# Properties

 Label 315.8.a.c.1.1 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 1.1 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)

 $$f(q)$$ $$=$$ $$q-14.6332 q^{2} +86.1320 q^{4} -125.000 q^{5} -343.000 q^{7} +612.665 q^{8} +O(q^{10})$$ $$q-14.6332 q^{2} +86.1320 q^{4} -125.000 q^{5} -343.000 q^{7} +612.665 q^{8} +1829.16 q^{10} +6473.63 q^{11} -11681.7 q^{13} +5019.20 q^{14} -19990.2 q^{16} -13460.5 q^{17} +34955.5 q^{19} -10766.5 q^{20} -94730.3 q^{22} -77831.4 q^{23} +15625.0 q^{25} +170941. q^{26} -29543.3 q^{28} +221135. q^{29} -23222.3 q^{31} +214100. q^{32} +196971. q^{34} +42875.0 q^{35} -422392. q^{37} -511512. q^{38} -76583.1 q^{40} -191818. q^{41} +310754. q^{43} +557587. q^{44} +1.13893e6 q^{46} +240747. q^{47} +117649. q^{49} -228645. q^{50} -1.00617e6 q^{52} +1.06654e6 q^{53} -809204. q^{55} -210144. q^{56} -3.23592e6 q^{58} -451838. q^{59} -831659. q^{61} +339818. q^{62} -574238. q^{64} +1.46021e6 q^{65} +2.26405e6 q^{67} -1.15938e6 q^{68} -627401. q^{70} +2.22036e6 q^{71} +4.99377e6 q^{73} +6.18096e6 q^{74} +3.01078e6 q^{76} -2.22046e6 q^{77} -2.72773e6 q^{79} +2.49877e6 q^{80} +2.80693e6 q^{82} +6.38392e6 q^{83} +1.68256e6 q^{85} -4.54734e6 q^{86} +3.96617e6 q^{88} +7.32978e6 q^{89} +4.00682e6 q^{91} -6.70378e6 q^{92} -3.52291e6 q^{94} -4.36943e6 q^{95} -2.38676e6 q^{97} -1.72159e6 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −14.6332 −1.29341 −0.646704 0.762741i $$-0.723853\pi$$
−0.646704 + 0.762741i $$0.723853\pi$$
$$3$$ 0 0
$$4$$ 86.1320 0.672906
$$5$$ −125.000 −0.447214
$$6$$ 0 0
$$7$$ −343.000 −0.377964
$$8$$ 612.665 0.423066
$$9$$ 0 0
$$10$$ 1829.16 0.578430
$$11$$ 6473.63 1.46647 0.733236 0.679974i $$-0.238009\pi$$
0.733236 + 0.679974i $$0.238009\pi$$
$$12$$ 0 0
$$13$$ −11681.7 −1.47470 −0.737351 0.675510i $$-0.763924\pi$$
−0.737351 + 0.675510i $$0.763924\pi$$
$$14$$ 5019.20 0.488863
$$15$$ 0 0
$$16$$ −19990.2 −1.22010
$$17$$ −13460.5 −0.664491 −0.332246 0.943193i $$-0.607806\pi$$
−0.332246 + 0.943193i $$0.607806\pi$$
$$18$$ 0 0
$$19$$ 34955.5 1.16917 0.584585 0.811333i $$-0.301257\pi$$
0.584585 + 0.811333i $$0.301257\pi$$
$$20$$ −10766.5 −0.300933
$$21$$ 0 0
$$22$$ −94730.3 −1.89675
$$23$$ −77831.4 −1.33385 −0.666926 0.745124i $$-0.732390\pi$$
−0.666926 + 0.745124i $$0.732390\pi$$
$$24$$ 0 0
$$25$$ 15625.0 0.200000
$$26$$ 170941. 1.90739
$$27$$ 0 0
$$28$$ −29543.3 −0.254335
$$29$$ 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. 1.15503
$$33$$ 0 0
$$34$$ 196971. 0.859459
$$35$$ 42875.0 0.169031
$$36$$ 0 0
$$37$$ −422392. −1.37091 −0.685456 0.728114i $$-0.740397\pi$$
−0.685456 + 0.728114i $$0.740397\pi$$
$$38$$ −511512. −1.51221
$$39$$ 0 0
$$40$$ −76583.1 −0.189201
$$41$$ −191818. −0.434657 −0.217329 0.976099i $$-0.569734\pi$$
−0.217329 + 0.976099i $$0.569734\pi$$
$$42$$ 0 0
$$43$$ 310754. 0.596042 0.298021 0.954559i $$-0.403673\pi$$
0.298021 + 0.954559i $$0.403673\pi$$
$$44$$ 557587. 0.986798
$$45$$ 0 0
$$46$$ 1.13893e6 1.72522
$$47$$ 240747. 0.338235 0.169117 0.985596i $$-0.445908\pi$$
0.169117 + 0.985596i $$0.445908\pi$$
$$48$$ 0 0
$$49$$ 117649. 0.142857
$$50$$ −228645. −0.258682
$$51$$ 0 0
$$52$$ −1.00617e6 −0.992336
$$53$$ 1.06654e6 0.984040 0.492020 0.870584i $$-0.336259\pi$$
0.492020 + 0.870584i $$0.336259\pi$$
$$54$$ 0 0
$$55$$ −809204. −0.655826
$$56$$ −210144. −0.159904
$$57$$ 0 0
$$58$$ −3.23592e6 −2.17771
$$59$$ −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. 0.181082
$$63$$ 0 0
$$64$$ −574238. −0.273818
$$65$$ 1.46021e6 0.659507
$$66$$ 0 0
$$67$$ 2.26405e6 0.919654 0.459827 0.888009i $$-0.347912\pi$$
0.459827 + 0.888009i $$0.347912\pi$$
$$68$$ −1.15938e6 −0.447140
$$69$$ 0 0
$$70$$ −627401. −0.218626
$$71$$ 2.22036e6 0.736241 0.368120 0.929778i $$-0.380001\pi$$
0.368120 + 0.929778i $$0.380001\pi$$
$$72$$ 0 0
$$73$$ 4.99377e6 1.50244 0.751222 0.660049i $$-0.229465\pi$$
0.751222 + 0.660049i $$0.229465\pi$$
$$74$$ 6.18096e6 1.77315
$$75$$ 0 0
$$76$$ 3.01078e6 0.786742
$$77$$ −2.22046e6 −0.554274
$$78$$ 0 0
$$79$$ −2.72773e6 −0.622452 −0.311226 0.950336i $$-0.600740\pi$$
−0.311226 + 0.950336i $$0.600740\pi$$
$$80$$ 2.49877e6 0.545647
$$81$$ 0 0
$$82$$ 2.80693e6 0.562189
$$83$$ 6.38392e6 1.22550 0.612751 0.790276i $$-0.290063\pi$$
0.612751 + 0.790276i $$0.290063\pi$$
$$84$$ 0 0
$$85$$ 1.68256e6 0.297170
$$86$$ −4.54734e6 −0.770926
