# Properties

 Label 175.8.a.b.1.2 Level $175$ Weight $8$ Character 175.1 Self dual yes Analytic conductor $54.667$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,8,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("175.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$54.6673794597$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.31662$$ of defining polynomial Character $$\chi$$ $$=$$ 175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.36675 q^{2} -24.7995 q^{3} -126.132 q^{4} +33.8947 q^{6} +343.000 q^{7} +347.335 q^{8} -1571.98 q^{9} +O(q^{10})$$ $$q-1.36675 q^{2} -24.7995 q^{3} -126.132 q^{4} +33.8947 q^{6} +343.000 q^{7} +347.335 q^{8} -1571.98 q^{9} -1432.37 q^{11} +3128.01 q^{12} +6136.30 q^{13} -468.795 q^{14} +15670.2 q^{16} +15858.5 q^{17} +2148.51 q^{18} -38567.5 q^{19} -8506.23 q^{21} +1957.69 q^{22} +63987.4 q^{23} -8613.73 q^{24} -8386.79 q^{26} +93220.9 q^{27} -43263.3 q^{28} +94236.6 q^{29} +275990. q^{31} -65876.1 q^{32} +35521.9 q^{33} -21674.6 q^{34} +198278. q^{36} -156532. q^{37} +52712.1 q^{38} -152177. q^{39} -303738. q^{41} +11625.9 q^{42} -636818. q^{43} +180667. q^{44} -87454.9 q^{46} -512021. q^{47} -388612. q^{48} +117649. q^{49} -393282. q^{51} -773984. q^{52} +201249. q^{53} -127410. q^{54} +119136. q^{56} +956454. q^{57} -128798. q^{58} -1.81196e6 q^{59} -982021. q^{61} -377210. q^{62} -539191. q^{63} -1.91575e6 q^{64} -48549.6 q^{66} +4.45336e6 q^{67} -2.00026e6 q^{68} -1.58686e6 q^{69} +725436. q^{71} -546005. q^{72} -2.17602e6 q^{73} +213940. q^{74} +4.86459e6 q^{76} -491301. q^{77} +207988. q^{78} -5.21525e6 q^{79} +1.12610e6 q^{81} +415135. q^{82} -6.07921e6 q^{83} +1.07291e6 q^{84} +870371. q^{86} -2.33702e6 q^{87} -497511. q^{88} -1.06137e7 q^{89} +2.10475e6 q^{91} -8.07086e6 q^{92} -6.84442e6 q^{93} +699805. q^{94} +1.63369e6 q^{96} -6.64483e6 q^{97} -160797. q^{98} +2.25166e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{2} + 30 q^{3} - 40 q^{4} - 768 q^{6} + 686 q^{7} + 960 q^{8} - 756 q^{9}+O(q^{10})$$ 2 * q - 16 * q^2 + 30 * q^3 - 40 * q^4 - 768 * q^6 + 686 * q^7 + 960 * q^8 - 756 * q^9 $$2 q - 16 q^{2} + 30 q^{3} - 40 q^{4} - 768 q^{6} + 686 q^{7} + 960 q^{8} - 756 q^{9} - 7906 q^{11} + 7848 q^{12} + 17818 q^{13} - 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 9792 q^{18} - 3612 q^{19} + 10290 q^{21} + 96688 q^{22} - 13844 q^{23} + 24960 q^{24} - 179328 q^{26} + 18090 q^{27} - 13720 q^{28} - 126898 q^{29} + 252768 q^{31} + 148224 q^{32} - 319230 q^{33} + 175296 q^{34} + 268560 q^{36} + 265860 q^{37} - 458800 q^{38} + 487974 q^{39} - 111920 q^{41} - 263424 q^{42} - 947572 q^{43} - 376920 q^{44} + 1051472 q^{46} - 271274 q^{47} - 1484064 q^{48} + 235298 q^{49} - 1130910 q^{51} + 232184 q^{52} + 1267792 q^{53} + 972000 q^{54} + 329280 q^{56} + 2871996 q^{57} + 3107120 q^{58} - 1360120 q^{59} - 1813680 q^{61} - 37392 q^{62} - 259308 q^{63} - 2489984 q^{64} + 5142624 q^{66} + 2189312 q^{67} - 3159640 q^{68} - 5851980 q^{69} - 1494928 q^{71} - 46080 q^{72} - 7169788 q^{73} - 5967024 q^{74} + 7875376 q^{76} - 2711758 q^{77} - 9159504 q^{78} - 7942974 q^{79} - 4775598 q^{81} - 2391792 q^{82} + 304712 q^{83} + 2691864 q^{84} + 5417712 q^{86} - 14455086 q^{87} - 4463680 q^{88} - 17943528 q^{89} + 6111574 q^{91} - 14774640 q^{92} - 8116992 q^{93} - 2823104 q^{94} + 13366272 q^{96} - 4258074 q^{97} - 1882384 q^{98} - 3030732 q^{99}+O(q^{100})$$ 2 * q - 16 * q^2 + 30 * q^3 - 40 * q^4 - 768 * q^6 + 686 * q^7 + 960 * q^8 - 756 * q^9 - 7906 * q^11 + 7848 * q^12 + 17818 * q^13 - 5488 * q^14 - 4320 * q^16 + 2398 * q^17 - 9792 * q^18 - 3612 * q^19 + 10290 * q^21 + 96688 * q^22 - 13844 * q^23 + 24960 * q^24 - 179328 * q^26 + 18090 * q^27 - 13720 * q^28 - 126898 * q^29 + 252768 * q^31 + 148224 * q^32 - 319230 * q^33 + 175296 * q^34 + 268560 * q^36 + 265860 * q^37 - 458800 * q^38 + 487974 * q^39 - 111920 * q^41 - 263424 * q^42 - 947572 * q^43 - 376920 * q^44 + 1051472 * q^46 - 271274 * q^47 - 1484064 * q^48 + 235298 * q^49 - 1130910 * q^51 + 232184 * q^52 + 1267792 * q^53 + 972000 * q^54 + 329280 * q^56 + 2871996 * q^57 + 3107120 * q^58 - 1360120 * q^59 - 1813680 * q^61 - 37392 * q^62 - 259308 * q^63 - 2489984 * q^64 + 5142624 * q^66 + 2189312 * q^67 - 3159640 * q^68 - 5851980 * q^69 - 1494928 * q^71 - 46080 * q^72 - 7169788 * q^73 - 5967024 * q^74 + 7875376 * q^76 - 2711758 * q^77 - 9159504 * q^78 - 7942974 * q^79 - 4775598 * q^81 - 2391792 * q^82 + 304712 * q^83 + 2691864 * q^84 + 5417712 * q^86 - 14455086 * q^87 - 4463680 * q^88 - 17943528 * q^89 + 6111574 * q^91 - 14774640 * q^92 - 8116992 * q^93 - 2823104 * q^94 + 13366272 * q^96 - 4258074 * q^97 - 1882384 * q^98 - 3030732 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.36675 −0.120805 −0.0604024 0.998174i $$-0.519238\pi$$
−0.0604024 + 0.998174i $$0.519238\pi$$
$$3$$ −24.7995 −0.530296 −0.265148 0.964208i $$-0.585421\pi$$
−0.265148 + 0.964208i $$0.585421\pi$$
$$4$$ −126.132 −0.985406
$$5$$ 0 0
$$6$$ 33.8947 0.0640623
$$7$$ 343.000 0.377964
$$8$$ 347.335 0.239847
$$9$$ −1571.98 −0.718786
$$10$$ 0 0
$$11$$ −1432.37 −0.324474 −0.162237 0.986752i $$-0.551871\pi$$
−0.162237 + 0.986752i $$0.551871\pi$$
$$12$$ 3128.01 0.522557
$$13$$ 6136.30 0.774649 0.387325 0.921943i $$-0.373399\pi$$
0.387325 + 0.921943i $$0.373399\pi$$
$$14$$ −468.795 −0.0456599
