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Properties

Label 475.2.b.b.324.3
Level $475$
Weight $2$
Character 475.324
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(324,475)] 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("475.324"); 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([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-8,0,-6] 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: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.3
Root \(-2.65109i\) of defining polynomial
Character \(\chi\) \(=\) 475.324
Dual form 475.2.b.b.324.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.27389i q^{2} -1.65109i q^{3} +0.377203 q^{4} -2.10331 q^{6} -3.65109i q^{7} -3.02830i q^{8} +0.273891 q^{9} +2.65109 q^{11} -0.622797i q^{12} +6.13161i q^{13} -4.65109 q^{14} -3.10331 q^{16} -2.34891i q^{17} -0.348907i q^{18} -1.00000 q^{19} -6.02830 q^{21} -3.37720i q^{22} +5.48052i q^{23} -5.00000 q^{24} +7.81100 q^{26} -5.40550i q^{27} -1.37720i q^{28} -0.651093 q^{29} -6.67939 q^{31} -2.10331i q^{32} -4.37720i q^{33} -2.99225 q^{34} +0.103312 q^{36} +8.70769i q^{37} +1.27389i q^{38} +10.1239 q^{39} +1.93273 q^{41} +7.67939i q^{42} -2.65884i q^{43} +1.00000 q^{44} +6.98158 q^{46} -3.71836i q^{47} +5.12386i q^{48} -6.33048 q^{49} -3.87826 q^{51} +2.31286i q^{52} +13.7544i q^{53} -6.88601 q^{54} -11.0566 q^{56} +1.65109i q^{57} +0.829422i q^{58} +7.84997 q^{59} -1.92498 q^{61} +8.50881i q^{62} -1.00000i q^{63} -8.88601 q^{64} -5.57608 q^{66} +4.44447i q^{67} -0.886014i q^{68} +9.04884 q^{69} +3.54778 q^{71} -0.829422i q^{72} -2.48052i q^{73} +11.0926 q^{74} -0.377203 q^{76} -9.67939i q^{77} -12.8967i q^{78} +15.1599 q^{79} -8.10331 q^{81} -2.46209i q^{82} -14.7282i q^{83} -2.27389 q^{84} -3.38708 q^{86} +1.07502i q^{87} -8.02830i q^{88} +5.06727 q^{89} +22.3871 q^{91} +2.06727i q^{92} +11.0283i q^{93} -4.73678 q^{94} -3.47277 q^{96} -3.22717i q^{97} +8.06434i q^{98} +0.726109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.27389i − 0.900777i −0.892833 − 0.450388i \(-0.851286\pi\)
0.892833 − 0.450388i \(-0.148714\pi\)
\(3\) − 1.65109i − 0.953259i −0.879104 − 0.476630i \(-0.841858\pi\)
0.879104 − 0.476630i \(-0.158142\pi\)
\(4\) 0.377203 0.188601
\(5\) 0 0
\(6\) −2.10331 −0.858674
\(7\) − 3.65109i − 1.37998i −0.723817 − 0.689992i \(-0.757614\pi\)
0.723817 − 0.689992i \(-0.242386\pi\)
\(8\) − 3.02830i − 1.07066i
\(9\) 0.273891 0.0912969
\(10\) 0 0
\(11\) 2.65109 0.799335 0.399667 − 0.916660i \(-0.369126\pi\)
0.399667 + 0.916660i \(0.369126\pi\)
\(12\) − 0.622797i − 0.179786i
\(13\) 6.13161i 1.70060i 0.526297 + 0.850301i \(0.323580\pi\)
−0.526297 + 0.850301i \(0.676420\pi\)
\(14\) −4.65109 −1.24306
\(15\) 0 0
\(16\) −3.10331 −0.775828
\(17\) − 2.34891i − 0.569694i −0.958573 − 0.284847i \(-0.908057\pi\)
0.958573 − 0.284847i \(-0.0919427\pi\)
\(18\) − 0.348907i − 0.0822381i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −6.02830 −1.31548
\(22\) − 3.37720i − 0.720022i
\(23\) 5.48052i 1.14277i 0.820683 + 0.571383i \(0.193593\pi\)
−0.820683 + 0.571383i \(0.806407\pi\)
\(24\) −5.00000 −1.02062
\(25\) 0 0
\(26\) 7.81100 1.53186
\(27\) − 5.40550i − 1.04029i
\(28\) − 1.37720i − 0.260267i
\(29\) −0.651093 −0.120905 −0.0604525 − 0.998171i \(-0.519254\pi\)
−0.0604525 + 0.998171i \(0.519254\pi\)
\(30\) 0 0
\(31\) −6.67939 −1.19965 −0.599827 − 0.800130i \(-0.704764\pi\)
