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Properties

Label 58.2.d.b.7.1
Level $58$
Weight $2$
Character 58.7
Analytic conductor $0.463$
Analytic rank $0$
Dimension $12$
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] = [58,2,Mod(7,58)] 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("58.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\), 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: \(0.463132331723\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
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: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 7.1
Root \(2.06920 - 0.996473i\) of defining polynomial
Character \(\chi\) \(=\) 58.7
Dual form 58.2.d.b.25.1

$q$-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+(-0.222521 - 0.974928i) q^{2} +(-2.06920 - 0.996473i) q^{3} +(-0.900969 + 0.433884i) q^{4} +(-0.788529 - 3.45477i) q^{5} +(-0.511050 + 2.23905i) q^{6} +(3.72857 + 1.79558i) q^{7} +(0.623490 + 0.781831i) q^{8} +(1.41815 + 1.77830i) q^{9} +(-3.19269 + 1.53752i) q^{10} +(-1.14832 + 1.43995i) q^{11} +2.29664 q^{12} +(2.09403 - 2.62583i) q^{13} +(0.920880 - 4.03464i) q^{14} +(-1.81096 + 7.93435i) q^{15} +(0.623490 - 0.781831i) q^{16} +3.52078 q^{17} +(1.41815 - 1.77830i) q^{18} +(-2.45556 + 1.18253i) q^{19} +(2.20941 + 2.77051i) q^{20} +(-5.92589 - 7.43083i) q^{21} +(1.65937 + 0.799109i) q^{22} +(0.679736 - 2.97812i) q^{23} +(-0.511050 - 2.23905i) q^{24} +(-6.80881 + 3.27895i) q^{25} +(-3.02595 - 1.45722i) q^{26} +(0.370748 + 1.62435i) q^{27} -4.13840 q^{28} +(0.127372 + 5.38366i) q^{29} +8.13840 q^{30} +(-0.196643 - 0.861548i) q^{31} +(-0.900969 - 0.433884i) q^{32} +(3.81096 - 1.83526i) q^{33} +(-0.783447 - 3.43250i) q^{34} +(3.26324 - 14.2972i) q^{35} +(-2.04929 - 0.986885i) q^{36} +(3.04846 + 3.82264i) q^{37} +(1.69930 + 2.13085i) q^{38} +(-6.94952 + 3.34671i) q^{39} +(2.20941 - 2.77051i) q^{40} -3.01488 q^{41} +(-5.92589 + 7.43083i) q^{42} +(-0.409830 + 1.79558i) q^{43} +(0.409830 - 1.79558i) q^{44} +(5.02538 - 6.30163i) q^{45} -3.05470 q^{46} +(1.25592 - 1.57487i) q^{47} +(-2.06920 + 0.996473i) q^{48} +(6.31365 + 7.91707i) q^{49} +(4.71184 + 5.90847i) q^{50} +(-7.28518 - 3.50836i) q^{51} +(-0.747349 + 3.27435i) q^{52} +(1.47479 + 6.46147i) q^{53} +(1.50113 - 0.722904i) q^{54} +(5.88016 + 2.83174i) q^{55} +(0.920880 + 4.03464i) q^{56} +6.25940 q^{57} +(5.22034 - 1.32216i) q^{58} +6.12406 q^{59} +(-1.81096 - 7.93435i) q^{60} +(-1.64476 - 0.792074i) q^{61} +(-0.796190 + 0.383425i) q^{62} +(2.09457 + 9.17693i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(-10.7228 - 5.16384i) q^{65} +(-2.63727 - 3.30703i) q^{66} +(-0.0862879 - 0.108202i) q^{67} +(-3.17211 + 