$$15$$ 0 0
$$16$$ 15670.2 0.956432
$$17$$ 15858.5 0.782871 0.391436 0.920205i $$-0.371979\pi$$
0.391436 + 0.920205i $$0.371979\pi$$
$$18$$ 2148.51 0.0868328
$$19$$ −38567.5 −1.28998 −0.644991 0.764190i $$-0.723139\pi$$
−0.644991 + 0.764190i $$0.723139\pi$$
$$20$$ 0 0
$$21$$ −8506.23 −0.200433
$$22$$ 1957.69 0.0391980
$$23$$ 63987.4 1.09660 0.548299 0.836282i $$-0.315276\pi$$
0.548299 + 0.836282i $$0.315276\pi$$
$$24$$ −8613.73 −0.127190
$$25$$ 0 0
$$26$$ −8386.79 −0.0935813
$$27$$ 93220.9 0.911466
$$28$$ −43263.3 −0.372449
$$29$$ 94236.6 0.717508 0.358754 0.933432i $$-0.383202\pi$$
0.358754 + 0.933432i $$0.383202\pi$$
$$30$$ 0 0
$$31$$ 275990. 1.66390 0.831951 0.554849i $$-0.187224\pi$$
0.831951 + 0.554849i $$0.187224\pi$$
$$32$$ −65876.1 −0.355388
$$33$$ 35521.9 0.172067
$$34$$ −21674.6 −0.0945746
$$35$$ 0 0
$$36$$ 198278. 0.708296
$$37$$ −156532. −0.508038 −0.254019 0.967199i $$-0.581753\pi$$
−0.254019 + 0.967199i $$0.581753\pi$$
$$38$$ 52712.1 0.155836
$$39$$ −152177. −0.410793
$$40$$ 0 0
$$41$$ −303738. −0.688266 −0.344133 0.938921i $$-0.611827\pi$$
−0.344133 + 0.938921i $$0.611827\pi$$
$$42$$ 11625.9 0.0242133
$$43$$ −636818. −1.22145 −0.610725 0.791843i $$-0.709122\pi$$
−0.610725 + 0.791843i $$0.709122\pi$$
$$44$$ 180667. 0.319738
$$45$$ 0 0
$$46$$ −87454.9 −0.132474
$$47$$ −512021. −0.719358 −0.359679 0.933076i $$-0.617114\pi$$
−0.359679 + 0.933076i $$0.617114\pi$$
$$48$$ −388612. −0.507192
$$49$$ 117649. 0.142857
$$50$$ 0 0
$$51$$ −393282. −0.415154
$$52$$ −773984. −0.763344
$$53$$ 201249. 0.185681 0.0928406 0.995681i $$-0.470405\pi$$
0.0928406 + 0.995681i $$0.470405\pi$$
$$54$$ −127410. −0.110109
$$55$$ 0 0
$$56$$ 119136. 0.0906535
$$57$$ 956454. 0.684072
$$58$$ −128798. −0.0866784
$$59$$ −1.81196e6 −1.14859 −0.574296 0.818648i $$-0.694724\pi$$
−0.574296 + 0.818648i $$0.694724\pi$$
$$60$$ 0 0
$$61$$ −982021. −0.553945 −0.276972 0.960878i $$-0.589331\pi$$
−0.276972 + 0.960878i $$0.589331\pi$$
$$62$$ −377210. −0.201007
$$63$$ −539191. −0.271676
$$64$$ −1.91575e6 −0.913499
$$65$$ 0 0
$$66$$ −48549.6 −0.0207865
$$67$$ 4.45336e6 1.80895 0.904474 0.426528i $$-0.140264\pi$$
0.904474 + 0.426528i $$0.140264\pi$$
$$68$$ −2.00026e6 −0.771446
$$69$$ −1.58686e6 −0.581522
$$70$$ 0 0
$$71$$ 725436. 0.240544 0.120272 0.992741i $$-0.461623\pi$$
0.120272 + 0.992741i $$0.461623\pi$$
$$72$$ −546005. −0.172398
$$73$$ −2.17602e6 −0.654685 −0.327343 0.944906i $$-0.606153\pi$$
−0.327343 + 0.944906i $$0.606153\pi$$
$$74$$ 213940. 0.0613735
$$75$$ 0 0
$$76$$ 4.86459e6 1.27116
$$77$$ −491301. −0.122639
$$78$$ 207988. 0.0496258
$$79$$ −5.21525e6 −1.19009 −0.595045 0.803692i $$-0.702866\pi$$
−0.595045 + 0.803692i $$0.702866\pi$$
$$80$$ 0 0
$$81$$ 1.12610e6 0.235439
$$82$$ 415135. 0.0831458
$$83$$ −6.07921e6 −1.16701 −0.583504 0.812110i $$-0.698319\pi$$
−0.583504 + 0.812110i $$0.698319\pi$$
$$84$$ 1.07291e6 0.197508
$$85$$ 0 0
$$86$$ 870371. 0.147557
$$87$$ −2.33702e6 −0.380492
$$88$$ −497511. −0.0778239
$$89$$ −1.06137e7 −1.59589 −0.797946 0.602729i $$-0.794080\pi$$
−0.797946 + 0.602729i $$0.794080\pi$$
$$90$$ 0 0
$$91$$ 2.10475e6 0.292790
$$92$$ −8.07086e6 −1.08059
$$93$$ −6.84442e6 −0.882361
$$94$$ 699805. 0.0869019
$$95$$ 0 0
$$96$$ 1.63369e6 0.188461
$$97$$ −6.64483e6 −0.739236 −0.369618 0.929184i $$-0.620511\pi$$
−0.369618 + 0.929184i $$0.620511\pi$$
$$98$$ −160797. −0.0172578
$$99$$ 2.25166e6 0.233227
$$100$$ 0 0
$$101$$ 1.07531e7 1.03851 0.519254 0.854620i $$-0.326210\pi$$
0.519254 + 0.854620i $$0.326210\pi$$
$$102$$ 537519. 0.0501526
$$103$$ 1.05886e7 0.954788 0.477394 0.878689i $$-0.341581\pi$$
0.477394 + 0.878689i $$0.341581\pi$$
$$104$$ 2.13135e6 0.185797
$$105$$ 0 0
$$106$$ −275057. −0.0224312
$$107$$ 8.37234e6 0.660699 0.330349 0.943859i $$-0.392833\pi$$
0.330349 + 0.943859i $$0.392833\pi$$
$$108$$ −1.17581e7 −0.898164
$$109$$ −1.95948e7 −1.44926 −0.724632 0.689136i $$-0.757990\pi$$
−0.724632 + 0.689136i $$0.757990\pi$$
$$110$$ 0 0
$$111$$ 3.88191e6 0.269411
$$112$$ 5.37487e6 0.361497
$$113$$ −1.36310e7 −0.888694 −0.444347 0.895855i $$-0.646564\pi$$
−0.444347 + 0.895855i $$0.646564\pi$$
$$114$$ −1.30723e6 −0.0826392
$$115$$ 0 0
$$116$$ −1.18863e7 −0.707037
$$117$$ −9.64617e6 −0.556807
$$118$$ 2.47649e6 0.138756
$$119$$ 5.43946e6 0.295898
$$120$$ 0 0
$$121$$ −1.74355e7 −0.894717
$$122$$ 1.34218e6 0.0669192
$$123$$ 7.53256e6 0.364985
$$124$$ −3.48112e7 −1.63962
$$125$$ 0 0
$$126$$ 736939. 0.0328197
$$127$$ −2.23763e7 −0.969336 −0.484668 0.874698i $$-0.661060\pi$$
−0.484668 + 0.874698i $$0.661060\pi$$
$$128$$ 1.10505e7 0.465743
$$129$$ 1.57928e7 0.647730
$$130$$ 0 0
$$131$$ −4.53330e6 −0.176183 −0.0880917 0.996112i $$-0.528077\pi$$
−0.0880917 + 0.996112i $$0.528077\pi$$
$$132$$ −4.48045e6 −0.169556
$$133$$ −1.32286e7 −0.487567
$$134$$ −6.08663e6 −0.218530
$$135$$ 0 0
$$136$$ 5.50821e6 0.187769
$$137$$ 5.07657e7 1.68674 0.843371 0.537332i $$-0.180568\pi$$
0.843371 + 0.537332i $$0.180568\pi$$
$$138$$ 2.16884e6 0.0702506
$$139$$ 1.05183e7 0.332195 0.166097 0.986109i $$-0.446883\pi$$
0.166097 + 0.986109i $$0.446883\pi$$
$$140$$ 0 0
$$141$$ 1.26979e7 0.381473
$$142$$ −991490. −0.0290589
$$143$$ −8.78942e6 −0.251353
$$144$$ −2.46333e7 −0.687470
$$145$$ 0 0
$$146$$ 2.97407e6 0.0790891
$$147$$ −2.91764e6 −0.0757566
$$148$$ 1.97437e7 0.500624
$$149$$ 5.43497e7 1.34600 0.673000 0.739642i $$-0.265005\pi$$