$$87$$ 0 0
$$88$$ 3.96617e6 0.620414
$$89$$ 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
$$92$$ −6.70378e6 −0.897557
$$93$$ 0 0
$$94$$ −3.52291e6 −0.437476
$$95$$ −4.36943e6 −0.522869
$$96$$ 0 0
$$97$$ −2.38676e6 −0.265526 −0.132763 0.991148i $$-0.542385\pi$$
−0.132763 + 0.991148i $$0.542385\pi$$
$$98$$ −1.72159e6 −0.184773
$$99$$ 0 0
$$100$$ 1.34581e6 0.134581
$$101$$ 1.92113e7 1.85538 0.927688 0.373356i $$-0.121793\pi$$
0.927688 + 0.373356i $$0.121793\pi$$
$$102$$ 0 0
$$103$$ −4.45359e6 −0.401587 −0.200793 0.979634i $$-0.564352\pi$$
−0.200793 + 0.979634i $$0.564352\pi$$
$$104$$ −7.15697e6 −0.623896
$$105$$ 0 0
$$106$$ −1.56070e7 −1.27277
$$107$$ −7.61385e6 −0.600843 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$108$$ 0 0
$$109$$ −1.99698e7 −1.47700 −0.738502 0.674251i $$-0.764467\pi$$
−0.738502 + 0.674251i $$0.764467\pi$$
$$110$$ 1.18413e7 0.848251
$$111$$ 0 0
$$112$$ 6.85663e6 0.461156
$$113$$ −2.57944e7 −1.68171 −0.840855 0.541261i $$-0.817947\pi$$
−0.840855 + 0.541261i $$0.817947\pi$$
$$114$$ 0 0
$$115$$ 9.72893e6 0.596517
$$116$$ 1.90468e7 1.13297
$$117$$ 0 0
$$118$$ 6.61185e6 0.370456
$$119$$ 4.61695e6 0.251154
$$120$$ 0 0
$$121$$ 2.24208e7 1.15054
$$122$$ 1.21699e7 0.606773
$$123$$ 0 0
$$124$$ −2.00018e6 −0.0942094
$$125$$ −1.95312e6 −0.0894427
$$126$$ 0 0
$$127$$ −6.75687e6 −0.292707 −0.146353 0.989232i $$-0.546754\pi$$
−0.146353 + 0.989232i $$0.546754\pi$$
$$128$$ −1.90018e7 −0.800868
$$129$$ 0 0
$$130$$ −2.13677e7 −0.853012
$$131$$ 2.21063e6 0.0859144 0.0429572 0.999077i $$-0.486322\pi$$
0.0429572 + 0.999077i $$0.486322\pi$$
$$132$$ 0 0
$$133$$ −1.19897e7 −0.441905
$$134$$ −3.31304e7 −1.18949
$$135$$ 0 0
$$136$$ −8.24677e6 −0.281124
$$137$$ 7.10581e6 0.236097 0.118049 0.993008i $$-0.462336\pi$$
0.118049 + 0.993008i $$0.462336\pi$$
$$138$$ 0 0
$$139$$ 9.21848e6 0.291144 0.145572 0.989348i $$-0.453498\pi$$
0.145572 + 0.989348i $$0.453498\pi$$
$$140$$ 3.69291e6 0.113742
$$141$$ 0 0
$$142$$ −3.24911e7 −0.952260
$$143$$ −7.56230e7 −2.16261
$$144$$ 0 0
$$145$$ −2.76418e7 −0.752972
$$146$$ −7.30751e7 −1.94328
$$147$$ 0 0
$$148$$ −3.63814e7 −0.922495
$$149$$ −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 $$0.774143\pi$$
$$152$$ 2.14160e7 0.494636
$$153$$ 0 0
$$154$$ 3.24925e7 0.716903
$$155$$ 2.90279e6 0.0626116
$$156$$ 0 0
$$157$$ 7.36596e7 1.51908 0.759540 0.650461i $$-0.225424\pi$$
0.759540 + 0.650461i $$0.225424\pi$$
$$158$$ 3.99155e7 0.805085
$$159$$ 0 0
$$160$$ −2.67625e7 −0.516544
$$161$$ 2.66962e7 0.504149
$$162$$ 0 0
$$163$$ −3.50642e7 −0.634172 −0.317086 0.948397i $$-0.602704\pi$$
−0.317086 + 0.948397i $$0.602704\pi$$
$$164$$ −1.65217e7 −0.292483
$$165$$ 0 0
$$166$$ −9.34175e7 −1.58508
$$167$$ −2.56950e6 −0.0426915 −0.0213458 0.999772i $$-0.506795\pi$$
−0.0213458 + 0.999772i $$0.506795\pi$$
$$168$$ 0 0
$$169$$ 7.37136e7 1.17475
$$170$$ −2.46213e7 −0.384362
$$171$$ 0 0
$$172$$ 2.67659e7 0.401081
$$173$$ −8.03463e7 −1.17979 −0.589895 0.807480i $$-0.700831\pi$$
−0.589895 + 0.807480i $$0.700831\pi$$
$$174$$ 0 0
$$175$$ −5.35938e6 −0.0755929
$$176$$ −1.29409e8 −1.78925
$$177$$ 0 0
$$178$$ −1.07259e8 −1.42548
$$179$$ −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 $$0.735090\pi$$
$$182$$ −5.86328e7 −0.720927
$$183$$ 0 0
$$184$$ −4.76846e7 −0.564307
$$185$$ 5.27990e7 0.613090
$$186$$ 0 0
$$187$$ −8.71382e7 −0.974458
$$188$$ 2.07360e7 0.227600
$$189$$ 0 0
$$190$$ 6.39390e7 0.676283
$$191$$ 2.21085e7 0.229584 0.114792 0.993390i $$-0.463380\pi$$
0.114792 + 0.993390i $$0.463380\pi$$
$$192$$ 0 0
$$193$$ −1.49793e8 −1.49983 −0.749913 0.661537i $$-0.769905\pi$$
−0.749913 + 0.661537i $$0.769905\pi$$
$$194$$ 3.49261e7 0.343434
$$195$$ 0 0
$$196$$ 1.01333e7 0.0961295
$$197$$ 5.70107e7 0.531281 0.265641 0.964072i $$-0.414417\pi$$
0.265641 + 0.964072i $$0.414417\pi$$
$$198$$ 0 0
$$199$$ −6.84161e7 −0.615421 −0.307711 0.951480i $$-0.599563\pi$$
−0.307711 + 0.951480i $$0.599563\pi$$
$$200$$ 9.57289e6 0.0846132
$$201$$ 0 0
$$202$$ −2.81124e8 −2.39976
$$203$$ −7.58492e7 −0.636377
$$204$$ 0 0
$$205$$ 2.39773e7 0.194385
$$206$$ 6.51704e7 0.519416
$$207$$ 0 0
$$208$$ 2.33519e8 1.79929
$$209$$ 2.26289e8 1.71455
$$210$$ 0 0
$$211$$ −1.36201e8 −0.998141 −0.499071 0.866561i $$-0.666325\pi$$
−0.499071 + 0.866561i $$0.666325\pi$$
$$212$$ 9.18635e7 0.662167
$$213$$ 0 0
