Properties

 Label 95.5.d.c.94.1 Level $95$ Weight $5$ Character 95.94 Self dual yes Analytic conductor $9.820$ Analytic rank $0$ Dimension $2$ CM discriminant -95 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,5,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 95.d (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$9.82014649297$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

 Embedding label 94.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 95.94

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.70820 q^{2} -4.47214 q^{3} +29.0000 q^{4} +25.0000 q^{5} +30.0000 q^{6} -87.2067 q^{8} -61.0000 q^{9} +O(q^{10})$$ $$q-6.70820 q^{2} -4.47214 q^{3} +29.0000 q^{4} +25.0000 q^{5} +30.0000 q^{6} -87.2067 q^{8} -61.0000 q^{9} -167.705 q^{10} -62.0000 q^{11} -129.692 q^{12} +67.0820 q^{13} -111.803 q^{15} +121.000 q^{16} +409.200 q^{18} +361.000 q^{19} +725.000 q^{20} +415.909 q^{22} +390.000 q^{24} +625.000 q^{25} -450.000 q^{26} +635.043 q^{27} +750.000 q^{30} +583.614 q^{32} +277.272 q^{33} -1769.00 q^{36} +2643.03 q^{37} -2421.66 q^{38} -300.000 q^{39} -2180.17 q^{40} -1798.00 q^{44} -1525.00 q^{45} -541.128 q^{48} +2401.00 q^{49} -4192.63 q^{50} +1945.38 q^{52} -791.568 q^{53} -4260.00 q^{54} -1550.00 q^{55} -1614.44 q^{57} -3242.30 q^{60} +7138.00 q^{61} -5851.00 q^{64} +1677.05 q^{65} -1860.00 q^{66} +6077.63 q^{67} +5319.61 q^{72} -17730.0 q^{74} -2795.08 q^{75} +10469.0 q^{76} +2012.46 q^{78} +3025.00 q^{80} +2101.00 q^{81} +5406.81 q^{88} +10230.0 q^{90} +9025.00 q^{95} -2610.00 q^{96} +15415.5 q^{97} -16106.4 q^{98} +3782.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 58 q^{4} + 50 q^{5} + 60 q^{6} - 122 q^{9}+O(q^{10})$$ 2 * q + 58 * q^4 + 50 * q^5 + 60 * q^6 - 122 * q^9 $$2 q + 58 q^{4} + 50 q^{5} + 60 q^{6} - 122 q^{9} - 124 q^{11} + 242 q^{16} + 722 q^{19} + 1450 q^{20} + 780 q^{24} + 1250 q^{25} - 900 q^{26} + 1500 q^{30} - 3538 q^{36} - 600 q^{39} - 3596 q^{44} - 3050 q^{45} + 4802 q^{49} - 8520 q^{54} - 3100 q^{55} + 14276 q^{61} - 11702 q^{64} - 3720 q^{66} - 35460 q^{74} + 20938 q^{76} + 6050 q^{80} + 4202 q^{81} + 18050 q^{95} - 5220 q^{96} + 7564 q^{99}+O(q^{100})$$ 2 * q + 58 * q^4 + 50 * q^5 + 60 * q^6 - 122 * q^9 - 124 * q^11 + 242 * q^16 + 722 * q^19 + 1450 * q^20 + 780 * q^24 + 1250 * q^25 - 900 * q^26 + 1500 * q^30 - 3538 * q^36 - 600 * q^39 - 3596 * q^44 - 3050 * q^45 + 4802 * q^49 - 8520 * q^54 - 3100 * q^55 + 14276 * q^61 - 11702 * q^64 - 3720 * q^66 - 35460 * q^74 + 20938 * q^76 + 6050 * q^80 + 4202 * q^81 + 18050 * q^95 - 5220 * q^96 + 7564 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −6.70820 −1.67705 −0.838525 0.544862i $$-0.816582\pi$$
−0.838525 + 0.544862i $$0.816582\pi$$
$$3$$ −4.47214 −0.496904 −0.248452 0.968644i $$-0.579922\pi$$
−0.248452 + 0.968644i $$0.579922\pi$$
$$4$$ 29.0000 1.81250
$$5$$ 25.0000 1.00000
$$6$$ 30.0000 0.833333
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ −87.2067 −1.36260
$$9$$ −61.0000 −0.753086
$$10$$ −167.705 −1.67705
$$11$$ −62.0000 −0.512397 −0.256198 0.966624i $$-0.582470\pi$$
−0.256198 + 0.966624i $$0.582470\pi$$
$$12$$ −129.692 −0.900638
$$13$$ 67.0820 0.396935 0.198468 0.980107i $$-0.436404\pi$$
0.198468 + 0.980107i $$0.436404\pi$$
$$14$$ 0 0
$$15$$ −111.803 −0.496904
$$16$$ 121.000 0.472656
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 409.200 1.26296
$$19$$ 361.000 1.00000
$$20$$ 725.000 1.81250
$$21$$ 0 0
$$22$$ 415.909 0.859315
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 390.000 0.677083
$$25$$ 625.000 1.00000
$$26$$ −450.000 −0.665680
$$27$$ 635.043 0.871116
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 750.000 0.833333
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 583.614 0.569935
