# Properties

 Label 245.2.a.b.1.1 Level $245$ Weight $2$ Character 245.1 Self dual yes Analytic conductor $1.956$ Analytic rank $0$ 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] = [245,2,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, 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("245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.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: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 245.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +3.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -6.00000 q^{6} +6.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +3.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -6.00000 q^{6} +6.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +6.00000 q^{12} +3.00000 q^{13} -3.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} -12.0000 q^{18} +6.00000 q^{19} -2.00000 q^{20} -2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} +9.00000 q^{27} -1.00000 q^{29} +6.00000 q^{30} +6.00000 q^{31} +8.00000 q^{32} +3.00000 q^{33} +6.00000 q^{34} +12.0000 q^{36} -12.0000 q^{38} +9.00000 q^{39} +6.00000 q^{41} -6.00000 q^{43} +2.00000 q^{44} -6.00000 q^{45} +8.00000 q^{46} -9.00000 q^{47} -12.0000 q^{48} -2.00000 q^{50} -9.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} -18.0000 q^{54} -1.00000 q^{55} +18.0000 q^{57} +2.00000 q^{58} -6.00000 q^{59} -6.00000 q^{60} -12.0000 q^{62} -8.00000 q^{64} -3.00000 q^{65} -6.00000 q^{66} -14.0000 q^{67} -6.00000 q^{68} -12.0000 q^{69} -8.00000 q^{71} +6.00000 q^{73} +3.00000 q^{75} +12.0000 q^{76} -18.0000 q^{78} -1.00000 q^{79} +4.00000 q^{80} +9.00000 q^{81} -12.0000 q^{82} +12.0000 q^{83} +3.00000 q^{85} +12.0000 q^{86} -3.00000 q^{87} +12.0000 q^{89} +12.0000 q^{90} -8.00000 q^{92} +18.0000 q^{93} +18.0000 q^{94} -6.00000 q^{95} +24.0000 q^{96} -15.0000 q^{97} +6.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 3.00000 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$4$$ 2.00000 1.00000
$$5$$ −1.00000 −0.447214
$$6$$ −6.00000 −2.44949
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 2.00000 0.632456
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 6.00000 1.73205
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ −3.00000 −0.774597
$$16$$ −4.00000 −1.00000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ −12.0000 −2.82843
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −6.00000 −1.17670
$$27$$ 9.00000 1.73205
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 6.00000 1.09545
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 3.00000 0.522233
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 12.0000 2.00000
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ −12.0000 −1.94666
$$39$$ 9.00000 1.44115
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −6.00000 −0.894427
$$46$$ 8.00000 1.17954
$$47$$ −9.00000 −1.31278 −0.656392 0.754420i $$-0.727918\pi$$
−0.656392 + 0.754420i $$0.727918\pi$$
$$48$$ −12.0000 −1.73205
$$49$$ 0 0
$$50$$ −2.00000 −0.282843
$$51$$ −9.00000 −1.26025
$$52$$ 6.00000 0.832050
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ −18.0000 −2.44949
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 18.0000 2.38416
$$58$$ 2.00000 0.262613
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ −6.00000 −0.774597
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −12.0000 −1.52400
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ −3.00000 −0.372104
$$66$$ −6.00000 −0.738549
$$67$$ −14.0000 −1.71037 −0.855186 0.518321i $$-0.826557\pi$$
−0.855186 + 0.518321i $$0.826557\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ −12.0000 −1.44463
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 3.00000 0.346410
$$76$$ 12.0000 1.37649
$$77$$ 0 0
$$78$$ −18.0000 −2.03810
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 9.00000 1.00000
$$82$$ −12.0000 −1.32518
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 12.0000 1.29399
