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 = [12,0] 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: 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: \( x^{12} + 6x^{10} + 29x^{8} + 40x^{6} + 43x^{4} + 7x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 201.2
Root \(0.579521 - 1.00376i\) of defining polynomial
Character \(\chi\) \(=\) 475.201
Dual form 475.2.e.g.26.2

$q$-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+(-0.431391 + 0.747190i) q^{2} +(-1.53957 + 2.66661i) q^{3} +(0.627804 + 1.08739i) q^{4} +(-1.32831 - 2.30070i) q^{6} +0.566520 q^{7} -2.80888 q^{8} +(-3.24054 - 5.61278i) q^{9} -1.91223 q^{11} -3.86619 q^{12} +(-0.0972656 - 0.168469i) q^{13} +(-0.244391 + 0.423298i) q^{14} +(-0.0438854 + 0.0760118i) q^{16} +(-2.64775 + 4.58603i) q^{17} +5.59175 q^{18} +(-2.36834 - 3.65936i) q^{19} +(-0.872196 + 1.51069i) q^{21} +(0.824918 - 1.42880i) q^{22} +(-1.68770 - 2.92318i) q^{23} +(4.32446 - 7.49018i) q^{24} +0.167838 q^{26} +10.7187 q^{27} +(0.355664 + 0.616027i) q^{28} +(4.36834 + 7.56619i) q^{29} +5.65662 q^{31} +(-2.84674 - 4.93070i) q^{32} +(2.94401 - 5.09917i) q^{33} +(-2.28442 - 3.95674i) q^{34} +(4.06885 - 7.04745i) q^{36} +0.955582 q^{37} +(3.75592 - 0.190988i) q^{38} +0.598988 q^{39} +(-5.02496 + 8.70349i) q^{41} +(-0.752514 - 1.30339i) q^{42} +(2.46622 - 4.27161i) q^{43} +(-1.20051 - 2.07934i) q^{44} +2.91223 q^{46} +(-4.41971 - 7.65516i) q^{47} +(-0.135129 - 0.234051i) q^{48} -6.67906 q^{49} +(-8.15277 - 14.1210i) q^{51} +(0.122128 - 0.211531i) q^{52} +(4.10305 + 7.10669i) q^{53} +(-4.62395 + 8.00892i) q^{54} -1.59128 q^{56} +(13.4043 - 0.681609i) q^{57} -7.53785 q^{58} +(-1.85713 + 3.21664i) q^{59} +(1.75561 + 3.04080i) q^{61} +(-2.44021 + 4.22657i) q^{62} +(-1.83583 - 3.17975i) q^{63} +4.73669 q^{64} +(2.54003 + 4.39947i) q^{66} +(-2.02182 - 3.50190i) q^{67} -6.64906 q^{68} +10.3933 q^{69} +(-2.59767 + 4.49929i) q^{71} +(9.10228 + 15.7656i) q^{72} +(-4.30205 + 7.45136i) q^{73} +(-0.412229 + 0.714002i) q^{74} +(2.49230 - 4.87268i) q^{76} -1.08332 q^{77} +(-0.258398 + 0.447558i) q^{78} +(-3.31324 + 5.73870i) q^{79} +(-6.78057 + 11.7443i) q^{81} +(-4.33544 - 7.50921i) q^{82} -4.51737 q^{83} -2.19027 q^{84} +(2.12780 + 3.68547i) q^{86} -26.9014 q^{87} +5.37122 q^{88} +(-1.68676 - 2.92155i) q^{89} +(-0.0551029 - 0.0954410i) q^{91} +(2.11909 - 3.67037i) q^{92} +(-8.70875 + 15.0840i) q^{93} +7.62648 q^{94} +17.5310 q^{96} +(-7.59611 + 13.1568i) q^{97} +(2.88128 - 4.99053i) q^{98} +(6.19665 + 10.7329i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4} - 12 q^{6} - 8 q^{9} + 4 q^{11} - 22 q^{14} - 14 q^{16} + 12 q^{19} - 20 q^{21} - 2 q^{24} - 44 q^{26} + 12 q^{29} + 60 q^{31} - 10 q^{34} + 14 q^{36} - 4 q^{39} - 12 q^{41} - 20 q^{44} + 8 q^{46}+ \cdots + 30 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.431391 + 0.747190i −0.305039 + 0.528343i −0.977270 − 0.211998i \(-0.932003\pi\)
0.672231 + 0.740342i \(0.265336\pi\)
\(3\) −1.53957 + 2.66661i −0.888870 + 1.53957i −0.0476573 + 0.998864i \(0.515176\pi\)
−0.841213 + 0.540704i \(0.818158\pi\)
\(4\) 0.627804 + 1.08739i 0.313902 + 0.543695i
\(5\) 0 0
\(6\) −1.32831 − 2.30070i −0.542280 − 0.939257i
\(7\) 0.566520 0.214124 0.107062 − 0.994252i \(-0.465856\pi\)
0.107062 + 0.994252i \(0.465856\pi\)
\(8\) −2.80888 −0.993088
\(9\) −3.24054 − 5.61278i −1.08018 − 1.87093i
\(10\) 0 0
\(11\) −1.91223 −0.576559 −0.288279 − 0.957546i \(-0.593083\pi\)
−0.288279 + 0.957546i \(0.593083\pi\)
\(12\) −3.86619 −1.11607
\(13\) −0.0972656 − 0.168469i −0.0269766 − 0.0467249i 0.852222 − 0.523180i \(-0.175255\pi\)
−0.879199 + 0.476456i \(0.841921\pi\)
\(14\) −0.244391 + 0.423298i −0.0653163 + 0.113131i
\(15\) 0 0
\(16\) −0.0438854 + 0.0760118i −0.0109714 + 0.0190029i
\(17\) −2.64775 + 4.58603i −0.642173 + 1.11228i 0.342774 + 0.939418i \(0.388633\pi\)
−0.984947 + 0.172858i \(0.944700\pi\)
\(18\) 5.59175 1.31799
\(19\) −2.36834 − 3.65936i −0.543335 − 0.839516i
\(20\) 0 0
\(21\) −0.872196 + 1.51069i −0.190329 + 0.329659i
\(22\) 0.824918 − 1.42880i 0.175873 − 0.304621i
\(23\) −1.68770 − 2.92318i −0.351910 − 0.609525i 0.634674 − 0.772780i \(-0.281134\pi\)
−0.986584 + 0.163254i \(0.947801\pi\)
\(24\) 4.32446 − 7.49018i 0.882726 − 1.52893i
\(25\) 0 0
\(26\) 0.167838 0.0329157
\(27\) 10.7187 2.06282
\(28\) 0.355664 + 0.616027i 0.0672141 + 0.116418i
\(29\) 4.36834 + 7.56619i 0.811181 + 1.40501i 0.912038 + 0.410106i \(0.134508\pi\)
−0.100857 + 0.994901i \(0.532158\pi\)
\(30\) 0 0
\(31\) 5.65662 1.01596 0.507980 − 0.861369i \(-0.330393\pi\)
0.507980 + 0.861369i \(0.330393\pi\)
\(32\) −2.84674 − 4.93070i −0.503238 − 0.871633i
\(33\) 2.94401 − 5.09917i 0.512486 − 0.887651i
\(34\) −2.28442 − 3.95674i −0.391776 − 0.678575i
\(35\) 0 0
\(36\) 4.06885 − 7.04745i 0.678142 − 1.17458i
