# Properties

 Label 315.8.a.c.1.2 Level $315$ Weight $8$ Character 315.1 Self dual yes Analytic conductor $98.401$ 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] = [315,8,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 315.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: $$98.4012830275$$ 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 $$\chi$$ $$=$$ 315.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.36675 q^{2} -126.132 q^{4} -125.000 q^{5} -343.000 q^{7} +347.335 q^{8} +O(q^{10})$$ $$q-1.36675 q^{2} -126.132 q^{4} -125.000 q^{5} -343.000 q^{7} +347.335 q^{8} +170.844 q^{10} +1432.37 q^{11} -6136.30 q^{13} +468.795 q^{14} +15670.2 q^{16} +15858.5 q^{17} -38567.5 q^{19} +15766.5 q^{20} -1957.69 q^{22} +63987.4 q^{23} +15625.0 q^{25} +8386.79 q^{26} +43263.3 q^{28} -94236.6 q^{29} +275990. q^{31} -65876.1 q^{32} -21674.6 q^{34} +42875.0 q^{35} +156532. q^{37} +52712.1 q^{38} -43416.9 q^{40} +303738. q^{41} +636818. q^{43} -180667. q^{44} -87454.9 q^{46} -512021. q^{47} +117649. q^{49} -21355.5 q^{50} +773984. q^{52} +201249. q^{53} -179046. q^{55} -119136. q^{56} +128798. q^{58} +1.81196e6 q^{59} -982021. q^{61} -377210. q^{62} -1.91575e6 q^{64} +767038. q^{65} -4.45336e6 q^{67} -2.00026e6 q^{68} -58599.4 q^{70} -725436. q^{71} +2.17602e6 q^{73} -213940. q^{74} +4.86459e6 q^{76} -491301. q^{77} -5.21525e6 q^{79} -1.95877e6 q^{80} -415135. q^{82} -6.07921e6 q^{83} -1.98231e6 q^{85} -870371. q^{86} +497511. q^{88} +1.06137e7 q^{89} +2.10475e6 q^{91} -8.07086e6 q^{92} +699805. q^{94} +4.82093e6 q^{95} +6.64483e6 q^{97} -160797. q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8}+O(q^{10})$$ 2 * q - 16 * q^2 - 40 * q^4 - 250 * q^5 - 686 * q^7 + 960 * q^8 $$2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8} + 2000 q^{10} + 7906 q^{11} - 17818 q^{13} + 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 3612 q^{19} + 5000 q^{20} - 96688 q^{22} - 13844 q^{23} + 31250 q^{25} + 179328 q^{26} + 13720 q^{28} + 126898 q^{29} + 252768 q^{31} + 148224 q^{32} + 175296 q^{34} + 85750 q^{35} - 265860 q^{37} - 458800 q^{38} - 120000 q^{40} + 111920 q^{41} + 947572 q^{43} + 376920 q^{44} + 1051472 q^{46} - 271274 q^{47} + 235298 q^{49} - 250000 q^{50} - 232184 q^{52} + 1267792 q^{53} - 988250 q^{55} - 329280 q^{56} - 3107120 q^{58} + 1360120 q^{59} - 1813680 q^{61} - 37392 q^{62} - 2489984 q^{64} + 2227250 q^{65} - 2189312 q^{67} - 3159640 q^{68} - 686000 q^{70} + 1494928 q^{71} + 7169788 q^{73} + 5967024 q^{74} + 7875376 q^{76} - 2711758 q^{77} - 7942974 q^{79} + 540000 q^{80} + 2391792 q^{82} + 304712 q^{83} - 299750 q^{85} - 5417712 q^{86} + 4463680 q^{88} + 17943528 q^{89} + 6111574 q^{91} - 14774640 q^{92} - 2823104 q^{94} + 451500 q^{95} + 4258074 q^{97} - 1882384 q^{98}+O(q^{100})$$ 2 * q - 16 * q^2 - 40 * q^4 - 250 * q^5 - 686 * q^7 + 960 * q^8 + 2000 * q^10 + 7906 * q^11 - 17818 * q^13 + 5488 * q^14 - 4320 * q^16 + 2398 * q^17 - 3612 * q^19 + 5000 * q^20 - 96688 * q^22 - 13844 * q^23 + 31250 * q^25 + 179328 * q^26 + 13720 * q^28 + 126898 * q^29 + 252768 * q^31 + 148224 * q^32 + 175296 * q^34 + 85750 * q^35 - 265860 * q^37 - 458800 * q^38 - 120000 * q^40 + 111920 * q^41 + 947572 * q^43 + 376920 * q^44 + 1051472 * q^46 - 271274 * q^47 + 235298 * q^49 - 250000 * q^50 - 232184 * q^52 + 1267792 * q^53 - 988250 * q^55 - 329280 * q^56 - 3107120 * q^58 + 1360120 * q^59 - 1813680 * q^61 - 37392 * q^62 - 2489984 * q^64 + 2227250 * q^65 - 2189312 * q^67 - 3159640 * q^68 - 686000 * q^70 + 1494928 * q^71 + 7169788 * q^73 + 5967024 * q^74 + 7875376 * q^76 - 2711758 * q^77 - 7942974 * q^79 + 540000 * q^80 + 2391792 * q^82 + 304712 * q^83 - 299750 * q^85 - 5417712 * q^86 + 4463680 * q^88 + 17943528 * q^89 + 6111574 * q^91 - 14774640 * q^92 - 2823104 * q^94 + 451500 * q^95 + 4258074 * q^97 - 1882384 * q^98

## 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 $$0.519238\pi$$
$$3$$ 0 0
$$4$$ −126.132 −0.985406
$$5$$ −125.000 −0.447214
$$6$$ 0 0
$$7$$ −343.000 −0.377964
$$8$$ 347.335 0.239847
$$9$$ 0 0
$$10$$ 170.844 0.0540256
$$11$$ 1432.37 0.324474 0.162237 0.986752i $$-0.448129\pi$$
0.162237 + 0.986752i $$0.448129\pi$$
$$12$$ 0 0
$$13$$ −6136.30 −0.774649 −0.387325 0.921943i $$-0.626601\pi$$
