Properties

 Label 175.8.b.c.99.1 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 99.1 Root $$1.65831 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.99 Dual form 175.8.b.c.99.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-14.6332i q^{2} -54.7995i q^{3} -86.1320 q^{4} -801.895 q^{6} +343.000i q^{7} -612.665i q^{8} -815.985 q^{9} +O(q^{10})$$ $$q-14.6332i q^{2} -54.7995i q^{3} -86.1320 q^{4} -801.895 q^{6} +343.000i q^{7} -612.665i q^{8} -815.985 q^{9} -6473.63 q^{11} +4719.99i q^{12} -11681.7i q^{13} +5019.20 q^{14} -19990.2 q^{16} -13460.5i q^{17} +11940.5i q^{18} -34955.5 q^{19} +18796.2 q^{21} +94730.3i q^{22} +77831.4i q^{23} -33573.7 q^{24} -170941. q^{26} -75130.9i q^{27} -29543.3i q^{28} +221135. q^{29} -23222.3 q^{31} +214100. i q^{32} +354752. i q^{33} -196971. q^{34} +70282.4 q^{36} +422392. i q^{37} +511512. i q^{38} -640151. q^{39} +191818. q^{41} -275050. i q^{42} +310754. i q^{43} +557587. q^{44} +1.13893e6 q^{46} +240747. i q^{47} +1.09545e6i q^{48} -117649. q^{49} -737628. q^{51} +1.00617e6i q^{52} -1.06654e6i q^{53} -1.09941e6 q^{54} +210144. q^{56} +1.91554e6i q^{57} -3.23592e6i q^{58} -451838. q^{59} -831659. q^{61} +339818. i q^{62} -279883. i q^{63} +574238. q^{64} +5.19117e6 q^{66} -2.26405e6i q^{67} +1.15938e6i q^{68} +4.26512e6 q^{69} -2.22036e6 q^{71} +499925. i q^{72} +4.99377e6i q^{73} +6.18096e6 q^{74} +3.01078e6 q^{76} -2.22046e6i q^{77} +9.36749e6i q^{78} +2.72773e6 q^{79} -5.90170e6 q^{81} -2.80693e6i q^{82} -6.38392e6i q^{83} -1.61896e6 q^{84} +4.54734e6 q^{86} -1.21181e7i q^{87} +3.96617e6i q^{88} +7.32978e6 q^{89} +4.00682e6 q^{91} -6.70378e6i q^{92} +1.27257e6i q^{93} +3.52291e6 q^{94} +1.17326e7 q^{96} +2.38676e6i q^{97} +1.72159e6i q^{98} +5.28239e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 14.6332i − 1.29341i −0.762741 0.646704i $$-0.776147\pi$$
0.762741 0.646704i $$-0.223853\pi$$
$$3$$ − 54.7995i − 1.17180i −0.810385 0.585898i $$-0.800742\pi$$
0.810385 0.585898i $$-0.199258\pi$$
$$4$$ −86.1320 −0.672906
$$5$$ 0 0
$$6$$ −801.895 −1.51561
$$7$$ 343.000i 0.377964i
$$8$$ − 612.665i − 0.423066i
$$9$$ −815.985 −0.373107
$$10$$ 0 0
$$11$$ −6473.63 −1.46647 −0.733236 0.679974i $$-0.761991\pi$$
−0.733236 + 0.679974i $$0.761991\pi$$
$$12$$ 4719.99i 0.788509i
$$13$$ − 11681.7i − 1.47470i −0.675510 0.737351i $$-0.736076\pi$$
0.675510 0.737351i $$-0.263924\pi$$
$$14$$ 5019.20 0.488863
$$15$$ 0 0
$$16$$ −19990.2 −1.22010
$$17$$ − 13460.5i − 0.664491i −0.943193 0.332246i $$-0.892194\pi$$
0.943193 0.332246i $$-0.107806\pi$$
$$18$$ 11940.5i 0.482580i
$$19$$ −34955.5 −1.16917 −0.584585 0.811333i $$-0.698743\pi$$
−0.584585 + 0.811333i $$0.698743\pi$$
$$20$$ 0 0
$$21$$ 18796.2 0.442897
$$22$$ 94730.3i 1.89675i
$$23$$ 77831.4i 1.33385i 0.745124 + 0.666926i $$0.232390\pi$$
−0.745124 + 0.666926i $$0.767610\pi$$
$$24$$ −33573.7 −0.495747
$$25$$ 0 0
$$26$$ −170941. −1.90739
$$27$$ − 75130.9i − 0.734591i
$$28$$ − 29543.3i − 0.254335i
$$29$$ 221135. 1.68370 0.841848 0.539715i $$-0.181468\pi$$
0.841848 + 0.539715i $$0.181468\pi$$
$$30$$ 0 0
$$31$$ −23222.3 −0.140004 −0.0700018 0.997547i $$-0.522301\pi$$
−0.0700018 + 0.997547i $$0.522301\pi$$
$$32$$ 214100.i 1.15503i
$$33$$ 354752.i 1.71841i
$$34$$ −196971. −0.859459
$$35$$ 0 0
$$36$$ 70282.4 0.251066
$$37$$ 422392.i 1.37091i 0.728114 + 0.685456i $$0.240397\pi$$
−0.728114 + 0.685456i $$0.759603\pi$$
$$38$$ 511512.i 1.51221i
$$39$$ −640151. −1.72805
$$40$$ 0 0
$$41$$ 191818. 0.434657 0.217329 0.976099i $$-0.430266\pi$$
0.217329 + 0.976099i $$0.430266\pi$$
$$42$$ − 275050.i − 0.572847i
$$43$$ 310754.i 0.596042i 0.954559 + 0.298021i $$0.0963266\pi$$
−0.954559 + 0.298021i $$0.903673\pi$$
$$44$$ 557587. 0.986798
$$45$$ 0 0
$$46$$ 1.13893e6 1.72522
$$47$$ 240747.i 0.338235i 0.985596 + 0.169117i $$0.0540917\pi$$
−0.985596 + 0.169117i $$0.945908\pi$$
$$48$$ 1.09545e6i 1.42971i
$$49$$ −117649. −0.142857
$$50$$ 0 0
$$51$$ −737628. −0.778649
$$52$$ 1.00617e6i 0.992336i
$$53$$ − 1.06654e6i − 0.984040i −0.870584 0.492020i $$-0.836259\pi$$
0.870584 0.492020i $$-0.163741\pi$$
$$54$$ −1.09941e6 −0.950126
$$55$$ 0 0
$$56$$ 210144. 0.159904
$$57$$ 1.91554e6i 1.37003i
$$58$$ − 3.23592e6i − 2.17771i
$$59$$ −451838. −0.286418 −0.143209 0.989692i $$-0.545742\pi$$
−0.143209 + 0.989692i $$0.545742\pi$$
$$60$$ 0 0
$$61$$ −831659. −0.469127 −0.234564 0.972101i $$-0.575366\pi$$
−0.234564 + 0.972101i $$0.575366\pi$$
$$62$$ 339818.i 0.181082i
$$63$$ − 279883.i − 0.141021i
$$64$$ 574238. 0.273818
$$65$$ 0 0
$$66$$ 5.19117e6 2.22260
$$67$$ − 2.26405e6i − 0.919654i −0.888009 0.459827i $$-0.847912\pi$$
0.888009 0.459827i $$-0.152088\pi$$
$$68$$ 1.15938e6i 0.447140i
$$69$$ 4.26512e6 1.56300
$$70$$ 0 0
$$71$$ −2.22036e6 −0.736241 −0.368120 0.929778i $$-0.619999\pi$$
−0.368120 + 0.929778i $$0.619999\pi$$
$$72$$ 499925.i 0.157849i
$$73$$ 4.99377e6i 1.50244i 0.660049 + 0.751222i $$0.270535\pi$$
−0.660049 + 0.751222i $$0.729465\pi$$
$$74$$ 6.18096e6 1.77315
$$75$$ 0 0
$$76$$ 3.01078e6 0.786742
$$77$$ − 2.22046e6i − 0.554274i
$$78$$ 9.36749e6i 2.23508i
