# Properties

 Label 20.3.d.b.19.1 Level $20$ Weight $3$ Character 20.19 Self dual yes Analytic conductor $0.545$ Analytic rank $0$ Dimension $1$ CM discriminant -20 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,3,Mod(19,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("20.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 20.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: $$0.544960528721$$ 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 19.1 Character $$\chi$$ $$=$$ 20.19

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -4.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -8.00000 q^{6} +4.00000 q^{7} +8.00000 q^{8} +7.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -4.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -8.00000 q^{6} +4.00000 q^{7} +8.00000 q^{8} +7.00000 q^{9} -10.0000 q^{10} -16.0000 q^{12} +8.00000 q^{14} +20.0000 q^{15} +16.0000 q^{16} +14.0000 q^{18} -20.0000 q^{20} -16.0000 q^{21} -44.0000 q^{23} -32.0000 q^{24} +25.0000 q^{25} +8.00000 q^{27} +16.0000 q^{28} -22.0000 q^{29} +40.0000 q^{30} +32.0000 q^{32} -20.0000 q^{35} +28.0000 q^{36} -40.0000 q^{40} +62.0000 q^{41} -32.0000 q^{42} +76.0000 q^{43} -35.0000 q^{45} -88.0000 q^{46} +4.00000 q^{47} -64.0000 q^{48} -33.0000 q^{49} +50.0000 q^{50} +16.0000 q^{54} +32.0000 q^{56} -44.0000 q^{58} +80.0000 q^{60} -58.0000 q^{61} +28.0000 q^{63} +64.0000 q^{64} -116.000 q^{67} +176.000 q^{69} -40.0000 q^{70} +56.0000 q^{72} -100.000 q^{75} -80.0000 q^{80} -95.0000 q^{81} +124.000 q^{82} +76.0000 q^{83} -64.0000 q^{84} +152.000 q^{86} +88.0000 q^{87} -142.000 q^{89} -70.0000 q^{90} -176.000 q^{92} +8.00000 q^{94} -128.000 q^{96} -66.0000 q^{98} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.00000
$$3$$ −4.00000 −1.33333 −0.666667 0.745356i $$-0.732280\pi$$
−0.666667 + 0.745356i $$0.732280\pi$$
$$4$$ 4.00000 1.00000
$$5$$ −5.00000 −1.00000
$$6$$ −8.00000 −1.33333
$$7$$ 4.00000 0.571429 0.285714 0.958315i $$-0.407769\pi$$
0.285714 + 0.958315i $$0.407769\pi$$
$$8$$ 8.00000 1.00000
$$9$$ 7.00000 0.777778
$$10$$ −10.0000 −1.00000
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −16.0000 −1.33333
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 8.00000 0.571429
$$15$$ 20.0000 1.33333
$$16$$ 16.0000 1.00000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 14.0000 0.777778
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ −20.0000 −1.00000
$$21$$ −16.0000 −0.761905
$$22$$ 0 0
$$23$$ −44.0000 −1.91304 −0.956522 0.291661i $$-0.905792\pi$$
−0.956522 + 0.291661i $$0.905792\pi$$
$$24$$ −32.0000 −1.33333
$$25$$ 25.0000 1.00000
$$26$$ 0 0
$$27$$ 8.00000 0.296296
$$28$$ 16.0000 0.571429
$$29$$ −22.0000 −0.758621 −0.379310 0.925270i $$-0.623839\pi$$
−0.379310 + 0.925270i $$0.623839\pi$$
$$30$$ 40.0000 1.33333
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 32.0000 1.00000
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −20.0000 −0.571429
$$36$$ 28.0000 0.777778
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −40.0000 −1.00000
$$41$$ 62.0000 1.51220 0.756098 0.654459i $$-0.227104\pi$$
0.756098 + 0.654459i $$0.227104\pi$$
$$42$$ −32.0000 −0.761905
$$43$$ 76.0000 1.76744 0.883721 0.468014i $$-0.155030\pi$$
0.883721 + 0.468014i $$0.155030\pi$$
$$44$$ 0 0
$$45$$ −35.0000 −0.777778
$$46$$ −88.0000 −1.91304
$$47$$ 4.00000 0.0851064 0.0425532 0.999094i $$-0.486451\pi$$
0.0425532 + 0.999094i $$0.486451\pi$$
$$48$$ −64.0000 −1.33333
$$49$$ −33.0000 −0.673469
$$50$$ 50.0000 1.00000
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 16.0000 0.296296
$$55$$ 0 0
$$56$$ 32.0000 0.571429
$$57$$ 0 0
$$58$$ −44.0000 −0.758621
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 80.0000 1.33333
