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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 99.11
Character \(\chi\) \(=\) 925.99
Dual form 925.2.ba.c.299.11

$q$-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+(1.53924 - 1.29158i) q^{2} +(-0.489982 + 0.583938i) q^{3} +(0.353796 - 2.00647i) q^{4} +1.53167i q^{6} +(0.593686 + 1.63114i) q^{7} +(-0.0376024 - 0.0651292i) q^{8} +(0.420044 + 2.38219i) q^{9} +(-0.381991 - 0.661628i) q^{11} +(0.998303 + 1.18973i) q^{12} +(0.00681312 - 0.0386391i) q^{13} +(3.02057 + 1.74393i) q^{14} +(3.68711 + 1.34200i) q^{16} +(0.758742 + 4.30304i) q^{17} +(3.72332 + 3.12424i) q^{18} +(-2.89648 + 3.45189i) q^{19} +(-1.24338 - 0.452553i) q^{21} +(-1.44252 - 0.525034i) q^{22} +(-0.913472 + 1.58218i) q^{23} +(0.0564559 + 0.00995470i) q^{24} +(-0.0394183 - 0.0682745i) q^{26} +(-3.57731 - 2.06536i) q^{27} +(3.48288 - 0.614127i) q^{28} +(4.07808 - 2.35448i) q^{29} -6.00718i q^{31} +(7.54998 - 2.74797i) q^{32} +(0.573518 + 0.101127i) q^{33} +(6.72559 + 5.64344i) q^{34} +4.92840 q^{36} +(5.76541 + 1.93907i) q^{37} +9.05431i q^{38} +(0.0192245 + 0.0229109i) q^{39} +(1.01550 - 5.75920i) q^{41} +(-2.49837 + 0.909331i) q^{42} +3.43382 q^{43} +(-1.46269 + 0.532374i) q^{44} +(0.637453 + 3.61517i) q^{46} +(3.14471 + 1.81560i) q^{47} +(-2.59026 + 1.49549i) q^{48} +(3.05416 - 2.56274i) q^{49} +(-2.88448 - 1.66535i) q^{51} +(-0.0751179 - 0.0273407i) q^{52} +(-4.08338 + 11.2190i) q^{53} +(-8.17392 + 1.44128i) q^{54} +(0.0839109 - 0.100001i) q^{56} +(-0.596466 - 3.38273i) q^{57} +(3.23615 - 8.89125i) q^{58} +(2.59304 - 7.12432i) q^{59} +(-5.71230 - 1.00723i) q^{61} +(-7.75873 - 9.24649i) q^{62} +(-3.63630 + 2.09942i) q^{63} +(4.14828 - 7.18504i) q^{64} +(1.01339 - 0.585084i) q^{66} +(-0.0977503 - 0.268567i) q^{67} +8.90238 q^{68} +(-0.476310 - 1.30865i) q^{69} +(-4.33845 - 3.64039i) q^{71} +(0.139355 - 0.116933i) q^{72} -4.13245i q^{73} +(11.3788 - 4.46177i) q^{74} +(5.90137 + 7.03298i) q^{76} +(0.852425 - 1.01588i) q^{77} +(0.0591823 + 0.0104354i) q^{78} +(0.799712 + 2.19719i) q^{79} +(-3.86030 + 1.40504i) q^{81} +(-5.87534 - 10.1764i) q^{82} +(-2.61385 + 0.460892i) q^{83} +(-1.34794 + 2.33470i) q^{84} +(5.28548 - 4.43504i) q^{86} +(-0.623315 + 3.53499i) q^{87} +(-0.0287276 + 0.0497576i) q^{88} +(5.87216 - 16.1336i) q^{89} +(0.0670707 - 0.0118264i) q^{91} +(2.85142 + 2.39263i) q^{92} +(3.50782 + 2.94341i) q^{93} +(7.18544 - 1.26699i) q^{94} +(-2.09471 + 5.75518i) q^{96} +(-7.43673 + 12.8808i) q^{97} +(1.39111 - 7.88935i) q^{98} +(1.41567 - 1.18789i) 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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53924 − 1.29158i 1.08841 − 0.913282i 0.0918155 − 0.995776i \(-0.470733\pi\)
0.996592 + 0.0824941i \(0.0262886\pi\)
\(3\) −0.489982 + 0.583938i −0.282891 + 0.337137i −0.888713 − 0.458464i \(-0.848400\pi\)
0.605822 + 0.795600i \(0.292844\pi\)
\(4\) 0.353796 − 2.00647i 0.176898 − 1.00324i
\(5\) 0 0
\(6\) 1.53167i 0.625301i
\(7\) 0.593686 + 1.63114i 0.224392 + 0.616513i 0.999890 − 0.0148376i \(-0.00472313\pi\)
