Properties

Label 78.6.b.a.25.6
Level 7878
Weight 66
Character 78.25
Analytic conductor 12.51012.510
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,6,Mod(25,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 78=2313 78 = 2 \cdot 3 \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 78.b (of order 22, degree 11, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 12.509937945412.5099379454
Analytic rank: 00
Dimension: 66
Coefficient field: Q[x]/(x6)\mathbb{Q}[x]/(x^{6} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x62x5+2x42946x3+131769x21332936x+6741792 x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 21132 2^{11}\cdot 3^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 25.6
Root 6.107586.10758i6.10758 - 6.10758i of defining polynomial
Character χ\chi == 78.25
Dual form 78.6.b.a.25.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+4.00000iq29.00000q316.0000q4+37.1158iq536.0000iq6+176.407iq764.0000iq8+81.0000q9148.463q10179.404iq11+144.000q12+(554.740252.104i)q13705.627q14334.042iq15+256.000q16933.141q17+324.000iq182335.97iq19593.853iq201587.66iq21+717.615q222792.36q23+576.000iq24+1747.42q25+(1008.422218.96i)q26729.000q272822.51iq28+1503.58q29+1336.17q30737.236iq31+1024.00iq32+1614.63iq333732.56iq346547.48q351296.00q363775.41iq37+9343.88q38+(4992.66+2268.94i)q39+2375.41q40+368.848iq41+6350.64q4220180.6q43+2870.46iq44+3006.38iq4511169.4iq46+20526.5iq472304.00q4814312.3q49+6989.67iq50+8398.27q51+(8875.84+4033.67i)q5225081.9q532916.00iq54+6658.72q55+11290.0q56+21023.7iq57+6014.33iq58+35326.3iq59+5344.68iq60+31741.4q61+2948.94q62+14288.9iq634096.00q64+(9357.0620589.6i)q656458.54q6646661.3iq67+14930.3q68+25131.2q6926189.9iq70+58973.4iq715184.00iq72+3411.20iq73+15101.6q7415726.8q75+37375.5iq76+31648.1q77+(9075.76+19970.6i)q7864977.3q79+9501.64iq80+6561.00q811475.39q82+12233.4iq83+25402.6iq8434634.3iq8580722.5iq8613532.2q8711481.8q8861754.7iq8912025.5q90+(44472.997859.9i)q91+44677.8q92+6635.12iq9382106.0q94+86701.4q959216.00iq9628030.7iq9757249.4iq9814531.7iq99+O(q100)q+4.00000i q^{2} -9.00000 q^{3} -16.0000 q^{4} +37.1158i q^{5} -36.0000i q^{6} +176.407i q^{7} -64.0000i q^{8} +81.0000 q^{9} -148.463 q^{10} -179.404i q^{11} +144.000 q^{12} +(-554.740 - 252.104i) q^{13} -705.627 q^{14} -334.042i q^{15} +256.000 q^{16} -933.141 q^{17} +324.000i q^{18} -2335.97i q^{19} -593.853i q^{20} -1587.66i q^{21} +717.615 q^{22} -2792.36 q^{23} +576.000i q^{24} +1747.42 q^{25} +(1008.42 - 2218.96i) q^{26} -729.000 q^{27} -2822.51i q^{28} +1503.58 q^{29} +1336.17 q^{30} -737.236i q^{31} +1024.00i q^{32} +1614.63i q^{33} -3732.56i q^{34} -6547.48 q^{35} -1296.00 q^{36} -3775.41i q^{37} +9343.88 q^{38} +(4992.66 + 2268.94i) q^{39} +2375.41 q^{40} +368.848i q^{41} +6350.64 q^{42} -20180.6 q^{43} +2870.46i q^{44} +3006.38i q^{45} -11169.4i q^{46} +20526.5i q^{47} -2304.00 q^{48} -14312.3 q^{49} +6989.67i q^{50} +8398.27 q^{51} +(8875.84 + 4033.67i) q^{52} -25081.9 q^{53} -2916.00i q^{54} +6658.72 q^{55} +11290.0 q^{56} +21023.7i q^{57} +6014.33i q^{58} +35326.3i q^{59} +5344.68i q^{60} +31741.4 q^{61} +2948.94 q^{62} +14288.9i q^{63} -4096.00 q^{64} +(9357.06 - 20589.6i) q^{65} -6458.54 q^{66} -46661.3i q^{67} +14930.3 q^{68} +25131.2 q^{69} -26189.9i q^{70} +58973.4i q^{71} -5184.00i q^{72} +3411.20i q^{73} +15101.6 q^{74} -15726.8 q^{75} +37375.5i q^{76} +31648.1 q^{77} +(-9075.76 + 19970.6i) q^{78} -64977.3 q^{79} +9501.64i q^{80} +6561.00 q^{81} -1475.39 q^{82} +12233.4i q^{83} +25402.6i q^{84} -34634.3i q^{85} -80722.5i q^{86} -13532.2 q^{87} -11481.8 q^{88} -61754.7i q^{89} -12025.5 q^{90} +(44472.9 - 97859.9i) q^{91} +44677.8 q^{92} +6635.12i q^{93} -82106.0 q^{94} +86701.4 q^{95} -9216.00i q^{96} -28030.7i q^{97} -57249.4i q^{98} -14531.7i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q54q396q4+486q9+320q10+864q12+530q131360q14+1536q16836q171296q22416q23+718q251360q264374q27+18788q292880q30++567168q95+O(q100) 6 q - 54 q^{3} - 96 q^{4} + 486 q^{9} + 320 q^{10} + 864 q^{12} + 530 q^{13} - 1360 q^{14} + 1536 q^{16} - 836 q^{17} - 1296 q^{22} - 416 q^{23} + 718 q^{25} - 1360 q^{26} - 4374 q^{27} + 18788 q^{29} - 2880 q^{30}+ \cdots + 567168 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/78Z)×\left(\mathbb{Z}/78\mathbb{Z}\right)^\times.

nn 5353 6767
χ(n)\chi(n) 11 1-1

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 4.00000i 0.707107i
33 −9.00000 −0.577350
44 −16.0000 −0.500000
55 37.1158i 0.663948i 0.943289 + 0.331974i 0.107715π0.107715\pi
−0.943289 + 0.331974i 0.892285π0.892285\pi
66 36.0000i 0.408248i
77 176.407i 1.36072i 0.732876 + 0.680362i 0.238177π0.238177\pi
−0.732876 + 0.680362i 0.761823π0.761823\pi
88 64.0000i 0.353553i
99 81.0000 0.333333
1010 −148.463 −0.469482
1111 179.404i 0.447044i −0.974699 0.223522i 0.928245π-0.928245\pi
0.974699 0.223522i 0.0717554π-0.0717554\pi
1212 144.000 0.288675
1313 −554.740 252.104i −0.910397 0.413735i
1414 −705.627 −0.962177
1515 334.042i 0.383330i
1616 256.000 0.250000
1717 −933.141 −0.783114 −0.391557 0.920154i 0.628063π-0.628063\pi
−0.391557 + 0.920154i 0.628063π0.628063\pi
1818 324.000i 0.235702i
1919 2335.97i 1.48451i −0.670117 0.742256i 0.733756π-0.733756\pi
0.670117 0.742256i 0.266244π-0.266244\pi
2020 593.853i 0.331974i
2121 1587.66i 0.785614i
2222 717.615 0.316108
2323 −2792.36 −1.10066 −0.550328 0.834948i 0.685497π-0.685497\pi
−0.550328 + 0.834948i 0.685497π0.685497\pi
2424 576.000i 0.204124i
2525 1747.42 0.559174
2626 1008.42 2218.96i 0.292555 0.643748i
2727 −729.000 −0.192450
2828 2822.51i 0.680362i
2929 1503.58 0.331996 0.165998 0.986126i 0.446916π-0.446916\pi
0.165998 + 0.986126i 0.446916π0.446916\pi
3030 1336.17 0.271055
3131 737.236i 0.137785i −0.997624 0.0688926i 0.978053π-0.978053\pi
0.997624 0.0688926i 0.0219466π-0.0219466\pi
3232 1024.00i 0.176777i
3333 1614.63i 0.258101i
3434 3732.56i 0.553745i
3535 −6547.48 −0.903450
3636 −1296.00 −0.166667
