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Properties

Label 49.7.h.a.3.17
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 3.17
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.17

$q$-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+(1.55186 - 3.95409i) q^{2} +(1.47718 + 0.110699i) q^{3} +(33.6888 + 31.2586i) q^{4} +(-51.2166 + 75.1210i) q^{5} +(2.73010 - 5.66911i) q^{6} +(-331.151 + 89.3766i) q^{7} +(420.812 - 202.652i) q^{8} +(-718.688 - 108.325i) q^{9} +(217.554 + 319.093i) q^{10} +(-2476.89 + 373.332i) q^{11} +(46.3042 + 49.9040i) q^{12} +(1811.95 - 1444.98i) q^{13} +(-160.498 + 1448.10i) q^{14} +(-83.9721 + 105.298i) q^{15} +(71.5377 + 954.604i) q^{16} +(-1701.58 + 5516.40i) q^{17} +(-1543.63 + 2673.65i) q^{18} +(-3472.86 + 2005.06i) q^{19} +(-4073.61 + 929.775i) q^{20} +(-499.064 + 95.3672i) q^{21} +(-2367.62 + 10373.2i) q^{22} +(10342.6 - 3190.27i) q^{23} +(644.049 - 252.770i) q^{24} +(2688.43 + 6850.01i) q^{25} +(-2901.68 - 9407.01i) q^{26} +(-2102.45 - 479.871i) q^{27} +(-13949.9 - 7340.33i) q^{28} +(6955.57 + 30474.3i) q^{29} +(286.043 + 495.441i) q^{30} +(-41091.9 - 23724.4i) q^{31} +(32449.8 + 10009.4i) q^{32} +(-3700.15 + 277.288i) q^{33} +(19171.7 + 15288.9i) q^{34} +(10246.4 - 29454.0i) q^{35} +(-20825.6 - 26114.5i) q^{36} +(27532.4 - 25546.3i) q^{37} +(2538.76 + 16843.6i) q^{38} +(2836.53 - 1933.92i) q^{39} +(-6329.12 + 41991.0i) q^{40} +(-16084.4 - 33399.6i) q^{41} +(-397.389 + 2121.34i) q^{42} +(34844.9 + 16780.4i) q^{43} +(-95113.4 - 64847.2i) q^{44} +(44946.2 - 48440.5i) q^{45} +(3435.71 - 45846.3i) q^{46} +(18829.3 + 7389.95i) q^{47} +1418.04i q^{48} +(101673. - 59194.3i) q^{49} +31257.6 q^{50} +(-3124.21 + 7960.36i) q^{51} +(106210. + 7959.37i) q^{52} +(13077.3 + 12133.9i) q^{53} +(-5160.17 + 7568.58i) q^{54} +(98813.1 - 205188. i) q^{55} +(-121240. + 104719. i) q^{56} +(-5352.00 + 2577.39i) q^{57} +(131292. + 19789.1i) q^{58} +(112897. + 165590. i) q^{59} +(-6120.38 + 922.500i) q^{60} +(-274770. - 296131. i) q^{61} +(-157578. + 125664. i) q^{62} +(247676. - 28362.0i) q^{63} +(51737.1 - 64876.3i) q^{64} +(15746.5 + 210122. i) q^{65} +(-4645.71 + 15061.0i) q^{66} +(-233896. + 405120. i) q^{67} +(-229760. + 132652. i) q^{68} +(15631.0 - 3567.68i) q^{69} +(-100563. - 86223.6i) q^{70} +(-49256.6 + 215807. i) q^{71} +(-324384. + 100059. i) q^{72} +(-172062. + 67529.5i) q^{73} +(-58285.9 - 148510. i) q^{74} +(3213.01 + 10416.3i) q^{75} +(-179672. - 41008.9i) q^{76} +(786858. - 345005. i) q^{77} +(-3244.96 - 14217.1i) q^{78} +(131003. + 226905. i) q^{79} +(-75374.7 - 43517.6i) q^{80} +(503249. + 155232. i) q^{81} +(-157026. + 11767.5i) q^{82} +(-552296. - 440441. i) q^{83} +(-19793.9 - 12387.2i) q^{84} +(-327248. - 410356. i) q^{85} +(120426. - 111739. i) q^{86} +(6901.15 + 45786.1i) q^{87} +(-966649. + 659050. i) q^{88} +(-197983. + 1.31353e6i) q^{89} +(-121788. - 252895. i) q^{90} +(-470880. + 640452. i) q^{91} +(448153. + 215819. i) q^{92} +(-58074.0 - 39594.2i) q^{93} +(58441.0 - 62984.4i) q^{94} +(27246.3 - 363577. i) q^{95} +(46826.2 + 18377.9i) q^{96} -516078. i q^{97} +(-76277.0 - 493884. i) q^{98} +1.82055e6 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.55186 − 3.95409i 0.193983 − 0.494261i −0.800479 − 0.599361i \(-0.795421\pi\)
0.994462 + 0.105101i \(0.0335165\pi\)
\(3\) 1.47718 + 0.110699i 0.0547104 + 0.00409998i 0.102057 − 0.994779i \(-0.467458\pi\)
−0.0473468 + 0.998879i \(0.515077\pi\)
\(4\) 33.6888 + 31.2586i 0.526388 + 0.488416i
\(5\) −51.2166 + 75.1210i −0.409733 + 0.600968i −0.974529 − 0.224263i \(-0.928003\pi\)
0.564796 + 0.825231i \(0.308955\pi\)
\(6\) 2.73010 − 5.66911i 0.0126394 − 0.0262459i
\(7\) −331.151 + 89.3766i −0.965454 + 0.260573i
\(8\) 420.812 − 202.652i 0.821898 − 0.395805i
\(9\) −718.688 − 108.325i −0.985854 − 0.148594i
\(10\) 217.554 + 319.093i 0.217554 + 0.319093i
\(11\) −2476.89 + 373.332i −1.86093 + 0.280490i −0.980911 − 0.194455i \(-0.937706\pi\)
−0.880015 + 0.474945i \(0.842468\pi\)
\(12\) 46.3042 + 49.9040i 0.0267964 + 0.0288796i
\(13\) 1811.95 − 1444.98i 0.824737 − 0.657706i −0.117344 − 0.993091i \(-0.537438\pi\)
0.942081 + 0.335385i \(0.108867\pi\)
\(14\) −160.498 + 1448.10i −0.0584907 + 0.527733i
\(15\) −83.9721 + 105.298i −0.0248806 + 0.0311993i
\(16\) 71.5377 + 954.604i 0.0174652 + 0.233058i
\(17\) −1701.58 + 5516.40i −0.346343 + 1.12282i 0.600094 + 0.799929i \(0.295130\pi\)
−0.946437 + 0.322888i \(0.895346\pi\)
\(18\) −1543.63 + 2673.65i −0.264683 + 0.458445i
\(19\) −3472.86 + 2005.06i −0.506322 + 0.292325i −0.731320 − 0.682034i \(-0.761096\pi\)
0.224999 + 0.974359i \(0.427762\pi\)
\(20\) −4073.61 + 929.775i −0.509201 + 0.116222i
\(21\) −499.064 + 95.3672i −0.0538888 + 0.0102977i
\(22\) −2367.62 + 10373.2i −0.222353 + 0.974194i
\(23\) 10342.6 − 3190.27i 0.850052 − 0.262206i 0.161033 − 0.986949i \(-0.448518\pi\)
0.689020 + 0.724743i \(0.258041\pi\)
\(24\) 644.049 − 252.770i 0.0465892 − 0.0182849i
\(25\) 2688.43 + 6850.01i 0.172060 + 0.438401i
\(26\) −2901.68 − 9407.01i −0.165093 − 0.535219i
