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Properties

Label 35.5.d.a.6.8
Level $35$
Weight $5$
Character 35.6
Analytic conductor $3.618$
Analytic rank $0$
Dimension $12$
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] = [35,5,Mod(6,35)] 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("35.6"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.d (of order \(2\), degree \(1\), 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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6.8
Root \(2.17355 + 2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 35.6
Dual form 35.5.d.a.6.7

$q$-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)
 
\(f(q)\) \(=\) \(q+1.17355 q^{2} +9.10378i q^{3} -14.6228 q^{4} +11.1803i q^{5} +10.6837i q^{6} +(-40.5002 + 27.5814i) q^{7} -35.9373 q^{8} -1.87877 q^{9} +13.1207i q^{10} +106.806 q^{11} -133.123i q^{12} +243.104i q^{13} +(-47.5290 + 32.3681i) q^{14} -101.783 q^{15} +191.790 q^{16} -148.252i q^{17} -2.20483 q^{18} -244.752i q^{19} -163.488i q^{20} +(-251.095 - 368.705i) q^{21} +125.342 q^{22} +621.143 q^{23} -327.166i q^{24} -125.000 q^{25} +285.294i q^{26} +720.302i q^{27} +(592.226 - 403.316i) q^{28} -774.981 q^{29} -119.448 q^{30} +403.109i q^{31} +800.073 q^{32} +972.334i q^{33} -173.981i q^{34} +(-308.369 - 452.806i) q^{35} +27.4729 q^{36} -1290.58 q^{37} -287.229i q^{38} -2213.16 q^{39} -401.792i q^{40} +2536.73i q^{41} +(-294.672 - 432.694i) q^{42} +2242.99 q^{43} -1561.79 q^{44} -21.0053i q^{45} +728.942 q^{46} +2494.61i q^{47} +1746.02i q^{48} +(879.536 - 2234.10i) q^{49} -146.694 q^{50} +1349.65 q^{51} -3554.85i q^{52} -281.688 q^{53} +845.310i q^{54} +1194.12i q^{55} +(1455.47 - 991.201i) q^{56} +2228.17 q^{57} -909.479 q^{58} -5822.15i q^{59} +1488.36 q^{60} -6268.19i q^{61} +473.068i q^{62} +(76.0908 - 51.8192i) q^{63} -2129.72 q^{64} -2717.98 q^{65} +1141.08i q^{66} +4128.43 q^{67} +2167.85i q^{68} +5654.74i q^{69} +(-361.886 - 531.390i) q^{70} -793.446 q^{71} +67.5182 q^{72} -1600.72i q^{73} -1514.56 q^{74} -1137.97i q^{75} +3578.96i q^{76} +(-4325.65 + 2945.84i) q^{77} -2597.25 q^{78} +9856.44 q^{79} +2144.28i q^{80} -6709.65 q^{81} +2976.98i q^{82} -617.012i q^{83} +(3671.70 + 5391.49i) q^{84} +1657.51 q^{85} +2632.26 q^{86} -7055.26i q^{87} -3838.31 q^{88} +7345.17i q^{89} -24.6508i q^{90} +(-6705.13 - 9845.75i) q^{91} -9082.83 q^{92} -3669.81 q^{93} +2927.55i q^{94} +2736.41 q^{95} +7283.69i q^{96} -67.6288i q^{97} +(1032.18 - 2621.83i) q^{98} -200.664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 122 q^{4} - 50 q^{7} - 186 q^{8} - 434 q^{9} + 126 q^{11} + 78 q^{14} + 50 q^{15} + 578 q^{16} + 734 q^{18} - 642 q^{21} + 2264 q^{22} - 756 q^{23} - 1500 q^{25} + 1414 q^{28} - 2190 q^{29}+ \cdots + 23084 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17355 0.293387 0.146694 − 0.989182i \(-0.453137\pi\)
