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Properties

Label 49.7.h.a.3.20
Level $49$
Weight $7$
Character 49.3
Analytic conductor $11.273$
Analytic rank $0$
Dimension $324$
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] = [49,7,Mod(3,49)] 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("49.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.h (of order \(42\), degree \(12\), 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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(27\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 3.20
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.20

$q$-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+(2.90184 - 7.39377i) q^{2} +(22.3918 + 1.67803i) q^{3} +(0.668154 + 0.619956i) q^{4} +(-33.9718 + 49.8275i) q^{5} +(77.3843 - 160.690i) q^{6} +(-28.4067 - 341.822i) q^{7} +(464.522 - 223.702i) q^{8} +(-222.282 - 33.5037i) q^{9} +(269.833 + 395.772i) q^{10} +(2577.53 - 388.500i) q^{11} +(13.9208 + 15.0031i) q^{12} +(1567.53 - 1250.06i) q^{13} +(-2609.78 - 781.880i) q^{14} +(-844.302 + 1058.72i) q^{15} +(-301.673 - 4025.55i) q^{16} +(-375.773 + 1218.23i) q^{17} +(-892.746 + 1546.28i) q^{18} +(-1111.92 + 641.969i) q^{19} +(-53.5893 + 12.2314i) q^{20} +(-62.4890 - 7701.66i) q^{21} +(4607.10 - 20185.0i) q^{22} +(-5120.00 + 1579.31i) q^{23} +(10776.9 - 4229.60i) q^{24} +(4379.76 + 11159.4i) q^{25} +(-4693.95 - 15217.4i) q^{26} +(-20880.0 - 4765.73i) q^{27} +(192.934 - 246.000i) q^{28} +(1857.80 + 8139.55i) q^{29} +(5377.91 + 9314.81i) q^{30} +(-32374.1 - 18691.2i) q^{31} +(891.810 + 275.087i) q^{32} +(58367.3 - 4374.03i) q^{33} +(7916.86 + 6313.49i) q^{34} +(17997.2 + 10196.9i) q^{35} +(-127.748 - 160.191i) q^{36} +(-59496.9 + 55205.1i) q^{37} +(1519.95 + 10084.2i) q^{38} +(37197.3 - 25360.7i) q^{39} +(-4634.15 + 30745.6i) q^{40} +(48421.1 + 100547. i) q^{41} +(-57125.6 - 21887.0i) q^{42} +(94622.5 + 45567.8i) q^{43} +(1963.04 + 1338.38i) q^{44} +(9220.74 - 9937.60i) q^{45} +(-3180.37 + 42439.1i) q^{46} +(-135041. - 52999.7i) q^{47} -90645.3i q^{48} +(-116035. + 19420.1i) q^{49} +95219.7 q^{50} +(-10458.5 + 26647.7i) q^{51} +(1822.33 + 136.565i) q^{52} +(13250.4 + 12294.6i) q^{53} +(-95827.2 + 140553. i) q^{54} +(-68205.4 + 141630. i) q^{55} +(-89661.8 - 152429. i) q^{56} +(-25975.2 + 12509.0i) q^{57} +(65573.0 + 9883.54i) q^{58} +(-119241. - 174895. i) q^{59} +(-1220.48 + 183.958i) q^{60} +(185091. + 199481. i) q^{61} +(-232143. + 185128. i) q^{62} +(-5137.97 + 76932.6i) q^{63} +(165705. - 207788. i) q^{64} +(9035.69 + 120573. i) q^{65} +(137032. - 444247. i) q^{66} +(190588. - 330108. i) q^{67} +(-1006.32 + 581.000i) q^{68} +(-117296. + 