# Properties

 Label 175.8.b.c.99.3 Level $175$ Weight $8$ Character 175.99 Analytic conductor $54.667$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,8,Mod(99,175)]

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([1, 0]))

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

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not 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: $$54.6673794597$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.3 Root $$-1.65831 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.99 Dual form 175.8.b.c.99.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.36675i q^{2} -24.7995i q^{3} +126.132 q^{4} +33.8947 q^{6} -343.000i q^{7} +347.335i q^{8} +1571.98 q^{9} +O(q^{10})$$ $$q+1.36675i q^{2} -24.7995i q^{3} +126.132 q^{4} +33.8947 q^{6} -343.000i q^{7} +347.335i q^{8} +1571.98 q^{9} -1432.37 q^{11} -3128.01i q^{12} +6136.30i q^{13} +468.795 q^{14} +15670.2 q^{16} -15858.5i q^{17} +2148.51i q^{18} +38567.5 q^{19} -8506.23 q^{21} -1957.69i q^{22} +63987.4i q^{23} +8613.73 q^{24} -8386.79 q^{26} -93220.9i q^{27} -43263.3i q^{28} -94236.6 q^{29} +275990. q^{31} +65876.1i q^{32} +35521.9i q^{33} +21674.6 q^{34} +198278. q^{36} +156532. i q^{37} +52712.1i q^{38} +152177. q^{39} -303738. q^{41} -11625.9i q^{42} -636818. i q^{43} -180667. q^{44} -87454.9 q^{46} +512021. i q^{47} -388612. i q^{48} -117649. q^{49} -393282. q^{51} +773984. i q^{52} +201249. i q^{53} +127410. q^{54} +119136. q^{56} -956454. i q^{57} -128798. i q^{58} +1.81196e6 q^{59} -982021. q^{61} +377210. i q^{62} -539191. i q^{63} +1.91575e6 q^{64} -48549.6 q^{66} -4.45336e6i q^{67} -2.00026e6i q^{68} +1.58686e6 q^{69} +725436. q^{71} +546005. i q^{72} -2.17602e6i q^{73} -213940. q^{74} +4.86459e6 q^{76} +491301. i q^{77} +207988. i q^{78} +5.21525e6 q^{79} +1.12610e6 q^{81} -415135. i q^{82} -6.07921e6i q^{83} -1.07291e6 q^{84} +870371. q^{86} +2.33702e6i q^{87} -497511. i q^{88} +1.06137e7 q^{89} +2.10475e6 q^{91} +8.07086e6i q^{92} -6.84442e6i q^{93} -699805. q^{94} +1.63369e6 q^{96} +6.64483e6i q^{97} -160797. i q^{98} -2.25166e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9}+O(q^{10})$$ 4 * q + 80 * q^4 - 1536 * q^6 + 1512 * q^9 $$4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9} - 15812 q^{11} + 10976 q^{14} - 8640 q^{16} + 7224 q^{19} + 20580 q^{21} - 49920 q^{24} - 358656 q^{26} + 253796 q^{29} + 505536 q^{31} - 350592 q^{34} + 537120 q^{36} - 975948 q^{39} - 223840 q^{41} + 753840 q^{44} + 2102944 q^{46} - 470596 q^{49} - 2261820 q^{51} - 1944000 q^{54} + 658560 q^{56} + 2720240 q^{59} - 3627360 q^{61} + 4979968 q^{64} + 10285248 q^{66} + 11703960 q^{69} - 2989856 q^{71} + 11934048 q^{74} + 15750752 q^{76} + 15885948 q^{79} - 9551196 q^{81} - 5383728 q^{84} + 10835424 q^{86} + 35887056 q^{89} + 12223148 q^{91} + 5646208 q^{94} + 26732544 q^{96} + 6061464 q^{99}+O(q^{100})$$ 4 * q + 80 * q^4 - 1536 * q^6 + 1512 * q^9 - 15812 * q^11 + 10976 * q^14 - 8640 * q^16 + 7224 * q^19 + 20580 * q^21 - 49920 * q^24 - 358656 * q^26 + 253796 * q^29 + 505536 * q^31 - 350592 * q^34 + 537120 * q^36 - 975948 * q^39 - 223840 * q^41 + 753840 * q^44 + 2102944 * q^46 - 470596 * q^49 - 2261820 * q^51 - 1944000 * q^54 + 658560 * q^56 + 2720240 * q^59 - 3627360 * q^61 + 4979968 * q^64 + 10285248 * q^66 + 11703960 * q^69 - 2989856 * q^71 + 11934048 * q^74 + 15750752 * q^76 + 15885948 * q^79 - 9551196 * q^81 - 5383728 * q^84 + 10835424 * q^86 + 35887056 * q^89 + 12223148 * q^91 + 5646208 * q^94 + 26732544 * q^96 + 6061464 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.36675i 0.120805i 0.998174 + 0.0604024i $$0.0192384\pi$$
−0.998174 + 0.0604024i $$0.980762\pi$$
$$3$$ − 24.7995i − 0.530296i −0.964208 0.265148i $$-0.914579\pi$$
0.964208 0.265148i $$-0.0854208\pi$$
$$4$$ 126.132 0.985406
$$5$$ 0 0
$$6$$ 33.8947 0.0640623
$$7$$ − 343.000i − 0.377964i
$$8$$ 347.335i 0.239847i
$$9$$ 1571.98 0.718786
$$10$$ 0 0
$$11$$ −1432.37 −0.324474 −0.162237 0.986752i $$-0.551871\pi$$
−0.162237 + 0.986752i $$0.551871\pi$$
$$12$$ − 3128.01i − 0.522557i
$$13$$ 6136.30i 0.774649i 0.921943 + 0.387325i $$0.126601\pi$$
−0.921943 + 0.387325i $$0.873399\pi$$
$$14$$ 468.795 0.0456599
$$15$$ 0 0
$$16$$ 15670.2 0.956432
$$17$$ − 15858.5i − 0.782871i −0.920205 0.391436i $$-0.871979\pi$$
0.920205 0.391436i $$-0.128021\pi$$
$$18$$ 2148.51i 0.0868328i
$$19$$ 38567.5 1.28998 0.644991 0.764190i $$-0.276861\pi$$
0.644991 + 0.764190i $$0.276861\pi$$
$$20$$ 0 0
$$21$$ −8506.23 −0.200433
$$22$$ − 1957.69i − 0.0391980i
$$23$$ 63987.4i 1.09660i 0.836282 + 0.548299i $$0.184724\pi$$
−0.836282 + 0.548299i $$0.815276\pi$$
$$24$$ 8613.73 0.127190
$$25$$ 0 0
$$26$$ −8386.79 −0.0935813
$$27$$ − 93220.9i − 0.911466i
$$28$$ − 43263.3i − 0.372449i
$$29$$ −94236.6 −0.717508 −0.358754 0.933432i $$-0.616798\pi$$
−0.358754 + 0.933432i $$0.616798\pi$$
$$30$$ 0 0
$$31$$ 275990. 1.66390 0.831951 0.554849i $$-0.187224\pi$$
0.831951 + 0.554849i $$0.187224\pi$$
$$32$$ 65876.1i 0.355388i
$$33$$ 35521.9i 0.172067i
$$34$$ 21674.6 0.0945746
$$35$$ 0 0
$$36$$ 198278. 0.708296
$$37$$ 156532.i 0.508038i 0.967199 + 0.254019i $$0.0817526\pi$$
−0.967199 + 0.254019i $$0.918247\pi$$
$$38$$ 52712.1i 0.155836i
$$39$$ 152177. 0.410793
$$40$$ 0 0
$$41$$ −303738. −0.688266 −0.344133 0.938921i $$-0.611827\pi$$
