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Properties

Label 250.2.e.c.99.3
Level $250$
Weight $2$
Character 250.99
Analytic conductor $1.996$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [250,2,Mod(49,250)] 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("250.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(250, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 250.e (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,4,0,-6,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 99.3
Root \(0.0566033 + 1.17421i\) of defining polynomial
Character \(\chi\) \(=\) 250.99
Dual form 250.2.e.c.149.3

$q$-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+(0.951057 - 0.309017i) q^{2} +(-1.74363 - 2.39991i) q^{3} +(0.809017 - 0.587785i) q^{4} +(-2.39991 - 1.74363i) q^{6} -1.83337i q^{7} +(0.587785 - 0.809017i) q^{8} +(-1.79224 + 5.51595i) q^{9} +(-0.566541 - 1.74363i) q^{11} +(-2.82126 - 0.916683i) q^{12} +(-2.29951 - 0.747156i) q^{13} +(-0.566541 - 1.74363i) q^{14} +(0.309017 - 0.951057i) q^{16} +(-1.63679 + 2.25284i) q^{17} +5.79981i q^{18} +(-1.35294 - 0.982966i) q^{19} +(-4.39991 + 3.19672i) q^{21} +(-1.07763 - 1.48322i) q^{22} +(7.38615 - 2.39991i) q^{23} -2.96645 q^{24} -2.41785 q^{26} +(7.89900 - 2.56654i) q^{27} +(-1.07763 - 1.48322i) q^{28} +(6.13597 - 4.45805i) q^{29} +(4.28304 + 3.11181i) q^{31} -1.00000i q^{32} +(-3.19672 + 4.39991i) q^{33} +(-0.860510 + 2.64838i) q^{34} +(1.79224 + 5.51595i) q^{36} +(1.25051 + 0.406315i) q^{37} +(-1.59047 - 0.516776i) q^{38} +(2.21640 + 6.82138i) q^{39} +(1.08621 - 3.34301i) q^{41} +(-3.19672 + 4.39991i) q^{42} -4.30550i q^{43} +(-1.48322 - 1.07763i) q^{44} +(6.28304 - 4.56489i) q^{46} +(1.07763 + 1.48322i) q^{47} +(-2.82126 + 0.916683i) q^{48} +3.63877 q^{49} +8.26057 q^{51} +(-2.29951 + 0.747156i) q^{52} +(3.83082 + 5.27267i) q^{53} +(6.71929 - 4.88185i) q^{54} +(-1.48322 - 1.07763i) q^{56} +4.96086i q^{57} +(4.45805 - 6.13597i) q^{58} +(2.79981 - 8.61694i) q^{59} +(0.799717 + 2.46127i) q^{61} +(5.03501 + 1.63597i) q^{62} +(10.1128 + 3.28583i) q^{63} +(-0.309017 - 0.951057i) q^{64} +(-1.68061 + 5.17240i) q^{66} +(-5.58402 + 7.68574i) q^{67} +2.78467i q^{68} +(-18.6383 - 13.5415i) q^{69} +(-0.247156 + 0.179569i) q^{71} +(3.40904 + 4.69215i) q^{72} +(-14.2164 + 4.61920i) q^{73} +1.31486 q^{74} -1.67232 q^{76} +(-3.19672 + 1.03868i) q^{77} +(4.21584 + 5.80261i) q^{78} +(-2.79981 + 2.03418i) q^{79} +(-5.85599 - 4.25462i) q^{81} -3.51505i q^{82} +(-3.74572 + 5.15555i) q^{83} +(-1.68061 + 5.17240i) q^{84} +(-1.33047 - 4.09478i) q^{86} +(-21.3978 - 6.95256i) q^{87} +(-1.74363 - 0.566541i) q^{88} +(1.02608 + 3.15794i) q^{89} +(-1.36981 + 4.21584i) q^{91} +(4.56489 - 6.28304i) q^{92} -15.7047i q^{93} +(1.48322 + 1.07763i) q^{94} +(-2.39991 + 1.74363i) q^{96} +(6.51864 + 8.97214i) q^{97} +(3.46068 - 1.12444i) q^{98} +10.6332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 2 q^{11} + 2 q^{14} - 4 q^{16} - 40 q^{19} - 38 q^{21} - 4 q^{24} + 44 q^{26} + 30 q^{29} - 18 q^{31} + 2 q^{34} - 2 q^{36} + 24 q^{39} - 18 q^{41} - 2 q^{44} + 14 q^{46}+ \cdots + 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/250\mathbb{Z}\right)^\times\).

