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Properties

Label 49.7.h.a.3.3
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 3.3
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.3

$q$-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+(-5.28100 + 13.4558i) q^{2} +(-41.7729 - 3.13045i) q^{3} +(-106.253 - 98.5887i) q^{4} +(0.579414 - 0.849844i) q^{5} +(262.725 - 545.555i) q^{6} +(-337.162 - 63.0124i) q^{7} +(1054.21 - 507.679i) q^{8} +(1014.32 + 152.884i) q^{9} +(8.37542 + 12.2845i) q^{10} +(-2127.07 + 320.604i) q^{11} +(4129.88 + 4450.96i) q^{12} +(45.0838 - 35.9531i) q^{13} +(2628.43 - 4204.01i) q^{14} +(-26.8642 + 33.6866i) q^{15} +(570.709 + 7615.58i) q^{16} +(948.346 - 3074.46i) q^{17} +(-7413.78 + 12841.0i) q^{18} +(-10101.1 + 5831.87i) q^{19} +(-145.350 + 33.1751i) q^{20} +(13887.0 + 3687.68i) q^{21} +(6919.08 - 30314.5i) q^{22} +(-6960.92 + 2147.16i) q^{23} +(-45626.6 + 17907.1i) q^{24} +(5708.07 + 14543.9i) q^{25} +(245.689 + 796.505i) q^{26} +(-12120.2 - 2766.36i) q^{27} +(29612.3 + 39935.7i) q^{28} +(-3707.77 - 16244.8i) q^{29} +(-311.410 - 539.377i) q^{30} +(49762.5 + 28730.4i) q^{31} +(-33929.0 - 10465.7i) q^{32} +(89857.5 - 6733.89i) q^{33} +(36361.0 + 28997.0i) q^{34} +(-248.907 + 250.025i) q^{35} +(-92702.1 - 116245. i) q^{36} +(23913.2 - 22188.2i) q^{37} +(-25128.4 - 166716. i) q^{38} +(-1995.83 + 1360.73i) q^{39} +(179.374 - 1190.07i) q^{40} +(-3630.92 - 7539.68i) q^{41} +(-122958. + 167385. i) q^{42} +(-44677.2 - 21515.4i) q^{43} +(257616. + 175640. i) q^{44} +(717.637 - 773.429i) q^{45} +(7868.94 - 105004. i) q^{46} +(-33686.2 - 13220.9i) q^{47} -319911. i q^{48} +(109708. + 42490.8i) q^{49} -225844. q^{50} +(-49239.6 + 125460. i) q^{51} +(-8334.87 - 624.612i) q^{52} +(-126863. - 117712. i) q^{53} +(101230. - 148478. i) q^{54} +(-959.990 + 1993.44i) q^{55} +(-387429. + 104742. i) q^{56} +(440209. - 211993. i) q^{57} +(238167. + 35897.9i) q^{58} +(139725. + 204939. i) q^{59} +(6175.53 - 930.811i) q^{60} +(-149336. - 160945. i) q^{61} +(-649385. + 517867. i) q^{62} +(-332356. - 115461. i) q^{63} +(15264.5 - 19141.0i) q^{64} +(-4.43237 - 59.1459i) q^{65} +(-383928. + 1.24466e6i) q^{66} +(246545. - 427028. i) q^{67} +(-403872. + 233176. i) q^{68} +(297500. - 67902.3i) q^{69} +(-2049.80 - 4669.62i) q^{70} +(96410.4 - 422402. i) q^{71} +(1.14692e6 - 353777. i) q^{72} +(-363298. + 142584. i) q^{73} +(172274. + 438946. i) q^{74} +(-192914. - 625410. i) q^{75} +(1.64823e6 + 376198. i) q^{76} +(737370. + 