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Properties

Label 58.2.d.a.45.1
Level $58$
Weight $2$
Character 58.45
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 45.1
Root \(-0.623490 - 0.781831i\) of defining polynomial
Character \(\chi\) \(=\) 58.45
Dual form 58.2.d.a.49.1

$q$-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 \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{7}\right)\)

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 −0.616255 − 0.787546i \(-0.711351\pi\)
0.904931 + 0.425559i \(0.139923\pi\)
\(4\) 0.623490 + 0.781831i 0.311745 + 0.390916i
\(5\) −2.92543 − 1.40881i −1.30829 − 0.630040i −0.355787 − 0.934567i \(-0.615787\pi\)
−0.952504 + 0.304527i \(0.901502\pi\)
\(6\) 0.722521 − 0.347948i 0.294968 − 0.142049i
\(7\) −1.62349 + 2.03579i −0.613621 + 0.769457i −0.987431 − 0.158048i \(-0.949480\pi\)
0.373810 + 0.927505i \(0.378051\pi\)
\(8\) 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 −0.383647 − 0.923480i \(-0.625332\pi\)
0.746337 − 0.665569i \(-0.231811\pi\)
\(12\) 0.801938 0.231499
\(13\) −0.425428 + 1.86392i −0.117992 + 0.516958i 0.881043 + 0.473037i \(0.156842\pi\)
−0.999035 + 0.0439215i \(0.986015\pi\)
\(14\) −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 0.264276 − 0.964447i \(-0.414867\pi\)
−0.999073 + 0.0430405i \(0.986296\pi\)
\(20\) −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 −0.233668 − 0.972316i \(-0.575073\pi\)
0.614498 + 0.788918i \(0.289359\pi\)
\(24\) 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 0.280000 − 0.960000i \(-0.409665\pi\)
−0.575981 + 0.817463i \(0.695380\pi\)
\(32\) −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 0.934111 + 0.356982i \(0.116194\pi\)
−0.686716 + 0.726926i \(0.740949\pi\)
\(38\) −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 −0.247095 − 0.968991i \(-0.579476\pi\)
0.603527 + 0.797343i \(0.293762\pi\)
\(44\) 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 −0.925281 − 0.379282i \(-0.876171\pi\)
0.998214 + 0.0597435i \(0.0190283\pi\)
\(48\) 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 −0.540525 − 0.841328i \(-0.681774\pi\)
−0.994789 + 0.101960i \(0.967489\pi\)
\(54\) 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 −0.285263 − 0.958449i \(-0.592081\pi\)
0.997896 + 0.0648356i \(0.0206523\pi\)
\(62\) −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 0.871563 + 0.490283i \(0.163107\pi\)
−0.572525 + 0.819887i \(0.694036\pi\)
\(68\) 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 −0.820211 − 0.572061i \(-0.806144\pi\)
0.987193 − 0.159532i \(-0.0509986\pi\)
\(72\) −2.12349 + 1.02262i −0.250256 + 0.120517i
\(73\) 8.39977 − 4.04512i 0.983119 − 0.473445i 0.127942 − 0.991782i \(-0.459163\pi\)
0.855177 + 0.518336i \(0.173449\pi\)
\(74\) −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 −0.976417 − 0.215891i \(-0.930734\pi\)
0.786050 − 0.618163i \(-0.212123\pi\)
\(80\) 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 0.971223 − 0.238171i \(-0.0765478\pi\)
−0.448317 + 0.893875i \(0.647976\pi\)
\(84\) −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 0.779018 − 0.627002i \(-0.215718\pi\)
−0.00450031 + 0.999990i \(0.501432\pi\)
\(90\) 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 0.848586 − 0.529057i \(-0.177454\pi\)
−0.704621 + 0.709584i \(0.748883\pi\)
\(98\) −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.45.1 ✓ 6
3.2 odd 2 522.2.k.c.451.1 6
4.3 odd 2 464.2.u.b.161.1 6
29.3 odd 28 1682.2.b.g.1681.4 6
29.7 even 7 1682.2.a.m.1.3 3
29.20 even 7 inner 58.2.d.a.49.1 yes 6
29.22 even 14 1682.2.a.n.1.1 3
29.26 odd 28 1682.2.b.g.1681.3 6
87.20 odd 14 522.2.k.c.397.1 6
116.107 odd 14 464.2.u.b.49.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.a.45.1 ✓ 6 1.1 even 1 trivial
58.2.d.a.49.1 yes 6 29.20 even 7 inner
464.2.u.b.49.1 6 116.107 odd 14
464.2.u.b.161.1 6 4.3 odd 2
522.2.k.c.397.1 6 87.20 odd 14
522.2.k.c.451.1 6 3.2 odd 2
1682.2.a.m.1.3 3 29.7 even 7
1682.2.a.n.1.1 3 29.22 even 14
1682.2.b.g.1681.3 6 29.26 odd 28
1682.2.b.g.1681.4 6 29.3 odd 28