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Properties

Label 58.2.d.a.49.1
Level $58$
Weight $2$
Character 58.49
Analytic conductor $0.463$
Analytic rank $0$
Dimension $6$
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 = [6] 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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 49.1
Root \(-0.623490 + 0.781831i\) of defining polynomial
Character \(\chi\) \(=\) 58.49
Dual form 58.2.d.a.45.1

$q$-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+(0.900969 - 0.433884i) q^{2} +(0.500000 + 0.626980i) q^{3} +(0.623490 - 0.781831i) q^{4} +(-2.92543 + 1.40881i) q^{5} +(0.722521 + 0.347948i) q^{6} +(-1.62349 - 2.03579i) q^{7} +(0.222521 - 0.974928i) q^{8} +(0.524459 - 2.29780i) q^{9} +(-2.02446 + 2.53859i) q^{10} +(1.20291 + 5.27028i) q^{11} +0.801938 q^{12} +(-0.425428 - 1.86392i) q^{13} +(-2.34601 - 1.12978i) q^{14} +(-2.34601 - 1.12978i) q^{15} +(-0.222521 - 0.974928i) q^{16} +2.85086 q^{17} +(-0.524459 - 2.29780i) q^{18} +(-3.20291 + 4.01632i) q^{19} +(-0.722521 + 3.16557i) q^{20} +(0.464656 - 2.03579i) q^{21} +(3.37047 + 4.22643i) q^{22} +(1.82640 + 0.879546i) q^{23} +(0.722521 - 0.347948i) q^{24} +(3.45593 - 4.33360i) q^{25} +(-1.19202 - 1.49475i) q^{26} +(3.87047 - 1.86392i) q^{27} -2.60388 q^{28} +(3.51357 - 4.08103i) q^{29} -2.60388 q^{30} +(-1.64795 + 0.793610i) q^{31} +(-0.623490 - 0.781831i) q^{32} +(-2.70291 + 3.38934i) q^{33} +(2.56853 - 1.23694i) q^{34} +(7.61745 + 3.66837i) q^{35} +(-1.46950 - 1.84270i) q^{36} +(1.50484 - 6.59315i) q^{37} +(-1.14310 + 5.00827i) q^{38} +(0.955927 - 1.19869i) q^{39} +(0.722521 + 3.16557i) q^{40} -9.78986 q^{41} +(-0.464656 - 2.03579i) q^{42} +(2.33728 + 1.12558i) q^{43} +(4.87047 + 2.34549i) q^{44} +(1.70291 + 7.46092i) q^{45} +2.02715 q^{46} +(0.500000 + 2.19064i) q^{47} +(0.500000 - 0.626980i) q^{48} +(0.0489173 - 0.214321i) q^{49} +(1.23341 - 5.40391i) q^{50} +(1.42543 + 1.78743i) q^{51} +(-1.72252 - 0.829522i) q^{52} +(-11.1773 + 5.38268i) q^{53} +(2.67845 - 3.35867i) q^{54} +(-10.9438 - 13.7231i) q^{55} +(-2.34601 + 1.12978i) q^{56} -4.11960 q^{57} +(1.39493 - 5.20136i) q^{58} -10.8509 q^{59} +(-2.34601 + 1.12978i) q^{60} +(5.56584 + 6.97935i) q^{61} +(-1.14042 + 1.43004i) q^{62} +(-5.52930 + 2.66277i) q^{63} +(-0.900969 - 0.433884i) q^{64} +(3.87047 + 4.85342i) q^{65} +(-0.964656 + 4.22643i) q^{66} +(2.44773 - 10.7242i) q^{67} +(1.77748 - 2.22889i) q^{68} +(0.361740 + 1.58489i) q^{69} +8.45473 q^{70} +(1.40701 + 6.16451i) q^{71} +(-2.12349 - 1.02262i) q^{72} +(8.39977 + 4.04512i) q^{73} +(-1.50484 - 6.59315i) q^{74} +4.44504 q^{75} +(1.14310 + 5.00827i) q^{76} +(8.77628 - 11.0051i) q^{77} +(0.341166 - 1.49475i) q^{78} +(-1.69202 + 7.41323i) q^{79} +(2.02446 + 2.53859i) q^{80} +(-3.26659 - 1.57311i) q^{81} +(-8.82036 + 4.24766i) q^{82} +(4.76391 - 5.97375i) q^{83} +(-1.30194 - 1.63258i) q^{84} +(-8.33997 + 4.01632i) q^{85} +2.59419 q^{86} +(4.31551 + 0.162426i) q^{87} +5.40581 q^{88} +(7.30678 - 3.51876i) q^{89} +(4.77144 + 5.98319i) q^{90} +(-3.10388 + 3.89214i) q^{91} +(1.82640 - 0.879546i) q^{92} +(-1.32155 - 0.636426i) q^{93} +(1.40097 + 1.75676i) q^{94} +(3.71164 - 16.2617i) q^{95} +(0.178448 - 0.781831i) q^{96} +(1.41789 - 1.77798i) q^{97} +(-0.0489173 - 0.214321i) q^{98} +12.7409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 4 q^{6} - 5 q^{7} + q^{8} - 6 q^{9} - 3 q^{10} - 6 q^{11} - 4 q^{12} + 11 q^{13} - 9 q^{14} - 9 q^{15} - q^{16} - 10 q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20}+ \cdots + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{6}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.900969 − 0.433884i 0.637081 − 0.306802i
\(3\) 0.500000 + 0.626980i 0.288675 + 0.361987i 0.904931 − 0.425559i \(-0.139923\pi\)
−0.616255 + 0.787546i \(0.711351\pi\)
\(4\) 0.623490 − 0.781831i 0.311745 − 0.390916i
\(5\) −2.92543 + 1.40881i −1.30829 + 0.630040i −0.952504 − 0.304527i \(-0.901502\pi\)
−0.355787 + 0.934567i \(0.615787\pi\)
\(6\) 0.722521 + 0.347948i 0.294968 + 0.142049i
\(7\) −1.62349 − 2.03579i −0.613621 − 0.769457i 0.373810 − 0.927505i \(-0.378051\pi\)
−0.987431 + 0.158048i \(0.949480\pi\)
\(8\) 0.222521 − 0.974928i 0.0786730 − 0.344689i
\(9\) 0.524459 − 2.29780i 0.174820 − 0.765935i
\(10\) −2.02446 + 2.53859i −0.640190 + 0.802773i
\(11\) 1.20291 + 5.27028i 0.362690 + 1.58905i 0.746337 + 0.665569i \(0.231811\pi\)
−0.383647 + 0.923480i \(0.625332\pi\)
\(12\) 0.801938 0.231499
\(13\) −0.425428 − 1.86392i −0.117992 − 0.516958i −0.999035 − 0.0439215i \(-0.986015\pi\)
0.881043 − 0.473037i \(-0.156842\pi\)
\(14\) −2.34601 − 1.12978i −0.626998 − 0.301946i
\(15\) −2.34601 − 1.12978i −0.605737 − 0.291708i
\(16\) −0.222521 − 0.974928i −0.0556302 − 0.243732i
\(17\) 2.85086 0.691434 0.345717 − 0.938339i \(-0.387636\pi\)
0.345717 + 0.938339i \(0.387636\pi\)
\(18\) −0.524459 − 2.29780i −0.123616 − 0.541597i
\(19\) −3.20291 + 4.01632i −0.734797 + 0.921407i −0.999073 − 0.0430405i \(-0.986296\pi\)
0.264276 + 0.964447i \(0.414867\pi\)
\(20\) −0.722521 + 3.16557i −0.161561 + 0.707843i
\(21\) 0.464656 − 2.03579i 0.101396 − 0.444246i
\(22\) 3.37047 + 4.22643i 0.718586 + 0.901079i
\(23\) 1.82640 + 0.879546i 0.380830 + 0.183398i 0.614498 − 0.788918i \(-0.289359\pi\)
−0.233668 + 0.972316i \(0.575073\pi\)
