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Properties

Label 35.5.l.a.32.4
Level $35$
Weight $5$
Character 35.32
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 32.4
Character \(\chi\) \(=\) 35.32
Dual form 35.5.l.a.23.4

$q$-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+(-0.894758 + 3.33928i) q^{2} +(-1.86540 - 6.96176i) q^{3} +(3.50620 + 2.02430i) q^{4} +(0.381117 + 24.9971i) q^{5} +24.9163 q^{6} +(9.66157 + 48.0380i) q^{7} +(-49.0093 + 49.0093i) q^{8} +(25.1617 - 14.5271i) q^{9} +(-83.8133 - 21.0937i) q^{10} +(-39.5345 + 68.4757i) q^{11} +(7.55226 - 28.1854i) 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} +(206.675 - 55.3784i) q^{17} +(25.9965 + 97.0203i) 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} +(-412.381 - 110.497i) q^{23} +(432.613 + 249.769i) q^{24} +(-624.709 + 19.0537i) q^{25} +(-91.1026 - 157.794i) q^{26} +(-560.876 - 560.876i) q^{27} +(-63.3682 + 187.989i) q^{28} +237.531i q^{29} +(9.49605 + 622.836i) q^{30} +(567.630 - 983.163i) q^{31} +(-487.357 + 130.587i) q^{32} +(550.459 + 147.495i) q^{33} +739.696i q^{34} +(-1197.13 + 259.819i) q^{35} +117.629 q^{36} +(-299.273 + 1116.90i) q^{37} +(636.151 + 2374.15i) q^{38} +(328.971 + 189.931i) q^{39} +(-1243.77 - 1206.41i) q^{40} +1403.97 q^{41} +(240.731 + 1196.93i) q^{42} +(2193.23 - 2193.23i) q^{43} +(-277.231 + 160.060i) q^{44} +(372.726 + 623.433i) q^{45} +(737.962 - 1278.19i) q^{46} +(-212.496 + 793.044i) 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} +(-206.111 + 55.2273i) q^{52} +(498.296 + 1859.67i) 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} +(-793.184 - 212.533i) q^{58} +(3064.87 + 1769.51i) q^{59} +(707.432 + 178.043i) q^{60} +(-433.722 - 751.229i) q^{61} +(2775.17 + 2775.17i) q^{62} +(940.957 + 1068.37i) q^{63} -4541.56i q^{64} +(-945.796 - 917.389i) q^{65} +(-985.055 + 1706.16i) q^{66} +(6625.21 - 1775.22i) q^{67} +(836.746 + 224.206i) q^{68} +3077.02i q^{69} +(203.531 - 4230.03i) q^{70} -486.669 q^{71} +(-521.194 + 1945.12i) q^{72} +(-488.055 - 1821.45i) q^{73} +(-3461.88 - 1998.72i) q^{74} +(1297.98 + 4313.53i) q^{75} +2878.46 q^{76} +(-3671.41 - 1237.58i) q^{77} +(-928.583 + 928.583i) q^{78} +(-5538.38 + 3197.58i) q^{79} +(3751.45 - 2242.84i) q^{80} +(-1681.73 + 2912.84i) q^{81} +(-1256.22 + 4688.26i) 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} +(1653.64 - 443.090i) q^{87} +(-1418.39 - 5293.51i) 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} +(-7903.40 - 2117.71i) q^{93} +(-2458.07 - 1419.17i) q^{94} +(9120.83 + 15255.8i) q^{95} +(1818.23 + 3149.26i) q^{96} +(-4804.80 - 4804.80i) q^{97} +(-1118.41 - 8224.75i) 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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.894758 + 3.33928i −0.223689 + 0.834820i 0.759236 + 0.650816i \(0.225573\pi\)