$$61$$ −58.0000 −0.950820 −0.475410 0.879764i $$-0.657700\pi$$
−0.475410 + 0.879764i $$0.657700\pi$$
$$62$$ 0 0
$$63$$ 28.0000 0.444444
$$64$$ 64.0000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −116.000 −1.73134 −0.865672 0.500612i $$-0.833108\pi$$
−0.865672 + 0.500612i $$0.833108\pi$$
$$68$$ 0 0
$$69$$ 176.000 2.55072
$$70$$ −40.0000 −0.571429
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 56.0000 0.777778
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ −100.000 −1.33333
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ −80.0000 −1.00000
$$81$$ −95.0000 −1.17284
$$82$$ 124.000 1.51220
$$83$$ 76.0000 0.915663 0.457831 0.889039i $$-0.348626\pi$$
0.457831 + 0.889039i $$0.348626\pi$$
$$84$$ −64.0000 −0.761905
$$85$$ 0 0
$$86$$ 152.000 1.76744
$$87$$ 88.0000 1.01149
$$88$$ 0 0
$$89$$ −142.000 −1.59551 −0.797753 0.602985i $$-0.793978\pi$$
−0.797753 + 0.602985i $$0.793978\pi$$
$$90$$ −70.0000 −0.777778
$$91$$ 0 0
$$92$$ −176.000 −1.91304
$$93$$ 0 0
$$94$$ 8.00000 0.0851064
$$95$$ 0 0
$$96$$ −128.000 −1.33333
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −66.0000 −0.673469
$$99$$ 0 0
$$100$$ 100.000 1.00000
$$101$$ 122.000 1.20792 0.603960 0.797014i $$-0.293589\pi$$
0.603960 + 0.797014i $$0.293589\pi$$
$$102$$ 0 0
$$103$$ −44.0000 −0.427184 −0.213592 0.976923i $$-0.568516\pi$$
−0.213592 + 0.976923i $$0.568516\pi$$
$$104$$ 0 0
$$105$$ 80.0000 0.761905
$$106$$ 0 0
$$107$$ 124.000 1.15888 0.579439 0.815015i $$-0.303272\pi$$
0.579439 + 0.815015i $$0.303272\pi$$
$$108$$ 32.0000 0.296296
$$109$$ 38.0000 0.348624 0.174312 0.984690i $$-0.444230\pi$$
0.174312 + 0.984690i $$0.444230\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 64.0000 0.571429
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 220.000 1.91304
$$116$$ −88.0000 −0.758621
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 160.000 1.33333
$$121$$ 121.000 1.00000
$$122$$ −116.000 −0.950820
$$123$$ −248.000 −2.01626
$$124$$ 0 0
$$125$$ −125.000 −1.00000
$$126$$ 56.0000 0.444444
$$127$$ −236.000 −1.85827 −0.929134 0.369744i $$-0.879446\pi$$
−0.929134 + 0.369744i $$0.879446\pi$$
$$128$$ 128.000 1.00000
$$129$$ −304.000 −2.35659
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −232.000 −1.73134
$$135$$ −40.0000 −0.296296
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 352.000 2.55072
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ −80.0000 −0.571429
$$141$$ −16.0000 −0.113475
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 112.000 0.777778
$$145$$ 110.000 0.758621
$$146$$ 0 0
$$147$$ 132.000 0.897959
$$148$$ 0 0
$$149$$ 278.000 1.86577 0.932886 0.360172i $$-0.117282\pi$$
0.932886 + 0.360172i $$0.117282\pi$$
$$150$$ −200.000 −1.33333
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ −160.000 −1.00000
$$161$$ −176.000 −1.09317
$$162$$ −190.000 −1.17284
$$163$$ −164.000 −1.00613 −0.503067 0.864247i $$-0.667795\pi$$
−0.503067 + 0.864247i $$0.667795\pi$$
$$164$$ 248.000 1.51220
$$165$$ 0 0
$$166$$ 152.000 0.915663
$$167$$ 244.000 1.46108 0.730539 0.682871i $$-0.239269\pi$$
0.730539 + 0.682871i $$0.239269\pi$$
$$168$$ −128.000 −0.761905
$$169$$ 169.000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 304.000 1.76744
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 176.000 1.01149
$$175$$ 100.000 0.571429
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −284.000 −1.59551
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ −140.000 −0.777778
$$181$$ −358.000 −1.97790 −0.988950 0.148248i $$-0.952637\pi$$
−0.988950 + 0.148248i $$0.952637\pi$$
$$182$$ 0 0
$$183$$ 232.000 1.26776
$$184$$ −352.000 −1.91304
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 16.0000 0.0851064