−0.344133 + 0.938921i $$0.611827\pi$$
$$42$$ − 11625.9i − 0.0242133i
$$43$$ − 636818.i − 1.22145i −0.791843 0.610725i $$-0.790878\pi$$
0.791843 0.610725i $$-0.209122\pi$$
$$44$$ −180667. −0.319738
$$45$$ 0 0
$$46$$ −87454.9 −0.132474
$$47$$ 512021.i 0.719358i 0.933076 + 0.359679i $$0.117114\pi$$
−0.933076 + 0.359679i $$0.882886\pi$$
$$48$$ − 388612.i − 0.507192i
$$49$$ −117649. −0.142857
$$50$$ 0 0
$$51$$ −393282. −0.415154
$$52$$ 773984.i 0.763344i
$$53$$ 201249.i 0.185681i 0.995681 + 0.0928406i $$0.0295947\pi$$
−0.995681 + 0.0928406i $$0.970405\pi$$
$$54$$ 127410. 0.110109
$$55$$ 0 0
$$56$$ 119136. 0.0906535
$$57$$ − 956454.i − 0.684072i
$$58$$ − 128798.i − 0.0866784i
$$59$$ 1.81196e6 1.14859 0.574296 0.818648i $$-0.305276\pi$$
0.574296 + 0.818648i $$0.305276\pi$$
$$60$$ 0 0
$$61$$ −982021. −0.553945 −0.276972 0.960878i $$-0.589331\pi$$
−0.276972 + 0.960878i $$0.589331\pi$$
$$62$$ 377210.i 0.201007i
$$63$$ − 539191.i − 0.271676i
$$64$$ 1.91575e6 0.913499
$$65$$ 0 0
$$66$$ −48549.6 −0.0207865
$$67$$ − 4.45336e6i − 1.80895i −0.426528 0.904474i $$-0.640264\pi$$
0.426528 0.904474i $$-0.359736\pi$$
$$68$$ − 2.00026e6i − 0.771446i
$$69$$ 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.i 0.172398i
$$73$$ − 2.17602e6i − 0.654685i −0.944906 0.327343i $$-0.893847\pi$$
0.944906 0.327343i $$-0.106153\pi$$
$$74$$ −213940. −0.0613735
$$75$$ 0 0
$$76$$ 4.86459e6 1.27116
$$77$$ 491301.i 0.122639i
$$78$$ 207988.i 0.0496258i
$$79$$ 5.21525e6 1.19009 0.595045 0.803692i $$-0.297134\pi$$
0.595045 + 0.803692i $$0.297134\pi$$
$$80$$ 0 0
$$81$$ 1.12610e6 0.235439
$$82$$ − 415135.i − 0.0831458i
$$83$$ − 6.07921e6i − 1.16701i −0.812110 0.583504i $$-0.801681\pi$$
0.812110 0.583504i $$-0.198319\pi$$
$$84$$ −1.07291e6 −0.197508
$$85$$ 0 0
$$86$$ 870371. 0.147557
$$87$$ 2.33702e6i 0.380492i
$$88$$ − 497511.i − 0.0778239i
$$89$$ 1.06137e7 1.59589 0.797946 0.602729i $$-0.205920\pi$$
0.797946 + 0.602729i $$0.205920\pi$$
$$90$$ 0 0
$$91$$ 2.10475e6 0.292790
$$92$$ 8.07086e6i 1.08059i
$$93$$ − 6.84442e6i − 0.882361i
$$94$$ −699805. −0.0869019
$$95$$ 0 0
$$96$$ 1.63369e6 0.188461
$$97$$ 6.64483e6i 0.739236i 0.929184 + 0.369618i $$0.120511\pi$$
−0.929184 + 0.369618i $$0.879489\pi$$
$$98$$ − 160797.i − 0.0172578i
$$99$$ −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.i − 0.0501526i
$$103$$ 1.05886e7i 0.954788i 0.878689 + 0.477394i $$0.158419\pi$$
−0.878689 + 0.477394i $$0.841581\pi$$
$$104$$ −2.13135e6 −0.185797
$$105$$ 0 0
$$106$$ −275057. −0.0224312
$$107$$ − 8.37234e6i − 0.660699i −0.943859 0.330349i $$-0.892833\pi$$
0.943859 0.330349i $$-0.107167\pi$$
$$108$$ − 1.17581e7i − 0.898164i
$$109$$ 1.95948e7 1.44926 0.724632 0.689136i $$-0.242010\pi$$
0.724632 + 0.689136i $$0.242010\pi$$
$$110$$ 0 0
$$111$$ 3.88191e6 0.269411
$$112$$ − 5.37487e6i − 0.361497i
$$113$$ − 1.36310e7i − 0.888694i −0.895855 0.444347i $$-0.853436\pi$$
0.895855 0.444347i $$-0.146564\pi$$
$$114$$ 1.30723e6 0.0826392
$$115$$ 0 0
$$116$$ −1.18863e7 −0.707037
$$117$$ 9.64617e6i 0.556807i
$$118$$ 2.47649e6i 0.138756i
$$119$$ −5.43946e6 −0.295898
$$120$$ 0 0
$$121$$ −1.74355e7 −0.894717
$$122$$ − 1.34218e6i − 0.0669192i
$$123$$ 7.53256e6i 0.364985i
$$124$$ 3.48112e7 1.63962
$$125$$ 0 0
$$126$$ 736939. 0.0328197
$$127$$ 2.23763e7i 0.969336i 0.874698 + 0.484668i $$0.161060\pi$$
−0.874698 + 0.484668i $$0.838940\pi$$
$$128$$ 1.10505e7i 0.465743i
$$129$$ −1.57928e7 −0.647730
$$130$$ 0 0
$$131$$ −4.53330e6 −0.176183 −0.0880917 0.996112i $$-0.528077\pi$$
−0.0880917 + 0.996112i $$0.528077\pi$$
$$132$$ 4.48045e6i 0.169556i
$$133$$ − 1.32286e7i − 0.487567i
$$134$$ 6.08663e6 0.218530
$$135$$ 0 0
$$136$$ 5.50821e6 0.187769
$$137$$ − 5.07657e7i − 1.68674i −0.537332 0.843371i $$-0.680568\pi$$
0.537332 0.843371i $$-0.319432\pi$$
$$138$$ 2.16884e6i 0.0702506i
$$139$$ −1.05183e7 −0.332195 −0.166097 0.986109i $$-0.553117\pi$$
−0.166097 + 0.986109i $$0.553117\pi$$
$$140$$ 0 0
$$141$$ 1.26979e7 0.381473
$$142$$ 991490.i 0.0290589i
$$143$$ − 8.78942e6i − 0.251353i
$$144$$ 2.46333e7 0.687470
$$145$$ 0 0
$$146$$ 2.97407e6 0.0790891
$$147$$ 2.91764e6i 0.0757566i
$$148$$ 1.97437e7i 0.500624i
$$149$$ −5.43497e7 −1.34600 −0.673000 0.739642i $$-0.734995\pi$$
−0.673000 + 0.739642i $$0.734995\pi$$
$$150$$ 0 0
$$151$$ −2.23258e7 −0.527700 −0.263850 0.964564i $$-0.584992\pi$$
−0.263850 + 0.964564i $$0.584992\pi$$
$$152$$ 1.33958e7i 0.309398i
$$153$$ − 2.49293e7i − 0.562717i
$$154$$ −671486. −0.0148154
$$155$$ 0 0
$$156$$ 1.91944e7 0.404798
$$157$$ − 4.37788e7i − 0.902848i −0.892310 0.451424i $$-0.850916\pi$$
0.892310 0.451424i $$-0.149084\pi$$
$$158$$ 7.12794e6i 0.143769i
$$159$$ 4.99087e6 0.0984661
$$160$$ 0 0
$$161$$ 2.19477e7 0.414475
$$162$$ 1.53910e6i 0.0284422i
$$163$$ − 4.05451e7i − 0.733300i −0.930359 0.366650i $$-0.880505\pi$$
0.930359 0.366650i $$-0.119495\pi$$
$$164$$ −3.83111e7 −0.678221
$$165$$ 0 0
$$166$$ 8.30876e6 0.140980
$$167$$ 9.73453e7i 1.61736i 0.588247 + 0.808682i $$0.299818\pi$$
−0.588247 + 0.808682i $$0.700182\pi$$
$$168$$ − 2.95451e6i − 0.0480732i
$$169$$ 2.50943e7 0.399919
$$170$$ 0 0
$$171$$ 6.06275e7 0.927221
$$172$$ − 8.03231e7i − 1.20362i
$$173$$ 5.10607e7i 0.749765i 0.927072 + 0.374882i $$0.122317\pi$$
−0.927072 + 0.374882i $$0.877683\pi$$
$$174$$ −3.19412e6 −0.0459652
$$175$$ 0 0
$$176$$ −2.24454e7 −0.310337