1.52761i) q^{68} +(-4.37412 + 5.48497i) q^{69} -14.6649 q^{70} +(-8.17273 + 10.2483i) q^{71} +(-0.506132 + 2.21751i) q^{72} +(3.42387 - 15.0009i) q^{73} +(3.04846 - 3.82264i) q^{74} +17.3562 q^{75} +(1.69930 - 2.13085i) q^{76} +(-6.86712 + 3.30703i) q^{77} +(4.80921 + 6.03056i) q^{78} +(-9.90051 - 12.4148i) q^{79} +(-3.19269 - 1.53752i) q^{80} +(2.36987 - 10.3831i) q^{81} +(0.670875 + 2.93929i) q^{82} +(0.0422914 - 0.0203665i) q^{83} +(8.56316 + 4.12380i) q^{84} +(-2.77623 - 12.1635i) q^{85} +1.84176 q^{86} +(5.10111 - 11.2668i) q^{87} -1.84176 q^{88} +(0.800961 + 3.50924i) q^{89} +(-7.26188 - 3.49714i) q^{90} +(12.5226 - 6.03056i) q^{91} +(0.679736 + 2.97812i) q^{92} +(-0.451617 + 1.97866i) q^{93} +(-1.81485 - 0.873987i) q^{94} +(6.02166 + 7.55092i) q^{95} +(1.43193 + 1.79558i) q^{96} +(-4.71887 + 2.27249i) q^{97} +(6.31365 - 7.91707i) q^{98} -4.18915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 11 q^{9} - 7 q^{10} - 2 q^{11} + 4 q^{12} + q^{13} + q^{14} - 9 q^{15} - 2 q^{16} - 12 q^{17} - 11 q^{18} - 6 q^{19} + 7 q^{20} - 13 q^{21}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.222521 − 0.974928i −0.157346 − 0.689378i
\(3\) −2.06920 − 0.996473i −1.19465 − 0.575314i −0.272505 − 0.962154i \(-0.587852\pi\)
−0.922147 + 0.386840i \(0.873566\pi\)
\(4\) −0.900969 + 0.433884i −0.450484 + 0.216942i
\(5\) −0.788529 − 3.45477i −0.352641 − 1.54502i −0.771058 − 0.636765i \(-0.780272\pi\)
0.418417 − 0.908255i \(-0.362585\pi\)
\(6\) −0.511050 + 2.23905i −0.208635 + 0.914090i
\(7\) 3.72857 + 1.79558i 1.40927 + 0.678666i 0.975018 − 0.222127i \(-0.0713001\pi\)
0.434247 + 0.900794i \(0.357014\pi\)
\(8\) 0.623490 + 0.781831i 0.220437 + 0.276419i
\(9\) 1.41815 + 1.77830i 0.472717 + 0.592768i
\(10\) −3.19269 + 1.53752i −1.00962 + 0.486206i
\(11\) −1.14832 + 1.43995i −0.346231 + 0.434160i −0.924206 − 0.381895i \(-0.875272\pi\)
0.577975 + 0.816055i \(0.303843\pi\)
\(12\) 2.29664 0.662982
\(13\) 2.09403 − 2.62583i 0.580778 − 0.728273i −0.401467 − 0.915873i \(-0.631500\pi\)
0.982245 + 0.187601i \(0.0600710\pi\)
\(14\) 0.920880 − 4.03464i 0.246115 − 1.07830i
\(15\) −1.81096 + 7.93435i −0.467589 + 2.04864i
\(16\) 0.623490 − 0.781831i 0.155872 − 0.195458i
\(17\) 3.52078 0.853914 0.426957 − 0.904272i \(-0.359586\pi\)
0.426957 + 0.904272i \(0.359586\pi\)
\(18\) 1.41815 − 1.77830i 0.334261 − 0.419150i
\(19\) −2.45556 + 1.18253i −0.563344 + 0.271292i −0.693807 − 0.720161i \(-0.744068\pi\)
0.130463 + 0.991453i \(0.458354\pi\)
\(20\) 2.20941 + 2.77051i 0.494039 + 0.619505i
\(21\) −5.92589 − 7.43083i −1.29313 − 1.62154i
\(22\) 1.65937 + 0.799109i 0.353778 + 0.170371i
\(23\) 0.679736 − 2.97812i 0.141735 − 0.620980i −0.853297 − 0.521425i \(-0.825401\pi\)
