# Properties

 Label 7942.2.a.p.1.1 Level $7942$ Weight $2$ Character 7942.1 Self dual yes Analytic conductor $63.417$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$63.4171892853$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7942.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{18} -3.00000 q^{20} -1.00000 q^{22} +2.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} +3.00000 q^{28} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -9.00000 q^{35} -3.00000 q^{36} +9.00000 q^{37} -3.00000 q^{40} -2.00000 q^{41} -8.00000 q^{43} -1.00000 q^{44} +9.00000 q^{45} +2.00000 q^{46} -12.0000 q^{47} +2.00000 q^{49} +4.00000 q^{50} -2.00000 q^{52} +13.0000 q^{53} +3.00000 q^{55} +3.00000 q^{56} -10.0000 q^{59} -2.00000 q^{62} -9.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -12.0000 q^{67} +4.00000 q^{68} -9.00000 q^{70} -2.00000 q^{71} -3.00000 q^{72} +12.0000 q^{73} +9.00000 q^{74} -3.00000 q^{77} +1.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +9.00000 q^{83} -12.0000 q^{85} -8.00000 q^{86} -1.00000 q^{88} -6.00000 q^{89} +9.00000 q^{90} -6.00000 q^{91} +2.00000 q^{92} -12.0000 q^{94} -5.00000 q^{97} +2.00000 q^{98} +3.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −3.00000 −1.00000
$$10$$ −3.00000 −0.948683
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ 0 0
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 3.00000 0.566947
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ −9.00000 −1.52128
$$36$$ −3.00000 −0.500000
$$37$$ 9.00000 1.47959 0.739795 0.672832i $$-0.234922\pi$$
0.739795 + 0.672832i $$0.234922\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 9.00000 1.34164
$$46$$ 2.00000 0.294884
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 4.00000 0.565685
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 13.0000 1.78569 0.892844 0.450367i $$-0.148707\pi$$
0.892844 + 0.450367i $$0.148707\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ −9.00000 −1.13389
$$64$$ 1.00000 0.125000
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ −9.00000 −1.07571
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 12.0000 1.40449 0.702247 0.711934i $$-0.252180\pi$$
0.702247 + 0.711934i $$0.252180\pi$$
$$74$$ 9.00000 1.04623
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 9.00000 1.00000
$$82$$ −2.00000 −0.220863
$$83$$ 9.00000 0.987878 0.493939 0.869496i $$-0.335557\pi$$
0.493939 + 0.869496i $$0.335557\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 9.00000 0.948683
$$91$$ −6.00000 −0.628971
$$92$$ 2.00000 0.208514
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ 2.00000 0.202031
$$99$$ 3.00000 0.301511
$$100$$ 4.00000 0.400000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 13.0000 1.26267
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 3.00000 0.286039
$$111$$ 0 0
$$112$$ 3.00000 0.283473
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ −10.0000 −0.920575
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ 3.00000 0.268328
$$126$$ −9.00000 −0.801784
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 6.00000 0.526235
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ −15.0000 −1.28154 −0.640768 0.767734i $$-0.721384\pi$$
−0.640768 + 0.767734i $$0.721384\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ −9.00000 −0.760639
$$141$$ 0 0
$$142$$ −2.00000 −0.167836
$$143$$ 2.00000 0.167248
