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Properties

Label 475.2.e.h.201.5
Level $475$
Weight $2$
Character 475.201
Analytic conductor $3.793$
Analytic rank $0$
Dimension $12$
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(26,475)] 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("475.26"); 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([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,2] 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: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 201.5
Root \(1.20634 + 2.08945i\) of defining polynomial
Character \(\chi\) \(=\) 475.201
Dual form 475.2.e.h.26.5

$q$-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+(1.08504 - 1.87935i) q^{2} +(-0.706345 + 1.22342i) q^{3} +(-1.35464 - 2.34630i) q^{4} +(1.53283 + 2.65494i) q^{6} +1.76171 q^{7} -1.53919 q^{8} +(0.502155 + 0.869757i) q^{9} +1.83810 q^{11} +3.82736 q^{12} +(-1.30242 - 2.25586i) q^{13} +(1.91153 - 3.31086i) q^{14} +(1.03919 - 1.79993i) q^{16} +(2.11787 - 3.66826i) q^{17} +2.17944 q^{18} +(4.01936 + 1.68664i) q^{19} +(-1.24437 + 2.15532i) q^{21} +(1.99442 - 3.45443i) q^{22} +(-1.10274 - 1.91001i) q^{23} +(1.08720 - 1.88308i) q^{24} -5.65274 q^{26} -5.65684 q^{27} +(-2.38647 - 4.13349i) q^{28} +(3.56413 + 6.17325i) q^{29} +0.303952 q^{31} +(-3.79432 - 6.57195i) q^{32} +(-1.29833 + 2.24878i) q^{33} +(-4.59597 - 7.96045i) q^{34} +(1.36047 - 2.35641i) q^{36} -3.90376 q^{37} +(7.53097 - 5.72370i) q^{38} +3.67984 q^{39} +(-4.11981 + 7.13572i) q^{41} +(2.70039 + 4.67722i) q^{42} +(1.17451 - 2.03431i) q^{43} +(-2.48996 - 4.31273i) q^{44} -4.78610 q^{46} +(-3.62738 - 6.28281i) q^{47} +(1.46805 + 2.54274i) q^{48} -3.89639 q^{49} +(2.99190 + 5.18211i) q^{51} +(-3.52862 + 6.11176i) q^{52} +(5.31020 + 9.19753i) q^{53} +(-6.13792 + 10.6312i) q^{54} -2.71160 q^{56} +(-4.90253 + 3.72603i) q^{57} +15.4689 q^{58} +(6.02692 - 10.4389i) q^{59} +(-5.26716 - 9.12299i) q^{61} +(0.329801 - 0.571231i) q^{62} +(0.884649 + 1.53226i) q^{63} -12.3112 q^{64} +(2.81749 + 4.88004i) q^{66} +(6.51579 + 11.2857i) q^{67} -11.4758 q^{68} +3.11567 q^{69} +(-5.91294 + 10.2415i) q^{71} +(-0.772911 - 1.33872i) q^{72} +(-4.58454 + 7.94066i) q^{73} +(-4.23574 + 7.33652i) q^{74} +(-1.48740 - 11.7154i) q^{76} +3.23819 q^{77} +(3.99278 - 6.91571i) q^{78} +(-3.94192 + 6.82761i) q^{79} +(2.48922 - 4.31145i) q^{81} +(8.94034 + 15.4851i) q^{82} -6.93584 q^{83} +6.74269 q^{84} +(-2.54879 - 4.41463i) q^{86} -10.0700 q^{87} -2.82918 q^{88} +(6.23646 + 10.8019i) q^{89} +(-2.29449 - 3.97417i) q^{91} +(-2.98764 + 5.17474i) q^{92} +(-0.214695 + 0.371862i) q^{93} -15.7435 q^{94} +10.7204 q^{96} +(3.87944 - 6.71939i) q^{97} +(-4.22775 + 7.32268i) q^{98} +(0.923009 + 1.59870i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 3 q^{3} - 2 q^{4} + q^{6} - 4 q^{7} - 12 q^{8} - 7 q^{9} - 2 q^{11} - 14 q^{12} + 5 q^{13} + 6 q^{14} + 6 q^{16} - 3 q^{17} - 14 q^{18} - 6 q^{19} - 3 q^{21} + 9 q^{22} - 6 q^{23} - 11 q^{24}+ \cdots + 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.08504 − 1.87935i 0.767241 − 1.32890i −0.171812 − 0.985130i \(-0.554962\pi\)
