# Properties

 Label 117.10.a.c.1.1 Level $117$ Weight $10$ Character 117.1 Self dual yes Analytic conductor $60.259$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,10,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 117.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: $$60.2591928312$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 1602x^{2} + 1544x + 342272$$ x^4 - x^3 - 1602*x^2 + 1544*x + 342272 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-36.8028$$ of defining polynomial Character $$\chi$$ $$=$$ 117.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-28.8028 q^{2} +317.603 q^{4} +258.914 q^{5} -8862.28 q^{7} +5599.18 q^{8} +O(q^{10})$$ $$q-28.8028 q^{2} +317.603 q^{4} +258.914 q^{5} -8862.28 q^{7} +5599.18 q^{8} -7457.46 q^{10} -36087.9 q^{11} -28561.0 q^{13} +255259. q^{14} -323885. q^{16} -327405. q^{17} +265525. q^{19} +82231.9 q^{20} +1.03943e6 q^{22} +2.42458e6 q^{23} -1.88609e6 q^{25} +822638. q^{26} -2.81469e6 q^{28} -3.99178e6 q^{29} -6.45220e6 q^{31} +6.46202e6 q^{32} +9.43018e6 q^{34} -2.29457e6 q^{35} -8.15498e6 q^{37} -7.64787e6 q^{38} +1.44971e6 q^{40} +720241. q^{41} -4.13245e7 q^{43} -1.14616e7 q^{44} -6.98348e7 q^{46} -3.67369e7 q^{47} +3.81864e7 q^{49} +5.43247e7 q^{50} -9.07106e6 q^{52} -1.67334e7 q^{53} -9.34367e6 q^{55} -4.96215e7 q^{56} +1.14975e8 q^{58} +5.89865e7 q^{59} +1.28785e8 q^{61} +1.85842e8 q^{62} -2.02955e7 q^{64} -7.39484e6 q^{65} -1.91470e8 q^{67} -1.03985e8 q^{68} +6.60901e7 q^{70} -3.40865e8 q^{71} +3.19890e8 q^{73} +2.34887e8 q^{74} +8.43316e7 q^{76} +3.19821e8 q^{77} +2.44328e8 q^{79} -8.38584e7 q^{80} -2.07450e7 q^{82} +4.38630e8 q^{83} -8.47697e7 q^{85} +1.19026e9 q^{86} -2.02063e8 q^{88} +7.90342e8 q^{89} +2.53116e8 q^{91} +7.70054e8 q^{92} +1.05813e9 q^{94} +6.87481e7 q^{95} -1.65696e8 q^{97} -1.09988e9 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 33 q^{2} + 1429 q^{4} - 471 q^{5} - 11241 q^{7} + 45543 q^{8}+O(q^{10})$$ 4 * q + 33 * q^2 + 1429 * q^4 - 471 * q^5 - 11241 * q^7 + 45543 * q^8 $$4 q + 33 q^{2} + 1429 q^{4} - 471 q^{5} - 11241 q^{7} + 45543 q^{8} - 67831 q^{10} + 40140 q^{11} - 114244 q^{13} + 277653 q^{14} + 726609 q^{16} - 78717 q^{17} + 209664 q^{19} - 870843 q^{20} + 1364090 q^{22} + 4257444 q^{23} - 2900157 q^{25} - 942513 q^{26} + 4035181 q^{28} + 1647936 q^{29} - 11366002 q^{31} + 29458959 q^{32} + 26257659 q^{34} + 13789797 q^{35} + 4636891 q^{37} - 25172466 q^{38} + 22536791 q^{40} - 13859538 q^{41} - 33368081 q^{43} - 66489222 q^{44} + 71369332 q^{46} + 3943005 q^{47} + 23294923 q^{49} + 4217748 q^{50} - 40813669 q^{52} + 171019326 q^{53} - 121160538 q^{55} + 281552967 q^{56} + 79964734 q^{58} + 63389388 q^{59} + 77050190 q^{61} + 95878740 q^{62} + 768962465 q^{64} + 13452231 q^{65} - 41174072 q^{67} + 717615423 q^{68} + 409056389 q^{70} - 252460989 q^{71} + 594415068 q^{73} + 957058539 q^{74} - 326897170 q^{76} - 561950454 q^{77} + 115998984 q^{79} + 509107233 q^{80} - 875148240 q^{82} + 79577862 q^{83} + 549463469 q^{85} + 589924887 q^{86} - 2327564370 q^{88} + 1152240276 q^{89} + 321054201 q^{91} + 4213481460 q^{92} + 1859909503 q^{94} + 1273705170 q^{95} + 1049098084 q^{97} - 420532254 q^{98}+O(q^{100})$$ 4 * q + 33 * q^2 + 1429 * q^4 - 471 * q^5 - 11241 * q^7 + 45543 * q^8 - 67831 * q^10 + 40140 * q^11 - 114244 * q^13 + 277653 * q^14 + 726609 * q^16 - 78717 * q^17 + 209664 * q^19 - 870843 * q^20 + 1364090 * q^22 + 4257444 * q^23 - 2900157 * q^25 - 942513 * q^26 + 4035181 * q^28 + 1647936 * q^29 - 11366002 * q^31 + 29458959 * q^32 + 26257659 * q^34 + 13789797 * q^35 + 4636891 * q^37 - 25172466 * q^38 + 22536791 * q^40 - 13859538 * q^41 - 33368081 * q^43 - 66489222 * q^44 + 71369332 * q^46 + 3943005 * q^47 + 23294923 * q^49 + 4217748 * q^50 - 40813669 * q^52 + 171019326 * q^53 - 121160538 * q^55 + 281552967 * q^56 + 79964734 * q^58 + 63389388 * q^59 + 77050190 * q^61 + 95878740 * q^62 + 768962465 * q^64 + 13452231 * q^65 - 41174072 * q^67 + 717615423 * q^68 + 409056389 * q^70 - 252460989 * q^71 + 594415068 * q^73 + 957058539 * q^74 - 326897170 * q^76 - 561950454 * q^77 + 115998984 * q^79 + 509107233 * q^80 - 875148240 * q^82 + 79577862 * q^83 + 549463469 * q^85 + 589924887 * q^86 - 2327564370 * q^88 + 1152240276 * q^89 + 321054201 * q^91 + 4213481460 * q^92 + 1859909503 * q^94 + 1273705170 * q^95 + 1049098084 * q^97 - 420532254 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −28.8028 −1.27292 −0.636459 0.771311i $$-0.719601\pi$$
−0.636459 + 0.771311i $$0.719601\pi$$
$$3$$ 0 0
$$4$$ 317.603 0.620318
$$5$$ 258.914 0.185264 0.0926319 0.995700i $$-0.470472\pi$$
0.0926319 + 0.995700i $$0.470472\pi$$
$$6$$ 0 0
$$7$$ −8862.28 −1.39510 −0.697548 0.716538i $$-0.745726\pi$$
−0.697548 + 0.716538i $$0.745726\pi$$
$$8$$ 5599.18 0.483303
$$9$$ 0 0
$$10$$ −7457.46 −0.235825
$$11$$ −36087.9 −0.743181 −0.371591 0.928397i $$-0.621188\pi$$
−0.371591 + 0.928397i $$0.621188\pi$$
$$12$$ 0 0
$$13$$ −28561.0 −0.277350
$$14$$ 255259. 1.77584
$$15$$ 0 0
$$16$$ −323885. −1.23552
$$17$$ −327405. −0.950747 −0.475373 0.879784i $$-0.657687\pi$$
