Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6,2,6,4,2,6] 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: \(59.1372077370\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.84770496.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 24x^{3} + 6x^{2} - 28x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.21099\) of defining polynomial
Character \(\chi\) \(=\) 7406.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.21099 q^{3} +1.00000 q^{4} -0.639974 q^{5} +2.21099 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.88850 q^{9} -0.639974 q^{10} +0.789006 q^{11} +2.21099 q^{12} -6.47694 q^{13} +1.00000 q^{14} -1.41498 q^{15} +1.00000 q^{16} +4.05151 q^{17} +1.88850 q^{18} +7.42742 q^{19} -0.639974 q^{20} +2.21099 q^{21} +0.789006 q^{22} +2.21099 q^{24} -4.59043 q^{25} -6.47694 q^{26} -2.45753 q^{27} +1.00000 q^{28} +5.05694 q^{29} -1.41498 q^{30} +10.4254 q^{31} +1.00000 q^{32} +1.74449 q^{33} +4.05151 q^{34} -0.639974 q^{35} +1.88850 q^{36} +5.20202 q^{37} +7.42742 q^{38} -14.3205 q^{39} -0.639974 q^{40} -6.05893 q^{41} +2.21099 q^{42} -4.15404 q^{43} +0.789006 q^{44} -1.20859 q^{45} +0.224280 q^{47} +2.21099 q^{48} +1.00000 q^{49} -4.59043 q^{50} +8.95787 q^{51} -6.47694 q^{52} +4.32603 q^{53} -2.45753 q^{54} -0.504943 q^{55} +1.00000 q^{56} +16.4220 q^{57} +5.05694 q^{58} +8.26992 q^{59} -1.41498 q^{60} +14.0811 q^{61} +10.4254 q^{62} +1.88850 q^{63} +1.00000 q^{64} +4.14508 q^{65} +1.74449 q^{66} -4.52691 q^{67} +4.05151 q^{68} -0.639974 q^{70} +12.8175 q^{71} +1.88850 q^{72} -4.45753 q^{73} +5.20202 q^{74} -10.1494 q^{75} +7.42742 q^{76} +0.789006 q^{77} -14.3205 q^{78} -5.84094 q^{79} -0.639974 q^{80} -11.0991 q^{81} -6.05893 q^{82} +13.7690 q^{83} +2.21099 q^{84} -2.59286 q^{85} -4.15404 q^{86} +11.1809 q^{87} +0.789006 q^{88} -1.01839 q^{89} -1.20859 q^{90} -6.47694 q^{91} +23.0505 q^{93} +0.224280 q^{94} -4.75335 q^{95} +2.21099 q^{96} +8.19926 q^{97} +1.00000 q^{98} +1.49003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} + 4 q^{10} + 16 q^{11} + 2 q^{12} + 6 q^{14} + 6 q^{16} + 2 q^{17} + 6 q^{18} + 18 q^{19} + 4 q^{20} + 2 q^{21} + 16 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.21099 1.27652 0.638259 − 0.769822i \(-0.279655\pi\)
0.638259 + 0.769822i \(0.279655\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.639974 −0.286205 −0.143103 − 0.989708i \(-0.545708\pi\)
−0.143103 + 0.989708i \(0.545708\pi\)
\(6\) 2.21099 0.902635
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.88850 0.629499
\(10\) −0.639974 −0.202378
\(11\) 0.789006 0.237894 0.118947 − 0.992901i \(-0.462048\pi\)
0.118947 + 0.992901i \(0.462048\pi\)
\(12\) 2.21099 0.638259
\(13\) −6.47694 −1.79638 −0.898190 − 0.439607i \(-0.855118\pi\)
−0.898190 + 0.439607i \(0.855118\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.41498 −0.365346
\(16\) 1.00000 0.250000
\(17\) 4.05151 0.982636 0.491318 − 0.870980i \(-0.336515\pi\)
0.491318 + 0.870980i \(0.336515\pi\)
\(18\) 1.88850 0.445123
\(19\) 7.42742 1.70397 0.851983 − 0.523569i \(-0.175400\pi\)
0.851983 + 0.523569i \(0.175400\pi\)
\(20\) −0.639974 −0.143103
\(21\) 2.21099 0.482479
\(22\) 0.789006 0.168217
\(23\) 0 0
\(24\) 2.21099 0.451317
\(25\) −4.59043 −0.918087
\(26\) −6.47694 −1.27023
