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Properties

Label 58.2.e.a.51.1
Level $58$
Weight $2$
Character 58.51
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(5,58)] 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("58.5"); 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([11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == 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 \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 51.1
Root \(0.781831 + 0.623490i\) of defining polynomial
Character \(\chi\) \(=\) 58.51
Dual form 58.2.e.a.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+(-0.974928 + 0.222521i) q^{2} +(0.626980 - 1.30194i) q^{3} +(0.900969 - 0.433884i) q^{4} +(-0.308457 - 1.35144i) q^{5} +(-0.321552 + 1.40881i) q^{6} +(1.78967 + 0.861862i) q^{7} +(-0.781831 + 0.623490i) q^{8} +(0.568532 + 0.712916i) q^{9} +(0.601447 + 1.24892i) q^{10} +(-2.70097 - 2.15395i) q^{11} -1.44504i q^{12} +(-3.81172 + 4.77974i) q^{13} +(-1.93659 - 0.442013i) q^{14} +(-1.95288 - 0.445733i) q^{15} +(0.623490 - 0.781831i) q^{16} +4.25956i q^{17} +(-0.712916 - 0.568532i) q^{18} +(-0.940908 - 1.95381i) q^{19} +(-0.864277 - 1.08377i) q^{20} +(2.24418 - 1.78967i) q^{21} +(3.11255 + 1.49893i) q^{22} +(0.127108 - 0.556895i) q^{23} +(0.321552 + 1.40881i) q^{24} +(2.77361 - 1.33570i) q^{25} +(2.65256 - 5.50809i) q^{26} +(5.51107 - 1.25786i) q^{27} +1.98639 q^{28} +(-4.17344 + 3.40329i) q^{29} +2.00311 q^{30} +(5.88393 - 1.34297i) q^{31} +(-0.433884 + 0.900969i) q^{32} +(-4.49776 + 2.16601i) q^{33} +(-0.947841 - 4.15276i) q^{34} +(0.612715 - 2.68448i) q^{35} +(0.821552 + 0.395639i) q^{36} +(-2.12207 + 1.69229i) q^{37} +(1.35208 + 1.69546i) q^{38} +(3.83306 + 7.95942i) q^{39} +(1.08377 + 0.864277i) q^{40} -9.14522i q^{41} +(-1.78967 + 2.24418i) q^{42} +(-10.7813 - 2.46076i) q^{43} +(-3.36805 - 0.768736i) q^{44} +(0.788095 - 0.988239i) q^{45} +0.571217i q^{46} +(-3.89807 - 3.10861i) q^{47} +(-0.626980 - 1.30194i) q^{48} +(-1.90430 - 2.38792i) q^{49} +(-2.40684 + 1.91939i) q^{50} +(5.54568 + 2.67066i) q^{51} +(-1.36039 + 5.96024i) q^{52} +(2.32017 + 10.1653i) q^{53} +(-5.09299 + 2.45265i) q^{54} +(-2.07780 + 4.31459i) q^{55} +(-1.93659 + 0.442013i) q^{56} -3.13368 q^{57} +(3.31150 - 4.24664i) q^{58} +9.01438 q^{59} +(-1.95288 + 0.445733i) q^{60} +(4.16364 - 8.64589i) q^{61} +(-5.43757 + 2.61859i) q^{62} +(0.403051 + 1.76588i) q^{63} +(0.222521 - 0.974928i) q^{64} +(7.63528 + 3.67696i) q^{65} +(3.90301 - 3.11255i) q^{66} +(-7.82616 - 9.81369i) q^{67} +(1.84815 + 3.83773i) q^{68} +(-0.645349 - 0.514648i) q^{69} +2.75352i q^{70} +(-2.66783 + 3.34535i) q^{71} +(-0.888992 - 0.202907i) q^{72} +(6.80675 + 