$$214$$ 1.11415e8 0.777135
$$215$$ −3.88442e7 −0.266558
$$216$$ 0 0
$$217$$ 7.96525e6 0.0529164
$$218$$ 2.92224e8 1.91037
$$219$$ 0 0
$$220$$ −6.96984e7 −0.441310
$$221$$ 1.57241e8 0.979927
$$222$$ 0 0
$$223$$ −1.47728e8 −0.892066 −0.446033 0.895017i $$-0.647164\pi$$
−0.446033 + 0.895017i $$0.647164\pi$$
$$224$$ −7.34363e7 −0.436559
$$225$$ 0 0
$$226$$ 3.77456e8 2.17514
$$227$$ −3.22427e8 −1.82954 −0.914768 0.403980i $$-0.867627\pi$$
−0.914768 + 0.403980i $$0.867627\pi$$
$$228$$ 0 0
$$229$$ 3.10033e8 1.70602 0.853010 0.521895i $$-0.174775\pi$$
0.853010 + 0.521895i $$0.174775\pi$$
$$230$$ −1.42366e8 −0.771540
$$231$$ 0 0
$$232$$ 1.35481e8 0.712315
$$233$$ −1.80410e8 −0.934361 −0.467181 0.884162i $$-0.654730\pi$$
−0.467181 + 0.884162i $$0.654730\pi$$
$$234$$ 0 0
$$235$$ −3.00934e7 −0.151263
$$236$$ −3.89177e7 −0.192733
$$237$$ 0 0
$$238$$ −6.75609e7 −0.324845
$$239$$ −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 $$0.331275\pi$$
$$242$$ −3.28089e8 −1.48812
$$243$$ 0 0
$$244$$ −7.16324e7 −0.315679
$$245$$ −1.47061e7 −0.0638877
$$246$$ 0 0
$$247$$ −4.08339e8 −1.72418
$$248$$ −1.42275e7 −0.0592308
$$249$$ 0 0
$$250$$ 2.85806e7 0.115686
$$251$$ 1.29875e8 0.518404 0.259202 0.965823i $$-0.416540\pi$$
0.259202 + 0.965823i $$0.416540\pi$$
$$252$$ 0 0
$$253$$ −5.03852e8 −1.95606
$$254$$ 9.88750e7 0.378589
$$255$$ 0 0
$$256$$ 3.51561e8 1.30967
$$257$$ 2.94635e8 1.08273 0.541363 0.840789i $$-0.317909\pi$$
0.541363 + 0.840789i $$0.317909\pi$$
$$258$$ 0 0
$$259$$ 1.44880e8 0.518156
$$260$$ 1.25771e8 0.443786
$$261$$ 0 0
$$262$$ −3.23486e7 −0.111122
$$263$$ −3.13955e8 −1.06420 −0.532098 0.846683i $$-0.678596\pi$$
−0.532098 + 0.846683i $$0.678596\pi$$
$$264$$ 0 0
$$265$$ −1.33318e8 −0.440076
$$266$$ 1.75449e8 0.571563
$$267$$ 0 0
$$268$$ 1.95007e8 0.618841
$$269$$ 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 $$0.541274\pi$$
$$272$$ 2.69077e8 0.810748
$$273$$ 0 0
$$274$$ −1.03981e8 −0.305371
$$275$$ 1.01151e8 0.293294
$$276$$ 0 0
$$277$$ 1.96909e8 0.556655 0.278327 0.960486i $$-0.410220\pi$$
0.278327 + 0.960486i $$0.410220\pi$$
$$278$$ −1.34896e8 −0.376568
$$279$$ 0 0
$$280$$ 2.62680e7 0.0715112
$$281$$ −3.32330e8 −0.893505 −0.446753 0.894658i $$-0.647420\pi$$
−0.446753 + 0.894658i $$0.647420\pi$$
$$282$$ 0 0
$$283$$ 4.40437e8 1.15513 0.577566 0.816344i $$-0.304003\pi$$
0.577566 + 0.816344i $$0.304003\pi$$
$$284$$ 1.91244e8 0.495421
$$285$$ 0 0
$$286$$ 1.10661e9 2.79714
$$287$$ 6.57937e7 0.164285
$$288$$ 0 0
$$289$$ −2.29154e8 −0.558451
$$290$$ 4.04490e8 0.973900
$$291$$ 0 0
$$292$$ 4.30123e8 1.01100
$$293$$ 3.05058e8 0.708510 0.354255 0.935149i $$-0.384735\pi$$
0.354255 + 0.935149i $$0.384735\pi$$
$$294$$ 0 0
$$295$$ 5.64797e7 0.128090
$$296$$ −2.58785e8 −0.579986
$$297$$ 0 0
$$298$$ 1.95058e8 0.426979
$$299$$ 9.09203e8 1.96703
$$300$$ 0 0
$$301$$ −1.06589e8 −0.225283
$$302$$ 9.39366e8 1.96250
$$303$$ 0 0
$$304$$ −6.98766e8 −1.42651
$$305$$ 1.03957e8 0.209800
$$306$$ 0 0
$$307$$ −2.41616e8 −0.476587 −0.238293 0.971193i $$-0.576588\pi$$
−0.238293 + 0.971193i $$0.576588\pi$$
$$308$$ −1.91252e8 −0.372975
$$309$$ 0 0
$$310$$ −4.24772e7 −0.0809823
$$311$$ 6.71768e7 0.126636 0.0633181 0.997993i $$-0.479832\pi$$
0.0633181 + 0.997993i $$0.479832\pi$$
$$312$$ 0 0
$$313$$ −6.75084e8 −1.24438 −0.622190 0.782867i $$-0.713757\pi$$
−0.622190 + 0.782867i $$0.713757\pi$$
$$314$$ −1.07788e9 −1.96479
$$315$$ 0 0
$$316$$ −2.34944e8 −0.418852
$$317$$ 4.65149e8 0.820135 0.410067 0.912055i $$-0.365505\pi$$
0.410067 + 0.912055i $$0.365505\pi$$
$$318$$ 0 0
$$319$$ 1.43154e9 2.46909
$$320$$ 7.17797e7 0.122455
$$321$$ 0 0
$$322$$ −3.90652e8 −0.652070
$$323$$ −4.70517e8 −0.776903
$$324$$ 0 0
$$325$$ −1.82527e8 −0.294940
$$326$$ 5.13103e8 0.820244
$$327$$ 0 0
$$328$$ −1.17520e8 −0.183889
$$329$$ −8.25762e7 −0.127841
$$330$$ 0 0
$$331$$ 2.69779e8 0.408893 0.204447 0.978878i $$-0.434460\pi$$
0.204447 + 0.978878i $$0.434460\pi$$
$$332$$ 5.49860e8 0.824648
$$333$$ 0 0
$$334$$ 3.76002e7 0.0552176
$$335$$ −2.83006e8 −0.411282
$$336$$ 0 0
$$337$$ −6.29093e8 −0.895385 −0.447693 0.894187i $$-0.647754\pi$$
−0.447693 + 0.894187i $$0.647754\pi$$
$$338$$ −1.07867e9 −1.51943
$$339$$ 0 0
$$340$$ 1.44922e8 0.199967
$$341$$ −1.50333e8 −0.205312
$$342$$ 0 0
$$343$$ −4.03536e7 −0.0539949
$$344$$ 1.90388e8 0.252165
$$345$$ 0 0
$$346$$ 1.17573e9 1.52595