$$33$$ 277.272 0.254612
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −1769.00 −1.36497
$$37$$ 2643.03 1.93063 0.965315 0.261088i $$-0.0840813\pi$$
0.965315 + 0.261088i $$0.0840813\pi$$
$$38$$ −2421.66 −1.67705
$$39$$ −300.000 −0.197239
$$40$$ −2180.17 −1.36260
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ −1798.00 −0.928719
$$45$$ −1525.00 −0.753086
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −541.128 −0.234865
$$49$$ 2401.00 1.00000
$$50$$ −4192.63 −1.67705
$$51$$ 0 0
$$52$$ 1945.38 0.719445
$$53$$ −791.568 −0.281797 −0.140899 0.990024i $$-0.544999\pi$$
−0.140899 + 0.990024i $$0.544999\pi$$
$$54$$ −4260.00 −1.46091
$$55$$ −1550.00 −0.512397
$$56$$ 0 0
$$57$$ −1614.44 −0.496904
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ −3242.30 −0.900638
$$61$$ 7138.00 1.91830 0.959151 0.282895i $$-0.0912949\pi$$
0.959151 + 0.282895i $$0.0912949\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −5851.00 −1.42847
$$65$$ 1677.05 0.396935
$$66$$ −1860.00 −0.426997
$$67$$ 6077.63 1.35389 0.676947 0.736031i $$-0.263302\pi$$
0.676947 + 0.736031i $$0.263302\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 5319.61 1.02616
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ −17730.0 −3.23776
$$75$$ −2795.08 −0.496904
$$76$$ 10469.0 1.81250
$$77$$ 0 0
$$78$$ 2012.46 0.330779
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 3025.00 0.472656
$$81$$ 2101.00 0.320226
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 5406.81 0.698194
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 10230.0 1.26296
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 9025.00 1.00000
$$96$$ −2610.00 −0.283203
$$97$$ 15415.5 1.63837 0.819187 0.573527i $$-0.194425\pi$$
0.819187 + 0.573527i $$0.194425\pi$$
$$98$$ −16106.4 −1.67705
$$99$$ 3782.00 0.385879
$$100$$ 18125.0 1.81250
$$101$$ 20098.0 1.97020 0.985100 0.171985i $$-0.0550182\pi$$
0.985100 + 0.171985i $$0.0550182\pi$$
$$102$$ 0 0
$$103$$ −14637.3 −1.37971 −0.689853 0.723949i $$-0.742325\pi$$
−0.689853 + 0.723949i $$0.742325\pi$$
$$104$$ −5850.00 −0.540865
$$105$$ 0 0
$$106$$ 5310.00 0.472588
$$107$$ −17320.6 −1.51285 −0.756423 0.654082i $$-0.773055\pi$$
−0.756423 + 0.654082i $$0.773055\pi$$
$$108$$ 18416.3 1.57890
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 10397.7 0.859315
$$111$$ −11820.0 −0.959338
$$112$$ 0 0
$$113$$ −23867.8 −1.86920 −0.934599 0.355703i $$-0.884241\pi$$
−0.934599 + 0.355703i $$0.884241\pi$$
$$114$$ 10830.0 0.833333
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −4092.00 −0.298926
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 9750.00 0.677083
$$121$$ −10797.0 −0.737450
$$122$$ −47883.2 −3.21709
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 15625.0 1.00000
$$126$$ 0 0
$$127$$ −14208.0 −0.880896 −0.440448 0.897778i $$-0.645180\pi$$
−0.440448 + 0.897778i $$0.645180\pi$$
$$128$$ 29911.9 1.82568
$$129$$ 0 0
$$130$$ −11250.0 −0.665680
$$131$$ −20398.0 −1.18863 −0.594313 0.804234i $$-0.702576\pi$$
−0.594313 + 0.804234i $$0.702576\pi$$
$$132$$ 8040.90 0.461484
$$133$$ 0 0
$$134$$ −40770.0 −2.27055
$$135$$ 15876.1 0.871116
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 1858.00 0.0961648 0.0480824 0.998843i $$-0.484689\pi$$
0.0480824 + 0.998843i $$0.484689\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −4159.09 −0.203388
$$144$$ −7381.00 −0.355951
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −10737.6 −0.496904
$$148$$ 76647.9 3.49927
$$149$$ 7618.00 0.343138 0.171569 0.985172i $$-0.445116\pi$$
0.171569 + 0.985172i $$0.445116\pi$$
$$150$$ 18750.0 0.833333
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ −31481.6 −1.36260