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 12.0000 0.993127
$$147$$ 0 0
$$148$$ 9.00000 0.739795
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ 3.00000 0.244137 0.122068 0.992522i $$-0.461047\pi$$
0.122068 + 0.992522i $$0.461047\pi$$
$$152$$ 0 0
$$153$$ −12.0000 −0.970143
$$154$$ −3.00000 −0.241747
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 9.00000 0.718278 0.359139 0.933284i $$-0.383070\pi$$
0.359139 + 0.933284i $$0.383070\pi$$
$$158$$ 1.00000 0.0795557
$$159$$ 0 0
$$160$$ −3.00000 −0.237171
$$161$$ 6.00000 0.472866
$$162$$ 9.00000 0.707107
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ −19.0000 −1.47026 −0.735132 0.677924i $$-0.762880\pi$$
−0.735132 + 0.677924i $$0.762880\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −12.0000 −0.920358
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ 12.0000 0.907115
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 9.00000 0.670820
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ 0 0
$$184$$ 2.00000 0.147442
$$185$$ −27.0000 −1.98508
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 0 0
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ −5.00000 −0.358979
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 3.00000 0.213201
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 6.00000 0.419058
$$206$$ −14.0000 −0.975426
$$207$$ −6.00000 −0.417029
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 13.0000 0.892844
$$213$$ 0 0
$$214$$ 9.00000 0.615227
$$215$$ 24.0000 1.63679
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ −2.00000 −0.135457
$$219$$ 0 0
$$220$$ 3.00000 0.202260
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 3.00000 0.200446
$$225$$ −12.0000 −0.800000
$$226$$ −10.0000 −0.665190
$$227$$ 25.0000 1.65931 0.829654 0.558278i $$-0.188538\pi$$
0.829654 + 0.558278i $$0.188538\pi$$
$$228$$ 0 0
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.00000 −0.131024 −0.0655122 0.997852i $$-0.520868\pi$$
−0.0655122 + 0.997852i $$0.520868\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 36.0000 2.34838
$$236$$ −10.0000 −0.650945
$$237$$ 0 0
$$238$$ 12.0000 0.777844
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ 3.00000 0.189737
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ −9.00000 −0.566947
$$253$$ −2.00000 −0.125739
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −11.0000 −0.686161 −0.343081 0.939306i $$-0.611470\pi$$
−0.343081 + 0.939306i $$0.611470\pi$$
$$258$$ 0 0
$$259$$ 27.0000 1.67770
$$260$$ 6.00000 0.372104
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ −17.0000 −1.04826 −0.524132 0.851637i $$-0.675610\pi$$
−0.524132 + 0.851637i $$0.675610\pi$$
$$264$$ 0 0
$$265$$ −39.0000 −2.39575
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −15.0000 −0.914566 −0.457283 0.889321i $$-0.651177\pi$$
−0.457283 + 0.889321i $$0.651177\pi$$
$$270$$ 0 0
$$271$$ −11.0000 −0.668202 −0.334101 0.942537i $$-0.608433\pi$$
−0.334101 + 0.942537i $$0.608433\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −15.0000 −0.906183
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 14.0000 0.841178 0.420589 0.907251i $$-0.361823\pi$$
0.420589 + 0.907251i $$0.361823\pi$$
$$278$$ 5.00000 0.299880
$$279$$ 6.00000 0.359211
$$280$$ −9.00000 −0.537853
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ −1.00000 −0.0594438 −0.0297219 0.999558i $$-0.509462\pi$$
