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Properties

Label 49.7.h.a.3.1
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 3.1
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.1

$q$-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+(-5.64737 + 14.3893i) q^{2} +(39.0389 + 2.92556i) q^{3} +(-128.243 - 118.992i) q^{4} +(88.5491 - 129.878i) q^{5} +(-262.564 + 545.219i) q^{6} +(-318.627 - 126.988i) q^{7} +(1545.11 - 744.086i) q^{8} +(794.620 + 119.770i) q^{9} +(1368.77 + 2007.62i) q^{10} +(2171.64 - 327.322i) q^{11} +(-4658.34 - 5020.49i) q^{12} +(361.549 - 288.326i) q^{13} +(3626.67 - 3867.65i) q^{14} +(3836.82 - 4811.22i) q^{15} +(1144.33 + 15270.1i) q^{16} +(-346.911 + 1124.66i) q^{17} +(-6210.91 + 10757.6i) q^{18} +(1453.01 - 838.897i) q^{19} +(-26810.2 + 6119.24i) q^{20} +(-12067.3 - 5889.65i) q^{21} +(-7554.14 + 33096.8i) q^{22} +(12463.6 - 3844.52i) q^{23} +(62496.3 - 24528.0i) q^{24} +(-3318.81 - 8456.18i) q^{25} +(2106.99 + 6830.70i) q^{26} +(2847.06 + 649.823i) q^{27} +(25751.0 + 54199.3i) q^{28} +(-6505.97 - 28504.5i) q^{29} +(47562.0 + 82379.8i) q^{30} +(-8560.88 - 4942.63i) q^{31} +(-121307. - 37418.3i) q^{32} +(85736.2 - 6425.04i) q^{33} +(-14223.8 - 11343.1i) q^{34} +(-44707.0 + 30137.8i) q^{35} +(-87652.6 - 109913. i) q^{36} +(3326.79 - 3086.81i) q^{37} +(3865.41 + 25645.3i) q^{38} +(14958.0 - 10198.2i) q^{39} +(40178.0 - 266564. i) q^{40} +(4779.34 + 9924.40i) q^{41} +(152896. - 140379. i) q^{42} +(-81872.2 - 39427.6i) q^{43} +(-317446. - 216431. i) q^{44} +(85918.2 - 92597.8i) q^{45} +(-15066.9 + 201054. i) q^{46} +(134188. + 52664.9i) q^{47} +599474. i q^{48} +(85397.0 + 80923.7i) q^{49} +140421. q^{50} +(-16833.3 + 42890.5i) q^{51} +(-80674.4 - 6045.71i) q^{52} +(114656. + 106385. i) q^{53} +(-25428.9 + 37297.3i) q^{54} +(149785. - 311032. i) q^{55} +(-586804. + 40874.8i) q^{56} +(59178.2 - 28498.7i) q^{57} +(446900. + 67359.4i) q^{58} +(15337.2 + 22495.5i) q^{59} +(-1.06454e6 + 160454. i) q^{60} +(-34663.8 - 37358.7i) q^{61} +(119467. - 95271.9i) q^{62} +(-237978. - 139069. i) q^{63} +(612452. - 767990. i) q^{64} +(-5432.23 - 72488.1i) q^{65} +(-391732. + 1.26996e6i) q^{66} +(-21617.5 + 37442.7i) q^{67} +(178314. - 102949. i) q^{68} +(497814. - 113623. i) q^{69} +(-181183. - 813500. i) q^{70} +(-127314. + 557798. i) q^{71} +(1.31689e6 - 406208. i) q^{72} +(-459018. + 180151. i) q^{73} +(25629.3 + 65302.4i) q^{74} +(-104824. - 339830. i) q^{75} +(-286160. - 65314.2i) q^{76} +(-733509. - 