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Properties

Label 875.2.bb.a.593.6
Level $875$
Weight $2$
Character 875.593
Analytic conductor $6.987$
Analytic rank $0$
Dimension $288$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [875,2,Mod(82,875)] 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("875.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(875, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([27, 50])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.bb (of order \(60\), degree \(16\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [288,-2] 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: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{60})\)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label 593.6
Character \(\chi\) \(=\) 875.593
Dual form 875.2.bb.a.332.6

$q$-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+(-1.08516 - 0.878741i) q^{2} +(1.75892 - 2.70850i) q^{3} +(-0.0104480 - 0.0491540i) q^{4} +(-4.28878 + 1.39351i) q^{6} +(-1.16570 + 2.37511i) q^{7} +(-1.29970 + 2.55081i) q^{8} +(-3.02197 - 6.78745i) q^{9} +(-0.600952 - 0.267561i) q^{11} +(-0.151511 - 0.0581596i) q^{12} +(-5.16984 + 0.818822i) q^{13} +(3.35207 - 1.55302i) q^{14} +(3.56006 - 1.58504i) q^{16} +(0.0289236 - 0.00151582i) q^{17} +(-2.68511 + 10.0210i) q^{18} +(0.311343 + 0.0661780i) q^{19} +(4.38262 + 7.33493i) q^{21} +(0.417009 + 0.818427i) q^{22} +(-2.99243 + 3.69534i) q^{23} +(4.62279 + 8.00691i) q^{24} +(6.32961 + 3.65440i) q^{26} +(-14.1299 - 2.23796i) q^{27} +(0.128925 + 0.0324836i) q^{28} +(-6.20126 - 2.01491i) q^{29} +(-3.03232 - 2.73031i) q^{31} +(0.274517 + 0.0735567i) q^{32} +(-1.78172 + 1.15706i) q^{33} +(-0.0327187 - 0.0237715i) q^{34} +(-0.302057 + 0.219457i) q^{36} +(-0.486654 + 1.26778i) q^{37} +(-0.279702 - 0.345403i) q^{38} +(-6.87556 + 15.4428i) q^{39} +(1.10223 - 1.51708i) q^{41} +(1.68968 - 11.8107i) q^{42} +(-2.47415 - 2.47415i) q^{43} +(-0.00687295 + 0.0323347i) q^{44} +(6.49449 - 1.38045i) q^{46} +(0.154192 - 2.94215i) q^{47} +(1.96878 - 12.4304i) q^{48} +(-4.28229 - 5.53732i) q^{49} +(0.0467688 - 0.0810060i) q^{51} +(0.0942629 + 0.245563i) q^{52} +(-0.558452 - 0.362663i) q^{53} +(13.3666 + 14.8451i) q^{54} +(-4.54339 - 6.06040i) q^{56} +(0.726871 - 0.726871i) q^{57} +(4.95874 + 7.63580i) q^{58} +(-0.345447 + 3.28671i) q^{59} +(10.5899 - 1.11304i) q^{61} +(0.891298 + 5.62743i) q^{62} +(19.6436 + 0.734614i) q^{63} +(-4.81442 - 6.62648i) q^{64} +(2.95020 + 0.310078i) q^{66} +(-0.747728 - 14.2675i) q^{67} +(-0.000376703 - 0.00140588i) q^{68} +(4.74539 + 14.6048i) q^{69} +(-0.171405 + 0.527530i) q^{71} +(21.2411 + 1.11320i) q^{72} +(0.756474 - 0.290383i) q^{73} +(1.64214 - 0.948091i) q^{74} -0.0159952i