3737 3775.41i 0.453377i −0.973967 0.226689i 0.927210π-0.927210\pi
0.973967 0.226689i 0.0727900π-0.0727900\pi
3838 9343.88 1.04971
3939 4992.66 + 2268.94i 0.525618 + 0.238870i
4040 2375.41 0.234741
4141 368.848i 0.0342680i 0.999853 + 0.0171340i 0.00545418π0.00545418\pi
−0.999853 + 0.0171340i 0.994546π0.994546\pi
4242 6350.64 0.555513
4343 −20180.6 −1.66442 −0.832211 0.554459i 0.812925π-0.812925\pi
−0.832211 + 0.554459i 0.812925π0.812925\pi
4444 2870.46i 0.223522i
4545 3006.38i 0.221316i
4646 11169.4i 0.778282i
4747 20526.5i 1.35541i 0.735335 + 0.677704i 0.237025π0.237025\pi
−0.735335 + 0.677704i 0.762975π0.762975\pi
4848 −2304.00 −0.144338
4949 −14312.3 −0.851570
5050 6989.67i 0.395395i
5151 8398.27 0.452131
5252 8875.84 + 4033.67i 0.455199 + 0.206867i
5353 −25081.9 −1.22651 −0.613253 0.789886i 0.710140π-0.710140\pi
−0.613253 + 0.789886i 0.710140π0.710140\pi
5454 2916.00i 0.136083i
5555 6658.72 0.296814
5656 11290.0 0.481089
5757 21023.7i 0.857083i
5858 6014.33i 0.234756i
5959 35326.3i 1.32120i 0.750738 + 0.660600i 0.229698π0.229698\pi
−0.750738 + 0.660600i 0.770302π0.770302\pi
6060 5344.68i 0.191665i
6161 31741.4 1.09220 0.546099 0.837721i 0.316112π-0.316112\pi
0.546099 + 0.837721i 0.316112π0.316112\pi
6262 2948.94 0.0974288
6363 14288.9i 0.453575i
6464 −4096.00 −0.125000
6565 9357.06 20589.6i 0.274698 0.604456i
6666 −6458.54 −0.182505
6767 46661.3i 1.26990i −0.772552 0.634951i 0.781020π-0.781020\pi
0.772552 0.634951i 0.218980π-0.218980\pi
6868 14930.3 0.391557
6969 25131.2 0.635465
7070 26189.9i 0.638835i
7171 58973.4i 1.38839i 0.719789 + 0.694193i 0.244238π0.244238\pi
−0.719789 + 0.694193i 0.755762π0.755762\pi
7272 5184.00i 0.117851i
7373 3411.20i 0.0749204i 0.999298 + 0.0374602i 0.0119267π0.0119267\pi
−0.999298 + 0.0374602i 0.988073π0.988073\pi
7474 15101.6 0.320586
7575 −15726.8 −0.322839
7676 37375.5i 0.742256i
7777 31648.1 0.608303
7878 −9075.76 + 19970.6i −0.168907 + 0.371668i
7979 −64977.3 −1.17137 −0.585685 0.810539i 0.699174π-0.699174\pi
−0.585685 + 0.810539i 0.699174π0.699174\pi
8080 9501.64i 0.165987i
8181 6561.00 0.111111
8282 −1475.39 −0.0242311
8383 12233.4i 0.194918i 0.995240 + 0.0974591i 0.0310715π0.0310715\pi
−0.995240 + 0.0974591i 0.968928π0.968928\pi
8484 25402.6i 0.392807i
8585 34634.3i 0.519947i
8686 80722.5i 1.17692i
8787 −13532.2 −0.191678
8888 −11481.8 −0.158054
8989 61754.7i 0.826409i −0.910638 0.413204i 0.864410π-0.864410\pi
0.910638 0.413204i 0.135590π-0.135590\pi
9090 −12025.5 −0.156494
9191 44472.9 97859.9i 0.562979 1.23880i
9292 44677.8 0.550328
9393 6635.12i 0.0795503i
9494 −82106.0 −0.958418
9595 86701.4 0.985638
9696 9216.00i 0.102062i
9797 28030.7i 0.302485i −0.988497 0.151243i 0.951673π-0.951673\pi
0.988497 0.151243i 0.0483274π-0.0483274\pi
9898 57249.4i 0.602151i
9999 14531.7i 0.149015i
100100 −27958.7 −0.279587
101101 102600. 1.00080 0.500398 0.865796i 0.333187π-0.333187\pi
0.500398 + 0.865796i 0.333187π0.333187\pi
102102 33593.1i 0.319705i
103103 −203803. −1.89286 −0.946430 0.322909i 0.895339π-0.895339\pi
−0.946430 + 0.322909i 0.895339π0.895339\pi
104104 −16134.7 + 35503.4i −0.146277 + 0.321874i
105105 58927.3 0.521607
106106 100327.i 0.867271i
107107 −147356. −1.24425 −0.622125 0.782918i 0.713731π-0.713731\pi
−0.622125 + 0.782918i 0.713731π0.713731\pi
108108 11664.0 0.0962250
109109 7846.65i 0.0632584i 0.999500 + 0.0316292i 0.0100696π0.0100696\pi
−0.999500 + 0.0316292i 0.989930π0.989930\pi
110110 26634.9i 0.209879i
111111 33978.7i 0.261758i
112112 45160.1i 0.340181i
113113 −192673. −1.41947 −0.709733 0.704471i 0.751184π-0.751184\pi
−0.709733 + 0.704471i 0.751184π0.751184\pi
114114 −84095.0 −0.606049
115115 103641.i 0.730778i
116116 −24057.3 −0.165998
117117 −44933.9 20420.5i −0.303466 0.137912i
118118 −141305. −0.934229
119119 164612.i 1.06560i
120120 −21378.7 −0.135528
121121 128865. 0.800152
122122 126966.i 0.772301i
123123 3319.64i 0.0197846i
124124 11795.8i 0.0688926i
125125 180844.i 1.03521i
126126 −57155.8 −0.320726
127127 255644. 1.40646 0.703229 0.710964i 0.251741π-0.251741\pi
0.703229 + 0.710964i 0.251741π0.251741\pi
128128 16384.0i 0.0883883i
129129 181626. 0.960955
130130 82358.5 + 37428.2i 0.427415 + 0.194241i
131131 −134257. −0.683530 −0.341765 0.939786i 0.611025π-0.611025\pi
−0.341765 + 0.939786i 0.611025π0.611025\pi
132132 25834.2i 0.129050i
133133 412081. 2.02001
134134 186645. 0.897956
135135 27057.4i 0.127777i
136136 59721.0i 0.276873i
137137 74373.4i 0.338545i −0.985569 0.169272i 0.945858π-0.945858\pi
0.985569 0.169272i 0.0541418π-0.0541418\pi
138138 100525.i 0.449341i
139139 36348.4 0.159569 0.0797844 0.996812i 0.474577π-0.474577\pi
0.0797844 + 0.996812i 0.474577π0.474577\pi
140140 104760. 0.451725
141141 184738.i 0.782545i
142142 −235893. −0.981737
143143 −45228.5 + 99522.5i −0.184958 + 0.406988i
144144 20736.0 0.0833333
145145 55806.7i 0.220428i
146146 −13644.8 −0.0529767
147147 128811. 0.491654
148148 60406.6i 0.226689i
149149 221253.i 0.816440i 0.912884 + 0.408220i 0.133850π0.133850\pi
−0.912884 + 0.408220i 0.866150π0.866150\pi
150150 62907.0i 0.228282i
151151 482448.i 1.72190i 0.508690 + 0.860950i 0.330130π0.330130\pi
−0.508690 + 0.860950i 0.669870π0.669870\pi
152152 −149502. −0.524854
153153 −75584.4 −0.261038
154154 126592.i 0.430135i
155155 27363.1 0.0914821
156156 −79882.5 36303.0i −0.262809 0.119435i
157157 −110726. −0.358508 −0.179254 0.983803i 0.557368π-0.557368\pi
−0.179254 + 0.983803i 0.557368π0.557368\pi
158158 259909.i 0.828283i
159159 225737. 0.708124
160160 −38006.6 −0.117370
161161 492591.i 1.49769i
162162 26244.0i 0.0785674i
163163 161064.i 0.474819i 0.971410 + 0.237410i 0.0762984π0.0762984\pi
−0.971410 + 0.237410i 0.923702π0.923702\pi
164164 5901.58i 0.0171340i
165165 −59928.5 −0.171365
166166 −48933.7 −0.137828
167167 466578.i 1.29459i 0.762238 + 0.647296i 0.224100π0.224100\pi
−0.762238 + 0.647296i 0.775900π0.775900\pi
168168 −101610. −0.277757
169169 244180. + 279705.i 0.657647 + 0.753326i