171479. i) q^{77} +(62271.0 + 272827. i) q^{78} +(-15902.7 - 27544.3i) q^{79} +(2.08457e6 + 1.20353e6i) q^{80} +(-450549. - 138976. i) q^{81} +(-169796. + 12724.4i) q^{82} +(-449830. - 358727. i) q^{83} +(846727. + 2.19122e6i) q^{84} +(115349. + 144643. i) q^{85} +(1.02970e6 - 955418. i) q^{86} +(-170594. - 1.13182e6i) q^{87} +(3.11187e6 - 2.12164e6i) q^{88} +(-136498. + 905608. i) q^{89} +(847202. + 1.75923e6i) q^{90} +(-151813. + 45955.8i) q^{91} +(-2.05584e6 - 990038. i) q^{92} +(-319748. - 218000. i) q^{93} +(-1.51562e6 + 1.63345e6i) q^{94} +(19708.9 - 262997. i) q^{95} +(-4.62622e6 - 1.81566e6i) q^{96} -96272.4i q^{97} +(-1.64670e6 + 771793. i) q^{98} +1.76483e6 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.64737 + 14.3893i −0.705921 + 1.79866i −0.112682 + 0.993631i \(0.535944\pi\)
−0.593239 + 0.805027i \(0.702151\pi\)
\(3\) 39.0389 + 2.92556i 1.44589 + 0.108354i 0.774415 − 0.632678i \(-0.218044\pi\)
0.671470 + 0.741032i \(0.265663\pi\)
\(4\) −128.243 − 118.992i −2.00379 − 1.85925i
\(5\) 88.5491 − 129.878i 0.708393 − 1.03902i −0.288283 − 0.957545i \(-0.593084\pi\)
0.996675 − 0.0814757i \(-0.0259633\pi\)
\(6\) −262.564 + 545.219i −1.21557 + 2.52416i
\(7\) −318.627 − 126.988i −0.928941 − 0.370228i
\(8\) 1545.11 − 744.086i 3.01780 − 1.45329i
\(9\) 794.620 + 119.770i 1.09001 + 0.164293i
\(10\) 1368.77 + 2007.62i 1.36877 + 2.00762i
\(11\) 2171.64 − 327.322i 1.63159 − 0.245922i 0.731567 − 0.681770i \(-0.238789\pi\)
0.900020 + 0.435848i \(0.143551\pi\)
\(12\) −4658.34 − 5020.49i −2.69580 − 2.90538i
\(13\) 361.549 − 288.326i 0.164565 − 0.131236i −0.537745 − 0.843108i \(-0.680724\pi\)
0.702310 + 0.711872i \(0.252152\pi\)
\(14\) 3626.67 − 3867.65i 1.32167 − 1.40949i
\(15\) 3836.82 − 4811.22i 1.13684 − 1.42555i
\(16\) 1144.33 + 15270.1i 0.279378 + 3.72804i
\(17\) −346.911 + 1124.66i −0.0706107 + 0.228914i −0.983999 − 0.178176i \(-0.942980\pi\)
0.913388 + 0.407090i \(0.133457\pi\)
\(18\) −6210.91 + 10757.6i −1.06497 + 1.84458i
\(19\) 1453.01 − 838.897i 0.211840 − 0.122306i −0.390326 − 0.920677i \(-0.627638\pi\)
0.602166 + 0.798371i \(0.294304\pi\)
\(20\) −26810.2 + 6119.24i −3.35127 + 0.764905i
\(21\) −12067.3 − 5889.65i −1.30303 − 0.635962i
\(22\) −7554.14 + 33096.8i −0.709442 + 3.10827i
\(23\) 12463.6 − 3844.52i 1.02438 − 0.315979i 0.263353 − 0.964700i \(-0.415172\pi\)
0.761027 + 0.648721i \(0.224696\pi\)
\(24\) 62496.3 − 24528.0i 4.52086 − 1.77431i
\(25\) −3318.81 − 8456.18i −0.212404 − 0.541196i
