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Properties

Label 1150.2.b.f.599.1
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,-6,0,0,-10,0,-14] 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: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) 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 599.1
Root \(-2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.f.599.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} -0.302776i q^{7} +1.00000i q^{8} -7.90833 q^{9} -5.30278 q^{11} +3.30278i q^{12} +0.302776i q^{13} -0.302776 q^{14} +1.00000 q^{16} -3.90833i q^{17} +7.90833i q^{18} +4.90833 q^{19} -1.00000 q^{21} +5.30278i q^{22} +1.00000i q^{23} +3.30278 q^{24} +0.302776 q^{26} +16.2111i q^{27} +0.302776i q^{28} -4.60555 q^{29} +2.90833 q^{31} -1.00000i q^{32} +17.5139i q^{33} -3.90833 q^{34} +7.90833 q^{36} +8.00000i q^{37} -4.90833i q^{38} +1.00000 q^{39} -9.90833 q^{41} +1.00000i q^{42} -5.21110i q^{43} +5.30278 q^{44} +1.00000 q^{46} +4.60555i q^{47} -3.30278i q^{48} +6.90833 q^{49} -12.9083 q^{51} -0.302776i q^{52} -3.21110i q^{53} +16.2111 q^{54} +0.302776 q^{56} -16.2111i q^{57} +4.60555i q^{58} +10.6056 q^{59} -6.51388 q^{61} -2.90833i q^{62} +2.39445i q^{63} -1.00000 q^{64} +17.5139 q^{66} -4.00000i q^{67} +3.90833i q^{68} +3.30278 q^{69} -12.6972 q^{71} -7.90833i q^{72} -15.8167i q^{73} +8.00000 q^{74} -4.90833 q^{76} +1.60555i q^{77} -1.00000i q^{78} -14.4222 q^{79} +29.8167 q^{81} +9.90833i q^{82} +3.21110i q^{83} +1.00000 q^{84} -5.21110 q^{86} +15.2111i q^{87} -5.30278i q^{88} +0.0916731 q^{91} -1.00000i q^{92} -9.60555i q^{93} +4.60555 q^{94} -3.30278 q^{96} +2.69722i q^{97} -6.90833i q^{98} +41.9361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 14 q^{11} + 6 q^{14} + 4 q^{16} - 2 q^{19} - 4 q^{21} + 6 q^{24} - 6 q^{26} - 4 q^{29} - 10 q^{31} + 6 q^{34} + 10 q^{36} + 4 q^{39} - 18 q^{41} + 14 q^{44} + 4 q^{46}+ \cdots + 74 q^{99}+O(q^{100}) \) 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
\(3\) − 3.30278i − 1.90686i −0.301617 − 0.953429i \(-0.597526\pi\)
0.301617 − 0.953429i \(-0.402474\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.30278 −1.34835
\(7\) − 0.302776i − 0.114438i −0.998362 − 0.0572192i \(-0.981777\pi\)
0.998362 − 0.0572192i \(-0.0182234\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −7.90833 −2.63611
\(10\) 0 0
\(11\) −5.30278 −1.59885 −0.799424 − 0.600768i \(-0.794862\pi\)
−0.799424 + 0.600768i \(0.794862\pi\)
\(12\) 3.30278i 0.953429i
\(13\) 0.302776i 0.0839749i 0.999118 + 0.0419874i \(0.0133689\pi\)
−0.999118 + 0.0419874i \(0.986631\pi\)
\(14\) −0.302776 −0.0809202
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.90833i − 0.947909i −0.880549 − 0.473954i \(-0.842826\pi\)
0.880549 − 0.473954i \(-0.157174\pi\)
\(18\) 7.90833i 1.86401i
\(19\) 4.90833 1.12605 0.563024 − 0.826441i \(-0.309638\pi\)
0.563024 + 0.826441i \(0.309638\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 5.30278i 1.13056i
\(23\) 1.00000i 0.208514i
\(24\) 3.30278 0.674176
\(25\) 0 0
\(26\) 0.302776 0.0593792
\(27\) 16.2111i 3.11983i
\(28\) 0.302776i 0.0572192i
\(29\) −4.60555 −0.855229 −0.427615 − 0.903961i \(-0.640646\pi\)
−0.427615 + 0.903961i \(0.640646\pi\)
\(30\) 0 0
