Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1058,6,Mod(1,1058)] 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("1058.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1058.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,8,-22,32,68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(169.686081362\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 46)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1058.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -19.4853 q^{3} +16.0000 q^{4} -36.7107 q^{5} -77.9411 q^{6} +204.083 q^{7} +64.0000 q^{8} +136.676 q^{9} -146.843 q^{10} +704.621 q^{11} -311.765 q^{12} -718.373 q^{13} +816.333 q^{14} +715.318 q^{15} +256.000 q^{16} -1842.84 q^{17} +546.705 q^{18} -14.9991 q^{19} -587.371 q^{20} -3976.62 q^{21} +2818.49 q^{22} -1247.06 q^{24} -1777.33 q^{25} -2873.49 q^{26} +2071.75 q^{27} +3265.33 q^{28} +2967.75 q^{29} +2861.27 q^{30} -3673.49 q^{31} +1024.00 q^{32} -13729.7 q^{33} -7371.35 q^{34} -7492.03 q^{35} +2186.82 q^{36} +12893.8 q^{37} -59.9965 q^{38} +13997.7 q^{39} -2349.48 q^{40} -1512.65 q^{41} -15906.5 q^{42} -19239.0 q^{43} +11273.9 q^{44} -5017.48 q^{45} -8321.34 q^{47} -4988.23 q^{48} +24843.0 q^{49} -7109.30 q^{50} +35908.2 q^{51} -11494.0 q^{52} -18650.6 q^{53} +8287.00 q^{54} -25867.1 q^{55} +13061.3 q^{56} +292.262 q^{57} +11871.0 q^{58} -25095.5 q^{59} +11445.1 q^{60} +51068.4 q^{61} -14694.0 q^{62} +27893.3 q^{63} +4096.00 q^{64} +26371.9 q^{65} -54919.0 q^{66} -9050.75 q^{67} -29485.4 q^{68} -29968.1 q^{70} +32393.3 q^{71} +8747.28 q^{72} -14836.9 q^{73} +51575.1 q^{74} +34631.7 q^{75} -239.986 q^{76} +143801. q^{77} +55990.8 q^{78} +36907.1 q^{79} -9397.93 q^{80} -73580.9 q^{81} -6050.60 q^{82} +58049.7 q^{83} -63625.9 q^{84} +67651.8 q^{85} -76955.8 q^{86} -57827.4 q^{87} +45095.8 q^{88} +91442.8 q^{89} -20069.9 q^{90} -146608. q^{91} +71579.0 q^{93} -33285.4 q^{94} +550.628 q^{95} -19952.9 q^{96} -28167.1 q^{97} +99371.9 q^{98} +96305.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 22 q^{3} + 32 q^{4} + 68 q^{5} - 88 q^{6} + 312 q^{7} + 128 q^{8} - 100 q^{9} + 272 q^{10} + 12 q^{11} - 352 q^{12} - 1482 q^{13} + 1248 q^{14} + 452 q^{15} + 512 q^{16} - 1768 q^{17} - 400 q^{18}+ \cdots + 260232 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −19.4853 −1.24998 −0.624991 − 0.780632i \(-0.714897\pi\)
−0.624991 + 0.780632i \(0.714897\pi\)
\(4\) 16.0000 0.500000
\(5\) −36.7107 −0.656701 −0.328350 − 0.944556i \(-0.606493\pi\)
−0.328350 + 0.944556i \(0.606493\pi\)
\(6\) −77.9411 −0.883870
\(7\) 204.083 1.57421 0.787104 − 0.616820i \(-0.211579\pi\)
0.787104 + 0.616820i \(0.211579\pi\)
\(8\) 64.0000 0.353553
\(9\) 136.676 0.562453
\(10\) −146.843 −0.464357
\(11\) 704.621 1.75580 0.877898 − 0.478847i \(-0.158945\pi\)
0.877898 + 0.478847i \(0.158945\pi\)
\(12\) −311.765 −0.624991
\(13\) −718.373 −1.17894 −0.589470 − 0.807791i \(-0.700663\pi\)
−0.589470 + 0.807791i \(0.700663\pi\)
\(14\) 816.333 1.11313
\(15\) 715.318 0.820863
\(16\) 256.000 0.250000
\(17\) −1842.84 −1.54655 −0.773276 − 0.634069i \(-0.781383\pi\)
−0.773276 + 0.634069i \(0.781383\pi\)
\(18\) 546.705 0.397715
\(19\) −14.9991 −0.00953196 −0.00476598 − 0.999989i \(-0.501517\pi\)
−0.00476598 + 0.999989i \(0.501517\pi\)
\(20\) −587.371 −0.328350
\(21\) −3976.62 −1.96773
\(22\) 2818.49 1.24154
\(23\) 0 0
\(24\) −1247.06 −0.441935