0.146694 + 0.989182i \(0.453137\pi\)
\(3\) 9.10378i 1.01153i 0.862671 + 0.505765i \(0.168790\pi\)
−0.862671 + 0.505765i \(0.831210\pi\)
\(4\) −14.6228 −0.913924
\(5\) 11.1803i 0.447214i
\(6\) 10.6837i 0.296770i
\(7\) −40.5002 + 27.5814i −0.826535 + 0.562885i
\(8\) −35.9373 −0.561521
\(9\) −1.87877 −0.0231947
\(10\) 13.1207i 0.131207i
\(11\) 106.806 0.882690 0.441345 − 0.897337i \(-0.354501\pi\)
0.441345 + 0.897337i \(0.354501\pi\)
\(12\) − 133.123i − 0.924462i
\(13\) 243.104i 1.43848i 0.694760 + 0.719242i \(0.255511\pi\)
−0.694760 + 0.719242i \(0.744489\pi\)
\(14\) −47.5290 + 32.3681i −0.242495 + 0.165143i
\(15\) −101.783 −0.452370
\(16\) 191.790 0.749181
\(17\) − 148.252i − 0.512982i −0.966547 − 0.256491i \(-0.917434\pi\)
0.966547 − 0.256491i \(-0.0825664\pi\)
\(18\) −2.20483 −0.00680505
\(19\) − 244.752i − 0.677984i −0.940789 − 0.338992i \(-0.889914\pi\)
0.940789 − 0.338992i \(-0.110086\pi\)
\(20\) − 163.488i − 0.408719i
\(21\) −251.095 − 368.705i −0.569376 − 0.836066i
\(22\) 125.342 0.258970
\(23\) 621.143 1.17418 0.587091 − 0.809521i \(-0.300273\pi\)
0.587091 + 0.809521i \(0.300273\pi\)
\(24\) − 327.166i − 0.567996i
\(25\) −125.000 −0.200000
\(26\) 285.294i 0.422033i
\(27\) 720.302i 0.988069i
\(28\) 592.226 − 403.316i 0.755390 − 0.514434i
\(29\) −774.981 −0.921500 −0.460750 − 0.887530i \(-0.652419\pi\)
−0.460750 + 0.887530i \(0.652419\pi\)
\(30\) −119.448 −0.132720
\(31\) 403.109i 0.419468i 0.977759 + 0.209734i \(0.0672598\pi\)
−0.977759 + 0.209734i \(0.932740\pi\)
\(32\) 800.073 0.781321
\(33\) 972.334i 0.892869i
\(34\) − 173.981i − 0.150502i
\(35\) −308.369 − 452.806i −0.251730 − 0.369638i
\(36\) 27.4729 0.0211982
\(37\) −1290.58 −0.942716 −0.471358 − 0.881942i \(-0.656236\pi\)
−0.471358 + 0.881942i \(0.656236\pi\)
\(38\) − 287.229i − 0.198912i
\(39\) −2213.16 −1.45507
\(40\) − 401.792i − 0.251120i
\(41\) 2536.73i 1.50906i 0.656264 + 0.754531i \(0.272136\pi\)
−0.656264 + 0.754531i \(0.727864\pi\)
\(42\) −294.672 − 432.694i −0.167048 − 0.245291i
\(43\) 2242.99 1.21308 0.606542 − 0.795052i \(-0.292556\pi\)
0.606542 + 0.795052i \(0.292556\pi\)
\(44\) −1561.79 −0.806712
\(45\) − 21.0053i − 0.0103730i
\(46\) 728.942 0.344490
\(47\) 2494.61i 1.12930i 0.825332 + 0.564648i \(0.190988\pi\)
−0.825332 + 0.564648i \(0.809012\pi\)
\(48\) 1746.02i 0.757819i
\(49\) 879.536 − 2234.10i 0.366321 − 0.930489i
\(50\) −146.694 −0.0586775
\(51\) 1349.65 0.518897
\(52\) − 3554.85i − 1.31466i
\(53\) −281.688 −0.100280 −0.0501402 − 0.998742i \(-0.515967\pi\)
−0.0501402 + 0.998742i \(0.515967\pi\)
\(54\) 845.310i 0.289887i
\(55\) 1194.12i 0.394751i
\(56\) 1455.47 − 991.201i 0.464117 − 0.316072i
\(57\) 2228.17 0.685802
\(58\) −909.479 −0.270356
