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Properties

Label 245.4.b.d.99.6
Level $245$
Weight $4$
Character 245.99
Analytic conductor $14.455$
Analytic rank $0$
Dimension $10$
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(99,245)] 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("245.99"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-36,-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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.6
Root \(0.555276i\) of defining polynomial
Character \(\chi\) \(=\) 245.99
Dual form 245.4.b.d.99.5

$q$-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)
 
\(f(q)\) \(=\) \(q+1.55528i q^{2} -4.96149i q^{3} +5.58112 q^{4} +(11.0923 - 1.40060i) q^{5} +7.71648 q^{6} +21.1224i q^{8} +2.38365 q^{9} +(2.17831 + 17.2515i) q^{10} +29.8232 q^{11} -27.6906i q^{12} +90.7316i q^{13} +(-6.94904 - 55.0341i) q^{15} +11.7978 q^{16} -29.5740i q^{17} +3.70723i q^{18} -62.3278 q^{19} +(61.9072 - 7.81689i) q^{20} +46.3833i q^{22} -90.6198i q^{23} +104.798 q^{24} +(121.077 - 31.0716i) q^{25} -141.113 q^{26} -145.787i q^{27} -193.070 q^{29} +(85.5932 - 10.8077i) q^{30} +152.123 q^{31} +187.328i q^{32} -147.967i q^{33} +45.9957 q^{34} +13.3034 q^{36} +102.453i q^{37} -96.9368i q^{38} +450.164 q^{39} +(29.5839 + 234.295i) q^{40} +266.744 q^{41} -387.125i q^{43} +166.447 q^{44} +(26.4400 - 3.33852i) q^{45} +140.939 q^{46} +152.298i q^{47} -58.5346i q^{48} +(48.3248 + 188.308i) q^{50} -146.731 q^{51} +506.384i q^{52} +81.5982i q^{53} +226.738 q^{54} +(330.807 - 41.7702i) q^{55} +309.238i q^{57} -300.277i q^{58} -235.884 q^{59} +(-38.7834 - 307.152i) q^{60} -510.453 q^{61} +236.593i q^{62} -196.964 q^{64} +(127.078 + 1006.42i) q^{65} +230.130 q^{66} -347.374i q^{67} -165.056i q^{68} -449.609 q^{69} +317.014 q^{71} +50.3483i q^{72} -709.901i q^{73} -159.343 q^{74} +(-154.161 - 600.720i) q^{75} -347.858 q^{76} +700.129i q^{78} -1062.95 q^{79} +(130.864 - 16.5240i) q^{80} -658.960 q^{81} +414.861i q^{82} +503.810i q^{83} +(-41.4212 - 328.043i) q^{85} +602.086 q^{86} +957.914i q^{87} +629.937i q^{88} +482.342 q^{89} +(5.19233 + 41.1215i) q^{90} -505.760i q^{92} -754.756i q^{93} -236.866 q^{94} +(-691.356 + 87.2960i) q^{95} +929.425 q^{96} -481.167i q^{97} +71.0879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 36 q^{4} - 6 q^{5} - 12 q^{6} - 46 q^{9} + 16 q^{10} + 84 q^{11} + 8 q^{15} + 148 q^{16} - 72 q^{19} + 68 q^{20} - 72 q^{24} - 362 q^{25} + 620 q^{26} + 88 q^{29} + 52 q^{30} - 120 q^{31} - 964 q^{34}+ \cdots - 5304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.55528i 0.549873i 0.961462 + 0.274937i \(0.0886568\pi\)
−0.961462 + 0.274937i \(0.911343\pi\)
\(3\) − 4.96149i − 0.954839i −0.878676 − 0.477419i \(-0.841572\pi\)
0.878676 − 0.477419i \(-0.158428\pi\)
\(4\) 5.58112 0.697640
\(5\) 11.0923 − 1.40060i 0.992122 − 0.125273i