$$79$$ 2.72773e6 0.622452 0.311226 0.950336i $$-0.399260\pi$$
0.311226 + 0.950336i $$0.399260\pi$$
$$80$$ 0 0
$$81$$ −5.90170e6 −1.23390
$$82$$ − 2.80693e6i − 0.562189i
$$83$$ − 6.38392e6i − 1.22550i −0.790276 0.612751i $$-0.790063\pi$$
0.790276 0.612751i $$-0.209937\pi$$
$$84$$ −1.61896e6 −0.298028
$$85$$ 0 0
$$86$$ 4.54734e6 0.770926
$$87$$ − 1.21181e7i − 1.97295i
$$88$$ 3.96617e6i 0.620414i
$$89$$ 7.32978e6 1.10211 0.551056 0.834468i $$-0.314225\pi$$
0.551056 + 0.834468i $$0.314225\pi$$
$$90$$ 0 0
$$91$$ 4.00682e6 0.557385
$$92$$ − 6.70378e6i − 0.897557i
$$93$$ 1.27257e6i 0.164056i
$$94$$ 3.52291e6 0.437476
$$95$$ 0 0
$$96$$ 1.17326e7 1.35346
$$97$$ 2.38676e6i 0.265526i 0.991148 + 0.132763i $$0.0423850\pi$$
−0.991148 + 0.132763i $$0.957615\pi$$
$$98$$ 1.72159e6i 0.184773i
$$99$$ 5.28239e6 0.547151
$$100$$ 0 0
$$101$$ −1.92113e7 −1.85538 −0.927688 0.373356i $$-0.878207\pi$$
−0.927688 + 0.373356i $$0.878207\pi$$
$$102$$ 1.07939e7i 1.00711i
$$103$$ − 4.45359e6i − 0.401587i −0.979634 0.200793i $$-0.935648\pi$$
0.979634 0.200793i $$-0.0643520\pi$$
$$104$$ −7.15697e6 −0.623896
$$105$$ 0 0
$$106$$ −1.56070e7 −1.27277
$$107$$ − 7.61385e6i − 0.600843i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971273\pi$$
$$108$$ 6.47118e6i 0.494311i
$$109$$ 1.99698e7 1.47700 0.738502 0.674251i $$-0.235533\pi$$
0.738502 + 0.674251i $$0.235533\pi$$
$$110$$ 0 0
$$111$$ 2.31469e7 1.60643
$$112$$ − 6.85663e6i − 0.461156i
$$113$$ 2.57944e7i 1.68171i 0.541261 + 0.840855i $$0.317947\pi$$
−0.541261 + 0.840855i $$0.682053\pi$$
$$114$$ 2.80306e7 1.77201
$$115$$ 0 0
$$116$$ −1.90468e7 −1.13297
$$117$$ 9.53209e6i 0.550222i
$$118$$ 6.61185e6i 0.370456i
$$119$$ 4.61695e6 0.251154
$$120$$ 0 0
$$121$$ 2.24208e7 1.15054
$$122$$ 1.21699e7i 0.606773i
$$123$$ − 1.05116e7i − 0.509330i
$$124$$ 2.00018e6 0.0942094
$$125$$ 0 0
$$126$$ −4.09560e6 −0.182398
$$127$$ 6.75687e6i 0.292707i 0.989232 + 0.146353i $$0.0467536\pi$$
−0.989232 + 0.146353i $$0.953246\pi$$
$$128$$ 1.90018e7i 0.800868i
$$129$$ 1.70292e7 0.698440
$$130$$ 0 0
$$131$$ −2.21063e6 −0.0859144 −0.0429572 0.999077i $$-0.513678\pi$$
−0.0429572 + 0.999077i $$0.513678\pi$$
$$132$$ − 3.05555e7i − 1.15633i
$$133$$ − 1.19897e7i − 0.441905i
$$134$$ −3.31304e7 −1.18949
$$135$$ 0 0
$$136$$ −8.24677e6 −0.281124
$$137$$ 7.10581e6i 0.236097i 0.993008 + 0.118049i $$0.0376639\pi$$
−0.993008 + 0.118049i $$0.962336\pi$$
$$138$$ − 6.24126e7i − 2.02160i
$$139$$ −9.21848e6 −0.291144 −0.145572 0.989348i $$-0.546502\pi$$
−0.145572 + 0.989348i $$0.546502\pi$$
$$140$$ 0 0
$$141$$ 1.31928e7 0.396342
$$142$$ 3.24911e7i 0.952260i
$$143$$ 7.56230e7i 2.16261i
$$144$$ 1.63117e7 0.455229
$$145$$ 0 0
$$146$$ 7.30751e7 1.94328
$$147$$ 6.44711e6i 0.167399i
$$148$$ − 3.63814e7i − 0.922495i
$$149$$ −1.33298e7 −0.330119 −0.165059 0.986284i $$-0.552782\pi$$
−0.165059 + 0.986284i $$0.552782\pi$$
$$150$$ 0 0
$$151$$ −6.41939e7 −1.51731 −0.758656 0.651492i $$-0.774143\pi$$
−0.758656 + 0.651492i $$0.774143\pi$$
$$152$$ 2.14160e7i 0.494636i
$$153$$ 1.09836e7i 0.247926i
$$154$$ −3.24925e7 −0.716903
$$155$$ 0 0
$$156$$ 5.51375e7 1.16282
$$157$$ − 7.36596e7i − 1.51908i −0.650461 0.759540i $$-0.725424\pi$$
0.650461 0.759540i $$-0.274576\pi$$
$$158$$ − 3.99155e7i − 0.805085i
$$159$$ −5.84460e7 −1.15309
$$160$$ 0 0
$$161$$ −2.66962e7 −0.504149
$$162$$ 8.63610e7i 1.59593i
$$163$$ − 3.50642e7i − 0.634172i −0.948397 0.317086i $$-0.897296\pi$$
0.948397 0.317086i $$-0.102704\pi$$
$$164$$ −1.65217e7 −0.292483
$$165$$ 0 0
$$166$$ −9.34175e7 −1.58508
$$167$$ − 2.56950e6i − 0.0426915i −0.999772 0.0213458i $$-0.993205\pi$$
0.999772 0.0213458i $$-0.00679508\pi$$
$$168$$ − 1.15158e7i − 0.187375i
$$169$$ −7.37136e7 −1.17475
$$170$$ 0 0
$$171$$ 2.85231e7 0.436225
$$172$$ − 2.67659e7i − 0.401081i
$$173$$ 8.03463e7i 1.17979i 0.807480 + 0.589895i $$0.200831\pi$$
−0.807480 + 0.589895i $$0.799169\pi$$
$$174$$ −1.77327e8 −2.55183
$$175$$ 0 0
$$176$$ 1.29409e8 1.78925
$$177$$ 2.47605e7i 0.335624i
$$178$$ − 1.07259e8i − 1.42548i
$$179$$ −9.99074e7 −1.30200 −0.651001 0.759077i $$-0.725651\pi$$
−0.651001 + 0.759077i $$0.725651\pi$$
$$180$$ 0 0
$$181$$ −1.07414e8 −1.34644 −0.673221 0.739442i $$-0.735090\pi$$
−0.673221 + 0.739442i $$0.735090\pi$$
$$182$$ − 5.86328e7i − 0.720927i
$$183$$ 4.55745e7i 0.549722i
$$184$$ 4.76846e7 0.564307
$$185$$ 0 0
$$186$$ 1.86218e7 0.212191
$$187$$ 8.71382e7i 0.974458i
$$188$$ − 2.07360e7i − 0.227600i
$$189$$ 2.57699e7 0.277649
$$190$$ 0 0
$$191$$ −2.21085e7 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$192$$ − 3.14679e7i − 0.320859i
$$193$$ − 1.49793e8i − 1.49983i −0.661537 0.749913i $$-0.730095\pi$$
0.661537 0.749913i $$-0.269905\pi$$
$$194$$ 3.49261e7 0.343434
$$195$$ 0 0
$$196$$ 1.01333e7 0.0961295
$$197$$ 5.70107e7i 0.531281i 0.964072 + 0.265641i $$0.0855835\pi$$
−0.964072 + 0.265641i $$0.914417\pi$$
$$198$$ − 7.72985e7i − 0.707690i
$$199$$ 6.84161e7 0.615421 0.307711 0.951480i $$-0.400437\pi$$
0.307711 + 0.951480i $$0.400437\pi$$
$$200$$ 0 0
$$201$$ −1.24069e8 −1.07765
$$202$$ 2.81124e8i 2.39976i
$$203$$ 7.58492e7i 0.636377i
$$204$$ 6.35333e7 0.523958
$$205$$ 0 0