170170 138537. 0.367658
171171 189214.i 0.494837i
172172 322890. 0.832211
173173 −240767. −0.611620 −0.305810 0.952093i 0.598927π-0.598927\pi
−0.305810 + 0.952093i 0.598927π0.598927\pi
174174 54129.0i 0.135537i
175175 308256.i 0.760881i
176176 45927.4i 0.111761i
177177 317937.i 0.762795i
178178 247019. 0.584359
179179 −266435. −0.621526 −0.310763 0.950487i 0.600585π-0.600585\pi
−0.310763 + 0.950487i 0.600585π0.600585\pi
180180 48102.1i 0.110658i
181181 64307.2 0.145903 0.0729513 0.997336i 0.476758π-0.476758\pi
0.0729513 + 0.997336i 0.476758π0.476758\pi
182182 391439. + 177892.i 0.875964 + 0.398086i
183183 −285673. −0.630581
184184 178711.i 0.389141i
185185 140127. 0.301019
186186 −26540.5 −0.0562505
187187 167409.i 0.350086i
188188 328424.i 0.677704i
189189 128601.i 0.261871i
190190 346806.i 0.696951i
191191 −925387. −1.83544 −0.917720 0.397228i 0.869972π-0.869972\pi
−0.917720 + 0.397228i 0.869972π0.869972\pi
192192 36864.0 0.0721688
193193 632306.i 1.22190i −0.791671 0.610948i 0.790789π-0.790789\pi
0.791671 0.610948i 0.209211π-0.209211\pi
194194 112123. 0.213889
195195 −84213.5 + 185307.i −0.158597 + 0.348983i
196196 228997. 0.425785
197197 100048.i 0.183671i 0.995774 + 0.0918357i 0.0292735π0.0292735\pi
−0.995774 + 0.0918357i 0.970727π0.970727\pi
198198 58126.8 0.105369
199199 −723076. −1.29435 −0.647174 0.762342i 0.724049π-0.724049\pi
−0.647174 + 0.762342i 0.724049π0.724049\pi
200200 111835.i 0.197698i
201201 419952.i 0.733178i
202202 410401.i 0.707669i
203203 265242.i 0.451754i
204204 −134372. −0.226066
205205 −13690.1 −0.0227521
206206 815214.i 1.33845i
207207 −226181. −0.366886
208208 −142013. 64538.7i −0.227599 0.103434i
209209 −419082. −0.663641
210210 235709.i 0.368832i
211211 259548. 0.401340 0.200670 0.979659i 0.435688π-0.435688\pi
0.200670 + 0.979659i 0.435688π0.435688\pi
212212 401310. 0.613253
213213 530760.i 0.801585i
214214 589423.i 0.879818i
215215 749020.i 1.10509i
216216 46656.0i 0.0680414i
217217 130053. 0.187488
218218 −31386.6 −0.0447304
219219 30700.8i 0.0432553i
220220 −106539. −0.148407
221221 517651. + 235249.i 0.712945 + 0.324002i
222222 −135915. −0.185091
223223 1.07196e6i 1.44349i −0.692156 0.721747i 0.743339π-0.743339\pi
0.692156 0.721747i 0.256661π-0.256661\pi
224224 −180641. −0.240544
225225 141541. 0.186391
226226 770692.i 1.00371i
227227 65210.0i 0.0839942i 0.999118 + 0.0419971i 0.0133720π0.0133720\pi
−0.999118 + 0.0419971i 0.986628π0.986628\pi
228228 336380.i 0.428541i
229229 191854.i 0.241759i 0.992667 + 0.120879i 0.0385715π0.0385715\pi
−0.992667 + 0.120879i 0.961429π0.961429\pi
230230 414563. 0.516738
231231 −284832. −0.351204
232232 96229.3i 0.117378i
233233 353920. 0.427087 0.213543 0.976934i 0.431500π-0.431500\pi
0.213543 + 0.976934i 0.431500π0.431500\pi
234234 81681.8 179736.i 0.0975182 0.214583i
235235 −761857. −0.899920
236236 565221.i 0.660600i
237237 584796. 0.676291
238238 658450. 0.753495
239239 1.15374e6i 1.30651i −0.757138 0.653255i 0.773403π-0.773403\pi
0.757138 0.653255i 0.226597π-0.226597\pi
240240 85514.8i 0.0958326i
241241 209753.i 0.232630i 0.993212 + 0.116315i 0.0371083π0.0371083\pi
−0.993212 + 0.116315i 0.962892π0.962892\pi
242242 515461.i 0.565793i
243243 −59049.0 −0.0641500
244244 −507862. −0.546099
245245 531214.i 0.565398i
246246 13278.5 0.0139898
247247 −588909. + 1.29586e6i −0.614194 + 1.35150i
248248 −47183.1 −0.0487144
249249 110101.i 0.112536i
250250 −723375. −0.732004
251251 427156. 0.427959 0.213979 0.976838i 0.431357π-0.431357\pi
0.213979 + 0.976838i 0.431357π0.431357\pi
252252 228623.i 0.226787i
253253 500960.i 0.492042i
254254 1.02258e6i 0.994515i
255255 311708.i 0.300191i
256256 65536.0 0.0625000
257257 207722. 0.196177 0.0980887 0.995178i 0.468727π-0.468727\pi
0.0980887 + 0.995178i 0.468727π0.468727\pi
258258 726503.i 0.679498i
259259 666008. 0.616922
260260 −149713. + 329434.i −0.137349 + 0.302228i
261261 121790. 0.110665
262262 537026.i 0.483328i
263263 −831712. −0.741453 −0.370726 0.928742i 0.620891π-0.620891\pi
−0.370726 + 0.928742i 0.620891π0.620891\pi
264264 103337. 0.0912524
265265 930933.i 0.814336i
266266 1.64832e6i 1.42836i
267267 555792.i 0.477127i
268268 746581.i 0.634951i
269269 1.67365e6 1.41021 0.705104 0.709104i 0.250901π-0.250901\pi
0.705104 + 0.709104i 0.250901π0.250901\pi
270270 108230. 0.0903518
271271 69792.6i 0.0577279i 0.999583 + 0.0288640i 0.00918896π0.00918896\pi
−0.999583 + 0.0288640i 0.990811π0.990811\pi
272272 −238884. −0.195779
273273 −400256. + 880739.i −0.325036 + 0.715221i
274274 297494. 0.239387
275275 313493.i 0.249975i
276276 −402100. −0.317732
277277 1.61380e6 1.26372 0.631859 0.775083i 0.282292π-0.282292\pi
0.631859 + 0.775083i 0.282292π0.282292\pi
278278 145394.i 0.112832i
279279 59716.1i 0.0459284i
280280 419039.i 0.319418i
281281 1.05655e6i 0.798224i 0.916902 + 0.399112i 0.130682π0.130682\pi
−0.916902 + 0.399112i 0.869318π0.869318\pi
282282 738954. 0.553343
283283 −512741. −0.380567 −0.190284 0.981729i 0.560941π-0.560941\pi
−0.190284 + 0.981729i 0.560941π0.560941\pi
284284 943574.i 0.694193i
285285 −780313. −0.569058
286286 −398090. 180914.i −0.287784 0.130785i
287287 −65067.4 −0.0466293
288288 82944.0i 0.0589256i
289289 −549105. −0.386732
290290 −223227. −0.155866
291291 252276.i 0.174640i
292292 54579.2i 0.0374602i
293293 2.71268e6i 1.84599i 0.384809 + 0.922996i 0.374267π0.374267\pi
−0.384809 + 0.922996i 0.625733π0.625733\pi
294294 515244.i 0.347652i
295295 −1.31117e6 −0.877207
296296 −241626. −0.160293
297297 130785.i 0.0860336i
298298 −885014. −0.577310
299299 1.54903e6 + 703967.i 1.00204 + 0.455380i
300300 251628. 0.161420
301301 3.56000e6i 2.26482i
302302 −1.92979e6 −1.21757
303303 −923403. −0.577810
304304 598009.i 0.371128i
305305 1.17811e6i 0.725162i
306306 302338.i 0.184582i
307307 2.44475e6i 1.48043i 0.672371 + 0.740215i 0.265276π0.265276\pi
−0.672371 + 0.740215i 0.734724π0.734724\pi
308308 −506369. −0.304152
309309 1.83423e6 1.09284
