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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(151,925)] 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("925.151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 151.3
Character \(\chi\) \(=\) 925.151
Dual form 925.2.bb.b.876.3

$q$-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+(-0.829311 + 0.988334i) q^{2} +(1.25448 - 1.05263i) q^{3} +(0.0582485 + 0.330344i) q^{4} +2.11280i q^{6} +(-2.68169 - 0.976055i) q^{7} +(-2.60945 - 1.50657i) q^{8} +(-0.0552654 + 0.313426i) q^{9} +(1.17577 - 2.03650i) q^{11} +(0.420801 + 0.353094i) q^{12} +(2.55154 - 0.449906i) q^{13} +(3.18862 - 1.84095i) q^{14} +(3.02262 - 1.10014i) q^{16} +(-6.62084 - 1.16743i) q^{17} +(-0.263937 - 0.314548i) q^{18} +(-4.53076 - 5.39955i) q^{19} +(-4.39154 + 1.59839i) q^{21} +(1.03766 + 2.85095i) q^{22} +(7.75287 - 4.47612i) q^{23} +(-4.85935 + 0.856835i) q^{24} +(-1.67136 + 2.89489i) q^{26} +(2.71699 + 4.70597i) q^{27} +(0.166229 - 0.942733i) q^{28} +(0.496364 + 0.286576i) q^{29} -4.07319i q^{31} +(0.641726 - 1.76313i) q^{32} +(-0.668701 - 3.79239i) q^{33} +(6.64455 - 5.57544i) q^{34} -0.106757 q^{36} +(5.55973 - 2.46767i) q^{37} +9.09397 q^{38} +(2.72726 - 3.25023i) q^{39} +(-1.54272 - 8.74921i) q^{41} +(2.06221 - 5.66587i) q^{42} +1.85159i q^{43} +(0.741232 + 0.269786i) q^{44} +(-2.00563 + 11.3745i) q^{46} +(-2.49863 - 4.32775i) q^{47} +(2.63375 - 4.56180i) q^{48} +(0.876457 + 0.735435i) q^{49} +(-9.53456 + 5.50478i) q^{51} +(0.297247 + 0.816680i) q^{52} +(-12.8323 + 4.67057i) q^{53} +(-6.90430 - 1.21742i) q^{54} +(5.52724 + 6.58711i) q^{56} +(-11.3675 - 2.00439i) q^{57} +(-0.694873 + 0.252913i) q^{58} +(-0.214282 - 0.588735i) q^{59} +(-2.25236 + 0.397151i) q^{61} +(4.02567 + 3.37794i) q^{62} +(0.454125 - 0.786568i) q^{63} +(4.42697 + 7.66773i) q^{64} +(4.30271 + 2.48417i) q^{66} +(11.5716 + 4.21171i) q^{67} -2.25516i q^{68} +(5.01409 - 13.7761i) q^{69} +(-2.29519 + 1.92589i) q^{71} +(0.616409 - 0.734608i) q^{72} -12.2346 q^{73} +(-2.17186 + 7.54134i) q^{74} +(1.51980 - 1.81123i) q^{76} +(-5.14079 + 4.31363i) q^{77} +(0.950560 + 5.39090i) q^{78} +(0.188725 - 0.518518i) q^{79} +(7.46485 + 2.71698i) q^{81} +(9.92654 + 5.73109i) q^{82} +(0.303143 - 1.71921i) q^{83} +(-0.783819 - 1.35761i) q^{84} +(-1.82999 - 1.53554i) q^{86} +(0.924336 - 0.162985i) q^{87} +(-6.13624 + 3.54276i) q^{88} +(-2.18420 - 6.00103i) q^{89} +(-7.28157 - 1.28394i) q^{91} +(1.93025 + 2.30039i) q^{92} +(-4.28756 - 5.10971i) q^{93} +(6.34940 + 1.11957i) q^{94} +(-1.05089 - 2.88730i) q^{96} +(2.90286 - 1.67596i) q^{97} +(-1.45371 + 0.256328i) q^{98} +(0.573311 + 0.481065i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.829311 + 0.988334i −0.586411 + 0.698858i −0.974912 − 0.222591i \(-0.928549\pi\)
0.388501 + 0.921448i \(0.372993\pi\)
\(3\) 1.25448 − 1.05263i 0.724272 − 0.607736i −0.204292 − 0.978910i \(-0.565489\pi\)
0.928563 + 0.371174i \(0.121045\pi\)
\(4\) 0.0582485 + 0.330344i 0.0291243 + 0.165172i
\(5\) 0 0
\(6\) 2.11280i 0.862546i