$$87$$ −3.00000 −0.321634
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 12.0000 1.26491
$$91$$ 0 0
$$92$$ −8.00000 −0.834058
$$93$$ 18.0000 1.86651
$$94$$ 18.0000 1.85656
$$95$$ −6.00000 −0.615587
$$96$$ 24.0000 2.44949
$$97$$ −15.0000 −1.52302 −0.761510 0.648154i $$-0.775541\pi$$
−0.761510 + 0.648154i $$0.775541\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 2.00000 0.200000
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 18.0000 1.78227
$$103$$ −9.00000 −0.886796 −0.443398 0.896325i $$-0.646227\pi$$
−0.443398 + 0.896325i $$0.646227\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 20.0000 1.94257
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 18.0000 1.73205
$$109$$ −15.0000 −1.43674 −0.718370 0.695662i $$-0.755111\pi$$
−0.718370 + 0.695662i $$0.755111\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ −36.0000 −3.37171
$$115$$ 4.00000 0.373002
$$116$$ −2.00000 −0.185695
$$117$$ 18.0000 1.66410
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 18.0000 1.62301
$$124$$ 12.0000 1.07763
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ −18.0000 −1.58481
$$130$$ 6.00000 0.526235
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 6.00000 0.522233
$$133$$ 0 0
$$134$$ 28.0000 2.41883
$$135$$ −9.00000 −0.774597
$$136$$ 0 0
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ 24.0000 2.04302
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −27.0000 −2.27381
$$142$$ 16.0000 1.34269
$$143$$ 3.00000 0.250873
$$144$$ −24.0000 −2.00000
$$145$$ 1.00000 0.0830455
$$146$$ −12.0000 −0.993127
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ −6.00000 −0.489898
$$151$$ 15.0000 1.22068 0.610341 0.792139i $$-0.291032\pi$$
0.610341 + 0.792139i $$0.291032\pi$$
$$152$$ 0 0
$$153$$ −18.0000 −1.45521
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ 18.0000 1.44115
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 2.00000 0.159111
$$159$$ −30.0000 −2.37915
$$160$$ −8.00000 −0.632456
$$161$$ 0 0
$$162$$ −18.0000 −1.41421
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 12.0000 0.937043
$$165$$ −3.00000 −0.233550
$$166$$ −24.0000 −1.86276
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ −6.00000 −0.460179
$$171$$ 36.0000 2.75299
$$172$$ −12.0000 −0.914991
$$173$$ −3.00000 −0.228086 −0.114043 0.993476i $$-0.536380\pi$$
−0.114043 + 0.993476i $$0.536380\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ −18.0000 −1.35296
$$178$$ −24.0000 −1.79888
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ −12.0000 −0.894427
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −36.0000 −2.63965
$$187$$ −3.00000 −0.219382
$$188$$ −18.0000 −1.31278
$$189$$ 0 0
$$190$$ 12.0000 0.870572
$$191$$ 17.0000 1.23008 0.615038 0.788497i $$-0.289140\pi$$
0.615038 + 0.788497i $$0.289140\pi$$
$$192$$ −24.0000 −1.73205
$$193$$ 12.0000 0.863779 0.431889 0.901927i $$-0.357847\pi$$
0.431889 + 0.901927i $$0.357847\pi$$
$$194$$ 30.0000 2.15387
$$195$$ −9.00000 −0.644503
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ 6.00000 0.425329 0.212664 0.977125i $$-0.431786\pi$$
0.212664 + 0.977125i $$0.431786\pi$$
$$200$$ 0 0
$$201$$ −42.0000 −2.96245
$$202$$ 36.0000 2.53295
$$203$$ 0 0
$$204$$ −18.0000 −1.26025
$$205$$ −6.00000 −0.419058
$$206$$ 18.0000 1.25412
$$207$$ −24.0000 −1.66812
$$208$$ −12.0000 −0.832050
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ −20.0000 −1.37361
$$213$$ −24.0000 −1.64445
$$214$$ 4.00000 0.273434
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 30.0000 2.03186
$$219$$ 18.0000 1.21633
$$220$$ −2.00000 −0.134840
$$221$$ −9.00000 −0.605406
$$222$$ 0 0
$$223$$ 3.00000 0.200895 0.100447 0.994942i $$-0.467973\pi$$
0.100447 + 0.994942i $$0.467973\pi$$
$$224$$ 0 0
$$225$$ 6.00000 0.400000
$$226$$ −16.0000 −1.06430
$$227$$ 3.00000 0.199117 0.0995585 0.995032i $$-0.468257\pi$$
0.0995585 + 0.995032i $$0.468257\pi$$
$$228$$ 36.0000 2.38416
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ −36.0000 −2.35339
$$235$$ 9.00000 0.587095
