Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [825,4,Mod(199,825)] 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("825.199"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,14,0,-6,0,0,-36,0,-44,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.j.199.4

$q$-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)
 
\(f(q)\) \(=\) \(q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} -11.0000 q^{11} -4.31534i q^{12} +49.1231i q^{13} +16.0000 q^{14} -50.4233 q^{16} +82.7083i q^{17} +23.0540i q^{18} +130.354 q^{19} +18.7386 q^{21} +28.1771i q^{22} +185.693i q^{23} -72.5312 q^{24} +125.831 q^{26} +27.0000i q^{27} +8.98485i q^{28} +8.90720 q^{29} +5.26137 q^{31} -64.2547i q^{32} +33.0000i q^{33} +211.862 q^{34} -12.9460 q^{36} -416.894i q^{37} -333.909i q^{38} +147.369 q^{39} -298.479 q^{41} -48.0000i q^{42} +513.633i q^{43} -15.8229 q^{44} +475.663 q^{46} +557.295i q^{47} +151.270i q^{48} +303.985 q^{49} +248.125 q^{51} +70.6610i q^{52} +168.064i q^{53} +69.1619 q^{54} +151.015 q^{56} -391.062i q^{57} -22.8163i q^{58} -618.773 q^{59} +786.405 q^{61} -13.4773i q^{62} -56.2159i q^{63} -567.978 q^{64} +84.5312 q^{66} -339.015i q^{67} +118.972i q^{68} +557.080 q^{69} +1120.71 q^{71} +217.594i q^{72} +123.430i q^{73} -1067.90 q^{74} +187.508 q^{76} -68.7083i q^{77} -377.494i q^{78} +309.835 q^{79} +81.0000 q^{81} +764.570i q^{82} +1021.22i q^{83} +26.9545 q^{84} +1315.70 q^{86} -26.7216i q^{87} +265.948i q^{88} +141.879 q^{89} -306.833 q^{91} +267.110i q^{92} -15.7841i q^{93} +1427.54 q^{94} -192.764 q^{96} +798.345i q^{97} -778.673i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 64 q^{14} - 78 q^{16} + 340 q^{19} - 24 q^{21} - 18 q^{24} + 124 q^{26} + 316 q^{29} + 120 q^{31} + 732 q^{34} - 126 q^{36} + 540 q^{39} + 76 q^{41} - 154 q^{44}+ \cdots + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.56155i − 0.905646i −0.891601 − 0.452823i \(-0.850417\pi\)
0.891601 − 0.452823i \(-0.149583\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 1.43845 0.179806
\(5\) 0 0
\(6\) −7.68466 −0.522875
\(7\) 6.24621i 0.337264i 0.985679 + 0.168632i \(0.0539349\pi\)
−0.985679 + 0.168632i \(0.946065\pi\)
\(8\) − 24.1771i − 1.06849i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) − 4.31534i − 0.103811i
\(13\) 49.1231i 1.04802i 0.851711 + 0.524011i \(0.175565\pi\)
−0.851711 + 0.524011i \(0.824435\pi\)
\(14\) 16.0000 0.305441
\(15\) 0 0
\(16\) −50.4233 −0.787864
\(17\) 82.7083i 1.17998i 0.807409 + 0.589992i \(0.200869\pi\)
−0.807409 + 0.589992i \(0.799131\pi\)
\(18\) 23.0540i 0.301882i
\(19\) 130.354 1.57396 0.786981 − 0.616977i \(-0.211643\pi\)
0.786981 + 0.616977i \(0.211643\pi\)
\(20\) 0 0
\(21\) 18.7386 0.194719
\(22\) 28.1771i 0.273062i
\(23\) 185.693i 1.68347i 0.539895 + 0.841733i \(0.318464\pi\)
−0.539895 + 0.841733i \(0.681536\pi\)
\(24\) −72.5312 −0.616891
\(25\) 0 0
\(26\) 125.831 0.949137
\(27\) 27.0000i 0.192450i
\(28\) 8.98485i 0.0606420i
\(29\) 8.90720 0.0570354 0.0285177 − 0.999593i \(-0.490921\pi\)
0.0285177 + 0.999593i \(0.490921\pi\)
\(30\) 0 0
\(31\) 5.26137 0.0304829 0.0152414 − 0.999884i \(-0.495148\pi\)
0.0152414 + 0.999884i \(0.495148\pi\)