−0.475373 + 0.879784i $$0.657687\pi$$
$$18$$ 0 0
$$19$$ 265525. 0.467428 0.233714 0.972305i $$-0.424912\pi$$
0.233714 + 0.972305i $$0.424912\pi$$
$$20$$ 82231.9 0.114923
$$21$$ 0 0
$$22$$ 1.03943e6 0.946008
$$23$$ 2.42458e6 1.80660 0.903299 0.429012i $$-0.141138\pi$$
0.903299 + 0.429012i $$0.141138\pi$$
$$24$$ 0 0
$$25$$ −1.88609e6 −0.965677
$$26$$ 822638. 0.353044
$$27$$ 0 0
$$28$$ −2.81469e6 −0.865404
$$29$$ −3.99178e6 −1.04803 −0.524017 0.851708i $$-0.675567\pi$$
−0.524017 + 0.851708i $$0.675567\pi$$
$$30$$ 0 0
$$31$$ −6.45220e6 −1.25482 −0.627408 0.778691i $$-0.715884\pi$$
−0.627408 + 0.778691i $$0.715884\pi$$
$$32$$ 6.46202e6 1.08942
$$33$$ 0 0
$$34$$ 9.43018e6 1.21022
$$35$$ −2.29457e6 −0.258461
$$36$$ 0 0
$$37$$ −8.15498e6 −0.715344 −0.357672 0.933847i $$-0.616429\pi$$
−0.357672 + 0.933847i $$0.616429\pi$$
$$38$$ −7.64787e6 −0.594997
$$39$$ 0 0
$$40$$ 1.44971e6 0.0895386
$$41$$ 720241. 0.0398062 0.0199031 0.999802i $$-0.493664\pi$$
0.0199031 + 0.999802i $$0.493664\pi$$
$$42$$ 0 0
$$43$$ −4.13245e7 −1.84332 −0.921658 0.388004i $$-0.873165\pi$$
−0.921658 + 0.388004i $$0.873165\pi$$
$$44$$ −1.14616e7 −0.461009
$$45$$ 0 0
$$46$$ −6.98348e7 −2.29965
$$47$$ −3.67369e7 −1.09815 −0.549076 0.835772i $$-0.685020\pi$$
−0.549076 + 0.835772i $$0.685020\pi$$
$$48$$ 0 0
$$49$$ 3.81864e7 0.946295
$$50$$ 5.43247e7 1.22923
$$51$$ 0 0
$$52$$ −9.07106e6 −0.172045
$$53$$ −1.67334e7 −0.291302 −0.145651 0.989336i $$-0.546528\pi$$
−0.145651 + 0.989336i $$0.546528\pi$$
$$54$$ 0 0
$$55$$ −9.34367e6 −0.137685
$$56$$ −4.96215e7 −0.674255
$$57$$ 0 0
$$58$$ 1.14975e8 1.33406
$$59$$ 5.89865e7 0.633751 0.316875 0.948467i $$-0.397366\pi$$
0.316875 + 0.948467i $$0.397366\pi$$
$$60$$ 0 0
$$61$$ 1.28785e8 1.19091 0.595456 0.803388i $$-0.296971\pi$$
0.595456 + 0.803388i $$0.296971\pi$$
$$62$$ 1.85842e8 1.59728
$$63$$ 0 0
$$64$$ −2.02955e7 −0.151213
$$65$$ −7.39484e6 −0.0513829
$$66$$ 0 0
$$67$$ −1.91470e8 −1.16082 −0.580410 0.814324i $$-0.697108\pi$$
−0.580410 + 0.814324i $$0.697108\pi$$
$$68$$ −1.03985e8 −0.589766
$$69$$ 0 0
$$70$$ 6.60901e7 0.328999
$$71$$ −3.40865e8 −1.59192 −0.795958 0.605351i $$-0.793033\pi$$
−0.795958 + 0.605351i $$0.793033\pi$$
$$72$$ 0 0
$$73$$ 3.19890e8 1.31840 0.659200 0.751968i $$-0.270895\pi$$
0.659200 + 0.751968i $$0.270895\pi$$
$$74$$ 2.34887e8 0.910574
$$75$$ 0 0
$$76$$ 8.43316e7 0.289954
$$77$$ 3.19821e8 1.03681
$$78$$ 0 0
$$79$$ 2.44328e8 0.705751 0.352876 0.935670i $$-0.385204\pi$$
0.352876 + 0.935670i $$0.385204\pi$$
$$80$$ −8.38584e7 −0.228898
$$81$$ 0 0
$$82$$ −2.07450e7 −0.0506700
$$83$$ 4.38630e8 1.01449 0.507244 0.861802i $$-0.330664\pi$$
0.507244 + 0.861802i $$0.330664\pi$$
$$84$$ 0 0
$$85$$ −8.47697e7 −0.176139
$$86$$ 1.19026e9 2.34639
$$87$$ 0 0
$$88$$ −2.02063e8 −0.359182
$$89$$ 7.90342e8 1.33524 0.667621 0.744501i $$-0.267312\pi$$
0.667621 + 0.744501i $$0.267312\pi$$
$$90$$ 0 0
$$91$$ 2.53116e8 0.386930
$$92$$ 7.70054e8 1.12067
$$93$$ 0 0
$$94$$ 1.05813e9 1.39786
$$95$$ 6.87481e7 0.0865974
$$96$$ 0 0
$$97$$ −1.65696e8 −0.190037 −0.0950185 0.995476i $$-0.530291\pi$$
−0.0950185 + 0.995476i $$0.530291\pi$$
$$98$$ −1.09988e9 −1.20456
$$99$$ 0 0
$$100$$ −5.99027e8 −0.599027
$$101$$ 7.41636e8 0.709161 0.354580 0.935026i $$-0.384624\pi$$
0.354580 + 0.935026i $$0.384624\pi$$
$$102$$ 0 0
$$103$$ 1.42175e9 1.24467 0.622336 0.782750i $$-0.286184\pi$$
0.622336 + 0.782750i $$0.286184\pi$$
$$104$$ −1.59918e8 −0.134044
$$105$$ 0 0
$$106$$ 4.81970e8 0.370803
$$107$$ 1.44196e9 1.06347 0.531736 0.846910i $$-0.321540\pi$$
0.531736 + 0.846910i $$0.321540\pi$$
$$108$$ 0 0
$$109$$ −8.42758e7 −0.0571852 −0.0285926 0.999591i $$-0.509103\pi$$
−0.0285926 + 0.999591i $$0.509103\pi$$
$$110$$ 2.69124e8 0.175261
$$111$$ 0 0
$$112$$ 2.87036e9 1.72367
$$113$$ −8.60241e8 −0.496326 −0.248163 0.968718i $$-0.579827\pi$$
−0.248163 + 0.968718i $$0.579827\pi$$
$$114$$ 0 0
$$115$$ 6.27758e8 0.334697
$$116$$ −1.26780e9 −0.650115
$$117$$ 0 0
$$118$$ −1.69898e9 −0.806712
$$119$$ 2.90155e9 1.32638
$$120$$ 0 0
$$121$$ −1.05561e9 −0.447681
$$122$$ −3.70936e9 −1.51593
$$123$$ 0 0
$$124$$ −2.04924e9 −0.778386
$$125$$ −9.94026e8 −0.364169
$$126$$ 0 0
$$127$$ −1.44120e9 −0.491596 −0.245798 0.969321i $$-0.579050\pi$$
−0.245798 + 0.969321i $$0.579050\pi$$
$$128$$ −2.72399e9 −0.896934
$$129$$ 0 0
$$130$$ 2.12992e8 0.0654062
$$131$$ 4.54928e9 1.34965 0.674827 0.737976i $$-0.264218\pi$$
0.674827 + 0.737976i $$0.264218\pi$$
$$132$$ 0 0
$$133$$ −2.35316e9 −0.652107
$$134$$ 5.51489e9 1.47763
$$135$$ 0 0
$$136$$ −1.83320e9 −0.459499
$$137$$ 1.99742e9 0.484426 0.242213 0.970223i $$-0.422127\pi$$
0.242213 + 0.970223i $$0.422127\pi$$
$$138$$ 0 0
$$139$$ 3.55325e8 0.0807345 0.0403672 0.999185i $$-0.487147\pi$$
0.0403672 + 0.999185i $$0.487147\pi$$
$$140$$ −7.28762e8 −0.160328
$$141$$ 0 0
$$142$$ 9.81789e9 2.02638
$$143$$ 1.03071e9 0.206121
$$144$$ 0 0
$$145$$ −1.03353e9 −0.194163
$$146$$ −9.21372e9 −1.67821
$$147$$ 0 0
$$148$$ −2.59005e9 −0.443741
$$149$$ −5.92546e9 −0.984881 −0.492440 0.870346i $$-0.663895\pi$$