$$177$$ − 4.49356e7i − 0.609094i
$$178$$ 1.45063e7i 0.192791i
$$179$$ −1.45811e8 −1.90023 −0.950113 0.311907i $$-0.899032\pi$$
−0.950113 + 0.311907i $$0.899032\pi$$
$$180$$ 0 0
$$181$$ −6.09656e7 −0.764205 −0.382102 0.924120i $$-0.624800\pi$$
−0.382102 + 0.924120i $$0.624800\pi$$
$$182$$ 2.87667e6i 0.0353704i
$$183$$ 2.43536e7i 0.293755i
$$184$$ −2.22251e7 −0.263015
$$185$$ 0 0
$$186$$ 9.35462e6 0.106593
$$187$$ 2.27151e7i 0.254021i
$$188$$ 6.45822e7i 0.708860i
$$189$$ −3.19748e7 −0.344502
$$190$$ 0 0
$$191$$ −1.52578e8 −1.58444 −0.792219 0.610237i $$-0.791074\pi$$
−0.792219 + 0.610237i $$0.791074\pi$$
$$192$$ − 4.75095e7i − 0.484425i
$$193$$ 1.39277e8i 1.39453i 0.716812 + 0.697267i $$0.245601\pi$$
−0.716812 + 0.697267i $$0.754399\pi$$
$$194$$ −9.08183e6 −0.0893033
$$195$$ 0 0
$$196$$ −1.48393e7 −0.140772
$$197$$ 6.52480e7i 0.608044i 0.952665 + 0.304022i $$0.0983297\pi$$
−0.952665 + 0.304022i $$0.901670\pi$$
$$198$$ − 3.07745e6i − 0.0281750i
$$199$$ −1.93503e6 −0.0174061 −0.00870307 0.999962i $$-0.502770\pi$$
−0.00870307 + 0.999962i $$0.502770\pi$$
$$200$$ 0 0
$$201$$ −1.10441e8 −0.959278
$$202$$ 1.46968e7i 0.125457i
$$203$$ 3.23232e7i 0.271192i
$$204$$ −4.96055e7 −0.409095
$$205$$ 0 0
$$206$$ −1.44719e7 −0.115343
$$207$$ 1.00587e8i 0.788219i
$$208$$ 9.61569e7i 0.740899i
$$209$$ −5.52427e7 −0.418565
$$210$$ 0 0
$$211$$ −5.17848e7 −0.379502 −0.189751 0.981832i $$-0.560768\pi$$
−0.189751 + 0.981832i $$0.560768\pi$$
$$212$$ 2.53839e7i 0.182971i
$$213$$ − 1.79904e7i − 0.127560i
$$214$$ 1.14429e7 0.0798156
$$215$$ 0 0
$$216$$ 3.23789e7 0.218612
$$217$$ − 9.46647e7i − 0.628896i
$$218$$ 2.67812e7i 0.175078i
$$219$$ −5.39642e7 −0.347177
$$220$$ 0 0
$$221$$ 9.73124e7 0.606450
$$222$$ 5.30560e6i 0.0325461i
$$223$$ 1.25065e8i 0.755209i 0.925967 + 0.377605i $$0.123252\pi$$
−0.925967 + 0.377605i $$0.876748\pi$$
$$224$$ 2.25955e7 0.134324
$$225$$ 0 0
$$226$$ 1.86301e7 0.107358
$$227$$ 1.92108e7i 0.109007i 0.998514 + 0.0545036i $$0.0173576\pi$$
−0.998514 + 0.0545036i $$0.982642\pi$$
$$228$$ − 1.20639e8i − 0.674089i
$$229$$ 1.05650e8 0.581360 0.290680 0.956820i $$-0.406118\pi$$
0.290680 + 0.956820i $$0.406118\pi$$
$$230$$ 0 0
$$231$$ 1.21840e7 0.0650353
$$232$$ − 3.27317e7i − 0.172092i
$$233$$ 2.31646e8i 1.19972i 0.800106 + 0.599859i $$0.204776\pi$$
−0.800106 + 0.599859i $$0.795224\pi$$
$$234$$ −1.31839e7 −0.0672649
$$235$$ 0 0
$$236$$ 2.28546e8 1.13183
$$237$$ − 1.29336e8i − 0.631101i
$$238$$ − 7.43438e6i − 0.0357458i
$$239$$ −1.09174e8 −0.517281 −0.258641 0.965974i $$-0.583275\pi$$
−0.258641 + 0.965974i $$0.583275\pi$$
$$240$$ 0 0
$$241$$ −8.25277e7 −0.379787 −0.189893 0.981805i $$-0.560814\pi$$
−0.189893 + 0.981805i $$0.560814\pi$$
$$242$$ − 2.38300e7i − 0.108086i
$$243$$ − 2.31801e8i − 1.03632i
$$244$$ −1.23864e8 −0.545861
$$245$$ 0 0
$$246$$ −1.02951e7 −0.0440919
$$247$$ 2.36662e8i 0.999283i
$$248$$ 9.58611e7i 0.399081i
$$249$$ −1.50761e8 −0.618860
$$250$$ 0 0
$$251$$ −2.40987e7 −0.0961912 −0.0480956 0.998843i $$-0.515315\pi$$
−0.0480956 + 0.998843i $$0.515315\pi$$
$$252$$ − 6.80092e7i − 0.267711i
$$253$$ − 9.16534e7i − 0.355817i
$$254$$ −3.05828e7 −0.117100
$$255$$ 0 0
$$256$$ 2.30112e8 0.857235
$$257$$ 9.75049e7i 0.358311i 0.983821 + 0.179156i $$0.0573366\pi$$
−0.983821 + 0.179156i $$0.942663\pi$$
$$258$$ − 2.15848e7i − 0.0782489i
$$259$$ 5.36904e7 0.192020
$$260$$ 0 0
$$261$$ −1.48139e8 −0.515735
$$262$$ − 6.19589e6i − 0.0212838i
$$263$$ − 2.98637e8i − 1.01228i −0.862452 0.506138i $$-0.831073\pi$$
0.862452 0.506138i $$-0.168927\pi$$
$$264$$ −1.23380e7 −0.0412697
$$265$$ 0 0
$$266$$ 1.80803e7 0.0589005
$$267$$ − 2.63216e8i − 0.846296i
$$268$$ − 5.61711e8i − 1.78255i
$$269$$ 3.90722e8 1.22387 0.611934 0.790909i $$-0.290392\pi$$
0.611934 + 0.790909i $$0.290392\pi$$
$$270$$ 0 0
$$271$$ 2.12098e8 0.647357 0.323678 0.946167i $$-0.395080\pi$$
0.323678 + 0.946167i $$0.395080\pi$$
$$272$$ − 2.48505e8i − 0.748763i
$$273$$ − 5.21968e7i − 0.155265i
$$274$$ 6.93841e7 0.203767
$$275$$ 0 0
$$276$$ 2.00153e8 0.573035
$$277$$ 1.86723e8i 0.527861i 0.964542 + 0.263930i $$0.0850189\pi$$
−0.964542 + 0.263930i $$0.914981\pi$$
$$278$$ − 1.43759e7i − 0.0401307i
$$279$$ 4.33853e8 1.19599
$$280$$ 0 0
$$281$$ −7.38791e8 −1.98632 −0.993161 0.116756i $$-0.962750\pi$$
−0.993161 + 0.116756i $$0.962750\pi$$
$$282$$ 1.73548e7i 0.0460838i
$$283$$ 3.11903e8i 0.818026i 0.912529 + 0.409013i $$0.134127\pi$$
−0.912529 + 0.409013i $$0.865873\pi$$
$$284$$ 9.15007e7 0.237034
$$285$$ 0 0
$$286$$ 1.20129e7 0.0303647
$$287$$ 1.04182e8i 0.260140i
$$288$$ 1.03556e8i 0.255448i
$$289$$ 1.58847e8 0.387113
$$290$$ 0 0
$$291$$ 1.64789e8 0.392014
$$292$$ − 2.74466e8i − 0.645131i
$$293$$ 5.05466e8i 1.17397i 0.809599 + 0.586983i $$0.199684\pi$$
−0.809599 + 0.586983i $$0.800316\pi$$
$$294$$ −3.98768e6 −0.00915176
$$295$$ 0 0
$$296$$ −5.43690e7 −0.121851
$$297$$ 1.33526e8i 0.295747i
$$298$$ − 7.42825e7i − 0.162603i
$$299$$ −3.92646e8 −0.849478
$$300$$ 0 0
$$301$$ −2.18429e8 −0.461665
$$302$$ − 3.05137e7i − 0.0637487i
$$303$$ − 2.66672e8i − 0.550717i
$$304$$ 6.04359e8 1.23378
$$305$$ 0 0
$$306$$ 3.40721e7 0.0679789
$$307$$ 4.67463e8i 0.922067i 0.887383 + 0.461034i $$0.152521\pi$$
−0.887383 + 0.461034i $$0.847479\pi$$
$$308$$ 6.19688e7i 0.120850i
$$309$$ 2.62591e8 0.506321
$$310$$ 0 0
$$311$$ 1.16022e7 0.0218714 0.0109357 0.999940i $$-0.496519\pi$$