$$347$$ 7.61715e8 0.978676 0.489338 0.872094i $$-0.337238\pi$$
0.489338 + 0.872094i $$0.337238\pi$$
$$348$$ 0 0
$$349$$ −3.31639e8 −0.417616 −0.208808 0.977957i $$-0.566958\pi$$
−0.208808 + 0.977957i $$0.566958\pi$$
$$350$$ 7.84251e7 0.0977725
$$351$$ 0 0
$$352$$ 1.38601e9 1.69381
$$353$$ −7.57419e8 −0.916484 −0.458242 0.888828i $$-0.651521\pi$$
−0.458242 + 0.888828i $$0.651521\pi$$
$$354$$ 0 0
$$355$$ −2.77545e8 −0.329257
$$356$$ 6.31329e8 0.741619
$$357$$ 0 0
$$358$$ 1.46197e9 1.68402
$$359$$ −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
$$362$$ 1.57182e9 1.74150
$$363$$ 0 0
$$364$$ 3.45116e8 0.375068
$$365$$ −6.24221e8 −0.671914
$$366$$ 0 0
$$367$$ 1.64615e9 1.73835 0.869177 0.494502i $$-0.164649\pi$$
0.869177 + 0.494502i $$0.164649\pi$$
$$368$$ 1.55586e9 1.62744
$$369$$ 0 0
$$370$$ −7.72620e8 −0.792976
$$371$$ −3.65824e8 −0.371932
$$372$$ 0 0
$$373$$ −1.16387e9 −1.16124 −0.580622 0.814173i $$-0.697191\pi$$
−0.580622 + 0.814173i $$0.697191\pi$$
$$374$$ 1.27512e9 1.26037
$$375$$ 0 0
$$376$$ 1.47497e8 0.143096
$$377$$ −2.58323e9 −2.48295
$$378$$ 0 0
$$379$$ 4.07762e8 0.384742 0.192371 0.981322i $$-0.438382\pi$$
0.192371 + 0.981322i $$0.438382\pi$$
$$380$$ −3.76348e8 −0.351841
$$381$$ 0 0
$$382$$ −3.23519e8 −0.296946
$$383$$ −7.10345e8 −0.646061 −0.323031 0.946389i $$-0.604702\pi$$
−0.323031 + 0.946389i $$0.604702\pi$$
$$384$$ 0 0
$$385$$ 2.77557e8 0.247879
$$386$$ 2.19196e9 1.93989
$$387$$ 0 0
$$388$$ −2.05576e8 −0.178674
$$389$$ −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
$$392$$ 7.20794e7 0.0604380
$$393$$ 0 0
$$394$$ −8.34252e8 −0.687164
$$395$$ 3.40966e8 0.278369
$$396$$ 0 0
$$397$$ 1.58231e9 1.26919 0.634593 0.772846i $$-0.281168\pi$$
0.634593 + 0.772846i $$0.281168\pi$$
$$398$$ 1.00115e9 0.795991
$$399$$ 0 0
$$400$$ −3.12346e8 −0.244021
$$401$$ 2.13647e9 1.65460 0.827298 0.561763i $$-0.189877\pi$$
0.827298 + 0.561763i $$0.189877\pi$$
$$402$$ 0 0
$$403$$ 2.71276e8 0.206464
$$404$$ 1.65471e9 1.24849
$$405$$ 0 0
$$406$$ 1.10992e9 0.823096
$$407$$ −2.73441e9 −2.01040
$$408$$ 0 0
$$409$$ −4.05635e8 −0.293159 −0.146580 0.989199i $$-0.546826\pi$$
−0.146580 + 0.989199i $$0.546826\pi$$
$$410$$ −3.50866e8 −0.251419
$$411$$ 0 0
$$412$$ −3.83596e8 −0.270230
$$413$$ 1.54980e8 0.108256
$$414$$ 0 0
$$415$$ −7.97990e8 −0.548061
$$416$$ −2.50105e9 −1.70332
$$417$$ 0 0
$$418$$ −3.31134e9 −2.21762
$$419$$ −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 $$0.587686\pi$$
$$422$$ 1.99306e9 1.29100
$$423$$ 0 0
$$424$$ 6.53434e8 0.416314
$$425$$ −2.10320e8 −0.132898
$$426$$ 0 0
$$427$$ 2.85259e8 0.177313
$$428$$ −6.55796e8 −0.404311
$$429$$ 0 0
$$430$$ 5.68418e8 0.344769
$$431$$ −8.26292e6 −0.00497122 −0.00248561 0.999997i $$-0.500791\pi$$
−0.00248561 + 0.999997i $$0.500791\pi$$
$$432$$ 0 0
$$433$$ −3.10619e9 −1.83874 −0.919369 0.393396i $$-0.871300\pi$$
−0.919369 + 0.393396i $$0.871300\pi$$
$$434$$ −1.16558e8 −0.0684426
$$435$$ 0 0
$$436$$ −1.72004e9 −0.993885
$$437$$ −2.72063e9 −1.55950
$$438$$ 0 0
$$439$$ −1.28295e9 −0.723742 −0.361871 0.932228i $$-0.617862\pi$$
−0.361871 + 0.932228i $$0.617862\pi$$
$$440$$ −4.95771e8 −0.277458
$$441$$ 0 0
$$442$$ −2.30095e9 −1.26745
$$443$$ 1.73263e9 0.946878 0.473439 0.880827i $$-0.343012\pi$$
0.473439 + 0.880827i $$0.343012\pi$$
$$444$$ 0 0
$$445$$ −9.16223e8 −0.492880
$$446$$ 2.16175e9 1.15381
$$447$$ 0 0
$$448$$ 1.96964e8 0.103493
$$449$$ −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
$$452$$ −2.22172e9 −1.13163
$$453$$ 0 0
$$454$$ 4.71816e9 2.36634
$$455$$ −5.00853e8 −0.249270
$$456$$ 0 0
$$457$$ 1.76868e9 0.866849 0.433425 0.901190i $$-0.357305\pi$$
0.433425 + 0.901190i $$0.357305\pi$$
$$458$$ −4.53679e9 −2.20658
$$459$$ 0 0
$$460$$ 8.37972e8 0.401400
$$461$$ 2.55825e8 0.121616 0.0608078 0.998149i $$-0.480632\pi$$
0.0608078 + 0.998149i $$0.480632\pi$$
$$462$$ 0 0
$$463$$ −4.19121e9 −1.96249 −0.981243 0.192777i $$-0.938250\pi$$
−0.981243 + 0.192777i $$0.938250\pi$$
$$464$$ −4.42052e9 −2.05428
$$465$$ 0 0
$$466$$ 2.63998e9 1.20851
$$467$$ 2.94239e9 1.33688 0.668438 0.743768i $$-0.266963\pi$$
0.668438 + 0.743768i $$0.266963\pi$$
$$468$$ 0 0
$$469$$ −7.76569e8 −0.347596
$$470$$ 4.40364e8 0.195645
$$471$$ 0 0
$$472$$ −2.76825e8 −0.121174
$$473$$ 2.01171e9 0.874079
$$474$$ 0 0
$$475$$ 5.46179e8 0.233834
$$476$$ 3.97667e8 0.169003
$$477$$ 0 0