$$236$$ −12.0000 −0.781133
$$237$$ −3.00000 −0.194871
$$238$$ 0 0
$$239$$ 7.00000 0.452792 0.226396 0.974035i $$-0.427306\pi$$
0.226396 + 0.974035i $$0.427306\pi$$
$$240$$ 12.0000 0.774597
$$241$$ −24.0000 −1.54598 −0.772988 0.634421i $$-0.781239\pi$$
−0.772988 + 0.634421i $$0.781239\pi$$
$$242$$ 20.0000 1.28565
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ −36.0000 −2.29528
$$247$$ 18.0000 1.14531
$$248$$ 0 0
$$249$$ 36.0000 2.28141
$$250$$ 2.00000 0.126491
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 4.00000 0.250982
$$255$$ 9.00000 0.563602
$$256$$ 16.0000 1.00000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 36.0000 2.24126
$$259$$ 0 0
$$260$$ −6.00000 −0.372104
$$261$$ −6.00000 −0.371391
$$262$$ −24.0000 −1.48272
$$263$$ −10.0000 −0.616626 −0.308313 0.951285i $$-0.599764\pi$$
−0.308313 + 0.951285i $$0.599764\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 0 0
$$267$$ 36.0000 2.20316
$$268$$ −28.0000 −1.71037
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 18.0000 1.09545
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 12.0000 0.727607
$$273$$ 0 0
$$274$$ −16.0000 −0.966595
$$275$$ 1.00000 0.0603023
$$276$$ −24.0000 −1.44463
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ 36.0000 2.15526
$$280$$ 0 0
$$281$$ −5.00000 −0.298275 −0.149137 0.988816i $$-0.547650\pi$$
−0.149137 + 0.988816i $$0.547650\pi$$
$$282$$ 54.0000 3.21565
$$283$$ 21.0000 1.24832 0.624160 0.781296i $$-0.285441\pi$$
0.624160 + 0.781296i $$0.285441\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ −18.0000 −1.06623
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ 48.0000 2.82843
$$289$$ −8.00000 −0.470588
$$290$$ −2.00000 −0.117444
$$291$$ −45.0000 −2.63795
$$292$$ 12.0000 0.702247
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ 6.00000 0.349334
$$296$$ 0 0
$$297$$ 9.00000 0.522233
$$298$$ −20.0000 −1.15857
$$299$$ −12.0000 −0.693978
$$300$$ 6.00000 0.346410
$$301$$ 0 0
$$302$$ −30.0000 −1.72631
$$303$$ −54.0000 −3.10222
$$304$$ −24.0000 −1.37649
$$305$$ 0 0
$$306$$ 36.0000 2.05798
$$307$$ −3.00000 −0.171219 −0.0856095 0.996329i $$-0.527284\pi$$
−0.0856095 + 0.996329i $$0.527284\pi$$
$$308$$ 0 0
$$309$$ −27.0000 −1.53598
$$310$$ 12.0000 0.681554
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 3.00000 0.169570 0.0847850 0.996399i $$-0.472980\pi$$
0.0847850 + 0.996399i $$0.472980\pi$$
$$314$$ −36.0000 −2.03160
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 60.0000 3.36463
$$319$$ −1.00000 −0.0559893
$$320$$ 8.00000 0.447214
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ −18.0000 −1.00155
$$324$$ 18.0000 1.00000
$$325$$ 3.00000 0.166410
$$326$$ −32.0000 −1.77232
$$327$$ −45.0000 −2.48851
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 6.00000 0.330289
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 24.0000 1.31717
$$333$$ 0 0
$$334$$ −6.00000 −0.328305
$$335$$ 14.0000 0.764902
$$336$$ 0 0
$$337$$ −24.0000 −1.30736 −0.653682 0.756770i $$-0.726776\pi$$
−0.653682 + 0.756770i $$0.726776\pi$$
$$338$$ 8.00000 0.435143
$$339$$ 24.0000 1.30350
$$340$$ 6.00000 0.325396
$$341$$ 6.00000 0.324918
$$342$$ −72.0000 −3.89331
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 6.00000 0.322562
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 27.0000 1.44115
$$352$$ 8.00000 0.426401
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 36.0000 1.91338
$$355$$ 8.00000 0.424596
$$356$$ 24.0000 1.27200
$$357$$ 0 0
$$358$$ −8.00000 −0.422813
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 12.0000 0.630706
$$363$$ −30.0000 −1.57459
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ −33.0000 −1.72259 −0.861293 0.508109i $$-0.830345\pi$$
−0.861293 + 0.508109i $$0.830345\pi$$
$$368$$ 16.0000 0.834058
$$369$$ 36.0000 1.87409
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 36.0000 1.86651
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 6.00000 0.310253
