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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); 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([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 299.12
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.c.99.12

$q$-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+(1.79899 + 1.50953i) q^{2} +(0.562780 + 0.670695i) q^{3} +(0.610385 + 3.46166i) q^{4} +2.05611i q^{6} +(-1.51982 + 4.17567i) q^{7} +(-1.77900 + 3.08132i) q^{8} +(0.387834 - 2.19951i) q^{9} +(-2.58382 + 4.47531i) q^{11} +(-1.97821 + 2.35754i) q^{12} +(-0.823207 - 4.66864i) q^{13} +(-9.03746 + 5.21778i) q^{14} +(-1.24561 + 0.453366i) q^{16} +(-0.331889 + 1.88224i) q^{17} +(4.01795 - 3.37146i) q^{18} +(-1.95658 - 2.33176i) q^{19} +(-3.65593 + 1.33065i) q^{21} +(-11.4039 + 4.15068i) q^{22} +(4.63208 + 8.02299i) q^{23} +(-3.06782 + 0.540939i) q^{24} +(5.56653 - 9.64151i) q^{26} +(3.96817 - 2.29102i) q^{27} +(-15.3824 - 2.71234i) q^{28} +(2.53625 + 1.46431i) q^{29} -5.61699i q^{31} +(3.76165 + 1.36913i) q^{32} +(-4.45569 + 0.785658i) q^{33} +(-3.43836 + 2.88513i) q^{34} +7.85071 q^{36} +(4.53051 + 4.05888i) q^{37} -7.14833i q^{38} +(2.66795 - 3.17954i) q^{39} +(0.132785 + 0.753063i) q^{41} +(-8.58565 - 3.12492i) q^{42} -0.369905 q^{43} +(-17.0691 - 6.21265i) q^{44} +(-3.77791 + 21.4256i) q^{46} +(9.80987 - 5.66373i) q^{47} +(-1.00508 - 0.580281i) q^{48} +(-9.76406 - 8.19302i) q^{49} +(-1.44919 + 0.836689i) q^{51} +(15.6588 - 5.69933i) q^{52} +(-1.58557 - 4.35631i) q^{53} +(10.5971 + 1.86855i) q^{54} +(-10.1628 - 12.1116i) q^{56} +(0.462776 - 2.62453i) q^{57} +(2.35228 + 6.46283i) q^{58} +(1.52849 + 4.19950i) q^{59} +(-2.71555 + 0.478824i) q^{61} +(8.47904 - 10.1049i) q^{62} +(8.59501 + 4.96233i) q^{63} +(6.02598 + 10.4373i) q^{64} +(-9.20173 - 5.31262i) q^{66} +(-1.54367 + 4.24119i) q^{67} -6.71825 q^{68} +(-2.77414 + 7.62190i) q^{69} +(-3.35412 + 2.81444i) q^{71} +(6.08746 + 5.10798i) q^{72} -2.68113i q^{73} +(2.02333 + 14.1408i) q^{74} +(6.87750 - 8.19628i) q^{76} +(-14.7605 - 17.5908i) q^{77} +(9.59925 - 1.69261i) q^{78} +(-0.687516 + 1.88893i) q^{79} +(-2.52648 - 0.919562i) q^{81} +(-0.897894 + 1.55520i) q^{82} +(1.82162 + 0.321200i) q^{83} +(-6.83778 - 11.8434i) q^{84} +(-0.665457 - 0.558384i) q^{86} +(0.445249 + 2.52514i) q^{87} +(-9.19324 - 15.9232i) q^{88} +(-2.11794 - 5.81901i) q^{89} +(20.7458 + 3.65805i) q^{91} +(-24.9456 + 20.9318i) q^{92} +(3.76729 - 3.16113i) q^{93} +(26.1975 + 4.61932i) q^{94} +(1.19871 + 3.29344i) q^{96} +(-1.42438 - 2.46710i) q^{97} +(-5.19783 - 29.4784i) q^{98} +(8.84141 + 7.41882i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{4} + 6 q^{7} + 18 q^{8} + 42 q^{12} + 18 q^{13} + 18 q^{14} - 18 q^{16} - 12 q^{17} - 24 q^{18} + 18 q^{19} - 24 q^{22} + 24 q^{23} + 30 q^{24} + 48 q^{28} - 18 q^{29} - 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.79899 + 1.50953i 1.27208 + 1.06740i 0.994285 + 0.106761i \(0.0340480\pi\)
0.277795 + 0.960640i \(0.410396\pi\)
\(3\) 0.562780 + 0.670695i 0.324921 + 0.387226i 0.903634 − 0.428305i \(-0.140889\pi\)