310310 109452.i 0.0646876i
311311 1.99183e6 1.16775 0.583877 0.811842i 0.301535π-0.301535\pi
0.583877 + 0.811842i 0.301535π0.301535\pi
312312 145212. 319530.i 0.0844533 0.185834i
313313 1.66396e6 0.960025 0.480013 0.877262i 0.340632π-0.340632\pi
0.480013 + 0.877262i 0.340632π0.340632\pi
314314 442903.i 0.253504i
315315 −530346. −0.301150
316316 1.03964e6 0.585685
317317 3.20482e6i 1.79125i −0.444813 0.895624i 0.646730π-0.646730\pi
0.444813 0.895624i 0.353270π-0.353270\pi
318318 902947.i 0.500719i
319319 269749.i 0.148417i
320320 152026.i 0.0829935i
321321 1.32620e6 0.718369
322322 1.97037e6 1.05903
323323 2.17979e6i 1.16254i
324324 −104976. −0.0555556
325325 −969362. 440532.i −0.509070 0.231350i
326326 −644254. −0.335748
327327 70619.9i 0.0365222i
328328 23606.3 0.0121156
329329 −3.62101e6 −1.84434
330330 239714.i 0.121174i
331331 3.53386e6i 1.77288i −0.462844 0.886440i 0.653171π-0.653171\pi
0.462844 0.886440i 0.346829π-0.346829\pi
332332 195735.i 0.0974591i
333333 305808.i 0.151126i
334334 −1.86631e6 −0.915415
335335 1.73187e6 0.843148
336336 406441.i 0.196404i
337337 1.10242e6 0.528777 0.264389 0.964416i 0.414830π-0.414830\pi
0.264389 + 0.964416i 0.414830π0.414830\pi
338338 −1.11882e6 + 976719.i −0.532682 + 0.465027i
339339 1.73406e6 0.819529
340340 554148.i 0.259973i
341341 −132263. −0.0615960
342342 756855. 0.349903
343343 440075.i 0.201972i
344344 1.29156e6i 0.588462i
345345 932766.i 0.421915i
346346 963068.i 0.432481i
347347 3.16432e6 1.41077 0.705386 0.708823i 0.250774π-0.250774\pi
0.705386 + 0.708823i 0.250774π0.250774\pi
348348 216516. 0.0958389
349349 1.97526e6i 0.868080i −0.900894 0.434040i 0.857088π-0.857088\pi
0.900894 0.434040i 0.142912π-0.142912\pi
350350 −1.23302e6 −0.538024
351351 404405. + 183784.i 0.175206 + 0.0796233i
352352 183710. 0.0790269
353353 2.24015e6i 0.956843i −0.878130 0.478422i 0.841209π-0.841209\pi
0.878130 0.478422i 0.158791π-0.158791\pi
354354 1.27175e6 0.539378
355355 −2.18884e6 −0.921815
356356 988075.i 0.413204i
357357 1.48151e6i 0.615226i
358358 1.06574e6i 0.439485i
359359 2.13770e6i 0.875407i 0.899119 + 0.437704i 0.144208π0.144208\pi
−0.899119 + 0.437704i 0.855792π0.855792\pi
360360 192408. 0.0782470
361361 −2.98066e6 −1.20377
362362 257229.i 0.103169i
363363 −1.15979e6 −0.461968
364364 −711567. + 1.56576e6i −0.281490 + 0.619400i
365365 −126609. −0.0497432
366366 1.14269e6i 0.445888i
367367 1.70502e6 0.660791 0.330395 0.943843i 0.392818π-0.392818\pi
0.330395 + 0.943843i 0.392818π0.392818\pi
368368 −714844. −0.275164
369369 29876.7i 0.0114227i
370370 560509.i 0.212852i
371371 4.42461e6i 1.66894i
372372 106162.i 0.0397751i
373373 −2.31503e6 −0.861558 −0.430779 0.902458i 0.641761π-0.641761\pi
−0.430779 + 0.902458i 0.641761π0.641761\pi
374374 −669636. −0.247548
375375 1.62759e6i 0.597679i
376376 1.31370e6 0.479209
377377 −834097. 379060.i −0.302248 0.137358i
378378 514402. 0.185171
379379 4.32354e6i 1.54612i −0.634336 0.773058i 0.718726π-0.718726\pi
0.634336 0.773058i 0.281274π-0.281274\pi
380380 −1.38722e6 −0.492819
381381 −2.30080e6 −0.812018
382382 3.70155e6i 1.29785i
383383 2.00351e6i 0.697901i 0.937141 + 0.348951i 0.113462π0.113462\pi
−0.937141 + 0.348951i 0.886538π0.886538\pi
384384 147456.i 0.0510310i
385385 1.17464e6i 0.403882i
386386 2.52922e6 0.864011
387387 −1.63463e6 −0.554807
388388 448490.i 0.151243i
389389 1.70696e6 0.571940 0.285970 0.958239i 0.407684π-0.407684\pi
0.285970 + 0.958239i 0.407684π0.407684\pi
390390 −741226. 336854.i −0.246768 0.112145i
391391 2.60567e6 0.861940
392392 915990.i 0.301076i
393393 1.20831e6 0.394636
394394 −400191. −0.129875
395395 2.41168e6i 0.777728i
396396 232507.i 0.0745073i
397397 1.22427e6i 0.389853i 0.980818 + 0.194927i 0.0624469π0.0624469\pi
−0.980818 + 0.194927i 0.937553π0.937553\pi
398398 2.89230e6i 0.915243i
399399 −3.70873e6 −1.16625
400400 447339. 0.139793
401401 4.15598e6i 1.29066i −0.763903 0.645332i 0.776719π-0.776719\pi
0.763903 0.645332i 0.223281π-0.223281\pi
402402 −1.67981e6 −0.518435
403403 −185860. + 408974.i −0.0570065 + 0.125439i
404404 −1.64161e6 −0.500398
405405 243517.i 0.0737720i
406406 −1.06097e6 −0.319439
407407 −677323. −0.202680
408408 537489.i 0.159852i
409409 6.52428e6i 1.92852i 0.264958 + 0.964260i 0.414642π0.414642\pi
−0.264958 + 0.964260i 0.585358π0.585358\pi
410410 54760.4i 0.0160882i
411411 669361.i 0.195459i
412412 3.26086e6 0.946430
413413 −6.23180e6 −1.79779
414414 904725.i 0.259427i
415415 −454053. −0.129416
416416 258155. 568054.i 0.0731387 0.160937i
417417 −327136. −0.0921271
418418 1.67633e6i 0.469265i
419419 4.77950e6 1.32999 0.664994 0.746849i 0.268434π-0.268434\pi
0.664994 + 0.746849i 0.268434π0.268434\pi
420420 −942837. −0.260803
421421 3.41558e6i 0.939203i 0.882878 + 0.469602i 0.155602π0.155602\pi
−0.882878 + 0.469602i 0.844398π0.844398\pi
422422 1.03819e6i 0.283790i
423423 1.66265e6i 0.451803i
424424 1.60524e6i 0.433636i
425425 −1.63059e6 −0.437897
426426 2.12304e6 0.566806
427427 5.59940e6i 1.48618i
428428 2.35769e6 0.622125
429429 407057. 895702.i 0.106785 0.234974i
430430 2.99608e6 0.781416
431431 6.21413e6i 1.61134i −0.592365 0.805670i 0.701806π-0.701806\pi
0.592365 0.805670i 0.298194π-0.298194\pi
432432 −186624. −0.0481125
433433 −5.83362e6 −1.49526 −0.747632 0.664113i 0.768809π-0.768809\pi
−0.747632 + 0.664113i 0.768809π0.768809\pi
434434 520214.i 0.132574i
435435 502260.i 0.127264i
436436 125546.i 0.0316292i
437437 6.52287e6i 1.63394i
438438 122803. 0.0305861
439439 −766922. −0.189928 −0.0949642 0.995481i 0.530274π-0.530274\pi
−0.0949642 + 0.995481i 0.530274π0.530274\pi
440440 426158.i 0.104939i
441441 −1.15930e6 −0.283857
442442 −940996. + 2.07060e6i −0.229104 + 0.504128i
443443 1.02042e6 0.247042 0.123521 0.992342i 0.460581π-0.460581\pi
0.123521 + 0.992342i 0.460581π0.460581\pi
444444 543659.i 0.130879i
445445 2.29207e6 0.548692
446446 4.28783e6 1.02071
447447 1.99128e6i 0.471372i
448448 722562.i 0.170091i
