# Properties

 Label 68.5.d.a.67.1 Level $68$ Weight $5$ Character 68.67 Self dual yes Analytic conductor $7.029$ Analytic rank $0$ Dimension $1$ CM discriminant -68 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,5,Mod(67,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, 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("68.67");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 68.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: $$7.02915748970$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 67.1 Character $$\chi$$ $$=$$ 68.67

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} -16.0000 q^{3} +16.0000 q^{4} -64.0000 q^{6} +64.0000 q^{7} +64.0000 q^{8} +175.000 q^{9} +O(q^{10})$$ $$q+4.00000 q^{2} -16.0000 q^{3} +16.0000 q^{4} -64.0000 q^{6} +64.0000 q^{7} +64.0000 q^{8} +175.000 q^{9} +208.000 q^{11} -256.000 q^{12} -274.000 q^{13} +256.000 q^{14} +256.000 q^{16} +289.000 q^{17} +700.000 q^{18} -1024.00 q^{21} +832.000 q^{22} -608.000 q^{23} -1024.00 q^{24} +625.000 q^{25} -1096.00 q^{26} -1504.00 q^{27} +1024.00 q^{28} +256.000 q^{31} +1024.00 q^{32} -3328.00 q^{33} +1156.00 q^{34} +2800.00 q^{36} +4384.00 q^{39} -4096.00 q^{42} +3328.00 q^{44} -2432.00 q^{46} -4096.00 q^{48} +1695.00 q^{49} +2500.00 q^{50} -4624.00 q^{51} -4384.00 q^{52} -4174.00 q^{53} -6016.00 q^{54} +4096.00 q^{56} +1024.00 q^{62} +11200.0 q^{63} +4096.00 q^{64} -13312.0 q^{66} +4624.00 q^{68} +9728.00 q^{69} -7904.00 q^{71} +11200.0 q^{72} -10000.0 q^{75} +13312.0 q^{77} +17536.0 q^{78} +2656.00 q^{79} +9889.00 q^{81} -16384.0 q^{84} +13312.0 q^{88} +542.000 q^{89} -17536.0 q^{91} -9728.00 q^{92} -4096.00 q^{93} -16384.0 q^{96} +6780.00 q^{98} +36400.0 q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 1.00000
$$3$$ −16.0000 −1.77778 −0.888889 0.458123i $$-0.848522\pi$$
−0.888889 + 0.458123i $$0.848522\pi$$
$$4$$ 16.0000 1.00000
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ −64.0000 −1.77778
$$7$$ 64.0000 1.30612 0.653061 0.757305i $$-0.273484\pi$$
0.653061 + 0.757305i $$0.273484\pi$$
$$8$$ 64.0000 1.00000
$$9$$ 175.000 2.16049
$$10$$ 0 0
$$11$$ 208.000 1.71901 0.859504 0.511129i $$-0.170772\pi$$
0.859504 + 0.511129i $$0.170772\pi$$
$$12$$ −256.000 −1.77778
$$13$$ −274.000 −1.62130 −0.810651 0.585530i $$-0.800887\pi$$
−0.810651 + 0.585530i $$0.800887\pi$$
$$14$$ 256.000 1.30612
$$15$$ 0 0
$$16$$ 256.000 1.00000
$$17$$ 289.000 1.00000
$$18$$ 700.000 2.16049
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ −1024.00 −2.32200
$$22$$ 832.000 1.71901
$$23$$ −608.000 −1.14934 −0.574669 0.818386i $$-0.694869\pi$$
−0.574669 + 0.818386i $$0.694869\pi$$
$$24$$ −1024.00 −1.77778
$$25$$ 625.000 1.00000
$$26$$ −1096.00 −1.62130
$$27$$ −1504.00 −2.06310
$$28$$ 1024.00 1.30612
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 256.000 0.266389 0.133195 0.991090i $$-0.457476\pi$$
0.133195 + 0.991090i $$0.457476\pi$$
$$32$$ 1024.00 1.00000
$$33$$ −3328.00 −3.05601
$$34$$ 1156.00 1.00000
$$35$$ 0 0
$$36$$ 2800.00 2.16049
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 4384.00 2.88231
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ −4096.00 −2.32200
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 3328.00 1.71901
$$45$$ 0 0
$$46$$ −2432.00 −1.14934
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −4096.00 −1.77778
$$49$$ 1695.00 0.705956
$$50$$ 2500.00 1.00000
$$51$$ −4624.00 −1.77778
$$52$$ −4384.00 −1.62130
$$53$$ −4174.00 −1.48594 −0.742969 0.669326i $$-0.766583\pi$$
−0.742969 + 0.669326i $$0.766583\pi$$
$$54$$ −6016.00 −2.06310
$$55$$ 0 0
$$56$$ 4096.00 1.30612
