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Properties

Label 1850.2.b.i.149.1
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
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] = [1850,2,Mod(149,1850)] 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("1850.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.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,-2] 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: \(14.7723243739\)
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 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.1
Root \(-2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.i.149.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} -2.60555i q^{7} +1.00000i q^{8} -7.90833 q^{9} -2.30278 q^{11} +3.30278i q^{12} -1.30278i q^{13} -2.60555 q^{14} +1.00000 q^{16} -6.00000i q^{17} +7.90833i q^{18} -2.00000 q^{19} -8.60555 q^{21} +2.30278i q^{22} -3.90833i q^{23} +3.30278 q^{24} -1.30278 q^{26} +16.2111i q^{27} +2.60555i q^{28} +3.90833 q^{29} -0.302776 q^{31} -1.00000i q^{32} +7.60555i q^{33} -6.00000 q^{34} +7.90833 q^{36} +1.00000i q^{37} +2.00000i q^{38} -4.30278 q^{39} +9.90833 q^{41} +8.60555i q^{42} -0.605551i q^{43} +2.30278 q^{44} -3.90833 q^{46} +4.60555i q^{47} -3.30278i q^{48} +0.211103 q^{49} -19.8167 q^{51} +1.30278i q^{52} +6.00000i q^{53} +16.2111 q^{54} +2.60555 q^{56} +6.60555i q^{57} -3.90833i q^{58} -10.6056 q^{59} +7.51388 q^{61} +0.302776i q^{62} +20.6056i q^{63} -1.00000 q^{64} +7.60555 q^{66} -3.51388i q^{67} +6.00000i q^{68} -12.9083 q^{69} +6.00000 q^{71} -7.90833i q^{72} +12.3028i q^{73} +1.00000 q^{74} +2.00000 q^{76} +6.00000i q^{77} +4.30278i q^{78} -9.11943 q^{79} +29.8167 q^{81} -9.90833i q^{82} -2.78890i q^{83} +8.60555 q^{84} -0.605551 q^{86} -12.9083i q^{87} -2.30278i q^{88} +9.21110 q^{89} -3.39445 q^{91} +3.90833i q^{92} +1.00000i q^{93} +4.60555 q^{94} -3.30278 q^{96} -16.4222i q^{97} -0.211103i q^{98} +18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 2 q^{11} + 4 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{21} + 6 q^{24} + 2 q^{26} - 6 q^{29} + 6 q^{31} - 24 q^{34} + 10 q^{36} - 10 q^{39} + 18 q^{41} + 2 q^{44} + 6 q^{46}+ \cdots + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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
\(7\) − 2.60555i − 0.984806i −0.870367 − 0.492403i \(-0.836119\pi\)
0.870367 − 0.492403i \(-0.163881\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −7.90833 −2.63611
\(10\) 0 0
\(11\) −2.30278 −0.694313 −0.347156 − 0.937807i \(-0.612853\pi\)
−0.347156 + 0.937807i \(0.612853\pi\)
\(12\) 3.30278i 0.953429i
\(13\) − 1.30278i − 0.361325i −0.983545 − 0.180662i \(-0.942176\pi\)
0.983545 − 0.180662i \(-0.0578242\pi\)
\(14\) −2.60555 −0.696363
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 − 0.727607i \(-0.740633\pi\)
0.685994 − 0.727607i \(-0.259367\pi\)
\(18\) 7.90833i 1.86401i
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −8.60555 −1.87789
\(22\) 2.30278i 0.490953i
\(23\) − 3.90833i − 0.814942i −0.913218 − 0.407471i \(-0.866411\pi\)
0.913218 − 0.407471i \(-0.133589\pi\)
\(24\) 3.30278 0.674176
\(25\) 0 0
\(26\) −1.30278 −0.255495
\(27\) 16.2111i 3.11983i
\(28\) 2.60555i 0.492403i
\(29\) 3.90833 0.725758 0.362879 − 0.931836i \(-0.381794\pi\)
0.362879 + 0.931836i \(0.381794\pi\)
\(30\) 0 0
\(31\) −0.302776 −0.0543801 −0.0271901 − 0.999630i \(-0.508656\pi\)
