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Properties

Label 475.2.l.e.251.5
Level $475$
Weight $2$
Character 475.251
Analytic conductor $3.793$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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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] = [475,2,Mod(101,475)] 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("475.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.l (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 251.5
Character \(\chi\) \(=\) 475.251
Dual form 475.2.l.e.176.5

$q$-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+(0.524915 + 0.191054i) q^{2} +(0.116387 + 0.660064i) q^{3} +(-1.29305 - 1.08500i) q^{4} +(-0.0650142 + 0.368714i) q^{6} +(-1.27745 + 2.21261i) q^{7} +(-1.03005 - 1.78411i) q^{8} +(2.39694 - 0.872414i) q^{9} +(2.45257 + 4.24797i) q^{11} +(0.565676 - 0.979779i) q^{12} +(-0.734199 + 4.16385i) q^{13} +(-1.09328 + 0.917371i) q^{14} +(0.386392 + 2.19134i) q^{16} +(5.20951 + 1.89611i) q^{17} +1.42487 q^{18} +(-0.481031 - 4.33228i) q^{19} +(-1.60914 - 0.585680i) q^{21} +(0.475800 + 2.69840i) q^{22} +(4.29918 + 3.60744i) q^{23} +(1.05774 - 0.887548i) q^{24} +(-1.18091 + 2.04540i) q^{26} +(1.86019 + 3.22195i) q^{27} +(4.05250 - 1.47499i) q^{28} +(-6.18499 + 2.25115i) q^{29} +(3.03750 - 5.26111i) q^{31} +(-0.931308 + 5.28171i) q^{32} +(-2.51849 + 2.11326i) q^{33} +(2.37229 + 1.99059i) q^{34} +(-4.04594 - 1.47260i) q^{36} -7.06139 q^{37} +(0.575196 - 2.36598i) q^{38} -2.83386 q^{39} +(0.153494 + 0.870507i) q^{41} +(-0.732768 - 0.614865i) q^{42} +(2.65299 - 2.22612i) q^{43} +(1.43775 - 8.15390i) q^{44} +(1.56749 + 2.71497i) q^{46} +(1.29340 - 0.470759i) q^{47} +(-1.40145 + 0.510087i) q^{48} +(0.236239 + 0.409178i) q^{49} +(-0.645232 + 3.65929i) q^{51} +(5.46714 - 4.58748i) q^{52} +(-8.76301 - 7.35304i) q^{53} +(0.360879 + 2.04665i) q^{54} +5.26337 q^{56} +(2.80359 - 0.821733i) q^{57} -3.67669 q^{58} +(1.16196 + 0.422919i) q^{59} +(-7.93284 - 6.65645i) q^{61} +(2.59959 - 2.18131i) q^{62} +(-1.13166 + 6.41796i) q^{63} +(0.727197 - 1.25954i) q^{64} +(-1.72574 + 0.628117i) q^{66} +(2.95724 - 1.07635i) q^{67} +(-4.67890 - 8.10409i) q^{68} +(-1.88077 + 3.25759i) q^{69} +(5.67478 - 4.76171i) q^{71} +(-4.02546 - 3.37776i) q^{72} +(1.83547 + 10.4095i) q^{73} +(-3.70663 - 1.34910i) q^{74} +(-4.07853 + 6.12379i) q^{76} -12.5321 q^{77} +(-1.48754 - 0.541419i) q^{78} +(2.04341 + 