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Properties

Label 49.7.h.a.33.6
Level $49$
Weight $7$
Character 49.33
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 33.6
Character \(\chi\) \(=\) 49.33
Dual form 49.7.h.a.3.6

$q$-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+(-3.59619 - 9.16295i) q^{2} +(6.31557 - 0.473286i) q^{3} +(-24.1117 + 22.3724i) q^{4} +(88.0567 + 129.155i) q^{5} +(-27.0487 - 56.1672i) q^{6} +(-147.850 - 309.499i) q^{7} +(-275.882 - 132.858i) q^{8} +(-681.195 + 102.674i) q^{9} +(866.776 - 1271.33i) q^{10} +(-1318.45 - 198.725i) q^{11} +(-141.690 + 152.706i) q^{12} +(-949.505 - 757.205i) q^{13} +(-2304.22 + 2467.76i) q^{14} +(617.256 + 774.014i) q^{15} +(-382.559 + 5104.89i) q^{16} +(-689.667 - 2235.85i) q^{17} +(3390.50 + 5872.52i) q^{18} +(2959.56 + 1708.70i) q^{19} +(-5012.71 - 1144.12i) q^{20} +(-1080.24 - 1884.68i) q^{21} +(2920.50 + 12795.6i) q^{22} +(-11223.1 - 3461.88i) q^{23} +(-1805.23 - 708.501i) q^{24} +(-3218.70 + 8201.12i) q^{25} +(-3523.63 + 11423.3i) q^{26} +(-8754.74 + 1998.21i) q^{27} +(10489.1 + 4154.78i) q^{28} +(-4908.74 + 21506.6i) q^{29} +(4872.48 - 8439.38i) q^{30} +(-7277.80 + 4201.84i) q^{31} +(29425.0 - 9076.42i) q^{32} +(-8420.82 - 631.054i) q^{33} +(-18006.8 + 14359.9i) q^{34} +(26954.3 - 46349.1i) q^{35} +(14127.7 - 17715.6i) q^{36} +(-45332.7 - 42062.6i) q^{37} +(5013.60 - 33263.1i) q^{38} +(-6355.04 - 4332.79i) q^{39} +(-7133.96 - 47330.7i) q^{40} +(17248.7 - 35817.3i) q^{41} +(-13384.5 + 16675.8i) q^{42} +(96082.4 - 46270.9i) q^{43} +(36236.0 - 24705.3i) q^{44} +(-73244.7 - 78939.0i) q^{45} +(8639.53 + 115287. i) q^{46} +(152057. - 59678.0i) q^{47} +32421.3i q^{48} +(-73929.9 + 91518.6i) q^{49} +86721.5 q^{50} +(-5413.84 - 13794.2i) q^{51} +(39834.6 - 2985.19i) q^{52} +(-154787. + 143622. i) q^{53} +(49793.2 + 73033.3i) q^{54} +(-90432.2 - 187784. i) q^{55} +(-330.208 + 105028. i) q^{56} +(19500.0 + 9390.70i) q^{57} +(214717. - 32363.3i) q^{58} +(-101329. + 148622. i) q^{59} +(-32199.6 - 4853.31i) q^{60} +(120141. - 129481. i) q^{61} +(64673.6 + 51575.5i) q^{62} +(132492. + 195649. i) q^{63} +(15288.5 + 19171.2i) q^{64} +(14186.9 - 189311. i) q^{65} +(24500.6 + 79429.0i) q^{66} +(-149585. - 259089. i) q^{67} +(66650.2 + 38480.5i) q^{68} +(-72518.9 - 16552.0i) q^{69} +(-521627. - 80300.6i) q^{70} +(17277.5 + 75697.9i) q^{71} +(201571. + 62176.3i) q^{72} +(-609730. - 239302. i) q^{73} +(-222392. + 566647. i) q^{74} +(-16446.4 + 53318.1i) q^{75} +(-109588. + 25012.6i) q^{76} +(133428. + 