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Properties

Label 841.2.d.h.605.2
Level $841$
Weight $2$
Character 841.605
Analytic conductor $6.715$
Analytic rank $0$
Dimension $12$
Inner twists $12$

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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] = [841,2,Mod(190,841)] 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("841.190"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.d (of order \(7\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-6,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.71541880999\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: 12.0.4413675765625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 605.2
Root \(-1.00883 - 1.26503i\) of defining polynomial
Character \(\chi\) \(=\) 841.605
Dual form 841.2.d.h.645.2

$q$-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+(0.497572 - 2.18001i) q^{2} +(-2.01463 + 0.970194i) q^{3} +(-2.70291 - 1.30165i) q^{4} +(-0.667563 + 2.92478i) q^{5} +(1.11260 + 4.87464i) q^{6} +(-1.80194 + 0.867767i) q^{7} +(-1.39417 + 1.74823i) q^{8} +(1.24698 - 1.56366i) q^{9} +(6.04388 + 2.91058i) q^{10} +(-1.39417 - 1.74823i) q^{11} +6.70820 q^{12} +(0.623490 + 0.781831i) q^{13} +(0.995144 + 4.36001i) q^{14} +(-1.49272 - 6.54002i) q^{15} +(-0.623490 - 0.781831i) q^{16} +4.47214 q^{17} +(-2.78833 - 3.49646i) q^{18} +(5.61141 - 7.03648i) q^{20} +(2.78833 - 3.49646i) q^{21} +(-4.50484 + 2.16942i) q^{22} +(-1.33513 - 5.84957i) q^{23} +(1.11260 - 4.87464i) q^{24} +(-3.60388 - 1.73553i) q^{25} +(2.01463 - 0.970194i) q^{26} +(0.497572 - 2.18001i) q^{27} +6.00000 q^{28} -15.0000 q^{30} +(1.49272 - 6.54002i) q^{31} +(-6.04388 + 2.91058i) q^{32} +(4.50484 + 2.16942i) q^{33} +(2.22521 - 9.74928i) q^{34} +(-1.33513 - 5.84957i) q^{35} +(-5.40581 + 2.60330i) q^{36} +(-2.01463 - 0.970194i) q^{39} +(-4.18250 - 5.24469i) q^{40} -4.47214 q^{41} +(-6.23490 - 7.81831i) q^{42} +(-1.49272 - 6.54002i) q^{43} +(1.49272 + 6.54002i) q^{44} +(3.74094 + 4.69099i) q^{45} -13.4164 q^{46} +(1.39417 + 1.74823i) q^{47} +(2.01463 + 0.970194i) q^{48} +(-1.87047 + 2.34549i) q^{49} +(-5.57666 + 6.99291i) q^{50} +(-9.00969 + 4.33884i) q^{51} +(-0.667563 - 2.92478i) q^{52} +(2.00269 - 8.77435i) q^{53} +(-4.50484 - 2.16942i) q^{54} +(6.04388 - 2.91058i) q^{55} +(0.995144 - 4.36001i) q^{56} +6.00000 q^{59} +(-4.47815 + 19.6200i) q^{60} +(12.0878 - 5.82116i) q^{61} +(-13.5145 - 6.50826i) q^{62} +(-0.890084 + 3.89971i) q^{63} +(2.89277 + 12.6741i) q^{64} +(-2.70291 + 1.30165i) q^{65} +(6.97083 - 8.74114i) q^{66} +(-4.98792 + 6.25465i) q^{67} +(-12.0878 - 5.82116i) q^{68} +(8.36499 + 10.4894i) q^{69} -13.4164 q^{70} +(0.995144 + 4.36001i) q^{72} +8.94427 q^{75} +(4.02926 + 1.94039i) q^{77} +(-3.11745 + 3.90916i) q^{78} +(4.18250 - 5.24469i) q^{79} +(2.70291 - 1.30165i) q^{80} +(2.44773 + 10.7242i) q^{81} +(-2.22521 + 9.74928i) q^{82} +(5.40581 + 2.60330i) q^{83} +(-12.0878 + 5.82116i) q^{84} +(-2.98543 + 13.0800i) q^{85} -15.0000 q^{86} +5.00000 q^{88} +(-0.995144 + 4.36001i) q^{89} +(12.0878 - 5.82116i) q^{90} +(-1.80194 - 0.867767i) q^{91} +(-4.00538 + 17.5487i) q^{92} +(3.33781 + 14.6239i) q^{93} +(4.50484 - 2.16942i) q^{94} +(9.35235 - 11.7275i) q^{96} +(-12.0878 - 5.82116i) q^{97} +(4.18250 + 5.24469i) q^{98} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{4} - 6 q^{5} + 10 q^{6} - 4 q^{7} - 4 q^{9} - 2 q^{13} + 2 q^{16} - 18 q^{20} - 10 q^{22} - 12 q^{23} + 10 q^{24} - 8 q^{25} + 72 q^{28} - 180 q^{30} + 10 q^{33} + 20 q^{34} - 12 q^{35} - 12 q^{36}+ \cdots - 30 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/841\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.497572 − 2.18001i 0.351836 − 1.54150i −0.421101 − 0.907014i \(-0.638356\pi\)
0.772937 − 0.634483i \(-0.218787\pi\)
\(3\) −2.01463 + 0.970194i −1.16315 + 0.560141i −0.912957 − 0.408056i \(-0.866207\pi\)
−0.250189 + 0.968197i \(0.580493\pi\)
\(4\) −2.70291 − 1.30165i −1.35145 − 0.650826i
\(5\) −0.667563 + 2.92478i −0.298543 + 1.30800i 0.573754 + 0.819027i \(0.305486\pi\)
−0.872297 + 0.488976i \(0.837371\pi\)
\(6\) 1.11260 + 4.87464i 0.454219 + 1.99006i
\(7\) −1.80194 + 0.867767i −0.681068 + 0.327985i −0.742233 − 0.670142i \(-0.766233\pi\)
0.0611641 + 0.998128i \(0.480519\pi\)
\(8\) −1.39417 + 1.74823i −0.492912 + 0.618092i
\(9\) 1.24698 − 1.56366i 0.415660 − 0.521221i
\(10\) 6.04388 + 2.91058i 1.91124 + 0.920406i
\(11\) −1.39417 − 1.74823i −0.420357 − 0.527111i 0.525892 − 0.850552i \(-0.323732\pi\)
−0.946248 + 0.323441i \(0.895160\pi\)
\(12\) 6.70820 1.93649
\(13\) 0.623490 + 0.781831i 0.172925 + 0.216841i 0.860740 − 0.509045i \(-0.170001\pi\)
−0.687815 + 0.725886i \(0.741430\pi\)
\(14\) 0.995144 + 4.36001i 0.265963 + 1.16526i
\(15\) −1.49272 − 6.54002i −0.385418 − 1.68862i
\(16\) −0.623490 − 0.781831i −0.155872 − 0.195458i
\(17\) 4.47214 1.08465 0.542326 − 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) −2.78833 − 3.49646i −0.657216 − 0.824123i
\(19\) 0 0 0.433884 − 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(20\) 5.61141 − 7.03648i 1.25475 − 1.57341i