0.0109357 + 0.999940i $$0.496519\pi$$
$$312$$ 5.28565e7i 0.0985274i
$$313$$ − 8.23197e8i − 1.51740i −0.651443 0.758698i $$-0.725836\pi$$
0.651443 0.758698i $$-0.274164\pi$$
$$314$$ 5.98346e7 0.109068
$$315$$ 0 0
$$316$$ 6.57810e8 1.17272
$$317$$ 3.89154e8i 0.686142i 0.939309 + 0.343071i $$0.111467\pi$$
−0.939309 + 0.343071i $$0.888533\pi$$
$$318$$ 6.82128e6i 0.0118952i
$$319$$ 1.34981e8 0.232812
$$320$$ 0 0
$$321$$ −2.07630e8 −0.350366
$$322$$ 2.99970e7i 0.0500706i
$$323$$ − 6.11621e8i − 1.00989i
$$324$$ 1.42037e8 0.232003
$$325$$ 0 0
$$326$$ 5.54150e7 0.0885861
$$327$$ − 4.85940e8i − 0.768539i
$$328$$ − 1.05499e8i − 0.165078i
$$329$$ 1.75623e8 0.271892
$$330$$ 0 0
$$331$$ 1.48582e8 0.225199 0.112600 0.993640i $$-0.464082\pi$$
0.112600 + 0.993640i $$0.464082\pi$$
$$332$$ − 7.66783e8i − 1.14998i
$$333$$ 2.46066e8i 0.365171i
$$334$$ −1.33047e8 −0.195385
$$335$$ 0 0
$$336$$ −1.33294e8 −0.191701
$$337$$ 1.23379e8i 0.175605i 0.996138 + 0.0878023i $$0.0279844\pi$$
−0.996138 + 0.0878023i $$0.972016\pi$$
$$338$$ 3.42977e7i 0.0483121i
$$339$$ −3.38041e8 −0.471271
$$340$$ 0 0
$$341$$ −3.95319e8 −0.539892
$$342$$ 8.28626e7i 0.112013i
$$343$$ 4.03536e7i 0.0539949i
$$344$$ 2.21189e8 0.292961
$$345$$ 0 0
$$346$$ −6.97872e7 −0.0905752
$$347$$ 1.31658e9i 1.69159i 0.533511 + 0.845793i $$0.320872\pi$$
−0.533511 + 0.845793i $$0.679128\pi$$
$$348$$ 2.94773e8i 0.374939i
$$349$$ −2.64521e8 −0.333097 −0.166549 0.986033i $$-0.553262\pi$$
−0.166549 + 0.986033i $$0.553262\pi$$
$$350$$ 0 0
$$351$$ 5.72032e8 0.706066
$$352$$ − 9.43586e7i − 0.115314i
$$353$$ − 1.30271e9i − 1.57629i −0.615490 0.788144i $$-0.711042\pi$$
0.615490 0.788144i $$-0.288958\pi$$
$$354$$ 6.14158e7 0.0735815
$$355$$ 0 0
$$356$$ 1.33873e9 1.57260
$$357$$ 1.34896e8i 0.156913i
$$358$$ − 1.99287e8i − 0.229556i
$$359$$ 1.03262e9 1.17790 0.588952 0.808168i $$-0.299541\pi$$
0.588952 + 0.808168i $$0.299541\pi$$
$$360$$ 0 0
$$361$$ 5.93578e8 0.664053
$$362$$ − 8.33248e7i − 0.0923196i
$$363$$ 4.32392e8i 0.474465i
$$364$$ 2.65476e8 0.288517
$$365$$ 0 0
$$366$$ −3.32853e7 −0.0354870
$$367$$ 1.13124e9i 1.19460i 0.802017 + 0.597302i $$0.203760\pi$$
−0.802017 + 0.597302i $$0.796240\pi$$
$$368$$ 1.00269e9i 1.04882i
$$369$$ −4.77472e8 −0.494716
$$370$$ 0 0
$$371$$ 6.90284e7 0.0701809
$$372$$ − 8.63300e8i − 0.869484i
$$373$$ − 5.38130e8i − 0.536916i −0.963291 0.268458i $$-0.913486\pi$$
0.963291 0.268458i $$-0.0865140\pi$$
$$374$$ −3.10459e7 −0.0306870
$$375$$ 0 0
$$376$$ −1.77843e8 −0.172536
$$377$$ − 5.78264e8i − 0.555817i
$$378$$ − 4.37015e7i − 0.0416175i
$$379$$ −7.83114e8 −0.738904 −0.369452 0.929250i $$-0.620455\pi$$
−0.369452 + 0.929250i $$0.620455\pi$$
$$380$$ 0 0
$$381$$ 5.54920e8 0.514035
$$382$$ − 2.08536e8i − 0.191408i
$$383$$ − 8.22468e8i − 0.748038i −0.927421 0.374019i $$-0.877980\pi$$
0.927421 0.374019i $$-0.122020\pi$$
$$384$$ 2.74047e8 0.246982
$$385$$ 0 0
$$386$$ −1.90357e8 −0.168466
$$387$$ − 1.00107e9i − 0.877961i
$$388$$ 8.38126e8i 0.728448i
$$389$$ 1.07007e9 0.921696 0.460848 0.887479i $$-0.347545\pi$$
0.460848 + 0.887479i $$0.347545\pi$$
$$390$$ 0 0
$$391$$ 1.01474e9 0.858495
$$392$$ − 4.08636e7i − 0.0342638i
$$393$$ 1.12424e8i 0.0934293i
$$394$$ −8.91777e7 −0.0734547
$$395$$ 0 0
$$396$$ −2.84006e8 −0.229823
$$397$$ − 9.64552e8i − 0.773676i −0.922148 0.386838i $$-0.873567\pi$$
0.922148 0.386838i $$-0.126433\pi$$
$$398$$ − 2.64471e6i − 0.00210275i
$$399$$ −3.28064e8 −0.258555
$$400$$ 0 0
$$401$$ −1.94810e9 −1.50871 −0.754357 0.656465i $$-0.772051\pi$$
−0.754357 + 0.656465i $$0.772051\pi$$
$$402$$ − 1.50945e8i − 0.115885i
$$403$$ 1.69356e9i 1.28894i
$$404$$ 1.35631e9 1.02335
$$405$$ 0 0
$$406$$ −4.41777e7 −0.0327614
$$407$$ − 2.24211e8i − 0.164845i
$$408$$ − 1.36601e8i − 0.0995732i
$$409$$ −8.63865e8 −0.624330 −0.312165 0.950028i $$-0.601054\pi$$
−0.312165 + 0.950028i $$0.601054\pi$$
$$410$$ 0 0
$$411$$ −1.25896e9 −0.894473
$$412$$ 1.33556e9i 0.940854i
$$413$$ − 6.21501e8i − 0.434127i
$$414$$ −1.37478e8 −0.0952206
$$415$$ 0 0
$$416$$ −4.04236e8 −0.275301
$$417$$ 2.60848e8i 0.176162i
$$418$$ − 7.55030e7i − 0.0505647i
$$419$$ −2.21337e9 −1.46996 −0.734978 0.678091i $$-0.762808\pi$$
−0.734978 + 0.678091i $$0.762808\pi$$
$$420$$ 0 0
$$421$$ −2.89866e9 −1.89326 −0.946631 0.322321i $$-0.895537\pi$$
−0.946631 + 0.322321i $$0.895537\pi$$
$$422$$ − 7.07769e7i − 0.0458456i
$$423$$ 8.04889e8i 0.517065i
$$424$$ −6.99008e7 −0.0445350
$$425$$ 0 0
$$426$$ 2.45884e7 0.0154098
$$427$$ 3.36833e8i 0.209371i
$$428$$ − 1.05602e9i − 0.651057i
$$429$$ −2.17973e8 −0.133292
$$430$$ 0 0
$$431$$ −2.42056e9 −1.45628 −0.728142 0.685426i $$-0.759616\pi$$
−0.728142 + 0.685426i $$0.759616\pi$$
$$432$$ − 1.46079e9i − 0.871754i
$$433$$ 2.26686e9i 1.34189i 0.741506 + 0.670946i $$0.234112\pi$$
−0.741506 + 0.670946i $$0.765888\pi$$
$$434$$ 1.29383e8 0.0759737
$$435$$ 0 0
$$436$$ 2.47153e9 1.42811
$$437$$ 2.46783e9i 1.41459i
$$438$$ − 7.37555e7i − 0.0419407i
$$439$$ −1.98911e9 −1.12210 −0.561052 0.827780i $$-0.689603\pi$$
−0.561052 + 0.827780i $$0.689603\pi$$
$$440$$ 0 0
$$441$$ −1.84942e8 −0.102684
$$442$$ 1.33002e8i 0.0732621i
$$443$$ 8.78038e8i 0.479844i 0.970792 + 0.239922i $$0.0771219\pi$$
−0.970792 + 0.239922i $$0.922878\pi$$
$$444$$ 4.89633e8 0.265479
$$445$$ 0 0
$$446$$ −1.70932e8 −0.0912329
$$447$$ 1.34785e9i 0.713779i