\(37\) 0.955582 0.157097 0.0785484 − 0.996910i \(-0.474971\pi\)
0.0785484 + 0.996910i \(0.474971\pi\)
\(38\) 3.75592 − 0.190988i 0.609291 − 0.0309824i
\(39\) 0.598988 0.0959149
\(40\) 0 0
\(41\) −5.02496 + 8.70349i −0.784768 + 1.35926i 0.144370 + 0.989524i \(0.453884\pi\)
−0.929138 + 0.369734i \(0.879449\pi\)
\(42\) −0.752514 − 1.30339i −0.116115 − 0.201118i
\(43\) 2.46622 − 4.27161i 0.376094 − 0.651415i −0.614396 − 0.788998i \(-0.710600\pi\)
0.990490 + 0.137583i \(0.0439335\pi\)
\(44\) −1.20051 − 2.07934i −0.180983 − 0.313472i
\(45\) 0 0
\(46\) 2.91223 0.429385
\(47\) −4.41971 − 7.65516i −0.644681 − 1.11662i −0.984375 − 0.176084i \(-0.943657\pi\)
0.339694 − 0.940536i \(-0.389676\pi\)
\(48\) −0.135129 − 0.234051i −0.0195042 − 0.0337823i
\(49\) −6.67906 −0.954151
\(50\) 0 0
\(51\) −8.15277 − 14.1210i −1.14162 − 1.97734i
\(52\) 0.122128 − 0.211531i 0.0169361 − 0.0293341i
\(53\) 4.10305 + 7.10669i 0.563597 + 0.976179i 0.997179 + 0.0750646i \(0.0239163\pi\)
−0.433581 + 0.901114i \(0.642750\pi\)
\(54\) −4.62395 + 8.00892i −0.629240 + 1.08988i
\(55\) 0 0
\(56\) −1.59128 −0.212644
\(57\) 13.4043 − 0.681609i 1.77545 − 0.0902813i
\(58\) −7.53785 −0.989768
\(59\) −1.85713 + 3.21664i −0.241777 + 0.418770i −0.961221 − 0.275781i \(-0.911064\pi\)
0.719443 + 0.694551i \(0.244397\pi\)
\(60\) 0 0
\(61\) 1.75561 + 3.04080i 0.224783 + 0.389335i 0.956254 − 0.292537i \(-0.0944995\pi\)
−0.731472 + 0.681872i \(0.761166\pi\)
\(62\) −2.44021 + 4.22657i −0.309907 + 0.536775i
\(63\) −1.83583 − 3.17975i −0.231293 − 0.400611i
\(64\) 4.73669 0.592086
\(65\) 0 0
\(66\) 2.54003 + 4.39947i 0.312657 + 0.541537i
\(67\) −2.02182 − 3.50190i −0.247005 − 0.427825i 0.715688 − 0.698420i \(-0.246113\pi\)
−0.962693 + 0.270594i \(0.912780\pi\)
\(68\) −6.64906 −0.806318
\(69\) 10.3933 1.25121
\(70\) 0 0
\(71\) −2.59767 + 4.49929i −0.308286 + 0.533967i −0.977988 − 0.208663i \(-0.933089\pi\)
0.669701 + 0.742631i \(0.266422\pi\)
\(72\) 9.10228 + 15.7656i 1.07271 + 1.85799i
\(73\) −4.30205 + 7.45136i −0.503516 + 0.872116i 0.496475 + 0.868051i \(0.334627\pi\)
−0.999992 + 0.00406505i \(0.998706\pi\)
\(74\) −0.412229 + 0.714002i −0.0479207 + 0.0830010i
\(75\) 0 0
\(76\) 2.49230 − 4.87268i 0.285886 − 0.558934i
\(77\) −1.08332 −0.123455
\(78\) −0.258398 + 0.447558i −0.0292578 + 0.0506760i
\(79\) −3.31324 + 5.73870i −0.372769 + 0.645654i −0.989990 − 0.141135i \(-0.954925\pi\)
