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Properties

Label 245.4.e.n.116.1
Level $245$
Weight $4$
Character 245.116
Analytic conductor $14.455$
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] = [245,4,Mod(116,245)] 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("245.116"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,3,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.5567659200.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 17x^{4} - 28x^{3} + 289x^{2} - 238x + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 116.1
Root \(-2.24283 + 3.88469i\) of defining polynomial
Character \(\chi\) \(=\) 245.116
Dual form 245.4.e.n.226.1

$q$-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+(-1.74283 - 3.01866i) q^{2} +(0.425119 - 0.736328i) q^{3} +(-2.07488 + 3.59380i) q^{4} +(2.50000 + 4.33013i) q^{5} -2.96363 q^{6} -13.4206 q^{8} +(13.1385 + 22.7566i) q^{9} +(8.71413 - 15.0933i) q^{10} +(3.45382 - 5.98219i) q^{11} +(1.76414 + 3.05559i) q^{12} +22.1364 q^{13} +4.25119 q^{15} +(39.9888 + 69.2626i) q^{16} +(44.1515 - 76.4726i) q^{17} +(45.7964 - 79.3217i) q^{18} +(18.4780 + 32.0048i) q^{19} -20.7488 q^{20} -24.0776 q^{22} +(47.7641 + 82.7299i) q^{23} +(-5.70535 + 9.88195i) q^{24} +(-12.5000 + 21.6506i) q^{25} +(-38.5799 - 66.8223i) q^{26} +45.2982 q^{27} +269.029 q^{29} +(-7.40909 - 12.8329i) q^{30} +(98.5570 - 170.706i) q^{31} +(85.7046 - 148.445i) q^{32} +(-2.93657 - 5.08629i) q^{33} -307.793 q^{34} -109.044 q^{36} +(-1.07273 - 1.85803i) q^{37} +(64.4078 - 111.558i) q^{38} +(9.41061 - 16.2997i) q^{39} +(-33.5515 - 58.1128i) q^{40} -174.127 q^{41} -17.0345 q^{43} +(14.3325 + 24.8247i) q^{44} +(-65.6927 + 113.783i) q^{45} +(166.489 - 288.368i) q^{46} +(-264.014 - 457.286i) q^{47} +68.0000 q^{48} +87.1413 q^{50} +(-37.5393 - 65.0200i) q^{51} +(-45.9304 + 79.5538i) q^{52} +(320.557 - 555.221i) q^{53} +(-78.9469 - 136.740i) q^{54} +34.5382 q^{55} +31.4214 q^{57} +(-468.871 - 812.109i) q^{58} +(-321.487 + 556.832i) q^{59} +(-8.82072 + 15.2779i) q^{60} +(71.4836 + 123.813i) q^{61} -687.070 q^{62} +42.3480 q^{64} +(55.3410 + 95.8534i) q^{65} +(-10.2359 + 17.7290i) q^{66} +(-239.398 + 414.650i) q^{67} +(183.218 + 317.343i) q^{68} +81.2218 q^{69} +105.550 q^{71} +(-176.327 - 305.407i) q^{72} +(493.256 - 854.344i) q^{73} +(-3.73917 + 6.47643i) q^{74} +(10.6280 + 18.4082i) q^{75} -153.358 q^{76} -65.6042 q^{78} +(549.930 + 952.507i) q^{79} +(-199.944 + 346.313i) q^{80} +(-335.484 + 581.075i) q^{81} +(303.474 + 525.632i) q^{82} +1236.62 q^{83} +441.515 q^{85} +(29.6882 + 51.4214i) q^{86} +(114.370 - 198.094i) q^{87} +(-46.3523 + 80.2845i) q^{88} +(-355.849 - 616.348i) q^{89} +457.964 q^{90} -396.420 q^{92} +(-83.7969 - 145.141i) q^{93} +(-920.262 + 1593.94i) q^{94} +(-92.3899 + 160.024i) q^{95} +(-72.8693 - 126.213i) q^{96} +636.553 q^{97} +181.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 2 q^{3} - 13 q^{4} + 15 q^{5} - 48 q^{6} - 30 q^{8} - 81 q^{9} - 15 q^{10} + 74 q^{11} - 152 q^{12} - 88 q^{13} + 20 q^{15} + 79 q^{16} - 52 q^{17} + 411 q^{18} + 168 q^{19} - 130 q^{20}+ \cdots - 6976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.74283 − 3.01866i −0.616182 − 1.06726i −0.990176 − 0.139827i \(-0.955345\pi\)
