# Properties

 Label 507.4.a.h.1.1 Level $507$ Weight $4$ Character 507.1 Self dual yes Analytic conductor $29.914$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.a (trivial)

## 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: yes Analytic conductor: $$29.9139683729$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.20905$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} -48.1187 q^{10} -25.8785 q^{11} +29.1483 q^{12} -47.3699 q^{14} +34.2966 q^{15} -47.3262 q^{16} -20.3276 q^{17} -37.8814 q^{18} -154.712 q^{19} +111.076 q^{20} +33.7629 q^{21} +108.924 q^{22} -180.418 q^{23} -21.6695 q^{24} +5.69520 q^{25} +27.0000 q^{27} +109.348 q^{28} -20.4522 q^{29} -144.356 q^{30} -266.424 q^{31} +256.984 q^{32} -77.6355 q^{33} +85.5599 q^{34} +128.661 q^{35} +87.4449 q^{36} -115.984 q^{37} +651.190 q^{38} -82.5765 q^{40} -391.184 q^{41} -142.110 q^{42} +151.407 q^{43} -251.438 q^{44} +102.890 q^{45} +759.390 q^{46} +467.365 q^{47} -141.979 q^{48} -216.341 q^{49} -23.9714 q^{50} -60.9828 q^{51} +79.9842 q^{53} -113.644 q^{54} -295.848 q^{55} -81.2915 q^{56} -464.136 q^{57} +86.0843 q^{58} +873.710 q^{59} +333.229 q^{60} -187.068 q^{61} +1121.39 q^{62} +101.289 q^{63} -703.047 q^{64} +326.772 q^{66} +609.204 q^{67} -197.505 q^{68} -541.255 q^{69} -541.542 q^{70} -248.038 q^{71} -65.0084 q^{72} -852.765 q^{73} +488.181 q^{74} +17.0856 q^{75} -1503.20 q^{76} -291.244 q^{77} -331.221 q^{79} -541.043 q^{80} +81.0000 q^{81} +1646.51 q^{82} +435.432 q^{83} +328.044 q^{84} -232.389 q^{85} -637.281 q^{86} -61.3566 q^{87} +186.924 q^{88} -259.233 q^{89} -433.068 q^{90} -1752.96 q^{92} -799.273 q^{93} -1967.16 q^{94} -1768.70 q^{95} +770.951 q^{96} -1225.17 q^{97} +910.589 q^{98} -232.907 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 9 * q^3 + 10 * q^4 - 4 * q^5 - 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 $$3 q - 2 q^{2} + 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 4 q^{10} + 16 q^{11} + 30 q^{12} - 176 q^{14} - 12 q^{15} - 110 q^{16} - 146 q^{17} - 18 q^{18} - 94 q^{19} + 244 q^{20} - 90 q^{21} - 56 q^{22} - 48 q^{23} + 18 q^{24} + 145 q^{25} + 81 q^{27} - 80 q^{28} - 2 q^{29} - 12 q^{30} - 302 q^{31} - 154 q^{32} + 48 q^{33} - 164 q^{34} + 80 q^{35} + 90 q^{36} - 374 q^{37} + 312 q^{38} - 516 q^{40} - 480 q^{41} - 528 q^{42} - 260 q^{43} - 712 q^{44} - 36 q^{45} + 1104 q^{46} + 24 q^{47} - 330 q^{48} + 447 q^{49} - 814 q^{50} - 438 q^{51} - 678 q^{53} - 54 q^{54} - 1552 q^{55} + 96 q^{56} - 282 q^{57} + 628 q^{58} + 1788 q^{59} + 732 q^{60} + 230 q^{61} + 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} - 460 q^{68} - 144 q^{69} - 1216 q^{70} + 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 435 q^{75} - 2392 q^{76} + 112 q^{77} - 24 q^{79} - 1100 q^{80} + 243 q^{81} + 564 q^{82} + 796 q^{83} - 240 q^{84} + 248 q^{85} - 1800 q^{86} - 6 q^{87} + 1608 q^{88} - 1436 q^{89} - 36 q^{90} - 1296 q^{92} - 906 q^{93} - 1920 q^{94} - 4032 q^{95} - 462 q^{96} - 3242 q^{97} + 5070 q^{98} + 144 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 9 * q^3 + 10 * q^4 - 4 * q^5 - 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 - 4 * q^10 + 16 * q^11 + 30 * q^12 - 176 * q^14 - 12 * q^15 - 110 * q^16 - 146 * q^17 - 18 * q^18 - 94 * q^19 + 244 * q^20 - 90 * q^21 - 56 * q^22 - 48 * q^23 + 18 * q^24 + 145 * q^25 + 81 * q^27 - 80 * q^28 - 2 * q^29 - 12 * q^30 - 302 * q^31 - 154 * q^32 + 48 * q^33 - 164 * q^34 + 80 * q^35 + 90 * q^36 - 374 * q^37 + 312 * q^38 - 516 * q^40 - 480 * q^41 - 528 * q^42 - 260 * q^43 - 712 * q^44 - 36 * q^45 + 1104 * q^46 + 24 * q^47 - 330 * q^48 + 447 * q^49 - 814 * q^50 - 438 * q^51 - 678 * q^53 - 54 * q^54 - 1552 * q^55 + 96 * q^56 - 282 * q^57 + 628 * q^58 + 1788 * q^59 + 732 * q^60 + 230 * q^61 + 1952 * q^62 - 270 * q^63 - 750 * q^64 - 168 * q^66 - 74 * q^67 - 460 * q^68 - 144 * q^69 - 1216 * q^70 + 948 * q^71 + 54 * q^72 + 222 * q^73 + 1724 * q^74 + 435 * q^75 - 2392 * q^76 + 112 * q^77 - 24 * q^79 - 1100 * q^80 + 243 * q^81 + 564 * q^82 + 796 * q^83 - 240 * q^84 + 248 * q^85 - 1800 * q^86 - 6 * q^87 + 1608 * q^88 - 1436 * q^89 - 36 * q^90 - 1296 * q^92 - 906 * q^93 - 1920 * q^94 - 4032 * q^95 - 462 * q^96 - 3242 * q^97 + 5070 * q^98 + 144 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.20905 −1.48812 −0.744062 0.668111i $$-0.767103\pi$$
−0.744062 + 0.668111i $$0.767103\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 9.71610 1.21451
$$5$$ 11.4322 1.02253 0.511264 0.859424i $$-0.329178\pi$$
0.511264 + 0.859424i $$0.329178\pi$$
$$6$$ −12.6271 −0.859169
$$7$$ 11.2543 0.607675 0.303838 0.952724i $$-0.401732\pi$$
0.303838 + 0.952724i $$0.401732\pi$$
$$8$$ −7.22315 −0.319221
$$9$$ 9.00000 0.333333
$$10$$ −48.1187 −1.52165
$$11$$ −25.8785 −0.709333 −0.354666 0.934993i $$-0.615406\pi$$
−0.354666 + 0.934993i $$0.615406\pi$$
$$12$$ 29.1483 0.701199
$$13$$ 0 0
$$14$$ −47.3699 −0.904296
$$15$$ 34.2966 0.590356
$$16$$ −47.3262 −0.739472
$$17$$ −20.3276 −0.290010 −0.145005 0.989431i $$-0.546320\pi$$
−0.145005 + 0.989431i $$0.546320\pi$$
$$18$$ −37.8814 −0.496041
$$19$$ −154.712 −1.86807 −0.934035 0.357181i $$-0.883738\pi$$
−0.934035 + 0.357181i $$0.883738\pi$$
$$20$$ 111.076 1.24187
$$21$$ 33.7629 0.350841
$$22$$ 108.924 1.05558
$$23$$ −180.418 −1.63565 −0.817823 0.575471i $$-0.804819\pi$$