1.55360i) q^{73} +(1.69229 - 2.12207i) q^{74} -4.44852i q^{75} +(-1.69546 - 1.35208i) q^{76} +(-2.97745 - 6.18273i) q^{77} +(-5.50809 - 6.90693i) q^{78} +(-9.09164 + 7.25034i) q^{79} +(-1.24892 - 0.601447i) q^{80} +(1.20895 - 5.29674i) q^{81} +(2.03500 + 8.91593i) q^{82} +(9.76203 - 4.70114i) q^{83} +(1.24543 - 2.58615i) q^{84} +(5.75653 - 1.31389i) q^{85} +11.0585 q^{86} +(1.81420 + 7.56736i) q^{87} +3.45467 q^{88} +(-2.78013 + 0.634546i) q^{89} +(-0.548431 + 1.13883i) q^{90} +(-10.9412 + 5.26901i) q^{91} +(-0.127108 - 0.556895i) q^{92} +(1.94065 - 8.50252i) q^{93} +(4.49206 + 2.16326i) q^{94} +(-2.35023 + 1.87425i) q^{95} +(0.900969 + 1.12978i) q^{96} +(-1.05219 - 2.18489i) q^{97} +(2.38792 + 1.90430i) q^{98} -3.15015i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{9} - 26 q^{13} - 14 q^{15} - 2 q^{16} + 2 q^{20} + 14 q^{21} + 4 q^{22} - 16 q^{23} + 12 q^{24} + 22 q^{25} + 14 q^{26} - 4 q^{28} + 18 q^{29} + 16 q^{30}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{13}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.974928 + 0.222521i −0.689378 + 0.157346i
\(3\) 0.626980 − 1.30194i 0.361987 − 0.751674i −0.637842 − 0.770167i \(-0.720173\pi\)
0.999829 + 0.0184933i \(0.00588693\pi\)
\(4\) 0.900969 − 0.433884i 0.450484 − 0.216942i
\(5\) −0.308457 − 1.35144i −0.137946 − 0.604381i −0.995885 − 0.0906292i \(-0.971112\pi\)
0.857939 − 0.513752i \(-0.171745\pi\)
\(6\) −0.321552 + 1.40881i −0.131273 + 0.575145i
\(7\) 1.78967 + 0.861862i 0.676433 + 0.325753i 0.740369 − 0.672201i \(-0.234651\pi\)
−0.0639355 + 0.997954i \(0.520365\pi\)
\(8\) −0.781831 + 0.623490i −0.276419 + 0.220437i
\(9\) 0.568532 + 0.712916i 0.189511 + 0.237639i
\(10\) 0.601447 + 1.24892i 0.190194 + 0.394942i
\(11\) −2.70097 − 2.15395i −0.814373 − 0.649441i 0.125069 − 0.992148i \(-0.460085\pi\)
−0.939442 + 0.342707i \(0.888656\pi\)
\(12\) − 1.44504i − 0.417148i
\(13\) −3.81172 + 4.77974i −1.05718 + 1.32566i −0.113963 + 0.993485i \(0.536355\pi\)
−0.943217 + 0.332177i \(0.892217\pi\)
\(14\) −1.93659 − 0.442013i −0.517574 − 0.118133i
\(15\) −1.95288 − 0.445733i −0.504233 − 0.115088i
\(16\) 0.623490 − 0.781831i 0.155872 − 0.195458i
\(17\) 4.25956i 1.03309i 0.856259 + 0.516547i \(0.172783\pi\)
−0.856259 + 0.516547i \(0.827217\pi\)
\(18\) −0.712916 − 0.568532i −0.168036 − 0.134004i
\(19\) −0.940908 − 1.95381i −0.215859 − 0.448236i 0.764719 − 0.644363i \(-0.222878\pi\)
−0.980578 + 0.196128i \(0.937163\pi\)
\(20\) −0.864277 − 1.08377i −0.193258 − 0.242338i
\(21\) 2.24418 − 1.78967i 0.489720 − 0.390539i
\(22\) 3.11255 + 1.49893i 0.663598 + 0.319572i
\(23\) 0.127108 − 0.556895i 0.0265038 − 0.116121i −0.959946 − 0.280186i \(-0.909604\pi\)
0.986449 + 0.164066i \(0.0524609\pi\)
