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Properties

Label 875.2.bb.c.782.9
Level $875$
Weight $2$
Character 875.782
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 = [288,8] 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 782.9
Character \(\chi\) \(=\) 875.782
Dual form 875.2.bb.c.668.9

$q$-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+(0.00351582 - 0.00434168i) q^{2} +(1.74506 + 1.13325i) q^{3} +(0.415817 + 1.95626i) q^{4} +(0.0110555 - 0.00359216i) q^{6} +(1.62930 + 2.08455i) q^{7} +(0.0199110 + 0.0101452i) q^{8} +(0.540752 + 1.21455i) q^{9} +(-3.82423 - 1.70266i) q^{11} +(-1.49132 + 3.88502i) q^{12} +(0.121872 + 0.769472i) q^{13} +(0.0147788 + 0.000255010i) q^{14} +(-3.65401 + 1.62687i) q^{16} +(0.213376 + 4.07146i) q^{17} +(0.00717437 + 0.00192237i) q^{18} +(3.90528 + 0.830092i) q^{19} +(0.480900 + 5.48408i) q^{21} +(-0.0208377 + 0.0106173i) q^{22} +(1.73616 + 1.40592i) q^{23} +(0.0232488 + 0.0402681i) q^{24} +(0.00376928 + 0.00217620i) q^{26} +(0.543751 - 3.43311i) q^{27} +(-3.40045 + 4.05414i) q^{28} +(5.75261 + 1.86914i) q^{29} +(-2.52494 - 2.27347i) q^{31} +(-0.0173510 + 0.0647547i) q^{32} +(-4.74396 - 7.30506i) q^{33} +(0.0184272 + 0.0133881i) q^{34} +(-2.15112 + 1.56288i) q^{36} +(-9.24337 - 3.54820i) q^{37} +(0.0173343 - 0.0140370i) q^{38} +(-0.659333 + 1.48089i) q^{39} +(6.32803 - 8.70979i) q^{41} +(0.0255009 + 0.0171932i) q^{42} +(-3.91852 + 3.91852i) q^{43} +(1.74067 - 8.18920i) q^{44} +(0.0122081 - 0.00259491i) q^{46} +(9.02298 + 0.472874i) q^{47} +(-8.22011 - 1.30194i) q^{48} +(-1.69074 + 6.79275i) q^{49} +(-4.24165 + 7.34675i) q^{51} +(-1.45461 + 0.558374i) q^{52} +(-0.266964 + 0.411088i) q^{53} +(-0.0129937 - 0.0144310i) q^{54} +(0.0112929 + 0.0580351i) q^{56} +(5.87423 + 5.87423i) q^{57} +(0.0283404 - 0.0184045i) q^{58} +(-0.130847 + 1.24493i) q^{59} +(-11.8065 + 1.24091i) q^{61} +(-0.0187479 + 0.00296938i) q^{62} +(-1.65074 + 3.10609i) q^{63} +(-4.70184 - 6.47153i) q^{64} +(-0.0483951 - 0.00508653i) q^{66} +(9.52378 - 0.499120i) q^{67} +(-7.87614 + 2.11040i) q^{68} +(1.43644 + 4.42092i) q^{69} +(3.35184 - 10.3159i) q^{71} +(-0.00155488 + 0.0296689i) q^{72} +(3.52568 + 9.18470i) q^{73} +(-0.0479032 + 0.0276569i) q^{74} +7.98492i q^{76} +(-2.68155 - 10.7460i) q^{77} +(0.00411143 + 0.00806914i) q^{78} +(4.84603 - 4.36339i) q^{79} +(7.50826 - 8.33877i) q^{81} +(-0.0155669 - 0.0580964i) q^{82} +(0.767252 - 1.50582i) q^{83} +(-10.5283 + 3.22114i) q^{84} +(0.00323614 + 0.0307898i) q^{86} +(7.92043 + 9.78092i) q^{87} +(-0.0588705 - 0.0726990i) q^{88} +(-0.115546 - 1.09935i) q^{89} +(-1.40544 + 