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 1024.00 0.266389
$$63$$ 11200.0 2.82187
$$64$$ 4096.00 1.00000
$$65$$ 0 0
$$66$$ −13312.0 −3.05601
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 4624.00 1.00000
$$69$$ 9728.00 2.04327
$$70$$ 0 0
$$71$$ −7904.00 −1.56794 −0.783971 0.620797i $$-0.786809\pi$$
−0.783971 + 0.620797i $$0.786809\pi$$
$$72$$ 11200.0 2.16049
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ −10000.0 −1.77778
$$76$$ 0 0
$$77$$ 13312.0 2.24524
$$78$$ 17536.0 2.88231
$$79$$ 2656.00 0.425573 0.212786 0.977099i $$-0.431746\pi$$
0.212786 + 0.977099i $$0.431746\pi$$
$$80$$ 0 0
$$81$$ 9889.00 1.50724
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −16384.0 −2.32200
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 13312.0 1.71901
$$89$$ 542.000 0.0684257 0.0342129 0.999415i $$-0.489108\pi$$
0.0342129 + 0.999415i $$0.489108\pi$$
$$90$$ 0 0
$$91$$ −17536.0 −2.11762
$$92$$ −9728.00 −1.14934
$$93$$ −4096.00 −0.473581
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −16384.0 −1.77778
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 6780.00 0.705956
$$99$$ 36400.0 3.71391
$$100$$ 10000.0 1.00000
$$101$$ −9586.00 −0.939712 −0.469856 0.882743i $$-0.655694\pi$$
−0.469856 + 0.882743i $$0.655694\pi$$
$$102$$ −18496.0 −1.77778
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ −17536.0 −1.62130
$$105$$ 0 0
$$106$$ −16696.0 −1.48594
$$107$$ −9776.00 −0.853874 −0.426937 0.904281i $$-0.640407\pi$$
−0.426937 + 0.904281i $$0.640407\pi$$
$$108$$ −24064.0 −2.06310
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 16384.0 1.30612
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −47950.0 −3.50281
$$118$$ 0 0
$$119$$ 18496.0 1.30612
$$120$$ 0 0
$$121$$ 28623.0 1.95499
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 4096.00 0.266389
$$125$$ 0 0
$$126$$ 44800.0 2.82187
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 16384.0 1.00000
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 32656.0 1.90292 0.951460 0.307773i $$-0.0995838\pi$$
0.951460 + 0.307773i $$0.0995838\pi$$
$$132$$ −53248.0 −3.05601
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 18496.0 1.00000
$$137$$ −36514.0 −1.94544 −0.972721 0.231978i $$-0.925480\pi$$
−0.972721 + 0.231978i $$0.925480\pi$$
$$138$$ 38912.0 2.04327
$$139$$ −36464.0 −1.88727 −0.943636 0.330984i $$-0.892619\pi$$
−0.943636 + 0.330984i $$0.892619\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −31616.0 −1.56794
$$143$$ −56992.0 −2.78703
$$144$$ 44800.0 2.16049
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −27120.0 −1.25503
$$148$$ 0 0
$$149$$ −43726.0 −1.96955 −0.984775 0.173831i $$-0.944385\pi$$
−0.984775 + 0.173831i $$0.944385\pi$$
$$150$$ −40000.0 −1.77778
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 50575.0 2.16049
$$154$$ 53248.0 2.24524
$$155$$ 0 0
$$156$$ 70144.0 2.88231
$$157$$ 39506.0 1.60274 0.801371 0.598167i $$-0.204104\pi$$
0.801371 + 0.598167i $$0.204104\pi$$
$$158$$ 10624.0 0.425573
$$159$$ 66784.0 2.64167
$$160$$ 0 0
$$161$$ −38912.0 −1.50118
$$162$$ 39556.0 1.50724
$$163$$ −28496.0 −1.07253 −0.536264 0.844050i $$-0.680165\pi$$
−0.536264 + 0.844050i $$0.680165\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −19328.0 −0.693033 −0.346517 0.938044i $$-0.612636\pi$$
−0.346517 + 0.938044i $$0.612636\pi$$
$$168$$ −65536.0 −2.32200
$$169$$ 46515.0 1.62862
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 40000.0 1.30612
$$176$$ 53248.0 1.71901
$$177$$ 0 0
$$178$$ 2168.00 0.0684257
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ −70144.0 −2.11762
$$183$$ 0 0
$$184$$ −38912.0 −1.14934
