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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [175,6,Mod(99,175)] 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("175.99"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-136] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.6.b.a.99.1

$q$-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)
 
\(f(q)\) \(=\) \(q+10.0000i q^{2} -14.0000i q^{3} -68.0000 q^{4} +140.000 q^{6} +49.0000i q^{7} -360.000i q^{8} +47.0000 q^{9} +232.000 q^{11} +952.000i q^{12} -140.000i q^{13} -490.000 q^{14} +1424.00 q^{16} +1722.00i q^{17} +470.000i q^{18} +98.0000 q^{19} +686.000 q^{21} +2320.00i q^{22} +1824.00i q^{23} -5040.00 q^{24} +1400.00 q^{26} -4060.00i q^{27} -3332.00i q^{28} -3418.00 q^{29} -7644.00 q^{31} +2720.00i q^{32} -3248.00i q^{33} -17220.0 q^{34} -3196.00 q^{36} +10398.0i q^{37} +980.000i q^{38} -1960.00 q^{39} -17962.0 q^{41} +6860.00i q^{42} +10880.0i q^{43} -15776.0 q^{44} -18240.0 q^{46} -9324.00i q^{47} -19936.0i q^{48} -2401.00 q^{49} +24108.0 q^{51} +9520.00i q^{52} +2262.00i q^{53} +40600.0 q^{54} +17640.0 q^{56} -1372.00i q^{57} -34180.0i q^{58} +2730.00 q^{59} +25648.0 q^{61} -76440.0i q^{62} +2303.00i q^{63} +18368.0 q^{64} +32480.0 q^{66} +48404.0i q^{67} -117096. i q^{68} +25536.0 q^{69} -58560.0 q^{71} -16920.0i q^{72} +68082.0i q^{73} -103980. q^{74} -6664.00 q^{76} +11368.0i q^{77} -19600.0i q^{78} -31784.0 q^{79} -45419.0 q^{81} -179620. i q^{82} -20538.0i q^{83} -46648.0 q^{84} -108800. q^{86} +47852.0i q^{87} -83520.0i q^{88} +50582.0 q^{89} +6860.00 q^{91} -124032. i q^{92} +107016. i q^{93} +93240.0 q^{94} +38080.0 q^{96} +58506.0i q^{97} -24010.0i q^{98} +10904.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 136 q^{4} + 280 q^{6} + 94 q^{9} + 464 q^{11} - 980 q^{14} + 2848 q^{16} + 196 q^{19} + 1372 q^{21} - 10080 q^{24} + 2800 q^{26} - 6836 q^{29} - 15288 q^{31} - 34440 q^{34} - 6392 q^{36} - 3920 q^{39}+ \cdots + 21808 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0000i 1.76777i 0.467707 + 0.883883i \(0.345080\pi\)
−0.467707 + 0.883883i \(0.654920\pi\)
\(3\) − 14.0000i − 0.898100i −0.893507 − 0.449050i \(-0.851762\pi\)
0.893507 − 0.449050i \(-0.148238\pi\)
\(4\) −68.0000 −2.12500
\(5\) 0 0
\(6\) 140.000 1.58763
\(7\) 49.0000i 0.377964i
\(8\) − 360.000i − 1.98874i
\(9\) 47.0000 0.193416
\(10\) 0 0
\(11\) 232.000 0.578104 0.289052 − 0.957313i \(-0.406660\pi\)
0.289052 + 0.957313i \(0.406660\pi\)
\(12\) 952.000i 1.90846i
\(13\) − 140.000i − 0.229757i −0.993380 − 0.114879i \(-0.963352\pi\)
0.993380 − 0.114879i \(-0.0366479\pi\)
\(14\) −490.000 −0.668153
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) 1722.00i 1.44514i 0.691296 + 0.722572i \(0.257040\pi\)
−0.691296 + 0.722572i \(0.742960\pi\)
\(18\) 470.000i 0.341914i
\(19\) 98.0000 0.0622791 0.0311395 − 0.999515i \(-0.490086\pi\)
0.0311395 + 0.999515i \(0.490086\pi\)
\(20\) 0 0
\(21\) 686.000 0.339450
\(22\) 2320.00i 1.02195i
\(23\) 1824.00i 0.718961i 0.933153 + 0.359480i \(0.117046\pi\)
−0.933153 + 0.359480i \(0.882954\pi\)
\(24\) −5040.00 −1.78609
\(25\) 0 0
\(26\) 1400.00 0.406158
\(27\) − 4060.00i − 1.07181i
\(28\) − 3332.00i − 0.803175i
\(29\) −3418.00 −0.754705 −0.377352 − 0.926070i \(-0.623165\pi\)
−0.377352 + 0.926070i \(0.623165\pi\)
\(30\) 0 0
\(31\) −7644.00 −1.42862 −0.714310 − 0.699830i \(-0.753259\pi\)
−0.714310 + 0.699830i \(0.753259\pi\)
\(32\) 2720.00i 0.469563i
\(33\) − 3248.00i − 0.519196i
\(34\) −17220.0 −2.55468
\(35\) 0 0
\(36\) −3196.00 −0.411008
\(37\) 10398.0i 1.24866i 0.781159 + 0.624332i \(0.214629\pi\)
