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Properties

Label 35.5.l.a.2.11
Level $35$
Weight $5$
Character 35.2
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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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(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), 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: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 2.11
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.11

$q$-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+(3.33928 - 0.894758i) q^{2} +(6.96176 + 1.86540i) q^{3} +(-3.50620 + 2.02430i) q^{4} +(21.4576 - 12.8286i) q^{5} +24.9163 q^{6} +(48.0380 + 9.66157i) q^{7} +(-49.0093 + 49.0093i) q^{8} +(-25.1617 - 14.5271i) q^{9} +(60.1743 - 62.0376i) q^{10} +(-39.5345 - 68.4757i) q^{11} +(-28.1854 + 7.55226i) q^{12} +(-37.2680 + 37.2680i) q^{13} +(169.057 - 10.7197i) q^{14} +(173.313 - 49.2827i) q^{15} +(-87.4155 + 151.408i) q^{16} +(-55.3784 + 206.675i) q^{17} +(-97.0203 - 25.9965i) q^{18} +(-615.723 - 355.488i) q^{19} +(-49.2655 + 88.4163i) q^{20} +(316.406 + 156.872i) q^{21} +(-193.286 - 193.286i) q^{22} +(110.497 + 412.381i) q^{23} +(-432.613 + 249.769i) q^{24} +(295.854 - 550.541i) q^{25} +(-91.1026 + 157.794i) q^{26} +(-560.876 - 560.876i) q^{27} +(-187.989 + 63.3682i) q^{28} +237.531i q^{29} +(534.644 - 319.642i) q^{30} +(567.630 + 983.163i) q^{31} +(130.587 - 487.357i) q^{32} +(-147.495 - 550.459i) q^{33} +739.696i q^{34} +(1154.72 - 408.947i) q^{35} +117.629 q^{36} +(1116.90 - 299.273i) q^{37} +(-2374.15 - 636.151i) q^{38} +(-328.971 + 189.931i) q^{39} +(-422.899 + 1680.34i) q^{40} +1403.97 q^{41} +(1196.93 + 240.731i) q^{42} +(2193.23 - 2193.23i) q^{43} +(277.231 + 160.060i) q^{44} +(-726.272 + 11.0731i) q^{45} +(737.962 + 1278.19i) q^{46} +(793.044 - 212.496i) q^{47} +(-891.002 + 891.002i) q^{48} +(2214.31 + 928.246i) q^{49} +(495.338 - 2103.13i) q^{50} +(-771.062 + 1335.52i) q^{51} +(55.2273 - 206.111i) q^{52} +(-1859.67 - 498.296i) q^{53} +(-2374.77 - 1371.07i) q^{54} +(-1726.76 - 962.150i) q^{55} +(-2827.82 + 1880.80i) q^{56} +(-3623.39 - 3623.39i) q^{57} +(212.533 + 793.184i) q^{58} +(-3064.87 + 1769.51i) q^{59} +(-507.905 + 523.633i) q^{60} +(-433.722 + 751.229i) q^{61} +(2775.17 + 2775.17i) q^{62} +(-1068.37 - 940.957i) q^{63} -4541.56i q^{64} +(-321.584 + 1277.78i) q^{65} +(-985.055 - 1706.16i) q^{66} +(-1775.22 + 6625.21i) q^{67} +(-224.206 - 836.746i) q^{68} +3077.02i q^{69} +(3490.04 - 2398.79i) q^{70} -486.669 q^{71} +(1945.12 - 521.194i) q^{72} +(1821.45 + 488.055i) q^{73} +(3461.88 - 1998.72i) q^{74} +(3086.64 - 3280.85i) q^{75} +2878.46 q^{76} +(-1237.58 - 3671.41i) q^{77} +(-928.583 + 928.583i) q^{78} +(5538.38 + 3197.58i) q^{79} +(66.6311 + 4370.27i) q^{80} +(-1681.73 - 2912.84i) q^{81} +(4688.26 - 1256.22i) q^{82} +(-325.522 + 325.522i) q^{83} +(-1426.94 + 90.4804i) q^{84} +(1463.07 + 5145.17i) q^{85} +(5361.41 - 9286.23i) q^{86} +(-443.090 + 1653.64i) q^{87} +(5293.51 + 1418.39i) q^{88} +(-3046.87 - 1759.11i) q^{89} +(-2415.32 + 686.814i) q^{90} +(-2150.35 + 1430.22i) q^{91} +(-1222.21 - 1222.21i) q^{92} +(2117.71 + 7903.40i) q^{93} +(2458.07 - 1419.17i) q^{94} +(-17772.3 + 270.965i) q^{95} +(1818.23 - 3149.26i) q^{96} +(-4804.80 - 4804.80i) q^{97} +(8224.75 + 1118.41i) q^{98} +2297.29i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.33928 − 0.894758i 0.834820 − 0.223689i 0.184005 − 0.982925i \(-0.441094\pi\)