$$448$$ − 6.57101e8i − 0.345270i
$$449$$ −1.53113e8 −0.0798270 −0.0399135 0.999203i $$-0.512708\pi$$
−0.0399135 + 0.999203i $$0.512708\pi$$
$$450$$ 0 0
$$451$$ 4.35064e8 0.223324
$$452$$ − 1.71930e9i − 0.875724i
$$453$$ 5.53668e8i 0.279837i
$$454$$ −2.62564e7 −0.0131686
$$455$$ 0 0
$$456$$ 3.32210e8 0.164072
$$457$$ − 2.39624e9i − 1.17442i −0.809435 0.587210i $$-0.800226\pi$$
0.809435 0.587210i $$-0.199774\pi$$
$$458$$ 1.44397e8i 0.0702311i
$$459$$ −1.47834e9 −0.713560
$$460$$ 0 0
$$461$$ 1.61913e9 0.769713 0.384856 0.922977i $$-0.374251\pi$$
0.384856 + 0.922977i $$0.374251\pi$$
$$462$$ 1.66525e7i 0.00785657i
$$463$$ − 1.16133e9i − 0.543778i −0.962329 0.271889i $$-0.912352\pi$$
0.962329 0.271889i $$-0.0876483\pi$$
$$464$$ −1.47670e9 −0.686247
$$465$$ 0 0
$$466$$ −3.16602e8 −0.144932
$$467$$ 2.83969e9i 1.29021i 0.764092 + 0.645107i $$0.223187\pi$$
−0.764092 + 0.645107i $$0.776813\pi$$
$$468$$ 1.21669e9i 0.548681i
$$469$$ −1.52750e9 −0.683718
$$470$$ 0 0
$$471$$ −1.08569e9 −0.478777
$$472$$ 6.29356e8i 0.275486i
$$473$$ 9.12156e8i 0.396328i
$$474$$ 1.76769e8 0.0762400
$$475$$ 0 0
$$476$$ −6.86090e8 −0.291579
$$477$$ 3.16360e8i 0.133465i
$$478$$ − 1.49214e8i − 0.0624901i
$$479$$ −2.38771e9 −0.992676 −0.496338 0.868129i $$-0.665322\pi$$
−0.496338 + 0.868129i $$0.665322\pi$$
$$480$$ 0 0
$$481$$ −9.60526e8 −0.393551
$$482$$ − 1.12795e8i − 0.0458801i
$$483$$ − 5.44292e8i − 0.219794i
$$484$$ −2.19917e9 −0.881660
$$485$$ 0 0
$$486$$ 3.16814e8 0.125192
$$487$$ − 2.20508e9i − 0.865113i −0.901607 0.432556i $$-0.857612\pi$$
0.901607 0.432556i $$-0.142388\pi$$
$$488$$ − 3.41090e8i − 0.132862i
$$489$$ −1.00550e9 −0.388866
$$490$$ 0 0
$$491$$ 4.28064e8 0.163201 0.0816006 0.996665i $$-0.473997\pi$$
0.0816006 + 0.996665i $$0.473997\pi$$
$$492$$ 9.50097e8i 0.359658i
$$493$$ 1.49445e9i 0.561716i
$$494$$ −3.23457e8 −0.120718
$$495$$ 0 0
$$496$$ 4.32482e9 1.59141
$$497$$ − 2.48824e8i − 0.0909171i
$$498$$ − 2.06053e8i − 0.0747613i
$$499$$ 2.95178e9 1.06349 0.531743 0.846906i $$-0.321537\pi$$
0.531743 + 0.846906i $$0.321537\pi$$
$$500$$ 0 0
$$501$$ 2.41412e9 0.857681
$$502$$ − 3.29369e7i − 0.0116204i
$$503$$ 5.22380e9i 1.83020i 0.403229 + 0.915099i $$0.367888\pi$$
−0.403229 + 0.915099i $$0.632112\pi$$
$$504$$ 1.87280e8 0.0651605
$$505$$ 0 0
$$506$$ 1.25267e8 0.0429844
$$507$$ − 6.22326e8i − 0.212075i
$$508$$ 2.82236e9i 0.955190i
$$509$$ 2.80532e9 0.942911 0.471455 0.881890i $$-0.343729\pi$$
0.471455 + 0.881890i $$0.343729\pi$$
$$510$$ 0 0
$$511$$ −7.46374e8 −0.247448
$$512$$ 1.72897e9i 0.569301i
$$513$$ − 3.59530e9i − 1.17577i
$$514$$ −1.33265e8 −0.0432857
$$515$$ 0 0
$$516$$ −1.99197e9 −0.638277
$$517$$ − 7.33401e8i − 0.233413i
$$518$$ 7.33814e7i 0.0231970i
$$519$$ 1.26628e9 0.397597
$$520$$ 0 0
$$521$$ 1.40563e8 0.0435450 0.0217725 0.999763i $$-0.493069\pi$$
0.0217725 + 0.999763i $$0.493069\pi$$
$$522$$ − 2.02468e8i − 0.0623032i
$$523$$ 1.81127e9i 0.553638i 0.960922 + 0.276819i $$0.0892803\pi$$
−0.960922 + 0.276819i $$0.910720\pi$$
$$524$$ −5.71794e8 −0.173612
$$525$$ 0 0
$$526$$ 4.08163e8 0.122288
$$527$$ − 4.37679e9i − 1.30262i
$$528$$ 5.56635e8i 0.164570i
$$529$$ −6.89567e8 −0.202526
$$530$$ 0 0
$$531$$ 2.84837e9 0.825592
$$532$$ − 1.66855e9i − 0.480452i
$$533$$ − 1.86383e9i − 0.533164i
$$534$$ 3.59750e8 0.102237
$$535$$ 0 0
$$536$$ 1.54681e9 0.433870
$$537$$ 3.61604e9i 1.00768i
$$538$$ 5.34019e8i 0.147849i
$$539$$ 1.68516e8 0.0463534
$$540$$ 0 0
$$541$$ −7.11633e9 −1.93226 −0.966130 0.258058i $$-0.916918\pi$$
−0.966130 + 0.258058i $$0.916918\pi$$
$$542$$ 2.89885e8i 0.0782038i
$$543$$ 1.51192e9i 0.405255i
$$544$$ 1.04469e9 0.278223
$$545$$ 0 0
$$546$$ 7.13400e7 0.0187568
$$547$$ − 6.02390e9i − 1.57370i −0.617144 0.786850i $$-0.711710\pi$$
0.617144 0.786850i $$-0.288290\pi$$
$$548$$ − 6.40318e9i − 1.66213i
$$549$$ −1.54372e9 −0.398168
$$550$$ 0 0
$$551$$ −3.63447e9 −0.925572
$$552$$ 5.51171e8i 0.139476i
$$553$$ − 1.78883e9i − 0.449812i
$$554$$ −2.55204e8 −0.0637681
$$555$$ 0 0
$$556$$ −1.32669e9 −0.327347
$$557$$ 3.55726e9i 0.872214i 0.899895 + 0.436107i $$0.143643\pi$$
−0.899895 + 0.436107i $$0.856357\pi$$
$$558$$ 5.92968e8i 0.144481i
$$559$$ 3.90771e9 0.946195
$$560$$ 0 0
$$561$$ 5.63324e8 0.134706
$$562$$ − 1.00974e9i − 0.239957i
$$563$$ − 2.51240e9i − 0.593347i −0.954979 0.296673i $$-0.904123\pi$$
0.954979 0.296673i $$-0.0958773\pi$$
$$564$$ 1.60161e9 0.375906
$$565$$ 0 0
$$566$$ −4.26293e8 −0.0988214
$$567$$ − 3.86252e8i − 0.0889877i
$$568$$ 2.51969e8i 0.0576937i
$$569$$ −3.02191e9 −0.687683 −0.343841 0.939028i $$-0.611728\pi$$
−0.343841 + 0.939028i $$0.611728\pi$$
$$570$$ 0 0
$$571$$ 4.13151e9 0.928716 0.464358 0.885648i $$-0.346285\pi$$
0.464358 + 0.885648i $$0.346285\pi$$
$$572$$ − 1.10863e9i − 0.247685i
$$573$$ 3.78386e9i 0.840221i
$$574$$ −1.42391e8 −0.0314262
$$575$$ 0 0
$$576$$ 3.01152e9 0.656610
$$577$$ − 3.66048e9i − 0.793274i −0.917976 0.396637i $$-0.870177\pi$$
0.917976 0.396637i $$-0.129823\pi$$
$$578$$ 2.17105e8i 0.0467651i
$$579$$ 3.45400e9 0.739516
$$580$$ 0 0
$$581$$ −2.08517e9 −0.441088
$$582$$ 2.25225e8i 0.0473572i
$$583$$ − 2.88262e8i − 0.0602487i
$$584$$ 7.55807e8 0.157024
$$585$$ 0 0
$$586$$ −6.90846e8 −0.141821
$$587$$ 8.93156e9i 1.82261i 0.411731 + 0.911305i $$0.364924\pi$$
−0.411731 + 0.911305i $$0.635076\pi$$
$$588$$ 3.68007e8i 0.0746510i