−0.817823 + 0.575471i $$0.804819\pi$$
$$24$$ −21.6695 −0.184302
$$25$$ 5.69520 0.0455616
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 109.348 0.738029
$$29$$ −20.4522 −0.130961 −0.0654806 0.997854i $$-0.520858\pi$$
−0.0654806 + 0.997854i $$0.520858\pi$$
$$30$$ −144.356 −0.878523
$$31$$ −266.424 −1.54359 −0.771794 0.635873i $$-0.780640\pi$$
−0.771794 + 0.635873i $$0.780640\pi$$
$$32$$ 256.984 1.41965
$$33$$ −77.6355 −0.409534
$$34$$ 85.5599 0.431571
$$35$$ 128.661 0.621364
$$36$$ 87.4449 0.404837
$$37$$ −115.984 −0.515340 −0.257670 0.966233i $$-0.582955\pi$$
−0.257670 + 0.966233i $$0.582955\pi$$
$$38$$ 651.190 2.77992
$$39$$ 0 0
$$40$$ −82.5765 −0.326412
$$41$$ −391.184 −1.49006 −0.745032 0.667029i $$-0.767566\pi$$
−0.745032 + 0.667029i $$0.767566\pi$$
$$42$$ −142.110 −0.522095
$$43$$ 151.407 0.536963 0.268482 0.963285i $$-0.413478\pi$$
0.268482 + 0.963285i $$0.413478\pi$$
$$44$$ −251.438 −0.861494
$$45$$ 102.890 0.340842
$$46$$ 759.390 2.43404
$$47$$ 467.365 1.45047 0.725236 0.688500i $$-0.241731\pi$$
0.725236 + 0.688500i $$0.241731\pi$$
$$48$$ −141.979 −0.426934
$$49$$ −216.341 −0.630731
$$50$$ −23.9714 −0.0678012
$$51$$ −60.9828 −0.167437
$$52$$ 0 0
$$53$$ 79.9842 0.207296 0.103648 0.994614i $$-0.466949\pi$$
0.103648 + 0.994614i $$0.466949\pi$$
$$54$$ −113.644 −0.286390
$$55$$ −295.848 −0.725312
$$56$$ −81.2915 −0.193983
$$57$$ −464.136 −1.07853
$$58$$ 86.0843 0.194887
$$59$$ 873.710 1.92792 0.963960 0.266045i $$-0.0857171\pi$$
0.963960 + 0.266045i $$0.0857171\pi$$
$$60$$ 333.229 0.716995
$$61$$ −187.068 −0.392649 −0.196325 0.980539i $$-0.562901\pi$$
−0.196325 + 0.980539i $$0.562901\pi$$
$$62$$ 1121.39 2.29705
$$63$$ 101.289 0.202558
$$64$$ −703.047 −1.37314
$$65$$ 0 0
$$66$$ 326.772 0.609437
$$67$$ 609.204 1.11084 0.555418 0.831571i $$-0.312558\pi$$
0.555418 + 0.831571i $$0.312558\pi$$
$$68$$ −197.505 −0.352221
$$69$$ −541.255 −0.944340
$$70$$ −541.542 −0.924667
$$71$$ −248.038 −0.414601 −0.207301 0.978277i $$-0.566468\pi$$
−0.207301 + 0.978277i $$0.566468\pi$$
$$72$$ −65.0084 −0.106407
$$73$$ −852.765 −1.36724 −0.683621 0.729838i $$-0.739596\pi$$
−0.683621 + 0.729838i $$0.739596\pi$$
$$74$$ 488.181 0.766890
$$75$$ 17.0856 0.0263050
$$76$$ −1503.20 −2.26880
$$77$$ −291.244 −0.431044
$$78$$ 0 0
$$79$$ −331.221 −0.471712 −0.235856 0.971788i $$-0.575789\pi$$
−0.235856 + 0.971788i $$0.575789\pi$$
$$80$$ −541.043 −0.756130
$$81$$ 81.0000 0.111111
$$82$$ 1646.51 2.21740
$$83$$ 435.432 0.575842 0.287921 0.957654i $$-0.407036\pi$$
0.287921 + 0.957654i $$0.407036\pi$$
$$84$$ 328.044 0.426101
$$85$$ −232.389 −0.296543
$$86$$ −637.281 −0.799067
$$87$$ −61.3566 −0.0756105
$$88$$ 186.924 0.226434
$$89$$ −259.233 −0.308749 −0.154375 0.988012i $$-0.549336\pi$$
−0.154375 + 0.988012i $$0.549336\pi$$
$$90$$ −433.068 −0.507216
$$91$$ 0 0
$$92$$ −1752.96 −1.98651
$$93$$ −799.273 −0.891191
$$94$$ −1967.16 −2.15848
$$95$$ −1768.70 −1.91015
$$96$$ 770.951 0.819634
$$97$$ −1225.17 −1.28245 −0.641223 0.767355i $$-0.721572\pi$$
−0.641223 + 0.767355i $$0.721572\pi$$
$$98$$ 910.589 0.938606
$$99$$ −232.907 −0.236444
$$100$$ 55.3351 0.0553351
$$101$$ 645.416 0.635855 0.317927 0.948115i $$-0.397013\pi$$
0.317927 + 0.948115i $$0.397013\pi$$
$$102$$ 256.680 0.249167
$$103$$ −511.137 −0.488969 −0.244484 0.969653i $$-0.578619\pi$$
−0.244484 + 0.969653i $$0.578619\pi$$
$$104$$ 0 0
$$105$$ 385.984 0.358745
$$106$$ −336.657 −0.308482
$$107$$ 608.195 0.549499 0.274750 0.961516i $$-0.411405\pi$$
0.274750 + 0.961516i $$0.411405\pi$$
$$108$$ 262.335 0.233733
$$109$$ 1300.04 1.14239 0.571197 0.820813i $$-0.306479\pi$$
0.571197 + 0.820813i $$0.306479\pi$$
$$110$$ 1245.24 1.07935
$$111$$ −347.951 −0.297532
$$112$$ −532.623 −0.449359
$$113$$ 42.1953 0.0351274 0.0175637 0.999846i $$-0.494409\pi$$
0.0175637 + 0.999846i $$0.494409\pi$$
$$114$$ 1953.57 1.60499
$$115$$ −2062.58 −1.67249
$$116$$ −198.716 −0.159054
$$117$$ 0 0
$$118$$ −3677.49 −2.86899
$$119$$ −228.773 −0.176232
$$120$$ −247.729 −0.188454
$$121$$ −661.303 −0.496847
$$122$$ 787.378 0.584311
$$123$$ −1173.55 −0.860289
$$124$$ −2588.61 −1.87471
$$125$$ −1363.92 −0.975939
$$126$$ −426.329 −0.301432
$$127$$ −311.018 −0.217310 −0.108655 0.994080i $$-0.534654\pi$$
−0.108655 + 0.994080i $$0.534654\pi$$
$$128$$ 903.291 0.623753
$$129$$ 454.222 0.310016
$$130$$ 0 0
$$131$$ 2000.98 1.33456 0.667278 0.744809i $$-0.267459\pi$$
0.667278 + 0.744809i $$0.267459\pi$$
$$132$$ −754.314 −0.497384
$$133$$ −1741.17 −1.13518
$$134$$ −2564.17 −1.65306
$$135$$ 308.669 0.196785
$$136$$ 146.829 0.0925773
$$137$$ −1038.53 −0.647644 −0.323822 0.946118i $$-0.604968\pi$$
−0.323822 + 0.946118i $$0.604968\pi$$
$$138$$ 2278.17 1.40529
$$139$$ −2858.46 −1.74426 −0.872128 0.489277i $$-0.837261\pi$$
−0.872128 + 0.489277i $$0.837261\pi$$
$$140$$ 1250.09 0.754655
$$141$$ 1402.09 0.837430
$$142$$ 1044.00 0.616978
$$143$$ 0 0
$$144$$ −425.936 −0.246491
$$145$$ −233.814 −0.133911
$$146$$ 3589.33 2.03462
$$147$$ −649.022 −0.364153
$$148$$ −1126.91 −0.625887
$$149$$ −743.479 −0.408780 −0.204390 0.978890i $$-0.565521\pi$$
−0.204390 + 0.978890i $$0.565521\pi$$
$$150$$ −71.9141 −0.0391451
$$151$$ −2277.24 −1.22728 −0.613640 0.789586i $$-0.710295\pi$$
−0.613640 + 0.789586i $$0.710295\pi$$
$$152$$ 1117.51 0.596328
$$153$$ −182.948 −0.0966700
$$154$$ 1225.86 0.641447
$$155$$ −3045.82 −1.57836