449449 4.18123e6i 0.978787i 0.872063 + 0.489394i 0.162782π0.162782\pi
−0.872063 + 0.489394i 0.837218π0.837218\pi
450450 566163.i 0.131798i
451451 66172.8 0.0153193
452452 3.08277e6 0.709733
453453 4.34203e6i 0.994139i
454454 −260840. −0.0593929
455455 3.63215e6 + 1.65065e6i 0.822498 + 0.373789i
456456 1.34552e6 0.303025
457457 3.58333e6i 0.802594i 0.915948 + 0.401297i 0.131441π0.131441\pi
−0.915948 + 0.401297i 0.868559π0.868559\pi
458458 −767417. −0.170949
459459 680260. 0.150710
460460 1.65825e6i 0.365389i
461461 1.67459e6i 0.366992i −0.983020 0.183496i 0.941259π-0.941259\pi
0.983020 0.183496i 0.0587414π-0.0587414\pi
462462 1.13933e6i 0.248339i
463463 6.62362e6i 1.43596i 0.696063 + 0.717981i 0.254934π0.254934\pi
−0.696063 + 0.717981i 0.745066π0.745066\pi
464464 384917. 0.0829989
465465 −246268. −0.0528172
466466 1.41568e6i 0.301996i
467467 3.80744e6 0.807868 0.403934 0.914788i 0.367642π-0.367642\pi
0.403934 + 0.914788i 0.367642π0.367642\pi
468468 718943. + 326727.i 0.151733 + 0.0689558i
469469 8.23138e6 1.72799
470470 3.04743e6i 0.636339i
471471 996531. 0.206985
472472 2.26089e6 0.467115
473473 3.62048e6i 0.744070i
474474 2.33918e6i 0.478210i
475475 4.08192e6i 0.830099i
476476 2.63380e6i 0.532801i
477477 −2.03163e6 −0.408836
478478 4.61495e6 0.923841
479479 4.85878e6i 0.967583i −0.875183 0.483792i 0.839259π-0.839259\pi
0.875183 0.483792i 0.160741π-0.160741\pi
480480 342059. 0.0677639
481481 −951798. + 2.09437e6i −0.187578 + 0.412754i
482482 −839013. −0.164494
483483 4.43332e6i 0.864692i
484484 −2.06184e6 −0.400076
485485 1.04038e6 0.200834
486486 236196.i 0.0453609i
487487 71255.1i 0.0136142i −0.999977 0.00680712i 0.997833π-0.997833\pi
0.999977 0.00680712i 0.00216679π-0.00216679\pi
488488 2.03145e6i 0.386150i
489489 1.44957e6i 0.274137i
490490 2.12486e6 0.399797
491491 −6.78617e6 −1.27034 −0.635172 0.772371i 0.719071π-0.719071\pi
−0.635172 + 0.772371i 0.719071π0.719071\pi
492492 53114.2i 0.00989231i
493493 −1.40305e6 −0.259990
494494 −5.18343e6 2.35563e6i −0.955651 0.434301i
495495 539356. 0.0989379
496496 188732.i 0.0344463i
497497 −1.04033e7 −1.88921
498498 440403. 0.0795750
499499 167487.i 0.0301114i 0.999887 + 0.0150557i 0.00479256π0.00479256\pi
−0.999887 + 0.0150557i 0.995207π0.995207\pi
500500 2.89350e6i 0.517605i
501501 4.19920e6i 0.747433i
502502 1.70862e6i 0.302613i
503503 1.00972e7 1.77942 0.889712 0.456522i 0.150905π-0.150905\pi
0.889712 + 0.456522i 0.150905π0.150905\pi
504504 914493. 0.160363
505505 3.80809e6i 0.664476i
506506 −2.00384e6 −0.347926
507507 −2.19762e6 2.51734e6i −0.379693 0.434933i
508508 −4.09031e6 −0.703229
509509 9.79146e6i 1.67515i 0.546324 + 0.837574i 0.316027π0.316027\pi
−0.546324 + 0.837574i 0.683973π0.683973\pi
510510 −1.24683e6 −0.212267
511511 −601758. −0.101946
512512 262144.i 0.0441942i
513513 1.70292e6i 0.285694i
514514 830887.i 0.138718i
515515 7.56433e6i 1.25676i
516516 −2.90601e6 −0.480477
517517 3.68253e6 0.605927
518518 2.66403e6i 0.436229i
519519 2.16690e6 0.353119
520520 −1.31774e6 598852.i −0.213708 0.0971205i
521521 −9.42352e6 −1.52096 −0.760481 0.649360i 0.775037π-0.775037\pi
−0.760481 + 0.649360i 0.775037π0.775037\pi
522522 487161.i 0.0782521i
523523 5.16320e6 0.825400 0.412700 0.910867i 0.364586π-0.364586\pi
0.412700 + 0.910867i 0.364586π0.364586\pi
524524 2.14811e6 0.341765
525525 2.77431e6i 0.439295i
526526 3.32685e6i 0.524286i
527527 687945.i 0.107901i
528528 413346.i 0.0645252i
529529 1.36094e6 0.211445
530530 3.72373e6 0.575823
531531 2.86143e6i 0.440400i
532532 −6.59330e6 −1.01001
533533 92988.3 204615.i 0.0141779 0.0311975i
534534 −2.22317e6 −0.337380
535535 5.46923e6i 0.826117i
536536 −2.98633e6 −0.448978
537537 2.39792e6 0.358838
538538 6.69458e6i 0.997167i
539539 2.56769e6i 0.380689i
540540 432919.i 0.0638884i
541541 1.03918e7i 1.52650i −0.646104 0.763250i 0.723603π-0.723603\pi
0.646104 0.763250i 0.276397π-0.276397\pi
542542 −279170. −0.0408198
543543 −578765. −0.0842369
544544 955536.i 0.138436i
545545 −291235. −0.0420002
546546 −3.52295e6 1.60103e6i −0.505738 0.229835i
547547 −7.33920e6 −1.04877 −0.524385 0.851481i 0.675705π-0.675705\pi
−0.524385 + 0.851481i 0.675705π0.675705\pi
548548 1.18997e6i 0.169272i
549549 2.57105e6 0.364066
550550 1.25397e6 0.176759
551551 3.51233e6i 0.492851i
552552 1.60840e6i 0.224671i
553553 1.14624e7i 1.59391i
554554 6.45520e6i 0.893584i
555555 −1.26115e6 −0.173793
556556 −581574. −0.0797844
557557 7.97266e6i 1.08884i −0.838812 0.544421i 0.816749π-0.816749\pi
0.838812 0.544421i 0.183251π-0.183251\pi
558558 238864. 0.0324763
559559 1.11950e7 + 5.08763e6i 1.51529 + 0.688630i
560560 −1.67615e6 −0.225862
561561 1.50668e6i 0.202122i
562562 −4.22621e6 −0.564430
563563 5.23721e6 0.696352 0.348176 0.937429i 0.386801π-0.386801\pi
0.348176 + 0.937429i 0.386801π0.386801\pi
564564 2.95581e6i 0.391273i
565565 7.15122e6i 0.942451i
566566 2.05096e6i 0.269102i
567567 1.15740e6i 0.151192i
568568 3.77430e6 0.490868
569569 −8.15975e6 −1.05657 −0.528283 0.849069i 0.677164π-0.677164\pi
−0.528283 + 0.849069i 0.677164π0.677164\pi
570570 3.12125e6i 0.402385i
571571 −6.61371e6 −0.848897 −0.424448 0.905452i 0.639532π-0.639532\pi
−0.424448 + 0.905452i 0.639532π0.639532\pi
572572 723656. 1.59236e6i 0.0924788 0.203494i
573573 8.32849e6 1.05969
574574 260269.i 0.0329719i
575575 −4.87942e6 −0.615458
576576 −331776. −0.0416667
577577 4.82867e6i 0.603793i 0.953341 + 0.301896i 0.0976196π0.0976196\pi
−0.953341 + 0.301896i 0.902380π0.902380\pi
578578 2.19642e6i 0.273461i
579579 5.69075e6i 0.705462i
580580 892907.i 0.110214i
581581 −2.15806e6 −0.265230
582582 −1.00910e6 −0.123489
583583 4.49978e6i 0.548302i
584584 218317. 0.0264883
585585 757922. 1.66776e6i 0.0915661 0.201485i
586586 −1.08507e7 −1.30531
587587 7.52814e6i 0.901763i 0.892584 + 0.450881i 0.148890π0.148890\pi
−0.892584 + 0.450881i 0.851110π0.851110\pi
588588 −2.06098e6 −0.245827
589589 −1.72216e6 −0.204544
590590 5.24466e6i 0.620279i