−0.599827 + 0.800130i \(0.704764\pi\)
\(32\) − 2.10331i − 0.371817i
\(33\) − 4.37720i − 0.761973i
\(34\) −2.99225 −0.513167
\(35\) 0 0
\(36\) 0.103312 0.0172187
\(37\) 8.70769i 1.43153i 0.698339 + 0.715767i \(0.253923\pi\)
−0.698339 + 0.715767i \(0.746077\pi\)
\(38\) 1.27389i 0.206652i
\(39\) 10.1239 1.62111
\(40\) 0 0
\(41\) 1.93273 0.301842 0.150921 − 0.988546i \(-0.451776\pi\)
0.150921 + 0.988546i \(0.451776\pi\)
\(42\) 7.67939i 1.18496i
\(43\) − 2.65884i − 0.405470i −0.979234 − 0.202735i \(-0.935017\pi\)
0.979234 − 0.202735i \(-0.0649830\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.98158 1.02938
\(47\) − 3.71836i − 0.542378i −0.962526 − 0.271189i \(-0.912583\pi\)
0.962526 − 0.271189i \(-0.0874169\pi\)
\(48\) 5.12386i 0.739565i
\(49\) −6.33048 −0.904355
\(50\) 0 0
\(51\) −3.87826 −0.543066
\(52\) 2.31286i 0.320736i
\(53\) 13.7544i 1.88931i 0.328061 + 0.944656i \(0.393605\pi\)
−0.328061 + 0.944656i \(0.606395\pi\)
\(54\) −6.88601 −0.937068
\(55\) 0 0
\(56\) −11.0566 −1.47750
\(57\) 1.65109i 0.218693i
\(58\) 0.829422i 0.108908i
\(59\) 7.84997 1.02198 0.510989 − 0.859587i \(-0.329279\pi\)
0.510989 + 0.859587i \(0.329279\pi\)
\(60\) 0 0
\(61\) −1.92498 −0.246469 −0.123234 − 0.992378i \(-0.539327\pi\)
−0.123234 + 0.992378i \(0.539327\pi\)
\(62\) 8.50881i 1.08062i
\(63\) − 1.00000i − 0.125988i
\(64\) −8.88601 −1.11075
\(65\) 0 0
\(66\) −5.57608 −0.686368
\(67\) 4.44447i 0.542978i 0.962442 + 0.271489i \(0.0875161\pi\)
−0.962442 + 0.271489i \(0.912484\pi\)
\(68\) − 0.886014i − 0.107445i
\(69\) 9.04884 1.08935
\(70\) 0 0
\(71\) 3.54778 0.421044 0.210522 − 0.977589i \(-0.432484\pi\)
0.210522 + 0.977589i \(0.432484\pi\)
\(72\) − 0.829422i − 0.0977483i
\(73\) − 2.48052i − 0.290322i −0.989408 − 0.145161i \(-0.953630\pi\)
0.989408 − 0.145161i \(-0.0463701\pi\)
\(74\) 11.0926 1.28949
\(75\) 0 0
\(76\) −0.377203 −0.0432681
\(77\) − 9.67939i − 1.10307i
\(78\) − 12.8967i − 1.46026i
\(79\) 15.1599 1.70562 0.852811 − 0.522219i \(-0.174896\pi\)
0.852811 + 0.522219i \(0.174896\pi\)
\(80\) 0 0
\(81\) −8.10331 −0.900368
\(82\) − 2.46209i − 0.271893i
\(83\) − 14.7282i − 1.61663i −0.588748 − 0.808317i \(-0.700379\pi\)
0.588748 − 0.808317i \(-0.299621\pi\)
\(84\) −2.27389 −0.248102
\(85\) 0 0
\(86\) −3.38708 −0.365238
\(87\) 1.07502i 0.115254i
\(88\) − 8.02830i − 0.855819i
\(89\) 5.06727 0.537129 0.268565 − 0.963262i \(-0.413451\pi\)
0.268565 + 0.963262i \(0.413451\pi\)
\(90\) 0 0
\(91\) 22.3871 2.34680
\(92\) 2.06727i 0.215527i
\(93\) 11.0283i 1.14358i
\(94\) −4.73678 −0.488562
\(95\) 0 0
\(96\) −3.47277 −0.354438
\(97\) − 3.22717i − 0.327670i −0.986488 − 0.163835i \(-0.947614\pi\)
0.986488 − 0.163835i \(-0.0523864\pi\)
\(98\) 8.06434i 0.814622i
\(99\) 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.b.b.324.3 6
5.2 odd 4 475.2.a.g.1.2 yes 3
5.3 odd 4 475.2.a.e.1.2 ✓ 3
5.4 even 2 inner 475.2.b.b.324.4 6
15.2 even 4 4275.2.a.ba.1.2 3
15.8 even 4 4275.2.a.bm.1.2 3
20.3 even 4 7600.2.a.cc.1.1 3
20.7 even 4 7600.2.a.bh.1.3 3
95.18 even 4 9025.2.a.bc.1.2 3
95.37 even 4 9025.2.a.y.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.2 ✓ 3 5.3 odd 4
475.2.a.g.1.2 yes 3 5.2 odd 4
475.2.b.b.324.3 6 1.1 even 1 trivial
475.2.b.b.324.4 6 5.4 even 2 inner
4275.2.a.ba.1.2 3 15.2 even 4
4275.2.a.bm.1.2 3 15.8 even 4
7600.2.a.bh.1.3 3 20.7 even 4
7600.2.a.cc.1.1 3 20.3 even 4
9025.2.a.y.1.2 3 95.37 even 4
9025.2.a.bc.1.2 3 95.18 even 4