−0.0297219 + 0.999558i $$0.509462\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ −6.00000 −0.354169
$$288$$ −3.00000 −0.176777
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 12.0000 0.702247
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 0 0
$$295$$ 30.0000 1.74667
$$296$$ 9.00000 0.523114
$$297$$ 0 0
$$298$$ −20.0000 −1.15857
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ 3.00000 0.172631
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −12.0000 −0.685994
$$307$$ 25.0000 1.42683 0.713413 0.700744i $$-0.247149\pi$$
0.713413 + 0.700744i $$0.247149\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ 6.00000 0.340777
$$311$$ 22.0000 1.24751 0.623753 0.781622i $$-0.285607\pi$$
0.623753 + 0.781622i $$0.285607\pi$$
$$312$$ 0 0
$$313$$ 19.0000 1.07394 0.536972 0.843600i $$-0.319568\pi$$
0.536972 + 0.843600i $$0.319568\pi$$
$$314$$ 9.00000 0.507899
$$315$$ 27.0000 1.52128
$$316$$ 1.00000 0.0562544
$$317$$ 26.0000 1.46031 0.730153 0.683284i $$-0.239449\pi$$
0.730153 + 0.683284i $$0.239449\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −3.00000 −0.167705
$$321$$ 0 0
$$322$$ 6.00000 0.334367
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ −8.00000 −0.443760
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ −36.0000 −1.98474
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 9.00000 0.493939
$$333$$ −27.0000 −1.47959
$$334$$ −19.0000 −1.03963
$$335$$ 36.0000 1.96689
$$336$$ 0 0
$$337$$ −28.0000 −1.52526 −0.762629 0.646837i $$-0.776092\pi$$
−0.762629 + 0.646837i $$0.776092\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ −12.0000 −0.650791
$$341$$ 2.00000 0.108306
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 19.0000 1.01997 0.509987 0.860182i $$-0.329650\pi$$
0.509987 + 0.860182i $$0.329650\pi$$
$$348$$ 0 0
$$349$$ −8.00000 −0.428230 −0.214115 0.976808i $$-0.568687\pi$$
−0.214115 + 0.976808i $$0.568687\pi$$
$$350$$ 12.0000 0.641427
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ −21.0000 −1.11772 −0.558859 0.829263i $$-0.688761\pi$$
−0.558859 + 0.829263i $$0.688761\pi$$
$$354$$ 0 0
$$355$$ 6.00000 0.318447
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ −23.0000 −1.21389 −0.606947 0.794742i $$-0.707606\pi$$
−0.606947 + 0.794742i $$0.707606\pi$$
$$360$$ 9.00000 0.474342
$$361$$ 0 0
$$362$$ −7.00000 −0.367912
$$363$$ 0 0
$$364$$ −6.00000 −0.314485
$$365$$ −36.0000 −1.88433
$$366$$ 0 0
$$367$$ −14.0000 −0.730794 −0.365397 0.930852i $$-0.619067\pi$$
−0.365397 + 0.930852i $$0.619067\pi$$
$$368$$ 2.00000 0.104257
$$369$$ 6.00000 0.312348
$$370$$ −27.0000 −1.40366
$$371$$ 39.0000 2.02478
$$372$$ 0 0
$$373$$ 8.00000 0.414224 0.207112 0.978317i $$-0.433593\pi$$
0.207112 + 0.978317i $$0.433593\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −18.0000 −0.920960
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 9.00000 0.458682
$$386$$ 16.0000 0.814379
$$387$$ 24.0000 1.21999
$$388$$ −5.00000 −0.253837
$$389$$ 37.0000 1.87597 0.937987 0.346670i $$-0.112688\pi$$
0.937987 + 0.346670i $$0.112688\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 2.00000 0.101015
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ −3.00000 −0.150946
$$396$$ 3.00000 0.150756
$$397$$ 33.0000 1.65622 0.828111 0.560564i $$-0.189416\pi$$
0.828111 + 0.560564i $$0.189416\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ −10.0000 −0.497519
$$405$$ −27.0000 −1.34164
$$406$$ 0 0