591591 900430.i 0.106043i
592592 966505.i 0.113344i
593593 8.22445e6i 0.960439i −0.877148 0.480220i 0.840557π-0.840557\pi
0.877148 0.480220i 0.159443π-0.159443\pi
594594 −523142. −0.0608350
595595 6.10972e6 0.707504
596596 3.54006e6i 0.408220i
597597 6.50769e6 0.747293
598598 −2.81587e6 + 6.19614e6i −0.322002 + 0.708546i
599599 −1.25959e6 −0.143437 −0.0717185 0.997425i 0.522848π-0.522848\pi
−0.0717185 + 0.997425i 0.522848π0.522848\pi
600600 1.00651e6i 0.114141i
601601 6.56541e6 0.741439 0.370720 0.928745i 0.379111π-0.379111\pi
0.370720 + 0.928745i 0.379111π0.379111\pi
602602 1.42400e7 1.60147
603603 3.77957e6i 0.423301i
604604 7.71916e6i 0.860950i
605605 4.78294e6i 0.531259i
606606 3.69361e6i 0.408573i
607607 −7.92228e6 −0.872727 −0.436364 0.899770i 0.643734π-0.643734\pi
−0.436364 + 0.899770i 0.643734π0.643734\pi
608608 2.39203e6 0.262427
609609 2.38718e6i 0.260821i
610610 −4.71243e6 −0.512767
611611 5.17482e6 1.13869e7i 0.560780 1.23396i
612612 1.20935e6 0.130519
613613 9.74465e6i 1.04741i −0.851901 0.523703i 0.824550π-0.824550\pi
0.851901 0.523703i 0.175450π-0.175450\pi
614614 −9.77898e6 −1.04682
615615 123211. 0.0131360
616616 2.02548e6i 0.215068i
617617 1.14334e7i 1.20910i 0.796569 + 0.604548i 0.206646π0.206646\pi
−0.796569 + 0.604548i 0.793354π0.793354\pi
618618 7.33693e6i 0.772757i
619619 1.07067e7i 1.12313i −0.827432 0.561566i 0.810199π-0.810199\pi
0.827432 0.561566i 0.189801π-0.189801\pi
620620 −437810. −0.0457411
621621 2.03563e6 0.211822
622622 7.96732e6i 0.825726i
623623 1.08939e7 1.12451
624624 1.27812e6 + 580849.i 0.131405 + 0.0597175i
625625 −1.25148e6 −0.128151
626626 6.65585e6i 0.678840i
627627 3.77174e6 0.383154
628628 1.77161e6 0.179254
629629 3.52299e6i 0.355046i
630630 2.12138e6i 0.212945i
631631 475976.i 0.0475895i −0.999717 0.0237948i 0.992425π-0.992425\pi
0.999717 0.0237948i 0.00757483π-0.00757483\pi
632632 4.15855e6i 0.414142i
633633 −2.33593e6 −0.231714
634634 1.28193e7 1.26660
635635 9.48844e6i 0.933814i
636636 −3.61179e6 −0.354062
637637 7.93963e6 + 3.60820e6i 0.775267 + 0.352324i
638638 1.07899e6 0.104946
639639 4.77684e6i 0.462795i
640640 608105. 0.0586852
641641 −1.90742e7 −1.83358 −0.916791 0.399368i 0.869230π-0.869230\pi
−0.916791 + 0.399368i 0.869230π0.869230\pi
642642 5.30481e6i 0.507963i
643643 1.16474e7i 1.11097i −0.831528 0.555483i 0.812534π-0.812534\pi
0.831528 0.555483i 0.187466π-0.187466\pi
644644 7.88146e6i 0.748845i
645645 6.74118e6i 0.638024i
646646 −8.71916e6 −0.822041
647647 1.83045e6 0.171908 0.0859540 0.996299i 0.472606π-0.472606\pi
0.0859540 + 0.996299i 0.472606π0.472606\pi
648648 419904.i 0.0392837i
649649 6.33768e6 0.590634
650650 1.76213e6 3.87745e6i 0.163589 0.359967i
651651 −1.17048e6 −0.108246
652652 2.57702e6i 0.237410i
653653 1.75829e7 1.61365 0.806824 0.590792i 0.201185π-0.201185\pi
0.806824 + 0.590792i 0.201185π0.201185\pi
654654 282479. 0.0258251
655655 4.98304e6i 0.453828i
656656 94425.2i 0.00856699i
657657 276307.i 0.0249735i
658658 1.44840e7i 1.30414i
659659 −1.84370e7 −1.65378 −0.826888 0.562367i 0.809891π-0.809891\pi
−0.826888 + 0.562367i 0.809891π0.809891\pi
660660 958855. 0.0856827
661661 1.08046e7i 0.961845i 0.876763 + 0.480922i 0.159698π0.159698\pi
−0.876763 + 0.480922i 0.840302π0.840302\pi
662662 1.41354e7 1.25362
663663 −4.65886e6 2.11724e6i −0.411619 0.187062i
664664 782939. 0.0689140
665665 1.52947e7i 1.34118i
666666 1.22323e6 0.106862
667667 −4.19855e6 −0.365413
668668 7.46525e6i 0.647296i
669669 9.64761e6i 0.833402i
670670 6.92749e6i 0.596196i
671671 5.69453e6i 0.488260i
672672 1.62576e6 0.138878
673673 −1.57676e7 −1.34193 −0.670964 0.741490i 0.734119π-0.734119\pi
−0.670964 + 0.741490i 0.734119π0.734119\pi
674674 4.40968e6i 0.373902i
675675 −1.27387e6 −0.107613
676676 −3.90688e6 4.47528e6i −0.328823 0.376663i
677677 2.86244e6 0.240029 0.120015 0.992772i 0.461706π-0.461706\pi
0.120015 + 0.992772i 0.461706π0.461706\pi
678678 6.93623e6i 0.579495i
679679 4.94480e6 0.411599
680680 −2.21659e6 −0.183829
681681 586890.i 0.0484941i
682682 529052.i 0.0435549i
683683 4.91047e6i 0.402783i 0.979511 + 0.201391i 0.0645463π0.0645463\pi
−0.979511 + 0.201391i 0.935454π0.935454\pi
684684 3.02742e6i 0.247419i
685685 2.76043e6 0.224776
686686 −1.76030e6 −0.142816
687687 1.72669e6i 0.139580i
688688 −5.16624e6 −0.416106
689689 1.39139e7 + 6.32325e6i 1.11661 + 0.507449i
690690 −3.73107e6 −0.298339
691691 1.79749e7i 1.43209i 0.698052 + 0.716047i 0.254051π0.254051\pi
−0.698052 + 0.716047i 0.745949π0.745949\pi
692692 3.85227e6 0.305810
693693 2.56349e6 0.202768
694694 1.26573e7i 0.997566i
695695 1.34910e6i 0.105945i
696696 866064.i 0.0677683i
697697 344188.i 0.0268357i
698698 7.90102e6 0.613825
699699 −3.18528e6 −0.246579
700700 4.93210e6i 0.380440i
701701 −8.74640e6 −0.672255 −0.336128 0.941816i 0.609117π-0.609117\pi
−0.336128 + 0.941816i 0.609117π0.609117\pi
702702 −735137. + 1.61762e6i −0.0563022 + 0.123889i
703703 −8.81925e6 −0.673044
704704 734838.i 0.0558805i
705705 6.85671e6 0.519569
706706 8.96061e6 0.676590
707707 1.80994e7i 1.36181i
708708 5.08699e6i 0.381398i
709709 2.43397e6i 0.181844i −0.995858 0.0909221i 0.971019π-0.971019\pi
0.995858 0.0909221i 0.0289814π-0.0289814\pi
710710 8.75538e6i 0.651822i
711711 −5.26316e6 −0.390457
712712 −3.95230e6 −0.292180
713713 2.05863e6i 0.151654i
714714 −5.92605e6 −0.435030
715715 −3.69386e6 1.67869e6i −0.270218 0.122802i
716716 4.26297e6 0.310763
717717 1.03836e7i 0.754313i
718718 −8.55079e6 −0.619006
719719 6.81620e6 0.491723 0.245861 0.969305i 0.420929π-0.420929\pi
0.245861 + 0.969305i 0.420929π0.420929\pi
720720 769633.i 0.0553290i
721721 3.59523e7i 2.57566i
722722 1.19226e7i 0.851196i
723723 1.88778e6i 0.134309i
724724 −1.02892e6 −0.0729513
725725 2.62739e6 0.185643
726726 4.63915e6i 0.326661i
727727 −4.33605e6 −0.304269 −0.152135 0.988360i 0.548615π-0.548615\pi
−0.152135 + 0.988360i 0.548615π0.548615\pi
728728 −6.26303e6 2.84627e6i −0.437982 0.199043i