$$189$$ 32.0000 0.169312
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ −256.000 −1.33333
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −132.000 −0.673469
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 200.000 1.00000
$$201$$ 464.000 2.30846
$$202$$ 244.000 1.20792
$$203$$ −88.0000 −0.433498
$$204$$ 0 0
$$205$$ −310.000 −1.51220
$$206$$ −88.0000 −0.427184
$$207$$ −308.000 −1.48792
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 160.000 0.761905
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 248.000 1.15888
$$215$$ −380.000 −1.76744
$$216$$ 64.0000 0.296296
$$217$$ 0 0
$$218$$ 76.0000 0.348624
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 436.000 1.95516 0.977578 0.210571i $$-0.0675325\pi$$
0.977578 + 0.210571i $$0.0675325\pi$$
$$224$$ 128.000 0.571429
$$225$$ 175.000 0.777778
$$226$$ 0 0
$$227$$ −356.000 −1.56828 −0.784141 0.620583i $$-0.786896\pi$$
−0.784141 + 0.620583i $$0.786896\pi$$
$$228$$ 0 0
$$229$$ −262.000 −1.14410 −0.572052 0.820217i $$-0.693853\pi$$
−0.572052 + 0.820217i $$0.693853\pi$$
$$230$$ 440.000 1.91304
$$231$$ 0 0
$$232$$ −176.000 −0.758621
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ −20.0000 −0.0851064
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 320.000 1.33333
$$241$$ 302.000 1.25311 0.626556 0.779376i $$-0.284464\pi$$
0.626556 + 0.779376i $$0.284464\pi$$
$$242$$ 242.000 1.00000
$$243$$ 308.000 1.26749
$$244$$ −232.000 −0.950820
$$245$$ 165.000 0.673469
$$246$$ −496.000 −2.01626
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −304.000 −1.22088
$$250$$ −250.000 −1.00000
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 112.000 0.444444
$$253$$ 0 0
$$254$$ −472.000 −1.85827
$$255$$ 0 0
$$256$$ 256.000 1.00000
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ −608.000 −2.35659
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −154.000 −0.590038
$$262$$ 0 0
$$263$$ −284.000 −1.07985 −0.539924 0.841714i $$-0.681547\pi$$
−0.539924 + 0.841714i $$0.681547\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 568.000 2.12734
$$268$$ −464.000 −1.73134
$$269$$ 38.0000 0.141264 0.0706320 0.997502i $$-0.477498\pi$$
0.0706320 + 0.997502i $$0.477498\pi$$
$$270$$ −80.0000 −0.296296
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 704.000 2.55072
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ −160.000 −0.571429
$$281$$ −418.000 −1.48754 −0.743772 0.668433i $$-0.766965\pi$$
−0.743772 + 0.668433i $$0.766965\pi$$
$$282$$ −32.0000 −0.113475
$$283$$ 316.000 1.11661 0.558304 0.829637i $$-0.311452\pi$$
0.558304 + 0.829637i $$0.311452\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 248.000 0.864111
$$288$$ 224.000 0.777778
$$289$$ 289.000 1.00000
$$290$$ 220.000 0.758621
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 264.000 0.897959
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 556.000 1.86577
$$299$$ 0 0
$$300$$ −400.000 −1.33333
$$301$$ 304.000 1.00997
$$302$$ 0 0
$$303$$ −488.000 −1.61056
$$304$$ 0 0
$$305$$ 290.000 0.950820
$$306$$ 0 0
$$307$$ −596.000 −1.94137 −0.970684 0.240359i $$-0.922735\pi$$
−0.970684 + 0.240359i $$0.922735\pi$$
$$308$$ 0 0
$$309$$ 176.000 0.569579
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ −140.000 −0.444444
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −320.000 −1.00000
$$321$$ −496.000 −1.54517
$$322$$ −352.000 −1.09317
$$323$$ 0 0
$$324$$ −380.000 −1.17284
$$325$$ 0 0
$$326$$ −328.000 −1.00613
$$327$$ −152.000 −0.464832
$$328$$ 496.000 1.51220
$$329$$ 16.0000 0.0486322
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 304.000 0.915663
$$333$$ 0 0
$$334$$ 488.000 1.46108
$$335$$ 580.000 1.73134
$$336$$ −256.000 −0.761905