$$478$$ 5.36292e9 2.24597
$$479$$ −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
$$482$$ −3.21535e9 −1.30787
$$483$$ 0 0
$$484$$ 1.93115e9 0.774206
$$485$$ 2.98345e8 0.118747
$$486$$ 0 0
$$487$$ −1.40131e9 −0.549771 −0.274886 0.961477i $$-0.588640\pi$$
−0.274886 + 0.961477i $$0.588640\pi$$
$$488$$ −5.09528e8 −0.198472
$$489$$ 0 0
$$490$$ 2.15198e8 0.0826329
$$491$$ 4.79712e9 1.82892 0.914462 0.404672i $$-0.132614\pi$$
0.914462 + 0.404672i $$0.132614\pi$$
$$492$$ 0 0
$$493$$ −2.97658e9 −1.11880
$$494$$ 5.97533e9 2.23007
$$495$$ 0 0
$$496$$ 4.64218e8 0.170819
$$497$$ −7.61585e8 −0.278273
$$498$$ 0 0
$$499$$ −2.01049e9 −0.724351 −0.362176 0.932110i $$-0.617966\pi$$
−0.362176 + 0.932110i $$0.617966\pi$$
$$500$$ −1.68227e8 −0.0601866
$$501$$ 0 0
$$502$$ −1.90050e9 −0.670509
$$503$$ −1.68618e9 −0.590766 −0.295383 0.955379i $$-0.595447\pi$$
−0.295383 + 0.955379i $$0.595447\pi$$
$$504$$ 0 0
$$505$$ −2.40141e9 −0.829749
$$506$$ 7.37300e9 2.52998
$$507$$ 0 0
$$508$$ −5.81983e8 −0.196964
$$509$$ −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
$$512$$ −2.71225e9 −0.893068
$$513$$ 0 0
$$514$$ −4.31147e9 −1.40041
$$515$$ 5.56698e8 0.179595
$$516$$ 0 0
$$517$$ 1.55851e9 0.496012
$$518$$ −2.12007e9 −0.670187
$$519$$ 0 0
$$520$$ 8.94621e8 0.279015
$$521$$ −3.28595e9 −1.01796 −0.508978 0.860779i $$-0.669977\pi$$
−0.508978 + 0.860779i $$0.669977\pi$$
$$522$$ 0 0
$$523$$ 1.29734e9 0.396549 0.198274 0.980147i $$-0.436466\pi$$
0.198274 + 0.980147i $$0.436466\pi$$
$$524$$ 1.90406e8 0.0578123
$$525$$ 0 0
$$526$$ 4.59418e9 1.37644
$$527$$ 3.12583e8 0.0930313
$$528$$ 0 0
$$529$$ 2.65291e9 0.779161
$$530$$ 1.95087e9 0.569198
$$531$$ 0 0
$$532$$ −1.03270e9 −0.297360
$$533$$ 2.24076e9 0.640990
$$534$$ 0 0
$$535$$ 9.51731e8 0.268705
$$536$$ 1.38710e9 0.389074
$$537$$ 0 0
$$538$$ −4.31308e9 −1.19412
$$539$$ 7.61617e8 0.209496
$$540$$ 0 0
$$541$$ 1.23726e9 0.335948 0.167974 0.985791i $$-0.446278\pi$$
0.167974 + 0.985791i $$0.446278\pi$$
$$542$$ 1.23986e9 0.334484
$$543$$ 0 0
$$544$$ −2.88189e9 −0.767505
$$545$$ 2.49623e9 0.660536
$$546$$ 0 0
$$547$$ −4.27288e9 −1.11626 −0.558130 0.829754i $$-0.688481\pi$$
−0.558130 + 0.829754i $$0.688481\pi$$
$$548$$ 6.12037e8 0.158871
$$549$$ 0 0
$$550$$ −1.48016e9 −0.379350
$$551$$ 7.72986e9 1.96853
$$552$$ 0 0
$$553$$ 9.35610e8 0.235265
$$554$$ −2.88141e9 −0.719982
$$555$$ 0 0
$$556$$ 7.94006e8 0.195912
$$557$$ 2.41921e9 0.593171 0.296586 0.955006i $$-0.404152\pi$$
0.296586 + 0.955006i $$0.404152\pi$$
$$558$$ 0 0
$$559$$ −3.63013e9 −0.878985
$$560$$ −8.57079e8 −0.206235
$$561$$ 0 0
$$562$$ 4.86306e9 1.15567
$$563$$ 3.03839e9 0.717570 0.358785 0.933420i $$-0.383191\pi$$
0.358785 + 0.933420i $$0.383191\pi$$
$$564$$ 0 0
$$565$$ 3.22430e9 0.752084
$$566$$ −6.44503e9 −1.49406
$$567$$ 0 0
$$568$$ 1.36034e9 0.311478
$$569$$ −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 $$0.432279\pi$$
$$572$$ −6.51356e9 −1.45523
$$573$$ 0 0
$$574$$ −9.62776e8 −0.212488
$$575$$ −1.21612e9 −0.266770
$$576$$ 0 0
$$577$$ 2.19182e9 0.474996 0.237498 0.971388i $$-0.423673\pi$$
0.237498 + 0.971388i $$0.423673\pi$$
$$578$$ 3.35327e9 0.722306
$$579$$ 0 0
$$580$$ −2.38085e9 −0.506679
$$581$$ −2.18969e9 −0.463196
$$582$$ 0 0
$$583$$ 6.90441e9 1.44307
$$584$$ 3.05951e9 0.635633
$$585$$ 0 0
$$586$$ −4.46399e9 −0.916392
$$587$$ 4.70415e9 0.959949 0.479974 0.877283i $$-0.340646\pi$$
0.479974 + 0.877283i $$0.340646\pi$$
$$588$$ 0 0
$$589$$ −8.11747e8 −0.163688
$$590$$ −8.26482e8 −0.165673
$$591$$ 0 0
$$592$$ 8.44368e9 1.67265
$$593$$ −3.66996e9 −0.722719 −0.361359 0.932427i $$-0.617687\pi$$
−0.361359 + 0.932427i $$0.617687\pi$$
$$594$$ 0 0
$$595$$ −5.77118e8 −0.112320
$$596$$ −1.14812e9 −0.222139
$$597$$ 0 0
$$598$$ −1.33046e10 −2.54418
$$599$$ 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 $$0.416995\pi$$
$$602$$ 1.55974e9 0.291383
$$603$$ 0 0
$$604$$ −5.52915e9 −1.02101
$$605$$ −2.80260e9 −0.514537
$$606$$ 0 0
$$607$$ −2.26388e9 −0.410859 −0.205430 0.978672i $$-0.565859\pi$$
−0.205430 + 0.978672i $$0.565859\pi$$
$$608$$ 7.48397e9 1.35042
$$609$$ 0 0
$$610$$ −1.52123e9 −0.271357
$$611$$ −2.81233e9 −0.498795
$$612$$ 0 0
$$613$$ −9.92731e9 −1.74068 −0.870342 0.492448i $$-0.836102\pi$$
−0.870342 + 0.492448i $$0.836102\pi$$
$$614$$ 3.53563e9 0.616422
$$615$$ 0 0
$$616$$ −1.36040e9 −0.234495