11.5887i) q^{79} +(3.95182 - 3.31597i) q^{81} +(-0.0857422 + 0.486268i) q^{82} +(1.15332 - 1.99761i) q^{83} +(1.44525 + 2.50324i) q^{84} +(1.81790 - 0.661662i) q^{86} +(-2.20576 - 3.82049i) q^{87} +(5.05255 - 8.75128i) q^{88} +(0.372102 - 2.11029i) q^{89} +(-8.27507 - 6.94361i) q^{91} +(-1.64499 - 9.32923i) q^{92} +(3.82620 + 1.39262i) q^{93} +0.768865 q^{94} -3.59466 q^{96} +(-4.52264 - 1.64610i) q^{97} +(0.0458306 + 0.259918i) q^{98} +(9.58464 + 8.04247i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 3 q^{3} - 6 q^{4} - 3 q^{6} - 18 q^{8} + 9 q^{9} - 24 q^{12} + 27 q^{13} + 12 q^{14} + 18 q^{16} + 36 q^{18} + 12 q^{19} + 15 q^{21} - 12 q^{22} + 12 q^{23} + 33 q^{24} - 18 q^{26} - 30 q^{27} + 42 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.524915 + 0.191054i 0.371171 + 0.135095i 0.520869 − 0.853637i \(-0.325608\pi\)
−0.149698 + 0.988732i \(0.547830\pi\)
\(3\) 0.116387 + 0.660064i 0.0671961 + 0.381088i 0.999796 + 0.0201773i \(0.00642307\pi\)
−0.932600 + 0.360911i \(0.882466\pi\)
\(4\) −1.29305 − 1.08500i −0.646527 − 0.542501i
\(5\) 0 0
\(6\) −0.0650142 + 0.368714i −0.0265419 + 0.150527i
\(7\) −1.27745 + 2.21261i −0.482831 + 0.836288i −0.999806 − 0.0197128i \(-0.993725\pi\)
0.516975 + 0.856001i \(0.327058\pi\)
\(8\) −1.03005 − 1.78411i −0.364179 − 0.630776i
\(9\) 2.39694 − 0.872414i 0.798980 − 0.290805i
\(10\) 0 0
\(11\) 2.45257 + 4.24797i 0.739477 + 1.28081i 0.952731 + 0.303815i \(0.0982604\pi\)
−0.213254 + 0.976997i \(0.568406\pi\)
\(12\) 0.565676 − 0.979779i 0.163297 − 0.282838i
\(13\) −0.734199 + 4.16385i −0.203630 + 1.15484i 0.695951 + 0.718090i \(0.254983\pi\)
−0.899581 + 0.436755i \(0.856128\pi\)
\(14\) −1.09328 + 0.917371i −0.292191 + 0.245178i
\(15\) 0 0
\(16\) 0.386392 + 2.19134i 0.0965980 + 0.547834i
\(17\) 5.20951 + 1.89611i 1.26349 + 0.459873i 0.884939 − 0.465707i \(-0.154200\pi\)
0.378552 + 0.925580i \(0.376422\pi\)
\(18\) 1.42487 0.335845
\(19\) −0.481031 − 4.33228i −0.110356 − 0.993892i
\(20\) 0 0
\(21\) −1.60914 − 0.585680i −0.351144 − 0.127806i
\(22\) 0.475800 + 2.69840i 0.101441 + 0.575300i
\(23\) 4.29918 + 3.60744i 0.896440 + 0.752203i 0.969491 − 0.245125i \(-0.0788292\pi\)
−0.0730510 + 0.997328i \(0.523274\pi\)
\(24\) 1.05774 − 0.887548i 0.215910 − 0.181170i
\(25\) 0 0
\(26\) −1.18091 + 2.04540i −0.231596 + 0.401135i
\(27\) 1.86019 + 3.22195i 0.357994 + 0.620064i
\(28\) 4.05250 − 1.47499i 0.765850 − 0.278747i
