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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 324.6
Character \(\chi\) \(=\) 925.324
Dual form 925.2.ba.c.374.6

$q$-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+(-0.0953810 - 0.540933i) q^{2} +(-0.728150 - 0.128393i) q^{3} +(1.59587 - 0.580851i) q^{4} +0.406126i q^{6} +(-1.52397 + 1.81619i) q^{7} +(-1.01569 - 1.75924i) q^{8} +(-2.30536 - 0.839082i) q^{9} +(2.36489 + 4.09611i) q^{11} +(-1.23661 + 0.218048i) q^{12} +(0.815139 - 0.296686i) q^{13} +(1.12779 + 0.651133i) q^{14} +(1.74719 - 1.46606i) q^{16} +(1.35505 + 0.493197i) q^{17} +(-0.234000 + 1.32708i) q^{18} +(6.76229 + 1.19237i) q^{19} +(1.34286 - 1.12679i) q^{21} +(1.99016 - 1.66994i) q^{22} +(3.45466 - 5.98365i) q^{23} +(0.513706 + 1.41140i) q^{24} +(-0.238236 - 0.412637i) q^{26} +(3.49189 + 2.01604i) q^{27} +(-1.37712 + 3.78361i) q^{28} +(8.47779 - 4.89465i) q^{29} -2.80209i q^{31} +(-4.07196 - 3.41678i) q^{32} +(-1.19609 - 3.28622i) q^{33} +(0.137540 - 0.780031i) q^{34} -4.16645 q^{36} +(-3.09431 + 5.23691i) q^{37} -3.77167i q^{38} +(-0.631636 + 0.111374i) q^{39} +(-0.948571 + 0.345252i) q^{41} +(-0.737603 - 0.618923i) q^{42} -5.15867 q^{43} +(6.15330 + 5.16323i) q^{44} +(-3.56626 - 1.29801i) q^{46} +(11.0972 + 6.40700i) q^{47} +(-1.46045 + 0.843189i) q^{48} +(0.239457 + 1.35803i) q^{49} +(-0.923355 - 0.533099i) q^{51} +(1.12853 - 0.946948i) q^{52} +(0.450573 + 0.536972i) q^{53} +(0.757484 - 2.08117i) q^{54} +(4.74299 + 0.836317i) q^{56} +(-4.77087 - 1.73645i) q^{57} +(-3.45630 - 4.11906i) q^{58} +(-3.21523 - 3.83177i) q^{59} +(-3.88070 - 10.6621i) q^{61} +(-1.51574 + 0.267267i) q^{62} +(5.03722 - 2.90824i) q^{63} +(0.820931 - 1.42189i) q^{64} +(-1.66354 + 0.960445i) q^{66} +(4.01637 - 4.78653i) q^{67} +2.44896 q^{68} +(-3.28377 + 3.91344i) q^{69} +(-1.27272 + 7.21798i) q^{71} +(0.865399 + 4.90792i) q^{72} +9.09139i q^{73} +(3.12796 + 1.17431i) q^{74} +(11.4844 - 2.02500i) q^{76} +(-11.0433 - 1.94724i) q^{77} +(0.120492 + 0.331049i) q^{78} +(-7.63743 + 9.10193i) q^{79} +(3.35426 + 2.81456i) q^{81} +(0.277234 + 0.480183i) q^{82} +(5.36381 - 14.7369i) q^{83} +(1.48854 - 2.57822i) q^{84} +(0.492039 + 2.79049i) q^{86} +(-6.80154 + 2.47556i) q^{87} +(4.80402 - 8.32080i) q^{88} +(-0.00507885 - 0.00605274i) q^{89} +(-0.703404 + 1.93259i) q^{91} +(2.03760 - 11.5558i) q^{92} +(-0.359768 + 2.04034i) q^{93} +(2.40729 - 6.61397i) q^{94} +(2.52631 + 3.01074i) q^{96} +(5.48742 - 9.50450i) q^{97} +(0.711762 - 0.259060i) q^{98} +(-2.01495 - 11.4274i) 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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0953810 − 0.540933i −0.0674446 − 0.382497i −0.999781 − 0.0209063i \(-0.993345\pi\)
0.932337 − 0.361591i \(-0.117766\pi\)
\(3\) −0.728150 − 0.128393i −0.420398 − 0.0741275i −0.0405522 − 0.999177i \(-0.512912\pi\)
−0.379845 + 0.925050i \(0.624023\pi\)
\(4\) 1.59587 − 0.580851i 0.797937 − 0.290425i
\(5\) 0 0
\(6\) 0.406126i 0.165800i
\(7\) −1.52397 + 1.81619i −0.576005 + 0.686456i −0.972852 − 0.231429i \(-0.925660\pi\)