0.673000 + 0.739642i $$0.265005\pi$$
$$150$$ 0 0
$$151$$ −2.23258e7 −0.527700 −0.263850 0.964564i $$-0.584992\pi$$
−0.263850 + 0.964564i $$0.584992\pi$$
$$152$$ −1.33958e7 −0.309398
$$153$$ −2.49293e7 −0.562717
$$154$$ 671486. 0.0148154
$$155$$ 0 0
$$156$$ 1.91944e7 0.404798
$$157$$ 4.37788e7 0.902848 0.451424 0.892310i $$-0.350916\pi$$
0.451424 + 0.892310i $$0.350916\pi$$
$$158$$ 7.12794e6 0.143769
$$159$$ −4.99087e6 −0.0984661
$$160$$ 0 0
$$161$$ 2.19477e7 0.414475
$$162$$ −1.53910e6 −0.0284422
$$163$$ −4.05451e7 −0.733300 −0.366650 0.930359i $$-0.619495\pi$$
−0.366650 + 0.930359i $$0.619495\pi$$
$$164$$ 3.83111e7 0.678221
$$165$$ 0 0
$$166$$ 8.30876e6 0.140980
$$167$$ −9.73453e7 −1.61736 −0.808682 0.588247i $$-0.799818\pi$$
−0.808682 + 0.588247i $$0.799818\pi$$
$$168$$ −2.95451e6 −0.0480732
$$169$$ −2.50943e7 −0.399919
$$170$$ 0 0
$$171$$ 6.06275e7 0.927221
$$172$$ 8.03231e7 1.20362
$$173$$ 5.10607e7 0.749765 0.374882 0.927072i $$-0.377683\pi$$
0.374882 + 0.927072i $$0.377683\pi$$
$$174$$ 3.19412e6 0.0459652
$$175$$ 0 0
$$176$$ −2.24454e7 −0.310337
$$177$$ 4.49356e7 0.609094
$$178$$ 1.45063e7 0.192791
$$179$$ 1.45811e8 1.90023 0.950113 0.311907i $$-0.100968\pi$$
0.950113 + 0.311907i $$0.100968\pi$$
$$180$$ 0 0
$$181$$ −6.09656e7 −0.764205 −0.382102 0.924120i $$-0.624800\pi$$
−0.382102 + 0.924120i $$0.624800\pi$$
$$182$$ −2.87667e6 −0.0353704
$$183$$ 2.43536e7 0.293755
$$184$$ 2.22251e7 0.263015
$$185$$ 0 0
$$186$$ 9.35462e6 0.106593
$$187$$ −2.27151e7 −0.254021
$$188$$ 6.45822e7 0.708860
$$189$$ 3.19748e7 0.344502
$$190$$ 0 0
$$191$$ −1.52578e8 −1.58444 −0.792219 0.610237i $$-0.791074\pi$$
−0.792219 + 0.610237i $$0.791074\pi$$
$$192$$ 4.75095e7 0.484425
$$193$$ 1.39277e8 1.39453 0.697267 0.716812i $$-0.254399\pi$$
0.697267 + 0.716812i $$0.254399\pi$$
$$194$$ 9.08183e6 0.0893033
$$195$$ 0 0
$$196$$ −1.48393e7 −0.140772
$$197$$ −6.52480e7 −0.608044 −0.304022 0.952665i $$-0.598330\pi$$
−0.304022 + 0.952665i $$0.598330\pi$$
$$198$$ −3.07745e6 −0.0281750
$$199$$ 1.93503e6 0.0174061 0.00870307 0.999962i $$-0.497230\pi$$
0.00870307 + 0.999962i $$0.497230\pi$$
$$200$$ 0 0
$$201$$ −1.10441e8 −0.959278
$$202$$ −1.46968e7 −0.125457
$$203$$ 3.23232e7 0.271192
$$204$$ 4.96055e7 0.409095
$$205$$ 0 0
$$206$$ −1.44719e7 −0.115343
$$207$$ −1.00587e8 −0.788219
$$208$$ 9.61569e7 0.740899
$$209$$ 5.52427e7 0.418565
$$210$$ 0 0
$$211$$ −5.17848e7 −0.379502 −0.189751 0.981832i $$-0.560768\pi$$
−0.189751 + 0.981832i $$0.560768\pi$$
$$212$$ −2.53839e7 −0.182971
$$213$$ −1.79904e7 −0.127560
$$214$$ −1.14429e7 −0.0798156
$$215$$ 0 0
$$216$$ 3.23789e7 0.218612
$$217$$ 9.46647e7 0.628896
$$218$$ 2.67812e7 0.175078
$$219$$ 5.39642e7 0.347177
$$220$$ 0 0
$$221$$ 9.73124e7 0.606450
$$222$$ −5.30560e6 −0.0325461
$$223$$ 1.25065e8 0.755209 0.377605 0.925967i $$-0.376748\pi$$
0.377605 + 0.925967i $$0.376748\pi$$
$$224$$ −2.25955e7 −0.134324
$$225$$ 0 0
$$226$$ 1.86301e7 0.107358
$$227$$ −1.92108e7 −0.109007 −0.0545036 0.998514i $$-0.517358\pi$$
−0.0545036 + 0.998514i $$0.517358\pi$$
$$228$$ −1.20639e8 −0.674089
$$229$$ −1.05650e8 −0.581360 −0.290680 0.956820i $$-0.593882\pi$$
−0.290680 + 0.956820i $$0.593882\pi$$
$$230$$ 0 0
$$231$$ 1.21840e7 0.0650353
$$232$$ 3.27317e7 0.172092
$$233$$ 2.31646e8 1.19972 0.599859 0.800106i $$-0.295224\pi$$
0.599859 + 0.800106i $$0.295224\pi$$
$$234$$ 1.31839e7 0.0672649
$$235$$ 0 0
$$236$$ 2.28546e8 1.13183
$$237$$ 1.29336e8 0.631101
$$238$$ −7.43438e6 −0.0357458
$$239$$ 1.09174e8 0.517281 0.258641 0.965974i $$-0.416725\pi$$
0.258641 + 0.965974i $$0.416725\pi$$
$$240$$ 0 0
$$241$$ −8.25277e7 −0.379787 −0.189893 0.981805i $$-0.560814\pi$$
−0.189893 + 0.981805i $$0.560814\pi$$
$$242$$ 2.38300e7 0.108086
$$243$$ −2.31801e8 −1.03632
$$244$$ 1.23864e8 0.545861
$$245$$ 0 0
$$246$$ −1.02951e7 −0.0440919
$$247$$ −2.36662e8 −0.999283
$$248$$ 9.58611e7 0.399081
$$249$$ 1.50761e8 0.618860
$$250$$ 0 0
$$251$$ −2.40987e7 −0.0961912 −0.0480956 0.998843i $$-0.515315\pi$$
−0.0480956 + 0.998843i $$0.515315\pi$$
$$252$$ 6.80092e7 0.267711
$$253$$ −9.16534e7 −0.355817
$$254$$ 3.05828e7 0.117100
$$255$$ 0 0
$$256$$ 2.30112e8 0.857235
$$257$$ −9.75049e7 −0.358311 −0.179156 0.983821i $$-0.557337\pi$$
−0.179156 + 0.983821i $$0.557337\pi$$
$$258$$ −2.15848e7 −0.0782489
$$259$$ −5.36904e7 −0.192020
$$260$$ 0 0
$$261$$ −1.48139e8 −0.515735
$$262$$ 6.19589e6 0.0212838
$$263$$ −2.98637e8 −1.01228 −0.506138 0.862452i $$-0.668927\pi$$
−0.506138 + 0.862452i $$0.668927\pi$$
$$264$$ 1.23380e7 0.0412697
$$265$$ 0 0
$$266$$ 1.80803e7 0.0589005
$$267$$ 2.63216e8 0.846296
$$268$$ −5.61711e8 −1.78255
$$269$$ −3.90722e8 −1.22387 −0.611934 0.790909i $$-0.709608\pi$$
−0.611934 + 0.790909i $$0.709608\pi$$
$$270$$ 0 0
$$271$$ 2.12098e8 0.647357 0.323678 0.946167i $$-0.395080\pi$$
0.323678 + 0.946167i $$0.395080\pi$$
$$272$$ 2.48505e8 0.748763
$$273$$ −5.21968e7 −0.155265
$$274$$ −6.93841e7 −0.203767
$$275$$ 0 0
$$276$$ 2.00153e8 0.573035
$$277$$ −1.86723e8 −0.527861 −0.263930 0.964542i $$-0.585019\pi$$
−0.263930 + 0.964542i $$0.585019\pi$$
$$278$$ −1.43759e7 −0.0401307
$$279$$ −4.33853e8 −1.19599
$$280$$ 0 0
$$281$$ −7.38791e8 −1.98632 −0.993161 0.116756i $$-0.962750\pi$$
−0.993161 + 0.116756i $$0.962750\pi$$
$$282$$ −1.73548e7 −0.0460838
$$283$$ 3.11903e8 0.818026 0.409013 0.912529i $$-0.365873\pi$$
