# Properties

 Label 87.7.d.b.86.1 Level $87$ Weight $7$ Character 87.86 Self dual yes Analytic conductor $20.015$ Analytic rank $0$ Dimension $1$ CM discriminant -87 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [87,7,Mod(86,87)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(87, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("87.86");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$87 = 3 \cdot 29$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 87.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: $$20.0147052749$$ 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 86.1 Character $$\chi$$ $$=$$ 87.86

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+13.0000 q^{2} -27.0000 q^{3} +105.000 q^{4} -351.000 q^{6} +338.000 q^{7} +533.000 q^{8} +729.000 q^{9} +O(q^{10})$$ $$q+13.0000 q^{2} -27.0000 q^{3} +105.000 q^{4} -351.000 q^{6} +338.000 q^{7} +533.000 q^{8} +729.000 q^{9} -806.000 q^{11} -2835.00 q^{12} +3002.00 q^{13} +4394.00 q^{14} +209.000 q^{16} +9778.00 q^{17} +9477.00 q^{18} -9126.00 q^{21} -10478.0 q^{22} -14391.0 q^{24} +15625.0 q^{25} +39026.0 q^{26} -19683.0 q^{27} +35490.0 q^{28} +24389.0 q^{29} -31395.0 q^{32} +21762.0 q^{33} +127114. q^{34} +76545.0 q^{36} -81054.0 q^{39} -132158. q^{41} -118638. q^{42} -84630.0 q^{44} -151502. q^{47} -5643.00 q^{48} -3405.00 q^{49} +203125. q^{50} -264006. q^{51} +315210. q^{52} -255879. q^{54} +180154. q^{56} +317057. q^{58} +246402. q^{63} -421511. q^{64} +282906. q^{66} +267098. q^{67} +1.02669e6 q^{68} +388557. q^{72} -421875. q^{75} -272428. q^{77} -1.05370e6 q^{78} +531441. q^{81} -1.71805e6 q^{82} -958230. q^{84} -658503. q^{87} -429598. q^{88} +71266.0 q^{89} +1.01468e6 q^{91} -1.96953e6 q^{94} +847665. q^{96} -44265.0 q^{98} -587574. q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$59$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 13.0000 1.62500 0.812500 0.582961i $$-0.198106\pi$$
0.812500 + 0.582961i $$0.198106\pi$$
$$3$$ −27.0000 −1.00000
$$4$$ 105.000 1.64062
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ −351.000 −1.62500
$$7$$ 338.000 0.985423 0.492711 0.870193i $$-0.336006\pi$$
0.492711 + 0.870193i $$0.336006\pi$$
$$8$$ 533.000 1.04102
$$9$$ 729.000 1.00000
$$10$$ 0 0
$$11$$ −806.000 −0.605560 −0.302780 0.953061i $$-0.597915\pi$$
−0.302780 + 0.953061i $$0.597915\pi$$
$$12$$ −2835.00 −1.64062
$$13$$ 3002.00 1.36641 0.683204 0.730227i $$-0.260586\pi$$
0.683204 + 0.730227i $$0.260586\pi$$
$$14$$ 4394.00 1.60131
$$15$$ 0 0
$$16$$ 209.000 0.0510254
$$17$$ 9778.00 1.99023 0.995115 0.0987225i $$-0.0314756\pi$$
0.995115 + 0.0987225i $$0.0314756\pi$$
$$18$$ 9477.00 1.62500
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ −9126.00 −0.985423
$$22$$ −10478.0 −0.984035
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ −14391.0 −1.04102
$$25$$ 15625.0 1.00000
$$26$$ 39026.0 2.22041
$$27$$ −19683.0 −1.00000
$$28$$ 35490.0 1.61671
$$29$$ 24389.0 1.00000
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ −31395.0 −0.958099
$$33$$ 21762.0 0.605560
$$34$$ 127114. 3.23412
$$35$$ 0 0
$$36$$ 76545.0 1.64062
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −81054.0 −1.36641
$$40$$ 0 0
$$41$$ −132158. −1.91753 −0.958764 0.284202i $$-0.908271\pi$$
−0.958764 + 0.284202i $$0.908271\pi$$
$$42$$ −118638. −1.60131
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ −84630.0 −0.993496
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −151502. −1.45923 −0.729617 0.683856i $$-0.760302\pi$$
−0.729617 + 0.683856i $$0.760302\pi$$
$$48$$ −5643.00 −0.0510254
$$49$$ −3405.00 −0.0289420
$$50$$ 203125. 1.62500
$$51$$ −264006. −1.99023
$$52$$ 315210. 2.24176