\(59\) − 5822.15i − 1.67255i −0.548309 − 0.836276i \(-0.684728\pi\)
0.548309 − 0.836276i \(-0.315272\pi\)
\(60\) 1488.36 0.413432
\(61\) − 6268.19i − 1.68454i −0.539053 − 0.842272i \(-0.681218\pi\)
0.539053 − 0.842272i \(-0.318782\pi\)
\(62\) 473.068i 0.123067i
\(63\) 76.0908 − 51.8192i 0.0191713 − 0.0130560i
\(64\) −2129.72 −0.519951
\(65\) −2717.98 −0.643309
\(66\) 1141.08i 0.261956i
\(67\) 4128.43 0.919678 0.459839 − 0.888002i \(-0.347907\pi\)
0.459839 + 0.888002i \(0.347907\pi\)
\(68\) 2167.85i 0.468826i
\(69\) 5654.74i 1.18772i
\(70\) −361.886 − 531.390i −0.0738544 − 0.108447i
\(71\) −793.446 −0.157399 −0.0786993 − 0.996898i \(-0.525077\pi\)
−0.0786993 + 0.996898i \(0.525077\pi\)
\(72\) 67.5182 0.0130243
\(73\) − 1600.72i − 0.300380i −0.988657 − 0.150190i \(-0.952011\pi\)
0.988657 − 0.150190i \(-0.0479885\pi\)
\(74\) −1514.56 −0.276581
\(75\) − 1137.97i − 0.202306i
\(76\) 3578.96i 0.619626i
\(77\) −4325.65 + 2945.84i −0.729575 + 0.496853i
\(78\) −2597.25 −0.426899
\(79\) 9856.44 1.57931 0.789653 − 0.613554i \(-0.210261\pi\)
0.789653 + 0.613554i \(0.210261\pi\)
\(80\) 2144.28i 0.335044i
\(81\) −6709.65 −1.02266
\(82\) 2976.98i 0.442740i
\(83\) − 617.012i − 0.0895648i −0.998997 − 0.0447824i \(-0.985741\pi\)
0.998997 − 0.0447824i \(-0.0142595\pi\)
\(84\) 3671.70 + 5391.49i 0.520366 + 0.764101i
\(85\) 1657.51 0.229413
\(86\) 2632.26 0.355903
\(87\) − 7055.26i − 0.932125i
\(88\) −3838.31 −0.495649
\(89\) 7345.17i 0.927304i 0.886018 + 0.463652i \(0.153461\pi\)
−0.886018 + 0.463652i \(0.846539\pi\)
\(90\) − 24.6508i − 0.00304331i
\(91\) −6705.13 − 9845.75i −0.809701 − 1.18896i
\(92\) −9082.83 −1.07311
\(93\) −3669.81 −0.424305
\(94\) 2927.55i 0.331321i
\(95\) 2736.41 0.303204
\(96\) 7283.69i 0.790330i
\(97\) − 67.6288i − 0.00718767i −0.999994 − 0.00359383i \(-0.998856\pi\)
0.999994 − 0.00359383i \(-0.00114396\pi\)
\(98\) 1032.18 − 2621.83i 0.107474 − 0.272994i
\(99\) −200.664 −0.0204738
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 35.5.d.a.6.8 yes 12
3.2 odd 2 315.5.h.a.181.5 12
4.3 odd 2 560.5.f.b.321.4 12
5.2 odd 4 175.5.c.d.174.16 24
5.3 odd 4 175.5.c.d.174.9 24
5.4 even 2 175.5.d.i.76.5 12
7.6 odd 2 inner 35.5.d.a.6.7 ✓ 12
21.20 even 2 315.5.h.a.181.6 12
28.27 even 2 560.5.f.b.321.9 12
35.13 even 4 175.5.c.d.174.15 24
35.27 even 4 175.5.c.d.174.10 24
35.34 odd 2 175.5.d.i.76.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.d.a.6.7 ✓ 12 7.6 odd 2 inner
35.5.d.a.6.8 yes 12 1.1 even 1 trivial
175.5.c.d.174.9 24 5.3 odd 4
175.5.c.d.174.10 24 35.27 even 4
175.5.c.d.174.15 24 35.13 even 4
175.5.c.d.174.16 24 5.2 odd 4
175.5.d.i.76.5 12 5.4 even 2
175.5.d.i.76.6 12 35.34 odd 2
315.5.h.a.181.5 12 3.2 odd 2
315.5.h.a.181.6 12 21.20 even 2
560.5.f.b.321.4 12 4.3 odd 2
560.5.f.b.321.9 12 28.27 even 2