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Properties

Label 1250.2.b.e.1249.5
Level $1250$
Weight $2$
Character 1250.1249
Analytic conductor $9.981$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [1250,2,Mod(1249,1250)] 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("1250.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1250, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1250.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8,0,2,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.98130025266\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14884000000.15
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 19x^{6} + 121x^{4} + 304x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.5
Root \(-2.96645i\) of defining polynomial
Character \(\chi\) \(=\) 1250.1249
Dual form 1250.2.b.e.1249.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.96645i q^{3} -1.00000 q^{4} +2.96645 q^{6} +1.83337i q^{7} -1.00000i q^{8} -5.79981 q^{9} -1.83337 q^{11} +2.96645i q^{12} +2.41785i q^{13} -1.83337 q^{14} +1.00000 q^{16} +2.78467i q^{17} -5.79981i q^{18} +1.67232 q^{19} +5.43858 q^{21} -1.83337i q^{22} +7.76626i q^{23} -2.96645 q^{24} -2.41785 q^{26} +8.30550i q^{27} -1.83337i q^{28} -7.58448 q^{29} -5.29413 q^{31} +1.00000i q^{32} +5.43858i q^{33} -2.78467 q^{34} +5.79981 q^{36} -1.31486i q^{37} +1.67232i q^{38} +7.17242 q^{39} +3.51505 q^{41} +5.43858i q^{42} +4.30550i q^{43} +1.83337 q^{44} -7.76626 q^{46} +1.83337i q^{47} -2.96645i q^{48} +3.63877 q^{49} +8.26057 q^{51} -2.41785i q^{52} +6.51738i q^{53} -8.30550 q^{54} +1.83337 q^{56} -4.96086i q^{57} -7.58448i q^{58} +9.06039 q^{59} +2.58794 q^{61} -5.29413i q^{62} -10.6332i q^{63} -1.00000 q^{64} -5.43858 q^{66} +9.50010i q^{67} -2.78467i q^{68} +23.0382 q^{69} +0.305502 q^{71} +5.79981i q^{72} -14.9480i q^{73} +1.31486 q^{74} -1.67232 q^{76} -3.36123i q^{77} +7.17242i q^{78} +3.46076 q^{79} +7.23840 q^{81} +3.51505i q^{82} +6.37261i q^{83} -5.43858 q^{84} -4.30550 q^{86} +22.4990i q^{87} +1.83337i q^{88} +3.32045 q^{89} -4.43280 q^{91} -7.76626i q^{92} +15.7047i q^{93} -1.83337 q^{94} +2.96645 q^{96} +11.0902i q^{97} +3.63877i q^{98} +10.6332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 2 q^{6} - 14 q^{9} - 4 q^{11} - 4 q^{14} + 8 q^{16} + 10 q^{19} - 14 q^{21} - 2 q^{24} + 22 q^{26} - 30 q^{29} - 24 q^{31} - 24 q^{34} + 14 q^{36} + 22 q^{39} + 26 q^{41} + 4 q^{44} - 8 q^{46}+ \cdots + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1250\mathbb{Z}\right)^\times\).