−0.578713 + 0.815531i \(0.696445\pi\)
\(4\) 0.610385 + 3.46166i 0.305192 + 1.73083i
\(5\) 0 0
\(6\) 2.05611i 0.839404i
\(7\) −1.51982 + 4.17567i −0.574438 + 1.57826i 0.222977 + 0.974824i \(0.428423\pi\)
−0.797415 + 0.603431i \(0.793800\pi\)
\(8\) −1.77900 + 3.08132i −0.628973 + 1.08941i
\(9\) 0.387834 − 2.19951i 0.129278 − 0.733172i
\(10\) 0 0
\(11\) −2.58382 + 4.47531i −0.779051 + 1.34936i 0.153439 + 0.988158i \(0.450965\pi\)
−0.932489 + 0.361197i \(0.882368\pi\)
\(12\) −1.97821 + 2.35754i −0.571060 + 0.680563i
\(13\) −0.823207 − 4.66864i −0.228317 − 1.29485i −0.856242 − 0.516575i \(-0.827207\pi\)
0.627926 − 0.778273i \(-0.283904\pi\)
\(14\) −9.03746 + 5.21778i −2.41536 + 1.39451i
\(15\) 0 0
\(16\) −1.24561 + 0.453366i −0.311403 + 0.113341i
\(17\) −0.331889 + 1.88224i −0.0804949 + 0.456509i 0.917743 + 0.397174i \(0.130009\pi\)
−0.998238 + 0.0593351i \(0.981102\pi\)
\(18\) 4.01795 − 3.37146i 0.947040 − 0.794661i
\(19\) −1.95658 − 2.33176i −0.448870 − 0.534942i 0.493398 − 0.869804i \(-0.335755\pi\)
−0.942267 + 0.334862i \(0.891310\pi\)
\(20\) 0 0
\(21\) −3.65593 + 1.33065i −0.797789 + 0.290371i
\(22\) −11.4039 + 4.15068i −2.43132 + 0.884928i
\(23\) 4.63208 + 8.02299i 0.965855 + 1.67291i 0.707300 + 0.706913i \(0.249913\pi\)
0.258555 + 0.965997i \(0.416754\pi\)
\(24\) −3.06782 + 0.540939i −0.626216 + 0.110419i
\(25\) 0 0
\(26\) 5.56653 − 9.64151i 1.09169 − 1.89086i
\(27\) 3.96817 − 2.29102i 0.763674 − 0.440907i
\(28\) −15.3824 − 2.71234i −2.90701 − 0.512584i
\(29\) 2.53625 + 1.46431i 0.470970 + 0.271915i 0.716646 − 0.697437i \(-0.245676\pi\)
−0.245676 + 0.969352i \(0.579010\pi\)
\(30\) 0 0
\(31\) − 5.61699i − 1.00884i −0.863458 − 0.504421i \(-0.831706\pi\)
0.863458 − 0.504421i \(-0.168294\pi\)
\(32\) 3.76165 + 1.36913i 0.664971 + 0.242030i
\(33\) −4.45569 + 0.785658i −0.775636 + 0.136766i
\(34\) −3.43836 + 2.88513i −0.589675 + 0.494796i
\(35\) 0 0
\(36\) 7.85071 1.30845
\(37\) 4.53051 + 4.05888i 0.744811 + 0.667276i
\(38\) − 7.14833i − 1.15961i
\(39\) 2.66795 − 3.17954i 0.427214 − 0.509134i
\(40\) 0 0
\(41\) 0.132785 + 0.753063i 0.0207376 + 0.117609i 0.993419 − 0.114534i \(-0.0365375\pi\)
−0.972682 + 0.232143i \(0.925426\pi\)
\(42\) −8.58565 − 3.12492i −1.32479 − 0.482186i
\(43\) −0.369905 −0.0564100 −0.0282050 − 0.999602i \(-0.508979\pi\)
−0.0282050 + 0.999602i \(0.508979\pi\)
\(44\) −17.0691 − 6.21265i −2.57327 − 0.936593i
\(45\) 0 0
\(46\) −3.77791 + 21.4256i −0.557022 + 3.15903i
\(47\) 9.80987 − 5.66373i 1.43092 − 0.826140i 0.433726 − 0.901045i \(-0.357199\pi\)
0.997191 + 0.0749052i \(0.0238654\pi\)
\(48\) −1.00508 − 0.580281i −0.145070 − 0.0837564i
\(49\) −9.76406 − 8.19302i −1.39487 − 1.17043i
\(50\) 0 0
\(51\) −1.44919 + 0.836689i −0.202927 + 0.117160i
\(52\) 15.6588 − 5.69933i 2.17148 − 0.790355i
\(53\) −1.58557 − 4.35631i −0.217794 − 0.598385i 0.781892 − 0.623414i \(-0.214255\pi\)
−0.999687 + 0.0250282i \(0.992032\pi\)
\(54\) 10.5971 + 1.86855i 1.44208 + 0.254277i
\(55\) 0 0
\(56\) −10.1628 − 12.1116i −1.35807 − 1.61848i
\(57\) 0.462776 − 2.62453i 0.0612962 − 0.347628i
\(58\) 2.35228 + 6.46283i 0.308869 + 0.848612i