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 338.000 1.00000
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −328.000 −0.956268
$$344$$ 608.000 1.76744
$$345$$ −880.000 −2.55072
$$346$$ 0 0
$$347$$ −116.000 −0.334294 −0.167147 0.985932i $$-0.553455\pi$$
−0.167147 + 0.985932i $$0.553455\pi$$
$$348$$ 352.000 1.01149
$$349$$ −22.0000 −0.0630372 −0.0315186 0.999503i $$-0.510034\pi$$
−0.0315186 + 0.999503i $$0.510034\pi$$
$$350$$ 200.000 0.571429
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −568.000 −1.59551
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ −280.000 −0.777778
$$361$$ 361.000 1.00000
$$362$$ −716.000 −1.97790
$$363$$ −484.000 −1.33333
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 464.000 1.26776
$$367$$ 724.000 1.97275 0.986376 0.164506i $$-0.0526031\pi$$
0.986376 + 0.164506i $$0.0526031\pi$$
$$368$$ −704.000 −1.91304
$$369$$ 434.000 1.17615
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 500.000 1.33333
$$376$$ 32.0000 0.0851064
$$377$$ 0 0
$$378$$ 64.0000 0.169312
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 944.000 2.47769
$$382$$ 0 0
$$383$$ −44.0000 −0.114883 −0.0574413 0.998349i $$-0.518294\pi$$
−0.0574413 + 0.998349i $$0.518294\pi$$
$$384$$ −512.000 −1.33333
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 532.000 1.37468
$$388$$ 0 0
$$389$$ −202.000 −0.519280 −0.259640 0.965705i $$-0.583604\pi$$
−0.259640 + 0.965705i $$0.583604\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −264.000 −0.673469
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 400.000 1.00000
$$401$$ −478.000 −1.19202 −0.596010 0.802977i $$-0.703248\pi$$
−0.596010 + 0.802977i $$0.703248\pi$$
$$402$$ 928.000 2.30846
$$403$$ 0 0
$$404$$ 488.000 1.20792
$$405$$ 475.000 1.17284
$$406$$ −176.000 −0.433498
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −802.000 −1.96088 −0.980440 0.196818i $$-0.936939\pi$$
−0.980440 + 0.196818i $$0.936939\pi$$
$$410$$ −620.000 −1.51220
$$411$$ 0 0
$$412$$ −176.000 −0.427184
$$413$$ 0 0
$$414$$ −616.000 −1.48792
$$415$$ −380.000 −0.915663
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 320.000 0.761905
$$421$$ −778.000 −1.84798 −0.923990 0.382415i $$-0.875092\pi$$
−0.923990 + 0.382415i $$0.875092\pi$$
$$422$$ 0 0
$$423$$ 28.0000 0.0661939
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −232.000 −0.543326
$$428$$ 496.000 1.15888
$$429$$ 0 0
$$430$$ −760.000 −1.76744
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 128.000 0.296296
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ −440.000 −1.01149
$$436$$ 152.000 0.348624
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ −231.000 −0.523810
$$442$$ 0 0
$$443$$ 796.000 1.79684 0.898420 0.439138i $$-0.144716\pi$$
0.898420 + 0.439138i $$0.144716\pi$$
$$444$$ 0 0
$$445$$ 710.000 1.59551
$$446$$ 872.000 1.95516
$$447$$ −1112.00 −2.48770
$$448$$ 256.000 0.571429
$$449$$ 398.000 0.886414 0.443207 0.896419i $$-0.353841\pi$$
0.443207 + 0.896419i $$0.353841\pi$$
$$450$$ 350.000 0.777778
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −712.000 −1.56828
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ −524.000 −1.14410
$$459$$ 0 0
$$460$$ 880.000 1.91304
$$461$$ 842.000 1.82646 0.913232 0.407440i $$-0.133578\pi$$
0.913232 + 0.407440i $$0.133578\pi$$
$$462$$ 0 0
$$463$$ −764.000 −1.65011 −0.825054 0.565054i $$-0.808855\pi$$
−0.825054 + 0.565054i $$0.808855\pi$$
$$464$$ −352.000 −0.758621
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 124.000 0.265525 0.132762 0.991148i $$-0.457615\pi$$
0.132762 + 0.991148i $$0.457615\pi$$
$$468$$ 0 0
$$469$$ −464.000 −0.989339
$$470$$ −40.0000 −0.0851064
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 640.000 1.33333
$$481$$ 0 0