q^{76} +(1.33602 - 1.11543i) q^{77} +(21.0312 - 10.7159i) q^{78} +(9.72376 - 8.75531i) q^{79} +(-16.0005 + 17.7704i) q^{81} +(-2.52921 + 0.677700i) q^{82} +(-6.95799 - 3.54527i) q^{83} +(0.314752 - 0.292059i) q^{84} +(0.510698 + 4.85897i) q^{86} +(-16.3649 + 13.2521i) q^{87} +(1.46355 - 1.18516i) q^{88} +(-1.21799 - 11.5884i) q^{89} +(4.08168 - 13.2334i) q^{91} +(0.212906 + 0.108481i) q^{92} +(-12.7287 + 3.41063i) q^{93} +(-2.75271 + 3.05720i) q^{94} +(0.682083 - 0.614150i) q^{96} +(3.47390 - 1.77004i) q^{97} +(-0.218921 + 9.77188i) q^{98} +4.88749i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q - 2 q^{2} - 6 q^{3} + 10 q^{4} + 10 q^{7} - 64 q^{8} + 10 q^{9} - 6 q^{11} + 6 q^{12} + 20 q^{14} - 30 q^{16} + 12 q^{17} + 14 q^{18} + 30 q^{19} - 12 q^{21} + 8 q^{22} - 30 q^{23} - 48 q^{26} + 58 q^{28}+ \cdots - 62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/875\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(626\)
\(\chi(n)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08516 − 0.878741i −0.767321 − 0.621364i 0.163625 − 0.986523i \(-0.447681\pi\)
−0.930945 + 0.365159i \(0.881015\pi\)
\(3\) 1.75892 − 2.70850i 1.01551 − 1.56375i 0.204382 − 0.978891i \(-0.434482\pi\)
0.811132 − 0.584863i \(-0.198852\pi\)
\(4\) −0.0104480 − 0.0491540i −0.00522401 − 0.0245770i
\(5\) 0 0
\(6\) −4.28878 + 1.39351i −1.75089 + 0.568897i
\(7\) −1.16570 + 2.37511i −0.440593 + 0.897707i
\(8\) −1.29970 + 2.55081i −0.459513 + 0.901846i
\(9\) −3.02197 − 6.78745i −1.00732 − 2.26248i
\(10\) 0 0
\(11\) −0.600952 − 0.267561i −0.181194 − 0.0806727i 0.314135 − 0.949378i \(-0.398286\pi\)
−0.495329 + 0.868706i \(0.664952\pi\)
\(12\) −0.151511 − 0.0581596i −0.0437375 − 0.0167892i
\(13\) −5.16984 + 0.818822i −1.43385 + 0.227100i −0.824529 − 0.565820i \(-0.808560\pi\)
−0.609326 + 0.792920i \(0.708560\pi\)
\(14\) 3.35207 − 1.55302i 0.895879 − 0.415061i
\(15\) 0 0
\(16\) 3.56006 − 1.58504i 0.890015 − 0.396260i
\(17\) 0.0289236 − 0.00151582i 0.00701501 − 0.000367641i −0.0488281 − 0.998807i \(-0.515549\pi\)
0.0558431 + 0.998440i \(0.482215\pi\)
\(18\) −2.68511 + 10.0210i −0.632886 + 2.36196i
\(19\) 0.311343 + 0.0661780i 0.0714270 + 0.0151823i 0.243487 − 0.969904i \(-0.421709\pi\)
−0.172060 + 0.985087i \(0.555042\pi\)
\(20\) 0 0
\(21\) 4.38262 + 7.33493i 0.956366 + 1.60061i
\(22\) 0.417009 + 0.818427i 0.0889067 + 0.174489i
\(23\) −2.99243 + 3.69534i −0.623964 + 0.770531i −0.986909 − 0.161279i \(-0.948438\pi\)
0.362945 + 0.931811i \(0.381771\pi\)
\(24\) 4.62279 + 8.00691i 0.943623 + 1.63440i
\(25\) 0 0
\(26\) 6.32961 + 3.65440i 1.24134 + 0.716687i
\(27\) −14.1299 − 2.23796i −2.71931 − 0.430696i
\(28\) 0.128925 + 0.0324836i 0.0243646 + 0.00613882i