$$185$$ 0 0
$$186$$ −16384.0 −0.473581
$$187$$ 60112.0 1.71901
$$188$$ 0 0
$$189$$ −96256.0 −2.69466
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ −65536.0 −1.77778
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 27120.0 0.705956
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 145600. 3.71391
$$199$$ 22048.0 0.556754 0.278377 0.960472i $$-0.410204\pi$$
0.278377 + 0.960472i $$0.410204\pi$$
$$200$$ 40000.0 1.00000
$$201$$ 0 0
$$202$$ −38344.0 −0.939712
$$203$$ 0 0
$$204$$ −73984.0 −1.77778
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −106400. −2.48314
$$208$$ −70144.0 −1.62130
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 87376.0 1.96258 0.981290 0.192537i $$-0.0616715\pi$$
0.981290 + 0.192537i $$0.0616715\pi$$
$$212$$ −66784.0 −1.48594
$$213$$ 126464. 2.78745
$$214$$ −39104.0 −0.853874
$$215$$ 0 0
$$216$$ −96256.0 −2.06310
$$217$$ 16384.0 0.347937
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −79186.0 −1.62130
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 65536.0 1.30612
$$225$$ 109375. 2.16049
$$226$$ 0 0
$$227$$ 103024. 1.99934 0.999670 0.0256849i $$-0.00817665\pi$$
0.999670 + 0.0256849i $$0.00817665\pi$$
$$228$$ 0 0
$$229$$ 74894.0 1.42816 0.714079 0.700065i $$-0.246846\pi$$
0.714079 + 0.700065i $$0.246846\pi$$
$$230$$ 0 0
$$231$$ −212992. −3.99153
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ −191800. −3.50281
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −42496.0 −0.756574
$$238$$ 73984.0 1.30612
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 114492. 1.95499
$$243$$ −36400.0 −0.616437
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 16384.0 0.266389
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 179200. 2.82187
$$253$$ −126464. −1.97572
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 65536.0 1.00000
$$257$$ −88834.0 −1.34497 −0.672486 0.740110i $$-0.734773\pi$$
−0.672486 + 0.740110i $$0.734773\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 130624. 1.90292
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ −212992. −3.05601
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8672.00 −0.121646
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 73984.0 1.00000
$$273$$ 280576. 3.76466
$$274$$ −146056. −1.94544
$$275$$ 130000. 1.71901
$$276$$ 155648. 2.04327
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ −145856. −1.88727
$$279$$ 44800.0 0.575532
$$280$$ 0 0
$$281$$ 118754. 1.50396 0.751979 0.659187i $$-0.229100\pi$$
0.751979 + 0.659187i $$0.229100\pi$$
$$282$$ 0 0
$$283$$ −159728. −1.99438 −0.997191 0.0749057i $$-0.976134\pi$$
−0.997191 + 0.0749057i $$0.976134\pi$$
$$284$$ −126464. −1.56794
$$285$$ 0 0
$$286$$ −227968. −2.78703
$$287$$ 0 0
$$288$$ 179200. 2.16049
$$289$$ 83521.0 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −73102.0 −0.851518 −0.425759 0.904837i $$-0.639993\pi$$
−0.425759 + 0.904837i $$0.639993\pi$$
$$294$$ −108480. −1.25503
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −312832. −3.54649
$$298$$ −174904. −1.96955
$$299$$ 166592. 1.86342
$$300$$ −160000. −1.77778
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 153376. 1.67060
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 202300. 2.16049
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 212992. 2.24524
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −126464. −1.30751 −0.653757 0.756705i $$-0.726808\pi$$
−0.653757 + 0.756705i $$0.726808\pi$$
$$312$$ 280576. 2.88231
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 158024. 1.60274
$$315$$ 0 0
$$316$$ 42496.0 0.425573
