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 35.2
Root \(0.433884 + 0.900969i\) of defining polynomial
Character \(\chi\) \(=\) 58.35
Dual form 58.2.e.a.5.2

$q$-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+(0.781831 + 0.623490i) q^{2} +(-0.240787 - 0.0549581i) q^{3} +(0.222521 + 0.974928i) q^{4} +(-0.892482 + 1.11914i) q^{5} +(-0.153989 - 0.193096i) q^{6} +(0.904459 - 3.96269i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(-2.64795 - 1.27518i) q^{9} +(-1.39554 + 0.318523i) q^{10} +(0.815852 + 1.69413i) q^{11} -0.246980i q^{12} +(-3.39903 + 1.63689i) q^{13} +(3.17783 - 2.53424i) q^{14} +(0.276404 - 0.220425i) q^{15} +(-0.900969 + 0.433884i) q^{16} -1.78568i q^{17} +(-1.27518 - 2.64795i) q^{18} +(3.79673 - 0.866579i) q^{19} +(-1.28967 - 0.621074i) q^{20} +(-0.435565 + 0.904459i) q^{21} +(-0.418416 + 1.83320i) q^{22} +(2.97489 + 3.73039i) q^{23} +(0.153989 - 0.193096i) q^{24} +(0.656661 + 2.87702i) q^{25} +(-3.67805 - 0.839492i) q^{26} +(1.14680 + 0.914542i) q^{27} +4.06460 q^{28} +(1.25499 + 5.23689i) q^{29} +0.353534 q^{30} +(-5.35487 - 4.27036i) q^{31} +(-0.974928 - 0.222521i) q^{32} +(-0.103340 - 0.452764i) q^{33} +(1.11335 - 1.39610i) q^{34} +(3.62759 + 4.54885i) q^{35} +(0.653989 - 2.86531i) q^{36} +(2.21049 - 4.59014i) q^{37} +(3.50870 + 1.68970i) q^{38} +(0.908404 - 0.207337i) q^{39} +(-0.621074 - 1.28967i) q^{40} +10.2017i q^{41} +(-0.904459 + 0.435565i) q^{42} +(2.35311 - 1.87654i) q^{43} +(-1.47011 + 1.17238i) q^{44} +(3.79035 - 1.82534i) q^{45} +4.77135i q^{46} +(-5.65094 - 11.7343i) q^{47} +(0.240787 - 0.0549581i) q^{48} +(-8.57812 - 4.13101i) q^{49} +(-1.28039 + 2.65877i) q^{50} +(-0.0981376 + 0.429969i) q^{51} +(-2.35220 - 2.94957i) q^{52} +(5.32989 - 6.68347i) q^{53} +(0.326396 + 1.43004i) q^{54} +(-2.62410 - 0.598934i) q^{55} +(3.17783 + 2.53424i) q^{56} -0.961830 q^{57} +(-2.28396 + 4.87684i) q^{58} -5.64006 q^{59} +(0.276404 + 0.220425i) q^{60} +(-11.9210 - 2.72090i) q^{61} +(-1.52408 - 6.67741i) q^{62} +(-7.44813 + 9.33966i) q^{63} +(-0.623490 - 0.781831i) q^{64} +(1.20167 - 5.26488i) q^{65} +(0.201499 - 0.418416i) q^{66} +(10.6285 + 5.11840i) q^{67} +(1.74091 - 0.397351i) q^{68} +(-0.511300 - 1.06173i) q^{69} +5.81820i q^{70} +(-3.36755 + 1.62173i) q^{71} +(2.29780 - 1.83244i) q^{72} +(3.62836 - 2.89352i) q^{73} +(4.59014 - 2.21049i) q^{74} -0.728839i q^{75} +(1.68970 + 3.50870i) q^{76} +(7.45124 - 1.70070i) q^{77} +(0.839492 + 0.404278i) q^{78} +(4.80754 - 9.98295i) q^{79} +(0.318523 - 1.39554i) q^{80} +(5.27144 + 6.61017i) q^{81} +(-6.36063 + 7.97598i) q^{82} +(0.807254 + 3.53681i) q^{83} +(-0.978705 - 0.223383i) q^{84} +(1.99842 + 1.59369i) q^{85} +3.00974 q^{86} +(-0.0143755 - 1.32995i) q^{87} -1.88035 q^{88} +(1.33024 + 1.06083i) q^{89} +(4.10150 + 0.936140i) q^{90} +(3.41220 + 14.9498i) q^{91} +(-2.97489 + 3.73039i) q^{92} +(1.05469 + 1.32254i) q^{93} +(2.89813 - 12.6975i) q^{94} +(-2.41869 + 5.02247i) q^{95} +(0.222521 + 0.107160i) q^{96} +(12.5268 - 2.85915i) q^{97} +(-4.13101 - 8.57812i) q^{98} -5.52634i 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 \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.781831 + 0.623490i 0.552838 + 0.440874i
\(3\) −0.240787 − 0.0549581i −0.139019 − 0.0317301i 0.152445 − 0.988312i \(-0.451285\pi\)
−0.291464 + 0.956582i \(0.594142\pi\)
\(4\) 0.222521 + 0.974928i 0.111260 + 0.487464i
\(5\) −0.892482 + 1.11914i −0.399130 + 0.500493i −0.940265 − 0.340442i \(-0.889423\pi\)
0.541135 + 0.840936i \(0.317995\pi\)
\(6\) −0.153989 − 0.193096i −0.0628659 − 0.0788313i
\(7\) 0.904459 − 3.96269i 0.341853 − 1.49776i −0.453305 − 0.891356i \(-0.649755\pi\)
0.795158 − 0.606402i \(-0.207388\pi\)
\(8\) −0.433884 + 0.900969i −0.153401 + 0.318541i
\(9\) −2.64795 − 1.27518i −0.882649 − 0.425062i
\(10\) −1.39554 + 0.318523i −0.441309 + 0.100726i
\(11\) 0.815852 + 1.69413i 0.245989 + 0.510800i 0.987006 − 0.160686i \(-0.0513706\pi\)
−0.741017 + 0.671486i \(0.765656\pi\)
\(12\) − 0.246980i − 0.0712969i
\(13\) −3.39903 + 1.63689i −0.942722 + 0.453991i −0.841129 − 0.540835i \(-0.818108\pi\)
−0.101593 + 0.994826i \(0.532394\pi\)
\(14\) 3.17783 − 2.53424i 0.849312 − 0.677304i
\(15\) 0.276404 − 0.220425i 0.0713672 − 0.0569135i
\(16\) −0.900969 + 0.433884i −0.225242 + 0.108471i
\(17\) − 1.78568i − 0.433091i −0.976273 − 0.216545i \(-0.930521\pi\)
0.976273 − 0.216545i \(-0.0694789\pi\)
\(18\) −1.27518 − 2.64795i −0.300564 − 0.624127i
\(19\) 3.79673 − 0.866579i 0.871029 − 0.198807i 0.236430 − 0.971648i \(-0.424022\pi\)
0.634599 + 0.772842i \(0.281165\pi\)
\(20\) −1.28967 − 0.621074i −0.288380 − 0.138876i
\(21\) −0.435565 + 0.904459i −0.0950480 + 0.197369i
\(22\) −0.418416 + 1.83320i −0.0892067 + 0.390840i
\(23\) 2.97489 + 3.73039i 0.620307 + 0.777841i 0.988388 − 0.151953i \(-0.0485563\pi\)
−0.368080 + 0.929794i \(0.619985\pi\)
\(24\) 0.153989 − 0.193096i 0.0314329 − 0.0394156i
\(25\) 0.656661 + 2.87702i 0.131332 + 0.575404i
\(26\) −3.67805 − 0.839492i −0.721325 − 0.164638i
\(27\) 1.14680 + 0.914542i 0.220702 + 0.176004i
\(28\) 4.06460 0.768138
\(29\) 1.25499 + 5.23689i 0.233045 + 0.972466i
\(30\) 0.353534 0.0645462
