Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6845,2,Mod(1,6845)] 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("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6845 = 5 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6845.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,-6,-6,30,36,-12,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.6576001836\)
Analytic rank: \(1\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.32
Character \(\chi\) \(=\) 6845.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00934 q^{2} -0.762276 q^{3} +2.03743 q^{4} +1.00000 q^{5} -1.53167 q^{6} +1.73582 q^{7} +0.0752048 q^{8} -2.41893 q^{9} +2.00934 q^{10} -0.763982 q^{11} -1.55308 q^{12} +0.0392352 q^{13} +3.48785 q^{14} -0.762276 q^{15} -3.92374 q^{16} -4.36942 q^{17} -4.86045 q^{18} +4.50612 q^{19} +2.03743 q^{20} -1.32318 q^{21} -1.53510 q^{22} +1.82694 q^{23} -0.0573268 q^{24} +1.00000 q^{25} +0.0788366 q^{26} +4.13073 q^{27} +3.53661 q^{28} -4.70896 q^{29} -1.53167 q^{30} -6.00718 q^{31} -8.03453 q^{32} +0.582366 q^{33} -8.77963 q^{34} +1.73582 q^{35} -4.92840 q^{36} +9.05431 q^{38} -0.0299081 q^{39} +0.0752048 q^{40} -5.84805 q^{41} -2.65871 q^{42} +3.43382 q^{43} -1.55656 q^{44} -2.41893 q^{45} +3.67094 q^{46} +3.63119 q^{47} +2.99098 q^{48} -3.98692 q^{49} +2.00934 q^{50} +3.33071 q^{51} +0.0799389 q^{52} -11.9390 q^{53} +8.30001 q^{54} -0.763982 q^{55} +0.130542 q^{56} -3.43491 q^{57} -9.46187 q^{58} +7.58154 q^{59} -1.55308 q^{60} -5.80042 q^{61} -12.0704 q^{62} -4.19884 q^{63} -8.29657 q^{64} +0.0392352 q^{65} +1.17017 q^{66} +0.285803 q^{67} -8.90238 q^{68} -1.39264 q^{69} +3.48785 q^{70} -5.66345 q^{71} -0.181915 q^{72} +4.13245 q^{73} -0.762276 q^{75} +9.18090 q^{76} -1.32614 q^{77} -0.0600953 q^{78} -2.33820 q^{79} -3.92374 q^{80} +4.10805 q^{81} -11.7507 q^{82} +2.65417 q^{83} -2.69588 q^{84} -4.36942 q^{85} +6.89970 q^{86} +3.58953 q^{87} -0.0574551 q^{88} -17.1690 q^{89} -4.86045 q^{90} +0.0681053 q^{91} +3.72227 q^{92} +4.57913 q^{93} +7.29629 q^{94} +4.50612 q^{95} +6.12453 q^{96} -14.8735 q^{97} -8.01106 q^{98} +1.84802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 6 q^{2} - 6 q^{3} + 30 q^{4} + 36 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8} + 30 q^{9} - 6 q^{10} - 12 q^{12} - 24 q^{13} + 24 q^{14} - 6 q^{15} + 18 q^{16} - 12 q^{17} - 30 q^{18} - 36 q^{19} + 30 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00934 1.42081 0.710407 − 0.703791i \(-0.248511\pi\)
0.710407 + 0.703791i \(0.248511\pi\)
\(3\) −0.762276 −0.440100 −0.220050 − 0.975489i \(-0.570622\pi\)
−0.220050 + 0.975489i \(0.570622\pi\)
\(4\) 2.03743 1.01871
\(5\) 1.00000 0.447214
\(6\) −1.53167 −0.625301
\(7\) 1.73582 0.656079 0.328040 − 0.944664i \(-0.393612\pi\)
0.328040 + 0.944664i \(0.393612\pi\)
\(8\) 0.0752048 0.0265889
\(9\) −2.41893 −0.806312
\(10\) 2.00934 0.635408
\(11\) −0.763982 −0.230349 −0.115175 − 0.993345i \(-0.536743\pi\)
−0.115175 + 0.993345i \(0.536743\pi\)
\(12\) −1.55308 −0.448336
\(13\) 0.0392352 0.0108819 0.00544094 − 0.999985i \(-0.498268\pi\)
0.00544094 + 0.999985i \(0.498268\pi\)
\(14\) 3.48785 0.932167
\(15\) −0.762276 −0.196819
\(16\) −3.92374 −0.980936
\(17\) −4.36942 −1.05974 −0.529870 − 0.848079i \(-0.677759\pi\)
−0.529870 + 0.848079i \(0.677759\pi\)
\(18\) −4.86045 −1.14562
\(19\) 4.50612 1.03378 0.516888 − 0.856053i \(-0.327090\pi\)
0.516888 + 0.856053i \(0.327090\pi\)
\(20\) 2.03743 0.455583
\(21\) −1.32318 −0.288741
\(22\) −1.53510 −0.327284
\(23\) 1.82694 0.380944 0.190472 − 0.981693i \(-0.438998\pi\)
0.190472 + 0.981693i \(0.438998\pi\)
\(24\) −0.0573268 −0.0117018
\(25\) 1.00000 0.200000
\(26\) 0.0788366 0.0154611
\(27\) 4.13073 0.794959
\(28\) 3.53661 0.668357