$$407$$ −9.00000 −0.446113
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 6.00000 0.296319
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ −30.0000 −1.47620
$$414$$ −6.00000 −0.294884
$$415$$ −27.0000 −1.32538
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 19.0000 0.926003 0.463002 0.886357i $$-0.346772\pi$$
0.463002 + 0.886357i $$0.346772\pi$$
$$422$$ 13.0000 0.632830
$$423$$ 36.0000 1.75038
$$424$$ 13.0000 0.631336
$$425$$ 16.0000 0.776114
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 9.00000 0.435031
$$429$$ 0 0
$$430$$ 24.0000 1.15738
$$431$$ 9.00000 0.433515 0.216757 0.976226i $$-0.430452\pi$$
0.216757 + 0.976226i $$0.430452\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ −6.00000 −0.288009
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −7.00000 −0.334092 −0.167046 0.985949i $$-0.553423\pi$$
−0.167046 + 0.985949i $$0.553423\pi$$
$$440$$ 3.00000 0.143019
$$441$$ −6.00000 −0.285714
$$442$$ −8.00000 −0.380521
$$443$$ −26.0000 −1.23530 −0.617649 0.786454i $$-0.711915\pi$$
−0.617649 + 0.786454i $$0.711915\pi$$
$$444$$ 0 0
$$445$$ 18.0000 0.853282
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ 3.00000 0.141737
$$449$$ −11.0000 −0.519122 −0.259561 0.965727i $$-0.583578\pi$$
−0.259561 + 0.965727i $$0.583578\pi$$
$$450$$ −12.0000 −0.565685
$$451$$ 2.00000 0.0941763
$$452$$ −10.0000 −0.470360
$$453$$ 0 0
$$454$$ 25.0000 1.17331
$$455$$ 18.0000 0.843853
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −5.00000 −0.233635
$$459$$ 0 0
$$460$$ −6.00000 −0.279751
$$461$$ −42.0000 −1.95614 −0.978068 0.208288i $$-0.933211\pi$$
−0.978068 + 0.208288i $$0.933211\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −2.00000 −0.0926482
$$467$$ −26.0000 −1.20314 −0.601568 0.798821i $$-0.705457\pi$$
−0.601568 + 0.798821i $$0.705457\pi$$
$$468$$ 6.00000 0.277350
$$469$$ −36.0000 −1.66233
$$470$$ 36.0000 1.66056
$$471$$ 0 0
$$472$$ −10.0000 −0.460287
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ −39.0000 −1.78569
$$478$$ −15.0000 −0.686084
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −18.0000 −0.820729
$$482$$ −22.0000 −1.00207
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 15.0000 0.681115
$$486$$ 0 0
$$487$$ 42.0000 1.90320 0.951601 0.307337i $$-0.0994378\pi$$
0.951601 + 0.307337i $$0.0994378\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −6.00000 −0.271052
$$491$$ −25.0000 −1.12823 −0.564117 0.825695i $$-0.690783\pi$$
−0.564117 + 0.825695i $$0.690783\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −9.00000 −0.404520
$$496$$ −2.00000 −0.0898027
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −9.00000 −0.400892
$$505$$ 30.0000 1.33498
$$506$$ −2.00000 −0.0889108
$$507$$ 0 0
$$508$$ −4.00000 −0.177471
$$509$$ 17.0000 0.753512 0.376756 0.926313i $$-0.377040\pi$$
0.376756 + 0.926313i $$0.377040\pi$$
$$510$$ 0 0
$$511$$ 36.0000 1.59255
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −11.0000 −0.485189
$$515$$ 42.0000 1.85074
$$516$$ 0 0
$$517$$ 12.0000 0.527759
$$518$$ 27.0000 1.18631
$$519$$ 0 0
$$520$$ 6.00000 0.263117
$$521$$ 15.0000 0.657162 0.328581 0.944476i $$-0.393430\pi$$
0.328581 + 0.944476i $$0.393430\pi$$
$$522$$ 0 0
$$523$$ −15.0000 −0.655904 −0.327952 0.944694i $$-0.606358\pi$$
−0.327952 + 0.944694i $$0.606358\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −17.0000 −0.741235