\(32\) − 64.2547i − 0.354961i
\(33\) 33.0000i 0.174078i
\(34\) 211.862 1.06865
\(35\) 0 0
\(36\) −12.9460 −0.0599353
\(37\) − 416.894i − 1.85235i −0.377094 − 0.926175i \(-0.623077\pi\)
0.377094 − 0.926175i \(-0.376923\pi\)
\(38\) − 333.909i − 1.42545i
\(39\) 147.369 0.605076
\(40\) 0 0
\(41\) −298.479 −1.13694 −0.568471 − 0.822703i \(-0.692465\pi\)
−0.568471 + 0.822703i \(0.692465\pi\)
\(42\) − 48.0000i − 0.176347i
\(43\) 513.633i 1.82159i 0.412863 + 0.910793i \(0.364529\pi\)
−0.412863 + 0.910793i \(0.635471\pi\)
\(44\) −15.8229 −0.0542135
\(45\) 0 0
\(46\) 475.663 1.52462
\(47\) 557.295i 1.72957i 0.502140 + 0.864786i \(0.332546\pi\)
−0.502140 + 0.864786i \(0.667454\pi\)
\(48\) 151.270i 0.454873i
\(49\) 303.985 0.886253
\(50\) 0 0
\(51\) 248.125 0.681264
\(52\) 70.6610i 0.188441i
\(53\) 168.064i 0.435574i 0.975996 + 0.217787i \(0.0698838\pi\)
−0.975996 + 0.217787i \(0.930116\pi\)
\(54\) 69.1619 0.174292
\(55\) 0 0
\(56\) 151.015 0.360362
\(57\) − 391.062i − 0.908728i
\(58\) − 22.8163i − 0.0516539i
\(59\) −618.773 −1.36538 −0.682689 − 0.730709i \(-0.739190\pi\)
−0.682689 + 0.730709i \(0.739190\pi\)
\(60\) 0 0
\(61\) 786.405 1.65064 0.825319 − 0.564667i \(-0.190996\pi\)
0.825319 + 0.564667i \(0.190996\pi\)
\(62\) − 13.4773i − 0.0276067i
\(63\) − 56.2159i − 0.112421i
\(64\) −567.978 −1.10933
\(65\) 0 0
\(66\) 84.5312 0.157653
\(67\) − 339.015i − 0.618169i −0.951035 − 0.309084i \(-0.899977\pi\)
0.951035 − 0.309084i \(-0.100023\pi\)
\(68\) 118.972i 0.212168i
\(69\) 557.080 0.971949
\(70\) 0 0
\(71\) 1120.71 1.87329 0.936645 − 0.350280i \(-0.113913\pi\)
0.936645 + 0.350280i \(0.113913\pi\)
\(72\) 217.594i 0.356162i
\(73\) 123.430i 0.197896i 0.995093 + 0.0989478i \(0.0315477\pi\)
−0.995093 + 0.0989478i \(0.968452\pi\)
\(74\) −1067.90 −1.67757
\(75\) 0 0
\(76\) 187.508 0.283008
\(77\) − 68.7083i − 0.101689i
\(78\) − 377.494i − 0.547985i
\(79\) 309.835 0.441255 0.220628 − 0.975358i \(-0.429189\pi\)
0.220628 + 0.975358i \(0.429189\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 764.570i 1.02967i
\(83\) 1021.22i 1.35053i 0.737577 + 0.675263i \(0.235970\pi\)
−0.737577 + 0.675263i \(0.764030\pi\)
\(84\) 26.9545 0.0350117
\(85\) 0 0
\(86\) 1315.70 1.64971
\(87\) − 26.7216i − 0.0329294i
\(88\) 265.948i 0.322161i
\(89\) 141.879 0.168979 0.0844894 − 0.996424i \(-0.473074\pi\)
0.0844894 + 0.996424i \(0.473074\pi\)
\(90\) 0 0
\(91\) −306.833 −0.353460
\(92\) 267.110i 0.302697i
\(93\) − 15.7841i − 0.0175993i
\(94\) 1427.54 1.56638
\(95\) 0 0
\(96\) −192.764 −0.204937
\(97\) 798.345i 0.835666i 0.908524 + 0.417833i \(0.137210\pi\)
−0.908524 + 0.417833i \(0.862790\pi\)
\(98\) − 778.673i − 0.802631i
\(99\) 99.0000 0.100504
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 825.4.c.j.199.1 4
5.2 odd 4 825.4.a.m.1.2 2
5.3 odd 4 165.4.a.c.1.1 ✓ 2
5.4 even 2 inner 825.4.c.j.199.4 4
15.2 even 4 2475.4.a.n.1.1 2
15.8 even 4 495.4.a.d.1.2 2
55.43 even 4 1815.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.1 ✓ 2 5.3 odd 4
495.4.a.d.1.2 2 15.8 even 4
825.4.a.m.1.2 2 5.2 odd 4
825.4.c.j.199.1 4 1.1 even 1 trivial
825.4.c.j.199.4 4 5.4 even 2 inner
1815.4.a.n.1.2 2 55.43 even 4
2475.4.a.n.1.1 2 15.2 even 4