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Properties

Label 58.2.d.a.7.1
Level $58$
Weight $2$
Character 58.7
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 7.1
Root \(0.900969 - 0.433884i\) of defining polynomial
Character \(\chi\) \(=\) 58.7
Dual form 58.2.d.a.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} +(0.500000 + 0.240787i) q^{3} +(-0.900969 + 0.433884i) q^{4} +(-0.0440730 - 0.193096i) q^{5} +(-0.123490 + 0.541044i) q^{6} +(-0.0990311 - 0.0476909i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(-1.67845 - 2.10471i) q^{9} +(0.178448 - 0.0859360i) q^{10} +(-0.832437 + 1.04384i) q^{11} -0.554958 q^{12} +(2.45593 - 3.07964i) q^{13} +(0.0244587 - 0.107160i) q^{14} +(0.0244587 - 0.107160i) q^{15} +(0.623490 - 0.781831i) q^{16} -2.91185 q^{17} +(1.67845 - 2.10471i) q^{18} +(-1.16756 + 0.562269i) q^{19} +(0.123490 + 0.154851i) q^{20} +(-0.0380322 - 0.0476909i) q^{21} +(-1.20291 - 0.579289i) q^{22} +(-1.73341 + 7.59455i) q^{23} +(-0.123490 - 0.541044i) q^{24} +(4.46950 - 2.15240i) q^{25} +(3.54892 + 1.70907i) q^{26} +(-0.702907 - 3.07964i) q^{27} +0.109916 q^{28} +(-1.39493 + 5.20136i) q^{29} +0.109916 q^{30} +(2.07942 + 9.11052i) q^{31} +(0.900969 + 0.433884i) q^{32} +(-0.667563 + 0.321481i) q^{33} +(-0.647948 - 2.83885i) q^{34} +(-0.00484434 + 0.0212244i) q^{35} +(2.42543 + 1.16802i) q^{36} +(-1.88740 - 2.36672i) q^{37} +(-0.807979 - 1.01317i) q^{38} +(1.96950 - 0.948461i) q^{39} +(-0.123490 + 0.154851i) q^{40} +3.76271 q^{41} +(0.0380322 - 0.0476909i) q^{42} +(1.48307 - 6.49777i) q^{43} +(0.297093 - 1.30165i) q^{44} +(-0.332437 + 0.416863i) q^{45} -7.78986 q^{46} +(0.500000 - 0.626980i) q^{47} +(0.500000 - 0.240787i) q^{48} +(-4.35690 - 5.46337i) q^{49} +(3.09299 + 3.87849i) q^{50} +(-1.45593 - 0.701137i) q^{51} +(-0.876510 + 3.84024i) q^{52} +(-1.85474 - 8.12615i) q^{53} +(2.84601 - 1.37057i) q^{54} +(0.238250 + 0.114735i) q^{55} +(0.0244587 + 0.107160i) q^{56} -0.719169 q^{57} +(-5.38135 - 0.202542i) q^{58} -5.08815 q^{59} +(0.0244587 + 0.107160i) q^{60} +(9.96346 + 4.79815i) q^{61} +(-8.41939 + 4.05456i) q^{62} +(0.0658433 + 0.288478i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(-0.702907 - 0.338502i) q^{65} +(-0.461968 - 0.579289i) q^{66} +(-6.85839 - 8.60015i) q^{67} +(2.62349 - 1.26341i) q^{68} +(-2.69537 + 3.37989i) q^{69} -0.0217703 q^{70} +(6.82640 - 8.56003i) q^{71} +(-0.599031 + 2.62453i) q^{72} +(-1.76875 + 7.74940i) q^{73} +(1.88740 - 2.36672i) q^{74} +2.75302 q^{75} +(0.807979 - 1.01317i) q^{76} +(0.132219 - 0.0636733i) q^{77} +(1.36294 + 1.70907i) q^{78} +(3.04892 + 3.82322i) q^{79} +(-0.178448 - 0.0859360i) q^{80} +(-1.40701 + 6.16451i) q^{81} +(0.837282 + 3.66837i) q^{82} +(10.5184 - 5.06540i) q^{83} +(0.0549581 + 0.0264664i) q^{84} +(0.128334 + 0.562269i) q^{85} +6.66487 q^{86} +(-1.94989 + 2.26480i) q^{87} +1.33513 q^{88} +(2.55765 + 11.2058i) q^{89} +(-0.480386 - 0.231342i) q^{90} +(-0.390084 + 0.187854i) q^{91} +(-1.73341 - 7.59455i) q^{92} +(-1.15399 + 5.05596i) q^{93} +(0.722521 + 0.347948i) q^{94} +(0.160030 + 0.200671i) q^{95} +(0.346011 + 0.433884i) q^{96} +(9.54288 - 4.59561i) q^{97} +(4.35690 - 5.46337i) q^{98} +3.59419 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 \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

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

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
\(3\) 0.500000 + 0.240787i 0.288675 + 0.139019i 0.572617 − 0.819823i \(-0.305928\pi\)
−0.283942 + 0.958841i \(0.591642\pi\)
\(4\) −0.900969 + 0.433884i −0.450484 + 0.216942i
\(5\) −0.0440730 − 0.193096i −0.0197100 − 0.0863553i 0.964116 − 0.265480i \(-0.0855305\pi\)
−0.983826 + 0.179125i \(0.942673\pi\)
\(6\) −0.123490 + 0.541044i −0.0504145 + 0.220880i
\(7\) −0.0990311 − 0.0476909i −0.0374302 − 0.0180255i 0.415075 − 0.909787i \(-0.363755\pi\)
−0.452505 + 0.891762i \(0.649470\pi\)
\(8\) −0.623490 − 0.781831i −0.220437 − 0.276419i
\(9\) −1.67845 − 2.10471i −0.559483 − 0.701569i
\(10\) 0.178448 − 0.0859360i 0.0564302 − 0.0271753i
\(11\) −0.832437 + 1.04384i −0.250989 + 0.314731i −0.891325 − 0.453364i \(-0.850224\pi\)
0.640336 + 0.768095i \(0.278795\pi\)
\(12\) −0.554958 −0.160203
\(13\) 2.45593 − 3.07964i 0.681152 − 0.854137i −0.314308 − 0.949321i \(-0.601772\pi\)
0.995460 + 0.0951839i \(0.0303439\pi\)
\(14\) 0.0244587 − 0.107160i 0.00653685 − 0.0286398i
\(15\) 0.0244587 − 0.107160i 0.00631520 − 0.0276687i
\(16\) 0.623490 − 0.781831i 0.155872 − 0.195458i
\(17\) −2.91185 −0.706228 −0.353114 − 0.935580i \(-0.614877\pi\)
−0.353114 + 0.935580i \(0.614877\pi\)
\(18\) 1.67845 − 2.10471i 0.395614 − 0.496084i
\(19\) −1.16756 + 0.562269i −0.267857 + 0.128993i −0.562992 − 0.826462i \(-0.690350\pi\)
0.295135 + 0.955456i \(0.404635\pi\)
\(20\) 0.123490 + 0.154851i 0.0276132 + 0.0346258i
\(21\) −0.0380322 − 0.0476909i −0.00829931 − 0.0104070i
\(22\) −1.20291 − 0.579289i −0.256461 − 0.123505i
\(23\) −1.73341 + 7.59455i −0.361440 + 1.58357i 0.388102 + 0.921616i \(0.373131\pi\)
−0.749542 + 0.661956i \(0.769726\pi\)
\(24\) −0.123490 − 0.541044i −0.0252073 − 0.110440i
\(25\) 4.46950 − 2.15240i 0.893900 − 0.430480i
\(26\) 3.54892 + 1.70907i 0.696000 + 0.335176i
\(27\) −0.702907 − 3.07964i −0.135274 − 0.592676i