$$206$$ −6.51704e7 −0.519416
$$207$$ − 6.35093e7i − 0.497669i
$$208$$ 2.33519e8i 1.79929i
$$209$$ 2.26289e8 1.71455
$$210$$ 0 0
$$211$$ −1.36201e8 −0.998141 −0.499071 0.866561i $$-0.666325\pi$$
−0.499071 + 0.866561i $$0.666325\pi$$
$$212$$ 9.18635e7i 0.662167i
$$213$$ 1.21675e8i 0.862724i
$$214$$ −1.11415e8 −0.777135
$$215$$ 0 0
$$216$$ −4.60301e7 −0.310780
$$217$$ − 7.96525e6i − 0.0529164i
$$218$$ − 2.92224e8i − 1.91037i
$$219$$ 2.73656e8 1.76056
$$220$$ 0 0
$$221$$ −1.57241e8 −0.979927
$$222$$ − 3.38714e8i − 2.07777i
$$223$$ − 1.47728e8i − 0.892066i −0.895017 0.446033i $$-0.852836\pi$$
0.895017 0.446033i $$-0.147164\pi$$
$$224$$ −7.34363e7 −0.436559
$$225$$ 0 0
$$226$$ 3.77456e8 2.17514
$$227$$ − 3.22427e8i − 1.82954i −0.403980 0.914768i $$-0.632373\pi$$
0.403980 0.914768i $$-0.367627\pi$$
$$228$$ − 1.64989e8i − 0.921901i
$$229$$ −3.10033e8 −1.70602 −0.853010 0.521895i $$-0.825225\pi$$
−0.853010 + 0.521895i $$0.825225\pi$$
$$230$$ 0 0
$$231$$ −1.21680e8 −0.649497
$$232$$ − 1.35481e8i − 0.712315i
$$233$$ 1.80410e8i 0.934361i 0.884162 + 0.467181i $$0.154730\pi$$
−0.884162 + 0.467181i $$0.845270\pi$$
$$234$$ 1.39485e8 0.711661
$$235$$ 0 0
$$236$$ 3.89177e7 0.192733
$$237$$ − 1.49478e8i − 0.729387i
$$238$$ − 6.75609e7i − 0.324845i
$$239$$ −3.66489e8 −1.73647 −0.868237 0.496150i $$-0.834747\pi$$
−0.868237 + 0.496150i $$0.834747\pi$$
$$240$$ 0 0
$$241$$ 2.19729e8 1.01118 0.505589 0.862775i $$-0.331275\pi$$
0.505589 + 0.862775i $$0.331275\pi$$
$$242$$ − 3.28089e8i − 1.48812i
$$243$$ 1.59099e8i 0.711286i
$$244$$ 7.16324e7 0.315679
$$245$$ 0 0
$$246$$ −1.53818e8 −0.658771
$$247$$ 4.08339e8i 1.72418i
$$248$$ 1.42275e7i 0.0592308i
$$249$$ −3.49836e8 −1.43604
$$250$$ 0 0
$$251$$ −1.29875e8 −0.518404 −0.259202 0.965823i $$-0.583460\pi$$
−0.259202 + 0.965823i $$0.583460\pi$$
$$252$$ 2.41069e7i 0.0948940i
$$253$$ − 5.03852e8i − 1.95606i
$$254$$ 9.88750e7 0.378589
$$255$$ 0 0
$$256$$ 3.51561e8 1.30967
$$257$$ 2.94635e8i 1.08273i 0.840789 + 0.541363i $$0.182091\pi$$
−0.840789 + 0.541363i $$0.817909\pi$$
$$258$$ − 2.49192e8i − 0.903369i
$$259$$ −1.44880e8 −0.518156
$$260$$ 0 0
$$261$$ −1.80443e8 −0.628199
$$262$$ 3.23486e7i 0.111122i
$$263$$ 3.13955e8i 1.06420i 0.846683 + 0.532098i $$0.178596\pi$$
−0.846683 + 0.532098i $$0.821404\pi$$
$$264$$ 2.17344e8 0.726999
$$265$$ 0 0
$$266$$ −1.75449e8 −0.571563
$$267$$ − 4.01668e8i − 1.29145i
$$268$$ 1.95007e8i 0.618841i
$$269$$ 2.94745e8 0.923238 0.461619 0.887078i $$-0.347269\pi$$
0.461619 + 0.887078i $$0.347269\pi$$
$$270$$ 0 0
$$271$$ −8.47290e7 −0.258607 −0.129303 0.991605i $$-0.541274\pi$$
−0.129303 + 0.991605i $$0.541274\pi$$
$$272$$ 2.69077e8i 0.810748i
$$273$$ − 2.19572e8i − 0.653142i
$$274$$ 1.03981e8 0.305371
$$275$$ 0 0
$$276$$ −3.67364e8 −1.05175
$$277$$ − 1.96909e8i − 0.556655i −0.960486 0.278327i $$-0.910220\pi$$
0.960486 0.278327i $$-0.0897800\pi$$
$$278$$ 1.34896e8i 0.376568i
$$279$$ 1.89491e7 0.0522364
$$280$$ 0 0
$$281$$ 3.32330e8 0.893505 0.446753 0.894658i $$-0.352580\pi$$
0.446753 + 0.894658i $$0.352580\pi$$
$$282$$ − 1.93054e8i − 0.512632i
$$283$$ 4.40437e8i 1.15513i 0.816344 + 0.577566i $$0.195997\pi$$
−0.816344 + 0.577566i $$0.804003\pi$$
$$284$$ 1.91244e8 0.495421
$$285$$ 0 0
$$286$$ 1.10661e9 2.79714
$$287$$ 6.57937e7i 0.164285i
$$288$$ − 1.74702e8i − 0.430948i
$$289$$ 2.29154e8 0.558451
$$290$$ 0 0
$$291$$ 1.30793e8 0.311143
$$292$$ − 4.30123e8i − 1.01100i
$$293$$ − 3.05058e8i − 0.708510i −0.935149 0.354255i $$-0.884735\pi$$
0.935149 0.354255i $$-0.115265\pi$$
$$294$$ 9.43421e7 0.216516
$$295$$ 0 0
$$296$$ 2.58785e8 0.579986
$$297$$ 4.86370e8i 1.07726i
$$298$$ 1.95058e8i 0.426979i
$$299$$ 9.09203e8 1.96703
$$300$$ 0 0
$$301$$ −1.06589e8 −0.225283
$$302$$ 9.39366e8i 1.96250i
$$303$$ 1.05277e9i 2.17412i
$$304$$ 6.98766e8 1.42651
$$305$$ 0 0
$$306$$ 1.60725e8 0.320670
$$307$$ 2.41616e8i 0.476587i 0.971193 + 0.238293i $$0.0765880\pi$$
−0.971193 + 0.238293i $$0.923412\pi$$
$$308$$ 1.91252e8i 0.372975i
$$309$$ −2.44054e8 −0.470578
$$310$$ 0 0
$$311$$ −6.71768e7 −0.126636 −0.0633181 0.997993i $$-0.520168\pi$$
−0.0633181 + 0.997993i $$0.520168\pi$$
$$312$$ 3.92198e8i 0.731079i
$$313$$ − 6.75084e8i − 1.24438i −0.782867 0.622190i $$-0.786243\pi$$
0.782867 0.622190i $$-0.213757\pi$$
$$314$$ −1.07788e9 −1.96479
$$315$$ 0 0
$$316$$ −2.34944e8 −0.418852
$$317$$ 4.65149e8i 0.820135i 0.912055 + 0.410067i $$0.134495\pi$$
−0.912055 + 0.410067i $$0.865505\pi$$
$$318$$ 8.55255e8i 1.49142i
$$319$$ −1.43154e9 −2.46909
$$320$$ 0 0
$$321$$ −4.17235e8 −0.704066
$$322$$ 3.90652e8i 0.652070i
$$323$$ 4.70517e8i 0.776903i
$$324$$ 5.08325e8 0.830298
$$325$$ 0 0
$$326$$ −5.13103e8 −0.820244
$$327$$ − 1.09434e9i − 1.73075i
$$328$$ − 1.17520e8i − 0.183889i
$$329$$ −8.25762e7 −0.127841
$$330$$ 0 0
$$331$$ 2.69779e8 0.408893 0.204447 0.978878i $$-0.434460\pi$$
0.204447 + 0.978878i $$0.434460\pi$$
$$332$$ 5.49860e8i 0.824648i
$$333$$ − 3.44665e8i − 0.511497i
$$334$$ −3.76002e7 −0.0552176
$$335$$ 0 0
$$336$$ −3.75740e8 −0.540381
$$337$$ 6.29093e8i 0.895385i 0.894187 + 0.447693i $$0.147754\pi$$
