↑

Properties

Label 475.2.j.d.49.1
Level $475$
Weight $2$
Character 475.49
Analytic conductor $3.793$
Analytic rank $0$
Dimension $24$
Inner twists $4$

Related objects

Downloads

Learn more

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(49,475)] 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("475.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.1
Character \(\chi\) \(=\) 475.49
Dual form 475.2.j.d.349.1

$q$-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+(-1.87935 - 1.08504i) q^{2} +(-1.22342 - 0.706345i) q^{3} +(1.35464 + 2.34630i) q^{4} +(1.53283 + 2.65494i) q^{6} -1.76171i q^{7} -1.53919i q^{8} +(-0.502155 - 0.869757i) q^{9} +1.83810 q^{11} -3.82736i q^{12} +(2.25586 - 1.30242i) q^{13} +(-1.91153 + 3.31086i) q^{14} +(1.03919 - 1.79993i) q^{16} +(-3.66826 - 2.11787i) q^{17} +2.17944i q^{18} +(-4.01936 - 1.68664i) q^{19} +(-1.24437 + 2.15532i) q^{21} +(-3.45443 - 1.99442i) q^{22} +(1.91001 - 1.10274i) q^{23} +(-1.08720 + 1.88308i) q^{24} -5.65274 q^{26} +5.65684i q^{27} +(4.13349 - 2.38647i) q^{28} +(-3.56413 - 6.17325i) q^{29} +0.303952 q^{31} +(-6.57195 + 3.79432i) q^{32} +(-2.24878 - 1.29833i) q^{33} +(4.59597 + 7.96045i) q^{34} +(1.36047 - 2.35641i) q^{36} +3.90376i q^{37} +(5.72370 + 7.53097i) q^{38} -3.67984 q^{39} +(-4.11981 + 7.13572i) q^{41} +(4.67722 - 2.70039i) q^{42} +(2.03431 + 1.17451i) q^{43} +(2.48996 + 4.31273i) q^{44} -4.78610 q^{46} +(-6.28281 + 3.62738i) q^{47} +(-2.54274 + 1.46805i) q^{48} +3.89639 q^{49} +(2.99190 + 5.18211i) q^{51} +(6.11176 + 3.52862i) q^{52} +(-9.19753 + 5.31020i) q^{53} +(6.13792 - 10.6312i) q^{54} -2.71160 q^{56} +(3.72603 + 4.90253i) q^{57} +15.4689i q^{58} +(-6.02692 + 10.4389i) q^{59} +(-5.26716 - 9.12299i) q^{61} +(-0.571231 - 0.329801i) q^{62} +(-1.53226 + 0.884649i) q^{63} +12.3112 q^{64} +(2.81749 + 4.88004i) q^{66} +(11.2857 - 6.51579i) q^{67} -11.4758i q^{68} -3.11567 q^{69} +(-5.91294 + 10.2415i) q^{71} +(-1.33872 + 0.772911i) q^{72} +(-7.94066 - 4.58454i) q^{73} +(4.23574 - 7.33652i) q^{74} +(-1.48740 - 11.7154i) q^{76} -3.23819i q^{77} +(6.91571 + 3.99278i) q^{78} +(3.94192 - 6.82761i) q^{79} +(2.48922 - 4.31145i) q^{81} +(15.4851 - 8.94034i) q^{82} -6.93584i q^{83} -6.74269 q^{84} +(-2.54879 - 4.41463i) q^{86} +10.0700i q^{87} -2.82918i q^{88} +(-6.23646 - 10.8019i) q^{89} +(-2.29449 - 3.97417i) q^{91} +(5.17474 + 2.98764i) q^{92} +(-0.371862 - 0.214695i) q^{93} +15.7435 q^{94} +10.7204 q^{96} +(-6.71939 - 3.87944i) q^{97} +(-7.32268 - 4.22775i) q^{98} +(-0.923009 - 1.59870i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{4} + 2 q^{6} + 14 q^{9} - 4 q^{11} - 12 q^{14} + 12 q^{16} + 12 q^{19} - 6 q^{21} + 22 q^{24} + 76 q^{26} + 6 q^{29} - 12 q^{31} - 2 q^{34} - 26 q^{36} - 32 q^{39} - 22 q^{41} + 42 q^{44} - 48 q^{46}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87935 − 1.08504i −1.32890 − 0.767241i −0.343771 − 0.939053i \(-0.611705\pi\)
−0.985130 + 0.171812i \(0.945038\pi\)
\(3\) −1.22342 − 0.706345i −0.706345 − 0.407808i 0.103361 − 0.994644i \(-0.467040\pi\)