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −8700.00 −0.357495
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 3540.00 0.140026
$$160$$ 14590.3 0.569935
$$161$$ 0 0
$$162$$ −14093.9 −0.537035
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 6931.81 0.254612
$$166$$ 0 0
$$167$$ 54269.4 1.94591 0.972953 0.231003i $$-0.0742008\pi$$
0.972953 + 0.231003i $$0.0742008\pi$$
$$168$$ 0 0
$$169$$ −24061.0 −0.842442
$$170$$ 0 0
$$171$$ −22021.0 −0.753086
$$172$$ 0 0
$$173$$ 29690.5 0.992031 0.496016 0.868314i $$-0.334796\pi$$
0.496016 + 0.868314i $$0.334796\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −7502.00 −0.242188
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ −44225.0 −1.36497
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ −31922.1 −0.953212
$$184$$ 0 0
$$185$$ 66075.8 1.93063
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −60541.5 −1.67705
$$191$$ 18242.0 0.500041 0.250021 0.968241i $$-0.419563\pi$$
0.250021 + 0.968241i $$0.419563\pi$$
$$192$$ 26166.5 0.709811
$$193$$ −70020.2 −1.87979 −0.939894 0.341466i $$-0.889077\pi$$
−0.939894 + 0.341466i $$0.889077\pi$$
$$194$$ −103410. −2.74764
$$195$$ −7500.00 −0.197239
$$196$$ 69629.0 1.81250
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ −25370.4 −0.647139
$$199$$ 24482.0 0.618217 0.309108 0.951027i $$-0.399969\pi$$
0.309108 + 0.951027i $$0.399969\pi$$
$$200$$ −54504.2 −1.36260
$$201$$ −27180.0 −0.672756
$$202$$ −134821. −3.30412
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 98190.0 2.31384
$$207$$ 0 0
$$208$$ 8116.93 0.187614
$$209$$ −22382.0 −0.512397
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ −22955.5 −0.510757
$$213$$ 0 0
$$214$$ 116190. 2.53712
$$215$$ 0 0
$$216$$ −55380.0 −1.18699
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ −44950.0 −0.928719
$$221$$ 0 0
$$222$$ 79291.0 1.60886
$$223$$ 49761.5 1.00065 0.500326 0.865837i $$-0.333213\pi$$
0.500326 + 0.865837i $$0.333213\pi$$
$$224$$ 0 0
$$225$$ −38125.0 −0.753086
$$226$$ 160110. 3.13474
$$227$$ 92372.0 1.79262 0.896311 0.443427i $$-0.146237\pi$$
0.896311 + 0.443427i $$0.146237\pi$$
$$228$$ −46818.8 −0.900638
$$229$$ 68098.0 1.29856 0.649282 0.760548i $$-0.275069\pi$$
0.649282 + 0.760548i $$0.275069\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 27450.0 0.501315
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −104638. −1.83187 −0.915933 0.401332i $$-0.868548\pi$$
−0.915933 + 0.401332i $$0.868548\pi$$
$$240$$ −13528.2 −0.234865
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 72428.5 1.23674
$$243$$ −60834.5 −1.03024
$$244$$ 207002. 3.47692
$$245$$ 60025.0 1.00000
$$246$$ 0 0
$$247$$ 24216.6 0.396935
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −104816. −1.67705
$$251$$ −92878.0 −1.47423 −0.737115 0.675767i $$-0.763813\pi$$
−0.737115 + 0.675767i $$0.763813\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 95310.0 1.47731
$$255$$ 0 0
$$256$$ −107039. −1.63329
$$257$$ 122317. 1.85192 0.925959 0.377623i $$-0.123258\pi$$
0.925959 + 0.377623i $$0.123258\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 48634.5 0.719445
$$261$$ 0 0
$$262$$ 136834. 1.99339
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ −24180.0 −0.346935
$$265$$ −19789.2 −0.281797
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 176251. 2.45393
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ −106500. −1.46091
$$271$$ −145262. −1.97794 −0.988971 0.148111i $$-0.952681\pi$$
−0.988971 + 0.148111i $$0.952681\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −38750.0 −0.512397
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ −12463.8 −0.161273
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ −40361.0 −0.496904
$$286$$ 27900.0 0.341092