1.50775i) q^{91} +(-2.02842 + 3.98100i) q^{92} +(-1.82975 - 6.82873i) q^{93} +(0.0337763 - 0.0375124i) q^{94} +(-0.103662 + 0.0933376i) q^{96} +(1.14422 + 2.24565i) q^{97} +(0.0235476 + 0.0312227i) q^{98} -5.56542i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q + 8 q^{2} + 24 q^{3} - 10 q^{4} + 10 q^{7} + 36 q^{8} - 10 q^{9} - 6 q^{11} + 36 q^{12} - 20 q^{14} - 30 q^{16} + 42 q^{17} + 14 q^{18} - 30 q^{19} - 12 q^{21} - 32 q^{22} + 40 q^{23} - 48 q^{26}+ \cdots - 222 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(626\)
\(\chi(n)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.00351582 − 0.00434168i 0.00248606 − 0.00307003i −0.775901 − 0.630854i \(-0.782705\pi\)
0.778387 + 0.627784i \(0.216038\pi\)
\(3\) 1.74506 + 1.13325i 1.00751 + 0.654284i 0.939197 − 0.343380i \(-0.111572\pi\)
0.0683126 + 0.997664i \(0.478238\pi\)
\(4\) 0.415817 + 1.95626i 0.207908 + 0.978132i
\(5\) 0 0
\(6\) 0.0110555 − 0.00359216i 0.00451340 − 0.00146649i
\(7\) 1.62930 + 2.08455i 0.615819 + 0.787888i
\(8\) 0.0199110 + 0.0101452i 0.00703960 + 0.00358685i
\(9\) 0.540752 + 1.21455i 0.180251 + 0.404849i
\(10\) 0 0
\(11\) −3.82423 − 1.70266i −1.15305 − 0.513370i −0.261013 − 0.965335i \(-0.584057\pi\)
−0.892036 + 0.451965i \(0.850723\pi\)
\(12\) −1.49132 + 3.88502i −0.430507 + 1.12151i
\(13\) 0.121872 + 0.769472i 0.0338013 + 0.213413i 0.998808 − 0.0488122i \(-0.0155436\pi\)
−0.965007 + 0.262225i \(0.915544\pi\)
\(14\) 0.0147788 0.000255010i 0.00394981 6.81542e-5i
\(15\) 0 0
\(16\) −3.65401 + 1.62687i −0.913503 + 0.406718i
\(17\) 0.213376 + 4.07146i 0.0517514 + 0.987475i 0.893910 + 0.448246i \(0.147951\pi\)
−0.842159 + 0.539229i \(0.818716\pi\)
\(18\) 0.00717437 + 0.00192237i 0.00169101 + 0.000453106i
\(19\) 3.90528 + 0.830092i 0.895932 + 0.190436i 0.632792 − 0.774321i \(-0.281909\pi\)
0.263140 + 0.964758i \(0.415242\pi\)
\(20\) 0 0
\(21\) 0.480900 + 5.48408i 0.104941 + 1.19672i
\(22\) −0.0208377 + 0.0106173i −0.00444261 + 0.00226363i
\(23\) 1.73616 + 1.40592i 0.362015 + 0.293154i 0.793020 − 0.609195i \(-0.208508\pi\)
−0.431005 + 0.902350i \(0.641841\pi\)
\(24\) 0.0232488 + 0.0402681i 0.00474564 + 0.00821969i
\(25\) 0 0
\(26\) 0.00376928 + 0.00217620i 0.000739218 + 0.000426787i
\(27\) 0.543751 − 3.43311i 0.104645 − 0.660702i
\(28\) −3.40045 + 4.05414i −0.642624 + 0.766161i
\(29\) 5.75261 + 1.86914i 1.06823 + 0.347090i 0.789799 − 0.613365i \(-0.210185\pi\)
0.278434 + 0.960455i \(0.410185\pi\)
\(30\) 0 0
\(31\) −2.52494 − 2.27347i −0.453493 − 0.408327i 0.410476 − 0.911871i \(-0.365362\pi\)
−0.863969 + 0.503544i \(0.832029\pi\)
\(32\) −0.0173510 + 0.0647547i −0.00306725 + 0.0114471i