$$482$$ 604.000 1.25311
$$483$$ 704.000 1.45756
$$484$$ 484.000 1.00000
$$485$$ 0 0
$$486$$ 616.000 1.26749
$$487$$ 484.000 0.993840 0.496920 0.867796i $$-0.334464\pi$$
0.496920 + 0.867796i $$0.334464\pi$$
$$488$$ −464.000 −0.950820
$$489$$ 656.000 1.34151
$$490$$ 330.000 0.673469
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ −992.000 −2.01626
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ −608.000 −1.22088
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ −500.000 −1.00000
$$501$$ −976.000 −1.94810
$$502$$ 0 0
$$503$$ 916.000 1.82107 0.910537 0.413428i $$-0.135669\pi$$
0.910537 + 0.413428i $$0.135669\pi$$
$$504$$ 224.000 0.444444
$$505$$ −610.000 −1.20792
$$506$$ 0 0
$$507$$ −676.000 −1.33333
$$508$$ −944.000 −1.85827
$$509$$ −982.000 −1.92927 −0.964637 0.263584i $$-0.915095\pi$$
−0.964637 + 0.263584i $$0.915095\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 512.000 1.00000
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 220.000 0.427184
$$516$$ −1216.00 −2.35659
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 722.000 1.38580 0.692898 0.721035i $$-0.256333\pi$$
0.692898 + 0.721035i $$0.256333\pi$$
$$522$$ −308.000 −0.590038
$$523$$ −164.000 −0.313576 −0.156788 0.987632i $$-0.550114\pi$$
−0.156788 + 0.987632i $$0.550114\pi$$
$$524$$ 0 0
$$525$$ −400.000 −0.761905
$$526$$ −568.000 −1.07985
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1407.00 2.65974
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 1136.00 2.12734
$$535$$ −620.000 −1.15888
$$536$$ −928.000 −1.73134
$$537$$ 0 0
$$538$$ 76.0000 0.141264
$$539$$ 0 0
$$540$$ −160.000 −0.296296
$$541$$ 362.000 0.669131 0.334566 0.942372i $$-0.391410\pi$$
0.334566 + 0.942372i $$0.391410\pi$$
$$542$$ 0 0
$$543$$ 1432.00 2.63720
$$544$$ 0 0
$$545$$ −190.000 −0.348624
$$546$$ 0 0
$$547$$ 1084.00 1.98172 0.990859 0.134900i $$-0.0430713\pi$$
0.990859 + 0.134900i $$0.0430713\pi$$
$$548$$ 0 0
$$549$$ −406.000 −0.739526
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 1408.00 2.55072
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −320.000 −0.571429
$$561$$ 0 0
$$562$$ −836.000 −1.48754
$$563$$ −1124.00 −1.99645 −0.998224 0.0595755i $$-0.981025\pi$$
−0.998224 + 0.0595755i $$0.981025\pi$$
$$564$$ −64.0000 −0.113475
$$565$$ 0 0
$$566$$ 632.000 1.11661
$$567$$ −380.000 −0.670194
$$568$$ 0 0
$$569$$ 158.000 0.277680 0.138840 0.990315i $$-0.455663\pi$$
0.138840 + 0.990315i $$0.455663\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 496.000 0.864111
$$575$$ −1100.00 −1.91304
$$576$$ 448.000 0.777778
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 578.000 1.00000
$$579$$ 0 0
$$580$$ 440.000 0.758621
$$581$$ 304.000 0.523236
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1076.00 −1.83305 −0.916525 0.399978i $$-0.869018\pi$$
−0.916525 + 0.399978i $$0.869018\pi$$
$$588$$ 528.000 0.897959
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1112.00 1.86577
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ −800.000 −1.33333
$$601$$ −418.000 −0.695507 −0.347754 0.937586i $$-0.613055\pi$$
−0.347754 + 0.937586i $$0.613055\pi$$
$$602$$ 608.000 1.00997
$$603$$ −812.000 −1.34660
$$604$$ 0 0
$$605$$ −605.000 −1.00000
$$606$$ −976.000 −1.61056
$$607$$ 964.000 1.58814 0.794069 0.607827i $$-0.207959\pi$$
0.794069 + 0.607827i $$0.207959\pi$$
$$608$$ 0 0
$$609$$ 352.000 0.577997
$$610$$ 580.000 0.950820
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ −1192.00 −1.94137
$$615$$ 1240.00 2.01626
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 352.000 0.569579
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ −352.000 −0.566828
$$622$$ 0 0
$$623$$ −568.000 −0.911717
$$624$$ 0 0
