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 = [5,0,-1,6,-5,2,7] 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: \(5\)
Coefficient field: 5.5.368464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \) 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 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.552543\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-0.180152 q^{2} -3.06709 q^{3} -1.96755 q^{4} -1.00000 q^{5} +0.552543 q^{6} +4.41500 q^{7} +0.714762 q^{8} +6.40702 q^{9} +0.180152 q^{10} +4.27171 q^{11} +6.03463 q^{12} +2.79978 q^{13} -0.795373 q^{14} +3.06709 q^{15} +3.80632 q^{16} +4.43948 q^{17} -1.15424 q^{18} -2.43507 q^{19} +1.96755 q^{20} -13.5412 q^{21} -0.769559 q^{22} -5.77387 q^{23} -2.19224 q^{24} +1.00000 q^{25} -0.504387 q^{26} -10.4496 q^{27} -8.68672 q^{28} -0.409254 q^{29} -0.552543 q^{30} -7.79180 q^{31} -2.11524 q^{32} -13.1017 q^{33} -0.799782 q^{34} -4.41500 q^{35} -12.6061 q^{36} +0.438683 q^{38} -8.58717 q^{39} -0.714762 q^{40} +0.757374 q^{41} +2.43948 q^{42} -2.19908 q^{43} -8.40479 q^{44} -6.40702 q^{45} +1.04018 q^{46} -4.26487 q^{47} -11.6743 q^{48} +12.4922 q^{49} -0.180152 q^{50} -13.6163 q^{51} -5.50870 q^{52} -0.137540 q^{53} +1.88253 q^{54} -4.27171 q^{55} +3.15568 q^{56} +7.46857 q^{57} +0.0737281 q^{58} +3.07119 q^{59} -6.03463 q^{60} +3.02909 q^{61} +1.40371 q^{62} +28.2870 q^{63} -7.23158 q^{64} -2.79978 q^{65} +2.36030 q^{66} -11.4482 q^{67} -8.73487 q^{68} +17.7090 q^{69} +0.795373 q^{70} -10.7144 q^{71} +4.57950 q^{72} +8.20680 q^{73} -3.06709 q^{75} +4.79111 q^{76} +18.8596 q^{77} +1.54700 q^{78} -7.11193 q^{79} -3.80632 q^{80} +12.8289 q^{81} -0.136443 q^{82} -11.3625 q^{83} +26.6429 q^{84} -4.43948 q^{85} +0.396170 q^{86} +1.25522 q^{87} +3.05326 q^{88} +16.2305 q^{89} +1.15424 q^{90} +12.3610 q^{91} +11.3603 q^{92} +23.8981 q^{93} +0.768326 q^{94} +2.43507 q^{95} +6.48763 q^{96} -18.3399 q^{97} -2.25051 q^{98} +27.3690 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} + 7 q^{7} + 6 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{13} - 4 q^{14} + q^{15} + 8 q^{16} + 8 q^{17} + 6 q^{18} - 14 q^{19} - 6 q^{20} - 9 q^{21} - 2 q^{22} - 2 q^{23}+ \cdots + 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.180152 −0.127387 −0.0636934 − 0.997970i \(-0.520288\pi\)
−0.0636934 + 0.997970i \(0.520288\pi\)
\(3\) −3.06709 −1.77078 −0.885392 − 0.464846i \(-0.846110\pi\)
−0.885392 + 0.464846i \(0.846110\pi\)
\(4\) −1.96755 −0.983773
\(5\) −1.00000 −0.447214
\(6\) 0.552543 0.225575
\(7\) 4.41500 1.66871 0.834357 − 0.551224i \(-0.185839\pi\)
0.834357 + 0.551224i \(0.185839\pi\)
\(8\) 0.714762 0.252707
\(9\) 6.40702 2.13567
\(10\) 0.180152 0.0569692
\(11\) 4.27171 1.28797 0.643985 − 0.765038i \(-0.277280\pi\)
0.643985 + 0.765038i \(0.277280\pi\)
\(12\) 6.03463 1.74205
\(13\) 2.79978 0.776520 0.388260 − 0.921550i \(-0.373076\pi\)
0.388260 + 0.921550i \(0.373076\pi\)
\(14\) −0.795373 −0.212572
\(15\) 3.06709 0.791918
\(16\) 3.80632 0.951581
\(17\) 4.43948 1.07673 0.538366 − 0.842711i \(-0.319042\pi\)
0.538366 + 0.842711i \(0.319042\pi\)
\(18\) −1.15424 −0.272057
\(19\) −2.43507 −0.558643 −0.279321 − 0.960198i \(-0.590110\pi\)
−0.279321 + 0.960198i \(0.590110\pi\)
\(20\) 1.96755 0.439956
\(21\) −13.5412 −2.95493
\(22\) −0.769559 −0.164071
\(23\) −5.77387 −1.20393 −0.601967 − 0.798521i \(-0.705616\pi\)
−0.601967 + 0.798521i \(0.705616\pi\)
\(24\) −2.19224 −0.447489
\(25\) 1.00000 0.200000
\(26\) −0.504387 −0.0989184
\(27\) −10.4496 −2.01103
\(28\) −8.68672 −1.64164