−0.894187 + 0.447693i $$0.852246\pi$$
$$338$$ 1.07867e9i 1.51943i
$$339$$ 1.41352e9 1.97062
$$340$$ 0 0
$$341$$ 1.50333e8 0.205312
$$342$$ − 4.17386e8i − 0.564218i
$$343$$ − 4.03536e7i − 0.0539949i
$$344$$ 1.90388e8 0.252165
$$345$$ 0 0
$$346$$ 1.17573e9 1.52595
$$347$$ 7.61715e8i 0.978676i 0.872094 + 0.489338i $$0.162762\pi$$
−0.872094 + 0.489338i $$0.837238\pi$$
$$348$$ 1.04375e9i 1.32761i
$$349$$ 3.31639e8 0.417616 0.208808 0.977957i $$-0.433042\pi$$
0.208808 + 0.977957i $$0.433042\pi$$
$$350$$ 0 0
$$351$$ −8.77657e8 −1.08330
$$352$$ − 1.38601e9i − 1.69381i
$$353$$ 7.57419e8i 0.916484i 0.888828 + 0.458242i $$0.151521\pi$$
−0.888828 + 0.458242i $$0.848479\pi$$
$$354$$ 3.62326e8 0.434099
$$355$$ 0 0
$$356$$ −6.31329e8 −0.741619
$$357$$ − 2.53006e8i − 0.294302i
$$358$$ 1.46197e9i 1.68402i
$$359$$ −1.46796e9 −1.67449 −0.837246 0.546827i $$-0.815836\pi$$
−0.837246 + 0.546827i $$0.815836\pi$$
$$360$$ 0 0
$$361$$ 3.28013e8 0.366958
$$362$$ 1.57182e9i 1.74150i
$$363$$ − 1.22865e9i − 1.34820i
$$364$$ −3.45116e8 −0.375068
$$365$$ 0 0
$$366$$ 6.66903e8 0.711015
$$367$$ − 1.64615e9i − 1.73835i −0.494502 0.869177i $$-0.664649\pi$$
0.494502 0.869177i $$-0.335351\pi$$
$$368$$ − 1.55586e9i − 1.62744i
$$369$$ −1.56521e8 −0.162174
$$370$$ 0 0
$$371$$ 3.65824e8 0.371932
$$372$$ − 1.09609e8i − 0.110394i
$$373$$ − 1.16387e9i − 1.16124i −0.814173 0.580622i $$-0.802809\pi$$
0.814173 0.580622i $$-0.197191\pi$$
$$374$$ 1.27512e9 1.26037
$$375$$ 0 0
$$376$$ 1.47497e8 0.143096
$$377$$ − 2.58323e9i − 2.48295i
$$378$$ − 3.77098e8i − 0.359114i
$$379$$ −4.07762e8 −0.384742 −0.192371 0.981322i $$-0.561618\pi$$
−0.192371 + 0.981322i $$0.561618\pi$$
$$380$$ 0 0
$$381$$ 3.70273e8 0.342993
$$382$$ 3.23519e8i 0.296946i
$$383$$ 7.10345e8i 0.646061i 0.946389 + 0.323031i $$0.104702\pi$$
−0.946389 + 0.323031i $$0.895298\pi$$
$$384$$ 1.04129e9 0.938454
$$385$$ 0 0
$$386$$ −2.19196e9 −1.93989
$$387$$ − 2.53571e8i − 0.222388i
$$388$$ − 2.05576e8i − 0.178674i
$$389$$ −1.95091e8 −0.168040 −0.0840202 0.996464i $$-0.526776\pi$$
−0.0840202 + 0.996464i $$0.526776\pi$$
$$390$$ 0 0
$$391$$ 1.04765e9 0.886333
$$392$$ 7.20794e7i 0.0604380i
$$393$$ 1.21141e8i 0.100674i
$$394$$ 8.34252e8 0.687164
$$395$$ 0 0
$$396$$ −4.54983e8 −0.368181
$$397$$ − 1.58231e9i − 1.26919i −0.772846 0.634593i $$-0.781168\pi$$
0.772846 0.634593i $$-0.218832\pi$$
$$398$$ − 1.00115e9i − 0.795991i
$$399$$ −6.57031e8 −0.517822
$$400$$ 0 0
$$401$$ −2.13647e9 −1.65460 −0.827298 0.561763i $$-0.810123\pi$$
−0.827298 + 0.561763i $$0.810123\pi$$
$$402$$ 1.81553e9i 1.39384i
$$403$$ 2.71276e8i 0.206464i
$$404$$ 1.65471e9 1.24849
$$405$$ 0 0
$$406$$ 1.10992e9 0.823096
$$407$$ − 2.73441e9i − 2.01040i
$$408$$ 4.51919e8i 0.329420i
$$409$$ 4.05635e8 0.293159 0.146580 0.989199i $$-0.453174\pi$$
0.146580 + 0.989199i $$0.453174\pi$$
$$410$$ 0 0
$$411$$ 3.89395e8 0.276658
$$412$$ 3.83596e8i 0.270230i
$$413$$ − 1.54980e8i − 0.108256i
$$414$$ −9.29347e8 −0.643690
$$415$$ 0 0
$$416$$ 2.50105e9 1.70332
$$417$$ 5.05168e8i 0.341161i
$$418$$ − 3.31134e9i − 2.21762i
$$419$$ −1.07475e9 −0.713771 −0.356886 0.934148i $$-0.616161\pi$$
−0.356886 + 0.934148i $$0.616161\pi$$
$$420$$ 0 0
$$421$$ −8.32900e8 −0.544009 −0.272004 0.962296i $$-0.587686\pi$$
−0.272004 + 0.962296i $$0.587686\pi$$
$$422$$ 1.99306e9i 1.29100i
$$423$$ − 1.96446e8i − 0.126198i
$$424$$ −6.53434e8 −0.416314
$$425$$ 0 0
$$426$$ 1.78050e9 1.11586
$$427$$ − 2.85259e8i − 0.177313i
$$428$$ 6.55796e8i 0.404311i
$$429$$ 4.14411e9 2.53414
$$430$$ 0 0
$$431$$ 8.26292e6 0.00497122 0.00248561 0.999997i $$-0.499209\pi$$
0.00248561 + 0.999997i $$0.499209\pi$$
$$432$$ 1.50188e9i 0.896277i
$$433$$ − 3.10619e9i − 1.83874i −0.393396 0.919369i $$-0.628700\pi$$
0.393396 0.919369i $$-0.371300\pi$$
$$434$$ −1.16558e8 −0.0684426
$$435$$ 0 0
$$436$$ −1.72004e9 −0.993885
$$437$$ − 2.72063e9i − 1.55950i
$$438$$ − 4.00448e9i − 2.27712i
$$439$$ 1.28295e9 0.723742 0.361871 0.932228i $$-0.382138\pi$$
0.361871 + 0.932228i $$0.382138\pi$$
$$440$$ 0 0
$$441$$ 9.59998e7 0.0533010
$$442$$ 2.30095e9i 1.26745i
$$443$$ − 1.73263e9i − 0.946878i −0.880827 0.473439i $$-0.843012\pi$$
0.880827 0.473439i $$-0.156988\pi$$
$$444$$ −1.99368e9 −1.08098
$$445$$ 0 0
$$446$$ −2.16175e9 −1.15381
$$447$$ 7.30464e8i 0.386832i
$$448$$ 1.96964e8i 0.103493i
$$449$$ −1.86903e9 −0.974439 −0.487219 0.873280i $$-0.661989\pi$$
−0.487219 + 0.873280i $$0.661989\pi$$
$$450$$ 0 0
$$451$$ −1.24176e9 −0.637413
$$452$$ − 2.22172e9i − 1.13163i
$$453$$ 3.51780e9i 1.77798i
$$454$$ −4.71816e9 −2.36634
$$455$$ 0 0
$$456$$ 1.17359e9 0.579613
$$457$$ − 1.76868e9i − 0.866849i −0.901190 0.433425i $$-0.857305\pi$$
0.901190 0.433425i $$-0.142695\pi$$
$$458$$ 4.53679e9i 2.20658i
$$459$$ −1.01130e9 −0.488129
$$460$$ 0 0
$$461$$ −2.55825e8 −0.121616 −0.0608078 0.998149i $$-0.519368\pi$$
−0.0608078 + 0.998149i $$0.519368\pi$$
$$462$$ 1.78057e9i 0.840065i
$$463$$ − 4.19121e9i − 1.96249i −0.192777 0.981243i $$-0.561750\pi$$
0.192777 0.981243i $$-0.438250\pi$$
$$464$$ −4.42052e9 −2.05428
$$465$$ 0 0
$$466$$ 2.63998e9 1.20851
$$467$$ 2.94239e9i 1.33688i 0.743768 + 0.668438i $$0.233037\pi$$