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ −24.0000 −1.23280 −0.616399 0.787434i $$-0.711409\pi$$
−0.616399 + 0.787434i $$0.711409\pi$$
$$380$$ −12.0000 −0.615587
$$381$$ −6.00000 −0.307389
$$382$$ −34.0000 −1.73959
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −24.0000 −1.22157
$$387$$ −36.0000 −1.82998
$$388$$ −30.0000 −1.52302
$$389$$ 13.0000 0.659126 0.329563 0.944134i $$-0.393099\pi$$
0.329563 + 0.944134i $$0.393099\pi$$
$$390$$ 18.0000 0.911465
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 36.0000 1.81596
$$394$$ 4.00000 0.201517
$$395$$ 1.00000 0.0503155
$$396$$ 12.0000 0.603023
$$397$$ 9.00000 0.451697 0.225849 0.974162i $$-0.427485\pi$$
0.225849 + 0.974162i $$0.427485\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −19.0000 −0.948815 −0.474407 0.880305i $$-0.657338\pi$$
−0.474407 + 0.880305i $$0.657338\pi$$
$$402$$ 84.0000 4.18954
$$403$$ 18.0000 0.896644
$$404$$ −36.0000 −1.79107
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ 12.0000 0.592638
$$411$$ 24.0000 1.18383
$$412$$ −18.0000 −0.886796
$$413$$ 0 0
$$414$$ 48.0000 2.35907
$$415$$ −12.0000 −0.589057
$$416$$ 24.0000 1.17670
$$417$$ 0 0
$$418$$ −12.0000 −0.586939
$$419$$ −6.00000 −0.293119 −0.146560 0.989202i $$-0.546820\pi$$
−0.146560 + 0.989202i $$0.546820\pi$$
$$420$$ 0 0
$$421$$ 1.00000 0.0487370 0.0243685 0.999703i $$-0.492242\pi$$
0.0243685 + 0.999703i $$0.492242\pi$$
$$422$$ −30.0000 −1.46038
$$423$$ −54.0000 −2.62557
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 48.0000 2.32561
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 9.00000 0.434524
$$430$$ −12.0000 −0.578691
$$431$$ 5.00000 0.240842 0.120421 0.992723i $$-0.461576\pi$$
0.120421 + 0.992723i $$0.461576\pi$$
$$432$$ −36.0000 −1.73205
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 0 0
$$435$$ 3.00000 0.143839
$$436$$ −30.0000 −1.43674
$$437$$ −24.0000 −1.14808
$$438$$ −36.0000 −1.72015
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 18.0000 0.856173
$$443$$ 34.0000 1.61539 0.807694 0.589601i $$-0.200715\pi$$
0.807694 + 0.589601i $$0.200715\pi$$
$$444$$ 0 0
$$445$$ −12.0000 −0.568855
$$446$$ −6.00000 −0.284108
$$447$$ 30.0000 1.41895
$$448$$ 0 0
$$449$$ 23.0000 1.08544 0.542719 0.839915i $$-0.317395\pi$$
0.542719 + 0.839915i $$0.317395\pi$$
$$450$$ −12.0000 −0.565685
$$451$$ 6.00000 0.282529
$$452$$ 16.0000 0.752577
$$453$$ 45.0000 2.11428
$$454$$ −6.00000 −0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 12.0000 0.560723
$$459$$ −27.0000 −1.26025
$$460$$ 8.00000 0.373002
$$461$$ 36.0000 1.67669 0.838344 0.545142i $$-0.183524\pi$$
0.838344 + 0.545142i $$0.183524\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 4.00000 0.185695
$$465$$ −18.0000 −0.834730
$$466$$ 16.0000 0.741186
$$467$$ −15.0000 −0.694117 −0.347059 0.937843i $$-0.612820\pi$$
−0.347059 + 0.937843i $$0.612820\pi$$
$$468$$ 36.0000 1.66410
$$469$$ 0 0
$$470$$ −18.0000 −0.830278
$$471$$ 54.0000 2.48819
$$472$$ 0 0
$$473$$ −6.00000 −0.275880
$$474$$ 6.00000 0.275589
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ −60.0000 −2.74721
$$478$$ −14.0000 −0.640345
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ −24.0000 −1.09545
$$481$$ 0 0
$$482$$ 48.0000 2.18634
$$483$$ 0 0
$$484$$ −20.0000 −0.909091
$$485$$ 15.0000 0.681115
$$486$$ 0 0
$$487$$ 36.0000 1.63132 0.815658 0.578535i $$-0.196375\pi$$
0.815658 + 0.578535i $$0.196375\pi$$
$$488$$ 0 0
$$489$$ 48.0000 2.17064
$$490$$ 0 0
$$491$$ 23.0000 1.03798 0.518988 0.854782i $$-0.326309\pi$$
0.518988 + 0.854782i $$0.326309\pi$$
$$492$$ 36.0000 1.62301
$$493$$ 3.00000 0.135113
$$494$$ −36.0000 −1.61972
$$495$$ −6.00000 −0.269680
$$496$$ −24.0000 −1.07763
$$497$$ 0 0
$$498$$ −72.0000 −3.22640
$$499$$ −27.0000 −1.20869 −0.604343 0.796724i $$-0.706564\pi$$
−0.604343 + 0.796724i $$0.706564\pi$$
$$500$$ −2.00000 −0.0894427
$$501$$ 9.00000 0.402090
$$502$$ 24.0000 1.07117
$$503$$ 9.00000 0.401290 0.200645 0.979664i $$-0.435696\pi$$
0.200645 + 0.979664i $$0.435696\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 8.00000 0.355643