\(29\) −4.70896 −0.874431 −0.437216 − 0.899357i \(-0.644035\pi\)
−0.437216 + 0.899357i \(0.644035\pi\)
\(30\) −1.53167 −0.279643
\(31\) −6.00718 −1.07892 −0.539461 − 0.842011i \(-0.681372\pi\)
−0.539461 + 0.842011i \(0.681372\pi\)
\(32\) −8.03453 −1.42032
\(33\) 0.582366 0.101377
\(34\) −8.77963 −1.50569
\(35\) 1.73582 0.293408
\(36\) −4.92840 −0.821401
\(37\) 0 0
\(38\) 9.05431 1.46880
\(39\) −0.0299081 −0.00478912
\(40\) 0.0752048 0.0118909
\(41\) −5.84805 −0.913311 −0.456656 − 0.889644i \(-0.650953\pi\)
−0.456656 + 0.889644i \(0.650953\pi\)
\(42\) −2.65871 −0.410247
\(43\) 3.43382 0.523653 0.261827 − 0.965115i \(-0.415675\pi\)
0.261827 + 0.965115i \(0.415675\pi\)
\(44\) −1.55656 −0.234660
\(45\) −2.41893 −0.360593
\(46\) 3.67094 0.541251
\(47\) 3.63119 0.529664 0.264832 − 0.964295i \(-0.414683\pi\)
0.264832 + 0.964295i \(0.414683\pi\)
\(48\) 2.99098 0.431710
\(49\) −3.98692 −0.569560
\(50\) 2.00934 0.284163
\(51\) 3.33071 0.466392
\(52\) 0.0799389 0.0110855
\(53\) −11.9390 −1.63995 −0.819973 − 0.572402i \(-0.806012\pi\)
−0.819973 + 0.572402i \(0.806012\pi\)
\(54\) 8.30001 1.12949
\(55\) −0.763982 −0.103015
\(56\) 0.130542 0.0174444
\(57\) −3.43491 −0.454965
\(58\) −9.46187 −1.24240
\(59\) 7.58154 0.987033 0.493516 − 0.869737i \(-0.335711\pi\)
0.493516 + 0.869737i \(0.335711\pi\)
\(60\) −1.55308 −0.200502
\(61\) −5.80042 −0.742668 −0.371334 − 0.928499i \(-0.621100\pi\)
−0.371334 + 0.928499i \(0.621100\pi\)
\(62\) −12.0704 −1.53295
\(63\) −4.19884 −0.529004
\(64\) −8.29657 −1.03707
\(65\) 0.0392352 0.00486653
\(66\) 1.17017 0.144038
\(67\) 0.285803 0.0349164 0.0174582 − 0.999848i \(-0.494443\pi\)
0.0174582 + 0.999848i \(0.494443\pi\)
\(68\) −8.90238 −1.07957
\(69\) −1.39264 −0.167654
\(70\) 3.48785 0.416878
\(71\) −5.66345 −0.672127 −0.336064 − 0.941839i \(-0.609096\pi\)
−0.336064 + 0.941839i \(0.609096\pi\)
\(72\) −0.181915 −0.0214389
\(73\) 4.13245 0.483667 0.241834 − 0.970318i \(-0.422251\pi\)
0.241834 + 0.970318i \(0.422251\pi\)
\(74\) 0 0
\(75\) −0.762276 −0.0880201
\(76\) 9.18090 1.05312
\(77\) −1.32614 −0.151127
\(78\) −0.0600953 −0.00680445
\(79\) −2.33820 −0.263068 −0.131534 − 0.991312i \(-0.541990\pi\)
−0.131534 + 0.991312i \(0.541990\pi\)
\(80\) −3.92374 −0.438688
\(81\) 4.10805 0.456450
\(82\) −11.7507 −1.29765
\(83\) 2.65417 0.291333 0.145667 − 0.989334i \(-0.453467\pi\)
0.145667 + 0.989334i \(0.453467\pi\)
\(84\) −2.69588 −0.294144
\(85\) −4.36942 −0.473930
\(86\) 6.89970 0.744014
\(87\) 3.58953 0.384838
\(88\) −0.0574551 −0.00612473
\(89\) −17.1690 −1.81992 −0.909958 − 0.414701i \(-0.863886\pi\)
−0.909958 + 0.414701i \(0.863886\pi\)
\(90\) −4.86045 −0.512336
\(91\) 0.0681053 0.00713938
\(92\) 3.72227 0.388073
\(93\) 4.57913 0.474834
\(94\) 7.29629 0.752555
\(95\) 4.50612 0.462318
\(96\) 6.12453 0.625082
\(97\) −14.8735 −1.51017 −0.755086 − 0.655626i \(-0.772405\pi\)
−0.755086 + 0.655626i \(0.772405\pi\)
\(98\) −8.01106 −0.809239
\(99\) 1.84802 0.185733
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 6845.2.a.t.1.32 36
37.22 odd 36 185.2.w.a.151.11 yes 72
37.32 odd 36 185.2.w.a.136.11 ✓ 72
37.36 even 2 6845.2.a.u.1.5 36
185.22 even 36 925.2.ba.c.299.11 72
185.32 even 36 925.2.ba.b.99.2 72
185.59 odd 36 925.2.bb.b.151.2 72
185.69 odd 36 925.2.bb.b.876.2 72
185.133 even 36 925.2.ba.b.299.2 72
185.143 even 36 925.2.ba.c.99.11 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.11 ✓ 72 37.32 odd 36
185.2.w.a.151.11 yes 72 37.22 odd 36
925.2.ba.b.99.2 72 185.32 even 36
925.2.ba.b.299.2 72 185.133 even 36
925.2.ba.c.99.11 72 185.143 even 36
925.2.ba.c.299.11 72 185.22 even 36
925.2.bb.b.151.2 72 185.59 odd 36
925.2.bb.b.876.2 72 185.69 odd 36
6845.2.a.t.1.32 36 1.1 even 1 trivial
6845.2.a.u.1.5 36 37.36 even 2