 2
175.4.b.e.99.2 6 35.13 even 4
175.4.b.e.99.5 6 35.27 even 4
245.4.a.l.1.3 3 1.1 even 1 trivial
245.4.e.m.116.1 6 7.5 odd 6
245.4.e.m.226.1 6 7.3 odd 6
245.4.e.n.116.1 6 7.2 even 3
245.4.e.n.226.1 6 7.4 even 3
315.4.a.p.1.1 3 21.20 even 2
560.4.a.u.1.2 3 28.27 even 2
1225.4.a.y.1.1 3 5.4 even 2
1575.4.a.ba.1.3 3 105.104 even 2
2205.4.a.bm.1.1 3 3.2 odd 2
2240.4.a.bt.1.2 3 56.13 odd 2
2240.4.a.bv.1.2 3 56.27 even 2