Properties

 Label 7942.2.a.q.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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{18} +1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} -4.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -8.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} +1.00000 q^{35} -3.00000 q^{36} +3.00000 q^{37} +1.00000 q^{40} -2.00000 q^{41} -8.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} -6.00000 q^{46} -6.00000 q^{49} -4.00000 q^{50} +2.00000 q^{52} -9.00000 q^{53} +1.00000 q^{55} +1.00000 q^{56} -8.00000 q^{58} -14.0000 q^{59} -10.0000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} +1.00000 q^{70} +6.00000 q^{71} -3.00000 q^{72} +3.00000 q^{74} +1.00000 q^{77} -11.0000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +7.00000 q^{83} -8.00000 q^{86} +1.00000 q^{88} +14.0000 q^{89} -3.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} +5.00000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})$$

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
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −3.00000 −1.00000
$$10$$ 1.00000 0.316228
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ 0 0
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ −8.00000 −1.48556 −0.742781 0.669534i $$-0.766494\pi$$
−0.742781 + 0.669534i $$0.766494\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ −3.00000 −0.500000
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −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
$$45$$ −3.00000 −0.447214
$$46$$ −6.00000 −0.884652
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ −4.00000 −0.565685
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −8.00000 −1.05045
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −10.0000 −1.27000
$$63$$ −3.00000 −0.377964
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 1.00000 0.119523
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 9.00000 1.00000
$$82$$ −2.00000 −0.220863
$$83$$ 7.00000 0.768350 0.384175 0.923260i $$-0.374486\pi$$
0.384175 + 0.923260i $$0.374486\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 2.00000 0.209657
$$92$$ −6.00000 −0.625543
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.00000 0.507673 0.253837 0.967247i $$-0.418307\pi$$
0.253837 + 0.967247i $$0.418307\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ −3.00000 −0.301511
$$100$$ −4.00000 −0.400000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ 17.0000 1.64345 0.821726 0.569883i $$-0.193011\pi$$
0.821726 + 0.569883i $$0.193011\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ −8.00000 −0.742781
$$117$$ −6.00000 −0.554700
$$118$$ −14.0000 −1.28880
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −10.0000 −0.898027
$$125$$ −9.00000 −0.804984
$$126$$ −3.00000 −0.267261
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −23.0000 −1.96502 −0.982511 0.186203i $$-0.940382\pi$$
−0.982511 + 0.186203i $$0.940382\pi$$
$$138$$ 0 0
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 2.00000 0.167248
$$144$$ −3.00000 −0.250000
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 3.00000 0.246598
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ −9.00000 −0.732410 −0.366205 0.930534i $$-0.619343\pi$$
−0.366205 + 0.930534i $$0.619343\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 1.00000 0.0805823
$$155$$ −10.0000 −0.803219
$$156$$ 0 0
$$157$$ 21.0000 1.67598 0.837991 0.545684i $$-0.183730\pi$$
0.837991 + 0.545684i $$0.183730\pi$$
$$158$$ −11.0000 −0.875113
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ −6.00000 −0.472866
$$162$$ 9.00000 0.707107
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 7.00000 0.543305
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ 22.0000 1.67263 0.836315 0.548250i $$-0.184706\pi$$
0.836315 + 0.548250i $$0.184706\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 14.0000 1.04934
$$179$$ −16.0000 −1.19590 −0.597948 0.801535i $$-0.704017\pi$$
−0.597948 + 0.801535i $$0.704017\pi$$
$$180$$ −3.00000 −0.223607
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 2.00000 0.148250
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 3.00000 0.220564