$$507$$ −12.0000 −0.532939
$$508$$ −4.00000 −0.177471
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ −18.0000 −0.797053
$$511$$ 0 0
$$512$$ −32.0000 −1.41421
$$513$$ 54.0000 2.38416
$$514$$ −36.0000 −1.58789
$$515$$ 9.00000 0.396587
$$516$$ −36.0000 −1.58481
$$517$$ −9.00000 −0.395820
$$518$$ 0 0
$$519$$ −9.00000 −0.395056
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 12.0000 0.525226
$$523$$ 24.0000 1.04945 0.524723 0.851273i $$-0.324169\pi$$
0.524723 + 0.851273i $$0.324169\pi$$
$$524$$ 24.0000 1.04844
$$525$$ 0 0
$$526$$ 20.0000 0.872041
$$527$$ −18.0000 −0.784092
$$528$$ −12.0000 −0.522233
$$529$$ −7.00000 −0.304348
$$530$$ −20.0000 −0.868744
$$531$$ −36.0000 −1.56227
$$532$$ 0 0
$$533$$ 18.0000 0.779667
$$534$$ −72.0000 −3.11574
$$535$$ 2.00000 0.0864675
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ −48.0000 −2.06943
$$539$$ 0 0
$$540$$ −18.0000 −0.774597
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ −48.0000 −2.06178
$$543$$ −18.0000 −0.772454
$$544$$ −24.0000 −1.02899
$$545$$ 15.0000 0.642529
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 16.0000 0.683486
$$549$$ 0 0
$$550$$ −2.00000 −0.0852803
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 56.0000 2.37921
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 4.00000 0.169485 0.0847427 0.996403i $$-0.472993\pi$$
0.0847427 + 0.996403i $$0.472993\pi$$
$$558$$ −72.0000 −3.04800
$$559$$ −18.0000 −0.761319
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 10.0000 0.421825
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ −54.0000 −2.27381
$$565$$ −8.00000 −0.336563
$$566$$ −42.0000 −1.76539
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −38.0000 −1.59304 −0.796521 0.604610i $$-0.793329\pi$$
−0.796521 + 0.604610i $$0.793329\pi$$
$$570$$ 36.0000 1.50787
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 51.0000 2.13056
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ −48.0000 −2.00000
$$577$$ 15.0000 0.624458 0.312229 0.950007i $$-0.398924\pi$$
0.312229 + 0.950007i $$0.398924\pi$$
$$578$$ 16.0000 0.665512
$$579$$ 36.0000 1.49611
$$580$$ 2.00000 0.0830455
$$581$$ 0 0
$$582$$ 90.0000 3.73062
$$583$$ −10.0000 −0.414158
$$584$$ 0 0
$$585$$ −18.0000 −0.744208
$$586$$ −18.0000 −0.743573
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 36.0000 1.48335
$$590$$ −12.0000 −0.494032
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ −45.0000 −1.84793 −0.923964 0.382479i $$-0.875070\pi$$
−0.923964 + 0.382479i $$0.875070\pi$$
$$594$$ −18.0000 −0.738549
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ 18.0000 0.736691
$$598$$ 24.0000 0.981433
$$599$$ 17.0000 0.694601 0.347301 0.937754i $$-0.387098\pi$$
0.347301 + 0.937754i $$0.387098\pi$$
$$600$$ 0 0
$$601$$ −30.0000 −1.22373 −0.611863 0.790964i $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ 0 0
$$603$$ −84.0000 −3.42074
$$604$$ 30.0000 1.22068
$$605$$ 10.0000 0.406558
$$606$$ 108.000 4.38720
$$607$$ 21.0000 0.852364 0.426182 0.904638i $$-0.359858\pi$$
0.426182 + 0.904638i $$0.359858\pi$$
$$608$$ 48.0000 1.94666
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −27.0000 −1.09230
$$612$$ −36.0000 −1.45521
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 6.00000 0.242140
$$615$$ −18.0000 −0.725830
$$616$$ 0 0
$$617$$ 4.00000 0.161034 0.0805170 0.996753i $$-0.474343\pi$$
0.0805170 + 0.996753i $$0.474343\pi$$
$$618$$ 54.0000 2.17220
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ −12.0000 −0.481932
$$621$$ −36.0000 −1.44463
$$622$$ 12.0000 0.481156
$$623$$ 0 0
$$624$$ −36.0000 −1.44115
$$625$$ 1.00000 0.0400000
$$626$$ −6.00000 −0.239808
$$627$$ 18.0000 0.718851
$$628$$ 36.0000 1.43656
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −11.0000 −0.437903 −0.218952 0.975736i $$-0.570264\pi$$
−0.218952 + 0.975736i $$0.570264\pi$$
$$632$$ 0 0
$$633$$ 45.0000 1.78859
$$634$$ 44.0000 1.74746
$$635$$ 2.00000 0.0793676
$$636$$ −60.0000 −2.37915
$$637$$ 0 0
$$638$$ 2.00000 0.0791808