\(27\) −2.45753 −0.472952
\(28\) 1.00000 0.188982
\(29\) 5.05694 0.939050 0.469525 − 0.882919i \(-0.344425\pi\)
0.469525 + 0.882919i \(0.344425\pi\)
\(30\) −1.41498 −0.258339
\(31\) 10.4254 1.87246 0.936230 − 0.351388i \(-0.114290\pi\)
0.936230 + 0.351388i \(0.114290\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.74449 0.303676
\(34\) 4.05151 0.694829
\(35\) −0.639974 −0.108175
\(36\) 1.88850 0.314749
\(37\) 5.20202 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(38\) 7.42742 1.20489
\(39\) −14.3205 −2.29311
\(40\) −0.639974 −0.101189
\(41\) −6.05893 −0.946245 −0.473123 − 0.880997i \(-0.656873\pi\)
−0.473123 + 0.880997i \(0.656873\pi\)
\(42\) 2.21099 0.341164
\(43\) −4.15404 −0.633485 −0.316743 − 0.948512i \(-0.602589\pi\)
−0.316743 + 0.948512i \(0.602589\pi\)
\(44\) 0.789006 0.118947
\(45\) −1.20859 −0.180166
\(46\) 0 0
\(47\) 0.224280 0.0327145 0.0163573 − 0.999866i \(-0.494793\pi\)
0.0163573 + 0.999866i \(0.494793\pi\)
\(48\) 2.21099 0.319130
\(49\) 1.00000 0.142857
\(50\) −4.59043 −0.649185
\(51\) 8.95787 1.25435
\(52\) −6.47694 −0.898190
\(53\) 4.32603 0.594227 0.297113 − 0.954842i \(-0.403976\pi\)
0.297113 + 0.954842i \(0.403976\pi\)
\(54\) −2.45753 −0.334427
\(55\) −0.504943 −0.0680865
\(56\) 1.00000 0.133631
\(57\) 16.4220 2.17514
\(58\) 5.05694 0.664009
\(59\) 8.26992 1.07665 0.538326 − 0.842737i \(-0.319057\pi\)
0.538326 + 0.842737i \(0.319057\pi\)
\(60\) −1.41498 −0.182673
\(61\) 14.0811 1.80290 0.901450 − 0.432883i \(-0.142504\pi\)
0.901450 + 0.432883i \(0.142504\pi\)
\(62\) 10.4254 1.32403
\(63\) 1.88850 0.237928
\(64\) 1.00000 0.125000
\(65\) 4.14508 0.514133
\(66\) 1.74449 0.214732
\(67\) −4.52691 −0.553050 −0.276525 − 0.961007i \(-0.589183\pi\)
−0.276525 + 0.961007i \(0.589183\pi\)
\(68\) 4.05151 0.491318
\(69\) 0 0
\(70\) −0.639974 −0.0764916
\(71\) 12.8175 1.52116 0.760580 − 0.649244i \(-0.224915\pi\)
0.760580 + 0.649244i \(0.224915\pi\)
\(72\) 1.88850 0.222561
\(73\) −4.45753 −0.521714 −0.260857 − 0.965377i \(-0.584005\pi\)
−0.260857 + 0.965377i \(0.584005\pi\)
\(74\) 5.20202 0.604722
\(75\) −10.1494 −1.17195
\(76\) 7.42742 0.851983
\(77\) 0.789006 0.0899155
\(78\) −14.3205 −1.62148
\(79\) −5.84094 −0.657157 −0.328579 − 0.944477i \(-0.606570\pi\)
−0.328579 + 0.944477i \(0.606570\pi\)
\(80\) −0.639974 −0.0715513
\(81\) −11.0991 −1.23323
\(82\) −6.05893 −0.669097
\(83\) 13.7690 1.51135 0.755675 − 0.654947i \(-0.227309\pi\)
0.755675 + 0.654947i \(0.227309\pi\)
\(84\) 2.21099 0.241239
\(85\) −2.59286 −0.281236
\(86\) −4.15404 −0.447942
\(87\) 11.1809 1.19871
\(88\) 0.789006 0.0841083
\(89\) −1.01839 −0.107949 −0.0539747 − 0.998542i \(-0.517189\pi\)
−0.0539747 + 0.998542i \(0.517189\pi\)
\(90\) −1.20859 −0.127396
\(91\) −6.47694 −0.678968
\(92\) 0 0
\(93\) 23.0505 2.39023
\(94\) 0.224280 0.0231327
\(95\) −4.75335 −0.487684
\(96\) 2.21099 0.225659
\(97\) 8.19926 0.832509 0.416254 − 0.909248i \(-0.363343\pi\)
0.416254 + 0.909248i \(0.363343\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.49003 0.149754
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 7406.2.a.bl.1.5 yes 6
23.22 odd 2 7406.2.a.bk.1.5 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bk.1.5 ✓ 6 23.22 odd 2
7406.2.a.bl.1.5 yes 6 1.1 even 1 trivial