$$589$$ 1.06442e10 2.14640
$$590$$ 0 0
$$591$$ 1.61812e9 0.322444
$$592$$ 2.45288e9i 0.485904i
$$593$$ − 8.00218e9i − 1.57586i −0.615766 0.787929i $$-0.711153\pi$$
0.615766 0.787929i $$-0.288847\pi$$
$$594$$ −1.82497e8 −0.0357276
$$595$$ 0 0
$$596$$ −6.85524e9 −1.32636
$$597$$ 4.79879e7i 0.00923041i
$$598$$ − 5.36649e8i − 0.102621i
$$599$$ −6.37081e9 −1.21116 −0.605579 0.795785i $$-0.707059\pi$$
−0.605579 + 0.795785i $$0.707059\pi$$
$$600$$ 0 0
$$601$$ 7.97677e9 1.49888 0.749439 0.662073i $$-0.230323\pi$$
0.749439 + 0.662073i $$0.230323\pi$$
$$602$$ − 2.98537e8i − 0.0557713i
$$603$$ − 7.00062e9i − 1.30025i
$$604$$ −2.81599e9 −0.519999
$$605$$ 0 0
$$606$$ 3.64474e8 0.0665293
$$607$$ 5.42119e9i 0.983863i 0.870634 + 0.491931i $$0.163709\pi$$
−0.870634 + 0.491931i $$0.836291\pi$$
$$608$$ 2.54067e9i 0.458444i
$$609$$ 8.01598e8 0.143812
$$610$$ 0 0
$$611$$ −3.14191e9 −0.557250
$$612$$ − 3.14438e9i − 0.554505i
$$613$$ − 8.21824e9i − 1.44101i −0.693450 0.720505i $$-0.743910\pi$$
0.693450 0.720505i $$-0.256090\pi$$
$$614$$ −6.38905e8 −0.111390
$$615$$ 0 0
$$616$$ −1.70646e8 −0.0294147
$$617$$ 8.15621e9i 1.39795i 0.715148 + 0.698973i $$0.246359\pi$$
−0.715148 + 0.698973i $$0.753641\pi$$
$$618$$ 3.58897e8i 0.0611660i
$$619$$ 6.46052e9 1.09484 0.547420 0.836858i $$-0.315610\pi$$
0.547420 + 0.836858i $$0.315610\pi$$
$$620$$ 0 0
$$621$$ 5.96497e9 0.999511
$$622$$ 1.58573e7i 0.00264218i
$$623$$ − 3.64051e9i − 0.603191i
$$624$$ 2.38464e9 0.392896
$$625$$ 0 0
$$626$$ 1.12511e9 0.183309
$$627$$ 1.36999e9i 0.221963i
$$628$$ − 5.52190e9i − 0.889672i
$$629$$ 2.48236e9 0.397729
$$630$$ 0 0
$$631$$ −8.82660e9 −1.39859 −0.699295 0.714833i $$-0.746503\pi$$
−0.699295 + 0.714833i $$0.746503\pi$$
$$632$$ 1.81144e9i 0.285439i
$$633$$ 1.28424e9i 0.201248i
$$634$$ −5.31876e8 −0.0828893
$$635$$ 0 0
$$636$$ 6.29509e8 0.0970291
$$637$$ − 7.21930e8i − 0.110664i
$$638$$ 1.84486e8i 0.0281249i
$$639$$ 1.14037e9 0.172900
$$640$$ 0 0
$$641$$ 8.54151e9 1.28095 0.640474 0.767980i $$-0.278738\pi$$
0.640474 + 0.767980i $$0.278738\pi$$
$$642$$ − 2.83778e8i − 0.0423259i
$$643$$ 1.20342e10i 1.78517i 0.450878 + 0.892585i $$0.351111\pi$$
−0.450878 + 0.892585i $$0.648889\pi$$
$$644$$ 2.76831e9 0.408426
$$645$$ 0 0
$$646$$ 8.35934e8 0.122000
$$647$$ 1.89174e8i 0.0274598i 0.999906 + 0.0137299i $$0.00437050\pi$$
−0.999906 + 0.0137299i $$0.995630\pi$$
$$648$$ 3.91133e8i 0.0564693i
$$649$$ −2.59539e9 −0.372688
$$650$$ 0 0
$$651$$ −2.34764e9 −0.333501
$$652$$ − 5.11403e9i − 0.722598i
$$653$$ − 8.70977e9i − 1.22408i −0.790826 0.612041i $$-0.790349\pi$$
0.790826 0.612041i $$-0.209651\pi$$
$$654$$ 6.64159e8 0.0928432
$$655$$ 0 0
$$656$$ −4.75963e9 −0.658279
$$657$$ − 3.42067e9i − 0.470579i
$$658$$ 2.40033e8i 0.0328458i
$$659$$ 7.48288e8 0.101852 0.0509260 0.998702i $$-0.483783\pi$$
0.0509260 + 0.998702i $$0.483783\pi$$
$$660$$ 0 0
$$661$$ 8.45586e9 1.13881 0.569407 0.822056i $$-0.307173\pi$$
0.569407 + 0.822056i $$0.307173\pi$$
$$662$$ 2.03074e8i 0.0272052i
$$663$$ − 2.41330e9i − 0.321598i
$$664$$ 2.11152e9 0.279903
$$665$$ 0 0
$$666$$ −3.36310e8 −0.0441144
$$667$$ − 6.02996e9i − 0.786817i
$$668$$ 1.22784e10i 1.59376i
$$669$$ 3.10154e9 0.400485
$$670$$ 0 0
$$671$$ 1.40661e9 0.179740
$$672$$ − 5.60357e8i − 0.0712316i
$$673$$ − 4.78543e9i − 0.605157i −0.953124 0.302578i $$-0.902153\pi$$
0.953124 0.302578i $$-0.0978474\pi$$
$$674$$ −1.68628e8 −0.0212139
$$675$$ 0 0
$$676$$ 3.16520e9 0.394083
$$677$$ − 1.29662e10i − 1.60603i −0.595958 0.803015i $$-0.703228\pi$$
0.595958 0.803015i $$-0.296772\pi$$
$$678$$ − 4.62018e8i − 0.0569318i
$$679$$ 2.27918e9 0.279405
$$680$$ 0 0
$$681$$ 4.76419e8 0.0578061
$$682$$ − 5.40302e8i − 0.0652216i
$$683$$ 9.15988e9i 1.10006i 0.835144 + 0.550031i $$0.185384\pi$$
−0.835144 + 0.550031i $$0.814616\pi$$
$$684$$ 7.64706e9 0.913689
$$685$$ 0 0
$$686$$ −5.51533e7 −0.00652285
$$687$$ − 2.62007e9i − 0.308293i
$$688$$ − 9.97905e9i − 1.16823i
$$689$$ −1.23492e9 −0.143838
$$690$$ 0 0
$$691$$ 1.05298e10 1.21407 0.607037 0.794673i $$-0.292358\pi$$
0.607037 + 0.794673i $$0.292358\pi$$
$$692$$ 6.44038e9i 0.738823i
$$693$$ 7.72318e8i 0.0881515i
$$694$$ −1.79944e9 −0.204352
$$695$$ 0 0
$$696$$ −8.11729e8 −0.0912596
$$697$$ 4.81683e9i 0.538824i
$$698$$ − 3.61534e8i − 0.0402398i
$$699$$ 5.74470e9 0.636205
$$700$$ 0 0
$$701$$ 1.27411e9 0.139699 0.0698497 0.997558i $$-0.477748\pi$$
0.0698497 + 0.997558i $$0.477748\pi$$
$$702$$ 7.81825e8i 0.0852962i
$$703$$ 6.03703e9i 0.655360i
$$704$$ −2.74405e9 −0.296406
$$705$$ 0 0
$$706$$ 1.78048e9 0.190423
$$707$$ − 3.68832e9i − 0.392519i
$$708$$ − 5.66782e9i − 0.600205i
$$709$$ 7.17795e9 0.756378 0.378189 0.925728i $$-0.376547\pi$$
0.378189 + 0.925728i $$0.376547\pi$$
$$710$$ 0 0
$$711$$ 8.19829e9 0.855421
$$712$$ 3.68653e9i 0.382769i
$$713$$ 1.76599e10i 1.82463i
$$714$$ −1.84369e8 −0.0189559
$$715$$ 0 0
$$716$$ −1.83914e10 −1.87249
$$717$$ 2.70746e9i 0.274312i
$$718$$ 1.41133e9i 0.142296i
$$719$$ −1.18502e10 −1.18898 −0.594488 0.804104i $$-0.702645\pi$$
−0.594488 + 0.804104i $$0.702645\pi$$
$$720$$ 0 0
$$721$$ 3.63188e9 0.360876
$$722$$ 8.11273e8i 0.0802208i
$$723$$ 2.04665e9i 0.201400i
$$724$$ −7.68971e9 −0.753052
$$725$$ 0 0
$$726$$ −5.90971e8 −0.0573176
$$727$$ − 4.67874e9i − 0.451605i −0.974173 0.225802i $$-0.927500\pi$$
0.974173 0.225802i $$-0.0725004\pi$$
$$728$$ 7.31054e8i 0.0702246i