\(29\) −6.20126 − 2.01491i −1.15155 − 0.374160i −0.329820 − 0.944044i \(-0.606988\pi\)
−0.821725 + 0.569884i \(0.806988\pi\)
\(30\) 0 0
\(31\) −3.03232 − 2.73031i −0.544620 − 0.490378i 0.350280 − 0.936645i \(-0.386087\pi\)
−0.894900 + 0.446267i \(0.852753\pi\)
\(32\) 0.274517 + 0.0735567i 0.0485283 + 0.0130031i
\(33\) −1.78172 + 1.15706i −0.310157 + 0.201418i
\(34\) −0.0327187 − 0.0237715i −0.00561120 − 0.00407678i
\(35\) 0 0
\(36\) −0.302057 + 0.219457i −0.0503428 + 0.0365762i
\(37\) −0.486654 + 1.26778i −0.0800054 + 0.208421i −0.967749 − 0.251918i \(-0.918939\pi\)
0.887743 + 0.460339i \(0.152272\pi\)
\(38\) −0.279702 − 0.345403i −0.0453737 − 0.0560318i
\(39\) −6.87556 + 15.4428i −1.10097 + 2.47282i
\(40\) 0 0
\(41\) 1.10223 − 1.51708i 0.172139 − 0.236929i −0.714227 − 0.699914i \(-0.753222\pi\)
0.886366 + 0.462985i \(0.153222\pi\)
\(42\) 1.68968 − 11.8107i 0.260724 − 1.82243i
\(43\) −2.47415 − 2.47415i −0.377304 − 0.377304i 0.492825 − 0.870129i \(-0.335964\pi\)
−0.870129 + 0.492825i \(0.835964\pi\)
\(44\) −0.00687295 + 0.0323347i −0.00103614 + 0.00487464i
\(45\) 0 0
\(46\) 6.49449 − 1.38045i 0.957560 − 0.203536i
\(47\) 0.154192 − 2.94215i 0.0224912 − 0.429157i −0.964058 − 0.265692i \(-0.914400\pi\)
0.986549 − 0.163465i \(-0.0522671\pi\)
\(48\) 1.96878 − 12.4304i 0.284169 − 1.79417i
\(49\) −4.28229 − 5.53732i −0.611756 − 0.791046i
\(50\) 0 0
\(51\) 0.0467688 − 0.0810060i 0.00654894 − 0.0113431i
\(52\) 0.0942629 + 0.245563i 0.0130719 + 0.0340535i
\(53\) −0.558452 − 0.362663i −0.0767092 − 0.0498156i 0.505715 − 0.862701i \(-0.331229\pi\)
−0.582424 + 0.812885i \(0.697896\pi\)
\(54\) 13.3666 + 14.8451i 1.81896 + 2.02016i
\(55\) 0 0
\(56\) −4.54339 − 6.06040i −0.607135 − 0.809855i
\(57\) 0.726871 − 0.726871i 0.0962765 − 0.0962765i
\(58\) 4.95874 + 7.63580i 0.651115 + 1.00263i
\(59\) −0.345447 + 3.28671i −0.0449734 + 0.427893i 0.948750 + 0.316028i \(0.102349\pi\)
−0.993723 + 0.111865i \(0.964317\pi\)
\(60\) 0 0
\(61\) 10.5899 − 1.11304i 1.35590 − 0.142511i 0.601471 − 0.798894i \(-0.294581\pi\)
0.754427 + 0.656384i \(0.227915\pi\)
\(62\) 0.891298 + 5.62743i 0.113195 + 0.714685i
\(63\) 19.6436 + 0.734614i 2.47487 + 0.0925527i
\(64\) −4.81442 − 6.62648i −0.601803 − 0.828310i
\(65\) 0 0
\(66\) 2.95020 + 0.310078i 0.363144 + 0.0381680i
\(67\) −0.747728 − 14.2675i −0.0913495 − 1.74305i −0.533223 − 0.845974i \(-0.679019\pi\)
0.441874 − 0.897077i \(-0.354314\pi\)
\(68\) −0.000376703 − 0.00140588i −4.56820e−5 − 0.000170488i
\(69\) 4.74539 + 14.6048i 0.571277 + 1.75821i
\(70\) 0 0
\(71\) −0.171405 + 0.527530i −0.0203420 + 0.0626063i −0.960712 − 0.277546i \(-0.910479\pi\)
0.940370 + 0.340153i \(0.110479\pi\)
\(72\) 21.2411 + 1.11320i 2.50329 + 0.131192i