−0.781159 + 0.624332i \(0.785371\pi\)
\(38\) 980.000i 0.110095i
\(39\) −1960.00 −0.206345
\(40\) 0 0
\(41\) −17962.0 −1.66876 −0.834382 − 0.551186i \(-0.814175\pi\)
−0.834382 + 0.551186i \(0.814175\pi\)
\(42\) 6860.00i 0.600069i
\(43\) 10880.0i 0.897342i 0.893697 + 0.448671i \(0.148102\pi\)
−0.893697 + 0.448671i \(0.851898\pi\)
\(44\) −15776.0 −1.22847
\(45\) 0 0
\(46\) −18240.0 −1.27096
\(47\) − 9324.00i − 0.615684i −0.951438 − 0.307842i \(-0.900393\pi\)
0.951438 − 0.307842i \(-0.0996068\pi\)
\(48\) − 19936.0i − 1.24892i
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) 24108.0 1.29788
\(52\) 9520.00i 0.488235i
\(53\) 2262.00i 0.110612i 0.998469 + 0.0553061i \(0.0176135\pi\)
−0.998469 + 0.0553061i \(0.982387\pi\)
\(54\) 40600.0 1.89471
\(55\) 0 0
\(56\) 17640.0 0.751672
\(57\) − 1372.00i − 0.0559329i
\(58\) − 34180.0i − 1.33414i
\(59\) 2730.00 0.102102 0.0510508 − 0.998696i \(-0.483743\pi\)
0.0510508 + 0.998696i \(0.483743\pi\)
\(60\) 0 0
\(61\) 25648.0 0.882529 0.441264 − 0.897377i \(-0.354530\pi\)
0.441264 + 0.897377i \(0.354530\pi\)
\(62\) − 76440.0i − 2.52547i
\(63\) 2303.00i 0.0731042i
\(64\) 18368.0 0.560547
\(65\) 0 0
\(66\) 32480.0 0.917817
\(67\) 48404.0i 1.31733i 0.752437 + 0.658664i \(0.228878\pi\)
−0.752437 + 0.658664i \(0.771122\pi\)
\(68\) − 117096.i − 3.07093i
\(69\) 25536.0 0.645699
\(70\) 0 0
\(71\) −58560.0 −1.37865 −0.689327 − 0.724450i \(-0.742094\pi\)
−0.689327 + 0.724450i \(0.742094\pi\)
\(72\) − 16920.0i − 0.384653i
\(73\) 68082.0i 1.49529i 0.664099 + 0.747645i \(0.268815\pi\)
−0.664099 + 0.747645i \(0.731185\pi\)
\(74\) −103980. −2.20735
\(75\) 0 0
\(76\) −6664.00 −0.132343
\(77\) 11368.0i 0.218503i
\(78\) − 19600.0i − 0.364770i
\(79\) −31784.0 −0.572982 −0.286491 − 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(80\) 0 0
\(81\) −45419.0 −0.769175
\(82\) − 179620.i − 2.94999i
\(83\) − 20538.0i − 0.327237i −0.986524 − 0.163619i \(-0.947683\pi\)
0.986524 − 0.163619i \(-0.0523167\pi\)
\(84\) −46648.0 −0.721331
\(85\) 0 0
\(86\) −108800. −1.58629
\(87\) 47852.0i 0.677801i
\(88\) − 83520.0i − 1.14970i
\(89\) 50582.0 0.676894 0.338447 − 0.940985i \(-0.390098\pi\)
0.338447 + 0.940985i \(0.390098\pi\)
\(90\) 0 0
\(91\) 6860.00 0.0868402
\(92\) − 124032.i − 1.52779i
\(93\) 107016.i 1.28304i
\(94\) 93240.0 1.08839
\(95\) 0 0
\(96\) 38080.0 0.421715
\(97\) 58506.0i 0.631351i 0.948867 + 0.315676i \(0.102231\pi\)
−0.948867 + 0.315676i \(0.897769\pi\)
\(98\) − 24010.0i − 0.252538i
\(99\) 10904.0 0.111814
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 175.6.b.a.99.2 2
5.2 odd 4 7.6.a.a.1.1 ✓ 1
5.3 odd 4 175.6.a.b.1.1 1
5.4 even 2 inner 175.6.b.a.99.1 2
15.2 even 4 63.6.a.e.1.1 1
20.7 even 4 112.6.a.g.1.1 1
35.2 odd 12 49.6.c.c.18.1 2
35.12 even 12 49.6.c.b.18.1 2
35.17 even 12 49.6.c.b.30.1 2
35.27 even 4 49.6.a.a.1.1 1
35.32 odd 12 49.6.c.c.30.1 2
40.27 even 4 448.6.a.c.1.1 1
40.37 odd 4 448.6.a.m.1.1 1
55.32 even 4 847.6.a.b.1.1 1
60.47 odd 4 1008.6.a.y.1.1 1
105.62 odd 4 441.6.a.k.1.1 1
140.27 odd 4 784.6.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 ✓ 1 5.2 odd 4
49.6.a.a.1.1 1 35.27 even 4
49.6.c.b.18.1 2 35.12 even 12
49.6.c.b.30.1 2 35.17 even 12
49.6.c.c.18.1 2 35.2 odd 12
49.6.c.c.30.1 2 35.32 odd 12
63.6.a.e.1.1 1 15.2 even 4
112.6.a.g.1.1 1 20.7 even 4
175.6.a.b.1.1 1 5.3 odd 4
175.6.b.a.99.1 2 5.4 even 2 inner
175.6.b.a.99.2 2 1.1 even 1 trivial
441.6.a.k.1.1 1 105.62 odd 4
448.6.a.c.1.1 1 40.27 even 4
448.6.a.m.1.1 1 40.37 odd 4
784.6.a.c.1.1 1 140.27 odd 4
847.6.a.b.1.1 1 55.32 even 4
1008.6.a.y.1.1 1 60.47 odd 4