−0.982925 + 0.184005i \(0.941094\pi\)
\(3\) −1.86540 − 6.96176i −0.207266 − 0.773528i −0.988747 − 0.149599i \(-0.952202\pi\)
0.781480 − 0.623930i \(-0.214465\pi\)
\(4\) 3.50620 + 2.02430i 0.219137 + 0.126519i
\(5\) 0.381117 + 24.9971i 0.0152447 + 0.999884i
\(6\) 24.9163 0.692121
\(7\) 9.66157 + 48.0380i 0.197175 + 0.980368i
\(8\) −49.0093 + 49.0093i −0.765770 + 0.765770i
\(9\) 25.1617 − 14.5271i 0.310639 − 0.179347i
\(10\) −83.8133 − 21.0937i −0.838133 − 0.210937i
\(11\) −39.5345 + 68.4757i −0.326731 + 0.565915i −0.981861 − 0.189601i \(-0.939281\pi\)
0.655130 + 0.755516i \(0.272614\pi\)
\(12\) 7.55226 − 28.1854i 0.0524463 − 0.195732i
\(13\) −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
\(17\) 206.675 − 55.3784i 0.715139 − 0.191621i 0.117137 − 0.993116i \(-0.462628\pi\)
0.598002 + 0.801495i \(0.295962\pi\)
\(18\) 25.9965 + 97.0203i 0.0802362 + 0.299445i
\(19\) 615.723 − 355.488i 1.70561 − 0.984732i 0.765763 − 0.643123i \(-0.222362\pi\)
0.939842 − 0.341609i \(-0.110972\pi\)
\(20\) −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
\(23\) −412.381 − 110.497i −0.779548 − 0.208879i −0.152963 − 0.988232i \(-0.548881\pi\)
−0.626585 + 0.779353i \(0.715548\pi\)
\(24\) 432.613 + 249.769i 0.751063 + 0.433627i
\(25\) −624.709 + 19.0537i −0.999535 + 0.0304858i
\(26\) −91.1026 − 157.794i −0.134767 − 0.233424i
\(27\) −560.876 − 560.876i −0.769377 − 0.769377i
\(28\) −63.3682 + 187.989i −0.0808268 + 0.239782i
\(29\) 237.531i 0.282439i 0.989978 + 0.141220i \(0.0451024\pi\)
−0.989978 + 0.141220i \(0.954898\pi\)
\(30\) 9.49605 + 622.836i 0.0105512 + 0.692040i
\(31\) 567.630 − 983.163i 0.590666 − 1.02306i −0.403477 − 0.914990i \(-0.632199\pi\)
0.994143 − 0.108073i \(-0.0344681\pi\)
\(32\) −487.357 + 130.587i −0.475935 + 0.127526i
\(33\) 550.459 + 147.495i 0.505472 + 0.135441i
\(34\) 739.696i 0.639876i
\(35\) −1197.13 + 259.819i −0.977249 + 0.212097i
\(36\) 117.629 0.0907634
\(37\) −299.273 + 1116.90i −0.218607 + 0.815853i 0.766258 + 0.642533i \(0.222116\pi\)
−0.984865 + 0.173321i \(0.944550\pi\)
\(38\) 636.151 + 2374.15i 0.440548 + 1.64415i
\(39\) 328.971 + 189.931i 0.216286 + 0.124873i
\(40\) −1243.77 − 1206.41i −0.777355 − 0.754007i
\(41\) 1403.97 0.835201 0.417601 − 0.908631i \(-0.362871\pi\)
0.417601 + 0.908631i \(0.362871\pi\)
\(42\) 240.731 + 1196.93i 0.136469 + 0.678533i
\(43\) 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
\(45\) 372.726 + 623.433i 0.184062 + 0.307868i
\(46\) 737.962 − 1278.19i 0.348753 − 0.604059i
\(47\) −212.496 + 793.044i −0.0961954 + 0.359006i −0.997198 − 0.0748076i \(-0.976166\pi\)
0.901003 + 0.433814i \(0.142832\pi\)
\(48\) −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
\(52\) −206.111 + 55.2273i −0.0762245 + 0.0204243i