$$156$$ 0 0
$$157$$ 3173.51 1.61321 0.806605 0.591091i $$-0.201303\pi$$
0.806605 + 0.591091i $$0.201303\pi$$
$$158$$ 1394.12 0.701966
$$159$$ 239.953 0.119682
$$160$$ 2937.89 1.45163
$$161$$ −2030.48 −0.993941
$$162$$ −340.933 −0.165347
$$163$$ 2314.65 1.11225 0.556126 0.831098i $$-0.312287\pi$$
0.556126 + 0.831098i $$0.312287\pi$$
$$164$$ −3800.78 −1.80970
$$165$$ −887.545 −0.418759
$$166$$ −1832.76 −0.856925
$$167$$ 2665.65 1.23517 0.617587 0.786502i $$-0.288110\pi$$
0.617587 + 0.786502i $$0.288110\pi$$
$$168$$ −243.874 −0.111996
$$169$$ 0 0
$$170$$ 978.138 0.441293
$$171$$ −1392.41 −0.622690
$$172$$ 1471.09 0.652148
$$173$$ −165.243 −0.0726198 −0.0363099 0.999341i $$-0.511560\pi$$
−0.0363099 + 0.999341i $$0.511560\pi$$
$$174$$ 258.253 0.112518
$$175$$ 64.0954 0.0276866
$$176$$ 1224.73 0.524532
$$177$$ 2621.13 1.11309
$$178$$ 1091.13 0.459457
$$179$$ 712.339 0.297446 0.148723 0.988879i $$-0.452484\pi$$
0.148723 + 0.988879i $$0.452484\pi$$
$$180$$ 999.688 0.413957
$$181$$ 2206.53 0.906133 0.453066 0.891477i $$-0.350330\pi$$
0.453066 + 0.891477i $$0.350330\pi$$
$$182$$ 0 0
$$183$$ −561.204 −0.226696
$$184$$ 1303.19 0.522132
$$185$$ −1325.95 −0.526949
$$186$$ 3364.18 1.32620
$$187$$ 526.048 0.205714
$$188$$ 4540.96 1.76162
$$189$$ 303.866 0.116947
$$190$$ 7444.54 2.84254
$$191$$ 1470.64 0.557129 0.278565 0.960417i $$-0.410141\pi$$
0.278565 + 0.960417i $$0.410141\pi$$
$$192$$ −2109.14 −0.792782
$$193$$ −369.560 −0.137832 −0.0689158 0.997622i $$-0.521954\pi$$
−0.0689158 + 0.997622i $$0.521954\pi$$
$$194$$ 5156.80 1.90844
$$195$$ 0 0
$$196$$ −2101.99 −0.766031
$$197$$ 4273.41 1.54552 0.772761 0.634697i $$-0.218875\pi$$
0.772761 + 0.634697i $$0.218875\pi$$
$$198$$ 980.315 0.351858
$$199$$ 4154.31 1.47985 0.739927 0.672687i $$-0.234860\pi$$
0.739927 + 0.672687i $$0.234860\pi$$
$$200$$ −41.1373 −0.0145442
$$201$$ 1827.61 0.641342
$$202$$ −2716.59 −0.946230
$$203$$ −230.175 −0.0795819
$$204$$ −592.515 −0.203355
$$205$$ −4472.09 −1.52363
$$206$$ 2151.40 0.727646
$$207$$ −1623.77 −0.545215
$$208$$ 0 0
$$209$$ 4003.71 1.32508
$$210$$ −1624.63 −0.533857
$$211$$ 1231.59 0.401830 0.200915 0.979609i $$-0.435608\pi$$
0.200915 + 0.979609i $$0.435608\pi$$
$$212$$ 777.134 0.251763
$$213$$ −744.114 −0.239370
$$214$$ −2559.92 −0.817723
$$215$$ 1730.92 0.549059
$$216$$ −195.025 −0.0614341
$$217$$ −2998.42 −0.938000
$$218$$ −5471.92 −1.70002
$$219$$ −2558.30 −0.789377
$$220$$ −2874.49 −0.880901
$$221$$ 0 0
$$222$$ 1464.54 0.442764
$$223$$ 2187.24 0.656809 0.328404 0.944537i $$-0.393489\pi$$
0.328404 + 0.944537i $$0.393489\pi$$
$$224$$ 2892.17 0.862684
$$225$$ 51.2568 0.0151872
$$226$$ −177.602 −0.0522739
$$227$$ −4138.67 −1.21010 −0.605051 0.796187i $$-0.706847\pi$$
−0.605051 + 0.796187i $$0.706847\pi$$
$$228$$ −4509.59 −1.30989
$$229$$ 835.354 0.241056 0.120528 0.992710i $$-0.461541\pi$$
0.120528 + 0.992710i $$0.461541\pi$$
$$230$$ 8681.50 2.48887
$$231$$ −873.733 −0.248863
$$232$$ 147.729 0.0418056
$$233$$ 3685.51 1.03625 0.518124 0.855305i $$-0.326630\pi$$
0.518124 + 0.855305i $$0.326630\pi$$
$$234$$ 0 0
$$235$$ 5343.01 1.48315
$$236$$ 8489.05 2.34148
$$237$$ −993.662 −0.272343
$$238$$ 962.917 0.262255
$$239$$ −3026.21 −0.819034 −0.409517 0.912303i $$-0.634303\pi$$
−0.409517 + 0.912303i $$0.634303\pi$$
$$240$$ −1623.13 −0.436552
$$241$$ −3265.58 −0.872839 −0.436420 0.899743i $$-0.643754\pi$$
−0.436420 + 0.899743i $$0.643754\pi$$
$$242$$ 2783.46 0.739370
$$243$$ 243.000 0.0641500
$$244$$ −1817.57 −0.476877
$$245$$ −2473.25 −0.644940
$$246$$ 4939.53 1.28022
$$247$$ 0 0
$$248$$ 1924.42 0.492746
$$249$$ 1306.30 0.332463
$$250$$ 5740.79 1.45232
$$251$$ −6363.16 −1.60016 −0.800078 0.599897i $$-0.795208\pi$$
−0.800078 + 0.599897i $$0.795208\pi$$
$$252$$ 984.131 0.246010
$$253$$ 4668.96 1.16022
$$254$$ 1309.09 0.323385
$$255$$ −697.168 −0.171209
$$256$$ 1822.38 0.444917
$$257$$ −6085.36 −1.47702 −0.738511 0.674242i $$-0.764471\pi$$
−0.738511 + 0.674242i $$0.764471\pi$$
$$258$$ −1911.84 −0.461342
$$259$$ −1305.31 −0.313159
$$260$$ 0 0
$$261$$ −184.070 −0.0436538
$$262$$ −8422.24 −1.98598
$$263$$ 123.227 0.0288916 0.0144458 0.999896i $$-0.495402\pi$$
0.0144458 + 0.999896i $$0.495402\pi$$
$$264$$ 560.773 0.130732
$$265$$ 914.395 0.211965
$$266$$ 7328.69 1.68929
$$267$$ −777.700 −0.178256
$$268$$ 5919.08 1.34913
$$269$$ −1935.79 −0.438763 −0.219381 0.975639i $$-0.570404\pi$$
−0.219381 + 0.975639i $$0.570404\pi$$
$$270$$ −1299.20 −0.292841
$$271$$ 4612.69 1.03395 0.516976 0.856000i $$-0.327058\pi$$
0.516976 + 0.856000i $$0.327058\pi$$
$$272$$ 962.028 0.214454
$$273$$ 0 0
$$274$$ 4371.20 0.963774
$$275$$ −147.383 −0.0323183
$$276$$ −5258.89 −1.14691
$$277$$ −5834.30 −1.26552 −0.632761 0.774347i $$-0.718078\pi$$
−0.632761 + 0.774347i $$0.718078\pi$$
$$278$$ 12031.4 2.59567
$$279$$ −2397.82 −0.514529
$$280$$ −929.341 −0.198353
$$281$$ −4691.91 −0.996071 −0.498036 0.867157i $$-0.665945\pi$$
−0.498036 + 0.867157i $$0.665945\pi$$
$$282$$ −5901.49 −1.24620
$$283$$ 3465.60 0.727945 0.363973 0.931410i $$-0.381420\pi$$
0.363973 + 0.931410i $$0.381420\pi$$
$$284$$ −2409.96 −0.503539
$$285$$ −5306.09 −1.10283
$$286$$ 0 0
$$287$$ −4402.50 −0.905475
$$288$$ 2312.85 0.473216
$$289$$ −4499.79 −0.915894
$$290$$ 984.133 0.199277
$$291$$ −3675.51 −0.740420
$$292$$ −8285.55 −1.66053