−0.387325 + 0.921943i $$0.626601\pi$$
$$14$$ 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 $$0.371979\pi$$
$$18$$ 0 0
$$19$$ −38567.5 −1.28998 −0.644991 0.764190i $$-0.723139\pi$$
−0.644991 + 0.764190i $$0.723139\pi$$
$$20$$ 15766.5 0.440687
$$21$$ 0 0
$$22$$ −1957.69 −0.0391980
$$23$$ 63987.4 1.09660 0.548299 0.836282i $$-0.315276\pi$$
0.548299 + 0.836282i $$0.315276\pi$$
$$24$$ 0 0
$$25$$ 15625.0 0.200000
$$26$$ 8386.79 0.0935813
$$27$$ 0 0
$$28$$ 43263.3 0.372449
$$29$$ −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 $$0.187224\pi$$
$$32$$ −65876.1 −0.355388
$$33$$ 0 0
$$34$$ −21674.6 −0.0945746
$$35$$ 42875.0 0.169031
$$36$$ 0 0
$$37$$ 156532. 0.508038 0.254019 0.967199i $$-0.418247\pi$$
0.254019 + 0.967199i $$0.418247\pi$$
$$38$$ 52712.1 0.155836
$$39$$ 0 0
$$40$$ −43416.9 −0.107263
$$41$$ 303738. 0.688266 0.344133 0.938921i $$-0.388173\pi$$
0.344133 + 0.938921i $$0.388173\pi$$
$$42$$ 0 0
$$43$$ 636818. 1.22145 0.610725 0.791843i $$-0.290878\pi$$
0.610725 + 0.791843i $$0.290878\pi$$
$$44$$ −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 $$0.617114\pi$$
$$48$$ 0 0
$$49$$ 117649. 0.142857
$$50$$ −21355.5 −0.0241610
$$51$$ 0 0
$$52$$ 773984. 0.763344
$$53$$ 201249. 0.185681 0.0928406 0.995681i $$-0.470405\pi$$
0.0928406 + 0.995681i $$0.470405\pi$$
$$54$$ 0 0
$$55$$ −179046. −0.145109
$$56$$ −119136. −0.0906535
$$57$$ 0 0
$$58$$ 128798. 0.0866784
$$59$$ 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. −0.201007
$$63$$ 0 0
$$64$$ −1.91575e6 −0.913499
$$65$$ 767038. 0.346434
$$66$$ 0 0
$$67$$ −4.45336e6 −1.80895 −0.904474 0.426528i $$-0.859736\pi$$
−0.904474 + 0.426528i $$0.859736\pi$$
$$68$$ −2.00026e6 −0.771446
$$69$$ 0 0
$$70$$ −58599.4 −0.0204197
$$71$$ −725436. −0.240544 −0.120272 0.992741i $$-0.538377\pi$$
−0.120272 + 0.992741i $$0.538377\pi$$
$$72$$ 0 0
$$73$$ 2.17602e6 0.654685 0.327343 0.944906i $$-0.393847\pi$$
0.327343 + 0.944906i $$0.393847\pi$$
$$74$$ −213940. −0.0613735
$$75$$ 0 0
$$76$$ 4.86459e6 1.27116
$$77$$ −491301. −0.122639
$$78$$ 0 0
$$79$$ −5.21525e6 −1.19009 −0.595045 0.803692i $$-0.702866\pi$$
−0.595045 + 0.803692i $$0.702866\pi$$
$$80$$ −1.95877e6 −0.427729
$$81$$ 0 0
$$82$$ −415135. −0.0831458
$$83$$ −6.07921e6 −1.16701 −0.583504 0.812110i $$-0.698319\pi$$
−0.583504 + 0.812110i $$0.698319\pi$$
$$84$$ 0 0
$$85$$ −1.98231e6 −0.350111
$$86$$ −870371. −0.147557
$$87$$ 0 0
$$88$$ 497511. 0.0778239
$$89$$ 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
$$92$$ −8.07086e6 −1.08059
$$93$$ 0 0
$$94$$ 699805. 0.0869019
$$95$$ 4.82093e6 0.576897
$$96$$ 0 0
$$97$$ 6.64483e6 0.739236 0.369618 0.929184i $$-0.379489\pi$$
0.369618 + 0.929184i $$0.379489\pi$$
$$98$$ −160797. −0.0172578
$$99$$ 0 0
$$100$$ −1.97081e6 −0.197081
$$101$$ −1.07531e7 −1.03851 −0.519254 0.854620i $$-0.673790\pi$$
−0.519254 + 0.854620i $$0.673790\pi$$
$$102$$ 0 0
$$103$$ −1.05886e7 −0.954788 −0.477394 0.878689i $$-0.658419\pi$$
−0.477394 + 0.878689i $$0.658419\pi$$
$$104$$ −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 $$0.392833\pi$$
$$108$$ 0 0
$$109$$ −1.95948e7 −1.44926 −0.724632 0.689136i $$-0.757990\pi$$
−0.724632 + 0.689136i $$0.757990\pi$$
$$110$$ 244711. 0.0175299
$$111$$ 0 0
$$112$$ −5.37487e6 −0.361497
$$113$$ −1.36310e7 −0.888694 −0.444347 0.895855i $$-0.646564\pi$$
−0.444347 + 0.895855i $$0.646564\pi$$
$$114$$ 0 0
$$115$$ −7.99843e6 −0.490413
$$116$$ 1.18863e7 0.707037
$$117$$ 0 0
$$118$$ −2.47649e6 −0.138756
$$119$$ −5.43946e6 −0.295898
$$120$$ 0 0
$$121$$ −1.74355e7 −0.894717
$$122$$ 1.34218e6 0.0669192
$$123$$ 0 0
$$124$$ −3.48112e7 −1.63962
$$125$$ −1.95312e6 −0.0894427
$$126$$ 0 0
$$127$$ 2.23763e7 0.969336 0.484668 0.874698i $$-0.338940\pi$$
0.484668 + 0.874698i $$0.338940\pi$$
$$128$$ 1.10505e7 0.465743
$$129$$ 0 0
$$130$$ −1.04835e6 −0.0418508
$$131$$ 4.53330e6 0.176183 0.0880917 0.996112i $$-0.471923\pi$$
0.0880917 + 0.996112i $$0.471923\pi$$
$$132$$ 0 0
$$133$$ 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 $$0.180568\pi$$
$$138$$ 0 0
$$139$$ 1.05183e7 0.332195 0.166097 0.986109i $$-0.446883\pi$$
0.166097 + 0.986109i $$0.446883\pi$$
$$140$$ −5.40791e6 −0.166564
$$141$$ 0 0
$$142$$ 991490. 0.0290589
$$143$$ −8.78942e6 −0.251353
$$144$$ 0 0