$$639$$ −48.0000 −1.89885
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 27.0000 1.06478 0.532388 0.846500i $$-0.321295\pi$$
0.532388 + 0.846500i $$0.321295\pi$$
$$644$$ 0 0
$$645$$ 18.0000 0.708749
$$646$$ 36.0000 1.41640
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ −6.00000 −0.235521
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ 32.0000 1.25322
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 90.0000 3.51928
$$655$$ −12.0000 −0.468879
$$656$$ −24.0000 −0.937043
$$657$$ 36.0000 1.40449
$$658$$ 0 0
$$659$$ −19.0000 −0.740135 −0.370067 0.929005i $$-0.620665\pi$$
−0.370067 + 0.929005i $$0.620665\pi$$
$$660$$ −6.00000 −0.233550
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ −24.0000 −0.932786
$$663$$ −27.0000 −1.04859
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.00000 0.154881
$$668$$ 6.00000 0.232147
$$669$$ 9.00000 0.347960
$$670$$ −28.0000 −1.08173
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 12.0000 0.462566 0.231283 0.972887i $$-0.425708\pi$$
0.231283 + 0.972887i $$0.425708\pi$$
$$674$$ 48.0000 1.84889
$$675$$ 9.00000 0.346410
$$676$$ −8.00000 −0.307692
$$677$$ −9.00000 −0.345898 −0.172949 0.984931i $$-0.555330\pi$$
−0.172949 + 0.984931i $$0.555330\pi$$
$$678$$ −48.0000 −1.84343
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 9.00000 0.344881
$$682$$ −12.0000 −0.459504
$$683$$ 8.00000 0.306111 0.153056 0.988218i $$-0.451089\pi$$
0.153056 + 0.988218i $$0.451089\pi$$
$$684$$ 72.0000 2.75299
$$685$$ −8.00000 −0.305664
$$686$$ 0 0
$$687$$ −18.0000 −0.686743
$$688$$ 24.0000 0.914991
$$689$$ −30.0000 −1.14291
$$690$$ −24.0000 −0.913664
$$691$$ 36.0000 1.36950 0.684752 0.728776i $$-0.259910\pi$$
0.684752 + 0.728776i $$0.259910\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −32.0000 −1.21470
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ 60.0000 2.27103
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 47.0000 1.77517 0.887583 0.460648i $$-0.152383\pi$$
0.887583 + 0.460648i $$0.152383\pi$$
$$702$$ −54.0000 −2.03810
$$703$$ 0 0
$$704$$ −8.00000 −0.301511
$$705$$ 27.0000 1.01688
$$706$$ 6.00000 0.225813
$$707$$ 0 0
$$708$$ −36.0000 −1.35296
$$709$$ −33.0000 −1.23934 −0.619671 0.784862i $$-0.712734\pi$$
−0.619671 + 0.784862i $$0.712734\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ −6.00000 −0.225018
$$712$$ 0 0
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ −3.00000 −0.112194
$$716$$ 8.00000 0.298974
$$717$$ 21.0000 0.784259
$$718$$ −64.0000 −2.38846
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 24.0000 0.894427
$$721$$ 0 0
$$722$$ −34.0000 −1.26535
$$723$$ −72.0000 −2.67771
$$724$$ −12.0000 −0.445976
$$725$$ −1.00000 −0.0371391
$$726$$ 60.0000 2.22681
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 12.0000 0.444140
$$731$$ 18.0000 0.665754
$$732$$ 0 0
$$733$$ 15.0000 0.554038 0.277019 0.960864i $$-0.410654\pi$$
0.277019 + 0.960864i $$0.410654\pi$$
$$734$$ 66.0000 2.43610
$$735$$ 0 0
$$736$$ −32.0000 −1.17954
$$737$$ −14.0000 −0.515697
$$738$$ −72.0000 −2.65036
$$739$$ −43.0000 −1.58178 −0.790890 0.611958i $$-0.790382\pi$$
−0.790890 + 0.611958i $$0.790382\pi$$
$$740$$ 0 0
$$741$$ 54.0000 1.98374
$$742$$ 0 0
$$743$$ −22.0000 −0.807102 −0.403551 0.914957i $$-0.632224\pi$$
−0.403551 + 0.914957i $$0.632224\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ −8.00000 −0.292901
$$747$$ 72.0000 2.63434
$$748$$ −6.00000 −0.219382
$$749$$ 0 0
$$750$$ 6.00000 0.219089
$$751$$ 15.0000 0.547358 0.273679 0.961821i $$-0.411759\pi$$
0.273679 + 0.961821i $$0.411759\pi$$
$$752$$ 36.0000 1.31278
$$753$$ −36.0000 −1.31191
$$754$$ 6.00000 0.218507
$$755$$ −15.0000 −0.545906
$$756$$ 0 0
$$757$$ 48.0000 1.74459 0.872295 0.488980i $$-0.162631\pi$$
0.872295 + 0.488980i $$0.162631\pi$$
$$758$$ 48.0000 1.74344
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 12.0000 0.434714
$$763$$ 0 0
$$764$$ 34.0000 1.23008
$$765$$ 18.0000 0.650791
$$766$$ 0 0
$$767$$ −18.0000 −0.649942
$$768$$ 48.0000 1.73205
