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(7,74)] 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("74.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 9.1
Root \(-2.14169 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 74.9
Dual form 74.2.f.b.33.1

$q$-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+(0.173648 + 0.984808i) q^{2} +(-0.238878 + 1.35474i) q^{3} +(-0.939693 + 0.342020i) q^{4} +(0.266044 + 0.223238i) q^{5} -1.37564 q^{6} +(-0.365982 - 0.307095i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(1.04081 + 0.378824i) q^{9} +(-0.173648 + 0.300767i) q^{10} +(-1.29268 - 2.23899i) q^{11} +(-0.238878 - 1.35474i) q^{12} +(4.21288 - 1.53336i) q^{13} +(0.238878 - 0.413749i) q^{14} +(-0.365982 + 0.307095i) q^{15} +(0.766044 - 0.642788i) q^{16} +(-1.88864 - 0.687407i) q^{17} +(-0.192334 + 1.09078i) q^{18} +(0.611631 - 3.46873i) q^{19} +(-0.326352 - 0.118782i) q^{20} +(0.503461 - 0.422454i) q^{21} +(1.98050 - 1.66184i) q^{22} +(-2.60486 + 4.51175i) q^{23} +(1.29268 - 0.470498i) q^{24} +(-0.847296 - 4.80526i) q^{25} +(2.24162 + 3.88261i) q^{26} +(-2.82530 + 4.89356i) q^{27} +(0.448944 + 0.163402i) q^{28} +(-0.114111 - 0.197646i) q^{29} +(-0.365982 - 0.307095i) q^{30} -6.74658 q^{31} +(0.766044 + 0.642788i) q^{32} +(3.34205 - 1.21641i) q^{33} +(0.349006 - 1.97931i) q^{34} +(-0.0288122 - 0.163402i) q^{35} -1.10761 q^{36} +(2.42322 + 5.57925i) q^{37} +3.52224 q^{38} +(1.07095 + 6.07365i) q^{39} +(0.0603074 - 0.342020i) q^{40} +(-10.0701 + 3.66521i) q^{41} +(0.503461 + 0.422454i) q^{42} +8.85889 q^{43} +(1.98050 + 1.66184i) q^{44} +(0.192334 + 0.333132i) q^{45} +(-4.89554 - 1.78183i) q^{46} +(-3.72890 + 6.45864i) q^{47} +(0.687821 + 1.19134i) q^{48} +(-1.17590 - 6.66887i) q^{49} +(4.58512 - 1.66885i) q^{50} +(1.38241 - 2.39441i) q^{51} +(-3.43437 + 2.88178i) q^{52} +(3.35194 - 2.81261i) q^{53} +(-5.30983 - 1.93262i) q^{54} +(0.155916 - 0.884246i) q^{55} +(-0.0829614 + 0.470498i) q^{56} +(4.55314 + 1.65721i) q^{57} +(0.174828 - 0.146698i) q^{58} +(3.43016 - 2.87825i) q^{59} +(0.238878 - 0.413749i) q^{60} +(1.35175 - 0.491998i) q^{61} +(-1.17153 - 6.64409i) q^{62} +(-0.264583 - 0.458271i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(1.46312 + 0.532531i) q^{65} +(1.77827 + 3.08005i) q^{66} +(-8.06111 - 6.76408i) q^{67} +2.00984 q^{68} +(-5.49002 - 4.60667i) q^{69} +(0.155916 - 0.0567489i) q^{70} +(1.54193 - 8.74474i) q^{71} +(-0.192334 - 1.09078i) q^{72} +7.18667 q^{73} +(-5.07370 + 3.35524i) q^{74} +6.71229 q^{75} +(0.611631 + 3.46873i) q^{76} +(-0.214485 + 1.21641i) q^{77} +(-5.79541 + 2.10936i) q^{78} +(2.70857 + 