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ −255879. −1.62500
$$55$$ 0 0
$$56$$ 180154. 1.02584
$$57$$ 0 0
$$58$$ 317057. 1.62500
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 246402. 0.985423
$$64$$ −421511. −1.60794
$$65$$ 0 0
$$66$$ 282906. 0.984035
$$67$$ 267098. 0.888068 0.444034 0.896010i $$-0.353547\pi$$
0.444034 + 0.896010i $$0.353547\pi$$
$$68$$ 1.02669e6 3.26522
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 388557. 1.04102
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ −421875. −1.00000
$$76$$ 0 0
$$77$$ −272428. −0.596732
$$78$$ −1.05370e6 −2.22041
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 531441. 1.00000
$$82$$ −1.71805e6 −3.11598
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −958230. −1.61671
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −658503. −1.00000
$$88$$ −429598. −0.630397
$$89$$ 71266.0 0.101091 0.0505455 0.998722i $$-0.483904\pi$$
0.0505455 + 0.998722i $$0.483904\pi$$
$$90$$ 0 0
$$91$$ 1.01468e6 1.34649
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −1.96953e6 −2.37125
$$95$$ 0 0
$$96$$ 847665. 0.958099
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −44265.0 −0.0470308
$$99$$ −587574. −0.605560
$$100$$ 1.64062e6 1.64062
$$101$$ −1.86561e6 −1.81074 −0.905369 0.424625i $$-0.860406\pi$$
−0.905369 + 0.424625i $$0.860406\pi$$
$$102$$ −3.43208e6 −3.23412
$$103$$ 1.65615e6 1.51561 0.757804 0.652482i $$-0.226272\pi$$
0.757804 + 0.652482i $$0.226272\pi$$
$$104$$ 1.60007e6 1.42245
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ −2.06672e6 −1.64062
$$109$$ −2.58957e6 −1.99963 −0.999813 0.0193314i $$-0.993846\pi$$
−0.999813 + 0.0193314i $$0.993846\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 70642.0 0.0502816
$$113$$ 2.77976e6 1.92651 0.963257 0.268580i $$-0.0865545\pi$$
0.963257 + 0.268580i $$0.0865545\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.56084e6 1.64062
$$117$$ 2.18846e6 1.36641
$$118$$ 0 0
$$119$$ 3.30496e6 1.96122
$$120$$ 0 0
$$121$$ −1.12192e6 −0.633297
$$122$$ 0 0
$$123$$ 3.56827e6 1.91753
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 3.20323e6 1.60131
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ −3.47036e6 −1.65480
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.24153e6 −1.44190 −0.720951 0.692986i $$-0.756295\pi$$
−0.720951 + 0.692986i $$0.756295\pi$$
$$132$$ 2.28501e6 0.993496
$$133$$ 0 0
$$134$$ 3.47227e6 1.44311
$$135$$ 0 0
$$136$$ 5.21167e6 2.07186
$$137$$ 4.95219e6 1.92591 0.962955 0.269662i $$-0.0869121\pi$$
0.962955 + 0.269662i $$0.0869121\pi$$
$$138$$ 0 0
$$139$$ −5.04405e6 −1.87817 −0.939086 0.343682i $$-0.888326\pi$$
−0.939086 + 0.343682i $$0.888326\pi$$
$$140$$ 0 0
$$141$$ 4.09055e6 1.45923
$$142$$ 0 0
$$143$$ −2.41961e6 −0.827442
$$144$$ 152361. 0.0510254
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 91935.0 0.0289420
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ −5.48438e6 −1.62500
$$151$$ 3.04920e6 0.885636 0.442818 0.896611i $$-0.353979\pi$$
0.442818 + 0.896611i $$0.353979\pi$$
$$152$$ 0 0
$$153$$ 7.12816e6 1.99023
$$154$$ −3.54156e6 −0.969690
$$155$$ 0 0
$$156$$ −8.51067e6 −2.24176
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 6.90873e6 1.62500
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ −1.38766e7 −3.14595
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ −4.86416e6 −1.02584
$$169$$ 4.18520e6 0.867073
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ −8.56054e6 −1.62500
$$175$$ 5.28125e6 0.985423
$$176$$ −168454. −0.0308989
$$177$$ 0 0
$$178$$ 926458. 0.164273
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ 1.16242e7 1.96033 0.980164 0.198190i $$-0.0635061\pi$$
