Properties

Label 2.26.a.b.1.2
Level $2$
Weight $26$
Character 2.1
Self dual yes
Analytic conductor $7.920$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2,26,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.91993559904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-162.829\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} +1.75788e6 q^{3} +1.67772e7 q^{4} -1.37041e8 q^{5} +7.20027e9 q^{6} -3.04153e10 q^{7} +6.87195e10 q^{8} +2.24285e12 q^{9} -5.61320e11 q^{10} +2.58704e12 q^{11} +2.94923e13 q^{12} -9.57327e13 q^{13} -1.24581e14 q^{14} -2.40901e14 q^{15} +2.81475e14 q^{16} -1.64685e15 q^{17} +9.18671e15 q^{18} +4.95030e15 q^{19} -2.29916e15 q^{20} -5.34664e16 q^{21} +1.05965e16 q^{22} -1.07650e16 q^{23} +1.20801e17 q^{24} -2.79243e17 q^{25} -3.92121e17 q^{26} +2.45323e18 q^{27} -5.10284e17 q^{28} -1.36741e18 q^{29} -9.86732e17 q^{30} -4.42000e18 q^{31} +1.15292e18 q^{32} +4.54771e18 q^{33} -6.74549e18 q^{34} +4.16814e18 q^{35} +3.76288e19 q^{36} +1.01944e19 q^{37} +2.02764e19 q^{38} -1.68287e20 q^{39} -9.41738e18 q^{40} +1.58687e20 q^{41} -2.18999e20 q^{42} +1.83575e20 q^{43} +4.34034e19 q^{44} -3.07362e20 q^{45} -4.40934e19 q^{46} +1.40203e21 q^{47} +4.94799e20 q^{48} -4.15978e20 q^{49} -1.14378e21 q^{50} -2.89496e21 q^{51} -1.60613e21 q^{52} -1.99903e21 q^{53} +1.00484e22 q^{54} -3.54531e20 q^{55} -2.09012e21 q^{56} +8.70202e21 q^{57} -5.60091e21 q^{58} -4.16691e21 q^{59} -4.04165e21 q^{60} +3.42128e22 q^{61} -1.81043e22 q^{62} -6.82170e22 q^{63} +4.72237e21 q^{64} +1.31193e22 q^{65} +1.86274e22 q^{66} +8.67051e22 q^{67} -2.76295e22 q^{68} -1.89236e22 q^{69} +1.70727e22 q^{70} -5.13159e22 q^{71} +1.54128e23 q^{72} +3.49147e22 q^{73} +4.17563e22 q^{74} -4.90875e23 q^{75} +8.30522e22 q^{76} -7.86857e22 q^{77} -6.89302e23 q^{78} +2.91588e23 q^{79} -3.85736e22 q^{80} +2.41214e24 q^{81} +6.49981e23 q^{82} -1.64916e24 q^{83} -8.97018e23 q^{84} +2.25686e23 q^{85} +7.51925e23 q^{86} -2.40374e24 q^{87} +1.77780e23 q^{88} +8.74435e23 q^{89} -1.25896e24 q^{90} +2.91174e24 q^{91} -1.80607e23 q^{92} -7.76982e24 q^{93} +5.74273e24 q^{94} -6.78393e23 q^{95} +2.02670e24 q^{96} +1.00608e25 q^{97} -1.70384e24 q^{98} +5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8192 q^{2} + 379848 q^{3} + 33554432 q^{4} + 741953100 q^{5} + 1555857408 q^{6} - 376536944 q^{7} + 137438953472 q^{8} + 3294531432666 q^{9} + 3039039897600 q^{10} + 8323034610264 q^{11} + 6372791943168 q^{12}+ \cdots + 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00 0.707107