−0.775498 + 0.631351i \(0.782501\pi\)
\(8\) −0.0376024 − 0.0651292i −0.0132945 − 0.0230267i
\(9\) 0.420044 + 2.38219i 0.140015 + 0.794062i
\(10\) 0 0
\(11\) −0.381991 − 0.661628i −0.115175 − 0.199488i 0.802675 − 0.596417i \(-0.203409\pi\)
−0.917850 + 0.396928i \(0.870076\pi\)
\(12\) 0.998303 + 1.18973i 0.288185 + 0.343446i
\(13\) 0.00681312 − 0.0386391i 0.00188962 − 0.0107166i −0.983848 − 0.179004i \(-0.942713\pi\)
0.985738 + 0.168287i \(0.0538236\pi\)
\(14\) 3.02057 + 1.74393i 0.807280 + 0.466084i
\(15\) 0 0
\(16\) 3.68711 + 1.34200i 0.921778 + 0.335500i
\(17\) 0.758742 + 4.30304i 0.184022 + 1.04364i 0.927206 + 0.374553i \(0.122204\pi\)
−0.743184 + 0.669088i \(0.766685\pi\)
\(18\) 3.72332 + 3.12424i 0.877595 + 0.736390i
\(19\) −2.89648 + 3.45189i −0.664498 + 0.791918i −0.988024 − 0.154301i \(-0.950687\pi\)
0.323526 + 0.946219i \(0.395132\pi\)
\(20\) 0 0
\(21\) −1.24338 − 0.452553i −0.271328 − 0.0987552i
\(22\) −1.44252 − 0.525034i −0.307546 − 0.111938i
\(23\) −0.913472 + 1.58218i −0.190472 + 0.329907i −0.945407 − 0.325893i \(-0.894335\pi\)
0.754935 + 0.655800i \(0.227669\pi\)
\(24\) 0.0564559 + 0.00995470i 0.0115240 + 0.00203199i
\(25\) 0 0
\(26\) −0.0394183 − 0.0682745i −0.00773057 − 0.0133897i
\(27\) −3.57731 − 2.06536i −0.688454 − 0.397479i
\(28\) 3.48288 − 0.614127i 0.658203 − 0.116059i
\(29\) 4.07808 − 2.35448i 0.757280 − 0.437216i −0.0710385 − 0.997474i \(-0.522631\pi\)
0.828318 + 0.560258i \(0.189298\pi\)
\(30\) 0 0
\(31\) − 6.00718i − 1.07892i −0.842011 − 0.539461i \(-0.818628\pi\)
0.842011 − 0.539461i \(-0.181372\pi\)
\(32\) 7.54998 − 2.74797i 1.33466 − 0.485777i
\(33\) 0.573518 + 0.101127i 0.0998367 + 0.0176039i
\(34\) 6.72559 + 5.64344i 1.15343 + 0.967842i
\(35\) 0 0
\(36\) 4.92840 0.821401
\(37\) 5.76541 + 1.93907i 0.947828 + 0.318782i
\(38\) 9.05431i 1.46880i
\(39\) 0.0192245 + 0.0229109i 0.00307839 + 0.00366868i
\(40\) 0 0
\(41\) 1.01550 − 5.75920i 0.158595 − 0.899436i −0.796830 − 0.604203i \(-0.793492\pi\)
0.955425 − 0.295233i \(-0.0953973\pi\)
\(42\) −2.49837 + 0.909331i −0.385506 + 0.140313i
\(43\) 3.43382 0.523653 0.261827 − 0.965115i \(-0.415675\pi\)
0.261827 + 0.965115i \(0.415675\pi\)
\(44\) −1.46269 + 0.532374i −0.220508 + 0.0802584i
\(45\) 0 0
\(46\) 0.637453 + 3.61517i 0.0939873 + 0.533028i
\(47\) 3.14471 + 1.81560i 0.458703 + 0.264832i 0.711499 − 0.702688i \(-0.248017\pi\)
−0.252796 + 0.967520i \(0.581350\pi\)
\(48\) −2.59026 + 1.49549i −0.373872 + 0.215855i
\(49\) 3.05416 − 2.56274i 0.436308 − 0.366106i
\(50\) 0 0
\(51\) −2.88448 − 1.66535i −0.403907 − 0.233196i
\(52\) −0.0751179 − 0.0273407i −0.0104170 − 0.00379147i
\(53\) −4.08338 + 11.2190i −0.560895 + 1.54105i 0.257434 + 0.966296i \(0.417123\pi\)
−0.818329 + 0.574750i \(0.805099\pi\)
\(54\) −8.17392 + 1.44128i −1.11233 + 0.196134i
\(55\) 0 0
\(56\) 0.0839109 − 0.100001i 0.0112131 − 0.0133632i
\(57\) −0.596466 − 3.38273i −0.0790039 − 0.448053i
\(58\) 3.23615 − 8.89125i 0.424927 − 1.16748i
\(59\) 2.59304 − 7.12432i 0.337585 − 0.927507i −0.648492 − 0.761221i \(-0.724600\pi\)