$$617$$ −2.27156e9 −0.389338 −0.194669 0.980869i $$-0.562363\pi$$
−0.194669 + 0.980869i $$0.562363\pi$$
$$618$$ 0 0
$$619$$ 6.90552e8 0.117025 0.0585126 0.998287i $$-0.481364\pi$$
0.0585126 + 0.998287i $$0.481364\pi$$
$$620$$ 2.50023e8 0.0421317
$$621$$ 0 0
$$622$$ −9.83015e8 −0.163792
$$623$$ −2.51412e9 −0.416560
$$624$$ 0 0
$$625$$ 2.44141e8 0.0400000
$$626$$ 9.87867e9 1.60949
$$627$$ 0 0
$$628$$ 6.34445e9 1.02220
$$629$$ 5.68560e9 0.910959
$$630$$ 0 0
$$631$$ −1.00992e10 −1.60023 −0.800116 0.599846i $$-0.795229\pi$$
−0.800116 + 0.599846i $$0.795229\pi$$
$$632$$ −1.67118e9 −0.263338
$$633$$ 0 0
$$634$$ −6.80665e9 −1.06077
$$635$$ 8.44609e8 0.130902
$$636$$ 0 0
$$637$$ −1.37434e9 −0.210672
$$638$$ −2.09482e10 −3.19355
$$639$$ 0 0
$$640$$ 2.37523e9 0.358159
$$641$$ −8.50418e8 −0.127535 −0.0637675 0.997965i $$-0.520312\pi$$
−0.0637675 + 0.997965i $$0.520312\pi$$
$$642$$ 0 0
$$643$$ 1.56624e9 0.232339 0.116169 0.993229i $$-0.462938\pi$$
0.116169 + 0.993229i $$0.462938\pi$$
$$644$$ 2.29940e9 0.339245
$$645$$ 0 0
$$646$$ 6.88520e9 1.00485
$$647$$ −1.06053e10 −1.53942 −0.769712 0.638391i $$-0.779600\pi$$
−0.769712 + 0.638391i $$0.779600\pi$$
$$648$$ 0 0
$$649$$ −2.92503e9 −0.420024
$$650$$ 2.67096e9 0.381478
$$651$$ 0 0
$$652$$ −3.02015e9 −0.426739
$$653$$ 1.83644e9 0.258096 0.129048 0.991638i $$-0.458808\pi$$
0.129048 + 0.991638i $$0.458808\pi$$
$$654$$ 0 0
$$655$$ −2.76328e8 −0.0384221
$$656$$ 3.83448e9 0.530327
$$657$$ 0 0
$$658$$ 1.20836e9 0.165350
$$659$$ 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 $$0.828451\pi$$
$$662$$ −3.94774e9 −0.528866
$$663$$ 0 0
$$664$$ 3.91121e9 0.518469
$$665$$ 1.49872e9 0.197626
$$666$$ 0 0
$$667$$ −1.72112e10 −2.24580
$$668$$ −2.21316e8 −0.0287274
$$669$$ 0 0
$$670$$ 4.14130e9 0.531955
$$671$$ −5.38386e9 −0.687962
$$672$$ 0 0
$$673$$ 1.02193e10 1.29231 0.646155 0.763206i $$-0.276376\pi$$
0.646155 + 0.763206i $$0.276376\pi$$
$$674$$ 9.20567e9 1.15810
$$675$$ 0 0
$$676$$ 6.34910e9 0.790494
$$677$$ −1.04119e10 −1.28965 −0.644823 0.764332i $$-0.723069\pi$$
−0.644823 + 0.764332i $$0.723069\pi$$
$$678$$ 0 0
$$679$$ 8.18659e8 0.100360
$$680$$ 1.03085e9 0.125722
$$681$$ 0 0
$$682$$ 2.19986e9 0.265552
$$683$$ −6.56705e9 −0.788675 −0.394338 0.918966i $$-0.629026\pi$$
−0.394338 + 0.918966i $$0.629026\pi$$
$$684$$ 0 0
$$685$$ −8.88226e8 −0.105586
$$686$$ 5.90504e8 0.0698375
$$687$$ 0 0
$$688$$ −6.21203e9 −0.727233
$$689$$ −1.24590e10 −1.45117
$$690$$ 0 0
$$691$$ 4.44242e9 0.512208 0.256104 0.966649i $$-0.417561\pi$$
0.256104 + 0.966649i $$0.417561\pi$$
$$692$$ −6.92039e9 −0.793888
$$693$$ 0 0
$$694$$ −1.11464e10 −1.26583
$$695$$ −1.15231e9 −0.130203
$$696$$ 0 0
$$697$$ 2.58197e9 0.288826
$$698$$ 4.85296e9 0.540148
$$699$$ 0 0
$$700$$ −4.61614e8 −0.0508669
$$701$$ 7.92343e9 0.868761 0.434380 0.900729i $$-0.356967\pi$$
0.434380 + 0.900729i $$0.356967\pi$$
$$702$$ 0 0
$$703$$ −1.47649e10 −1.60283
$$704$$ −3.71741e9 −0.401546
$$705$$ 0 0
$$706$$ 1.10835e10 1.18539
$$707$$ −6.58948e9 −0.701266
$$708$$ 0 0
$$709$$ 9.27216e9 0.977055 0.488528 0.872548i $$-0.337534\pi$$
0.488528 + 0.872548i $$0.337534\pi$$
$$710$$ 4.06139e9 0.425864
$$711$$ 0 0
$$712$$ 4.49070e9 0.466266
$$713$$ 1.80743e9 0.186744
$$714$$ 0 0
$$715$$ 9.45288e9 0.967148
$$716$$ −8.60522e9 −0.876126
$$717$$ 0 0
$$718$$ 2.14810e10 2.16580
$$719$$ 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
$$722$$ −4.79990e9 −0.474626
$$723$$ 0 0
$$724$$ −9.25181e9 −0.906029
$$725$$ 3.45523e9 0.336739
$$726$$ 0 0
$$727$$ −9.22691e9 −0.890606 −0.445303 0.895380i $$-0.646904\pi$$
−0.445303 + 0.895380i $$0.646904\pi$$
$$728$$ 2.45484e9 0.235811
$$729$$ 0 0
$$730$$ 9.13438e9 0.869059
$$731$$ −4.18290e9 −0.396065
$$732$$ 0 0
$$733$$ 4.96865e9 0.465988 0.232994 0.972478i $$-0.425148\pi$$
0.232994 + 0.972478i $$0.425148\pi$$
$$734$$ −2.40885e10 −2.24840
$$735$$ 0 0
$$736$$ −1.66637e10 −1.54063
$$737$$ 1.46566e10 1.34865
$$738$$ 0 0
$$739$$ −1.96084e9 −0.178726 −0.0893628 0.995999i $$-0.528483\pi$$
−0.0893628 + 0.995999i $$0.528483\pi$$
$$740$$ 4.54768e9 0.412552
$$741$$ 0 0
$$742$$ 5.35320e9 0.481060
$$743$$ −9.56947e9 −0.855908 −0.427954 0.903801i $$-0.640765\pi$$
−0.427954 + 0.903801i $$0.640765\pi$$
$$744$$ 0 0
$$745$$ 1.66622e9 0.147634
$$746$$ 1.70312e10 1.50196
$$747$$ 0 0
$$748$$ −7.50539e9 −0.655719
$$749$$ 2.61155e9 0.227097