\(n\) \(627\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 2.96645i − 1.71268i −0.516413 − 0.856340i \(-0.672733\pi\)
0.516413 − 0.856340i \(-0.327267\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.96645 1.21105
\(7\) 1.83337i 0.692947i 0.938060 + 0.346474i \(0.112621\pi\)
−0.938060 + 0.346474i \(0.887379\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −5.79981 −1.93327
\(10\) 0 0
\(11\) −1.83337 −0.552781 −0.276390 − 0.961045i \(-0.589138\pi\)
−0.276390 + 0.961045i \(0.589138\pi\)
\(12\) 2.96645i 0.856340i
\(13\) 2.41785i 0.670590i 0.942113 + 0.335295i \(0.108836\pi\)
−0.942113 + 0.335295i \(0.891164\pi\)
\(14\) −1.83337 −0.489988
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.78467i 0.675381i 0.941257 + 0.337691i \(0.109646\pi\)
−0.941257 + 0.337691i \(0.890354\pi\)
\(18\) − 5.79981i − 1.36703i
\(19\) 1.67232 0.383657 0.191829 − 0.981428i \(-0.438558\pi\)
0.191829 + 0.981428i \(0.438558\pi\)
\(20\) 0 0
\(21\) 5.43858 1.18680
\(22\) − 1.83337i − 0.390875i
\(23\) 7.76626i 1.61938i 0.586860 + 0.809689i \(0.300364\pi\)
−0.586860 + 0.809689i \(0.699636\pi\)
\(24\) −2.96645 −0.605524
\(25\) 0 0
\(26\) −2.41785 −0.474179
\(27\) 8.30550i 1.59839i
\(28\) − 1.83337i − 0.346474i
\(29\) −7.58448 −1.40840 −0.704201 − 0.710000i \(-0.748695\pi\)
−0.704201 + 0.710000i \(0.748695\pi\)
\(30\) 0 0
\(31\) −5.29413 −0.950853 −0.475427 − 0.879755i \(-0.657706\pi\)
−0.475427 + 0.879755i \(0.657706\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.43858i 0.946736i
\(34\) −2.78467 −0.477567
\(35\) 0 0
\(36\) 5.79981 0.966636
\(37\) − 1.31486i − 0.216162i −0.994142 − 0.108081i \(-0.965529\pi\)
0.994142 − 0.108081i \(-0.0344706\pi\)
\(38\) 1.67232i 0.271286i
\(39\) 7.17242 1.14851
\(40\) 0 0
\(41\) 3.51505 0.548958 0.274479 − 0.961593i \(-0.411495\pi\)
0.274479 + 0.961593i \(0.411495\pi\)
\(42\) 5.43858i 0.839192i
\(43\) 4.30550i 0.656583i 0.944576 + 0.328291i \(0.106473\pi\)
−0.944576 + 0.328291i \(0.893527\pi\)
\(44\) 1.83337 0.276390
\(45\) 0 0
\(46\) −7.76626 −1.14507
\(47\) 1.83337i 0.267424i 0.991020 + 0.133712i \(0.0426897\pi\)
−0.991020 + 0.133712i \(0.957310\pi\)
\(48\) − 2.96645i − 0.428170i
\(49\) 3.63877 0.519824
\(50\) 0 0
\(51\) 8.26057 1.15671
\(52\) − 2.41785i − 0.335295i
\(53\) 6.51738i 0.895231i 0.894226 + 0.447615i \(0.147727\pi\)
−0.894226 + 0.447615i \(0.852273\pi\)
\(54\) −8.30550 −1.13024
\(55\) 0 0
\(56\) 1.83337 0.244994
\(57\) − 4.96086i − 0.657082i
\(58\) − 7.58448i − 0.995891i
\(59\) 9.06039 1.17956 0.589781 − 0.807563i \(-0.299214\pi\)
0.589781 + 0.807563i \(0.299214\pi\)
\(60\) 0 0
\(61\) 2.58794 0.331351 0.165676 − 0.986180i \(-0.447019\pi\)
0.165676 + 0.986180i \(0.447019\pi\)