$$145$$ 1.17796e7 0.320879
$$146$$ −2.97407e6 −0.0790891
$$147$$ 0 0
$$148$$ −1.97437e7 −0.500624
$$149$$ −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 $$0.584992\pi$$
$$152$$ −1.33958e7 −0.309398
$$153$$ 0 0
$$154$$ 671486. 0.0148154
$$155$$ −3.44988e7 −0.744120
$$156$$ 0 0
$$157$$ −4.37788e7 −0.902848 −0.451424 0.892310i $$-0.649084\pi$$
−0.451424 + 0.892310i $$0.649084\pi$$
$$158$$ 7.12794e6 0.143769
$$159$$ 0 0
$$160$$ 8.23451e6 0.158934
$$161$$ −2.19477e7 −0.414475
$$162$$ 0 0
$$163$$ 4.05451e7 0.733300 0.366650 0.930359i $$-0.380505\pi$$
0.366650 + 0.930359i $$0.380505\pi$$
$$164$$ −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 $$0.799818\pi$$
$$168$$ 0 0
$$169$$ −2.50943e7 −0.399919
$$170$$ 2.70932e6 0.0422951
$$171$$ 0 0
$$172$$ −8.03231e7 −1.20362
$$173$$ 5.10607e7 0.749765 0.374882 0.927072i $$-0.377683\pi$$
0.374882 + 0.927072i $$0.377683\pi$$
$$174$$ 0 0
$$175$$ −5.35938e6 −0.0755929
$$176$$ 2.24454e7 0.310337
$$177$$ 0 0
$$178$$ −1.45063e7 −0.192791
$$179$$ −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 $$0.624800\pi$$
$$182$$ −2.87667e6 −0.0353704
$$183$$ 0 0
$$184$$ 2.22251e7 0.263015
$$185$$ −1.95665e7 −0.227202
$$186$$ 0 0
$$187$$ 2.27151e7 0.254021
$$188$$ 6.45822e7 0.708860
$$189$$ 0 0
$$190$$ −6.58901e6 −0.0696920
$$191$$ 1.52578e8 1.58444 0.792219 0.610237i $$-0.208926\pi$$
0.792219 + 0.610237i $$0.208926\pi$$
$$192$$ 0 0
$$193$$ −1.39277e8 −1.39453 −0.697267 0.716812i $$-0.745601\pi$$
−0.697267 + 0.716812i $$0.745601\pi$$
$$194$$ −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 $$0.598330\pi$$
$$198$$ 0 0
$$199$$ 1.93503e6 0.0174061 0.00870307 0.999962i $$-0.497230\pi$$
0.00870307 + 0.999962i $$0.497230\pi$$
$$200$$ 5.42711e6 0.0479693
$$201$$ 0 0
$$202$$ 1.46968e7 0.125457
$$203$$ 3.23232e7 0.271192
$$204$$ 0 0
$$205$$ −3.79673e7 −0.307802
$$206$$ 1.44719e7 0.115343
$$207$$ 0 0
$$208$$ −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
$$213$$ 0 0
$$214$$ −1.14429e7 −0.0798156
$$215$$ −7.96023e7 −0.546249
$$216$$ 0 0
$$217$$ −9.46647e7 −0.628896
$$218$$ 2.67812e7 0.175078
$$219$$ 0 0
$$220$$ 2.25834e7 0.142991
$$221$$ −9.73124e7 −0.606450
$$222$$ 0 0
$$223$$ −1.25065e8 −0.755209 −0.377605 0.925967i $$-0.623252\pi$$
−0.377605 + 0.925967i $$0.623252\pi$$
$$224$$ 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 $$0.517358\pi$$
$$228$$ 0 0
$$229$$ −1.05650e8 −0.581360 −0.290680 0.956820i $$-0.593882\pi$$
−0.290680 + 0.956820i $$0.593882\pi$$
$$230$$ 1.09319e7 0.0592443
$$231$$ 0 0
$$232$$ −3.27317e7 −0.172092
$$233$$ 2.31646e8 1.19972 0.599859 0.800106i $$-0.295224\pi$$
0.599859 + 0.800106i $$0.295224\pi$$
$$234$$ 0 0
$$235$$ 6.40026e7 0.321707
$$236$$ −2.28546e8 −1.13183
$$237$$ 0 0
$$238$$ 7.43438e6 0.0357458
$$239$$ −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 $$0.560814\pi$$
$$242$$ 2.38300e7 0.108086
$$243$$ 0 0
$$244$$ 1.23864e8 0.545861
$$245$$ −1.47061e7 −0.0638877
$$246$$ 0 0
$$247$$ 2.36662e8 0.999283
$$248$$ 9.58611e7 0.399081
$$249$$ 0 0
$$250$$ 2.66943e6 0.0108051
$$251$$ 2.40987e7 0.0961912 0.0480956 0.998843i $$-0.484685\pi$$
0.0480956 + 0.998843i $$0.484685\pi$$
$$252$$ 0 0
$$253$$ 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 $$0.557337\pi$$
$$258$$ 0 0
$$259$$ −5.36904e7 −0.192020
$$260$$ −9.67480e7 −0.341378
$$261$$ 0 0
$$262$$ −6.19589e6 −0.0212838
$$263$$ −2.98637e8 −1.01228 −0.506138 0.862452i $$-0.668927\pi$$
−0.506138 + 0.862452i $$0.668927\pi$$
$$264$$ 0 0
$$265$$ −2.51561e7 −0.0830392
$$266$$ −1.80803e7 −0.0589005
$$267$$ 0 0
$$268$$ 5.61711e8 1.78255
$$269$$ 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 $$0.395080\pi$$
$$272$$ 2.48505e8 0.748763
$$273$$ 0 0
$$274$$ −6.93841e7 −0.203767
$$275$$ 2.23807e7 0.0648947
$$276$$ 0 0
$$277$$ 1.86723e8 0.527861 0.263930 0.964542i $$-0.414981\pi$$
0.263930 + 0.964542i $$0.414981\pi$$
$$278$$ −1.43759e7 −0.0401307
$$279$$ 0 0
$$280$$ 1.48920e7 0.0405415
$$281$$ 7.38791e8 1.98632 0.993161 0.116756i $$-0.0372496\pi$$
0.993161 + 0.116756i $$0.0372496\pi$$
$$282$$ 0 0
$$283$$ −3.11903e8 −0.818026 −0.409013 0.912529i $$-0.634127\pi$$
−0.409013 + 0.912529i $$0.634127\pi$$
$$284$$ 9.15007e7 0.237034
$$285$$ 0 0
$$286$$ 1.20129e7 0.0303647
$$287$$ −1.04182e8 −0.260140