−0.492440 + 0.870346i $$0.663895\pi$$
$$150$$ 0 0
$$151$$ −1.07568e10 −1.68378 −0.841891 0.539647i $$-0.818558\pi$$
−0.841891 + 0.539647i $$0.818558\pi$$
$$152$$ 1.48672e9 0.225909
$$153$$ 0 0
$$154$$ −9.21176e9 −1.31977
$$155$$ −1.67056e9 −0.232472
$$156$$ 0 0
$$157$$ 8.87634e9 1.16597 0.582983 0.812485i $$-0.301886\pi$$
0.582983 + 0.812485i $$0.301886\pi$$
$$158$$ −7.03734e9 −0.898363
$$159$$ 0 0
$$160$$ 1.67311e9 0.201829
$$161$$ −2.14873e10 −2.52038
$$162$$ 0 0
$$163$$ 2.75052e9 0.305190 0.152595 0.988289i $$-0.451237\pi$$
0.152595 + 0.988289i $$0.451237\pi$$
$$164$$ 2.28751e8 0.0246925
$$165$$ 0 0
$$166$$ −1.26338e10 −1.29136
$$167$$ 9.17849e9 0.913160 0.456580 0.889682i $$-0.349074\pi$$
0.456580 + 0.889682i $$0.349074\pi$$
$$168$$ 0 0
$$169$$ 8.15731e8 0.0769231
$$170$$ 2.44161e9 0.224210
$$171$$ 0 0
$$172$$ −1.31248e10 −1.14344
$$173$$ −9.75049e8 −0.0827598 −0.0413799 0.999143i $$-0.513175\pi$$
−0.0413799 + 0.999143i $$0.513175\pi$$
$$174$$ 0 0
$$175$$ 1.67150e10 1.34721
$$176$$ 1.16883e10 0.918218
$$177$$ 0 0
$$178$$ −2.27641e10 −1.69965
$$179$$ 8.88356e9 0.646768 0.323384 0.946268i $$-0.395180\pi$$
0.323384 + 0.946268i $$0.395180\pi$$
$$180$$ 0 0
$$181$$ −5.22791e9 −0.362055 −0.181028 0.983478i $$-0.557942\pi$$
−0.181028 + 0.983478i $$0.557942\pi$$
$$182$$ −7.29045e9 −0.492530
$$183$$ 0 0
$$184$$ 1.35757e10 0.873134
$$185$$ −2.11144e9 −0.132527
$$186$$ 0 0
$$187$$ 1.18154e10 0.706577
$$188$$ −1.16678e10 −0.681204
$$189$$ 0 0
$$190$$ −1.98014e9 −0.110231
$$191$$ −9.23471e9 −0.502080 −0.251040 0.967977i $$-0.580773\pi$$
−0.251040 + 0.967977i $$0.580773\pi$$
$$192$$ 0 0
$$193$$ 1.07065e10 0.555441 0.277720 0.960662i $$-0.410421\pi$$
0.277720 + 0.960662i $$0.410421\pi$$
$$194$$ 4.77250e9 0.241901
$$195$$ 0 0
$$196$$ 1.21281e10 0.587004
$$197$$ 2.77001e10 1.31034 0.655169 0.755482i $$-0.272597\pi$$
0.655169 + 0.755482i $$0.272597\pi$$
$$198$$ 0 0
$$199$$ 3.32299e10 1.50207 0.751035 0.660262i $$-0.229555\pi$$
0.751035 + 0.660262i $$0.229555\pi$$
$$200$$ −1.05606e10 −0.466715
$$201$$ 0 0
$$202$$ −2.13612e10 −0.902703
$$203$$ 3.53763e10 1.46211
$$204$$ 0 0
$$205$$ 1.86481e8 0.00737465
$$206$$ −4.09503e10 −1.58436
$$207$$ 0 0
$$208$$ 9.25048e9 0.342673
$$209$$ −9.58225e9 −0.347383
$$210$$ 0 0
$$211$$ −1.49261e10 −0.518411 −0.259205 0.965822i $$-0.583461\pi$$
−0.259205 + 0.965822i $$0.583461\pi$$
$$212$$ −5.31458e9 −0.180700
$$213$$ 0 0
$$214$$ −4.15325e10 −1.35371
$$215$$ −1.06995e10 −0.341500
$$216$$ 0 0
$$217$$ 5.71812e10 1.75059
$$218$$ 2.42738e9 0.0727920
$$219$$ 0 0
$$220$$ −2.96758e9 −0.0854083
$$221$$ 9.35101e9 0.263690
$$222$$ 0 0
$$223$$ 2.23373e10 0.604865 0.302433 0.953171i $$-0.402201\pi$$
0.302433 + 0.953171i $$0.402201\pi$$
$$224$$ −5.72683e10 −1.51984
$$225$$ 0 0
$$226$$ 2.47774e10 0.631782
$$227$$ −5.20726e9 −0.130165 −0.0650824 0.997880i $$-0.520731\pi$$
−0.0650824 + 0.997880i $$0.520731\pi$$
$$228$$ 0 0
$$229$$ 1.35573e10 0.325773 0.162887 0.986645i $$-0.447920\pi$$
0.162887 + 0.986645i $$0.447920\pi$$
$$230$$ −1.80812e10 −0.426042
$$231$$ 0 0
$$232$$ −2.23507e10 −0.506518
$$233$$ −2.06224e8 −0.00458393 −0.00229196 0.999997i $$-0.500730\pi$$
−0.00229196 + 0.999997i $$0.500730\pi$$
$$234$$ 0 0
$$235$$ −9.51170e9 −0.203448
$$236$$ 1.87343e10 0.393127
$$237$$ 0 0
$$238$$ −8.35729e10 −1.68838
$$239$$ −1.89887e10 −0.376448 −0.188224 0.982126i $$-0.560273\pi$$
−0.188224 + 0.982126i $$0.560273\pi$$
$$240$$ 0 0
$$241$$ −6.11533e10 −1.16773 −0.583866 0.811850i $$-0.698461\pi$$
−0.583866 + 0.811850i $$0.698461\pi$$
$$242$$ 3.04045e10 0.569861
$$243$$ 0 0
$$244$$ 4.09024e10 0.738745
$$245$$ 9.88700e9 0.175314
$$246$$ 0 0
$$247$$ −7.58366e9 −0.129641
$$248$$ −3.61270e10 −0.606457
$$249$$ 0 0
$$250$$ 2.86308e10 0.463557
$$251$$ 3.20685e9 0.0509973 0.0254986 0.999675i $$-0.491883\pi$$
0.0254986 + 0.999675i $$0.491883\pi$$
$$252$$ 0 0
$$253$$ −8.74981e10 −1.34263
$$254$$ 4.15107e10 0.625761
$$255$$ 0 0
$$256$$ 8.88499e10 1.29294
$$257$$ 2.80963e10 0.401744 0.200872 0.979617i $$-0.435622\pi$$
0.200872 + 0.979617i $$0.435622\pi$$
$$258$$ 0 0
$$259$$ 7.22717e10 0.997975
$$260$$ −2.34862e9 −0.0318738
$$261$$ 0 0
$$262$$ −1.31032e11 −1.71800
$$263$$ 1.25524e11 1.61780 0.808902 0.587944i $$-0.200062\pi$$
0.808902 + 0.587944i $$0.200062\pi$$
$$264$$ 0 0
$$265$$ −4.33252e9 −0.0539677
$$266$$ 6.77776e10 0.830078
$$267$$ 0 0
$$268$$ −6.08116e10 −0.720078
$$269$$ 1.78558e10 0.207918 0.103959 0.994582i $$-0.466849\pi$$
0.103959 + 0.994582i $$0.466849\pi$$
$$270$$ 0 0
$$271$$ 1.25204e11 1.41013 0.705063 0.709144i $$-0.250919\pi$$
0.705063 + 0.709144i $$0.250919\pi$$
$$272$$ 1.06041e11 1.17467
$$273$$ 0 0
$$274$$ −5.75314e10 −0.616634
$$275$$ 6.80650e10 0.717673
$$276$$ 0 0
$$277$$ −5.55594e10 −0.567020 −0.283510 0.958969i $$-0.591499\pi$$
−0.283510 + 0.958969i $$0.591499\pi$$
$$278$$ −1.02344e10 −0.102768
$$279$$ 0 0
$$280$$ −1.28477e10 −0.124915
$$281$$ −1.01237e11 −0.968635 −0.484317 0.874892i $$-0.660932\pi$$
−0.484317 + 0.874892i $$0.660932\pi$$
$$282$$ 0 0
$$283$$ −1.08041e11 −1.00127 −0.500635 0.865658i $$-0.666900\pi$$