\(53\) 498.296 + 1859.67i 0.177393 + 0.662038i 0.996132 + 0.0878725i \(0.0280068\pi\)
−0.818739 + 0.574166i \(0.805327\pi\)
\(54\) 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
\(58\) −793.184 − 212.533i −0.235786 − 0.0631787i
\(59\) 3064.87 + 1769.51i 0.880458 + 0.508333i 0.870809 − 0.491621i \(-0.163595\pi\)
0.00964857 + 0.999953i \(0.496929\pi\)
\(60\) 707.432 + 178.043i 0.196509 + 0.0494563i
\(61\) −433.722 − 751.229i −0.116561 − 0.201889i 0.801842 − 0.597536i \(-0.203854\pi\)
−0.918403 + 0.395647i \(0.870520\pi\)
\(62\) 2775.17 + 2775.17i 0.721948 + 0.721948i
\(63\) 940.957 + 1068.37i 0.237077 + 0.269177i
\(64\) − 4541.56i − 1.10878i
\(65\) −945.796 − 917.389i −0.223857 − 0.217134i
\(66\) −985.055 + 1706.16i −0.226137 + 0.391682i
\(67\) 6625.21 − 1775.22i 1.47588 − 0.395460i 0.570935 − 0.820995i \(-0.306581\pi\)
0.904942 + 0.425535i \(0.139914\pi\)
\(68\) 836.746 + 224.206i 0.180957 + 0.0484873i
\(69\) 3077.02i 0.646296i
\(70\) 203.531 − 4230.03i 0.0415369 − 0.863271i
\(71\) −486.669 −0.0965422 −0.0482711 − 0.998834i \(-0.515371\pi\)
−0.0482711 + 0.998834i \(0.515371\pi\)
\(72\) −521.194 + 1945.12i −0.100539 + 0.375217i
\(73\) −488.055 − 1821.45i −0.0915848 − 0.341799i 0.904895 − 0.425635i \(-0.139949\pi\)
−0.996480 + 0.0838364i \(0.973283\pi\)
\(74\) −3461.88 − 1998.72i −0.632191 − 0.364995i
\(75\) 1297.98 + 4313.53i 0.230752 + 0.766850i
\(76\) 2878.46 0.498349
\(77\) −3671.41 − 1237.58i −0.619229 − 0.208733i
\(78\) −928.583 + 928.583i −0.152627 + 0.152627i
\(79\) −5538.38 + 3197.58i −0.887418 + 0.512351i −0.873097 − 0.487546i \(-0.837892\pi\)
−0.0143211 + 0.999897i \(0.504559\pi\)
\(80\) 3751.45 − 2242.84i 0.586164 − 0.350443i
\(81\) −1681.73 + 2912.84i −0.256322 + 0.443962i
\(82\) −1256.22 + 4688.26i −0.186826 + 0.697243i
\(83\) −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
\(87\) 1653.64 − 443.090i 0.218475 − 0.0585401i
\(88\) −1418.39 − 5293.51i −0.183160 − 0.683562i
\(89\) 3046.87 − 1759.11i 0.384657 − 0.222082i −0.295186 − 0.955440i \(-0.595381\pi\)
0.679842 + 0.733358i \(0.262048\pi\)
\(90\) −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
\(93\) −7903.40 − 2117.71i −0.913793 − 0.244850i
\(94\) −2458.07 − 1419.17i −0.278188 − 0.160612i
\(95\) 9120.83 + 15255.8i 1.01062 + 1.69040i
\(96\) 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 \(0.867596\pi\)
\(98\) −1118.41 − 8224.75i −0.116452 − 0.856388i
\(99\) 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.32.4 yes 56
5.3 odd 4 inner 35.5.l.a.18.11 yes 56
7.2 even 3 inner 35.5.l.a.2.11 ✓ 56
35.23 odd 12 inner 35.5.l.a.23.4 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.11 ✓ 56 7.2 even 3 inner
35.5.l.a.18.11 yes 56 5.3 odd 4 inner
35.5.l.a.23.4 yes 56 35.23 odd 12 inner
35.5.l.a.32.4 yes 56 1.1 even 1 trivial