0.939053 − 0.343771i \(-0.111705\pi\)
\(3\) −0.706345 + 1.22342i −0.407808 + 0.706345i −0.994644 − 0.103361i \(-0.967040\pi\)
0.586836 + 0.809706i \(0.300373\pi\)
\(4\) −1.35464 − 2.34630i −0.677319 − 1.17315i
\(5\) 0 0
\(6\) 1.53283 + 2.65494i 0.625775 + 1.08387i
\(7\) 1.76171 0.665862 0.332931 − 0.942951i \(-0.391962\pi\)
0.332931 + 0.942951i \(0.391962\pi\)
\(8\) −1.53919 −0.544185
\(9\) 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 \(0.410625\pi\)
\(12\) 3.82736 1.10486
\(13\) −1.30242 − 2.25586i −0.361227 − 0.625664i 0.626936 − 0.779071i \(-0.284309\pi\)
−0.988163 + 0.153407i \(0.950975\pi\)
\(14\) 1.91153 − 3.31086i 0.510877 − 0.884865i
\(15\) 0 0
\(16\) 1.03919 − 1.79993i 0.259797 − 0.449982i
\(17\) 2.11787 − 3.66826i 0.513659 − 0.889684i −0.486215 − 0.873839i \(-0.661623\pi\)
0.999874 − 0.0158451i \(-0.00504385\pi\)
\(18\) 2.17944 0.513698
\(19\) 4.01936 + 1.68664i 0.922104 + 0.386943i
\(20\) 0 0
\(21\) −1.24437 + 2.15532i −0.271544 + 0.470328i
\(22\) 1.99442 − 3.45443i 0.425211 − 0.736487i
\(23\) −1.10274 − 1.91001i −0.229938 − 0.398264i 0.727851 − 0.685735i \(-0.240519\pi\)
−0.957790 + 0.287470i \(0.907186\pi\)
\(24\) 1.08720 − 1.88308i 0.221923 − 0.384382i
\(25\) 0 0
\(26\) −5.65274 −1.10859
\(27\) −5.65684 −1.08866
\(28\) −2.38647 − 4.13349i −0.451001 − 0.781157i
\(29\) 3.56413 + 6.17325i 0.661842 + 1.14634i 0.980131 + 0.198351i \(0.0635585\pi\)
−0.318289 + 0.947994i \(0.603108\pi\)
\(30\) 0 0
\(31\) 0.303952 0.0545913 0.0272957 − 0.999627i \(-0.491310\pi\)
0.0272957 + 0.999627i \(0.491310\pi\)
\(32\) −3.79432 − 6.57195i −0.670747 − 1.16177i
\(33\) −1.29833 + 2.24878i −0.226010 + 0.391462i
\(34\) −4.59597 − 7.96045i −0.788202 − 1.36521i
\(35\) 0 0
\(36\) 1.36047 − 2.35641i 0.226746 − 0.392735i
\(37\) −3.90376 −0.641774 −0.320887 − 0.947118i \(-0.603981\pi\)
−0.320887 + 0.947118i \(0.603981\pi\)
\(38\) 7.53097 − 5.72370i 1.22168 − 0.928506i
\(39\) 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 \(0.944187\pi\)
\(42\) 2.70039 + 4.67722i 0.416680 + 0.721711i
\(43\) 1.17451 − 2.03431i 0.179111 − 0.310229i −0.762465 − 0.647029i \(-0.776011\pi\)
0.941576 + 0.336800i \(0.109345\pi\)
\(44\) −2.48996 − 4.31273i −0.375375 − 0.650169i
\(45\) 0 0
\(46\) −4.78610 −0.705672
\(47\) −3.62738 − 6.28281i −0.529108 − 0.916441i −0.999424 − 0.0339433i \(-0.989193\pi\)
0.470316 − 0.882498i \(-0.344140\pi\)
\(48\) 1.46805 + 2.54274i 0.211895 + 0.367013i
\(49\) −3.89639 −0.556627
\(50\) 0 0
\(51\) 2.99190 + 5.18211i 0.418949 + 0.725641i
\(52\) −3.52862 + 6.11176i −0.489332 + 0.847548i
\(53\) 5.31020 + 9.19753i 0.729412 + 1.26338i 0.957132 + 0.289652i \(0.0935395\pi\)
−0.227720 + 0.973727i \(0.573127\pi\)
\(54\) −6.13792 + 10.6312i −0.835265 + 1.44672i
\(55\) 0 0
\(56\) −2.71160 −0.362353
