Properties

 Label 725.2.a.a.1.1 Level $725$ Weight $2$ Character 725.1 Self dual yes Analytic conductor $5.789$ 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] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, 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("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$5.78915414654$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} -6.00000 q^{11} -2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} -2.00000 q^{19} -6.00000 q^{22} -2.00000 q^{23} -2.00000 q^{26} -2.00000 q^{28} -1.00000 q^{29} +2.00000 q^{31} +5.00000 q^{32} +2.00000 q^{34} +3.00000 q^{36} -10.0000 q^{37} -2.00000 q^{38} +2.00000 q^{41} -8.00000 q^{43} +6.00000 q^{44} -2.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} +2.00000 q^{52} +6.00000 q^{53} -6.00000 q^{56} -1.00000 q^{58} -8.00000 q^{59} -6.00000 q^{61} +2.00000 q^{62} -6.00000 q^{63} +7.00000 q^{64} -2.00000 q^{67} -2.00000 q^{68} -12.0000 q^{71} +9.00000 q^{72} +6.00000 q^{73} -10.0000 q^{74} +2.00000 q^{76} -12.0000 q^{77} -10.0000 q^{79} +9.00000 q^{81} +2.00000 q^{82} +14.0000 q^{83} -8.00000 q^{86} +18.0000 q^{88} +18.0000 q^{89} -4.00000 q^{91} +2.00000 q^{92} +12.0000 q^{94} -2.00000 q^{97} -3.00000 q^{98} +18.0000 q^{99} +O(q^{100})$$

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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −6.00000 −0.801784
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 2.00000 0.254000
$$63$$ −6.00000 −0.755929
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 9.00000 1.06066
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 2.00000 0.220863
$$83$$ 14.0000 1.53670 0.768350 0.640030i $$-0.221078\pi$$
0.768350 + 0.640030i $$0.221078\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 18.0000 1.91881
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 2.00000 0.208514
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 18.0000 1.80907
$$100$$ 0 0
$$101$$ 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 $$0.404481\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.00000 0.0928477
$$117$$ 6.00000 0.554700
$$118$$ −8.00000 −0.736460
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ −6.00000 −0.543214
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ 12.0000 1.00349
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −6.00000 −0.485071
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 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
$$166$$ 14.0000 1.08661
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ 8.00000 0.609994
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ 0 0
$$178$$ 18.0000 1.34916
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.0000 −0.877527
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −22.0000 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$192$$ 0 0
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 18.0000 1.27920
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 6.00000 0.418040
$$207$$ 6.00000 0.417029
$$208$$ 2.00000 0.138675
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −6.00000 −0.410152
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ −14.0000 −0.948200
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 10.0000 0.668153
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ −22.0000 −1.46019 −0.730096 0.683345i $$-0.760525\pi$$
−0.730096 + 0.683345i $$0.760525\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −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
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 25.0000 1.60706
$$243$$ 0 0
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ −6.00000 −0.381000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 12.0000 0.754434
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 30.0000 1.87135 0.935674 0.352865i $$-0.114792\pi$$
0.935674 + 0.352865i $$0.114792\pi$$
$$258$$ 0 0
$$259$$ −20.0000 −1.24274
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 14.0000 0.864923
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ 0 0
$$268$$ 2.00000 0.122169
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 0 0
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ −22.0000 −1.30776 −0.653882 0.756596i $$-0.726861\pi$$
−0.653882 + 0.756596i $$0.726861\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 4.00000 0.236113
$$288$$ −15.0000 −0.883883
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −6.00000 −0.351123
$$293$$ 2.00000 0.116841 0.0584206 0.998292i $$-0.481394\pi$$
0.0584206 + 0.998292i $$0.481394\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 30.0000 1.74371
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ −4.00000 −0.230174
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −22.0000 −1.24751 −0.623753 0.781622i $$-0.714393\pi$$