\(26\) 2106.99 + 6830.70i 0.119879 + 0.388638i
\(27\) 2847.06 + 649.823i 0.144646 + 0.0330144i
\(28\) 25751.0 + 54199.3i 1.17306 + 2.46899i
\(29\) −6505.97 − 28504.5i −0.266758 − 1.16874i −0.913760 − 0.406255i \(-0.866834\pi\)
0.647001 − 0.762489i \(-0.276023\pi\)
\(30\) 47562.0 + 82379.8i 1.76156 + 3.05110i
\(31\) −8560.88 − 4942.63i −0.287365 − 0.165910i 0.349388 − 0.936978i \(-0.386389\pi\)
−0.636753 + 0.771068i \(0.719723\pi\)
\(32\) −121307. − 37418.3i −3.70200 − 1.14191i
\(33\) 85736.2 − 6425.04i 2.38573 − 0.178786i
\(34\) −14223.8 − 11343.1i −0.361893 − 0.288600i
\(35\) −44707.0 + 30137.8i −1.04273 + 0.702922i
\(36\) −87652.6 − 109913.i −1.87870 − 2.35581i
\(37\) 3326.79 − 3086.81i 0.0656780 − 0.0609403i −0.646651 − 0.762786i \(-0.723831\pi\)
0.712329 + 0.701846i \(0.247640\pi\)
\(38\) 3865.41 + 25645.3i 0.0704441 + 0.467366i
\(39\) 14958.0 − 10198.2i 0.252162 − 0.171921i
\(40\) 40178.0 − 266564.i 0.627781 − 4.16506i
\(41\) 4779.34 + 9924.40i 0.0693452 + 0.143997i 0.932760 − 0.360498i \(-0.117393\pi\)
−0.863415 + 0.504495i \(0.831679\pi\)
\(42\) 152896. − 140379.i 2.06371 − 1.89476i
\(43\) −81872.2 − 39427.6i −1.02975 − 0.495900i −0.158818 − 0.987308i \(-0.550768\pi\)
−0.870930 + 0.491408i \(0.836483\pi\)
\(44\) −317446. − 216431.i −3.72659 − 2.54075i
\(45\) 85918.2 − 92597.8i 0.942861 − 1.01616i
\(46\) −15066.9 + 201054.i −0.154793 + 2.06556i
\(47\) 134188. + 52664.9i 1.29247 + 0.507257i 0.909192 − 0.416377i \(-0.136700\pi\)
0.383277 + 0.923633i \(0.374796\pi\)
\(48\) 599474.i 5.42059i
\(49\) 85397.0 + 80923.7i 0.725862 + 0.687840i
\(50\) 140421. 1.12337
\(51\) −16833.3 + 42890.5i −0.126899 + 0.323333i
\(52\) −80674.4 − 6045.71i −0.573754 − 0.0429969i
\(53\) 114656. + 106385.i 0.770138 + 0.714584i 0.963841 − 0.266476i \(-0.0858595\pi\)
−0.193703 + 0.981060i \(0.562050\pi\)
\(54\) −25428.9 + 37297.3i −0.161490 + 0.236863i
\(55\) 149785. − 311032.i 0.900286 − 1.86946i
\(56\) −586804. + 40874.8i −3.34140 + 0.232751i
\(57\) 59178.2 − 28498.7i 0.319549 − 0.153887i
\(58\) 446900. + 67359.4i 2.29048 + 0.345234i
\(59\) 15337.2 + 22495.5i 0.0746773 + 0.109532i 0.861760 − 0.507315i \(-0.169362\pi\)
−0.787083 + 0.616847i \(0.788410\pi\)
\(60\) −1.06454e6 + 160454.i −4.92843 + 0.742842i
\(61\) −34663.8 − 37358.7i −0.152717 − 0.164590i 0.652059 − 0.758168i \(-0.273905\pi\)
−0.804776 + 0.593578i \(0.797715\pi\)
\(62\) 119467. − 95271.9i 0.501272 − 0.399751i
\(63\) −237978. − 139069.i −0.951732 − 0.556172i
\(64\) 612452. − 767990.i 2.33632 − 2.92965i