$$769$$ 42.0000 1.51456 0.757279 0.653091i $$-0.226528\pi$$
0.757279 + 0.653091i $$0.226528\pi$$
$$770$$ 0 0
$$771$$ 54.0000 1.94476
$$772$$ 24.0000 0.863779
$$773$$ 3.00000 0.107903 0.0539513 0.998544i $$-0.482818\pi$$
0.0539513 + 0.998544i $$0.482818\pi$$
$$774$$ 72.0000 2.58799
$$775$$ 6.00000 0.215526
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −26.0000 −0.932145
$$779$$ 36.0000 1.28983
$$780$$ −18.0000 −0.644503
$$781$$ −8.00000 −0.286263
$$782$$ −24.0000 −0.858238
$$783$$ −9.00000 −0.321634
$$784$$ 0 0
$$785$$ −18.0000 −0.642448
$$786$$ −72.0000 −2.56815
$$787$$ 39.0000 1.39020 0.695100 0.718913i $$-0.255360\pi$$
0.695100 + 0.718913i $$0.255360\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ −30.0000 −1.06803
$$790$$ −2.00000 −0.0711568
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −18.0000 −0.638796
$$795$$ 30.0000 1.06399
$$796$$ 12.0000 0.425329
$$797$$ −27.0000 −0.956389 −0.478195 0.878254i $$-0.658709\pi$$
−0.478195 + 0.878254i $$0.658709\pi$$
$$798$$ 0 0
$$799$$ 27.0000 0.955191
$$800$$ 8.00000 0.282843
$$801$$ 72.0000 2.54399
$$802$$ 38.0000 1.34183
$$803$$ 6.00000 0.211735
$$804$$ −84.0000 −2.96245
$$805$$ 0 0
$$806$$ −36.0000 −1.26805
$$807$$ 72.0000 2.53452
$$808$$ 0 0
$$809$$ −35.0000 −1.23053 −0.615267 0.788319i $$-0.710952\pi$$
−0.615267 + 0.788319i $$0.710952\pi$$
$$810$$ 18.0000 0.632456
$$811$$ 48.0000 1.68551 0.842754 0.538299i $$-0.180933\pi$$
0.842754 + 0.538299i $$0.180933\pi$$
$$812$$ 0 0
$$813$$ 72.0000 2.52515
$$814$$ 0 0
$$815$$ −16.0000 −0.560456
$$816$$ 36.0000 1.26025
$$817$$ −36.0000 −1.25948
$$818$$ −60.0000 −2.09785
$$819$$ 0 0
$$820$$ −12.0000 −0.419058
$$821$$ −23.0000 −0.802706 −0.401353 0.915924i $$-0.631460\pi$$
−0.401353 + 0.915924i $$0.631460\pi$$
$$822$$ −48.0000 −1.67419
$$823$$ 50.0000 1.74289 0.871445 0.490493i $$-0.163183\pi$$
0.871445 + 0.490493i $$0.163183\pi$$
$$824$$ 0 0
$$825$$ 3.00000 0.104447
$$826$$ 0 0
$$827$$ −16.0000 −0.556375 −0.278187 0.960527i $$-0.589734\pi$$
−0.278187 + 0.960527i $$0.589734\pi$$
$$828$$ −48.0000 −1.66812
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 24.0000 0.833052
$$831$$ −84.0000 −2.91393
$$832$$ −24.0000 −0.832050
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −3.00000 −0.103819
$$836$$ 12.0000 0.415029
$$837$$ 54.0000 1.86651
$$838$$ 12.0000 0.414533
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ −2.00000 −0.0689246
$$843$$ −15.0000 −0.516627
$$844$$ 30.0000 1.03264
$$845$$ 4.00000 0.137604
$$846$$ 108.000 3.71312
$$847$$ 0 0
$$848$$ 40.0000 1.37361
$$849$$ 63.0000 2.16215
$$850$$ 6.00000 0.205798
$$851$$ 0 0
$$852$$ −48.0000 −1.64445
$$853$$ 30.0000 1.02718 0.513590 0.858036i $$-0.328315\pi$$
0.513590 + 0.858036i $$0.328315\pi$$
$$854$$ 0 0
$$855$$ −36.0000 −1.23117
$$856$$ 0 0
$$857$$ 54.0000 1.84460 0.922302 0.386469i $$-0.126305\pi$$
0.922302 + 0.386469i $$0.126305\pi$$
$$858$$ −18.0000 −0.614510
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ 12.0000 0.409197
$$861$$ 0 0
$$862$$ −10.0000 −0.340601
$$863$$ −34.0000 −1.15737 −0.578687 0.815550i $$-0.696435\pi$$
−0.578687 + 0.815550i $$0.696435\pi$$
$$864$$ 72.0000 2.44949
$$865$$ 3.00000 0.102003
$$866$$ 60.0000 2.03888
$$867$$ −24.0000 −0.815083
$$868$$ 0 0
$$869$$ −1.00000 −0.0339227
$$870$$ −6.00000 −0.203419
$$871$$ −42.0000 −1.42312
$$872$$ 0 0
$$873$$ −90.0000 −3.04604
$$874$$ 48.0000 1.62362
$$875$$ 0 0
$$876$$ 36.0000 1.21633
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 27.0000 0.910687
$$880$$ 4.00000 0.134840
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ −18.0000 −0.605406
$$885$$ 18.0000 0.605063
$$886$$ −68.0000 −2.28450
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 24.0000 0.804482
$$891$$ 9.00000 0.301511
$$892$$ 6.00000 0.200895
$$893$$ −54.0000 −1.80704
$$894$$ −60.0000 −2.00670
$$895$$ −4.00000 −0.133705
$$896$$ 0 0
$$897$$ −36.0000 −1.20201
$$898$$ −46.0000 −1.53504
$$899$$ −6.00000 −0.200111
$$900$$ 12.0000 0.400000
$$901$$ 30.0000 0.999445