26772.1i) q^{69} +(127618. - 103477. i) q^{70} +(8510.00 - 37284.7i) q^{71} +(-110750. + 34161.8i) q^{72} +(76172.7 - 29895.6i) q^{73} +(235523. + 600103. i) q^{74} +(79344.6 + 257229. i) q^{75} +(-1140.93 - 260.409i) q^{76} +(-206017. - 870019. i) q^{77} +(-79570.6 - 348621. i) q^{78} +(160296. + 277642. i) q^{79} +(210831. + 121724. i) q^{80} +(-302950. - 93447.7i) q^{81} +(883934. - 66241.7i) q^{82} +(-575581. - 459010. i) q^{83} +(4732.94 - 5184.63i) q^{84} +(-47935.6 - 60109.3i) q^{85} +(611497. - 567387. i) q^{86} +(27941.0 + 185376. i) q^{87} +(1.11041e6 - 757065. i) q^{88} +(-37428.5 + 248322. i) q^{89} +(-46719.2 - 97013.4i) q^{90} +(-471826. - 500305. i) q^{91} +(-4400.06 - 2118.96i) q^{92} +(-693549. - 472853. i) q^{93} +(-783736. + 844666. i) q^{94} +(5786.33 - 77213.2i) q^{95} +(19507.6 + 7656.17i) q^{96} +1.40243e6i q^{97} +(-193128. + 914291. i) q^{98} -585955. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 324 q - 13 q^{2} - 11 q^{3} + 819 q^{4} - 179 q^{5} + 770 q^{6} + 392 q^{7} + 828 q^{8} - 1160 q^{9} - 2594 q^{10} - 5305 q^{11} + 7497 q^{12} - 14 q^{13} - 11403 q^{14} - 6196 q^{15} + 27903 q^{16} - 5107 q^{17}+ \cdots - 4449616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.90184 − 7.39377i 0.362730 − 0.924221i −0.626706 − 0.779255i \(-0.715597\pi\)
0.989437 − 0.144966i \(-0.0463073\pi\)
\(3\) 22.3918 + 1.67803i 0.829325 + 0.0621493i 0.482625 − 0.875827i \(-0.339683\pi\)
0.346699 + 0.937976i \(0.387302\pi\)
\(4\) 0.668154 + 0.619956i 0.0104399 + 0.00968682i
\(5\) −33.9718 + 49.8275i −0.271775 + 0.398620i −0.937726 − 0.347375i \(-0.887073\pi\)
0.665952 + 0.745995i \(0.268026\pi\)
\(6\) 77.3843 − 160.690i 0.358261 − 0.743936i
\(7\) −28.4067 − 341.822i −0.0828184 − 0.996565i
\(8\) 464.522 − 223.702i 0.907270 − 0.436918i
\(9\) −222.282 − 33.5037i −0.304914 − 0.0459584i
\(10\) 269.833 + 395.772i 0.269833 + 0.395772i
\(11\) 2577.53 − 388.500i 1.93653 − 0.291886i 0.938030 − 0.346553i \(-0.112648\pi\)
0.998505 + 0.0546675i \(0.0174099\pi\)
\(12\) 13.9208 + 15.0031i 0.00805604 + 0.00868235i
\(13\) 1567.53 − 1250.06i 0.713485 − 0.568985i −0.198058 − 0.980190i \(-0.563463\pi\)
0.911543 + 0.411205i \(0.134892\pi\)
\(14\) −2609.78 − 781.880i −0.951087 − 0.284942i
\(15\) −844.302 + 1058.72i −0.250163 + 0.313695i
\(16\) −301.673 − 4025.55i −0.0736507 − 0.982800i
\(17\) −375.773 + 1218.23i −0.0764855 + 0.247960i −0.985739 − 0.168284i \(-0.946178\pi\)
0.909253 + 0.416244i \(0.136654\pi\)
\(18\) −892.746 + 1546.28i −0.153077 + 0.265137i
\(19\) −1111.92 + 641.969i −0.162111 + 0.0935951i −0.578861 − 0.815426i \(-0.696503\pi\)
0.416749 + 0.909021i \(0.363169\pi\)
\(20\) −53.5893 + 12.2314i −0.00669866 + 0.00152893i