$$288$$ 0 0
$$289$$ −1.58847e8 −0.387113
$$290$$ −1.60997e7 −0.0387638
$$291$$ 0 0
$$292$$ −2.74466e8 −0.645131
$$293$$ 5.05466e8 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$294$$ 0 0
$$295$$ −2.26495e8 −0.513666
$$296$$ 5.43690e7 0.121851
$$297$$ 0 0
$$298$$ 7.42825e7 0.162603
$$299$$ −3.92646e8 −0.849478
$$300$$ 0 0
$$301$$ −2.18429e8 −0.461665
$$302$$ 3.05137e7 0.0637487
$$303$$ 0 0
$$304$$ −6.04359e8 −1.23378
$$305$$ 1.22753e8 0.247732
$$306$$ 0 0
$$307$$ 4.67463e8 0.922067 0.461034 0.887383i $$-0.347479\pi$$
0.461034 + 0.887383i $$0.347479\pi$$
$$308$$ 6.19688e7 0.120850
$$309$$ 0 0
$$310$$ 4.71512e7 0.0898933
$$311$$ −1.16022e7 −0.0218714 −0.0109357 0.999940i $$-0.503481\pi$$
−0.0109357 + 0.999940i $$0.503481\pi$$
$$312$$ 0 0
$$313$$ 8.23197e8 1.51740 0.758698 0.651443i $$-0.225836\pi$$
0.758698 + 0.651443i $$0.225836\pi$$
$$314$$ 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 $$0.611467\pi$$
$$318$$ 0 0
$$319$$ −1.34981e8 −0.232812
$$320$$ 2.39468e8 0.408529
$$321$$ 0 0
$$322$$ 2.99970e7 0.0500706
$$323$$ −6.11621e8 −1.00989
$$324$$ 0 0
$$325$$ −9.58797e7 −0.154930
$$326$$ −5.54150e7 −0.0885861
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 1.33047e8 0.195385
$$335$$ 5.56670e8 0.808986
$$336$$ 0 0
$$337$$ 1.23379e8 0.175605 0.0878023 0.996138i $$-0.472016\pi$$
0.0878023 + 0.996138i $$0.472016\pi$$
$$338$$ 3.42977e7 0.0483121
$$339$$ 0 0
$$340$$ 2.50033e8 0.345001
$$341$$ 3.95319e8 0.539892
$$342$$ 0 0
$$343$$ −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 $$0.820872\pi$$
$$348$$ 0 0
$$349$$ 2.64521e8 0.333097 0.166549 0.986033i $$-0.446738\pi$$
0.166549 + 0.986033i $$0.446738\pi$$
$$350$$ 7.32493e6 0.00913199
$$351$$ 0 0
$$352$$ −9.43586e7 −0.115314
$$353$$ −1.30271e9 −1.57629 −0.788144 0.615490i $$-0.788958\pi$$
−0.788144 + 0.615490i $$0.788958\pi$$
$$354$$ 0 0
$$355$$ 9.06795e7 0.107575
$$356$$ −1.33873e9 −1.57260
$$357$$ 0 0
$$358$$ 1.99287e8 0.229556
$$359$$ 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
$$362$$ 8.33248e7 0.0923196
$$363$$ 0 0
$$364$$ −2.65476e8 −0.288517
$$365$$ −2.72002e8 −0.292784
$$366$$ 0 0
$$367$$ 1.13124e9 1.19460 0.597302 0.802017i $$-0.296240\pi$$
0.597302 + 0.802017i $$0.296240\pi$$
$$368$$ 1.00269e9 1.04882
$$369$$ 0 0
$$370$$ 2.67425e7 0.0274470
$$371$$ −6.90284e7 −0.0701809
$$372$$ 0 0
$$373$$ 5.38130e8 0.536916 0.268458 0.963291i $$-0.413486\pi$$
0.268458 + 0.963291i $$0.413486\pi$$
$$374$$ −3.10459e7 −0.0306870
$$375$$ 0 0
$$376$$ −1.77843e8 −0.172536
$$377$$ 5.78264e8 0.555817
$$378$$ 0 0
$$379$$ 7.83114e8 0.738904 0.369452 0.929250i $$-0.379545\pi$$
0.369452 + 0.929250i $$0.379545\pi$$
$$380$$ −6.08074e8 −0.568478
$$381$$ 0 0
$$382$$ −2.08536e8 −0.191408
$$383$$ −8.22468e8 −0.748038 −0.374019 0.927421i $$-0.622020\pi$$
−0.374019 + 0.927421i $$0.622020\pi$$
$$384$$ 0 0
$$385$$ 6.14127e7 0.0548460
$$386$$ 1.90357e8 0.168466
$$387$$ 0 0
$$388$$ −8.38126e8 −0.728448
$$389$$ 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
$$392$$ 4.08636e7 0.0342638
$$393$$ 0 0
$$394$$ 8.91777e7 0.0734547
$$395$$ 6.51906e8 0.532225
$$396$$ 0 0
$$397$$ −9.64552e8 −0.773676 −0.386838 0.922148i $$-0.626433\pi$$
−0.386838 + 0.922148i $$0.626433\pi$$
$$398$$ −2.64471e6 −0.00210275
$$399$$ 0 0
$$400$$ 2.44846e8 0.191286
$$401$$ 1.94810e9 1.50871 0.754357 0.656465i $$-0.227949\pi$$
0.754357 + 0.656465i $$0.227949\pi$$
$$402$$ 0 0
$$403$$ −1.69356e9 −1.28894
$$404$$ 1.35631e9 1.02335
$$405$$ 0 0
$$406$$ −4.41777e7 −0.0327614
$$407$$ 2.24211e8 0.164845
$$408$$ 0 0
$$409$$ 8.63865e8 0.624330 0.312165 0.950028i $$-0.398946\pi$$
0.312165 + 0.950028i $$0.398946\pi$$
$$410$$ 5.18918e7 0.0371839
$$411$$ 0 0
$$412$$ 1.33556e9 0.940854
$$413$$ −6.21501e8 −0.434127
$$414$$ 0 0
$$415$$ 7.59901e8 0.521902
$$416$$ 4.04236e8 0.275301
$$417$$ 0 0
$$418$$ 7.55030e7 0.0505647
$$419$$ −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 $$0.895537\pi$$
$$422$$ 7.07769e7 0.0458456
$$423$$ 0 0
$$424$$ 6.99008e7 0.0445350
$$425$$ 2.47789e8 0.156574
$$426$$ 0 0
$$427$$ 3.36833e8 0.209371
$$428$$ −1.05602e9 −0.651057
$$429$$ 0 0
$$430$$ 1.08796e8 0.0659895
$$431$$ 2.42056e9 1.45628 0.728142 0.685426i $$-0.240384\pi$$
0.728142 + 0.685426i $$0.240384\pi$$
