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Properties

Label 475.2.a.e.1.2
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
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] = [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 = [3,-2] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.273891\) 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)
 
\(f(q)\) \(=\) \(q-1.27389 q^{2} +1.65109 q^{3} -0.377203 q^{4} -2.10331 q^{6} -3.65109 q^{7} +3.02830 q^{8} -0.273891 q^{9} +2.65109 q^{11} -0.622797 q^{12} -6.13161 q^{13} +4.65109 q^{14} -3.10331 q^{16} -2.34891 q^{17} +0.348907 q^{18} +1.00000 q^{19} -6.02830 q^{21} -3.37720 q^{22} -5.48052 q^{23} +5.00000 q^{24} +7.81100 q^{26} -5.40550 q^{27} +1.37720 q^{28} +0.651093 q^{29} -6.67939 q^{31} -2.10331 q^{32} +4.37720 q^{33} +2.99225 q^{34} +0.103312 q^{36} +8.70769 q^{37} -1.27389 q^{38} -10.1239 q^{39} +1.93273 q^{41} +7.67939 q^{42} +2.65884 q^{43} -1.00000 q^{44} +6.98158 q^{46} -3.71836 q^{47} -5.12386 q^{48} +6.33048 q^{49} -3.87826 q^{51} +2.31286 q^{52} -13.7544 q^{53} +6.88601 q^{54} -11.0566 q^{56} +1.65109 q^{57} -0.829422 q^{58} -7.84997 q^{59} -1.92498 q^{61} +8.50881 q^{62} +1.00000 q^{63} +8.88601 q^{64} -5.57608 q^{66} +4.44447 q^{67} +0.886014 q^{68} -9.04884 q^{69} +3.54778 q^{71} -0.829422 q^{72} +2.48052 q^{73} -11.0926 q^{74} -0.377203 q^{76} -9.67939 q^{77} +12.8967 q^{78} -15.1599 q^{79} -8.10331 q^{81} -2.46209 q^{82} +14.7282 q^{83} +2.27389 q^{84} -3.38708 q^{86} +1.07502 q^{87} +8.02830 q^{88} -5.06727 q^{89} +22.3871 q^{91} +2.06727 q^{92} -11.0283 q^{93} +4.73678 q^{94} -3.47277 q^{96} -3.22717 q^{97} -8.06434 q^{98} -0.726109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} + 7 q^{14} - 6 q^{16} - 14 q^{17} + 8 q^{18} + 3 q^{19} - 6 q^{21} - 5 q^{22} - 8 q^{23} + 15 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27389 −0.900777 −0.450388 − 0.892833i \(-0.648714\pi\)
−0.450388 + 0.892833i \(0.648714\pi\)
\(3\) 1.65109 0.953259 0.476630 − 0.879104i \(-0.341858\pi\)
0.476630 + 0.879104i \(0.341858\pi\)
\(4\) −0.377203 −0.188601
\(5\) 0 0
\(6\) −2.10331 −0.858674
\(7\) −3.65109 −1.37998 −0.689992 − 0.723817i \(-0.742386\pi\)
−0.689992 + 0.723817i \(0.742386\pi\)
\(8\) 3.02830 1.07066
\(9\) −0.273891 −0.0912969
\(10\) 0 0
\(11\) 2.65109 0.799335 0.399667 − 0.916660i \(-0.369126\pi\)
0.399667 + 0.916660i \(0.369126\pi\)
\(12\) −0.622797 −0.179786
\(13\) −6.13161 −1.70060 −0.850301 − 0.526297i \(-0.823580\pi\)
−0.850301 + 0.526297i \(0.823580\pi\)
\(14\) 4.65109 1.24306
\(15\) 0 0
\(16\) −3.10331 −0.775828
\(17\) −2.34891 −0.569694 −0.284847 − 0.958573i \(-0.591943\pi\)
−0.284847 + 0.958573i \(0.591943\pi\)
\(18\) 0.348907 0.0822381
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −6.02830 −1.31548
\(22\) −3.37720 −0.720022
\(23\) −5.48052 −1.14277 −0.571383 − 0.820683i \(-0.693593\pi\)
−0.571383 + 0.820683i \(0.693593\pi\)
\(24\) 5.00000 1.02062
\(25\) 0 0
\(26\) 7.81100 1.53186
\(27\) −5.40550 −1.04029
\(28\) 1.37720 0.260267
\(29\) 0.651093 0.120905 0.0604525 − 0.998171i \(-0.480746\pi\)
0.0604525 + 0.998171i \(0.480746\pi\)
\(30\) 0 0
\(31\) −6.67939 −1.19965 −0.599827 − 0.800130i \(-0.704764\pi\)
−0.599827 + 0.800130i \(0.704764\pi\)
\(32\) −2.10331 −0.371817