\(21\) −62.4890 − 7701.66i −0.00674755 − 0.831623i
\(22\) 4607.10 − 20185.0i 0.432673 − 1.89566i
\(23\) −5120.00 + 1579.31i −0.420811 + 0.129803i −0.497927 − 0.867219i \(-0.665905\pi\)
0.0771158 + 0.997022i \(0.475429\pi\)
\(24\) 10776.9 − 4229.60i 0.779575 − 0.305961i
\(25\) 4379.76 + 11159.4i 0.280304 + 0.714204i
\(26\) −4693.95 − 15217.4i −0.267066 − 0.865806i
\(27\) −20880.0 − 4765.73i −1.06082 − 0.242124i
\(28\) 192.934 − 246.000i 0.00878892 − 0.0112063i
\(29\) 1857.80 + 8139.55i 0.0761736 + 0.333739i 0.998628 − 0.0523679i \(-0.0166769\pi\)
−0.922454 + 0.386106i \(0.873820\pi\)
\(30\) 5377.91 + 9314.81i 0.199182 + 0.344993i
\(31\) −32374.1 − 18691.2i −1.08671 − 0.627411i −0.154009 − 0.988069i \(-0.549219\pi\)
−0.932698 + 0.360659i \(0.882552\pi\)
\(32\) 891.810 + 275.087i 0.0272159 + 0.00839499i
\(33\) 58367.3 − 4374.03i 1.62416 − 0.121714i
\(34\) 7916.86 + 6313.49i 0.201426 + 0.160632i
\(35\) 17997.2 + 10196.9i 0.419759 + 0.237828i
\(36\) −127.748 − 160.191i −0.00273808 − 0.00343345i
\(37\) −59496.9 + 55205.1i −1.17460 + 1.08987i −0.180259 + 0.983619i \(0.557693\pi\)
−0.994340 + 0.106249i \(0.966116\pi\)
\(38\) 1519.95 + 10084.2i 0.0276999 + 0.183777i
\(39\) 37197.3 − 25360.7i 0.627073 − 0.427531i
\(40\) −4634.15 + 30745.6i −0.0724086 + 0.480399i
\(41\) 48421.1 + 100547.i 0.702559 + 1.45888i 0.880111 + 0.474767i \(0.157468\pi\)
−0.177553 + 0.984111i \(0.556818\pi\)
\(42\) −57125.6 − 21887.0i −0.771051 − 0.295418i
\(43\) 94622.5 + 45567.8i 1.19012 + 0.573130i 0.920843 − 0.389934i \(-0.127502\pi\)
0.269273 + 0.963064i \(0.413217\pi\)
\(44\) 1963.04 + 1338.38i 0.0230447 + 0.0157116i
\(45\) 9220.74 − 9937.60i 0.101188 − 0.109055i
\(46\) −3180.37 + 42439.1i −0.0326741 + 0.436006i
\(47\) −135041. − 52999.7i −1.30069 − 0.510482i −0.388950 − 0.921259i \(-0.627162\pi\)
−0.911736 + 0.410777i \(0.865257\pi\)
\(48\) − 90645.3i − 0.819637i
\(49\) −116035. + 19420.1i −0.986282 + 0.165068i
\(50\) 95219.7 0.761757
\(51\) −10458.5 + 26647.7i −0.0788419 + 0.200886i
\(52\) 1822.33 + 136.565i 0.0129604 + 0.000971246i
\(53\) 13250.4 + 12294.6i 0.0890024 + 0.0825821i 0.723419 − 0.690409i \(-0.242569\pi\)
−0.634417 + 0.772991i \(0.718760\pi\)
\(54\) −95827.2 + 140553.i −0.608566 + 0.892602i
\(55\) −68205.4 + 141630.i −0.409950 + 0.851269i
\(56\) −89661.8 − 152429.i −0.510556 − 0.867968i
\(57\) −25975.2 + 12509.0i −0.140260 + 0.0675456i
\(58\) 65573.0 + 9883.54i 0.336079 + 0.0506557i
\(59\) −119241. − 174895.i −0.580591 − 0.851571i 0.417784 − 0.908546i \(-0.362807\pi\)
−0.998376 + 0.0569750i \(0.981854\pi\)
\(60\) −1220.48 + 183.958i −0.00565039 + 0.000851659i
\(61\) 185091. + 199481.i 0.815449 + 0.878845i 0.994472 − 0.105000i \(-0.0334841\pi\)