$$729$$ −3.28577e9 −0.314116
$$730$$ 0 0
$$731$$ −1.00990e10 −0.956238
$$732$$ 3.07177e9i 0.289468i
$$733$$ − 1.28552e9i − 0.120563i −0.998181 0.0602817i $$-0.980800\pi$$
0.998181 0.0602817i $$-0.0191999\pi$$
$$734$$ −1.54612e9 −0.144314
$$735$$ 0 0
$$736$$ −4.21524e9 −0.389718
$$737$$ 6.37884e9i 0.586956i
$$738$$ − 6.52585e8i − 0.0597641i
$$739$$ 5.26720e9 0.480091 0.240046 0.970762i $$-0.422838\pi$$
0.240046 + 0.970762i $$0.422838\pi$$
$$740$$ 0 0
$$741$$ 5.86909e9 0.529916
$$742$$ 9.43446e7i 0.00847819i
$$743$$ − 4.15012e9i − 0.371193i −0.982626 0.185596i $$-0.940578\pi$$
0.982626 0.185596i $$-0.0594217\pi$$
$$744$$ 2.37731e9 0.211631
$$745$$ 0 0
$$746$$ 7.35489e8 0.0648620
$$747$$ − 9.55643e9i − 0.838829i
$$748$$ 2.86511e9i 0.250314i
$$749$$ −2.87171e9 −0.249721
$$750$$ 0 0
$$751$$ −6.37970e9 −0.549618 −0.274809 0.961499i $$-0.588615\pi$$
−0.274809 + 0.961499i $$0.588615\pi$$
$$752$$ 8.02346e9i 0.688017i
$$753$$ 5.97635e8i 0.0510098i
$$754$$ 7.90343e8 0.0671453
$$755$$ 0 0
$$756$$ −4.03304e9 −0.339474
$$757$$ − 1.19658e10i − 1.00255i −0.865289 0.501274i $$-0.832865\pi$$
0.865289 0.501274i $$-0.167135\pi$$
$$758$$ − 1.07032e9i − 0.0892631i
$$759$$ −2.27296e9 −0.188688
$$760$$ 0 0
$$761$$ −2.00959e10 −1.65296 −0.826479 0.562967i $$-0.809660\pi$$
−0.826479 + 0.562967i $$0.809660\pi$$
$$762$$ 7.58437e8i 0.0620979i
$$763$$ − 6.72100e9i − 0.547770i
$$764$$ −1.92450e10 −1.56131
$$765$$ 0 0
$$766$$ 1.12411e9 0.0903665
$$767$$ 1.11187e10i 0.889756i
$$768$$ − 5.70667e9i − 0.454588i
$$769$$ −2.46683e10 −1.95613 −0.978064 0.208304i $$-0.933206\pi$$
−0.978064 + 0.208304i $$0.933206\pi$$
$$770$$ 0 0
$$771$$ 2.41807e9 0.190011
$$772$$ 1.75673e10i 1.37418i
$$773$$ − 8.88824e9i − 0.692130i −0.938211 0.346065i $$-0.887518\pi$$
0.938211 0.346065i $$-0.112482\pi$$
$$774$$ 1.36821e9 0.106062
$$775$$ 0 0
$$776$$ −2.30798e9 −0.177303
$$777$$ − 1.33149e9i − 0.101828i
$$778$$ 1.46252e9i 0.111345i
$$779$$ −1.17144e10 −0.887850
$$780$$ 0 0
$$781$$ −1.03909e9 −0.0780502
$$782$$ 1.38690e9i 0.103710i
$$783$$ 8.78483e9i 0.653984i
$$784$$ −1.84358e9 −0.136633
$$785$$ 0 0
$$786$$ −1.53655e8 −0.0112867
$$787$$ − 4.65006e9i − 0.340053i −0.985439 0.170027i $$-0.945615\pi$$
0.985439 0.170027i $$-0.0543854\pi$$
$$788$$ 8.22986e9i 0.599171i
$$789$$ −7.40606e9 −0.536806
$$790$$ 0 0
$$791$$ −4.67542e9 −0.335895
$$792$$ − 7.82079e8i − 0.0559387i
$$793$$ − 6.02598e9i − 0.429113i
$$794$$ 1.31830e9 0.0934638
$$795$$ 0 0
$$796$$ −2.44070e8 −0.0171521
$$797$$ 1.42890e10i 0.999762i 0.866094 + 0.499881i $$0.166623\pi$$
−0.866094 + 0.499881i $$0.833377\pi$$
$$798$$ − 4.48381e8i − 0.0312347i
$$799$$ 8.11987e9 0.563165
$$800$$ 0 0
$$801$$ 1.66846e10 1.14711
$$802$$ − 2.66257e9i − 0.182260i
$$803$$ 3.11685e9i 0.212428i
$$804$$ −1.39302e10 −0.945279
$$805$$ 0 0
$$806$$ −2.31467e9 −0.155710
$$807$$ − 9.68971e9i − 0.649013i
$$808$$ 3.73494e9i 0.249083i
$$809$$ 4.92320e9 0.326909 0.163455 0.986551i $$-0.447736\pi$$
0.163455 + 0.986551i $$0.447736\pi$$
$$810$$ 0 0
$$811$$ 2.35801e10 1.55229 0.776145 0.630555i $$-0.217173\pi$$
0.776145 + 0.630555i $$0.217173\pi$$
$$812$$ 4.07698e9i 0.267235i
$$813$$ − 5.25992e9i − 0.343291i
$$814$$ 3.06440e8 0.0199141
$$815$$ 0 0
$$816$$ −6.16280e9 −0.397066
$$817$$ − 2.45605e10i − 1.57565i
$$818$$ − 1.18069e9i − 0.0754221i
$$819$$ 3.30864e9 0.210453
$$820$$ 0 0
$$821$$ −2.86630e10 −1.80768 −0.903838 0.427875i $$-0.859262\pi$$
−0.903838 + 0.427875i $$0.859262\pi$$
$$822$$ − 1.72069e9i − 0.108057i
$$823$$ 2.76897e10i 1.73148i 0.500490 + 0.865742i $$0.333153\pi$$
−0.500490 + 0.865742i $$0.666847\pi$$
$$824$$ −3.67778e9 −0.229003
$$825$$ 0 0
$$826$$ 8.49437e8 0.0524447
$$827$$ − 1.27176e10i − 0.781873i −0.920418 0.390936i $$-0.872151\pi$$
0.920418 0.390936i $$-0.127849\pi$$
$$828$$ 1.26873e10i 0.776716i
$$829$$ 1.50770e10 0.919127 0.459563 0.888145i $$-0.348006\pi$$
0.459563 + 0.888145i $$0.348006\pi$$
$$830$$ 0 0
$$831$$ 4.63064e9 0.279923
$$832$$ 1.17556e10i 0.707641i
$$833$$ 1.86573e9i 0.111839i
$$834$$ −3.56514e8 −0.0212812
$$835$$ 0 0
$$836$$ −6.96787e9 −0.412457
$$837$$ − 2.57281e10i − 1.51659i
$$838$$ − 3.02512e9i − 0.177578i
$$839$$ −4.59511e9 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$840$$ 0 0
$$841$$ −8.36934e9 −0.485182
$$842$$ − 3.96175e9i − 0.228715i
$$843$$ 1.83216e10i 1.05334i
$$844$$ −6.53172e9 −0.373963
$$845$$ 0 0
$$846$$ −1.10008e9 −0.0624639
$$847$$ 5.98038e9i 0.338171i
$$848$$ 3.15361e9i 0.177591i
$$849$$ 7.73504e9 0.433796
$$850$$ 0 0
$$851$$ −1.00161e10 −0.557114
$$852$$ − 2.26917e9i − 0.125698i
$$853$$ 1.13971e9i 0.0628740i 0.999506 + 0.0314370i $$0.0100084\pi$$
−0.999506 + 0.0314370i $$0.989992\pi$$
$$854$$ −4.60367e8 −0.0252931
$$855$$ 0 0
$$856$$ 2.90801e9 0.158466
$$857$$ 7.79419e9i 0.422998i 0.977378 + 0.211499i $$0.0678345\pi$$
−0.977378 + 0.211499i $$0.932166\pi$$
$$858$$ − 2.97915e8i − 0.0161023i
$$859$$ −1.27280e10 −0.685147 −0.342573 0.939491i $$-0.611299\pi$$
−0.342573 + 0.939491i $$0.611299\pi$$
$$860$$ 0 0
$$861$$ 2.58367e9 0.137951
$$862$$ − 3.30831e9i − 0.175926i
$$863$$ − 2.53204e9i − 0.134101i −0.997750 0.0670507i $$-0.978641\pi$$
0.997750 0.0670507i $$-0.0213589\pi$$
$$864$$ 6.14103e9 0.323924
$$865$$ 0 0
$$866$$ −3.09823e9 −0.162107
$$867$$ − 3.93933e9i − 0.205284i
$$868$$ − 1.19402e10i − 0.619718i
$$869$$ −7.47014e9 −0.386153