$$625$$ 625.000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ −280.000 −0.444444
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 1180.00 1.85827
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −640.000 −1.00000
$$641$$ −1138.00 −1.77535 −0.887676 0.460470i $$-0.847681\pi$$
−0.887676 + 0.460470i $$0.847681\pi$$
$$642$$ −992.000 −1.54517
$$643$$ −404.000 −0.628305 −0.314152 0.949373i $$-0.601720\pi$$
−0.314152 + 0.949373i $$0.601720\pi$$
$$644$$ −704.000 −1.09317
$$645$$ 1520.00 2.35659
$$646$$ 0 0
$$647$$ −956.000 −1.47759 −0.738794 0.673931i $$-0.764605\pi$$
−0.738794 + 0.673931i $$0.764605\pi$$
$$648$$ −760.000 −1.17284
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −656.000 −1.00613
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ −304.000 −0.464832
$$655$$ 0 0
$$656$$ 992.000 1.51220
$$657$$ 0 0
$$658$$ 32.0000 0.0486322
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ −298.000 −0.450832 −0.225416 0.974263i $$-0.572374\pi$$
−0.225416 + 0.974263i $$0.572374\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 608.000 0.915663
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 968.000 1.45127
$$668$$ 976.000 1.46108
$$669$$ −1744.00 −2.60688
$$670$$ 1160.00 1.73134
$$671$$ 0 0
$$672$$ −512.000 −0.761905
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 200.000 0.296296
$$676$$ 676.000 1.00000
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 1424.00 2.09104
$$682$$ 0 0
$$683$$ 556.000 0.814056 0.407028 0.913416i $$-0.366565\pi$$
0.407028 + 0.913416i $$0.366565\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −656.000 −0.956268
$$687$$ 1048.00 1.52547
$$688$$ 1216.00 1.76744
$$689$$ 0 0
$$690$$ −1760.00 −2.55072
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −232.000 −0.334294
$$695$$ 0 0
$$696$$ 704.000 1.01149
$$697$$ 0 0
$$698$$ −44.0000 −0.0630372
$$699$$ 0 0
$$700$$ 400.000 0.571429
$$701$$ 902.000 1.28673 0.643367 0.765558i $$-0.277537\pi$$
0.643367 + 0.765558i $$0.277537\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 80.0000 0.113475
$$706$$ 0 0
$$707$$ 488.000 0.690240
$$708$$ 0 0
$$709$$ 698.000 0.984485 0.492243 0.870458i $$-0.336177\pi$$
0.492243 + 0.870458i $$0.336177\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −1136.00 −1.59551
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ −560.000 −0.777778
$$721$$ −176.000 −0.244105
$$722$$ 722.000 1.00000
$$723$$ −1208.00 −1.67082
$$724$$ −1432.00 −1.97790
$$725$$ −550.000 −0.758621
$$726$$ −968.000 −1.33333
$$727$$ −1436.00 −1.97524 −0.987620 0.156863i $$-0.949862\pi$$
−0.987620 + 0.156863i $$0.949862\pi$$
$$728$$ 0 0
$$729$$ −377.000 −0.517147
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 928.000 1.26776
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 1448.00 1.97275
$$735$$ −660.000 −0.897959
$$736$$ −1408.00 −1.91304
$$737$$ 0 0
$$738$$ 868.000 1.17615
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −764.000 −1.02826 −0.514132 0.857711i $$-0.671886\pi$$
−0.514132 + 0.857711i $$0.671886\pi$$
$$744$$ 0 0
$$745$$ −1390.00 −1.86577
$$746$$ 0 0
$$747$$ 532.000 0.712182
$$748$$ 0 0
$$749$$ 496.000 0.662216
$$750$$ 1000.00 1.33333
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 64.0000 0.0851064
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 128.000 0.169312
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 242.000 0.318003 0.159001 0.987278i $$-0.449173\pi$$
0.159001 + 0.987278i $$0.449173\pi$$
$$762$$ 1888.00 2.47769
$$763$$ 152.000 0.199214
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −88.0000 −0.114883
$$767$$ 0 0
$$768$$ −1024.00 −1.33333
$$769$$ −1342.00 −1.74512 −0.872562 0.488504i $$-0.837543\pi$$
−0.872562 + 0.488504i $$0.837543\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 1064.00 1.37468