\(33\) −4.74396 − 7.30506i −0.825817 − 1.27165i
\(34\) 0.0184272 + 0.0133881i 0.00316024 + 0.00229605i
\(35\) 0 0
\(36\) −2.15112 + 1.56288i −0.358521 + 0.260480i
\(37\) −9.24337 − 3.54820i −1.51960 − 0.583320i −0.551564 − 0.834132i \(-0.685969\pi\)
−0.968036 + 0.250813i \(0.919302\pi\)
\(38\) 0.0173343 − 0.0140370i 0.00281199 − 0.00227710i
\(39\) −0.659333 + 1.48089i −0.105578 + 0.237131i
\(40\) 0 0
\(41\) 6.32803 − 8.70979i 0.988272 − 1.36024i 0.0560206 − 0.998430i \(-0.482159\pi\)
0.932252 − 0.361810i \(-0.117841\pi\)
\(42\) 0.0255009 + 0.0171932i 0.00393487 + 0.00265296i
\(43\) −3.91852 + 3.91852i −0.597569 + 0.597569i −0.939665 − 0.342096i \(-0.888863\pi\)
0.342096 + 0.939665i \(0.388863\pi\)
\(44\) 1.74067 − 8.18920i 0.262416 − 1.23457i
\(45\) 0 0
\(46\) 0.0122081 − 0.00259491i 0.00179998 − 0.000382599i
\(47\) 9.02298 + 0.472874i 1.31614 + 0.0689758i 0.697434 − 0.716649i \(-0.254325\pi\)
0.618703 + 0.785625i \(0.287658\pi\)
\(48\) −8.22011 − 1.30194i −1.18647 − 0.187919i
\(49\) −1.69074 + 6.79275i −0.241534 + 0.970392i
\(50\) 0 0
\(51\) −4.24165 + 7.34675i −0.593949 + 1.02875i
\(52\) −1.45461 + 0.558374i −0.201719 + 0.0774326i
\(53\) −0.266964 + 0.411088i −0.0366703 + 0.0564673i −0.856530 − 0.516097i \(-0.827385\pi\)
0.819860 + 0.572564i \(0.194051\pi\)
\(54\) −0.0129937 − 0.0144310i −0.00176822 − 0.00196381i
\(55\) 0 0
\(56\) 0.0112929 + 0.0580351i 0.00150908 + 0.00775526i
\(57\) 5.87423 + 5.87423i 0.778060 + 0.778060i
\(58\) 0.0283404 − 0.0184045i 0.00372127 − 0.00241662i
\(59\) −0.130847 + 1.24493i −0.0170348 + 0.162076i −0.999732 − 0.0231431i \(-0.992633\pi\)
0.982697 + 0.185219i \(0.0592993\pi\)
\(60\) 0 0
\(61\) −11.8065 + 1.24091i −1.51166 + 0.158882i −0.823768 − 0.566927i \(-0.808132\pi\)
−0.687896 + 0.725810i \(0.741465\pi\)
\(62\) −0.0187479 + 0.00296938i −0.00238099 + 0.000377112i
\(63\) −1.65074 + 3.10609i −0.207974 + 0.391331i
\(64\) −4.70184 − 6.47153i −0.587730 − 0.808941i
\(65\) 0 0
\(66\) −0.0483951 − 0.00508653i −0.00595703 − 0.000626109i
\(67\) 9.52378 − 0.499120i 1.16352 − 0.0609772i 0.539219 − 0.842165i \(-0.318719\pi\)
0.624296 + 0.781188i \(0.285386\pi\)
\(68\) −7.87614 + 2.11040i −0.955122 + 0.255924i
\(69\) 1.43644 + 4.42092i 0.172928 + 0.532216i
\(70\) 0 0
\(71\) 3.35184 − 10.3159i 0.397790 − 1.22427i −0.528977 − 0.848636i \(-0.677424\pi\)
0.926767 − 0.375637i \(-0.122576\pi\)
\(72\) −0.00155488 + 0.0296689i −0.000183244 + 0.00349651i
\(73\) 3.52568 + 9.18470i 0.412649 + 1.07499i 0.969655 + 0.244479i \(0.0786169\pi\)
−0.557005 + 0.830509i \(0.688050\pi\)
\(74\) −0.0479032 + 0.0276569i −0.00556863 + 0.00321505i