\(59\) 1.52849 + 4.19950i 0.198993 + 0.546728i 0.998548 − 0.0538642i \(-0.0171538\pi\)
−0.799556 + 0.600592i \(0.794932\pi\)
\(60\) 0 0
\(61\) −2.71555 + 0.478824i −0.347690 + 0.0613071i −0.344766 − 0.938689i \(-0.612042\pi\)
−0.00292379 + 0.999996i \(0.500931\pi\)
\(62\) 8.47904 − 10.1049i 1.07684 − 1.28333i
\(63\) 8.59501 + 4.96233i 1.08287 + 0.625195i
\(64\) 6.02598 + 10.4373i 0.753248 + 1.30466i
\(65\) 0 0
\(66\) −9.20173 − 5.31262i −1.13265 − 0.653938i
\(67\) −1.54367 + 4.24119i −0.188589 + 0.518144i −0.997569 − 0.0696928i \(-0.977798\pi\)
0.808979 + 0.587837i \(0.200020\pi\)
\(68\) −6.71825 −0.814707
\(69\) −2.77414 + 7.62190i −0.333968 + 0.917568i
\(70\) 0 0
\(71\) −3.35412 + 2.81444i −0.398061 + 0.334013i −0.819744 − 0.572731i \(-0.805884\pi\)
0.421683 + 0.906744i \(0.361440\pi\)
\(72\) 6.08746 + 5.10798i 0.717414 + 0.601982i
\(73\) − 2.68113i − 0.313803i −0.987614 − 0.156902i \(-0.949849\pi\)
0.987614 − 0.156902i \(-0.0501505\pi\)
\(74\) 2.02333 + 14.1408i 0.235207 + 1.64384i
\(75\) 0 0
\(76\) 6.87750 − 8.19628i 0.788903 − 0.940178i
\(77\) −14.7605 − 17.5908i −1.68211 − 2.00466i
\(78\) 9.59925 − 1.69261i 1.08690 − 0.191650i
\(79\) −0.687516 + 1.88893i −0.0773516 + 0.212522i −0.972341 − 0.233566i \(-0.924961\pi\)
0.894989 + 0.446087i \(0.147183\pi\)
\(80\) 0 0
\(81\) −2.52648 − 0.919562i −0.280720 − 0.102174i
\(82\) −0.897894 + 1.55520i −0.0991558 + 0.171743i
\(83\) 1.82162 + 0.321200i 0.199949 + 0.0352563i 0.272725 − 0.962092i \(-0.412075\pi\)
−0.0727768 + 0.997348i \(0.523186\pi\)
\(84\) −6.83778 − 11.8434i −0.746063 − 1.29222i
\(85\) 0 0
\(86\) −0.665457 − 0.558384i −0.0717580 − 0.0602121i
\(87\) 0.445249 + 2.52514i 0.0477358 + 0.270723i
\(88\) −9.19324 − 15.9232i −0.980003 − 1.69741i
\(89\) −2.11794 − 5.81901i −0.224502 − 0.616813i 0.775391 − 0.631482i \(-0.217553\pi\)
−0.999892 + 0.0146684i \(0.995331\pi\)
\(90\) 0 0
\(91\) 20.7458 + 3.65805i 2.17475 + 0.383468i
\(92\) −24.9456 + 20.9318i −2.60075 + 2.18229i
\(93\) 3.76729 − 3.16113i 0.390650 − 0.327794i
\(94\) 26.1975 + 4.61932i 2.70206 + 0.476446i
\(95\) 0 0
\(96\) 1.19871 + 3.29344i 0.122343 + 0.336135i
\(97\) −1.42438 − 2.46710i −0.144624 − 0.250496i 0.784609 − 0.619991i \(-0.212864\pi\)
−0.929233 + 0.369496i \(0.879531\pi\)
\(98\) −5.19783 − 29.4784i −0.525060 − 2.97776i
\(99\) 8.84141 + 7.41882i 0.888595 + 0.745620i
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.ba.c.299.12 72
5.2 odd 4 185.2.w.a.151.1 yes 72
5.3 odd 4 925.2.bb.b.151.12 72
5.4 even 2 925.2.ba.b.299.1 72
37.25 even 18 925.2.ba.b.99.1 72
185.32 even 36 6845.2.a.u.1.2 36
185.42 even 36 6845.2.a.t.1.35 36
185.62 odd 36 185.2.w.a.136.1 ✓ 72
185.99 even 18 inner 925.2.ba.c.99.12 72
185.173 odd 36 925.2.bb.b.876.12 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.1 ✓ 72 185.62 odd 36
185.2.w.a.151.1 yes 72 5.2 odd 4
925.2.ba.b.99.1 72 37.25 even 18
925.2.ba.b.299.1 72 5.4 even 2
925.2.ba.c.99.12 72 185.99 even 18 inner
925.2.ba.c.299.12 72 1.1 even 1 trivial
925.2.bb.b.151.12 72 5.3 odd 4
925.2.bb.b.876.12 72 185.173 odd 36
6845.2.a.t.1.35 36 185.42 even 36
6845.2.a.u.1.2 36 185.32 even 36