−0.743768 + 0.668438i $$0.766963\pi$$
$$468$$ − 8.21018e8i − 0.370247i
$$469$$ 7.76569e8 0.347596
$$470$$ 0 0
$$471$$ −4.03651e9 −1.78005
$$472$$ 2.76825e8i 0.121174i
$$473$$ − 2.01171e9i − 0.874079i
$$474$$ −2.18735e9 −0.943396
$$475$$ 0 0
$$476$$ −3.97667e8 −0.169003
$$477$$ 8.70283e8i 0.367152i
$$478$$ 5.36292e9i 2.24597i
$$479$$ −4.57933e9 −1.90383 −0.951914 0.306366i $$-0.900887\pi$$
−0.951914 + 0.306366i $$0.900887\pi$$
$$480$$ 0 0
$$481$$ 4.93425e9 2.02169
$$482$$ − 3.21535e9i − 1.30787i
$$483$$ 1.46294e9i 0.590760i
$$484$$ −1.93115e9 −0.774206
$$485$$ 0 0
$$486$$ 2.32813e9 0.919984
$$487$$ 1.40131e9i 0.549771i 0.961477 + 0.274886i $$0.0886400\pi$$
−0.961477 + 0.274886i $$0.911360\pi$$
$$488$$ 5.09528e8i 0.198472i
$$489$$ −1.92150e9 −0.743121
$$490$$ 0 0
$$491$$ −4.79712e9 −1.82892 −0.914462 0.404672i $$-0.867386\pi$$
−0.914462 + 0.404672i $$0.867386\pi$$
$$492$$ 9.05381e8i 0.342731i
$$493$$ − 2.97658e9i − 1.11880i
$$494$$ 5.97533e9 2.23007
$$495$$ 0 0
$$496$$ 4.64218e8 0.170819
$$497$$ − 7.61585e8i − 0.278273i
$$498$$ 5.11923e9i 1.85739i
$$499$$ 2.01049e9 0.724351 0.362176 0.932110i $$-0.382034\pi$$
0.362176 + 0.932110i $$0.382034\pi$$
$$500$$ 0 0
$$501$$ −1.40807e8 −0.0500258
$$502$$ 1.90050e9i 0.670509i
$$503$$ 1.68618e9i 0.590766i 0.955379 + 0.295383i $$0.0954472\pi$$
−0.955379 + 0.295383i $$0.904553\pi$$
$$504$$ −1.71474e8 −0.0596613
$$505$$ 0 0
$$506$$ −7.37300e9 −2.52998
$$507$$ 4.03947e9i 1.37656i
$$508$$ − 5.81983e8i − 0.196964i
$$509$$ −1.63477e9 −0.549470 −0.274735 0.961520i $$-0.588590\pi$$
−0.274735 + 0.961520i $$0.588590\pi$$
$$510$$ 0 0
$$511$$ −1.71286e9 −0.567871
$$512$$ − 2.71225e9i − 0.893068i
$$513$$ 2.62624e9i 0.858862i
$$514$$ 4.31147e9 1.40041
$$515$$ 0 0
$$516$$ −1.46676e9 −0.469985
$$517$$ − 1.55851e9i − 0.496012i
$$518$$ 2.12007e9i 0.670187i
$$519$$ 4.40294e9 1.38247
$$520$$ 0 0
$$521$$ 3.28595e9 1.01796 0.508978 0.860779i $$-0.330023\pi$$
0.508978 + 0.860779i $$0.330023\pi$$
$$522$$ 2.64046e9i 0.812518i
$$523$$ 1.29734e9i 0.396549i 0.980147 + 0.198274i $$0.0635337\pi$$
−0.980147 + 0.198274i $$0.936466\pi$$
$$524$$ 1.90406e8 0.0578123
$$525$$ 0 0
$$526$$ 4.59418e9 1.37644
$$527$$ 3.12583e8i 0.0930313i
$$528$$ − 7.09155e9i − 2.09663i
$$529$$ −2.65291e9 −0.779161
$$530$$ 0 0
$$531$$ 3.68693e8 0.106865
$$532$$ 1.03270e9i 0.297360i
$$533$$ − 2.24076e9i − 0.640990i
$$534$$ −5.87771e9 −1.67038
$$535$$ 0 0
$$536$$ −1.38710e9 −0.389074
$$537$$ 5.47487e9i 1.52568i
$$538$$ − 4.31308e9i − 1.19412i
$$539$$ 7.61617e8 0.209496
$$540$$ 0 0
$$541$$ 1.23726e9 0.335948 0.167974 0.985791i $$-0.446278\pi$$
0.167974 + 0.985791i $$0.446278\pi$$
$$542$$ 1.23986e9i 0.334484i
$$543$$ 5.88625e9i 1.57775i
$$544$$ 2.88189e9 0.767505
$$545$$ 0 0
$$546$$ −3.21305e9 −0.844779
$$547$$ 4.27288e9i 1.11626i 0.829754 + 0.558130i $$0.188481\pi$$
−0.829754 + 0.558130i $$0.811519\pi$$
$$548$$ − 6.12037e8i − 0.158871i
$$549$$ 6.78621e8 0.175035
$$550$$ 0 0
$$551$$ −7.72986e9 −1.96853
$$552$$ − 2.61309e9i − 0.661253i
$$553$$ 9.35610e8i 0.235265i
$$554$$ −2.88141e9 −0.719982
$$555$$ 0 0
$$556$$ 7.94006e8 0.195912
$$557$$ 2.41921e9i 0.593171i 0.955006 + 0.296586i $$0.0958480\pi$$
−0.955006 + 0.296586i $$0.904152\pi$$
$$558$$ − 2.77286e8i − 0.0675630i
$$559$$ 3.63013e9 0.878985
$$560$$ 0 0
$$561$$ 4.77513e9 1.14187
$$562$$ − 4.86306e9i − 1.15567i
$$563$$ − 3.03839e9i − 0.717570i −0.933420 0.358785i $$-0.883191\pi$$
0.933420 0.358785i $$-0.116809\pi$$
$$564$$ −1.13632e9 −0.266701
$$565$$ 0 0
$$566$$ 6.44503e9 1.49406
$$567$$ − 2.02428e9i − 0.466370i
$$568$$ 1.36034e9i 0.311478i
$$569$$ −2.85539e9 −0.649788 −0.324894 0.945750i $$-0.605329\pi$$
−0.324894 + 0.945750i $$0.605329\pi$$
$$570$$ 0 0
$$571$$ 1.87867e9 0.422304 0.211152 0.977453i $$-0.432279\pi$$
0.211152 + 0.977453i $$0.432279\pi$$
$$572$$ − 6.51356e9i − 1.45523i
$$573$$ 1.21153e9i 0.269026i
$$574$$ 9.62776e8 0.212488
$$575$$ 0 0
$$576$$ −4.68569e8 −0.102163
$$577$$ − 2.19182e9i − 0.474996i −0.971388 0.237498i $$-0.923673\pi$$
0.971388 0.237498i $$-0.0763273\pi$$
$$578$$ − 3.35327e9i − 0.722306i
$$579$$ −8.20858e9 −1.75749
$$580$$ 0 0
$$581$$ 2.18969e9 0.463196
$$582$$ − 1.91393e9i − 0.402435i
$$583$$ 6.90441e9i 1.44307i
$$584$$ 3.05951e9 0.635633
$$585$$ 0 0
$$586$$ −4.46399e9 −0.916392
$$587$$ 4.70415e9i 0.959949i 0.877283 + 0.479974i $$0.159354\pi$$
−0.877283 + 0.479974i $$0.840646\pi$$
$$588$$ − 5.55302e8i − 0.112644i
$$589$$ 8.11747e8 0.163688
$$590$$ 0 0
$$591$$ 3.12416e9 0.622554
$$592$$ − 8.44368e9i − 1.67265i
$$593$$ 3.66996e9i 0.722719i 0.932427 + 0.361359i $$0.117687\pi$$
−0.932427 + 0.361359i $$0.882313\pi$$
$$594$$ 7.11718e9 1.39333
$$595$$ 0 0
$$596$$ 1.14812e9 0.222139
$$597$$ − 3.74917e9i − 0.721149i
$$598$$ − 1.33046e10i − 2.54418i
$$599$$ 5.46935e9 1.03978 0.519890 0.854233i $$-0.325973\pi$$
0.519890 + 0.854233i $$0.325973\pi$$
$$600$$ 0 0
$$601$$ 2.74417e9 0.515645 0.257822 0.966192i $$-0.416995\pi$$
0.257822 + 0.966192i $$0.416995\pi$$
$$602$$ 1.55974e9i 0.291383i
$$603$$ 1.84743e9i 0.343129i
$$604$$ 5.52915e9 1.02101
$$605$$ 0 0
$$606$$ 1.54054e10 2.81203