\(31\) −5.35487 − 4.27036i −0.961762 − 0.766980i 0.0107233 − 0.999943i \(-0.496587\pi\)
−0.972486 + 0.232963i \(0.925158\pi\)
\(32\) −0.974928 − 0.222521i −0.172345 − 0.0393365i
\(33\) −0.103340 − 0.452764i −0.0179892 − 0.0788160i
\(34\) 1.11335 − 1.39610i 0.190938 − 0.239429i
\(35\) 3.62759 + 4.54885i 0.613174 + 0.768896i
\(36\) 0.653989 − 2.86531i 0.108998 − 0.477552i
\(37\) 2.21049 − 4.59014i 0.363403 − 0.754614i −0.636458 − 0.771312i \(-0.719601\pi\)
0.999861 + 0.0166978i \(0.00531532\pi\)
\(38\) 3.50870 + 1.68970i 0.569187 + 0.274106i
\(39\) 0.908404 − 0.207337i 0.145461 − 0.0332005i
\(40\) −0.621074 − 1.28967i −0.0982005 − 0.203915i
\(41\) 10.2017i 1.59323i 0.604485 + 0.796616i \(0.293379\pi\)
−0.604485 + 0.796616i \(0.706621\pi\)
\(42\) −0.904459 + 0.435565i −0.139561 + 0.0672091i
\(43\) 2.35311 − 1.87654i 0.358846 − 0.286170i −0.427427 − 0.904050i \(-0.640580\pi\)
0.786272 + 0.617880i \(0.212008\pi\)
\(44\) −1.47011 + 1.17238i −0.221628 + 0.176742i
\(45\) 3.79035 − 1.82534i 0.565033 − 0.272105i
\(46\) 4.77135i 0.703498i
\(47\) −5.65094 − 11.7343i −0.824274 − 1.71162i −0.693796 − 0.720172i \(-0.744063\pi\)
−0.130479 − 0.991451i \(-0.541651\pi\)
\(48\) 0.240787 − 0.0549581i 0.0347547 − 0.00793252i
\(49\) −8.57812 − 4.13101i −1.22545 − 0.590144i
\(50\) −1.28039 + 2.65877i −0.181075 + 0.376006i
\(51\) −0.0981376 + 0.429969i −0.0137420 + 0.0602077i
\(52\) −2.35220 − 2.94957i −0.326192 − 0.409032i
\(53\) 5.32989 − 6.68347i 0.732116 − 0.918045i −0.266839 − 0.963741i \(-0.585979\pi\)
0.998955 + 0.0456963i \(0.0145507\pi\)
\(54\) 0.326396 + 1.43004i 0.0444169 + 0.194603i
\(55\) −2.62410 − 0.598934i −0.353834 − 0.0807602i
\(56\) 3.17783 + 2.53424i 0.424656 + 0.338652i
\(57\) −0.961830 −0.127397
\(58\) −2.28396 + 4.87684i −0.299898 + 0.640360i
\(59\) −5.64006 −0.734273 −0.367137 − 0.930167i \(-0.619662\pi\)
−0.367137 + 0.930167i \(0.619662\pi\)
\(60\) 0.276404 + 0.220425i 0.0356836 + 0.0284567i
\(61\) −11.9210 − 2.72090i −1.52633 − 0.348376i −0.624696 − 0.780868i \(-0.714777\pi\)
−0.901637 + 0.432493i \(0.857634\pi\)
\(62\) −1.52408 − 6.67741i −0.193558 − 0.848032i
\(63\) −7.44813 + 9.33966i −0.938376 + 1.17669i
\(64\) −0.623490 − 0.781831i −0.0779362 − 0.0977289i
\(65\) 1.20167 − 5.26488i 0.149049 − 0.653028i
\(66\) 0.201499 − 0.418416i 0.0248028 − 0.0515035i
\(67\) 10.6285 + 5.11840i 1.29847 + 0.625312i 0.950071 − 0.312034i \(-0.101010\pi\)
0.348402 + 0.937345i \(0.386724\pi\)
\(68\) 1.74091 − 0.397351i 0.211116 − 0.0481859i
\(69\) −0.511300 − 1.06173i −0.0615533 − 0.127817i