437441. i) q^{77} +(-16847.2 + 73812.4i) q^{78} +(413726. - 716594. i) q^{79} +(-693011. + 400110. i) q^{80} +(425544. - 131263. i) q^{81} +(-390222. - 29243.1i) q^{82} +(170769. - 136184. i) q^{83} +(68211.2 + 21275.4i) q^{84} +(228042. - 285956. i) q^{85} +(-769508. - 713999. i) q^{86} +(-20822.7 + 138150. i) q^{87} +(337335. + 229991. i) q^{88} +(61163.0 + 405790. i) q^{89} +(-459912. + 955017. i) q^{90} +(-93969.8 + 405823. i) q^{91} +(348059. - 167616. i) q^{92} +(-43974.8 + 29981.5i) q^{93} +(-1.09365e6 - 1.17868e6i) q^{94} +(39920.8 + 532706. i) q^{95} +(181540. - 71249.2i) q^{96} -716182. i q^{97} +(1.10445e6 + 348297. i) q^{98} +918527. 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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{41}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.59619 − 9.16295i −0.449524 − 1.14537i −0.958628 − 0.284663i \(-0.908118\pi\)
0.509104 − 0.860705i \(-0.329977\pi\)
\(3\) 6.31557 − 0.473286i 0.233910 − 0.0175291i 0.0427406 − 0.999086i \(-0.486391\pi\)
0.191169 + 0.981557i \(0.438772\pi\)
\(4\) −24.1117 + 22.3724i −0.376745 + 0.349568i
\(5\) 88.0567 + 129.155i 0.704454 + 1.03324i 0.997031 + 0.0770020i \(0.0245348\pi\)
−0.292577 + 0.956242i \(0.594513\pi\)
\(6\) −27.0487 − 56.1672i −0.125225 − 0.260033i
\(7\) −147.850 − 309.499i −0.431049 − 0.902329i
\(8\) −275.882 − 132.858i −0.538832 − 0.259488i
\(9\) −681.195 + 102.674i −0.934424 + 0.140842i
\(10\) 866.776 − 1271.33i 0.866776 − 1.27133i
\(11\) −1318.45 − 198.725i −0.990572 − 0.149305i −0.366288 − 0.930502i \(-0.619371\pi\)
−0.624285 + 0.781197i \(0.714609\pi\)
\(12\) −141.690 + 152.706i −0.0819967 + 0.0883715i
\(13\) −949.505 − 757.205i −0.432183 − 0.344654i 0.383111 − 0.923702i \(-0.374853\pi\)
−0.815293 + 0.579048i \(0.803424\pi\)
\(14\) −2304.22 + 2467.76i −0.839732 + 0.899328i
\(15\) 617.256 + 774.014i 0.182891 + 0.229338i
\(16\) −382.559 + 5104.89i −0.0933981 + 1.24631i
\(17\) −689.667 − 2235.85i −0.140376 − 0.455088i 0.857898 − 0.513821i \(-0.171770\pi\)
−0.998274 + 0.0587326i \(0.981294\pi\)
\(18\) 3390.50 + 5872.52i 0.581362 + 1.00695i
\(19\) 2959.56 + 1708.70i 0.431485 + 0.249118i 0.699979 − 0.714163i \(-0.253193\pi\)
−0.268494 + 0.963281i \(0.586526\pi\)
\(20\) −5012.71 − 1144.12i −0.626589 − 0.143015i
\(21\) −1080.24 − 1884.68i −0.116644 − 0.203508i
\(22\) 2920.50 + 12795.6i 0.274277 + 1.20169i
\(23\) −11223.1 − 3461.88i −0.922424 − 0.284530i −0.203062 − 0.979166i \(-0.565089\pi\)
−0.719362 + 0.694636i \(0.755566\pi\)