0.617222 + 0.786789i \(0.288258\pi\)
\(80\) 0 0
\(81\) −6.78057 + 11.7443i −0.753397 + 1.30492i
\(82\) −4.33544 − 7.50921i −0.478770 − 0.829253i
\(83\) −4.51737 −0.495845 −0.247923 − 0.968780i \(-0.579748\pi\)
−0.247923 + 0.968780i \(0.579748\pi\)
\(84\) −2.19027 −0.238978
\(85\) 0 0
\(86\) 2.12780 + 3.68547i 0.229447 + 0.397414i
\(87\) −26.9014 −2.88414
\(88\) 5.37122 0.572574
\(89\) −1.68676 − 2.92155i −0.178796 − 0.309684i 0.762672 − 0.646785i \(-0.223887\pi\)
−0.941468 + 0.337101i \(0.890554\pi\)
\(90\) 0 0
\(91\) −0.0551029 − 0.0954410i −0.00577635 − 0.0100049i
\(92\) 2.11909 − 3.67037i 0.220930 − 0.382663i
\(93\) −8.70875 + 15.0840i −0.903056 + 1.56414i
\(94\) 7.62648 0.786612
\(95\) 0 0
\(96\) 17.5310 1.78925
\(97\) −7.59611 + 13.1568i −0.771268 + 1.33588i 0.165600 + 0.986193i \(0.447044\pi\)
−0.936868 + 0.349683i \(0.886289\pi\)
\(98\) 2.88128 − 4.99053i 0.291053 − 0.504119i
\(99\) 6.19665 + 10.7329i 0.622787 + 1.07870i
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.g.201.2 12
5.2 odd 4 95.2.i.b.49.2 ✓ 12
5.3 odd 4 95.2.i.b.49.5 yes 12
5.4 even 2 inner 475.2.e.g.201.5 12
15.2 even 4 855.2.be.d.334.5 12
15.8 even 4 855.2.be.d.334.2 12
19.7 even 3 inner 475.2.e.g.26.2 12
19.8 odd 6 9025.2.a.bt.1.2 6
19.11 even 3 9025.2.a.bu.1.5 6
95.7 odd 12 95.2.i.b.64.5 yes 12
95.8 even 12 1805.2.b.g.1084.5 6
95.27 even 12 1805.2.b.g.1084.2 6
95.49 even 6 9025.2.a.bu.1.2 6
95.64 even 6 inner 475.2.e.g.26.5 12
95.68 odd 12 1805.2.b.f.1084.2 6
95.83 odd 12 95.2.i.b.64.2 yes 12
95.84 odd 6 9025.2.a.bt.1.5 6
95.87 odd 12 1805.2.b.f.1084.5 6
285.83 even 12 855.2.be.d.64.5 12
285.197 even 12 855.2.be.d.64.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.b.49.2 ✓ 12 5.2 odd 4
95.2.i.b.49.5 yes 12 5.3 odd 4
95.2.i.b.64.2 yes 12 95.83 odd 12
95.2.i.b.64.5 yes 12 95.7 odd 12
475.2.e.g.26.2 12 19.7 even 3 inner
475.2.e.g.26.5 12 95.64 even 6 inner
475.2.e.g.201.2 12 1.1 even 1 trivial
475.2.e.g.201.5 12 5.4 even 2 inner
855.2.be.d.64.2 12 285.197 even 12
855.2.be.d.64.5 12 285.83 even 12
855.2.be.d.334.2 12 15.8 even 4
855.2.be.d.334.5 12 15.2 even 4
1805.2.b.f.1084.2 6 95.68 odd 12
1805.2.b.f.1084.5 6 95.87 odd 12
1805.2.b.g.1084.2 6 95.27 even 12
1805.2.b.g.1084.5 6 95.8 even 12
9025.2.a.bt.1.2 6 19.8 odd 6
9025.2.a.bt.1.5 6 95.84 odd 6
9025.2.a.bu.1.2 6 95.49 even 6
9025.2.a.bu.1.5 6 19.11 even 3