\(70\) 5.81820i 0.695407i
\(71\) −3.36755 + 1.62173i −0.399654 + 0.192463i −0.622905 − 0.782298i \(-0.714048\pi\)
0.223250 + 0.974761i \(0.428333\pi\)
\(72\) 2.29780 − 1.83244i 0.270799 − 0.215955i
\(73\) 3.62836 − 2.89352i 0.424667 − 0.338661i −0.387722 − 0.921776i \(-0.626738\pi\)
0.812389 + 0.583116i \(0.198167\pi\)
\(74\) 4.59014 − 2.21049i 0.533593 − 0.256965i
\(75\) − 0.728839i − 0.0841590i
\(76\) 1.68970 + 3.50870i 0.193822 + 0.402476i
\(77\) 7.45124 − 1.70070i 0.849147 − 0.193812i
\(78\) 0.839492 + 0.404278i 0.0950537 + 0.0457754i
\(79\) 4.80754 − 9.98295i 0.540890 − 1.12317i −0.434088 − 0.900870i \(-0.642929\pi\)
0.974978 − 0.222299i \(-0.0713563\pi\)
\(80\) 0.318523 − 1.39554i 0.0356120 − 0.156026i
\(81\) 5.27144 + 6.61017i 0.585715 + 0.734464i
\(82\) −6.36063 + 7.97598i −0.702415 + 0.880800i
\(83\) 0.807254 + 3.53681i 0.0886077 + 0.388216i 0.999713 − 0.0239581i \(-0.00762682\pi\)
−0.911105 + 0.412174i \(0.864770\pi\)
\(84\) −0.978705 − 0.223383i −0.106785 − 0.0243731i
\(85\) 1.99842 + 1.59369i 0.216759 + 0.172860i
\(86\) 3.00974 0.324548
\(87\) −0.0143755 − 1.32995i −0.00154122 − 0.142585i
\(88\) −1.88035 −0.200446
\(89\) 1.33024 + 1.06083i 0.141005 + 0.112448i 0.691452 − 0.722422i \(-0.256971\pi\)
−0.550447 + 0.834870i \(0.685543\pi\)
\(90\) 4.10150 + 0.936140i 0.432336 + 0.0986778i
\(91\) 3.41220 + 14.9498i 0.357696 + 1.56717i
\(92\) −2.97489 + 3.73039i −0.310154 + 0.388920i
\(93\) 1.05469 + 1.32254i 0.109367 + 0.137141i
\(94\) 2.89813 − 12.6975i 0.298919 − 1.30965i
\(95\) −2.41869 + 5.02247i −0.248153 + 0.515294i
\(96\) 0.222521 + 0.107160i 0.0227109 + 0.0109370i
\(97\) 12.5268 − 2.85915i 1.27190 − 0.290303i 0.467288 − 0.884105i \(-0.345231\pi\)
0.804611 + 0.593803i \(0.202374\pi\)
\(98\) −4.13101 − 8.57812i −0.417295 − 0.866521i
\(99\) − 5.52634i − 0.555418i
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.35.2 yes 12
3.2 odd 2 522.2.n.a.325.1 12
4.3 odd 2 464.2.y.c.209.2 12
29.5 even 14 inner 58.2.e.a.5.2 ✓ 12
29.11 odd 28 1682.2.a.s.1.6 6
29.13 even 14 1682.2.b.j.1681.1 12
29.16 even 7 1682.2.b.j.1681.11 12
29.18 odd 28 1682.2.a.r.1.2 6
87.5 odd 14 522.2.n.a.469.1 12
116.63 odd 14 464.2.y.c.353.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.5.2 ✓ 12 29.5 even 14 inner
58.2.e.a.35.2 yes 12 1.1 even 1 trivial
464.2.y.c.209.2 12 4.3 odd 2
464.2.y.c.353.2 12 116.63 odd 14
522.2.n.a.325.1 12 3.2 odd 2
522.2.n.a.469.1 12 87.5 odd 14
1682.2.a.r.1.2 6 29.18 odd 28
1682.2.a.s.1.6 6 29.11 odd 28
1682.2.b.j.1681.1 12 29.13 even 14
1682.2.b.j.1681.11 12 29.16 even 7