\(24\) −1805.23 − 708.501i −0.130587 − 0.0512516i
\(25\) −3218.70 + 8201.12i −0.205997 + 0.524871i
\(26\) −3523.63 + 11423.3i −0.200479 + 0.649938i
\(27\) −8754.74 + 1998.21i −0.444787 + 0.101520i
\(28\) 10489.1 + 4154.78i 0.477821 + 0.189267i
\(29\) −4908.74 + 21506.6i −0.201269 + 0.881816i 0.768897 + 0.639373i \(0.220806\pi\)
−0.970166 + 0.242443i \(0.922051\pi\)
\(30\) 4872.48 − 8439.38i 0.180462 − 0.312570i
\(31\) −7277.80 + 4201.84i −0.244295 + 0.141044i −0.617149 − 0.786846i \(-0.711713\pi\)
0.372854 + 0.927890i \(0.378379\pi\)
\(32\) 29425.0 − 9076.42i 0.897981 − 0.276990i
\(33\) −8420.82 − 631.054i −0.234322 − 0.0175600i
\(34\) −18006.8 + 14359.9i −0.458141 + 0.365355i
\(35\) 26954.3 − 46349.1i 0.628671 − 1.08103i
\(36\) 14127.7 − 17715.6i 0.302806 − 0.379706i
\(37\) −45332.7 − 42062.6i −0.894967 − 0.830408i 0.0912567 − 0.995827i \(-0.470912\pi\)
−0.986223 + 0.165420i \(0.947102\pi\)
\(38\) 5013.60 − 33263.1i 0.0913690 − 0.606194i
\(39\) −6355.04 − 4332.79i −0.107133 − 0.0730422i
\(40\) −7133.96 − 47330.7i −0.111468 − 0.739543i
\(41\) 17248.7 − 35817.3i 0.250268 − 0.519686i −0.737552 − 0.675291i \(-0.764018\pi\)
0.987819 + 0.155604i \(0.0497325\pi\)
\(42\) −13384.5 + 16675.8i −0.180657 + 0.225081i
\(43\) 96082.4 − 46270.9i 1.20848 − 0.581972i 0.282397 − 0.959298i \(-0.408870\pi\)
0.926081 + 0.377325i \(0.123156\pi\)
\(44\) 36236.0 − 24705.3i 0.425385 − 0.290023i
\(45\) −73244.7 − 78939.0i −0.803783 − 0.866272i
\(46\) 8639.53 + 115287.i 0.0887599 + 1.18442i
\(47\) 152057. − 59678.0i 1.46458 − 0.574805i 0.506737 − 0.862101i \(-0.330852\pi\)
0.957842 + 0.287296i \(0.0927563\pi\)
\(48\) 32421.3i 0.293162i
\(49\) −73929.9 + 91518.6i −0.628394 + 0.777896i
\(50\) 86721.5 0.693772
\(51\) −5413.84 − 13794.2i −0.0408126 − 0.103989i
\(52\) 39834.6 − 2985.19i 0.283303 − 0.0212306i
\(53\) −154787. + 143622.i −1.03970 + 0.964701i −0.999405 − 0.0344869i \(-0.989020\pi\)
−0.0402951 + 0.999188i \(0.512830\pi\)
\(54\) 49793.2 + 73033.3i 0.316220 + 0.463809i
\(55\) −90432.2 − 187784.i −0.543544 − 1.12868i
\(56\) −330.208 + 105028.i −0.00188028 + 0.598056i
\(57\) 19500.0 + 9390.70i 0.105295 + 0.0507076i
\(58\) 214717. − 32363.3i 1.10048 − 0.165871i
\(59\) −101329. + 148622.i −0.493376 + 0.723650i −0.989489 − 0.144608i \(-0.953808\pi\)
0.496113 + 0.868258i \(0.334760\pi\)
\(60\) −32199.6 − 4853.31i −0.149072 − 0.0224690i
\(61\) 120141. − 129481.i 0.529300 − 0.570450i −0.410636 − 0.911799i \(-0.634693\pi\)
0.939937 + 0.341349i \(0.110884\pi\)
\(62\) 64673.6 + 51575.5i 0.271364 + 0.216406i