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −404.000 −0.519280
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −176.000 −0.224777
$$784$$ −528.000 −0.673469
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −116.000 −0.147395 −0.0736976 0.997281i $$-0.523480\pi$$
−0.0736976 + 0.997281i $$0.523480\pi$$
$$788$$ 0 0
$$789$$ 1136.00 1.43980
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 800.000 1.00000
$$801$$ −994.000 −1.24095
$$802$$ −956.000 −1.19202
$$803$$ 0 0
$$804$$ 1856.00 2.30846
$$805$$ 880.000 1.09317
$$806$$ 0 0
$$807$$ −152.000 −0.188352
$$808$$ 976.000 1.20792
$$809$$ 1298.00 1.60445 0.802225 0.597022i $$-0.203649\pi$$
0.802225 + 0.597022i $$0.203649\pi$$
$$810$$ 950.000 1.17284
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ −352.000 −0.433498
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 820.000 1.00613
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −1604.00 −1.96088
$$819$$ 0 0
$$820$$ −1240.00 −1.51220
$$821$$ 662.000 0.806334 0.403167 0.915126i $$-0.367909\pi$$
0.403167 + 0.915126i $$0.367909\pi$$
$$822$$ 0 0
$$823$$ 1396.00 1.69623 0.848117 0.529810i $$-0.177737\pi$$
0.848117 + 0.529810i $$0.177737\pi$$
$$824$$ −352.000 −0.427184
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −596.000 −0.720677 −0.360339 0.932822i $$-0.617339\pi$$
−0.360339 + 0.932822i $$0.617339\pi$$
$$828$$ −1232.00 −1.48792
$$829$$ 1478.00 1.78287 0.891435 0.453148i $$-0.149699\pi$$
0.891435 + 0.453148i $$0.149699\pi$$
$$830$$ −760.000 −0.915663
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −1220.00 −1.46108
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 640.000 0.761905
$$841$$ −357.000 −0.424495
$$842$$ −1556.00 −1.84798
$$843$$ 1672.00 1.98339
$$844$$ 0 0
$$845$$ −845.000 −1.00000
$$846$$ 56.0000 0.0661939
$$847$$ 484.000 0.571429
$$848$$ 0 0
$$849$$ −1264.00 −1.48881
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ −464.000 −0.543326
$$855$$ 0 0
$$856$$ 992.000 1.15888
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ −1520.00 −1.76744
$$861$$ −992.000 −1.15215
$$862$$ 0 0
$$863$$ 1636.00 1.89571 0.947856 0.318698i $$-0.103246\pi$$
0.947856 + 0.318698i $$0.103246\pi$$
$$864$$ 256.000 0.296296
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −1156.00 −1.33333
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −880.000 −1.01149
$$871$$ 0 0
$$872$$ 304.000 0.348624
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −500.000 −0.571429
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −1618.00 −1.83655 −0.918275 0.395944i $$-0.870417\pi$$
−0.918275 + 0.395944i $$0.870417\pi$$
$$882$$ −462.000 −0.523810
$$883$$ 1276.00 1.44507 0.722537 0.691332i $$-0.242976\pi$$
0.722537 + 0.691332i $$0.242976\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 1592.00 1.79684
$$887$$ 964.000 1.08681 0.543405 0.839471i $$-0.317135\pi$$
0.543405 + 0.839471i $$0.317135\pi$$
$$888$$ 0 0
$$889$$ −944.000 −1.06187
$$890$$ 1420.00 1.59551
$$891$$ 0 0
$$892$$ 1744.00 1.95516
$$893$$ 0 0
$$894$$ −2224.00 −2.48770
$$895$$ 0 0
$$896$$ 512.000 0.571429
$$897$$ 0 0
$$898$$ 796.000 0.886414
$$899$$ 0 0
$$900$$ 700.000 0.777778
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −1216.00 −1.34662
$$904$$ 0 0
$$905$$ 1790.00 1.97790
$$906$$ 0 0
$$907$$ −1796.00 −1.98015 −0.990077 0.140525i $$-0.955121\pi$$
−0.990077 + 0.140525i $$0.955121\pi$$
$$908$$ −1424.00 −1.56828
$$909$$ 854.000 0.939494
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −1160.00 −1.26776
$$916$$ −1048.00 −1.14410
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 1760.00 1.91304
$$921$$ 2384.00 2.58849
$$922$$ 1684.00 1.82646