−0.179023 + 0.983845i \(0.557294\pi\)
\(62\) −232143. + 185128.i −0.974048 + 0.776777i
\(63\) −5137.97 + 76932.6i −0.0205480 + 0.307673i
\(64\) 165705. − 207788.i 0.632115 − 0.792647i
\(65\) 9035.69 + 120573.i 0.0329019 + 0.439046i
\(66\) 137032. − 444247.i 0.476640 − 1.54523i
\(67\) 190588. − 330108.i 0.633682 − 1.09757i −0.353111 − 0.935581i \(-0.614876\pi\)
0.986793 − 0.161987i \(-0.0517903\pi\)
\(68\) −1006.32 + 581.000i −0.00320044 + 0.00184778i
\(69\) −117296. + 26772.1i −0.357056 + 0.0814957i
\(70\) 127618. − 103477.i 0.372065 − 0.301683i
\(71\) 8510.00 − 37284.7i 0.0237769 − 0.104173i −0.961647 − 0.274290i \(-0.911557\pi\)
0.985424 + 0.170117i \(0.0544144\pi\)
\(72\) −110750. + 34161.8i −0.296719 + 0.0915258i
\(73\) 76172.7 − 29895.6i 0.195808 − 0.0768490i −0.265411 − 0.964135i \(-0.585508\pi\)
0.461219 + 0.887286i \(0.347412\pi\)
\(74\) 235523. + 600103.i 0.581217 + 1.48092i
\(75\) 79344.6 + 257229.i 0.188076 + 0.609728i
\(76\) −1140.93 − 260.409i −0.00259907 − 0.000593220i
\(77\) −206017. − 870019.i −0.451264 − 1.90571i
\(78\) −79570.6 − 348621.i −0.167675 − 0.734633i
\(79\) 160296. + 277642.i 0.325119 + 0.563123i 0.981536 − 0.191275i \(-0.0612622\pi\)
−0.656417 + 0.754398i \(0.727929\pi\)
\(80\) 210831. + 121724.i 0.411780 + 0.237741i
\(81\) −302950. − 93447.7i −0.570054 − 0.175838i
\(82\) 883934. − 66241.7i 1.60317 − 0.120141i
\(83\) −575581. − 459010.i −1.00663 − 0.802765i −0.0262097 − 0.999656i \(-0.508344\pi\)
−0.980425 + 0.196892i \(0.936915\pi\)
\(84\) 4732.94 − 5184.63i 0.00798533 − 0.00874742i
\(85\) −47935.6 − 60109.3i −0.0780551 − 0.0978780i
\(86\) 611497. − 567387.i 0.961389 − 0.892039i
\(87\) 27941.0 + 185376.i 0.0424311 + 0.281512i
\(88\) 1.11041e6 − 757065.i 1.62943 − 1.11093i
\(89\) −37428.5 + 248322.i −0.0530924 + 0.352245i 0.946473 + 0.322782i \(0.104618\pi\)
−0.999566 + 0.0294634i \(0.990620\pi\)
\(90\) −46719.2 − 97013.4i −0.0640867 − 0.133077i
\(91\) −471826. − 500305.i −0.626121 − 0.663912i
\(92\) −4400.06 − 2118.96i −0.00565060 − 0.00272119i
\(93\) −693549. − 472853.i −0.862240 − 0.587865i
\(94\) −783736. + 844666.i −0.943596 + 1.01695i
\(95\) 5786.33 − 77213.2i 0.00674889 − 0.0900577i
\(96\) 19507.6 + 7656.17i 0.0220491 + 0.00865362i
\(97\) 1.40243e6i 1.53662i 0.640077 + 0.768311i \(0.278903\pi\)
−0.640077 + 0.768311i \(0.721097\pi\)
\(98\) −193128. + 914291.i −0.205195 + 0.971418i
\(99\) −585955. −0.603891
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 49.7.h.a.3.20 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.20 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.20 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.20 yes 324 49.33 odd 42 inner