0.980164 + 0.198190i $$0.0635061\pi$$
$$182$$ 1.31908e7 2.18805
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −7.88107e6 −1.20520
$$188$$ −1.59077e7 −2.39405
$$189$$ −6.65285e6 −0.985423
$$190$$ 0 0
$$191$$ −8.42296e6 −1.20883 −0.604414 0.796670i $$-0.706593\pi$$
−0.604414 + 0.796670i $$0.706593\pi$$
$$192$$ 1.13808e7 1.60794
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −357525. −0.0474830
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ −7.63846e6 −0.984035
$$199$$ −5.46993e6 −0.694101 −0.347051 0.937846i $$-0.612817\pi$$
−0.347051 + 0.937846i $$0.612817\pi$$
$$200$$ 8.32812e6 1.04102
$$201$$ −7.21165e6 −0.888068
$$202$$ −2.42529e7 −2.94245
$$203$$ 8.24348e6 0.985423
$$204$$ −2.77206e7 −3.26522
$$205$$ 0 0
$$206$$ 2.15299e7 2.46286
$$207$$ 0 0
$$208$$ 627418. 0.0697215
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.04910e7 −1.04102
$$217$$ 0 0
$$218$$ −3.36645e7 −3.24939
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2.93536e7 2.71947
$$222$$ 0 0
$$223$$ −1.27910e7 −1.15343 −0.576715 0.816945i $$-0.695666\pi$$
−0.576715 + 0.816945i $$0.695666\pi$$
$$224$$ −1.06115e7 −0.944133
$$225$$ 1.13906e7 1.00000
$$226$$ 3.61369e7 3.13059
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 7.35556e6 0.596732
$$232$$ 1.29993e7 1.04102
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 2.84500e7 2.22041
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 4.29645e7 3.18698
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −2.12038e7 −1.51483 −0.757413 0.652936i $$-0.773537\pi$$
−0.757413 + 0.652936i $$0.773537\pi$$
$$242$$ −1.45850e7 −1.02911
$$243$$ −1.43489e7 −1.00000
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 4.63875e7 3.11598
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −3.03718e7 −1.92066 −0.960329 0.278870i $$-0.910040\pi$$
−0.960329 + 0.278870i $$0.910040\pi$$
$$252$$ 2.58722e7 1.61671
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −1.81380e7 −1.08111
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 1.77796e7 1.00000
$$262$$ −4.21398e7 −2.34309
$$263$$ 1.26271e7 0.694123 0.347062 0.937842i $$-0.387179\pi$$
0.347062 + 0.937842i $$0.387179\pi$$
$$264$$ 1.15991e7 0.630397
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.92418e6 −0.101091
$$268$$ 2.80453e7 1.45699
$$269$$ −7.92469e6 −0.407123 −0.203562 0.979062i $$-0.565252\pi$$
−0.203562 + 0.979062i $$0.565252\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 2.04360e6 0.101552
$$273$$ −2.73963e7 −1.34649
$$274$$ 6.43785e7 3.12960
$$275$$ −1.25938e7 −0.605560
$$276$$ 0 0
$$277$$ −4.24167e7 −1.99571 −0.997854 0.0654718i $$-0.979145\pi$$
−0.997854 + 0.0654718i $$0.979145\pi$$
$$278$$ −6.55727e7 −3.05203
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 5.31772e7 2.37125
$$283$$ −4.20180e7 −1.85386 −0.926928 0.375240i $$-0.877560\pi$$
−0.926928 + 0.375240i $$0.877560\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −3.14550e7 −1.34459
$$287$$ −4.46694e7 −1.88958
$$288$$ −2.28870e7 −0.958099
$$289$$ 7.14717e7 2.96102
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 3.17371e7 1.26172 0.630862 0.775895i $$-0.282702\pi$$
0.630862 + 0.775895i $$0.282702\pi$$
$$294$$ 1.19516e6 0.0470308
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.58645e7 0.605560
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −4.42969e7 −1.64062
$$301$$ 0 0
$$302$$ 3.96396e7 1.43916
$$303$$ 5.03714e7 1.81074
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 9.26661e7 3.23412
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ −2.86049e7 −0.979014
$$309$$ −4.47159e7 −1.51561
$$310$$ 0 0
$$311$$ 5.26143e7 1.74913 0.874566 0.484907i $$-0.161147\pi$$
