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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 299.2
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.b.99.2

$q$-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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53924 − 1.29158i −1.08841 − 0.913282i −0.0918155 − 0.995776i \(-0.529267\pi\)
−0.996592 + 0.0824941i \(0.973711\pi\)
\(3\) 0.489982 + 0.583938i 0.282891 + 0.337137i 0.888713 − 0.458464i \(-0.151600\pi\)
−0.605822 + 0.795600i \(0.707156\pi\)
\(4\) 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 \(-0.995277\pi\)
0.775498 + 0.631351i \(0.217499\pi\)
\(8\) 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 −0.917850 − 0.396928i \(-0.870076\pi\)
0.802675 + 0.596417i \(0.203409\pi\)
\(12\) −0.998303 + 1.18973i −0.288185 + 0.343446i
\(13\) −0.00681312 − 0.0386391i −0.00188962 − 0.0107166i 0.983848 − 0.179004i \(-0.0572875\pi\)
−0.985738 + 0.168287i \(0.946176\pi\)
\(14\) 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 0.743184 + 0.669088i \(0.233315\pi\)
−0.927206 + 0.374553i \(0.877796\pi\)
\(18\) −3.72332 + 3.12424i −0.877595 + 0.736390i
\(19\) −2.89648 − 3.45189i −0.664498 − 0.791918i 0.323526 − 0.946219i \(-0.395132\pi\)
−0.988024 + 0.154301i \(0.950687\pi\)
\(20\) 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 \(-0.105665\pi\)
−0.754935 + 0.655800i \(0.772331\pi\)
\(24\) 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 0.828318 − 0.560258i \(-0.189298\pi\)
−0.0710385 + 0.997474i \(0.522631\pi\)
\(30\) 0 0
\(31\) 6.00718i 1.07892i 0.842011 + 0.539461i \(0.181372\pi\)
−0.842011 + 0.539461i \(0.818628\pi\)
\(32\) −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 0.955425 + 0.295233i \(0.0953973\pi\)
−0.796830 + 0.604203i \(0.793492\pi\)
\(42\) 2.49837 + 0.909331i 0.385506 + 0.140313i
\(43\) −3.43382 −0.523653 −0.261827 − 0.965115i \(-0.584325\pi\)
−0.261827 + 0.965115i \(0.584325\pi\)
\(44\) −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 \(-0.751983\pi\)
0.252796 + 0.967520i \(0.418650\pi\)
\(48\) 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 0.818329 + 0.574750i \(0.194901\pi\)
−0.257434 + 0.966296i \(0.582877\pi\)
\(54\) −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 0.986078 + 0.166286i \(0.0531776\pi\)
−0.648492 + 0.761221i \(0.724600\pi\)
\(60\) 0 0
\(61\) −5.71230 + 1.00723i −0.731385 + 0.128963i −0.526925 − 0.849912i \(-0.676655\pi\)
−0.204460 + 0.978875i \(0.565544\pi\)
\(62\) 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 \(-0.883331\pi\)
0.945520 + 0.325563i \(0.105554\pi\)
\(68\) −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 −0.862842 − 0.505473i \(-0.831318\pi\)
0.347963 + 0.937508i \(0.386874\pi\)
\(72\) −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 −0.886541 − 0.462650i \(-0.846899\pi\)
0.976516 + 0.215447i \(0.0691208\pi\)
\(80\) 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 \(-0.397912\pi\)
−0.0283424 + 0.999598i \(0.509023\pi\)
\(84\) −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 0.700916 + 0.713244i \(0.252775\pi\)
−0.0784679 + 0.996917i \(0.525003\pi\)
\(90\) 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 0.945332 + 0.326111i \(0.105738\pi\)
−0.190246 + 0.981736i \(0.560928\pi\)
\(98\) −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.b.299.2 72
5.2 odd 4 185.2.w.a.151.11 yes 72
5.3 odd 4 925.2.bb.b.151.2 72
5.4 even 2 925.2.ba.c.299.11 72
37.25 even 18 925.2.ba.c.99.11 72
185.32 even 36 6845.2.a.t.1.32 36
185.42 even 36 6845.2.a.u.1.5 36
185.62 odd 36 185.2.w.a.136.11 ✓ 72
185.99 even 18 inner 925.2.ba.b.99.2 72
185.173 odd 36 925.2.bb.b.876.2 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.11 ✓ 72 185.62 odd 36
185.2.w.a.151.11 yes 72 5.2 odd 4
925.2.ba.b.99.2 72 185.99 even 18 inner
925.2.ba.b.299.2 72 1.1 even 1 trivial
925.2.ba.c.99.11 72 37.25 even 18
925.2.ba.c.299.11 72 5.4 even 2
925.2.bb.b.151.2 72 5.3 odd 4
925.2.bb.b.876.2 72 185.173 odd 36
6845.2.a.t.1.32 36 185.32 even 36
6845.2.a.u.1.5 36 185.42 even 36