\(24\) 0.321552 + 1.40881i 0.0656365 + 0.287572i
\(25\) 2.77361 − 1.33570i 0.554721 − 0.267140i
\(26\) 2.65256 − 5.50809i 0.520209 − 1.08023i
\(27\) 5.51107 − 1.25786i 1.06061 − 0.242076i
\(28\) 1.98639 0.375392
\(29\) −4.17344 + 3.40329i −0.774989 + 0.631975i
\(30\) 2.00311 0.365716
\(31\) 5.88393 − 1.34297i 1.05678 − 0.241204i 0.341388 − 0.939923i \(-0.389103\pi\)
0.715397 + 0.698718i \(0.246246\pi\)
\(32\) −0.433884 + 0.900969i −0.0767005 + 0.159270i
\(33\) −4.49776 + 2.16601i −0.782960 + 0.377054i
\(34\) −0.947841 − 4.15276i −0.162553 − 0.712193i
\(35\) 0.612715 − 2.68448i 0.103568 − 0.453760i
\(36\) 0.821552 + 0.395639i 0.136925 + 0.0659398i
\(37\) −2.12207 + 1.69229i −0.348866 + 0.278211i −0.782207 − 0.623019i \(-0.785906\pi\)
0.433341 + 0.901230i \(0.357335\pi\)
\(38\) 1.35208 + 1.69546i 0.219337 + 0.275039i
\(39\) 3.83306 + 7.95942i 0.613780 + 1.27453i
\(40\) 1.08377 + 0.864277i 0.171359 + 0.136654i
\(41\) − 9.14522i − 1.42824i −0.700021 − 0.714122i \(-0.746826\pi\)
0.700021 − 0.714122i \(-0.253174\pi\)
\(42\) −1.78967 + 2.24418i −0.276153 + 0.346285i
\(43\) −10.7813 − 2.46076i −1.64413 − 0.375262i −0.702440 − 0.711743i \(-0.747906\pi\)
−0.941690 + 0.336481i \(0.890763\pi\)
\(44\) −3.36805 − 0.768736i −0.507753 − 0.115891i
\(45\) 0.788095 − 0.988239i 0.117482 − 0.147318i
\(46\) 0.571217i 0.0842213i
\(47\) −3.89807 − 3.10861i −0.568592 − 0.453437i 0.296513 − 0.955029i \(-0.404176\pi\)
−0.865104 + 0.501592i \(0.832748\pi\)
\(48\) −0.626980 − 1.30194i −0.0904968 − 0.187919i
\(49\) −1.90430 − 2.38792i −0.272043 − 0.341131i
\(50\) −2.40684 + 1.91939i −0.340379 + 0.271443i
\(51\) 5.54568 + 2.67066i 0.776551 + 0.373967i
\(52\) −1.36039 + 5.96024i −0.188652 + 0.826537i
\(53\) 2.32017 + 10.1653i 0.318700 + 1.39632i 0.839834 + 0.542843i \(0.182652\pi\)
−0.521134 + 0.853475i \(0.674491\pi\)
\(54\) −5.09299 + 2.45265i −0.693068 + 0.333764i
\(55\) −2.07780 + 4.31459i −0.280170 + 0.581780i
\(56\) −1.93659 + 0.442013i −0.258787 + 0.0590665i
\(57\) −3.13368 −0.415066
\(58\) 3.31150 − 4.24664i 0.434822 − 0.557611i
\(59\) 9.01438 1.17357 0.586786 − 0.809742i \(-0.300393\pi\)
0.586786 + 0.809742i \(0.300393\pi\)
\(60\) −1.95288 + 0.445733i −0.252116 + 0.0575439i
\(61\) 4.16364 − 8.64589i 0.533100 − 1.10699i −0.444356 − 0.895850i \(-0.646567\pi\)
0.977455 − 0.211142i \(-0.0677184\pi\)
\(62\) −5.43757 + 2.61859i −0.690572 + 0.332562i
\(63\) 0.403051 + 1.76588i 0.0507797 + 0.222480i
\(64\) 0.222521 − 0.974928i 0.0278151 − 0.121866i
\(65\) 7.63528 + 3.67696i 0.947040 + 0.456070i
\(66\) 3.90301 − 3.11255i 0.480428 − 0.383128i
\(67\) −7.82616 − 9.81369i −0.956118 − 1.19893i −0.979955 − 0.199218i \(-0.936160\pi\)