−0.623753 + 0.781622i $$0.714393\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −4.00000 −0.222911
$$323$$ −4.00000 −0.222566
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −18.0000 −0.989369 −0.494685 0.869072i $$-0.664716\pi$$
−0.494685 + 0.869072i $$0.664716\pi$$
$$332$$ −14.0000 −0.768350
$$333$$ 30.0000 1.64399
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 6.00000 0.324443
$$343$$ −20.0000 −1.07990
$$344$$ 24.0000 1.29399
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 6.00000 0.322097 0.161048 0.986947i $$-0.448512\pi$$
0.161048 + 0.986947i $$0.448512\pi$$
$$348$$ 0 0
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −30.0000 −1.59901
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 22.0000 1.16112 0.580558 0.814219i $$-0.302835\pi$$
0.580558 + 0.814219i $$0.302835\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 6.00000 0.315353
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ 2.00000 0.104257
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ −18.0000 −0.932005 −0.466002 0.884783i $$-0.654306\pi$$
−0.466002 + 0.884783i $$0.654306\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ −36.0000 −1.85656
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ 6.00000 0.308199 0.154100 0.988055i $$-0.450752\pi$$
0.154100 + 0.988055i $$0.450752\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −22.0000 −1.12562
$$383$$ −14.0000 −0.715367 −0.357683 0.933843i $$-0.616433\pi$$
−0.357683 + 0.933843i $$0.616433\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 24.0000 1.21999
$$388$$ 2.00000 0.101535
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 9.00000 0.454569
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ −18.0000 −0.904534
$$397$$ 30.0000 1.50566 0.752828 0.658217i $$-0.228689\pi$$
0.752828 + 0.658217i $$0.228689\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ 60.0000 2.97409
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −6.00000 −0.295599
$$413$$ −16.0000 −0.787309
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ 12.0000 0.586939
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 14.0000 0.681509
$$423$$ −36.0000 −1.75038
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −12.0000 −0.580721
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ −4.00000 −0.190261
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ 0 0
$$448$$ 14.0000 0.661438
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ −22.0000 −1.03251
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0000 1.77757 0.888783 0.458329i $$-0.151552\pi$$
0.888783 + 0.458329i $$0.151552\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 24.0000 1.10469
$$473$$ 48.0000 2.20704
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ −18.0000 −0.824163
$$478$$ −12.0000 −0.548867
$$479$$ −14.0000 −0.639676 −0.319838 0.947472i $$-0.603629\pi$$
−0.319838 + 0.947472i $$0.603629\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ −25.0000 −1.13636
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 34.0000 1.54069 0.770344 0.637629i $$-0.220085\pi$$
0.770344 + 0.637629i $$0.220085\pi$$
$$488$$ 18.0000 0.814822
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −2.00000 −0.0902587 −0.0451294 0.998981i $$-0.514370\pi$$
−0.0451294 + 0.998981i $$0.514370\pi$$
$$492$$ 0 0
$$493$$ −2.00000 −0.0900755
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ −24.0000 −1.07655
$$498$$ 0 0
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −6.00000 −0.267793
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 18.0000 0.801784
$$505$$ 0 0
$$506$$ 12.0000 0.533465
$$507$$ 0 0
$$508$$ −16.0000 −0.709885
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 30.0000 1.32324
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −72.0000 −3.16656
$$518$$ −20.0000 −0.878750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 3.00000 0.131306
$$523$$ −42.0000 −1.83653 −0.918266 0.395964i $$-0.870410\pi$$
−0.918266 + 0.395964i $$0.870410\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ 4.00000 0.173422
$$533$$ −4.00000 −0.173259
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.00000 0.259161
$$537$$ 0 0
$$538$$ 26.0000 1.12094
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −14.0000 −0.601907 −0.300954 0.953639i $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ 0 0
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 18.0000 0.768221
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ −20.0000 −0.850487
$$554$$ −18.0000 −0.764747
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 22.0000 0.928014
$$563$$ −20.0000 −0.842900 −0.421450 0.906852i $$-0.638479\pi$$
−0.421450 + 0.906852i $$0.638479\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ 18.0000 0.755929
$$568$$ 36.0000 1.51053