$$607$$ 2.26388e9i 0.410859i 0.978672 + 0.205430i $$0.0658591\pi$$
−0.978672 + 0.205430i $$0.934141\pi$$
$$608$$ − 7.48397e9i − 1.35042i
$$609$$ 4.15650e9 0.745705
$$610$$ 0 0
$$611$$ 2.81233e9 0.498795
$$612$$ − 9.46035e8i − 0.166831i
$$613$$ − 9.92731e9i − 1.74068i −0.492448 0.870342i $$-0.663898\pi$$
0.492448 0.870342i $$-0.336102\pi$$
$$614$$ 3.53563e9 0.616422
$$615$$ 0 0
$$616$$ −1.36040e9 −0.234495
$$617$$ − 2.27156e9i − 0.389338i −0.980869 0.194669i $$-0.937637\pi$$
0.980869 0.194669i $$-0.0623632\pi$$
$$618$$ 3.57131e9i 0.608650i
$$619$$ −6.90552e8 −0.117025 −0.0585126 0.998287i $$-0.518636\pi$$
−0.0585126 + 0.998287i $$0.518636\pi$$
$$620$$ 0 0
$$621$$ 5.84755e9 0.979836
$$622$$ 9.83015e8i 0.163792i
$$623$$ 2.51412e9i 0.416560i
$$624$$ 1.27967e10 2.10840
$$625$$ 0 0
$$626$$ −9.87867e9 −1.60949
$$627$$ − 1.24005e10i − 2.00911i
$$628$$ 6.34445e9i 1.02220i
$$629$$ 5.68560e9 0.910959
$$630$$ 0 0
$$631$$ −1.00992e10 −1.60023 −0.800116 0.599846i $$-0.795229\pi$$
−0.800116 + 0.599846i $$0.795229\pi$$
$$632$$ − 1.67118e9i − 0.263338i
$$633$$ 7.46375e9i 1.16962i
$$634$$ 6.80665e9 1.06077
$$635$$ 0 0
$$636$$ 5.03407e9 0.775925
$$637$$ 1.37434e9i 0.210672i
$$638$$ 2.09482e10i 3.19355i
$$639$$ 1.81178e9 0.274697
$$640$$ 0 0
$$641$$ 8.50418e8 0.127535 0.0637675 0.997965i $$-0.479688\pi$$
0.0637675 + 0.997965i $$0.479688\pi$$
$$642$$ 6.10550e9i 0.910644i
$$643$$ 1.56624e9i 0.232339i 0.993229 + 0.116169i $$0.0370615\pi$$
−0.993229 + 0.116169i $$0.962938\pi$$
$$644$$ 2.29940e9 0.339245
$$645$$ 0 0
$$646$$ 6.88520e9 1.00485
$$647$$ − 1.06053e10i − 1.53942i −0.638391 0.769712i $$-0.720400\pi$$
0.638391 0.769712i $$-0.279600\pi$$
$$648$$ 3.61576e9i 0.522020i
$$649$$ 2.92503e9 0.420024
$$650$$ 0 0
$$651$$ −4.36492e8 −0.0620073
$$652$$ 3.02015e9i 0.426739i
$$653$$ − 1.83644e9i − 0.258096i −0.991638 0.129048i $$-0.958808\pi$$
0.991638 0.129048i $$-0.0411921\pi$$
$$654$$ −1.60137e10 −2.23856
$$655$$ 0 0
$$656$$ −3.83448e9 −0.530327
$$657$$ − 4.07484e9i − 0.560573i
$$658$$ 1.20836e9i 0.165350i
$$659$$ 5.03583e9 0.685444 0.342722 0.939437i $$-0.388651\pi$$
0.342722 + 0.939437i $$0.388651\pi$$
$$660$$ 0 0
$$661$$ −1.27453e10 −1.71651 −0.858254 0.513225i $$-0.828451\pi$$
−0.858254 + 0.513225i $$0.828451\pi$$
$$662$$ − 3.94774e9i − 0.528866i
$$663$$ 8.61674e9i 1.14827i
$$664$$ −3.91121e9 −0.518469
$$665$$ 0 0
$$666$$ −5.04357e9 −0.661574
$$667$$ 1.72112e10i 2.24580i
$$668$$ 2.21316e8i 0.0287274i
$$669$$ −8.09544e9 −1.04532
$$670$$ 0 0
$$671$$ 5.38386e9 0.687962
$$672$$ 4.02427e9i 0.511558i
$$673$$ 1.02193e10i 1.29231i 0.763206 + 0.646155i $$0.223624\pi$$
−0.763206 + 0.646155i $$0.776376\pi$$
$$674$$ 9.20567e9 1.15810
$$675$$ 0 0
$$676$$ 6.34910e9 0.790494
$$677$$ − 1.04119e10i − 1.28965i −0.764332 0.644823i $$-0.776931\pi$$
0.764332 0.644823i $$-0.223069\pi$$
$$678$$ − 2.06844e10i − 2.54882i
$$679$$ −8.18659e8 −0.100360
$$680$$ 0 0
$$681$$ −1.76688e10 −2.14384
$$682$$ − 2.19986e9i − 0.265552i
$$683$$ 6.56705e9i 0.788675i 0.918966 + 0.394338i $$0.129026\pi$$
−0.918966 + 0.394338i $$0.870974\pi$$
$$684$$ −2.45675e9 −0.293539
$$685$$ 0 0
$$686$$ −5.90504e8 −0.0698375
$$687$$ 1.69897e10i 1.99911i
$$688$$ − 6.21203e9i − 0.727233i
$$689$$ −1.24590e10 −1.45117
$$690$$ 0 0
$$691$$ 4.44242e9 0.512208 0.256104 0.966649i $$-0.417561\pi$$
0.256104 + 0.966649i $$0.417561\pi$$
$$692$$ − 6.92039e9i − 0.793888i
$$693$$ 1.81186e9i 0.206804i
$$694$$ 1.11464e10 1.26583
$$695$$ 0 0
$$696$$ −7.42431e9 −0.834688
$$697$$ − 2.58197e9i − 0.288826i
$$698$$ − 4.85296e9i − 0.540148i
$$699$$ 9.88638e9 1.09488
$$700$$ 0 0
$$701$$ −7.92343e9 −0.868761 −0.434380 0.900729i $$-0.643033\pi$$
−0.434380 + 0.900729i $$0.643033\pi$$
$$702$$ 1.28430e10i 1.40115i
$$703$$ − 1.47649e10i − 1.60283i
$$704$$ −3.71741e9 −0.401546
$$705$$ 0 0
$$706$$ 1.10835e10 1.18539
$$707$$ − 6.58948e9i − 0.701266i
$$708$$ − 2.13267e9i − 0.225843i
$$709$$ −9.27216e9 −0.977055 −0.488528 0.872548i $$-0.662466\pi$$
−0.488528 + 0.872548i $$0.662466\pi$$
$$710$$ 0 0
$$711$$ −2.22578e9 −0.232241
$$712$$ − 4.49070e9i − 0.466266i
$$713$$ − 1.80743e9i − 0.186744i
$$714$$ −3.70230e9 −0.380652
$$715$$ 0 0
$$716$$ 8.60522e9 0.876126
$$717$$ 2.00834e10i 2.03479i
$$718$$ 2.14810e10i 2.16580i
$$719$$ 5.48244e9 0.550076 0.275038 0.961433i $$-0.411310\pi$$
0.275038 + 0.961433i $$0.411310\pi$$
$$720$$ 0 0
$$721$$ 1.52758e9 0.151786
$$722$$ − 4.79990e9i − 0.474626i
$$723$$ − 1.20410e10i − 1.18489i
$$724$$ 9.25181e9 0.906029
$$725$$ 0 0
$$726$$ −1.79791e10 −1.74377
$$727$$ 9.22691e9i 0.890606i 0.895380 + 0.445303i $$0.146904\pi$$
−0.895380 + 0.445303i $$0.853096\pi$$
$$728$$ − 2.45484e9i − 0.235811i
$$729$$ −4.18848e9 −0.400415
$$730$$ 0 0
$$731$$ 4.18290e9 0.396065
$$732$$ − 3.92542e9i − 0.369911i
$$733$$ 4.96865e9i 0.465988i 0.972478 + 0.232994i $$0.0748523\pi$$
−0.972478 + 0.232994i $$0.925148\pi$$
$$734$$ −2.40885e10 −2.24840
$$735$$ 0 0
$$736$$ −1.66637e10 −1.54063
$$737$$ 1.46566e10i 1.34865i
$$738$$ 2.29041e9i 0.209757i
$$739$$ 1.96084e9 0.178726 0.0893628 0.995999i $$-0.471517\pi$$
0.0893628 + 0.995999i $$0.471517\pi$$