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ −39.0000 −1.69405
$$531$$ 30.0000 1.30189
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ −27.0000 −1.16731
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ −15.0000 −0.646696
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −20.0000 −0.859867 −0.429934 0.902861i $$-0.641463\pi$$
−0.429934 + 0.902861i $$0.641463\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ 0 0
$$544$$ 4.00000 0.171499
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ 31.0000 1.32546 0.662732 0.748857i $$-0.269397\pi$$
0.662732 + 0.748857i $$0.269397\pi$$
$$548$$ −15.0000 −0.640768
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 3.00000 0.127573
$$554$$ 14.0000 0.594803
$$555$$ 0 0
$$556$$ 5.00000 0.212047
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 16.0000 0.676728
$$560$$ −9.00000 −0.380319
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −41.0000 −1.72794 −0.863972 0.503540i $$-0.832031\pi$$
−0.863972 + 0.503540i $$0.832031\pi$$
$$564$$ 0 0
$$565$$ 30.0000 1.26211
$$566$$ −1.00000 −0.0420331
$$567$$ 27.0000 1.13389
$$568$$ −2.00000 −0.0839181
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 13.0000 0.544033 0.272017 0.962293i $$-0.412309\pi$$
0.272017 + 0.962293i $$0.412309\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ 8.00000 0.333623
$$576$$ −3.00000 −0.125000
$$577$$ 41.0000 1.70685 0.853426 0.521214i $$-0.174521\pi$$
0.853426 + 0.521214i $$0.174521\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 27.0000 1.12015
$$582$$ 0 0
$$583$$ −13.0000 −0.538405
$$584$$ 12.0000 0.496564
$$585$$ −18.0000 −0.744208
$$586$$ 24.0000 0.991431
$$587$$ −2.00000 −0.0825488 −0.0412744 0.999148i $$-0.513142\pi$$
−0.0412744 + 0.999148i $$0.513142\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 30.0000 1.23508
$$591$$ 0 0
$$592$$ 9.00000 0.369898
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ −36.0000 −1.47586
$$596$$ −20.0000 −0.819232
$$597$$ 0 0
$$598$$ −4.00000 −0.163572
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ −24.0000 −0.978167
$$603$$ 36.0000 1.46603
$$604$$ 3.00000 0.122068
$$605$$ −3.00000 −0.121967
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ −12.0000 −0.485071
$$613$$ −44.0000 −1.77714 −0.888572 0.458738i $$-0.848302\pi$$
−0.888572 + 0.458738i $$0.848302\pi$$
$$614$$ 25.0000 1.00892
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ 33.0000 1.32853 0.664265 0.747497i $$-0.268745\pi$$
0.664265 + 0.747497i $$0.268745\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 6.00000 0.240966
$$621$$ 0 0
$$622$$ 22.0000 0.882120
$$623$$ −18.0000 −0.721155
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 19.0000 0.759393
$$627$$ 0 0
$$628$$ 9.00000 0.359139
$$629$$ 36.0000 1.43541
$$630$$ 27.0000 1.07571
$$631$$ −10.0000 −0.398094 −0.199047 0.979990i $$-0.563785\pi$$
−0.199047 + 0.979990i $$0.563785\pi$$
$$632$$ 1.00000 0.0397779
$$633$$ 0 0
$$634$$ 26.0000 1.03259
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ −4.00000 −0.158486
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ −3.00000 −0.118585
$$641$$ 25.0000 0.987441 0.493720 0.869621i $$-0.335637\pi$$
0.493720 + 0.869621i $$0.335637\pi$$
$$642$$ 0 0
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ 9.00000 0.353553
$$649$$ 10.0000 0.392534
$$650$$ −8.00000 −0.313786
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 36.0000 1.40664
