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Properties

Label 74.2.a.a.1.2
Level $74$
Weight $2$
Character 74.1
Self dual yes
Analytic conductor $0.591$
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] = [74,2,Mod(1,74)] 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("74.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 74.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} -2.30278 q^{5} -3.30278 q^{6} -2.60555 q^{7} -1.00000 q^{8} +7.90833 q^{9} +2.30278 q^{10} -2.30278 q^{11} +3.30278 q^{12} +1.30278 q^{13} +2.60555 q^{14} -7.60555 q^{15} +1.00000 q^{16} -6.00000 q^{17} -7.90833 q^{18} +2.00000 q^{19} -2.30278 q^{20} -8.60555 q^{21} +2.30278 q^{22} +3.90833 q^{23} -3.30278 q^{24} +0.302776 q^{25} -1.30278 q^{26} +16.2111 q^{27} -2.60555 q^{28} -3.90833 q^{29} +7.60555 q^{30} -0.302776 q^{31} -1.00000 q^{32} -7.60555 q^{33} +6.00000 q^{34} +6.00000 q^{35} +7.90833 q^{36} +1.00000 q^{37} -2.00000 q^{38} +4.30278 q^{39} +2.30278 q^{40} +9.90833 q^{41} +8.60555 q^{42} +0.605551 q^{43} -2.30278 q^{44} -18.2111 q^{45} -3.90833 q^{46} +4.60555 q^{47} +3.30278 q^{48} -0.211103 q^{49} -0.302776 q^{50} -19.8167 q^{51} +1.30278 q^{52} -6.00000 q^{53} -16.2111 q^{54} +5.30278 q^{55} +2.60555 q^{56} +6.60555 q^{57} +3.90833 q^{58} +10.6056 q^{59} -7.60555 q^{60} +7.51388 q^{61} +0.302776 q^{62} -20.6056 q^{63} +1.00000 q^{64} -3.00000 q^{65} +7.60555 q^{66} -3.51388 q^{67} -6.00000 q^{68} +12.9083 q^{69} -6.00000 q^{70} +6.00000 q^{71} -7.90833 q^{72} -12.3028 q^{73} -1.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} +6.00000 q^{77} -4.30278 q^{78} +9.11943 q^{79} -2.30278 q^{80} +29.8167 q^{81} -9.90833 q^{82} +2.78890 q^{83} -8.60555 q^{84} +13.8167 q^{85} -0.605551 q^{86} -12.9083 q^{87} +2.30278 q^{88} -9.21110 q^{89} +18.2111 q^{90} -3.39445 q^{91} +3.90833 q^{92} -1.00000 q^{93} -4.60555 q^{94} -4.60555 q^{95} -3.30278 q^{96} -16.4222 q^{97} +0.211103 q^{98} -18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 5 q^{9} + q^{10} - q^{11} + 3 q^{12} - q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} - 12 q^{17} - 5 q^{18} + 4 q^{19} - q^{20} - 10 q^{21}+ \cdots - 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.30278 1.90686 0.953429 − 0.301617i \(-0.0975264\pi\)
0.953429 + 0.301617i \(0.0975264\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.30278 −1.02983 −0.514916 − 0.857240i \(-0.672177\pi\)
−0.514916 + 0.857240i \(0.672177\pi\)
\(6\) −3.30278 −1.34835
\(7\) −2.60555 −0.984806 −0.492403 − 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) −1.00000 −0.353553
\(9\) 7.90833 2.63611
\(10\) 2.30278 0.728202
\(11\) −2.30278 −0.694313 −0.347156 − 0.937807i \(-0.612853\pi\)
−0.347156 + 0.937807i \(0.612853\pi\)
\(12\) 3.30278 0.953429
\(13\) 1.30278 0.361325 0.180662 − 0.983545i \(-0.442176\pi\)
0.180662 + 0.983545i \(0.442176\pi\)
\(14\) 2.60555 0.696363
\(15\) −7.60555 −1.96374
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −7.90833 −1.86401
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −2.30278 −0.514916
\(21\) −8.60555 −1.87789
\(22\) 2.30278 0.490953
\(23\) 3.90833 0.814942 0.407471 − 0.913218i \(-0.366411\pi\)
0.407471 + 0.913218i \(0.366411\pi\)
\(24\) −3.30278 −0.674176
\(25\) 0.302776 0.0605551
\(26\) −1.30278 −0.255495
\(27\) 16.2111 3.11983
\(28\) −2.60555 −0.492403
\(29\) −3.90833 −0.725758 −0.362879 − 0.931836i \(-0.618206\pi\)