$$750$$ 0 0
$$751$$ 8.11719e9 0.699304 0.349652 0.936880i $$-0.386300\pi$$
0.349652 + 0.936880i $$0.386300\pi$$
$$752$$ −4.81257e9 −0.412681
$$753$$ 0 0
$$754$$ 3.78010e10 3.21147
$$755$$ 8.02424e9 0.678562
$$756$$ 0 0
$$757$$ −9.11117e9 −0.763376 −0.381688 0.924291i $$-0.624657\pi$$
−0.381688 + 0.924291i $$0.624657\pi$$
$$758$$ −5.96689e9 −0.497629
$$759$$ 0 0
$$760$$ −2.67700e9 −0.221208
$$761$$ −1.71359e10 −1.40948 −0.704742 0.709464i $$-0.748937\pi$$
−0.704742 + 0.709464i $$0.748937\pi$$
$$762$$ 0 0
$$763$$ 6.84965e9 0.558255
$$764$$ 1.90425e9 0.154489
$$765$$ 0 0
$$766$$ 1.03947e10 0.835621
$$767$$ 5.27823e9 0.422381
$$768$$ 0 0
$$769$$ −1.82316e10 −1.44572 −0.722858 0.690997i $$-0.757172\pi$$
−0.722858 + 0.690997i $$0.757172\pi$$
$$770$$ −4.06156e9 −0.320609
$$771$$ 0 0
$$772$$ −1.29020e10 −1.00924
$$773$$ 1.45965e10 1.13663 0.568316 0.822810i $$-0.307595\pi$$
0.568316 + 0.822810i $$0.307595\pi$$
$$774$$ 0 0
$$775$$ −3.62849e8 −0.0280007
$$776$$ −1.46228e9 −0.112335
$$777$$ 0 0
$$778$$ 2.85481e9 0.217345
$$779$$ −6.70510e9 −0.508188
$$780$$ 0 0
$$781$$ 1.43738e10 1.07968
$$782$$ −1.53305e10 −1.14639
$$783$$ 0 0
$$784$$ −2.35182e9 −0.174300
$$785$$ −9.20745e9 −0.679353
$$786$$ 0 0
$$787$$ 1.42358e10 1.04104 0.520522 0.853848i $$-0.325737\pi$$
0.520522 + 0.853848i $$0.325737\pi$$
$$788$$ 4.91045e9 0.357503
$$789$$ 0 0
$$790$$ −4.98944e9 −0.360045
$$791$$ 8.84748e9 0.635627
$$792$$ 0 0
$$793$$ 9.71519e9 0.691823
$$794$$ −2.31544e10 −1.64158
$$795$$ 0 0
$$796$$ −5.89281e9 −0.414121
$$797$$ −1.03325e10 −0.722935 −0.361467 0.932385i $$-0.617724\pi$$
−0.361467 + 0.932385i $$0.617724\pi$$
$$798$$ 0 0
$$799$$ −3.24057e9 −0.224754
$$800$$ 3.34531e9 0.231005
$$801$$ 0 0
$$802$$ −3.12635e10 −2.14007
$$803$$ 3.23278e10 2.20329
$$804$$ 0 0
$$805$$ −3.33702e9 −0.225462
$$806$$ −3.96965e9 −0.267042
$$807$$ 0 0
$$808$$ 1.17701e10 0.784947
$$809$$ 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 $$0.469828\pi$$
$$812$$ −6.53304e9 −0.428222
$$813$$ 0 0
$$814$$ 4.00133e10 2.60027
$$815$$ 4.38303e9 0.283611
$$816$$ 0 0
$$817$$ 1.08626e10 0.696875
$$818$$ 5.93575e9 0.379175
$$819$$ 0 0
$$820$$ 2.06521e9 0.130803
$$821$$ 1.42014e10 0.895633 0.447817 0.894125i $$-0.352202\pi$$
0.447817 + 0.894125i $$0.352202\pi$$
$$822$$ 0 0
$$823$$ −2.79354e10 −1.74685 −0.873424 0.486961i $$-0.838105\pi$$
−0.873424 + 0.486961i $$0.838105\pi$$
$$824$$ −2.72856e9 −0.169898
$$825$$ 0 0
$$826$$ −2.26787e9 −0.140019
$$827$$ 1.48560e8 0.00913341 0.00456670 0.999990i $$-0.498546\pi$$
0.00456670 + 0.999990i $$0.498546\pi$$
$$828$$ 0 0
$$829$$ 3.98861e9 0.243154 0.121577 0.992582i $$-0.461205\pi$$
0.121577 + 0.992582i $$0.461205\pi$$
$$830$$ 1.16772e10 0.708868
$$831$$ 0 0
$$832$$ 6.70807e9 0.403800
$$833$$ −1.58361e9 −0.0949273
$$834$$ 0 0
$$835$$ 3.21188e8 0.0190922
$$836$$ 1.94907e10 1.15373
$$837$$ 0 0
$$838$$ 1.57271e10 0.923198
$$839$$ −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
$$842$$ 1.21880e10 0.703625
$$843$$ 0 0
$$844$$ −1.17313e10 −0.671655
$$845$$ −9.21419e9 −0.525362
$$846$$ 0 0
$$847$$ −7.69033e9 −0.434863
$$848$$ −2.13204e10 −1.20063
$$849$$ 0 0
$$850$$ 3.07767e9 0.171892
$$851$$ 3.28754e10 1.82859
$$852$$ 0 0
$$853$$ 1.54310e9 0.0851282 0.0425641 0.999094i $$-0.486447\pi$$
0.0425641 + 0.999094i $$0.486447\pi$$
$$854$$ −4.17427e9 −0.229339
$$855$$ 0 0
$$856$$ −4.66474e9 −0.254196
$$857$$ 1.29972e10 0.705369 0.352684 0.935742i $$-0.385269\pi$$
0.352684 + 0.935742i $$0.385269\pi$$
$$858$$ 0 0
$$859$$ −1.98316e10 −1.06754 −0.533768 0.845631i $$-0.679224\pi$$
−0.533768 + 0.845631i $$0.679224\pi$$
$$860$$ −3.34573e9 −0.179369
$$861$$ 0 0
$$862$$ 1.20913e8 0.00642982
$$863$$ −4.94264e9 −0.261771 −0.130886 0.991397i $$-0.541782\pi$$
−0.130886 + 0.991397i $$0.541782\pi$$
$$864$$ 0 0
$$865$$ 1.00433e10 0.527618
$$866$$ 4.54536e10 2.37824
$$867$$ 0 0
$$868$$ 6.86063e8 0.0356078
$$869$$ −1.76583e10 −0.912809
$$870$$ 0 0
$$871$$ −2.64480e10 −1.35621
$$872$$ −1.22348e10 −0.624870
$$873$$ 0 0
$$874$$ 3.98117e10 2.01707
$$875$$ 6.69922e8 0.0338062
$$876$$ 0 0
$$877$$ −7.37011e9 −0.368957 −0.184478 0.982837i $$-0.559060\pi$$
−0.184478 + 0.982837i $$0.559060\pi$$
$$878$$ 1.87737e10 0.936094
$$879$$ 0 0
$$880$$ 1.61761e10 0.800176
$$881$$ −9.74385e9 −0.480081 −0.240041 0.970763i $$-0.577161\pi$$
−0.240041 + 0.970763i $$0.577161\pi$$