\(n\) \(127\)
\(\chi(n)\) \(e\left(\frac{9}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.951057 − 0.309017i 0.672499 − 0.218508i
\(3\) −1.74363 − 2.39991i −1.00669 − 1.38559i −0.921131 − 0.389254i \(-0.872733\pi\)
−0.0855571 − 0.996333i \(-0.527267\pi\)
\(4\) 0.809017 − 0.587785i 0.404508 − 0.293893i
\(5\) 0 0
\(6\) −2.39991 − 1.74363i −0.979758 − 0.711836i
\(7\) − 1.83337i − 0.692947i −0.938060 − 0.346474i \(-0.887379\pi\)
0.938060 − 0.346474i \(-0.112621\pi\)
\(8\) 0.587785 − 0.809017i 0.207813 − 0.286031i
\(9\) −1.79224 + 5.51595i −0.597414 + 1.83865i
\(10\) 0 0
\(11\) −0.566541 − 1.74363i −0.170819 − 0.525726i 0.828599 − 0.559842i \(-0.189138\pi\)
−0.999418 + 0.0341166i \(0.989138\pi\)
\(12\) −2.82126 − 0.916683i −0.814428 − 0.264624i
\(13\) −2.29951 − 0.747156i −0.637769 − 0.207224i −0.0277557 − 0.999615i \(-0.508836\pi\)
−0.610014 + 0.792391i \(0.708836\pi\)
\(14\) −0.566541 − 1.74363i −0.151414 − 0.466006i
\(15\) 0 0
\(16\) 0.309017 − 0.951057i 0.0772542 − 0.237764i
\(17\) −1.63679 + 2.25284i −0.396979 + 0.546395i −0.959983 − 0.280060i \(-0.909646\pi\)
0.563003 + 0.826455i \(0.309646\pi\)
\(18\) 5.79981i 1.36703i
\(19\) −1.35294 − 0.982966i −0.310385 − 0.225508i 0.421677 − 0.906746i \(-0.361442\pi\)
−0.732062 + 0.681238i \(0.761442\pi\)
\(20\) 0 0
\(21\) −4.39991 + 3.19672i −0.960138 + 0.697581i
\(22\) −1.07763 − 1.48322i −0.229750 − 0.316224i
\(23\) 7.38615 − 2.39991i 1.54012 − 0.500415i 0.588710 − 0.808344i \(-0.299636\pi\)
0.951409 + 0.307929i \(0.0996359\pi\)
\(24\) −2.96645 −0.605524
\(25\) 0 0
\(26\) −2.41785 −0.474179
\(27\) 7.89900 − 2.56654i 1.52016 − 0.493931i
\(28\) −1.07763 − 1.48322i −0.203652 − 0.280303i
\(29\) 6.13597 − 4.45805i 1.13942 − 0.827838i 0.152383 − 0.988321i \(-0.451305\pi\)
0.987039 + 0.160483i \(0.0513052\pi\)
\(30\) 0 0
\(31\) 4.28304 + 3.11181i 0.769256 + 0.558897i 0.901735 − 0.432288i \(-0.142294\pi\)
−0.132479 + 0.991186i \(0.542294\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) −3.19672 + 4.39991i −0.556477 + 0.765925i
\(34\) −0.860510 + 2.64838i −0.147576 + 0.454193i
\(35\) 0 0
\(36\) 1.79224 + 5.51595i 0.298707 + 0.919325i
\(37\) 1.25051 + 0.406315i 0.205582 + 0.0667977i 0.409998 − 0.912086i \(-0.365529\pi\)
−0.204416 + 0.978884i \(0.565529\pi\)
\(38\) −1.59047 − 0.516776i −0.258009 − 0.0838321i