0.874566 + 0.484907i $$0.161147\pi$$
$$312$$ −4.32018e7 −1.42245
$$313$$ −5.25314e7 −1.71311 −0.856557 0.516052i $$-0.827401\pi$$
−0.856557 + 0.516052i $$0.827401\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.13678e6 0.255432 0.127716 0.991811i $$-0.459235\pi$$
0.127716 + 0.991811i $$0.459235\pi$$
$$318$$ 0 0
$$319$$ −1.96575e7 −0.605560
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 5.58013e7 1.64062
$$325$$ 4.69062e7 1.36641
$$326$$ 0 0
$$327$$ 6.99185e7 1.99963
$$328$$ −7.04402e7 −1.99618
$$329$$ −5.12077e7 −1.43796
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −1.90733e6 −0.0502816
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 5.44075e7 1.40899
$$339$$ −7.50536e7 −1.92651
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −4.09163e7 −1.01394
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ −6.91428e7 −1.64062
$$349$$ −8.06657e7 −1.89763 −0.948817 0.315825i $$-0.897719\pi$$
−0.948817 + 0.315825i $$0.897719\pi$$
$$350$$ 6.86562e7 1.60131
$$351$$ −5.90884e7 −1.36641
$$352$$ 2.53044e7 0.580186
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 7.48293e6 0.165852
$$357$$ −8.92340e7 −1.96122
$$358$$ 0 0
$$359$$ 8.18203e7 1.76839 0.884194 0.467120i $$-0.154708\pi$$
0.884194 + 0.467120i $$0.154708\pi$$
$$360$$ 0 0
$$361$$ 4.70459e7 1.00000
$$362$$ 1.51115e8 3.18553
$$363$$ 3.02920e7 0.633297
$$364$$ 1.06541e8 2.20909
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ −9.63432e7 −1.91753
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 45866.0 0.000883821 0 0.000441911 1.00000i $$-0.499859\pi$$
0.000441911 1.00000i $$0.499859\pi$$
$$374$$ −1.02454e8 −1.95846
$$375$$ 0 0
$$376$$ −8.07506e7 −1.51908
$$377$$ 7.32158e7 1.36641
$$378$$ −8.64871e7 −1.60131
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −1.09498e8 −1.96435
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 9.36998e7 1.65480
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −7.00237e6 −0.118959 −0.0594794 0.998230i $$-0.518944\pi$$
−0.0594794 + 0.998230i $$0.518944\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.81486e6 −0.0301291
$$393$$ 8.75212e7 1.44190
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −6.16953e7 −0.993496
$$397$$ 6.89034e7 1.10121 0.550603 0.834767i $$-0.314398\pi$$
0.550603 + 0.834767i $$0.314398\pi$$
$$398$$ −7.11091e7 −1.12791
$$399$$ 0 0
$$400$$ 3.26562e6 0.0510254
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ −9.37514e7 −1.44311
$$403$$ 0 0
$$404$$ −1.95889e8 −2.97074
$$405$$ 0 0
$$406$$ 1.07165e8 1.60131
$$407$$ 0 0
$$408$$ −1.40715e8 −2.07186
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ −1.33709e8 −1.92591
$$412$$ 1.73895e8 2.48654
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −9.42478e7 −1.30916
$$417$$ 1.36189e8 1.87817
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ −1.10445e8 −1.45923
$$424$$ 0 0
$$425$$ 1.52781e8 1.99023
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 6.53295e7 0.827442
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −4.11375e6 −0.0510254
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.71905e8 −3.28064
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 1.18161e8 1.39663 0.698315 0.715790i $$-0.253933\pi$$
0.698315 + 0.715790i $$0.253933\pi$$
$$440$$ 0 0
$$441$$ −2.48224e6 −0.0289420
$$442$$ 3.81596e8 4.41913
$$443$$ 1.14561e7 0.131773 0.0658865 0.997827i $$-0.479012\pi$$
0.0658865 + 0.997827i $$0.479012\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −1.66283e8 −1.87432
$$447$$ 0 0
$$448$$ −1.42471e8 −1.58450
$$449$$ −1.67070e8 −1.84570 −0.922848 0.385165i $$-0.874145\pi$$
−0.922848 + 0.385165i $$0.874145\pi$$
$$450$$ 1.48078e8 1.62500