$$287$$ 0 0
$$288$$ −35600.4 −0.429211
$$289$$ 83521.0 1.00000
$$290$$ 0 0
$$291$$ −68940.0 −0.814114
$$292$$ 0 0
$$293$$ −171663. −1.99959 −0.999796 0.0202081i $$-0.993567\pi$$
−0.999796 + 0.0202081i $$0.993567\pi$$
$$294$$ 72030.0 0.833333
$$295$$ 0 0
$$296$$ −230490. −2.63068
$$297$$ −39372.7 −0.446357
$$298$$ −51103.1 −0.575459
$$299$$ 0 0
$$300$$ −81057.5 −0.900638
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −89881.0 −0.979000
$$304$$ 43681.0 0.472656
$$305$$ 178450. 1.91830
$$306$$ 0 0
$$307$$ −186260. −1.97625 −0.988127 0.153638i $$-0.950901\pi$$
−0.988127 + 0.153638i $$0.950901\pi$$
$$308$$ 0 0
$$309$$ 65460.0 0.685581
$$310$$ 0 0
$$311$$ −62222.0 −0.643314 −0.321657 0.946856i $$-0.604240\pi$$
−0.321657 + 0.946856i $$0.604240\pi$$
$$312$$ 26162.0 0.268758
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 65324.5 0.650066 0.325033 0.945703i $$-0.394625\pi$$
0.325033 + 0.945703i $$0.394625\pi$$
$$318$$ −23747.0 −0.234831
$$319$$ 0 0
$$320$$ −146275. −1.42847
$$321$$ 77460.0 0.751740
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 60929.0 0.580409
$$325$$ 41926.3 0.396935
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ −46500.0 −0.426997
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ −161225. −1.45393
$$334$$ −364050. −3.26338
$$335$$ 151941. 1.35389
$$336$$ 0 0
$$337$$ −223504. −1.96800 −0.984001 0.178165i $$-0.942984\pi$$
−0.984001 + 0.178165i $$0.942984\pi$$
$$338$$ 161406. 1.41282
$$339$$ 106740. 0.928812
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 147721. 1.26296
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −199170. −1.66369
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ 24722.0 0.202970 0.101485 0.994837i $$-0.467641\pi$$
0.101485 + 0.994837i $$0.467641\pi$$
$$350$$ 0 0
$$351$$ 42600.0 0.345776
$$352$$ −36184.1 −0.292033
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −253262. −1.96508 −0.982542 0.186041i $$-0.940434\pi$$
−0.982542 + 0.186041i $$0.940434\pi$$
$$360$$ 132990. 1.02616
$$361$$ 130321. 1.00000
$$362$$ 0 0
$$363$$ 48285.7 0.366442
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 214140. 1.59858
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ −443250. −3.23776
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 223316. 1.60510 0.802551 0.596584i $$-0.203476\pi$$
0.802551 + 0.596584i $$0.203476\pi$$
$$374$$ 0 0
$$375$$ −69877.1 −0.496904
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 261725. 1.81250
$$381$$ 63540.0 0.437721
$$382$$ −122371. −0.838594
$$383$$ 276660. 1.88603 0.943015 0.332751i $$-0.107977\pi$$
0.943015 + 0.332751i $$0.107977\pi$$
$$384$$ −133770. −0.907186
$$385$$ 0 0
$$386$$ 469710. 3.15250
$$387$$ 0 0
$$388$$ 447048. 2.96955
$$389$$ 247922. 1.63838 0.819192 0.573519i $$-0.194422\pi$$
0.819192 + 0.573519i $$0.194422\pi$$
$$390$$ 50311.5 0.330779
$$391$$ 0 0
$$392$$ −209383. −1.36260
$$393$$ 91222.6 0.590633
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 109678. 0.699406
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ −164230. −1.03678
$$399$$ 0 0
$$400$$ 75625.0 0.472656
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 182329. 1.12825
$$403$$ 0 0
$$404$$ 582842. 3.57099
$$405$$ 52525.0 0.320226
$$406$$ 0 0
$$407$$ −163868. −0.989248
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −424482. −2.50072
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 39150.0 0.226227
$$417$$ −8309.23 −0.0477847
$$418$$ 150143. 0.859315
$$419$$ −141358. −0.805179 −0.402589 0.915381i $$-0.631890\pi$$
−0.402589 + 0.915381i $$0.631890\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 69030.0 0.383978
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −502297. −2.74203
$$429$$ 18600.0 0.101064
