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Properties

Label 1150.2.b.g.599.3
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $4$
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] = [1150,2,Mod(599,1150)] 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("1150.599"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,2,0,0,-10,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) 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 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.3
Root \(-2.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.g.599.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.79129i q^{3} -1.00000 q^{4} +2.79129 q^{6} +1.79129i q^{7} -1.00000i q^{8} -4.79129 q^{9} -0.791288 q^{11} +2.79129i q^{12} +5.79129i q^{13} -1.79129 q^{14} +1.00000 q^{16} -0.791288i q^{17} -4.79129i q^{18} -5.79129 q^{19} +5.00000 q^{21} -0.791288i q^{22} +1.00000i q^{23} -2.79129 q^{24} -5.79129 q^{26} +5.00000i q^{27} -1.79129i q^{28} -7.58258 q^{29} -3.37386 q^{31} +1.00000i q^{32} +2.20871i q^{33} +0.791288 q^{34} +4.79129 q^{36} +4.00000i q^{37} -5.79129i q^{38} +16.1652 q^{39} -6.79129 q^{41} +5.00000i q^{42} +11.1652i q^{43} +0.791288 q^{44} -1.00000 q^{46} +4.41742i q^{47} -2.79129i q^{48} +3.79129 q^{49} -2.20871 q^{51} -5.79129i q^{52} +6.00000i q^{53} -5.00000 q^{54} +1.79129 q^{56} +16.1652i q^{57} -7.58258i q^{58} +13.5826 q^{59} +10.3739 q^{61} -3.37386i q^{62} -8.58258i q^{63} -1.00000 q^{64} -2.20871 q^{66} -11.1652i q^{67} +0.791288i q^{68} +2.79129 q^{69} +8.37386 q^{71} +4.79129i q^{72} +12.7477i q^{73} -4.00000 q^{74} +5.79129 q^{76} -1.41742i q^{77} +16.1652i q^{78} -8.00000 q^{79} -0.417424 q^{81} -6.79129i q^{82} -6.00000i q^{83} -5.00000 q^{84} -11.1652 q^{86} +21.1652i q^{87} +0.791288i q^{88} -15.1652 q^{89} -10.3739 q^{91} -1.00000i q^{92} +9.41742i q^{93} -4.41742 q^{94} +2.79129 q^{96} +7.95644i q^{97} +3.79129i q^{98} +3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{6} - 10 q^{9} + 6 q^{11} + 2 q^{14} + 4 q^{16} - 14 q^{19} + 20 q^{21} - 2 q^{24} - 14 q^{26} - 12 q^{29} + 14 q^{31} - 6 q^{34} + 10 q^{36} + 28 q^{39} - 18 q^{41} - 6 q^{44} - 4 q^{46}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 2.79129i − 1.61155i −0.592221 − 0.805775i \(-0.701749\pi\)
0.592221 − 0.805775i \(-0.298251\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.79129 1.13954
\(7\) 1.79129i 0.677043i 0.940959 + 0.338522i \(0.109927\pi\)
−0.940959 + 0.338522i \(0.890073\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −4.79129 −1.59710
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 − 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) 2.79129i 0.805775i
\(13\) 5.79129i 1.60621i 0.595835 + 0.803107i \(0.296821\pi\)
−0.595835 + 0.803107i \(0.703179\pi\)
\(14\) −1.79129 −0.478742
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 0.791288i − 0.191915i −0.995385 − 0.0959577i \(-0.969409\pi\)
0.995385 − 0.0959577i \(-0.0305914\pi\)
\(18\) − 4.79129i − 1.12932i
\(19\) −5.79129 −1.32861 −0.664306 − 0.747460i \(-0.731273\pi\)
−0.664306 + 0.747460i \(0.731273\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) − 0.791288i − 0.168703i
\(23\) 1.00000i 0.208514i
\(24\) −2.79129 −0.569769
\(25\) 0 0
\(26\) −5.79129 −1.13576
\(27\) 5.00000i 0.962250i