$$432$$ 0 0
$$433$$ −2.26686e9 −1.34189 −0.670946 0.741506i $$-0.734112\pi$$
−0.670946 + 0.741506i $$0.734112\pi$$
$$434$$ 1.29383e8 0.0759737
$$435$$ 0 0
$$436$$ 2.47153e9 1.42811
$$437$$ −2.46783e9 −1.41459
$$438$$ 0 0
$$439$$ 1.98911e9 1.12210 0.561052 0.827780i $$-0.310397\pi$$
0.561052 + 0.827780i $$0.310397\pi$$
$$440$$ −6.21888e7 −0.0348039
$$441$$ 0 0
$$442$$ 1.33002e8 0.0732621
$$443$$ 8.78038e8 0.479844 0.239922 0.970792i $$-0.422878\pi$$
0.239922 + 0.970792i $$0.422878\pi$$
$$444$$ 0 0
$$445$$ −1.32672e9 −0.713705
$$446$$ 1.70932e8 0.0912329
$$447$$ 0 0
$$448$$ 6.57101e8 0.345270
$$449$$ −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
$$452$$ 1.71930e9 0.875724
$$453$$ 0 0
$$454$$ 2.62564e7 0.0131686
$$455$$ −2.63094e8 −0.130940
$$456$$ 0 0
$$457$$ −2.39624e9 −1.17442 −0.587210 0.809435i $$-0.699774\pi$$
−0.587210 + 0.809435i $$0.699774\pi$$
$$458$$ 1.44397e8 0.0702311
$$459$$ 0 0
$$460$$ 1.00886e9 0.483256
$$461$$ −1.61913e9 −0.769713 −0.384856 0.922977i $$-0.625749\pi$$
−0.384856 + 0.922977i $$0.625749\pi$$
$$462$$ 0 0
$$463$$ 1.16133e9 0.543778 0.271889 0.962329i $$-0.412352\pi$$
0.271889 + 0.962329i $$0.412352\pi$$
$$464$$ −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 $$0.723187\pi$$
$$468$$ 0 0
$$469$$ 1.52750e9 0.683718
$$470$$ −8.74756e7 −0.0388637
$$471$$ 0 0
$$472$$ 6.29356e8 0.275486
$$473$$ 9.12156e8 0.396328
$$474$$ 0 0
$$475$$ −6.02617e8 −0.257996
$$476$$ 6.86090e8 0.291579
$$477$$ 0 0
$$478$$ 1.49214e8 0.0624901
$$479$$ −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
$$482$$ 1.12795e8 0.0458801
$$483$$ 0 0
$$484$$ 2.19917e9 0.881660
$$485$$ −8.30604e8 −0.330596
$$486$$ 0 0
$$487$$ −2.20508e9 −0.865113 −0.432556 0.901607i $$-0.642388\pi$$
−0.432556 + 0.901607i $$0.642388\pi$$
$$488$$ −3.41090e8 −0.132862
$$489$$ 0 0
$$490$$ 2.00996e7 0.00771794
$$491$$ −4.28064e8 −0.163201 −0.0816006 0.996665i $$-0.526003\pi$$
−0.0816006 + 0.996665i $$0.526003\pi$$
$$492$$ 0 0
$$493$$ −1.49445e9 −0.561716
$$494$$ −3.23457e8 −0.120718
$$495$$ 0 0
$$496$$ 4.32482e9 1.59141
$$497$$ 2.48824e8 0.0909171
$$498$$ 0 0
$$499$$ −2.95178e9 −1.06349 −0.531743 0.846906i $$-0.678463\pi$$
−0.531743 + 0.846906i $$0.678463\pi$$
$$500$$ 2.46352e8 0.0881374
$$501$$ 0 0
$$502$$ −3.29369e7 −0.0116204
$$503$$ 5.22380e9 1.83020 0.915099 0.403229i $$-0.132112\pi$$
0.915099 + 0.403229i $$0.132112\pi$$
$$504$$ 0 0
$$505$$ 1.34414e9 0.464435
$$506$$ −1.25267e8 −0.0429844
$$507$$ 0 0
$$508$$ −2.82236e9 −0.955190
$$509$$ 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
$$512$$ −1.72897e9 −0.569301
$$513$$ 0 0
$$514$$ 1.33265e8 0.0432857
$$515$$ 1.32357e9 0.426994
$$516$$ 0 0
$$517$$ −7.33401e8 −0.233413
$$518$$ 7.33814e7 0.0231970
$$519$$ 0 0
$$520$$ 2.66419e8 0.0830909
$$521$$ −1.40563e8 −0.0435450 −0.0217725 0.999763i $$-0.506931\pi$$
−0.0217725 + 0.999763i $$0.506931\pi$$
$$522$$ 0 0
$$523$$ −1.81127e9 −0.553638 −0.276819 0.960922i $$-0.589280\pi$$
−0.276819 + 0.960922i $$0.589280\pi$$
$$524$$ −5.71794e8 −0.173612
$$525$$ 0 0
$$526$$ 4.08163e8 0.122288
$$527$$ 4.37679e9 1.30262
$$528$$ 0 0
$$529$$ 6.89567e8 0.202526
$$530$$ 3.43821e7 0.0100315
$$531$$ 0 0
$$532$$ −1.66855e9 −0.480452
$$533$$ −1.86383e9 −0.533164
$$534$$ 0 0
$$535$$ −1.04654e9 −0.295474
$$536$$ −1.54681e9 −0.433870
$$537$$ 0 0
$$538$$ −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
$$543$$ 0 0
$$544$$ −1.04469e9 −0.278223
$$545$$ 2.44935e9 0.648130
$$546$$ 0 0
$$547$$ −6.02390e9 −1.57370 −0.786850 0.617144i $$-0.788290\pi$$
−0.786850 + 0.617144i $$0.788290\pi$$
$$548$$ −6.40318e9 −1.66213
$$549$$ 0 0
$$550$$ −3.05888e7 −0.00783959
$$551$$ 3.63447e9 0.925572
$$552$$ 0 0
$$553$$ 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 $$0.643643\pi$$
$$558$$ 0 0
$$559$$ −3.90771e9 −0.946195
$$560$$ 6.71859e8 0.161666
$$561$$ 0 0
$$562$$ −1.00974e9 −0.239957
$$563$$ −2.51240e9 −0.593347 −0.296673 0.954979i $$-0.595877\pi$$
−0.296673 + 0.954979i $$0.595877\pi$$
$$564$$ 0 0
$$565$$ 1.70387e9 0.397436
$$566$$ 4.26293e8 0.0988214
$$567$$ 0 0
$$568$$ −2.51969e8 −0.0576937
$$569$$ −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 $$0.346285\pi$$
$$572$$ 1.10863e9 0.247685
$$573$$ 0 0
$$574$$ 1.42391e8 0.0314262