$$293$$ −2677.31 −0.533822 −0.266911 0.963721i $$-0.586003\pi$$
−0.266911 + 0.963721i $$0.586003\pi$$
$$294$$ 2731.77 0.541904
$$295$$ 9988.43 1.97135
$$296$$ 837.767 0.164507
$$297$$ −698.720 −0.136511
$$298$$ 3129.34 0.608315
$$299$$ 0 0
$$300$$ 166.005 0.0319477
$$301$$ 1703.98 0.326299
$$302$$ 9585.02 1.82634
$$303$$ 1936.25 0.367111
$$304$$ 7321.93 1.38139
$$305$$ −2138.60 −0.401494
$$306$$ 770.039 0.143857
$$307$$ −471.915 −0.0877316 −0.0438658 0.999037i $$-0.513967\pi$$
−0.0438658 + 0.999037i $$0.513967\pi$$
$$308$$ −2829.76 −0.523508
$$309$$ −1533.41 −0.282306
$$310$$ 12820.0 2.34880
$$311$$ −1518.52 −0.276872 −0.138436 0.990371i $$-0.544207\pi$$
−0.138436 + 0.990371i $$0.544207\pi$$
$$312$$ 0 0
$$313$$ 4049.86 0.731348 0.365674 0.930743i $$-0.380839\pi$$
0.365674 + 0.930743i $$0.380839\pi$$
$$314$$ −13357.5 −2.40066
$$315$$ 1157.95 0.207121
$$316$$ −3218.17 −0.572900
$$317$$ −3253.96 −0.576532 −0.288266 0.957550i $$-0.593079\pi$$
−0.288266 + 0.957550i $$0.593079\pi$$
$$318$$ −1009.97 −0.178102
$$319$$ 529.272 0.0928951
$$320$$ −8037.37 −1.40407
$$321$$ 1824.59 0.317254
$$322$$ 8546.40 1.47911
$$323$$ 3144.92 0.541759
$$324$$ 787.004 0.134946
$$325$$ 0 0
$$326$$ −9742.46 −1.65517
$$327$$ 3900.11 0.659561
$$328$$ 2825.58 0.475660
$$329$$ 5259.86 0.881415
$$330$$ 3735.72 0.623165
$$331$$ 3422.45 0.568322 0.284161 0.958777i $$-0.408285\pi$$
0.284161 + 0.958777i $$0.408285\pi$$
$$332$$ 4230.71 0.699368
$$333$$ −1043.85 −0.171780
$$334$$ −11219.8 −1.83809
$$335$$ 6964.54 1.13586
$$336$$ −1597.87 −0.259437
$$337$$ −9301.67 −1.50354 −0.751772 0.659423i $$-0.770801\pi$$
−0.751772 + 0.659423i $$0.770801\pi$$
$$338$$ 0 0
$$339$$ 126.586 0.0202808
$$340$$ −2257.92 −0.360155
$$341$$ 6894.66 1.09492
$$342$$ 5860.71 0.926640
$$343$$ −6294.99 −0.990955
$$344$$ −1093.64 −0.171410
$$345$$ −6187.74 −0.965613
$$346$$ 695.518 0.108067
$$347$$ 216.898 0.0335554 0.0167777 0.999859i $$-0.494659\pi$$
0.0167777 + 0.999859i $$0.494659\pi$$
$$348$$ −596.147 −0.0918299
$$349$$ 4809.84 0.737721 0.368861 0.929485i $$-0.379748\pi$$
0.368861 + 0.929485i $$0.379748\pi$$
$$350$$ −269.781 −0.0412011
$$351$$ 0 0
$$352$$ −6650.35 −1.00700
$$353$$ 2859.64 0.431170 0.215585 0.976485i $$-0.430834\pi$$
0.215585 + 0.976485i $$0.430834\pi$$
$$354$$ −11032.5 −1.65641
$$355$$ −2835.62 −0.423941
$$356$$ −2518.74 −0.374980
$$357$$ −686.319 −0.101747
$$358$$ −2998.27 −0.442636
$$359$$ −3686.04 −0.541899 −0.270949 0.962594i $$-0.587338\pi$$
−0.270949 + 0.962594i $$0.587338\pi$$
$$360$$ −743.188 −0.108804
$$361$$ 17076.8 2.48969
$$362$$ −9287.39 −1.34844
$$363$$ −1983.91 −0.286855
$$364$$ 0 0
$$365$$ −9748.98 −1.39804
$$366$$ 2362.14 0.337352
$$367$$ −3470.59 −0.493633 −0.246816 0.969062i $$-0.579384\pi$$
−0.246816 + 0.969062i $$0.579384\pi$$
$$368$$ 8538.52 1.20951
$$369$$ −3520.65 −0.496688
$$370$$ 5580.98 0.784166
$$371$$ 900.166 0.125968
$$372$$ −7765.82 −1.08236
$$373$$ −11963.4 −1.66070 −0.830352 0.557240i $$-0.811860\pi$$
−0.830352 + 0.557240i $$0.811860\pi$$
$$374$$ −2214.16 −0.306127
$$375$$ −4091.75 −0.563459
$$376$$ −3375.85 −0.463021
$$377$$ 0 0
$$378$$ −1278.99 −0.174032
$$379$$ −345.604 −0.0468403 −0.0234202 0.999726i $$-0.507456\pi$$
−0.0234202 + 0.999726i $$0.507456\pi$$
$$380$$ −17184.8 −2.31990
$$381$$ −933.055 −0.125464
$$382$$ −6189.99 −0.829078
$$383$$ 3386.40 0.451793 0.225897 0.974151i $$-0.427469\pi$$
0.225897 + 0.974151i $$0.427469\pi$$
$$384$$ 2709.87 0.360124
$$385$$ −3329.56 −0.440754
$$386$$ 1555.49 0.205110
$$387$$ 1362.67 0.178988
$$388$$ −11903.9 −1.55755
$$389$$ −1629.88 −0.212438 −0.106219 0.994343i $$-0.533874\pi$$
−0.106219 + 0.994343i $$0.533874\pi$$
$$390$$ 0 0
$$391$$ 3667.47 0.474353
$$392$$ 1562.66 0.201343
$$393$$ 6002.95 0.770506
$$394$$ −17987.0 −2.29993
$$395$$ −3786.58 −0.482338
$$396$$ −2262.94 −0.287165
$$397$$ −7938.94 −1.00364 −0.501819 0.864973i $$-0.667336\pi$$
−0.501819 + 0.864973i $$0.667336\pi$$
$$398$$ −17485.7 −2.20221
$$399$$ −5223.52 −0.655396
$$400$$ −269.532 −0.0336915
$$401$$ −214.402 −0.0267001 −0.0133500 0.999911i $$-0.504250\pi$$
−0.0133500 + 0.999911i $$0.504250\pi$$
$$402$$ −7692.51 −0.954396
$$403$$ 0 0
$$404$$ 6270.93 0.772253
$$405$$ 926.008 0.113614
$$406$$ 968.819 0.118428
$$407$$ 3001.48 0.365548
$$408$$ 440.488 0.0534495
$$409$$ 4783.73 0.578338 0.289169 0.957278i $$-0.406621\pi$$
0.289169 + 0.957278i $$0.406621\pi$$
$$410$$ 18823.2 2.26735
$$411$$ −3115.58 −0.373917
$$412$$ −4966.25 −0.593859
$$413$$ 9832.99 1.17155
$$414$$ 6834.51 0.811347
$$415$$ 4977.95 0.588815
$$416$$ 0 0
$$417$$ −8575.39 −1.00705
$$418$$ −16851.8 −1.97189
$$419$$ −9903.67 −1.15472 −0.577358 0.816491i $$-0.695916\pi$$
−0.577358 + 0.816491i $$0.695916\pi$$
$$420$$ 3750.26 0.435700
$$421$$ 12120.6 1.40314 0.701572 0.712598i $$-0.252482\pi$$
0.701572 + 0.712598i $$0.252482\pi$$
$$422$$ −5183.82 −0.597973
$$423$$ 4206.28 0.483491
$$424$$ −577.738 −0.0661732
$$425$$ −115.770 −0.0132133
$$426$$ 3132.01 0.356213
$$427$$ −2105.32 −0.238603
$$428$$ 5909.28 0.667374
$$429$$ 0 0
$$430$$ −7285.53 −0.817068
$$431$$ 13672.6 1.52805 0.764023 0.645189i $$-0.223221\pi$$
0.764023 + 0.645189i $$0.223221\pi$$
$$432$$ −1277.81 −0.142311
$$433$$ 7113.10 0.789455 0.394727 0.918798i $$-0.370839\pi$$
0.394727 + 0.918798i $$0.370839\pi$$
$$434$$ 12620.5 1.39586
$$435$$ −701.441 −0.0773138