$$656$$ −2.00000 −0.0780869
$$657$$ −36.0000 −1.40449
$$658$$ −36.0000 −1.40343
$$659$$ 1.00000 0.0389545 0.0194772 0.999810i $$-0.493800\pi$$
0.0194772 + 0.999810i $$0.493800\pi$$
$$660$$ 0 0
$$661$$ −1.00000 −0.0388955 −0.0194477 0.999811i $$-0.506191\pi$$
−0.0194477 + 0.999811i $$0.506191\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ −27.0000 −1.04623
$$667$$ 0 0
$$668$$ −19.0000 −0.735132
$$669$$ 0 0
$$670$$ 36.0000 1.39080
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ −28.0000 −1.07852
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −8.00000 −0.307465 −0.153732 0.988113i $$-0.549129\pi$$
−0.153732 + 0.988113i $$0.549129\pi$$
$$678$$ 0 0
$$679$$ −15.0000 −0.575647
$$680$$ −12.0000 −0.460179
$$681$$ 0 0
$$682$$ 2.00000 0.0765840
$$683$$ −30.0000 −1.14792 −0.573959 0.818884i $$-0.694593\pi$$
−0.573959 + 0.818884i $$0.694593\pi$$
$$684$$ 0 0
$$685$$ 45.0000 1.71936
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ −26.0000 −0.990521
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 9.00000 0.341882
$$694$$ 19.0000 0.721230
$$695$$ −15.0000 −0.568982
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ −8.00000 −0.302804
$$699$$ 0 0
$$700$$ 12.0000 0.453557
$$701$$ −24.0000 −0.906467 −0.453234 0.891392i $$-0.649730\pi$$
−0.453234 + 0.891392i $$0.649730\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ −30.0000 −1.12827
$$708$$ 0 0
$$709$$ 31.0000 1.16423 0.582115 0.813107i $$-0.302225\pi$$
0.582115 + 0.813107i $$0.302225\pi$$
$$710$$ 6.00000 0.225176
$$711$$ −3.00000 −0.112509
$$712$$ −6.00000 −0.224860
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ −6.00000 −0.224387
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ −23.0000 −0.858352
$$719$$ 4.00000 0.149175 0.0745874 0.997214i $$-0.476236\pi$$
0.0745874 + 0.997214i $$0.476236\pi$$
$$720$$ 9.00000 0.335410
$$721$$ −42.0000 −1.56416
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −7.00000 −0.260153
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 22.0000 0.815935 0.407967 0.912996i $$-0.366238\pi$$
0.407967 + 0.912996i $$0.366238\pi$$
$$728$$ −6.00000 −0.222375
$$729$$ −27.0000 −1.00000
$$730$$ −36.0000 −1.33242
$$731$$ −32.0000 −1.18356
$$732$$ 0 0
$$733$$ −24.0000 −0.886460 −0.443230 0.896408i $$-0.646168\pi$$
−0.443230 + 0.896408i $$0.646168\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ 12.0000 0.442026
$$738$$ 6.00000 0.220863
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ −27.0000 −0.992540
$$741$$ 0 0
$$742$$ 39.0000 1.43174
$$743$$ −19.0000 −0.697042 −0.348521 0.937301i $$-0.613316\pi$$
−0.348521 + 0.937301i $$0.613316\pi$$
$$744$$ 0 0
$$745$$ 60.0000 2.19823
$$746$$ 8.00000 0.292901
$$747$$ −27.0000 −0.987878
$$748$$ −4.00000 −0.146254
$$749$$ 27.0000 0.986559
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −9.00000 −0.327544
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −34.0000 −1.23494
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −44.0000 −1.59500 −0.797499 0.603320i $$-0.793844\pi$$
−0.797499 + 0.603320i $$0.793844\pi$$
$$762$$ 0 0
$$763$$ −6.00000 −0.217215
$$764$$ −18.0000 −0.651217
$$765$$ 36.0000 1.30158
$$766$$ 24.0000 0.867155
$$767$$ 20.0000 0.722158
$$768$$ 0 0
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 9.00000 0.324337
$$771$$ 0 0
$$772$$ 16.0000 0.575853
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 24.0000 0.862662
