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Properties

Label 475.2.a.j.1.6
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [475,2,Mod(1,475)] 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("475.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.66064384.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 9x^{4} + 13x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.30397\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.41987 q^{2} -0.537080 q^{3} +3.85577 q^{4} -1.29966 q^{6} +3.18676 q^{7} +4.49073 q^{8} -2.71155 q^{9} +4.15544 q^{11} -2.07086 q^{12} -2.07086 q^{13} +7.71155 q^{14} +3.15544 q^{16} -5.79470 q^{17} -6.56159 q^{18} -1.00000 q^{19} -1.71155 q^{21} +10.0556 q^{22} +2.60794 q^{23} -2.41188 q^{24} -5.01121 q^{26} +3.06756 q^{27} +12.2874 q^{28} +6.00000 q^{29} +2.59933 q^{31} -1.34571 q^{32} -2.23180 q^{33} -14.0224 q^{34} -10.4551 q^{36} -4.30266 q^{37} -2.41987 q^{38} +1.11222 q^{39} -0.599328 q^{41} -4.14172 q^{42} -3.18676 q^{43} +16.0224 q^{44} +6.31087 q^{46} -11.7086 q^{47} -1.69472 q^{48} +3.15544 q^{49} +3.11222 q^{51} -7.98476 q^{52} -11.7503 q^{53} +7.42309 q^{54} +14.3109 q^{56} +0.537080 q^{57} +14.5192 q^{58} -1.71155 q^{59} -8.75476 q^{61} +6.29004 q^{62} -8.64104 q^{63} -9.56732 q^{64} -5.40067 q^{66} -4.76228 q^{67} -22.3430 q^{68} -1.40067 q^{69} +13.7115 q^{71} -12.1768 q^{72} +2.72714 q^{73} -10.4119 q^{74} -3.85577 q^{76} +13.2424 q^{77} +2.69142 q^{78} -1.40067 q^{79} +6.48711 q^{81} -1.45030 q^{82} +7.07154 q^{83} -6.59933 q^{84} -7.71155 q^{86} -3.22248 q^{87} +18.6609 q^{88} +16.5353 q^{89} -6.59933 q^{91} +10.0556 q^{92} -1.39605 q^{93} -28.3333 q^{94} +0.722754 q^{96} -2.07086 q^{97} +7.63575 q^{98} -11.2677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51}+ \cdots - 30 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41987 1.71111 0.855553 − 0.517715i \(-0.173217\pi\)
0.855553 + 0.517715i \(0.173217\pi\)
\(3\) −0.537080 −0.310083 −0.155042 − 0.987908i \(-0.549551\pi\)
−0.155042 + 0.987908i \(0.549551\pi\)
\(4\) 3.85577 1.92789
\(5\) 0 0
\(6\) −1.29966 −0.530586
\(7\) 3.18676 1.20448 0.602241 − 0.798314i \(-0.294275\pi\)
0.602241 + 0.798314i \(0.294275\pi\)
\(8\) 4.49073 1.58771
\(9\) −2.71155 −0.903848
\(10\) 0 0
\(11\) 4.15544 1.25291 0.626456 − 0.779457i \(-0.284505\pi\)
0.626456 + 0.779457i \(0.284505\pi\)
\(12\) −2.07086 −0.597805
\(13\) −2.07086 −0.574353 −0.287176 − 0.957878i \(-0.592717\pi\)
−0.287176 + 0.957878i \(0.592717\pi\)
\(14\) 7.71155 2.06100
\(15\) 0 0
\(16\) 3.15544 0.788859
\(17\) −5.79470 −1.40542 −0.702710 − 0.711476i \(-0.748027\pi\)
−0.702710 + 0.711476i \(0.748027\pi\)
\(18\) −6.56159 −1.54658
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.71155 −0.373490
\(22\) 10.0556 2.14386
\(23\) 2.60794 0.543793 0.271896 − 0.962327i \(-0.412349\pi\)
0.271896 + 0.962327i \(0.412349\pi\)
\(24\) −2.41188 −0.492323
\(25\) 0 0
\(26\) −5.01121 −0.982779
\(27\) 3.06756 0.590352
\(28\) 12.2874 2.32210
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.59933 0.466853 0.233427 − 0.972374i \(-0.425006\pi\)
0.233427 + 0.972374i \(0.425006\pi\)
\(32\) −1.34571 −0.237890
\(33\) −2.23180 −0.388507
\(34\) −14.0224 −2.40482
\(35\) 0 0
\(36\) −10.4551 −1.74252
\(37\) −4.30266 −0.707353 −0.353677 − 0.935368i \(-0.615069\pi\)
−0.353677 + 0.935368i \(0.615069\pi\)
\(38\) −2.41987 −0.392555
\(39\) 1.11222 0.178097
\(40\) 0 0