\(25\) −1777.33 −0.568744
\(26\) −2873.49 −0.833636
\(27\) 2071.75 0.546925
\(28\) 3265.33 0.787104
\(29\) 2967.75 0.655287 0.327644 − 0.944801i \(-0.393745\pi\)
0.327644 + 0.944801i \(0.393745\pi\)
\(30\) 2861.27 0.580438
\(31\) −3673.49 −0.686554 −0.343277 − 0.939234i \(-0.611537\pi\)
−0.343277 + 0.939234i \(0.611537\pi\)
\(32\) 1024.00 0.176777
\(33\) −13729.7 −2.19471
\(34\) −7371.35 −1.09358
\(35\) −7492.03 −1.03378
\(36\) 2186.82 0.281227
\(37\) 12893.8 1.54837 0.774187 − 0.632958i \(-0.218159\pi\)
0.774187 + 0.632958i \(0.218159\pi\)
\(38\) −59.9965 −0.00674011
\(39\) 13997.7 1.47365
\(40\) −2349.48 −0.232179
\(41\) −1512.65 −0.140533 −0.0702666 − 0.997528i \(-0.522385\pi\)
−0.0702666 + 0.997528i \(0.522385\pi\)
\(42\) −15906.5 −1.39140
\(43\) −19239.0 −1.58676 −0.793378 − 0.608729i \(-0.791680\pi\)
−0.793378 + 0.608729i \(0.791680\pi\)
\(44\) 11273.9 0.877898
\(45\) −5017.48 −0.369364
\(46\) 0 0
\(47\) −8321.34 −0.549476 −0.274738 − 0.961519i \(-0.588591\pi\)
−0.274738 + 0.961519i \(0.588591\pi\)
\(48\) −4988.23 −0.312495
\(49\) 24843.0 1.47813
\(50\) −7109.30 −0.402163
\(51\) 35908.2 1.93316
\(52\) −11494.0 −0.589470
\(53\) −18650.6 −0.912018 −0.456009 − 0.889975i \(-0.650721\pi\)
−0.456009 + 0.889975i \(0.650721\pi\)
\(54\) 8287.00 0.386734
\(55\) −25867.1 −1.15303
\(56\) 13061.3 0.556567
\(57\) 292.262 0.0119148
\(58\) 11871.0 0.463358
\(59\) −25095.5 −0.938569 −0.469284 − 0.883047i \(-0.655488\pi\)
−0.469284 + 0.883047i \(0.655488\pi\)
\(60\) 11445.1 0.410432
\(61\) 51068.4 1.75723 0.878613 − 0.477534i \(-0.158469\pi\)
0.878613 + 0.477534i \(0.158469\pi\)
\(62\) −14694.0 −0.485467
\(63\) 27893.3 0.885419
\(64\) 4096.00 0.125000
\(65\) 26371.9 0.774210
\(66\) −54919.0 −1.55190
\(67\) −9050.75 −0.246319 −0.123159 − 0.992387i \(-0.539303\pi\)
−0.123159 + 0.992387i \(0.539303\pi\)
\(68\) −29485.4 −0.773276
\(69\) 0 0
\(70\) −29968.1 −0.730995
\(71\) 32393.3 0.762622 0.381311 − 0.924447i \(-0.375473\pi\)
0.381311 + 0.924447i \(0.375473\pi\)
\(72\) 8747.28 0.198857
\(73\) −14836.9 −0.325863 −0.162931 − 0.986637i \(-0.552095\pi\)
−0.162931 + 0.986637i \(0.552095\pi\)
\(74\) 51575.1 1.09487
\(75\) 34631.7 0.710920
\(76\) −239.986 −0.00476598
\(77\) 143801. 2.76399
\(78\) 55990.8 1.04203
\(79\) 36907.1 0.665339 0.332669 − 0.943044i \(-0.392051\pi\)
0.332669 + 0.943044i \(0.392051\pi\)
\(80\) −9397.93 −0.164175
\(81\) −73580.9 −1.24610
\(82\) −6050.60 −0.0993720
\(83\) 58049.7 0.924921 0.462460 − 0.886640i \(-0.346967\pi\)
0.462460 + 0.886640i \(0.346967\pi\)
\(84\) −63625.9 −0.983866
\(85\) 67651.8 1.01562
\(86\) −76955.8 −1.12201
\(87\) −57827.4 −0.819097
\(88\) 45095.8 0.620768
\(89\) 91442.8 1.22370 0.611850 − 0.790974i \(-0.290426\pi\)
0.611850 + 0.790974i \(0.290426\pi\)
\(90\) −20069.9 −0.261179
\(91\) −146608. −1.85590
\(92\) 0 0
\(93\) 71579.0 0.858180
\(94\) −33285.4 −0.388538
\(95\) 550.628 0.00625964
\(96\) −19952.9 −0.220968
\(97\) −28167.1 −0.303957 −0.151979 − 0.988384i \(-0.548564\pi\)
−0.151979 + 0.988384i \(0.548564\pi\)
\(98\) 99371.9 1.04520
\(99\) 96305.0 0.987554
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 1058.6.a.d.1.1 2
23.22 odd 2 46.6.a.b.1.1 ✓ 2
69.68 even 2 414.6.a.f.1.1 2
92.91 even 2 368.6.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.6.a.b.1.1 ✓ 2 23.22 odd 2
368.6.a.b.1.2 2 92.91 even 2
414.6.a.f.1.1 2 69.68 even 2
1058.6.a.d.1.1 2 1.1 even 1 trivial