$$575$$ 9.99804e8 0.219320
$$576$$ 0 0
$$577$$ −3.66048e9 −0.793274 −0.396637 0.917976i $$-0.629823\pi$$
−0.396637 + 0.917976i $$0.629823\pi$$
$$578$$ 2.17105e8 0.0467651
$$579$$ 0 0
$$580$$ −1.48578e9 −0.316196
$$581$$ 2.08517e9 0.441088
$$582$$ 0 0
$$583$$ 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 $$0.864924\pi$$
$$588$$ 0 0
$$589$$ −1.06442e10 −2.14640
$$590$$ 3.09562e8 0.0620534
$$591$$ 0 0
$$592$$ 2.45288e9 0.485904
$$593$$ −8.00218e9 −1.57586 −0.787929 0.615766i $$-0.788847\pi$$
−0.787929 + 0.615766i $$0.788847\pi$$
$$594$$ 0 0
$$595$$ 6.79932e8 0.132329
$$596$$ 6.85524e9 1.32636
$$597$$ 0 0
$$598$$ 5.36649e8 0.102621
$$599$$ −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 $$0.230323\pi$$
$$602$$ 2.98537e8 0.0557713
$$603$$ 0 0
$$604$$ 2.81599e9 0.519999
$$605$$ 2.17944e9 0.400130
$$606$$ 0 0
$$607$$ 5.42119e9 0.983863 0.491931 0.870634i $$-0.336291\pi$$
0.491931 + 0.870634i $$0.336291\pi$$
$$608$$ 2.54067e9 0.458444
$$609$$ 0 0
$$610$$ −1.67772e8 −0.0299272
$$611$$ 3.14191e9 0.557250
$$612$$ 0 0
$$613$$ 8.21824e9 1.44101 0.720505 0.693450i $$-0.243910\pi$$
0.720505 + 0.693450i $$0.243910\pi$$
$$614$$ −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 $$0.746359\pi$$
$$618$$ 0 0
$$619$$ −6.46052e9 −1.09484 −0.547420 0.836858i $$-0.684390\pi$$
−0.547420 + 0.836858i $$0.684390\pi$$
$$620$$ 4.35140e9 0.733260
$$621$$ 0 0
$$622$$ 1.58573e7 0.00264218
$$623$$ −3.64051e9 −0.603191
$$624$$ 0 0
$$625$$ 2.44141e8 0.0400000
$$626$$ −1.12511e9 −0.183309
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 5.31876e8 0.0828893
$$635$$ −2.79703e9 −0.433500
$$636$$ 0 0
$$637$$ −7.21930e8 −0.110664
$$638$$ 1.84486e8 0.0281249
$$639$$ 0 0
$$640$$ −1.38131e9 −0.208287
$$641$$ −8.54151e9 −1.28095 −0.640474 0.767980i $$-0.721262\pi$$
−0.640474 + 0.767980i $$0.721262\pi$$
$$642$$ 0 0
$$643$$ −1.20342e10 −1.78517 −0.892585 0.450878i $$-0.851111\pi$$
−0.892585 + 0.450878i $$0.851111\pi$$
$$644$$ 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 $$0.504370\pi$$
$$648$$ 0 0
$$649$$ 2.59539e9 0.372688
$$650$$ 1.31044e8 0.0187163
$$651$$ 0 0
$$652$$ −5.11403e9 −0.722598
$$653$$ −8.70977e9 −1.22408 −0.612041 0.790826i $$-0.709651\pi$$
−0.612041 + 0.790826i $$0.709651\pi$$
$$654$$ 0 0
$$655$$ −5.66662e8 −0.0787916
$$656$$ 4.75963e9 0.658279
$$657$$ 0 0
$$658$$ −2.40033e8 −0.0328458
$$659$$ 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 $$0.307173\pi$$
$$662$$ −2.03074e8 −0.0272052
$$663$$ 0 0
$$664$$ −2.11152e9 −0.279903
$$665$$ −1.65358e9 −0.218047
$$666$$ 0 0
$$667$$ −6.02996e9 −0.786817
$$668$$ 1.22784e10 1.59376
$$669$$ 0 0
$$670$$ −7.60829e8 −0.0977294
$$671$$ −1.40661e9 −0.179740
$$672$$ 0 0
$$673$$ 4.78543e9 0.605157 0.302578 0.953124i $$-0.402153\pi$$
0.302578 + 0.953124i $$0.402153\pi$$
$$674$$ −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 $$0.203228\pi$$
$$678$$ 0 0
$$679$$ −2.27918e9 −0.279405
$$680$$ −6.88526e8 −0.0839729
$$681$$ 0 0
$$682$$ −5.40302e8 −0.0652216
$$683$$ 9.15988e9 1.10006 0.550031 0.835144i $$-0.314616\pi$$
0.550031 + 0.835144i $$0.314616\pi$$
$$684$$ 0 0
$$685$$ −6.34572e9 −0.754334
$$686$$ 5.51533e7 0.00652285
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 1.79944e9 0.204352
$$695$$ −1.31478e9 −0.148562
$$696$$ 0 0
$$697$$ 4.81683e9 0.538824
$$698$$ −3.61534e8 −0.0402398
$$699$$ 0 0
$$700$$ 6.75989e8 0.0744897
$$701$$ −1.27411e9 −0.139699 −0.0698497 0.997558i $$-0.522252\pi$$
−0.0698497 + 0.997558i $$0.522252\pi$$
$$702$$ 0 0
$$703$$ −6.03703e9 −0.655360
$$704$$ −2.74405e9 −0.296406
$$705$$ 0 0
$$706$$ 1.78048e9 0.190423
$$707$$ 3.68832e9 0.392519
$$708$$ 0 0
$$709$$ −7.17795e9 −0.756378 −0.378189 0.925728i $$-0.623453\pi$$
−0.378189 + 0.925728i $$0.623453\pi$$
$$710$$ −1.23936e8 −0.0129955
$$711$$ 0 0
$$712$$ 3.68653e9 0.382769
$$713$$ 1.76599e10 1.82463
$$714$$ 0 0
$$715$$ 1.09868e9 0.112409
$$716$$ 1.83914e10 1.87249
$$717$$ 0 0
$$718$$ −1.41133e9 −0.142296
$$719$$ −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
$$722$$ −8.11273e8 −0.0802208
$$723$$ 0 0
$$724$$ 7.68971e9 0.753052
$$725$$ −1.47245e9 −0.143502
$$726$$ 0 0
$$727$$ −4.67874e9 −0.451605 −0.225802 0.974173i $$-0.572500\pi$$
−0.225802 + 0.974173i $$0.572500\pi$$
$$728$$ 7.31054e8 0.0702246