0.995032 − 0.0995556i \(-0.0317421\pi\)
\(24\) −0.511050 − 2.23905i −0.104318 − 0.457045i
\(25\) −6.80881 + 3.27895i −1.36176 + 0.655790i
\(26\) −3.02595 − 1.45722i −0.593439 − 0.285785i
\(27\) 0.370748 + 1.62435i 0.0713504 + 0.312607i
\(28\) −4.13840 −0.782083
\(29\) 0.127372 + 5.38366i 0.0236524 + 0.999720i
\(30\) 8.13840 1.48586
\(31\) −0.196643 − 0.861548i −0.0353181 − 0.154739i 0.954194 − 0.299188i \(-0.0967159\pi\)
−0.989512 + 0.144450i \(0.953859\pi\)
\(32\) −0.900969 − 0.433884i −0.159270 − 0.0767005i
\(33\) 3.81096 − 1.83526i 0.663404 − 0.319478i
\(34\) −0.783447 − 3.43250i −0.134360 − 0.588670i
\(35\) 3.26324 − 14.2972i 0.551589 − 2.41667i
\(36\) −2.04929 − 0.986885i −0.341548 − 0.164481i
\(37\) 3.04846 + 3.82264i 0.501163 + 0.628439i 0.966491 − 0.256700i \(-0.0826351\pi\)
−0.465328 + 0.885138i \(0.654064\pi\)
\(38\) 1.69930 + 2.13085i 0.275663 + 0.345670i
\(39\) −6.94952 + 3.34671i −1.11281 + 0.535903i
\(40\) 2.20941 − 2.77051i 0.349338 − 0.438056i
\(41\) −3.01488 −0.470846 −0.235423 − 0.971893i \(-0.575647\pi\)
−0.235423 + 0.971893i \(0.575647\pi\)
\(42\) −5.92589 + 7.43083i −0.914384 + 1.14660i
\(43\) −0.409830 + 1.79558i −0.0624985 + 0.273824i −0.996516 − 0.0834039i \(-0.973421\pi\)
0.934017 + 0.357228i \(0.116278\pi\)
\(44\) 0.409830 − 1.79558i 0.0617842 − 0.270694i
\(45\) 5.02538 − 6.30163i 0.749139 − 0.939391i
\(46\) −3.05470 −0.450392
\(47\) 1.25592 − 1.57487i 0.183194 − 0.229718i −0.681751 − 0.731584i \(-0.738781\pi\)
0.864946 + 0.501866i \(0.167353\pi\)
\(48\) −2.06920 + 0.996473i −0.298663 + 0.143828i
\(49\) 6.31365 + 7.91707i 0.901950 + 1.13101i
\(50\) 4.71184 + 5.90847i 0.666355 + 0.835583i
\(51\) −7.28518 − 3.50836i −1.02013 − 0.491269i
\(52\) −0.747349 + 3.27435i −0.103639 + 0.454071i
\(53\) 1.47479 + 6.46147i 0.202578 + 0.887551i 0.969360 + 0.245644i \(0.0789993\pi\)
−0.766783 + 0.641907i \(0.778144\pi\)
\(54\) 1.50113 − 0.722904i 0.204277 − 0.0983748i
\(55\) 5.88016 + 2.83174i 0.792881 + 0.381831i
\(56\) 0.920880 + 4.03464i 0.123058 + 0.539151i
\(57\) 6.25940 0.829077
\(58\) 5.22034 − 1.32216i 0.685464 − 0.173607i
\(59\) 6.12406 0.797285 0.398642 − 0.917106i \(-0.369481\pi\)
0.398642 + 0.917106i \(0.369481\pi\)
\(60\) −1.81096 − 7.93435i −0.233794 − 1.02432i
\(61\) −1.64476 − 0.792074i −0.210590 − 0.101415i 0.325616 − 0.945502i \(-0.394428\pi\)
−0.536205 + 0.844088i \(0.680143\pi\)
\(62\) −0.796190 + 0.383425i −0.101116 + 0.0486950i
\(63\) 2.09457 + 9.17693i 0.263892 + 1.15618i
\(64\) −0.222521 + 0.974928i −0.0278151 + 0.121866i
\(65\) −10.7228 − 5.16384i −1.33000 − 0.640495i
\(66\) −2.63727 − 3.30703i −0.324625 − 0.407067i
\(67\) −0.0862879 − 0.108202i −0.0105417 − 0.0132189i 0.776533 − 0.630077i \(-0.216977\pi\)