−0.500635 + 0.865658i $$0.666900\pi$$
$$284$$ −1.08260e11 −0.987495
$$285$$ 0 0
$$286$$ −2.96873e10 −0.262376
$$287$$ −6.38298e9 −0.0555335
$$288$$ 0 0
$$289$$ −1.13940e10 −0.0960810
$$290$$ 2.97685e10 0.247153
$$291$$ 0 0
$$292$$ 1.01598e11 0.817828
$$293$$ −3.53224e9 −0.0279992 −0.0139996 0.999902i $$-0.504456\pi$$
−0.0139996 + 0.999902i $$0.504456\pi$$
$$294$$ 0 0
$$295$$ 1.52724e10 0.117411
$$296$$ −4.56612e10 −0.345728
$$297$$ 0 0
$$298$$ 1.70670e11 1.25367
$$299$$ −6.92485e10 −0.501060
$$300$$ 0 0
$$301$$ 3.66229e11 2.57160
$$302$$ 3.09826e11 2.14332
$$303$$ 0 0
$$304$$ −8.59996e10 −0.577518
$$305$$ 3.33441e10 0.220633
$$306$$ 0 0
$$307$$ −1.82031e11 −1.16956 −0.584781 0.811191i $$-0.698820\pi$$
−0.584781 + 0.811191i $$0.698820\pi$$
$$308$$ 1.01576e11 0.643152
$$309$$ 0 0
$$310$$ 4.81170e10 0.295918
$$311$$ 1.51907e11 0.920781 0.460390 0.887717i $$-0.347709\pi$$
0.460390 + 0.887717i $$0.347709\pi$$
$$312$$ 0 0
$$313$$ 6.10194e10 0.359351 0.179675 0.983726i $$-0.442495\pi$$
0.179675 + 0.983726i $$0.442495\pi$$
$$314$$ −2.55664e11 −1.48418
$$315$$ 0 0
$$316$$ 7.75994e10 0.437790
$$317$$ −6.89227e10 −0.383350 −0.191675 0.981458i $$-0.561392\pi$$
−0.191675 + 0.981458i $$0.561392\pi$$
$$318$$ 0 0
$$319$$ 1.44055e11 0.778880
$$320$$ −5.25478e9 −0.0280143
$$321$$ 0 0
$$322$$ 6.18896e11 3.20823
$$323$$ −8.69341e10 −0.444405
$$324$$ 0 0
$$325$$ 5.38686e10 0.267831
$$326$$ −7.92227e10 −0.388481
$$327$$ 0 0
$$328$$ 4.03276e9 0.0192385
$$329$$ 3.25573e11 1.53203
$$330$$ 0 0
$$331$$ 4.70422e10 0.215408 0.107704 0.994183i $$-0.465650\pi$$
0.107704 + 0.994183i $$0.465650\pi$$
$$332$$ 1.39310e11 0.629306
$$333$$ 0 0
$$334$$ −2.64366e11 −1.16238
$$335$$ −4.95744e10 −0.215058
$$336$$ 0 0
$$337$$ −3.76711e11 −1.59101 −0.795507 0.605944i $$-0.792795\pi$$
−0.795507 + 0.605944i $$0.792795\pi$$
$$338$$ −2.34954e10 −0.0979167
$$339$$ 0 0
$$340$$ −2.69231e10 −0.109262
$$341$$ 2.32846e11 0.932556
$$342$$ 0 0
$$343$$ 1.92062e10 0.0749235
$$344$$ −2.31383e11 −0.890880
$$345$$ 0 0
$$346$$ 2.80842e10 0.105346
$$347$$ −1.68331e11 −0.623278 −0.311639 0.950201i $$-0.600878\pi$$
−0.311639 + 0.950201i $$0.600878\pi$$
$$348$$ 0 0
$$349$$ −9.36126e10 −0.337769 −0.168884 0.985636i $$-0.554016\pi$$
−0.168884 + 0.985636i $$0.554016\pi$$
$$350$$ −4.81441e11 −1.71489
$$351$$ 0 0
$$352$$ −2.33201e11 −0.809634
$$353$$ 3.49043e11 1.19644 0.598222 0.801330i $$-0.295874\pi$$
0.598222 + 0.801330i $$0.295874\pi$$
$$354$$ 0 0
$$355$$ −8.82548e10 −0.294924
$$356$$ 2.51015e11 0.828276
$$357$$ 0 0
$$358$$ −2.55872e11 −0.823282
$$359$$ −5.83173e11 −1.85299 −0.926493 0.376312i $$-0.877192\pi$$
−0.926493 + 0.376312i $$0.877192\pi$$
$$360$$ 0 0
$$361$$ −2.52184e11 −0.781512
$$362$$ 1.50579e11 0.460866
$$363$$ 0 0
$$364$$ 8.03903e10 0.240020
$$365$$ 8.28239e10 0.244252
$$366$$ 0 0
$$367$$ 5.28604e11 1.52101 0.760507 0.649329i $$-0.224950\pi$$
0.760507 + 0.649329i $$0.224950\pi$$
$$368$$ −7.85286e11 −2.23209
$$369$$ 0 0
$$370$$ 6.08154e10 0.168696
$$371$$ 1.48296e11 0.406394
$$372$$ 0 0
$$373$$ −4.19405e11 −1.12187 −0.560937 0.827858i $$-0.689559\pi$$
−0.560937 + 0.827858i $$0.689559\pi$$
$$374$$ −3.40316e11 −0.899414
$$375$$ 0 0
$$376$$ −2.05697e11 −0.530740
$$377$$ 1.14009e11 0.290672
$$378$$ 0 0
$$379$$ 4.24128e11 1.05590 0.527948 0.849277i $$-0.322962\pi$$
0.527948 + 0.849277i $$0.322962\pi$$
$$380$$ 2.18346e10 0.0537180
$$381$$ 0 0
$$382$$ 2.65986e11 0.639107
$$383$$ −6.52696e10 −0.154995 −0.0774973 0.996993i $$-0.524693\pi$$
−0.0774973 + 0.996993i $$0.524693\pi$$
$$384$$ 0 0
$$385$$ 8.28062e10 0.192083
$$386$$ −3.08376e11 −0.707030
$$387$$ 0 0
$$388$$ −5.26254e10 −0.117883
$$389$$ 5.91803e11 1.31040 0.655200 0.755455i $$-0.272584\pi$$
0.655200 + 0.755455i $$0.272584\pi$$
$$390$$ 0 0
$$391$$ −7.93819e11 −1.71762
$$392$$ 2.13813e11 0.457347
$$393$$ 0 0
$$394$$ −7.97842e11 −1.66795
$$395$$ 6.32600e10 0.130750
$$396$$ 0 0
$$397$$ −3.05131e11 −0.616495 −0.308248 0.951306i $$-0.599743\pi$$
−0.308248 + 0.951306i $$0.599743\pi$$
$$398$$ −9.57115e11 −1.91201
$$399$$ 0 0
$$400$$ 6.10876e11 1.19312
$$401$$ −2.56722e11 −0.495808 −0.247904 0.968785i $$-0.579742\pi$$
−0.247904 + 0.968785i $$0.579742\pi$$
$$402$$ 0 0
$$403$$ 1.84281e11 0.348023
$$404$$ 2.35546e11 0.439906
$$405$$ 0 0
$$406$$ −1.01894e12 −1.86114
$$407$$ 2.94296e11 0.531631
$$408$$ 0 0
$$409$$ −4.57003e11 −0.807540 −0.403770 0.914860i $$-0.632300\pi$$
−0.403770 + 0.914860i $$0.632300\pi$$
$$410$$ −5.37117e9 −0.00938732
$$411$$ 0 0
$$412$$ 4.51551e11 0.772093
$$413$$ −5.22755e11 −0.884144
$$414$$ 0 0
$$415$$ 1.13567e11 0.187948
$$416$$ −1.84562e11 −0.302150
$$417$$ 0 0
$$418$$ 2.75996e11 0.442190
$$419$$ 1.65068e11 0.261637 0.130818 0.991406i $$-0.458240\pi$$
0.130818 + 0.991406i $$0.458240\pi$$
$$420$$ 0 0
$$421$$ −9.10255e10 −0.141219 −0.0706096 0.997504i $$-0.522494\pi$$
−0.0706096 + 0.997504i $$0.522494\pi$$
$$422$$ 4.29913e11 0.659894
$$423$$ 0 0
$$424$$ −9.36935e10 −0.140787
$$425$$ 6.17514e11 0.918114
$$426$$ 0 0
$$427$$ −1.14133e12 −1.66144
$$428$$ 4.57971e11 0.659691
$$429$$ 0 0