\(33\) 4.37720 0.761973
\(34\) 2.99225 0.513167
\(35\) 0 0
\(36\) 0.103312 0.0172187
\(37\) 8.70769 1.43153 0.715767 − 0.698339i \(-0.246077\pi\)
0.715767 + 0.698339i \(0.246077\pi\)
\(38\) −1.27389 −0.206652
\(39\) −10.1239 −1.62111
\(40\) 0 0
\(41\) 1.93273 0.301842 0.150921 − 0.988546i \(-0.451776\pi\)
0.150921 + 0.988546i \(0.451776\pi\)
\(42\) 7.67939 1.18496
\(43\) 2.65884 0.405470 0.202735 − 0.979234i \(-0.435017\pi\)
0.202735 + 0.979234i \(0.435017\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.98158 1.02938
\(47\) −3.71836 −0.542378 −0.271189 − 0.962526i \(-0.587417\pi\)
−0.271189 + 0.962526i \(0.587417\pi\)
\(48\) −5.12386 −0.739565
\(49\) 6.33048 0.904355
\(50\) 0 0
\(51\) −3.87826 −0.543066
\(52\) 2.31286 0.320736
\(53\) −13.7544 −1.88931 −0.944656 − 0.328061i \(-0.893605\pi\)
−0.944656 + 0.328061i \(0.893605\pi\)
\(54\) 6.88601 0.937068
\(55\) 0 0
\(56\) −11.0566 −1.47750
\(57\) 1.65109 0.218693
\(58\) −0.829422 −0.108908
\(59\) −7.84997 −1.02198 −0.510989 − 0.859587i \(-0.670721\pi\)
−0.510989 + 0.859587i \(0.670721\pi\)
\(60\) 0 0
\(61\) −1.92498 −0.246469 −0.123234 − 0.992378i \(-0.539327\pi\)
−0.123234 + 0.992378i \(0.539327\pi\)
\(62\) 8.50881 1.08062
\(63\) 1.00000 0.125988
\(64\) 8.88601 1.11075
\(65\) 0 0
\(66\) −5.57608 −0.686368
\(67\) 4.44447 0.542978 0.271489 − 0.962442i \(-0.412484\pi\)
0.271489 + 0.962442i \(0.412484\pi\)
\(68\) 0.886014 0.107445
\(69\) −9.04884 −1.08935
\(70\) 0 0
\(71\) 3.54778 0.421044 0.210522 − 0.977589i \(-0.432484\pi\)
0.210522 + 0.977589i \(0.432484\pi\)
\(72\) −0.829422 −0.0977483
\(73\) 2.48052 0.290322 0.145161 − 0.989408i \(-0.453630\pi\)
0.145161 + 0.989408i \(0.453630\pi\)
\(74\) −11.0926 −1.28949
\(75\) 0 0
\(76\) −0.377203 −0.0432681
\(77\) −9.67939 −1.10307
\(78\) 12.8967 1.46026
\(79\) −15.1599 −1.70562 −0.852811 − 0.522219i \(-0.825104\pi\)
−0.852811 + 0.522219i \(0.825104\pi\)
\(80\) 0 0
\(81\) −8.10331 −0.900368
\(82\) −2.46209 −0.271893
\(83\) 14.7282 1.61663 0.808317 − 0.588748i \(-0.200379\pi\)
0.808317 + 0.588748i \(0.200379\pi\)
\(84\) 2.27389 0.248102
\(85\) 0 0
\(86\) −3.38708 −0.365238
\(87\) 1.07502 0.115254
\(88\) 8.02830 0.855819
\(89\) −5.06727 −0.537129 −0.268565 − 0.963262i \(-0.586549\pi\)
−0.268565 + 0.963262i \(0.586549\pi\)
\(90\) 0 0
\(91\) 22.3871 2.34680
\(92\) 2.06727 0.215527
\(93\) −11.0283 −1.14358
\(94\) 4.73678 0.488562
\(95\) 0 0
\(96\) −3.47277 −0.354438
\(97\) −3.22717 −0.327670 −0.163835 − 0.986488i \(-0.552386\pi\)
−0.163835 + 0.986488i \(0.552386\pi\)
\(98\) −8.06434 −0.814622
\(99\) −0.726109 −0.0729767
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.e.1.2 ✓ 3
3.2 odd 2 4275.2.a.bm.1.2 3
4.3 odd 2 7600.2.a.cc.1.1 3
5.2 odd 4 475.2.b.b.324.3 6
5.3 odd 4 475.2.b.b.324.4 6
5.4 even 2 475.2.a.g.1.2 yes 3
15.14 odd 2 4275.2.a.ba.1.2 3
19.18 odd 2 9025.2.a.bc.1.2 3
20.19 odd 2 7600.2.a.bh.1.3 3
95.94 odd 2 9025.2.a.y.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.2 ✓ 3 1.1 even 1 trivial
475.2.a.g.1.2 yes 3 5.4 even 2
475.2.b.b.324.3 6 5.2 odd 4
475.2.b.b.324.4 6 5.3 odd 4
4275.2.a.ba.1.2 3 15.14 odd 2
4275.2.a.bm.1.2 3 3.2 odd 2
7600.2.a.bh.1.3 3 20.19 odd 2
7600.2.a.cc.1.1 3 4.3 odd 2
9025.2.a.y.1.2 3 95.94 odd 2
9025.2.a.bc.1.2 3 19.18 odd 2