\(62\) − 5.29413i − 0.672355i
\(63\) − 10.6332i − 1.33965i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.43858 −0.669443
\(67\) 9.50010i 1.16062i 0.814395 + 0.580311i \(0.197069\pi\)
−0.814395 + 0.580311i \(0.802931\pi\)
\(68\) − 2.78467i − 0.337691i
\(69\) 23.0382 2.77347
\(70\) 0 0
\(71\) 0.305502 0.0362564 0.0181282 − 0.999836i \(-0.494229\pi\)
0.0181282 + 0.999836i \(0.494229\pi\)
\(72\) 5.79981i 0.683515i
\(73\) − 14.9480i − 1.74954i −0.484542 − 0.874768i \(-0.661014\pi\)
0.484542 − 0.874768i \(-0.338986\pi\)
\(74\) 1.31486 0.152850
\(75\) 0 0
\(76\) −1.67232 −0.191829
\(77\) − 3.36123i − 0.383048i
\(78\) 7.17242i 0.812117i
\(79\) 3.46076 0.389366 0.194683 − 0.980866i \(-0.437632\pi\)
0.194683 + 0.980866i \(0.437632\pi\)
\(80\) 0 0
\(81\) 7.23840 0.804266
\(82\) 3.51505i 0.388172i
\(83\) 6.37261i 0.699484i 0.936846 + 0.349742i \(0.113731\pi\)
−0.936846 + 0.349742i \(0.886269\pi\)
\(84\) −5.43858 −0.593398
\(85\) 0 0
\(86\) −4.30550 −0.464274
\(87\) 22.4990i 2.41214i
\(88\) 1.83337i 0.195437i
\(89\) 3.32045 0.351967 0.175984 − 0.984393i \(-0.443689\pi\)
0.175984 + 0.984393i \(0.443689\pi\)
\(90\) 0 0
\(91\) −4.43280 −0.464684
\(92\) − 7.76626i − 0.809689i
\(93\) 15.7047i 1.62851i
\(94\) −1.83337 −0.189097
\(95\) 0 0
\(96\) 2.96645 0.302762
\(97\) 11.0902i 1.12604i 0.826445 + 0.563018i \(0.190360\pi\)
−0.826445 + 0.563018i \(0.809640\pi\)
\(98\) 3.63877i 0.367571i
\(99\) 10.6332 1.06867
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 1250.2.b.e.1249.5 8
5.2 odd 4 1250.2.a.f.1.1 4
5.3 odd 4 1250.2.a.l.1.4 4
5.4 even 2 inner 1250.2.b.e.1249.4 8
20.3 even 4 10000.2.a.t.1.1 4
20.7 even 4 10000.2.a.x.1.4 4
25.2 odd 20 250.2.d.d.101.2 8
25.9 even 10 250.2.e.c.99.3 16
25.11 even 5 250.2.e.c.149.3 16
25.12 odd 20 250.2.d.d.151.2 8
25.13 odd 20 50.2.d.b.31.1 yes 8
25.14 even 10 250.2.e.c.149.2 16
25.16 even 5 250.2.e.c.99.2 16
25.23 odd 20 50.2.d.b.21.1 ✓ 8
75.23 even 20 450.2.h.e.271.1 8
75.38 even 20 450.2.h.e.181.1 8
100.23 even 20 400.2.u.d.321.2 8
100.63 even 20 400.2.u.d.81.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.2.d.b.21.1 ✓ 8 25.23 odd 20
50.2.d.b.31.1 yes 8 25.13 odd 20
250.2.d.d.101.2 8 25.2 odd 20
250.2.d.d.151.2 8 25.12 odd 20
250.2.e.c.99.2 16 25.16 even 5
250.2.e.c.99.3 16 25.9 even 10
250.2.e.c.149.2 16 25.14 even 10
250.2.e.c.149.3 16 25.11 even 5
400.2.u.d.81.2 8 100.63 even 20
400.2.u.d.321.2 8 100.23 even 20
450.2.h.e.181.1 8 75.38 even 20
450.2.h.e.271.1 8 75.23 even 20
1250.2.a.f.1.1 4 5.2 odd 4
1250.2.a.l.1.4 4 5.3 odd 4
1250.2.b.e.1249.4 8 5.4 even 2 inner
1250.2.b.e.1249.5 8 1.1 even 1 trivial
10000.2.a.t.1.1 4 20.3 even 4
10000.2.a.x.1.4 4 20.7 even 4