$$430$$ 3.08176e11 0.434701
$$431$$ −7.94605e11 −1.10918 −0.554592 0.832122i $$-0.687126\pi$$
−0.554592 + 0.832122i $$0.687126\pi$$
$$432$$ 0 0
$$433$$ −8.26325e11 −1.12968 −0.564840 0.825200i $$-0.691062\pi$$
−0.564840 + 0.825200i $$0.691062\pi$$
$$434$$ −1.64698e12 −2.22836
$$435$$ 0 0
$$436$$ −2.67662e10 −0.0354730
$$437$$ 6.43787e11 0.844453
$$438$$ 0 0
$$439$$ 1.21562e12 1.56210 0.781050 0.624469i $$-0.214684\pi$$
0.781050 + 0.624469i $$0.214684\pi$$
$$440$$ −5.23169e10 −0.0665434
$$441$$ 0 0
$$442$$ −2.69335e11 −0.335655
$$443$$ 8.73136e11 1.07712 0.538561 0.842586i $$-0.318968\pi$$
0.538561 + 0.842586i $$0.318968\pi$$
$$444$$ 0 0
$$445$$ 2.04631e11 0.247372
$$446$$ −6.43377e11 −0.769943
$$447$$ 0 0
$$448$$ 1.79864e11 0.210957
$$449$$ −1.22107e12 −1.41785 −0.708927 0.705282i $$-0.750821\pi$$
−0.708927 + 0.705282i $$0.750821\pi$$
$$450$$ 0 0
$$451$$ −2.59920e10 −0.0295832
$$452$$ −2.73215e11 −0.307880
$$453$$ 0 0
$$454$$ 1.49984e11 0.165689
$$455$$ 6.55352e10 0.0716842
$$456$$ 0 0
$$457$$ 6.32643e11 0.678478 0.339239 0.940700i $$-0.389830\pi$$
0.339239 + 0.940700i $$0.389830\pi$$
$$458$$ −3.90490e11 −0.414682
$$459$$ 0 0
$$460$$ 1.99378e11 0.207619
$$461$$ −8.09117e11 −0.834367 −0.417184 0.908822i $$-0.636983\pi$$
−0.417184 + 0.908822i $$0.636983\pi$$
$$462$$ 0 0
$$463$$ 7.69685e11 0.778393 0.389196 0.921155i $$-0.372753\pi$$
0.389196 + 0.921155i $$0.372753\pi$$
$$464$$ 1.29288e12 1.29487
$$465$$ 0 0
$$466$$ 5.93984e9 0.00583496
$$467$$ −1.59708e12 −1.55382 −0.776909 0.629613i $$-0.783213\pi$$
−0.776909 + 0.629613i $$0.783213\pi$$
$$468$$ 0 0
$$469$$ 1.69686e12 1.61946
$$470$$ 2.73964e11 0.258972
$$471$$ 0 0
$$472$$ 3.30276e11 0.306294
$$473$$ 1.49132e12 1.36992
$$474$$ 0 0
$$475$$ −5.00804e11 −0.451384
$$476$$ 9.21542e11 0.822780
$$477$$ 0 0
$$478$$ 5.46929e11 0.479187
$$479$$ −1.61308e12 −1.40006 −0.700028 0.714115i $$-0.746829\pi$$
−0.700028 + 0.714115i $$0.746829\pi$$
$$480$$ 0 0
$$481$$ 2.32914e11 0.198401
$$482$$ 1.76139e12 1.48643
$$483$$ 0 0
$$484$$ −3.35265e11 −0.277705
$$485$$ −4.29009e10 −0.0352070
$$486$$ 0 0
$$487$$ 9.29058e11 0.748449 0.374225 0.927338i $$-0.377909\pi$$
0.374225 + 0.927338i $$0.377909\pi$$
$$488$$ 7.21089e11 0.575572
$$489$$ 0 0
$$490$$ −2.84774e11 −0.223160
$$491$$ 2.30662e12 1.79106 0.895528 0.445005i $$-0.146798\pi$$
0.895528 + 0.445005i $$0.146798\pi$$
$$492$$ 0 0
$$493$$ 1.30693e12 0.996415
$$494$$ 2.18431e11 0.165022
$$495$$ 0 0
$$496$$ 2.08977e12 1.55036
$$497$$ 3.02084e12 2.22088
$$498$$ 0 0
$$499$$ −1.31410e11 −0.0948800 −0.0474400 0.998874i $$-0.515106\pi$$
−0.0474400 + 0.998874i $$0.515106\pi$$
$$500$$ −3.15706e11 −0.225901
$$501$$ 0 0
$$502$$ −9.23664e10 −0.0649153
$$503$$ 1.88419e12 1.31241 0.656203 0.754584i $$-0.272162\pi$$
0.656203 + 0.754584i $$0.272162\pi$$
$$504$$ 0 0
$$505$$ 1.92020e11 0.131382
$$506$$ 2.52019e12 1.70906
$$507$$ 0 0
$$508$$ −4.57730e11 −0.304946
$$509$$ 1.00390e11 0.0662919 0.0331459 0.999451i $$-0.489447\pi$$
0.0331459 + 0.999451i $$0.489447\pi$$
$$510$$ 0 0
$$511$$ −2.83495e12 −1.83930
$$512$$ −1.16445e12 −0.748866
$$513$$ 0 0
$$514$$ −8.09252e11 −0.511387
$$515$$ 3.68110e11 0.230593
$$516$$ 0 0
$$517$$ 1.32576e12 0.816126
$$518$$ −2.08163e12 −1.27034
$$519$$ 0 0
$$520$$ −4.14051e10 −0.0248335
$$521$$ 1.04516e11 0.0621462 0.0310731 0.999517i $$-0.490108\pi$$
0.0310731 + 0.999517i $$0.490108\pi$$
$$522$$ 0 0
$$523$$ 1.17694e12 0.687855 0.343927 0.938996i $$-0.388243\pi$$
0.343927 + 0.938996i $$0.388243\pi$$
$$524$$ 1.44487e12 0.837215
$$525$$ 0 0
$$526$$ −3.61544e12 −2.05933
$$527$$ 2.11248e12 1.19301
$$528$$ 0 0
$$529$$ 4.07744e12 2.26379
$$530$$ 1.24789e11 0.0686964
$$531$$ 0 0
$$532$$ −7.47370e11 −0.404514
$$533$$ −2.05708e10 −0.0110403
$$534$$ 0 0
$$535$$ 3.73343e11 0.197023
$$536$$ −1.07208e12 −0.561028
$$537$$ 0 0
$$538$$ −5.14296e11 −0.264663
$$539$$ −1.37807e12 −0.703269
$$540$$ 0 0
$$541$$ −2.98502e12 −1.49817 −0.749083 0.662476i $$-0.769506\pi$$
−0.749083 + 0.662476i $$0.769506\pi$$
$$542$$ −3.60624e12 −1.79497
$$543$$ 0 0
$$544$$ −2.11570e12 −1.03576
$$545$$ −2.18202e10 −0.0105943
$$546$$ 0 0
$$547$$ 3.78628e12 1.80830 0.904148 0.427219i $$-0.140507\pi$$
0.904148 + 0.427219i $$0.140507\pi$$
$$548$$ 6.34387e11 0.300498
$$549$$ 0 0
$$550$$ −1.96047e12 −0.913539
$$551$$ −1.05992e12 −0.489880
$$552$$ 0 0
$$553$$ −2.16530e12 −0.984591
$$554$$ 1.60027e12 0.721770
$$555$$ 0 0
$$556$$ 1.12852e11 0.0500811
$$557$$ −1.00551e12 −0.442627 −0.221313 0.975203i $$-0.571034\pi$$
−0.221313 + 0.975203i $$0.571034\pi$$
$$558$$ 0 0
$$559$$ 1.18027e12 0.511244
$$560$$ 7.43177e11 0.319335
$$561$$ 0 0
$$562$$ 2.91591e12 1.23299
$$563$$ −2.76515e11 −0.115993 −0.0579964 0.998317i $$-0.518471\pi$$
−0.0579964 + 0.998317i $$0.518471\pi$$
$$564$$ 0 0
$$565$$ −2.22728e11 −0.0919512
$$566$$ 3.11190e12 1.27453
$$567$$ 0 0
$$568$$ −1.90857e12 −0.769378
$$569$$ −8.39220e11 −0.335638 −0.167819 0.985818i $$-0.553672\pi$$
−0.167819 + 0.985818i $$0.553672\pi$$
$$570$$ 0 0
$$571$$ −1.02285e11 −0.0402672 −0.0201336 0.999797i $$-0.506409\pi$$
−0.0201336 + 0.999797i $$0.506409\pi$$