2.27276i) q^{79} +0.347296 q^{80} +(-3.40919 - 2.86065i) q^{81} +(-5.35818 - 9.28064i) q^{82} +(-1.32932 - 0.483834i) q^{83} +(-0.328611 + 0.569170i) q^{84} +(-0.349006 - 0.604496i) q^{85} +(1.53833 + 8.72431i) q^{86} +(0.295018 - 0.107378i) q^{87} +(-1.29268 + 2.23899i) q^{88} +(2.92954 - 2.45818i) q^{89} +(-0.294673 + 0.247260i) q^{90} +(-2.01273 - 0.732572i) q^{91} +(0.904658 - 5.13057i) q^{92} +(1.61161 - 9.13989i) q^{93} +(-7.00804 - 2.55072i) q^{94} +(0.937074 - 0.786298i) q^{95} +(-1.05380 + 0.884246i) q^{96} +(-9.24175 + 16.0072i) q^{97} +(6.36336 - 2.31607i) q^{98} +(-0.497253 - 2.82006i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} - 3 q^{17} + 6 q^{18} - 3 q^{19} - 6 q^{20} - 33 q^{21} - 3 q^{22} - 21 q^{23} - 3 q^{24} - 6 q^{25}+ \cdots - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.173648 + 0.984808i 0.122788 + 0.696364i
\(3\) −0.238878 + 1.35474i −0.137916 + 0.782162i 0.834868 + 0.550450i \(0.185544\pi\)
−0.972784 + 0.231712i \(0.925567\pi\)
\(4\) −0.939693 + 0.342020i −0.469846 + 0.171010i
\(5\) 0.266044 + 0.223238i 0.118979 + 0.0998350i 0.700336 − 0.713814i \(-0.253034\pi\)
−0.581357 + 0.813649i \(0.697478\pi\)
\(6\) −1.37564 −0.561604
\(7\) −0.365982 − 0.307095i −0.138328 − 0.116071i 0.570998 − 0.820951i \(-0.306556\pi\)
−0.709326 + 0.704880i \(0.751001\pi\)
\(8\) −0.500000 − 0.866025i −0.176777 − 0.306186i
\(9\) 1.04081 + 0.378824i 0.346937 + 0.126275i
\(10\) −0.173648 + 0.300767i −0.0549124 + 0.0951110i
\(11\) −1.29268 − 2.23899i −0.389758 − 0.675081i 0.602659 − 0.797999i \(-0.294108\pi\)
−0.992417 + 0.122918i \(0.960775\pi\)
\(12\) −0.238878 − 1.35474i −0.0689581 − 0.391081i
\(13\) 4.21288 − 1.53336i 1.16844 − 0.425278i 0.316336 − 0.948647i \(-0.397547\pi\)
0.852106 + 0.523369i \(0.175325\pi\)
\(14\) 0.238878 − 0.413749i 0.0638428 − 0.110579i
\(15\) −0.365982 + 0.307095i −0.0944962 + 0.0792917i
\(16\) 0.766044 − 0.642788i 0.191511 − 0.160697i
\(17\) −1.88864 − 0.687407i −0.458062 − 0.166721i 0.102675 − 0.994715i \(-0.467260\pi\)
−0.560737 + 0.827994i \(0.689482\pi\)
\(18\) −0.192334 + 1.09078i −0.0453335 + 0.257099i
\(19\) 0.611631 − 3.46873i 0.140318 − 0.795782i −0.830690 − 0.556735i \(-0.812054\pi\)
0.971008 − 0.239047i \(-0.0768350\pi\)
\(20\) −0.326352 − 0.118782i −0.0729745 − 0.0265605i
\(21\) 0.503461 − 0.422454i 0.109864 − 0.0921869i
\(22\) 1.98050 − 1.66184i 0.422245 − 0.354305i
\(23\) −2.60486 + 4.51175i −0.543151 + 0.940765i 0.455570 + 0.890200i \(0.349435\pi\)
−0.998721 + 0.0505649i \(0.983898\pi\)
\(24\) 1.29268 − 0.470498i 0.263867 − 0.0960399i
\(25\) −0.847296 − 4.80526i −0.169459 − 0.961051i