\(7\) −2.68169 − 0.976055i −1.01358 − 0.368914i −0.218775 − 0.975775i \(-0.570206\pi\)
−0.794808 + 0.606861i \(0.792428\pi\)
\(8\) −2.60945 − 1.50657i −0.922580 − 0.532652i
\(9\) −0.0552654 + 0.313426i −0.0184218 + 0.104475i
\(10\) 0 0
\(11\) 1.17577 − 2.03650i 0.354509 − 0.614027i −0.632525 − 0.774540i \(-0.717981\pi\)
0.987034 + 0.160513i \(0.0513148\pi\)
\(12\) 0.420801 + 0.353094i 0.121475 + 0.101930i
\(13\) 2.55154 − 0.449906i 0.707671 − 0.124781i 0.191783 − 0.981437i \(-0.438573\pi\)
0.515888 + 0.856656i \(0.327462\pi\)
\(14\) 3.18862 − 1.84095i 0.852195 − 0.492015i
\(15\) 0 0
\(16\) 3.02262 − 1.10014i 0.755654 − 0.275035i
\(17\) −6.62084 − 1.16743i −1.60579 − 0.283144i −0.702342 − 0.711840i \(-0.747862\pi\)
−0.903449 + 0.428696i \(0.858973\pi\)
\(18\) −0.263937 − 0.314548i −0.0622105 − 0.0741396i
\(19\) −4.53076 − 5.39955i −1.03943 − 1.23874i −0.970495 − 0.241121i \(-0.922485\pi\)
−0.0689331 − 0.997621i \(-0.521960\pi\)
\(20\) 0 0
\(21\) −4.39154 + 1.59839i −0.958312 + 0.348797i
\(22\) 1.03766 + 2.85095i 0.221230 + 0.607824i
\(23\) 7.75287 − 4.47612i 1.61659 − 0.933336i 0.628790 − 0.777575i \(-0.283550\pi\)
0.987795 − 0.155761i \(-0.0497829\pi\)
\(24\) −4.85935 + 0.856835i −0.991911 + 0.174901i
\(25\) 0 0
\(26\) −1.67136 + 2.89489i −0.327782 + 0.567734i
\(27\) 2.71699 + 4.70597i 0.522886 + 0.905665i
\(28\) 0.166229 − 0.942733i 0.0314144 − 0.178160i
\(29\) 0.496364 + 0.286576i 0.0921726 + 0.0532159i 0.545378 − 0.838190i \(-0.316386\pi\)
−0.453205 + 0.891406i \(0.649720\pi\)
\(30\) 0 0
\(31\) − 4.07319i − 0.731566i −0.930700 − 0.365783i \(-0.880801\pi\)
0.930700 − 0.365783i \(-0.119199\pi\)
\(32\) 0.641726 − 1.76313i 0.113442 − 0.311680i
\(33\) −0.668701 − 3.79239i −0.116406 − 0.660171i
\(34\) 6.64455 − 5.57544i 1.13953 − 0.956180i
\(35\) 0 0
\(36\) −0.106757 −0.0177929
\(37\) 5.55973 − 2.46767i 0.914014 − 0.405683i
\(38\) 9.09397 1.47524
\(39\) 2.72726 − 3.25023i 0.436712 − 0.520453i
\(40\) 0 0
\(41\) −1.54272 − 8.74921i −0.240933 − 1.36640i −0.829752 − 0.558133i \(-0.811518\pi\)
0.588819 − 0.808265i \(-0.299593\pi\)
\(42\) 2.06221 − 5.66587i 0.318205 − 0.874262i
\(43\) 1.85159i 0.282365i 0.989984 + 0.141182i \(0.0450904\pi\)
−0.989984 + 0.141182i \(0.954910\pi\)
\(44\) 0.741232 + 0.269786i 0.111745 + 0.0406718i
\(45\) 0 0
\(46\) −2.00563 + 11.3745i −0.295715 + 1.67708i
\(47\) −2.49863 − 4.32775i −0.364462 − 0.631267i 0.624227 − 0.781243i \(-0.285414\pi\)
−0.988690 + 0.149975i \(0.952081\pi\)
\(48\) 2.63375 − 4.56180i 0.380150 − 0.658439i
\(49\) 0.876457 + 0.735435i 0.125208 + 0.105062i
\(50\) 0 0
\(51\) −9.53456 + 5.50478i −1.33511 + 0.770824i
\(52\) 0.297247 + 0.816680i 0.0412208 + 0.113253i
\(53\) −12.8323 + 4.67057i −1.76265 + 0.641552i −0.999986 − 0.00524360i \(-0.998331\pi\)
−0.762663 + 0.646796i \(0.776109\pi\)
\(54\) −6.90430 − 1.21742i −0.939557 − 0.165669i
\(55\) 0 0
\(56\) 5.52724 + 6.58711i 0.738609 + 0.880239i
\(57\) −11.3675 − 2.00439i −1.50566 − 0.265488i
\(58\) −0.694873 + 0.252913i −0.0912413 + 0.0332091i
\(59\) −0.214282 − 0.588735i −0.0278971 − 0.0766467i 0.924962 − 0.380059i \(-0.124096\pi\)