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 267136. 2.64167
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 156416. 1.51800
$$322$$ −155648. −1.50118
$$323$$ 0 0
$$324$$ 158224. 1.50724
$$325$$ −171250. −1.62130
$$326$$ −113984. −1.07253
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ −77312.0 −0.693033
$$335$$ 0 0
$$336$$ −262144. −2.32200
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 186060. 1.62862
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 53248.0 0.457925
$$342$$ 0 0
$$343$$ −45184.0 −0.384058
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −240656. −1.99865 −0.999327 0.0366737i $$-0.988324\pi$$
−0.999327 + 0.0366737i $$0.988324\pi$$
$$348$$ 0 0
$$349$$ −236206. −1.93928 −0.969639 0.244541i $$-0.921363\pi$$
−0.969639 + 0.244541i $$0.921363\pi$$
$$350$$ 160000. 1.30612
$$351$$ 412096. 3.34491
$$352$$ 212992. 1.71901
$$353$$ −103294. −0.828945 −0.414472 0.910062i $$-0.636034\pi$$
−0.414472 + 0.910062i $$0.636034\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 8672.00 0.0684257
$$357$$ −295936. −2.32200
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 130321. 1.00000
$$362$$ 0 0
$$363$$ −457968. −3.47554
$$364$$ −280576. −2.11762
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 269344. 1.99975 0.999874 0.0158877i $$-0.00505741\pi$$
0.999874 + 0.0158877i $$0.00505741\pi$$
$$368$$ −155648. −1.14934
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −267136. −1.94082
$$372$$ −65536.0 −0.473581
$$373$$ 228686. 1.64370 0.821849 0.569706i $$-0.192943\pi$$
0.821849 + 0.569706i $$0.192943\pi$$
$$374$$ 240448. 1.71901
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −385024. −2.69466
$$379$$ −73424.0 −0.511163 −0.255582 0.966787i $$-0.582267\pi$$
−0.255582 + 0.966787i $$0.582267\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −262144. −1.77778
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −79858.0 −0.527739 −0.263870 0.964558i $$-0.584999\pi$$
−0.263870 + 0.964558i $$0.584999\pi$$
$$390$$ 0 0
$$391$$ −175712. −1.14934
$$392$$ 108480. 0.705956
$$393$$ −522496. −3.38297
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 582400. 3.71391
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 88192.0 0.556754
$$399$$ 0 0
$$400$$ 160000. 1.00000
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ −70144.0 −0.431897
$$404$$ −153376. −0.939712
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −295936. −1.77778
$$409$$ −292126. −1.74632 −0.873160 0.487435i $$-0.837933\pi$$
−0.873160 + 0.487435i $$0.837933\pi$$
$$410$$ 0 0
$$411$$ 584224. 3.45856
$$412$$ 0 0
$$413$$ 0 0
$$414$$ −425600. −2.48314
$$415$$ 0 0
$$416$$ −280576. −1.62130
$$417$$ 583424. 3.35515
$$418$$ 0 0
$$419$$ 138928. 0.791337 0.395669 0.918393i $$-0.370513\pi$$
0.395669 + 0.918393i $$0.370513\pi$$
$$420$$ 0 0
$$421$$ 324494. 1.83081 0.915403 0.402538i $$-0.131872\pi$$
0.915403 + 0.402538i $$0.131872\pi$$
$$422$$ 349504. 1.96258
$$423$$ 0 0
$$424$$ −267136. −1.48594
$$425$$ 180625. 1.00000
$$426$$ 505856. 2.78745
$$427$$ 0 0
$$428$$ −156416. −0.853874
$$429$$ 911872. 4.95472
$$430$$ 0 0
$$431$$ −62624.0 −0.337121 −0.168561 0.985691i $$-0.553912\pi$$
−0.168561 + 0.985691i $$0.553912\pi$$
$$432$$ −385024. −2.06310
$$433$$ −374722. −1.99863 −0.999317 0.0369452i $$-0.988237\pi$$
−0.999317 + 0.0369452i $$0.988237\pi$$
$$434$$ 65536.0 0.347937
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 204256. 1.05985 0.529927 0.848043i $$-0.322219\pi$$
0.529927 + 0.848043i $$0.322219\pi$$
$$440$$ 0 0
$$441$$ 296625. 1.52521
$$442$$ −316744. −1.62130
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 699616. 3.50142