\(6\) 7.71648 0.525040
\(7\) 0 0
\(8\) 21.1224i 0.933486i
\(9\) 2.38365 0.0882832
\(10\) 2.17831 + 17.2515i 0.0688843 + 0.545541i
\(11\) 29.8232 0.817457 0.408728 − 0.912656i \(-0.365972\pi\)
0.408728 + 0.912656i \(0.365972\pi\)
\(12\) − 27.6906i − 0.666133i
\(13\) 90.7316i 1.93572i 0.251481 + 0.967862i \(0.419083\pi\)
−0.251481 + 0.967862i \(0.580917\pi\)
\(14\) 0 0
\(15\) −6.94904 − 55.0341i −0.119616 − 0.947317i
\(16\) 11.7978 0.184341
\(17\) − 29.5740i − 0.421927i −0.977494 − 0.210963i \(-0.932340\pi\)
0.977494 − 0.210963i \(-0.0676600\pi\)
\(18\) 3.70723i 0.0485446i
\(19\) −62.3278 −0.752577 −0.376289 − 0.926503i \(-0.622800\pi\)
−0.376289 + 0.926503i \(0.622800\pi\)
\(20\) 61.9072 − 7.81689i 0.692144 − 0.0873955i
\(21\) 0 0
\(22\) 46.3833i 0.449498i
\(23\) − 90.6198i − 0.821545i −0.911738 − 0.410772i \(-0.865259\pi\)
0.911738 − 0.410772i \(-0.134741\pi\)
\(24\) 104.798 0.891329
\(25\) 121.077 − 31.0716i 0.968613 − 0.248572i
\(26\) −141.113 −1.06440
\(27\) − 145.787i − 1.03913i
\(28\) 0 0
\(29\) −193.070 −1.23628 −0.618141 − 0.786067i \(-0.712114\pi\)
−0.618141 + 0.786067i \(0.712114\pi\)
\(30\) 85.5932 − 10.8077i 0.520904 − 0.0657734i
\(31\) 152.123 0.881357 0.440679 − 0.897665i \(-0.354738\pi\)
0.440679 + 0.897665i \(0.354738\pi\)
\(32\) 187.328i 1.03485i
\(33\) − 147.967i − 0.780539i
\(34\) 45.9957 0.232006
\(35\) 0 0
\(36\) 13.3034 0.0615899
\(37\) 102.453i 0.455222i 0.973752 + 0.227611i \(0.0730915\pi\)
−0.973752 + 0.227611i \(0.926908\pi\)
\(38\) − 96.9368i − 0.413822i
\(39\) 450.164 1.84830
\(40\) 29.5839 + 234.295i 0.116941 + 0.926133i
\(41\) 266.744 1.01606 0.508030 − 0.861340i \(-0.330374\pi\)
0.508030 + 0.861340i \(0.330374\pi\)
\(42\) 0 0
\(43\) − 387.125i − 1.37293i −0.727163 − 0.686465i \(-0.759162\pi\)
0.727163 − 0.686465i \(-0.240838\pi\)
\(44\) 166.447 0.570290
\(45\) 26.4400 − 3.33852i 0.0875877 − 0.0110595i
\(46\) 140.939 0.451745
\(47\) 152.298i 0.472660i 0.971673 + 0.236330i \(0.0759446\pi\)
−0.971673 + 0.236330i \(0.924055\pi\)
\(48\) − 58.5346i − 0.176016i
\(49\) 0 0
\(50\) 48.3248 + 188.308i 0.136683 + 0.532614i
\(51\) −146.731 −0.402872
\(52\) 506.384i 1.35044i
\(53\) 81.5982i 0.211479i 0.994394 + 0.105739i \(0.0337209\pi\)
−0.994394 + 0.105739i \(0.966279\pi\)
\(54\) 226.738 0.571392
\(55\) 330.807 − 41.7702i 0.811017 − 0.102405i
\(56\) 0 0
\(57\) 309.238i 0.718590i
\(58\) − 300.277i − 0.679798i
\(59\) −235.884 −0.520499 −0.260250 − 0.965541i \(-0.583805\pi\)
−0.260250 + 0.965541i \(0.583805\pi\)
\(60\) −38.7834 − 307.152i −0.0834486 − 0.660886i
\(61\) −510.453 −1.07142 −0.535712 − 0.844401i \(-0.679957\pi\)
−0.535712 + 0.844401i \(0.679957\pi\)
\(62\) 236.593i 0.484635i
\(63\) 0 0
\(64\) −196.964 −0.384696
\(65\) 127.078 + 1006.42i 0.242494 + 1.92048i
\(66\) 230.130 0.429198
\(67\) − 347.374i − 0.633410i −0.948524 − 0.316705i \(-0.897423\pi\)