\(27\) −2102.45 − 479.871i −0.106816 − 0.0243800i
\(28\) −13949.9 − 7340.33i −0.635471 − 0.334381i
\(29\) 6955.57 + 30474.3i 0.285193 + 1.24951i 0.891038 + 0.453929i \(0.149978\pi\)
−0.605845 + 0.795583i \(0.707165\pi\)
\(30\) 286.043 + 495.441i 0.0105942 + 0.0183497i
\(31\) −41091.9 − 23724.4i −1.37934 − 0.796362i −0.387260 − 0.921970i \(-0.626579\pi\)
−0.992080 + 0.125608i \(0.959912\pi\)
\(32\) 32449.8 + 10009.4i 0.990289 + 0.305464i
\(33\) −3700.15 + 277.288i −0.102962 + 0.00771594i
\(34\) 19171.7 + 15288.9i 0.487780 + 0.388991i
\(35\) 10246.4 − 29454.0i 0.238982 − 0.686973i
\(36\) −20825.6 − 26114.5i −0.446366 − 0.559725i
\(37\) 27532.4 − 25546.3i 0.543549 − 0.504340i −0.359926 − 0.932981i \(-0.617198\pi\)
0.903475 + 0.428641i \(0.141007\pi\)
\(38\) 2538.76 + 16843.6i 0.0462669 + 0.306961i
\(39\) 2836.53 − 1933.92i 0.0478183 − 0.0326020i
\(40\) −6329.12 + 41991.0i −0.0988925 + 0.656109i
\(41\) −16084.4 − 33399.6i −0.233375 − 0.484607i 0.751088 − 0.660202i \(-0.229529\pi\)
−0.984463 + 0.175595i \(0.943815\pi\)
\(42\) −397.389 + 2121.34i −0.00536374 + 0.0286327i
\(43\) 34844.9 + 16780.4i 0.438262 + 0.211056i 0.639986 − 0.768386i \(-0.278940\pi\)
−0.201724 + 0.979442i \(0.564654\pi\)
\(44\) −95113.4 − 64847.2i −1.11656 − 0.761261i
\(45\) 44946.2 − 48440.5i 0.493237 − 0.531583i
\(46\) 3435.71 − 45846.3i 0.0352974 − 0.471011i
\(47\) 18829.3 + 7389.95i 0.181359 + 0.0711783i 0.454283 − 0.890857i \(-0.349895\pi\)
−0.272924 + 0.962036i \(0.587991\pi\)
\(48\) 1418.04i 0.0128223i
\(49\) 101673. − 59194.3i 0.864203 − 0.503143i
\(50\) 31257.6 0.250061
\(51\) −3124.21 + 7960.36i −0.0235521 + 0.0600098i
\(52\) 106210. + 7959.37i 0.755366 + 0.0566068i
\(53\) 13077.3 + 12133.9i 0.0878395 + 0.0815031i 0.722871 − 0.690983i \(-0.242822\pi\)
−0.635031 + 0.772487i \(0.719013\pi\)
\(54\) −5160.17 + 7568.58i −0.0327705 + 0.0480655i
\(55\) 98813.1 − 205188.i 0.593918 − 1.23328i
\(56\) −121240. + 104719.i −0.690368 + 0.596296i
\(57\) −5352.00 + 2577.39i −0.0288996 + 0.0139173i
\(58\) 131292. + 19789.1i 0.672907 + 0.101424i
\(59\) 112897. + 165590.i 0.549702 + 0.806265i 0.996069 − 0.0885853i \(-0.0282346\pi\)
−0.446366 + 0.894850i \(0.647282\pi\)
\(60\) −6120.38 + 922.500i −0.0283351 + 0.00427083i
\(61\) −274770. − 296131.i −1.21054 − 1.30465i −0.939689 − 0.342031i \(-0.888885\pi\)
−0.270852 − 0.962621i \(-0.587305\pi\)
\(62\) −157578. + 125664.i −0.661179 + 0.527273i
\(63\) 247676. − 28362.0i 0.990517 − 0.113427i
\(64\) 51737.1 − 64876.3i 0.197362 − 0.247484i