\(65\) −5432.23 − 72488.1i −0.0197805 − 0.263953i
\(66\) −391732. + 1.26996e6i −1.36257 + 4.41733i
\(67\) −21617.5 + 37442.7i −0.0718757 + 0.124492i −0.899723 − 0.436461i \(-0.856232\pi\)
0.827848 + 0.560953i \(0.189565\pi\)
\(68\) 178314. − 102949.i 0.567098 − 0.327414i
\(69\) 497814. − 113623.i 1.51537 − 0.345874i
\(70\) −181183. − 813500.i −0.528231 − 2.37172i
\(71\) −127314. + 557798.i −0.355713 + 1.55848i 0.408034 + 0.912967i \(0.366214\pi\)
−0.763748 + 0.645515i \(0.776643\pi\)
\(72\) 1.31689e6 − 406208.i 3.52820 − 1.08831i
\(73\) −459018. + 180151.i −1.17994 + 0.463094i −0.872567 − 0.488494i \(-0.837546\pi\)
−0.307377 + 0.951588i \(0.599451\pi\)
\(74\) 25629.3 + 65302.4i 0.0632473 + 0.161151i
\(75\) −104824. − 339830.i −0.248471 − 0.805522i
\(76\) −286160. − 65314.2i −0.651881 − 0.148788i
\(77\) −733509. − 171479.i −1.60670 − 0.375612i
\(78\) 62271.0 + 272827.i 0.131221 + 0.574915i
\(79\) −15902.7 − 27544.3i −0.0322545 − 0.0558665i 0.849448 − 0.527673i \(-0.176935\pi\)
−0.881702 + 0.471807i \(0.843602\pi\)
\(80\) 2.08457e6 + 1.20353e6i 4.07142 + 2.35064i
\(81\) −450549. − 138976.i −0.847788 − 0.261508i
\(82\) −169796. + 12724.4i −0.307953 + 0.0230779i
\(83\) −449830. − 358727.i −0.786709 − 0.627379i 0.145476 − 0.989362i \(-0.453529\pi\)
−0.932185 + 0.361983i \(0.882100\pi\)
\(84\) 846727. + 2.19122e6i 1.42858 + 3.69699i
\(85\) 115349. + 144643.i 0.187827 + 0.235527i
\(86\) 1.02970e6 − 955418.i 1.61888 − 1.50210i
\(87\) −170594. − 1.13182e6i −0.259064 − 1.71877i
\(88\) 3.11187e6 − 2.12164e6i 4.56640 − 3.11332i
\(89\) −136498. + 905608.i −0.193623 + 1.28461i 0.655008 + 0.755622i \(0.272665\pi\)
−0.848631 + 0.528985i \(0.822573\pi\)
\(90\) 847202. + 1.75923e6i 1.16214 + 2.41322i
\(91\) −151813. + 45955.8i −0.201458 + 0.0609840i
\(92\) −2.05584e6 − 990038.i −2.64013 − 1.27142i
\(93\) −319748. − 218000.i −0.397519 − 0.271024i
\(94\) −1.51562e6 + 1.63345e6i −1.82476 + 1.96663i
\(95\) 19708.9 − 262997.i 0.0229875 − 0.306747i
\(96\) −4.62622e6 − 1.81566e6i −5.22893 − 2.05220i
\(97\) − 96272.4i − 0.105484i −0.998608 − 0.0527420i \(-0.983204\pi\)
0.998608 − 0.0527420i \(-0.0167961\pi\)
\(98\) −1.64670e6 + 771793.i −1.74959 + 0.820017i
\(99\) 1.76483e6 1.81886
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.3.1 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.1 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.1 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.1 yes 324 49.33 odd 42 inner