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ 0 0
$$574$$ 4.00000 0.166957
$$575$$ 0 0
$$576$$ −21.0000 −0.875000
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 28.0000 1.16164
$$582$$ 0 0
$$583$$ −36.0000 −1.49097
$$584$$ −18.0000 −0.744845
$$585$$ 0 0
$$586$$ 2.00000 0.0826192
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 4.00000 0.163572
$$599$$ −18.0000 −0.735460 −0.367730 0.929933i $$-0.619865\pi$$
−0.367730 + 0.929933i $$0.619865\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ −16.0000 −0.652111
$$603$$ 6.00000 0.244339
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ −10.0000 −0.405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 6.00000 0.242536
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 36.0000 1.45048
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ 26.0000 1.04503 0.522514 0.852631i $$-0.324994\pi$$
0.522514 + 0.852631i $$0.324994\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −22.0000 −0.882120
$$623$$ 36.0000 1.44231
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 30.0000 1.19334
$$633$$ 0 0
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.00000 0.237729
$$638$$ 6.00000 0.237542
$$639$$ 36.0000 1.42414
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ −27.0000 −1.06066
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 34.0000 1.33052 0.665261 0.746611i $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ −18.0000 −0.702247
$$658$$ 24.0000 0.935617
$$659$$ −26.0000 −1.01282 −0.506408 0.862294i $$-0.669027\pi$$
−0.506408 + 0.862294i $$0.669027\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −18.0000 −0.699590
$$663$$ 0 0
$$664$$ −42.0000 −1.62992
$$665$$ 0 0
$$666$$ 30.0000 1.16248
$$667$$ 2.00000 0.0774403
$$668$$ 18.0000 0.696441
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 36.0000 1.38976
$$672$$ 0 0
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −12.0000 −0.459504
$$683$$ −18.0000 −0.688751 −0.344375 0.938832i $$-0.611909\pi$$
−0.344375 + 0.938832i $$0.611909\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 36.0000 1.36753
$$694$$ 6.00000 0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 34.0000 1.28692
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 20.0000 0.754314
$$704$$ −42.0000 −1.58293
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ 20.0000 0.752177
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ 30.0000 1.12509
$$712$$ −54.0000 −2.02374
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 22.0000 0.821033
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ −15.0000 −0.558242
$$723$$ 0 0
$$724$$ −6.00000 −0.222988
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 12.0000 0.444750
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ 0 0
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ −10.0000 −0.368605
$$737$$ 12.0000 0.442026
$$738$$ −6.00000 −0.220863
$$739$$ 46.0000 1.69214 0.846069 0.533074i $$-0.178963\pi$$
0.846069 + 0.533074i $$0.178963\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −18.0000 −0.659027
$$747$$ −42.0000 −1.53670
$$748$$ 12.0000 0.438763
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 0 0
$$754$$ 2.00000 0.0728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 6.00000 0.217930
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ −28.0000 −1.01367
$$764$$ 22.0000 0.795932
$$765$$ 0 0
$$766$$ −14.0000 −0.505841
$$767$$ 16.0000 0.577727
$$768$$ 0 0
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −10.0000 −0.359908
$$773$$ −10.0000 −0.359675 −0.179838 0.983696i $$-0.557557\pi$$
−0.179838 + 0.983696i $$0.557557\pi$$
$$774$$ 24.0000 0.862662
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ 72.0000 2.57636
$$782$$ −4.00000 −0.143040
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −18.0000 −0.641631 −0.320815 0.947142i $$-0.603957\pi$$
−0.320815 + 0.947142i $$0.603957\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ −54.0000 −1.91881
$$793$$ 12.0000 0.426132
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ 34.0000 1.20434 0.602171 0.798367i $$-0.294303\pi$$
0.602171 + 0.798367i $$0.294303\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ −54.0000 −1.90800
$$802$$ −14.0000 −0.494357
$$803$$ −36.0000 −1.27041
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ −30.0000 −1.05540
$$809$$ −14.0000 −0.492214 −0.246107 0.969243i $$-0.579151\pi$$
−0.246107 + 0.969243i $$0.579151\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 0 0
$$814$$ 60.0000 2.10300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ −22.0000 −0.769212
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ −18.0000 −0.627060
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ 48.0000 1.66912 0.834562 0.550914i $$-0.185721\pi$$
0.834562 + 0.550914i $$0.185721\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −14.0000 −0.485363