$$448$$ 262144. 1.30612
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 437500. 2.16049
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 412096. 1.99934
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 147806. 0.707717 0.353859 0.935299i $$-0.384869\pi$$
0.353859 + 0.935299i $$0.384869\pi$$
$$458$$ 299576. 1.42816
$$459$$ −434656. −2.06310
$$460$$ 0 0
$$461$$ 180242. 0.848114 0.424057 0.905636i $$-0.360606\pi$$
0.424057 + 0.905636i $$0.360606\pi$$
$$462$$ −851968. −3.99153
$$463$$ 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 $$\pi$$
$$468$$ −767200. −3.50281
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −632096. −2.84932
$$472$$ 0 0
$$473$$ 0 0
$$474$$ −169984. −0.756574
$$475$$ 0 0
$$476$$ 295936. 1.30612
$$477$$ −730450. −3.21036
$$478$$ 0 0
$$479$$ −422432. −1.84114 −0.920568 0.390583i $$-0.872274\pi$$
−0.920568 + 0.390583i $$0.872274\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 622592. 2.66876
$$484$$ 457968. 1.95499
$$485$$ 0 0
$$486$$ −145600. −0.616437
$$487$$ −473888. −1.99810 −0.999051 0.0435486i $$-0.986134\pi$$
−0.999051 + 0.0435486i $$0.986134\pi$$
$$488$$ 0 0
$$489$$ 455936. 1.90672
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 65536.0 0.266389
$$497$$ −505856. −2.04793
$$498$$ 0 0
$$499$$ −140144. −0.562825 −0.281413 0.959587i $$-0.590803\pi$$
−0.281413 + 0.959587i $$0.590803\pi$$
$$500$$ 0 0
$$501$$ 309248. 1.23206
$$502$$ 0 0
$$503$$ 488032. 1.92891 0.964456 0.264244i $$-0.0851225\pi$$
0.964456 + 0.264244i $$0.0851225\pi$$
$$504$$ 716800. 2.82187
$$505$$ 0 0
$$506$$ −505856. −1.97572
$$507$$ −744240. −2.89532
$$508$$ 0 0
$$509$$ 38354.0 0.148039 0.0740193 0.997257i $$-0.476417\pi$$
0.0740193 + 0.997257i $$0.476417\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 262144. 1.00000
$$513$$ 0 0
$$514$$ −355336. −1.34497
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 522496. 1.90292
$$525$$ −640000. −2.32200
$$526$$ 0 0
$$527$$ 73984.0 0.266389
$$528$$ −851968. −3.05601
$$529$$ 89823.0 0.320979
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −34688.0 −0.121646
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 352560. 1.21354
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 295936. 1.00000
$$545$$ 0 0
$$546$$ 1.12230e6 3.76466
$$547$$ 329104. 1.09991 0.549957 0.835193i $$-0.314644\pi$$
0.549957 + 0.835193i $$0.314644\pi$$
$$548$$ −584224. −1.94544
$$549$$ 0 0
$$550$$ 520000. 1.71901
$$551$$ 0 0
$$552$$ 622592. 2.04327
$$553$$ 169984. 0.555850
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −583424. −1.88727
$$557$$ 32366.0 0.104323 0.0521613 0.998639i $$-0.483389\pi$$
0.0521613 + 0.998639i $$0.483389\pi$$
$$558$$ 179200. 0.575532
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −961792. −3.05601
$$562$$ 475016. 1.50396
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −638912. −1.99438
$$567$$ 632896. 1.96864
$$568$$ −505856. −1.56794
$$569$$ −331678. −1.02445 −0.512227 0.858850i $$-0.671179\pi$$
−0.512227 + 0.858850i $$0.671179\pi$$
$$570$$ 0 0
$$571$$ 650416. 1.99489 0.997445 0.0714371i $$-0.0227585\pi$$
0.997445 + 0.0714371i $$0.0227585\pi$$
$$572$$ −911872. −2.78703
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −380000. −1.14934
$$576$$ 716800. 2.16049
$$577$$ −573442. −1.72242 −0.861208 0.508253i $$-0.830291\pi$$
−0.861208 + 0.508253i $$0.830291\pi$$
$$578$$ 334084. 1.00000
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −868192. −2.55434
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −292408. −0.851518
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ −433920. −1.25503