\(29\) −6.18499 + 2.25115i −1.14852 + 0.418029i −0.844985 − 0.534790i \(-0.820391\pi\)
−0.303539 + 0.952819i \(0.598168\pi\)
\(30\) 0 0
\(31\) 3.03750 − 5.26111i 0.545552 − 0.944923i −0.453020 − 0.891500i \(-0.649653\pi\)
0.998572 − 0.0534231i \(-0.0170132\pi\)
\(32\) −0.931308 + 5.28171i −0.164633 + 0.933683i
\(33\) −2.51849 + 2.11326i −0.438412 + 0.367871i
\(34\) 2.37229 + 1.99059i 0.406845 + 0.341383i
\(35\) 0 0
\(36\) −4.04594 − 1.47260i −0.674324 − 0.245434i
\(37\) −7.06139 −1.16089 −0.580443 − 0.814301i \(-0.697120\pi\)
−0.580443 + 0.814301i \(0.697120\pi\)
\(38\) 0.575196 − 2.36598i 0.0933091 − 0.383813i
\(39\) −2.83386 −0.453781
\(40\) 0 0
\(41\) 0.153494 + 0.870507i 0.0239717 + 0.135950i 0.994445 − 0.105259i \(-0.0335671\pi\)
−0.970473 + 0.241209i \(0.922456\pi\)
\(42\) −0.732768 − 0.614865i −0.113068 − 0.0948757i
\(43\) 2.65299 − 2.22612i 0.404577 − 0.339480i −0.417683 − 0.908593i \(-0.637158\pi\)
0.822260 + 0.569113i \(0.192713\pi\)
\(44\) 1.43775 − 8.15390i 0.216749 − 1.22925i
\(45\) 0 0
\(46\) 1.56749 + 2.71497i 0.231114 + 0.400301i
\(47\) 1.29340 − 0.470759i 0.188662 − 0.0686672i −0.245962 − 0.969280i \(-0.579104\pi\)
0.434623 + 0.900612i \(0.356882\pi\)
\(48\) −1.40145 + 0.510087i −0.202282 + 0.0736247i
\(49\) 0.236239 + 0.409178i 0.0337484 + 0.0584540i
\(50\) 0 0
\(51\) −0.645232 + 3.65929i −0.0903505 + 0.512403i
\(52\) 5.46714 − 4.58748i 0.758156 − 0.636169i
\(53\) −8.76301 − 7.35304i −1.20369 − 1.01002i −0.999517 − 0.0310838i \(-0.990104\pi\)
−0.204176 − 0.978934i \(-0.565451\pi\)
\(54\) 0.360879 + 2.04665i 0.0491094 + 0.278513i
\(55\) 0 0
\(56\) 5.26337 0.703348
\(57\) 2.80359 − 0.821733i 0.371345 − 0.108841i
\(58\) −3.67669 −0.482773
\(59\) 1.16196 + 0.422919i 0.151274 + 0.0550593i 0.416547 − 0.909114i \(-0.363240\pi\)
−0.265273 + 0.964173i \(0.585462\pi\)
\(60\) 0 0
\(61\) −7.93284 − 6.65645i −1.01570 − 0.852271i −0.0266159 − 0.999646i \(-0.508473\pi\)
−0.989081 + 0.147375i \(0.952918\pi\)
\(62\) 2.59959 − 2.18131i 0.330148 − 0.277027i
\(63\) −1.13166 + 6.41796i −0.142576 + 0.808587i
\(64\) 0.727197 − 1.25954i 0.0908996 − 0.157443i
\(65\) 0 0
\(66\) −1.72574 + 0.628117i −0.212424 + 0.0773159i
\(67\) 2.95724 − 1.07635i 0.361284 − 0.131497i −0.154999 − 0.987915i \(-0.549537\pi\)
0.516283 + 0.856418i \(0.327315\pi\)
\(68\) −4.67890 − 8.10409i −0.567400 − 0.982765i
\(69\) −1.88077 + 3.25759i −0.226418 + 0.392168i
\(70\) 0 0
\(71\) 5.67478 − 4.76171i 0.673473 − 0.565111i −0.240618 − 0.970620i \(-0.577350\pi\)