\(21\) 2.78833 − 3.49646i 0.608464 − 0.762989i
\(22\) −4.50484 + 2.16942i −0.960436 + 0.462522i
\(23\) −1.33513 − 5.84957i −0.278393 − 1.21972i −0.899825 − 0.436251i \(-0.856306\pi\)
0.621432 − 0.783468i \(-0.286551\pi\)
\(24\) 1.11260 − 4.87464i 0.227109 − 0.995032i
\(25\) −3.60388 − 1.73553i −0.720775 − 0.347107i
\(26\) 2.01463 − 0.970194i 0.395101 − 0.190271i
\(27\) 0.497572 − 2.18001i 0.0957578 − 0.419542i
\(28\) 6.00000 1.13389
\(29\) 0 0
\(30\) −15.0000 −2.73861
\(31\) 1.49272 − 6.54002i 0.268100 − 1.17462i −0.644121 − 0.764923i \(-0.722777\pi\)
0.912221 − 0.409698i \(-0.134366\pi\)
\(32\) −6.04388 + 2.91058i −1.06842 + 0.514523i
\(33\) 4.50484 + 2.16942i 0.784193 + 0.377647i
\(34\) 2.22521 − 9.74928i 0.381620 − 1.67199i
\(35\) −1.33513 − 5.84957i −0.225677 − 0.988757i
\(36\) −5.40581 + 2.60330i −0.900969 + 0.433884i
\(37\) 0 0 −0.781831 − 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(38\) 0 0
\(39\) −2.01463 − 0.970194i −0.322599 − 0.155355i
\(40\) −4.18250 − 5.24469i −0.661311 − 0.829258i
\(41\) −4.47214 −0.698430 −0.349215 − 0.937043i \(-0.613552\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) −6.23490 − 7.81831i −0.962066 − 1.20639i
\(43\) −1.49272 − 6.54002i −0.227637 − 0.997343i −0.951560 − 0.307462i \(-0.900520\pi\)
0.723923 − 0.689881i \(-0.242337\pi\)
\(44\) 1.49272 + 6.54002i 0.225035 + 0.985944i
\(45\) 3.74094 + 4.69099i 0.557666 + 0.699291i
\(46\) −13.4164 −1.97814
\(47\) 1.39417 + 1.74823i 0.203360 + 0.255005i 0.873045 − 0.487640i \(-0.162142\pi\)
−0.669685 + 0.742645i \(0.733571\pi\)
\(48\) 2.01463 + 0.970194i 0.290786 + 0.140035i
\(49\) −1.87047 + 2.34549i −0.267210 + 0.335071i
\(50\) −5.57666 + 6.99291i −0.788659 + 0.988947i
\(51\) −9.00969 + 4.33884i −1.26161 + 0.607559i
\(52\) −0.667563 − 2.92478i −0.0925743 − 0.405595i
\(53\) 2.00269 − 8.77435i 0.275090 − 1.20525i −0.628828 − 0.777545i \(-0.716465\pi\)
0.903918 − 0.427705i \(-0.140678\pi\)
\(54\) −4.50484 − 2.16942i −0.613032 − 0.295220i
\(55\) 6.04388 − 2.91058i 0.814957 − 0.392463i
\(56\) 0.995144 − 4.36001i 0.132982 − 0.582631i
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −4.47815 + 19.6200i −0.578126 + 2.53294i
\(61\) 12.0878 − 5.82116i 1.54768 − 0.745323i 0.551627 − 0.834091i \(-0.314007\pi\)
0.996052 + 0.0887673i \(0.0282928\pi\)
\(62\) −13.5145 − 6.50826i −1.71635 − 0.826549i
\(63\) −0.890084 + 3.89971i −0.112140 + 0.491317i
\(64\) 2.89277 + 12.6741i 0.361597 + 1.58426i
\(65\) −2.70291 + 1.30165i −0.335254 + 0.161450i
\(66\) 6.97083 − 8.74114i 0.858050 − 1.07596i