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 76840.2 0.411738
$$433$$ 86898.1 0.463484 0.231742 0.972777i $$-0.425558\pi$$
0.231742 + 0.972777i $$0.425558\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 135170. 0.698194
$$441$$ −146461. −0.753086
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ −342780. −1.73880
$$445$$ 0 0
$$446$$ −333810. −1.67815
$$447$$ −34068.7 −0.170506
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 255750. 1.26296
$$451$$ 0 0
$$452$$ −692166. −3.38792
$$453$$ 0 0
$$454$$ −619650. −3.00632
$$455$$ 0 0
$$456$$ 140790. 0.677083
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ −456815. −2.17776
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −67438.0 −0.317324 −0.158662 0.987333i $$-0.550718\pi$$
−0.158662 + 0.987333i $$0.550718\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ −118668. −0.541804
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 225625. 1.00000
$$476$$ 0 0
$$477$$ 48285.7 0.212218
$$478$$ 701933. 3.07213
$$479$$ 349138. 1.52169 0.760845 0.648934i $$-0.224785\pi$$
0.760845 + 0.648934i $$0.224785\pi$$
$$480$$ −65250.0 −0.283203
$$481$$ 177300. 0.766335
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −313113. −1.33663
$$485$$ 385386. 1.63837
$$486$$ 408090. 1.72776
$$487$$ 470071. 1.98201 0.991004 0.133835i $$-0.0427292\pi$$
0.991004 + 0.133835i $$0.0427292\pi$$
$$488$$ −622481. −2.61389
$$489$$ 0 0
$$490$$ −402660. −1.67705
$$491$$ −393358. −1.63164 −0.815821 0.578304i $$-0.803715\pi$$
−0.815821 + 0.578304i $$0.803715\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −162450. −0.665680
$$495$$ 94550.0 0.385879
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 461218. 1.85227 0.926137 0.377188i $$-0.123109\pi$$
0.926137 + 0.377188i $$0.123109\pi$$
$$500$$ 453125. 1.81250
$$501$$ −242700. −0.966928
$$502$$ 623045. 2.47236
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 502450. 1.97020
$$506$$ 0 0
$$507$$ 107604. 0.418613
$$508$$ −412031. −1.59662
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 239449. 0.913427
$$513$$ 229251. 0.871116
$$514$$ −820530. −3.10576
$$515$$ −365933. −1.37971
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −132780. −0.492944
$$520$$ −146250. −0.540865
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 242636. 0.887057 0.443528 0.896260i $$-0.353727\pi$$
0.443528 + 0.896260i $$0.353727\pi$$
$$524$$ −591542. −2.15438
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 33550.0 0.120344
$$529$$ 279841. 1.00000
$$530$$ 132750. 0.472588
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −433015. −1.51285
$$536$$ −530010. −1.84482
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −148862. −0.512397
$$540$$ 460406. 1.57890
$$541$$ −472862. −1.61562 −0.807811 0.589441i $$-0.799348\pi$$
−0.807811 + 0.589441i $$0.799348\pi$$
$$542$$ 974447. 3.31711
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −584030. −1.95191 −0.975956 0.217968i $$-0.930057\pi$$
−0.975956 + 0.217968i $$0.930057\pi$$
$$548$$ 0 0
$$549$$ −435418. −1.44465
$$550$$ 259943. 0.859315
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −295500. −0.959338
$$556$$ 53882.0 0.174299
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 545525. 1.72107 0.860533 0.509395i $$-0.170131\pi$$
0.860533 + 0.509395i $$0.170131\pi$$
$$564$$ 0 0
$$565$$ −596695. −1.86920
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 270750. 0.833333
$$571$$ −479102. −1.46945 −0.734727 0.678363i $$-0.762690\pi$$
−0.734727 + 0.678363i $$0.762690\pi$$
$$572$$ −120614. −0.368641
$$573$$ −81580.7 −0.248472
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 356911. 1.07576
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ −560276. −1.67705