$$572$$ 3.27356e11 0.127861
$$573$$ 0 0
$$574$$ 1.83848e11 0.0706896
$$575$$ −4.57297e12 −1.74459
$$576$$ 0 0
$$577$$ −1.44021e12 −0.540921 −0.270461 0.962731i $$-0.587176\pi$$
−0.270461 + 0.962731i $$0.587176\pi$$
$$578$$ 3.28181e11 0.122303
$$579$$ 0 0
$$580$$ −3.28251e11 −0.120443
$$581$$ −3.88726e12 −1.41531
$$582$$ 0 0
$$583$$ 6.03874e11 0.216490
$$584$$ 1.79112e12 0.637187
$$585$$ 0 0
$$586$$ 1.01738e11 0.0356407
$$587$$ 2.13185e11 0.0741116 0.0370558 0.999313i $$-0.488202\pi$$
0.0370558 + 0.999313i $$0.488202\pi$$
$$588$$ 0 0
$$589$$ −1.71322e12 −0.586536
$$590$$ −4.39889e11 −0.149455
$$591$$ 0 0
$$592$$ 2.64128e12 0.883825
$$593$$ 1.76049e12 0.584640 0.292320 0.956321i $$-0.405573\pi$$
0.292320 + 0.956321i $$0.405573\pi$$
$$594$$ 0 0
$$595$$ 7.51253e11 0.245731
$$596$$ −1.88194e12 −0.610940
$$597$$ 0 0
$$598$$ 1.99455e12 0.637808
$$599$$ 2.06753e12 0.656191 0.328096 0.944644i $$-0.393593\pi$$
0.328096 + 0.944644i $$0.393593\pi$$
$$600$$ 0 0
$$601$$ −1.24403e12 −0.388951 −0.194476 0.980907i $$-0.562301\pi$$
−0.194476 + 0.980907i $$0.562301\pi$$
$$602$$ −1.05484e13 −3.27344
$$603$$ 0 0
$$604$$ −3.41639e12 −1.04448
$$605$$ −2.73312e11 −0.0829391
$$606$$ 0 0
$$607$$ −1.31511e12 −0.393200 −0.196600 0.980484i $$-0.562990\pi$$
−0.196600 + 0.980484i $$0.562990\pi$$
$$608$$ 1.71583e12 0.509223
$$609$$ 0 0
$$610$$ −9.60406e11 −0.280847
$$611$$ 1.04924e12 0.304573
$$612$$ 0 0
$$613$$ −3.08252e12 −0.881726 −0.440863 0.897574i $$-0.645328\pi$$
−0.440863 + 0.897574i $$0.645328\pi$$
$$614$$ 5.24301e12 1.48876
$$615$$ 0 0
$$616$$ 1.79074e12 0.501094
$$617$$ 2.29075e10 0.00636348 0.00318174 0.999995i $$-0.498987\pi$$
0.00318174 + 0.999995i $$0.498987\pi$$
$$618$$ 0 0
$$619$$ −4.45350e12 −1.21925 −0.609626 0.792689i $$-0.708680\pi$$
−0.609626 + 0.792689i $$0.708680\pi$$
$$620$$ −5.30576e11 −0.144207
$$621$$ 0 0
$$622$$ −4.37535e12 −1.17208
$$623$$ −7.00424e12 −1.86279
$$624$$ 0 0
$$625$$ 3.42640e12 0.898210
$$626$$ −1.75753e12 −0.457423
$$627$$ 0 0
$$628$$ 2.81915e12 0.723270
$$629$$ 2.66998e12 0.680111
$$630$$ 0 0
$$631$$ 3.83831e12 0.963847 0.481923 0.876213i $$-0.339938\pi$$
0.481923 + 0.876213i $$0.339938\pi$$
$$632$$ 1.36804e12 0.341092
$$633$$ 0 0
$$634$$ 1.98517e12 0.487973
$$635$$ −3.73148e11 −0.0910749
$$636$$ 0 0
$$637$$ −1.09064e12 −0.262455
$$638$$ −4.14919e12 −0.991449
$$639$$ 0 0
$$640$$ −7.05279e11 −0.166169
$$641$$ −9.09238e11 −0.212724 −0.106362 0.994327i $$-0.533920\pi$$
−0.106362 + 0.994327i $$0.533920\pi$$
$$642$$ 0 0
$$643$$ 8.23839e12 1.90061 0.950305 0.311321i $$-0.100771\pi$$
0.950305 + 0.311321i $$0.100771\pi$$
$$644$$ −6.82444e12 −1.56344
$$645$$ 0 0
$$646$$ 2.50395e12 0.565691
$$647$$ 8.73559e12 1.95985 0.979925 0.199365i $$-0.0638880\pi$$
0.979925 + 0.199365i $$0.0638880\pi$$
$$648$$ 0 0
$$649$$ −2.12870e12 −0.470992
$$650$$ −1.55157e12 −0.340926
$$651$$ 0 0
$$652$$ 8.73573e11 0.189315
$$653$$ 6.13232e11 0.131982 0.0659911 0.997820i $$-0.478979\pi$$
0.0659911 + 0.997820i $$0.478979\pi$$
$$654$$ 0 0
$$655$$ 1.17787e12 0.250042
$$656$$ −2.33275e11 −0.0491815
$$657$$ 0 0
$$658$$ −9.37742e12 −1.95015
$$659$$ 8.83233e12 1.82428 0.912139 0.409882i $$-0.134430\pi$$
0.912139 + 0.409882i $$0.134430\pi$$
$$660$$ 0 0
$$661$$ −5.61801e12 −1.14466 −0.572329 0.820024i $$-0.693960\pi$$
−0.572329 + 0.820024i $$0.693960\pi$$
$$662$$ −1.35495e12 −0.274196
$$663$$ 0 0
$$664$$ 2.45597e12 0.490306
$$665$$ −6.09265e11 −0.120812
$$666$$ 0 0
$$667$$ −9.67839e12 −1.89338
$$668$$ 2.91511e12 0.566450
$$669$$ 0 0
$$670$$ 1.42788e12 0.273751
$$671$$ −4.64757e12 −0.885064
$$672$$ 0 0
$$673$$ 4.50151e12 0.845844 0.422922 0.906166i $$-0.361004\pi$$
0.422922 + 0.906166i $$0.361004\pi$$
$$674$$ 1.08503e13 2.02523
$$675$$ 0 0
$$676$$ 2.59079e11 0.0477168
$$677$$ −4.33098e12 −0.792387 −0.396193 0.918167i $$-0.629669\pi$$
−0.396193 + 0.918167i $$0.629669\pi$$
$$678$$ 0 0
$$679$$ 1.46844e12 0.265120
$$680$$ −4.74641e11 −0.0851285
$$681$$ 0 0
$$682$$ −6.70664e12 −1.18707
$$683$$ −6.25107e12 −1.09916 −0.549580 0.835441i $$-0.685212\pi$$
−0.549580 + 0.835441i $$0.685212\pi$$
$$684$$ 0 0
$$685$$ 5.17161e11 0.0897466
$$686$$ −5.53193e11 −0.0953715
$$687$$ 0 0
$$688$$ 1.33844e13 2.27746
$$689$$ 4.77923e11 0.0807926
$$690$$ 0 0
$$691$$ 7.34082e12 1.22488 0.612440 0.790517i $$-0.290188\pi$$
0.612440 + 0.790517i $$0.290188\pi$$
$$692$$ −3.09679e11 −0.0513374
$$693$$ 0 0
$$694$$ 4.84841e12 0.793381
$$695$$ 9.19986e10 0.0149572
$$696$$ 0 0
$$697$$ −2.35810e11 −0.0378456
$$698$$ 2.69631e12 0.429952
$$699$$ 0 0
$$700$$ 5.30875e12 0.835701
$$701$$ 1.71509e11 0.0268259 0.0134130 0.999910i $$-0.495730\pi$$
0.0134130 + 0.999910i $$0.495730\pi$$
$$702$$ 0 0
$$703$$ −2.16535e12 −0.334372
$$704$$ 7.32421e11 0.112379
$$705$$ 0 0
$$706$$ −1.00534e13 −1.52297
$$707$$ −6.57259e12 −0.989348
$$708$$ 0 0
$$709$$ −1.00896e13 −1.49956 −0.749780 0.661687i $$-0.769841\pi$$
−0.749780 + 0.661687i $$0.769841\pi$$
$$710$$ 2.54199e12 0.375414
$$711$$ 0 0
$$712$$ 4.42527e12 0.645327
$$713$$ −1.56439e13 −2.26695
$$714$$ 0 0
$$715$$ 2.66865e11 0.0381868