$$451$$ 1.06519e8 1.16118
$$452$$ 2.91875e8 3.16069
$$453$$ −8.23285e7 −0.885636
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.58727e8 1.66304 0.831520 0.555495i $$-0.187471\pi$$
0.831520 + 0.555495i $$0.187471\pi$$
$$458$$ 0 0
$$459$$ −1.92460e8 −1.99023
$$460$$ 0 0
$$461$$ −9.73084e6 −0.0993225 −0.0496612 0.998766i $$-0.515814\pi$$
−0.0496612 + 0.998766i $$0.515814\pi$$
$$462$$ 9.56222e7 0.969690
$$463$$ −1.97688e8 −1.99177 −0.995883 0.0906462i $$-0.971107\pi$$
−0.995883 + 0.0906462i $$0.971107\pi$$
$$464$$ 5.09730e6 0.0510254
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1.80543e8 1.77268 0.886341 0.463034i $$-0.153239\pi$$
0.886341 + 0.463034i $$0.153239\pi$$
$$468$$ 2.29788e8 2.24176
$$469$$ 9.02791e7 0.875122
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 3.47021e8 3.21762
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 2.19478e8 1.99703 0.998514 0.0545017i $$-0.0173570\pi$$
0.998514 + 0.0545017i $$0.0173570\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −2.75649e8 −2.46159
$$483$$ 0 0
$$484$$ −1.17802e8 −1.03900
$$485$$ 0 0
$$486$$ −1.86536e8 −1.62500
$$487$$ −2.00089e8 −1.73235 −0.866176 0.499738i $$-0.833429\pi$$
−0.866176 + 0.499738i $$0.833429\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 3.55132e7 0.300017 0.150008 0.988685i $$-0.452070\pi$$
0.150008 + 0.988685i $$0.452070\pi$$
$$492$$ 3.74668e8 3.14595
$$493$$ 2.38476e8 1.99023
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2.46649e8 1.98507 0.992537 0.121941i $$-0.0389119\pi$$
0.992537 + 0.121941i $$0.0389119\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −3.94834e8 −3.12107
$$503$$ 1.93831e8 1.52307 0.761534 0.648125i $$-0.224446\pi$$
0.761534 + 0.648125i $$0.224446\pi$$
$$504$$ 1.31332e8 1.02584
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.13000e8 −0.867073
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.36910e7 −0.102006
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.22111e8 0.883653
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 2.31135e8 1.62500
$$523$$ −8.09344e7 −0.565754 −0.282877 0.959156i $$-0.591289\pi$$
−0.282877 + 0.959156i $$0.591289\pi$$
$$524$$ −3.40360e8 −2.36562
$$525$$ −1.42594e8 −0.985423
$$526$$ 1.64152e8 1.12795
$$527$$ 0 0
$$528$$ 4.54826e6 0.0308989
$$529$$ 1.48036e8 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.96738e8 −2.62013
$$534$$ −2.50144e7 −0.164273
$$535$$ 0 0
$$536$$ 1.42363e8 0.924493
$$537$$ 0 0
$$538$$ −1.03021e8 −0.661575
$$539$$ 2.74443e6 0.0175261
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ −3.13854e8 −1.96033
$$544$$ −3.06980e8 −1.90684
$$545$$ 0 0
$$546$$ −3.56151e8 −2.18805
$$547$$ −3.17273e8 −1.93852 −0.969262 0.246031i $$-0.920874\pi$$
−0.969262 + 0.246031i $$0.920874\pi$$
$$548$$ 5.19980e8 3.15970
$$549$$ 0 0
$$550$$ −1.63719e8 −0.984035
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −5.51417e8 −3.24303
$$555$$ 0 0
$$556$$ −5.29626e8 −3.08138
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 2.12789e8 1.20520
$$562$$ 0 0
$$563$$ −3.56237e8 −1.99625 −0.998123 0.0612458i $$-0.980493\pi$$
−0.998123 + 0.0612458i $$0.980493\pi$$
$$564$$ 4.29508e8 2.39405
$$565$$ 0 0
$$566$$ −5.46234e8 −3.01251
$$567$$ 1.79627e8 0.985423
$$568$$ 0 0
$$569$$ −2.27313e8 −1.23392 −0.616962 0.786993i $$-0.711637\pi$$
−0.616962 + 0.786993i $$0.711637\pi$$
$$570$$ 0 0
$$571$$ −2.76063e8 −1.48286 −0.741431 0.671029i $$-0.765852\pi$$
−0.741431 + 0.671029i $$0.765852\pi$$
$$572$$ −2.54059e8 −1.35752
$$573$$ 2.27420e8 1.20883
$$574$$ −5.80702e8 −3.07056
$$575$$ 0 0
$$576$$ −3.07282e8 −1.60794
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 9.29132e8 4.81165