0.409013 + 0.912529i $$0.365873\pi$$
$$284$$ −9.15007e7 −0.237034
$$285$$ 0 0
$$286$$ 1.20129e7 0.0303647
$$287$$ −1.04182e8 −0.260140
$$288$$ 1.03556e8 0.255448
$$289$$ −1.58847e8 −0.387113
$$290$$ 0 0
$$291$$ 1.64789e8 0.392014
$$292$$ 2.74466e8 0.645131
$$293$$ 5.05466e8 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$294$$ 3.98768e6 0.00915176
$$295$$ 0 0
$$296$$ −5.43690e7 −0.121851
$$297$$ −1.33526e8 −0.295747
$$298$$ −7.42825e7 −0.162603
$$299$$ 3.92646e8 0.849478
$$300$$ 0 0
$$301$$ −2.18429e8 −0.461665
$$302$$ 3.05137e7 0.0637487
$$303$$ −2.66672e8 −0.550717
$$304$$ −6.04359e8 −1.23378
$$305$$ 0 0
$$306$$ 3.40721e7 0.0679789
$$307$$ −4.67463e8 −0.922067 −0.461034 0.887383i $$-0.652521\pi$$
−0.461034 + 0.887383i $$0.652521\pi$$
$$308$$ 6.19688e7 0.120850
$$309$$ −2.62591e8 −0.506321
$$310$$ 0 0
$$311$$ 1.16022e7 0.0218714 0.0109357 0.999940i $$-0.496519\pi$$
0.0109357 + 0.999940i $$0.496519\pi$$
$$312$$ −5.28565e7 −0.0985274
$$313$$ −8.23197e8 −1.51740 −0.758698 0.651443i $$-0.774164\pi$$
−0.758698 + 0.651443i $$0.774164\pi$$
$$314$$ −5.98346e7 −0.109068
$$315$$ 0 0
$$316$$ 6.57810e8 1.17272
$$317$$ −3.89154e8 −0.686142 −0.343071 0.939309i $$-0.611467\pi$$
−0.343071 + 0.939309i $$0.611467\pi$$
$$318$$ 6.82128e6 0.0118952
$$319$$ −1.34981e8 −0.232812
$$320$$ 0 0
$$321$$ −2.07630e8 −0.350366
$$322$$ −2.99970e7 −0.0500706
$$323$$ −6.11621e8 −1.00989
$$324$$ −1.42037e8 −0.232003
$$325$$ 0 0
$$326$$ 5.54150e7 0.0885861
$$327$$ 4.85940e8 0.768539
$$328$$ −1.05499e8 −0.165078
$$329$$ −1.75623e8 −0.271892
$$330$$ 0 0
$$331$$ 1.48582e8 0.225199 0.112600 0.993640i $$-0.464082\pi$$
0.112600 + 0.993640i $$0.464082\pi$$
$$332$$ 7.66783e8 1.14998
$$333$$ 2.46066e8 0.365171
$$334$$ 1.33047e8 0.195385
$$335$$ 0 0
$$336$$ −1.33294e8 −0.191701
$$337$$ −1.23379e8 −0.175605 −0.0878023 0.996138i $$-0.527984\pi$$
−0.0878023 + 0.996138i $$0.527984\pi$$
$$338$$ 3.42977e7 0.0483121
$$339$$ 3.38041e8 0.471271
$$340$$ 0 0
$$341$$ −3.95319e8 −0.539892
$$342$$ −8.28626e7 −0.112013
$$343$$ 4.03536e7 0.0539949
$$344$$ −2.21189e8 −0.292961
$$345$$ 0 0
$$346$$ −6.97872e7 −0.0905752
$$347$$ −1.31658e9 −1.69159 −0.845793 0.533511i $$-0.820872\pi$$
−0.845793 + 0.533511i $$0.820872\pi$$
$$348$$ 2.94773e8 0.374939
$$349$$ 2.64521e8 0.333097 0.166549 0.986033i $$-0.446738\pi$$
0.166549 + 0.986033i $$0.446738\pi$$
$$350$$ 0 0
$$351$$ 5.72032e8 0.706066
$$352$$ 9.43586e7 0.115314
$$353$$ −1.30271e9 −1.57629 −0.788144 0.615490i $$-0.788958\pi$$
−0.788144 + 0.615490i $$0.788958\pi$$
$$354$$ −6.14158e7 −0.0735815
$$355$$ 0 0
$$356$$ 1.33873e9 1.57260
$$357$$ −1.34896e8 −0.156913
$$358$$ −1.99287e8 −0.229556
$$359$$ −1.03262e9 −1.17790 −0.588952 0.808168i $$-0.700459\pi$$
−0.588952 + 0.808168i $$0.700459\pi$$
$$360$$ 0 0
$$361$$ 5.93578e8 0.664053
$$362$$ 8.33248e7 0.0923196
$$363$$ 4.32392e8 0.474465
$$364$$ −2.65476e8 −0.288517
$$365$$ 0 0
$$366$$ −3.32853e7 −0.0354870
$$367$$ −1.13124e9 −1.19460 −0.597302 0.802017i $$-0.703760\pi$$
−0.597302 + 0.802017i $$0.703760\pi$$
$$368$$ 1.00269e9 1.04882
$$369$$ 4.77472e8 0.494716
$$370$$ 0 0
$$371$$ 6.90284e7 0.0701809
$$372$$ 8.63300e8 0.869484
$$373$$ −5.38130e8 −0.536916 −0.268458 0.963291i $$-0.586514\pi$$
−0.268458 + 0.963291i $$0.586514\pi$$
$$374$$ 3.10459e7 0.0306870
$$375$$ 0 0
$$376$$ −1.77843e8 −0.172536
$$377$$ 5.78264e8 0.555817
$$378$$ −4.37015e7 −0.0416175
$$379$$ 7.83114e8 0.738904 0.369452 0.929250i $$-0.379545\pi$$
0.369452 + 0.929250i $$0.379545\pi$$
$$380$$ 0 0
$$381$$ 5.54920e8 0.514035
$$382$$ 2.08536e8 0.191408
$$383$$ −8.22468e8 −0.748038 −0.374019 0.927421i $$-0.622020\pi$$
−0.374019 + 0.927421i $$0.622020\pi$$
$$384$$ −2.74047e8 −0.246982
$$385$$ 0 0
$$386$$ −1.90357e8 −0.168466
$$387$$ 1.00107e9 0.877961
$$388$$ 8.38126e8 0.728448
$$389$$ −1.07007e9 −0.921696 −0.460848 0.887479i $$-0.652455\pi$$
−0.460848 + 0.887479i $$0.652455\pi$$
$$390$$ 0 0
$$391$$ 1.01474e9 0.858495
$$392$$ 4.08636e7 0.0342638
$$393$$ 1.12424e8 0.0934293
$$394$$ 8.91777e7 0.0734547
$$395$$ 0 0
$$396$$ −2.84006e8 −0.229823
$$397$$ 9.64552e8 0.773676 0.386838 0.922148i $$-0.373567\pi$$
0.386838 + 0.922148i $$0.373567\pi$$
$$398$$ −2.64471e6 −0.00210275
$$399$$ 3.28064e8 0.258555
$$400$$ 0 0
$$401$$ −1.94810e9 −1.50871 −0.754357 0.656465i $$-0.772051\pi$$
−0.754357 + 0.656465i $$0.772051\pi$$
$$402$$ 1.50945e8 0.115885
$$403$$ 1.69356e9 1.28894
$$404$$ −1.35631e9 −1.02335
$$405$$ 0 0
$$406$$ −4.41777e7 −0.0327614
$$407$$ 2.24211e8 0.164845
$$408$$ −1.36601e8 −0.0995732
$$409$$ 8.63865e8 0.624330 0.312165 0.950028i $$-0.398946\pi$$
0.312165 + 0.950028i $$0.398946\pi$$
$$410$$ 0 0
$$411$$ −1.25896e9 −0.894473
$$412$$ −1.33556e9 −0.940854
$$413$$ −6.21501e8 −0.434127
$$414$$ 1.37478e8 0.0952206
$$415$$ 0 0
$$416$$ −4.04236e8 −0.275301
$$417$$ −2.60848e8 −0.176162
$$418$$ −7.55030e7 −0.0505647
$$419$$ 2.21337e9 1.46996 0.734978 0.678091i $$-0.237192\pi$$
0.734978 + 0.678091i $$0.237192\pi$$
$$420$$ 0 0
$$421$$ −2.89866e9 −1.89326 −0.946631 0.322321i $$-0.895537\pi$$
−0.946631 + 0.322321i $$0.895537\pi$$
$$422$$ 7.07769e7 0.0458456
$$423$$ 8.04889e8 0.517065
$$424$$ 6.99008e7 0.0445350
$$425$$ 0 0
$$426$$ 2.45884e7 0.0154098
$$427$$ −3.36833e8 −0.209371
$$428$$ −1.05602e9 −0.651057
$$429$$ 2.17973e8 0.133292
$$430$$ 0 0