$$716$$ 2.82145e12 0.401202
$$717$$ 0 0
$$718$$ 1.67970e13 2.35870
$$719$$ −8.18068e12 −1.14159 −0.570794 0.821093i $$-0.693365\pi$$
−0.570794 + 0.821093i $$0.693365\pi$$
$$720$$ 0 0
$$721$$ −1.25999e13 −1.73644
$$722$$ 7.26362e12 0.994800
$$723$$ 0 0
$$724$$ −1.66040e12 −0.224589
$$725$$ 7.52885e12 1.01206
$$726$$ 0 0
$$727$$ −9.98421e11 −0.132559 −0.0662795 0.997801i $$-0.521113\pi$$
−0.0662795 + 0.997801i $$0.521113\pi$$
$$728$$ 1.41724e12 0.187005
$$729$$ 0 0
$$730$$ −2.38556e12 −0.310912
$$731$$ 1.35298e13 1.75253
$$732$$ 0 0
$$733$$ −2.10457e12 −0.269274 −0.134637 0.990895i $$-0.542987\pi$$
−0.134637 + 0.990895i $$0.542987\pi$$
$$734$$ −1.52253e13 −1.93613
$$735$$ 0 0
$$736$$ 1.56677e13 1.96814
$$737$$ 6.90977e12 0.862700
$$738$$ 0 0
$$739$$ −1.28931e13 −1.59023 −0.795113 0.606461i $$-0.792588\pi$$
−0.795113 + 0.606461i $$0.792588\pi$$
$$740$$ −6.70599e11 −0.0822092
$$741$$ 0 0
$$742$$ −4.27135e12 −0.517306
$$743$$ −2.72357e11 −0.0327861 −0.0163930 0.999866i $$-0.505218\pi$$
−0.0163930 + 0.999866i $$0.505218\pi$$
$$744$$ 0 0
$$745$$ −1.53418e12 −0.182463
$$746$$ 1.20801e13 1.42805
$$747$$ 0 0
$$748$$ 3.75259e12 0.438303
$$749$$ −1.27790e13 −1.48365
$$750$$ 0 0
$$751$$ −1.40530e13 −1.61209 −0.806045 0.591855i $$-0.798396\pi$$
−0.806045 + 0.591855i $$0.798396\pi$$
$$752$$ 1.18985e13 1.35679
$$753$$ 0 0
$$754$$ −3.28379e12 −0.370002
$$755$$ −2.78508e12 −0.311944
$$756$$ 0 0
$$757$$ 8.53220e12 0.944342 0.472171 0.881507i $$-0.343470\pi$$
0.472171 + 0.881507i $$0.343470\pi$$
$$758$$ −1.22161e13 −1.34407
$$759$$ 0 0
$$760$$ 3.84933e11 0.0418528
$$761$$ −1.13835e13 −1.23039 −0.615197 0.788373i $$-0.710924\pi$$
−0.615197 + 0.788373i $$0.710924\pi$$
$$762$$ 0 0
$$763$$ 7.46876e11 0.0797789
$$764$$ −2.93297e12 −0.311450
$$765$$ 0 0
$$766$$ 1.87995e12 0.197295
$$767$$ −1.68471e12 −0.175771
$$768$$ 0 0
$$769$$ 1.94858e12 0.200933 0.100466 0.994940i $$-0.467967\pi$$
0.100466 + 0.994940i $$0.467967\pi$$
$$770$$ −2.38505e12 −0.244506
$$771$$ 0 0
$$772$$ 3.40040e12 0.344550
$$773$$ 1.09355e13 1.10161 0.550807 0.834633i $$-0.314320\pi$$
0.550807 + 0.834633i $$0.314320\pi$$
$$774$$ 0 0
$$775$$ 1.21694e13 1.21175
$$776$$ −9.27760e11 −0.0918455
$$777$$ 0 0
$$778$$ −1.70456e13 −1.66803
$$779$$ 1.91242e11 0.0186065
$$780$$ 0 0
$$781$$ 1.23011e13 1.18308
$$782$$ 2.28642e13 2.18638
$$783$$ 0 0
$$784$$ −1.23680e13 −1.16917
$$785$$ 2.29821e12 0.216011
$$786$$ 0 0
$$787$$ 7.50366e12 0.697247 0.348623 0.937263i $$-0.386649\pi$$
0.348623 + 0.937263i $$0.386649\pi$$
$$788$$ 8.79764e12 0.812827
$$789$$ 0 0
$$790$$ −1.82207e12 −0.166434
$$791$$ 7.62369e12 0.692423
$$792$$ 0 0
$$793$$ −3.67822e12 −0.330300
$$794$$ 8.78865e12 0.784747
$$795$$ 0 0
$$796$$ 1.05539e13 0.931762
$$797$$ −1.65844e12 −0.145592 −0.0727962 0.997347i $$-0.523192\pi$$
−0.0727962 + 0.997347i $$0.523192\pi$$
$$798$$ 0 0
$$799$$ 1.20278e13 1.04406
$$800$$ −1.21880e13 −1.05202
$$801$$ 0 0
$$802$$ 7.39433e12 0.631123
$$803$$ −1.15441e13 −0.979810
$$804$$ 0 0
$$805$$ −5.56337e12 −0.466935
$$806$$ −5.30782e12 −0.443005
$$807$$ 0 0
$$808$$ 4.15256e12 0.342740
$$809$$ 7.07921e12 0.581054 0.290527 0.956867i $$-0.406169\pi$$
0.290527 + 0.956867i $$0.406169\pi$$
$$810$$ 0 0
$$811$$ −1.84596e13 −1.49840 −0.749201 0.662343i $$-0.769562\pi$$
−0.749201 + 0.662343i $$0.769562\pi$$
$$812$$ 1.12356e13 0.906973
$$813$$ 0 0
$$814$$ −8.47657e12 −0.676722
$$815$$ 7.12147e11 0.0565406
$$816$$ 0 0
$$817$$ −1.09727e13 −0.861616
$$818$$ 1.31630e13 1.02793
$$819$$ 0 0
$$820$$ 5.92268e10 0.00457463
$$821$$ −2.40377e13 −1.84650 −0.923250 0.384200i $$-0.874477\pi$$
−0.923250 + 0.384200i $$0.874477\pi$$
$$822$$ 0 0
$$823$$ −1.26494e12 −0.0961104 −0.0480552 0.998845i $$-0.515302\pi$$
−0.0480552 + 0.998845i $$0.515302\pi$$
$$824$$ 7.96062e12 0.601554
$$825$$ 0 0
$$826$$ 1.50568e13 1.12544
$$827$$ 9.81229e12 0.729450 0.364725 0.931115i $$-0.381163\pi$$
0.364725 + 0.931115i $$0.381163\pi$$
$$828$$ 0 0
$$829$$ 1.26128e13 0.927504 0.463752 0.885965i $$-0.346503\pi$$
0.463752 + 0.885965i $$0.346503\pi$$
$$830$$ −3.27107e12 −0.239242
$$831$$ 0 0
$$832$$ 5.79658e11 0.0419389
$$833$$ −1.25024e13 −0.899687
$$834$$ 0 0
$$835$$ 2.37644e12 0.169175
$$836$$ −3.04335e12 −0.215488
$$837$$ 0 0
$$838$$ −4.75441e12 −0.333042
$$839$$ 8.53632e12 0.594760 0.297380 0.954759i $$-0.403887\pi$$
0.297380 + 0.954759i $$0.403887\pi$$
$$840$$ 0 0
$$841$$ 1.42715e12 0.0983759
$$842$$ 2.62179e12 0.179760
$$843$$ 0 0
$$844$$ −4.74056e12 −0.321580
$$845$$ 2.11204e11 0.0142511
$$846$$ 0 0
$$847$$ 9.35511e12 0.624559
$$848$$ 5.41970e12 0.359910
$$849$$ 0 0
$$850$$ −1.77862e13 −1.16868
$$851$$ −1.97724e13 −1.29234
$$852$$ 0 0
$$853$$ −1.12025e13 −0.724507 −0.362254 0.932080i $$-0.617993\pi$$
−0.362254 + 0.932080i $$0.617993\pi$$
$$854$$ 3.28734e13 2.11487
$$855$$ 0 0
$$856$$ 8.07379e12 0.513979
$$857$$ 2.23241e13 1.41371 0.706856 0.707357i $$-0.250113\pi$$
0.706856 + 0.707357i $$0.250113\pi$$
$$858$$ 0 0
$$859$$ −2.11323e13 −1.32428 −0.662138 0.749382i $$-0.730351\pi$$
−0.662138 + 0.749382i $$0.730351\pi$$
$$860$$ −3.39819e12 −0.211838