\(3\) 1.75788e6 1.90974 0.954868 − 0.297031i \(-0.0959963\pi\)
0.954868 + 0.297031i \(0.0959963\pi\)
\(4\) 1.67772e7 0.500000
\(5\) −1.37041e8 −0.251030 −0.125515 − 0.992092i \(-0.540058\pi\)
−0.125515 + 0.992092i \(0.540058\pi\)
\(6\) 7.20027e9 1.35039
\(7\) −3.04153e10 −0.830552 −0.415276 − 0.909696i \(-0.636315\pi\)
−0.415276 + 0.909696i \(0.636315\pi\)
\(8\) 6.87195e10 0.353553
\(9\) 2.24285e12 2.64709
\(10\) −5.61320e11 −0.177505
\(11\) 2.58704e12 0.248539 0.124270 − 0.992248i \(-0.460341\pi\)
0.124270 + 0.992248i \(0.460341\pi\)
\(12\) 2.94923e13 0.954868
\(13\) −9.57327e13 −1.13964 −0.569821 − 0.821769i \(-0.692988\pi\)
−0.569821 + 0.821769i \(0.692988\pi\)
\(14\) −1.24581e14 −0.587289
\(15\) −2.40901e14 −0.479400
\(16\) 2.81475e14 0.250000
\(17\) −1.64685e15 −0.685555 −0.342777 − 0.939417i \(-0.611368\pi\)
−0.342777 + 0.939417i \(0.611368\pi\)
\(18\) 9.18671e15 1.87178
\(19\) 4.95030e15 0.513111 0.256555 − 0.966530i \(-0.417412\pi\)
0.256555 + 0.966530i \(0.417412\pi\)
\(20\) −2.29916e15 −0.125515
\(21\) −5.34664e16 −1.58613
\(22\) 1.05965e16 0.175744
\(23\) −1.07650e16 −0.102427 −0.0512136 − 0.998688i \(-0.516309\pi\)
−0.0512136 + 0.998688i \(0.516309\pi\)
\(24\) 1.20801e17 0.675194
\(25\) −2.79243e17 −0.936984
\(26\) −3.92121e17 −0.805849
\(27\) 2.45323e18 3.14551
\(28\) −5.10284e17 −0.415276
\(29\) −1.36741e18 −0.717668 −0.358834 − 0.933401i \(-0.616826\pi\)
−0.358834 + 0.933401i \(0.616826\pi\)
\(30\) −9.86732e17 −0.338987
\(31\) −4.42000e18 −1.00786 −0.503931 − 0.863744i \(-0.668113\pi\)
−0.503931 + 0.863744i \(0.668113\pi\)
\(32\) 1.15292e18 0.176777
\(33\) 4.54771e18 0.474645
\(34\) −6.74549e18 −0.484760
\(35\) 4.16814e18 0.208493
\(36\) 3.76288e19 1.32355
\(37\) 1.01944e19 0.254590 0.127295 − 0.991865i \(-0.459371\pi\)
0.127295 + 0.991865i \(0.459371\pi\)
\(38\) 2.02764e19 0.362824
\(39\) −1.68287e20 −2.17642
\(40\) −9.41738e18 −0.0887524
\(41\) 1.58687e20 1.09835 0.549177 − 0.835706i \(-0.314941\pi\)
0.549177 + 0.835706i \(0.314941\pi\)
\(42\) −2.18999e20 −1.12157
\(43\) 1.83575e20 0.700582 0.350291 − 0.936641i \(-0.386083\pi\)
0.350291 + 0.936641i \(0.386083\pi\)
\(44\) 4.34034e19 0.124270
\(45\) −3.07362e20 −0.664499
\(46\) −4.40934e19 −0.0724270
\(47\) 1.40203e21 1.76009 0.880045 − 0.474890i \(-0.157512\pi\)
0.880045 + 0.474890i \(0.157512\pi\)
\(48\) 4.94799e20 0.477434
\(49\) −4.15978e20 −0.310184
\(50\) −1.14378e21 −0.662548
\(51\) −2.89496e21 −1.30923
\(52\) −1.60613e21 −0.569821