\(28\) 0.109916 0.0207722
\(29\) −1.39493 + 5.20136i −0.259032 + 0.965869i
\(30\) 0.109916 0.0200679
\(31\) 2.07942 + 9.11052i 0.373474 + 1.63630i 0.716942 + 0.697133i \(0.245541\pi\)
−0.343467 + 0.939165i \(0.611601\pi\)
\(32\) 0.900969 + 0.433884i 0.159270 + 0.0767005i
\(33\) −0.667563 + 0.321481i −0.116208 + 0.0559627i
\(34\) −0.647948 − 2.83885i −0.111122 − 0.486858i
\(35\) −0.00484434 + 0.0212244i −0.000818843 + 0.00358758i
\(36\) 2.42543 + 1.16802i 0.404238 + 0.194671i
\(37\) −1.88740 − 2.36672i −0.310286 − 0.389086i 0.602098 − 0.798422i \(-0.294332\pi\)
−0.912384 + 0.409336i \(0.865760\pi\)
\(38\) −0.807979 − 1.01317i −0.131071 − 0.164358i
\(39\) 1.96950 − 0.948461i 0.315372 − 0.151875i
\(40\) −0.123490 + 0.154851i −0.0195255 + 0.0244841i
\(41\) 3.76271 0.587636 0.293818 − 0.955861i \(-0.405074\pi\)
0.293818 + 0.955861i \(0.405074\pi\)
\(42\) 0.0380322 − 0.0476909i 0.00586850 − 0.00735886i
\(43\) 1.48307 − 6.49777i 0.226167 − 0.990901i −0.726567 − 0.687095i \(-0.758885\pi\)
0.952734 − 0.303806i \(-0.0982574\pi\)
\(44\) 0.297093 − 1.30165i 0.0447885 − 0.196231i
\(45\) −0.332437 + 0.416863i −0.0495568 + 0.0621423i
\(46\) −7.78986 −1.14855
\(47\) 0.500000 − 0.626980i 0.0729325 − 0.0914545i −0.744027 − 0.668150i \(-0.767087\pi\)
0.816959 + 0.576695i \(0.195658\pi\)
\(48\) 0.500000 − 0.240787i 0.0721688 − 0.0347547i
\(49\) −4.35690 − 5.46337i −0.622414 − 0.780482i
\(50\) 3.09299 + 3.87849i 0.437415 + 0.548501i
\(51\) −1.45593 − 0.701137i −0.203871 − 0.0981789i
\(52\) −0.876510 + 3.84024i −0.121550 + 0.532546i
\(53\) −1.85474 − 8.12615i −0.254768 − 1.11621i −0.926760 − 0.375654i \(-0.877418\pi\)
0.671992 − 0.740558i \(-0.265439\pi\)
\(54\) 2.84601 − 1.37057i 0.387293 − 0.186510i
\(55\) 0.238250 + 0.114735i 0.0321257 + 0.0154709i
\(56\) 0.0244587 + 0.107160i 0.00326843 + 0.0143199i
\(57\) −0.719169 −0.0952562
\(58\) −5.38135 − 0.202542i −0.706606 − 0.0265951i
\(59\) −5.08815 −0.662420 −0.331210 − 0.943557i \(-0.607457\pi\)
−0.331210 + 0.943557i \(0.607457\pi\)
\(60\) 0.0244587 + 0.107160i 0.00315760 + 0.0138344i
\(61\) 9.96346 + 4.79815i 1.27569 + 0.614340i 0.944279 − 0.329146i \(-0.106761\pi\)
0.331411 + 0.943486i \(0.392475\pi\)
\(62\) −8.41939 + 4.05456i −1.06926 + 0.514930i
\(63\) 0.0658433 + 0.288478i 0.00829547 + 0.0363448i
\(64\) −0.222521 + 0.974928i −0.0278151 + 0.121866i
\(65\) −0.702907 − 0.338502i −0.0871848 − 0.0419860i
\(66\) −0.461968 − 0.579289i −0.0568643 − 0.0713056i
\(67\) −6.85839 − 8.60015i −0.837885 − 1.05068i −0.997977 − 0.0635730i \(-0.979750\pi\)
0.160092 − 0.987102i \(-0.448821\pi\)
\(68\) 2.62349 − 1.26341i 0.318145 − 0.153210i