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 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
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 5.00000 0.358979
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ −8.00000 −0.561490
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 6.00000 0.418040
$$207$$ 18.0000 1.25109
$$208$$ 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 $$0.352322\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ 0 0
$$214$$ 17.0000 1.16210
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ −10.0000 −0.678844
$$218$$ −10.0000 −0.677285
$$219$$ 0 0
$$220$$ 1.00000 0.0674200
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 12.0000 0.800000
$$226$$ 2.00000 0.133038
$$227$$ 25.0000 1.65931 0.829654 0.558278i $$-0.188538\pi$$
0.829654 + 0.558278i $$0.188538\pi$$
$$228$$ 0 0
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 0 0
$$232$$ −8.00000 −0.525226
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ −14.0000 −0.911322
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5.00000 −0.323423 −0.161712 0.986838i $$-0.551701\pi$$
−0.161712 + 0.986838i $$0.551701\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −10.0000 −0.635001
$$249$$ 0 0
$$250$$ −9.00000 −0.569210
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ −3.00000 −0.188982
$$253$$ −6.00000 −0.377217
$$254$$ 20.0000 1.25491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −21.0000 −1.30994 −0.654972 0.755653i $$-0.727320\pi$$
−0.654972 + 0.755653i $$0.727320\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ 2.00000 0.124035
$$261$$ 24.0000 1.48556
$$262$$ −20.0000 −1.23560
$$263$$ −3.00000 −0.184988 −0.0924940 0.995713i $$-0.529484\pi$$
−0.0924940 + 0.995713i $$0.529484\pi$$
$$264$$ 0 0
$$265$$ −9.00000 −0.552866
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ −21.0000 −1.28039 −0.640196 0.768211i $$-0.721147\pi$$
−0.640196 + 0.768211i $$0.721147\pi$$
$$270$$ 0 0
$$271$$ 15.0000 0.911185 0.455593 0.890188i $$-0.349427\pi$$
0.455593 + 0.890188i $$0.349427\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −23.0000 −1.38948
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −5.00000 −0.299880
$$279$$ 30.0000 1.79605
$$280$$ 1.00000 0.0597614
$$281$$ 20.0000 1.19310 0.596550 0.802576i $$-0.296538\pi$$
0.596550 + 0.802576i $$0.296538\pi$$
$$282$$ 0 0
$$283$$ −7.00000 −0.416107 −0.208053 0.978117i $$-0.566713\pi$$
−0.208053 + 0.978117i $$0.566713\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ −2.00000 −0.118056
$$288$$ −3.00000 −0.176777
$$289$$ −17.0000 −1.00000
$$290$$ −8.00000 −0.469776
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ −14.0000 −0.815112
$$296$$ 3.00000 0.174371
$$297$$ 0 0
$$298$$ −4.00000 −0.231714
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −9.00000 −0.517892
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −23.0000 −1.31268 −0.656340 0.754466i $$-0.727896\pi$$
−0.656340 + 0.754466i $$0.727896\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ 0 0
$$310$$ −10.0000 −0.567962
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ 0 0
$$313$$ −21.0000 −1.18699 −0.593495 0.804838i $$-0.702252\pi$$
−0.593495 + 0.804838i $$0.702252\pi$$
$$314$$ 21.0000 1.18510
$$315$$ −3.00000 −0.169031
$$316$$ −11.0000 −0.618798
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 1.00000 0.0559017
$$321$$ 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
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 7.00000 0.384175
$$333$$ −9.00000 −0.493197
$$334$$ 9.00000 0.492458
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −8.00000 −0.435788 −0.217894 0.975972i $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ −3.00000 −0.161048 −0.0805242 0.996753i $$-0.525659\pi$$
−0.0805242 + 0.996753i $$0.525659\pi$$
$$348$$ 0 0
$$349$$ 28.0000 1.49881 0.749403 0.662114i $$-0.230341\pi$$
0.749403 + 0.662114i $$0.230341\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ 11.0000 0.585471 0.292735 0.956193i $$-0.405434\pi$$
0.292735 + 0.956193i $$0.405434\pi$$
$$354$$ 0 0
$$355$$ 6.00000 0.318447
$$356$$ 14.0000 0.741999
$$357$$ 0 0
$$358$$ −16.0000 −0.845626