$$579$$ 313140. 0.934074
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 462464. 1.36531
$$583$$ 49077.2 0.144392
$$584$$ 0 0
$$585$$ −102300. −0.298926
$$586$$ 1.15155e6 3.35342
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ −311390. −0.900638
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 319807. 0.912524
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 264120. 0.748563
$$595$$ 0 0
$$596$$ 220922. 0.621937
$$597$$ −109487. −0.307194
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 243750. 0.677083
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ −370736. −1.01960
$$604$$ 0 0
$$605$$ −269925. −0.737450
$$606$$ 602940. 1.64183
$$607$$ −309369. −0.839652 −0.419826 0.907605i $$-0.637909\pi$$
−0.419826 + 0.907605i $$0.637909\pi$$
$$608$$ 210685. 0.569935
$$609$$ 0 0
$$610$$ −1.19708e6 −3.21709
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 1.24947e6 3.31428
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ −439119. −1.14976
$$619$$ −766142. −1.99953 −0.999765 0.0216731i $$-0.993101\pi$$
−0.999765 + 0.0216731i $$0.993101\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 417398. 1.07887
$$623$$ 0 0
$$624$$ −36300.0 −0.0932261
$$625$$ 390625. 1.00000
$$626$$ 0 0
$$627$$ 100095. 0.254612
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 504178. 1.26627 0.633133 0.774043i $$-0.281768\pi$$
0.633133 + 0.774043i $$0.281768\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −438210. −1.09019
$$635$$ −355199. −0.880896
$$636$$ 102660. 0.253797
$$637$$ 161064. 0.396935
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 747797. 1.82568
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ −519617. −1.26071
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ −183221. −0.436341
$$649$$ 0 0
$$650$$ −281250. −0.665680
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ −509950. −1.18863
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 201023. 0.461484
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 1.08153e6 2.43832
$$667$$ 0 0
$$668$$ 1.57381e6 3.52695
$$669$$ −222540. −0.497228
$$670$$ −1.01925e6 −2.27055
$$671$$ −442556. −0.982931
$$672$$ 0 0
$$673$$ −540775. −1.19395 −0.596976 0.802259i $$-0.703631\pi$$
−0.596976 + 0.802259i $$0.703631\pi$$
$$674$$ 1.49931e6 3.30044
$$675$$ 396902. 0.871116
$$676$$ −697769. −1.52693
$$677$$ 533289. 1.16355 0.581775 0.813350i $$-0.302358\pi$$
0.581775 + 0.813350i $$0.302358\pi$$
$$678$$ −716034. −1.55767
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −413100. −0.890761
$$682$$ 0 0
$$683$$ −641774. −1.37575 −0.687877 0.725828i $$-0.741457\pi$$
−0.687877 + 0.725828i $$0.741457\pi$$
$$684$$ −638609. −1.36497
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −304544. −0.645262
$$688$$ 0 0
$$689$$ −53100.0 −0.111855
$$690$$ 0 0
$$691$$ 954658. 1.99936 0.999682 0.0252304i $$-0.00803194\pi$$
0.999682 + 0.0252304i $$0.00803194\pi$$
$$692$$ 861025. 1.79806
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 46450.0 0.0961648
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −165840. −0.340392
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −75422.0 −0.153484 −0.0767418 0.997051i $$-0.524452\pi$$
−0.0767418 + 0.997051i $$0.524452\pi$$
$$702$$ −285769. −0.579885
$$703$$ 954135. 1.93063
$$704$$ 362762. 0.731942
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −964558. −1.91883 −0.959414 0.282003i $$-0.909001\pi$$
−0.959414 + 0.282003i $$0.909001\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −103977. −0.203388
$$716$$ 0 0
$$717$$ 467955. 0.910261
$$718$$ 1.69893e6 3.29555
$$719$$ 522898. 1.01148 0.505742 0.862685i $$-0.331219\pi$$
0.505742 + 0.862685i $$0.331219\pi$$