0.373994 − 0.927431i \(-0.377988\pi\)
\(3\) 0.425119 − 0.736328i 0.0818142 − 0.141706i −0.822215 − 0.569177i \(-0.807262\pi\)
0.904029 + 0.427471i \(0.140595\pi\)
\(4\) −2.07488 + 3.59380i −0.259360 + 0.449225i
\(5\) 2.50000 + 4.33013i 0.223607 + 0.387298i
\(6\) −2.96363 −0.201650
\(7\) 0 0
\(8\) −13.4206 −0.593112
\(9\) 13.1385 + 22.7566i 0.486613 + 0.842838i
\(10\) 8.71413 − 15.0933i 0.275565 − 0.477292i
\(11\) 3.45382 − 5.98219i 0.0946696 − 0.163972i −0.814801 − 0.579741i \(-0.803154\pi\)
0.909471 + 0.415768i \(0.136487\pi\)
\(12\) 1.76414 + 3.05559i 0.0424387 + 0.0735060i
\(13\) 22.1364 0.472272 0.236136 − 0.971720i \(-0.424119\pi\)
0.236136 + 0.971720i \(0.424119\pi\)
\(14\) 0 0
\(15\) 4.25119 0.0731769
\(16\) 39.9888 + 69.2626i 0.624825 + 1.08223i
\(17\) 44.1515 − 76.4726i 0.629901 − 1.09102i −0.357671 − 0.933848i \(-0.616429\pi\)
0.987571 − 0.157172i \(-0.0502377\pi\)
\(18\) 45.7964 − 79.3217i 0.599684 − 1.03868i
\(19\) 18.4780 + 32.0048i 0.223113 + 0.386442i 0.955752 − 0.294175i \(-0.0950449\pi\)
−0.732639 + 0.680618i \(0.761712\pi\)
\(20\) −20.7488 −0.231979
\(21\) 0 0
\(22\) −24.0776 −0.233335
\(23\) 47.7641 + 82.7299i 0.433022 + 0.750017i 0.997132 − 0.0756833i \(-0.0241138\pi\)
−0.564110 + 0.825700i \(0.690780\pi\)
\(24\) −5.70535 + 9.88195i −0.0485250 + 0.0840477i
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) −38.5799 − 66.8223i −0.291005 − 0.504036i
\(27\) 45.2982 0.322876
\(28\) 0 0
\(29\) 269.029 1.72267 0.861336 − 0.508035i \(-0.169628\pi\)
0.861336 + 0.508035i \(0.169628\pi\)
\(30\) −7.40909 − 12.8329i −0.0450903 − 0.0780986i
\(31\) 98.5570 − 170.706i 0.571012 − 0.989021i −0.425451 − 0.904982i \(-0.639884\pi\)
0.996462 − 0.0840395i \(-0.0267822\pi\)
\(32\) 85.7046 − 148.445i 0.473455 − 0.820049i
\(33\) −2.93657 − 5.08629i −0.0154906 − 0.0268306i
\(34\) −307.793 −1.55253
\(35\) 0 0
\(36\) −109.044 −0.504832
\(37\) −1.07273 − 1.85803i −0.00476638 − 0.00825561i 0.863632 − 0.504122i \(-0.168184\pi\)
−0.868399 + 0.495867i \(0.834851\pi\)
\(38\) 64.4078 − 111.558i 0.274956 − 0.476238i
\(39\) 9.41061 − 16.2997i 0.0386385 − 0.0669239i
\(40\) −33.5515 − 58.1128i −0.132624 − 0.229711i
\(41\) −174.127 −0.663271 −0.331636 − 0.943408i \(-0.607600\pi\)
−0.331636 + 0.943408i \(0.607600\pi\)
\(42\) 0 0
\(43\) −17.0345 −0.0604125 −0.0302062 − 0.999544i \(-0.509616\pi\)
−0.0302062 + 0.999544i \(0.509616\pi\)
\(44\) 14.3325 + 24.8247i 0.0491070 + 0.0850558i
\(45\) −65.6927 + 113.783i −0.217620 + 0.376929i
\(46\) 166.489 − 288.368i 0.533641 − 0.924293i
\(47\) −264.014 − 457.286i −0.819371 − 1.41919i −0.906146 − 0.422964i \(-0.860990\pi\)
0.0867752 − 0.996228i \(-0.472344\pi\)
\(48\) 68.0000 0.204478
\(49\) 0 0
\(50\) 87.1413 0.246473
\(51\) −37.5393 − 65.0200i −0.103070 − 0.178522i
\(52\) −45.9304 + 79.5538i −0.122488 + 0.212156i
\(53\) 320.557 − 555.221i 0.830790 − 1.43897i −0.0666227 − 0.997778i \(-0.521222\pi\)
0.897413 − 0.441192i \(-0.145444\pi\)
\(54\) −78.9469 − 136.740i −0.198950 − 0.344592i
\(55\) 34.5382 0.0846750
\(56\) 0 0
\(57\) 31.4214 0.0730151
\(58\) −468.871 − 812.109i −1.06148 − 1.83854i