729729 531441. 0.0370370
730730 506437.i 0.0351737i
731731 1.88314e7 1.30343
732732 4.57076e6 0.315290
733733 2.01689e7i 1.38651i 0.720693 + 0.693254i 0.243824π0.243824\pi
−0.720693 + 0.693254i 0.756176π0.756176\pi
734734 6.82008e6i 0.467250i
735735 4.78093e6i 0.326433i
736736 2.85938e6i 0.194570i
737737 −8.37122e6 −0.567702
738738 −119507. −0.00807704
739739 2.05423e7i 1.38369i 0.722048 + 0.691843i 0.243201π0.243201\pi
−0.722048 + 0.691843i 0.756799π0.756799\pi
740740 −2.24204e6 −0.150509
741741 5.30018e6 1.16627e7i 0.354605 0.780286i
742742 1.76984e7 1.18012
743743 1.48776e7i 0.988689i 0.869266 + 0.494345i 0.164592π0.164592\pi
−0.869266 + 0.494345i 0.835408π0.835408\pi
744744 424648. 0.0281253
745745 −8.21200e6 −0.542074
746746 9.26012e6i 0.609213i
747747 990907.i 0.0649728i
748748 2.67855e6i 0.175043i
749749 2.59946e7i 1.69308i
750750 6.51037e6 0.422623
751751 −2.94950e7 −1.90831 −0.954154 0.299315i 0.903242π-0.903242\pi
−0.954154 + 0.299315i 0.903242π0.903242\pi
752752 5.25478e6i 0.338852i
753753 −3.84440e6 −0.247082
754754 1.51624e6 3.33639e6i 0.0971269 0.213722i
755755 −1.79064e7 −1.14325
756756 2.05761e6i 0.130936i
757757 2.01715e7 1.27938 0.639690 0.768633i 0.279063π-0.279063\pi
0.639690 + 0.768633i 0.279063π0.279063\pi
758758 1.72942e7 1.09327
759759 4.50864e6i 0.284080i
760760 5.54889e6i 0.348476i
761761 2.06956e7i 1.29544i 0.761880 + 0.647719i 0.224277π0.224277\pi
−0.761880 + 0.647719i 0.775723π0.775723\pi
762762 9.20319e6i 0.574184i
763763 −1.38420e6 −0.0860772
764764 1.48062e7 0.917720
765765 2.80538e6i 0.173316i
766766 −8.01403e6 −0.493491
767767 8.90592e6 1.95969e7i 0.546626 1.20282i
768768 −589824. −0.0360844
769769 1.69130e7i 1.03135i −0.856785 0.515674i 0.827542π-0.827542\pi
0.856785 0.515674i 0.172458π-0.172458\pi
770770 −4.69857e6 −0.285587
771771 −1.86950e6 −0.113263
772772 1.01169e7i 0.610948i
773773 3.10969e7i 1.87184i −0.352212 0.935920i 0.614570π-0.614570\pi
0.352212 0.935920i 0.385430π-0.385430\pi
774774 6.53852e6i 0.392308i
775775 1.28826e6i 0.0770458i
776776 −1.79396e6 −0.106945
777777 −5.99407e6 −0.356180
778778 6.82786e6i 0.404423i
779779 861619. 0.0508712
780780 1.34742e6 2.96490e6i 0.0792986 0.174491i
781781 1.05800e7 0.620669
782782 1.04227e7i 0.609483i
783783 −1.09611e6 −0.0638926
784784 −3.66396e6 −0.212893
785785 4.10967e6i 0.238031i
786786 4.83324e6i 0.279050i
787787 628913.i 0.0361954i −0.999836 0.0180977i 0.994239π-0.994239\pi
0.999836 0.0180977i 0.00576099π-0.00576099\pi
788788 1.60076e6i 0.0918357i
789789 7.48541e6 0.428078
790790 9.64674e6 0.549937
791791 3.39888e7i 1.93150i
792792 −930030. −0.0526846
793793 −1.76082e7 8.00215e6i −0.994335 0.451881i
794794 −4.89708e6 −0.275668
795795 8.37840e6i 0.470157i
796796 1.15692e7 0.647174
797797 −2.43866e7 −1.35990 −0.679948 0.733260i 0.737998π-0.737998\pi
−0.679948 + 0.733260i 0.737998π0.737998\pi
798798 1.48349e7i 0.824666i
799799 1.91541e7i 1.06144i
800800 1.78936e6i 0.0988489i
801801 5.00213e6i 0.275470i
802802 1.66239e7 0.912637
803803 611982. 0.0334927
804804 6.71923e6i 0.366589i
805805 1.82829e7 0.994388
806806 −1.63590e6 743442.i −0.0886989 0.0403097i
807807 −1.50628e7 −0.814183
808808 6.56642e6i 0.353835i
809809 −2.86033e7 −1.53654 −0.768272 0.640123i 0.778883π-0.778883\pi
−0.768272 + 0.640123i 0.778883π0.778883\pi
810810 −974067. −0.0521647
811811 5.97923e6i 0.319222i 0.987180 + 0.159611i 0.0510240π0.0510240\pi
−0.987180 + 0.159611i 0.948976π0.948976\pi
812812 4.24387e6i 0.225877i
813813 628133.i 0.0333292i
814814 2.70929e6i 0.143316i
815815 −5.97800e6 −0.315255
816816 2.14996e6 0.113033
817817 4.71414e7i 2.47085i
818818 −2.60971e7 −1.36367
819819 3.60231e6 7.92665e6i 0.187660 0.412933i
820820 219042. 0.0113761
821821 1.46642e7i 0.759279i −0.925134 0.379640i 0.876048π-0.876048\pi
0.925134 0.379640i 0.123952π-0.123952\pi
822822 −2.67744e6 −0.138210
823823 1.80149e7 0.927113 0.463557 0.886067i 0.346573π-0.346573\pi
0.463557 + 0.886067i 0.346573π0.346573\pi
824824 1.30434e7i 0.669227i
825825 2.82144e6i 0.144323i
826826 2.49272e7i 1.27123i
827827 9.93027e6i 0.504891i 0.967611 + 0.252445i 0.0812348π0.0812348\pi
−0.967611 + 0.252445i 0.918765π0.918765\pi
828828 3.61890e6 0.183443
829829 2.97991e7 1.50597 0.752986 0.658037i 0.228613π-0.228613\pi
0.752986 + 0.658037i 0.228613π0.228613\pi
830830 1.81621e6i 0.0915106i
831831 −1.45242e7 −0.729608
832832 2.27221e6 + 1.03262e6i 0.113800 + 0.0517169i
833833 1.33554e7 0.666877
834834 1.30854e6i 0.0651437i
835835 −1.73174e7 −0.859542
836836 6.70532e6 0.331821
837837 537445.i 0.0265168i
838838 1.91180e7i 0.940443i
839839 1.35283e7i 0.663496i −0.943368 0.331748i 0.892362π-0.892362\pi
0.943368 0.331748i 0.107638π-0.107638\pi
840840 3.77135e6i 0.184416i
841841 −1.82504e7 −0.889779
842842 −1.36623e7 −0.664117
843843 9.50897e6i 0.460855i
844844 −4.15277e6 −0.200670
845845 −1.03815e7 + 9.06293e6i −0.500169 + 0.436643i
846846 −6.65058e6 −0.319473
847847 2.27327e7i 1.08879i
848848 −6.42096e6 −0.306627
849849 4.61467e6 0.219721
850850 6.52235e6i 0.309640i
851851 1.05423e7i 0.499013i
852852 8.49217e6i 0.400792i
853853 9.03698e6i 0.425257i −0.977133 0.212628i 0.931798π-0.931798\pi
0.977133 0.212628i 0.0682023π-0.0682023\pi
854854 −2.23976e7 −1.05089
855855 7.02282e6 0.328546
856856 9.43078e6i 0.439909i
857857 −9.55364e6 −0.444341 −0.222171 0.975008i 0.571314π-0.571314\pi
−0.222171 + 0.975008i 0.571314π0.571314\pi
858858 3.58281e6 + 1.62823e6i 0.166152 + 0.0755086i
859859 3.73649e6 0.172775 0.0863874 0.996262i 0.472468π-0.472468\pi
0.0863874 + 0.996262i 0.472468π0.472468\pi
860860 1.19843e7i 0.552545i
861861 585606. 0.0269214
862862 2.48565e7 1.13939
863863 3.44318e7i 1.57374i 0.617120 + 0.786869i 0.288299π0.288299\pi
−0.617120 + 0.786869i 0.711701π0.711701\pi
864864 746496.i 0.0340207i
865865 8.93626e6i 0.406084i
866866 2.33345e7i 1.05731i
867867 4.94194e6 0.223280
868868 −2.08085e6 −0.0937438
869869 1.16572e7i 0.523653i