$$870$$ 0 0
$$871$$ 2.73272e10 1.40130
$$872$$ 6.80595e9i 0.347601i
$$873$$ 1.04456e10i 0.531352i
$$874$$ −3.37291e9 −0.170889
$$875$$ 0 0
$$876$$ −6.80661e9 −0.342110
$$877$$ 5.00988e9i 0.250800i 0.992106 + 0.125400i $$0.0400215\pi$$
−0.992106 + 0.125400i $$0.959979\pi$$
$$878$$ − 2.71862e9i − 0.135556i
$$879$$ 1.25353e10 0.622550
$$880$$ 0 0
$$881$$ 9.46900e9 0.466539 0.233270 0.972412i $$-0.425058\pi$$
0.233270 + 0.972412i $$0.425058\pi$$
$$882$$ − 2.52770e8i − 0.0124047i
$$883$$ − 1.11146e10i − 0.543289i −0.962398 0.271644i $$-0.912433\pi$$
0.962398 0.271644i $$-0.0875675\pi$$
$$884$$ 1.22742e10 0.597600
$$885$$ 0 0
$$886$$ −1.20006e9 −0.0579675
$$887$$ − 7.27986e9i − 0.350260i −0.984545 0.175130i $$-0.943965\pi$$
0.984545 0.175130i $$-0.0560346\pi$$
$$888$$ 1.34832e9i 0.0646173i
$$889$$ 7.67506e9 0.366375
$$890$$ 0 0
$$891$$ −1.61298e9 −0.0763938
$$892$$ 1.57746e10i 0.744188i
$$893$$ 1.97473e10i 0.927959i
$$894$$ −1.84217e9 −0.0862280
$$895$$ 0 0
$$896$$ 3.79032e9 0.176034
$$897$$ 9.73743e9i 0.450475i
$$898$$ − 2.09267e8i − 0.00964348i
$$899$$ −2.60084e10 −1.19386
$$900$$ 0 0
$$901$$ 3.19150e9 0.145365
$$902$$ 5.94624e8i 0.0269786i
$$903$$ 5.41692e9i 0.244819i
$$904$$ 4.73451e9 0.213150
$$905$$ 0 0
$$906$$ −7.56726e8 −0.0338057
$$907$$ − 1.39503e10i − 0.620809i −0.950605 0.310405i $$-0.899535\pi$$
0.950605 0.310405i $$-0.100465\pi$$
$$908$$ 2.42310e9i 0.107416i
$$909$$ 1.69038e10 0.746465
$$910$$ 0 0
$$911$$ 2.98148e8 0.0130653 0.00653263 0.999979i $$-0.497921\pi$$
0.00653263 + 0.999979i $$0.497921\pi$$
$$912$$ − 1.49878e10i − 0.654268i
$$913$$ 8.70765e9i 0.378663i
$$914$$ 3.27506e9 0.141876
$$915$$ 0 0
$$916$$ 1.33259e10 0.572876
$$917$$ 1.55492e9i 0.0665910i
$$918$$ − 2.02053e9i − 0.0862015i
$$919$$ −2.67202e10 −1.13563 −0.567814 0.823157i $$-0.692211\pi$$
−0.567814 + 0.823157i $$0.692211\pi$$
$$920$$ 0 0
$$921$$ 1.15928e10 0.488969
$$922$$ 2.21295e9i 0.0929850i
$$923$$ 4.45149e9i 0.186337i
$$924$$ 1.53680e9 0.0640861
$$925$$ 0 0
$$926$$ 1.58725e9 0.0656910
$$927$$ 1.66451e10i 0.686288i
$$928$$ − 6.20794e9i − 0.254994i
$$929$$ 3.66336e10 1.49908 0.749540 0.661959i $$-0.230275\pi$$
0.749540 + 0.661959i $$0.230275\pi$$
$$930$$ 0 0
$$931$$ −4.53742e9 −0.184283
$$932$$ 2.92180e10i 1.18221i
$$933$$ − 2.87728e8i − 0.0115983i
$$934$$ −3.88115e9 −0.155864
$$935$$ 0 0
$$936$$ −3.35045e9 −0.133548
$$937$$ 1.28088e10i 0.508649i 0.967119 + 0.254325i $$0.0818531\pi$$
−0.967119 + 0.254325i $$0.918147\pi$$
$$938$$ − 2.08772e9i − 0.0825965i
$$939$$ −2.04149e10 −0.804669
$$940$$ 0 0
$$941$$ −1.20663e10 −0.472073 −0.236037 0.971744i $$-0.575849\pi$$
−0.236037 + 0.971744i $$0.575849\pi$$
$$942$$ − 1.48387e9i − 0.0578386i
$$943$$ − 1.94354e10i − 0.754751i
$$944$$ 2.83937e10 1.09855
$$945$$ 0 0
$$946$$ −1.24669e9 −0.0478784
$$947$$ 8.36023e9i 0.319885i 0.987126 + 0.159942i $$0.0511308\pi$$
−0.987126 + 0.159942i $$0.948869\pi$$
$$948$$ − 1.63133e10i − 0.621890i
$$949$$ 1.33527e10 0.507151
$$950$$ 0 0
$$951$$ 9.65082e9 0.363859
$$952$$ − 1.88931e9i − 0.0709700i
$$953$$ − 4.49530e10i − 1.68242i −0.540710 0.841209i $$-0.681844\pi$$
0.540710 0.841209i $$-0.318156\pi$$
$$954$$ −4.32386e8 −0.0161232
$$955$$ 0 0
$$956$$ −1.37703e10 −0.509732
$$957$$ − 3.34747e9i − 0.123459i
$$958$$ − 3.26341e9i − 0.119920i
$$959$$ −1.74126e10 −0.637528
$$960$$ 0 0
$$961$$ 4.86580e10 1.76857
$$962$$ − 1.31280e9i − 0.0475429i
$$963$$ − 1.31612e10i − 0.474901i
$$964$$ −1.04094e10 −0.374244
$$965$$ 0 0
$$966$$ 7.43911e8 0.0265522
$$967$$ 1.34247e8i 0.00477432i 0.999997 + 0.00238716i $$0.000759858\pi$$
−0.999997 + 0.00238716i $$0.999240\pi$$
$$968$$ − 6.05596e9i − 0.214595i
$$969$$ −1.51679e10 −0.535541
$$970$$ 0 0
$$971$$ −3.00377e10 −1.05293 −0.526465 0.850197i $$-0.676483\pi$$
−0.526465 + 0.850197i $$0.676483\pi$$
$$972$$ − 2.92375e10i − 1.02119i
$$973$$ 3.60777e9i 0.125558i
$$974$$ 3.01379e9 0.104510
$$975$$ 0 0
$$976$$ −1.53884e10 −0.529810
$$977$$ 4.52860e10i 1.55358i 0.629761 + 0.776789i $$0.283153\pi$$
−0.629761 + 0.776789i $$0.716847\pi$$
$$978$$ − 1.37426e9i − 0.0469769i
$$979$$ −1.52028e10 −0.517825
$$980$$ 0 0
$$981$$ 3.08027e10 1.04171
$$982$$ 5.85057e8i 0.0197155i
$$983$$ − 4.61443e10i − 1.54946i −0.632290 0.774731i $$-0.717885\pi$$
0.632290 0.774731i $$-0.282115\pi$$
$$984$$ −2.61632e9 −0.0875404
$$985$$ 0 0
$$986$$ −2.04254e9 −0.0678580
$$987$$ − 4.35537e9i − 0.144183i
$$988$$ 2.98506e10i 0.984700i
$$989$$ 4.07484e10 1.33944
$$990$$ 0 0
$$991$$ −1.05400e10 −0.344018 −0.172009 0.985095i $$-0.555026\pi$$
−0.172009 + 0.985095i $$0.555026\pi$$
$$992$$ 1.81812e10i 0.591331i
$$993$$ − 3.68475e9i − 0.119422i
$$994$$ 3.40081e8 0.0109832
$$995$$ 0 0
$$996$$ −1.90158e10 −0.609828
$$997$$ − 5.00734e10i − 1.60020i −0.599868 0.800099i $$-0.704780\pi$$
0.599868 0.800099i $$-0.295220\pi$$
$$998$$ 4.03435e9i 0.128474i
$$999$$ 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.b.c.99.3 4
5.2 odd 4 175.8.a.b.1.2 2
5.3 odd 4 35.8.a.a.1.1 2
5.4 even 2 inner 175.8.b.c.99.2 4
15.8 even 4 315.8.a.c.1.2 2
20.3 even 4 560.8.a.i.1.1 2
35.13 even 4 245.8.a.b.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.1 2 5.3 odd 4
175.8.a.b.1.2 2 5.2 odd 4
175.8.b.c.99.2 4 5.4 even 2 inner
175.8.b.c.99.3 4 1.1 even 1 trivial
245.8.a.b.1.1 2 35.13 even 4
315.8.a.c.1.2 2 15.8 even 4
560.8.a.i.1.1 2 20.3 even 4