−0.787074 + 0.616858i \(0.788405\pi\)
\(68\) −3.17211 + 1.52761i −0.384675 + 0.185250i
\(69\) −4.37412 + 5.48497i −0.526582 + 0.660313i
\(70\) −14.6649 −1.75279
\(71\) −8.17273 + 10.2483i −0.969925 + 1.21625i 0.00640935 + 0.999979i \(0.497960\pi\)
−0.976334 + 0.216268i \(0.930612\pi\)
\(72\) −0.506132 + 2.21751i −0.0596482 + 0.261336i
\(73\) 3.42387 − 15.0009i 0.400733 − 1.75573i −0.223709 − 0.974656i \(-0.571817\pi\)
0.624442 − 0.781071i \(-0.285326\pi\)
\(74\) 3.04846 − 3.82264i 0.354376 − 0.444373i
\(75\) 17.3562 2.00412
\(76\) 1.69930 − 2.13085i 0.194923 − 0.244426i
\(77\) −6.86712 + 3.30703i −0.782581 + 0.376871i
\(78\) 4.80921 + 6.03056i 0.544536 + 0.682827i
\(79\) −9.90051 − 12.4148i −1.11389 − 1.39678i −0.908390 − 0.418123i \(-0.862688\pi\)
−0.205504 − 0.978656i \(-0.565883\pi\)
\(80\) −3.19269 − 1.53752i −0.356953 − 0.171900i
\(81\) 2.36987 − 10.3831i 0.263319 − 1.15367i
\(82\) 0.670875 + 2.93929i 0.0740857 + 0.324591i
\(83\) 0.0422914 − 0.0203665i 0.00464209 − 0.00223551i −0.431561 − 0.902084i \(-0.642037\pi\)
0.436203 + 0.899848i \(0.356323\pi\)
\(84\) 8.56316 + 4.12380i 0.934317 + 0.449943i
\(85\) −2.77623 − 12.1635i −0.301125 − 1.31931i
\(86\) 1.84176 0.198602
\(87\) 5.10111 − 11.2668i 0.546897 − 1.20793i
\(88\) −1.84176 −0.196332
\(89\) 0.800961 + 3.50924i 0.0849017 + 0.371979i 0.999474 − 0.0324449i \(-0.0103293\pi\)
−0.914572 + 0.404423i \(0.867472\pi\)
\(90\) −7.26188 − 3.49714i −0.765470 − 0.368631i
\(91\) 12.5226 − 6.03056i 1.31272 − 0.632175i
\(92\) 0.679736 + 2.97812i 0.0708673 + 0.310490i
\(93\) −0.451617 + 1.97866i −0.0468305 + 0.205178i
\(94\) −1.81485 − 0.873987i −0.187188 − 0.0901449i
\(95\) 6.02166 + 7.55092i 0.617809 + 0.774708i
\(96\) 1.43193 + 1.79558i 0.146146 + 0.183261i
\(97\) −4.71887 + 2.27249i −0.479129 + 0.230736i −0.657829 − 0.753167i \(-0.728525\pi\)
0.178700 + 0.983904i \(0.442811\pi\)
\(98\) 6.31365 − 7.91707i 0.637775 − 0.799745i
\(99\) −4.18915 −0.421025
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 58.2.d.b.7.1 ✓ 12
3.2 odd 2 522.2.k.h.181.2 12
4.3 odd 2 464.2.u.h.65.2 12
29.2 odd 28 1682.2.b.i.1681.2 12
29.5 even 14 1682.2.a.q.1.2 6
29.24 even 7 1682.2.a.t.1.5 6
29.25 even 7 inner 58.2.d.b.25.1 yes 12
29.27 odd 28 1682.2.b.i.1681.11 12
87.83 odd 14 522.2.k.h.199.2 12
116.83 odd 14 464.2.u.h.257.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.1 ✓ 12 1.1 even 1 trivial
58.2.d.b.25.1 yes 12 29.25 even 7 inner
464.2.u.h.65.2 12 4.3 odd 2
464.2.u.h.257.2 12 116.83 odd 14
522.2.k.h.181.2 12 3.2 odd 2
522.2.k.h.199.2 12 87.83 odd 14
1682.2.a.q.1.2 6 29.5 even 14
1682.2.a.t.1.5 6 29.24 even 7
1682.2.b.i.1681.2 12 29.2 odd 28
1682.2.b.i.1681.11 12 29.27 odd 28