\(39\) 2.21640 + 6.82138i 0.354908 + 1.09229i
\(40\) 0 0
\(41\) 1.08621 − 3.34301i 0.169637 − 0.522090i −0.829711 − 0.558194i \(-0.811495\pi\)
0.999348 + 0.0361034i \(0.0114946\pi\)
\(42\) −3.19672 + 4.39991i −0.493265 + 0.678920i
\(43\) − 4.30550i − 0.656583i −0.944576 − 0.328291i \(-0.893527\pi\)
0.944576 − 0.328291i \(-0.106473\pi\)
\(44\) −1.48322 − 1.07763i −0.223604 − 0.162458i
\(45\) 0 0
\(46\) 6.28304 − 4.56489i 0.926383 − 0.673057i
\(47\) 1.07763 + 1.48322i 0.157188 + 0.216350i 0.880346 − 0.474332i \(-0.157310\pi\)
−0.723158 + 0.690682i \(0.757310\pi\)
\(48\) −2.82126 + 0.916683i −0.407214 + 0.132312i
\(49\) 3.63877 0.519824
\(50\) 0 0
\(51\) 8.26057 1.15671
\(52\) −2.29951 + 0.747156i −0.318885 + 0.103612i
\(53\) 3.83082 + 5.27267i 0.526203 + 0.724257i 0.986546 − 0.163485i \(-0.0522734\pi\)
−0.460342 + 0.887741i \(0.652273\pi\)
\(54\) 6.71929 − 4.88185i 0.914380 − 0.664336i
\(55\) 0 0
\(56\) −1.48322 − 1.07763i −0.198204 − 0.144004i
\(57\) 4.96086i 0.657082i
\(58\) 4.45805 − 6.13597i 0.585370 − 0.805693i
\(59\) 2.79981 − 8.61694i 0.364505 − 1.12183i −0.585786 − 0.810466i \(-0.699214\pi\)
0.950291 − 0.311364i \(-0.100786\pi\)
\(60\) 0 0
\(61\) 0.799717 + 2.46127i 0.102393 + 0.315134i 0.989110 − 0.147180i \(-0.0470195\pi\)
−0.886717 + 0.462313i \(0.847019\pi\)
\(62\) 5.03501 + 1.63597i 0.639447 + 0.207769i
\(63\) 10.1128 + 3.28583i 1.27409 + 0.413976i
\(64\) −0.309017 − 0.951057i −0.0386271 − 0.118882i
\(65\) 0 0
\(66\) −1.68061 + 5.17240i −0.206869 + 0.636679i
\(67\) −5.58402 + 7.68574i −0.682196 + 0.938963i −0.999958 − 0.00920814i \(-0.997069\pi\)
0.317761 + 0.948171i \(0.397069\pi\)
\(68\) 2.78467i 0.337691i
\(69\) −18.6383 − 13.5415i −2.24379 − 1.63021i
\(70\) 0 0
\(71\) −0.247156 + 0.179569i −0.0293320 + 0.0213110i −0.602355 − 0.798229i \(-0.705771\pi\)
0.573023 + 0.819540i \(0.305771\pi\)
\(72\) 3.40904 + 4.69215i 0.401760 + 0.552975i
\(73\) −14.2164 + 4.61920i −1.66391 + 0.540636i −0.981685 − 0.190509i \(-0.938986\pi\)
−0.682222 + 0.731145i \(0.738986\pi\)
\(74\) 1.31486 0.152850
\(75\) 0 0
\(76\) −1.67232 −0.191829
\(77\) −3.19672 + 1.03868i −0.364300 + 0.118368i
\(78\) 4.21584 + 5.80261i 0.477350 + 0.657016i
\(79\) −2.79981 + 2.03418i −0.315004 + 0.228864i −0.734041 − 0.679106i \(-0.762368\pi\)