$$436$$ 12631.3 1.38745
$$437$$ 27912.9 3.05550
$$438$$ 10768.0 1.17469
$$439$$ −6022.04 −0.654707 −0.327353 0.944902i $$-0.606157\pi$$
−0.327353 + 0.944902i $$0.606157\pi$$
$$440$$ 2136.96 0.231535
$$441$$ −1947.07 −0.210244
$$442$$ 0 0
$$443$$ −12994.4 −1.39364 −0.696821 0.717245i $$-0.745403\pi$$
−0.696821 + 0.717245i $$0.745403\pi$$
$$444$$ −3380.72 −0.361356
$$445$$ −2963.61 −0.315704
$$446$$ −9206.20 −0.977413
$$447$$ −2230.44 −0.236009
$$448$$ −7912.30 −0.834422
$$449$$ −10984.3 −1.15452 −0.577260 0.816560i $$-0.695878\pi$$
−0.577260 + 0.816560i $$0.695878\pi$$
$$450$$ −215.742 −0.0226004
$$451$$ 10123.2 1.05695
$$452$$ 409.973 0.0426627
$$453$$ −6831.72 −0.708570
$$454$$ 17419.9 1.80078
$$455$$ 0 0
$$456$$ 3352.52 0.344290
$$457$$ −9834.10 −1.00661 −0.503304 0.864109i $$-0.667882\pi$$
−0.503304 + 0.864109i $$0.667882\pi$$
$$458$$ −3516.05 −0.358721
$$459$$ −548.845 −0.0558124
$$460$$ −20040.2 −2.03126
$$461$$ −3401.42 −0.343644 −0.171822 0.985128i $$-0.554965\pi$$
−0.171822 + 0.985128i $$0.554965\pi$$
$$462$$ 3677.59 0.370339
$$463$$ −1739.42 −0.174596 −0.0872979 0.996182i $$-0.527823\pi$$
−0.0872979 + 0.996182i $$0.527823\pi$$
$$464$$ 967.925 0.0968422
$$465$$ −9137.45 −0.911267
$$466$$ −15512.5 −1.54207
$$467$$ −7958.82 −0.788630 −0.394315 0.918975i $$-0.629018\pi$$
−0.394315 + 0.918975i $$0.629018\pi$$
$$468$$ 0 0
$$469$$ 6856.16 0.675028
$$470$$ −22489.0 −2.20711
$$471$$ 9520.54 0.931387
$$472$$ −6310.94 −0.615433
$$473$$ −3918.20 −0.380886
$$474$$ 4182.37 0.405280
$$475$$ −881.114 −0.0851122
$$476$$ −2222.78 −0.214036
$$477$$ 719.858 0.0690986
$$478$$ 12737.5 1.21882
$$479$$ −8431.98 −0.804315 −0.402158 0.915570i $$-0.631740\pi$$
−0.402158 + 0.915570i $$0.631740\pi$$
$$480$$ 8813.66 0.838097
$$481$$ 0 0
$$482$$ 13745.0 1.29889
$$483$$ −6091.45 −0.573852
$$484$$ −6425.29 −0.603427
$$485$$ −14006.4 −1.31133
$$486$$ −1022.80 −0.0954632
$$487$$ 11684.7 1.08723 0.543617 0.839334i $$-0.317055\pi$$
0.543617 + 0.839334i $$0.317055\pi$$
$$488$$ 1351.22 0.125342
$$489$$ 6943.94 0.642159
$$490$$ 10410.0 0.959750
$$491$$ −3954.70 −0.363489 −0.181745 0.983346i $$-0.558174\pi$$
−0.181745 + 0.983346i $$0.558174\pi$$
$$492$$ −11402.3 −1.04483
$$493$$ 415.744 0.0379801
$$494$$ 0 0
$$495$$ −2662.63 −0.241771
$$496$$ 12608.8 1.14144
$$497$$ −2791.49 −0.251943
$$498$$ −5498.27 −0.494746
$$499$$ 5690.37 0.510493 0.255246 0.966876i $$-0.417843\pi$$
0.255246 + 0.966876i $$0.417843\pi$$
$$500$$ −13251.9 −1.18529
$$501$$ 7996.95 0.713128
$$502$$ 26782.8 2.38123
$$503$$ 10859.1 0.962595 0.481298 0.876557i $$-0.340166\pi$$
0.481298 + 0.876557i $$0.340166\pi$$
$$504$$ −731.623 −0.0646609
$$505$$ 7378.53 0.650178
$$506$$ −19651.9 −1.72655
$$507$$ 0 0
$$508$$ −3021.88 −0.263926
$$509$$ 18558.6 1.61610 0.808049 0.589115i $$-0.200524\pi$$
0.808049 + 0.589115i $$0.200524\pi$$
$$510$$ 2934.41 0.254780
$$511$$ −9597.27 −0.830838
$$512$$ −14896.8 −1.28584
$$513$$ −4177.22 −0.359510
$$514$$ 25613.6 2.19799
$$515$$ −5843.42 −0.499984
$$516$$ 4413.27 0.376518
$$517$$ −12094.7 −1.02887
$$518$$ 5494.13 0.466020
$$519$$ −495.730 −0.0419271
$$520$$ 0 0
$$521$$ 17297.5 1.45454 0.727271 0.686350i $$-0.240788\pi$$
0.727271 + 0.686350i $$0.240788\pi$$
$$522$$ 774.759 0.0649622
$$523$$ −5016.11 −0.419386 −0.209693 0.977767i $$-0.567247\pi$$
−0.209693 + 0.977767i $$0.567247\pi$$
$$524$$ 19441.8 1.62084
$$525$$ 192.286 0.0159849
$$526$$ −518.667 −0.0429942
$$527$$ 5415.77 0.447656
$$528$$ 3674.19 0.302839
$$529$$ 20383.8 1.67533
$$530$$ −3848.73 −0.315431
$$531$$ 7863.39 0.642640
$$532$$ −16917.4 −1.37869
$$533$$ 0 0
$$534$$ 3273.38 0.265268
$$535$$ 6953.01 0.561878
$$536$$ −4400.37 −0.354603
$$537$$ 2137.02 0.171730
$$538$$ 8147.83 0.652933
$$539$$ 5598.57 0.447398
$$540$$ 2999.06 0.238998
$$541$$ −17642.3 −1.40204 −0.701018 0.713144i $$-0.747271\pi$$
−0.701018 + 0.713144i $$0.747271\pi$$
$$542$$ −19415.0 −1.53865
$$543$$ 6619.59 0.523156
$$544$$ −5223.86 −0.411712
$$545$$ 14862.3 1.16813
$$546$$ 0 0
$$547$$ −18414.9 −1.43943 −0.719713 0.694271i $$-0.755727\pi$$
−0.719713 + 0.694271i $$0.755727\pi$$
$$548$$ −10090.4 −0.786571
$$549$$ −1683.61 −0.130883
$$550$$ 620.343 0.0480937
$$551$$ 3164.20 0.244645
$$552$$ 3909.57 0.301453
$$553$$ −3727.66 −0.286648
$$554$$ 24556.9 1.88325
$$555$$ −3977.84 −0.304234
$$556$$ −27773.1 −2.11842
$$557$$ −8179.15 −0.622193 −0.311096 0.950378i $$-0.600696\pi$$
−0.311096 + 0.950378i $$0.600696\pi$$
$$558$$ 10092.5 0.765683
$$559$$ 0 0
$$560$$ −6089.05 −0.459481
$$561$$ 1578.14 0.118769
$$562$$ 19748.5 1.48228
$$563$$ −1880.07 −0.140738 −0.0703690 0.997521i $$-0.522418\pi$$
−0.0703690 + 0.997521i $$0.522418\pi$$
$$564$$ 13622.9 1.01707
$$565$$ 482.385 0.0359187
$$566$$ −14586.9 −1.08327
$$567$$ 911.598 0.0675194
$$568$$ 1791.62 0.132350
$$569$$ 10118.3 0.745485 0.372743 0.927935i $$-0.378417\pi$$
0.372743 + 0.927935i $$0.378417\pi$$
$$570$$ 22333.6 1.64114
$$571$$ 23428.9 1.71711 0.858555 0.512721i $$-0.171362\pi$$
0.858555 + 0.512721i $$0.171362\pi$$
$$572$$ 0 0
$$573$$ 4411.92 0.321659
$$574$$ 18530.3 1.34746
$$575$$ −1027.52 −0.0745225
$$576$$ −6327.42 −0.457713
$$577$$ −20508.1 −1.47966 −0.739831 0.672793i $$-0.765094\pi$$
−0.739831 + 0.672793i $$0.765094\pi$$
$$578$$ 18939.8 1.36296
$$579$$ −1108.68 −0.0795771
$$580$$ −2271.76 −0.162637
$$581$$ 4900.49 0.349925