$$902$$ −12.0000 −0.399556
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 6.00000 0.199447
$$906$$ −90.0000 −2.99005
$$907$$ 24.0000 0.796907 0.398453 0.917189i $$-0.369547\pi$$
0.398453 + 0.917189i $$0.369547\pi$$
$$908$$ 6.00000 0.199117
$$909$$ −108.000 −3.58213
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ −72.0000 −2.38416
$$913$$ 12.0000 0.397142
$$914$$ −12.0000 −0.396925
$$915$$ 0 0
$$916$$ −12.0000 −0.396491
$$917$$ 0 0
$$918$$ 54.0000 1.78227
$$919$$ −9.00000 −0.296883 −0.148441 0.988921i $$-0.547426\pi$$
−0.148441 + 0.988921i $$0.547426\pi$$
$$920$$ 0 0
$$921$$ −9.00000 −0.296560
$$922$$ −72.0000 −2.37119
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ −54.0000 −1.77359
$$928$$ −8.00000 −0.262613
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 36.0000 1.18049
$$931$$ 0 0
$$932$$ −16.0000 −0.524097
$$933$$ −18.0000 −0.589294
$$934$$ 30.0000 0.981630
$$935$$ 3.00000 0.0981105
$$936$$ 0 0
$$937$$ −27.0000 −0.882052 −0.441026 0.897494i $$-0.645385\pi$$
−0.441026 + 0.897494i $$0.645385\pi$$
$$938$$ 0 0
$$939$$ 9.00000 0.293704
$$940$$ 18.0000 0.587095
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ −108.000 −3.51883
$$943$$ −24.0000 −0.781548
$$944$$ 24.0000 0.781133
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ −2.00000 −0.0649913 −0.0324956 0.999472i $$-0.510346\pi$$
−0.0324956 + 0.999472i $$0.510346\pi$$
$$948$$ −6.00000 −0.194871
$$949$$ 18.0000 0.584305
$$950$$ −12.0000 −0.389331
$$951$$ −66.0000 −2.14020
$$952$$ 0 0
$$953$$ −38.0000 −1.23094 −0.615470 0.788160i $$-0.711034\pi$$
−0.615470 + 0.788160i $$0.711034\pi$$
$$954$$ 120.000 3.88514
$$955$$ −17.0000 −0.550107
$$956$$ 14.0000 0.452792
$$957$$ −3.00000 −0.0969762
$$958$$ 48.0000 1.55081
$$959$$ 0 0
$$960$$ 24.0000 0.774597
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ −48.0000 −1.54598
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ −54.0000 −1.73473
$$970$$ −30.0000 −0.963242
$$971$$ −18.0000 −0.577647 −0.288824 0.957382i $$-0.593264\pi$$
−0.288824 + 0.957382i $$0.593264\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −72.0000 −2.30703
$$975$$ 9.00000 0.288231
$$976$$ 0 0
$$977$$ 34.0000 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$978$$ −96.0000 −3.06974
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ −90.0000 −2.87348
$$982$$ −46.0000 −1.46792
$$983$$ 45.0000 1.43528 0.717639 0.696416i $$-0.245223\pi$$
0.717639 + 0.696416i $$0.245223\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ −6.00000 −0.191079
$$987$$ 0 0
$$988$$ 36.0000 1.14531
$$989$$ 24.0000 0.763156
$$990$$ 12.0000 0.381385
$$991$$ −36.0000 −1.14358 −0.571789 0.820401i $$-0.693750\pi$$
−0.571789 + 0.820401i $$0.693750\pi$$
$$992$$ 48.0000 1.52400
$$993$$ 36.0000 1.14243
$$994$$ 0 0
$$995$$ −6.00000 −0.190213
$$996$$ 72.0000 2.28141
$$997$$ −3.00000 −0.0950110 −0.0475055 0.998871i $$-0.515127\pi$$
−0.0475055 + 0.998871i $$0.515127\pi$$
$$998$$ 54.0000 1.70934
$$999$$ 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 245.2.a.b.1.1 yes 1
3.2 odd 2 2205.2.a.l.1.1 1
4.3 odd 2 3920.2.a.a.1.1 1
5.2 odd 4 1225.2.b.a.99.1 2
5.3 odd 4 1225.2.b.a.99.2 2
5.4 even 2 1225.2.a.h.1.1 1
7.2 even 3 245.2.e.c.116.1 2
7.3 odd 6 245.2.e.d.226.1 2
7.4 even 3 245.2.e.c.226.1 2
7.5 odd 6 245.2.e.d.116.1 2
7.6 odd 2 245.2.a.a.1.1 1
21.20 even 2 2205.2.a.j.1.1 1
28.27 even 2 3920.2.a.bj.1.1 1
35.13 even 4 1225.2.b.b.99.2 2
35.27 even 4 1225.2.b.b.99.1 2
35.34 odd 2 1225.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.a.1.1 1 7.6 odd 2
245.2.a.b.1.1 yes 1 1.1 even 1 trivial
245.2.e.c.116.1 2 7.2 even 3
245.2.e.c.226.1 2 7.4 even 3
245.2.e.d.116.1 2 7.5 odd 6
245.2.e.d.226.1 2 7.3 odd 6
1225.2.a.h.1.1 1 5.4 even 2
1225.2.a.j.1.1 1 35.34 odd 2
1225.2.b.a.99.1 2 5.2 odd 4
1225.2.b.a.99.2 2 5.3 odd 4
1225.2.b.b.99.1 2 35.27 even 4
1225.2.b.b.99.2 2 35.13 even 4
2205.2.a.j.1.1 1 21.20 even 2
2205.2.a.l.1.1 1 3.2 odd 2
3920.2.a.a.1.1 1 4.3 odd 2
3920.2.a.bj.1.1 1 28.27 even 2