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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 176.8
Character \(\chi\) \(=\) 925.176
Dual form 925.2.bb.b.226.8

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.540933 − 0.0953810i 0.382497 − 0.0674446i 0.0209063 − 0.999781i \(-0.493345\pi\)
0.361591 + 0.932337i \(0.382234\pi\)
\(3\) −0.128393 + 0.728150i −0.0741275 + 0.420398i 0.925050 + 0.379845i \(0.124023\pi\)
−0.999177 + 0.0405522i \(0.987088\pi\)
\(4\) −1.59587 + 0.580851i −0.797937 + 0.290425i
\(5\) 0 0
\(6\) 0.406126i 0.165800i
\(7\) −1.81619 − 1.52397i −0.686456 − 0.576005i 0.231429 − 0.972852i \(-0.425660\pi\)
−0.917885 + 0.396847i \(0.870104\pi\)
\(8\) −1.75924 + 1.01569i −0.621984 + 0.359102i
\(9\) 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 \(0.580650\pi\)
\(12\) −0.218048 − 1.23661i −0.0629451 − 0.356979i
\(13\) −0.296686 − 0.815139i −0.0822859 − 0.226079i 0.891726 − 0.452576i \(-0.149495\pi\)
−0.974012 + 0.226497i \(0.927273\pi\)
\(14\) −1.12779 − 0.651133i −0.301416 − 0.174023i
\(15\) 0 0
\(16\) 1.74719 − 1.46606i 0.436797 − 0.366516i
\(17\) −0.493197 + 1.35505i −0.119618 + 0.328647i −0.985022 − 0.172427i \(-0.944839\pi\)
0.865405 + 0.501074i \(0.167061\pi\)
\(18\) 1.32708 + 0.234000i 0.312795 + 0.0551542i
\(19\) −6.76229 − 1.19237i −1.55137 − 0.273549i −0.668700 − 0.743533i \(-0.733149\pi\)
−0.882675 + 0.469984i \(0.844260\pi\)
\(20\) 0 0
\(21\) 1.34286 − 1.12679i 0.293036 − 0.245887i
\(22\) 1.66994 + 1.99016i 0.356032 + 0.424303i
\(23\) −5.98365 − 3.45466i −1.24768 − 0.720347i −0.277031 − 0.960861i \(-0.589351\pi\)
−0.970646 + 0.240514i \(0.922684\pi\)
\(24\) −0.513706 − 1.41140i −0.104860 − 0.288100i
\(25\) 0 0
\(26\) −0.238236 − 0.412637i −0.0467219 − 0.0809248i
\(27\) −2.01604 + 3.49189i −0.387988 + 0.672014i
\(28\) 3.78361 + 1.37712i 0.715035 + 0.260251i
\(29\) −8.47779 + 4.89465i −1.57429 + 0.908914i −0.578651 + 0.815575i \(0.696421\pi\)
−0.995634 + 0.0933391i \(0.970246\pi\)
\(30\) 0 0
\(31\) − 2.80209i − 0.503271i −0.967822 − 0.251635i \(-0.919032\pi\)
0.967822 − 0.251635i \(-0.0809684\pi\)
\(32\) 3.41678 − 4.07196i 0.604007 − 0.719828i
\(33\) −3.28622 + 1.19609i −0.572057 + 0.208212i
\(34\) −0.137540 + 0.780031i −0.0235880 + 0.133774i
\(35\) 0 0
\(36\) −4.16645 −0.694408
\(37\) −5.23691 − 3.09431i −0.860943 − 0.508702i
\(38\) −3.77167 −0.611846
\(39\) 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 \(0.414006\pi\)
\(42\) 0.618923 − 0.737603i 0.0955019 − 0.113815i
\(43\) 5.15867i 0.786690i 0.919391 + 0.393345i \(0.128682\pi\)
−0.919391 + 0.393345i \(0.871318\pi\)
\(44\) −6.15330 − 5.16323i −0.927645 − 0.778387i
\(45\) 0 0
\(46\) −3.56626 − 1.29801i −0.525816 − 0.191382i
\(47\) −6.40700 + 11.0972i −0.934557 + 1.61870i −0.159135 + 0.987257i \(0.550870\pi\)
−0.775422 + 0.631443i \(0.782463\pi\)
\(48\) 0.843189 + 1.46045i 0.121704 + 0.210797i
\(49\) −0.239457 − 1.35803i −0.0342081 − 0.194004i
\(50\) 0 0
\(51\) −0.923355 − 0.533099i −0.129296 − 0.0746488i
\(52\) 0.946948 + 1.12853i 0.131318 + 0.156499i
\(53\) 0.536972 − 0.450573i 0.0737588 − 0.0618909i −0.605163 − 0.796102i \(-0.706892\pi\)
0.678922 + 0.734211i \(0.262448\pi\)