0.986078 − 0.166286i \(-0.0531776\pi\)
\(60\) 0 0
\(61\) −5.71230 − 1.00723i −0.731385 − 0.128963i −0.204460 − 0.978875i \(-0.565544\pi\)
−0.526925 + 0.849912i \(0.676655\pi\)
\(62\) −7.75873 − 9.24649i −0.985359 − 1.17431i
\(63\) −3.63630 + 2.09942i −0.458131 + 0.264502i
\(64\) 4.14828 − 7.18504i 0.518535 − 0.898130i
\(65\) 0 0
\(66\) 1.01339 − 0.585084i 0.124740 − 0.0720188i
\(67\) −0.0977503 − 0.268567i −0.0119421 − 0.0328106i 0.933578 − 0.358373i \(-0.116669\pi\)
−0.945520 + 0.325563i \(0.894446\pi\)
\(68\) 8.90238 1.07957
\(69\) −0.476310 − 1.30865i −0.0573410 − 0.157543i
\(70\) 0 0
\(71\) −4.33845 − 3.64039i −0.514879 − 0.432035i 0.347963 − 0.937508i \(-0.386874\pi\)
−0.862842 + 0.505473i \(0.831318\pi\)
\(72\) 0.139355 − 0.116933i 0.0164232 − 0.0137807i
\(73\) − 4.13245i − 0.483667i −0.970318 − 0.241834i \(-0.922251\pi\)
0.970318 − 0.241834i \(-0.0777488\pi\)
\(74\) 11.3788 − 4.46177i 1.32276 − 0.518670i
\(75\) 0 0
\(76\) 5.90137 + 7.03298i 0.676933 + 0.806738i
\(77\) 0.852425 − 1.01588i 0.0971428 − 0.115770i
\(78\) 0.0591823 + 0.0104354i 0.00670108 + 0.00118158i
\(79\) 0.799712 + 2.19719i 0.0899746 + 0.247203i 0.976516 − 0.215447i \(-0.0691208\pi\)
−0.886541 + 0.462650i \(0.846899\pi\)
\(80\) 0 0
\(81\) −3.86030 + 1.40504i −0.428923 + 0.156115i
\(82\) −5.87534 − 10.1764i −0.648823 − 1.12379i
\(83\) −2.61385 + 0.460892i −0.286907 + 0.0505895i −0.315250 − 0.949009i \(-0.602088\pi\)
0.0283424 + 0.999598i \(0.490977\pi\)
\(84\) −1.34794 + 2.33470i −0.147072 + 0.254736i
\(85\) 0 0
\(86\) 5.28548 − 4.43504i 0.569948 − 0.478243i
\(87\) −0.623315 + 3.53499i −0.0668264 + 0.378991i
\(88\) −0.0287276 + 0.0497576i −0.00306237 + 0.00530418i
\(89\) 5.87216 − 16.1336i 0.622448 − 1.71016i −0.0784679 − 0.996917i \(-0.525003\pi\)
0.700916 − 0.713244i \(-0.252775\pi\)
\(90\) 0 0
\(91\) 0.0670707 − 0.0118264i 0.00703092 − 0.00123974i
\(92\) 2.85142 + 2.39263i 0.297281 + 0.249449i
\(93\) 3.50782 + 2.94341i 0.363744 + 0.305217i
\(94\) 7.18544 − 1.26699i 0.741122 − 0.130680i
\(95\) 0 0
\(96\) −2.09471 + 5.75518i −0.213791 + 0.587385i
\(97\) −7.43673 + 12.8808i −0.755086 + 1.30785i 0.190246 + 0.981736i \(0.439072\pi\)
−0.945332 + 0.326111i \(0.894262\pi\)
\(98\) 1.39111 − 7.88935i 0.140523 − 0.796945i
\(99\) 1.41567 − 1.18789i 0.142280 − 0.119387i
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.99.11 72
5.2 odd 4 185.2.w.a.136.11 ✓ 72
5.3 odd 4 925.2.bb.b.876.2 72
5.4 even 2 925.2.ba.b.99.2 72
37.3 even 18 925.2.ba.b.299.2 72
185.3 odd 36 925.2.bb.b.151.2 72
185.22 even 36 6845.2.a.t.1.32 36
185.52 even 36 6845.2.a.u.1.5 36
185.77 odd 36 185.2.w.a.151.11 yes 72
185.114 even 18 inner 925.2.ba.c.299.11 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.11 ✓ 72 5.2 odd 4
185.2.w.a.151.11 yes 72 185.77 odd 36
925.2.ba.b.99.2 72 5.4 even 2
925.2.ba.b.299.2 72 37.3 even 18
925.2.ba.c.99.11 72 1.1 even 1 trivial
925.2.ba.c.299.11 72 185.114 even 18 inner
925.2.bb.b.151.2 72 185.3 odd 36
925.2.bb.b.876.2 72 5.3 odd 4
6845.2.a.t.1.32 36 185.22 even 36
6845.2.a.u.1.5 36 185.52 even 36