$$882$$ 0 0
$$883$$ −2.79540e10 −1.36641 −0.683206 0.730226i $$-0.739415\pi$$
−0.683206 + 0.730226i $$0.739415\pi$$
$$884$$ 1.35435e10 0.659399
$$885$$ 0 0
$$886$$ −2.53541e10 −1.22470
$$887$$ 2.08176e10 1.00161 0.500804 0.865561i $$-0.333038\pi$$
0.500804 + 0.865561i $$0.333038\pi$$
$$888$$ 0 0
$$889$$ 2.31761e9 0.110633
$$890$$ 1.34073e10 0.637495
$$891$$ 0 0
$$892$$ −1.27241e10 −0.600276
$$893$$ 8.41542e9 0.395454
$$894$$ 0 0
$$895$$ 1.24884e10 0.582273
$$896$$ 6.51763e9 0.302700
$$897$$ 0 0
$$898$$ 2.73500e10 1.26035
$$899$$ −5.13526e9 −0.235724
$$900$$ 0 0
$$901$$ −1.43562e10 −0.653886
$$902$$ 1.81710e10 0.824435
$$903$$ 0 0
$$904$$ −1.58033e10 −0.711474
$$905$$ 1.34268e10 0.602147
$$906$$ 0 0
$$907$$ 3.61602e10 1.60918 0.804591 0.593830i $$-0.202385\pi$$
0.804591 + 0.593830i $$0.202385\pi$$
$$908$$ −2.77713e10 −1.23111
$$909$$ 0 0
$$910$$ 7.32910e9 0.322408
$$911$$ −1.79811e10 −0.787954 −0.393977 0.919120i $$-0.628901\pi$$
−0.393977 + 0.919120i $$0.628901\pi$$
$$912$$ 0 0
$$913$$ 4.13272e10 1.79717
$$914$$ −2.58816e10 −1.12119
$$915$$ 0 0
$$916$$ 2.67038e10 1.14799
$$917$$ −7.58245e8 −0.0324726
$$918$$ 0 0
$$919$$ −1.01373e10 −0.430841 −0.215421 0.976521i $$-0.569112\pi$$
−0.215421 + 0.976521i $$0.569112\pi$$
$$920$$ 5.96057e9 0.252366
$$921$$ 0 0
$$922$$ −3.74355e9 −0.157299
$$923$$ −2.59376e10 −1.08574
$$924$$ 0 0
$$925$$ −6.59987e9 −0.274182
$$926$$ 6.13311e10 2.53830
$$927$$ 0 0
$$928$$ 4.73449e10 1.94471
$$929$$ 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
$$932$$ −1.55391e10 −0.628738
$$933$$ 0 0
$$934$$ −4.30567e10 −1.72913
$$935$$ 1.08923e10 0.435791
$$936$$ 0 0
$$937$$ 1.16627e10 0.463138 0.231569 0.972818i $$-0.425614\pi$$
0.231569 + 0.972818i $$0.425614\pi$$
$$938$$ 1.13637e10 0.449584
$$939$$ 0 0
$$940$$ −2.59200e9 −0.101786
$$941$$ −3.03134e10 −1.18596 −0.592982 0.805216i $$-0.702049\pi$$
−0.592982 + 0.805216i $$0.702049\pi$$
$$942$$ 0 0
$$943$$ 1.49295e10 0.579768
$$944$$ 9.03231e9 0.349460
$$945$$ 0 0
$$946$$ −2.94378e10 −1.13054
$$947$$ −2.84339e10 −1.08796 −0.543979 0.839099i $$-0.683083\pi$$
−0.543979 + 0.839099i $$0.683083\pi$$
$$948$$ 0 0
$$949$$ −5.83357e10 −2.21566
$$950$$ −7.99238e9 −0.302443
$$951$$ 0 0
$$952$$ 2.82864e9 0.106255
$$953$$ 4.09127e10 1.53120 0.765602 0.643315i $$-0.222441\pi$$
0.765602 + 0.643315i $$0.222441\pi$$
$$954$$ 0 0
$$955$$ −2.76356e9 −0.102673
$$956$$ −3.15664e10 −1.16848
$$957$$ 0 0
$$958$$ 6.70105e10 2.46243
$$959$$ −2.43729e9 −0.0892365
$$960$$ 0 0
$$961$$ −2.69733e10 −0.980399
$$962$$ −7.22042e10 −2.61487
$$963$$ 0 0
$$964$$ 1.89257e10 0.680428
$$965$$ 1.87241e10 0.670742
$$966$$ 0 0
$$967$$ −4.11324e10 −1.46282 −0.731411 0.681937i $$-0.761138\pi$$
−0.731411 + 0.681937i $$0.761138\pi$$
$$968$$ 1.37364e10 0.486754
$$969$$ 0 0
$$970$$ −4.36576e9 −0.153588
$$971$$ −2.85539e10 −1.00092 −0.500459 0.865760i $$-0.666835\pi$$
−0.500459 + 0.865760i $$0.666835\pi$$
$$972$$ 0 0
$$973$$ −3.16194e9 −0.110042
$$974$$ 2.05057e10 0.711079
$$975$$ 0 0
$$976$$ 1.66250e10 0.572384
$$977$$ −3.64395e10 −1.25009 −0.625045 0.780589i $$-0.714919\pi$$
−0.625045 + 0.780589i $$0.714919\pi$$
$$978$$ 0 0
$$979$$ 4.74503e10 1.61622
$$980$$ −1.26667e9 −0.0429904
$$981$$ 0 0
$$982$$ −7.01975e10 −2.36555
$$983$$ 3.71636e10 1.24790 0.623951 0.781463i $$-0.285526\pi$$
0.623951 + 0.781463i $$0.285526\pi$$
$$984$$ 0 0
$$985$$ −7.12634e9 −0.237596
$$986$$ 4.35570e10 1.44707
$$987$$ 0 0
$$988$$ −3.51711e10 −1.16021
$$989$$ −2.41864e10 −0.795032
$$990$$ 0 0
$$991$$ 1.80657e10 0.589655 0.294827 0.955551i $$-0.404738\pi$$
0.294827 + 0.955551i $$0.404738\pi$$
$$992$$ −4.97190e9 −0.161708
$$993$$ 0 0
$$994$$ 1.11445e10 0.359921
$$995$$ 8.55201e9 0.275225
$$996$$ 0 0
$$997$$ −2.62408e10 −0.838580 −0.419290 0.907852i $$-0.637721\pi$$
−0.419290 + 0.907852i $$0.637721\pi$$
$$998$$ 2.94199e10 0.936882
$$999$$ 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.1.1 2
3.2 odd 2 35.8.a.a.1.2 2
12.11 even 2 560.8.a.i.1.2 2
15.2 even 4 175.8.b.c.99.4 4
15.8 even 4 175.8.b.c.99.1 4
15.14 odd 2 175.8.a.b.1.1 2
21.20 even 2 245.8.a.b.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.2 2 3.2 odd 2
175.8.a.b.1.1 2 15.14 odd 2
175.8.b.c.99.1 4 15.8 even 4
175.8.b.c.99.4 4 15.2 even 4
245.8.a.b.1.2 2 21.20 even 2
315.8.a.c.1.1 2 1.1 even 1 trivial
560.8.a.i.1.2 2 12.11 even 2