$$720$$ −184525. −0.355951
$$721$$ 0 0
$$722$$ −874220. −1.67705
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ −323910. −0.614541
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 101879. 0.191703
$$730$$ 0 0
$$731$$ 0 0
$$732$$ −925741. −1.72770
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ −268440. −0.496904
$$736$$ 0 0
$$737$$ −376813. −0.693731
$$738$$ 0 0
$$739$$ 873362. 1.59921 0.799605 0.600527i $$-0.205042\pi$$
0.799605 + 0.600527i $$0.205042\pi$$
$$740$$ 1.91620e6 3.49927
$$741$$ −108300. −0.197239
$$742$$ 0 0
$$743$$ 1.09624e6 1.98577 0.992884 0.119085i $$-0.0379962\pi$$
0.992884 + 0.119085i $$0.0379962\pi$$
$$744$$ 0 0
$$745$$ 190450. 0.343138
$$746$$ −1.49805e6 −2.69184
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 468750. 0.833333
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 415363. 0.732551
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ −787040. −1.36260
$$761$$ 902578. 1.55853 0.779265 0.626694i $$-0.215592\pi$$
0.779265 + 0.626694i $$0.215592\pi$$
$$762$$ −426239. −0.734080
$$763$$ 0 0
$$764$$ 529018. 0.906325
$$765$$ 0 0
$$766$$ −1.85589e6 −3.16297
$$767$$ 0 0
$$768$$ 478693. 0.811586
$$769$$ −714542. −1.20830 −0.604150 0.796870i $$-0.706487\pi$$
−0.604150 + 0.796870i $$0.706487\pi$$
$$770$$ 0 0
$$771$$ −547020. −0.920226
$$772$$ −2.03059e6 −3.40712
$$773$$ −697371. −1.16709 −0.583546 0.812080i $$-0.698335\pi$$
−0.583546 + 0.812080i $$0.698335\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −1.34433e6 −2.23245
$$777$$ 0 0
$$778$$ −1.66311e6 −2.74765
$$779$$ 0 0
$$780$$ −217500. −0.357495
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 290521. 0.472656
$$785$$ 0 0
$$786$$ −611940. −0.990521
$$787$$ −404142. −0.652507 −0.326253 0.945282i $$-0.605786\pi$$
−0.326253 + 0.945282i $$0.605786\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −329816. −0.525800
$$793$$ 478832. 0.761441
$$794$$ 0 0
$$795$$ 88500.0 0.140026
$$796$$ 709978. 1.12052
$$797$$ −694796. −1.09381 −0.546903 0.837196i $$-0.684193\pi$$
−0.546903 + 0.837196i $$0.684193\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 364759. 0.569935
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ −788220. −1.21937
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −1.75268e6 −2.68460
$$809$$ −59038.0 −0.0902058 −0.0451029 0.998982i $$-0.514362\pi$$
−0.0451029 + 0.998982i $$0.514362\pi$$
$$810$$ −352348. −0.537035
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 649631. 0.982847
$$814$$ 1.09926e6 1.65902
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.33320e6 −1.97792 −0.988959 0.148189i $$-0.952656\pi$$
−0.988959 + 0.148189i $$0.952656\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 1.27647e6 1.87999
$$825$$ 173295. 0.254612
$$826$$ 0 0
$$827$$ 887053. 1.29700 0.648498 0.761217i $$-0.275398\pi$$
0.648498 + 0.761217i $$0.275398\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −392497. −0.567009
$$833$$ 0 0
$$834$$ 55740.0 0.0801373
$$835$$ 1.35673e6 1.94591
$$836$$ −649078. −0.928719
$$837$$ 0 0
$$838$$ 948258. 1.35033
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 707281. 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −601525. −0.842442
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −95779.7 −0.133193
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ −550525. −0.753086
$$856$$ 1.51047e6 2.06141
$$857$$ 1.12071e6 1.52592 0.762962 0.646444i $$-0.223745\pi$$
0.762962 + 0.646444i $$0.223745\pi$$
$$858$$ −124773. −0.169490
$$859$$ 1.42104e6 1.92584 0.962921 0.269784i $$-0.0869523\pi$$
0.962921 + 0.269784i $$0.0869523\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −1.44214e6 −1.93636 −0.968181 0.250249i $$-0.919487\pi$$
−0.968181 + 0.250249i $$0.919487\pi$$