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 4.12582e8 2.05030
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 9.65318e6 0.0474830
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 2.06238e8 0.984035
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.47688e8 0.694101
$$598$$ 0 0
$$599$$ 3.79855e8 1.76741 0.883706 0.468042i $$-0.155040\pi$$
0.883706 + 0.468042i $$0.155040\pi$$
$$600$$ −2.24859e8 −1.04102
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 1.94714e8 0.888068
$$604$$ 3.20166e8 1.45300
$$605$$ 0 0
$$606$$ 6.54828e8 2.94245
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ −2.22574e8 −0.985423
$$610$$ 0 0
$$611$$ −4.54809e8 −1.99391
$$612$$ 7.48457e8 3.26522
$$613$$ 4.09911e8 1.77954 0.889771 0.456407i $$-0.150864\pi$$
0.889771 + 0.456407i $$0.150864\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −1.45204e8 −0.621208
$$617$$ −1.20584e8 −0.513375 −0.256687 0.966494i $$-0.582631\pi$$
−0.256687 + 0.966494i $$0.582631\pi$$
$$618$$ −5.81307e8 −2.46286
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 6.83986e8 2.84234
$$623$$ 2.40879e7 0.0996173
$$624$$ −1.69403e7 −0.0697215
$$625$$ 2.44141e8 1.00000
$$626$$ −6.82909e8 −2.78381
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −4.04012e7 −0.160807 −0.0804037 0.996762i $$-0.525621\pi$$
−0.0804037 + 0.996762i $$0.525621\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 1.05778e8 0.415076
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.02218e7 −0.0395466
$$638$$ −2.55548e8 −0.984035
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 8.63571e7 0.327887 0.163943 0.986470i $$-0.447579\pi$$
0.163943 + 0.986470i $$0.447579\pi$$
$$642$$ 0 0
$$643$$ 4.09322e8 1.53968 0.769842 0.638234i $$-0.220335\pi$$
0.769842 + 0.638234i $$0.220335\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 2.83258e8 1.04102
$$649$$ 0 0
$$650$$ 6.09781e8 2.22041
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 5.51406e8 1.98031 0.990153 0.139990i $$-0.0447069\pi$$
0.990153 + 0.139990i $$0.0447069\pi$$
$$654$$ 9.08940e8 3.24939
$$655$$ 0 0
$$656$$ −2.76210e7 −0.0978427
$$657$$ 0 0
$$658$$ −6.65700e8 −2.33669
$$659$$ −1.02798e8 −0.359192 −0.179596 0.983740i $$-0.557479\pi$$
−0.179596 + 0.983740i $$0.557479\pi$$
$$660$$ 0 0
$$661$$ 4.76905e7 0.165130 0.0825652 0.996586i $$-0.473689\pi$$
0.0825652 + 0.996586i $$0.473689\pi$$
$$662$$ 0 0
$$663$$ −7.92546e8 −2.71947
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 3.45358e8 1.15343
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.86511e8 0.944133
$$673$$ −1.81732e8 −0.596192 −0.298096 0.954536i $$-0.596351\pi$$
−0.298096 + 0.954536i $$0.596351\pi$$
$$674$$ 0 0
$$675$$ −3.07547e8 −1.00000
$$676$$ 4.39445e8 1.42254
$$677$$ 6.14494e8 1.98039 0.990197 0.139676i $$-0.0446060\pi$$
0.990197 + 0.139676i $$0.0446060\pi$$
$$678$$ −9.75696e8 −3.13059
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −5.31911e8 −1.64766
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 5.54226e8 1.67978 0.839890 0.542757i $$-0.182619\pi$$
0.839890 + 0.542757i $$0.182619\pi$$
$$692$$ 0 0
$$693$$ −1.98600e8 −0.596732
$$694$$ 0 0
$$695$$ 0 0
$$696$$ −3.50982e8 −1.04102
$$697$$ −1.29224e9 −3.81632
$$698$$ −1.04865e9 −3.08366
$$699$$ 0 0
$$700$$ 5.54531e8 1.61671
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ −7.68149e8 −2.22041
$$703$$ 0 0
$$704$$ 3.39738e8 0.973702
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6.30575e8 −1.78434
$$708$$ 0 0
$$709$$ −3.91915e7 −0.109965 −0.0549824 0.998487i $$-0.517510\pi$$
−0.0549824 + 0.998487i $$0.517510\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 3.79848e7 0.105237
$$713$$ 0 0
$$714$$ −1.16004e9 −3.18698