\(24\) 0.722521 − 0.347948i 0.147484 − 0.0710245i
\(25\) 3.45593 − 4.33360i 0.691185 − 0.866719i
\(26\) −1.19202 − 1.49475i −0.233775 − 0.293144i
\(27\) 3.87047 − 1.86392i 0.744872 − 0.358712i
\(28\) −2.60388 −0.492086
\(29\) 3.51357 − 4.08103i 0.652454 − 0.757828i
\(30\) −2.60388 −0.475400
\(31\) −1.64795 + 0.793610i −0.295980 + 0.142537i −0.575981 − 0.817463i \(-0.695380\pi\)
0.280000 + 0.960000i \(0.409665\pi\)
\(32\) −0.623490 − 0.781831i −0.110218 − 0.138210i
\(33\) −2.70291 + 3.38934i −0.470516 + 0.590008i
\(34\) 2.56853 − 1.23694i 0.440500 − 0.212133i
\(35\) 7.61745 + 3.66837i 1.28758 + 0.620068i
\(36\) −1.46950 − 1.84270i −0.244917 − 0.307116i
\(37\) 1.50484 − 6.59315i 0.247395 − 1.08391i −0.686716 − 0.726926i \(-0.740949\pi\)
0.934111 − 0.356982i \(-0.116194\pi\)
\(38\) −1.14310 + 5.00827i −0.185436 + 0.812448i
\(39\) 0.955927 − 1.19869i 0.153071 − 0.191945i
\(40\) 0.722521 + 3.16557i 0.114241 + 0.500521i
\(41\) −9.78986 −1.52892 −0.764459 − 0.644672i \(-0.776994\pi\)
−0.764459 + 0.644672i \(0.776994\pi\)
\(42\) −0.464656 − 2.03579i −0.0716980 − 0.314129i
\(43\) 2.33728 + 1.12558i 0.356432 + 0.171649i 0.603527 − 0.797343i \(-0.293762\pi\)
−0.247095 + 0.968991i \(0.579476\pi\)
\(44\) 4.87047 + 2.34549i 0.734251 + 0.353597i
\(45\) 1.70291 + 7.46092i 0.253854 + 1.11221i
\(46\) 2.02715 0.298887
\(47\) 0.500000 + 2.19064i 0.0729325 + 0.319538i 0.998214 − 0.0597435i \(-0.0190283\pi\)
−0.925281 + 0.379282i \(0.876171\pi\)
\(48\) 0.500000 − 0.626980i 0.0721688 − 0.0904968i
\(49\) 0.0489173 − 0.214321i 0.00698819 − 0.0306173i
\(50\) 1.23341 − 5.40391i 0.174430 − 0.764228i
\(51\) 1.42543 + 1.78743i 0.199600 + 0.250290i
\(52\) −1.72252 − 0.829522i −0.238871 − 0.115034i
\(53\) −11.1773 + 5.38268i −1.53531 + 0.739368i −0.994789 − 0.101960i \(-0.967489\pi\)
−0.540525 + 0.841328i \(0.681774\pi\)
\(54\) 2.67845 − 3.35867i 0.364491 − 0.457057i
\(55\) −10.9438 − 13.7231i −1.47567 − 1.85043i
\(56\) −2.34601 + 1.12978i −0.313499 + 0.150973i
\(57\) −4.11960 −0.545655
\(58\) 1.39493 − 5.20136i 0.183163 − 0.682972i
\(59\) −10.8509 −1.41266 −0.706331 − 0.707882i \(-0.749651\pi\)
−0.706331 + 0.707882i \(0.749651\pi\)
\(60\) −2.34601 + 1.12978i −0.302869 + 0.145854i
\(61\) 5.56584 + 6.97935i 0.712633 + 0.893614i 0.997896 − 0.0648356i \(-0.0206523\pi\)
−0.285263 + 0.958449i \(0.592081\pi\)
\(62\) −1.14042 + 1.43004i −0.144833 + 0.181615i
\(63\) −5.52930 + 2.66277i −0.696627 + 0.335478i
\(64\) −0.900969 − 0.433884i −0.112621 − 0.0542355i
\(65\) 3.87047 + 4.85342i 0.480073 + 0.601992i
\(66\) −0.964656 + 4.22643i −0.118741 + 0.520238i
\(67\) 2.44773 − 10.7242i 0.299038 − 1.31017i −0.572525 − 0.819887i \(-0.694036\pi\)