$$729$$ 0 0
$$730$$ 3.71759e8 0.0353697
$$731$$ 1.00990e10 0.956238
$$732$$ 0 0
$$733$$ 1.28552e9 0.120563 0.0602817 0.998181i $$-0.480800\pi$$
0.0602817 + 0.998181i $$0.480800\pi$$
$$734$$ −1.54612e9 −0.144314
$$735$$ 0 0
$$736$$ −4.21524e9 −0.389718
$$737$$ −6.37884e9 −0.586956
$$738$$ 0 0
$$739$$ −5.26720e9 −0.480091 −0.240046 0.970762i $$-0.577162\pi$$
−0.240046 + 0.970762i $$0.577162\pi$$
$$740$$ 2.46796e9 0.223886
$$741$$ 0 0
$$742$$ 9.43446e7 0.00847819
$$743$$ −4.15012e9 −0.371193 −0.185596 0.982626i $$-0.559422\pi$$
−0.185596 + 0.982626i $$0.559422\pi$$
$$744$$ 0 0
$$745$$ 6.79371e9 0.601950
$$746$$ −7.35489e8 −0.0648620
$$747$$ 0 0
$$748$$ −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
$$753$$ 0 0
$$754$$ −7.90343e8 −0.0671453
$$755$$ 2.79072e9 0.235995
$$756$$ 0 0
$$757$$ −1.19658e10 −1.00255 −0.501274 0.865289i $$-0.667135\pi$$
−0.501274 + 0.865289i $$0.667135\pi$$
$$758$$ −1.07032e9 −0.0892631
$$759$$ 0 0
$$760$$ 1.67448e9 0.138367
$$761$$ 2.00959e10 1.65296 0.826479 0.562967i $$-0.190340\pi$$
0.826479 + 0.562967i $$0.190340\pi$$
$$762$$ 0 0
$$763$$ 6.72100e9 0.547770
$$764$$ −1.92450e10 −1.56131
$$765$$ 0 0
$$766$$ 1.12411e9 0.0903665
$$767$$ −1.11187e10 −0.889756
$$768$$ 0 0
$$769$$ 2.46683e10 1.95613 0.978064 0.208304i $$-0.0667944\pi$$
0.978064 + 0.208304i $$0.0667944\pi$$
$$770$$ −8.39358e7 −0.00662567
$$771$$ 0 0
$$772$$ 1.75673e10 1.37418
$$773$$ −8.88824e9 −0.692130 −0.346065 0.938211i $$-0.612482\pi$$
−0.346065 + 0.938211i $$0.612482\pi$$
$$774$$ 0 0
$$775$$ 4.31235e9 0.332781
$$776$$ 2.30798e9 0.177303
$$777$$ 0 0
$$778$$ −1.46252e9 −0.111345
$$779$$ −1.17144e10 −0.887850
$$780$$ 0 0
$$781$$ −1.03909e9 −0.0780502
$$782$$ −1.38690e9 −0.103710
$$783$$ 0 0
$$784$$ 1.84358e9 0.136633
$$785$$ 5.47235e9 0.403766
$$786$$ 0 0
$$787$$ −4.65006e9 −0.340053 −0.170027 0.985439i $$-0.554385\pi$$
−0.170027 + 0.985439i $$0.554385\pi$$
$$788$$ 8.22986e9 0.599171
$$789$$ 0 0
$$790$$ −8.90993e8 −0.0642953
$$791$$ 4.67542e9 0.335895
$$792$$ 0 0
$$793$$ 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 $$0.666623\pi$$
$$798$$ 0 0
$$799$$ −8.11987e9 −0.563165
$$800$$ −1.02931e9 −0.0710776
$$801$$ 0 0
$$802$$ −2.66257e9 −0.182260
$$803$$ 3.11685e9 0.212428
$$804$$ 0 0
$$805$$ 2.74346e9 0.185359
$$806$$ 2.31467e9 0.155710
$$807$$ 0 0
$$808$$ −3.73494e9 −0.249083
$$809$$ 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 $$0.217173\pi$$
$$812$$ −4.07698e9 −0.267235
$$813$$ 0 0
$$814$$ −3.06440e8 −0.0199141
$$815$$ −5.06813e9 −0.327942
$$816$$ 0 0
$$817$$ −2.45605e10 −1.57565
$$818$$ −1.18069e9 −0.0754221
$$819$$ 0 0
$$820$$ 4.78889e9 0.303310
$$821$$ 2.86630e10 1.80768 0.903838 0.427875i $$-0.140738\pi$$
0.903838 + 0.427875i $$0.140738\pi$$
$$822$$ 0 0
$$823$$ −2.76897e10 −1.73148 −0.865742 0.500490i $$-0.833153\pi$$
−0.865742 + 0.500490i $$0.833153\pi$$
$$824$$ −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 $$0.372151\pi$$
$$828$$ 0 0
$$829$$ −1.50770e10 −0.919127 −0.459563 0.888145i $$-0.651994\pi$$
−0.459563 + 0.888145i $$0.651994\pi$$
$$830$$ −1.03860e9 −0.0630483
$$831$$ 0 0
$$832$$ 1.17556e10 0.707641
$$833$$ 1.86573e9 0.111839
$$834$$ 0 0
$$835$$ 1.21682e10 0.723307
$$836$$ 6.96787e9 0.412457
$$837$$ 0 0
$$838$$ 3.02512e9 0.177578
$$839$$ −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
$$842$$ 3.96175e9 0.228715
$$843$$ 0 0
$$844$$ 6.53172e9 0.373963
$$845$$ 3.13679e9 0.178849
$$846$$ 0 0
$$847$$ 5.98038e9 0.338171
$$848$$ 3.15361e9 0.177591
$$849$$ 0 0
$$850$$ −3.38665e8 −0.0189149
$$851$$ 1.00161e10 0.557114
$$852$$ 0 0
$$853$$ −1.13971e9 −0.0628740 −0.0314370 0.999506i $$-0.510008\pi$$
−0.0314370 + 0.999506i $$0.510008\pi$$
$$854$$ −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 $$0.567834\pi$$
$$858$$ 0 0
$$859$$ 1.27280e10 0.685147 0.342573 0.939491i $$-0.388701\pi$$
0.342573 + 0.939491i $$0.388701\pi$$
$$860$$ 1.00404e10 0.538277
$$861$$ 0 0
$$862$$ −3.30831e9 −0.175926
$$863$$ −2.53204e9 −0.134101 −0.0670507 0.997750i $$-0.521359\pi$$
−0.0670507 + 0.997750i $$0.521359\pi$$
$$864$$ 0 0
$$865$$ −6.38258e9 −0.335305
$$866$$ 3.09823e9 0.162107
$$867$$ 0 0
$$868$$ 1.19402e10 0.619718
$$869$$ −7.47014e9 −0.386153
$$870$$ 0 0
$$871$$ 2.73272e10 1.40130