$$582$$ 15470.4 1.10184
$$583$$ −2069.87 −0.147042
$$584$$ 6159.65 0.436452
$$585$$ 0 0
$$586$$ 11268.9 0.794394
$$587$$ 5968.43 0.419665 0.209833 0.977737i $$-0.432708\pi$$
0.209833 + 0.977737i $$0.432708\pi$$
$$588$$ −6305.97 −0.442268
$$589$$ 41219.0 2.88353
$$590$$ −42041.8 −2.93361
$$591$$ 12820.2 0.892308
$$592$$ 5489.06 0.381080
$$593$$ 14659.5 1.01517 0.507584 0.861602i $$-0.330539\pi$$
0.507584 + 0.861602i $$0.330539\pi$$
$$594$$ 2940.95 0.203146
$$595$$ −2615.38 −0.180202
$$596$$ −7223.72 −0.496468
$$597$$ 12462.9 0.854394
$$598$$ 0 0
$$599$$ 23635.9 1.61225 0.806125 0.591746i $$-0.201561\pi$$
0.806125 + 0.591746i $$0.201561\pi$$
$$600$$ −123.412 −0.00839711
$$601$$ −11527.0 −0.782356 −0.391178 0.920315i $$-0.627932\pi$$
−0.391178 + 0.920315i $$0.627932\pi$$
$$602$$ −7172.15 −0.485573
$$603$$ 5482.83 0.370279
$$604$$ −22125.9 −1.49055
$$605$$ −7560.15 −0.508039
$$606$$ −8149.77 −0.546306
$$607$$ −5098.56 −0.340930 −0.170465 0.985364i $$-0.554527\pi$$
−0.170465 + 0.985364i $$0.554527\pi$$
$$608$$ −39758.4 −2.65200
$$609$$ −690.525 −0.0459466
$$610$$ 9001.47 0.597473
$$611$$ 0 0
$$612$$ −1777.55 −0.117407
$$613$$ −1516.39 −0.0999128 −0.0499564 0.998751i $$-0.515908\pi$$
−0.0499564 + 0.998751i $$0.515908\pi$$
$$614$$ 1986.31 0.130556
$$615$$ −13416.3 −0.879668
$$616$$ 2103.70 0.137598
$$617$$ −18539.3 −1.20966 −0.604832 0.796353i $$-0.706760\pi$$
−0.604832 + 0.796353i $$0.706760\pi$$
$$618$$ 6454.20 0.420107
$$619$$ −25684.9 −1.66779 −0.833897 0.551920i $$-0.813895\pi$$
−0.833897 + 0.551920i $$0.813895\pi$$
$$620$$ −29593.5 −1.91694
$$621$$ −4871.30 −0.314780
$$622$$ 6391.51 0.412020
$$623$$ −2917.49 −0.187619
$$624$$ 0 0
$$625$$ −16304.5 −1.04349
$$626$$ −17046.1 −1.08834
$$627$$ 12011.1 0.765038
$$628$$ 30834.2 1.95926
$$629$$ 2357.67 0.149454
$$630$$ −4873.88 −0.308222
$$631$$ 22410.9 1.41389 0.706945 0.707269i $$-0.250073\pi$$
0.706945 + 0.707269i $$0.250073\pi$$
$$632$$ 2392.46 0.150580
$$633$$ 3694.77 0.231997
$$634$$ 13696.1 0.857950
$$635$$ −3555.62 −0.222206
$$636$$ 2331.40 0.145356
$$637$$ 0 0
$$638$$ −2227.73 −0.138239
$$639$$ −2232.34 −0.138200
$$640$$ 10326.6 0.637804
$$641$$ 6827.81 0.420721 0.210361 0.977624i $$-0.432536\pi$$
0.210361 + 0.977624i $$0.432536\pi$$
$$642$$ −7679.77 −0.472113
$$643$$ 23264.3 1.42684 0.713418 0.700738i $$-0.247146\pi$$
0.713418 + 0.700738i $$0.247146\pi$$
$$644$$ −19728.4 −1.20715
$$645$$ 5192.76 0.316999
$$646$$ −13237.1 −0.806204
$$647$$ 14745.9 0.896014 0.448007 0.894030i $$-0.352134\pi$$
0.448007 + 0.894030i $$0.352134\pi$$
$$648$$ −585.075 −0.0354690
$$649$$ −22610.3 −1.36754
$$650$$ 0 0
$$651$$ −8995.26 −0.541554
$$652$$ 22489.3 1.35084
$$653$$ 10909.0 0.653755 0.326878 0.945067i $$-0.394003\pi$$
0.326878 + 0.945067i $$0.394003\pi$$
$$654$$ −16415.8 −0.981509
$$655$$ 22875.7 1.36462
$$656$$ 18513.2 1.10186
$$657$$ −7674.89 −0.455747
$$658$$ −22139.0 −1.31166
$$659$$ −4182.99 −0.247263 −0.123631 0.992328i $$-0.539454\pi$$
−0.123631 + 0.992328i $$0.539454\pi$$
$$660$$ −8623.47 −0.508588
$$661$$ −2224.23 −0.130881 −0.0654406 0.997856i $$-0.520845\pi$$
−0.0654406 + 0.997856i $$0.520845\pi$$
$$662$$ −14405.2 −0.845734
$$663$$ 0 0
$$664$$ −3145.19 −0.183821
$$665$$ −19905.4 −1.16075
$$666$$ 4393.63 0.255630
$$667$$ 3689.95 0.214206
$$668$$ 25899.7 1.50013
$$669$$ 6561.72 0.379209
$$670$$ −29314.1 −1.69030
$$671$$ 4841.04 0.278519
$$672$$ 8676.51 0.498071
$$673$$ −24152.5 −1.38337 −0.691687 0.722197i $$-0.743132\pi$$
−0.691687 + 0.722197i $$0.743132\pi$$
$$674$$ 39151.2 2.23746
$$675$$ 153.770 0.00876833
$$676$$ 0 0
$$677$$ −15310.7 −0.869187 −0.434593 0.900627i $$-0.643108\pi$$
−0.434593 + 0.900627i $$0.643108\pi$$
$$678$$ −532.806 −0.0301804
$$679$$ −13788.4 −0.779310
$$680$$ 1678.58 0.0946628
$$681$$ −12416.0 −0.698652
$$682$$ −29020.0 −1.62937
$$683$$ −11399.6 −0.638646 −0.319323 0.947646i $$-0.603455\pi$$
−0.319323 + 0.947646i $$0.603455\pi$$
$$684$$ −13528.8 −0.756265
$$685$$ −11872.6 −0.662233
$$686$$ 26495.9 1.47466
$$687$$ 2506.06 0.139174
$$688$$ −7165.54 −0.397069
$$689$$ 0 0
$$690$$ 26044.5 1.43695
$$691$$ 3323.23 0.182955 0.0914773 0.995807i $$-0.470841\pi$$
0.0914773 + 0.995807i $$0.470841\pi$$
$$692$$ −1605.52 −0.0881976
$$693$$ −2621.20 −0.143681
$$694$$ −912.936 −0.0499345
$$695$$ −32678.5 −1.78355
$$696$$ 443.188 0.0241365
$$697$$ 7951.82 0.432133
$$698$$ −20244.8 −1.09782
$$699$$ 11056.5 0.598279
$$700$$ 622.758 0.0336257
$$701$$ −12670.4 −0.682673 −0.341336 0.939941i $$-0.610880\pi$$
−0.341336 + 0.939941i $$0.610880\pi$$
$$702$$ 0 0
$$703$$ 17944.0 0.962692
$$704$$ 18193.8 0.974012
$$705$$ 16029.0 0.856295
$$706$$ −12036.4 −0.641635
$$707$$ 7263.71 0.386393
$$708$$ 25467.2 1.35186
$$709$$ −13075.2 −0.692594 −0.346297 0.938125i $$-0.612561\pi$$
−0.346297 + 0.938125i $$0.612561\pi$$
$$710$$ 11935.3 0.630877
$$711$$ −2980.99 −0.157237
$$712$$ 1872.48 0.0985592
$$713$$ 48067.8 2.52476
$$714$$ 2888.75 0.151413
$$715$$ 0 0
$$716$$ 6921.16 0.361251
$$717$$ −9078.62 −0.472869
$$718$$ 15514.7 0.806412
$$719$$ −2988.41 −0.155005 −0.0775026 0.996992i $$-0.524695\pi$$
−0.0775026 + 0.996992i $$0.524695\pi$$
$$720$$ −4869.38 −0.252043
$$721$$ −5752.48 −0.297134
$$722$$ −71877.0 −3.70496
$$723$$ −9796.73 −0.503934
$$724$$ 21438.9 1.10051
$$725$$ −116.479 −0.00596680
$$726$$ 8350.37 0.426875