$$775$$ −8.00000 −0.287368
$$776$$ −5.00000 −0.179490
$$777$$ 0 0
$$778$$ 37.0000 1.32651
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 2.00000 0.0715656
$$782$$ 8.00000 0.286079
$$783$$ 0 0
$$784$$ 2.00000 0.0714286
$$785$$ −27.0000 −0.963671
$$786$$ 0 0
$$787$$ 11.0000 0.392108 0.196054 0.980593i $$-0.437187\pi$$
0.196054 + 0.980593i $$0.437187\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 0 0
$$790$$ −3.00000 −0.106735
$$791$$ −30.0000 −1.06668
$$792$$ 3.00000 0.106600
$$793$$ 0 0
$$794$$ 33.0000 1.17113
$$795$$ 0 0
$$796$$ −14.0000 −0.496217
$$797$$ 23.0000 0.814702 0.407351 0.913272i $$-0.366453\pi$$
0.407351 + 0.913272i $$0.366453\pi$$
$$798$$ 0 0
$$799$$ −48.0000 −1.69812
$$800$$ 4.00000 0.141421
$$801$$ 18.0000 0.635999
$$802$$ 30.0000 1.05934
$$803$$ −12.0000 −0.423471
$$804$$ 0 0
$$805$$ −18.0000 −0.634417
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ −27.0000 −0.948683
$$811$$ −1.00000 −0.0351147 −0.0175574 0.999846i $$-0.505589\pi$$
−0.0175574 + 0.999846i $$0.505589\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −9.00000 −0.315450
$$815$$ −12.0000 −0.420342
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −10.0000 −0.349642
$$819$$ 18.0000 0.628971
$$820$$ 6.00000 0.209529
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ 0 0
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ −30.0000 −1.04383
$$827$$ 24.0000 0.834562 0.417281 0.908778i $$-0.362983\pi$$
0.417281 + 0.908778i $$0.362983\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ −27.0000 −0.937184
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ 8.00000 0.277184
$$834$$ 0 0
$$835$$ 57.0000 1.97257
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −4.00000 −0.138178
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 19.0000 0.654783
$$843$$ 0 0
$$844$$ 13.0000 0.447478
$$845$$ 27.0000 0.928828
$$846$$ 36.0000 1.23771
$$847$$ 3.00000 0.103081
$$848$$ 13.0000 0.446422
$$849$$ 0 0
$$850$$ 16.0000 0.548795
$$851$$ 18.0000 0.617032
$$852$$ 0 0
$$853$$ 18.0000 0.616308 0.308154 0.951336i $$-0.400289\pi$$
0.308154 + 0.951336i $$0.400289\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 9.00000 0.307614
$$857$$ −48.0000 −1.63965 −0.819824 0.572615i $$-0.805929\pi$$
−0.819824 + 0.572615i $$0.805929\pi$$
$$858$$ 0 0
$$859$$ −6.00000 −0.204717 −0.102359 0.994748i $$-0.532639\pi$$
−0.102359 + 0.994748i $$0.532639\pi$$
$$860$$ 24.0000 0.818393
$$861$$ 0 0
$$862$$ 9.00000 0.306541
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ 0 0
$$865$$ 54.0000 1.83606
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ −6.00000 −0.203653
$$869$$ −1.00000 −0.0339227
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ −2.00000 −0.0677285
$$873$$ 15.0000 0.507673
$$874$$ 0 0
$$875$$ 9.00000 0.304256
$$876$$ 0 0
$$877$$ 42.0000 1.41824 0.709120 0.705088i $$-0.249093\pi$$
0.709120 + 0.705088i $$0.249093\pi$$
$$878$$ −7.00000 −0.236239
$$879$$ 0 0
$$880$$ 3.00000 0.101130
$$881$$ 26.0000 0.875962 0.437981 0.898984i $$-0.355694\pi$$
0.437981 + 0.898984i $$0.355694\pi$$
$$882$$ −6.00000 −0.202031
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ −26.0000 −0.873487
$$887$$ 9.00000 0.302190 0.151095 0.988519i $$-0.451720\pi$$
0.151095 + 0.988519i $$0.451720\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 18.0000 0.603361
$$891$$ −9.00000 −0.301511
$$892$$ −4.00000 −0.133930