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −1528.00 −1.65011
$$927$$ −308.000 −0.332255
$$928$$ −704.000 −0.758621
$$929$$ −562.000 −0.604952 −0.302476 0.953157i $$-0.597813\pi$$
−0.302476 + 0.953157i $$0.597813\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 248.000 0.265525
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ −928.000 −0.989339
$$939$$ 0 0
$$940$$ −80.0000 −0.0851064
$$941$$ −118.000 −0.125399 −0.0626993 0.998032i $$-0.519971\pi$$
−0.0626993 + 0.998032i $$0.519971\pi$$
$$942$$ 0 0
$$943$$ −2728.00 −2.89290
$$944$$ 0 0
$$945$$ −160.000 −0.169312
$$946$$ 0 0
$$947$$ 1804.00 1.90496 0.952482 0.304596i $$-0.0985216\pi$$
0.952482 + 0.304596i $$0.0985216\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 1280.00 1.33333
$$961$$ 961.000 1.00000
$$962$$ 0 0
$$963$$ 868.000 0.901350
$$964$$ 1208.00 1.25311
$$965$$ 0 0
$$966$$ 1408.00 1.45756
$$967$$ 244.000 0.252327 0.126163 0.992009i $$-0.459734\pi$$
0.126163 + 0.992009i $$0.459734\pi$$
$$968$$ 968.000 1.00000
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 1232.00 1.26749
$$973$$ 0 0
$$974$$ 968.000 0.993840
$$975$$ 0 0
$$976$$ −928.000 −0.950820
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 1312.00 1.34151
$$979$$ 0 0
$$980$$ 660.000 0.673469
$$981$$ 266.000 0.271152
$$982$$ 0 0
$$983$$ −284.000 −0.288911 −0.144456 0.989511i $$-0.546143\pi$$
−0.144456 + 0.989511i $$0.546143\pi$$
$$984$$ −1984.00 −2.01626
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −64.0000 −0.0648430
$$988$$ 0 0
$$989$$ −3344.00 −3.38119
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −1216.00 −1.22088
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 0 0
$$999$$ 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 20.3.d.b.19.1 yes 1
3.2 odd 2 180.3.f.a.19.1 1
4.3 odd 2 20.3.d.a.19.1 1
5.2 odd 4 100.3.b.c.51.2 2
5.3 odd 4 100.3.b.c.51.1 2
5.4 even 2 20.3.d.a.19.1 1
8.3 odd 2 320.3.h.a.319.1 1
8.5 even 2 320.3.h.b.319.1 1
12.11 even 2 180.3.f.b.19.1 1
15.2 even 4 900.3.c.h.451.1 2
15.8 even 4 900.3.c.h.451.2 2
15.14 odd 2 180.3.f.b.19.1 1
16.3 odd 4 1280.3.e.c.639.2 2
16.5 even 4 1280.3.e.b.639.2 2
16.11 odd 4 1280.3.e.c.639.1 2
16.13 even 4 1280.3.e.b.639.1 2
20.3 even 4 100.3.b.c.51.2 2
20.7 even 4 100.3.b.c.51.1 2
20.19 odd 2 CM 20.3.d.b.19.1 yes 1
40.3 even 4 1600.3.b.f.1151.1 2
40.13 odd 4 1600.3.b.f.1151.2 2
40.19 odd 2 320.3.h.b.319.1 1
40.27 even 4 1600.3.b.f.1151.2 2
40.29 even 2 320.3.h.a.319.1 1
40.37 odd 4 1600.3.b.f.1151.1 2
60.23 odd 4 900.3.c.h.451.1 2
60.47 odd 4 900.3.c.h.451.2 2
60.59 even 2 180.3.f.a.19.1 1
80.19 odd 4 1280.3.e.b.639.1 2
80.29 even 4 1280.3.e.c.639.2 2
80.59 odd 4 1280.3.e.b.639.2 2
80.69 even 4 1280.3.e.c.639.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.d.a.19.1 1 4.3 odd 2
20.3.d.a.19.1 1 5.4 even 2
20.3.d.b.19.1 yes 1 1.1 even 1 trivial
20.3.d.b.19.1 yes 1 20.19 odd 2 CM
100.3.b.c.51.1 2 5.3 odd 4
100.3.b.c.51.1 2 20.7 even 4
100.3.b.c.51.2 2 5.2 odd 4
100.3.b.c.51.2 2 20.3 even 4
180.3.f.a.19.1 1 3.2 odd 2
180.3.f.a.19.1 1 60.59 even 2
180.3.f.b.19.1 1 12.11 even 2
180.3.f.b.19.1 1 15.14 odd 2
320.3.h.a.319.1 1 8.3 odd 2
320.3.h.a.319.1 1 40.29 even 2
320.3.h.b.319.1 1 8.5 even 2
320.3.h.b.319.1 1 40.19 odd 2
900.3.c.h.451.1 2 15.2 even 4
900.3.c.h.451.1 2 60.23 odd 4
900.3.c.h.451.2 2 15.8 even 4
900.3.c.h.451.2 2 60.47 odd 4
1280.3.e.b.639.1 2 16.13 even 4
1280.3.e.b.639.1 2 80.19 odd 4
1280.3.e.b.639.2 2 16.5 even 4
1280.3.e.b.639.2 2 80.59 odd 4
1280.3.e.c.639.1 2 16.11 odd 4
1280.3.e.c.639.1 2 80.69 even 4
1280.3.e.c.639.2 2 16.3 odd 4
1280.3.e.c.639.2 2 80.29 even 4
1600.3.b.f.1151.1 2 40.3 even 4
1600.3.b.f.1151.1 2 40.37 odd 4
1600.3.b.f.1151.2 2 40.13 odd 4
1600.3.b.f.1151.2 2 40.27 even 4