25936.2i) q^{77} +(-7769.74 - 34041.4i) q^{78} +(115072. + 199310. i) q^{79} +(6802.73 + 3927.56i) q^{80} +(-216931. - 66914.3i) q^{81} +(120627. - 9039.75i) q^{82} +(392949. + 313367. i) q^{83} +(-1.11198e6 - 1.76093e6i) q^{84} +(-2063.33 - 2587.33i) q^{85} +(525447. - 487543. i) q^{86} +(104031. + 690199. i) q^{87} +(-2.07961e6 + 1.41785e6i) q^{88} +(-100896. + 669400. i) q^{89} +(6617.24 + 13740.8i) q^{90} +(-17466.0 + 9281.20i) q^{91} +(951307. + 458126. i) q^{92} +(-1.98879e6 - 1.35593e6i) q^{93} +(355794. - 383455. i) q^{94} +(-896.535 + 11963.4i) q^{95} +(1.38455e6 + 543397. i) q^{96} +487119. i q^{97} +(-1.15111e6 + 1.25181e6i) q^{98} -2.20654e6 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.28100 + 13.4558i −0.660125 + 1.68197i 0.0690234 + 0.997615i \(0.478012\pi\)
−0.729148 + 0.684356i \(0.760084\pi\)
\(3\) −41.7729 − 3.13045i −1.54714 − 0.115942i −0.726414 − 0.687258i \(-0.758814\pi\)
−0.820731 + 0.571315i \(0.806433\pi\)
\(4\) −106.253 − 98.5887i −1.66021 − 1.54045i
\(5\) 0.579414 − 0.849844i 0.00463531 − 0.00679875i −0.823914 − 0.566715i \(-0.808214\pi\)
0.828549 + 0.559916i \(0.189167\pi\)
\(6\) 262.725 − 545.555i 1.21632 − 2.52572i
\(7\) −337.162 − 63.0124i −0.982981 − 0.183710i
\(8\) 1054.21 − 507.679i 2.05900 − 0.991561i
\(9\) 1014.32 + 152.884i 1.39138 + 0.209717i
\(10\) 8.37542 + 12.2845i 0.00837542 + 0.0122845i
\(11\) −2127.07 + 320.604i −1.59810 + 0.240875i −0.886847 − 0.462062i \(-0.847110\pi\)
−0.711252 + 0.702937i \(0.751871\pi\)
\(12\) 4129.88 + 4450.96i 2.38998 + 2.57579i
\(13\) 45.0838 − 35.9531i 0.0205206 − 0.0163646i −0.613176 − 0.789946i \(-0.710108\pi\)
0.633696 + 0.773582i \(0.281537\pi\)
\(14\) 2628.43 − 4204.01i 0.957884 − 1.53207i
\(15\) −26.8642 + 33.6866i −0.00795976 + 0.00998122i
\(16\) 570.709 + 7615.58i 0.139333 + 1.85927i
\(17\) 948.346 − 3074.46i 0.193028 − 0.625781i −0.806381 − 0.591397i \(-0.798577\pi\)
0.999409 − 0.0343844i \(-0.0109470\pi\)
\(18\) −7413.78 + 12841.0i −1.27122 + 2.20183i
\(19\) −10101.1 + 5831.87i −1.47268 + 0.850251i −0.999528 − 0.0307300i \(-0.990217\pi\)
−0.473151 + 0.880981i \(0.656883\pi\)
\(20\) −145.350 + 33.1751i −0.0181687 + 0.00414689i
\(21\) 13887.0 + 3687.68i 1.49951 + 0.398195i
\(22\) 6919.08 − 30314.5i 0.649801 − 2.84696i
\(23\) −6960.92 + 2147.16i −0.572115 + 0.176474i −0.567296 − 0.823514i \(-0.692010\pi\)
−0.00481949 + 0.999988i \(0.501534\pi\)
\(24\) −45626.6 + 17907.1i −3.30053 + 1.29536i
\(25\) 5708.07 + 14543.9i 0.365316 + 0.930811i