0.650816 + 0.759236i \(0.274427\pi\)
\(3\) 6.96176 + 1.86540i 0.773528 + 0.207266i 0.623930 − 0.781480i \(-0.285535\pi\)
0.149599 + 0.988747i \(0.452202\pi\)
\(4\) −3.50620 + 2.02430i −0.219137 + 0.126519i
\(5\) 21.4576 − 12.8286i 0.858302 − 0.513144i
\(6\) 24.9163 0.692121
\(7\) 48.0380 + 9.66157i 0.980368 + 0.197175i
\(8\) −49.0093 + 49.0093i −0.765770 + 0.765770i
\(9\) −25.1617 − 14.5271i −0.310639 − 0.179347i
\(10\) 60.1743 − 62.0376i 0.601743 − 0.620376i
\(11\) −39.5345 − 68.4757i −0.326731 − 0.565915i 0.655130 − 0.755516i \(-0.272614\pi\)
−0.981861 + 0.189601i \(0.939281\pi\)
\(12\) −28.1854 + 7.55226i −0.195732 + 0.0524463i
\(13\) −37.2680 + 37.2680i −0.220521 + 0.220521i −0.808718 − 0.588197i \(-0.799838\pi\)
0.588197 + 0.808718i \(0.299838\pi\)
\(14\) 169.057 − 10.7197i 0.862537 − 0.0546924i
\(15\) 173.313 − 49.2827i 0.770279 − 0.219034i
\(16\) −87.4155 + 151.408i −0.341467 + 0.591438i
\(17\) −55.3784 + 206.675i −0.191621 + 0.715139i 0.801495 + 0.598002i \(0.204038\pi\)
−0.993116 + 0.117137i \(0.962628\pi\)
\(18\) −97.0203 − 25.9965i −0.299445 − 0.0802362i
\(19\) −615.723 − 355.488i −1.70561 − 0.984732i −0.939842 − 0.341609i \(-0.889028\pi\)
−0.765763 − 0.643123i \(-0.777638\pi\)
\(20\) −49.2655 + 88.4163i −0.123164 + 0.221041i
\(21\) 316.406 + 156.872i 0.717475 + 0.355718i
\(22\) −193.286 − 193.286i −0.399351 − 0.399351i
\(23\) 110.497 + 412.381i 0.208879 + 0.779548i 0.988232 + 0.152963i \(0.0488814\pi\)
−0.779353 + 0.626585i \(0.784452\pi\)
\(24\) −432.613 + 249.769i −0.751063 + 0.433627i
\(25\) 295.854 − 550.541i 0.473366 − 0.880866i
\(26\) −91.1026 + 157.794i −0.134767 + 0.233424i
\(27\) −560.876 − 560.876i −0.769377 − 0.769377i
\(28\) −187.989 + 63.3682i −0.239782 + 0.0808268i
\(29\) 237.531i 0.282439i 0.989978 + 0.141220i \(0.0451024\pi\)
−0.989978 + 0.141220i \(0.954898\pi\)
\(30\) 534.644 − 319.642i 0.594049 − 0.355158i
\(31\) 567.630 + 983.163i 0.590666 + 1.02306i 0.994143 + 0.108073i \(0.0344681\pi\)
−0.403477 + 0.914990i \(0.632199\pi\)
\(32\) 130.587 − 487.357i 0.127526 − 0.475935i
\(33\) −147.495 − 550.459i −0.135441 − 0.505472i
\(34\) 739.696i 0.639876i
\(35\) 1154.72 − 408.947i 0.942632 − 0.333835i
\(36\) 117.629 0.0907634
\(37\) 1116.90 − 299.273i 0.815853 − 0.218607i 0.173321 − 0.984865i \(-0.444550\pi\)
0.642533 + 0.766258i \(0.277884\pi\)
\(38\) −2374.15 − 636.151i −1.64415 − 0.440548i
\(39\) −328.971 + 189.931i −0.216286 + 0.124873i
\(40\) −422.899 + 1680.34i −0.264312 + 1.05021i
\(41\) 1403.97 0.835201 0.417601 − 0.908631i \(-0.362871\pi\)
0.417601 + 0.908631i \(0.362871\pi\)
\(42\) 1196.93 + 240.731i 0.678533 + 0.136469i
\(43\) 2193.23 − 2193.23i 1.18617 − 1.18617i 0.208055 − 0.978117i \(-0.433287\pi\)
0.978117 − 0.208055i \(-0.0667133\pi\)
\(44\) 277.231 + 160.060i 0.143198 + 0.0826754i
\(45\) −726.272 + 11.0731i −0.358653 + 0.00546819i
\(46\) 737.962 + 1278.19i 0.348753 + 0.604059i
\(47\) 793.044 − 212.496i 0.359006 − 0.0961954i −0.0748076 − 0.997198i \(-0.523834\pi\)
0.433814 + 0.901003i \(0.357168\pi\)
\(48\) −891.002 + 891.002i −0.386719 + 0.386719i
\(49\) 2214.31 + 928.246i 0.922244 + 0.386608i
\(50\) 495.338 − 2103.13i 0.198135 − 0.841252i
\(51\) −771.062 + 1335.52i −0.296448 + 0.513463i
\(52\) 55.2273 − 206.111i 0.0204243 − 0.0762245i