\(67\) −4.98792 + 6.25465i −0.609371 + 0.764127i −0.986806 − 0.161909i \(-0.948235\pi\)
0.377434 + 0.926036i \(0.376806\pi\)
\(68\) −12.0878 − 5.82116i −1.46586 − 0.705919i
\(69\) 8.36499 + 10.4894i 1.00703 + 1.26277i
\(70\) −13.4164 −1.60357
\(71\) 0 0 0.781831 − 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(72\) 0.995144 + 4.36001i 0.117279 + 0.513832i
\(73\) 0 0 0.974928 − 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(74\) 0 0
\(75\) 8.94427 1.03280
\(76\) 0 0
\(77\) 4.02926 + 1.94039i 0.459176 + 0.221128i
\(78\) −3.11745 + 3.90916i −0.352982 + 0.442625i
\(79\) 4.18250 − 5.24469i 0.470568 − 0.590073i −0.488742 − 0.872428i \(-0.662544\pi\)
0.959310 + 0.282355i \(0.0911156\pi\)
\(80\) 2.70291 − 1.30165i 0.302194 − 0.145529i
\(81\) 2.44773 + 10.7242i 0.271970 + 1.19158i
\(82\) −2.22521 + 9.74928i −0.245733 + 1.07663i
\(83\) 5.40581 + 2.60330i 0.593365 + 0.285750i 0.706368 − 0.707845i \(-0.250333\pi\)
−0.113003 + 0.993595i \(0.536047\pi\)
\(84\) −12.0878 + 5.82116i −1.31888 + 0.635141i
\(85\) −2.98543 + 13.0800i −0.323816 + 1.41873i
\(86\) −15.0000 −1.61749
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) −0.995144 + 4.36001i −0.105485 + 0.462160i 0.894404 + 0.447260i \(0.147600\pi\)
−0.999889 + 0.0149001i \(0.995257\pi\)
\(90\) 12.0878 − 5.82116i 1.27416 − 0.613604i
\(91\) −1.80194 − 0.867767i −0.188894 − 0.0909667i
\(92\) −4.00538 + 17.5487i −0.417589 + 1.82958i
\(93\) 3.33781 + 14.6239i 0.346115 + 1.51643i
\(94\) 4.50484 − 2.16942i 0.464639 − 0.223758i
\(95\) 0 0
\(96\) 9.35235 − 11.7275i 0.954520 − 1.19693i
\(97\) −12.0878 − 5.82116i −1.22733 − 0.591049i −0.295984 − 0.955193i \(-0.595647\pi\)
−0.931343 + 0.364144i \(0.881362\pi\)
\(98\) 4.18250 + 5.24469i 0.422496 + 0.529793i
\(99\) −4.47214 −0.449467
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 841.2.d.h.605.2 12
29.2 odd 28 841.2.e.g.651.2 12
29.3 odd 28 841.2.e.g.196.2 12
29.4 even 14 inner 841.2.d.h.574.2 12
29.5 even 14 inner 841.2.d.h.190.1 12
29.6 even 14 841.2.a.b.1.2 2
29.7 even 7 inner 841.2.d.h.645.2 12
29.8 odd 28 841.2.e.g.63.2 12
29.9 even 14 inner 841.2.d.h.778.2 12
29.10 odd 28 841.2.e.g.267.1 12
29.11 odd 28 841.2.e.g.270.1 12
29.12 odd 4 841.2.e.g.236.2 12
29.13 even 14 inner 841.2.d.h.571.1 12
29.14 odd 28 29.2.b.a.28.2 yes 2
29.15 odd 28 29.2.b.a.28.1 ✓ 2
29.16 even 7 inner 841.2.d.h.571.2 12
29.17 odd 4 841.2.e.g.236.1 12
29.18 odd 28 841.2.e.g.270.2 12
29.19 odd 28 841.2.e.g.267.2 12
29.20 even 7 inner 841.2.d.h.778.1 12
29.21 odd 28 841.2.e.g.63.1 12
29.22 even 14 inner 841.2.d.h.645.1 12