\(26\) 2.24162 + 3.88261i 0.439619 + 0.761442i
\(27\) −2.82530 + 4.89356i −0.543729 + 0.941767i
\(28\) 0.448944 + 0.163402i 0.0848424 + 0.0308801i
\(29\) −0.114111 − 0.197646i −0.0211899 − 0.0367019i 0.855236 − 0.518239i \(-0.173412\pi\)
−0.876426 + 0.481537i \(0.840079\pi\)
\(30\) −0.365982 − 0.307095i −0.0668189 − 0.0560677i
\(31\) −6.74658 −1.21172 −0.605861 − 0.795571i \(-0.707171\pi\)
−0.605861 + 0.795571i \(0.707171\pi\)
\(32\) 0.766044 + 0.642788i 0.135419 + 0.113630i
\(33\) 3.34205 − 1.21641i 0.581776 − 0.211749i
\(34\) 0.349006 − 1.97931i 0.0598540 − 0.339449i
\(35\) −0.0288122 − 0.163402i −0.00487015 − 0.0276200i
\(36\) −1.10761 −0.184601
\(37\) 2.42322 + 5.57925i 0.398376 + 0.917222i
\(38\) 3.52224 0.571383
\(39\) 1.07095 + 6.07365i 0.171489 + 0.972563i
\(40\) 0.0603074 − 0.342020i 0.00953543 − 0.0540781i
\(41\) −10.0701 + 3.66521i −1.57268 + 0.572410i −0.973597 − 0.228275i \(-0.926691\pi\)
−0.599086 + 0.800685i \(0.704469\pi\)
\(42\) 0.503461 + 0.422454i 0.0776857 + 0.0651860i
\(43\) 8.85889 1.35097 0.675484 − 0.737374i \(-0.263935\pi\)
0.675484 + 0.737374i \(0.263935\pi\)
\(44\) 1.98050 + 1.66184i 0.298572 + 0.250532i
\(45\) 0.192334 + 0.333132i 0.0286715 + 0.0496604i
\(46\) −4.89554 − 1.78183i −0.721807 − 0.262716i
\(47\) −3.72890 + 6.45864i −0.543916 + 0.942090i 0.454758 + 0.890615i \(0.349726\pi\)
−0.998674 + 0.0514750i \(0.983608\pi\)
\(48\) 0.687821 + 1.19134i 0.0992785 + 0.171955i
\(49\) −1.17590 − 6.66887i −0.167986 − 0.952696i
\(50\) 4.58512 − 1.66885i 0.648434 − 0.236011i
\(51\) 1.38241 − 2.39441i 0.193577 − 0.335285i
\(52\) −3.43437 + 2.88178i −0.476261 + 0.399631i
\(53\) 3.35194 − 2.81261i 0.460424 − 0.386342i −0.382863 − 0.923805i \(-0.625062\pi\)
0.843287 + 0.537464i \(0.180617\pi\)
\(54\) −5.30983 − 1.93262i −0.722576 − 0.262996i
\(55\) 0.155916 − 0.884246i 0.0210238 − 0.119232i
\(56\) −0.0829614 + 0.470498i −0.0110862 + 0.0628729i
\(57\) 4.55314 + 1.65721i 0.603078 + 0.219502i
\(58\) 0.174828 − 0.146698i 0.0229560 − 0.0192624i
\(59\) 3.43016 − 2.87825i 0.446569 − 0.374716i −0.391592 − 0.920139i \(-0.628076\pi\)
0.838161 + 0.545423i \(0.183631\pi\)
\(60\) 0.238878 − 0.413749i 0.0308390 − 0.0534147i
\(61\) 1.35175 − 0.491998i 0.173074 − 0.0629939i −0.254030 − 0.967196i \(-0.581756\pi\)
0.427104 + 0.904203i \(0.359534\pi\)
\(62\) −1.17153 − 6.64409i −0.148785 − 0.843800i
\(63\) −0.264583 − 0.458271i −0.0333343 − 0.0577367i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 1.46312 + 0.532531i 0.181477 + 0.0660523i
\(66\) 1.77827 + 3.08005i 0.218890 + 0.379128i
\(67\) −8.06111 − 6.76408i −0.984822 − 0.826363i −1.18267e−5 − 1.00000i \(-0.500004\pi\)