$$727$$ −5507.46 −0.280963 −0.140482 0.990083i $$-0.544865\pi$$
−0.140482 + 0.990083i $$0.544865\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 41033.9 2.08046
$$731$$ −3077.75 −0.155725
$$732$$ −5452.71 −0.275325
$$733$$ 36585.2 1.84353 0.921764 0.387751i $$-0.126748\pi$$
0.921764 + 0.387751i $$0.126748\pi$$
$$734$$ 14607.9 0.734587
$$735$$ −7419.75 −0.372356
$$736$$ −46364.6 −2.32204
$$737$$ −15765.3 −0.787953
$$738$$ 14818.6 0.739133
$$739$$ −6425.89 −0.319865 −0.159933 0.987128i $$-0.551128\pi$$
−0.159933 + 0.987128i $$0.551128\pi$$
$$740$$ −12883.0 −0.639986
$$741$$ 0 0
$$742$$ −3788.84 −0.187457
$$743$$ −20411.0 −1.00782 −0.503908 0.863757i $$-0.668105\pi$$
−0.503908 + 0.863757i $$0.668105\pi$$
$$744$$ 5773.27 0.284487
$$745$$ −8499.60 −0.417988
$$746$$ 50354.6 2.47133
$$747$$ 3918.89 0.191947
$$748$$ 5111.13 0.249842
$$749$$ 6844.81 0.333917
$$750$$ 17222.4 0.838496
$$751$$ −24259.5 −1.17875 −0.589375 0.807860i $$-0.700626\pi$$
−0.589375 + 0.807860i $$0.700626\pi$$
$$752$$ −22118.6 −1.07258
$$753$$ −19089.5 −0.923850
$$754$$ 0 0
$$755$$ −26033.9 −1.25493
$$756$$ 2952.39 0.142034
$$757$$ 9295.39 0.446297 0.223148 0.974785i $$-0.428367\pi$$
0.223148 + 0.974785i $$0.428367\pi$$
$$758$$ 1454.66 0.0697042
$$759$$ 14006.9 0.669851
$$760$$ 12775.6 0.609761
$$761$$ 21974.7 1.04676 0.523378 0.852101i $$-0.324672\pi$$
0.523378 + 0.852101i $$0.324672\pi$$
$$762$$ 3927.27 0.186706
$$763$$ 14631.0 0.694204
$$764$$ 14288.9 0.676641
$$765$$ −2091.50 −0.0988476
$$766$$ −14253.5 −0.672324
$$767$$ 0 0
$$768$$ 5467.13 0.256873
$$769$$ −22987.4 −1.07795 −0.538977 0.842320i $$-0.681189\pi$$
−0.538977 + 0.842320i $$0.681189\pi$$
$$770$$ 14014.3 0.655897
$$771$$ −18256.1 −0.852759
$$772$$ −3590.68 −0.167398
$$773$$ −31970.9 −1.48760 −0.743799 0.668404i $$-0.766978\pi$$
−0.743799 + 0.668404i $$0.766978\pi$$
$$774$$ −5735.53 −0.266356
$$775$$ −1517.34 −0.0703283
$$776$$ 8849.59 0.409384
$$777$$ −3915.94 −0.180803
$$778$$ 6860.25 0.316133
$$779$$ 60520.8 2.78354
$$780$$ 0 0
$$781$$ 6418.85 0.294090
$$782$$ −15436.6 −0.705896
$$783$$ −552.209 −0.0252035
$$784$$ 10238.6 0.466408
$$785$$ 36280.2 1.64955
$$786$$ −25266.7 −1.14661
$$787$$ −6087.26 −0.275715 −0.137857 0.990452i $$-0.544022\pi$$
−0.137857 + 0.990452i $$0.544022\pi$$
$$788$$ 41520.9 1.87706
$$789$$ 369.680 0.0166805
$$790$$ 15937.9 0.717779
$$791$$ 474.878 0.0213460
$$792$$ 1682.32 0.0754780
$$793$$ 0 0
$$794$$ 33415.4 1.49354
$$795$$ 2743.19 0.122378
$$796$$ 40363.6 1.79730
$$797$$ 23080.0 1.02577 0.512883 0.858458i $$-0.328577\pi$$
0.512883 + 0.858458i $$0.328577\pi$$
$$798$$ 21986.1 0.975311
$$799$$ −9500.41 −0.420651
$$800$$ 1463.57 0.0646813
$$801$$ −2333.10 −0.102916
$$802$$ 902.429 0.0397330
$$803$$ 22068.3 0.969829
$$804$$ 17757.3 0.778918
$$805$$ −23212.9 −1.01633
$$806$$ 0 0
$$807$$ −5807.37 −0.253320
$$808$$ −4661.94 −0.202978
$$809$$ −32377.8 −1.40710 −0.703550 0.710646i $$-0.748403\pi$$
−0.703550 + 0.710646i $$0.748403\pi$$
$$810$$ −3897.61 −0.169072
$$811$$ −26352.8 −1.14103 −0.570513 0.821288i $$-0.693256\pi$$
−0.570513 + 0.821288i $$0.693256\pi$$
$$812$$ −2236.40 −0.0966532
$$813$$ 13838.1 0.596952
$$814$$ −12633.4 −0.543980
$$815$$ 26461.5 1.13731
$$816$$ 2886.08 0.123815
$$817$$ −23424.5 −1.00308
$$818$$ −20135.0 −0.860638
$$819$$ 0 0
$$820$$ −43451.3 −1.85047
$$821$$ −35355.3 −1.50294 −0.751468 0.659770i $$-0.770654\pi$$
−0.751468 + 0.659770i $$0.770654\pi$$
$$822$$ 13113.6 0.556435
$$823$$ −12663.3 −0.536347 −0.268173 0.963371i $$-0.586420\pi$$
−0.268173 + 0.963371i $$0.586420\pi$$
$$824$$ 3692.02 0.156089
$$825$$ −442.149 −0.0186590
$$826$$ −41387.6 −1.74341
$$827$$ 16295.2 0.685176 0.342588 0.939486i $$-0.388697\pi$$
0.342588 + 0.939486i $$0.388697\pi$$
$$828$$ −15776.7 −0.662170
$$829$$ 13638.9 0.571411 0.285705 0.958318i $$-0.407772\pi$$
0.285705 + 0.958318i $$0.407772\pi$$
$$830$$ −20952.4 −0.876229
$$831$$ −17502.9 −0.730649
$$832$$ 0 0
$$833$$ 4397.69 0.182918
$$834$$ 36094.2 1.49861
$$835$$ 30474.2 1.26300
$$836$$ 38900.5 1.60933
$$837$$ −7193.46 −0.297064
$$838$$ 41685.0 1.71836
$$839$$ 1890.31 0.0777838 0.0388919 0.999243i $$-0.487617\pi$$
0.0388919 + 0.999243i $$0.487617\pi$$
$$840$$ −2788.02 −0.114519
$$841$$ −23970.7 −0.982849
$$842$$ −51016.4 −2.08805
$$843$$ −14075.7 −0.575082
$$844$$ 11966.3 0.488028
$$845$$ 0 0
$$846$$ −17704.5 −0.719494
$$847$$ −7442.50 −0.301921
$$848$$ −3785.35 −0.153289
$$849$$ 10396.8 0.420279
$$850$$ 487.280 0.0196630
$$851$$ 20925.6 0.842914
$$852$$ −7229.89 −0.290718
$$853$$ −1620.21 −0.0650351 −0.0325175 0.999471i $$-0.510352\pi$$
−0.0325175 + 0.999471i $$0.510352\pi$$
$$854$$ 8861.39 0.355071
$$855$$ −15918.3 −0.636718
$$856$$ −4393.08 −0.175412
$$857$$ −14508.4 −0.578292 −0.289146 0.957285i $$-0.593371\pi$$
−0.289146 + 0.957285i $$0.593371\pi$$
$$858$$ 0 0
$$859$$ 29639.8 1.17730 0.588648 0.808389i $$-0.299660\pi$$
0.588648 + 0.808389i $$0.299660\pi$$
$$860$$ 16817.8 0.666839
$$861$$ −13207.5 −0.522776
$$862$$ −57548.8 −2.27392
$$863$$ −21528.8 −0.849186 −0.424593 0.905384i $$-0.639583\pi$$
−0.424593 + 0.905384i $$0.639583\pi$$
$$864$$ 6938.56 0.273211
$$865$$ −1889.10 −0.0742557
$$866$$ −29939.4 −1.17481
$$867$$ −13499.4 −0.528792
$$868$$ −29132.9 −1.13921
$$869$$ 8571.50 0.334601
$$870$$ 2952.40 0.115053
$$871$$ 0 0
$$872$$ −9390.36 −0.364676