29.23 even 7 841.2.a.b.1.1 2
29.24 even 7 inner 841.2.d.h.190.2 12
29.25 even 7 inner 841.2.d.h.574.1 12
29.26 odd 28 841.2.e.g.196.1 12
29.27 odd 28 841.2.e.g.651.1 12
29.28 even 2 inner 841.2.d.h.605.1 12
87.14 even 28 261.2.c.a.28.1 2
87.23 odd 14 7569.2.a.i.1.2 2
87.35 odd 14 7569.2.a.i.1.1 2
87.44 even 28 261.2.c.a.28.2 2
116.15 even 28 464.2.e.a.289.1 2
116.43 even 28 464.2.e.a.289.2 2
145.14 odd 28 725.2.c.c.376.1 2
145.43 even 28 725.2.d.a.724.3 4
145.44 odd 28 725.2.c.c.376.2 2
145.72 even 28 725.2.d.a.724.2 4
145.73 even 28 725.2.d.a.724.1 4
145.102 even 28 725.2.d.a.724.4 4
203.160 even 28 1421.2.b.b.1275.1 2
203.188 even 28 1421.2.b.b.1275.2 2
232.43 even 28 1856.2.e.f.1217.1 2
232.101 odd 28 1856.2.e.g.1217.2 2
232.131 even 28 1856.2.e.f.1217.2 2
232.189 odd 28 1856.2.e.g.1217.1 2
348.131 odd 28 4176.2.o.k.289.1 2
348.275 odd 28 4176.2.o.k.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.2.b.a.28.1 ✓ 2 29.15 odd 28
29.2.b.a.28.2 yes 2 29.14 odd 28
261.2.c.a.28.1 2 87.14 even 28
261.2.c.a.28.2 2 87.44 even 28
464.2.e.a.289.1 2 116.15 even 28
464.2.e.a.289.2 2 116.43 even 28
725.2.c.c.376.1 2 145.14 odd 28
725.2.c.c.376.2 2 145.44 odd 28
725.2.d.a.724.1 4 145.73 even 28
725.2.d.a.724.2 4 145.72 even 28
725.2.d.a.724.3 4 145.43 even 28
725.2.d.a.724.4 4 145.102 even 28
841.2.a.b.1.1 2 29.23 even 7
841.2.a.b.1.2 2 29.6 even 14
841.2.d.h.190.1 12 29.5 even 14 inner
841.2.d.h.190.2 12 29.24 even 7 inner
841.2.d.h.571.1 12 29.13 even 14 inner
841.2.d.h.571.2 12 29.16 even 7 inner
841.2.d.h.574.1 12 29.25 even 7 inner
841.2.d.h.574.2 12 29.4 even 14 inner
841.2.d.h.605.1 12 29.28 even 2 inner
841.2.d.h.605.2 12 1.1 even 1 trivial
841.2.d.h.645.1 12 29.22 even 14 inner
841.2.d.h.645.2 12 29.7 even 7 inner
841.2.d.h.778.1 12 29.20 even 7 inner
841.2.d.h.778.2 12 29.9 even 14 inner
841.2.e.g.63.1 12 29.21 odd 28
841.2.e.g.63.2 12 29.8 odd 28
841.2.e.g.196.1 12 29.26 odd 28
841.2.e.g.196.2 12 29.3 odd 28
841.2.e.g.236.1 12 29.17 odd 4
841.2.e.g.236.2 12 29.12 odd 4
841.2.e.g.267.1 12 29.10 odd 28
841.2.e.g.267.2 12 29.19 odd 28
841.2.e.g.270.1 12 29.11 odd 28
841.2.e.g.270.2 12 29.18 odd 28
841.2.e.g.651.1 12 29.27 odd 28
841.2.e.g.651.2 12 29.2 odd 28
1421.2.b.b.1275.1 2 203.160 even 28
1421.2.b.b.1275.2 2 203.188 even 28
1856.2.e.f.1217.1 2 232.43 even 28
1856.2.e.f.1217.2 2 232.131 even 28
1856.2.e.g.1217.1 2 232.189 odd 28
1856.2.e.g.1217.2 2 232.101 odd 28
4176.2.o.k.289.1 2 348.131 odd 28
4176.2.o.k.289.2 2 348.275 odd 28
7569.2.a.i.1.1 2 87.35 odd 14
7569.2.a.i.1.2 2 87.23 odd 14