$$740$$ 0 0
$$741$$ 2.23768e10 2.02038
$$742$$ − 5.35320e9i − 0.481060i
$$743$$ 9.56947e9i 0.855908i 0.903801 + 0.427954i $$0.140765\pi$$
−0.903801 + 0.427954i $$0.859235\pi$$
$$744$$ 7.79660e8 0.0694064
$$745$$ 0 0
$$746$$ −1.70312e10 −1.50196
$$747$$ 5.20918e9i 0.457244i
$$748$$ − 7.50539e9i − 0.655719i
$$749$$ 2.61155e9 0.227097
$$750$$ 0 0
$$751$$ 8.11719e9 0.699304 0.349652 0.936880i $$-0.386300\pi$$
0.349652 + 0.936880i $$0.386300\pi$$
$$752$$ − 4.81257e9i − 0.412681i
$$753$$ 7.11710e9i 0.607464i
$$754$$ −3.78010e10 −3.21147
$$755$$ 0 0
$$756$$ −2.21961e9 −0.186832
$$757$$ 9.11117e9i 0.763376i 0.924291 + 0.381688i $$0.124657\pi$$
−0.924291 + 0.381688i $$0.875343\pi$$
$$758$$ 5.96689e9i 0.497629i
$$759$$ −2.76109e10 −2.29210
$$760$$ 0 0
$$761$$ 1.71359e10 1.40948 0.704742 0.709464i $$-0.251063\pi$$
0.704742 + 0.709464i $$0.251063\pi$$
$$762$$ − 5.41830e9i − 0.443630i
$$763$$ 6.84965e9i 0.558255i
$$764$$ 1.90425e9 0.154489
$$765$$ 0 0
$$766$$ 1.03947e10 0.835621
$$767$$ 5.27823e9i 0.422381i
$$768$$ − 1.92654e10i − 1.53466i
$$769$$ 1.82316e10 1.44572 0.722858 0.690997i $$-0.242828\pi$$
0.722858 + 0.690997i $$0.242828\pi$$
$$770$$ 0 0
$$771$$ 1.61458e10 1.26873
$$772$$ 1.29020e10i 1.00924i
$$773$$ − 1.45965e10i − 1.13663i −0.822810 0.568316i $$-0.807595\pi$$
0.822810 0.568316i $$-0.192405\pi$$
$$774$$ −3.71056e9 −0.287638
$$775$$ 0 0
$$776$$ 1.46228e9 0.112335
$$777$$ 7.93937e9i 0.607173i
$$778$$ 2.85481e9i 0.217345i
$$779$$ −6.70510e9 −0.508188
$$780$$ 0 0
$$781$$ 1.43738e10 1.07968
$$782$$ − 1.53305e10i − 1.14639i
$$783$$ − 1.66141e10i − 1.23683i
$$784$$ 2.35182e9 0.174300
$$785$$ 0 0
$$786$$ 1.77269e9 0.130213
$$787$$ − 1.42358e10i − 1.04104i −0.853848 0.520522i $$-0.825737\pi$$
0.853848 0.520522i $$-0.174263\pi$$
$$788$$ − 4.91045e9i − 0.357503i
$$789$$ 1.72046e10 1.24702
$$790$$ 0 0
$$791$$ −8.84748e9 −0.635627
$$792$$ − 3.23633e9i − 0.231481i
$$793$$ 9.71519e9i 0.691823i
$$794$$ −2.31544e10 −1.64158
$$795$$ 0 0
$$796$$ −5.89281e9 −0.414121
$$797$$ − 1.03325e10i − 0.722935i −0.932385 0.361467i $$-0.882276\pi$$
0.932385 0.361467i $$-0.117724\pi$$
$$798$$ 9.61450e9i 0.669756i
$$799$$ 3.24057e9 0.224754
$$800$$ 0 0
$$801$$ −5.98099e9 −0.411206
$$802$$ 3.12635e10i 2.14007i
$$803$$ − 3.23278e10i − 2.20329i
$$804$$ 1.06863e10 0.725155
$$805$$ 0 0
$$806$$ 3.96965e9 0.267042
$$807$$ − 1.61519e10i − 1.08185i
$$808$$ 1.17701e10i 0.784947i
$$809$$ 1.58064e10 1.04957 0.524787 0.851233i $$-0.324145\pi$$
0.524787 + 0.851233i $$0.324145\pi$$
$$810$$ 0 0
$$811$$ 2.87547e9 0.189294 0.0946469 0.995511i $$-0.469828\pi$$
0.0946469 + 0.995511i $$0.469828\pi$$
$$812$$ − 6.53304e9i − 0.428222i
$$813$$ 4.64311e9i 0.303034i
$$814$$ −4.00133e10 −2.60027
$$815$$ 0 0
$$816$$ 1.47453e10 0.950032
$$817$$ − 1.08626e10i − 0.696875i
$$818$$ − 5.93575e9i − 0.379175i
$$819$$ −3.26951e9 −0.207964
$$820$$ 0 0
$$821$$ −1.42014e10 −0.895633 −0.447817 0.894125i $$-0.647798\pi$$
−0.447817 + 0.894125i $$0.647798\pi$$
$$822$$ − 5.69811e9i − 0.357832i
$$823$$ − 2.79354e10i − 1.74685i −0.486961 0.873424i $$-0.661895\pi$$
0.486961 0.873424i $$-0.338105\pi$$
$$824$$ −2.72856e9 −0.169898
$$825$$ 0 0
$$826$$ −2.26787e9 −0.140019
$$827$$ 1.48560e8i 0.00913341i 0.999990 + 0.00456670i $$0.00145363\pi$$
−0.999990 + 0.00456670i $$0.998546\pi$$
$$828$$ 5.47018e9i 0.334885i
$$829$$ −3.98861e9 −0.243154 −0.121577 0.992582i $$-0.538795\pi$$
−0.121577 + 0.992582i $$0.538795\pi$$
$$830$$ 0 0
$$831$$ −1.07905e10 −0.652286
$$832$$ − 6.70807e9i − 0.403800i
$$833$$ 1.58361e9i 0.0949273i
$$834$$ 7.39225e9 0.441261
$$835$$ 0 0
$$836$$ −1.94907e10 −1.15373
$$837$$ 1.74471e9i 0.102845i
$$838$$ 1.57271e10i 0.923198i
$$839$$ −2.43693e9 −0.142455 −0.0712273 0.997460i $$-0.522692\pi$$
−0.0712273 + 0.997460i $$0.522692\pi$$
$$840$$ 0 0
$$841$$ 3.16506e10 1.83483
$$842$$ 1.21880e10i 0.703625i
$$843$$ − 1.82115e10i − 1.04701i
$$844$$ 1.17313e10 0.671655
$$845$$ 0 0
$$846$$ −2.87464e9 −0.163225
$$847$$ 7.69033e9i 0.434863i
$$848$$ 2.13204e10i 1.20063i
$$849$$ 2.41357e10 1.35358
$$850$$ 0 0
$$851$$ −3.28754e10 −1.82859
$$852$$ − 1.04801e10i − 0.580533i
$$853$$ 1.54310e9i 0.0851282i 0.999094 + 0.0425641i $$0.0135527\pi$$
−0.999094 + 0.0425641i $$0.986447\pi$$
$$854$$ −4.17427e9 −0.229339
$$855$$ 0 0
$$856$$ −4.66474e9 −0.254196
$$857$$ 1.29972e10i 0.705369i 0.935742 + 0.352684i $$0.114731\pi$$
−0.935742 + 0.352684i $$0.885269\pi$$
$$858$$ − 6.06417e10i − 3.27768i
$$859$$ 1.98316e10 1.06754 0.533768 0.845631i $$-0.320776\pi$$
0.533768 + 0.845631i $$0.320776\pi$$
$$860$$ 0 0
$$861$$ 3.60546e9 0.192509
$$862$$ − 1.20913e8i − 0.00642982i
$$863$$ 4.94264e9i 0.261771i 0.991397 + 0.130886i $$0.0417820\pi$$
−0.991397 + 0.130886i $$0.958218\pi$$
$$864$$ 1.60855e10 0.848472
$$865$$ 0 0
$$866$$ −4.54536e10 −2.37824
$$867$$ − 1.25575e10i − 0.654391i
$$868$$ 6.86063e8i 0.0356078i
$$869$$ −1.76583e10 −0.912809
$$870$$ 0 0
$$871$$ −2.64480e10 −1.35621
$$872$$ − 1.22348e10i − 0.624870i
$$873$$ − 1.94756e9i − 0.0990697i
$$874$$ −3.98117e10 −2.01707
$$875$$ 0 0
$$876$$ −2.35705e10 −1.18469