\(65\) 15746.5 + 210122.i 0.0573382 + 0.765125i
\(66\) −4645.71 + 15061.0i −0.0161592 + 0.0523869i
\(67\) −233896. + 405120.i −0.777677 + 1.34698i 0.155601 + 0.987820i \(0.450269\pi\)
−0.933278 + 0.359156i \(0.883065\pi\)
\(68\) −229760. + 132652.i −0.730713 + 0.421877i
\(69\) 15631.0 − 3567.68i 0.0475818 − 0.0108602i
\(70\) −100563. − 86223.6i −0.293185 − 0.251381i
\(71\) −49256.6 + 215807.i −0.137622 + 0.602964i 0.858331 + 0.513096i \(0.171502\pi\)
−0.995954 + 0.0898675i \(0.971356\pi\)
\(72\) −324384. + 100059.i −0.869086 + 0.268077i
\(73\) −172062. + 67529.5i −0.442301 + 0.173590i −0.576031 − 0.817427i \(-0.695399\pi\)
0.133731 + 0.991018i \(0.457304\pi\)
\(74\) −58285.9 − 148510.i −0.143836 − 0.366489i
\(75\) 3213.01 + 10416.3i 0.00761602 + 0.0246905i
\(76\) −179672. − 41008.9i −0.409298 − 0.0934195i
\(77\) 786858. − 345005.i 1.72355 − 0.755707i
\(78\) −3244.96 − 14217.1i −0.00683793 − 0.0299589i
\(79\) 131003. + 226905.i 0.265706 + 0.460216i 0.967748 − 0.251919i \(-0.0810616\pi\)
−0.702042 + 0.712135i \(0.747728\pi\)
\(80\) −75374.7 − 43517.6i −0.147216 − 0.0849953i
\(81\) 503249. + 155232.i 0.946953 + 0.292096i
\(82\) −157026. + 11767.5i −0.284793 + 0.0213423i
\(83\) −552296. − 440441.i −0.965912 − 0.770289i 0.00735361 − 0.999973i \(-0.497659\pi\)
−0.973265 + 0.229684i \(0.926231\pi\)
\(84\) −19793.9 − 12387.2i −0.0333959 − 0.0208995i
\(85\) −327248. − 410356.i −0.532869 − 0.668197i
\(86\) 120426. − 111739.i 0.189332 − 0.175675i
\(87\) 6901.15 + 45786.1i 0.0104801 + 0.0695306i
\(88\) −966649. + 659050.i −1.41847 + 0.967098i
\(89\) −197983. + 1.31353e6i −0.280840 + 1.86325i 0.191033 + 0.981584i \(0.438816\pi\)
−0.471873 + 0.881667i \(0.656422\pi\)
\(90\) −121788. − 252895.i −0.167061 − 0.346906i
\(91\) −470880. + 640452.i −0.624865 + 0.849889i
\(92\) 448153. + 215819.i 0.575523 + 0.277157i
\(93\) −58074.0 − 39594.2i −0.0721992 − 0.0492246i
\(94\) 58441.0 − 62984.4i 0.0703613 − 0.0758315i
\(95\) 27246.3 − 363577.i 0.0317788 − 0.424059i
\(96\) 46826.2 + 18377.9i 0.0529267 + 0.0207722i
\(97\) − 516078.i − 0.565458i −0.959200 − 0.282729i \(-0.908760\pi\)
0.959200 − 0.282729i \(-0.0912397\pi\)
\(98\) −76277.0 − 493884.i −0.0810430 − 0.524743i
\(99\) 1.82055e6 1.87628
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.3.17 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.17 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.17 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.17 yes 324 49.33 odd 42 inner