$$431$$ −2.42056e9 −1.45628 −0.728142 0.685426i $$-0.759616\pi$$
−0.728142 + 0.685426i $$0.759616\pi$$
$$432$$ 1.46079e9 0.871754
$$433$$ 2.26686e9 1.34189 0.670946 0.741506i $$-0.265888\pi$$
0.670946 + 0.741506i $$0.265888\pi$$
$$434$$ −1.29383e8 −0.0759737
$$435$$ 0 0
$$436$$ 2.47153e9 1.42811
$$437$$ −2.46783e9 −1.41459
$$438$$ −7.37555e7 −0.0419407
$$439$$ 1.98911e9 1.12210 0.561052 0.827780i $$-0.310397\pi$$
0.561052 + 0.827780i $$0.310397\pi$$
$$440$$ 0 0
$$441$$ −1.84942e8 −0.102684
$$442$$ −1.33002e8 −0.0732621
$$443$$ 8.78038e8 0.479844 0.239922 0.970792i $$-0.422878\pi$$
0.239922 + 0.970792i $$0.422878\pi$$
$$444$$ −4.89633e8 −0.265479
$$445$$ 0 0
$$446$$ −1.70932e8 −0.0912329
$$447$$ −1.34785e9 −0.713779
$$448$$ −6.57101e8 −0.345270
$$449$$ 1.53113e8 0.0798270 0.0399135 0.999203i $$-0.487292\pi$$
0.0399135 + 0.999203i $$0.487292\pi$$
$$450$$ 0 0
$$451$$ 4.35064e8 0.223324
$$452$$ 1.71930e9 0.875724
$$453$$ 5.53668e8 0.279837
$$454$$ 2.62564e7 0.0131686
$$455$$ 0 0
$$456$$ 3.32210e8 0.164072
$$457$$ 2.39624e9 1.17442 0.587210 0.809435i $$-0.300226\pi$$
0.587210 + 0.809435i $$0.300226\pi$$
$$458$$ 1.44397e8 0.0702311
$$459$$ 1.47834e9 0.713560
$$460$$ 0 0
$$461$$ 1.61913e9 0.769713 0.384856 0.922977i $$-0.374251\pi$$
0.384856 + 0.922977i $$0.374251\pi$$
$$462$$ −1.66525e7 −0.00785657
$$463$$ −1.16133e9 −0.543778 −0.271889 0.962329i $$-0.587648\pi$$
−0.271889 + 0.962329i $$0.587648\pi$$
$$464$$ 1.47670e9 0.686247
$$465$$ 0 0
$$466$$ −3.16602e8 −0.144932
$$467$$ −2.83969e9 −1.29021 −0.645107 0.764092i $$-0.723187\pi$$
−0.645107 + 0.764092i $$0.723187\pi$$
$$468$$ 1.21669e9 0.548681
$$469$$ 1.52750e9 0.683718
$$470$$ 0 0
$$471$$ −1.08569e9 −0.478777
$$472$$ −6.29356e8 −0.275486
$$473$$ 9.12156e8 0.396328
$$474$$ −1.76769e8 −0.0762400
$$475$$ 0 0
$$476$$ −6.86090e8 −0.291579
$$477$$ −3.16360e8 −0.133465
$$478$$ −1.49214e8 −0.0624901
$$479$$ 2.38771e9 0.992676 0.496338 0.868129i $$-0.334678\pi$$
0.496338 + 0.868129i $$0.334678\pi$$
$$480$$ 0 0
$$481$$ −9.60526e8 −0.393551
$$482$$ 1.12795e8 0.0458801
$$483$$ −5.44292e8 −0.219794
$$484$$ 2.19917e9 0.881660
$$485$$ 0 0
$$486$$ 3.16814e8 0.125192
$$487$$ 2.20508e9 0.865113 0.432556 0.901607i $$-0.357612\pi$$
0.432556 + 0.901607i $$0.357612\pi$$
$$488$$ −3.41090e8 −0.132862
$$489$$ 1.00550e9 0.388866
$$490$$ 0 0
$$491$$ 4.28064e8 0.163201 0.0816006 0.996665i $$-0.473997\pi$$
0.0816006 + 0.996665i $$0.473997\pi$$
$$492$$ −9.50097e8 −0.359658
$$493$$ 1.49445e9 0.561716
$$494$$ 3.23457e8 0.120718
$$495$$ 0 0
$$496$$ 4.32482e9 1.59141
$$497$$ 2.48824e8 0.0909171
$$498$$ −2.06053e8 −0.0747613
$$499$$ −2.95178e9 −1.06349 −0.531743 0.846906i $$-0.678463\pi$$
−0.531743 + 0.846906i $$0.678463\pi$$
$$500$$ 0 0
$$501$$ 2.41412e9 0.857681
$$502$$ 3.29369e7 0.0116204
$$503$$ 5.22380e9 1.83020 0.915099 0.403229i $$-0.132112\pi$$
0.915099 + 0.403229i $$0.132112\pi$$
$$504$$ −1.87280e8 −0.0651605
$$505$$ 0 0
$$506$$ 1.25267e8 0.0429844
$$507$$ 6.22326e8 0.212075
$$508$$ 2.82236e9 0.955190
$$509$$ −2.80532e9 −0.942911 −0.471455 0.881890i $$-0.656271\pi$$
−0.471455 + 0.881890i $$0.656271\pi$$
$$510$$ 0 0
$$511$$ −7.46374e8 −0.247448
$$512$$ −1.72897e9 −0.569301
$$513$$ −3.59530e9 −1.17577
$$514$$ 1.33265e8 0.0432857
$$515$$ 0 0
$$516$$ −1.99197e9 −0.638277
$$517$$ 7.33401e8 0.233413
$$518$$ 7.33814e7 0.0231970
$$519$$ −1.26628e9 −0.397597
$$520$$ 0 0
$$521$$ 1.40563e8 0.0435450 0.0217725 0.999763i $$-0.493069\pi$$
0.0217725 + 0.999763i $$0.493069\pi$$
$$522$$ 2.02468e8 0.0623032
$$523$$ 1.81127e9 0.553638 0.276819 0.960922i $$-0.410720\pi$$
0.276819 + 0.960922i $$0.410720\pi$$
$$524$$ 5.71794e8 0.173612
$$525$$ 0 0
$$526$$ 4.08163e8 0.122288
$$527$$ 4.37679e9 1.30262
$$528$$ 5.56635e8 0.164570
$$529$$ 6.89567e8 0.202526
$$530$$ 0 0
$$531$$ 2.84837e9 0.825592
$$532$$ 1.66855e9 0.480452
$$533$$ −1.86383e9 −0.533164
$$534$$ −3.59750e8 −0.102237
$$535$$ 0 0
$$536$$ 1.54681e9 0.433870
$$537$$ −3.61604e9 −1.00768
$$538$$ 5.34019e8 0.147849
$$539$$ −1.68516e8 −0.0463534
$$540$$ 0 0
$$541$$ −7.11633e9 −1.93226 −0.966130 0.258058i $$-0.916918\pi$$
−0.966130 + 0.258058i $$0.916918\pi$$
$$542$$ −2.89885e8 −0.0782038
$$543$$ 1.51192e9 0.405255
$$544$$ −1.04469e9 −0.278223
$$545$$ 0 0
$$546$$ 7.13400e7 0.0187568
$$547$$ 6.02390e9 1.57370 0.786850 0.617144i $$-0.211710\pi$$
0.786850 + 0.617144i $$0.211710\pi$$
$$548$$ −6.40318e9 −1.66213
$$549$$ 1.54372e9 0.398168
$$550$$ 0 0
$$551$$ −3.63447e9 −0.925572
$$552$$ −5.51171e8 −0.139476
$$553$$ −1.78883e9 −0.449812
$$554$$ 2.55204e8 0.0637681
$$555$$ 0 0
$$556$$ −1.32669e9 −0.327347
$$557$$ −3.55726e9 −0.872214 −0.436107 0.899895i $$-0.643643\pi$$
−0.436107 + 0.899895i $$0.643643\pi$$
$$558$$ 5.92968e8 0.144481
$$559$$ −3.90771e9 −0.946195
$$560$$ 0 0
$$561$$ 5.63324e8 0.134706
$$562$$ 1.00974e9 0.239957
$$563$$ −2.51240e9 −0.593347 −0.296673 0.954979i $$-0.595877\pi$$
−0.296673 + 0.954979i $$0.595877\pi$$
$$564$$ −1.60161e9 −0.375906
$$565$$ 0 0
$$566$$ −4.26293e8 −0.0988214
$$567$$ 3.86252e8 0.0889877
$$568$$ 2.51969e8 0.0576937
$$569$$ 3.02191e9 0.687683 0.343841 0.939028i $$-0.388272\pi$$
0.343841 + 0.939028i $$0.388272\pi$$
$$570$$ 0 0
$$571$$ 4.13151e9 0.928716 0.464358 0.885648i $$-0.346285\pi$$
0.464358 + 0.885648i $$0.346285\pi$$
$$572$$ 1.10863e9 0.247685
$$573$$ 3.78386e9 0.840221
$$574$$ 1.42391e8 0.0314262
$$575$$ 0 0