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 1.06366e9 2.87363
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 5.59777e8 1.49351
$$722$$ 6.11596e8 1.62500
$$723$$ 5.72503e8 1.51483
$$724$$ 1.22054e9 3.21616
$$725$$ 3.81078e8 1.00000
$$726$$ 3.93796e8 1.02911
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 5.40822e8 1.40172
$$729$$ 3.87420e8 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.15281e8 −0.537778
$$738$$ −1.25246e9 −3.11598
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 7.55700e8 1.84240 0.921198 0.389093i $$-0.127212\pi$$
0.921198 + 0.389093i $$0.127212\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 596258. 0.00143621
$$747$$ 0 0
$$748$$ −8.27512e8 −1.97729
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ −3.16639e7 −0.0744580
$$753$$ 8.20040e8 1.92066
$$754$$ 9.51805e8 2.22041
$$755$$ 0 0
$$756$$ −6.98550e8 −1.61671
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ −8.75276e8 −1.97048
$$764$$ −8.84411e8 −1.98323
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 4.89726e8 1.08111
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 5.51471e7 0.119394 0.0596972 0.998217i $$-0.480986\pi$$
0.0596972 + 0.998217i $$0.480986\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −9.10309e7 −0.193308
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −4.80049e8 −1.00000
$$784$$ −711645. −0.00147678
$$785$$ 0 0
$$786$$ 1.13778e9 2.34309
$$787$$ −8.86902e8 −1.81950 −0.909749 0.415158i $$-0.863726\pi$$
−0.909749 + 0.415158i $$0.863726\pi$$
$$788$$ 0 0
$$789$$ −3.40932e8 −0.694123
$$790$$ 0 0
$$791$$ 9.39560e8 1.89843
$$792$$ −3.13177e8 −0.630397
$$793$$ 0 0
$$794$$ 8.95744e8 1.78946
$$795$$ 0 0
$$796$$ −5.74343e8 −1.13876
$$797$$ −4.12151e8 −0.814106 −0.407053 0.913405i $$-0.633444\pi$$
−0.407053 + 0.913405i $$0.633444\pi$$
$$798$$ 0 0
$$799$$ −1.48139e9 −2.90421
$$800$$ −4.90547e8 −0.958099
$$801$$ 5.19529e7 0.101091
$$802$$ 0 0
$$803$$ 0 0
$$804$$ −7.57223e8 −1.45699
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 2.13967e8 0.407123
$$808$$ −9.94368e8 −1.88501
$$809$$ 3.59615e8 0.679191 0.339596 0.940572i $$-0.389710\pi$$
0.339596 + 0.940572i $$0.389710\pi$$
$$810$$ 0 0
$$811$$ −8.89508e8 −1.66758 −0.833791 0.552080i $$-0.813834\pi$$
−0.833791 + 0.552080i $$0.813834\pi$$
$$812$$ 8.65566e8 1.61671
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −5.51773e7 −0.101552
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 7.39699e8 1.34649
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ −1.73822e9 −3.12960
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 8.82726e8 1.57777
$$825$$ 3.40031e8 0.605560
$$826$$ 0 0
$$827$$ −4.06219e8 −0.718196 −0.359098 0.933300i $$-0.616916\pi$$
−0.359098 + 0.933300i $$0.616916\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 1.14525e9 1.99571
$$832$$ −1.26538e9 −2.19710
$$833$$ −3.32941e7 −0.0576013
$$834$$ 1.77046e9 3.05203
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −1.07472e9 −1.81974 −0.909869 0.414895i $$-0.863818\pi$$
−0.909869 + 0.414895i $$0.863818\pi$$
$$840$$ 0 0
$$841$$ 5.94823e8 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ −1.43578e9 −2.37125
$$847$$ −3.79211e8 −0.624066
$$848$$ 0 0
$$849$$ 1.13449e9 1.85386
$$850$$ 1.98616e9 3.23412
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 8.49284e8 1.34459
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 1.20607e9 1.88958
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 6.17948e8 0.958099
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −1.92974e9 −2.96102
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 8.01828e8 1.21346
$$872$$ −1.38024e9 −2.08164