\(75\) 0 0
\(76\) 7.98492i 0.915933i
\(77\) −2.68155 − 10.7460i −0.305591 − 1.22462i
\(78\) 0.00411143 + 0.00806914i 0.000465528 + 0.000913651i
\(79\) 4.84603 − 4.36339i 0.545221 − 0.490919i −0.349873 − 0.936797i \(-0.613775\pi\)
0.895094 + 0.445878i \(0.147109\pi\)
\(80\) 0 0
\(81\) 7.50826 − 8.33877i 0.834252 − 0.926530i
\(82\) −0.0155669 − 0.0580964i −0.00171907 − 0.00641567i
\(83\) 0.767252 − 1.50582i 0.0842169 − 0.165285i −0.845057 − 0.534676i \(-0.820434\pi\)
0.929274 + 0.369391i \(0.120434\pi\)
\(84\) −10.5283 + 3.22114i −1.14874 + 0.351455i
\(85\) 0 0
\(86\) 0.00323614 + 0.0307898i 0.000348962 + 0.00332015i
\(87\) 7.92043 + 9.78092i 0.849160 + 1.04862i
\(88\) −0.0588705 − 0.0726990i −0.00627561 − 0.00774974i
\(89\) −0.115546 − 1.09935i −0.0122479 − 0.116531i 0.986690 − 0.162614i \(-0.0519927\pi\)
−0.998938 + 0.0460834i \(0.985326\pi\)
\(90\) 0 0
\(91\) −1.40544 + 1.50775i −0.147330 + 0.158056i
\(92\) −2.02842 + 3.98100i −0.211477 + 0.415048i
\(93\) −1.82975 − 6.82873i −0.189737 − 0.708107i
\(94\) 0.0337763 − 0.0375124i 0.00348376 − 0.00386910i
\(95\) 0 0
\(96\) −0.103662 + 0.0933376i −0.0105800 + 0.00952623i
\(97\) 1.14422 + 2.24565i 0.116177 + 0.228011i 0.941772 − 0.336253i \(-0.109160\pi\)
−0.825594 + 0.564264i \(0.809160\pi\)
\(98\) 0.0235476 + 0.0312227i 0.00237867 + 0.00315397i
\(99\) − 5.56542i − 0.559346i
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.c.782.9 288
5.2 odd 4 875.2.bb.b.593.9 288
5.3 odd 4 875.2.bb.a.593.10 288
5.4 even 2 175.2.x.a.152.10 yes 288
7.3 odd 6 inner 875.2.bb.c.157.9 288
25.9 even 10 875.2.bb.b.82.10 288
25.12 odd 20 175.2.x.a.138.10 yes 288
25.13 odd 20 inner 875.2.bb.c.418.9 288
25.16 even 5 875.2.bb.a.82.9 288
35.3 even 12 875.2.bb.a.843.9 288
35.17 even 12 875.2.bb.b.843.10 288
35.24 odd 6 175.2.x.a.52.10 yes 288
175.38 even 60 inner 875.2.bb.c.668.9 288
175.59 odd 30 875.2.bb.b.332.9 288
175.66 odd 30 875.2.bb.a.332.10 288
175.87 even 60 175.2.x.a.38.10 ✓ 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.x.a.38.10 ✓ 288 175.87 even 60
175.2.x.a.52.10 yes 288 35.24 odd 6
175.2.x.a.138.10 yes 288 25.12 odd 20
175.2.x.a.152.10 yes 288 5.4 even 2
875.2.bb.a.82.9 288 25.16 even 5
875.2.bb.a.332.10 288 175.66 odd 30
875.2.bb.a.593.10 288 5.3 odd 4
875.2.bb.a.843.9 288 35.3 even 12
875.2.bb.b.82.10 288 25.9 even 10
875.2.bb.b.332.9 288 175.59 odd 30
875.2.bb.b.593.9 288 5.2 odd 4
875.2.bb.b.843.10 288 35.17 even 12
875.2.bb.c.157.9 288 7.3 odd 6 inner
875.2.bb.c.418.9 288 25.13 odd 20 inner
875.2.bb.c.668.9 288 175.38 even 60 inner
875.2.bb.c.782.9 288 1.1 even 1 trivial