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 546626. 1.55446 0.777232 0.629214i $$-0.216623\pi$$
0.777232 + 0.629214i $$0.216623\pi$$
$$594$$ −1.25133e6 −3.54649
$$595$$ 0 0
$$596$$ −699616. −1.96955
$$597$$ −352768. −0.989784
$$598$$ 666368. 1.86342
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ −640000. −1.77778
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 613504. 1.67060
$$607$$ −38336.0 −0.104047 −0.0520235 0.998646i $$-0.516567\pi$$
−0.0520235 + 0.998646i $$0.516567\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 809200. 2.16049
$$613$$ 741746. 1.97394 0.986971 0.160900i $$-0.0514397\pi$$
0.986971 + 0.160900i $$0.0514397\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 851968. 2.24524
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 594928. 1.55268 0.776342 0.630312i $$-0.217073\pi$$
0.776342 + 0.630312i $$0.217073\pi$$
$$620$$ 0 0
$$621$$ 914432. 2.37120
$$622$$ −505856. −1.30751
$$623$$ 34688.0 0.0893723
$$624$$ 1.12230e6 2.88231
$$625$$ 390625. 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 632096. 1.60274
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 169984. 0.425573
$$633$$ −1.39802e6 −3.48903
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 1.06854e6 2.64167
$$637$$ −464430. −1.14457
$$638$$ 0 0
$$639$$ −1.38320e6 −3.38753
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 625664. 1.51800
$$643$$ 614704. 1.48677 0.743386 0.668863i $$-0.233219\pi$$
0.743386 + 0.668863i $$0.233219\pi$$
$$644$$ −622592. −1.50118
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 632896. 1.50724
$$649$$ 0 0
$$650$$ −685000. −1.62130
$$651$$ −262144. −0.618554
$$652$$ −455936. −1.07253
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 394034. 0.901843 0.450921 0.892564i $$-0.351095\pi$$
0.450921 + 0.892564i $$0.351095\pi$$
$$662$$ 0 0
$$663$$ 1.26698e6 2.88231
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −309248. −0.693033
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −1.04858e6 −2.32200
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ −940000. −2.06310
$$676$$ 744240. 1.62862
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −1.64838e6 −3.55438
$$682$$ 212992. 0.457925
$$683$$ 875824. 1.87748 0.938740 0.344626i $$-0.111994\pi$$
0.938740 + 0.344626i $$0.111994\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −180736. −0.384058
$$687$$ −1.19830e6 −2.53895
$$688$$ 0 0
$$689$$ 1.14368e6 2.40915
$$690$$ 0 0
$$691$$ −646064. −1.35307 −0.676534 0.736412i $$-0.736519\pi$$
−0.676534 + 0.736412i $$0.736519\pi$$
$$692$$ 0 0
$$693$$ 2.32960e6 4.85082
$$694$$ −962624. −1.99865
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −944824. −1.93928
$$699$$ 0 0
$$700$$ 640000. 1.30612
$$701$$ 845102. 1.71978 0.859890 0.510479i $$-0.170532\pi$$
0.859890 + 0.510479i $$0.170532\pi$$
$$702$$ 1.64838e6 3.34491
$$703$$ 0 0
$$704$$ 851968. 1.71901
$$705$$ 0 0
$$706$$ −413176. −0.828945
$$707$$ −613504. −1.22738
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 464800. 0.919447
$$712$$ 34688.0 0.0684257
$$713$$ −155648. −0.306171
$$714$$ −1.18374e6 −2.32200
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −232544. −0.449829 −0.224914 0.974379i $$-0.572210\pi$$
−0.224914 + 0.974379i $$0.572210\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 521284. 1.00000
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ −1.83187e6 −3.47554
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ −1.12230e6 −2.11762
$$729$$ −218609. −0.411351
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −110254. −0.205204 −0.102602 0.994722i $$-0.532717\pi$$