$$359$$ 3.00000 0.158334 0.0791670 0.996861i $$-0.474774\pi$$
0.0791670 + 0.996861i $$0.474774\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ 0 0
$$362$$ −5.00000 −0.262794
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 6.00000 0.312348
$$370$$ 3.00000 0.155963
$$371$$ −9.00000 −0.467257
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −16.0000 −0.824042
$$378$$ 0 0
$$379$$ 2.00000 0.102733 0.0513665 0.998680i $$-0.483642\pi$$
0.0513665 + 0.998680i $$0.483642\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −18.0000 −0.920960
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ 1.00000 0.0509647
$$386$$ 4.00000 0.203595
$$387$$ 24.0000 1.21999
$$388$$ 5.00000 0.253837
$$389$$ −23.0000 −1.16615 −0.583073 0.812420i $$-0.698150\pi$$
−0.583073 + 0.812420i $$0.698150\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −6.00000 −0.303046
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ −11.0000 −0.553470
$$396$$ −3.00000 −0.150756
$$397$$ −27.0000 −1.35509 −0.677546 0.735481i $$-0.736956\pi$$
−0.677546 + 0.735481i $$0.736956\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ −20.0000 −0.996271
$$404$$ 10.0000 0.497519
$$405$$ 9.00000 0.447214
$$406$$ −8.00000 −0.397033
$$407$$ 3.00000 0.148704
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ 0 0
$$412$$ 6.00000 0.295599
$$413$$ −14.0000 −0.688895
$$414$$ 18.0000 0.884652
$$415$$ 7.00000 0.343616
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −15.0000 −0.731055 −0.365528 0.930800i $$-0.619111\pi$$
−0.365528 + 0.930800i $$0.619111\pi$$
$$422$$ 13.0000 0.632830
$$423$$ 0 0
$$424$$ −9.00000 −0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 17.0000 0.821726
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ 5.00000 0.240842 0.120421 0.992723i $$-0.461576\pi$$
0.120421 + 0.992723i $$0.461576\pi$$
$$432$$ 0 0
$$433$$ 30.0000 1.44171 0.720854 0.693087i $$-0.243750\pi$$
0.720854 + 0.693087i $$0.243750\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 29.0000 1.38409 0.692047 0.721852i $$-0.256709\pi$$
0.692047 + 0.721852i $$0.256709\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ 22.0000 1.04525 0.522626 0.852562i $$-0.324953\pi$$
0.522626 + 0.852562i $$0.324953\pi$$
$$444$$ 0 0
$$445$$ 14.0000 0.663664
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ −5.00000 −0.235965 −0.117982 0.993016i $$-0.537643\pi$$
−0.117982 + 0.993016i $$0.537643\pi$$
$$450$$ 12.0000 0.565685
$$451$$ −2.00000 −0.0941763
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 25.0000 1.17331
$$455$$ 2.00000 0.0937614
$$456$$ 0 0
$$457$$ 38.0000 1.77757 0.888783 0.458329i $$-0.151552\pi$$
0.888783 + 0.458329i $$0.151552\pi$$
$$458$$ −1.00000 −0.0467269
$$459$$ 0 0
$$460$$ −6.00000 −0.279751
$$461$$ 26.0000 1.21094 0.605470 0.795868i $$-0.292985\pi$$
0.605470 + 0.795868i $$0.292985\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −14.0000 −0.644402
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 27.0000 1.23625
$$478$$ −5.00000 −0.228695
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 5.00000 0.227038
$$486$$ 0 0
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −6.00000 −0.271052
$$491$$ −15.0000 −0.676941 −0.338470 0.940977i $$-0.609909\pi$$
−0.338470 + 0.940977i $$0.609909\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −3.00000 −0.134840
$$496$$ −10.0000 −0.449013
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 10.0000 0.444994
$$506$$ −6.00000 −0.266733
$$507$$ 0 0
$$508$$ 20.0000 0.887357
$$509$$ −21.0000 −0.930809 −0.465404 0.885098i $$-0.654091\pi$$
−0.465404 + 0.885098i $$0.654091\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −21.0000 −0.926270
$$515$$ 6.00000 0.264392
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 3.00000 0.131812
$$519$$ 0 0
$$520$$ 2.00000 0.0877058
$$521$$ −31.0000 −1.35813 −0.679067 0.734076i $$-0.737616\pi$$
−0.679067 + 0.734076i $$0.737616\pi$$
$$522$$ 24.0000 1.05045
$$523$$ −7.00000 −0.306089 −0.153044 0.988219i $$-0.548908\pi$$
−0.153044 + 0.988219i $$0.548908\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −3.00000 −0.130806
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −9.00000 −0.390935