\(73\) 0.756474 − 0.290383i 0.0885386 − 0.0339868i −0.313696 − 0.949524i \(-0.601567\pi\)
0.402234 + 0.915537i \(0.368234\pi\)
\(74\) 1.64214 − 0.948091i 0.190895 − 0.110213i
\(75\) 0 0
\(76\) − 0.0159952i − 0.00183477i
\(77\) 1.33602 − 1.11543i 0.152253 − 0.127115i
\(78\) 21.0312 − 10.7159i 2.38132 − 1.21334i
\(79\) 9.72376 − 8.75531i 1.09401 − 0.985050i 0.0940639 − 0.995566i \(-0.470014\pi\)
0.999945 + 0.0105164i \(0.00334752\pi\)
\(80\) 0 0
\(81\) −16.0005 + 17.7704i −1.77783 + 1.97448i
\(82\) −2.52921 + 0.677700i −0.279305 + 0.0748395i
\(83\) −6.95799 − 3.54527i −0.763739 − 0.389144i 0.0283125 − 0.999599i \(-0.490987\pi\)
−0.792051 + 0.610455i \(0.790987\pi\)
\(84\) 0.314752 − 0.292059i 0.0343422 − 0.0318662i
\(85\) 0 0
\(86\) 0.510698 + 4.85897i 0.0550700 + 0.523956i
\(87\) −16.3649 + 13.2521i −1.75450 + 1.42077i
\(88\) 1.46355 − 1.18516i 0.156015 − 0.126339i
\(89\) −1.21799 − 11.5884i −0.129107 − 1.22837i −0.846762 − 0.531972i \(-0.821451\pi\)
0.717655 − 0.696399i \(-0.245216\pi\)
\(90\) 0 0
\(91\) 4.08168 − 13.2334i 0.427876 − 1.38724i
\(92\) 0.212906 + 0.108481i 0.0221969 + 0.0113099i
\(93\) −12.7287 + 3.41063i −1.31990 + 0.353666i
\(94\) −2.75271 + 3.05720i −0.283921 + 0.315326i
\(95\) 0 0
\(96\) 0.682083 − 0.614150i 0.0696148 − 0.0626814i
\(97\) 3.47390 − 1.77004i 0.352721 − 0.179720i −0.268648 − 0.963238i \(-0.586577\pi\)
0.621369 + 0.783518i \(0.286577\pi\)
\(98\) −0.218921 + 9.77188i −0.0221144 + 0.987109i
\(99\) 4.88749i 0.491211i
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 875.2.bb.a.593.6 288
5.2 odd 4 875.2.bb.c.782.13 288
5.3 odd 4 175.2.x.a.152.6 yes 288
5.4 even 2 875.2.bb.b.593.13 288
7.3 odd 6 inner 875.2.bb.a.843.13 288
25.9 even 10 175.2.x.a.138.6 yes 288
25.12 odd 20 inner 875.2.bb.a.82.13 288
25.13 odd 20 875.2.bb.b.82.6 288
25.16 even 5 875.2.bb.c.418.13 288
35.3 even 12 175.2.x.a.52.6 yes 288
35.17 even 12 875.2.bb.c.157.13 288
35.24 odd 6 875.2.bb.b.843.6 288
175.38 even 60 875.2.bb.b.332.13 288
175.59 odd 30 175.2.x.a.38.6 ✓ 288
175.66 odd 30 875.2.bb.c.668.13 288
175.87 even 60 inner 875.2.bb.a.332.6 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.x.a.38.6 ✓ 288 175.59 odd 30
175.2.x.a.52.6 yes 288 35.3 even 12
175.2.x.a.138.6 yes 288 25.9 even 10
175.2.x.a.152.6 yes 288 5.3 odd 4
875.2.bb.a.82.13 288 25.12 odd 20 inner
875.2.bb.a.332.6 288 175.87 even 60 inner
875.2.bb.a.593.6 288 1.1 even 1 trivial
875.2.bb.a.843.13 288 7.3 odd 6 inner
875.2.bb.b.82.6 288 25.13 odd 20
875.2.bb.b.332.13 288 175.38 even 60
875.2.bb.b.593.13 288 5.4 even 2
875.2.bb.b.843.6 288 35.24 odd 6
875.2.bb.c.157.13 288 35.17 even 12
875.2.bb.c.418.13 288 25.16 even 5
875.2.bb.c.668.13 288 175.66 odd 30
875.2.bb.c.782.13 288 5.2 odd 4