\(29\) −0.409254 −0.0759967 −0.0379983 − 0.999278i \(-0.512098\pi\)
−0.0379983 + 0.999278i \(0.512098\pi\)
\(30\) −0.552543 −0.100880
\(31\) −7.79180 −1.39945 −0.699724 − 0.714413i \(-0.746694\pi\)
−0.699724 + 0.714413i \(0.746694\pi\)
\(32\) −2.11524 −0.373926
\(33\) −13.1017 −2.28072
\(34\) −0.799782 −0.137161
\(35\) −4.41500 −0.746272
\(36\) −12.6061 −2.10102
\(37\) 0 0
\(38\) 0.438683 0.0711638
\(39\) −8.58717 −1.37505
\(40\) −0.714762 −0.113014
\(41\) 0.757374 0.118282 0.0591410 − 0.998250i \(-0.481164\pi\)
0.0591410 + 0.998250i \(0.481164\pi\)
\(42\) 2.43948 0.376420
\(43\) −2.19908 −0.335357 −0.167679 − 0.985842i \(-0.553627\pi\)
−0.167679 + 0.985842i \(0.553627\pi\)
\(44\) −8.40479 −1.26707
\(45\) −6.40702 −0.955103
\(46\) 1.04018 0.153366
\(47\) −4.26487 −0.622095 −0.311048 − 0.950394i \(-0.600680\pi\)
−0.311048 + 0.950394i \(0.600680\pi\)
\(48\) −11.6743 −1.68504
\(49\) 12.4922 1.78461
\(50\) −0.180152 −0.0254774
\(51\) −13.6163 −1.90666
\(52\) −5.50870 −0.763919
\(53\) −0.137540 −0.0188926 −0.00944630 − 0.999955i \(-0.503007\pi\)
−0.00944630 + 0.999955i \(0.503007\pi\)
\(54\) 1.88253 0.256179
\(55\) −4.27171 −0.575998
\(56\) 3.15568 0.421695
\(57\) 7.46857 0.989236
\(58\) 0.0737281 0.00968098
\(59\) 3.07119 0.399835 0.199918 − 0.979813i \(-0.435933\pi\)
0.199918 + 0.979813i \(0.435933\pi\)
\(60\) −6.03463 −0.779068
\(61\) 3.02909 0.387835 0.193918 − 0.981018i \(-0.437881\pi\)
0.193918 + 0.981018i \(0.437881\pi\)
\(62\) 1.40371 0.178271
\(63\) 28.2870 3.56383
\(64\) −7.23158 −0.903948
\(65\) −2.79978 −0.347270
\(66\) 2.36030 0.290533
\(67\) −11.4482 −1.39862 −0.699310 − 0.714819i \(-0.746509\pi\)
−0.699310 + 0.714819i \(0.746509\pi\)
\(68\) −8.73487 −1.05926
\(69\) 17.7090 2.13191
\(70\) 0.795373 0.0950652
\(71\) −10.7144 −1.27156 −0.635781 − 0.771870i \(-0.719322\pi\)
−0.635781 + 0.771870i \(0.719322\pi\)
\(72\) 4.57950 0.539699
\(73\) 8.20680 0.960534 0.480267 − 0.877122i \(-0.340540\pi\)
0.480267 + 0.877122i \(0.340540\pi\)
\(74\) 0 0
\(75\) −3.06709 −0.354157
\(76\) 4.79111 0.549578
\(77\) 18.8596 2.14925
\(78\) 1.54700 0.175163
\(79\) −7.11193 −0.800155 −0.400077 − 0.916481i \(-0.631017\pi\)
−0.400077 + 0.916481i \(0.631017\pi\)
\(80\) −3.80632 −0.425560
\(81\) 12.8289 1.42543
\(82\) −0.136443 −0.0150676
\(83\) −11.3625 −1.24719 −0.623597 − 0.781746i \(-0.714329\pi\)
−0.623597 + 0.781746i \(0.714329\pi\)
\(84\) 26.6429 2.90698
\(85\) −4.43948 −0.481529
\(86\) 0.396170 0.0427201
\(87\) 1.25522 0.134574
\(88\) 3.05326 0.325479
\(89\) 16.2305 1.72043 0.860214 − 0.509933i \(-0.170330\pi\)
0.860214 + 0.509933i \(0.170330\pi\)
\(90\) 1.15424 0.121668
\(91\) 12.3610 1.29579
\(92\) 11.3603 1.18440
\(93\) 23.8981 2.47812
\(94\) 0.768326 0.0792468
\(95\) 2.43507 0.249833
\(96\) 6.48763 0.662141
\(97\) −18.3399 −1.86213 −0.931066 − 0.364850i \(-0.881120\pi\)
−0.931066 + 0.364850i \(0.881120\pi\)
\(98\) −2.25051 −0.227336
\(99\) 27.3690 2.75068
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.g.1.3 5
37.36 even 2 185.2.a.d.1.3 ✓ 5
111.110 odd 2 1665.2.a.q.1.3 5
148.147 odd 2 2960.2.a.ba.1.5 5
185.73 odd 4 925.2.b.g.149.5 10
185.147 odd 4 925.2.b.g.149.6 10
185.184 even 2 925.2.a.h.1.3 5
259.258 odd 2 9065.2.a.j.1.3 5
555.554 odd 2 8325.2.a.cc.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.d.1.3 ✓ 5 37.36 even 2
925.2.a.h.1.3 5 185.184 even 2
925.2.b.g.149.5 10 185.73 odd 4
925.2.b.g.149.6 10 185.147 odd 4
1665.2.a.q.1.3 5 111.110 odd 2
2960.2.a.ba.1.5 5 148.147 odd 2
6845.2.a.g.1.3 5 1.1 even 1 trivial
8325.2.a.cc.1.3 5 555.554 odd 2
9065.2.a.j.1.3 5 259.258 odd 2