−0.102602 + 0.994722i $$0.532717\pi$$
$$734$$ 1.07738e6 1.99975
$$735$$ 0 0
$$736$$ −622592. −1.14934
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −1.06854e6 −1.94082
$$743$$ 891904. 1.61562 0.807812 0.589440i $$-0.200651\pi$$
0.807812 + 0.589440i $$0.200651\pi$$
$$744$$ −262144. −0.473581
$$745$$ 0 0
$$746$$ 914744. 1.64370
$$747$$ 0 0
$$748$$ 961792. 1.71901
$$749$$ −625664. −1.11526
$$750$$ 0 0
$$751$$ −846752. −1.50133 −0.750665 0.660683i $$-0.770267\pi$$
−0.750665 + 0.660683i $$0.770267\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −1.54010e6 −2.69466
$$757$$ −705202. −1.23061 −0.615307 0.788288i $$-0.710968\pi$$
−0.615307 + 0.788288i $$0.710968\pi$$
$$758$$ −293696. −0.511163
$$759$$ 2.02342e6 3.51239
$$760$$ 0 0
$$761$$ 320414. 0.553276 0.276638 0.960974i $$-0.410780\pi$$
0.276638 + 0.960974i $$0.410780\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −1.04858e6 −1.77778
$$769$$ 1.07929e6 1.82510 0.912551 0.408963i $$-0.134110\pi$$
0.912551 + 0.408963i $$0.134110\pi$$
$$770$$ 0 0
$$771$$ 1.42134e6 2.39106
$$772$$ 0 0
$$773$$ 264206. 0.442164 0.221082 0.975255i $$-0.429041\pi$$
0.221082 + 0.975255i $$0.429041\pi$$
$$774$$ 0 0
$$775$$ 160000. 0.266389
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −319432. −0.527739
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −1.64403e6 −2.69531
$$782$$ −702848. −1.14934
$$783$$ 0 0
$$784$$ 433920. 0.705956
$$785$$ 0 0
$$786$$ −2.08998e6 −3.38297
$$787$$ 757264. 1.22264 0.611319 0.791384i $$-0.290639\pi$$
0.611319 + 0.791384i $$0.290639\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 2.32960e6 3.71391
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 352768. 0.556754
$$797$$ 477266. 0.751353 0.375676 0.926751i $$-0.377410\pi$$
0.375676 + 0.926751i $$0.377410\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 640000. 1.00000
$$801$$ 94850.0 0.147833
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −280576. −0.431897
$$807$$ 0 0
$$808$$ −613504. −0.939712
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 1.29746e6 1.97265 0.986327 0.164800i $$-0.0526978\pi$$
0.986327 + 0.164800i $$0.0526978\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −1.18374e6 −1.77778
$$817$$ 0 0
$$818$$ −1.16850e6 −1.74632
$$819$$ −3.06880e6 −4.57510
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 2.33690e6 3.45856
$$823$$ −1.33002e6 −1.96362 −0.981809 0.189869i $$-0.939194\pi$$
−0.981809 + 0.189869i $$0.939194\pi$$
$$824$$ 0 0
$$825$$ −2.08000e6 −3.05601
$$826$$ 0 0
$$827$$ −877808. −1.28348 −0.641739 0.766923i $$-0.721787\pi$$
−0.641739 + 0.766923i $$0.721787\pi$$
$$828$$ −1.70240e6 −2.48314
$$829$$ −280366. −0.407959 −0.203979 0.978975i $$-0.565388\pi$$
−0.203979 + 0.978975i $$0.565388\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.12230e6 −1.62130
$$833$$ 489855. 0.705956
$$834$$ 2.33370e6 3.35515
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −385024. −0.549588
$$838$$ 555712. 0.791337
$$839$$ −1.39270e6 −1.97849 −0.989247 0.146252i $$-0.953279\pi$$
−0.989247 + 0.146252i $$0.953279\pi$$
$$840$$ 0 0
$$841$$ 707281. 1.00000
$$842$$ 1.29798e6 1.83081
$$843$$ −1.90006e6 −2.67370
$$844$$ 1.39802e6 1.96258
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.83187e6 2.55346
$$848$$ −1.06854e6 −1.48594
$$849$$ 2.55565e6 3.54557
$$850$$ 722500. 1.00000
$$851$$ 0 0
$$852$$ 2.02342e6 2.78745
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −625664. −0.853874
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 3.64749e6 4.95472
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −250496. −0.337121