\(57\) −4.90253 + 3.72603i −0.649356 + 0.493525i
\(58\) 15.4689 2.03117
\(59\) 6.02692 − 10.4389i 0.784638 − 1.35903i −0.144578 − 0.989493i \(-0.546182\pi\)
0.929215 − 0.369539i \(-0.120484\pi\)
\(60\) 0 0
\(61\) −5.26716 − 9.12299i −0.674390 − 1.16808i −0.976647 − 0.214852i \(-0.931073\pi\)
0.302256 − 0.953227i \(-0.402260\pi\)
\(62\) 0.329801 − 0.571231i 0.0418847 − 0.0725464i
\(63\) 0.884649 + 1.53226i 0.111455 + 0.193046i
\(64\) −12.3112 −1.53891
\(65\) 0 0
\(66\) 2.81749 + 4.88004i 0.346809 + 0.600691i
\(67\) 6.51579 + 11.2857i 0.796031 + 1.37877i 0.922183 + 0.386755i \(0.126404\pi\)
−0.126152 + 0.992011i \(0.540263\pi\)
\(68\) −11.4758 −1.39164
\(69\) 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 \(0.919075\pi\)
\(72\) −0.772911 − 1.33872i −0.0910884 − 0.157770i
\(73\) −4.58454 + 7.94066i −0.536580 + 0.929384i 0.462505 + 0.886617i \(0.346951\pi\)
−0.999085 + 0.0427676i \(0.986382\pi\)
\(74\) −4.23574 + 7.33652i −0.492395 + 0.852854i
\(75\) 0 0
\(76\) −1.48740 − 11.7154i −0.170616 − 1.34385i
\(77\) 3.23819 0.369026
\(78\) 3.99278 − 6.91571i 0.452094 − 0.783049i
\(79\) −3.94192 + 6.82761i −0.443501 + 0.768167i −0.997946 − 0.0640536i \(-0.979597\pi\)
0.554445 + 0.832220i \(0.312930\pi\)
\(80\) 0 0
\(81\) 2.48922 − 4.31145i 0.276580 − 0.479050i
\(82\) 8.94034 + 15.4851i 0.987296 + 1.71005i
\(83\) −6.93584 −0.761307 −0.380654 − 0.924718i \(-0.624301\pi\)
−0.380654 + 0.924718i \(0.624301\pi\)
\(84\) 6.74269 0.735688
\(85\) 0 0
\(86\) −2.54879 − 4.41463i −0.274843 − 0.476041i
\(87\) −10.0700 −1.07962
\(88\) −2.82918 −0.301592
\(89\) 6.23646 + 10.8019i 0.661063 + 1.14500i 0.980337 + 0.197333i \(0.0632279\pi\)
−0.319273 + 0.947663i \(0.603439\pi\)
\(90\) 0 0
\(91\) −2.29449 − 3.97417i −0.240528 − 0.416606i
\(92\) −2.98764 + 5.17474i −0.311483 + 0.539504i
\(93\) −0.214695 + 0.371862i −0.0222628 + 0.0385603i
\(94\) −15.7435 −1.62381
\(95\) 0 0
\(96\) 10.7204 1.09414
\(97\) 3.87944 − 6.71939i 0.393898 − 0.682251i −0.599062 − 0.800703i \(-0.704460\pi\)
0.992960 + 0.118452i \(0.0377931\pi\)
\(98\) −4.22775 + 7.32268i −0.427067 + 0.739703i
\(99\) 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.e.h.201.5 yes 12
5.2 odd 4 475.2.j.d.49.12 24
5.3 odd 4 475.2.j.d.49.1 24
5.4 even 2 475.2.e.f.201.2 yes 12
19.7 even 3 inner 475.2.e.h.26.5 yes 12
19.8 odd 6 9025.2.a.by.1.5 6
19.11 even 3 9025.2.a.br.1.2 6
95.7 odd 12 475.2.j.d.349.1 24
95.49 even 6 9025.2.a.bz.1.5 6
95.64 even 6 475.2.e.f.26.2 ✓ 12
95.83 odd 12 475.2.j.d.349.12 24
95.84 odd 6 9025.2.a.bs.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.e.f.26.2 ✓ 12 95.64 even 6
475.2.e.f.201.2 yes 12 5.4 even 2
475.2.e.h.26.5 yes 12 19.7 even 3 inner
475.2.e.h.201.5 yes 12 1.1 even 1 trivial
475.2.j.d.49.1 24 5.3 odd 4
475.2.j.d.49.12 24 5.2 odd 4
475.2.j.d.349.1 24 95.7 odd 12
475.2.j.d.349.12 24 95.83 odd 12
9025.2.a.br.1.2 6 19.11 even 3
9025.2.a.bs.1.2 6 95.84 odd 6
9025.2.a.by.1.5 6 19.8 odd 6
9025.2.a.bz.1.5 6 95.49 even 6