0.396847 + 0.917885i \(0.370104\pi\)
\(8\) −1.01569 − 1.75924i −0.359102 − 0.621984i
\(9\) −2.30536 − 0.839082i −0.768453 − 0.279694i
\(10\) 0 0
\(11\) 2.36489 + 4.09611i 0.713042 + 1.23502i 0.963710 + 0.266951i \(0.0860163\pi\)
−0.250668 + 0.968073i \(0.580650\pi\)
\(12\) −1.23661 + 0.218048i −0.356979 + 0.0629451i
\(13\) 0.815139 − 0.296686i 0.226079 − 0.0822859i −0.226497 − 0.974012i \(-0.572727\pi\)
0.452576 + 0.891726i \(0.350505\pi\)
\(14\) 1.12779 + 0.651133i 0.301416 + 0.174023i
\(15\) 0 0
\(16\) 1.74719 − 1.46606i 0.436797 − 0.366516i
\(17\) 1.35505 + 0.493197i 0.328647 + 0.119618i 0.501074 − 0.865405i \(-0.332939\pi\)
−0.172427 + 0.985022i \(0.555161\pi\)
\(18\) −0.234000 + 1.32708i −0.0551542 + 0.312795i
\(19\) 6.76229 + 1.19237i 1.55137 + 0.273549i 0.882675 − 0.469984i \(-0.155740\pi\)
0.668700 + 0.743533i \(0.266851\pi\)
\(20\) 0 0
\(21\) 1.34286 − 1.12679i 0.293036 − 0.245887i
\(22\) 1.99016 − 1.66994i 0.424303 − 0.356032i
\(23\) 3.45466 − 5.98365i 0.720347 − 1.24768i −0.240514 − 0.970646i \(-0.577316\pi\)
0.960861 − 0.277031i \(-0.0893506\pi\)
\(24\) 0.513706 + 1.41140i 0.104860 + 0.288100i
\(25\) 0 0
\(26\) −0.238236 − 0.412637i −0.0467219 − 0.0809248i
\(27\) 3.49189 + 2.01604i 0.672014 + 0.387988i
\(28\) −1.37712 + 3.78361i −0.260251 + 0.715035i
\(29\) 8.47779 − 4.89465i 1.57429 − 0.908914i 0.578651 − 0.815575i \(-0.303579\pi\)
0.995634 − 0.0933391i \(-0.0297541\pi\)
\(30\) 0 0
\(31\) − 2.80209i − 0.503271i −0.967822 − 0.251635i \(-0.919032\pi\)
0.967822 − 0.251635i \(-0.0809684\pi\)
\(32\) −4.07196 − 3.41678i −0.719828 − 0.604007i
\(33\) −1.19609 − 3.28622i −0.208212 − 0.572057i
\(34\) 0.137540 − 0.780031i 0.0235880 − 0.133774i
\(35\) 0 0
\(36\) −4.16645 −0.694408
\(37\) −3.09431 + 5.23691i −0.508702 + 0.860943i
\(38\) − 3.77167i − 0.611846i
\(39\) −0.631636 + 0.111374i −0.101143 + 0.0178342i
\(40\) 0 0
\(41\) −0.948571 + 0.345252i −0.148142 + 0.0539193i −0.415027 − 0.909809i \(-0.636228\pi\)
0.266885 + 0.963728i \(0.414006\pi\)
\(42\) −0.737603 − 0.618923i −0.113815 − 0.0955019i
\(43\) −5.15867 −0.786690 −0.393345 − 0.919391i \(-0.628682\pi\)
−0.393345 + 0.919391i \(0.628682\pi\)
\(44\) 6.15330 + 5.16323i 0.927645 + 0.778387i
\(45\) 0 0
\(46\) −3.56626 − 1.29801i −0.525816 − 0.191382i
\(47\) 11.0972 + 6.40700i 1.61870 + 0.934557i 0.987257 + 0.159135i \(0.0508705\pi\)
0.631443 + 0.775422i \(0.282463\pi\)
\(48\) −1.46045 + 0.843189i −0.210797 + 0.121704i
\(49\) 0.239457 + 1.35803i 0.0342081 + 0.194004i
\(50\) 0 0
\(51\) −0.923355 − 0.533099i −0.129296 − 0.0746488i
\(52\) 1.12853 − 0.946948i 0.156499 − 0.131318i
\(53\) 0.450573 + 0.536972i 0.0618909 + 0.0737588i 0.796102 − 0.605163i \(-0.206892\pi\)
−0.734211 + 0.678922i \(0.762448\pi\)
\(54\) 0.757484 − 2.08117i 0.103081 − 0.283211i
\(55\) 0 0
\(56\) 4.74299 + 0.836317i 0.633809 + 0.111758i
\(57\) −4.77087 − 1.73645i −0.631917 − 0.229999i
\(58\) −3.45630 − 4.11906i −0.453834 − 0.540859i
\(59\) −3.21523 − 3.83177i −0.418588 − 0.498853i 0.515006 − 0.857187i \(-0.327790\pi\)