\(26\) 245.689 + 796.505i 0.0139787 + 0.0453178i
\(27\) −12120.2 − 2766.36i −0.615771 − 0.140546i
\(28\) 29612.3 + 39935.7i 1.34896 + 1.81923i
\(29\) −3707.77 − 16244.8i −0.152026 − 0.666071i −0.992295 − 0.123900i \(-0.960460\pi\)
0.840268 − 0.542171i \(-0.182397\pi\)
\(30\) −311.410 − 539.377i −0.0115337 − 0.0199769i
\(31\) 49762.5 + 28730.4i 1.67039 + 0.964398i 0.967420 + 0.253176i \(0.0814751\pi\)
0.702967 + 0.711223i \(0.251858\pi\)
\(32\) −33929.0 − 10465.7i −1.03543 − 0.319389i
\(33\) 89857.5 − 6733.89i 2.50042 − 0.187380i
\(34\) 36361.0 + 28997.0i 0.925123 + 0.737761i
\(35\) −248.907 + 250.025i −0.00580542 + 0.00583149i
\(36\) −92702.1 − 116245.i −1.98693 − 2.49153i
\(37\) 23913.2 − 22188.2i 0.472098 − 0.438043i −0.407895 − 0.913029i \(-0.633737\pi\)
0.879994 + 0.474986i \(0.157547\pi\)
\(38\) −25128.4 − 166716.i −0.457946 − 3.03827i
\(39\) −1995.83 + 1360.73i −0.0336457 + 0.0229392i
\(40\) 179.374 − 1190.07i 0.00280272 − 0.0185948i
\(41\) −3630.92 − 7539.68i −0.0526824 − 0.109396i 0.872966 − 0.487781i \(-0.162194\pi\)
−0.925648 + 0.378385i \(0.876479\pi\)
\(42\) −122958. + 167385.i −1.65962 + 2.25928i
\(43\) −44677.2 − 21515.4i −0.561928 − 0.270610i 0.131283 − 0.991345i \(-0.458090\pi\)
−0.693211 + 0.720735i \(0.743805\pi\)
\(44\) 257616. + 175640.i 3.02423 + 2.06189i
\(45\) 717.637 − 773.429i 0.00787530 − 0.00848756i
\(46\) 7868.94 − 105004.i 0.0808431 − 1.07878i
\(47\) −33686.2 − 13220.9i −0.324458 − 0.127341i 0.197521 − 0.980299i \(-0.436711\pi\)
−0.521979 + 0.852958i \(0.674806\pi\)
\(48\) − 319911.i − 2.89272i
\(49\) 109708. + 42490.8i 0.932502 + 0.361166i
\(50\) −225844. −1.80675
\(51\) −49239.6 + 125460.i −0.371197 + 0.945794i
\(52\) −8334.87 − 624.612i −0.0592774 − 0.00444223i
\(53\) −126863. − 117712.i −0.852134 − 0.790665i 0.127419 − 0.991849i \(-0.459331\pi\)
−0.979553 + 0.201184i \(0.935521\pi\)
\(54\) 101230. − 148478.i 0.642880 − 0.942931i
\(55\) −959.990 + 1993.44i −0.00577004 + 0.0119816i
\(56\) −387429. + 104742.i −2.20611 + 0.596428i
\(57\) 440209. − 211993.i 2.37703 − 1.14472i
\(58\) 238167. + 35897.9i 1.22067 + 0.183986i
\(59\) 139725. + 204939.i 0.680328 + 0.997858i 0.998754 + 0.0499101i \(0.0158935\pi\)
−0.318426 + 0.947948i \(0.603154\pi\)
\(60\) 6175.53 − 930.811i 0.0285904 − 0.00430931i
\(61\) −149336. − 160945.i −0.657921 − 0.709070i 0.312655 − 0.949867i \(-0.398782\pi\)
−0.970576 + 0.240797i \(0.922591\pi\)
\(62\) −649385. + 517867.i −2.72475 + 2.17292i
\(63\) −332356. − 115461.i −1.32918 − 0.461758i