0.871563 − 0.490283i \(-0.163107\pi\)
\(68\) 1.77748 − 2.22889i 0.215551 − 0.270292i
\(69\) 0.361740 + 1.58489i 0.0435484 + 0.190798i
\(70\) 8.45473 1.01053
\(71\) 1.40701 + 6.16451i 0.166981 + 0.731593i 0.987193 + 0.159532i \(0.0509986\pi\)
−0.820211 + 0.572061i \(0.806144\pi\)
\(72\) −2.12349 − 1.02262i −0.250256 − 0.120517i
\(73\) 8.39977 + 4.04512i 0.983119 + 0.473445i 0.855177 − 0.518336i \(-0.173449\pi\)
0.127942 + 0.991782i \(0.459163\pi\)
\(74\) −1.50484 − 6.59315i −0.174935 − 0.766439i
\(75\) 4.44504 0.513269
\(76\) 1.14310 + 5.00827i 0.131123 + 0.574488i
\(77\) 8.77628 − 11.0051i 1.00015 − 1.25415i
\(78\) 0.341166 − 1.49475i 0.0386295 − 0.169247i
\(79\) −1.69202 + 7.41323i −0.190367 + 0.834054i 0.786050 + 0.618163i \(0.212123\pi\)
−0.976417 + 0.215891i \(0.930734\pi\)
\(80\) 2.02446 + 2.53859i 0.226341 + 0.283823i
\(81\) −3.26659 − 1.57311i −0.362955 − 0.174790i
\(82\) −8.82036 + 4.24766i −0.974045 + 0.469076i
\(83\) 4.76391 − 5.97375i 0.522907 − 0.655704i −0.448317 − 0.893875i \(-0.647976\pi\)
0.971223 + 0.238171i \(0.0765478\pi\)
\(84\) −1.30194 − 1.63258i −0.142053 − 0.178129i
\(85\) −8.33997 + 4.01632i −0.904597 + 0.435631i
\(86\) 2.59419 0.279738
\(87\) 4.31551 + 0.162426i 0.462671 + 0.0174139i
\(88\) 5.40581 0.576262
\(89\) 7.30678 − 3.51876i 0.774517 − 0.372988i −0.00450031 − 0.999990i \(-0.501432\pi\)
0.779018 + 0.627002i \(0.215718\pi\)
\(90\) 4.77144 + 5.98319i 0.502954 + 0.630684i
\(91\) −3.10388 + 3.89214i −0.325375 + 0.408007i
\(92\) 1.82640 − 0.879546i 0.190415 − 0.0916990i
\(93\) −1.32155 − 0.636426i −0.137039 − 0.0659943i
\(94\) 1.40097 + 1.75676i 0.144499 + 0.181196i
\(95\) 3.71164 − 16.2617i 0.380806 − 1.66842i
\(96\) 0.178448 − 0.781831i 0.0182128 − 0.0797953i
\(97\) 1.41789 − 1.77798i 0.143965 − 0.180527i −0.704621 − 0.709584i \(-0.748883\pi\)
0.848586 + 0.529057i \(0.177454\pi\)
\(98\) −0.0489173 − 0.214321i −0.00494140 − 0.0216497i
\(99\) 12.7409 1.28051
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.a.49.1 yes 6
3.2 odd 2 522.2.k.c.397.1 6
4.3 odd 2 464.2.u.b.49.1 6
29.4 even 14 1682.2.a.n.1.1 3
29.10 odd 28 1682.2.b.g.1681.3 6
29.16 even 7 inner 58.2.d.a.45.1 ✓ 6
29.19 odd 28 1682.2.b.g.1681.4 6
29.25 even 7 1682.2.a.m.1.3 3
87.74 odd 14 522.2.k.c.451.1 6
116.103 odd 14 464.2.u.b.161.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.a.45.1 ✓ 6 29.16 even 7 inner
58.2.d.a.49.1 yes 6 1.1 even 1 trivial
464.2.u.b.49.1 6 4.3 odd 2
464.2.u.b.161.1 6 116.103 odd 14
522.2.k.c.397.1 6 3.2 odd 2
522.2.k.c.451.1 6 87.74 odd 14
1682.2.a.m.1.3 3 29.25 even 7
1682.2.a.n.1.1 3 29.4 even 14
1682.2.b.g.1681.3 6 29.10 odd 28
1682.2.b.g.1681.4 6 29.19 odd 28