\(54\) −0.757484 + 2.08117i −0.103081 + 0.283211i
\(55\) 0 0
\(56\) 4.74299 + 0.836317i 0.633809 + 0.111758i
\(57\) 1.73645 − 4.77087i 0.229999 − 0.631917i
\(58\) −4.11906 + 3.45630i −0.540859 + 0.453834i
\(59\) 3.21523 + 3.83177i 0.418588 + 0.498853i 0.933594 − 0.358333i \(-0.116655\pi\)
−0.515006 + 0.857187i \(0.672210\pi\)
\(60\) 0 0
\(61\) −3.88070 − 10.6621i −0.496872 − 1.36515i −0.894282 − 0.447504i \(-0.852313\pi\)
0.397410 − 0.917641i \(-0.369909\pi\)
\(62\) −0.267267 − 1.51574i −0.0339429 − 0.192500i
\(63\) −2.90824 − 5.03722i −0.366404 − 0.634630i
\(64\) −0.820931 + 1.42189i −0.102616 + 0.177737i
\(65\) 0 0
\(66\) −1.66354 + 0.960445i −0.204768 + 0.118223i
\(67\) 4.78653 + 4.01637i 0.584768 + 0.490678i 0.886509 − 0.462712i \(-0.153123\pi\)
−0.301741 + 0.953390i \(0.597568\pi\)
\(68\) − 2.44896i − 0.296980i
\(69\) 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 \(0.912334\pi\)
\(72\) −4.90792 + 0.865399i −0.578404 + 0.101988i
\(73\) 9.09139 1.06407 0.532033 − 0.846723i \(-0.321428\pi\)
0.532033 + 0.846723i \(0.321428\pi\)
\(74\) −3.12796 − 1.17431i −0.363617 − 0.136511i
\(75\) 0 0
\(76\) 11.4844 − 2.02500i 1.31735 − 0.232284i
\(77\) 1.94724 − 11.0433i 0.221908 − 1.25851i
\(78\) 0.331049 − 0.120492i 0.0374840 − 0.0136430i
\(79\) 7.63743 − 9.10193i 0.859278 − 1.02405i −0.140147 − 0.990131i \(-0.544757\pi\)
0.999425 − 0.0339168i \(-0.0107981\pi\)
\(80\) 0 0
\(81\) 3.35426 + 2.81456i 0.372696 + 0.312729i
\(82\) −0.480183 + 0.277234i −0.0530273 + 0.0306153i
\(83\) −14.7369 − 5.36381i −1.61759 − 0.588755i −0.634669 − 0.772784i \(-0.718864\pi\)
−0.982921 + 0.184029i \(0.941086\pi\)
\(84\) −1.48854 + 2.57822i −0.162413 + 0.281307i
\(85\) 0 0
\(86\) 0.492039 + 2.79049i 0.0530580 + 0.300907i
\(87\) −2.47556 − 6.80154i −0.265408 − 0.729202i
\(88\) −8.32080 − 4.80402i −0.887000 − 0.512110i
\(89\) 0.00507885 + 0.00605274i 0.000538357 + 0.000641589i 0.766314 − 0.642467i \(-0.222089\pi\)
−0.765775 + 0.643108i \(0.777644\pi\)
\(90\) 0 0
\(91\) −0.703404 + 1.93259i −0.0737368 + 0.202590i
\(92\) 11.5558 + 2.03760i 1.20477 + 0.212434i
\(93\) 2.04034 + 0.359768i 0.211574 + 0.0373062i
\(94\) −2.40729 + 6.61397i −0.248293 + 0.682179i
\(95\) 0 0
\(96\) 2.52631 + 3.01074i 0.257840 + 0.307282i
\(97\) 9.50450 + 5.48742i 0.965035 + 0.557163i 0.897719 − 0.440568i \(-0.145223\pi\)
0.0673162 + 0.997732i \(0.478556\pi\)
\(98\) −0.259060 − 0.711762i −0.0261690 − 0.0718989i
\(99\) 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.bb.b.176.8 72
5.2 odd 4 925.2.ba.b.324.7 72
5.3 odd 4 925.2.ba.c.324.6 72
5.4 even 2 185.2.w.a.176.5 yes 72
37.4 even 18 inner 925.2.bb.b.226.8 72
185.4 even 18 185.2.w.a.41.5 ✓ 72
185.39 odd 36 6845.2.a.t.1.17 36
185.78 odd 36 925.2.ba.b.374.7 72
185.109 odd 36 6845.2.a.u.1.20 36
185.152 odd 36 925.2.ba.c.374.6 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.5 ✓ 72 185.4 even 18
185.2.w.a.176.5 yes 72 5.4 even 2
925.2.ba.b.324.7 72 5.2 odd 4
925.2.ba.b.374.7 72 185.78 odd 36
925.2.ba.c.324.6 72 5.3 odd 4
925.2.ba.c.374.6 72 185.152 odd 36
925.2.bb.b.176.8 72 1.1 even 1 trivial
925.2.bb.b.226.8 72 37.4 even 18 inner
6845.2.a.t.1.17 36 185.39 odd 36
6845.2.a.u.1.20 36 185.109 odd 36