$$576$$ 3.01152e9 0.656610
$$577$$ 3.66048e9 0.793274 0.396637 0.917976i $$-0.370177\pi$$
0.396637 + 0.917976i $$0.370177\pi$$
$$578$$ 2.17105e8 0.0467651
$$579$$ −3.45400e9 −0.739516
$$580$$ 0 0
$$581$$ −2.08517e9 −0.441088
$$582$$ −2.25225e8 −0.0473572
$$583$$ −2.88262e8 −0.0602487
$$584$$ −7.55807e8 −0.157024
$$585$$ 0 0
$$586$$ −6.90846e8 −0.141821
$$587$$ −8.93156e9 −1.82261 −0.911305 0.411731i $$-0.864924\pi$$
−0.911305 + 0.411731i $$0.864924\pi$$
$$588$$ 3.68007e8 0.0746510
$$589$$ −1.06442e10 −2.14640
$$590$$ 0 0
$$591$$ 1.61812e9 0.322444
$$592$$ −2.45288e9 −0.485904
$$593$$ −8.00218e9 −1.57586 −0.787929 0.615766i $$-0.788847\pi$$
−0.787929 + 0.615766i $$0.788847\pi$$
$$594$$ 1.82497e8 0.0357276
$$595$$ 0 0
$$596$$ −6.85524e9 −1.32636
$$597$$ −4.79879e7 −0.00923041
$$598$$ −5.36649e8 −0.102621
$$599$$ 6.37081e9 1.21116 0.605579 0.795785i $$-0.292941\pi$$
0.605579 + 0.795785i $$0.292941\pi$$
$$600$$ 0 0
$$601$$ 7.97677e9 1.49888 0.749439 0.662073i $$-0.230323\pi$$
0.749439 + 0.662073i $$0.230323\pi$$
$$602$$ 2.98537e8 0.0557713
$$603$$ −7.00062e9 −1.30025
$$604$$ 2.81599e9 0.519999
$$605$$ 0 0
$$606$$ 3.64474e8 0.0665293
$$607$$ −5.42119e9 −0.983863 −0.491931 0.870634i $$-0.663709\pi$$
−0.491931 + 0.870634i $$0.663709\pi$$
$$608$$ 2.54067e9 0.458444
$$609$$ −8.01598e8 −0.143812
$$610$$ 0 0
$$611$$ −3.14191e9 −0.557250
$$612$$ 3.14438e9 0.554505
$$613$$ −8.21824e9 −1.44101 −0.720505 0.693450i $$-0.756090\pi$$
−0.720505 + 0.693450i $$0.756090\pi$$
$$614$$ 6.38905e8 0.111390
$$615$$ 0 0
$$616$$ −1.70646e8 −0.0294147
$$617$$ −8.15621e9 −1.39795 −0.698973 0.715148i $$-0.746359\pi$$
−0.698973 + 0.715148i $$0.746359\pi$$
$$618$$ 3.58897e8 0.0611660
$$619$$ −6.46052e9 −1.09484 −0.547420 0.836858i $$-0.684390\pi$$
−0.547420 + 0.836858i $$0.684390\pi$$
$$620$$ 0 0
$$621$$ 5.96497e9 0.999511
$$622$$ −1.58573e7 −0.00264218
$$623$$ −3.64051e9 −0.603191
$$624$$ −2.38464e9 −0.392896
$$625$$ 0 0
$$626$$ 1.12511e9 0.183309
$$627$$ −1.36999e9 −0.221963
$$628$$ −5.52190e9 −0.889672
$$629$$ −2.48236e9 −0.397729
$$630$$ 0 0
$$631$$ −8.82660e9 −1.39859 −0.699295 0.714833i $$-0.746503\pi$$
−0.699295 + 0.714833i $$0.746503\pi$$
$$632$$ −1.81144e9 −0.285439
$$633$$ 1.28424e9 0.201248
$$634$$ 5.31876e8 0.0828893
$$635$$ 0 0
$$636$$ 6.29509e8 0.0970291
$$637$$ 7.21930e8 0.110664
$$638$$ 1.84486e8 0.0281249
$$639$$ −1.14037e9 −0.172900
$$640$$ 0 0
$$641$$ 8.54151e9 1.28095 0.640474 0.767980i $$-0.278738\pi$$
0.640474 + 0.767980i $$0.278738\pi$$
$$642$$ 2.83778e8 0.0423259
$$643$$ 1.20342e10 1.78517 0.892585 0.450878i $$-0.148889\pi$$
0.892585 + 0.450878i $$0.148889\pi$$
$$644$$ −2.76831e9 −0.408426
$$645$$ 0 0
$$646$$ 8.35934e8 0.122000
$$647$$ −1.89174e8 −0.0274598 −0.0137299 0.999906i $$-0.504370\pi$$
−0.0137299 + 0.999906i $$0.504370\pi$$
$$648$$ 3.91133e8 0.0564693
$$649$$ 2.59539e9 0.372688
$$650$$ 0 0
$$651$$ −2.34764e9 −0.333501
$$652$$ 5.11403e9 0.722598
$$653$$ −8.70977e9 −1.22408 −0.612041 0.790826i $$-0.709651\pi$$
−0.612041 + 0.790826i $$0.709651\pi$$
$$654$$ −6.64159e8 −0.0928432
$$655$$ 0 0
$$656$$ −4.75963e9 −0.658279
$$657$$ 3.42067e9 0.470579
$$658$$ 2.40033e8 0.0328458
$$659$$ −7.48288e8 −0.101852 −0.0509260 0.998702i $$-0.516217\pi$$
−0.0509260 + 0.998702i $$0.516217\pi$$
$$660$$ 0 0
$$661$$ 8.45586e9 1.13881 0.569407 0.822056i $$-0.307173\pi$$
0.569407 + 0.822056i $$0.307173\pi$$
$$662$$ −2.03074e8 −0.0272052
$$663$$ −2.41330e9 −0.321598
$$664$$ −2.11152e9 −0.279903
$$665$$ 0 0
$$666$$ −3.36310e8 −0.0441144
$$667$$ 6.02996e9 0.786817
$$668$$ 1.22784e10 1.59376
$$669$$ −3.10154e9 −0.400485
$$670$$ 0 0
$$671$$ 1.40661e9 0.179740
$$672$$ 5.60357e8 0.0712316
$$673$$ −4.78543e9 −0.605157 −0.302578 0.953124i $$-0.597847\pi$$
−0.302578 + 0.953124i $$0.597847\pi$$
$$674$$ 1.68628e8 0.0212139
$$675$$ 0 0
$$676$$ 3.16520e9 0.394083
$$677$$ 1.29662e10 1.60603 0.803015 0.595958i $$-0.203228\pi$$
0.803015 + 0.595958i $$0.203228\pi$$
$$678$$ −4.62018e8 −0.0569318
$$679$$ −2.27918e9 −0.279405
$$680$$ 0 0
$$681$$ 4.76419e8 0.0578061
$$682$$ 5.40302e8 0.0652216
$$683$$ 9.15988e9 1.10006 0.550031 0.835144i $$-0.314616\pi$$
0.550031 + 0.835144i $$0.314616\pi$$
$$684$$ −7.64706e9 −0.913689
$$685$$ 0 0
$$686$$ −5.51533e7 −0.00652285
$$687$$ 2.62007e9 0.308293
$$688$$ −9.97905e9 −1.16823
$$689$$ 1.23492e9 0.143838
$$690$$ 0 0
$$691$$ 1.05298e10 1.21407 0.607037 0.794673i $$-0.292358\pi$$
0.607037 + 0.794673i $$0.292358\pi$$
$$692$$ −6.44038e9 −0.738823
$$693$$ 7.72318e8 0.0881515
$$694$$ 1.79944e9 0.204352
$$695$$ 0 0
$$696$$ −8.11729e8 −0.0912596
$$697$$ −4.81683e9 −0.538824
$$698$$ −3.61534e8 −0.0402398
$$699$$ −5.74470e9 −0.636205
$$700$$ 0 0
$$701$$ 1.27411e9 0.139699 0.0698497 0.997558i $$-0.477748\pi$$
0.0698497 + 0.997558i $$0.477748\pi$$
$$702$$ −7.81825e8 −0.0852962
$$703$$ 6.03703e9 0.655360
$$704$$ 2.74405e9 0.296406
$$705$$ 0 0
$$706$$ 1.78048e9 0.190423
$$707$$ 3.68832e9 0.392519
$$708$$ −5.66782e9 −0.600205
$$709$$ −7.17795e9 −0.756378 −0.378189 0.925728i $$-0.623453\pi$$
−0.378189 + 0.925728i $$0.623453\pi$$
$$710$$ 0 0
$$711$$ 8.19829e9 0.855421
$$712$$ −3.68653e9 −0.382769
$$713$$ 1.76599e10 1.82463
$$714$$ 1.84369e8 0.0189559
$$715$$ 0 0
$$716$$ −1.83914e10 −1.87249
$$717$$ −2.70746e9 −0.274312
$$718$$ 1.41133e9 0.142296
$$719$$ 1.18502e10 1.18898 0.594488 0.804104i $$-0.297355\pi$$