−0.0271901 + 0.999630i \(0.508656\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 7.60555i 1.32396i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 7.90833 1.31805
\(37\) 1.00000i 0.164399i
\(38\) 2.00000i 0.324443i
\(39\) −4.30278 −0.688996
\(40\) 0 0
\(41\) 9.90833 1.54742 0.773710 − 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(42\) 8.60555i 1.32787i
\(43\) − 0.605551i − 0.0923457i −0.998933 − 0.0461729i \(-0.985297\pi\)
0.998933 − 0.0461729i \(-0.0147025\pi\)
\(44\) 2.30278 0.347156
\(45\) 0 0
\(46\) −3.90833 −0.576251
\(47\) 4.60555i 0.671789i 0.941900 + 0.335894i \(0.109039\pi\)
−0.941900 + 0.335894i \(0.890961\pi\)
\(48\) − 3.30278i − 0.476715i
\(49\) 0.211103 0.0301575
\(50\) 0 0
\(51\) −19.8167 −2.77489
\(52\) 1.30278i 0.180662i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 16.2111 2.20605
\(55\) 0 0
\(56\) 2.60555 0.348181
\(57\) 6.60555i 0.874927i
\(58\) − 3.90833i − 0.513188i
\(59\) −10.6056 −1.38073 −0.690363 − 0.723464i \(-0.742549\pi\)
−0.690363 + 0.723464i \(0.742549\pi\)
\(60\) 0 0
\(61\) 7.51388 0.962054 0.481027 − 0.876706i \(-0.340264\pi\)
0.481027 + 0.876706i \(0.340264\pi\)
\(62\) 0.302776i 0.0384525i
\(63\) 20.6056i 2.59606i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 7.60555 0.936179
\(67\) − 3.51388i − 0.429289i −0.976692 − 0.214644i \(-0.931141\pi\)
0.976692 − 0.214644i \(-0.0688592\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −12.9083 −1.55398
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) − 7.90833i − 0.932005i
\(73\) 12.3028i 1.43993i 0.694010 + 0.719965i \(0.255842\pi\)
−0.694010 + 0.719965i \(0.744158\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 6.00000i 0.683763i
\(78\) 4.30278i 0.487193i
\(79\) −9.11943 −1.02602 −0.513008 − 0.858384i \(-0.671469\pi\)
−0.513008 + 0.858384i \(0.671469\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) − 9.90833i − 1.09419i
\(83\) − 2.78890i − 0.306121i −0.988217 − 0.153061i \(-0.951087\pi\)
0.988217 − 0.153061i \(-0.0489130\pi\)
\(84\) 8.60555 0.938943
\(85\) 0 0
\(86\) −0.605551 −0.0652983
\(87\) − 12.9083i − 1.38392i
\(88\) − 2.30278i − 0.245477i
\(89\) 9.21110 0.976375 0.488187 − 0.872739i \(-0.337658\pi\)
0.488187 + 0.872739i \(0.337658\pi\)
\(90\) 0 0
\(91\) −3.39445 −0.355835
\(92\) 3.90833i 0.407471i
\(93\) 1.00000i 0.103695i
\(94\) 4.60555 0.475026
\(95\) 0 0
\(96\) −3.30278 −0.337088
\(97\) − 16.4222i − 1.66742i −0.552201 − 0.833711i \(-0.686212\pi\)
0.552201 − 0.833711i \(-0.313788\pi\)
\(98\) − 0.211103i − 0.0213246i
\(99\) 18.2111 1.83028
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 1850.2.b.i.149.1 4
5.2 odd 4 1850.2.a.u.1.1 2
5.3 odd 4 74.2.a.a.1.2 ✓ 2
5.4 even 2 inner 1850.2.b.i.149.4 4
15.8 even 4 666.2.a.j.1.2 2
20.3 even 4 592.2.a.f.1.1 2
35.13 even 4 3626.2.a.a.1.1 2
40.3 even 4 2368.2.a.ba.1.2 2
40.13 odd 4 2368.2.a.s.1.1 2
55.43 even 4 8954.2.a.p.1.2 2
60.23 odd 4 5328.2.a.bf.1.2 2
185.73 odd 4 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 ✓ 2 5.3 odd 4
592.2.a.f.1.1 2 20.3 even 4
666.2.a.j.1.2 2 15.8 even 4
1850.2.a.u.1.1 2 5.2 odd 4
1850.2.b.i.149.1 4 1.1 even 1 trivial
1850.2.b.i.149.4 4 5.4 even 2 inner
2368.2.a.s.1.1 2 40.13 odd 4
2368.2.a.ba.1.2 2 40.3 even 4
2738.2.a.l.1.2 2 185.73 odd 4
3626.2.a.a.1.1 2 35.13 even 4
5328.2.a.bf.1.2 2 60.23 odd 4
8954.2.a.p.1.2 2 55.43 even 4