$$531$$ 42.0000 1.82264
$$532$$ 0 0
$$533$$ −4.00000 −0.173259
$$534$$ 0 0
$$535$$ 17.0000 0.734974
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ −21.0000 −0.905374
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ 15.0000 0.644305
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −10.0000 −0.428353
$$546$$ 0 0
$$547$$ −1.00000 −0.0427569 −0.0213785 0.999771i $$-0.506805\pi$$
−0.0213785 + 0.999771i $$0.506805\pi$$
$$548$$ −23.0000 −0.982511
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −11.0000 −0.467768
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −5.00000 −0.212047
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 30.0000 1.27000
$$559$$ −16.0000 −0.676728
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 20.0000 0.843649
$$563$$ −17.0000 −0.716465 −0.358232 0.933632i $$-0.616620\pi$$
−0.358232 + 0.933632i $$0.616620\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ −7.00000 −0.294232
$$567$$ 9.00000 0.377964
$$568$$ 6.00000 0.251754
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −37.0000 −1.54840 −0.774201 0.632940i $$-0.781848\pi$$
−0.774201 + 0.632940i $$0.781848\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ −2.00000 −0.0834784
$$575$$ 24.0000 1.00087
$$576$$ −3.00000 −0.125000
$$577$$ 25.0000 1.04076 0.520382 0.853934i $$-0.325790\pi$$
0.520382 + 0.853934i $$0.325790\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ 0 0
$$580$$ −8.00000 −0.332182
$$581$$ 7.00000 0.290409
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ 0 0
$$585$$ −6.00000 −0.248069
$$586$$ 0 0
$$587$$ 26.0000 1.07313 0.536567 0.843857i $$-0.319721\pi$$
0.536567 + 0.843857i $$0.319721\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −14.0000 −0.576371
$$591$$ 0 0
$$592$$ 3.00000 0.123299
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ 0 0
$$598$$ −12.0000 −0.490716
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ −36.0000 −1.46603
$$604$$ −9.00000 −0.366205
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 4.00000 0.161558 0.0807792 0.996732i $$-0.474259\pi$$
0.0807792 + 0.996732i $$0.474259\pi$$
$$614$$ −23.0000 −0.928204
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ −15.0000 −0.603877 −0.301939 0.953327i $$-0.597634\pi$$
−0.301939 + 0.953327i $$0.597634\pi$$
$$618$$ 0 0
$$619$$ 46.0000 1.84890 0.924448 0.381308i $$-0.124526\pi$$
0.924448 + 0.381308i $$0.124526\pi$$
$$620$$ −10.0000 −0.401610
$$621$$ 0 0
$$622$$ −10.0000 −0.400963
$$623$$ 14.0000 0.560898
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ −21.0000 −0.839329
$$627$$ 0 0
$$628$$ 21.0000 0.837991
$$629$$ 0 0
$$630$$ −3.00000 −0.119523
$$631$$ −22.0000 −0.875806 −0.437903 0.899022i $$-0.644279\pi$$
−0.437903 + 0.899022i $$0.644279\pi$$
$$632$$ −11.0000 −0.437557
$$633$$ 0 0
$$634$$ 22.0000 0.873732
$$635$$ 20.0000 0.793676
$$636$$ 0 0
$$637$$ −12.0000 −0.475457
$$638$$ −8.00000 −0.316723
$$639$$ −18.0000 −0.712069
$$640$$ 1.00000 0.0395285
$$641$$ 39.0000 1.54041 0.770204 0.637798i $$-0.220155\pi$$
0.770204 + 0.637798i $$0.220155\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 9.00000 0.353553
$$649$$ −14.0000 −0.549548
$$650$$ −8.00000 −0.313786
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ −20.0000 −0.781465
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −15.0000 −0.584317 −0.292159 0.956370i $$-0.594373\pi$$
−0.292159 + 0.956370i $$0.594373\pi$$
$$660$$ 0 0
$$661$$ −11.0000 −0.427850 −0.213925 0.976850i $$-0.568625\pi$$
−0.213925 + 0.976850i $$0.568625\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ 7.00000 0.271653
$$665$$ 0 0
$$666$$ −9.00000 −0.348743
$$667$$ 48.0000 1.85857
$$668$$ 9.00000 0.348220
$$669$$ 0 0
$$670$$ 12.0000 0.463600
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ 0 0
$$679$$ 5.00000 0.191882
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −10.0000 −0.382920
$$683$$ 26.0000 0.994862 0.497431 0.867503i $$-0.334277\pi$$
0.497431 + 0.867503i $$0.334277\pi$$
$$684$$ 0 0
$$685$$ −23.0000 −0.878785
$$686$$ −13.0000 −0.496342
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ −18.0000 −0.685745
$$690$$ 0 0
$$691$$ −50.0000 −1.90209 −0.951045 0.309053i $$-0.899988\pi$$