0.0238376 − 0.999716i \(-0.492412\pi\)
\(68\) 1.84815 + 3.83773i 0.224121 + 0.465393i
\(69\) −0.645349 − 0.514648i −0.0776909 − 0.0619564i
\(70\) 2.75352i 0.329108i
\(71\) −2.66783 + 3.34535i −0.316613 + 0.397020i −0.914517 − 0.404548i \(-0.867429\pi\)
0.597904 + 0.801568i \(0.296000\pi\)
\(72\) −0.888992 − 0.202907i −0.104769 − 0.0239128i
\(73\) 6.80675 + 1.55360i 0.796670 + 0.181835i 0.601435 − 0.798922i \(-0.294596\pi\)
0.195235 + 0.980756i \(0.437453\pi\)
\(74\) 1.69229 − 2.12207i 0.196725 − 0.246686i
\(75\) − 4.44852i − 0.513671i
\(76\) −1.69546 − 1.35208i −0.194482 − 0.155094i
\(77\) −2.97745 − 6.18273i −0.339312 − 0.704588i
\(78\) −5.50809 − 6.90693i −0.623669 − 0.782056i
\(79\) −9.09164 + 7.25034i −1.02289 + 0.815727i −0.983024 − 0.183478i \(-0.941264\pi\)
−0.0398656 + 0.999205i \(0.512693\pi\)
\(80\) −1.24892 − 0.601447i −0.139633 − 0.0672438i
\(81\) 1.20895 − 5.29674i 0.134327 − 0.588527i
\(82\) 2.03500 + 8.91593i 0.224729 + 0.984600i
\(83\) 9.76203 − 4.70114i 1.07152 − 0.516018i 0.186925 − 0.982374i \(-0.440148\pi\)
0.884597 + 0.466357i \(0.154434\pi\)
\(84\) 1.24543 − 2.58615i 0.135887 − 0.282173i
\(85\) 5.75653 − 1.31389i 0.624383 − 0.142511i
\(86\) 11.0585 1.19247
\(87\) 1.81420 + 7.56736i 0.194503 + 0.811306i
\(88\) 3.45467 0.368269
\(89\) −2.78013 + 0.634546i −0.294693 + 0.0672617i −0.367311 − 0.930098i \(-0.619722\pi\)
0.0726184 + 0.997360i \(0.476864\pi\)
\(90\) −0.548431 + 1.13883i −0.0578097 + 0.120043i
\(91\) −10.9412 + 5.26901i −1.14695 + 0.552342i
\(92\) −0.127108 − 0.556895i −0.0132519 − 0.0580603i
\(93\) 1.94065 − 8.50252i 0.201236 − 0.881671i
\(94\) 4.49206 + 2.16326i 0.463321 + 0.223124i
\(95\) −2.35023 + 1.87425i −0.241129 + 0.192294i
\(96\) 0.900969 + 1.12978i 0.0919548 + 0.115308i
\(97\) −1.05219 − 2.18489i −0.106833 − 0.221842i 0.840698 − 0.541504i \(-0.182145\pi\)
−0.947532 + 0.319662i \(0.896431\pi\)
\(98\) 2.38792 + 1.90430i 0.241216 + 0.192363i
\(99\) − 3.15015i − 0.316602i
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.e.a.51.1 yes 12
3.2 odd 2 522.2.n.a.109.2 12
4.3 odd 2 464.2.y.c.225.1 12
29.2 odd 28 1682.2.a.r.1.3 6
29.4 even 14 inner 58.2.e.a.33.1 ✓ 12
29.5 even 14 1682.2.b.j.1681.10 12
29.24 even 7 1682.2.b.j.1681.4 12
29.27 odd 28 1682.2.a.s.1.3 6
87.62 odd 14 522.2.n.a.91.2 12
116.91 odd 14 464.2.y.c.33.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.33.1 ✓ 12 29.4 even 14 inner
58.2.e.a.51.1 yes 12 1.1 even 1 trivial
464.2.y.c.33.1 12 116.91 odd 14
464.2.y.c.225.1 12 4.3 odd 2
522.2.n.a.91.2 12 87.62 odd 14
522.2.n.a.109.2 12 3.2 odd 2
1682.2.a.r.1.3 6 29.2 odd 28
1682.2.a.s.1.3 6 29.27 odd 28
1682.2.b.j.1681.4 12 29.24 even 7
1682.2.b.j.1681.10 12 29.5 even 14