\(41\) −0.599328 −0.0935993 −0.0467997 − 0.998904i \(-0.514902\pi\)
−0.0467997 + 0.998904i \(0.514902\pi\)
\(42\) −4.14172 −0.639081
\(43\) −3.18676 −0.485976 −0.242988 − 0.970029i \(-0.578128\pi\)
−0.242988 + 0.970029i \(0.578128\pi\)
\(44\) 16.0224 2.41547
\(45\) 0 0
\(46\) 6.31087 0.930487
\(47\) −11.7086 −1.70787 −0.853937 − 0.520376i \(-0.825792\pi\)
−0.853937 + 0.520376i \(0.825792\pi\)
\(48\) −1.69472 −0.244612
\(49\) 3.15544 0.450777
\(50\) 0 0
\(51\) 3.11222 0.435798
\(52\) −7.98476 −1.10729
\(53\) −11.7503 −1.61403 −0.807017 − 0.590529i \(-0.798919\pi\)
−0.807017 + 0.590529i \(0.798919\pi\)
\(54\) 7.42309 1.01015
\(55\) 0 0
\(56\) 14.3109 1.91237
\(57\) 0.537080 0.0711380
\(58\) 14.5192 1.90647
\(59\) −1.71155 −0.222824 −0.111412 − 0.993774i \(-0.535537\pi\)
−0.111412 + 0.993774i \(0.535537\pi\)
\(60\) 0 0
\(61\) −8.75476 −1.12093 −0.560466 − 0.828177i \(-0.689378\pi\)
−0.560466 + 0.828177i \(0.689378\pi\)
\(62\) 6.29004 0.798836
\(63\) −8.64104 −1.08867
\(64\) −9.56732 −1.19591
\(65\) 0 0
\(66\) −5.40067 −0.664777
\(67\) −4.76228 −0.581805 −0.290902 − 0.956753i \(-0.593956\pi\)
−0.290902 + 0.956753i \(0.593956\pi\)
\(68\) −22.3430 −2.70949
\(69\) −1.40067 −0.168621
\(70\) 0 0
\(71\) 13.7115 1.62726 0.813631 − 0.581382i \(-0.197488\pi\)
0.813631 + 0.581382i \(0.197488\pi\)
\(72\) −12.1768 −1.43505
\(73\) 2.72714 0.319188 0.159594 − 0.987183i \(-0.448982\pi\)
0.159594 + 0.987183i \(0.448982\pi\)
\(74\) −10.4119 −1.21036
\(75\) 0 0
\(76\) −3.85577 −0.442287
\(77\) 13.2424 1.50911
\(78\) 2.69142 0.304743
\(79\) −1.40067 −0.157588 −0.0787939 − 0.996891i \(-0.525107\pi\)
−0.0787939 + 0.996891i \(0.525107\pi\)
\(80\) 0 0
\(81\) 6.48711 0.720790
\(82\) −1.45030 −0.160158
\(83\) 7.07154 0.776203 0.388101 − 0.921617i \(-0.373131\pi\)
0.388101 + 0.921617i \(0.373131\pi\)
\(84\) −6.59933 −0.720046
\(85\) 0 0
\(86\) −7.71155 −0.831557
\(87\) −3.22248 −0.345486
\(88\) 18.6609 1.98926
\(89\) 16.5353 1.75274 0.876370 − 0.481639i \(-0.159958\pi\)
0.876370 + 0.481639i \(0.159958\pi\)
\(90\) 0 0
\(91\) −6.59933 −0.691798
\(92\) 10.0556 1.04837
\(93\) −1.39605 −0.144763
\(94\) −28.3333 −2.92236
\(95\) 0 0
\(96\) 0.722754 0.0737658
\(97\) −2.07086 −0.210264 −0.105132 − 0.994458i \(-0.533526\pi\)
−0.105132 + 0.994458i \(0.533526\pi\)
\(98\) 7.63575 0.771327
\(99\) −11.2677 −1.13244
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 475.2.a.j.1.6 6
3.2 odd 2 4275.2.a.br.1.1 6
4.3 odd 2 7600.2.a.ck.1.4 6
5.2 odd 4 95.2.b.b.39.6 yes 6
5.3 odd 4 95.2.b.b.39.1 ✓ 6
5.4 even 2 inner 475.2.a.j.1.1 6
15.2 even 4 855.2.c.d.514.1 6
15.8 even 4 855.2.c.d.514.6 6
15.14 odd 2 4275.2.a.br.1.6 6
19.18 odd 2 9025.2.a.bx.1.1 6
20.3 even 4 1520.2.d.h.609.4 6
20.7 even 4 1520.2.d.h.609.3 6
20.19 odd 2 7600.2.a.ck.1.3 6
95.18 even 4 1805.2.b.e.1084.6 6
95.37 even 4 1805.2.b.e.1084.1 6
95.94 odd 2 9025.2.a.bx.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.1 ✓ 6 5.3 odd 4
95.2.b.b.39.6 yes 6 5.2 odd 4
475.2.a.j.1.1 6 5.4 even 2 inner
475.2.a.j.1.6 6 1.1 even 1 trivial
855.2.c.d.514.1 6 15.2 even 4
855.2.c.d.514.6 6 15.8 even 4
1520.2.d.h.609.3 6 20.7 even 4
1520.2.d.h.609.4 6 20.3 even 4
1805.2.b.e.1084.1 6 95.37 even 4
1805.2.b.e.1084.6 6 95.18 even 4
4275.2.a.br.1.1 6 3.2 odd 2
4275.2.a.br.1.6 6 15.14 odd 2
7600.2.a.ck.1.3 6 20.19 odd 2
7600.2.a.ck.1.4 6 4.3 odd 2
9025.2.a.bx.1.1 6 19.18 odd 2
9025.2.a.bx.1.6 6 95.94 odd 2