−0.951045 + 0.309053i $$0.899988\pi$$
$$692$$ 22.0000 0.836315
$$693$$ −3.00000 −0.113961
$$694$$ −3.00000 −0.113878
$$695$$ −5.00000 −0.189661
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 28.0000 1.05982
$$699$$ 0 0
$$700$$ −4.00000 −0.151186
$$701$$ −36.0000 −1.35970 −0.679851 0.733351i $$-0.737955\pi$$
−0.679851 + 0.733351i $$0.737955\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 11.0000 0.413990
$$707$$ 10.0000 0.376089
$$708$$ 0 0
$$709$$ 43.0000 1.61490 0.807449 0.589937i $$-0.200847\pi$$
0.807449 + 0.589937i $$0.200847\pi$$
$$710$$ 6.00000 0.225176
$$711$$ 33.0000 1.23760
$$712$$ 14.0000 0.524672
$$713$$ 60.0000 2.24702
$$714$$ 0 0
$$715$$ 2.00000 0.0747958
$$716$$ −16.0000 −0.597948
$$717$$ 0 0
$$718$$ 3.00000 0.111959
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 6.00000 0.223452
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −5.00000 −0.185824
$$725$$ 32.0000 1.18845
$$726$$ 0 0
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −36.0000 −1.32969 −0.664845 0.746981i $$-0.731502\pi$$
−0.664845 + 0.746981i $$0.731502\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 12.0000 0.442026
$$738$$ 6.00000 0.220863
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 3.00000 0.110282
$$741$$ 0 0
$$742$$ −9.00000 −0.330400
$$743$$ 41.0000 1.50414 0.752072 0.659081i $$-0.229055\pi$$
0.752072 + 0.659081i $$0.229055\pi$$
$$744$$ 0 0
$$745$$ −4.00000 −0.146549
$$746$$ −4.00000 −0.146450
$$747$$ −21.0000 −0.768350
$$748$$ 0 0
$$749$$ 17.0000 0.621166
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −16.0000 −0.582686
$$755$$ −9.00000 −0.327544
$$756$$ 0 0
$$757$$ −50.0000 −1.81728 −0.908640 0.417579i $$-0.862879\pi$$
−0.908640 + 0.417579i $$0.862879\pi$$
$$758$$ 2.00000 0.0726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −36.0000 −1.30500 −0.652499 0.757789i $$-0.726280\pi$$
−0.652499 + 0.757789i $$0.726280\pi$$
$$762$$ 0 0
$$763$$ −10.0000 −0.362024
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ −28.0000 −1.01102
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 1.00000 0.0360375
$$771$$ 0 0
$$772$$ 4.00000 0.143963
$$773$$ 38.0000 1.36677 0.683383 0.730061i $$-0.260508\pi$$
0.683383 + 0.730061i $$0.260508\pi$$
$$774$$ 24.0000 0.862662
$$775$$ 40.0000 1.43684
$$776$$ 5.00000 0.179490
$$777$$ 0 0
$$778$$ −23.0000 −0.824590
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ 21.0000 0.749522
$$786$$ 0 0
$$787$$ 11.0000 0.392108 0.196054 0.980593i $$-0.437187\pi$$
0.196054 + 0.980593i $$0.437187\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ −11.0000 −0.391362
$$791$$ 2.00000 0.0711118
$$792$$ −3.00000 −0.106600
$$793$$ 0 0
$$794$$ −27.0000 −0.958194
$$795$$ 0 0
$$796$$ 10.0000 0.354441
$$797$$ 21.0000 0.743858 0.371929 0.928261i $$-0.378696\pi$$
0.371929 + 0.928261i $$0.378696\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −4.00000 −0.141421
$$801$$ −42.0000 −1.48400
$$802$$ 10.0000 0.353112
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −6.00000 −0.211472
$$806$$ −20.0000 −0.704470
$$807$$ 0 0
$$808$$ 10.0000 0.351799
$$809$$ −44.0000 −1.54696 −0.773479 0.633822i $$-0.781485\pi$$
−0.773479 + 0.633822i $$0.781485\pi$$
$$810$$ 9.00000 0.316228
$$811$$ 23.0000 0.807639 0.403820 0.914839i $$-0.367682\pi$$
0.403820 + 0.914839i $$0.367682\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 0 0
$$814$$ 3.00000 0.105150
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 10.0000 0.349642
$$819$$ −6.00000 −0.209657
$$820$$ −2.00000 −0.0698430
$$821$$ −20.0000 −0.698005 −0.349002 0.937122i $$-0.613479\pi$$
−0.349002 + 0.937122i $$0.613479\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ −14.0000 −0.487122
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ 18.0000 0.625543
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 7.00000 0.242974
$$831$$ 0 0
$$832$$ 2.00000 0.0693375
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 9.00000 0.311458
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ −15.0000 −0.516934
$$843$$ 0 0
$$844$$ 13.0000 0.447478
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 1.00000 0.0343604
$$848$$ −9.00000 −0.309061