−0.933594 + 0.358333i \(0.883345\pi\)
\(60\) 0 0
\(61\) −3.88070 − 10.6621i −0.496872 − 1.36515i −0.894282 − 0.447504i \(-0.852313\pi\)
0.397410 − 0.917641i \(-0.369909\pi\)
\(62\) −1.51574 + 0.267267i −0.192500 + 0.0339429i
\(63\) 5.03722 − 2.90824i 0.634630 − 0.366404i
\(64\) 0.820931 − 1.42189i 0.102616 − 0.177737i
\(65\) 0 0
\(66\) −1.66354 + 0.960445i −0.204768 + 0.118223i
\(67\) 4.01637 − 4.78653i 0.490678 − 0.584768i −0.462712 − 0.886509i \(-0.653123\pi\)
0.953390 + 0.301741i \(0.0975679\pi\)
\(68\) 2.44896 0.296980
\(69\) −3.28377 + 3.91344i −0.395319 + 0.471123i
\(70\) 0 0
\(71\) −1.27272 + 7.21798i −0.151045 + 0.856616i 0.811269 + 0.584673i \(0.198777\pi\)
−0.962313 + 0.271943i \(0.912334\pi\)
\(72\) 0.865399 + 4.90792i 0.101988 + 0.578404i
\(73\) 9.09139i 1.06407i 0.846723 + 0.532033i \(0.178572\pi\)
−0.846723 + 0.532033i \(0.821428\pi\)
\(74\) 3.12796 + 1.17431i 0.363617 + 0.136511i
\(75\) 0 0
\(76\) 11.4844 − 2.02500i 1.31735 − 0.232284i
\(77\) −11.0433 − 1.94724i −1.25851 − 0.221908i
\(78\) 0.120492 + 0.331049i 0.0136430 + 0.0374840i
\(79\) −7.63743 + 9.10193i −0.859278 + 1.02405i 0.140147 + 0.990131i \(0.455243\pi\)
−0.999425 + 0.0339168i \(0.989202\pi\)
\(80\) 0 0
\(81\) 3.35426 + 2.81456i 0.372696 + 0.312729i
\(82\) 0.277234 + 0.480183i 0.0306153 + 0.0530273i
\(83\) 5.36381 − 14.7369i 0.588755 − 1.61759i −0.184029 − 0.982921i \(-0.558914\pi\)
0.772784 − 0.634669i \(-0.218864\pi\)
\(84\) 1.48854 − 2.57822i 0.162413 − 0.281307i
\(85\) 0 0
\(86\) 0.492039 + 2.79049i 0.0530580 + 0.300907i
\(87\) −6.80154 + 2.47556i −0.729202 + 0.265408i
\(88\) 4.80402 − 8.32080i 0.512110 − 0.887000i
\(89\) −0.00507885 − 0.00605274i −0.000538357 − 0.000641589i 0.765775 − 0.643108i \(-0.222356\pi\)
−0.766314 + 0.642467i \(0.777911\pi\)
\(90\) 0 0
\(91\) −0.703404 + 1.93259i −0.0737368 + 0.202590i
\(92\) 2.03760 − 11.5558i 0.212434 − 1.20477i
\(93\) −0.359768 + 2.04034i −0.0373062 + 0.211574i
\(94\) 2.40729 − 6.61397i 0.248293 − 0.682179i
\(95\) 0 0
\(96\) 2.52631 + 3.01074i 0.257840 + 0.307282i
\(97\) 5.48742 − 9.50450i 0.557163 − 0.965035i −0.440568 − 0.897719i \(-0.645223\pi\)
0.997732 − 0.0673162i \(-0.0214436\pi\)
\(98\) 0.711762 − 0.259060i 0.0718989 − 0.0261690i
\(99\) −2.01495 − 11.4274i −0.202510 − 1.14849i
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.324.6 72
5.2 odd 4 925.2.bb.b.176.8 72
5.3 odd 4 185.2.w.a.176.5 yes 72
5.4 even 2 925.2.ba.b.324.7 72
37.4 even 18 925.2.ba.b.374.7 72
185.4 even 18 inner 925.2.ba.c.374.6 72
185.78 odd 36 185.2.w.a.41.5 ✓ 72
185.113 even 36 6845.2.a.t.1.17 36
185.152 odd 36 925.2.bb.b.226.8 72
185.183 even 36 6845.2.a.u.1.20 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.5 ✓ 72 185.78 odd 36
185.2.w.a.176.5 yes 72 5.3 odd 4
925.2.ba.b.324.7 72 5.4 even 2
925.2.ba.b.374.7 72 37.4 even 18
925.2.ba.c.324.6 72 1.1 even 1 trivial
925.2.ba.c.374.6 72 185.4 even 18 inner
925.2.bb.b.176.8 72 5.2 odd 4
925.2.bb.b.226.8 72 185.152 odd 36
6845.2.a.t.1.17 36 185.113 even 36
6845.2.a.u.1.20 36 185.183 even 36