$$873$$ −11026.5 −0.427482
$$874$$ −117487. −4.54696
$$875$$ −15349.9 −0.593054
$$876$$ −24856.7 −0.958708
$$877$$ −14865.3 −0.572366 −0.286183 0.958175i $$-0.592387\pi$$
−0.286183 + 0.958175i $$0.592387\pi$$
$$878$$ 25347.1 0.974285
$$879$$ −8031.92 −0.308202
$$880$$ 14001.4 0.536348
$$881$$ −21336.0 −0.815921 −0.407961 0.913000i $$-0.633760\pi$$
−0.407961 + 0.913000i $$0.633760\pi$$
$$882$$ 8195.30 0.312869
$$883$$ 37538.2 1.43065 0.715323 0.698794i $$-0.246280\pi$$
0.715323 + 0.698794i $$0.246280\pi$$
$$884$$ 0 0
$$885$$ 29965.3 1.13816
$$886$$ 54694.1 2.07391
$$887$$ 34575.0 1.30881 0.654406 0.756144i $$-0.272919\pi$$
0.654406 + 0.756144i $$0.272919\pi$$
$$888$$ 2513.30 0.0949784
$$889$$ −3500.29 −0.132054
$$890$$ 12474.0 0.469807
$$891$$ −2096.16 −0.0788148
$$892$$ 21251.4 0.797703
$$893$$ −72306.9 −2.70958
$$894$$ 9388.02 0.351211
$$895$$ 8143.61 0.304146
$$896$$ 10165.9 0.379039
$$897$$ 0 0
$$898$$ 46233.3 1.71807
$$899$$ 5448.96 0.202150
$$900$$ 498.016 0.0184450
$$901$$ −1625.89 −0.0601178
$$902$$ −42609.2 −1.57287
$$903$$ 5111.95 0.188389
$$904$$ −304.783 −0.0112134
$$905$$ 25225.5 0.926545
$$906$$ 28755.0 1.05444
$$907$$ −10424.8 −0.381641 −0.190820 0.981625i $$-0.561115\pi$$
−0.190820 + 0.981625i $$0.561115\pi$$
$$908$$ −40211.7 −1.46968
$$909$$ 5808.75 0.211952
$$910$$ 0 0
$$911$$ 10961.8 0.398661 0.199331 0.979932i $$-0.436123\pi$$
0.199331 + 0.979932i $$0.436123\pi$$
$$912$$ 21965.8 0.797543
$$913$$ −11268.3 −0.408464
$$914$$ 41392.2 1.49796
$$915$$ −6415.80 −0.231803
$$916$$ 8116.38 0.292765
$$917$$ 22519.7 0.810976
$$918$$ 2310.12 0.0830558
$$919$$ −10779.2 −0.386914 −0.193457 0.981109i $$-0.561970\pi$$
−0.193457 + 0.981109i $$0.561970\pi$$
$$920$$ 14898.3 0.533895
$$921$$ −1415.74 −0.0506519
$$922$$ 14316.7 0.511384
$$923$$ 0 0
$$924$$ −8489.28 −0.302248
$$925$$ −660.549 −0.0234797
$$926$$ 7321.32 0.259820
$$927$$ −4600.23 −0.162990
$$928$$ −5255.88 −0.185919
$$929$$ −5429.07 −0.191735 −0.0958675 0.995394i $$-0.530563\pi$$
−0.0958675 + 0.995394i $$0.530563\pi$$
$$930$$ 38460.0 1.35608
$$931$$ 33470.5 1.17825
$$932$$ 35808.8 1.25854
$$933$$ −4555.55 −0.159852
$$934$$ 33499.1 1.17358
$$935$$ 6013.88 0.210348
$$936$$ 0 0
$$937$$ −21300.1 −0.742631 −0.371315 0.928507i $$-0.621093\pi$$
−0.371315 + 0.928507i $$0.621093\pi$$
$$938$$ −28857.9 −1.00453
$$939$$ 12149.6 0.422244
$$940$$ 51913.2 1.80130
$$941$$ −26851.2 −0.930207 −0.465103 0.885256i $$-0.653983\pi$$
−0.465103 + 0.885256i $$0.653983\pi$$
$$942$$ −40072.4 −1.38602
$$943$$ 70576.7 2.43722
$$944$$ −41349.4 −1.42564
$$945$$ 3473.86 0.119582
$$946$$ 16491.9 0.566805
$$947$$ 8021.68 0.275258 0.137629 0.990484i $$-0.456052\pi$$
0.137629 + 0.990484i $$0.456052\pi$$
$$948$$ −9654.52 −0.330764
$$949$$ 0 0
$$950$$ 3708.65 0.126658
$$951$$ −9761.88 −0.332861
$$952$$ 1652.46 0.0562569
$$953$$ 35715.0 1.21398 0.606990 0.794709i $$-0.292377\pi$$
0.606990 + 0.794709i $$0.292377\pi$$
$$954$$ −3029.92 −0.102827
$$955$$ 16812.6 0.569680
$$956$$ −29402.9 −0.994726
$$957$$ 1587.82 0.0536330
$$958$$ 35490.6 1.19692
$$959$$ −11687.9 −0.393557
$$960$$ −24112.1 −0.810641
$$961$$ 41190.9 1.38266
$$962$$ 0 0
$$963$$ 5473.76 0.183166
$$964$$ −31728.7 −1.06007
$$965$$ −4224.88 −0.140936
$$966$$ 25639.2 0.853963
$$967$$ −53338.8 −1.77380 −0.886898 0.461965i $$-0.847145\pi$$
−0.886898 + 0.461965i $$0.847145\pi$$
$$968$$ 4776.69 0.158604
$$969$$ 9434.77 0.312785
$$970$$ 58953.6 1.95143
$$971$$ 23112.9 0.763882 0.381941 0.924187i $$-0.375256\pi$$
0.381941 + 0.924187i $$0.375256\pi$$
$$972$$ 2361.01 0.0779110
$$973$$ −32170.0 −1.05994
$$974$$ −49181.3 −1.61794
$$975$$ 0 0
$$976$$ 8853.22 0.290353
$$977$$ 52874.6 1.73143 0.865715 0.500538i $$-0.166864\pi$$
0.865715 + 0.500538i $$0.166864\pi$$
$$978$$ −29227.4 −0.955612
$$979$$ 6708.57 0.219006
$$980$$ −24030.4 −0.783287
$$981$$ 11700.3 0.380798
$$982$$ 16645.5 0.540917
$$983$$ −45173.1 −1.46572 −0.732858 0.680381i $$-0.761814\pi$$
−0.732858 + 0.680381i $$0.761814\pi$$
$$984$$ 8476.73 0.274622
$$985$$ 48854.5 1.58034
$$986$$ −1749.89 −0.0565190
$$987$$ 15779.6 0.508885
$$988$$ 0 0
$$989$$ −27316.7 −0.878281
$$990$$ 11207.2 0.359785
$$991$$ 60485.6 1.93884 0.969418 0.245414i $$-0.0789239\pi$$
0.969418 + 0.245414i $$0.0789239\pi$$
$$992$$ −68466.7 −2.19135
$$993$$ 10267.3 0.328121
$$994$$ 11749.5 0.374922
$$995$$ 47492.8 1.51319
$$996$$ 12692.1 0.403780
$$997$$ −18108.1 −0.575214 −0.287607 0.957749i $$-0.592860\pi$$
−0.287607 + 0.957749i $$0.592860\pi$$
$$998$$ −23951.1 −0.759677
$$999$$ −3131.56 −0.0991773
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.4.a.h.1.1 3
3.2 odd 2 1521.4.a.u.1.3 3
13.5 odd 4 507.4.b.g.337.6 6
13.8 odd 4 507.4.b.g.337.1 6
13.12 even 2 39.4.a.c.1.3 3
39.38 odd 2 117.4.a.f.1.1 3
52.51 odd 2 624.4.a.t.1.1 3
65.64 even 2 975.4.a.l.1.1 3
91.90 odd 2 1911.4.a.k.1.3 3
104.51 odd 2 2496.4.a.bp.1.3 3
104.77 even 2 2496.4.a.bl.1.3 3
156.155 even 2 1872.4.a.bk.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 13.12 even 2
117.4.a.f.1.1 3 39.38 odd 2
507.4.a.h.1.1 3 1.1 even 1 trivial
507.4.b.g.337.1 6 13.8 odd 4
507.4.b.g.337.6 6 13.5 odd 4
624.4.a.t.1.1 3 52.51 odd 2
975.4.a.l.1.1 3 65.64 even 2
1521.4.a.u.1.3 3 3.2 odd 2
1872.4.a.bk.1.3 3 156.155 even 2
1911.4.a.k.1.3 3 91.90 odd 2
2496.4.a.bl.1.3 3 104.77 even 2
2496.4.a.bp.1.3 3 104.51 odd 2