0.419037 + 0.907969i \(0.362368\pi\)
\(80\) 0 0
\(81\) −5.85599 − 4.25462i −0.650665 − 0.472736i
\(82\) − 3.51505i − 0.388172i
\(83\) −3.74572 + 5.15555i −0.411147 + 0.565895i −0.963498 − 0.267717i \(-0.913731\pi\)
0.552351 + 0.833612i \(0.313731\pi\)
\(84\) −1.68061 + 5.17240i −0.183370 + 0.564355i
\(85\) 0 0
\(86\) −1.33047 − 4.09478i −0.143469 − 0.441551i
\(87\) −21.3978 − 6.95256i −2.29408 − 0.745393i
\(88\) −1.74363 − 0.566541i −0.185872 − 0.0603935i
\(89\) 1.02608 + 3.15794i 0.108764 + 0.334741i 0.990595 − 0.136824i \(-0.0436894\pi\)
−0.881832 + 0.471565i \(0.843689\pi\)
\(90\) 0 0
\(91\) −1.36981 + 4.21584i −0.143595 + 0.441940i
\(92\) 4.56489 − 6.28304i 0.475923 − 0.655052i
\(93\) − 15.7047i − 1.62851i
\(94\) 1.48322 + 1.07763i 0.152983 + 0.111149i
\(95\) 0 0
\(96\) −2.39991 + 1.74363i −0.244939 + 0.177959i
\(97\) 6.51864 + 8.97214i 0.661867 + 0.910982i 0.999541 − 0.0302807i \(-0.00964013\pi\)
−0.337674 + 0.941263i \(0.609640\pi\)
\(98\) 3.46068 − 1.12444i 0.349581 − 0.113586i
\(99\) 10.6332 1.06867
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 250.2.e.c.99.3 16
5.2 odd 4 50.2.d.b.31.1 yes 8
5.3 odd 4 250.2.d.d.151.2 8
5.4 even 2 inner 250.2.e.c.99.2 16
15.2 even 4 450.2.h.e.181.1 8
20.7 even 4 400.2.u.d.81.2 8
25.2 odd 20 1250.2.a.l.1.4 4
25.3 odd 20 250.2.d.d.101.2 8
25.4 even 10 inner 250.2.e.c.149.3 16
25.11 even 5 1250.2.b.e.1249.4 8
25.14 even 10 1250.2.b.e.1249.5 8
25.21 even 5 inner 250.2.e.c.149.2 16
25.22 odd 20 50.2.d.b.21.1 ✓ 8
25.23 odd 20 1250.2.a.f.1.1 4
75.47 even 20 450.2.h.e.271.1 8
100.23 even 20 10000.2.a.x.1.4 4
100.27 even 20 10000.2.a.t.1.1 4
100.47 even 20 400.2.u.d.321.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.2.d.b.21.1 ✓ 8 25.22 odd 20
50.2.d.b.31.1 yes 8 5.2 odd 4
250.2.d.d.101.2 8 25.3 odd 20
250.2.d.d.151.2 8 5.3 odd 4
250.2.e.c.99.2 16 5.4 even 2 inner
250.2.e.c.99.3 16 1.1 even 1 trivial
250.2.e.c.149.2 16 25.21 even 5 inner
250.2.e.c.149.3 16 25.4 even 10 inner
400.2.u.d.81.2 8 20.7 even 4
400.2.u.d.321.2 8 100.47 even 20
450.2.h.e.181.1 8 15.2 even 4
450.2.h.e.271.1 8 75.47 even 20
1250.2.a.f.1.1 4 25.23 odd 20
1250.2.a.l.1.4 4 25.2 odd 20
1250.2.b.e.1249.4 8 25.11 even 5
1250.2.b.e.1249.5 8 25.14 even 10
10000.2.a.t.1.1 4 100.27 even 20
10000.2.a.x.1.4 4 100.23 even 20