0.594488 + 0.804104i $$0.297355\pi$$
$$720$$ 0 0
$$721$$ 3.63188e9 0.360876
$$722$$ −8.11273e8 −0.0802208
$$723$$ 2.04665e9 0.201400
$$724$$ 7.68971e9 0.753052
$$725$$ 0 0
$$726$$ −5.90971e8 −0.0573176
$$727$$ 4.67874e9 0.451605 0.225802 0.974173i $$-0.427500\pi$$
0.225802 + 0.974173i $$0.427500\pi$$
$$728$$ 7.31054e8 0.0702246
$$729$$ 3.28577e9 0.314116
$$730$$ 0 0
$$731$$ −1.00990e10 −0.956238
$$732$$ −3.07177e9 −0.289468
$$733$$ −1.28552e9 −0.120563 −0.0602817 0.998181i $$-0.519200\pi$$
−0.0602817 + 0.998181i $$0.519200\pi$$
$$734$$ 1.54612e9 0.144314
$$735$$ 0 0
$$736$$ −4.21524e9 −0.389718
$$737$$ −6.37884e9 −0.586956
$$738$$ −6.52585e8 −0.0597641
$$739$$ −5.26720e9 −0.480091 −0.240046 0.970762i $$-0.577162\pi$$
−0.240046 + 0.970762i $$0.577162\pi$$
$$740$$ 0 0
$$741$$ 5.86909e9 0.529916
$$742$$ −9.43446e7 −0.00847819
$$743$$ −4.15012e9 −0.371193 −0.185596 0.982626i $$-0.559422\pi$$
−0.185596 + 0.982626i $$0.559422\pi$$
$$744$$ −2.37731e9 −0.211631
$$745$$ 0 0
$$746$$ 7.35489e8 0.0648620
$$747$$ 9.55643e9 0.838829
$$748$$ 2.86511e9 0.250314
$$749$$ 2.87171e9 0.249721
$$750$$ 0 0
$$751$$ −6.37970e9 −0.549618 −0.274809 0.961499i $$-0.588615\pi$$
−0.274809 + 0.961499i $$0.588615\pi$$
$$752$$ −8.02346e9 −0.688017
$$753$$ 5.97635e8 0.0510098
$$754$$ −7.90343e8 −0.0671453
$$755$$ 0 0
$$756$$ −4.03304e9 −0.339474
$$757$$ 1.19658e10 1.00255 0.501274 0.865289i $$-0.332865\pi$$
0.501274 + 0.865289i $$0.332865\pi$$
$$758$$ −1.07032e9 −0.0892631
$$759$$ 2.27296e9 0.188688
$$760$$ 0 0
$$761$$ −2.00959e10 −1.65296 −0.826479 0.562967i $$-0.809660\pi$$
−0.826479 + 0.562967i $$0.809660\pi$$
$$762$$ −7.58437e8 −0.0620979
$$763$$ −6.72100e9 −0.547770
$$764$$ 1.92450e10 1.56131
$$765$$ 0 0
$$766$$ 1.12411e9 0.0903665
$$767$$ −1.11187e10 −0.889756
$$768$$ −5.70667e9 −0.454588
$$769$$ 2.46683e10 1.95613 0.978064 0.208304i $$-0.0667944\pi$$
0.978064 + 0.208304i $$0.0667944\pi$$
$$770$$ 0 0
$$771$$ 2.41807e9 0.190011
$$772$$ −1.75673e10 −1.37418
$$773$$ −8.88824e9 −0.692130 −0.346065 0.938211i $$-0.612482\pi$$
−0.346065 + 0.938211i $$0.612482\pi$$
$$774$$ −1.36821e9 −0.106062
$$775$$ 0 0
$$776$$ −2.30798e9 −0.177303
$$777$$ 1.33149e9 0.101828
$$778$$ 1.46252e9 0.111345
$$779$$ 1.17144e10 0.887850
$$780$$ 0 0
$$781$$ −1.03909e9 −0.0780502
$$782$$ −1.38690e9 −0.103710
$$783$$ 8.78483e9 0.653984
$$784$$ 1.84358e9 0.136633
$$785$$ 0 0
$$786$$ −1.53655e8 −0.0112867
$$787$$ 4.65006e9 0.340053 0.170027 0.985439i $$-0.445615\pi$$
0.170027 + 0.985439i $$0.445615\pi$$
$$788$$ 8.22986e9 0.599171
$$789$$ 7.40606e9 0.536806
$$790$$ 0 0
$$791$$ −4.67542e9 −0.335895
$$792$$ 7.82079e8 0.0559387
$$793$$ −6.02598e9 −0.429113
$$794$$ −1.31830e9 −0.0934638
$$795$$ 0 0
$$796$$ −2.44070e8 −0.0171521
$$797$$ −1.42890e10 −0.999762 −0.499881 0.866094i $$-0.666623\pi$$
−0.499881 + 0.866094i $$0.666623\pi$$
$$798$$ −4.48381e8 −0.0312347
$$799$$ −8.11987e9 −0.563165
$$800$$ 0 0
$$801$$ 1.66846e10 1.14711
$$802$$ 2.66257e9 0.182260
$$803$$ 3.11685e9 0.212428
$$804$$ 1.39302e10 0.945279
$$805$$ 0 0
$$806$$ −2.31467e9 −0.155710
$$807$$ 9.68971e9 0.649013
$$808$$ 3.73494e9 0.249083
$$809$$ −4.92320e9 −0.326909 −0.163455 0.986551i $$-0.552264\pi$$
−0.163455 + 0.986551i $$0.552264\pi$$
$$810$$ 0 0
$$811$$ 2.35801e10 1.55229 0.776145 0.630555i $$-0.217173\pi$$
0.776145 + 0.630555i $$0.217173\pi$$
$$812$$ −4.07698e9 −0.267235
$$813$$ −5.25992e9 −0.343291
$$814$$ −3.06440e8 −0.0199141
$$815$$ 0 0
$$816$$ −6.16280e9 −0.397066
$$817$$ 2.45605e10 1.57565
$$818$$ −1.18069e9 −0.0754221
$$819$$ −3.30864e9 −0.210453
$$820$$ 0 0
$$821$$ −2.86630e10 −1.80768 −0.903838 0.427875i $$-0.859262\pi$$
−0.903838 + 0.427875i $$0.859262\pi$$
$$822$$ 1.72069e9 0.108057
$$823$$ 2.76897e10 1.73148 0.865742 0.500490i $$-0.166847\pi$$
0.865742 + 0.500490i $$0.166847\pi$$
$$824$$ 3.67778e9 0.229003
$$825$$ 0 0
$$826$$ 8.49437e8 0.0524447
$$827$$ 1.27176e10 0.781873 0.390936 0.920418i $$-0.372151\pi$$
0.390936 + 0.920418i $$0.372151\pi$$
$$828$$ 1.26873e10 0.776716
$$829$$ −1.50770e10 −0.919127 −0.459563 0.888145i $$-0.651994\pi$$
−0.459563 + 0.888145i $$0.651994\pi$$
$$830$$ 0 0
$$831$$ 4.63064e9 0.279923
$$832$$ −1.17556e10 −0.707641
$$833$$ 1.86573e9 0.111839
$$834$$ 3.56514e8 0.0212812
$$835$$ 0 0
$$836$$ −6.96787e9 −0.412457
$$837$$ 2.57281e10 1.51659
$$838$$ −3.02512e9 −0.177578
$$839$$ 4.59511e9 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$840$$ 0 0
$$841$$ −8.36934e9 −0.485182
$$842$$ 3.96175e9 0.228715
$$843$$ 1.83216e10 1.05334
$$844$$ 6.53172e9 0.373963
$$845$$ 0 0
$$846$$ −1.10008e9 −0.0624639
$$847$$ −5.98038e9 −0.338171
$$848$$ 3.15361e9 0.177591
$$849$$ −7.73504e9 −0.433796
$$850$$ 0 0
$$851$$ −1.00161e10 −0.557114
$$852$$ 2.26917e9 0.125698
$$853$$ 1.13971e9 0.0628740 0.0314370 0.999506i $$-0.489992\pi$$
0.0314370 + 0.999506i $$0.489992\pi$$
$$854$$ 4.60367e8 0.0252931
$$855$$ 0 0
$$856$$ 2.90801e9 0.158466
$$857$$ −7.79419e9 −0.422998 −0.211499 0.977378i $$-0.567834\pi$$
−0.211499 + 0.977378i $$0.567834\pi$$
$$858$$ −2.97915e8 −0.0161023
$$859$$ 1.27280e10 0.685147 0.342573 0.939491i $$-0.388701\pi$$
0.342573 + 0.939491i $$0.388701\pi$$
$$860$$ 0 0
$$861$$ 2.58367e9 0.137951
$$862$$ 3.30831e9 0.175926
$$863$$ −2.53204e9 −0.134101 −0.0670507 0.997750i $$-0.521359\pi$$
−0.0670507 + 0.997750i $$0.521359\pi$$