$$864$$ 370620. 0.496480
$$865$$ 742263. 0.992031
$$866$$ −582930. −0.777286
$$867$$ −373517. −0.496904
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 407700. 0.537408
$$872$$ 0 0
$$873$$ −940343. −1.23384
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.53627e6 −1.99742 −0.998709 0.0507905i $$-0.983826\pi$$
−0.998709 + 0.0507905i $$0.983826\pi$$
$$878$$ 0 0
$$879$$ 767700. 0.993605
$$880$$ −187550. −0.242188
$$881$$ 1.26018e6 1.62360 0.811802 0.583933i $$-0.198487\pi$$
0.811802 + 0.583933i $$0.198487\pi$$
$$882$$ 982490. 1.26296
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 646523. 0.821745 0.410872 0.911693i $$-0.365224\pi$$
0.410872 + 0.911693i $$0.365224\pi$$
$$888$$ 1.03078e6 1.30720
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −130262. −0.164083
$$892$$ 1.44308e6 1.81368
$$893$$ 0 0
$$894$$ 228540. 0.285948
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ −1.10562e6 −1.36497
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 2.08143e6 2.54698
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 671746. 0.816565 0.408282 0.912856i $$-0.366128\pi$$
0.408282 + 0.912856i $$0.366128\pi$$
$$908$$ 2.67879e6 3.24913
$$909$$ −1.22598e6 −1.48373
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ −195347. −0.234865
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −798053. −0.953212
$$916$$ 1.97484e6 2.35365
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 813602. 0.963343 0.481672 0.876352i $$-0.340030\pi$$
0.481672 + 0.876352i $$0.340030\pi$$
$$920$$ 0 0
$$921$$ 832980. 0.982009
$$922$$ 452388. 0.532168
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 1.65190e6 1.93063
$$926$$ 0 0
$$927$$ 892875. 1.03904
$$928$$ 0 0
$$929$$ −955198. −1.10678 −0.553391 0.832922i $$-0.686666\pi$$
−0.553391 + 0.832922i $$0.686666\pi$$
$$930$$ 0 0
$$931$$ 866761. 1.00000
$$932$$ 0 0
$$933$$ 278265. 0.319665
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 356850. 0.407318
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −1.51354e6 −1.67705
$$951$$ −292140. −0.323020
$$952$$ 0 0
$$953$$ −1.68922e6 −1.85995 −0.929973 0.367628i $$-0.880170\pi$$
−0.929973 + 0.367628i $$0.880170\pi$$
$$954$$ −323910. −0.355900
$$955$$ 456050. 0.500041
$$956$$ −3.03450e6 −3.32026
$$957$$ 0 0
$$958$$ −2.34209e6 −2.55195
$$959$$ 0 0
$$960$$ 654162. 0.709811
$$961$$ 923521. 1.00000
$$962$$ −1.18936e6 −1.28518
$$963$$ 1.05656e6 1.13930
$$964$$ 0 0
$$965$$ −1.75051e6 −1.87979
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 941570. 1.00485
$$969$$ 0 0
$$970$$ −2.58525e6 −2.74764
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ −1.76420e6 −1.86730
$$973$$ 0 0
$$974$$ −3.15333e6 −3.32393
$$975$$ −187500. −0.197239
$$976$$ 863698. 0.906697
$$977$$ −907848. −0.951095 −0.475548 0.879690i $$-0.657750\pi$$
−0.475548 + 0.879690i $$0.657750\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 1.74072e6 1.81250
$$981$$ 0 0
$$982$$ 2.63873e6 2.73635
$$983$$ −1.93243e6 −1.99985 −0.999925 0.0122789i $$-0.996091\pi$$
−0.999925 + 0.0122789i $$0.996091\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 702282. 0.719445
$$989$$ 0 0
$$990$$ −634261. −0.647139
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 612050. 0.618217
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ −3.09394e6 −3.10636
$$999$$ 1.67844e6 1.68180
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 95.5.d.c.94.1 2
5.4 even 2 inner 95.5.d.c.94.2 yes 2
19.18 odd 2 inner 95.5.d.c.94.2 yes 2
95.94 odd 2 CM 95.5.d.c.94.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
95.5.d.c.94.1 2 1.1 even 1 trivial
95.5.d.c.94.1 2 95.94 odd 2 CM
95.5.d.c.94.2 yes 2 5.4 even 2 inner
95.5.d.c.94.2 yes 2 19.18 odd 2 inner