\(63\) 132492. + 195649.i 0.529868 + 0.782448i
\(64\) 15288.5 + 19171.2i 0.0583211 + 0.0731323i
\(65\) 14186.9 − 189311.i 0.0516591 − 0.689343i
\(66\) 24500.6 + 79429.0i 0.0852206 + 0.276278i
\(67\) −149585. − 259089.i −0.497352 − 0.861438i 0.502644 − 0.864494i \(-0.332361\pi\)
−0.999995 + 0.00305547i \(0.999027\pi\)
\(68\) 66650.2 + 38480.5i 0.211970 + 0.122381i
\(69\) −72518.9 − 16552.0i −0.220752 − 0.0503851i
\(70\) −521627. − 80300.6i −1.52078 − 0.234113i
\(71\) 17277.5 + 75697.9i 0.0482733 + 0.211499i 0.993312 − 0.115458i \(-0.0368337\pi\)
−0.945039 + 0.326958i \(0.893977\pi\)
\(72\) 201571. + 62176.3i 0.540045 + 0.166582i
\(73\) −609730. − 239302.i −1.56736 − 0.615144i −0.586517 − 0.809937i \(-0.699501\pi\)
−0.980845 + 0.194792i \(0.937597\pi\)
\(74\) −222392. + 566647.i −0.548814 + 1.39835i
\(75\) −16446.4 + 53318.1i −0.0389842 + 0.126384i
\(76\) −109588. + 25012.6i −0.249644 + 0.0569795i
\(77\) 133428. + 437441.i 0.292263 + 0.958179i
\(78\) −16847.2 + 73812.4i −0.0355013 + 0.155541i
\(79\) 413726. − 716594.i 0.839134 − 1.45342i −0.0514865 − 0.998674i \(-0.516396\pi\)
0.890620 − 0.454748i \(-0.150271\pi\)
\(80\) −693011. + 400110.i −1.35354 + 0.781465i
\(81\) 425544. − 131263.i 0.800736 − 0.246994i
\(82\) −390222. − 29243.1i −0.707733 − 0.0530373i
\(83\) 170769. − 136184.i 0.298658 − 0.238172i −0.462682 − 0.886524i \(-0.653113\pi\)
0.761341 + 0.648352i \(0.224542\pi\)
\(84\) 68211.2 + 21275.4i 0.115085 + 0.0358956i
\(85\) 228042. − 285956.i 0.371328 − 0.465631i
\(86\) −769508. − 713999.i −1.20981 − 1.12254i
\(87\) −20822.7 + 138150.i −0.0316213 + 0.209794i
\(88\) 337335. + 229991.i 0.495010 + 0.337492i
\(89\) 61163.0 + 405790.i 0.0867598 + 0.575614i 0.989400 + 0.145217i \(0.0463881\pi\)
−0.902640 + 0.430396i \(0.858374\pi\)
\(90\) −459912. + 955017.i −0.630881 + 1.31004i
\(91\) −93969.8 + 405823.i −0.124699 + 0.538533i
\(92\) 348059. − 167616.i 0.446981 − 0.215255i
\(93\) −43974.8 + 29981.5i −0.0546707 + 0.0372739i
\(94\) −1.09365e6 − 1.17868e6i −1.31673 − 1.41909i
\(95\) 39920.8 + 532706.i 0.0465616 + 0.621322i
\(96\) 181540. − 71249.2i 0.205191 − 0.0805316i
\(97\) − 716182.i − 0.784708i −0.919814 − 0.392354i \(-0.871661\pi\)
0.919814 − 0.392354i \(-0.128339\pi\)
\(98\) 1.10445e6 + 348297.i 1.17345 + 0.370059i
\(99\) 918527. 0.946643
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.33.6 yes 324
49.3 odd 42 inner 49.7.h.a.3.6 ✓ 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.6 ✓ 324 49.3 odd 42 inner
49.7.h.a.33.6 yes 324 1.1 even 1 trivial