−0.952859 + 0.303412i \(0.901874\pi\)
\(60\) 0 0
\(61\) −2.25236 + 0.397151i −0.288385 + 0.0508500i −0.315969 − 0.948769i \(-0.602330\pi\)
0.0275844 + 0.999619i \(0.491218\pi\)
\(62\) 4.02567 + 3.37794i 0.511260 + 0.428998i
\(63\) 0.454125 − 0.786568i 0.0572144 − 0.0990982i
\(64\) 4.42697 + 7.66773i 0.553371 + 0.958467i
\(65\) 0 0
\(66\) 4.30271 + 2.48417i 0.529627 + 0.305780i
\(67\) 11.5716 + 4.21171i 1.41369 + 0.514542i 0.932211 − 0.361914i \(-0.117877\pi\)
0.481481 + 0.876456i \(0.340099\pi\)
\(68\) − 2.25516i − 0.273478i
\(69\) 5.01409 − 13.7761i 0.603625 − 1.65845i
\(70\) 0 0
\(71\) −2.29519 + 1.92589i −0.272389 + 0.228562i −0.768742 − 0.639559i \(-0.779117\pi\)
0.496353 + 0.868121i \(0.334672\pi\)
\(72\) 0.616409 − 0.734608i 0.0726445 − 0.0865743i
\(73\) −12.2346 −1.43196 −0.715978 − 0.698123i \(-0.754019\pi\)
−0.715978 + 0.698123i \(0.754019\pi\)
\(74\) −2.17186 + 7.54134i −0.252474 + 0.876662i
\(75\) 0 0
\(76\) 1.51980 − 1.81123i 0.174333 − 0.207762i
\(77\) −5.14079 + 4.31363i −0.585847 + 0.491584i
\(78\) 0.950560 + 5.39090i 0.107630 + 0.610399i
\(79\) 0.188725 − 0.518518i 0.0212332 − 0.0583378i −0.928623 − 0.371024i \(-0.879007\pi\)
0.949856 + 0.312686i \(0.101229\pi\)
\(80\) 0 0
\(81\) 7.46485 + 2.71698i 0.829428 + 0.301887i
\(82\) 9.92654 + 5.73109i 1.09620 + 0.632893i
\(83\) 0.303143 − 1.71921i 0.0332743 − 0.188708i −0.963640 − 0.267203i \(-0.913900\pi\)
0.996914 + 0.0784954i \(0.0250116\pi\)
\(84\) −0.783819 − 1.35761i −0.0855216 − 0.148128i
\(85\) 0 0
\(86\) −1.82999 − 1.53554i −0.197333 − 0.165582i
\(87\) 0.924336 − 0.162985i 0.0990992 − 0.0174739i
\(88\) −6.13624 + 3.54276i −0.654126 + 0.377660i
\(89\) −2.18420 − 6.00103i −0.231524 − 0.636108i 0.768469 − 0.639887i \(-0.221019\pi\)
−0.999993 + 0.00377986i \(0.998797\pi\)
\(90\) 0 0
\(91\) −7.28157 − 1.28394i −0.763316 − 0.134593i
\(92\) 1.93025 + 2.30039i 0.201243 + 0.239832i
\(93\) −4.28756 − 5.10971i −0.444599 − 0.529852i
\(94\) 6.34940 + 1.11957i 0.654891 + 0.115475i
\(95\) 0 0
\(96\) −1.05089 − 2.88730i −0.107256 − 0.294684i
\(97\) 2.90286 − 1.67596i 0.294740 − 0.170168i −0.345337 − 0.938479i \(-0.612235\pi\)
0.640078 + 0.768310i \(0.278902\pi\)
\(98\) −1.45371 + 0.256328i −0.146847 + 0.0258931i
\(99\) 0.573311 + 0.481065i 0.0576199 + 0.0483489i
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 925.2.bb.b.151.3 72
5.2 odd 4 925.2.ba.b.299.3 72
5.3 odd 4 925.2.ba.c.299.10 72
5.4 even 2 185.2.w.a.151.10 yes 72
37.25 even 18 inner 925.2.bb.b.876.3 72
185.62 odd 36 925.2.ba.c.99.10 72
185.69 odd 36 6845.2.a.t.1.27 36
185.79 odd 36 6845.2.a.u.1.10 36
185.99 even 18 185.2.w.a.136.10 ✓ 72
185.173 odd 36 925.2.ba.b.99.3 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.10 ✓ 72 185.99 even 18
185.2.w.a.151.10 yes 72 5.4 even 2
925.2.ba.b.99.3 72 185.173 odd 36
925.2.ba.b.299.3 72 5.2 odd 4
925.2.ba.c.99.10 72 185.62 odd 36
925.2.ba.c.299.10 72 5.3 odd 4
925.2.bb.b.151.3 72 1.1 even 1 trivial
925.2.bb.b.876.3 72 37.25 even 18 inner
6845.2.a.t.1.27 36 185.69 odd 36
6845.2.a.u.1.10 36 185.79 odd 36