−0.362879 + 0.931836i \(0.618206\pi\)
\(30\) 7.60555 1.38858
\(31\) −0.302776 −0.0543801 −0.0271901 − 0.999630i \(-0.508656\pi\)
−0.0271901 + 0.999630i \(0.508656\pi\)
\(32\) −1.00000 −0.176777
\(33\) −7.60555 −1.32396
\(34\) 6.00000 1.02899
\(35\) 6.00000 1.01419
\(36\) 7.90833 1.31805
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 4.30278 0.688996
\(40\) 2.30278 0.364101
\(41\) 9.90833 1.54742 0.773710 − 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(42\) 8.60555 1.32787
\(43\) 0.605551 0.0923457 0.0461729 − 0.998933i \(-0.485297\pi\)
0.0461729 + 0.998933i \(0.485297\pi\)
\(44\) −2.30278 −0.347156
\(45\) −18.2111 −2.71475
\(46\) −3.90833 −0.576251
\(47\) 4.60555 0.671789 0.335894 − 0.941900i \(-0.390961\pi\)
0.335894 + 0.941900i \(0.390961\pi\)
\(48\) 3.30278 0.476715
\(49\) −0.211103 −0.0301575
\(50\) −0.302776 −0.0428189
\(51\) −19.8167 −2.77489
\(52\) 1.30278 0.180662
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −16.2111 −2.20605
\(55\) 5.30278 0.715026
\(56\) 2.60555 0.348181
\(57\) 6.60555 0.874927
\(58\) 3.90833 0.513188
\(59\) 10.6056 1.38073 0.690363 − 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) −7.60555 −0.981872
\(61\) 7.51388 0.962054 0.481027 − 0.876706i \(-0.340264\pi\)
0.481027 + 0.876706i \(0.340264\pi\)
\(62\) 0.302776 0.0384525
\(63\) −20.6056 −2.59606
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) 7.60555 0.936179
\(67\) −3.51388 −0.429289 −0.214644 − 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(68\) −6.00000 −0.727607
\(69\) 12.9083 1.55398
\(70\) −6.00000 −0.717137
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −7.90833 −0.932005
\(73\) −12.3028 −1.43993 −0.719965 − 0.694010i \(-0.755842\pi\)
−0.719965 + 0.694010i \(0.755842\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) 6.00000 0.683763
\(78\) −4.30278 −0.487193
\(79\) 9.11943 1.02602 0.513008 − 0.858384i \(-0.328531\pi\)
0.513008 + 0.858384i \(0.328531\pi\)
\(80\) −2.30278 −0.257458
\(81\) 29.8167 3.31296
\(82\) −9.90833 −1.09419
\(83\) 2.78890 0.306121 0.153061 − 0.988217i \(-0.451087\pi\)
0.153061 + 0.988217i \(0.451087\pi\)
\(84\) −8.60555 −0.938943
\(85\) 13.8167 1.49863
\(86\) −0.605551 −0.0652983
\(87\) −12.9083 −1.38392
\(88\) 2.30278 0.245477
\(89\) −9.21110 −0.976375 −0.488187 − 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(90\) 18.2111 1.91962
\(91\) −3.39445 −0.355835
\(92\) 3.90833 0.407471
\(93\) −1.00000 −0.103695
\(94\) −4.60555 −0.475026
\(95\) −4.60555 −0.472520
\(96\) −3.30278 −0.337088
\(97\) −16.4222 −1.66742 −0.833711 − 0.552201i \(-0.813788\pi\)
−0.833711 + 0.552201i \(0.813788\pi\)
\(98\) 0.211103 0.0213246
\(99\) −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 74.2.a.a.1.2 ✓ 2
3.2 odd 2 666.2.a.j.1.2 2
4.3 odd 2 592.2.a.f.1.1 2
5.2 odd 4 1850.2.b.i.149.1 4
5.3 odd 4 1850.2.b.i.149.4 4
5.4 even 2 1850.2.a.u.1.1 2
7.6 odd 2 3626.2.a.a.1.1 2
8.3 odd 2 2368.2.a.ba.1.2 2
8.5 even 2 2368.2.a.s.1.1 2
11.10 odd 2 8954.2.a.p.1.2 2
12.11 even 2 5328.2.a.bf.1.2 2
37.36 even 2 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 ✓ 2 1.1 even 1 trivial
592.2.a.f.1.1 2 4.3 odd 2
666.2.a.j.1.2 2 3.2 odd 2
1850.2.a.u.1.1 2 5.4 even 2
1850.2.b.i.149.1 4 5.2 odd 4
1850.2.b.i.149.4 4 5.3 odd 4
2368.2.a.s.1.1 2 8.5 even 2
2368.2.a.ba.1.2 2 8.3 odd 2
2738.2.a.l.1.2 2 37.36 even 2
3626.2.a.a.1.1 2 7.6 odd 2
5328.2.a.bf.1.2 2 12.11 even 2
8954.2.a.p.1.2 2 11.10 odd 2