\(59\) −321.487 + 556.832i −0.709391 + 1.22870i 0.255692 + 0.966758i \(0.417697\pi\)
−0.965083 + 0.261944i \(0.915637\pi\)
\(60\) −8.82072 + 15.2779i −0.0189792 + 0.0328729i
\(61\) 71.4836 + 123.813i 0.150042 + 0.259880i 0.931242 − 0.364400i \(-0.118726\pi\)
−0.781201 + 0.624280i \(0.785393\pi\)
\(62\) −687.070 −1.40739
\(63\) 0 0
\(64\) 42.3480 0.0827109
\(65\) 55.3410 + 95.8534i 0.105603 + 0.182910i
\(66\) −10.2359 + 17.7290i −0.0190901 + 0.0330650i
\(67\) −239.398 + 414.650i −0.436525 + 0.756083i −0.997419 − 0.0718045i \(-0.977124\pi\)
0.560894 + 0.827888i \(0.310458\pi\)
\(68\) 183.218 + 317.343i 0.326742 + 0.565934i
\(69\) 81.2218 0.141710
\(70\) 0 0
\(71\) 105.550 0.176430 0.0882150 − 0.996101i \(-0.471884\pi\)
0.0882150 + 0.996101i \(0.471884\pi\)
\(72\) −176.327 − 305.407i −0.288616 − 0.499897i
\(73\) 493.256 − 854.344i 0.790839 − 1.36977i −0.134609 − 0.990899i \(-0.542978\pi\)
0.925448 − 0.378875i \(-0.123689\pi\)
\(74\) −3.73917 + 6.47643i −0.00587391 + 0.0101739i
\(75\) 10.6280 + 18.4082i 0.0163628 + 0.0283413i
\(76\) −153.358 −0.231466
\(77\) 0 0
\(78\) −65.6042 −0.0952335
\(79\) 549.930 + 952.507i 0.783190 + 1.35652i 0.930074 + 0.367371i \(0.119742\pi\)
−0.146885 + 0.989154i \(0.546925\pi\)
\(80\) −199.944 + 346.313i −0.279430 + 0.483987i
\(81\) −335.484 + 581.075i −0.460197 + 0.797085i
\(82\) 303.474 + 525.632i 0.408696 + 0.707882i
\(83\) 1236.62 1.63538 0.817691 − 0.575657i \(-0.195254\pi\)
0.817691 + 0.575657i \(0.195254\pi\)
\(84\) 0 0
\(85\) 441.515 0.563400
\(86\) 29.6882 + 51.4214i 0.0372251 + 0.0644757i
\(87\) 114.370 − 198.094i 0.140939 − 0.244114i
\(88\) −46.3523 + 80.2845i −0.0561496 + 0.0972540i
\(89\) −355.849 − 616.348i −0.423819 − 0.734076i 0.572490 − 0.819912i \(-0.305977\pi\)
−0.996309 + 0.0858354i \(0.972644\pi\)
\(90\) 457.964 0.536374
\(91\) 0 0
\(92\) −396.420 −0.449235
\(93\) −83.7969 − 145.141i −0.0934337 − 0.161832i
\(94\) −920.262 + 1593.94i −1.00976 + 1.74896i
\(95\) −92.3899 + 160.024i −0.0997790 + 0.172822i
\(96\) −72.8693 − 126.213i −0.0774708 − 0.134183i
\(97\) 636.553 0.666311 0.333156 − 0.942872i \(-0.391887\pi\)
0.333156 + 0.942872i \(0.391887\pi\)
\(98\) 0 0
\(99\) 181.513 0.184270
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 245.4.e.n.116.1 6
7.2 even 3 inner 245.4.e.n.226.1 6
7.3 odd 6 35.4.a.c.1.3 ✓ 3
7.4 even 3 245.4.a.l.1.3 3
7.5 odd 6 245.4.e.m.226.1 6
7.6 odd 2 245.4.e.m.116.1 6
21.11 odd 6 2205.4.a.bm.1.1 3
21.17 even 6 315.4.a.p.1.1 3
28.3 even 6 560.4.a.u.1.2 3
35.3 even 12 175.4.b.e.99.2 6
35.4 even 6 1225.4.a.y.1.1 3
35.17 even 12 175.4.b.e.99.5 6
35.24 odd 6 175.4.a.f.1.1 3
56.3 even 6 2240.4.a.bv.1.2 3
56.45 odd 6 2240.4.a.bt.1.2 3
105.59 even 6 1575.4.a.ba.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.3 ✓ 3 7.3 odd 6
175.4.a.f.1.1 3 35.24 odd 6
175.4.b.e.99.2 6 35.3 even 12
175.4.b.e.99.5 6 35.17 even 12
245.4.a.l.1.3 3 7.4 even 3
245.4.e.m.116.1 6 7.6 odd 2
245.4.e.m.226.1 6 7.5 odd 6
245.4.e.n.116.1 6 1.1 even 1 trivial
245.4.e.n.226.1 6 7.2 even 3 inner
315.4.a.p.1.1 3 21.17 even 6
560.4.a.u.1.2 3 28.3 even 6
1225.4.a.y.1.1 3 35.4 even 6
1575.4.a.ba.1.3 3 105.59 even 6
2205.4.a.bm.1.1 3 21.11 odd 6
2240.4.a.bt.1.2 3 56.45 odd 6
2240.4.a.bv.1.2 3 56.3 even 6