$$861$$ 0 0
$$862$$ 2.28869e13 1.41190
$$863$$ −1.17061e13 −0.718395 −0.359197 0.933262i $$-0.616950\pi$$
−0.359197 + 0.933262i $$0.616950\pi$$
$$864$$ 0 0
$$865$$ −2.52454e11 −0.0153324
$$866$$ 2.38005e13 1.43799
$$867$$ 0 0
$$868$$ 1.81609e13 1.08592
$$869$$ −8.81730e12 −0.524501
$$870$$ 0 0
$$871$$ 5.46859e12 0.321954
$$872$$ −4.71876e11 −0.0276378
$$873$$ 0 0
$$874$$ −1.85429e13 −1.07492
$$875$$ 8.80934e12 0.508051
$$876$$ 0 0
$$877$$ 1.17836e13 0.672638 0.336319 0.941748i $$-0.390818\pi$$
0.336319 + 0.941748i $$0.390818\pi$$
$$878$$ −3.50134e13 −1.98842
$$879$$ 0 0
$$880$$ 3.02627e12 0.170113
$$881$$ 1.93741e13 1.08350 0.541751 0.840539i $$-0.317762\pi$$
0.541751 + 0.840539i $$0.317762\pi$$
$$882$$ 0 0
$$883$$ −1.50199e13 −0.831466 −0.415733 0.909487i $$-0.636475\pi$$
−0.415733 + 0.909487i $$0.636475\pi$$
$$884$$ 2.96991e12 0.163572
$$885$$ 0 0
$$886$$ −2.51488e13 −1.37109
$$887$$ −1.23661e13 −0.670776 −0.335388 0.942080i $$-0.608867\pi$$
−0.335388 + 0.942080i $$0.608867\pi$$
$$888$$ 0 0
$$889$$ 1.27723e13 0.685824
$$890$$ −5.89394e12 −0.314884
$$891$$ 0 0
$$892$$ 7.09439e12 0.375209
$$893$$ −9.75457e12 −0.513306
$$894$$ 0 0
$$895$$ 2.30008e12 0.119823
$$896$$ 2.41408e13 1.25131
$$897$$ 0 0
$$898$$ 3.51702e13 1.80481
$$899$$ 2.57558e13 1.31509
$$900$$ 0 0
$$901$$ 5.47860e12 0.276954
$$902$$ 7.48644e11 0.0376570
$$903$$ 0 0
$$904$$ −4.81664e12 −0.239876
$$905$$ −1.35358e12 −0.0670757
$$906$$ 0 0
$$907$$ 9.76708e12 0.479217 0.239608 0.970870i $$-0.422981\pi$$
0.239608 + 0.970870i $$0.422981\pi$$
$$908$$ −1.65384e12 −0.0807436
$$909$$ 0 0
$$910$$ −1.88760e12 −0.0912480
$$911$$ 1.91556e13 0.921430 0.460715 0.887548i $$-0.347593\pi$$
0.460715 + 0.887548i $$0.347593\pi$$
$$912$$ 0 0
$$913$$ −1.58293e13 −0.753949
$$914$$ −1.82219e13 −0.863647
$$915$$ 0 0
$$916$$ 4.30586e12 0.202083
$$917$$ −4.03170e13 −1.88290
$$918$$ 0 0
$$919$$ −2.82464e13 −1.30630 −0.653150 0.757229i $$-0.726553\pi$$
−0.653150 + 0.757229i $$0.726553\pi$$
$$920$$ 3.51493e12 0.161760
$$921$$ 0 0
$$922$$ 2.33049e13 1.06208
$$923$$ 9.73546e12 0.441518
$$924$$ 0 0
$$925$$ 1.53810e13 0.690792
$$926$$ −2.21691e13 −0.990830
$$927$$ 0 0
$$928$$ −2.57950e13 −1.14175
$$929$$ −1.81017e13 −0.797348 −0.398674 0.917093i $$-0.630529\pi$$
−0.398674 + 0.917093i $$0.630529\pi$$
$$930$$ 0 0
$$931$$ 1.01395e13 0.442324
$$932$$ −6.54974e10 −0.00284350
$$933$$ 0 0
$$934$$ 4.60004e13 1.97788
$$935$$ 3.05916e12 0.130903
$$936$$ 0 0
$$937$$ 2.51202e13 1.06462 0.532310 0.846550i $$-0.321324\pi$$
0.532310 + 0.846550i $$0.321324\pi$$
$$938$$ −4.88745e13 −2.06143
$$939$$ 0 0
$$940$$ −3.02095e12 −0.126202
$$941$$ −8.09156e11 −0.0336418 −0.0168209 0.999859i $$-0.505355\pi$$
−0.0168209 + 0.999859i $$0.505355\pi$$
$$942$$ 0 0
$$943$$ 1.74628e12 0.0719138
$$944$$ −1.91048e13 −0.783014
$$945$$ 0 0
$$946$$ −4.29541e13 −1.74379
$$947$$ 1.25569e13 0.507349 0.253675 0.967290i $$-0.418361\pi$$
0.253675 + 0.967290i $$0.418361\pi$$
$$948$$ 0 0
$$949$$ −9.13636e12 −0.365658
$$950$$ 1.44246e13 0.574575
$$951$$ 0 0
$$952$$ 1.62463e13 0.641045
$$953$$ 3.64201e13 1.43029 0.715144 0.698977i $$-0.246361\pi$$
0.715144 + 0.698977i $$0.246361\pi$$
$$954$$ 0 0
$$955$$ −2.39100e12 −0.0930173
$$956$$ −6.03087e12 −0.233518
$$957$$ 0 0
$$958$$ 4.64612e13 1.78216
$$959$$ −1.77017e13 −0.675821
$$960$$ 0 0
$$961$$ 1.51913e13 0.574564
$$962$$ −6.70859e12 −0.252548
$$963$$ 0 0
$$964$$ −1.94225e13 −0.724366
$$965$$ 2.77205e12 0.102903
$$966$$ 0 0
$$967$$ 2.96691e13 1.09115 0.545575 0.838062i $$-0.316311\pi$$
0.545575 + 0.838062i $$0.316311\pi$$
$$968$$ −5.91055e12 −0.216366
$$969$$ 0 0
$$970$$ 1.23567e12 0.0448156
$$971$$ 1.05417e13 0.380563 0.190281 0.981730i $$-0.439060\pi$$
0.190281 + 0.981730i $$0.439060\pi$$
$$972$$ 0 0
$$973$$ −3.14899e12 −0.112632
$$974$$ −2.67595e13 −0.952714
$$975$$ 0 0
$$976$$ −4.17114e13 −1.47140
$$977$$ 2.40729e13 0.845285 0.422643 0.906296i $$-0.361103\pi$$
0.422643 + 0.906296i $$0.361103\pi$$
$$978$$ 0 0
$$979$$ −2.85218e13 −0.992328
$$980$$ 3.14014e12 0.108751
$$981$$ 0 0
$$982$$ −6.64372e13 −2.27987
$$983$$ 3.57135e13 1.21995 0.609974 0.792421i $$-0.291180\pi$$
0.609974 + 0.792421i $$0.291180\pi$$
$$984$$ 0 0
$$985$$ 7.17195e12 0.242758
$$986$$ −3.76432e13 −1.26835
$$987$$ 0 0
$$988$$ −2.40859e12 −0.0804187
$$989$$ −1.00195e14 −3.33013
$$990$$ 0 0
$$991$$ 2.48780e13 0.819376 0.409688 0.912226i $$-0.365638\pi$$
0.409688 + 0.912226i $$0.365638\pi$$
$$992$$ −4.16943e13 −1.36702
$$993$$ 0 0
$$994$$ −8.70089e13 −2.82699
$$995$$ 8.60369e12 0.278279
$$996$$ 0 0
$$997$$ −3.31717e13 −1.06326 −0.531630 0.846977i $$-0.678420\pi$$
−0.531630 + 0.846977i $$0.678420\pi$$
$$998$$ 3.78497e12 0.120774
$$999$$ 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 117.10.a.c.1.1 4
3.2 odd 2 13.10.a.a.1.4 4
12.11 even 2 208.10.a.g.1.4 4
15.14 odd 2 325.10.a.a.1.1 4
39.38 odd 2 169.10.a.a.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.a.1.4 4 3.2 odd 2
117.10.a.c.1.1 4 1.1 even 1 trivial
169.10.a.a.1.1 4 39.38 odd 2
208.10.a.g.1.4 4 12.11 even 2
325.10.a.a.1.1 4 15.14 odd 2