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −18.0000 −0.617032
$$852$$ 0 0
$$853$$ −30.0000 −1.02718 −0.513590 0.858036i $$-0.671685\pi$$
−0.513590 + 0.858036i $$0.671685\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 17.0000 0.581048
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ 5.00000 0.170301
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ 22.0000 0.748022
$$866$$ 30.0000 1.01944
$$867$$ 0 0
$$868$$ −10.0000 −0.339422
$$869$$ −11.0000 −0.373149
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ −10.0000 −0.338643
$$873$$ −15.0000 −0.507673
$$874$$ 0 0
$$875$$ −9.00000 −0.304256
$$876$$ 0 0
$$877$$ −18.0000 −0.607817 −0.303908 0.952701i $$-0.598292\pi$$
−0.303908 + 0.952701i $$0.598292\pi$$
$$878$$ 29.0000 0.978703
$$879$$ 0 0
$$880$$ 1.00000 0.0337100
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 18.0000 0.606092
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 22.0000 0.739104
$$887$$ −35.0000 −1.17518 −0.587592 0.809157i $$-0.699924\pi$$
−0.587592 + 0.809157i $$0.699924\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 14.0000 0.469281
$$891$$ 9.00000 0.301511
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −5.00000 −0.166852
$$899$$ 80.0000 2.66815
$$900$$ 12.0000 0.400000
$$901$$ 0 0
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ −5.00000 −0.166206
$$906$$ 0 0
$$907$$ −38.0000 −1.26177 −0.630885 0.775877i $$-0.717308\pi$$
−0.630885 + 0.775877i $$0.717308\pi$$
$$908$$ 25.0000 0.829654
$$909$$ −30.0000 −0.995037
$$910$$ 2.00000 0.0662994
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 7.00000 0.231666
$$914$$ 38.0000 1.25693
$$915$$ 0 0
$$916$$ −1.00000 −0.0330409
$$917$$ −20.0000 −0.660458
$$918$$ 0 0
$$919$$ −39.0000 −1.28649 −0.643246 0.765660i $$-0.722413\pi$$
−0.643246 + 0.765660i $$0.722413\pi$$
$$920$$ −6.00000 −0.197814
$$921$$ 0 0
$$922$$ 26.0000 0.856264
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ −12.0000 −0.394558
$$926$$ 16.0000 0.525793
$$927$$ −18.0000 −0.591198
$$928$$ −8.00000 −0.262613
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ 6.00000 0.196326
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ 16.0000 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$938$$ 12.0000 0.391814
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 0 0
$$943$$ 12.0000 0.390774
$$944$$ −14.0000 −0.455661
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 10.0000 0.324956 0.162478 0.986712i $$-0.448051\pi$$
0.162478 + 0.986712i $$0.448051\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ 27.0000 0.874157
$$955$$ −18.0000 −0.582466
$$956$$ −5.00000 −0.161712
$$957$$ 0 0
$$958$$ −32.0000 −1.03387
$$959$$ −23.0000 −0.742709
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 6.00000 0.193448
$$963$$ −51.0000 −1.64345
$$964$$ 10.0000 0.322078
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ −1.00000 −0.0321578 −0.0160789 0.999871i $$-0.505118\pi$$
−0.0160789 + 0.999871i $$0.505118\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ 5.00000 0.160540
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ −5.00000 −0.160293
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 33.0000 1.05576 0.527882 0.849318i $$-0.322986\pi$$
0.527882 + 0.849318i $$0.322986\pi$$
$$978$$ 0 0
$$979$$ 14.0000 0.447442
$$980$$ −6.00000 −0.191663
$$981$$ 30.0000 0.957826
$$982$$ −15.0000 −0.478669
$$983$$ 14.0000 0.446531 0.223265 0.974758i $$-0.428328\pi$$
0.223265 + 0.974758i $$0.428328\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 48.0000 1.52631
$$990$$ −3.00000 −0.0953463
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ −10.0000 −0.317500
$$993$$ 0 0
$$994$$ 6.00000 0.190308
$$995$$ 10.0000 0.317021
$$996$$ 0 0
$$997$$ 52.0000 1.64686 0.823428 0.567420i $$-0.192059\pi$$
0.823428 + 0.567420i $$0.192059\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 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.q.1.1 1
19.7 even 3 418.2.e.b.353.1 yes 2
19.11 even 3 418.2.e.b.45.1 2
19.18 odd 2 7942.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.e.b.45.1 2 19.11 even 3
418.2.e.b.353.1 yes 2 19.7 even 3
7942.2.a.f.1.1 1 19.18 odd 2
7942.2.a.q.1.1 1 1.1 even 1 trivial