−0.809706 + 0.586836i \(0.800373\pi\)
\(4\) 1.35464 + 2.34630i 0.677319 + 1.17315i
\(5\) 0 0
\(6\) 1.53283 + 2.65494i 0.625775 + 1.08387i
\(7\) − 1.76171i − 0.665862i −0.942951 − 0.332931i \(-0.891962\pi\)
0.942951 − 0.332931i \(-0.108038\pi\)
\(8\) − 1.53919i − 0.544185i
\(9\) −0.502155 − 0.869757i −0.167385 − 0.289919i
\(10\) 0 0
\(11\) 1.83810 0.554208 0.277104 − 0.960840i \(-0.410625\pi\)
0.277104 + 0.960840i \(0.410625\pi\)
\(12\) − 3.82736i − 1.10486i
\(13\) 2.25586 − 1.30242i 0.625664 − 0.361227i −0.153407 − 0.988163i \(-0.549025\pi\)
0.779071 + 0.626936i \(0.215691\pi\)
\(14\) −1.91153 + 3.31086i −0.510877 + 0.884865i
\(15\) 0 0
\(16\) 1.03919 − 1.79993i 0.259797 − 0.449982i
\(17\) −3.66826 − 2.11787i −0.889684 − 0.513659i −0.0158451 − 0.999874i \(-0.505044\pi\)
−0.873839 + 0.486215i \(0.838377\pi\)
\(18\) 2.17944i 0.513698i
\(19\) −4.01936 − 1.68664i −0.922104 − 0.386943i
\(20\) 0 0
\(21\) −1.24437 + 2.15532i −0.271544 + 0.470328i
\(22\) −3.45443 − 1.99442i −0.736487 − 0.425211i
\(23\) 1.91001 − 1.10274i 0.398264 − 0.229938i −0.287470 − 0.957790i \(-0.592814\pi\)
0.685735 + 0.727851i \(0.259481\pi\)
\(24\) −1.08720 + 1.88308i −0.221923 + 0.384382i
\(25\) 0 0
\(26\) −5.65274 −1.10859
\(27\) 5.65684i 1.08866i
\(28\) 4.13349 − 2.38647i 0.781157 − 0.451001i
\(29\) −3.56413 − 6.17325i −0.661842 − 1.14634i −0.980131 − 0.198351i \(-0.936442\pi\)
0.318289 − 0.947994i \(-0.396892\pi\)
\(30\) 0 0
\(31\) 0.303952 0.0545913 0.0272957 − 0.999627i \(-0.491310\pi\)
0.0272957 + 0.999627i \(0.491310\pi\)
\(32\) −6.57195 + 3.79432i −1.16177 + 0.670747i
\(33\) −2.24878 − 1.29833i −0.391462 − 0.226010i
\(34\) 4.59597 + 7.96045i 0.788202 + 1.36521i
\(35\) 0 0
\(36\) 1.36047 − 2.35641i 0.226746 − 0.392735i
\(37\) 3.90376i 0.641774i 0.947118 + 0.320887i \(0.103981\pi\)
−0.947118 + 0.320887i \(0.896019\pi\)
\(38\) 5.72370 + 7.53097i 0.928506 + 1.22168i
\(39\) −3.67984 −0.589246
\(40\) 0 0
\(41\) −4.11981 + 7.13572i −0.643406 + 1.11441i 0.341261 + 0.939969i \(0.389146\pi\)
−0.984667 + 0.174444i \(0.944187\pi\)
\(42\) 4.67722 − 2.70039i 0.721711 − 0.416680i
\(43\) 2.03431 + 1.17451i 0.310229 + 0.179111i 0.647029 − 0.762465i \(-0.276011\pi\)
−0.336800 + 0.941576i \(0.609345\pi\)
\(44\) 2.48996 + 4.31273i 0.375375 + 0.650169i
\(45\) 0 0
\(46\) −4.78610 −0.705672
\(47\) −6.28281 + 3.62738i −0.916441 + 0.529108i −0.882498 − 0.470316i \(-0.844140\pi\)
−0.0339433 + 0.999424i \(0.510807\pi\)
\(48\) −2.54274 + 1.46805i −0.367013 + 0.211895i
\(49\) 3.89639 0.556627
\(50\) 0 0
\(51\) 2.99190 + 5.18211i 0.418949 + 0.725641i
\(52\) 6.11176 + 3.52862i 0.847548 + 0.489332i
\(53\) −9.19753 + 5.31020i −1.26338 + 0.729412i −0.973727 − 0.227720i \(-0.926873\pi\)
−0.289652 + 0.957132i \(0.593540\pi\)
\(54\) 6.13792 − 10.6312i 0.835265 − 1.44672i
\(55\) 0 0
\(56\) −2.71160 −0.362353
\(57\) 3.72603 + 4.90253i 0.493525 + 0.649356i
\(58\) 15.4689i 2.03117i
\(59\) −6.02692 + 10.4389i −0.784638 + 1.35903i 0.144578 + 0.989493i \(0.453818\pi\)