\(28\) − 1.79129i − 0.338522i
\(29\) −7.58258 −1.40805 −0.704024 − 0.710176i \(-0.748615\pi\)
−0.704024 + 0.710176i \(0.748615\pi\)
\(30\) 0 0
\(31\) −3.37386 −0.605964 −0.302982 − 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.20871i 0.384487i
\(34\) 0.791288 0.135705
\(35\) 0 0
\(36\) 4.79129 0.798548
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) − 5.79129i − 0.939471i
\(39\) 16.1652 2.58850
\(40\) 0 0
\(41\) −6.79129 −1.06062 −0.530310 − 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(42\) 5.00000i 0.771517i
\(43\) 11.1652i 1.70267i 0.524623 + 0.851335i \(0.324206\pi\)
−0.524623 + 0.851335i \(0.675794\pi\)
\(44\) 0.791288 0.119291
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.41742i 0.644348i 0.946681 + 0.322174i \(0.104414\pi\)
−0.946681 + 0.322174i \(0.895586\pi\)
\(48\) − 2.79129i − 0.402888i
\(49\) 3.79129 0.541613
\(50\) 0 0
\(51\) −2.20871 −0.309282
\(52\) − 5.79129i − 0.803107i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.79129 0.239371
\(57\) 16.1652i 2.14113i
\(58\) − 7.58258i − 0.995641i
\(59\) 13.5826 1.76830 0.884150 − 0.467202i \(-0.154738\pi\)
0.884150 + 0.467202i \(0.154738\pi\)
\(60\) 0 0
\(61\) 10.3739 1.32824 0.664119 − 0.747627i \(-0.268807\pi\)
0.664119 + 0.747627i \(0.268807\pi\)
\(62\) − 3.37386i − 0.428481i
\(63\) − 8.58258i − 1.08130i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.20871 −0.271874
\(67\) − 11.1652i − 1.36404i −0.731333 − 0.682020i \(-0.761102\pi\)
0.731333 − 0.682020i \(-0.238898\pi\)
\(68\) 0.791288i 0.0959577i
\(69\) 2.79129 0.336032
\(70\) 0 0
\(71\) 8.37386 0.993795 0.496897 − 0.867809i \(-0.334473\pi\)
0.496897 + 0.867809i \(0.334473\pi\)
\(72\) 4.79129i 0.564659i
\(73\) 12.7477i 1.49201i 0.665941 + 0.746004i \(0.268030\pi\)
−0.665941 + 0.746004i \(0.731970\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 5.79129 0.664306
\(77\) − 1.41742i − 0.161530i
\(78\) 16.1652i 1.83034i
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) − 6.79129i − 0.749972i
\(83\) − 6.00000i − 0.658586i −0.944228 − 0.329293i \(-0.893190\pi\)
0.944228 − 0.329293i \(-0.106810\pi\)
\(84\) −5.00000 −0.545545
\(85\) 0 0
\(86\) −11.1652 −1.20397
\(87\) 21.1652i 2.26914i
\(88\) 0.791288i 0.0843516i
\(89\) −15.1652 −1.60750 −0.803751 − 0.594965i \(-0.797166\pi\)
−0.803751 + 0.594965i \(0.797166\pi\)
\(90\) 0 0
\(91\) −10.3739 −1.08748
\(92\) − 1.00000i − 0.104257i
\(93\) 9.41742i 0.976541i
\(94\) −4.41742 −0.455623
\(95\) 0 0
\(96\) 2.79129 0.284885
\(97\) 7.95644i 0.807854i 0.914791 + 0.403927i \(0.132355\pi\)
−0.914791 + 0.403927i \(0.867645\pi\)
\(98\) 3.79129i 0.382978i
\(99\) 3.79129 0.381039
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 1150.2.b.g.599.3 4
5.2 odd 4 230.2.a.a.1.1 ✓ 2
5.3 odd 4 1150.2.a.o.1.2 2
5.4 even 2 inner 1150.2.b.g.599.2 4
15.2 even 4 2070.2.a.x.1.1 2
20.3 even 4 9200.2.a.bs.1.1 2
20.7 even 4 1840.2.a.n.1.2 2
40.27 even 4 7360.2.a.bk.1.1 2
40.37 odd 4 7360.2.a.bq.1.2 2
115.22 even 4 5290.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 ✓ 2 5.2 odd 4
1150.2.a.o.1.2 2 5.3 odd 4
1150.2.b.g.599.2 4 5.4 even 2 inner
1150.2.b.g.599.3 4 1.1 even 1 trivial
1840.2.a.n.1.2 2 20.7 even 4
2070.2.a.x.1.1 2 15.2 even 4
5290.2.a.e.1.1 2 115.22 even 4
7360.2.a.bk.1.1 2 40.27 even 4
7360.2.a.bq.1.2 2 40.37 odd 4
9200.2.a.bs.1.1 2 20.3 even 4