\(69\) −2.69537 + 3.37989i −0.324485 + 0.406891i
\(70\) −0.0217703 −0.00260204
\(71\) 6.82640 − 8.56003i 0.810144 − 1.01589i −0.189279 − 0.981923i \(-0.560615\pi\)
0.999423 − 0.0339653i \(-0.0108136\pi\)
\(72\) −0.599031 + 2.62453i −0.0705965 + 0.309303i
\(73\) −1.76875 + 7.74940i −0.207017 + 0.906999i 0.759523 + 0.650481i \(0.225432\pi\)
−0.966539 + 0.256518i \(0.917425\pi\)
\(74\) 1.88740 − 2.36672i 0.219405 − 0.275125i
\(75\) 2.75302 0.317891
\(76\) 0.807979 − 1.01317i 0.0926815 − 0.116219i
\(77\) 0.132219 − 0.0636733i 0.0150678 − 0.00725625i
\(78\) 1.36294 + 1.70907i 0.154322 + 0.193514i
\(79\) 3.04892 + 3.82322i 0.343030 + 0.430146i 0.923183 − 0.384362i \(-0.125578\pi\)
−0.580153 + 0.814508i \(0.697007\pi\)
\(80\) −0.178448 − 0.0859360i −0.0199511 − 0.00960794i
\(81\) −1.40701 + 6.16451i −0.156334 + 0.684946i
\(82\) 0.837282 + 3.66837i 0.0924623 + 0.405104i
\(83\) 10.5184 − 5.06540i 1.15455 − 0.556000i 0.244150 − 0.969737i \(-0.421491\pi\)
0.910396 + 0.413737i \(0.135777\pi\)
\(84\) 0.0549581 + 0.0264664i 0.00599642 + 0.00288773i
\(85\) 0.128334 + 0.562269i 0.0139198 + 0.0609866i
\(86\) 6.66487 0.718692
\(87\) −1.94989 + 2.26480i −0.209050 + 0.242812i
\(88\) 1.33513 0.142325
\(89\) 2.55765 + 11.2058i 0.271110 + 1.18781i 0.908705 + 0.417439i \(0.137072\pi\)
−0.637595 + 0.770372i \(0.720071\pi\)
\(90\) −0.480386 − 0.231342i −0.0506371 − 0.0243855i
\(91\) −0.390084 + 0.187854i −0.0408919 + 0.0196925i
\(92\) −1.73341 − 7.59455i −0.180720 − 0.791786i
\(93\) −1.15399 + 5.05596i −0.119663 + 0.524278i
\(94\) 0.722521 + 0.347948i 0.0745223 + 0.0358881i
\(95\) 0.160030 + 0.200671i 0.0164187 + 0.0205884i
\(96\) 0.346011 + 0.433884i 0.0353146 + 0.0442831i
\(97\) 9.54288 − 4.59561i 0.968932 − 0.466613i 0.118648 − 0.992936i \(-0.462144\pi\)
0.850285 + 0.526323i \(0.176430\pi\)
\(98\) 4.35690 − 5.46337i 0.440113 − 0.551884i
\(99\) 3.59419 0.361229
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.7.1 ✓ 6
3.2 odd 2 522.2.k.c.181.1 6
4.3 odd 2 464.2.u.b.65.1 6
29.2 odd 28 1682.2.b.g.1681.5 6
29.5 even 14 1682.2.a.n.1.2 3
29.24 even 7 1682.2.a.m.1.2 3
29.25 even 7 inner 58.2.d.a.25.1 yes 6
29.27 odd 28 1682.2.b.g.1681.2 6
87.83 odd 14 522.2.k.c.199.1 6
116.83 odd 14 464.2.u.b.257.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.a.7.1 ✓ 6 1.1 even 1 trivial
58.2.d.a.25.1 yes 6 29.25 even 7 inner
464.2.u.b.65.1 6 4.3 odd 2
464.2.u.b.257.1 6 116.83 odd 14
522.2.k.c.181.1 6 3.2 odd 2
522.2.k.c.199.1 6 87.83 odd 14
1682.2.a.m.1.2 3 29.24 even 7
1682.2.a.n.1.2 3 29.5 even 14
1682.2.b.g.1681.2 6 29.27 odd 28
1682.2.b.g.1681.5 6 29.2 odd 28