$$872$$ −6.80595e9 −0.347601
$$873$$ 0 0
$$874$$ 3.37291e9 0.170889
$$875$$ 6.69922e8 0.0338062
$$876$$ 0 0
$$877$$ 5.00988e9 0.250800 0.125400 0.992106i $$-0.459979\pi$$
0.125400 + 0.992106i $$0.459979\pi$$
$$878$$ −2.71862e9 −0.135556
$$879$$ 0 0
$$880$$ −2.80568e9 −0.138787
$$881$$ −9.46900e9 −0.466539 −0.233270 0.972412i $$-0.574942\pi$$
−0.233270 + 0.972412i $$0.574942\pi$$
$$882$$ 0 0
$$883$$ 1.11146e10 0.543289 0.271644 0.962398i $$-0.412433\pi$$
0.271644 + 0.962398i $$0.412433\pi$$
$$884$$ 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 $$0.443965\pi$$
$$888$$ 0 0
$$889$$ −7.67506e9 −0.366375
$$890$$ 1.81329e9 0.0862190
$$891$$ 0 0
$$892$$ 1.57746e10 0.744188
$$893$$ 1.97473e10 0.927959
$$894$$ 0 0
$$895$$ 1.82264e10 0.849807
$$896$$ −3.79032e9 −0.176034
$$897$$ 0 0
$$898$$ 2.09267e8 0.00964348
$$899$$ −2.60084e10 −1.19386
$$900$$ 0 0
$$901$$ 3.19150e9 0.145365
$$902$$ −5.94624e8 −0.0269786
$$903$$ 0 0
$$904$$ −4.73451e9 −0.213150
$$905$$ 7.62070e9 0.341763
$$906$$ 0 0
$$907$$ −1.39503e10 −0.620809 −0.310405 0.950605i $$-0.600465\pi$$
−0.310405 + 0.950605i $$0.600465\pi$$
$$908$$ 2.42310e9 0.107416
$$909$$ 0 0
$$910$$ 3.59584e8 0.0158181
$$911$$ −2.98148e8 −0.0130653 −0.00653263 0.999979i $$-0.502079\pi$$
−0.00653263 + 0.999979i $$0.502079\pi$$
$$912$$ 0 0
$$913$$ −8.70765e9 −0.378663
$$914$$ 3.27506e9 0.141876
$$915$$ 0 0
$$916$$ 1.33259e10 0.572876
$$917$$ −1.55492e9 −0.0665910
$$918$$ 0 0
$$919$$ 2.67202e10 1.13563 0.567814 0.823157i $$-0.307789\pi$$
0.567814 + 0.823157i $$0.307789\pi$$
$$920$$ −2.77813e9 −0.117624
$$921$$ 0 0
$$922$$ 2.21295e9 0.0929850
$$923$$ 4.45149e9 0.186337
$$924$$ 0 0
$$925$$ 2.44581e9 0.101608
$$926$$ −1.58725e9 −0.0656910
$$927$$ 0 0
$$928$$ 6.20794e9 0.254994
$$929$$ 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
$$932$$ −2.92180e10 −1.18221
$$933$$ 0 0
$$934$$ 3.88115e9 0.155864
$$935$$ −2.83939e9 −0.113602
$$936$$ 0 0
$$937$$ 1.28088e10 0.508649 0.254325 0.967119i $$-0.418147\pi$$
0.254325 + 0.967119i $$0.418147\pi$$
$$938$$ −2.08772e9 −0.0825965
$$939$$ 0 0
$$940$$ −8.07278e9 −0.317012
$$941$$ 1.20663e10 0.472073 0.236037 0.971744i $$-0.424151\pi$$
0.236037 + 0.971744i $$0.424151\pi$$
$$942$$ 0 0
$$943$$ 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 $$0.551131\pi$$
$$948$$ 0 0
$$949$$ −1.33527e10 −0.507151
$$950$$ 8.23627e8 0.0311672
$$951$$ 0 0
$$952$$ −1.88931e9 −0.0709700
$$953$$ −4.49530e10 −1.68242 −0.841209 0.540710i $$-0.818156\pi$$
−0.841209 + 0.540710i $$0.818156\pi$$
$$954$$ 0 0
$$955$$ −1.90722e10 −0.708582
$$956$$ 1.37703e10 0.509732
$$957$$ 0 0
$$958$$ 3.26341e9 0.119920
$$959$$ −1.74126e10 −0.637528
$$960$$ 0 0
$$961$$ 4.86580e10 1.76857
$$962$$ 1.31280e9 0.0475429
$$963$$ 0 0
$$964$$ 1.04094e10 0.374244
$$965$$ 1.74096e10 0.623654
$$966$$ 0 0
$$967$$ 1.34247e8 0.00477432 0.00238716 0.999997i $$-0.499240\pi$$
0.00238716 + 0.999997i $$0.499240\pi$$
$$968$$ −6.05596e9 −0.214595
$$969$$ 0 0
$$970$$ 1.13523e9 0.0399376
$$971$$ 3.00377e10 1.05293 0.526465 0.850197i $$-0.323517\pi$$
0.526465 + 0.850197i $$0.323517\pi$$
$$972$$ 0 0
$$973$$ −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 $$0.783153\pi$$
$$978$$ 0 0
$$979$$ 1.52028e10 0.517825
$$980$$ 1.85491e9 0.0629553
$$981$$ 0 0
$$982$$ 5.85057e8 0.0197155
$$983$$ −4.61443e10 −1.54946 −0.774731 0.632290i $$-0.782115\pi$$
−0.774731 + 0.632290i $$0.782115\pi$$
$$984$$ 0 0
$$985$$ 8.15600e9 0.271926
$$986$$ 2.04254e9 0.0678580
$$987$$ 0 0
$$988$$ −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
$$993$$ 0 0
$$994$$ −3.40081e8 −0.0109832
$$995$$ −2.41879e8 −0.00778427
$$996$$ 0 0
$$997$$ −5.00734e10 −1.60020 −0.800099 0.599868i $$-0.795220\pi$$
−0.800099 + 0.599868i $$0.795220\pi$$
$$998$$ 4.03435e9 0.128474
$$999$$ 0 0
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 315.8.a.c.1.2 2
3.2 odd 2 35.8.a.a.1.1 2
12.11 even 2 560.8.a.i.1.1 2
15.2 even 4 175.8.b.c.99.3 4
15.8 even 4 175.8.b.c.99.2 4
15.14 odd 2 175.8.a.b.1.2 2
21.20 even 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 3.2 odd 2
175.8.a.b.1.2 2 15.14 odd 2
175.8.b.c.99.2 4 15.8 even 4
175.8.b.c.99.3 4 15.2 even 4
245.8.a.b.1.1 2 21.20 even 2
315.8.a.c.1.2 2 1.1 even 1 trivial
560.8.a.i.1.1 2 12.11 even 2