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ 0 0
$$838$$ 4.00000 0.138178
$$839$$ −14.0000 −0.483334 −0.241667 0.970359i $$-0.577694\pi$$
−0.241667 + 0.970359i $$0.577694\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 2.00000 0.0689246
$$843$$ 0 0
$$844$$ −14.0000 −0.481900
$$845$$ 0 0
$$846$$ −36.0000 −1.23771
$$847$$ 50.0000 1.71802
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 20.0000 0.685591
$$852$$ 0 0
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ −12.0000 −0.410632
$$855$$ 0 0
$$856$$ 18.0000 0.615227
$$857$$ −58.0000 −1.98124 −0.990621 0.136637i $$-0.956370\pi$$
−0.990621 + 0.136637i $$0.956370\pi$$
$$858$$ 0 0
$$859$$ 18.0000 0.614152 0.307076 0.951685i $$-0.400649\pi$$
0.307076 + 0.951685i $$0.400649\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 42.0000 1.42970 0.714848 0.699280i $$-0.246496\pi$$
0.714848 + 0.699280i $$0.246496\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ −4.00000 −0.135769
$$869$$ 60.0000 2.03536
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 42.0000 1.42230
$$873$$ 6.00000 0.203069
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 38.0000 1.28317 0.641584 0.767052i $$-0.278277\pi$$
0.641584 + 0.767052i $$0.278277\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 9.00000 0.303046
$$883$$ 50.0000 1.68263 0.841317 0.540542i $$-0.181781\pi$$
0.841317 + 0.540542i $$0.181781\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ −54.0000 −1.80907
$$892$$ −14.0000 −0.468755
$$893$$ −24.0000 −0.803129
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ 10.0000 0.333704
$$899$$ −2.00000 −0.0667037
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ −12.0000 −0.399556
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 22.0000 0.730096
$$909$$ −30.0000 −0.995037
$$910$$ 0 0
$$911$$ 14.0000 0.463841 0.231920 0.972735i $$-0.425499\pi$$
0.231920 + 0.972735i $$0.425499\pi$$
$$912$$ 0 0
$$913$$ −84.0000 −2.77999
$$914$$ 38.0000 1.25693
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 28.0000 0.924641
$$918$$ 0 0
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 18.0000 0.592798
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −22.0000 −0.722965
$$927$$ −18.0000 −0.591198
$$928$$ −5.00000 −0.164133
$$929$$ 2.00000 0.0656179 0.0328089 0.999462i $$-0.489555\pi$$
0.0328089 + 0.999462i $$0.489555\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 18.0000 0.589610
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ −18.0000 −0.588348
$$937$$ −18.0000 −0.588034 −0.294017 0.955800i $$-0.594992\pi$$
−0.294017 + 0.955800i $$0.594992\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −26.0000 −0.847576 −0.423788 0.905761i $$-0.639300\pi$$
−0.423788 + 0.905761i $$0.639300\pi$$
$$942$$ 0 0
$$943$$ −4.00000 −0.130258
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ −60.0000 −1.94974 −0.974869 0.222779i $$-0.928487\pi$$
−0.974869 + 0.222779i $$0.928487\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −12.0000 −0.388922
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ −14.0000 −0.452319
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 20.0000 0.644826
$$963$$ 18.0000 0.580042
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 20.0000 0.643157 0.321578 0.946883i $$-0.395787\pi$$
0.321578 + 0.946883i $$0.395787\pi$$
$$968$$ −75.0000 −2.41059
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 34.0000 1.08943
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ −108.000 −3.45169
$$980$$ 0 0
$$981$$ 42.0000 1.34096
$$982$$ −2.00000 −0.0638226
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.00000 −0.0636930
$$987$$ 0 0
$$988$$ −4.00000 −0.127257
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 28.0000 0.889449 0.444725 0.895667i $$-0.353302\pi$$
0.444725 + 0.895667i $$0.353302\pi$$
$$992$$ 10.0000 0.317500
$$993$$ 0 0
$$994$$ −24.0000 −0.761234
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 58.0000 1.83688 0.918439 0.395562i $$-0.129450\pi$$
0.918439 + 0.395562i $$0.129450\pi$$
$$998$$ −12.0000 −0.379853
$$999$$ 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 725.2.a.a.1.1 1
3.2 odd 2 6525.2.a.d.1.1 1
5.2 odd 4 725.2.b.a.349.2 2
5.3 odd 4 725.2.b.a.349.1 2
5.4 even 2 145.2.a.a.1.1 1
15.14 odd 2 1305.2.a.f.1.1 1
20.19 odd 2 2320.2.a.e.1.1 1
35.34 odd 2 7105.2.a.b.1.1 1
40.19 odd 2 9280.2.a.o.1.1 1
40.29 even 2 9280.2.a.l.1.1 1
145.144 even 2 4205.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.a.1.1 1 5.4 even 2
725.2.a.a.1.1 1 1.1 even 1 trivial
725.2.b.a.349.1 2 5.3 odd 4
725.2.b.a.349.2 2 5.2 odd 4
1305.2.a.f.1.1 1 15.14 odd 2
2320.2.a.e.1.1 1 20.19 odd 2
4205.2.a.a.1.1 1 145.144 even 2
6525.2.a.d.1.1 1 3.2 odd 2
7105.2.a.b.1.1 1 35.34 odd 2
9280.2.a.l.1.1 1 40.29 even 2
9280.2.a.o.1.1 1 40.19 odd 2