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ −1.54010e6 −2.06310
$$865$$ 0 0
$$866$$ −1.49889e6 −1.99863
$$867$$ −1.33634e6 −1.77778
$$868$$ 262144. 0.347937
$$869$$ 552448. 0.731563
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 817024. 1.05985
$$879$$ 1.16963e6 1.51381
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 1.18650e6 1.52521
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ −1.26698e6 −1.62130
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 798304. 1.01466 0.507331 0.861752i $$-0.330632\pi$$
0.507331 + 0.861752i $$0.330632\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 2.05691e6 2.59096
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 2.79846e6 3.50142
$$895$$ 0 0
$$896$$ 1.04858e6 1.30612
$$897$$ −2.66547e6 −3.31275
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 1.75000e6 2.16049
$$901$$ −1.20629e6 −1.48594
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −1.15525e6 −1.40430 −0.702151 0.712028i $$-0.747777\pi$$
−0.702151 + 0.712028i $$0.747777\pi$$
$$908$$ 1.64838e6 1.99934
$$909$$ −1.67755e6 −2.03024
$$910$$ 0 0
$$911$$ 58816.0 0.0708694 0.0354347 0.999372i $$-0.488718\pi$$
0.0354347 + 0.999372i $$0.488718\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 591224. 0.707717
$$915$$ 0 0
$$916$$ 1.19830e6 1.42816
$$917$$ 2.08998e6 2.48545
$$918$$ −1.73862e6 −2.06310
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 720968. 0.848114
$$923$$ 2.16570e6 2.54211
$$924$$ −3.40787e6 −3.99153
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 2.02342e6 2.32447
$$934$$ 0 0
$$935$$ 0 0
$$936$$ −3.06880e6 −3.50281
$$937$$ −163294. −0.185991 −0.0929953 0.995667i $$-0.529644\pi$$
−0.0929953 + 0.995667i $$0.529644\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ −2.52838e6 −2.84932
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.61243e6 1.79797 0.898983 0.437984i $$-0.144307\pi$$
0.898983 + 0.437984i $$0.144307\pi$$
$$948$$ −679936. −0.756574
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.18374e6 1.30612
$$953$$ −1.09731e6 −1.20822 −0.604109 0.796902i $$-0.706471\pi$$
−0.604109 + 0.796902i $$0.706471\pi$$
$$954$$ −2.92180e6 −3.21036
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −1.68973e6 −1.84114
$$959$$ −2.33690e6 −2.54099
$$960$$ 0 0
$$961$$ −857985. −0.929037
$$962$$ 0 0
$$963$$ −1.71080e6 −1.84479
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 2.49037e6 2.66876
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 1.83187e6 1.95499
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ −582400. −0.616437
$$973$$ −2.33370e6 −2.46501
$$974$$ −1.89555e6 −1.99810
$$975$$ 2.74000e6 2.88231
$$976$$ 0 0
$$977$$ −10174.0 −0.0106587 −0.00532933 0.999986i $$-0.501696\pi$$
−0.00532933 + 0.999986i $$0.501696\pi$$
$$978$$ 1.82374e6 1.90672
$$979$$ 112736. 0.117624
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1.76118e6 1.82263 0.911313 0.411714i $$-0.135070\pi$$
0.911313 + 0.411714i $$0.135070\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −1.89718e6 −1.93180 −0.965900 0.258916i $$-0.916635\pi$$
−0.965900 + 0.258916i $$0.916635\pi$$
$$992$$ 262144. 0.266389
$$993$$ 0 0
$$994$$ −2.02342e6 −2.04793
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ −560576. −0.562825
$$999$$ 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 68.5.d.a.67.1 1
4.3 odd 2 68.5.d.b.67.1 yes 1
17.16 even 2 68.5.d.b.67.1 yes 1
68.67 odd 2 CM 68.5.d.a.67.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
68.5.d.a.67.1 1 1.1 even 1 trivial
68.5.d.a.67.1 1 68.67 odd 2 CM
68.5.d.b.67.1 yes 1 4.3 odd 2
68.5.d.b.67.1 yes 1 17.16 even 2