−0.929215 + 0.369539i \(0.879516\pi\)
\(60\) 0 0
\(61\) −5.26716 − 9.12299i −0.674390 − 1.16808i −0.976647 − 0.214852i \(-0.931073\pi\)
0.302256 − 0.953227i \(-0.402260\pi\)
\(62\) −0.571231 − 0.329801i −0.0725464 − 0.0418847i
\(63\) −1.53226 + 0.884649i −0.193046 + 0.111455i
\(64\) 12.3112 1.53891
\(65\) 0 0
\(66\) 2.81749 + 4.88004i 0.346809 + 0.600691i
\(67\) 11.2857 − 6.51579i 1.37877 − 0.796031i 0.386755 − 0.922183i \(-0.373596\pi\)
0.992011 + 0.126152i \(0.0402627\pi\)
\(68\) − 11.4758i − 1.39164i
\(69\) −3.11567 −0.375083
\(70\) 0 0
\(71\) −5.91294 + 10.2415i −0.701737 + 1.21544i 0.266119 + 0.963940i \(0.414259\pi\)
−0.967856 + 0.251504i \(0.919075\pi\)
\(72\) −1.33872 + 0.772911i −0.157770 + 0.0910884i
\(73\) −7.94066 − 4.58454i −0.929384 − 0.536580i −0.0427676 − 0.999085i \(-0.513618\pi\)
−0.886617 + 0.462505i \(0.846951\pi\)
\(74\) 4.23574 − 7.33652i 0.492395 − 0.852854i
\(75\) 0 0
\(76\) −1.48740 − 11.7154i −0.170616 − 1.34385i
\(77\) − 3.23819i − 0.369026i
\(78\) 6.91571 + 3.99278i 0.783049 + 0.452094i
\(79\) 3.94192 − 6.82761i 0.443501 − 0.768167i −0.554445 − 0.832220i \(-0.687070\pi\)
0.997946 + 0.0640536i \(0.0204029\pi\)
\(80\) 0 0
\(81\) 2.48922 − 4.31145i 0.276580 − 0.479050i
\(82\) 15.4851 − 8.94034i 1.71005 − 0.987296i
\(83\) − 6.93584i − 0.761307i −0.924718 − 0.380654i \(-0.875699\pi\)
0.924718 − 0.380654i \(-0.124301\pi\)
\(84\) −6.74269 −0.735688
\(85\) 0 0
\(86\) −2.54879 − 4.41463i −0.274843 − 0.476041i
\(87\) 10.0700i 1.07962i
\(88\) − 2.82918i − 0.301592i
\(89\) −6.23646 − 10.8019i −0.661063 − 1.14500i −0.980337 − 0.197333i \(-0.936772\pi\)
0.319273 − 0.947663i \(-0.396561\pi\)
\(90\) 0 0
\(91\) −2.29449 − 3.97417i −0.240528 − 0.416606i
\(92\) 5.17474 + 2.98764i 0.539504 + 0.311483i
\(93\) −0.371862 − 0.214695i −0.0385603 − 0.0222628i
\(94\) 15.7435 1.62381
\(95\) 0 0
\(96\) 10.7204 1.09414
\(97\) −6.71939 − 3.87944i −0.682251 − 0.393898i 0.118452 − 0.992960i \(-0.462207\pi\)
−0.800703 + 0.599062i \(0.795540\pi\)
\(98\) −7.32268 − 4.22775i −0.739703 − 0.427067i
\(99\) −0.923009 − 1.59870i −0.0927659 − 0.160675i
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.j.d.49.1 24
5.2 odd 4 475.2.e.h.201.5 yes 12
5.3 odd 4 475.2.e.f.201.2 yes 12
5.4 even 2 inner 475.2.j.d.49.12 24
19.7 even 3 inner 475.2.j.d.349.12 24
95.7 odd 12 475.2.e.h.26.5 yes 12
95.8 even 12 9025.2.a.bs.1.2 6
95.27 even 12 9025.2.a.by.1.5 6
95.64 even 6 inner 475.2.j.d.349.1 24
95.68 odd 12 9025.2.a.bz.1.5 6
95.83 odd 12 475.2.e.f.26.2 ✓ 12
95.87 odd 12 9025.2.a.br.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.e.f.26.2 ✓ 12 95.83 odd 12
475.2.e.f.201.2 yes 12 5.3 odd 4
475.2.e.h.26.5 yes 12 95.7 odd 12
475.2.e.h.201.5 yes 12 5.2 odd 4
475.2.j.d.49.1 24 1.1 even 1 trivial
475.2.j.d.49.12 24 5.4 even 2 inner
475.2.j.d.349.1 24 95.64 even 6 inner
475.2.j.d.349.12 24 19.7 even 3 inner
9025.2.a.br.1.2 6 95.87 odd 12
9025.2.a.bs.1.2 6 95.8 even 12
9025.2.a.by.1.5 6 95.27 even 12
9025.2.a.bz.1.5 6 95.68 odd 12