$$864$$ −6.14103e9 −0.323924
$$865$$ 0 0
$$866$$ −3.09823e9 −0.162107
$$867$$ 3.93933e9 0.205284
$$868$$ −1.19402e10 −0.619718
$$869$$ 7.47014e9 0.386153
$$870$$ 0 0
$$871$$ 2.73272e10 1.40130
$$872$$ −6.80595e9 −0.347601
$$873$$ 1.04456e10 0.531352
$$874$$ 3.37291e9 0.170889
$$875$$ 0 0
$$876$$ −6.80661e9 −0.342110
$$877$$ −5.00988e9 −0.250800 −0.125400 0.992106i $$-0.540021\pi$$
−0.125400 + 0.992106i $$0.540021\pi$$
$$878$$ −2.71862e9 −0.135556
$$879$$ −1.25353e10 −0.622550
$$880$$ 0 0
$$881$$ 9.46900e9 0.466539 0.233270 0.972412i $$-0.425058\pi$$
0.233270 + 0.972412i $$0.425058\pi$$
$$882$$ 2.52770e8 0.0124047
$$883$$ −1.11146e10 −0.543289 −0.271644 0.962398i $$-0.587567\pi$$
−0.271644 + 0.962398i $$0.587567\pi$$
$$884$$ −1.22742e10 −0.597600
$$885$$ 0 0
$$886$$ −1.20006e9 −0.0579675
$$887$$ 7.27986e9 0.350260 0.175130 0.984545i $$-0.443965\pi$$
0.175130 + 0.984545i $$0.443965\pi$$
$$888$$ 1.34832e9 0.0646173
$$889$$ −7.67506e9 −0.366375
$$890$$ 0 0
$$891$$ −1.61298e9 −0.0763938
$$892$$ −1.57746e10 −0.744188
$$893$$ 1.97473e10 0.927959
$$894$$ 1.84217e9 0.0862280
$$895$$ 0 0
$$896$$ 3.79032e9 0.176034
$$897$$ −9.73743e9 −0.450475
$$898$$ −2.09267e8 −0.00964348
$$899$$ 2.60084e10 1.19386
$$900$$ 0 0
$$901$$ 3.19150e9 0.145365
$$902$$ −5.94624e8 −0.0269786
$$903$$ 5.41692e9 0.244819
$$904$$ −4.73451e9 −0.213150
$$905$$ 0 0
$$906$$ −7.56726e8 −0.0338057
$$907$$ 1.39503e10 0.620809 0.310405 0.950605i $$-0.399535\pi$$
0.310405 + 0.950605i $$0.399535\pi$$
$$908$$ 2.42310e9 0.107416
$$909$$ −1.69038e10 −0.746465
$$910$$ 0 0
$$911$$ 2.98148e8 0.0130653 0.00653263 0.999979i $$-0.497921\pi$$
0.00653263 + 0.999979i $$0.497921\pi$$
$$912$$ 1.49878e10 0.654268
$$913$$ 8.70765e9 0.378663
$$914$$ −3.27506e9 −0.141876
$$915$$ 0 0
$$916$$ 1.33259e10 0.572876
$$917$$ −1.55492e9 −0.0665910
$$918$$ −2.02053e9 −0.0862015
$$919$$ 2.67202e10 1.13563 0.567814 0.823157i $$-0.307789\pi$$
0.567814 + 0.823157i $$0.307789\pi$$
$$920$$ 0 0
$$921$$ 1.15928e10 0.488969
$$922$$ −2.21295e9 −0.0929850
$$923$$ 4.45149e9 0.186337
$$924$$ −1.53680e9 −0.0640861
$$925$$ 0 0
$$926$$ 1.58725e9 0.0656910
$$927$$ −1.66451e10 −0.686288
$$928$$ −6.20794e9 −0.254994
$$929$$ −3.66336e10 −1.49908 −0.749540 0.661959i $$-0.769725\pi$$
−0.749540 + 0.661959i $$0.769725\pi$$
$$930$$ 0 0
$$931$$ −4.53742e9 −0.184283
$$932$$ −2.92180e10 −1.18221
$$933$$ −2.87728e8 −0.0115983
$$934$$ 3.88115e9 0.155864
$$935$$ 0 0
$$936$$ −3.35045e9 −0.133548
$$937$$ −1.28088e10 −0.508649 −0.254325 0.967119i $$-0.581853\pi$$
−0.254325 + 0.967119i $$0.581853\pi$$
$$938$$ −2.08772e9 −0.0825965
$$939$$ 2.04149e10 0.804669
$$940$$ 0 0
$$941$$ −1.20663e10 −0.472073 −0.236037 0.971744i $$-0.575849\pi$$
−0.236037 + 0.971744i $$0.575849\pi$$
$$942$$ 1.48387e9 0.0578386
$$943$$ −1.94354e10 −0.754751
$$944$$ −2.83937e10 −1.09855
$$945$$ 0 0
$$946$$ −1.24669e9 −0.0478784
$$947$$ −8.36023e9 −0.319885 −0.159942 0.987126i $$-0.551131\pi$$
−0.159942 + 0.987126i $$0.551131\pi$$
$$948$$ −1.63133e10 −0.621890
$$949$$ −1.33527e10 −0.507151
$$950$$ 0 0
$$951$$ 9.65082e9 0.363859
$$952$$ 1.88931e9 0.0709700
$$953$$ −4.49530e10 −1.68242 −0.841209 0.540710i $$-0.818156\pi$$
−0.841209 + 0.540710i $$0.818156\pi$$
$$954$$ 4.32386e8 0.0161232
$$955$$ 0 0
$$956$$ −1.37703e10 −0.509732
$$957$$ 3.34747e9 0.123459
$$958$$ −3.26341e9 −0.119920
$$959$$ 1.74126e10 0.637528
$$960$$ 0 0
$$961$$ 4.86580e10 1.76857
$$962$$ 1.31280e9 0.0475429
$$963$$ −1.31612e10 −0.474901
$$964$$ 1.04094e10 0.374244
$$965$$ 0 0
$$966$$ 7.43911e8 0.0265522
$$967$$ −1.34247e8 −0.00477432 −0.00238716 0.999997i $$-0.500760\pi$$
−0.00238716 + 0.999997i $$0.500760\pi$$
$$968$$ −6.05596e9 −0.214595
$$969$$ 1.51679e10 0.535541
$$970$$ 0 0
$$971$$ −3.00377e10 −1.05293 −0.526465 0.850197i $$-0.676483\pi$$
−0.526465 + 0.850197i $$0.676483\pi$$
$$972$$ 2.92375e10 1.02119
$$973$$ 3.60777e9 0.125558
$$974$$ −3.01379e9 −0.104510
$$975$$ 0 0
$$976$$ −1.53884e10 −0.529810
$$977$$ −4.52860e10 −1.55358 −0.776789 0.629761i $$-0.783153\pi$$
−0.776789 + 0.629761i $$0.783153\pi$$
$$978$$ −1.37426e9 −0.0469769
$$979$$ 1.52028e10 0.517825
$$980$$ 0 0
$$981$$ 3.08027e10 1.04171
$$982$$ −5.85057e8 −0.0197155
$$983$$ −4.61443e10 −1.54946 −0.774731 0.632290i $$-0.782115\pi$$
−0.774731 + 0.632290i $$0.782115\pi$$
$$984$$ 2.61632e9 0.0875404
$$985$$ 0 0
$$986$$ −2.04254e9 −0.0678580
$$987$$ 4.35537e9 0.144183
$$988$$ 2.98506e10 0.984700
$$989$$ −4.07484e10 −1.33944
$$990$$ 0 0
$$991$$ −1.05400e10 −0.344018 −0.172009 0.985095i $$-0.555026\pi$$
−0.172009 + 0.985095i $$0.555026\pi$$
$$992$$ −1.81812e10 −0.591331
$$993$$ −3.68475e9 −0.119422
$$994$$ −3.40081e8 −0.0109832
$$995$$ 0 0
$$996$$ −1.90158e10 −0.609828
$$997$$ 5.00734e10 1.60020 0.800099 0.599868i $$-0.204780\pi$$
0.800099 + 0.599868i $$0.204780\pi$$
$$998$$ 4.03435e9 0.128474
$$999$$ −1.45920e10 −0.463059
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 175.8.a.b.1.2 2
5.2 odd 4 175.8.b.c.99.2 4
5.3 odd 4 175.8.b.c.99.3 4
5.4 even 2 35.8.a.a.1.1 2
15.14 odd 2 315.8.a.c.1.2 2
20.19 odd 2 560.8.a.i.1.1 2
35.34 odd 2 245.8.a.b.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.1 2 5.4 even 2
175.8.a.b.1.2 2 1.1 even 1 trivial
175.8.b.c.99.2 4 5.2 odd 4
175.8.b.c.99.3 4 5.3 odd 4
245.8.a.b.1.1 2 35.34 odd 2
315.8.a.c.1.2 2 15.14 odd 2
560.8.a.i.1.1 2 20.19 odd 2