\(31\) 2.90833 0.522351 0.261175 − 0.965291i \(-0.415890\pi\)
0.261175 + 0.965291i \(0.415890\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 17.5139i 3.04877i
\(34\) −3.90833 −0.670273
\(35\) 0 0
\(36\) 7.90833 1.31805
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) − 4.90833i − 0.796236i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.90833 −1.54742 −0.773710 − 0.633540i \(-0.781601\pi\)
−0.773710 + 0.633540i \(0.781601\pi\)
\(42\) 1.00000i 0.154303i
\(43\) − 5.21110i − 0.794686i −0.917670 − 0.397343i \(-0.869932\pi\)
0.917670 − 0.397343i \(-0.130068\pi\)
\(44\) 5.30278 0.799424
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 4.60555i 0.671789i 0.941900 + 0.335894i \(0.109039\pi\)
−0.941900 + 0.335894i \(0.890961\pi\)
\(48\) − 3.30278i − 0.476715i
\(49\) 6.90833 0.986904
\(50\) 0 0
\(51\) −12.9083 −1.80753
\(52\) − 0.302776i − 0.0419874i
\(53\) − 3.21110i − 0.441079i −0.975378 − 0.220539i \(-0.929218\pi\)
0.975378 − 0.220539i \(-0.0707818\pi\)
\(54\) 16.2111 2.20605
\(55\) 0 0
\(56\) 0.302776 0.0404601
\(57\) − 16.2111i − 2.14721i
\(58\) 4.60555i 0.604739i
\(59\) 10.6056 1.38073 0.690363 − 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) 0 0
\(61\) −6.51388 −0.834017 −0.417008 − 0.908903i \(-0.636921\pi\)
−0.417008 + 0.908903i \(0.636921\pi\)
\(62\) − 2.90833i − 0.369358i
\(63\) 2.39445i 0.301672i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 17.5139 2.15581
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 3.90833i 0.473954i
\(69\) 3.30278 0.397607
\(70\) 0 0
\(71\) −12.6972 −1.50688 −0.753442 − 0.657515i \(-0.771608\pi\)
−0.753442 + 0.657515i \(0.771608\pi\)
\(72\) − 7.90833i − 0.932005i
\(73\) − 15.8167i − 1.85120i −0.378504 − 0.925600i \(-0.623561\pi\)
0.378504 − 0.925600i \(-0.376439\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.90833 −0.563024
\(77\) 1.60555i 0.182970i
\(78\) − 1.00000i − 0.113228i
\(79\) −14.4222 −1.62262 −0.811312 − 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 9.90833i 1.09419i
\(83\) 3.21110i 0.352464i 0.984349 + 0.176232i \(0.0563909\pi\)
−0.984349 + 0.176232i \(0.943609\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −5.21110 −0.561928
\(87\) 15.2111i 1.63080i
\(88\) − 5.30278i − 0.565278i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0.0916731 0.00960995
\(92\) − 1.00000i − 0.104257i
\(93\) − 9.60555i − 0.996049i
\(94\) 4.60555 0.475026
\(95\) 0 0
\(96\) −3.30278 −0.337088
\(97\) 2.69722i 0.273862i 0.990581 + 0.136931i \(0.0437238\pi\)
−0.990581 + 0.136931i \(0.956276\pi\)
\(98\) − 6.90833i − 0.697846i
\(99\) 41.9361 4.21473
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.f.599.1 4
5.2 odd 4 1150.2.a.m.1.1 2
5.3 odd 4 230.2.a.b.1.2 ✓ 2
5.4 even 2 inner 1150.2.b.f.599.4 4
15.8 even 4 2070.2.a.w.1.1 2
20.3 even 4 1840.2.a.j.1.1 2
20.7 even 4 9200.2.a.ca.1.2 2
40.3 even 4 7360.2.a.bu.1.2 2
40.13 odd 4 7360.2.a.bc.1.1 2
115.68 even 4 5290.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.2 ✓ 2 5.3 odd 4
1150.2.a.m.1.1 2 5.2 odd 4
1150.2.b.f.599.1 4 1.1 even 1 trivial
1150.2.b.f.599.4 4 5.4 even 2 inner
1840.2.a.j.1.1 2 20.3 even 4
2070.2.a.w.1.1 2 15.8 even 4
5290.2.a.j.1.2 2 115.68 even 4
7360.2.a.bc.1.1 2 40.13 odd 4
7360.2.a.bu.1.2 2 40.3 even 4
9200.2.a.ca.1.2 2 20.7 even 4