0.914091 + 0.405509i \(0.132906\pi\)
\(72\) −4.02546 − 3.37776i −0.474404 − 0.398073i
\(73\) 1.83547 + 10.4095i 0.214825 + 1.21833i 0.881209 + 0.472726i \(0.156730\pi\)
−0.666384 + 0.745609i \(0.732159\pi\)
\(74\) −3.70663 − 1.34910i −0.430887 − 0.156830i
\(75\) 0 0
\(76\) −4.07853 + 6.12379i −0.467839 + 0.702447i
\(77\) −12.5321 −1.42817
\(78\) −1.48754 − 0.541419i −0.168430 − 0.0613036i
\(79\) 2.04341 + 11.5887i 0.229901 + 1.30383i 0.853091 + 0.521763i \(0.174725\pi\)
−0.623189 + 0.782071i \(0.714163\pi\)
\(80\) 0 0
\(81\) 3.95182 − 3.31597i 0.439091 − 0.368441i
\(82\) −0.0857422 + 0.486268i −0.00946864 + 0.0536993i
\(83\) 1.15332 − 1.99761i 0.126594 − 0.219267i −0.795761 − 0.605611i \(-0.792929\pi\)
0.922355 + 0.386344i \(0.126262\pi\)
\(84\) 1.44525 + 2.50324i 0.157689 + 0.273126i
\(85\) 0 0
\(86\) 1.81790 − 0.661662i 0.196029 − 0.0713489i
\(87\) −2.20576 − 3.82049i −0.236482 − 0.409599i
\(88\) 5.05255 − 8.75128i 0.538604 − 0.932889i
\(89\) 0.372102 − 2.11029i 0.0394427 − 0.223691i −0.958715 − 0.284370i \(-0.908216\pi\)
0.998157 + 0.0606793i \(0.0193267\pi\)
\(90\) 0 0
\(91\) −8.27507 − 6.94361i −0.867463 − 0.727888i
\(92\) −1.64499 − 9.32923i −0.171503 − 0.972639i
\(93\) 3.82620 + 1.39262i 0.396758 + 0.144408i
\(94\) 0.768865 0.0793024
\(95\) 0 0
\(96\) −3.59466 −0.366878
\(97\) −4.52264 − 1.64610i −0.459204 − 0.167137i 0.102052 − 0.994779i \(-0.467459\pi\)
−0.561256 + 0.827642i \(0.689682\pi\)
\(98\) 0.0458306 + 0.259918i 0.00462959 + 0.0262557i
\(99\) 9.58464 + 8.04247i 0.963293 + 0.808299i
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 475.2.l.e.251.5 yes 42
5.2 odd 4 475.2.u.d.99.6 84
5.3 odd 4 475.2.u.d.99.9 84
5.4 even 2 475.2.l.d.251.3 yes 42
19.5 even 9 inner 475.2.l.e.176.5 yes 42
19.9 even 9 9025.2.a.cs.1.8 21
19.10 odd 18 9025.2.a.cp.1.14 21
95.9 even 18 9025.2.a.cq.1.14 21
95.24 even 18 475.2.l.d.176.3 ✓ 42
95.29 odd 18 9025.2.a.cr.1.8 21
95.43 odd 36 475.2.u.d.24.6 84
95.62 odd 36 475.2.u.d.24.9 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.l.d.176.3 ✓ 42 95.24 even 18
475.2.l.d.251.3 yes 42 5.4 even 2
475.2.l.e.176.5 yes 42 19.5 even 9 inner
475.2.l.e.251.5 yes 42 1.1 even 1 trivial
475.2.u.d.24.6 84 95.43 odd 36
475.2.u.d.24.9 84 95.62 odd 36
475.2.u.d.99.6 84 5.2 odd 4
475.2.u.d.99.9 84 5.3 odd 4
9025.2.a.cp.1.14 21 19.10 odd 18
9025.2.a.cq.1.14 21 95.9 even 18
9025.2.a.cr.1.8 21 95.29 odd 18
9025.2.a.cs.1.8 21 19.9 even 9