0.948524 − 0.316705i \(-0.102577\pi\)
\(68\) − 165.056i − 0.294353i
\(69\) −449.609 −0.784443
\(70\) 0 0
\(71\) 317.014 0.529896 0.264948 − 0.964263i \(-0.414645\pi\)
0.264948 + 0.964263i \(0.414645\pi\)
\(72\) 50.3483i 0.0824112i
\(73\) − 709.901i − 1.13819i −0.822273 − 0.569093i \(-0.807294\pi\)
0.822273 − 0.569093i \(-0.192706\pi\)
\(74\) −159.343 −0.250315
\(75\) −154.161 − 600.720i −0.237347 − 0.924869i
\(76\) −347.858 −0.525028
\(77\) 0 0
\(78\) 700.129i 1.01633i
\(79\) −1062.95 −1.51382 −0.756908 − 0.653522i \(-0.773291\pi\)
−0.756908 + 0.653522i \(0.773291\pi\)
\(80\) 130.864 − 16.5240i 0.182888 − 0.0230929i
\(81\) −658.960 −0.903923
\(82\) 414.861i 0.558704i
\(83\) 503.810i 0.666270i 0.942879 + 0.333135i \(0.108106\pi\)
−0.942879 + 0.333135i \(0.891894\pi\)
\(84\) 0 0
\(85\) −41.4212 − 328.043i −0.0528560 − 0.418603i
\(86\) 602.086 0.754937
\(87\) 957.914i 1.18045i
\(88\) 629.937i 0.763085i
\(89\) 482.342 0.574474 0.287237 − 0.957860i \(-0.407263\pi\)
0.287237 + 0.957860i \(0.407263\pi\)
\(90\) 5.19233 + 41.1215i 0.00608133 + 0.0481621i
\(91\) 0 0
\(92\) − 505.760i − 0.573142i
\(93\) − 754.756i − 0.841554i
\(94\) −236.866 −0.259903
\(95\) −691.356 + 87.2960i −0.746649 + 0.0942777i
\(96\) 929.425 0.988115
\(97\) − 481.167i − 0.503661i −0.967771 − 0.251831i \(-0.918967\pi\)
0.967771 − 0.251831i \(-0.0810326\pi\)
\(98\) 0 0
\(99\) 71.0879 0.0721677
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.b.d.99.6 10
5.2 odd 4 1225.4.a.be.1.3 5
5.3 odd 4 1225.4.a.bh.1.3 5
5.4 even 2 inner 245.4.b.d.99.5 10
7.2 even 3 245.4.j.f.214.6 20
7.3 odd 6 245.4.j.e.79.5 20
7.4 even 3 245.4.j.f.79.5 20
7.5 odd 6 245.4.j.e.214.6 20
7.6 odd 2 35.4.b.a.29.6 yes 10
21.20 even 2 315.4.d.c.64.5 10
28.27 even 2 560.4.g.f.449.3 10
35.4 even 6 245.4.j.f.79.6 20
35.9 even 6 245.4.j.f.214.5 20
35.13 even 4 175.4.a.j.1.3 5
35.19 odd 6 245.4.j.e.214.5 20
35.24 odd 6 245.4.j.e.79.6 20
35.27 even 4 175.4.a.i.1.3 5
35.34 odd 2 35.4.b.a.29.5 ✓ 10
105.62 odd 4 1575.4.a.bq.1.3 5
105.83 odd 4 1575.4.a.bn.1.3 5
105.104 even 2 315.4.d.c.64.6 10
140.139 even 2 560.4.g.f.449.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.b.a.29.5 ✓ 10 35.34 odd 2
35.4.b.a.29.6 yes 10 7.6 odd 2
175.4.a.i.1.3 5 35.27 even 4
175.4.a.j.1.3 5 35.13 even 4
245.4.b.d.99.5 10 5.4 even 2 inner
245.4.b.d.99.6 10 1.1 even 1 trivial
245.4.j.e.79.5 20 7.3 odd 6
245.4.j.e.79.6 20 35.24 odd 6
245.4.j.e.214.5 20 35.19 odd 6
245.4.j.e.214.6 20 7.5 odd 6
245.4.j.f.79.5 20 7.4 even 3
245.4.j.f.79.6 20 35.4 even 6
245.4.j.f.214.5 20 35.9 even 6
245.4.j.f.214.6 20 7.2 even 3
315.4.d.c.64.5 10 21.20 even 2
315.4.d.c.64.6 10 105.104 even 2
560.4.g.f.449.3 10 28.27 even 2
560.4.g.f.449.8 10 140.139 even 2
1225.4.a.be.1.3 5 5.2 odd 4
1225.4.a.bh.1.3 5 5.3 odd 4
1575.4.a.bn.1.3 5 105.83 odd 4
1575.4.a.bq.1.3 5 105.62 odd 4