\(53\) −1859.67 − 498.296i −0.662038 − 0.177393i −0.0878725 − 0.996132i \(-0.528007\pi\)
−0.574166 + 0.818739i \(0.694673\pi\)
\(54\) −2374.77 − 1371.07i −0.814393 − 0.470190i
\(55\) −1726.76 − 962.150i −0.570830 − 0.318066i
\(56\) −2827.82 + 1880.80i −0.901728 + 0.599746i
\(57\) −3623.39 − 3623.39i −1.11523 − 1.11523i
\(58\) 212.533 + 793.184i 0.0631787 + 0.235786i
\(59\) −3064.87 + 1769.51i −0.880458 + 0.508333i −0.870809 − 0.491621i \(-0.836405\pi\)
−0.00964857 + 0.999953i \(0.503071\pi\)
\(60\) −507.905 + 523.633i −0.141085 + 0.145454i
\(61\) −433.722 + 751.229i −0.116561 + 0.201889i −0.918403 − 0.395647i \(-0.870520\pi\)
0.801842 + 0.597536i \(0.203854\pi\)
\(62\) 2775.17 + 2775.17i 0.721948 + 0.721948i
\(63\) −1068.37 − 940.957i −0.269177 − 0.237077i
\(64\) − 4541.56i − 1.10878i
\(65\) −321.584 + 1277.78i −0.0761146 + 0.302433i
\(66\) −985.055 − 1706.16i −0.226137 − 0.391682i
\(67\) −1775.22 + 6625.21i −0.395460 + 1.47588i 0.425535 + 0.904942i \(0.360086\pi\)
−0.820995 + 0.570935i \(0.806581\pi\)
\(68\) −224.206 − 836.746i −0.0484873 − 0.180957i
\(69\) 3077.02i 0.646296i
\(70\) 3490.04 − 2398.79i 0.712253 − 0.489549i
\(71\) −486.669 −0.0965422 −0.0482711 − 0.998834i \(-0.515371\pi\)
−0.0482711 + 0.998834i \(0.515371\pi\)
\(72\) 1945.12 − 521.194i 0.375217 − 0.100539i
\(73\) 1821.45 + 488.055i 0.341799 + 0.0915848i 0.425635 − 0.904895i \(-0.360051\pi\)
−0.0838364 + 0.996480i \(0.526717\pi\)
\(74\) 3461.88 − 1998.72i 0.632191 − 0.364995i
\(75\) 3086.64 − 3280.85i 0.548736 − 0.583262i
\(76\) 2878.46 0.498349
\(77\) −1237.58 − 3671.41i −0.208733 − 0.619229i
\(78\) −928.583 + 928.583i −0.152627 + 0.152627i
\(79\) 5538.38 + 3197.58i 0.887418 + 0.512351i 0.873097 − 0.487546i \(-0.162108\pi\)
0.0143211 + 0.999897i \(0.495441\pi\)
\(80\) 66.6311 + 4370.27i 0.0104111 + 0.682854i
\(81\) −1681.73 − 2912.84i −0.256322 − 0.443962i
\(82\) 4688.26 − 1256.22i 0.697243 − 0.186826i
\(83\) −325.522 + 325.522i −0.0472525 + 0.0472525i −0.730338 − 0.683086i \(-0.760638\pi\)
0.683086 + 0.730338i \(0.260638\pi\)
\(84\) −1426.94 + 90.4804i −0.202231 + 0.0128232i
\(85\) 1463.07 + 5145.17i 0.202501 + 0.712134i
\(86\) 5361.41 − 9286.23i 0.724906 − 1.25557i
\(87\) −443.090 + 1653.64i −0.0585401 + 0.218475i
\(88\) 5293.51 + 1418.39i 0.683562 + 0.183160i
\(89\) −3046.87 − 1759.11i −0.384657 − 0.222082i 0.295186 − 0.955440i \(-0.404619\pi\)
−0.679842 + 0.733358i \(0.737952\pi\)
\(90\) −2415.32 + 686.814i −0.298188 + 0.0847918i
\(91\) −2150.35 + 1430.22i −0.259673 + 0.172711i
\(92\) −1222.21 − 1222.21i −0.144401 − 0.144401i
\(93\) 2117.71 + 7903.40i 0.244850 + 0.913793i
\(94\) 2458.07 − 1419.17i 0.278188 − 0.160612i
\(95\) −17772.3 + 270.965i −1.96923 + 0.0300239i
\(96\) 1818.23 − 3149.26i 0.197290 − 0.341717i
\(97\) −4804.80 − 4804.80i −0.510660 − 0.510660i 0.404068 − 0.914729i \(-0.367596\pi\)
−0.914729 + 0.404068i \(0.867596\pi\)
\(98\) 8224.75 + 1118.41i 0.856388 + 0.116452i
\(99\) 2297.29i 0.234393i
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.l.a.2.11 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.4 yes 56
7.4 even 3 inner 35.5.l.a.32.4 yes 56
35.18 odd 12 inner 35.5.l.a.18.11 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.11 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.11 yes 56 35.18 odd 12 inner
35.5.l.a.23.4 yes 56 5.3 odd 4 inner
35.5.l.a.32.4 yes 56 7.4 even 3 inner