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −36.0000 −1.20335
$$896$$ 3.00000 0.100223
$$897$$ 0 0
$$898$$ −11.0000 −0.367075
$$899$$ 0 0
$$900$$ −12.0000 −0.400000
$$901$$ 52.0000 1.73237
$$902$$ 2.00000 0.0665927
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ 21.0000 0.698064
$$906$$ 0 0
$$907$$ 38.0000 1.26177 0.630885 0.775877i $$-0.282692\pi$$
0.630885 + 0.775877i $$0.282692\pi$$
$$908$$ 25.0000 0.829654
$$909$$ 30.0000 0.995037
$$910$$ 18.0000 0.596694
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ −9.00000 −0.297857
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −5.00000 −0.165205
$$917$$ −36.0000 −1.18882
$$918$$ 0 0
$$919$$ 51.0000 1.68233 0.841167 0.540775i $$-0.181869\pi$$
0.841167 + 0.540775i $$0.181869\pi$$
$$920$$ −6.00000 −0.197814
$$921$$ 0 0
$$922$$ −42.0000 −1.38320
$$923$$ 4.00000 0.131662
$$924$$ 0 0
$$925$$ 36.0000 1.18367
$$926$$ −16.0000 −0.525793
$$927$$ 42.0000 1.37946
$$928$$ 0 0
$$929$$ 2.00000 0.0656179 0.0328089 0.999462i $$-0.489555\pi$$
0.0328089 + 0.999462i $$0.489555\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −2.00000 −0.0655122
$$933$$ 0 0
$$934$$ −26.0000 −0.850746
$$935$$ 12.0000 0.392442
$$936$$ 6.00000 0.196116
$$937$$ −40.0000 −1.30674 −0.653372 0.757037i $$-0.726646\pi$$
−0.653372 + 0.757037i $$0.726646\pi$$
$$938$$ −36.0000 −1.17544
$$939$$ 0 0
$$940$$ 36.0000 1.17419
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ −4.00000 −0.130258
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 34.0000 1.10485 0.552426 0.833562i $$-0.313702\pi$$
0.552426 + 0.833562i $$0.313702\pi$$
$$948$$ 0 0
$$949$$ −24.0000 −0.779073
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 12.0000 0.388922
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ −39.0000 −1.26267
$$955$$ 54.0000 1.74740
$$956$$ −15.0000 −0.485135
$$957$$ 0 0
$$958$$ −8.00000 −0.258468
$$959$$ −45.0000 −1.45313
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −18.0000 −0.580343
$$963$$ −27.0000 −0.870063
$$964$$ −22.0000 −0.708572
$$965$$ −48.0000 −1.54517
$$966$$ 0 0
$$967$$ 53.0000 1.70437 0.852183 0.523245i $$-0.175279\pi$$
0.852183 + 0.523245i $$0.175279\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ 15.0000 0.481621
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 0 0
$$973$$ 15.0000 0.480878
$$974$$ 42.0000 1.34577
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −9.00000 −0.287936 −0.143968 0.989582i $$-0.545986\pi$$
−0.143968 + 0.989582i $$0.545986\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ −6.00000 −0.191663
$$981$$ 6.00000 0.191565
$$982$$ −25.0000 −0.797782
$$983$$ 18.0000 0.574111 0.287055 0.957914i $$-0.407324\pi$$
0.287055 + 0.957914i $$0.407324\pi$$
$$984$$ 0 0
$$985$$ 66.0000 2.10293
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ −9.00000 −0.286039
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ −6.00000 −0.190308
$$995$$ 42.0000 1.33149
$$996$$ 0 0
$$997$$ 52.0000 1.64686 0.823428 0.567420i $$-0.192059\pi$$
0.823428 + 0.567420i $$0.192059\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ 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 7942.2.a.p.1.1 1
19.8 odd 6 418.2.e.e.45.1 2
19.12 odd 6 418.2.e.e.353.1 yes 2
19.18 odd 2 7942.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.e.e.45.1 2 19.8 odd 6
418.2.e.e.353.1 yes 2 19.12 odd 6
7942.2.a.e.1.1 1 19.18 odd 2
7942.2.a.p.1.1 1 1.1 even 1 trivial