\(64\) 15264.5 − 19141.0i 0.0582294 − 0.0730173i
\(65\) −4.43237 − 59.1459i −1.61397e−5 − 0.000215370i
\(66\) −383928. + 1.24466e6i −1.33542 + 4.32932i
\(67\) 246545. − 427028.i 0.819731 − 1.41982i −0.0861489 − 0.996282i \(-0.527456\pi\)
0.905880 − 0.423534i \(-0.139211\pi\)
\(68\) −403872. + 233176.i −1.28445 + 0.741578i
\(69\) 297500. − 67902.3i 0.905606 − 0.206699i
\(70\) −2049.80 − 4669.62i −0.00597609 − 0.0136140i
\(71\) 96410.4 − 422402.i 0.269370 − 1.18019i −0.641378 − 0.767225i \(-0.721637\pi\)
0.910748 − 0.412962i \(-0.135506\pi\)
\(72\) 1.14692e6 − 353777.i 3.07280 − 0.947834i
\(73\) −363298. + 142584.i −0.933886 + 0.366523i −0.782996 − 0.622027i \(-0.786309\pi\)
−0.150890 + 0.988550i \(0.548214\pi\)
\(74\) 172274. + 438946.i 0.425132 + 1.08322i
\(75\) −192914. − 625410.i −0.457277 − 1.48245i
\(76\) 1.64823e6 + 376198.i 3.75472 + 0.856991i
\(77\) 737370. + 25936.2i 1.61515 + 0.0568111i
\(78\) −7769.74 − 34041.4i −0.0163728 − 0.0717339i
\(79\) 115072. + 199310.i 0.233392 + 0.404247i 0.958804 − 0.284068i \(-0.0916840\pi\)
−0.725412 + 0.688315i \(0.758351\pi\)
\(80\) 6802.73 + 3927.56i 0.0132866 + 0.00767101i
\(81\) −216931. − 66914.3i −0.408194 − 0.125911i
\(82\) 120627. − 9039.75i 0.218778 − 0.0163951i
\(83\) 392949. + 313367.i 0.687230 + 0.548048i 0.903660 − 0.428250i \(-0.140870\pi\)
−0.216430 + 0.976298i \(0.569441\pi\)
\(84\) −1.11198e6 − 1.76093e6i −1.87611 − 2.97101i
\(85\) −2063.33 − 2587.33i −0.00335978 − 0.00421304i
\(86\) 525447. − 487543.i 0.826101 − 0.766510i
\(87\) 104031. + 690199.i 0.157981 + 1.04813i
\(88\) −2.07961e6 + 1.41785e6i −3.05164 + 2.08057i
\(89\) −100896. + 669400.i −0.143121 + 0.949546i 0.795836 + 0.605512i \(0.207032\pi\)
−0.938957 + 0.344034i \(0.888206\pi\)
\(90\) 6617.24 + 13740.8i 0.00907714 + 0.0188489i
\(91\) −17466.0 + 9281.20i −0.0231777 + 0.0123163i
\(92\) 951307. + 458126.i 1.22168 + 0.588330i
\(93\) −1.98879e6 − 1.35593e6i −2.47252 − 1.68573i
\(94\) 355794. − 383455.i 0.428366 − 0.461669i
\(95\) −896.535 + 11963.4i −0.00104567 + 0.0139536i
\(96\) 1.38455e6 + 543397.i 1.56493 + 0.614191i
\(97\) 487119.i 0.533728i 0.963734 + 0.266864i \(0.0859874\pi\)
−0.963734 + 0.266864i \(0.914013\pi\)
\(98\) −1.15111e6 + 1.25181e6i −1.22304 + 1.33003i
\(99\) −2.20654e6 −2.27408
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.3.3 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.3 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.3 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.3 yes 324 49.33 odd 42 inner