870870 2.00904e6 0.0899892
871871 −1.17635e7 + 2.58849e7i −0.525403 + 1.15612i
872872 502186. 0.0223652
873873 2.27048e6i 0.100828i
874874 −2.60915e7 −1.15537
875875 −3.19020e7 −1.40863
876876 491213.i 0.0216276i
877877 2.12195e7i 0.931613i −0.884887 0.465807i 0.845764π-0.845764\pi
0.884887 0.465807i 0.154236π-0.154236\pi
878878 3.06769e6i 0.134300i
879879 2.44141e7i 1.06578i
880880 1.70463e6 0.0742034
881881 3.83954e7 1.66663 0.833315 0.552798i 0.186440π-0.186440\pi
0.833315 + 0.552798i 0.186440π0.186440\pi
882882 4.63720e6i 0.200717i
883883 3.58049e7 1.54540 0.772699 0.634773i 0.218906π-0.218906\pi
0.772699 + 0.634773i 0.218906π0.218906\pi
884884 −8.28241e6 3.76398e6i −0.356472 0.162001i
885885 1.18005e7 0.506456
886886 4.08168e6i 0.174685i
887887 2.52692e7 1.07841 0.539203 0.842176i 0.318726π-0.318726\pi
0.539203 + 0.842176i 0.318726π0.318726\pi
888888 2.17464e6 0.0925453
889889 4.50973e7i 1.91380i
890890 9.16830e6i 0.387984i
891891 1.17707e6i 0.0496715i
892892 1.71513e7i 0.721747i
893893 4.79493e7 2.01212
894894 7.96512e6 0.333310
895895 9.88896e6i 0.412661i
896896 2.89025e6 0.120272
897897 −1.39413e7 6.33570e6i −0.578525 0.262914i
898898 −1.67249e7 −0.692107
899899 1.10850e6i 0.0457441i
900900 −2.26465e6 −0.0931956
901901 2.34049e7 0.960495
902902 264691.i 0.0108324i
903903 3.20400e7i 1.30759i
904904 1.23311e7i 0.501857i
905905 2.38681e6i 0.0968717i
906906 1.73681e7 0.702962
907907 −2.49639e7 −1.00761 −0.503806 0.863817i 0.668067π-0.668067\pi
−0.503806 + 0.863817i 0.668067π0.668067\pi
908908 1.04336e6i 0.0419971i
909909 8.31063e6 0.333599
910910 −6.60259e6 + 1.45286e7i −0.264308 + 0.581594i
911911 −807347. −0.0322303 −0.0161151 0.999870i 0.505130π-0.505130\pi
−0.0161151 + 0.999870i 0.505130π0.505130\pi
912912 5.38208e6i 0.214271i
913913 2.19472e6 0.0871370
914914 −1.43333e7 −0.567520
915915 1.06030e7i 0.418673i
916916 3.06967e6i 0.120879i
917917 2.36838e7i 0.930095i
918918 2.72104e6i 0.106568i
919919 −2.92665e6 −0.114309 −0.0571547 0.998365i 0.518203π-0.518203\pi
−0.0571547 + 0.998365i 0.518203π0.518203\pi
920920 −6.63301e6 −0.258369
921921 2.20027e7i 0.854726i
922922 6.69836e6 0.259502
923923 1.48674e7 3.27149e7i 0.574423 1.26398i
924924 4.55732e6 0.175602
925925 6.59722e6i 0.253517i
926926 −2.64945e7 −1.01538
927927 −1.65081e7 −0.630953
928928 1.53967e6i 0.0586891i
929929 2.03515e7i 0.773672i 0.922148 + 0.386836i 0.126432π0.126432\pi
−0.922148 + 0.386836i 0.873568π0.873568\pi
930930 985072.i 0.0373474i
931931 3.34332e7i 1.26417i
932932 −5.66273e6 −0.213543
933933 −1.79265e7 −0.674203
934934 1.52297e7i 0.571249i
935935 −6.21352e6 −0.232439
936936 −1.30691e6 + 2.87577e6i −0.0487591 + 0.107291i
937937 −659937. −0.0245558 −0.0122779 0.999925i 0.503908π-0.503908\pi
−0.0122779 + 0.999925i 0.503908π0.503908\pi
938938 3.29255e7i 1.22187i
939939 −1.49757e7 −0.554271
940940 1.21897e7 0.449960
941941 3.40074e7i 1.25199i −0.779828 0.625994i 0.784694π-0.784694\pi
0.779828 0.625994i 0.215306π-0.215306\pi
942942 3.98612e6i 0.146360i
943943 1.02996e6i 0.0377173i
944944 9.04354e6i 0.330300i
945945 4.77311e6 0.173869
946946 −1.44819e7 −0.526137
947947 1.44758e7i 0.524527i −0.964996 0.262263i 0.915531π-0.915531\pi
0.964996 0.262263i 0.0844689π-0.0844689\pi
948948 −9.35673e6 −0.338145
949949 859978. 1.89233e6i 0.0309972 0.0682073i
950950 1.63277e7 0.586969
951951 2.88434e7i 1.03418i
952952 −1.05352e7 −0.376747
953953 1.88172e7 0.671156 0.335578 0.942012i 0.391068π-0.391068\pi
0.335578 + 0.942012i 0.391068π0.391068\pi
954954 8.12652e6i 0.289090i
955955 3.43465e7i 1.21864i
956956 1.84598e7i 0.653255i
957957 2.42774e6i 0.0856883i
958958 1.94351e7 0.684185
959959 1.31200e7 0.460666
960960 1.36824e6i 0.0479163i
961961 2.80856e7 0.981015
962962 −8.37748e6 3.80719e6i −0.291861 0.132638i
963963 −1.19358e7 −0.414750
964964 3.35605e6i 0.116315i
965965 2.34685e7 0.811275
966966 −1.77333e7 −0.611430
967967 4.30702e7i 1.48119i 0.671952 + 0.740595i 0.265456π0.265456\pi
−0.671952 + 0.740595i 0.734544π0.734544\pi
968968 8.24738e6i 0.282896i
969969 1.96181e7i 0.671194i
970970 4.16152e6i 0.142011i
971971 2.85474e7 0.971668 0.485834 0.874051i 0.338516π-0.338516\pi
0.485834 + 0.874051i 0.338516π0.338516\pi
972972 944784. 0.0320750
973973 6.41210e6i 0.217129i
974974 285020. 0.00962673
975975 8.72426e6 + 3.96479e6i 0.293912 + 0.133570i
976976 8.12580e6 0.273050
977977 4.23484e7i 1.41939i 0.704510 + 0.709694i 0.251167π0.251167\pi
−0.704510 + 0.709694i 0.748833π0.748833\pi
978978 5.79829e6 0.193844
979979 −1.10790e7 −0.369441
980980 8.49942e6i 0.282699i
981981 635579.i 0.0210861i
982982 2.71447e7i 0.898269i
983983 1.39064e7i 0.459020i −0.973306 0.229510i 0.926288π-0.926288\pi
0.973306 0.229510i 0.0737124π-0.0737124\pi
984984 −212457. −0.00699492
985985 −3.71335e6 −0.121948
986986 5.61222e6i 0.183841i
987987 3.25891e7 1.06483
988988 9.42254e6 2.07337e7i 0.307097 0.675748i
989989 5.63516e7 1.83196
990990 2.15742e6i 0.0699597i
991991 −3.49486e6 −0.113043 −0.0565217 0.998401i 0.518001π-0.518001\pi
−0.0565217 + 0.998401i 0.518001π0.518001\pi
992992 754930. 0.0243572
993993 3.18047e7i 1.02357i
994994 4.16132e7i 1.33587i
995995 2.68376e7i 0.859380i
996996 1.76161e6i 0.0562681i
997997 77983.9 0.00248466 0.00124233 0.999999i 0.499605π-0.499605\pi
0.00124233 + 0.999999i 0.499605π0.499605\pi
998998 −669949. −0.0212920
999999 2.75227e6i 0.0872525i
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 78.6.b.a.25.6 yes 6
3.2 odd 2 234.6.b.c.181.1 6
4.3 odd 2 624.6.c.d.337.5 6
13.5 odd 4 1014.6.a.q.1.3 3
13.8 odd 4 1014.6.a.o.1.1 3
13.12 even 2 inner 78.6.b.a.25.1 6
39.38 odd 2 234.6.b.c.181.6 6
52.51 odd 2 624.6.c.d.337.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.6.b.a.25.1 6 13.12 even 2 inner
78.6.b.a.25.6 yes 6 1.1 even 1 trivial
234.6.b.c.181.1 6 3.2 odd 2
234.6.b.c.181.6 6 39.38 odd 2
624.6.c.d.337.2 6 52.51 odd 2
624.6.c.d.337.5 6 4.3 odd 2
1014.6.a.o.1.1 3 13.8 odd 4
1014.6.a.q.1.3 3 13.5 odd 4