$$877$$ 7.37011e9i 0.368957i 0.982837 + 0.184478i $$0.0590596\pi$$
−0.982837 + 0.184478i $$0.940940\pi$$
$$878$$ − 1.87737e10i − 0.936094i
$$879$$ −1.67170e10 −0.830229
$$880$$ 0 0
$$881$$ 9.74385e9 0.480081 0.240041 0.970763i $$-0.422839\pi$$
0.240041 + 0.970763i $$0.422839\pi$$
$$882$$ − 1.40479e9i − 0.0689400i
$$883$$ − 2.79540e10i − 1.36641i −0.730226 0.683206i $$-0.760585\pi$$
0.730226 0.683206i $$-0.239415\pi$$
$$884$$ 1.35435e10 0.659399
$$885$$ 0 0
$$886$$ −2.53541e10 −1.22470
$$887$$ 2.08176e10i 1.00161i 0.865561 + 0.500804i $$0.166962\pi$$
−0.865561 + 0.500804i $$0.833038\pi$$
$$888$$ − 1.41813e10i − 0.679626i
$$889$$ −2.31761e9 −0.110633
$$890$$ 0 0
$$891$$ 3.82054e10 1.80948
$$892$$ 1.27241e10i 0.600276i
$$893$$ − 8.41542e9i − 0.395454i
$$894$$ 1.06891e10 0.500332
$$895$$ 0 0
$$896$$ −6.51763e9 −0.302700
$$897$$ − 4.98239e10i − 2.30496i
$$898$$ 2.73500e10i 1.26035i
$$899$$ −5.13526e9 −0.235724
$$900$$ 0 0
$$901$$ −1.43562e10 −0.653886
$$902$$ 1.81710e10i 0.824435i
$$903$$ 5.84100e9i 0.263986i
$$904$$ 1.58033e10 0.711474
$$905$$ 0 0
$$906$$ 5.14768e10 2.29965
$$907$$ − 3.61602e10i − 1.60918i −0.593830 0.804591i $$-0.702385\pi$$
0.593830 0.804591i $$-0.297615\pi$$
$$908$$ 2.77713e10i 1.23111i
$$909$$ 1.56761e10 0.692254
$$910$$ 0 0
$$911$$ 1.79811e10 0.787954 0.393977 0.919120i $$-0.371099\pi$$
0.393977 + 0.919120i $$0.371099\pi$$
$$912$$ − 3.82920e10i − 1.67158i
$$913$$ 4.13272e10i 1.79717i
$$914$$ −2.58816e10 −1.12119
$$915$$ 0 0
$$916$$ 2.67038e10 1.14799
$$917$$ − 7.58245e8i − 0.0324726i
$$918$$ 1.47986e10i 0.631351i
$$919$$ 1.01373e10 0.430841 0.215421 0.976521i $$-0.430888\pi$$
0.215421 + 0.976521i $$0.430888\pi$$
$$920$$ 0 0
$$921$$ 1.32405e10 0.558463
$$922$$ 3.74355e9i 0.157299i
$$923$$ 2.59376e10i 1.08574i
$$924$$ 1.04805e10 0.437050
$$925$$ 0 0
$$926$$ −6.13311e10 −2.53830
$$927$$ 3.63406e9i 0.149835i
$$928$$ 4.73449e10i 1.94471i
$$929$$ 2.39605e10 0.980485 0.490242 0.871586i $$-0.336908\pi$$
0.490242 + 0.871586i $$0.336908\pi$$
$$930$$ 0 0
$$931$$ 4.11248e9 0.167024
$$932$$ − 1.55391e10i − 0.628738i
$$933$$ 3.68125e9i 0.148392i
$$934$$ 4.30567e10 1.72913
$$935$$ 0 0
$$936$$ 5.83998e9 0.232780
$$937$$ − 1.16627e10i − 0.463138i −0.972818 0.231569i $$-0.925614\pi$$
0.972818 0.231569i $$-0.0743859\pi$$
$$938$$ − 1.13637e10i − 0.449584i
$$939$$ −3.69943e10 −1.45816
$$940$$ 0 0
$$941$$ 3.03134e10 1.18596 0.592982 0.805216i $$-0.297951\pi$$
0.592982 + 0.805216i $$0.297951\pi$$
$$942$$ 5.90672e10i 2.30233i
$$943$$ 1.49295e10i 0.579768i
$$944$$ 9.03231e9 0.349460
$$945$$ 0 0
$$946$$ −2.94378e10 −1.13054
$$947$$ − 2.84339e10i − 1.08796i −0.839099 0.543979i $$-0.816917\pi$$
0.839099 0.543979i $$-0.183083\pi$$
$$948$$ 1.28748e10i 0.490809i
$$949$$ 5.83357e10 2.21566
$$950$$ 0 0
$$951$$ 2.54900e10 0.961031
$$952$$ − 2.82864e9i − 0.106255i
$$953$$ − 4.09127e10i − 1.53120i −0.643315 0.765602i $$-0.722441\pi$$
0.643315 0.765602i $$-0.277559\pi$$
$$954$$ 1.27351e10 0.474878
$$955$$ 0 0
$$956$$ 3.15664e10 1.16848
$$957$$ 7.84479e10i 2.89327i
$$958$$ 6.70105e10i 2.46243i
$$959$$ −2.43729e9 −0.0892365
$$960$$ 0 0
$$961$$ −2.69733e10 −0.980399
$$962$$ − 7.22042e10i − 2.61487i
$$963$$ 6.21278e9i 0.224179i
$$964$$ −1.89257e10 −0.680428
$$965$$ 0 0
$$966$$ 2.14075e10 0.764094
$$967$$ 4.11324e10i 1.46282i 0.681937 + 0.731411i $$0.261138\pi$$
−0.681937 + 0.731411i $$0.738862\pi$$
$$968$$ − 1.37364e10i − 0.486754i
$$969$$ 2.57841e10 0.910372
$$970$$ 0 0
$$971$$ 2.85539e10 1.00092 0.500459 0.865760i $$-0.333165\pi$$
0.500459 + 0.865760i $$0.333165\pi$$
$$972$$ − 1.37035e10i − 0.478629i
$$973$$ − 3.16194e9i − 0.110042i
$$974$$ 2.05057e10 0.711079
$$975$$ 0 0
$$976$$ 1.66250e10 0.572384
$$977$$ − 3.64395e10i − 1.25009i −0.780589 0.625045i $$-0.785081\pi$$
0.780589 0.625045i $$-0.214919\pi$$
$$978$$ 2.81178e10i 0.961159i
$$979$$ −4.74503e10 −1.61622
$$980$$ 0 0
$$981$$ −1.62951e10 −0.551081
$$982$$ 7.01975e10i 2.36555i
$$983$$ − 3.71636e10i − 1.24790i −0.781463 0.623951i $$-0.785526\pi$$
0.781463 0.623951i $$-0.214474\pi$$
$$984$$ −6.44006e9 −0.215480
$$985$$ 0 0
$$986$$ −4.35570e10 −1.44707
$$987$$ 4.52513e9i 0.149803i
$$988$$ − 3.51711e10i − 1.16021i
$$989$$ −2.41864e10 −0.795032
$$990$$ 0 0
$$991$$ 1.80657e10 0.589655 0.294827 0.955551i $$-0.404738\pi$$
0.294827 + 0.955551i $$0.404738\pi$$
$$992$$ − 4.97190e9i − 0.161708i
$$993$$ − 1.47838e10i − 0.479140i
$$994$$ −1.11445e10 −0.359921
$$995$$ 0 0
$$996$$ 3.01321e10 0.966320
$$997$$ 2.62408e10i 0.838580i 0.907852 + 0.419290i $$0.137721\pi$$
−0.907852 + 0.419290i $$0.862279\pi$$
$$998$$ − 2.94199e10i − 0.936882i
$$999$$ 3.17347e10 1.00706
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.99.1 4
5.2 odd 4 35.8.a.a.1.2 2
5.3 odd 4 175.8.a.b.1.1 2
5.4 even 2 inner 175.8.b.c.99.4 4
15.2 even 4 315.8.a.c.1.1 2
20.7 even 4 560.8.a.i.1.2 2
35.27 even 4 245.8.a.b.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.2 2 5.2 odd 4
175.8.a.b.1.1 2 5.3 odd 4
175.8.b.c.99.1 4 1.1 even 1 trivial
175.8.b.c.99.4 4 5.4 even 2 inner
245.8.a.b.1.2 2 35.27 even 4
315.8.a.c.1.1 2 15.2 even 4
560.8.a.i.1.2 2 20.7 even 4