\(53\) −1.99903e21 −0.558946 −0.279473 − 0.960154i \(-0.590160\pi\)
−0.279473 + 0.960154i \(0.590160\pi\)
\(54\) 1.00484e22 2.22421
\(55\) −3.54531e20 −0.0623908
\(56\) −2.09012e21 −0.293644
\(57\) 8.70202e21 0.979906
\(58\) −5.60091e21 −0.507468
\(59\) −4.16691e21 −0.304905 −0.152452 − 0.988311i \(-0.548717\pi\)
−0.152452 + 0.988311i \(0.548717\pi\)
\(60\) −4.04165e21 −0.239700
\(61\) 3.42128e22 1.65031 0.825156 − 0.564904i \(-0.191087\pi\)
0.825156 + 0.564904i \(0.191087\pi\)
\(62\) −1.81043e22 −0.712666
\(63\) −6.82170e22 −2.19855
\(64\) 4.72237e21 0.125000
\(65\) 1.31193e22 0.286084
\(66\) 1.86274e22 0.335624
\(67\) 8.67051e22 1.29452 0.647261 − 0.762268i \(-0.275914\pi\)
0.647261 + 0.762268i \(0.275914\pi\)
\(68\) −2.76295e22 −0.342777
\(69\) −1.89236e22 −0.195609
\(70\) 1.70727e22 0.147427
\(71\) −5.13159e22 −0.371127 −0.185564 − 0.982632i \(-0.559411\pi\)
−0.185564 + 0.982632i \(0.559411\pi\)
\(72\) 1.54128e23 0.935888
\(73\) 3.49147e22 0.178432 0.0892159 − 0.996012i \(-0.471564\pi\)
0.0892159 + 0.996012i \(0.471564\pi\)
\(74\) 4.17563e22 0.180022
\(75\) −4.90875e23 −1.78939
\(76\) 8.30522e22 0.256555
\(77\) −7.86857e22 −0.206425
\(78\) −6.89302e23 −1.53896
\(79\) 2.91588e23 0.555176 0.277588 − 0.960700i \(-0.410465\pi\)
0.277588 + 0.960700i \(0.410465\pi\)
\(80\) −3.85736e22 −0.0627574
\(81\) 2.41214e24 3.36000
\(82\) 6.49981e23 0.776654
\(83\) −1.64916e24 −1.69351 −0.846753 − 0.531986i \(-0.821446\pi\)
−0.846753 + 0.531986i \(0.821446\pi\)
\(84\) −8.97018e23 −0.793067
\(85\) 2.25686e23 0.172095
\(86\) 7.51925e23 0.495386
\(87\) −2.40374e24 −1.37056
\(88\) 1.77780e23 0.0878720
\(89\) 8.74435e23 0.375277 0.187639 − 0.982238i \(-0.439917\pi\)
0.187639 + 0.982238i \(0.439917\pi\)
\(90\) −1.25896e24 −0.469871
\(91\) 2.91174e24 0.946532
\(92\) −1.80607e23 −0.0512136
\(93\) −7.76982e24 −1.92475
\(94\) 5.74273e24 1.24457
\(95\) −6.78393e23 −0.128806
\(96\) 2.02670e24 0.337597
\(97\) 1.00608e25 1.47227 0.736134 − 0.676835i \(-0.236649\pi\)
0.736134 + 0.676835i \(0.236649\pi\)
\(98\) −1.70384e24 −0.219333
\(99\) 5.80235e24 0.657907
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 2.26.a.b.1.2 ✓ 2
3.2 odd 2 18.26.a.e.1.2 2
4.3 odd 2 16.26.a.c.1.1 2
5.2 odd 4 50.26.b.e.49.3 4
5.3 odd 4 50.26.b.e.49.2 4
5.4 even 2 50.26.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.2 ✓ 2 1.1 even 1 trivial
16.26.a.c.1.1 2 4.3 odd 2
18.26.a.e.1.2 2 3.2 odd 2
50.26.a.c.1.1 2 5.4 even 2
50.26.b.e.49.2 4 5.3 odd 4
50.26.b.e.49.3 4 5.2 odd 4