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7.87836e8 −1.16798 −0.583992 0.811759i $$-0.698510\pi$$
−0.583992 + 0.811759i $$0.698510\pi$$
$$878$$ 1.53610e9 2.26952
$$879$$ −8.56901e8 −1.26172
$$880$$ 0 0
$$881$$ −1.05140e9 −1.53759 −0.768796 0.639494i $$-0.779144\pi$$
−0.768796 + 0.639494i $$0.779144\pi$$
$$882$$ −3.22692e7 −0.0470308
$$883$$ 6.46270e8 0.938710 0.469355 0.883009i $$-0.344486\pi$$
0.469355 + 0.883009i $$0.344486\pi$$
$$884$$ 3.08212e9 4.46163
$$885$$ 0 0
$$886$$ 1.48930e8 0.214131
$$887$$ −1.38515e9 −1.98484 −0.992420 0.122893i $$-0.960783\pi$$
−0.992420 + 0.122893i $$0.960783\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.28341e8 −0.605560
$$892$$ −1.34306e9 −1.89235
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.17298e9 −1.63068
$$897$$ 0 0
$$898$$ −2.17191e9 −2.99926
$$899$$ 0 0
$$900$$ 1.19602e9 1.64062
$$901$$ 0 0
$$902$$ 1.38475e9 1.88691
$$903$$ 0 0
$$904$$ 1.48161e9 2.00553
$$905$$ 0 0
$$906$$ −1.07027e9 −1.43916
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ −1.36003e9 −1.81074
$$910$$ 0 0
$$911$$ −1.66419e8 −0.220114 −0.110057 0.993925i $$-0.535103\pi$$
−0.110057 + 0.993925i $$0.535103\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 2.06345e9 2.70244
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1.09564e9 −1.42088
$$918$$ −2.50198e9 −3.23412
$$919$$ −4.23881e8 −0.546131 −0.273066 0.961995i $$-0.588038\pi$$
−0.273066 + 0.961995i $$0.588038\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −1.26501e8 −0.161399
$$923$$ 0 0
$$924$$ 7.72333e8 0.979014
$$925$$ 0 0
$$926$$ −2.56995e9 −3.23662
$$927$$ 1.20733e9 1.51561
$$928$$ −7.65693e8 −0.958099
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −1.42059e9 −1.74913
$$934$$ 2.34706e9 2.88061
$$935$$ 0 0
$$936$$ 1.16645e9 1.42245
$$937$$ 8.02759e8 0.975812 0.487906 0.872896i $$-0.337761\pi$$
0.487906 + 0.872896i $$0.337761\pi$$
$$938$$ 1.17363e9 1.42207
$$939$$ 1.41835e9 1.71311
$$940$$ 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
$$947$$ −9.43083e8 −1.11045 −0.555226 0.831700i $$-0.687368\pi$$
−0.555226 + 0.831700i $$0.687368\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −2.19693e8 −0.255432
$$952$$ 1.76155e9 2.04166
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 5.30753e8 0.605560
$$958$$ 2.85321e9 3.24517
$$959$$ 1.67384e9 1.89784
$$960$$ 0 0
$$961$$ 8.87504e8 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −2.22640e9 −2.48526
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ −5.97986e8 −0.659273
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2.42617e8 0.265011 0.132505 0.991182i $$-0.457698\pi$$
0.132505 + 0.991182i $$0.457698\pi$$
$$972$$ −1.50664e9 −1.64062
$$973$$ −1.70489e9 −1.85079
$$974$$ −2.60116e9 −2.81507
$$975$$ −1.26647e9 −1.36641
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ −5.74404e7 −0.0612166
$$980$$ 0 0
$$981$$ −1.88780e9 −1.99963
$$982$$ 4.61672e8 0.487528
$$983$$ 9.65580e8 1.01655 0.508274 0.861195i $$-0.330284\pi$$
0.508274 + 0.861195i $$0.330284\pi$$
$$984$$ 1.90189e9 1.99618
$$985$$ 0 0
$$986$$ 3.10018e9 3.23412
$$987$$ 1.38261e9 1.43796
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 1.51849e9 1.56023 0.780117 0.625633i $$-0.215159\pi$$
0.780117 + 0.625633i $$0.215159\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 3.20643e9 3.22575
$$999$$ 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 87.7.d.b.86.1 yes 1
3.2 odd 2 87.7.d.a.86.1 1
29.28 even 2 87.7.d.a.86.1 1
87.86 odd 2 CM 87.7.d.b.86.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
87.7.d.a.86.1 1 3.2 odd 2
87.7.d.a.86.1 1 29.28 even 2
87.7.d.b.86.1 yes 1 1.1 even 1 trivial
87.7.d.b.86.1 yes 1 87.86 odd 2 CM