−0.984810 + 0.173637i \(0.944448\pi\)
\(68\) 2.00984 0.243729
\(69\) −5.49002 − 4.60667i −0.660921 − 0.554578i
\(70\) 0.155916 − 0.0567489i 0.0186356 − 0.00678280i
\(71\) 1.54193 − 8.74474i 0.182994 − 1.03781i −0.745512 − 0.666492i \(-0.767795\pi\)
0.928506 − 0.371318i \(-0.121094\pi\)
\(72\) −0.192334 − 1.09078i −0.0226668 − 0.128550i
\(73\) 7.18667 0.841136 0.420568 − 0.907261i \(-0.361831\pi\)
0.420568 + 0.907261i \(0.361831\pi\)
\(74\) −5.07370 + 3.35524i −0.589805 + 0.390038i
\(75\) 6.71229 0.775069
\(76\) 0.611631 + 3.46873i 0.0701589 + 0.397891i
\(77\) −0.214485 + 1.21641i −0.0244429 + 0.138622i
\(78\) −5.79541 + 2.10936i −0.656201 + 0.238838i
\(79\) 2.70857 + 2.27276i 0.304738 + 0.255705i 0.782313 − 0.622885i \(-0.214040\pi\)
−0.477575 + 0.878591i \(0.658484\pi\)
\(80\) 0.347296 0.0388289
\(81\) −3.40919 − 2.86065i −0.378799 − 0.317850i
\(82\) −5.35818 − 9.28064i −0.591712 − 1.02487i
\(83\) −1.32932 − 0.483834i −0.145912 − 0.0531077i 0.268032 − 0.963410i \(-0.413627\pi\)
−0.413944 + 0.910302i \(0.635849\pi\)
\(84\) −0.328611 + 0.569170i −0.0358544 + 0.0621016i
\(85\) −0.349006 − 0.604496i −0.0378550 − 0.0655668i
\(86\) 1.53833 + 8.72431i 0.165882 + 0.940766i
\(87\) 0.295018 − 0.107378i 0.0316292 − 0.0115121i
\(88\) −1.29268 + 2.23899i −0.137800 + 0.238677i
\(89\) 2.92954 − 2.45818i 0.310531 − 0.260567i −0.474180 − 0.880428i \(-0.657256\pi\)
0.784711 + 0.619861i \(0.212811\pi\)
\(90\) −0.294673 + 0.247260i −0.0310612 + 0.0260635i
\(91\) −2.01273 − 0.732572i −0.210991 − 0.0767944i
\(92\) 0.904658 − 5.13057i 0.0943172 − 0.534899i
\(93\) 1.61161 − 9.13989i 0.167116 − 0.947762i
\(94\) −7.00804 − 2.55072i −0.722824 − 0.263086i
\(95\) 0.937074 − 0.786298i 0.0961417 − 0.0806725i
\(96\) −1.05380 + 0.884246i −0.107553 + 0.0902480i
\(97\) −9.24175 + 16.0072i −0.938358 + 1.62528i −0.169824 + 0.985474i \(0.554320\pi\)
−0.768534 + 0.639809i \(0.779013\pi\)
\(98\) 6.36336 − 2.31607i 0.642797 − 0.233959i
\(99\) −0.497253 − 2.82006i −0.0499758 − 0.283427i
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.f.b.9.1 ✓ 12
3.2 odd 2 666.2.x.g.379.1 12
4.3 odd 2 592.2.bc.d.305.2 12
37.12 even 9 2738.2.a.t.1.3 6
37.25 even 18 2738.2.a.q.1.3 6
37.33 even 9 inner 74.2.f.b.33.1 yes 12
111.107 odd 18 666.2.x.g.181.1 12
148.107 odd 18 592.2.bc.d.33.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.9.1 ✓ 12 1.1 even 1 trivial
74.2.f.b.33.1 yes 12 37.33 even 9 inner
592.2.bc.d.33.2 12 148.107 odd 18
592.2.bc.d.305.2 12 4.3 odd 2
666.2.x.g.181.1 12 111.107 odd 18
666.2.x.g.379.1 12 3.2 odd 2
2738.2.a.q.1.3 6 37.25 even 18
2738.2.a.t.1.3 6 37.12 even 9